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Actuarial Notes for
Spring 2014 CAS Exam5

Syllabus Section A
Ratemaking, Classification Analysis,
Miscellaneous Ratemaking Topics

Volume 1a

Table of Contents
Exam 5 – Volume 1a: Ratemaking – Part 1
Syllabus Section/Title

Author

Page

A. Chapter 1: Introduction ............................................... Modlin, Werner ......................................................................... 1
A. Chapter 2: Rating Manuals .......................................... Modlin, Werner ....................................................................... 15
A. Chapter 3: Ratemaking Data ...................................... Modlin, Werner ....................................................................... 32
A. Chapter 4: Exposures ................................................... Modlin, Werner ........................................................................ 46
A. Chapter 5: Premium ..................................................... Modlin, Werner ........................................................................ 73
A. Chapter 6: Losses and LAE ....................................... Modlin, Werner ...................................................................... 152
A. Chapter 7: Other Expenses and Profit ...................... Modlin, Werner ...................................................................... 211
A. Chapter 8: Overall Indication .................................... Modlin, Werner ...................................................................... 232

A. Statement of Principles Re PC Ins Ratemaking ........ CAS .......................................................................................... 263
A. Actuarial Standard No. 13 – Trending Proc. ........... CAS .......................................................................................... 278
A. Statement of Principles Re Class Ratemaking .......... CAS .......................................................................................... 284
ISO Personal Auto Manual................................................... ISO ........................................................................................ 309

Notes:
The predecessor papers to the CAS 2011 syllabus reading “Basic Ratemaking” by Werner, G. and Modlin, C. were numerous.
Past CAS questions and our solutions to those questions associated with those readings that are within this volume, remain
relevant to understanding the content covered in these chapters.

For those purchasing our online review course, streamline your study of any chapter, by logging into m.ALL10.com
Our chapter/article commentary is found under the section titled “Online Study Guide”, and can be accessed by clicking on
the ‘light bulb’ icon in our E-Learning Center.

Chapter 1 - Introduction
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Sec
1
2
3
4
5

Description
Introduction and Rating Manuals
Basic Insurance Terms
Fundamental Insurance Equation
Basic Insurance Ratios
Key Concepts

Pages
1-1
1-5
5-7
7 - 11
11 - 11

Introduction and Rating Manuals

1-1

Insurance and Non-insurance Product Pricing:
The price of a product should reflect its costs as well as an acceptable profit. This leads to the following
relationship between price, cost, and profit:
Price = Cost + Profit.
For non-insurers, production cost is known before the product is sold, and thus the price can be set so
that the desired profit per unit of product can be obtained.
For insurers, the ultimate cost of an insurance policy is not known before the product is sold, which
introduces complexity for the insurer when setting prices.
Rating Manuals
In general, premiums are based on a rate per unit of risk exposed.
 Rating manuals contains information to classify and calculate the premium for a given risk.
 Chapter 2 contains more detailed information and specific examples of rating manuals.
The ratemaking process allows one to modify existing rating manuals or create new ones.

Basic Insurance Terms

1-5

Exposure
An exposure is a unit of risk that underlies the premium. Different exposures are used when making rates
for different lines of business (e.g. annual payroll in hundreds of dollars is the typical exposure unit for
U.S. workers compensation insurance).
Four ways insurers measure exposures are as follows:
 Written exposures are the total exposures arising from policies issued during a specified time
period (e.g. a calendar year or quarter).
 Earned exposures are the portion of written exposures for which coverage has already been
provided (as of a certain point in time).
 Unearned exposures are the portion of written exposures for which coverage has not yet been
provided (as of that point in time).
 In-force exposures are the number of units exposed to loss at a given point in time.
See chapter 4 for more examples on how exposure measures are used for ratemaking.

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Chapter 1 - Introduction
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Premium
Four types of premiums are as follows:

Written premium: Total premium from policies issued during a specified period.

Earned premium: The portion of written premium for which coverage has already been provided
(as of a certain point in time).

Unearned premium: The portion of written premium for which coverage has yet to be provided.

In-force premium: The full-term premium for policies in effect at a given point in time.
See chapter 5 for examples of premium measures and how they are used for ratemaking.
Claim
A claim is a demand for indemnification for the financial consequences of an event covered by a policy.
 The claimant can be an insured or a third party alleging damages covered by a policy.
 The date of loss or accident date (a.k.a. occurrence date) is the date of the loss event.
 Claims not known by the insurer are unreported claims or incurred but not reported (IBNR) claims.
After the claim is reported to the insurer, the claim is a reported claim.
Until the claim is settled, the reported claim is an open claim.
Once the claim is settled, it is a closed claim.
If further activity occurs after the claim is closed, the claim may be re-opened.
Loss
Loss is the amount paid or payable to the claimant under the policy.
The authors use the term claim to refer to the demand for compensation, and loss to refer to the
amount of compensation.
Paid losses are amounts that have been paid to claimants.
Case reserves are estimates of the amount needed to settle a claim and excludes any payments already made.
Reported loss (or case incurred loss) is the sum of paid losses and the current case reserve for a claim:
Reported Losses = Paid Losses + Case Reserve.
Ultimate loss is the amount to close and settle all claims for a defined group of policies.
Two reasons why reported losses and ultimate losses are different:
1. When there are unreported claims, the estimated amount to settle these claims is known as incurred
but not reported (IBNR) reserve.
2. The incurred but not enough reported (IBNER) reserve (a.k.a. development on known claims) is the
difference between the aggregate reported losses at the time the losses are evaluated and the
aggregate amount estimated to ultimately settle these reported claims.
Ultimate Losses = Reported Losses + IBNR Reserve + IBNER Reserve.
Loss Adjustment Expense (LAE)
LAE represent insurer expenses in settling claims, and can be separated into:
Allocated loss adjustment expenses (ALAE) and unallocated loss adjustment expenses (ULAE):
LAE = ALAE + ULAE.
ALAE are directly attributable to a specific claim (e.g. fees for outside legal counsel hired to defend a claim).
ULAE cannot be directly assigned to a specific claim (e.g. salaries of claims department personnel
not assignable to a specific claim).
See Chapter 6 to see how loss and LAE data are used in the ratemaking purposes.

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Chapter 1 - Introduction
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Underwriting Expenses (U/W expenses)
U/W expenses (a.k.a. operational and administrative expenses) are related to acquiring and servicing policies.
Four categories for classifying these expenses are:
1. Commissions and brokerage are:
 amounts paid to insurance agents or brokers as compensation for generating business.
 paid as a percentage of premium written.
 vary between new and renewal business
 based on the quality of the business written or the volume of business written or both.
2. Other acquisition costs (other than commissions and brokerage expenses) include costs
associated with media advertisements and mailings to prospective insureds.
3. General expenses include the remaining expenses associated with the insurance operations and
other miscellaneous costs (e.g. costs associated with the general upkeep of the home office).
4. Taxes, licenses, and fees include all taxes and miscellaneous fees paid by the insurer excluding
federal income taxes (e.g. premium taxes and licensing fees)
Underwriting Profit (UW Profit)
Since premiums may be insufficient to pay claims and expenses, capital must be maintained to support
this risk, and the insurer is entitled to earn a reasonable expected return (profit) on that capital.
Two main sources of profit for insurers are UW profit and investment income (II).
1. UW profit (i.e. operating income) is the total profit from all policies (a.k.a. income minus outgo).
2. II is generated from funds invested in securities held by the insurer.
See chapter 7 to see how UW expense provisions are derived and how it’s incorporated in the ratemaking
process.

Fundamental Insurance Equation

5-7

Price = Cost + Profit. As it applies to the insurance industry:
 Premium is the “price” of the insurance product.
 “Cost” is the sum of the losses, LAE, and UW expenses.
 UW profit is income minus the outgo from issuing policies.
Note: Profit is also derived from II
The prior formula transformed into the fundamental insurance equation is:
Premium = Losses + LAE + UW Expenses + UW Profit.
The goal of ratemaking: To assure that the fundamental insurance equation is balanced (e.g. rates should be
set so premium is expected to cover all costs and achieve the target UW profit).
 This goal is stated in the 2nd principle of the CAS “Statement of Principles Regarding P&C Ratemaking”
which states “A rate provides for all costs associated with the transfer of risk.”
 Two key points in achieving balance in the fundamental equation are:
1. Ratemaking is prospective.
2. Balance should be attained at the aggregate and individual levels.

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Chapter 1 - Introduction
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
1. Ratemaking is Prospective
Ratemaking involves estimating the components of the fundamental insurance equation to determine whether
or not the estimated premium is likely to achieve the target profit during the period the rates will be in effect.
While ratemaking uses historical experience to estimate future expected costs, this does not mean
premiums are set to recoup past losses.
Recall that the first principle in the CAS “Statement of Principles Regarding P&C Insurance
Ratemaking” states that “A rate is an estimate of the expected value of future costs”
Factors that impact the components of the fundamental insurance equation and may necessitate a
restatement of the historical experience are:
 Rate changes
 Operational changes
 Inflationary pressures
 Changes in the mix of business written
 Law changes
2. Overall and Individual Balance
The fundamental insurance equation must be in balance at both an overall level as well as at an
individual/segment level when considering rate adequacy.
If proposed rates are either too high or too low to achieve the targeted profit, decreasing or increasing
rates uniformly should be considered.
Two methods for calculating the overall adequacy of current rates are discussed in Chapter 8.
Principle 3 of the CAS “Statement of Principles Regarding P&C Insurance Ratemaking” states “A rate
provides for the costs associated with an individual risk transfer”
Failure to recognize differences in risk will lead to rates that are not equitable.
Chapters 9 - 11 discuss how insurers vary rates to recognize differences between insureds.

Basic Insurance Ratios

7 - 11

Insurers, insurance regulators, rating agencies, and investors rely on a set of basic ratios to monitor and
evaluate the appropriateness of an insurer’s rates.
Frequency (a measure of the rate at which claims occur): Frequency 

Number of Claims
Number of Exposures

Assume the number of claims is 100,000 and the number of earned exposures is 2,000,000.
Then frequency is 5% (= 100,000 / 2,000,000).
Analyzing changes in claims frequency can help identify:
 industry trends associated with the incidence of claims
 utilization of insurance coverage.
 the effectiveness of specific underwriting actions.

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Chapter 1 - Introduction
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Severity (a measure of the average cost of claims):

Severity 

Total Losses
Number of Claims

Assume total loss dollars are $300,000,000 and the number of claims is 100,000.
Then severity is $3,000 (= $300,000,000 / 100,000).
Values used in the numerator and denominator do vary: For example:
 Paid severity is calculated using paid losses on closed claims divided by closed claims.
 Reported severity is calculated using reported losses and reported claims.
 ALAE may be included or excluded from the numerator.
Analyzing changes in severity:
 provides information about loss trends and
 highlights the impact of any changes in claims handling procedures.
Pure Premium (or Loss Cost or Burning Cost): (a measure of the average loss per exposure)

Pure Premium =

Total Losses
= Frequency x Severity
Number of Exposures

Pure premiums are the portion of the risk’s expected costs that is “purely” attributable to loss.
Assume total loss dollars are $300,000,000 and the number of exposures is 2,000,000.
Then pure premium is $150 (= $300,000,000 / 2,000,000) = 5.0% x $3,000.
Pure premium is often calculated using reported losses (or ultimate losses) and earned exposures, and
reported losses may or may not include ALAE and/or ULAE.
Changes in pure premium show industry trends in overall loss costs due to changes in both frequency and
severity.
Average Premium
While the pure premium focuses on the loss portion of the fundamental insurance equation, the average
premium focuses on the premium side of the ratio. Average Premium =

Total Premium
No. of Exposures

Let total premium equal $400,000,000 and total exposures equal 2,000,000
Then average premium is $200 (=$400,000,000 / 2,000,000).
Note: premium and exposures must be on the same basis (e.g., written, earned, or in-force).
Changes in average premium, adjusted for rate changes, show changes in the mix of business written (e.g.,
shifts toward higher or lower risk characteristics reflected in rates).

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Chapter 1 - Introduction
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Loss Ratio (a measure of the portion of each premium dollar used to pay losses):

Loss Ratio =

Total Losses
Pure Premium
=
Total Premium Average Premium

Assume total loss dollars equal $300,000,000 and total premium equal 400,000,000.
Then the loss ratio is 75% (= $300,000,000 / $400,000,000).
The ratio is typically total reported losses to total earned premium. However, other variations include LAE in
the calculation of loss ratios (commonly referred to as loss and LAE ratios).
The loss and LAE ratio is a measure of the adequacy of overall rates.
LAE Ratio (a measure of claim-related expense to total losses):

LAE Ratio 

Total Loss Adjustment Expenses
Total Losses

LAE includes both allocated and unallocated loss adjustment expenses.
Insurers differ as to whether paid or reported (incurred) figures are used.
The Loss and LAE ratio equals the Loss ratio x [1.0 + LAE ratio].
Insurers may use this ratio to:
 determine if costs associated with claim settlement procedures are stable or not.
 compare its ratio to those of other insurers as a benchmark for its claims settlement procedures.
Underwriting Expense Ratio (a measure of the portion of each premium dollar to pay for UW expenses)

UW Expense Ratio =

Total UW Expenses
Total Premium

U/W expenses are divided into expenses incurred at the onset of the policy (e.g. commissions, other
acquisition, taxes, licenses, and fees) and expenses incurred throughout the policy (e.g. general expenses).
i. Expenses incurred at the onset of the policy are related to written premium and expenses incurred
throughout the policy are related to earned premium.
ii. This is done to better match expense payments to premiums associated with expenses and to better
estimate what % of future policy premium should be charged to pay for these costs.
Individual expense category ratios are summed to compute the overall UW expense ratio.
Insurers review the UW expense ratio:
 over time and compare actual changes in the ratio to expected changes based on inflation.
 to compare its ratio to other insurer ratios as a benchmark for policy acquisition and service expenses.
Operating Expense Ratio (OER is the portion of the premium dollar to pay for LAE and UW expenses)

OER = UW Expense Ratio +

LAE
Total Earned Premium

OER is used to monitor operational expenditures and is key to determining overall profitability.

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Chapter 1 - Introduction
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Combined Ratio (a combination of the loss and expense ratios)

Combined Ratio = Loss Ratio +

LAE
Underwriting Expenses
+
Earned Premium
Written Premium

i. The loss ratio should not include LAE or it will be double counted.
ii. For insurers that compare UW expenses incurred at the onset of the policy to earned premium rather
than to written premium, the Combined Ratio = Loss Ratio + OER.
The combined ratio measures the profitability of a book of business.
Retention Ratio (a measure of the rate at which existing insureds renew their policies upon expiration)

Retention Ratio =

Number of Policies Renewed
Number of Potential Renewal Policies

If 100,000 policies are anticipated to renew in a given month and 85,000 of the insureds choose to renew,
then the retention ratio is 85% (= 85,000 / 100,000).
Retention ratios are:
 used to gauge the competitiveness of rates and are closely examined following rate changes or major
changes in service.
 a key parameter in projecting future premium volume.
Close Ratio (a.k.a. hit ratio, quote-to-close ratio, or conversion rate is a measure of the rate at which
prospective insureds accept a new business quote)

Close Ratio 

Number of Accepted Quotes
Number of Quotes

Example: If an insurer makes 300,000 quotes in a month and generates 60,000 new policies from those
quotes, then the close ratio is 20% (= 60,000 / 300,000).
Close ratios and changes in the close ratios are monitored by product management and marketing departments.
Closed ratios are used to determine the competitiveness of rates for new business.

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Chapter 1 - Introduction
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.

Key Concepts

11 - 11

1. Relationship between price, cost and profit
2. Rating manuals
3. Basic insurance terms
a. Exposure
b. Premium
c. Claim
d. Loss
e. Loss adjustment expense
f. Underwriting expense
g. Underwriting profit
4. Goal of ratemaking
a. Fundamental insurance equation
b. Ratemaking is prospective
c. Overall and individual balance
5. Basic insurance ratios
a. Frequency
b. Severity
c. Pure premium
d. Average premium
e. Loss ratio
f. Loss adjustment expense ratio
g. Underwriting expense ratio
h. Operating expense ratio
i. Combined ratio
j. Retention ratio
k. Close ratio

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Chapter 1 - Introduction
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
The predecessor papers to the current syllabus reading “Basic Ratemaking” by Werner, G.
and Modlin, C. were numerous. While past CAS questions were drawn from prior syllabus
readings, the ones shown below remain relevant to the content covered in this chapter.
Questions from the 1990 exam
4. (1 point) According to the Study Note Reading - Foundations of Casualty Actuarial Science, Chapter 1,
“Ratemaking," which of the following are true?
1. The description of the goal of the ratemaking process includes consideration of generating a reasonablereturn on funds provided by investors.
2. Regulatory review generally requires that rates shall not be inadequate, excessive or unfairly
discriminatory between risks of like kind and quality.
3. The two basic approaches used in manual ratemaking are the pure premium method and the loss ratio
method. (see chapter 8)
A. 1.
B. 2
C. 1, 3
D. 2, 3
E. 1, 2, 3

Questions from the 2008 exam
13. (2.0 points) Define the following terms.
a. Written premium
b. Earned premium
c. Unearned premium
d. In-force premium

Questions from the 2010 exam
11. (2 points)
a. (0.75 point) Explain how the standard economic formula, Price = Cost + Profit, relates to the fundamental
insurance equation.
b. (1.25 points) Company ABC replaced inexperienced adjusters with experienced adjusters who have a
greater knowledge of the product. Explain the impact of this change on each component of the
fundamental insurance equation.
12. (1 point) Given the following information:
• 2008 earned premium = $200,000
• 2008 incurred losses = $125,000
• Loss adjustment expense ratio = 0.14
• Underwriting expense ratio = 0.25
Calculate the combined ratio.

Questions from the 2011 exam
8. (1.25 points) Given the following information:
Calendar Year 2010
Written premium
$280.00
Earned premium
$308.00
Commissions
$33.60
Taxes, licenses and fees
$9.80
General expenses
$36.96
LAE ratio (to loss)
8.2%
Combined ratio
100%
Calculate the 2010 operating expense ratio.

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Chapter 1 - Introduction
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2012 exam
10. (2.5 points) The fundamental insurance equation is:
Premium = Losses + Loss Adjustment Expense + Underwriting Expenses + Underwriting Profit
a. (1 point) Werner and Modlin state that "It is important to consider the [fundamental insurance]
equation at the individual or segment level" in addition to the aggregate level.
Discuss two reasons it would be acceptable to maintain an imbalance in the fundamental
insurance equation at the individual or segment level.
b. (1.5 points) Reconcile an imbalance in the fundamental insurance equation with the following
quote from the Statement of Principles Regarding Property & Casualty Insurance Ratemaking:
"A rate provides for the costs associated with an individual risk transfer."

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Answer E.

Solutions to questions from the 2008:
Model Solution - Question 13
a. Written Premium are the dollar amounts charged by an insurer for policies written during a specific time period.
The total policy premium is included in the written premium.
b. Earned Premium is the amount of the policy premiums that have been exposed to risk during a specified time
period. Earned Premium is directly proportional to the portion of the policy period covered by the insurer during
the specified time period.
c. Unearned Premium is the portion of policy premium that has yet to be exposed to risk as it covers a future time
period during which the policy will be in-effect.
d. In-force Premium is the total written premium of all policies in effect at a specific point in time.

Solutions to questions from the 2010:
Question 11
a. Explain how the standard economic formula, Price = Cost + Profit, relates to the fundamental insurance
equation.
Premium = Loss + Loss adjustment expense + UW expense + UW profit
↑
↑
Price

Cost

Profit

b. Explain the impact of using experienced adjusters on each component of the fundamental insurance equation.
* Losses will decrease due to better (more judicious) claims adjusting
* Loss adjustment expenses will increase due to a larger fee paid to more experienced claims adjusters
* UW expense will remain the same as they cover the costs incurred at the onset of the policy (e.g.
commissions, other acquisition, taxes, licenses, and fees) and expenses incurred throughout the policy
(e.g. general expenses), which are not impacted by the use of more experienced adjusters
Comments: The following only makes sense if the reduction in losses is greater than the increase in LAE
(which is a reasonable assumption since losses comprise a very large percentage of premiums).
* Premium will decrease if the UW profit is to remain the same
* UW profit will increase if the Premium is to remain the same

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Chapter 1 - Introduction
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2010 (continued):
Question 12: Calculate the combined ratio, using the given data in the problem.
Step 1: Write an equation to determine the combined ratio

Combined Ratio = Loss Ratio +

LAE
Underwriting Expenses
+
= Loss Ratio  OER
Earned Premium
Written Premium

Total Losses
Total Premium
Total Loss Adjustment Expenses
LAE Ratio 
Total Losses
Total UW Expenses
UW Expense Ratio =
Total Premium
LAE
OER = UW Expense Ratio +
Total Earned Premium
Loss Ratio =

Step 2: Using equations in Step 1, and the data given in the problem, solve for the components of the
combined ratio
Loss ratio = 125,000/200,000 = 0.625
LAE = LAE ratio * Incurred Losses = 0.14 x 125,000 = 17,500
Operating expense ratio = OER = UW expense ratio + LAE/Earned Premium
= .25 + 17,500/200,000 = .3375
Combined ratio = Loss ratio + OER = 0.625 + .3375 = .9625 = 96.25%

Solutions to questions from the 2011:
8. Calculate the 2010 operating expense ratio.
Question 8 – Model Solution 1
Combined ratio = Loss Ratio + LAE/EPremium + UW Expense Ratio
OER = LAE/EPremium + UW Expense Ratio
UW Expense Ratio = TaxesLicFee/WP + Comm/WP + General/EP
= (9.80 + 33.6)/280 + 36.96/308 = .275
LR * (1+LAE ratio) = 1 - UW Expense Ratio = 1 - .275 = .725
CR = 1.0 = L/EP + .082L/EP + .275; since .082 = LAE/L, LAE = .082L
Solve for L:
L = LR*EP/(1+LAE). L= .725*308/1.082 = 206.377
Solve for LAE: LAE = .082 * L = .082 * 206.377 = 16.923
OER = 16.923/308 +.275 = .32994
Question 8 – Model Solution 2
Combined ratio = Loss Ratio + OER = LR * (1+LAE ratio) + U/W Expense Ratio
Solve for the LR: 100% = LR * (1+8.2%) + (33.60 + 9.80)/280 + 36.96/308; LR = 67%
OER = Combined Ratio – Loss Ratio = 100% - 67% = 33%

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Chapter 1 - Introduction
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2011
Question 8 – Model Solution 3
OER = LAE/E Premium + UW Expense Ratio
Underwriting expense ratio = 33.60/280 + 9.8/280 + 36.96/308 = 0.275
Combined ratio = Loss Ratio (1 + 0.082) + UW Expense/Written premium
UW Expense/Written Premium = [33.60 + 9.8 + 36.96]/280 = 0.287
Combined ratio = LR(1.082) + 0.287
Solve for LR: LR = 0.65896
CR = 1.0 = 0.65896 + LAE/Earned premium + 0.287
Solve for LAE/EP: LAE/Earned Premium = 0.054
So operating expense ratio = 0.054 + 0.275 = 0.329

Questions from the 2012 exam
10a. (1 point) Werner and Modlin state that "It is important to consider the [fundamental insurance]
equation at the individual or segment level" in addition to the aggregate level.
Discuss two reasons it would be acceptable to maintain an imbalance in the fundamental insurance
equation at the individual or segment level.
Question 10 Model - Solution 1 – part a
1. Maintain competitive position. If changing rates would hurt your competitive position then it may be
acceptable to take less of a change and have an unbalanced Fund. Ins Equation -> In other words hurting
retention enough to offset increase.
2. If the relative cost of the change outweighs the benefit. If the operational cost of changing rating
algorithms or data collection processes outweigh the change in premiums associated with the change then
it could be appropriate to have an unbalanced Fund. Ins Equation

Question 10 Model - Solution 2 – part a
1. It might due to a regulatory constraint. The regulator restrict the rate change (e.g. capped at +/- 25%)
2. Marketing Constraint. If the company’s marketing objective is to increase the market share on age
group 50-55 drivers, it may reduce rate to attract this group of insureds. Company may have look at the
long term profitability of the book using an asset share pricing technique.
Examiners Comments
This part of question was generally answered well. Common answers that received credit included marketing
considerations (riding the market cycle, competitor pressure), regulatory considerations (e.g. cap on rate
changes, restrictions on rating variables), and an asset share pricing approach that anticipates future profits at
the expense of initial costs.

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Chapter 1 - Introduction
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2012 exam
10b. (1.5 points) Reconcile an imbalance in the fundamental insurance equation with the following quote
from the Statement of Principles Regarding Property & Casualty Insurance Ratemaking: "A rate
provides for the costs associated with an individual risk transfer."
Question 10 - Model Solution – part b
An actuarially sound indication many not always be implemented since an insurance company needs
to balance other objectives, such as marketing, then actuarially balancing premium and loss.
The actuary is allowed to deviate from this principle under influence of management, with the proper
disclosure.
Additionally asset sharing pricing techniques have demonstrated that under certain circumstances, it is
ultimately profitable to write business that currently produce a net loss.
Examiners Comments
Part b was not answered well.
By far the most common response was a mathematical balancing of the fundamental insurance
equation, either by raising the premium or lowering expenses. However, the question was asking
candidates to justify their reasoning for an imbalanced fundamental insurance equation from part A in light
of the actuarial standards of practice.
Successful candidates acknowledged that actuarial rate indications can balance the fundamental
insurance equation but that management may decide to choose premiums that differ from actuarial
indications, or that regulatory restrictions supersede all actuarial standards of practice.

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Chapter 2 – Rating Manuals
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Sec
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Description
Rating Manuals and Rules
Rate Pages
Rating Algorithms
Underwriting Guidelines
Homeowners Rating Manual Example
Medical Malpractice rating Manual Example
U. S. Workers Compensation Rating Manual Example
Key Concepts

Pages
13 - 14
14 - 15
15 - 16
16 - 17
17 - 23
23 - 28
29 - 34
34 - 34

Rating Manuals and Rules

13 - 14

Rating manuals are used by insurers to classify risks and calculate the premium for a given risk.
This chapter describes what is contained in rate manuals and gives examples of different rating components
for various lines of business.
For most lines of business, the following is necessary to calculate the premium for a given risk:
 Rules
Found in the insurer’s rating manual
 Rate pages (i.e. base rates, rating tables, and fees)
Found in the insurer’s rating manual
 Rating algorithm
Found in the insurer’s rating manual
 Underwriting guidelines
Found in the insurer’s UW manual
RULES
Rating manual rules:
 contain qualitative information to apply to the quantitative rating algorithms contained in the manual.
 begin with definitions of the risk being insured (e.g. rules for a homeowners insurer may define what is
considered a primary residence)
 provides a summary of policy forms offered to the insured (if more than one form is offered)
 summarize what is covered (e.g. types of liability or damage)
 outline limitations or exclusion of coverage.
 outline premium determination considerations (e.g. minimum premium, down payments, and refunds in
the event of cancellation).
Rules define how to classify a risk before the rating algorithm can be applied.
Class ratemaking groups risks with similar characteristics (represented by rating variables) and varies the
rate accordingly.
Rules also contain optional insurance coverage information (a.k.a. endorsements or riders), which:
 describe the optional coverage, any restrictions on such coverage, and any applicable classification rules.
 may contain the rating algorithm for the optional coverage as well.
In addition to rules, insurers use UW guidelines to specify additional acceptability criteria (e.g. an insurer may
choose not to write a risk with two or more convictions of driving under the influence).
UW guidelines are usually found in a separate underwriting manual.

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Chapter 2 – Rating Manuals
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
2

Rate Pages

14 - 15

Rate pages contain inputs (e.g. base rates, rating tables, and fees) to calculate premium.
A base risk is a risk profile pre-defined by the insurer.
The base risk can be a set of common risk characteristics or can be chosen based on marketing objectives.
Example 1: The base risk for personal auto collision coverage may be an adult, married male, with a $500
deductible, who lives in a very populated area, etc.
 The insurer may have an objective to encourage new insureds to purchase a deductible of $500 or
higher (even though it may have more policies with a $250 deductible).
If the base is set at the $500 deductible, it will be used in the initial premium quote. But if the insured
requests a comparison quote with a $250 deductible, a higher premium will result (relative to using a
base set at a $250 deductible), which may deter the insured psychologically.
Example 2: A multi-product discount for homeowners who have an auto policy with the same insurer.
 If the insurer sets the base equal to those who qualify for the discount, then there will be an increase in
premium for those who do not qualify for the discount.
Although the premium charged is the same whether buying a single or multi-product discount, a
discount has more positive appeal than an increase in premium.
The base rate is the rate that applies to the base risk (and is usually not the average rate).
If the product contains multiple coverages priced separately (as in personal auto insurance), then there is a
separate base risk, base rate, and rating tables for each coverage.
Rates for all risk profiles, other than the base profile, will vary from the base rate.
The rate variation for different risk characteristics occurs by modifying the base rate (e.g. applying
multipliers, addends, etc. in the rating algorithm).
 Characteristics are rating variables (a.k.a. discounts/surcharges or credits/debits) and the rate
variations are contained in rating tables.
 The variations from the base rate are referred to as relativities, factors, or multipliers (if applied to the
rating algorithm multiplicatively) or addends (if applied to the base rate or some other figure in an
additive or subtractive manner).
Rating Variables for various lines of insurance are as follows:
Type of Insurance
Rating Variables
Personal Automobile
Driver Age and Gender, Model Year, Accident History
Homeowners
Amount of Insurance, Age of Home, Construction Type
Workers Compensation
Occupation Class Code
Commercial General Liability
Classification, Territory, Limit of Liability
Medical Malpractice
Specialty, Territory, Limit of Liability
Commercial Automobile
Driver Class, Territory, Limit of Liability
Rate pages contain all the components needed to calculate rates.

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Chapter 2 – Rating Manuals
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Expenses:
The premium charged must consider expenses incurred in acquiring and servicing policies.
 Some expenses vary by the amount of premium (e.g. commission is usually a % of the premium)
 Some expenses are fixed regardless of the premium (e.g. the cost of issuing a policy).
An insurer may include an explicit expense fee in the rating algorithm to account for fixed expenses and
incorporate a provision within the base rate to account for variable expenses.
Otherwise, an insurer may incorporate all expenses via a provision within the base rates.
In this case, the insurer may have a minimum premium so that the premium charged is adequate to cover
expenses and an amount for minimal expected losses.

Rating Algorithms

15 - 16

Rating algorithms describes how to combine the components in the rules and rate pages to calculate the
premium charged for any risk not pre-printed in a rate table.
The algorithm includes instructions such as:
 the order in which rating variables should be applied
 how rating variables are applied in calculating premium (e.g. multiplicative, additive, or some unique
mathematical expression)
 maximum and minimum premiums (or in some cases the maximum discount or surcharge to be applied)
 specifics with how rounding takes place.
Separate rating algorithms by coverage may apply (if the product contains multiple coverages).
A few examples are included in this chapter for illustrative purposes.

Underwriting Guidelines

16 - 17

UW guidelines criteria are used to specify:
 Decisions to accept, decline, or refer risks. (e.g. risks with a certain set of characteristics (e.g., a
household with two or more losses in the last 12 months) may not be eligible for insurance or the
application must be referred to a senior underwriter).
 Company placement.
An insurance group may have one of its companies provide personal auto insurance to preferred/low-risk
drivers and another to provide insurance to nonstandard/high-risk drivers.
Establishing separate companies to achieve this purpose is due to either:
i. regulatory issues (cannot get approval for the full spectrum of rates within one company) or
ii. different distribution systems (one company selling through agents and another selling directly to
the consumer).
 Tier placement. Jurisdictions may permit insurers to charge different rates within a single company to
risks with different underwriting characteristics.
i. UW guidelines specify the rules to assign the insured to the correct tier.
ii. The rating algorithm and rate pages specify how the tier placement affects the premium calculation.
 Schedule rating credits/debits (used in commercial lines products to vary premium from manual rates).
SR applies credits and debits depending on the presence or absence of characteristics.
i. SR may be specific and no judgment is required or permitted.
ii. SR may allow the underwriter to use subjective factors in applying credits or debits.

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Chapter 2 – Rating Manuals
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Note: While UW criteria has been historically subjective in nature, there has been a trend over time (especially
for personal lines products) to designate new explanatory variables as UW criteria, which can then be
used for placement into rating tiers or separate companies.
The trend to designate new explanatory variables as UW criteria has given some companies a
competitive advantage by reducing the transparency of the rating algorithm.
Examples of Underwriting Characteristics used in Various Lines of Insurance
Type of Insurance
Underwriting Characteristics
Personal Automobile
Insurance Credit Score, Homeownership, Prior Bodily Injury Limits
Homeowners
Insurance Credit Score, Prior Loss Information, Age of Home
Workers Compensation
Safety Programs, Number of Employees, Prior Loss Information
Commercial General Liability
Insurance Credit Score, Years in Business, Number of Employees
Medical Malpractice
Patient Complaint History, Years Since Residency,
Number of Weekly Patients
Commercial Automobile
Driver Tenure, Average Driver Age, Earnings Stability

Homeowners Rating Manual Example

17 - 23

The following is an example of a rating algorithm for a homeowners policy issued by the Wicked Good
Insurance Company (Wicked Good or WGIC).
WGIC’s homeowners rating manual is used to calculate the premium for a homeowners insurance policy.
The following are excerpts from WGIC’s homeowners rating manual.
Base Rates
The exposure base for homeowners insurance is a home insured for one year.
The base rate (an all-peril base rate) for WGIC is shown below.
Coverage
Base Rate
All Perils Combined
$500

Rating and Underwriting Characteristics
Amount of Insurance (AOL)
AOI:
 is a key rating variable for homeowners insurance.
 represents the amount of coverage purchased to cover damage to the dwelling and is the maximum
amount the insurer expects to pay to repair or replace the home.
The table below shows rate relativities to apply to WGIC’s base rate depending on the AOI purchased.

Note that the base rate corresponds to a home with an amount of insurance of $200,000, and thus has a AOI
rate relativity of 1.00.

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Chapter 2 – Rating Manuals
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Amount of Insurance (AOI) Rating Table
AOI (in thousands)
$ 80
$ 95
:::
$170
$185
$200
$215
:::
$410
$425
$440
$455
$470
$485
$500
Additional $15K

Rate Relativity
0.56
0.63
:::
0.91
0.96
1.00
1.04
:::
1.51
1.54
1.57
1.60
1.63
1.66
1.69
0.03

If a policyholder purchases $425,000 of insurance for his home, a rate relativity of 1.54 is applied to the base
rate. Straight-line interpolation is used for values not listed in the table.
Territory
The location of the home is a key rating variable.
 Homeowners insurers group similar geographic units (e.g. zip codes) to form rating territories.
 WGIC grouped zip codes into five distinct rating territories (with rate relativities shown below).
 Territory 3 is the base territory (and thus has a relativity of 1.00) and all other territories are expressed
relative to Territory 3.
Territory Rate Relativity
1
0.80
2
0.90
3
4
5

1.00
1.10
1.15

Protection Class and Construction Type
WGIC’s homeowners rates vary by fire protection class and construction type.
 Class 1 indicates the highest quality protection while class 10 refers to the lowest quality protection.
Within each class, there is a separate relativity based on construction type (frame and masonry).
Frame construction is more susceptible to loss than masonry and therefore frame relativities are
higher than the masonry relativities across every protection class.
 The base rate for this two-way variable is Protection Class 1-4 Frame (although Protection Class 5
Masonry coincidentally has a relativity of 1.00).

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Chapter 2 – Rating Manuals
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Protection Class / Construction Type Rating Table
Protection Class
Construction Type
Frame
Masonry
1-4
1.00
0.90
5
1.05
1.00
6
1.10
1.05
7
1.15
1.10
8
1.25
1.15
9
2.10
1.75
10
2.30
1.90
Underwriting Tier
WGIC uses UW characteristics (used to place insurance policies into one of four distinct underwriting tiers
based on the overall riskiness of the exposure to loss) that are not explicitly shown in the rating manual.
Underwriting Tier Rating Table
Tier
Rate Relativity
A
0.80
B
0.95
C
1.00
D
1.45
Tier D is considered the most risky and has the highest rate relativity.
Deductible
Policyholders choose their deductible. Rate relativities for each deductible are shown in the table below.
Deductible
Rate Relativity
$250
1.00
$500
0.95
$1,000
0.85
$5,000
0.70
Miscellaneous Credits
Wicked Good offers the following discounts:
Miscellaneous Credit
Credit Amount
New Home Discount
20%
5-Year Claims-Free Discount
10%
Multi-Policy Discount
7%
Insurers offering a large number of discounts will have a maximum discount percentage that can be used,
however Wicked Good does not limit the overall cumulative discount.

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Chapter 2 – Rating Manuals
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Additional Optional Coverages
The basic homeowners policy includes:
i. a $100,000 limit for liability coverage and a $500 limit for medical coverage (this split limit is often
expressed as $100,000/$500).
ii. a $2,500 inside limit to jewelry losses within the contents coverage.
The following tables show the additional premium charged if the policyholder elects to purchase additional
higher limits:
Jewelry Coverage Rate
Limit
Additive
$ 2,500
Included
$ 5,000
$35
$10,000
$60
Liability/Medical Rate
Limit
Additive
$100,000/$500
Included
$300,000/$1,000
$25
$500,000/$2,500
$45
Expense Fee
WGIC has an explicit expense fee to cover fixed expenses incurred in the acquiring and servicing policies.
The expense fee is $50 per policy as shown in the table below.
Policy Fee
$50
Homeowners Rating Algorithm for WGIC
The rating algorithm to calculate the final premium for a homeowners policy for WGIC is:
Total Premium =
All-Peril Base Rate x AOI Relativity
x Territory Relativity
x Protection Class / Construction Type Relativity
x Underwriting Tier Relativity
x Deductible Credit
x [1.0 - New Home Discount – Claims-Free Discount]
x [1.0 - Multi-Policy Discount]
+ Increased Jewelry Coverage Rate
+ Increased Liability/Medical Coverage Rate
+ Policy Fee.
Rounding is common and WGIC rounds to the penny after each step and to the whole dollar at the final step.

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Chapter 2 – Rating Manuals
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Homeowners Rate Calculation Example for WGIC
WGIC is preparing a renewal quote for a homeowner with the following risk characteristics:
• Amount of insurance = $215,000
• The insured lives in Territory 4.
• The home is frame construction located in Fire Protection Class 7.
• Based on the insured’s credit score, tenure with the company, and loss history, the policy is in UW Tier C.
• The insured opts for a $1,000 deductible.
• The home falls under the definition of a new home as defined in Wicked Good’s rating rules.
• The insured is eligible for the five-year claims-free discount.
• There is no corresponding auto or excess liability policy written with WGIC.
• The insured is eligible for the five-year claims-free discount.
• There is no corresponding auto or excess liability policy written with WGIC.
• The policyholder opts to increase coverage for jewelry to $5,000 and to increase liability/medical
coverage limits to $300,000/$1,000.
Entries from Rating Manual
Base Rate
$500
AOI Relativity
1.04
Territory Relativity
1.10
Protection Class / Construction Type Relativity 1.15
Underwriting Tier Relativity
1.00
Deductible Credit
0.85
New Home Discount
20%
Claims-Free Discount
10%
Multi-Policy Discount
0%
Increased Jewelry Coverage Rate
$35
Increased Liability/Medical Coverage Rate
$25
Expense Fee
$50
The rating algorithm from the rating manual can be applied to calculate the final premium for the policy:

$501  $500 *1.04 *1.10 *1.15 *1.00 * 0.85 *[1.0 - 0.20 - 0.10]*[1.0 - 0]  $35  $25  $50.

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Chapter 2 – Rating Manuals
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Medical Malpractice rating Manual Example

23 - 28

The following a rating algorithm for a medical malpractice (MM) policy issued by WGIC for its Nurses
Professional Liability program. WGIC’s rating manual (with excerpts shown below) is used to calculate the
premium.
Base Rates
The exposure base for MM insurance is a medical professional insured for one year.
Wicked Good’s rating manual shows base rates for annual MM coverage for its nurses program, which
vary depending on whether the professional is employed or operates his or her own practice.
Base Rates
Annual Rate Per
Nurse
Employed
$2,500
Self-Employed
$3,000
Rating and Underwriting Characteristics
Specialty Factor
Wicked Good varies malpractice premium based on specialties shown in the table below.
Specialty Rating Table
Rate
Specialty
Relativity
Psychiatric
0.80
Family Practice
1.00
Pediatrics
1.10
Obstetrics
1.30
All Other Specialties
1.05
Nurses practicing in obstetrics have the highest rate relativity due to higher exposure to loss.
Part-time Status
Professionals who work 20 hours or less per week are part-time professionals, and WG has determined
that the rate should be 50% of the base rate shown in the table below.
Part-time Rating Table
Rate Relativity
Full-time
1.00
Part-time
0.50
Territory
Rate relativities also apply to the base rate to calculate the rate for a nurse in a specific territory.
Territory Rate Relativity
1
0.80
2
1.00
3
1.25
4
1.50

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Chapter 2 – Rating Manuals
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Claims-free Discount
Individual insureds who have been with WGIC for at least three consecutive years preceding the effective
date of the current policy may qualify for a claims-free discount.
 To qualify, the individual insured cannot have cumulative reported losses in X/S of $5,000 over the prior 3
years.
 The amount of the claims-free discount is 15%.
Schedule Rating (SR)
Commercial lines insurers incorporate SR into their rating algorithms to adjust manual premium based on
objective criteria or underwriter judgment.
WGIC’s schedule rating plan includes the following credits and debits.
A. Continuing Education – A credit of up to 25% for attendance at approved continuing education
courses and seminars. The total hours spent at courses and seminars must be at least 15 hours in
the prior 12 months.
B. Procedure – A debit of up to 25% for nurses who have professional licenses and/or scope of
practice in high-risk exposure areas such as invasive surgery or pediatric care.
C. Workplace Setting – A debit of up to 25% for nurses that work in high-risk workplace settings (e.g.
surgical centers and nursing homes).
A maximum aggregate schedule rating credit or debit of 25% is used by WGIG.
Limit Factors
WGIC offers different per claim and annual aggregate limits for its Nurse’s Professional Liability program.
The following are relativities corresponding to each limit option:
Limit Rating Table
Limit Option
Rate Relativity
$100K/$300K
0.60
$500K/$1M
0.80
$1M/$3M
1.00
$2M/$4M
1.15
WGIC pays all ALAE in addition to the limit shown.
Deductible
Deductible options available to the insured reduce premium and the associated credit are shown below.
Deductible Rating Table
Deductible
(Per Claim)
Credit
None
0%
$1,000
5%
$5,000
8%

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Chapter 2 – Rating Manuals
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Claims-made Factor
WGIC writes claims-made MM policies as opposed to occurrence policies.
 For CM policies, the coverage trigger is the date the claim is reported rather than the date the event
occurs.
 A policyholder who buys a CM policy for the first time is only offered coverage for claims occurring
after the start of the policy and reported during the year.
 When the CM policy is renewed, coverage is provided for claims occurring after the original inception
date and reported during the policy period.
 Also, an extended reporting endorsement covers claims that occur during the coverage period but are
reported after the policy terminates (e.g. a doctor who retires may purchase an extended reporting
endorsement to cover claims reported after the MM policy terminates).
The extended reporting endorsement factors adjust the premium based Years of Prior Claims-made
Coverage. See Chapter 16 for more details on CM coverage.
Claims-Made Maturity Factors
Maturity
Factor
1st Year
0.200
2nd Year
0.400
3rd Year
0.800
4th Year
0.900
5th Year
0.950
6th Year
0.975
Mature
1.000
Extended Reporting Endorsement Factors
Years of Prior Factor
Claims-made
Coverage
12 Month
0.940
24 Month
1.700
36 Month
2.000
48 Month
2.250
60 Month
2.400
Group Credit
The size of the credit depends on the number of nurses that are insured under the policy.
Group Credit
Number of
Credit
Nurses
1
0%
2 – 14
5%
15+
10%
The final premium (including the group credit) should be calculated for each nurse and aggregated for all
professionals to determine the premium for the group policy.

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Chapter 2 – Rating Manuals
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Minimum Premium
The rating manual specifies that the minimum premium for each nurse, after all discounts, is $100.
Medical Malpractice Rating Algorithm for WGIC
 Rating variables are applied multiplicatively, not additively, in consecutive order.
 Premium is rounded to the nearest penny after each step and to the nearest dollar amount at the end to
determine the final premium per professional.
Total Premium per Professional = [Max of Min Premium in the rating manual of $100 or
(Base Rate per Nurse
x Specialty Relativity
x Part-time Status Relativity
x Territory Relativity
x (1.0 - Claims-free Discount)
x (1.0 +/- Schedule Rating Debit/Credit) x Limit Relativity
x (1.0 - Deductible Credit)
x Claims-made Factor
x (1.0 - Group Credit ))]
The total policy premium for a policy with multiple professionals is the sum of the premium for the
individual professionals on the policy.
Medical Malpractice Rate Calculation Example for WGIC
A practice of five nurses applied for MM coverage with WGIC.
Quoted premium was $6,500 for a single policy covering the five professionals.
The practice has recently added a psychiatric nurse, and has requested a new quote from WGIC to cover
all six professionals on a single policy. Assume the following characteristics:
 The new nurse is an employed professional who works 15 hours per week.
 He was previously covered by an occurrence policy and is applying for a CM policy with WGIC.
 He practices in Wicked Good’s Territory 3.
 He attended five hours of approved continuing education courses in the prior 12 months.
 He holds a professional license in senior care, which is considered high risk. He also works in a
senior care facility. The underwriter has chosen to apply debits of 25% for each of these criteria,
but the maximum aggregate debit allowable is 25%.
 The policy has $1M/$3M of coverage with a $1,000 deductible per claim.

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Chapter 2 – Rating Manuals
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
The following rating tables from WGIC’s rating manual is used to calculate the premium
Entries from Rating Manual
Employed Annual Rate
$2,500
Specialty Relativity
0.80
Part-time Status Relativity
0.50
Territory 3 Relativity
1.25
Schedule Rating (subject to 25% maximum)
0%+25%+25% (capped at 25%)
Limit Relativity for $1M/$3M
1.00
Credit for $1000 Deductible
5%
Claims-made Factor
0.20
Group Credit
5%
Minimum Premium
$100
Using the rating manual’s rating algorithm, the premium for the individual nurse is calculated as follows:
$282 = $2,500 x 0.80 x 0.50 x 1.25 x [1.00 + 0.25] x 1.00 x [1.00 - 0.05] x 0.20 x [1.00 - 0.05].
Since this premium is greater than the minimum premium per nurse of $100, it applies
The total premium for the six individuals combined is $6,782 = $6,500 + $282.

U. S. Workers Compensation Rating Manual Example

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Workers compensation (WC) insurance is a heavily regulated line of business, and insurers are required
to submit statistical information on WC losses and premium in detail to the National Council on
Compensation Insurance (NCCI), which collects and aggregates the data for ratemaking purposes.
NCCI is the licensed rating and statistical organization for most states, but several states have
independent bureaus or operate as monopolistic plans.
NCCI provides WC insurers with loss cost (the portion of the rates that covers the expected future
losses and LAE for a policy) estimates.
WC insurers calculate their own rates by adjusting the NCCI loss costs to account for their UW
expenses and any perceived difference in loss potential.
The WC ratemaking process produces a rate manual showing the manual premium for each risk.
The premium collected by the insurer is net premium (manual premium adjusted for premium discounts,
individual risk rating modifications (e.g. schedule rating, experience rating), and expense constants).
WGIC writes WC insurance for small companies with 50 employees or less, relies on NCCI for the overall loss
costs and rating tables, but is able to determine its expense provision needed to profitably write business.

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Chapter 2 – Rating Manuals
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Class Rate
The classification system groups employers with similar operations and similar loss exposures based on job
duties performed by the employees.
The table below shows class rates for specific operations (in this case, retirement centers) that WGIC writes, and
are based on the NCCI class rates, adjusted for WGIC’s expenses and perceived differences in loss potential.
Class Rates
Rate per
$100 of
Class
Payroll
8810-Clerical
0.49
8825-Food Service Employees
2.77
8824-Health Care Employees
3.99
8826-All Other Employees
3.79
To calculate manual premium:
 determine which classes best describe the activities of the company seeking insurance.
 estimate the amount of exposure ($100s of payroll) expected for each class during the policy period
using the insured’s data.
 multiply the rate per $100 of payroll by the estimated payroll for each class, and aggregate across all
classes for which the prospective insured has exposures to compute manual premium.

Rating and Underwriting Characteristics
Experience Rating (ER)
Manual rates are averages reflecting the usual conditions found in each class.
Manual rates are adjusted using ER to reflect that each risk within a class is different to some extent in
terms of loss potential.
 ER applies for larger policies (which are believed to have more stable loss experience) and NCCI
designates minimum aggregate manual premium for a company to be eligible for ER.
 Regulators mandate that ER be used if the employer meets the industry eligibility requirements.
When using ER, manual premium is adjusted upward if the actual losses for the company are higher than
expected and vice versa. See Chapter 15 for more information on ER.
WGIC only insures small companies and thus ER is not applicable to its insureds.
Schedule Rating (SR)
WGIC has a set of credits and debits that require the underwriter to apply judgment in the UW process.
The underwriter uses judgment (based on experience and internal guidelines) to select a value between the
maximum and minimum for each attribute that may apply for an insured’s workplace operations.
The range of schedule credits and debits that WG’s underwriters can apply is shown below:
 The overall maximum credit or debit that an underwriter can apply to a single policy is 25%.
 The policy must have an annual manual premium of at least $1,000 to qualify for schedule rating.

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Premises

Classification
Peculiarities

+/-10%

Schedule Rating
Range of Modification
Medical
Safety
Employees —
Facilities
Devices
Selection,
Training,
Supervision
+/-5%
-5% - 0%
+/-10%

Management —
Safety
Organization
+/-5%

Premium Credits
Additional premium credits can be offered to insureds for other factors that may reduce the risk of a WC
claim or limit the cost of a claim once an injury has occurred.
 These credits are not subject to any overall maximum credit.
Premium Credits
Factor
Credit
Pre-employment Drug Screening
5%
Employee Assistance Program
10%
Return-to-Work Program
5%

Expenses
Expense Constant
 A fixed fee (expense constant, and in WG’s case equal to $150 per policy) can be added to all policies to
cover expenses common to all WC policies.
 This fee does not vary by policy size and covers expenses that are not included in the manual rate.
Premium Discount (for administrative expenses that vary with policy size)
 Not all expenses increase uniformly as the premium increases (e.g. a company with $200,000 of payroll
may not generate twice the administrative expenses for the insurer as a $100,000 payroll insured).
 WC insurers reduce the premium for large insureds by using premium discounts to adjust for expense
savings.
Since WG writes only policies for small companies, it does not offer premium discounts.
Minimum Premium
The WC rating manual specifies that the minimum premium for any policy is $1,500.

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Chapter 2 – Rating Manuals
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Workers Compensation Rating Algorithm for WGIC
The rating algorithm to calculate the final premium for a given policy using the aforementioned rating manual
variables is as follows:
Total Premium = Higher of
N

[ (Classi rate x $ Payroll for classi / 100)

where N  number of classes

i 1

x (1.0+ Schedule Rating Factor)
x (1.0- Pre-Employment Drug Screening Credit)
x (1.0- Employee Assistance Program Credit)
x (1.0- Return-to-Work Program Credit)
+ Expense Constant]
and, the Minimum Premium specified in the rating manual ($1,500 in WGs case).
Premium is rounded to the nearest penny after each step and to the nearest dollar amount at the end to
determine the total premium (as stated in the manual)
ER factors and premium discounts do not appear in WGIC’s rating algorithm because these rating
variables do not apply to its book of business.
Workers Compensation Rate Calculation Example for WGIC
A retirement living center with the following employee classes groups has requested a quote.
Payroll by Class
Class
Payroll
8810 – Clerical
$35,000
8825 - Food Service Employees
$75,000
8824 - Health Care Employees
$100,000
8826 - All Other Employees & Salespersons, Drivers
$25,000
 The center has trained its entire staff in first aid and first aid equipment is available in the building.
 The center has been inspected by Wicked Good and the premises are clean and well-maintained.
 The center requires all employees to be drug-tested prior to employment.
Steps in computing manual premium.
Step 1: Compute aggregate manual premium.
Manual Premium by Class
Class
Payroll
Payroll/$100

8810 Clerical
8825 - Food Service Employees
8824 - Health Care Employees
8826 - All Other Employees
Total

(1)
$35,000
$75,000
$100,000
$25,000
$235,000

(2)=(1)/100
$350
$750
$1,000
$250

Rate per $100 of Class Manual
Payroll
Premium
(3)
(4)=(2)*(3)
0.49
$171.50
2.77
$2,077.50
3.99
$3,990.00
3.79
$947.50
$7,186.50

Total manual premium for the policy is $7,186.50 = $171.50 + $2,077.50 + $3,990.00 + $947.50.

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Chapter 2 – Rating Manuals
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Step 2: Underwriter determination of the following credits that should apply based on the retirement living
center’s characteristics:
Schedule Rating Modifications
Modification
Premises
Classification
Medical
Safety
Employees —
Management
Peculiarities
Facilities
Devices
Selection,
—Safety
Training,
Organization
Supervision
-10%
0%
0%
-2.5%
-5%
0%
The total credit (reduction to manual premium) for SR is 10% + 2.5% + 5% = 17.5%.
 The credit takes into account the first aid equipment, staff training, and cleanliness of the premises.
 Since the credit is less than the maximum allowable credit of 25%, the entire 17.5% credit is applied to
the manual premium.
The schedule rating factor applied to manual premium is 0.825 =1.000 - 0.175.
Step 3: Determine the following other factors that apply to the policy:
Entries from Wicked Good’s Rating Manual
Entries from Rating Manual
Pre-employment Drug Screening Credit
5%
Employee Assistance Program Credit
0%
Return-to-Work Program Credit
0%
Expense Constant
$150
The Employee Assistance Program credit and Return-to-Work credit do not apply to the policy because the
center does not have those programs.
Thus, the total premium for the policy is $5,782 = $7,186.50 x 0.825 x (1.0 - 0.05) x (1.0 - 0) x (1.0 - 0) + $150.
Since $5,782 is greater than the minimum premium per policy of $1,500, the total premium for the policy is $5,782.

Key Concepts

34 - 34

1. Basic components of a rate manual
a. Rules
b. Rate pages
c. Rating algorithm
d. Underwriting guidelines
2. Simple rating examples
a. Homeowners
b. Medical malpractice
c. U.S. workers compensation

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Chapter 3 – Ratemaking Data
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Sec
1
2
3
4

Description
Introduction and Internal Data
Data Aggregation
External Data
Key Concepts

Pages
36 - 42
42 - 44
44 - 47
47 - 47

Introduction and Internal Data

36 - 42

The quality of the final rates depends on the quality and quantity of data available.
Ratemaking involves analyzing rate adequacy for various insurance products.
Insurers use internal historical data or industry historical data to compute rates.
Collection and maintenance of relevant and consistent historical data is critical to the process.
Use of relevant external or internal data that has some relationship to a new product offering is key
when pricing a new insurance product.
This chapter focuses on:
 describing high-level specifications for ratemaking data
 discussing various data aggregation methods
 providing insights on external data.
INTERNAL DATA
Data requirements depend upon the type of ratemaking analyses being performed. Examples:
 A full multivariate classification analysis requires historical detail about each item being priced
(e.g. an individual risk, policy, or class of policies).
 Conducting an overall analysis of the adequacy of rates does not require a detailed
understanding of the individual characteristics for each policy
Two types of internal data involved in a ratemaking analysis are:
 risk information (e.g. exposures, premium, claim counts, losses, and claim or policy characteristics).
 accounting information (e.g. UW expenses and ULAE, and often available only at an aggregate level).
Data retrieval processes for ratemaking analysis vary from insurer to insurer.
Actuaries may have access to:
 a database specifically designed for ratemaking analyses.
 general databases containing detailed transactional information and then manipulate the data to
make it appropriate for ratemaking analysis.
The following sections describe a particular set of database specifications for risk information and
accounting information. The actuary should review the:
 key coverages of the individual insurance product and the type of ratemaking analysis to be
performed to conclude whether existing data specifications are adequate.
 available data for appropriateness for its intended purpose, reasonableness and
comprehensiveness of the data elements.

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Chapter 3 – Ratemaking Data
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Risk Data
Insurer databases record policy exposure and premium separately from losses in a claims database, however the
ratemaking analysis ultimately requires linking this information for ratemaking purposes.
Policy Database
A policy database captures records (i.e. individual policies or some subdivision of the policy) and fields
(i.e. explanatory information about the record).
A record is defined in a product’s policy database depending upon what exposure measure is used and
how premium is calculated.
Examples of policy database organization for different lines of business:
 In homeowners insurance, a record may be a home for an annual policy period.
 In U.S. WC insurance, rating is based on the payroll of industry classes so separate records are
maintained at the class level.
 In personal auto insurance, separate records are created for:
i. each coverage (though this could be handled via a coverage indicator field in the database).
ii. each auto on a policy (if multiple autos are insured on one policy) or separate records may be
maintained for individual operators on each auto.
Example: An auto policy insuring two drivers on two cars for six coverages could involve 24
records (or four records if coverage is handled as a field).
In addition, records are also subdivided according to any changes in the risk(s) during the policy period (i.e. if
a policy is amended during the policy term, separate records are created for the partial policy periods before
and after the change). See the examples provided later in this summary to better illustrate this.
Fields often present for each record in the policy database are:
• Policy identifier
• Risk identifier(s): When there are multiple risks on a policy, unique risk identifiers are required (e.g.
vehicle number and operator number may be necessary for personal auto databases).
• Relevant dates: While each record contains the effective and expiration dates for the policy or
coverage, separate records are maintained for individual risks and/or individual coverages on the
policy, and the start date of each risk/coverage is recorded.
(e.g. if collision coverage for a new car is added to an existing auto policy, a record is added with
the relevant start date noted).
• Premium: If the line of business has multiple coverages, premium is recorded by coverage as a
separate record or via a coverage indicator field.
(e.g. personal auto databases track premium separately for bodily injury, property damage,
comprehensive, collision and earned and in-force premium can be calculated from the data on record).
• Exposure: Is typically the written exposure but it can be recorded by coverage.
• Characteristics: Include rating variables, UW variables, etc. Some characteristics describe the
policy as a whole (e.g., the policy origination year), while others describe individual risks (e.g.
make/model of automobile) and consequently vary between different records on the same policy.

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Chapter 3 – Ratemaking Data
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Example: Homeowners policies used to construct a policy database:
 Policy A is written on 1/1/2010 with an annual premium of $1,100. The home is located in Territory 1
and the insured has a $250 deductible. The policy remains unchanged for the full term of the policy.
 Policy B is written on 4/1/2010 with an annual premium of $600. The home is located in Territory 2 and
the insured has a deductible of $250. The policy is canceled on 12/31/2010.
 Policy C is written on 7/1/2010 with an annual premium of $1,000. The home is located in Territory 3
and has a deductible of $500. On 1/1/2011, the insured decreases the deductible to $250. The full
annual term premium after the deductible change is $1,200.
Policy database construction:
Policy A can be represented with one record since expired at its original expiration date and had no changes.
Policy B is represented by two records because it was canceled before the policy expired.
The first record for contains information known at policy inception (e.g. one exposure and $600 in WP).
The second record represents an adjustment for the cancellation such that when aggregated, the two records
show a result net of cancellation. As the policy was canceled 75% of the way through the policy period, the
second record should show -0.25 exposure and -$150 (=25% x -$600) of written premium.
Policy C is represented by three records since it has a mid-term adjustment
The first record includes all the information at policy inception.
The second record negates the portion of the original policy that is unearned at the time of the amendment
(i.e. -0.50 exposure and -$500 premium and deductible equal to $500).
The third record represents the information applicable to the portion of the policy written with the new
deductible (i.e. +0.50 exposure and +$600 premium and deductible equal to $250).
Policy Database
Original
Original
Transaction
Effective Termination Effective
Policy
Date
Date
Date
A
B
B
C
C
C

01/01/10
04/01/10
04/01/10
07/01/10
07/01/10
07/01/10

12/31/10
03/31/11
03/31/11
06/30/11
06/30/11
06/30/11

01/01/10
04/01/10
12/31/10
07/01/10
01/01/11
01/01/11

Ded

Other Written Written
Terr Chars Exposure Premium

$250
$250
$250
$500
$500
$250

1
2
2
3
3
3

…
…
…
…
…
…

1.00
1.00
-0.25
1.00
-0.50
0.50

$1,100
$600
-$150
$1,000
-$500
$600

This is ordered by policy rather than transaction effective date.

In a more sophisticated data capture, information for:
 Policy B would be aggregated to one record that shows a “net” exposure of 0.75 and “net” written
premium of $450.
 Policy C would be aggregated to two records representing before and after the deductible change.
The first record would reflect the period of time with the $500 deductible and would have a “net”
exposure of 0.50 and “net” written premium of $500.
The second record reflecting the period of time with the $250 deductible would be identical to the third
record in the original example. The exposure is 0.50 and written premium is $600. This type of
transaction aggregation is required for statistical ratemaking analysis (e.g. GLMS see Chapter 10).

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Chapter 3 – Ratemaking Data
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Claims Database
Each record represents a transaction tied to a specific claim (e.g. a payment or a change in reserve).
Claims involving multiple coverages or causes of loss may be represented as separate records or via indicator
fields.
Fields often present for each record in a claims database are as follows:
• Policy identifier
• Risk identifier(s): If relevant, the claim database contains a way to identify the risk that had the claim.
This will be necessary to match the claim to the corresponding record in the policy database.
• Claim identifier: The claim database contains a unique identifier for each specific claim. This same
identifier is used if the claim has multiple claim transaction records.
• Claimant identifier: The claim database contains a unique identifier for each specific claimant on a
particular claim.
• Relevant loss dates: includes fields for the date of loss, the date the company was notified of the loss
(i.e. the report date), and the date of the transaction for the specific record (e.g. date of a loss payment,
reserve change, or claim status change).
• Claim status: Tracks whether the claim is open (i.e. still an active claim) or closed (i.e. has been
settled). For some policies, it may be common for claims to be re-opened, and it may be advantageous
to add the re-opened and re-closed status descriptions.
• Claim count: Identifies the number of claims by coverage associated with the loss occurrence.
Alternatively, if each record or a collection of records defines a single claim by coverage, aggregating
claim counts can be accomplished without this explicit field.
• Paid loss: Captures the payments made for each claim record. If there are multiple coverages, perils or
types of loss, the loss payments can be tracked in separate fields or separate records.
If the product is susceptible to catastrophic losses (e.g. hurricanes for property coverage), then
catastrophic payments are tracked separately either through a separate record or an indicator included
on the record.
• Event identifier: Identifies any extraordinary event (e.g. catastrophe) involving this particular claim.
• Case reserve: Includes the case reserve or the change in the case reserve at the time the transaction
is recorded (e.g. if a payment of $500 is made at a particular date, and this triggers a simultaneous
change in the case reserve, a record is established for this transaction and the paid loss and case
reserve fields are populated)
The case reserve is recorded in separate fields or records by coverage, peril or type of loss and by
catastrophe or non-catastrophe claim, if applicable (as with paid losses).
• Allocated loss adjustment expense:
If ALAE can be subdivided into finer categorization, additional fields may be used accordingly.
Insurers may not set ALAE reserves and only payments are tracked on the database.
If a case reserve for ALAE is set, it is maintained in the database, captured separately by coverage or
peril and by catastrophe or non-catastrophe, if applicable.
ULAE cannot be assigned to a specific claim and are handled elsewhere.
• Salvage/subrogation: If an insurer replaces property, it assumes ownership of the damaged property,
which may then be reconditioned and sold to offset part of the payments made for the loss; these
recoveries are called salvage. When an insurer pays for an insured’s loss, the company receives the
rights to subrogate (i.e. to recover any damages from a third party who was at fault to the loss event).
Any salvage or subrogation that offsets the loss is tracked and linked to the original claim, if possible.
• Claim characteristics: Insurers may collect characteristics associated with the claims (e.g. type of
injury, physician information). While studying the impacts of these characteristics on average claim size
may be interesting for certain purposes (e.g. loss reserve studies), only characteristics known for every
policyholder at the time of policy quotation are usable in the rating algorithm. V

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BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Example: Homeowners policies used to construct a claims database:
The following example policies can help clarify the data requirements.
• Policy A: A covered loss occurs on 1/1/2010. The claim is reported to the insurer on 1/15/2010,
and an initial case reserve of $10,000 is established. An initial payment of $1,000 is made on
3/1/2010, with a corresponding $1,000 reduction in the case reserve. A final payment of $9,000 is
made on 5/1/2010, and the claim is closed.
• Policy B: No claim activity.
• Policy C: A covered loss occurs on 10/1/2010, is reported on 10/15/2010, and a case reserve of
$18,000 is established. The insurer makes a payment of $2,000 on 12/15/2010, and reduces the
case reserve to $17,000. An additional payment of $7,000 is made on 3/1/2011, and the case
reserve is reduced to $15,000. The claim is closed on 3/1/2012, when the insurer makes a final
payment of $15,000 and receives a $1,000 salvage recovery by selling damaged property.
• Policy C: A second loss occurs on 2/1/2011. The claim is reported on 2/15/2011, and an initial
reserve of $15,000 is set. On 12/1/2011, the company pays a law firm $1,000 for fees related to
the handling of the claim. The claim is closed on that date with no loss payments made.
Claims database construction:
The claim from Policy A generates 3 separate records:
 one when the claim is reported and the initial reserve is set,
 one when the first payment is made,
 one when the last payment is made.
There are no claim records for Policy B as no claims were reported.
The two claims from Policy C generate six records:
 For claim 1, one record when the claim is reported and the initial reserve is set, and three for the
three different dates that payments and reserve adjustments are made.
 For claim 2, one record on the date it is reported and the initial reserve is set and a subsequent
record on the date the claim is closed.
Claim Database
Policy

Claim

Accident

Report

Transaction

Claim

Loss

Case

Paid

Salvage/

Number

Date

Status

Chars

Payment

Reserve

ALAE

Subro

01/10/10

01/15/10

Open

…

$10,000

01/10/10

01/15/10

03/01/10

Open

…

$1,000

$9,000

01/10/10

01/15/10

05/01/10

Closed

…

$9,000

10/01/10

10/15/10

Open

…

$18,000

10/01/10

10/15/10

12/15/10

Open

…

$2,000

$17,000

10/01/10

10/15/10

03/01/11

Open

…

$7,000

$15,000

10/01/10

10/15/10

03/01/12

Closed

…

$15,000

$1,000

02/01/11

02/15/11

Open

…

$15,000

02/01/11

02/15/11

12/01/11

Closed

…

$1,000

This is ordered by policy rather than transaction date.

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Chapter 3 – Ratemaking Data
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Accounting Information
Some required data for ratemaking is not specific to any one policy.
 The salary of the CEO is an expense that cannot be allocated to line of business or individual policy.
 UW expenses and ULAE fall into this category and should be tracked at the aggregate level.
UW expenses (incurred in acquiring and servicing policies) include general expenses, other acquisition
expenses, commissions and brokerage, and taxes, licenses, and fees.
 Commissions can be assigned to specific policies.
 General expenses (e.g. costs associated with the company’s buildings, and other acquisition expenses
like advertising costs) cannot be assigned to a specific claim and are tracked at the aggregate level.
Loss adjustment expenses (LAE) are expenses incurred in the process of settling claims.
 Allocated loss adjustment expenses (ALAE) are directly attributable to a specific claim and are captured
on the claim record.
 Unallocated loss adjustment expenses (ULAE) cannot be assigned to a specific claim, and include
items like the cost of a claim center or salaries of employees responsible for maintaining claims
records. Since ULAE cannot be assigned to a specific claim, these are tracked at the aggregate level.
Insurers track UW and ULAE expenses paid by calendar year.
Subdivision to line of business (LOB) and state may be approximated.
Aggregate figures are used to determine expense provisions used in the ratemaking process.

Data Aggregation

42 - 44

Policy, claim, and accounting databases must be aggregated for ratemaking purposes.
Three objectives when aggregating data for ratemaking purposes are:
1. Accurately matching losses and premium for the policy
2. Using the most recent data available
3. Minimizing the cost of data collection and retrieval.
Four data aggregation methods are calendar year (CY), AY (AY), policy year (PY), and report year (RY).
 Each method differs in how well it achieves the above listed objectives.
 Annual accounting periods are used although other periods (e.g. monthly, quarterly) can be used too.
The annual period does not need to be a CY (e.g. 1/1 to 12/31) but could be a fiscal year
(e.g. 7/1/ to 6/30), however CY, by definition needs to be 1/1/XX – 12/31/XX.

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Chapter 3 – Ratemaking Data
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
CY aggregation captures premium and loss transactions during a 12-month CY (without regard to policy
effective date, accident date, or report date of the claim).
 CY earned premium (EP) and earned exposure are those earned during a 12 month period.
At CY end, all premium and exposures are fixed.
 CY paid losses include all loss paid during the CY regardless of occurrence date or report date.
 CY Reported losses = paid losses + the change in case reserves during that twelve-month CY.
At the end of the CY, all reported losses are fixed.
Advantage of CY aggregation: data is quickly available at CY end. CY data is used for financial reporting
so there is no additional expense to aggregate the data this way for ratemaking purposes.
Disadvantage of CY aggregation: the mismatch in timing between premium and losses.
CY EP come from policies in force during the year (written either in the previous or the current CY).
Losses, however, may include payments and reserve changes on claims from policies issued years ago.
CY year aggregation for ratemaking analysis may be most appropriate for lines of business or individual
coverages in which losses are reported and settled relatively quickly (e.g. homeowners).
AY aggregation of premium and exposures follows the same precept as CY premium and exposures, and thus
the method is often referred to as CY-AY or FY-AY.
AY aggregation of losses considers losses for accidents that have occurred during a twelve-month period,
regardless of when the policy was issued or the claim was reported.
AY paid losses include loss payments only for those claims that occurred during the year.
AY reported losses = loss payments + plus case reserves only for those claims that occurred during the year.
At AY end, reported losses change as additional claims are reported, claims are paid, or reserves are changed.
Advantage: AY aggregation provides a better match of premium and losses than CY aggregation.
Losses on accidents occurring during the year are compared to EP on policies during the same year.
Since the AY is not closed (fixed) at year end, future development on known losses needs to be estimated.
Selecting a valuation date several months after year end allows the emergence of some development in the
data which may improve the estimation of ultimate losses.
PY aggregation (a.k.a. UW year) considers all premium and loss transactions on policies that were written
during a 12-month period, regardless of when the claim occurred or was reported, reserved, or paid.
 All premium and exposures earned on policies written during the year are part of that policy year’s
earned premium and earned exposures.
 Premium and exposures are fixed after the expiration date of all policies written during the year.
 PY paid losses include payments made on those claims covered by policies written during the year.
 PY reported losses = payments + case reserves only for those claims covered by policies written
during the year.
At PY end, losses change as additional claims occur, claims are paid, or reserves are changed.
Advantage: PY aggregation represents the best match between losses and premium (since losses on
policies written during the year are compared with premium earned on those same policies).
Disadvantage: Data takes longer to develop than both CY and AY, since PY exposures for a product with an
annual policy term are not fully earned until 24 months after the start of the PY.

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Chapter 3 – Ratemaking Data
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
RY aggregation is:
 similar to CY-AY except losses are aggregated according to when the claim was reported (as opposed
to when the claim occurred).
 used for commercial lines products using claims-made policies (e.g. medical malpractice).
See Chapter 16.
Overall versus Classification Analysis
When reviewing the adequacy of the overall rate level, the premium, losses, and exposures can be highly
summarized (aggregated by CY, AY, PY, or RY for the product and location (e.g. state) being analyzed).
If a class analysis is being performed, then the data must be at a more refined level.
 For a univariate classification analysis, the data can be aggregated by year (AY or PY) for each level
(e.g. territory) of the rating variable being studied.
 For a multivariate analysis, it is preferable to organize data at the individual policy or risk level.
Limited Data
Actuaries are sometimes required to perform ratemaking analysis and work with the data that is available and
use actuarial judgment to overcome the data deficiencies (e.g. if EP by territory normally used for an analysis of
auto territorial relativities is not available actuary may use in-force premium by territory to estimate the earned
premium by territory).

External Data

44 - 47

When pricing an existing line of business, it is helpful to supplement internal data with external data.
When pricing a new line of business, using external data may be necessary.
The most commonly used sources of external information are described below.
A. Statistical Plans
U.S. property and casualty (P&C) insurance is regulated at the state level, and regulators require insurers to file
statistical data that is consistent in format and summary-based.
Examples:
1. The Texas Private Passenger Automobile Statistical Plan.
 TX used a benchmark rate system for setting personal auto premiums from which insurers could
deviate.
 The benchmark rates were determined based on an analysis of statistical data provided by insurers
writing in Texas, with data aggregated by territory, deductible, and driver class.
 The data was also publicly available and was used by insurers to supplement internal analyses.
2. National Council for Compensation Insurance (NCCI) and Insurance Services Office, Inc (ISO) are two
organizations that meet the U.S. industry’s need for aggregated data.
 These organizations collect, summarize and analyze the aggregated data and make the results of the
analysis available to the participating insurers.
 Participating insurers may be able to request the aggregated data to perform their own independent
analysis.
 These statistical plans collect data at the transactional level, allowing insurers and actuaries to have the
flexibility to perform in-depth analysis at both the overall and segment levels.
State regulators may initiate ad hoc data calls to address a specific need (e.g. several state regulators have
requested closed claim information on medical malpractice claims, and medical malpractice insurers may
request the data to supplement their own data.

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Chapter 3 – Ratemaking Data
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
B. Other Aggregated Industry Data
Many insurers voluntarily report data to various organizations to be aggregated and used by the insurance
industry and by regulators, public policy makers, or the general public. Examples:
1. A large percentage of U.S. personal lines insurers report quarterly loss data for the “Fast Track Monitoring
System”, used by insurers and U.S. state regulators to analyze loss trends.
2. The Highway Loss Data Institute (HLDI) sponsored by U.S. personal auto insurance insurers:
 compiles member insurer data and provides detailed loss information by type of car to member
insurers and public policy makers.
 provides highly summarized information useful to insurers as well as the general public (e.g. information
on which make and model cars have the highest incident of auto injury).
C. Competitor Rate Filings/Manuals
Competitor rate filings may be available to the public (depending on the jurisdiction).
U.S. insurers may be required to submit rate filings (which include actuarial justification for rate changes and the
manual pages needed to rate a policy) to the appropriate regulatory body when changing rates.
 A filed rate change may only involve a change to base rates only. However, the filing may still include
helpful information related to overall indicated loss cost levels and trends in losses and expenses.
 However, if the insurer is making changes to rating variable differentials (e.g. driver age relativities) the
filing may also include information about the indicated relationships between the different levels for
each rating variable undergoing a change.
Insurers may be required to include the manual pages necessary to rate policies. Recall that a manual contains
the rules, rating structures, and rating algorithms used to estimate the overall average premium level charged
and the premium differences due to different characteristics.
 However it can be very difficult to get a complete copy of a competitor’s rate manual.
i. Insurers do not file a complete manual with each change, but rather file only the pages that are
changing (it may take several filings to piece together a complete manual).
ii. Insurers often create underwriting tiers, which have a significant impact on the final premium, and the
rating manual without the underwriting rules is incomplete information.
 An insurer must take great care when relying on information from a competitor’s rate filing.
Each company has different insureds, goals, expense levels, and operating procedures, and if
differences are material, competitor information may not be relevant (e.g. a personal automobile insurer
specializing in writing preferred or super-preferred drivers t has different rates and rating variables than
a non-standard personal automobile insurer).

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Chapter 3 – Ratemaking Data
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
D. Other Third-Party Data (not specific to insurance)
The most commonly used types are:
1. Economic data (e.g. Consumer Price Index (CPI))
Insurers may examine the CPI at the component level (e.g. medical cost and construction cost indices) to
find trends relevant to the insurance product being priced.
2. Geo-demographic data (i.e. average characteristics of a particular area).
i. Population density can be a predictor of accident frequency.
ii. Weather indices, theft indices, and average annual miles driven.
3. Credit data is used by insurers to evaluate the insurance loss experience of risks with different credit scores.
Insurers feel credit is an important predictor of risk and began to vary rates accordingly.
4. Other information related to different insurance products include:
• Personal automobile insurance: vehicle characteristics, department of motor vehicle records
• Homeowners insurance: distance to fire station
• Earthquake insurance: type of soil
• Medical malpractice: characteristics of hospital in which doctor practices
• Commercial general liability: type of owner (proprietor, stock)
• Workers compensation: OSHA inspection data.

Key Concepts

47 - 47

1. Internal data
a. Policy database
b. Claim database
c. Accounting data
2. Data aggregation
a. Calendar year (CY)
b. Accident year (AY)
c. Policy year (PY)
d. Report year (RY)
3. External data
a. Data calls and statistical plans
b. Other insurance industry aggregated data
c. Competitor information
d. Other third-party data

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Questions from the 2006 exam:
32. (2 points)
a. (1.5 points) For both premium and loss data, describe the following methods for grouping ratemaking
experience:


Policy Year



Calendar Year



Accident Year

b (0.5 point) For purposes of ratemaking, which method in part a. above is most responsive and which
method is least responsive?

Questions from the 2007 exam:
53. (2.5 points)
a. (1.5 points) Briefly define policy year, calendar year, and accident year loss experience.
b. (0.5 point) Which of the three performs the best with respect to responsiveness? Explain.
c. (0.5 point) Which of the three performs the best with respect to matching premiums and losses?
Explain.

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Chapter 3 – Ratemaking Data
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
The predecessor papers to the current syllabus reading “Basic Ratemaking” by Werner, G.
and Modlin, C. were numerous. While past CAS questions were drawn from prior syllabus
readings, the ones shown below remain relevant to the content covered in this chapter.
Questions from the 1993 Exam:
Question 49.
a. Calendar year data (premiums and losses) for ratemaking is readily available from annual statement
page 14. However, it is susceptible to changes in reserve level adequacy from year to year.
Calendar/accident year data is also readily available after the end of the year. However, AY losses at the
end of the 1st year are immature and may require substantial development to determine an estimate of its
ultimate value.
Since policy year data is not available until two calendar years after the date of the 1st policy written, the
data is more mature than the prior types mentioned. However, its delay in availability makes it less
responsive to identifying any form of change in the experience.
Report year data is convenient for claims made pricing, since the number of claims reported are frozen at
the end of the report period. Not very useful for pricing occurrence coverage.
b. CY data is used in Auto Physical Damage ratemaking (Chernick), off the current syllabus), CY/ AY data is
used in Automobile ratemaking (Stern, off the current syllabus), PY data is used in Commercial General
Liability (Graves, off the syllabus), and RY data is used in CM ratemaking (Marker/Moh, off the syllabus).
c. CY data is appropriate due to the short tailed nature of auto physical damage, CY/AY data is appropriate
for auto liability since it is responsive to change and since development does not exceed 63 months, PY
data is stable and more mature, which is appropriate for long-tailed liability lines, and RY data is
appropriate for traditional claims made analysis.

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Chapter 3 – Ratemaking Data
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2006 exam:
32. (2 points)
a. (1.5 points) For both premium and loss data, describe the following methods for grouping
ratemaking experience: Policy Year
Calendar Year
Accident Year
b (0.5 point) For purposes of ratemaking, which method in part a. above is most responsive and
which method is least responsive?
Initial comments
Review of the following comments made by different authors is helpful prior to answering the question.
McClenahan on PY: Policy year data is based upon the year in which the policy giving rise to exposures,
premiums, claims and losses is effective.
Graves on PY: For the premises and operations lines of insurance, policy year data is used for ratemaking.
The main reason for this is that these lines of insurance tend to have long pay-out patterns
(tails). Claims are not reported to insurers as quickly as in other lines. This creates a problem
when trying to match incurred losses with the premiums from which they arise. This task of
matching incurred losses to earned premiums is achieved through the use of policy year data.
McClenahan on AY: Generally insurers maintain claim data based upon accident date—the date of the
occurrence which gave rise to the claim, and report date—the date the insurer receives
notice of the claim. Claim data can then be aggregated based upon these dates. For
example, the total of all claims with accident dates during 2001 is the accident year 2001
claim count:
Feldblum on RM: Ratemaking should balance the considerations of stability, responsiveness, and equity.
Policy year experience, being the most homogeneous, represents stability; calendar year
experience, being the most recent, represents responsiveness.
Feldblum on CY: Development factors are needed for policy year premium, but not necessarily for calendar
year premium. Calendar year premiums include audit premiums from past policies. If the
premium volume is steady, then the current year’s audits, which actually relate to past
exposures, are about equal to next year’s audits, which relate to the current exposures.
Tiller on ratemaking responsiveness when using experience rating:
The length of the experience rating period usually ranges from two to five years. The shorter the period, the
more responsive the plan will be to changes that truly affect loss (and ALAE) experience, such as changes in
the risk control program, and the more subject to unusual fluctuations in loss (and ALAE) experience.
Conversely, a longer period will result in less responsiveness to changes and to unusual or catastrophic
occurrences.
CAS Model Solution
Part a.
Policy Year – Group premium and losses based upon policies issued during a given block of time.
Calendar Year – Experience for a give block of time.
Premiums = written premium during the period + unearned premium reserve at beginning of period –
unearned premium reserve at end of period.
Losses = paid losses during period + reserves at end of period – reserves at beginning of period.
Accident Year – Premiums are the same as calendar year. Losses are grouped based upon accidents
occurring during the period.
Part b. Calendar Year data is the most responsive because it is the most mature. Policy year is the least
responsive because it is the least mature.

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Chapter 3 – Ratemaking Data
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2007 exam:
53. (2.5 points)
a.
(1.5 points) Briefly define policy year, calendar year, and accident year loss experience.
b.
(0.5 point) Which of the three performs the best with respect to responsiveness? Explain.
c.
(0.5 point) Which of the three performs the best with respect to matching premiums and
losses? Explain.
CAS Model Solution
a. PY: Losses are allocated to the year in which the policy was written.
CY: Losses are allocated to the year in which payments were made and reserves were changed.
AY: Losses are allocated to the year in which the accident occurred.
b. Calendar year is the most recent and responsive because there is no delay due to developing losses.
c. Policy year matches premiums and losses best because the losses are generated by the same
policies for which premium was collected.

Exam 5, V1a

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Chapter 4 – Exposures
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Sec
1
2
3
4
5
1

Description
Criteria For Exposure Bases
Exposures For Large Commercial Risks
Aggregation of Exposures
Exposure Trend
Key Concepts

Pages
49 - 51
51 -51
51 – 61
61 - 62
63 - 63

Criteria For Exposure Bases

49 - 51

Base rates are expressed as a rate per exposure (see chapter 2). Premium is calculated as the base rate
multiplied by the number of exposures and adjusted by the effect of rating variables and other fees.
CRITERIA FOR EXPOSURE BASES (EB)
A good exposure base should meet the following 3 criteria. It should:
1. be directly proportional to expected loss
2. be practical
3. consider preexisting exposure bases used within the industry.
1. Proportional to Expected Loss
The expected loss of a policy with two exposures should be twice the expected loss of a policy with one
exposure.
This does not mean that the exposure base is the only item by which losses vary.
Expected loss varies by factors used as rating or underwriting variables to reflect risk level differences.
The factor with the most direct relationship to the losses should be selected as the exposure base
(which makes it more easily understood by the insured).
Example: Should homeowners insurance exposure base be number of house years or amount of
insurance?
i. The expected loss for one home insured for 2 years is two times the expected loss of the same
home insured for 1 year.
ii. The expected loss for homes also varies by amount of insurance purchased.
While the expected loss for a $200,000 home is higher than that for a $100,000 home, it may not
necessarily be two times higher.
Since the EB should be the factor most directly proportional to the expected loss, number of house
years is the preferred EB, and amount of insurance should be used as a rating variable.
The exposure base should be responsive to any change in exposure to risk. For some insurance
lines, the exposure base can be responsive to even small changes in exposure.
Example:
Payroll is the commonly used exposure base for WC insurance. As the number of workers increases
(decreases) or the average number of hours worked increases (decreases), both payroll and the risk of
loss increase (decrease) too.
Thus, the EB (i.e., payroll) moves in proportion to expected losses, and the premium will change with
this exposure base change as well.

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Chapter 4 – Exposures
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
2. Practical
The exposure base should be practical, meaning it should be:
1. objective
2. relatively easy to use and
3. inexpensive to obtain and verify.
The EB will be consistently measured by meeting these criteria.
A well-defined and objective exposure should not be able to be manipulated (by policyholders and
producers/underwriters).
Moral Hazard Example:
Asking a personal auto policyholder to state their estimated annual miles driven provides opportunity for
dishonesty more so than the use of car-years as the exposure base.
However, advances in technology may change the choice of EB for personal auto insurance.
Example: Onboard diagnostic devices can accurately track driving patterns and transmit this data to insurers.
Thus, some commercial long haul trucking carriers have implemented miles driven as an EB.
For products liability, products currently in use is the exposure base that is most proportional to expected loss.
However, it is difficult for most firms to accurately track how many of their products are actually being used
during the period covered by the insurance policy.
Therefore, gross sales is used as the EB as it is a reasonable and practical proxy for products in use.
3. Historical Precedence
If there is a more accurate or practical EB than the one currently in use (e.g. miles driven versus car years),
consider the following before implementing it.
1. Any change in the EB can lead to large premium swings for individual insureds.
2. A change in EB will require a change in the rating algorithm, which may require a significant effort to adjust
the rating systems, manuals, etc.
3. Since ratemaking analysis is based on several years of data, a change in EB may necessitate significant
data adjustments for future analyses.
Example: WC has historically used payroll as an EB.
In the 1980s, there was pressure to change the EB to hours worked for medical coverage to correct
perceived inadequacies of the EB for union companies with higher pay scales.
 Although hours worked made intuitive sense, the EB was not changed at that time, given concerns
regarding the transition.
 Instead, the rating variables and rating algorithm were adjusted to address the inequities (note that the
debate over the choice of WC EB continues to reemerge).
EBs currently used for different lines of business are shown below:
Line of Business
Typical Exposure Bases
Personal Automobile
Earned Car Year
Homeowners
Earned House Year
Workers Compensation
Payroll
Commercial General Liability
Sales Revenue, Payroll , Square Footage, Number of Units
Commercial Business Property
Amount of Insurance Coverage
Physician's Professional Liability
Number of Physician Years
Professional Liability
Number of Professionals (e.g., Lawyers or Accountants)
Personal Articles Floater
Value of Item

Exam 5, V1a

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Chapter 4 – Exposures
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.

Exposures For Large Commercial Risks

51 -51

Large commercial risks present challenges for the use more conventional EBs. The amount of exposure for
each separate coverage is difficult to track.
Thus, ratemaking is often done via composite rating and loss-rated composite rating.
In composite rating, the premium is initially calculated using estimates for each exposure measure along with
relevant rating algorithms for each coverage (e.g. commercial multi-peril policies use different exposure
measures for each coverage part (e.g. sales revenue for general liability, amount of insurance or
property value for commercial business property)).
Since these individual exposure estimates are expected to change over the policy term, a proxy measure is
used to gauge the overall change in exposure to loss (e.g. if property value is chosen as the proxy exposure
measure, a 20% increase in property value during the policy term would trigger a premium adjustment of
20% for the whole policy’s premium), rather than auditing each exposure measure.
In loss-rated composite rating, premium is calculated based on the risk’s historical loss experience, with the
implicit exposure base being the risk itself (See Chapter 15 for more detail).

Aggregation of Exposures

51 – 61

Methods of Aggregation for Annual Terms
Two methods to aggregate exposures are CY (the same as Calendar-AY) and PY.
Recall the 4 common methods of data aggregation are CY, AY, PY, and RY.
Homeowners policies are used to demonstrate these concepts for which there is one exposure per policy with
an annual policy period. Base data for the example:
Policies
Policy Effective
Expiration Exposure
Date
Date
A
10/01/10
09/30/11
1.00
B
01/01/11
12/31/11
1.00
C
04/01/11
03/31/12
1.00
D
07/01/11
06/30/12
1.00
E
10/01/11
09/30/12
1.00
F
01/01/12
12/31/12
1.00
Note: Examples using semi-annual terms are provided later in this chapter.

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Chapter 4 – Expo
osures
BASIC RATTEMAKING – WERNER, G
G. AND MOD
DLIN, C.
These policies are reprresented picto
orially below.

The x-axis
s represents time
t
and the y-axis represents the perccentage of the
e policy term tthat has expirred (this
representation is not applicable to products
p
like warranty
w
that don’t earn evvenly).
Each diag
gonal line represents a diffe
erent policy.
 At
A policy incep
ption, 0% of th
he policy term
m has expired,, and that point is on the lo
ower x-axis att the
efffective date.
 At
A policy expira
ation, 100% of
o the policy te
erm has expirred, and that point is locate
ed on the upp
per xax
xis at the exp
piration date.
 The line conne
ecting the effe
ective and exp
piration pointss depicts the % of the policcy term expire
ed at
ea
ach date.

2-month CY w
CY and AY
A Aggregatiion consider all
a exposures
s during the 12
without regard
d to the date of policy
issuance. Since CY an
nd AY exposu
ures are gene
erally the sam
me (excluding policies that undergo audiits), the text
t
CY expo
osure.
uses the term
 At
A the end of th
he CY, all exp
posures are fiixed.
 Since CY captures transacttions occurring on or after tthe first day o
of the year, an
nd on or before the last
da
ay of the yearr, CY is repre
esented graph
hically as a sq
quare (as sho
own below).
Calend
dar Year Agg
gregation

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Chapter 4 – Expo
osures
BASIC RATTEMAKING – WERNER, G
G. AND MOD
DLIN, C.
PY (a.k.a. UW year) ag
ggregation considers
c
all exposures
e
on
n policies with effective dattes during the
e year.
PY is reprresented grap
phically using a parallelog
gram starting with a policy written on the
e first day of tthe PY and
ending with a policy wrritten on the la
ast day of the
e PY.
Y
Aggreg
gation
Policy Year

Since PY
P data takes longer to cap
pture, most ra
atemaking ana
alysis focusess on CY expo
osures.

Four typ
pes of expos
sures
1. Written exposures
s arise from policies issued
d (i.e. underw
written or writte
en) during a sspecified perio
od of time
(e.g. a calendar qua
arter or a CY)).
CY 2011 written exp
posures are the sum of the
e exposures ffor all policiess that had an effective date
e in 2011.
 Since polic
cies B, C, D and E all have
e effective dattes (shown ass large circless on the horizo
ontal axis)
in 2011; the
eir entire exposure contrib
butes to CY 20
011 written exxposure.
 However, policies
p
A and
d F have effec
ctive dates in years 2010 a
and 2012, and
d thus do not contribute
to CY 2011
1 written expo
osure.
CY Wrritten Exposu
ures

Distribu
ution of Calen
ndar Year Written Expos ures a/o 12/3
31/12
W ritten Exposu
ures
Effective Expiration
Date
Policy
y
Date
Exposure C
CY 2010 CY 2011 CY 20
012
10/01/10 09/30/11
0..00
A
1.00
0.00
0
1.00
01/01/11 12/31/11
1.00
B
0.00
1..00
0.00
0
04/01/11 03/31/12
1.00
C
0.00
1..00
0.00
0
07/01/11 06/30/12
1.00
D
0.00
1..00
0.00
0
10/01/11 09/30/12
1.00
E
0.00
1..00
0.00
0
01/01/12 12/31/12
1.00
F
0.00
0..00
1.00
0
Total
6.00
1.00
4..00
1.00
0

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Chapter 4 – Exposures
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Policy contribution to CY:
 Each policy contributes a written exposure to a single CY in this example.
 However, if a policy cancels midterm, the policy will contribute a written exposure to two different CYs
if the policy cancellation date is in a different CY year than the original policy effective date.
Example:
If Policy D is cancelled on 3/31/2012 (i.e. after 75% of the policy has expired), then Policy D will
contribute 1 written exposure to CY 2011 and -0.25 written exposure to CY 2012.
PY Written Exposure

Distribution of PY Written Exposures a/o 12/31/12
Written Exposures
Effective Expiration
Date
Policy
Date
Exposure PY 2010 PY 2011 PY 2012
A
10/01/10 09/30/11
1.00
1.00
0.00
0.00
B
01/01/11 12/31/11
1.00
0.00
1.00
0.00
C
04/01/11 03/31/12
1.00
0.00
1.00
0.00
D
07/01/11 06/30/12
1.00
0.00
1.00
0.00
E
10/01/11 09/30/12
1.00
0.00
1.00
0.00
F
01/01/12 12/31/12
1.00
0.00
0.00
1.00
Total
6.00
1.00
4.00
1.00
In case of cancellation, the original written exposure and the written exposure due to the cancellation are all
booked in the same PY (since PY written exposures are aggregated by policy effective dates).
This contrasts with CY written exposure and cancellation exposure which can apply to two different CYs
depending on when the cancellation occurs.
2. Earned exposures are the portion of written exposures for which coverage has already been provided as of
a certain point in time.
Assume the probability of a claim is evenly distributed throughout the year.
If all policies are written on 1/1 for one year, earned exposures as of 5/31/XX are 5/12 of written
exposures.

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Chapter 4 – Expo
osures
BASIC RATTEMAKING – WERNER, G
G. AND MOD
DLIN, C.
To better understand th
he difference between CY and PY earn
ned exposuress, look at the CY diagram:
CY Ea
arned Expos
sure

For Po
olicy C, 75% of
o the policy period
p
is earne
ed in 2011 an
nd 25% of the
e policy period
d is earned in
n 2012.
Policy C contributes
s 0.75 (75% * 1.00) of earn
ned exposure
e to CY 2011 a
and 0.25 earn
ned exposure
e to CY 2012.
Distriibution of Ca
alendar Year Earned Exposures a/o 1 2/31/12
Earned Exposures
Effective Expiration
Polic
Date
Exposure CY 2010 CY
Y 2011 CY 2012
cy
Date
1.00
0.75
0..00
10/01/10 09/30/11
A
0.25
1.00
01/01/11 12/31/11
0.00
B
1.00
0..00
1.00
04/01/11 03/31/12
0.00
C
0.75
0..25
1.00
07/01/11 06/30/12
0.00
D
0.50
0..50
1.00
10/01/11 09/30/12
0.00
E
0.25
0..75
1.00
01/01/12 12/31/12
0.00
F
0.00
1..00
Tota
al
6.00
0.25
3.25
2..50

Conside
er PY Earned
d Exposure





Exam 5, V1a

Earned exposure is ass
signed to the year
y
the policcy was written
n and increases over time.
At the end of a PY (i.e. 24 months affter the start o
of a PY having annual policcies), PY earned and
written exp
posures are equivalent.
e
Unlike CY
Y earned exposure, expos
sure for one policy cannot be earned
d in two diffe
erent PYs.

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Chapter 4 – Expo
osures
BASIC RATTEMAKING – WERNER, G
G. AND MOD
DLIN, C.
Distriibution of PY
Y Earned Exp
posures a/o 12/31/12
1
Earned Exposures
e Expiration
Effective
Date
Policy
Exposure PY 2010 PY
Date
Y 2011 PY 2
2012
1.00
A
10/01/10
0 09/30/11
1.00
0.00
0.00
1.00
0.00
1.00
0.00
B
01/01/11
1 12/31/11
1.00
0.00
1.00
0.00
C
04/01/11
1 03/31/12
1.00
0.00
1.00
0.00
D
07/01/11
1 06/30/12
1.00
0.00
1.00
0.00
E
10/01/11
1 09/30/12
1.00
0.00
0.00
F
01/01/12
2 12/31/12
1.00
Tottal
6.00
1.00
4.00
1.00
Note: An even earrning pattern assumption is
s not approprriate for lines such as warrranty and thosse
affected by seasonal
s
flucttuations in wrritings (e.g. bo
oat owners insurance).
Earning patte
ern assumptio
ons are usually based on h
historical exp
perience.
3. Unearrned exposurres are the po
ortion of writte
en exposuress for which co
overage has n
not yet been p
provided as
of that point in time (and applies to individual policies and g
groups of poliicies).
Written
n Exposures = Earned Exp
posures + Une
earned Expossures.
For gro
oups of policie
es, the formula depends on the method
d of data aggrregation.
* For PY
P aggregatio
on as of a cerrtain point in time,
t
the form
mula above ap
pplies.
* For CY
C aggregatio
on, the formu
ula becomes
CY Unearned
U
Exp
posures = CY
Y Written Expo
osures – CY Earned Expo
osures + Unea
arned Exposu
ures
as off the beginnin
ng of CY.
4. In-forc
ce exposures
s are the num
mber of insure
ed units expossed to having
g a claim at a given point in
n time.
Examp
ple: The in-fo
orce exposure
e as of 6/15/20
011 is the sum
m of full-term exposures fo
or all policies that have
an incep
ption date on or
o before 6/15
5/2011 and a n expiration d
date after 6/15
5/2011.
A vertica
al line drawn at
a the valuatio
on date will in
ntersect the po
olicies that arre in-force on that date.
Policies A, B, and C are
a in effect on
o 6/15/11 and
d each contributes to 6/15
5/11 in-force e
exposures.
In-Fo
orce Exposu
ure

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Chapter 4 – Exposures
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
In-force Exposure by Date
In-Force Exposure a/o
Policy
A
B
C
D
E
F
Total

Effective
Date
10/01/10
01/01/11
04/01/11
07/01/11
10/01/11
01/01/12

Expiration
Date
09/30/11
12/31/11
03/31/12
06/30/12
09/30/12
12/31/12

Exposure 01/01/11 06/15/11
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.00
1.00
1.00
0.00
0.00
1.00
0.00
0.00
1.00
0.00
0.00
6.00
2.00
3.00

01/01/12
0.00
0.00
1.00
1.00
1.00
1.00
4.00

Policy Terms Other Than Annual
When policy terms are shorter or longer than a year, then aggregation for each type of exposure is calculated
differently.
If the policies are six-month policies, each policy would represent one-half of an exposure
Six-Month Policies
Effective
Expiration
Date
Date
Policy
Exposure
A
10/01/10
03/31/11
0.50
B
01/01/11
06/30/11
0.50
C
04/01/11
09/30/11
0.50
D
07/01/11
12/31/11
0.50
E
10/01/11
03/31/12
0.50
F
01/01/12
06/30/12
0.50
Example Policies

Exam 5, V1a

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Chapter 4 – Exposures
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
CY Written Exposures a/o 12/31/12
Written Exposures
Policy
A
B
C
D
E
F
Total

Effective
Date
10/01/10
01/01/11
04/01/11
07/01/11
10/01/11
01/01/12

Expiration
Date
Exposure CY 2010 CY 2011 CY 2012
03/31/11
0.00
0.00
0.50
0.50
06/30/11
0.50
0.00
0.50
0.00
09/30/11
0.50
0.00
0.50
0.00
12/31/11
0.50
0.00
0.50
0.00
03/31/12
0.50
0.00
0.50
0.00
06/30/12
0.00
0.00
0.50
0.50
3.00
0.50
2.00
0.50

CY Earned Exposures a/o 12/31/12
Earned Exposure
Effective
Date
Policy
A
10/01/10
B
01/01/11
C
04/01/11
D
07/01/11
E
10/01/11
F
01/01/12
Total

Expiration
Date
03/31/11
06/30/11
09/30/11
12/31/11
03/31/12
06/30/12

Exposure CY 2010 CY 2011 CY 2012
0.50
0.00
0.25
0.25
0.50
0.00
0.50
0.00
0.50
0.00
0.50
0.00
0.50
0.00
0.50
0.00
0.50
0.00
0.25
0.25
0.50
0.00
0.00
0.50
3.00
0.25
2.00
0.75

Policy Written Exposures a/o 12/31/12
Effective Expiration
Policy
Date
Date
Exposure
10/1/2010 3/31/2011
0.50
A
1/1/2011 6/30/2011
B
0.50
4/1/2011 9/30/2011
C
0.50
7/1/2011 12/31/2011
D
0.50
10/1/2011 3/31/2012
E
0.50
1/1/2012 6/30/2012
F
0.50
Total
3.00

Written Exposures
PY 2010 PY 2011 PY 2012
0.50
0.00
0.00
0.00
0.50
0.00
0.00
0.50
0.00
0.00
0.50
0.00
0.00
0.50
0.00
0.00
0.00
0.50
0.50
2.00
0.50

Policy Year Earned Exposures a/o 12/31/12
Effective Expiration
Policy
Date
Date
Exposure
10/1/2010 3/31/2011
0.50
A
1/1/2011 6/30/2011
B
0.50
4/1/2011 9/30/2011
C
0.50
7/1/2011 12/31/2011
D
0.50
10/1/2011 3/31/2012
E
0.50
1/1/2012 6/30/2012
F
0.50
Total
3.00

Earned Exposures
PY 2010 PY 2011 PY 2012
0.50
0.00
0.00
0.00
0.50
0.00
0.00
0.50
0.00
0.00
0.50
0.00
0.00
0.50
0.00
0.00
0.00
0.50
0.50
2.00
0.50

Exam 5, V1a

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Chapter 4 – Exposures
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Assuming insured units are “number of homes” insured at a point in time, each semi-annual policy
contributes one in-force exposure.
In-force Exposure by Date
Effective Expiration No. of Houses
Policy
Date
Date
Insured
10/1/2010 3/31/2011
A
1.00
1/1/2011 6/30/2011
B
1.00
4/1/2011 9/30/2011
C
1.00
7/1/2011 12/31/2011
D
1.00
10/1/2011 3/31/2012
E
1.00
1/1/2012 6/30/2012
F
1.00
Total
6.00

In-Force Exposures a/o
CY 2010 CY 2011 CY 2012
1.00
0.00
0.00
1.00
1.00
0.00
0.00
1.00
0.00
0.00
0.00
0.00
0.00
0.00
1.00
0.00
0.00
1.00
2.00
2.00
2.00

Calculation of Blocks of Exposures
Insurers may have policy information summarized on a monthly or quarterly basis and need to calculate
exposures for a block of policies using this summarized data. In such a case:
 it is customary to treat all policies as if they were written on the mid-point of the period.
 when summarizing on a monthly basis, all policies are assumed to be written on the 15th of the month.
(i.e. this is known as “15th of the month” rule or the “24ths” method.)
 this approximation applies as long as policies are written uniformly during each time period.
 if this approach is applied to longer periods (e.g. quarters or years), the assumption of uniform writings is
less likely to be reasonable.
To demonstrate how the rule applies, assume an insurer begins writing annual policies in 2010 and writes 240
exposures each month.
It is reasonable to assume that some of the 240 exposures written in July were in-force as of the first day of
the month.
However, the “15th of the month” rule assumes that none of the exposures from the July policies contribute
to in-force exposures as of 7/1/2010 because the rule assumes all the July policies are written on 7/15.
(see the table below and look at in-force exposures as of 7/1/2010 and at 7/10/2010 written exposures).

Exam 5, V1a

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Chapter 4 – Exposures
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Aggregate In-force Calculation
Written
Assumed
Month
Effective
Date
Exposure
Jan 10
240
01/15/10
Feb 10
240
02/15/10
Mar 10
240
03/15/10
Apr 10
240
04/15/10
May 10
240
05/15/10
June 10
240
06/15/10
240
July 10
07/15/10
Aug 10
240
08/15/10
Sep 10
240
09/15/10
Cot 10
240
10/15/10
Nov10
240
11/15/10
Dec 10
240
12/15/10
Total
2,880

07/01/10
240
240
240
240
240
240
0
0
0
0
0
0
1,440

01/01/11
240
240
240
240
240
240
240
240
240
240
240
240
2,880

07/01/11
0
0
0
0
0
0
240
240
240
240
240
240
1,440

Earned Exposure %’s calculation:
Since policies for a given month are assumed to be written on the 15th of the month, the written exposures for
annual policies will be earned over a 13-month calendar period:
 1/24 of the exposure will be earned in the second half of the month in which it was written
 1/12 (or 2/24) of the exposure will be earned in each of the next 11 months (i.e. months 2-12) and
 1/24 of the exposure will be earned in the first half of month 13.
Distribution of earned exposures to CYs 2010 and 2011:
1
2
3
4
5
(6) = (2) x (4) (7) = (2) x (5)
Earned %
Earned Exposures
Written Exposures
Assumed
Month
Written
Effective date
2010
2011
2010
2011
Jan 10
Feb 10
Mar 10
Apr 10
May 10
Jun 10
Jul 10
Aug-10
Sep-10
Oct 10
Nov 10
Dec 10
Total

240
240
240
240
240
240
240
240
240
240
240
240
2,881

01/15/10
02/15/10
03/15/10
04/15/10
05/15/10
06/15/10
07/15/10
08/15/10
09/15/10
10/15/10
11/15/10
12/15/10

23/24
21/24
19/24
17/24
15/24
13/24
11/24
9/24
7/24
5/24
3/24
1/24

1/24
3/24
5/24
7/24
9/24
11/24
13/24
15/24
17/24
19/24
21/24
23/24

230
210
190
170
150
130
110
90
70
50
30
10
1,440

10
30
50
70
90
110
130
150
170
190
210
230
1,440

(4) = Portion of exposure earned in 2010. (5) = Portion of exposure earned in 2011.
The same principles apply when using the “15th of the month” rule on PY aggregation.

Exam 5, V1a

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Chapter 4 – Exposures
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
4

Exposure Trend

61 - 62

For some lines of business, the exposure measure is inflation sensitive (e.g. payroll and sales revenue are
influenced by inflationary pressures).
These trends can be measured via internal insurance company data (e.g. WC payroll) or via industry indices
(e.g. average wage index).
The way in which exposure trend impacts the calculation of the overall rate level indication depends on:
 whether the loss ratio or pure premium method is employed and
 how loss trends are calculated
These are discussed in Chapters 5 and 6.

Key Concepts

63 - 63

1. Definition of an exposure
2. Criteria of a good exposure base
a. Proportional to expected loss
b. Practical
c. Considers historical precedence
3. Exposure bases for large commercial risks
4. Exposure aggregation
a. Calendar year v. policy year
b. Written, earned, unearned, in-force
5. Calculation for blocks of exposure (“15th of the month” rule)
6. Exposure trend

Exam 5, V1a

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Chapter 4 – Exposures
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
The predecessor papers to the current syllabus reading “Basic Ratemaking” by Werner, G.
and Modlin, C. were numerous. While past CAS questions were drawn from prior syllabus
readings, the ones shown below remain relevant to the content covered in this chapter.

Section 1: Criteria for Exposure Bases
Questions from the 1992 exam
53. In the Study Note Reading "Exposure Bases Revisited", Bouska discusses Causes and Controversy
Involved in Changing Exposure Bases.
(a) (1 point)
What are the three desirable traits of an exposure base?
(b) (1.5 points) Discuss the issues surrounding Workers Compensation with regard to using hours
worked versus payroll.

Question from the 1995 exam
36. According to McClenahan, chapter 2, “Ratemaking," Foundations of Casualty Actuarial Science, the
specific exposure unit used for a given type of insurance should depend on several factors.
(a) (2 points) List and briefly describe the four factors he discusses.
(b) (1 point) Based on the four factors in (a), discuss the use of the following exposure units for automobile
ratemaking: 1) car years 2) miles driven per year.

Question from the 1997 exam
25. A. (1 point) According to the "Statement of Principles Regarding Property and Casualty
Ratemaking," what are three desirable features for exposure units to have?
C. (2 points) According to Bouska, "Exposure Bases Revisited," the standard exposure bases are
often not used for large risks. Briefly describe two alternative rating plans used for large risks that
modify the usual exposure base.

Questions from the 2009 exam
17. (2 points) An insurance company is considering changing the personal automobile exposure base
from earned car years to number of miles driven.
a. (1 point) Identify four desirable characteristics of an exposure base.
b. (1 point) Discuss whether or not the change to a miles-driven exposure base should be made,
referencing each of the four characteristics identified in part a, above.

Questions from the 2010 exam
16. (2 points)
a. (1 point) Identify and briefly describe two criteria for a good exposure base.
b. (0.5 point) Evaluate "market value of the house" as an exposure base for homeowners insurance
using the two criteria identified in part a. above.
c. (0.5 point) Provide two reasons why a change in exposure base may be difficult.

Questions from the 2011 exam
2. (1.5 points) An insurer is considering changing the exposure base used to price personal auto from
earned car years to annual miles driven. Evaluate the merits of this change based on each of three
different criteria of a good exposure base.

Questions from the 2012 exam:
2. (1.5 points) An insurance company is considering changing its exposure base for workers
compensation from payroll to number of employees. Evaluate the merits of this change based on each
of three different criteria of a good exposure base.

Exam 5, V1a

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Chapter 4 – Exposures
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Section 2: Computing Exposures
Questions from the 2000 exam
38. (4 points) Based on McClenahan, "Ratemaking," chapter 2 of Foundations of Casualty Actuarial Science,
and the following data, answer the questions below.
Personal Automobile Liability Data:
Calendar Year 1997
Calendar Year 1998
Number of Autos
Number of Autos
Written on
Written on
Effective Date
Effective Date
Effective Date
Effective Date
January 1, 1997
100
January 1, 1998
900
April 1, 1997
300
April 1, 1998
1,100
July 1, 1997
500
July 1, 1998
1,300
October 1, 1997
700
October 1, 1998
1,500
Assume:
• All policies are twelve-month policies.
• Written premium per car during calendar year 1997 is $500.
• A uniform rate increase of 15% was introduced effective July 1, 1998.
a. (1/2 point)
b. (1 point)
c. (1/2 point)
d. (1 point)
e. (1 point)

Calculate the number of in-force exposures on January 1, 1998. (chapter 4)
Calculate the number of earned exposures for calendar year 1998. (chapter 4)
List the two methods McClenahan describes that are used to adjust earned premiums to a
current rate level basis. (chapter 5)
Which of the two methods listed in part c. above would be more appropriate to use for this
company's personal automobile liability business? Briefly explain why. (chapter 5)
Using your selected method from part d. above, calculate the on-level earned premium for
calendar year 1998. (chapter 5)

Questions from the 2010 exam:
17. (2 points) Given the following activity on five annual personal automobile policies as of June 30, 2009:
Policy
1
2
3
4
5

Effective Date
July 1, 2007
October 1, 2007
January 1, 2008
March 1, 2008
July 1, 2008

Original Expiration
Date
June 30, 2008
September 30, 2008
December 31, 2008
February 28, 2009
June 30, 2009

Mid-term Cancellation
Date
N/A
March 31, 2008
N/A
June 30, 2008
N/A

The exposure base is earned car years.
a. (0.5 point) Calculate the 2008 calendar year written exposure.
b. (0.5 point) Calculate the 2008 calendar year earned exposure.
c. (0.5 point) Calculate the 2007 policy year written exposure.
d. (0.5 point) Calculate the in-force exposure as of April 1, 2008.

Exam 5, V1a

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Chapter 4 – Exposures
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2011 exam:
3. (1.25 points) Given the following:
•
Each policy insures only one car
•
Policies are earned evenly throughout the year
Policy
A
B
C
D
E
F

Effective Date
February 1, 2009
May 1, 2009
August 1, 2009
November 1, 2009
January 1, 2010
July 1, 2010

Original Expiration Date Cancellation Date
July 31, 2009
October 31, 2009
January 31, 2010
April 30, 2010
January 31, 2010
June 30, 2010
December 31, 2010

a. (0.25 point) Calculate the written car years in calendar year 2010.
b. (0.25 point) Calculate the written car years in policy year 2010.
c. (0.25 point) Calculate the earned car years in calendar year 2010.
d. (0.25 point) Calculate the earned car years in policy year 2010.
e. (0.25 point) Calculate the number of in-force policies as of January 1, 2010.

Questions from the 2012 exam:
3. (1.5 points) Given the following information:


An insurance company started writing business on January 1, 2011.



All policies are one-year term.
Policy Effective Dates
January 1 through March 31
April 1 through June 30
July 1 through September 30
October 1 through December 31

Exposures
100
200
300
400

a. (1 point) Calculate the 2011 earned exposures assuming policies are written uniformly during each
quarter.
b. (0.5 point) Discuss the appropriateness of the assumption in part a. above given the exposure data.

Exam 5, V1a

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Chapter 4 – Exposures
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
The predecessor papers to the current syllabus reading “Basic Ratemaking” by Werner, G. and
Modlin, C. were numerous. While past CAS questions were drawn from prior syllabus
readings, the ones shown below remain relevant to the content covered in this chapter.

Section 1: Criteria for Exposure Bases
Solutions to questions from the 1992 exam
53. (a) 1. An accurate measure of the exposure to loss.
2. Easy to determine for the insurer.
3. Difficult to manipulate by the insured.
Present Day Update: While the above 3 criteria were the right answers in 1992, the current
reading by Werner and Modlin, list them differently:
1. Proportional to expected loss: The selected EB should be the factor most directly
proportional to loss and be responsive to any change in exposure to risk.
2. Practical – Objective and Easy to Obtain/verify
3. Historical Precedence – changes in historical EB can cause large premium swings,
changes in rating algorithms, and necessitate adjustments to historical data analyses.
(b) It was caused by discontent among insureds over the inequities in the rating mechanism.
If a unionized company pays more per employee, it will have higher payroll and pay more for its WC coverage.
1. To the extent that the unionized company's indemnity losses are higher, the premium difference is correct.
2. To the extent that losses are from medical payments, or are capped by max benefits, use of
payroll is not justified.

Solutions to questions from the 1995 exam
Question 36.
a1. Reasonableness: the exposure unit should be a reasonable measure of the exposure to loss.
2. Ease of Determination: the exposure unit must be subject to accurate determination.
3. Responsiveness to Change: It should react to change in the true exposure to loss.
4. Historical Practice: A change in an exposure unit could render the prior history unusable.
Present Day Update: The list according to Werner and Modlin is a little different:
1. Proportional to expected loss: The selected EB should be the factor most directly
proportional to loss and be responsive to any change in exposure to risk.
2. Practical – Objective and Easy to Obtain/verify
3. Historical Precedence – changes in historical EB can cause large premium swings, changes
in rating algorithms, and necessitate adjustments to historical data analyses.
b. Reasonableness: Car-years are a reasonable measure of the exposure to loss, but doesn’t
differentiate by type of vehicle. It is easy to determine and somewhat responsive to change. Historically,
it has been the industry measure for some time.
Reasonableness: Miles driven are a reasonable measure of the exposure to loss, but doesn’t account for
the location of the driving (urban or rural). It is not easy to determine since it subject to audit by the
insurance company. It is responsive to change, since the relative exposure to loss increases as miles
driven increases. It would be difficult to implement and would render the prior history unusable.

Exam 5, V1a

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Chapter 4 – Exposures
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 1997 exam
Question 25.
A Exposure units should:
1. Vary with the hazard.
2. Be practical.
3. Be verifiable.
Present Day Update: The list according to Werner and Modlin is:
1. Proportional to expected loss: The selected EB should be the factor most directly proportional
to loss and be responsive to any change in exposure to risk.
2. Practical – Objective and Easy to Obtain/verify
3. Historical Precedence – changes in historical EB can cause large premium swings, changes in
rating algorithms, and necessitate adjustments to historical data analyses.
B. Question no longer applicable to the content in this chapter.
C. Large Risks are usually subject to either Composite Rating or Loss Rating.
1. Composite Rating is used to simplify the rating for insureds with multiple exposures (hundreds of
vehicles in their auto fleets or many insured locations).


First, a proxy exposure base (such as receipts or mileage for long haul trucking) is selected.



Next, the rate per proxy unit is determined by dividing the risk’s premium, calculated
normally, by proxy exposure base.
The simplified equation for charged premium = (Number of expected proxy units) * (Rate per proxy unit).
After policy expiration, the firm’s receipts are audited, so that the actual number of actual proxy units can
be used to determine the firm’s final premium.
2. Under Loss rating, the exposure base is the risk itself, and the rate is its expected losses.
The equation for charged premium = Expected Losses + Expense Load, for a very large risk.

Exam 5, V1a

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Chapter 4 – Exposures
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2009 exam
Question 17
a1. varies with the hazard (WM would say be directly proportional to expected loss)
2. verifiable (WM would say this is a characteristic of being practical)
3. not subject to manipulation (WM would say this is a characteristic of being practical)
4. practical
Present Day Update: The Werner and Modlin text uses the following list:
1. Proportional to expected loss: The selected EB should be the factor most directly
proportional to loss and be responsive to any change in exposure to risk.
2. Practical – Objective and Easy to Obtain/verify
3. Historical Precedence – changes in historical EB can cause large premium swings,
changes in rating algorithms, and necessitate adjustments to historical data analyses.
b1. Miles driven certainly varies with the hazard; the more you drive the more likely you are to get in an
accident.
2. Verifiable - may not be easy to verify. Someone would have to inspect each car at the end of the year to
read the odometer.
3. Certainly subject to manipulation. If the insured was asked how many miles driven in a year without
verification, he could easily lie. Even if the number was verified, there are still ways to turn the numbers
on an odometer back.
4. Miles driven is practical and intuitive. Most insured would understand that miles driven would be directly
correlated to probability of accidents.
Overall, the change to miles driven should not be made since the downsides of costly verification and
possibility of manipulation out weigh the benefits of varying with the hazard and practicality.

Solutions to questions from the 2010 exam
Question 16
a. (1 point) Identify and briefly describe two criteria for a good exposure base.
b. (0.5 point) Evaluate "market value of the house" as an exposure base for homeowners insurance using
the two criteria identified in part a. above.
c. (0.5 point) Provide two reasons why a change in exposure base may be difficult.
a1. 1. Directly proportional to loss. The exposure should have direct relationship to loss and vary proportionally
to it (i.e. the expected loss of a policy with two exposures should be twice the expected loss of a similar
policy with one exposure).
a2. Practical. Exposure should be
• Objective, not subjective, and definitively measurable
• Verifiable. Can be checked
b1. No. A house with $ 200K market value does not have 2 times expected loss than house with $100K market
value.
b2. No. Market value is somewhat subjective. No definite measure.
c1. Rates are likely to change substantially when an exposure base changes. Insured may not be happy with
changes.
c2. System limitations: hard to build new system based on new exposure, and may not even have data for it.

Exam 5, V1a

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Chapter 4 – Exposures
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2011 exam:
Question 2 – Model Solution 1
Car year to annual miles driven, 3 criteria:
1. Proportional to expected loss:
Should select variable with the most direct relationship to loss. Should adjust based on modifications
to exposure of the risk to a loss.
Annual miles driven seems a better choice, since the more you drive, the more at risk you are to have
a loss.
2. Practical: Should be objective, well-defined, and relatively easy to obtain and verify.
Miles driven are objective and a well-defined exposure, but can be expensive to send inspectors to
verify odometer. Also, if ask client, it is subject to moral hazard.
3. Historical precedent: Car years have historically been used. Changing to miles driven could cause: significant variation in premium
-need to modify systems
-need to collect new data (cost of survey or inspections)
Based on the 3 criteria, the costs of implementing this new structure and practical issues overweight
the benefits of the 1st one. Should keep earned car years as exposure base.
Question 2 – Model Solution 2
Exposure base should be:
1. proportional to loss
2. practical (verifiable, objective, easy to admin)
3. Have historical precedence
Annual miles driven satisfies 1 in that it is proportional to loss. More miles driven = more exposure.
Annual miles driven does not satisfy 2 in that it is difficult to verify and can be easily manipulated.
Annual miles driven does not satisfy 3 since it hasn’t been used in the past. Changing the exposure base
may cause prem. swings. Also, the data needed may not be readily available to create a database.

Exam 5, V1a

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Chapter 4 – Exposures
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2012 exam:
2. (1.5 points) An insurance company is considering changing its exposure base for workers
compensation from payroll to number of employees. Evaluate the merits of this change based on each
of three different criteria of a good exposure base.
Question 2 (Exam 5A Question 2)
1. Directly proportional to expected loss: Number of employees does reflect exposure to loss, but
payroll is more reflective of exposure loss. For example, having twice as many employees does
not mean that the expected losses will double, but only that frequency of loss would double
(severity would depend on the payroll distribution). Payroll is responsive to changes in both
frequency and severity.
2. Practical: Numbers of employees is a well-defined and objective measure. However, it may not be
as easy to obtain as payroll information because payroll is tracked for numerous financial reports
whereas number of employees is not. It may be harder to administer because insured could
manipulate information regarding number of employees more easily than that regarding payroll.
3. Considers historical precedence: Number of employees does not meet this criteria because payroll
has been used historically as the exposure base for WC. Changing to numbers of employees may
lead to the following issues:
1. Lead to large premium swings.
2. Require significant systems changes.
3. Require a change in rating algorithm.
4. Necessitate significant data adjustments for future ratemaking analysis.
CONCLUSION: Given these constraints, I would NOT recommend changing the exposure base to
number of employees.
Examiner Comments
Candidates scored well on this question. Some candidates lost points for either not supporting the reason or
restating the criteria as the reason.

Exam 5, V1a

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Chapter 4 – Exposures
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Section 2: Computing Exposures
Solutions to questions from the 2000 exam
Question 38.
Parts a and b. the number of in-force exposures on January 1, 1998, and earned exposures for CY 1998.
Number of Autos
Number of
Written on
Inforce Exposures
1998 Earned
1998 Earned
Effective Date
Effective Date
Factor
Exposures
on 1/1/98
(1)
(2)
(3)
(4)=(1)*(3)
January 1, 1997
100
0
0.0
0
April 1, 1997
300
300
.25
75
July 1, 1997
500
500
.50
250
October 1, 1997
700
700
.75
525
January 1, 1998
900
900
1.0
900
April 1, 1998
1,100
0
.75
825
July 1, 1998
1,300
0
.50
650
October 1, 1998
1,500
0
.25
375
Total
2,400
3,600
* In-force exposures are the number of insured units exposed to having a claim at a given point in time.
Inforce exposure counts a full car year for each 12 month policy in force as of 1/1/98, regardless of the length of
the remaining term.
* Earned exposures are the portion of written exposures for which coverage has already been provided as of a
certain point in time. For example:
3 of the 12 months of coverage for the 300 exposures written on 4/1/97 occur during CY 1998. Assuming
there are no policy cancellations, this portion (3/12) of the total exposures written will be earned during CY
1998, and thus the 1998 Earned Factor is .25.
Parts c., d. and e. See Chapter 5.

Exam 5, V1a

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Chapter 4 – Exposures
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2010 exam:
Question 17. Compute CY, PY and In-force Exposures
Initial comments:
* CY captures transactions occurring on or after the first day of the CY, and on or before the last day of the CY.
* Ex. CY 2011 written exposures are the sum of the exposures for all policies that had an effective date in 2011.
* Earned exposures are the portion of written exposures for which coverage has already been provided as of a
certain point in time.
* PY (a.k.a. UW year) aggregation considers all exposures on policies with effective dates during the year.
* In-force exposures are the number of insured units exposed to having a claim at a given point in time.
* If a policy cancels midterm, the policy will contribute written exposure to two different CYs if the date of the
cancellation is in a different calendar year than the original effective date ( positively or negatively, respectively)
CAS Model Solution “Un-Edited” shown below.
A. Policy

08 CY WE

B. Policy

08 CY EE

1
2
3
4
5

0
-0.5
1
1-2/3
1

1
2
3
4
5

0.5
0.25
1.0
0.333
0.5

1.833

Exam 5, V1a

2.583

C. Policy

07 PY WE

D. Policy

In-Force 4/1/08

1
2
3
4
5

1.0
0.5
0
0
0
1.5

1
2
3
4
5

1
0
1
1
0
3

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 2014 by All 10, Inc.

Chapter 4 – Exposures
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2011 exam:
Question 3
Policy
Effective Date
Original Expiration Date Cancellation Date
A
February 1, 2009
July 31, 2009
B
May 1, 2009
October 31, 2009
C
August 1, 2009
January 31, 2010
D
November 1, 2009
April 30, 2010
January 31, 2010
E
January 1, 2010
June 30, 2010
F
July 1, 2010
December 31, 2010
a. (0.25 point) Calculate the written car years in calendar year 2010.
b. (0.25 point) Calculate the written car years in policy year 2010.
c. (0.25 point) Calculate the earned car years in calendar year 2010.
d. (0.25 point) Calculate the earned car years in policy year 2010.
e. (0.25 point) Calculate the number of in-force policies as of January 1, 2010.
Initial comments:
 Since we are asked to compute CY and PY written car years, CY and PY earned car years and in-force
policies for six different policies, it is best to set up a table similar to the one below to answer the
question in the most efficient way possible.
 Since the given policies are six-month policies, each would represent one-half of a written exposure.
 Since insured units are defined as number of autos insured at a point in time, each semi-annual policy
can contribute to one in-force exposure.
 Since the exposures needing to be calculated are associated with 2010, it is clear that policy A and
policy B contribute 0 exposures to questions a., b. c. d. and e.
Definitions of the type of exposures being asked to compute are as follows:
Written exposures arise from policies issued (i.e. underwritten or written) during a specified period of time
(e.g. a calendar quarter or a CY). CY 2011 written exposures are the sum of the exposures for all policies that
had an effective date in 2011.
If a policy cancels midterm, the policy will contribute a written exposure to two different CYs if the policy
cancellation date is in a different CY year than the original policy effective date.
Policy D is cancelled on 1/31/2010, one half way through its policy period. Policy D will contribute 1/2 written
exposure to CY 2009 and -(1/2)*(1/2) = -0.25 written exposure to CY 2010.
Earned exposures are the portion of written exposures for which coverage has already been provided as of a
certain point in time.
The % of Policy C earned in CY 2010 is 1/6 (January only). Thus, Policy C contributes 1/2*1/6 = 1/12 earned
exposures to CY 2010.
The % of Policy D earned in CY 2010 is 1/6 (January only). Thus, Policy D contributes 1/2*1/6 = 1/12 earned
exposures to CY 2010.
Note: Unlike CY earned exposure, exposure for one policy cannot be earned in two different PYs.
In-force exposures are the number of insured units exposed to having a claim at a given point in time.
Policies A and B are not exposed to loss as of 1/1/2010 (due to policy expiration). Policy F is not exposed to
loss as of 1/1/2010 (since it is not effective until 7/1/2010).

Exam 5, V1a

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Chapter 4 – Exposures
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2011 exam:
Question 3 – CAS Model Solution
(a)
(b)
(c)
(d)
Policy
A
0
0
0
0
B
0
0
0
0
C
0
0
1/12
0
D
-1/4
0
1/12
0
E
1/2 1/2 1/2 1/2
F
1/2 1/2 1/2 1/2
Total
.75
1 14/12 1

(e)
0
0
1
1
1
0
3

Assume that a full policy = ½ car year (semi annual)
(a) .75 = -1/4 + 1/2 + 1/2
(b) 1 = 1/2 + 1/2
(c) 14/12 = 1/12 + 1/12 + 1/2 + 1/2
(d) 1 = 1/2 + 1/2
(e) 3 = 1 + 1 + 1 (recall that each semi-annual policy can contribute to one in-force exposure).

Exam 5, V1a

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Chapter 4 – Exposures
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2012 exam:
3a. (1 point) Calculate the 2011 earned exposures assuming policies are written uniformly during each quarter.
3b. (0.5 point) Discuss the appropriateness of the assumption in part a. above given the exposure data.
Question 3 – Model Solution 1 (Exam 5A Question 3)
a. Pol Eff dates
(1)
1/1 thru 3/31
4/1 thru 6/30
7/1 thru 9/30
10/1 thru 12/31

Avg eff date
(2)
2/15
5/15
8/15
11/15

% yr rem
(3)
0.875
0.625
0.375
0.125

exp
(4)
100
200
300
400

EE
(5)=(3)*(4)
87.5
125.0
112.5
50.0
375.0

2011 Earned Exposures: 375.0
3/12=.25/2=.125. [6/12+3/12]/2 = [.5+.25]/2=.375. [9/12+6/12]/2 = [.75+.5]/2=.625.
[12/12+9/12]/2 = [1.0+.75]/2=.875.
b The assumption of uniform writings throughout the quarter seems inappropriate, given that there is such
a dramatic increase in writings from one quarter to the next. It’s more likely that writings increase
throughout the quarter as well.
Question 3 – Model Solution 2 (Exam 5A Question 3)
Proportion Earned
Jan– 23/24
F - 21/24
100

M - 19/24
A - 17/24
M - 15/24
200
J - 13/24
J- 11/24
A - 9/24
300
S - 7/24
O - 5/24
N - 3/24
400
D - 1/24
2011 Earned Exposure = Avg No. of Policies Written per month * monthly Proportion Earned by year end
= 100/3 [(23 +21+19) /24] + 200/3[(17+15+13)/24] + 300/3 [(11+9+7)/24] + 400/3 [(5+3+1)/24]
= 87.5 + 125 + 112.5 + 50 = 375
b. Exposure is increasing each quarter. It is likely that this is the case within quarter ie March has more
exposure than January. We assume uniform exposure which does not appear correct with this
increasing observed exposure trend.

Exam 5, V1a

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Chapter 4 – Exposures
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2012 exam
Question 3 – Model Solution 3 (Exam 5A Question 3)
a
Policy eff dates exposures
Average written
1/1 – 3/31
100
2/15
4/1 – 6/30
200
5/15
7/1 – 9/30
300
8/15
10/1 – 12/31
400
11/15

Earned year
10.5/12
7.5/12
4.5/12
1.5/12

earned

87.5
125
112.5
50.
375
(Answer for a))

b. Appropriate to assume that policies are written uniformly during each quarter?
→ As written exposures are steadily increasing.
It won’t be appropriate to assume policies are uniformly written during the year.
→ Quarterly periods are fairly granular enough to assume that polices are written uniformly in the period.

Examiners Comments
Candidates scored well on this question. Some candidates used the same assumptions but
applied/calculated on a monthly basis. This was given full credit as well. Common mistakes include
making the exposures uniform throughout the year and effective at the beginning of the month instead of
uniform throughout the quarter.

Exam 5, V1a

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Sec
1
2
3
1

Description
Premium Aggregation
Adjustments To Premium
Key Concepts

Pages
63 - 70
70 - 87
88 - 88

Premium Aggregation

63 - 70

The goal of ratemaking is to balance the fundamental insurance equation:
Premium = Losses + LAE + UW Expenses + UW Profit.
The ratemaking process begins with applying a series of adjustments to historical premium.
1. Bring historical premium to the rate level currently in effect.
Without this adjustment, any rate changes during or after the historical period with not be fully reflected
in the premium and will distort the projections
2. Develop premium to ultimate levels if the premium is still changing.
3. Project the historical premium to the premium level expected in the future.
This accounts for changes in the mix of business that have occurred or are expected to occur after the
historical experience period.
Appendices A, C, and D provide examples from various lines of business of the premium adjustments
made in ratemaking analysis.
Two approaches to evaluate the adequacy of rates underlying an insurer’s premium are the:
 Pure premium approach and
 Loss ratio approach.
The loss ratio approach requires that premium to be collected during a future time period be
estimated (this is not the case when using the pure premium approach). When using the pure
premium approach, the adjustments in this chapter are not needed.
This chapter covers:
 ways to define and aggregate premium
 techniques used to adjust historical premium to current rate level
 techniques used to develop historical premium to ultimate level
 techniques used to measure and apply premium trend

Exam 5, V1a

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Chapte
er 5 – Prem ium
BASIC RATTEMAKING – WERNER, G
G. AND MOD
DLIN, C.
Methods of Aggregattion for Annu
ual Terms
Two meth
hods to aggregate premium
ms are CY (the same as Ca
alendar-AY) a
and PY.
Recall the 4 common
n methods of data aggrega
ation are CY, AY, PY, and RY.
Homeow
wners policies
s are used to demonstrate
e these conce
epts
Effective Expiration
Policy
y
Date
Date
Premium
A
10/01/10
09/30/11
$200
B
01/01/11
12/31/11
$250
C
04/01/11
03/31/12
$300
D
07/01/11
06/30/12
$400
E
10/01/11
09/30/12
$350
F
01/01/12
12/31/12
$225
These policies
p
are illustrated belo
ow.

The x-axis
s represents time
t
and the y-axis represents the perccentage of the
e policy term tthat has expirred (this
representation is not applicable to products
p
like warranty
w
that don’t earn evvenly).
CY and AY
A Aggregatiion consider all
a premium trransactions d
during the 12--month CY wiithout regard to the
date of po
olicy issuance
e (since CY an
nd AY premiu
ums are equivvalent, the texxt uses the term CY premiu
um).
 At
A the end of th
he CY, CY prremiums are fixed.
f
 Since CY captures transacttions occurring on or after tthe first day o
of the year, an
nd on or before the last
da
ay of the yearr, CY is repre
esented graph
hically as a sq
quare (as sho
own below).
CY Ag
ggregation

Exam 5, V1a

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Chapte
er 5 – Prem ium
BASIC RATTEMAKING – WERNER, G
G. AND MOD
DLIN, C.
PY (a.k.a. UW year) ag
ggregation considers
c
all premiums
p
on policies with effective date
es during the year.
PY is reprresented grap
phically using a parallelog
gram starting with a policy written on the
e first day of tthe PY and
ending with a policy wrritten on the la
ast day of the
e PY.
gregation
PY Agg

Since a PY takes 24 months to co
omplete, and CY premium is fixed at 12
2 months, mosst ratemaking
g analysis
focuses
s on CY premiums (and AY
Y losses).

Four typ
pes of premium
1. Written premium arise
a
from policies issued (i.e.
(
underwrittten) during a specified perriod of time (e
e.g. a
calendar quarter or a CY).
CY 2011 written pre
emium is the sum
s
of premiums for policiies having an
n effective datte in 2011.
 Since polic
cies B, C, D and E all have
e effective dattes (shown ass large circless on the horizo
ontal axis) in
2011, theirr entire premiu
um contribute
es to CY 2011
1 written prem
mium.
 However, policies
p
A and
d F have effec
ctive dates in years 2010 a
and 2012, and
d thus do not contribute to
CY 2011 written
w
premium.
CY Wrritten Premiu
um

The dis
stribution of written
w
premiu
um to each ca
alendar year iss shown belo
ow:
Calend
dar Year Written Premium
m a/o 12/31/1
12
Written Premium
m
Efffective
Expiration
E
Premium
Date
Date
Policy
P
CY 2010
CY 2011
CY 2012
A
10
0/01/10
09/30/11
0
$200.00
$200.00
$ 250.00
$250.00
B
01
1/01/11
12/31/11
1
$ 300.00
C
04
4/01/11
03/31/12
0
$300.00
$ 400.00
$400.00
D
07
7/01/11
06/30/12
0
$ 350.00
$350.00
E
10
0/01/11
09/30/12
0
$225.00
$225.00
F
01
1/01/12
12/31/12
1
$1,300.00
$1725.00
$200.00
Total
T
$225.00

Exam 5, V1a

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Policy contribution to CY:
 Each policy contributes written premium to a single CY in this example.
 However, if a policy cancels midterm, the policy will contribute written premium to two different CYs if
the policy cancellation date is in a different CY year than the original policy effective date.
If Policy D is cancelled on 3/31/2012 (i.e. after 75% of the policy has expired), then Policy D will
contribute $400 of written premium to CY 2011 and -$100= (-$400 *.25) of written premium to CY 2012.
PY Written Premium

Distribution of PY Written Premium a/o 12/31/12
Effective
Expiration
Policy
Date
Date
Premium
A
10/01/10
09/30/11
$200.00
B
01/01/11
12/31/11
$250.00
C
04/01/11
03/31/12
$300.00
D
07/01/11
06/30/12
$400.00
E
10/01/11
09/30/12
$350.00
F
01/01/12
12/31/12
$225.00
Total
$ 1,725.00

PY 2010
$200.00

Written Premium
PY 2011
PY 2012
$250.00
$300.00
$400.00
$350.00

$200.00

$1,300.00

$225.00
$225.00

In case of cancellation, the original written premium and the written premium due to the cancellation are
booked to the same PY (since PY written premium are aggregated by policy effective dates).
This contrasts with CY written premium and cancellation premium which can apply to two different CYs
depending on when the cancellation occurs.

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Chapte
er 5 – Prem ium
BASIC RATTEMAKING – WERNER, G
G. AND MOD
DLIN, C.
2. Earned
d premium are
a the portion
n of written prremium for wh
hich coverage
e has been prrovided and the insurer
is entittled to retain as
a of a certain
n point in time
e.
To bettter understan
nd the differen
nce between CY
C and PY ea
arned exposu
ure, look at th
he CY diagram
m:
CY Ea
arned premiu
um

For Po
olicy C, 75% of
o the policy period
p
is earne
ed in 2011 an
nd 25% of the
e policy period
d is earned in
n 2012.
Policy C contributes
s $225 (75% * $300) of earrned premium
m to CY 2011 and $75 earn
ned premium to CY 2012.
Distriibution of CY
Y Earned Pre
emium a/o 12
2/31/12
Effective
E
Ex
xpiration
Date
Date
Policy
Premium
A
10/01/10
09
9/30/11
$200.00
$
B
01/01/11
0
12
2/31/11
$250.00
$
04/01/11
0
03
3/31/12
$300.00
$
C
D
07/01/11
0
06
6/30/12
$400.00
$
E
10/01/11
09
9/30/12
$350.00
$
F
01/01/12
0
12
2/31/12
$225.00
$
Total

Earned Premium
CY 2011
CY 2012
$150.00
$250.00
$75.00
$225.00
$200.00
$200.00
$87.50
$262.50
$225.00
$50.00
$912.50
$762.50

CY
Y 2010
$50.00

arned Premiu
um:
PY Ea






Exam 5, V1a

Earned pre
emium is assigned to the year
y
the policyy was written and increase
es over time.
At the end of a PY (i.e. 24 months affter the start o
of a PY having annual policcies), PY earned and
written pre
emium are equivalent.
Unlike CY
Y earned prem
mium, premiium for one p
policy canno
ot be earned in two differrent PYs.
Premiums for lines subjject to premiu
um audits con
ntinue to deve
elop after the end of the po
olicy period.

Page 77

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Chapte
er 5 – Prem ium
BASIC RATTEMAKING – WERNER, G
G. AND MOD
DLIN, C.
PY Earne
ed Premium a/o
a 12/31/12
Effective
E
Expiration
E
Earrned Premium
m
Policy
Date
Date
Premium
PY 2010
PY 2011
PY 2012
A
10/01/10
1
09/30/11
0
$200.00
$200.00
$250.00
$250.00 $
B
01/01/11
0
12/31/11
1
$300.00
$300.00
C
04/01/11
0
03/31/12
0
$400.00
$400.00
D
07/01/11
0
06/30/12
0
$350.00
$350.00
E
10/01/11
1
09/30/12
0
$225.00
F
01/01/12
0
12/31/12
1
$225.00
$200.00
$1,300.00
Total
$1,725.00
$
$225.00
3. Unearrned premium
m is the portio
on of written premium
p
for w
which coverag
ge has not ye
et been provid
ded as of
that po
oint in time (an
nd applies to individual policies and gro
oups of policie
es).
Written
n Premium = Earned Prem
mium + Unearn
ned Premium
m (ok when PY
Y aggregation
n is used)
CY Unearned Prem
mium = CY WP
P – CY EP + Unearned Pre
emium as of tthe beginning
g of the CY.
4. In-forc
ce premiums
s are the number of insured
d units expossed to having a claim at a g
given point in time.
Examp
ple: The in-fo
orce premium as of 6/15/20
011 is the sum
m of full-term premium for all policies that have an
inception
n date on or before
b
6/15/20
011 and an exxpiration date
e after 6/15/20
011.
A vertica
al line drawn at
a the valuatio
on date will in
ntersect the po
olicies that arre in-force on that date.
Policies A, B, and C are
a in effect on
o 6/15/11 and
d each contributes to the 6
6/15/11 in-forcce exposures
s.
In-Fo
orce Premium
m

In-fo
orce Premium
m by Date
In-F
Force Premium
m a/o
Poliicy
A
B
C
D
E
F
Tottal

Effective
Date
10/01/10
01/01/11
04/01/11
07/01/11
10/01/11
01/01/12

Expiration
Date
09/30/11
12/31/11
03/31/12
06/30/12
09/30/12
12/31/12

Premium

01
1/01/11 06/15
5/11

$200.00 $2
200.00
$250.00 $2
250.00
$300.00
$400.00
$350.00
$225.00
$1,725.00 $4
450.00

01/01/1 2

$200
0.00
$250
0.00
$300
0.00

$--$300.0
00
$400.0
00
$350.0
00
$225.0
00
$750
0.00 $1,275.0
00

Calcullation of in-fo
orce premium
m (in case off a mid-term adjustment)):
 Assume
A
Policy
y D is changed on 1/1/2012
2 and full-term
m premium in
ncreases from
m $400 to $800.
 The policyhold
der will pay $6
600 (=$400 x 0.5 + $800 x 0.5).
00 for an in-fo
orce date betw
ween 7/1/2011 and 12/31//2011 and $80
00 for an
 The in-force prremium is $40
in
n-force date between 1/1/2012 and 6/30
0/2012.
 The in-force prremium is the
e best estimatte of the insurrer’s mix of bu
usiness as of a given date. The most
ecent in-force premium is used
u
to measure the impacct of a rate ch
hange on an e
existing portfo
olio.
re

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Policy Terms Other Than Annual
When policy terms are not annual the concepts are the same. See chapter 4 for the techniques involved.
Caution is needed when interpreting in-force premium when considering portfolios with policies of different
terms.
Calculation of Blocks of Policies
Insurers may have policy information summarized on a monthly or quarterly basis and need to calculate
exposures for the block of policies using this summarized data. In such a case,
 it is customary to treat all policies as if they were written on the mid-point of the period.
 when summarizing on a monthly basis, all policies are assumed to be written on the 15th of the month.
(i.e. this is known as “15th of the month” rule )
 this approximation applies as long as policies are written uniformly during each time period.
 if this approach is applied to longer periods (e.g. quarters or years), the assumption of uniform writings is
less likely to be reasonable.

Adjustments To Premium

70 - 87

To project future premium, historical premium must be:
 brought to current rate level. This involves adjusting premium for rate increases (decreases) that
occurred during or after the historical experience period.
This is known as adjusting the premium “to current rate level” or putting the premium “on-level”.
Two current rate level methods are extension of exposures and the parallelogram method.
 developed to ultimate. This is relevant when an analyzing incomplete policy years or premium that
has yet to undergo audit.
 adjusted for actual or expected distributional changes. This is done through premium trending,
and both the one-step and two-step trending are discussed in this section.
Current Rate Level
Consider a case in which all policies were written at a rate of $200 during the historical period.
 After the historical period, there was a 5% rate increase so the current rate in effect is $210.
 Assume the “true” indicated rate for the future ratemaking time period is $220.
i. If the historical rate (i.e. $200) is compared to the indicated rate (i.e. $220) without considering the
5% increase already implemented, the conclusion that rates need to be increased by 10% is
reached, resulting in a new indicated rate of $231 (= $210 x 1.10), which is excessive.
ii. If instead, historical premium were restated to the present rate level of $210 and compared to the
indicated rate, the correct rate need of 4.8% (= $220/210 - 1.00) is reached.
The extension of exposures method and the parallelogram method bring premium to the current rate level are
discussed below.

Exam 5, V1a

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Simple Example
Assume policies have annual terms and premium is calculated according to the following rating algorithm:
Premium = Exposure x Rate per Exposure x Class Factor + Policy Fee.
The class factor has three values, or levels (X, Y, and Z), each with a distinct rate differential. The following
three rate changes occurred during or after the historical experience period.
• 7/1/2010: the base rate was increased and resulted in an overall average rate level increase of 5%.*
• 1/1/2011: the base rate and policy fee were adjusted resulting in an overall average rate level
increase of 10%.
• 4/1/2012: the policy fee and class Y and Z rate relativities were changed resulting in an overall
average rate level decrease of -1%.
* The reader may be confused by the overall average rate changes provided in this example [e.g., how a 5.6% (=950/900-1.00)
change in rate per exposure results in an overall average rate change of 5.0%]. The overall average rate change considers the
average change in the total premium per policy, which is a function of the rate per exposure, the number of exposures per
policy, the applicable class factors, and the policy fee. These detailed inputs have not been provided; the overall average rate
change should be taken as a given for the purpose of illustrating premium at current rate level techniques.

Rate Change History
Rate
Level
Effective
Group
Date
1
Initial
2
07/01/10
3
01/01/11
4
04/01/12

Overall
Average
Rate change
-5.0%
10.0%
-1.0%

Rate
Per
Exposure
$900
$950
$1,045
$1,045

X
1.00
1.00
1.00
1.00

Class Factor
Y
0.60
0.60
0.60
0.70

Z
1.10
1.10
1.10
1.10

Policy
Fee
$1,000
$1,000
$1,100
$1,090

Method 1: Extension of Exposures
This method rerates every policy to restate historical premium to the amount that would be charged under the
current rates.
Advantage: It is the most accurate current rate leveling method, given the level of current computing power to
perform the number of calculations required to rerate each policy.
Disadvantage: The rating variables, risk characteristics and rating algorithm needed to rerate each policy
during the historical period are often not readily available.
Assume the following:
 We wish to adjust the historical premium for PY 2011 to the current rate level.
 One such policy was effective on 3/1/2011 and had 10 class Y exposures.
 The actual premium charged for the policy was based on the rates effective on 1/1/2011, and was
$7,370 (= 10 x $1,045 x 0.60 + $1,100).
To put the premium on-level:
 Substitute the current base rate, class factor, and policy fee in the calculations; this results in an onlevel premium of $8,405 (= 10 x $1,045 x 0.70 + $1,090).
 Perform the same calculation for every policy written in 2011 and then aggregate across all policies.
Notes: Policies with the exact same rating characteristics can be grouped for the purposes of the extension of
exposures technique, but is only relevant in lines with simple rating algorithms and few rating variables.
In commercial lines products, where subjective debits and credits can be applied to manual premium,
complicates the use of the extension of exposures technique since it may be difficult to determine what
debits and credits would be applied under today’s schedule rating guidelines.

Exam 5, V1a

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Method 2: Parallelogram Method (a.k.a. the geometric method)
The parallelogram method:
 is performed on a group of policies
 is less accurate than extension of exposures.
 assumes that premium is written evenly throughout the time period
 involves adjusting aggregated historical premium by an average factor to put the premium on-level.
 application varies by policy term, method of aggregation (CY vs. PY), and whether the rate change
affects policies midterm or only policies with effective dates occurring after the change.
Standard Calculations
The objective: Replace the average rate level for a given historical year with the current rate level.
The major steps are as follows:
1. Determine the timing and amount of the rate changes during and after the experience period and group
the policies into rate level groups according to the timing of each rate change.
2. Calculate the portion of the year’s earned premium corresponding to each rate level group.
3. Calculate the cumulative rate level index for each rate level group.
4. Calculate the weighted average cumulative rate level index for each year.
5. Calculate the on-level factor as the ratio of the current cumulative rate level index and the average
cumulative rate level index for the appropriate year.
6. Apply the on-level factor to the earned premium for the appropriate year.
For the parallelogram method, exact rates are not required.
Step 1: Obtain the effective date and overall rate changes for the policies under consideration.
Recall that annual policies have been issued and rate changes apply to policies effective on or after the
date (i.e. do not apply to policies in mid-term).
Rate
Overall
Level
Average
Effective
Group
Rate
Date
1
Initial
2
07/01/10
5.0%
3
01/01/11
10.0%
4
04/01/12
-1.0%
Step 2: View these rate changes in graphical format.
Assume the actuary is trying to adjust each CY’s EP premium to current rate level.
 CYs are represented by squares.
 Each rate change is represented by a diagonal line, the slope of which depends on the term
of the policy (which is annual in this case)
 The numbers 1, 2, 3, and 4 represent the rate level group in effect.

Exam 5, V1a

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Rate Changes assuming CY EP with Annual Policies

Next calculate the portion of each CY’s EP (the area within the square) that corresponds to each rate level.
For CY 2011, there are three areas representing EP on policies written:
 after 1/1/2010 and prior to the 7/1/2010 rate change (area of rate level group 1 in CY 2011).
 on or after 7/1/2010 and before 1/1/2011 (area of rate level group 2 in CY 2011).
 on or after 1/1/2011 and before 1/1/2012 (area of rate level group 3 in CY 2011).
Geometry and the assumption that the policies written are uniformly distributed are used to calculate the
portion of the square represented by each rate level area.
Note: The following geometric formulae may be used in the parallelogram method:
Area of a triangle: ½ x base x height
Area of a parallelogram: base x height
Area of a trapezoid: ½ x (base1 + base 2) x height

Area 1 in CY 2011 is a triangle with area equal to ½ x base x height.
The base and height are both 6 months (1/1/2011 to 6/30/2011) so the area (in months) is 18 (= ½ x 6 x 6).
This area’s portion of the entire CY square is 0.125 (=18 /(12 x 12)).
Simplify by restating the base and height as portions of a year (0.125 = ½ x ½ x ½).
In some areas (e.g. area 2 in CY 2011), it is easier to calculate as 1.0 - the sum of the remaining areas.
CY 2011 rate levels area are shown below:
Area 1 in CY 2011:
0.125
=0.50 x 0.50 x 0.50
Area 2 in CY 2011:
0.375
=1.00 - (0.125 + 0.500)
Area 3 in CY 2011:
0.500
=0.50 x 1.00 x 1.0

Exam 5, V1a

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Step 3: Calculate the cumulative rate level index for each rate level group.
 The first rate level group is assigned a rate level of 1.00.
 The cumulative rate level index of each subsequent group is the prior group’s cumulative rate
level index multiplied by the rate level for that group.
i. the cumulative rate level index for the second rate level group is 1.05 (= 1.00 x 1.05).
ii. the cumulative rate level index for the third rate level group is 1.155 (= 1.05 x 1.10).
1
2
3
4
Overall
Average Rate Level Cumulative
Rate
Effective
Rate
Level
Date
Index
Rate Level
Change
Group
Index
1
Initial
-1.00
1.0000
2
7/1/10
5.0%
1.05
1.0500
3
1/1/11
10.0%
1.10
1.1550
4
4/1/12
-1.0%
0.99
1.1435
(4)=
(Previous Row 4) x (3)
Step 4: Calculate the average rate level index for each year (i.e. the weighted average of the cumulative
rate level indices in Step 3, using the areas calculated in Step 2 as weights).
The average rate level index for CY 2011 is 1.0963 =1.000 x 0.125 + 1.0500 x 0.375 + 1.1550 x 0.500.
Step 5: Calculate the on-level factor as follows:

On - Level Factor for Historical Period 



Current Cumulative Rate Level Index
Average Rate Level Index for Historical Period

The numerator is the most recent cumulative rate level index
The denominator is the result of Step 4.

The on-level factor for CY 2011 EP (assuming annual policies) is 1.0431 

1.1435
1.0963

Step 6: The on-level factor is applied to the CY 2011 EP to bring it to current rate level.
CY 2011 EP at current rate level= CY 2011 EP x 1.0431.

Exam 5, V1a

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Standard CY Calculations for Six-Month Policies
If the policy term is six months (common in personal automobile coverage), then the rate level groups can
be depicted as follows:

Step 2: The areas for CY 2011 are:
Area 1 in CY 2011: N/A
Area 2 in CY 2011: 0.250
Area 3 in CY 2011: 0.750

= 0.50 x 0.50 x 1.00
= 1.00 - 0.250

Step 3: The cumulative rate level indices are the same as those used for the annual policies.
Step 4: The average rate level index for CY 2001 assuming semi-annual policies:
1.1288 = 1.0500 x 0.250 + 1.1550 x 0.750
Step 5: The on-level factor to adjust CY 2011 EP to current rate level is: 1.0130 =

1.1435
(and is
1.1288

smaller than for annual policies because the semi-annual rate changes earn more quickly).
Standard PY Calculations for Annual Policies

Since PY 2011 only had one rate level applied to the whole year, PY 2012 will be reviewed.
The area of each parallelogram is base x height.
Area 3 in Policy Year 2012 has a base of 3 months (or 0.25 of a year) and the height is 12 months (or 1.00 year).
Step 2: The relevant areas for PY 2012 are as follows:
• Area 3 in PY 2012: 0.25 = 0.25 x 1.00
• Area 4 in PY 2012: 0.75 = 0.75 x 1.00
Step 3: The cumulative rate level indices are the same as those used in the CY example.
Step 4: The average rate level index for PY 2012 is: 1.1464 = 1.1550 x 0.25 + 1.1435 x 0.75.
Step 5: The on-level factor to adjust PY 2012 EP to current rate level is 0.9975 

Exam 5, V1a

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1.1435
1.1464

 2014 by All 10, Inc.

Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Rate Changes Mandated by Law
Rate changes mandated by law changes apply the rate change to all policies on or after a specific date
(including those in-force).
The rate level change is represented as a vertical line.
Assume a law change mandates a rate decrease of 5% on 7/1/2011 applicable to all policies.

The vertical line splits rate level groups 2 and 3 into two pieces each.
The -5% law change impacts rate level indices associated with the portion of areas 2b, 3b, and 4.
The areas for CY 2011 are as follows:
• Area 1 in CY 2011: 0.125 =
• Area 2a in CY 2011: 0.250 =
• Area 2b in CY 2011: 0.125 =
• Area 3a in CY 2011: 0.125 =
• Area 3b in CY 2011: 0.375 =

0.50 x 0.50 x 0.50
0.50 - 0.125 - 0.125
0.50 x 0.50 x 0.50
0.50 x 0.50 x 0.50
0.50 - 0.125

The cumulative rate level indices associated with each group are as follows:
Step 3 (with Benefit Change)
Rate Level
Cumulative Rate
Group
Level Index
1
2a
2b
3a
3b
4

1.0000
1.0500
0.9975
1.1550
1.0973
1.0863

CY 2011 on-level factor:

1.0171 

Exam 5, V1a

1.0863
1.0000 x 0.125  1.0500 x 0.250  0.9975 x 0.125  1.1550 x 0.125  1.0973 x 0.375

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 2014 by All 10, Inc.

Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Comments on the Parallelogram Method
Two problems with the parallelogram method:
1. The method is not useful if the assumption that policies are evenly written throughout the year is not true.
Example: Boat owners policies are usually purchased prior to the start of boat season and thus are not
uniformly written throughout the year.
Ways to partially circumvent the need for uniform writings:
a. Use a more refined period of time than a year (e.g. quarters or months).
b. Calculate the actual distribution of writings and use these to determine more accurate weightings to
compute the historical average rate level.
Aggregate policies based on which rate level was applicable rather than based on a time period, and
the premium for each rate level group is adjusted together based on subsequent rate changes.
2. Premium for certain classes will not be on-level if the implemented rate changes vary by class.
Even if the overall premium may be adjusted to a current rate level, adjusted premium will not be
appropriate for class ratemaking.
This major shortcoming has caused insurers to favor of the extension of exposures approach.
Premium Development
When working with an incomplete year of data or when premiums for a line of business are subject to premium
audits, premium development methods are used for ratemaking purposes.
To incorporate responsiveness into the ratemaking analysis, the actuary may choose to use data for a year that
is not yet complete (more common for PY analysis due to the time it takes for the PY to close).
Assume a ratemaking analysis is performed on PY 2011 data as of 12/31/2011.
 While WP is known, it is not known which policies may have changes or will be cancelled during the
policy term.
 To estimate how premium will develop to ultimate, historical patterns of premium development are
analyzed to understand the effect of cancellations and mid-term adjustment on PY premium.
For Lines that utilize premium audits:
 The insured will pay premium based on an estimate of the total exposure.
 Once the policy period is complete and the actual exposure is known, the final premium is calculated.
For example, WC premium depends on payroll and the final WC premium is determined by payroll
audits that occur 3 - 6 months after the policy expires.
Premium development depends on several factors including:
 The type of plan (permitted by the jurisdiction or offered by the carrier).
 The stability between the original premium estimate and the final audited premium.
 Internal company operations (e.g. auditing procedures, marketing strategy, accounting policy, etc.).

Exam 5, V1a

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
PY Premium Development Example:
 A WC carrier writes one policy per month in 2011.
 Estimated premium for each policy is booked at policy inception for $500,000.
 Premium develops upward by 8% at the first audit (6 months after the policy expires).
At 12/31/2012, the six policies written in the first half of 2011 have completed their audits, but the six policies
written in the second half of the year have not.
PY 2011 premium as of 12/31/2012 is: $6,240,000 = 6 x $500,000 x 1.08 + 6 x $500,000
At 12/31/2013, all twelve policies have completed their final audits and premium is final.
PY 2011 premium as of 12/31/2013 is: $6,480,000 = 12 x $500,000 x 1.08
From 12/31/2012 (24 months after the start of the PY) to 12/31/2013 (36 months after the start of the PY), the
premium development factor is 1.0385 (= $6.48 million / $6.24 million).
Premium development does not typically apply to CY premium since CY premium is fixed. However, some
actuaries may adjust CY premium if audit patterns are changing and a CY analysis is being performed.
Note: Rates changes, Inflationary changes and Policy Characteristic Distributional changes impact the
average premium level
Exposure Trend
The average premium level can change over time due to inflation in lines of business with exposure bases that
are inflation-sensitive, like payroll (for WC and GL) or receipts (GL).
Trends are used to project inflation-sensitive exposures (and thus premium) and are determined using internal
company data (e.g. WC payroll data) or industry or government indices (e.g. average wage index).
Premium Trend
The average premium level can change over time due to changes in the characteristics of the policies written
(a.k.a. distributional changes) and the resulting change in average premium level is known as premium trend.
Examples that can cause changes in the average premium level:
• A rating characteristic can cause average premium to change (e.g. HO premium varies based on
the amount of insurance purchased, which is indexed and increases automatically with inflation;
therefore, average premium increases as well).
• Moving all existing insureds to a higher deductible (e.g. if an insurer moves each insured to a
higher deductible upon renewal, and renewals are spread throughout the year, there will be a decrease
in average premium over the entire transition period).
Trend is not necessary once the transition is complete.
• Acquiring the entire portfolio of another insurer writing higher policy limits (e.g. a HO insurer
acquires a book of business that includes predominantly high-valued homes, the acquisition will cause a
very abrupt increase in the average premium due to the increase in average home values).
After the books are consolidated, no additional shifts in the business are expected.

Exam 5, V1a

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
To adjust for premium trend, the actuary needs to:
 determine how to measure any changes that have occurred
 decide whether observed distributional shifts were caused by a one-time event or a shift that is
expected to continue in the future
 judgmentally incorporate any additional shifts that are reasonably expected to happen in the future.
Actuaries examine changes in historical average premium per exposure to determine premium trend.
Average premium should be calculated on an exposure basis rather than a policy basis, using the exposure
base underlying the rate.
A decision to use earned or written premium must be made.
Written premium is a leading indicator of trends that will emerge in earned premium and the trends observed
in written premium are appropriate to apply to historical earned premium.
Assuming adequate data is available, the actuary will use quarterly average written premium (as
opposed to annual average written premium) to make the statistic as responsive as possible.
Data used to estimate premium trend due to distributional changes: Change in Average WP
(1)

(2)

(3)

(4)
Average
Written
Premium at
Rate Level

(5)

Quarter

Written
Premium at
Current Rate

Written
Exposures

1Q09

$323,189.17

453

$713.44

2Q09

$328,324.81

458

$716.87

3Q09

$333,502.30

463

$720.31

4Q09

$338,721.94

468

$723.76

1Q 10

$343,666.70

472

$728.11

2.1%

2Q10

$348,696.47

477

$731.02

2.0%

3Q10

$353,027.03

481

$733.94

1.9%

4Q10

$358,098.58

485

$738.35

2.0%

1Q11

$361,754.88

488

$741.30

1.8%

2Q11

$367,654.15

493

$745.75

2.0%

3Q11

$372,305.01

497

$749.10

2.1%

4Q11

$377,253.00

501

$753.00

2.0%

Annual
Change

(4) = (2) / (3)
(5) = (4) / (Prior Year4) - 1.0
Changes in the quarterly average WP are used to determine the amount historical premium needs to be
adjusted for premium trend.
Note the premium used has been adjusted to the current rate level (if this is not done, the data will show
an abrupt change in the average written premium corresponding to the effective date of the rate change).

Exam 5, V1a

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Chapte
er 5 – Prem ium
BASIC RATTEMAKING – WERNER, G
G. AND MOD
DLIN, C.
Two meth
hods for adjus
sting historica
al data for premium trend: o
one-step and two-step tren
nding.
One-Step
p Trending
The trend factor adjustts historical prremium to acc
count for exp ected premiu
um levels from
m distributiona
al
shifts in premium writin
ngs.
The Proce
ess: Using th
he annual cha
anges from the prior table, the actuary m
may select a ttrend factor o
of 2%
(the am
mount average
e premium is expected to cchange annua
ally).
Next:
Determine th
he trend perio
od.
Assume: WP is used as the basis of the trend selection
s
and EP for the ovverall rate leve
el indications
Compute:: The trend period
p
as the length of time
e from the ave
erage written date of policies with premiium
earned during the historical period to the average w
written date fo
or policies tha
at will be in efffect
during the time the rates
s will be in effe
ect. *
* Some insurers determine the
e trend period as the average date
e of premium ea
arned in the expe
erience period to the
e of premium earned in the projec
cted period. This simply shifts botth dates by the ssame amount, so
o the
average date
trend period is
i the same length.

Example: Assume CY
Y 2011 EP is being
b
used to estimate the rate need forr annual policcies that are to be
in effect from 1/1/2013 – 12/31/2013.
The historica
al and projected periods ca
an be represe
ented as follow
ws:

Historical period: CY
Y 2011 EP co
ontains premiu
um from policcies written 1//1/2010 to 12//31/2011.
Th
hus, the avera
age written da
ate for premiu
um earned is 1/1/2011.
Projecte
ed period: Po
olicies will be written from 1/1/2013
1
– 12
2/31/2013.
Th
hus, the avera
age written da
ate during the
e projected pe
eriod is 6/30/2
2013.
Therefo
ore, the trend period is 2.5 years (i.e. 1/1
1/2011 - 6/30 /2013).
The adjus
stment to acco
ount for prem
mium trend is: 1.0508 (= (1..0 + 0.02)2.5).
Trend Period
P
for 1-S
Step Trendin
ng

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Items affecting the length of the trend period:
1. If the historical period consists of policies with terms other than 12 months, the “trend from” date will
be different than discussed above.
Example: If the policies in the prior example were six-month policies, then the “trend from” date is 4/1/2011.
The “trend to” date is unchanged.
Trend Period for 1-Step Trending with 6-Month Policies

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Two-Step Trending
Two-step trending is used when the insurer expects premium trend to change over time.
Adjust the historical premium to the level present at the end of the historical period, and then apply a separate
adjustment to project premium into the future.
Two step trending may be used by a homeowners’ insurer that observes large increases in amount of
insurance during the experience period that are not expected to continue into the future.
Step 1: Adjust the historical premium to the current trend level using the following adjustment factor:

Current Premium Trend Factor =

Latest Average WP at Current Rate Level
Historical Average EP at Current Rate Level

If average EP for CY 2011 is $740.00 and the average WP for the latest available quarter (Calendar
Quarter 4Q 2011) is $753.00, then the current premium trend factor is 1.0176 (= 753.00/740.00).
The latest average WP is for the fourth quarter of 2011; thus, the average written date is
11/15/2011 (this will be “trend from” date for the second step in the process).
If the average been based on the average WP for CY 2011 (as opposed to the fourth quarter), then the
average written date would have been 6/30/2011.
When average premium is volatile, select a current trend versus using the actual change in average premium.
The current trend factor is calculated by trending (1.0 + selected current trend) from the average written
date of premium earned in the experience period (i.e. 1/1/2011) to the average written date of the latest
period in the trend data (i.e. 11/15/2001).
Step 2: Compute the projected premium trend factor.
Select the amount the average premium is expected to change annually from the “trend from” date to the
projected period.
The “trend from” date is 11/15/2011.
The “trend to” date is the average written date during the period the proposed rates are to be in effect,
which is still 6/30/2013.
Thus, the projected trend period is 1.625 years long (11/15/2011 to 6/30/2013).
Given a projected annual premium trend of 2%, the projected trend factor is 1.0327 (= (1.0 + 0.02)1.625).
Trend Period for 2-Step Trending

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
The total premium trend factor for two-step trending is the product of the current trend factor and the
projected trend factor (i.e. 1.0509 (= 1.0176 x 1.0327)).
That number is applied to the average historical EP at current rate level to adjust it to the projected level:
CY11 EP at projected rate level = CY11 EP at current rate level x Current Trend Factor x Projected
Trend Factor.
Two-Step Trending
(1) CY 2011 Earned Premium at Current Rate Level
(2) CY 2011 Earned Exposures
(3) CY 2011 Average Earned Premium at Current Rate Level
(4) 4th Quarter of 2011 Average Written Premium at Current Rate Level
(5)Step 1 Factor
(6) Selected Projected Premium Trend
(7) Projected Trend Period
(8) Step 2 Factor
(9) Total Premium Trend Factor
(10) Projected Premium at Current Rate Level

$1,440,788
1,947
$740.00
$753.00
1.0176
2.0%
1.6250
1.0327
1.0509
$1,514,124

(3) = (1) / (2)
(5) = (4) / (3)
(7)
(8) = (1.0 + (6))
(9) = (5) x (8)
(10)= (1) x (9)

Appendices A-D provide realistic examples of ratemaking analysis, including the premium adjustments,
intended to reinforce the concepts covered in this chapter.

Key Concepts

88 - 88

1. Premium aggregation
a. Calendar year v. policy year
b. In-force v. written v. earned v. unearned premium
2. Premium at current rate level
a. Extension of exposures
b. Parallelogram method
3. Premium development
4. Exposure trend
5. Premium trend
a. One-step trending
b. Two-step trending

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
The predecessor papers to the current syllabus reading “Basic Ratemaking” by Werner, G. and
Modlin, C. were numerous. While past CAS questions were drawn from prior syllabus
readings, the ones shown below remain relevant to the content covered in this chapter.

Section 1: Premium Aggregation – In General
Questions from the 1989 Exam:
43. (3 points) You are given the following data.
Personal Lines Automobile - State A
Rate level history:
+10% effective 7/1/86
+10% effective 7/1/88
Assume that exposures are uniformly distributed throughout the year.
Using the parallelogram method described in McClenahan's chapter on ratemaking (Study Note 16) and
"A Refined Model for Premium Adjustment" by Miller and Davis (note: the latter is no longer on the
syllabus), calculate the on-level factors needed to bring calendar year 1987 and 1988 earned premiums
to current rate level.
a. (1.5 points)

Assume policies are annual (each policy has a 12 month term.)

b. (1.5 points) Assume policies are semiannual (each policy has a six month term.)

Questions from the 1991 exam
For the next three questions use the parallelogram method as described in Chapter 2 of the CAS textbook
Foundations of Casualty Actuarial Science and assume exposures are written uniformly throughout the year.
You are given the following data:
Effective Date
7/1/88
1/1/89
7/1/89
7/1/90
1/1/91

Rate Change
+ 8.0 %
+ 10.0 %
+ 5.0 %
+ 2.0 %
+ 2.0 %

14. Assume all policies have a six month term. The on-level factor for calendar year 1989 earned premium is
in which of the following ranges?
A. < 1.05 B. > 1.05 but < 1.09 C. > 1.09 but < 1.13 D. > 1.13 but < 1.17 E. >1.17
15. Assume all policies have a six month term. The on-level factor for policy year 1989 earned premium is in
which of the following ranges?
A. < 1.05 B. > 1.05 but < 1.09 C. > 1.09 but < 1.13 D. > 1.13 but < 1.17 E. > 1.17
16. Assume all policies have a twelve month term. The on-level factor for calendar year 1989 earned
premium is in which of the following ranges?
A. < 1.05 B. > 1.05 but < 1.09 C. > 1.09 but < 1.13 D. > 1.13 but < 1.17 E. > 1.17

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 1994 exam
1. An insurer writes the following policies during 1992:
Effective
Date
May 1
August 1
November 1

Policy
Term
6 months
12 months
6 months

Premium
$6,000
$12,000
$2,400

What is the insurer's unearned premium reserve on December 31, 1992?
A. <$6,000

B. >$6,000 but <$7,000

C. >$7,000 but <$8,000

D.> $8,000, but < $9,000 E. > $9,000.

Questions from the 1996 exam
Question 30. (4 points) You are given:
Wisconsin Personal Automobile Bodily Injury
20/40 Basic Limits
Calendar/
Accident
Year
1992
1993
1994
Combined

Ultimate
Loss &
ALAE
325,000
575,000
800,000
1,700,000

Written
Premium
750,000
1,000,000
1,250,000
3,000,000

Earned
Premium
375,000
875,000
1,125,000
2,375,000

Rate Level History
Effective
% Rate
Date
Change
1/1/91
+7.0%
10/1/93
+5.0%
7/1/94
+3.0%
1/1/95
+5.0%

Questions from the 1997 exam
19. You are given:
Effective Date
4/1/94
7/1/95
4/1/96

Rate Change
+5.0%
+13.0%
-3.0%

• All policies are 12 month policies.
• Policies are written uniformly throughout the year.
Using the parallelogram method described by McClenahan, "Ratemaking," chapter 2 of Foundations of
Casualty Actuarial Science, in what range does the on-level premium factor fall, to bring calendar year 1995
earned premium to current rate level?
A. < 1.07

Exam 5, V1a

B. > 1.07 but < 1.09

C. > 1.09 but < 1.11

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D. > 1.11 but < 1.13

E. > 1.13

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 1998 exam
41. (2 points)
You are given the following information for your company's private passenger automobile line of business.
Calendar
Year
1994
1995
1996

Earned
Premium
$1,000
$1,200
$1,400

Overall
Rate Change
+5.0%
+10.0%
-5.0%
+15.0%

Effective
Date
9/1/94
1/1/95
1/1/96
4/1/97

Assume all policies are semi-annual and that all months have the same number of days.
Using the parallelogram method as described in McClenahan, "Ratemaking," chapter 2 of Foundations of
Casualty Actuarial Science, compute the calendar year 1995 earned premium at present rates.

Questions from the 1999 exam
58. (2 points) Using the Loss Ratio method described in McClenanhan's "Ratemaking" chapter 2 of
Foundations of Casualty Actuarial Science, you have performed a rate review for your company's
Homeowners line of business which issues annual policies. You have calculated a Rate Level Adjustment
Factor (RLAF) of 1.080 for Calendar Year 1998 Earned Premium. The only rate change in the past few
years was one that you assumed to be effective 1/1/98. However, upon further review, you realize that the
effective date is incorrect and that the rate change was actually implemented effective 3/1/98.
Recalculate the RLAF using the 3/1/98 effective date. Assume that all months have an equal number
of days and that premium writings are evenly distributed through the year.

Questions from the 2000 exam
38. (4 points) Based on McClenahan, "Ratemaking," chapter 2 of Foundations of Casualty Actuarial Science,
and the following data, answer the questions below. Personal Automobile Liability Data:
Calendar Year 1997
Calendar Year 1998
No. of Autos Written on
No. of Autos Written on
Effective Date
Effective Date
Effective Date
Effective Date
January 1, 1997
100
January 1, 1998
900
April 1, 1997
300
April 1, 1998
1,100
July 1, 1997
500
July 1, 1998
1,300
October 1, 1997
700
October 1, 1998
1,500
Assume:
• All policies are twelve-month policies.
• Written premium per car during calendar year 1997 is $500.
• A uniform rate increase of 15% was introduced effective July 1, 1998.
a. (1/2 point)
Calculate the number of in-force exposures on January 1, 1998. (chapter 4)
b. (1 point)
Calculate the number of earned exposures for calendar year 1998. (chapter 4)
c. (1/2 point)
List the two methods McClenahan describes that are used to adjust earned premiums to a
current rate level basis. (chapter 5)
d. (1 point)
Which of the two methods listed in part c. above would be more appropriate to use for this
company's personal automobile liability business? Briefly explain why. (chapter 5)
e. (1 point)
Using your selected method from part d. above, calculate the on-level earned premium for
calendar year 1998. (chapter 5)

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2000 exam
40. (4 points) Using the techniques described by McClenahan in "Ratemaking," chapter 2 of Foundations of
Casualty Actuarial Science, and the following data, answer the questions below.
You are given the following information for your company's homeowners business in a single state:
Calendar/
Ultimate Loss
Accident Year
and ALAE
Written Premium
Earned Premium
1997
635,000
1,000,000
975,000
1998
595,000
1,050,000
1,000,000
Effective Date
July 1, 1996
January 1, 1998
July 1, 1999

Rate Change
+4.0%
+1.8%
+3.0%

Target Loss and ALAE Ratio
Proposed effective date
Effective period for rates
Credibility
Alternative indication
Policy period
Severity trend
Frequency trend

0.670
July 1, 2000
One year
0.60
0.0%
Twelve months
+3.0%
+1.0%

a. (1 1/2 points) Calculate the on-level factors for each of the two calendar years 1997 and 1998. (chapter 5)
b. (1 1/2 points) Calculate the trended projected ultimate on-level loss and ALAE ratio for the combined
experience period 1997-1998. (chapter 6)
c. (1 point) Calculate the credibility-weighted indicated rate level change. (chapter 8)

Questions from the 2001 exam
Question 38. (2 points) Using the parallelogram method described by McClenahan in “Ratemaking,”
chapter 2, Foundations of Casualty Actuarial Science, determine the calendar year 1999
on-level earned premium. Show all work.
Calendar Year

Earned Premium

Effective Date

Rate Change

1997

$10,000

July 1, 1997

+5.2%

1998

$11,500

No Change

1999

$14,000

April 1, 19999

+7.4%

Exam 5, V1a



All policies are 2-year policies.



Policies are written uniformly throughout the year.

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2002 exam
17. (4 points) Based on McClenahan, "Ratemaking," chapter 2 of Foundations of Casualty Actuarial
Science, and the following data, answer the questions below. Show all work.
Projected rates to be effective January 1, 2003 and in effect for 1 year.
Target loss and ALAE ratio is 65%.
Experience is from the accident period January 1, 2000 to June 30, 2001.
Developed accident period loss and ALAE is $21,500.
Annual trend factor is 3%.
All policies have one-year terms and are written uniformly throughout the year.
The rate on January 1, 1999 was $120 per exposure.
Effective Date
January 1, 2000
January 1, 2001
Year
1998
1999
2000
2001

Rate Change
+10%
-15%
Written Exposures
200
200
200
200

a. (1 point) Calculate the experience period trended developed loss and ALAE. (chapter 6)
b. (2 points) Calculate the experience period on-level earned premium. (chapter 5)
c. (1 point) Calculate the indicated statewide rate level change. (chapter 8)

Questions from the 2003 exam
10. A 12-month policy is written on March 1, 2002 for a premium of $900. As of December 31, 2002,
which of the following is true?

A.
B.
C.
D.
E.

Exam 5, V1a

Calendar Year
2002 Written
Premium
$900
$750
$900
$750
$900

Calendar Year
2002 Earned
Premium
$900
$750
$750
$750
$750

Inforce
Premium
$900
$900
$750
$750
$900

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2005 exam:
38. (1.5 points) The parallelogram method is used to adjust calendar year 2003 earned premium to
current rate level. Given the following information, will the parallelogram method understate,
overstate, or accurately state the on-level factor applied to calendar year 2003 earned
premium? Explain your answer.
• There was a 10% rate increase effective on January 1, 2003.
• The written exposures grew 5% each month in 2003.

Questions from the 2006 exam:
28. (3 points) Company XYZ reduced rates 8% effective May 1, 2004, which was their first rate
change since January 1, 2000. Assume all policies have annual terms.
a. (1 point) Using the parallelogram method, calculate the 2005 on-level factor. Show all work.
b. (0.5 point) Assume that this change was for a boatowners line and that 50% of the policies are
written uniformly throughout May and June, with the other 50% written uniformly throughout the rest
of the year. Is the calculation above reasonable for this line? Explain.
c. (1.5 points) Based on the assumptions given in part b. above, calculate the 2005 on-level factor.
Show all work.

Questions from the 2007 exam:
34. (2.0 points) You are given the following information for four policies with annual policy terms:
Policy
Effective Date
Premium
A
January 1, 2004
$1,200
B
July 1, 2004
2,400
C
November 1, 2004
3,600
D
April 1, 2005
600
Based on these four policies, calculate:
a. (0.5 point) 2004 written premium.
b. (0.5 point) 2004 earned premium.
c. (0.5 point) 2004 policy year premium.
d. (0.5 point) Premium in-force as of March 31, 2005.
Show all work.

Questions from the 2008:
14. (2.5 points) Assume a -8% rate change was implemented effective March 1, 2005 and that all policies have
annual terms.
a. (1.0 point) Calculate the on-level factors for calendar years 2005 and 2006 earned premiums using the
parallelogram method.
b. (1.0 point) Calculate the on-level factors for policy years 2005 and 2006 earned premiums using the
parallelogram method.
c. (0.5 point) Briefly describe the extension of exposure method and briefly explain why it may be preferable
to the parallelogram method for determining on-level premiums.

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2009 exam:
18. (2 points) The following is the premium associated with five annual policies, where premium is earned
uniformly throughout the year:
Policy
1
2
3
4
5

Effective Date
January 1, 2007
April 1, 2007
July 1, 2007
October 1, 2007
January 1, 2008

Premium
$750
$1,200
$900
$800
$850

a. (0.5 point) Calculate the total calendar year 2007 written premium.
b. (0.5 point) Calculate the total calendar year 2008 earned premium.
c. (0.5 point) Calculate the total policy year 2007 earned premium as of March 31, 2008.
d. (0.5 point) Calculate the total in-force premium as of July 1, 2008.

Questions from the 2011 exam:
4. (1.5 points) Company ABC began writing annual personal automobile policies on January 1, 2010,
using the following rating structure:
•
Policy Premium = Base Rate x Class Factor + Policy Fee
•
Base Rate = $1,000
•
Policy Fee = $50
Class
Class Factor
Teens
2.00
Adults
1.00
On July 1, 2010, the company increased the base rate to $1,100 and revised the class factor for adults to 0.90.
Company ABC writes 10 policies per quarter, each with an effective date of the beginning of the quarter.
The company writes an even distribution of teen and adult classes each quarter.
a. (1 point) Calculate the calendar year 2010 earned premium.
b. (0.5 point) Calculate the on-level factor that applies to the calendar year 2010 earned premium to
bring premiums to current rate level.

Questions from the 2012 exam:
4. (2 points) Explain whether the following statements are correct or incorrect.
a. (0.5 point) Calendar year 2011 written premium will be fixed (i.e. not change) at December 31, 2011.
b. (0.5 point) Calendar year 2011 earned premium will be fully earned (i.e. not change) at December 31,
2011.
c.

(0.5 point) Policy year 2011 written premium will be fixed (i.e. not change) at December 31, 2011.

d. (0.5 point) Policy year 2011 earned premium will be fully earned (i.e. not change) at December 31, 2011.

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2012 exam:
5. (1 point)
a. (0.5 point) Discuss whether or not it is appropriate to perform a classification ratemaking analysis
using premiums adjusted with aggregate on-level factors.
b. (0.5 point) State one advantage and one disadvantage of the parallelogram method relative to the
extension of exposures method.

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Section 2: Premium Aggregation – For Workers’ Compensation
Questions from the 1994 exam
48. (3 points) Answer this question using the Feldblum Study Note Reading, "Workers Compensation
Ratemaking," and the information below.
The adjustments to rates that affect the experience period are shown below.
• Experience rate change of 10% on 7/1/92.
• Law amendment change of 2% on 1/1/93.
• Experience rate change of 15% on 7/1/93.
• Law amendment change of 3% on 1/1/94.
Premium writings are evenly distributed throughout the year.
(a) (1.5 points) What adjustment factor is needed to bring calendar year 1993 premiums to current level?
(Show a diagram representing the appropriate time periods.)
(b) (1.5 points) What adjustment factor is needed to bring policy year 1993 premiums to current level?
(Show a diagram representing the appropriate time periods.)

Questions from the 1996 exam
Question 36. (3 points)
Rate
Implementation
Change
Date
Type of Change
+8%
5/1/94
Experience
+15%
7/1/95
Law Amendment
-10%
7/1/95
Experience
+5%
4/1/96
Experience
• Policies are written uniformly throughout the year.
According to Feldblum, "Workers' Compensation Ratemaking:"
(a) (2 points) Calculate the premium adjustment factor to bring policy year 1995 premium to current rate level.
(b) (1 point) How are experience rate changes and law amendment rate changes different in their
purpose and their effect?

Questions from the 1997 exam
12. You are given:
• Full estimated policy premium is booked at inception.
• Premium develops upward by 7% at final audit, six months after the policy expires.
• All policies are written for an annual period.
• Premium is written uniformly throughout the year.
Based on Feldblum, "Workers' Compensation Ratemaking," in what range does the policy year premium
development factor fall for 24 to 36 months?
A. < 1.01

Exam 5, V1a

B. > 1.01 but < 1.02

C. > 1.02 but < 1.03 D. > 1.03 but < 1.04

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E. > 1.04

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 1999 exam
37. (2 points) Based on Feldblum, 'Workers' Compensation Ratemaking," answer the following.
a. (1 point) Using the information shown below, calculate the policy year premium development factor from 24 to
36 months.


Initial estimates of policy year premium are $1 million per month from January through June and
$1.1 million per month for the remainder of 1 year.



Final audit occurs six months after policy expiration.



Premium develops upward by 20% at the final audit.



All policies are annual.

b. (1 point) Feldblum states that while development factors are necessary for policy year data, premium
development factors may not need to be applied to calendar year premiums. Explain why.

Questions from the 2001 exam
Question 15. Based on Feldblum, “Workers’ Compensation Ratemaking,” and the following information,
compute the policy year reported premium development factor from 12 to 24 months.


Final audit occurs 3 months after policy expiration.



On average, audits result in 15% additional premium.



Premium writings are even throughout the year.



All policies are annual.

A. < 1.050

B. > 1.050 but < 1.075

C. > 1.075 but < 1.100

D. > 1.100 but < 1.125

E. > 1.125

Question 47. (3 points) Feldblum, “Workers’ Compensation Ratemaking,” describes three different types
of experience periods by which insurance data is compiled.
a. (1½ points) Describe how premiums and losses are compiled under each of the three experience periods:
 Policy Year
 Calendar Year
 Calendar/Accident Year
b. (1½ points) State one advantage and one disadvantage associated with each type of experience period.

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2002 exam
27. (6 points) Based on Feldblum, "Workers' Compensation Ratemaking," and the information shown
below, answer the following questions. Show all work.
• Through the use of deviations and schedule rating, your company has been charging 25% below its
manual rates for workers compensation.
• Policy year 2000 earned premium as of December 31, 2001 = $90 million.
• Policy year 2000 reported loss as of December 31, 2001 = $40 million.
• Written premium is distributed uniformly by month.
• Policy term is 12 months.
• Policy audits occur 6 months after expiration and produce a 10% increase in premium.
• The following rate changes have been implemented:
Date
Amount
July 1, 1999
- 6.0%
July 1, 2000
+10.0%
July 1, 2001
+ 7.0%
• There was a 5% increase in the benefit levels effective January 1, 2001. There was no rate change
to account for this.
• Loss development factor = 1.80.
• Annual loss trend = 8%.
• Annual wage trend = 4%.
• The effective date for this analysis is July 1, 2002.
• Rates will be effective for a period of one year.
• Loss adjustment expense = 20% of loss.
• The target loss and loss adjustment expense ratio is 72%.
a. (2 points) What is the policy year 2000 earned premium after all appropriate adjustments for
premium development, current rate level, premium trend, and benefit changes? (chapter 5)
b. (2 points) What are the policy year 2000 losses after the appropriate adjustments for loss development, loss
trend, and benefit changes? (see chapter 6, but will be computed in this chapter)
c. (½ point) What is the projected loss and loss adjustment expense ratio for policy year 2000?
(See chapter 6), but this will be computed in this chapter)
d. (½ point) What is the indicated rate change based on experience from policy year 2000?
(See chapter 8 for the computations needed to answer this question)
e. (1 point) What should the ratio of charged to manual premium be in order to produce the target
loss and loss adjustment expense ratio? (See chapter 8)

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2003 exam
33. (2 points) Using the information shown below, calculate the factor needed to adjust policy year 2002
written premium to current level. Show all work.
• Policies are written uniformly throughout the year and have a term of 12 months.
• The law amendment change affects all policies in force.
Assume the following rate changes:
• Law amendment change on July 1, 2002 = +10%
• Experience rate change on October 1, 2002 = +5%
• Experience rate change on January 1, 2003 = +7%

Questions from the 2004 exam
11. Given the following data, calculate the policy year 2001 premium development factor from 24 to 36 months.
• Full estimated policy year premium is booked at inception, $10 million a month in 2001.
• Premium develops upward by 5% at the final audit, three months after the policy expires.
• All policies are annual.
A. < 1.010 B. > 1.010 but < 1.015 C. > 1.015 but < 1.020 D. > 1.020 but < 1.025 E. > 1.025
31. (4 points) Given the following information, answer the questions below. Show all work.
• Policies are written uniformly throughout the year.
• Polices have a term of 12 months.
• The law amendment change affects all policies in force.
Assume the following rate changes:
• Experience rate change on October 1, 2001 =+7%
• Experience rate change on July 1, 2002 =+10%
• Law amendment change on July 1, 2003 = -5%
a. (2 points) Calculate the factor needed to adjust calendar year 2002 earned premium to current level.
b. (2 points) Calculate the factor needed to adjust policy year 2002 earned premium to current level.

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2007 exam
37. (2.0 points) Assume the following information about a worker's compensation insurer:
 All policies are annual.
 April 1, 2004: The company implemented a 10% experience rate change.
 October 1, 2004: The company implemented a 5% rate change due to a law change that
impacted all in-force policies.
a. (1.0 point) Draw the diagram underlying the calculation of the current rate level factor used to adjust
policy year 2004 premium to current rate level.
 Label the starting and ending dates of the historical period.
 Label the rate change and law change.
 Calculate the relative rate level of each area and label the diagram.
 Do not calculate the percentage each area represents of the year.
b. (1.0 point) Draw the diagram underlying the calculation of the current rate level factor used to adjust
calendar year 2004 earned premium to current rate level.
 Label the starting and ending, dates of the historical period.
 Label the rate change and law change.
 Calculate the relative rate level of each area and label the diagram.
 Do not calculate the percentage each area represents of the year.
Show all work.

Questions from the 2009 exam
19. (2.5 points) Given the following information:
• All policies are semi-annual.
• A +5% rate change was implemented effective October 1, 2007.
• A benefit change of +10% was enacted affecting premium on all outstanding policies on July 1, 2008.
a. (0.75 point) Draw and label a diagram of the parallelogram method for calendar year 2008 earned
premium.
b. (1.25 points) Calculate the on-level factor for calendar year 2008 earned premium.
c. (0.5 point) Explain why the parallelogram method may not be appropriate for calculating on-level
factors for snowmobile insurance.

Questions from the 2010 exam
19. (3 points) Given the following information for Company XYZ book of business in State X:
• All policies are semi-annual.
• A law change is effective on July 1, 2008 and applies to all in-force and future policies.
The estimated overall premium impact of the law change is +10%.
• A 5% overall rate increase is implemented on October 1, 2008.
• 2008 calendar year earned premium is $1,000,000.
a. (1 point) Draw and fully label a diagram for calendar year 2008 earned premium reflecting the parallelogram
method.
b. (1 point) Calculate the on-level factor for calendar year 2008 earned premium.
c. (1 point) Draw and fully label a diagram for policy year 2008 earned premium reflecting the parallelogram
method.

Exam 5, V1a

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Section 3: Premium Aggregation – Using the One and Two Step Procedures
Questions from the 2003 exam
11. Given the information below, determine the written premium trend period.
• Experience period is April 1, 2001 to March 31, 2002
• Planned effective date is April 1, 2003
• Policies have a 6-month term
• Rates are reviewed every 18 months
• Historical premium is earned premium
A. < 1.8 years
B.  1.8 years, but < 2.1 years
D.  2.4 years, but < 2.7 years
E.  2.7 years

C.  2.1 years, but < 2.4 years

Questions from the 2004 exam:
35. (3 points) You are given the following information. Using a two-step trending procedure as described in
Jones, "An Introduction to Premium Trend," answer the questions below. Show all work.
• The experience period is January 1, 2001 through December 31, 2003.
• Planned effective date is July 1, 2005.
• Rates are reviewed annually.
• Policies have a 6-month term.
• The trend will apply to calendar-accident year 2002 earned premium at current rate level.
a. (1 point) Calculate the beginning and ending dates for each of the Step 1 and Step 2 trend periods,
assuming the selected trend is based on average written premium.
b. (1 point) Calculate the beginning and ending dates for each of the Step 1 and Step 2 trend periods,
assuming the selected trend is based on average earned premium.
c. (1 point) Describe a situation when it may be more appropriate to use a two-step trending procedure,
rather than a one-step trending procedure.

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2005 exam:
37. (4 points)
Given the information below, answer the following questions. Show all work.
Calendar/Accident Year
Average Written Premium
2002
$1,000.00
2003
$933.33
2004
$882.00
•
•
•
•
•

The planned effective date for a rate change is January 1, 2006.
Rates are reviewed every 18 months.
All policies are annual, and are written uniformly throughout the year.
A 20% rate decrease was implemented effective July 1, 2003.
A separate analysis has determined that a shift in the limit distribution from 2002-2004 has
resulted in a +3% annual premium trend. This shift is not expected to continue past 2004.
a. (3.5 points) Using two-step trending, determine the total premium trend factors for each year
above.
b. (0.5 point) Why is two-step trending a more suitable procedure for trending premium than for
trending loss frequency or severity?

Questions from the 2006 exam:
26. (3.5 points) As the actuary for Company XYZ, you are performing a physical damage rate review
for State X. Use the following information to answer the questions below.

Experience period consists of calendar year premium for 2002 through 2004.

Current level earned premium for calendar year 2002 is $42,500,000.

Planned effective date of rate revision is June 1, 2006.

Anticipate annual rate revisions every 12 months.
Each year, insureds purchase newer, more expensive vehicles, resulting in upward premium drift.
Historically, the premium drift has averaged 5% through 2004. However, given current trends and
expectations regarding future car sales, the insurer expects a 3% premium drift in the future.
The insurer uses exponential premium trend.
a. (1.5 points) Assume all policies have a six-month term. Use 2-step trending with average written
premium to calculate the trended premium for calendar year 2002. Show all work.
b. (1.5 points) Assume all policies have an annual term. Use 2-step trending with average written
premium to calculate the trended premium for calendar year 2002. Show all work.
c. (0.5 point) Explain one advantage of using 2-step trending in this example over 1-step trending.
27. (1 point)
a. (0.5 point) Explain why using average premiums is better than total premiums when analyzing
premium trend.
b. (0.5 point) Give one argument for using average earned premiums in the premium trend analysis
and one argument for using average written premiums.

Exam 5, V1a

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2007 exam:
36. (3.0 points) You are given the following information:
 All policies are annual.
 The future policy period begins January 1, 2007.
 The future annual premium trend is 3% per year.
 The proposed rates will be in effect for one year.
Calendar
Earned
Average Written
Year
Exposures
Premium
At Current Rate Level
2003
1.000
$3,777
2004
1,050
3,688
2005
1,100
3,998

Average Earned
Premium
At Current Rate Level
$3,605
3,749
3,899

Calculate the trended premium for each year, using the two-step trending method. Show all work.

Questions from the 2008 exam:
15. (2.0 points)
a. (0.75 point) Question no longer applicable to the content covered in this chapter.
b. (1.25 points) You are given the following information.

Accident
Year
2004
2005
2006
2007

Average Earned
Premium at Current
Rate Level
$ 98
$102
$106
$110

Average Written
Premium at Current
Rate Level
$100
$104
$108
$112

 The projected premium trend is 4%.
 The proposed effective date of new rates is January 1, 2009.
 The proposed rates will remain in effect for one year.
 All policies are semi-annual.
Calculate the premium trend factor needed to project 2006 calendar/accident year earned premium to
prospective rate levels, using the two-step trending procedure.

Exam 5, V1a

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2010 exam:
18. (2 points) Given the following information:
Calendar
Year
2008
2009

Earned

Written

Exposures Exposures
1,000
1,100
1,200
1,300

On-Level
Earned

On-Level
Written

Premium
$ 487,500
$ 615,000

Premium
$ 550,000
$ 682,500

•
All policies are annual.
•
Proposed effective date is January 1, 2011.
•
Rates are expected to be in effect for one year.
•
Projected premium trend is 5%.
Calculate the calendar year 2008 earned premium at prospective levels using two-step trending.

Questions from the 2011 exam:
5. (2.25 points) Given the following information:
•
Policy term: six months
•
Proposed rates in effect: January 1, 2012, to June 30, 2013
•
Selected projected premium trend: 5%
Calendar
Average Earned Premium
Average Written Premium
Year
at Current Rate Level
at Current Rate Level
2009
$375
$380
2010
$390
$395
a. (2 points) Calculate the total premium trend factor for each of calendar years 2009 and 2010 using
two-step trending.
b. (0.25 point) Briefly discuss when it is appropriate to use two-step trending.

Questions from the 2012 exam:
6. (2 points) Given the following information for a Homeowners company:


The 4th Calendar Quarter of 2011 (4Q11) Average Written Premium is $560.



The proposed effective date of the next rate change is July 1, 2012.



Assume a +5% prospective annual premium trend.

 Rate review is performed every 2 years.
Calendar Year Ending
Earned Exposures (House-Years) Earned Premium at Current Rates
December 31, 2009
10,000
$5,000,000
December 31, 2010
10,000
$5,250,000
December 31, 2011
10,000
$5,512,500
a. (1 point) Use the two-step trending method to calculate the projected earned premium for the
calendar year ending December 31, 2009.
b. (1 point) After completing the analysis, the actuary determines that the assumed annual increase in
the amount of insurance to account for inflation was materially reduced post-January 1, 2012.
Discuss any necessary adjustments to the completed analysis in part a. above

Exam 5, V1a

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
The predecessor papers to the current syllabus reading “Basic Ratemaking” by Werner, G.
and Modlin, C. were numerous. While past CAS questions were drawn from prior syllabus
readings, the ones shown below remain relevant to the content covered in this chapter.

Section 1: Premium Aggregation – In General
Solutions to questions from the 1989 Exam:
Question 43.
Step 1: Draw a unit square for each calendar year and diagonal lines at points in time representing historical
rate changes.
Step 2: Calculate the numerator of the on-level factor. This is the product of all rate changes.
Step 3: Calculate the average rate level factor for each calendar year. This is a weighted average of the rate
level factors in each calendar year. The weights will be relative proportions of each square. First
calculate the area of all triangles (area = .5*base*height) within a unit square and then determine the
remaining proportion of the square by subtracting the sum of the areas of the triangles from 1.0.
Step 4: Divide the result of step 1 by the result of step 3:
1.0

.10

1.0
.50

1.10
1.10

0.0
7/86

1.21
1987

7/88

1989

On- Level Factor
a. Assuming annual policies:

CY 1988

b. Assuming semi-annual policies: CY 1987:
CY 1988

Exam 5, V1a

1.1*1.1
1.21

1.112
.125*(1).875(1.1) 1.0875

CY 1987:

1.1*1.1
1.21

1.086
.875*(1.1).125(1.21) 1.11375
1.1*1.1 1.21

1.1
1.1
1.1
1.1*1.1
1.21

1.073
.75*(1.1).25(1.21) 1.1275

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 1991 exam
Note: View the earning of CY EP using a unit square. View the earning of PY EP using a parallelogram.
Compute on-level factors as follows: [Current rate level factor / average rate level factor (during the
period in question).

Rate Changes
+0.08
1.0
% of

+0.10

0.0 1988

+.02

1.00
1.08

Exposure

+.05
1.272

1.188

1.247

1989

1.298

1990

1991

Step 1: Current rate level factor=1.08 * 1.10 * 1.05 * 1.02 * 1.02 = 1.298. This is the numerator for each onlevel factor.
Step 2: Calculate the denominators for each on-level factor. The denominators are the average rate level
factor for each calendar/ policy year. This is a weighted average of the rate level factors in each
calendar / policy year. The weights will be relative proportions of each square / parallelogram. First
calculate the area of all triangles (area = .5*base*height) within a unit square / parallelogram and
then determine the remaining proportion of the square by subtracting the sum of the areas of the
triangles from 1.0.
Question
14
15
16

Average rate level factor
.25(1.08)+.50*(1.188)+.25*(1.247) = 1.176.
.50(1.188)+.50*(1.247) = 1.218
.125(1.00)+.375*(1.08)+.375*(1.188)+.125(1.247) = 1.131

On-level factor
1.298/1.176 = 1.104
1.298/1.218 = 1.066
1.298/1.131 = 1.147

Answer
C
B
D

Solutions to questions from the 1994 exam
Question 1.
The premium for the policy effective 5/1 is fully earned by 11/1/92. There is no unearned premium at 12/31/92.
5/12 ths of the premium for the policy effective 8/1 is earned by 12/31/92.
The unearned premium is = (7/12) * $12,000 = $7,000.
2/6 ths of the premium for the policy effective 11/1 is earned by 12/31/92.
The unearned premium is = (4/6) * $2,400 = $1,600.
Thus, the total unearned premium = $7,000 + 1,600 = 8,600.

Exam 5, V1a

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Answer D.

 2014 by All 10, Inc.

Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 1996 exam
Question 30
(a) To calculate the on-level earned premium for the experience period 1992-1994, CY on-level factors must be
computed first.
1.05
1.00
1.00

1.00

1.082
1.05

1/91
(i)
(ii)
(iii)

(iv)
(v)
(vi)
(vii)

1/92

1/93

1/94

1.136
1.082
1/95

The rate change in 1991 is not relevant to the calculation.
Calculate the numerator of the on-level factor. This is equal to (1.05)(1.03)(1.05) = 1.136
Calculate the average rate level factor for the calendar year. This is a weighted average of the rate
level factors in the calendar year. The weights will be relative proportions of the square. First
calculate the area of all triangles (area = .5*base*height) within a unit square and then determine the
remaining proportion of the square by subtracting the sum of the areas of the triangles from 1.0.
For CY 1992, the average rate level factor = 1.00. The on-level factor = 1.136 / 1.00 = 1.136.
For CY 1993, the average rate level factor = (1/2)(.25)(.25)*1.05 + (1.0 - .0325)*1.00 = 1.002.
The on-level factor = 1.136 / 1.002 = 1.134
For CY 1994, the average rate level factor = (1/2)(.75)(.75)*1.00 + (1/2)(.5)(.5)*1.082+ (1.0 .40625)*1.05 = 1.04
The on-level factor = 1.136 / 1.04 = 1.092
Thus, the on-level premium is computed
On level
On level
as
CY
EP
factor
EP
1992 375,000
1.1355
425,812
1993 875,000
1.1337
991,987
1994 1,125,000
1.0920
1,228,500
Total
2,646,299

Solutions to questions from the 1997 exam
Question 19.
(a) To facilitate the calculation of CY on-level factors, setup a diagram similar to the one below:

Calculate the numerator of the on-level factor. This is equal to (1.05)*(1.13)*(1-.03) = 1.150905.
Calculate the average rate level factor for the calendar year. This is a weighted average of the rate level
factors in the calendar year. The weights will be relative proportions of the square.
First calculate the area of all triangles (area = .50 * base * height) within a unit square and then determine the
remaining proportion of the square by subtracting the sum of the areas of the triangles from 1.0.
For CY 1995, the average rate level factor = (1/2)(3/12)(3/12)*1.0 + (1/2)(1/2)(1/2)*1.1865+ (1.0 - .15625)*1.05
= .03125 + .1483125 + .8859375 = 1.0655
The on-level factor = 1.150905 / 1.0655 = 1.0801549.

Exam 5, V1a

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Answer B.

 2014 by All 10, Inc.

Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 1998 exam
Question 41.
Note: View the earning of CY EP using a unit square. View the earning of PY EP using a parallelogram.
Compute on-level factors as follows: [Current rate level factor / average rate level factor (during the
period in question).

Rate Changes
+ 0.05 + .10
0.5

0.028

-0.05

0.15

0.25
0.222
0.75

0
9/1/94

1/1/95

1/1/96

1/1/97

Step 1: Current rate level factor =1.05 * 1.10 * .95 * 1.15 = 1.262. This is the numerator for each on-level
factor.
Step 2: Calculate the denominators for each on-level factor. The denominators are the average rate level factor
for each calendar/ policy year. This is a weighted average of the rate level factors in each calendar /
policy year. The weights will be relative proportions of each square / parallelogram. Note: It may be
convenient to think of CY 95 with a base of 12 units and a height of 6 units. To compute the relative
proportion of the unit square, calculate the areas of as many triangles as possible, and then compute the
remaining area by subtracting the sum of the areas of the two triangles from 1.0.
Shape
Dotted Triangle
Bold Triangle
Difference
Remainder

Area
(1/2) * (2/12) * (2/6) = .028
(1/2) * (6/12) * (6/6) = .25
.25 - .028 = .222
1 - .028 - .222 = .75

Rate Level
1.0
1.05
1.155

Step 3: Compute EP at present rates by multiplying EP by the CY on-level factor.
a. The weighted rate level for 1995 is 1.0 * (.028) + 1.05 * (.222) +1.155 * (.75) = 1.127
b. The 1995 CY on-level factor is 1.262 / 1.127 = 1.120
c. CY 1995 On-Level EP = $1,200 * 1.120 = $1,344
Quicker Solution:
1.00

1.05
1.155

9/1

1/1 1995

The dotted line refers to the 6 month term.
Focus on only the 1995 square.
As above, numerator is 1.00 * 1.05 * 1.155 = 1.262
Note that small area is ½ * 2/12 * 4/12 = 1/36
Denominator is 1.155(.75) + 1.00(1/36) * 1.05 (1-0.75-1/36) = 1.127
1.262/1.127 = 1.12 (on-level factor for 1995)

1.12 * 1200 = 1,344.

Exam 5, V1a

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 1999 exam:
Question 58. Given:
 The Company issues annual policies, calculated an RLAF of 1.080 for CY 1998 earned premium

It was assumed that the only rate change that took place in the last few years was effective 1/1/98, but
it was later determined that it was actually effective 3/1/98.
 It is assumed that all months have an equal number of days and that premium writings are evenly
distributed through the year.
Step 1: Based on the given information, construct a diagram similar to the one below:
To recalculate the RLAF using the 3/1/98 effective date, first calculate the rate change at 1/1/98.
X%
1.00
1+X

1/97

1/98

3/1

1/99

(during the period in question) RLAF 

Current Rate Level Factor
Avg Rate Level Factor

Since we are assuming only one rate change effective 1/1/98, the current rate level factor is 1+X.
The average rate level factor for the calendar year is the weighted average of the rate level factors in the
calendar year. The weights will be relative proportions of the square. Solve for X.
1 X
Thus, 1.08 
, .54 + .54(1+X)= (1+X). .08 = .46X;
X = .174
[(.50*1.00)  (.50*1 X )]
Step 2: To recalculate the RLAF using the 3/1/98 effective date, re-compute the average rate level factor.
1.174
1.174
RLAF 

1.1071
[.50(.10/12)(.10/12)*1.174  (1.0 .50(10/12)(10/12))*1.00] 1.0604

Solutions to questions from the 2000 exam:
Question 38.
c. List two methods used to adjust earned premiums to a current rate level basis.
1. Extension of
The best method. Re-rate each policy using current rates.
Exposure:
2. Parallelogram:
a. Assumes exposures are uniformly written over the Calendar Year (CY)
b. Each CY of EP is viewed as a unit square, 1 year wide, 100% of
exposure high.
d. The more appropriate method to use for this company's personal automobile liability business would be the
extension of exposures method. The company's writings show an increasing trend in written exposures which
violates the parallelogram method's assumption that exposures are uniformly written over the calendar year.
e. Using your selected method from part d. above, calculate the on-level earned premium for calendar year 1998.
When using the extension of exposure technique, on-level earned premium equals current rate per unit of
exposure * number of earned exposures. In this example:
the current rate per unit of exposures is $500 * 1.15 = $575
the number of earned exposures in 1998 = 3,600
Thus, on-level earned premium for calendar year 1998 equals $575 * 3,600 = $2,070,000

Exam 5, V1a

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2000 exam:
Question 40.
a. Calculate the on-level factors for each of the two calendar years 1997 and 1998.
Step 1: Draw a unit square for each calendar year and diagonal lines at points in time representing historical
rate changes.
Step 2: Calculate the numerator of the on-level factor. This is the product of all rate changes.
Step 3: Calculate the average rate level factor for each calendar year. This is a weighted average of the rate
level factors in each calendar year. The weights will be relative proportions of each square. First
calculate the area of all triangles (area = .50 * base * height) within a unit square and then determine
the remaining proportion of the square by subtracting the sum of the areas of the triangles from 1.0.
Step 4: Divide the result of step 1 by the result of step 3:
Rate Changes
+0.04

+0.018

+0.03

1.0
1.00*1.04=1.04

1.04*1.018=1.0587
1.0587*1.03=1.0905

7/1/96

1/1/97

1/1/98

7/1/99

On-level factor for CY 1997:
1.04*1.018*1.03
1.0905

1.0536
(1/ 2)*(6/12)*(6/12)*(1)  (1.0  36/ 288)*(1.04) 1.035
On-level factor for CY 1997 equals 1.0536 * 975,000 = 1,027,260
On-level factor for CY 1998:
1.04*1.018*1.03
1.0905

1.0392
(1/ 2)*(12/12)*(12/12)*(1.04)  (1/ 2)*(1)*(1)*(1.0587) 1.0494
On-level factor for CY 1998 equals 1.0392 * 1,000,000 = 1,039,200
Quicker Solution:
Numerator is 1.04 * 1.018 * 1.03 = 1.0905
1997 Denominator : (1/8) 1.00 + (7/8) 1.04 = 1.035 On-level factor = 1.0905/1.035 = 1.054
1998 Denominator: (1/2) 1.04 + (1/2) 1.0587 = 1.049 On-level factor = 1.0905/1.049 = 1.039

Exam 5, V1a

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2001 exam:
Question 38. (2 points) Using the parallelogram method described by McClenahan in “Ratemaking,”
determine the calendar year 1999 on-level earned premium. Show all work.
Step 1: Draw a rectangle (normally a unit square if 1-year policies were issued) for each calendar year and
diagonal lines at points in time representing historical rate changes.
Step 2: Calculate the numerator of the on-level factor. This is the product of all rate changes.
Step 3: Calculate the average rate level factor for calendar year 1999. This is a weighted average of the rate
level factors in calendar year 1999. The weights will be relative proportions of each rectangle. First
calculate the area of all triangles (area = .5 * base * height) within a unit rectangle and then
determine the remaining proportion of the rectangle by subtracting the sum of the areas of the
triangles from 1.0. Note: Since 2-year policies are issued, the ratio of the height to the base is 2:1.
Step 4: Divide the result of step 1 by the result of step 3:
Rate Changes
+0.052

+0.074

2
1.00

1.052
1.052 * 1.074

0
7/1/97

1/1/1998

4/1/99

1/1/2000

Area of triangle: 1/2 * base * height
Rate level
Area
1.00
1/2 * 6/12 * 6/24 =
0.0625
1.129848
1/2 * 9/12 * 9/24 =
0.140625
1.052
1.0 - 0625 - .140625 = 0.7968750

On-level factor for CY 1997:
1.052*1.074
1.129848

1.0661987
(1/ 2)*(6/12)*(6/ 24)*(1.0)  (1/ 2)*(9/12)*(9/ 24)*(1.129848)  (.796875)*(1.052) 1.0596974
On-level earned premium for CY 1999 equals 1.0661987 * $14,000 = $14,927

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2002 exam:
Question 17.
b. (2 points) Calculate the experience period on-level earned premium.
Step 1: Draw a rectangle (normally a unit square for a calendar year if 1-year policies were issued) for each
period and diagonal lines at points in time representing historical rate changes.
+10%
-15%
Rate Level: 1.00
.50* b*h =
.50*12*12
= 72

.50* b*h =
.50*6*12
= 36

Area = (12*18) – (72
+36)
=216

Rate Level: 1.10
1/1/2000

.935 = 1.10*.85

1/1/2001

6/30/200

1/1/2002

No of Earned Exposures:200

100
Step 2: Calculate the rate level at various levels during the experience period. This is the product of all
rate changes at a given point in time (i.e. 1.00; 1.00 * 1.10 = 1.10; 1.10 * .85 = .935).
Step 3: Calculate the on-level factor for the experience period. This is the current rate level divided by the
weighted average of the rate level factors in the experience period. The weights will be relative
proportions of each rectangle or triangle. First calculate the area of all triangles (area = .5 * base *
height) within a unit rectangle and then determine the remaining proportion of the rectangle by
subtracting the sum of the areas of the triangles from 1.0.
AvgRateLevel Factor 

.50*12*12*1.0.50*6*6*.935(2167236)*1.10 1.0529
12*18

Experience Period On-level Factor = .935/1.0529=.888
Step 4: Calculate the experience period on-level earned premium.
Exposures
Writtten in
CY
1999
2000
2000
2001

Exposures
Earned in
Experience Period
100
100
75
25

Rate
Level
1.000
1.100
1.100
0.935

Rate
120
120
120
120

Earned
Premium
12,000
13,200
9,900
2,805
37,905

Experience
Period
Onlevel
Factor
0.888
0.888
0.888
0.888

Experience
Period
Earned
Premium
10,656
11,722
8,791
2,491
33,660

Question 17.
Alternatively, on-level EP = Current Rate * Earned Exposures = ($120*1.1*.85) * (200+100) = 33,660.

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2003 exam:
10. A 12-month policy is written on March 1, 2002 for a premium of $900. As of December 31, 2002,
which of the following is true?
Step 1: Answering this question is best understood in terms of exposures
Written exposures are those units of exposures on policies written during the period in question,
Earned exposures are the exposure units actually exposed to loss during the period, and
Inforce exposures are those exposure units exposed to loss at a given point in time.….
Step 2: Based on the definitions in Step 1, only earned premium differs from written premium and inforce
premium and therefore needs to be computed.
Thus, earned premium at 12/31/02 equals $900 * 10/12 = $750.
Answer E.

Solutions to questions from the 2005 exam:
38. (1.5 points) The parallelogram method is used to adjust calendar year 2003 earned premium to
current rate level. Given the following information, will the parallelogram method understate,
overstate, or accurately state the on-level factor applied to calendar year 2003 earned
premium? Explain your answer.
• There was a 10% rate increase effective on January 1, 2003.
• The written exposures grew 5% each month in 2003.
The parallelogram method assumes a uniform distribution of policies is written over an entire calendar year.
Using the parallelogram method, the on-level factor for CY 2003 is computed as
Current Rate Level
1.10

1.048
Average Rate Level .50*(1.0) .50*(1.1)
However, if exposures are growing 5% each month, more weight should be given to the current rate level factor,
1.10.
For example, the on-level factor could be computed as

1.10
, where z is less than 50%.
z *(1.0)  (1 z )*(1.1)

This would produce a lower on-level factor compared to that produced by the traditional method.
Hence, the parallelogram method would overstate the on-level factor applied to CY 2003 premiums.

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2006 exam:
28. (3 points) Company XYZ reduced rates 8% effective May 1, 2004, which was their first rate
change since January 1, 2000. Assume all policies have annual terms.
a. (1 point) Using the parallelogram method, calculate the 2005 on-level factor. Show all work.
b. (0.5 point) Assume that this change was for a boatowners line and that 50% of the policies are
written uniformly throughout May and June, with the other 50% written uniformly throughout the rest
of the year. Is the calculation above reasonable for this line? Explain.
c. (1.5 points) Based on the assumptions given in part b. above, calculate the 2005 on-level factor.
Show all work.
a. The parallelogram method assumes a uniform distribution of policies is written over an entire calendar year.
Step 1: Draw a unit square to represent a calendar year, since 1-year policies were issued, for each period
under consideration and draw diagonal lines at points in time representing historical rate changes.
Rate Change
-.08

%
of
Exposure

1.00

1.00
1.0*(1.0-.08)=.92

5/1

2004

1/1

2005

Step 2: Calculate the rate level at points in time when the rate level change during the experience period.
This is the product of all rate changes at a given point in time (i.e. 1.0; 1.0 * (1.0* -.08) = .92)
Step 3: Calculate the on-level factor for the experience period. This is the current rate level divided by the
weighted average of the rate level factors in the experience period. The weights will be relative
proportions of each square or triangle. First calculate the area of all triangles (area = .5 * base *
height) within a unit square and then determine the remaining proportion of the square by
subtracting the sum of the areas of the triangles from 1.0.
Current Rate Level Factor
OLF 
Avg Rate Level Factor
OLF 

.92
.92
.92


.9952
[.50(4/12)(4/12)*1.00  (1.0 [.50(4/12)(4/12)*1.00])*.92] [.0556*1.0 .9444*.92] .9244

b. No, the calculation is not reasonable because the parallelogram method assumes uniform distribution
of written policies throughout the year. Since 50% of the total policies written during CY 2004
occurred in May and June, more weight will be given to the current rate level in the calculation of the
average rate level factor for 2005, raising the on-level factor closer to 1.0.
c. Initial comments:
We must determine the % of policies written between January and April (inclusive 2004) and the proportion of
those policies, by month, earned in CY 2005 as a % of total policies earned in 2005.
Since 50% of the policies were written in May and June of 2004, and assuming uniform writings in all other
months, 50% policies of the remaining policies were written evenly throughout the remaining 10 months of CY
2004. This implies that on average, 5% of the total policies written during 2004 were written during each month,
other than during the months of May and June.

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BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2006 exam:
Question 28 (part c. continued):
Now, consider a policy year divided into twenty four equal parts, with the first month and the last month of
the policy year earning only 1/24 of the premium (earned premium is spread over thirteen months).
Thus, we assume that the average policy for each month was written in the middle of the month, such
that only 1/24th of the January 2004 policies were still unearned as of 1/1/2005, 3/24th of the February
2004 policies were still unearned as of 1/1/2005, 5/24th of the March 2004 policies were still unearned as
of 1/1/2005 and 7/24th of the April 2004 policies were still unearned as of 1/1/2005.
Therefore, the proportion of CY 2005 earned exposures from policies written in 2004 at a 1.00 rate level
can be computed as follows:
January 2004 policies:

.05 * (1/24) = 0.0021

February 2004 policies: .05 * (3/24) = 0.0063
March 2004 policies:

.05 * (5/24) = 0.0104

April 2004 policies:

.05 * (7/24) = 0.0146

Total = 0.0021 + 0.0063 + 0.0104 + 0.0146 = 0.0334
Average Rate Level for 2005 = 0.0334(1.00) + .9666(0.92) = 0.9227
Current Rate Level = 0.92
On-level Factor for 2005 = 0.92/0.9227 = 0.9971
**Finally compare .9227 to .9244, which was computed in part a, and commented on in part b.**
Solutions to questions from the 2007 exam:
34. Calculate:
a. (0.5 point) 2004 written premium.
b. (0.5 point) 2004 earned premium.
c. (0.5 point) 2004 policy year premium.
d. (0.5 point) Premium in-force as of March 31, 2005.
Model Solution
a. WP includes all premium written during a calendar period. Thus, 2004 WP = 1,200+ 2,400 + 3,600 = 7,200
b. EP includes that portion of calendar year written premium which has been earned as of 12/31 of the calendar
year. 2004 EP = 1,200 + 2,400(1/2) + 3,600(1/6) = 3,000
c. PY premium includes all premium associated with policies issued during a given time period. Policy year data
is based upon the year in which the policy giving rise to exposures, premiums, claims and losses is effective.
Thus, 2004 PY Premium = 1,200 + 2,400 + 3,600 = 7,200
d. In-force premium includes the full-term premium for each policy that has not expired at a point in time.
All individual policy premiums are aggregated to arrive at a total in-force premium for the insurer.
Inforce Premium as of 3/31/05 = 2,400 + 3,600 = 6,000

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2008:
Model Solution - Question 14
14. (2.5 points) Assume a -8% rate change was implemented effective March 1, 2005 and that all policies have
annual terms.
a. (1.0 point) Calculate the on-level factors for calendar years 2005 and 2006 earned premiums using the
parallelogram method.
Initial comments. Note that the question fails to state whether policies are uniformly written throughout the policy
period. When computing on-level factors using the parallelogram method, such an assumption must be made.
Therefore if the question does not state that polices are uniformly written throughout the policy period, it is wise to
state that on your answer sheet prior to solving the problem.
a. Calculate the on-level factors for CYs 2005 and 2006 earned premiums using the parallelogram method.
Step 1: Draw a unit square to represent a calendar year, since 1-year policies were issued, for each period
under consideration and draw diagonal lines at points in time representing historical rate changes.
Rate Change
-.08

%
of
Exposure

1.00

1.00
1.0*(1.0-.08)=.92

3/1

2005

1/1

2006

Current Rate Level Factor
Avg Rate Level Factor

CY 05 OLF 

.92
.92

.9463
[.50(5/6)(5/6)*0.92  (1.0 [.50(5/6)(5/6)*1.00]] [.3194 .6528]

CY 06 OLF 

.92
.92

.9988
[.50(1/6)(1/6)*1.00  (1.0 [.50(1/6)(1/6)*.92]] [.0139 .9072]

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2008 (continued):
Model Solution - Question 14 (continued):
b. (1.0 point) Calculate the on-level factors for policy years 2005 and 2006 earned premiums using the
parallelogram method.
Step 1: Draw a parallelogram to represent a policy year, since 1-year policies were issued. For PYs 2005
and 2006, draw diagonal lines at points in time representing historical rate changes.
Rate Change
-.08

%
of
Exposure

0.92
1.00

0.92
0.92
0.92

3/1

2005

0.92

1/1

2006

Step 2: Calculate the on-level factor for the experience periods. This is the current rate level divided by the
weighted average of the rate level factors in the experience period. Calculate the average rate level
factor for the policy year. This is a weighted average of the rate level factors in the policy year. The
weights will be relative proportions of the parallelogram.
Note for the period 1/1 – 3/1, the rate level factor is 1.0. The relative area of the parallelogram at a
1.0 rate level is 1.0 * (1/6)(1.0) = 1/6.
The remaining area of the parallelogram at a 0.92 rate level is .92 * [1.0 - (1/6)(1.0)] = .92 * (5/6) = .7667.
The average rate level factor for the policy year = (1/6)*1.0 + (5/6)*.92 =.9333
.92
.92
PY 05 OLF 

.9857
[.1667 .7667] .9334
Note: Upon review of the above diagram, the PY 2006 parallelogram shows a 0.92 rate level
throughout the entire policy period. Therefore:
.92*1.0
PY 06 OLF 
1.00
1*.92
c. (0.5 point) Briefly describe the extension of exposure method and briefly explain why it may be preferable to
the parallelogram method for determining on-level premiums.
Extension of exposure method re-rates each policy at current rate level. This may be preferable to the
parallelogram method since it does not require policies to be written uniformly throughout policy period.

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2009 exam:
Question#: 18
a. WP includes all premium written during a calendar period.
Thus, CY 2007 WP = 750 + 1,200 + 900 + 800 = $3,650
b. EP includes that portion of calendar year written premium which has been earned as of 12/31 of the calendar
year. CY 2008 EP = 1,200 (3/12) + 900 (6/12) + 800 (9/12) + 850 = 300 + 450 + 6 00 + 850 = $2,200
c. PY EP premium includes all premium associated with policies, issued during a given time period, as of a
given evaluation date. Thus, PY 2007 earned premium as of 3/31/08
= 750 + 1,200+ 900 (9/12)+ 800(6/12) = 750 + 1,200+ 675 + 400 = $3,025
d. In-force premium includes the full-term premium for each policy that has not expired at a point in time.
All individual policy premiums are aggregated to arrive at a total in-force premium for the insurer.
In - force premium as of 7/1/08 = 800 + 850 = $1,650

Solutions to questions from the 2011 exam:
4a. (1 point) Calculate the calendar year 2010 earned premium.
4b. (0.5 point) Calculate the on-level factor that applies to the calendar year 2010 earned premium to
bring premiums to current rate level.
Question 4 – Model Solution 1
Givens: Policy Premium = Base Rate x Class Factor + Policy Fee; Base Rate = $1,000; Policy Fee = $50
Class Teens: Class factor = 2.00; Class Adults: Class factor = 1.00
ABC writes 10 policies per quarter, each with an effective date of the beginning of the quarter.
On 7/1, the company increased the base rate to $1,100 and revised the class factor for adults to 0.90.
The company writes an even distribution of teen and adult classes each quarter.
a. 10 pols issued per quarter equally = 5 adult and 5 teen policies issued each quarter
Quarter 1: Adult = 1000 * (1) + 50 = 1050; * 5 policies = 5,250
Teens = 1000 * (2) + 50 = 2050; * 5 policies = 10,250
Quarter 2: same as quarter 1
Quarter 3: Adult = 1100 * (.90) + 50 = 1040; * 5 policies = 5,200
Teens = 1100 * (2) + 50 = 2250; * 5 policies = 11,250
Quarter 4: same as quarter 3
2010 EP = (5,250 + 10,250) + (5,250 + 10,250) * .75 + (5200 + 11250) * .5 + (5200+11250) * .25
= 15,500 + 11,625 + 8,225 + 4,112.50 = 39,462.50
b. EP for 2010 if all @ CRL = [Latest EP for Adult and Teens] * % earned per quarter
= (5200 + 11250)(1 + .75 + .5 + .25) = (16450)*(2.5) = 41,125
OLF = EP @CRL/CY 2010 EP = 41,125/39,462.5 = 1.0421286
Question 4 – Model Solution 2
a. Q1 EP: (1000 * 2 + 50) * 5 + (1,000 * 1 + 50) * 5= 15,500
Q2 EP: 15,500 * 3/4 = 11625
Q3 EP: [(1,100 * 2 + 50) * 5 + (1,100 * .9 + 50) * 5] * 1/2 = 16,450 * 1/2 = 8,225
Q4 EP: 16450 * 1/4 = 4112.5
2010 EP = 15,500 + 11,625 + 8,225 + 4,112.5 = 39,462.5
b. 16450 * (1 + ¾ + ½ + ¼ ) = 41,125
On level factor = 41,125/ 39,462.5 = 1.042

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2012 exam:
4a.
4b.
4c.
4d.

(0.5 point) Calendar year 2011 written premium will be fixed (i.e. not change) at December 31, 2011.
(0.5 point) Calendar year 2011 earned premium will be fully earned (i.e. not change) at 12/31/ 2011.
(0.5 point) Policy year 2011 written premium will be fixed (i.e. not change) at December 31, 2011.
(0.5 point) Policy year 2011 earned premium will be fully earned (i.e. not change) at December 31, 2011.

Question 4 – Model Solution 1 (Exam 5A Question 4)
a. True, because calendar year written premium is based off of transactions that occur in that year.
For example, if a policy that was effective in 2011 is cancelled sometime in 2012 before expiration,
this would not impact calendar year 2011 written premium, but would be reflected in calendar year
2012 written premium.
b. True, because calendar year earned premium comes from policy transactions that are effective
before 1/1/2012. Similar to part (a), if a policy that was effective in 2011 is cancelled in 2012 (prior to
expiration), this would not impact CY 2011 Earned Premium, but would be reflected in CY 2012
Earned Premium.
c. False, because Policy Year 2011 written premium is based off all transactions for policies that
were effective in 2011. So, if a policy written in 2011 is cancelled in 2012 prior to expiration, this
would be reflected in PY 2011 written premium (it would not impact PY 2012 written premium).
d. False, because Policy Year 2011 earned premium accounts for all transactions for policies that
were effective in 2011 (regardless of transaction date). Same would hold true for Earned Prem as
holds true for written premium in the example from part (c).
Question 4 – Model Solution 2 (Exam 5A Question 4)
a. True – CY WP is fixed at year end.
CY WP includes all transactions in the calendar period.
b. True – CY EP is fixed at year end.
CY EP = CY WP + Starting UEPR – Ending UEPR. All these are fixed at year end.
c. False – PY11 WP is not fixed @ 12/31/2011.
Endorsements and audit premiums in CY2012 and (possibly) beyond will change WP.
d. False – PY11 EP cannot be fully earned at 12/31/2011.
A policy written 12/1/2011 is only 1/12 earned a/o 12/31/11.
Question 4 – Model Solution 3 (Exam 5A Question 4)
a. Yes. Includes new prem written + midterm adjustments during calendar year 2011.
b. True, calendar year earned premium is premium associated with coverage provided during
calendar year 2011.
c. Policy year 2011 written premium will not be fixed as of 12/31/2011, because any midterm changes
associated with policies effective during 2011, even if change happens in 2012 or later, should be
included. E.g. policy effective 7/1/2011, add a new vehicle on 4/1/2012, this contributes to PY 2011
written.
d. PY 2011 earned prem will not be fixed as of 12/31/11. This is the earned premium associate with all
policies with effective dates in 2011. If they are annual policies, all coverage has not been provided

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2012 exam:
Examiners Comments - Exam 5 Question 4 (Exam 5A Question 4)
a. Many candidates answered this correctly. However, some just repeated the question explaining that
calendar year 2011 written premium will be fixed at 12/31/11, which isn’t enough for the explanation.
There were also candidates who mentioned this includes premium written in 2011 and any
cancellations, which isn’t enough of an explanation as need to give some indication as to when
cancellation occurred to differentiate from policy year premium. Many candidates mentioned that any
transactions occurring for in 2012 will count towards calendar year 2012 written premium, which is
enough of an explanation.
b. Many candidates answered this correctly. However, some just repeated the question explaining that
calendar year 2011 earned premium will be fixed at 12/31/11, which isn’t enough for the explanation.
Some candidates mentioned what is earned afterwards in 2012 will go towards calendar year 2012
earned premium, which is enough of an explanation. Similar to a), occasionally a candidate would
explain that calendar year data is fixed, which is not enough of an explanation, because need to
indicate when it is fixed (i. e. at end of year).
c. Of all the parts, part c. was the one most frequently answered incorrectly. Many candidates answered
this correctly. However, there were also a significant amount of candidates who did not indicate when
the cancellation or midterm adjustment occurred, which is not enough of an explanation as it does not
differentiate from calendar year premium. Many times a candidate would say this part is correct
because it only includes premium written during the year, which receives 0 points. Occasionally a
candidate would say this is fixed at 12/31/12, which isn’t enough of an explanation to receive full credit
as it is not necessarily true (i.e. audits).
d. Many candidates answered this correctly. Some candidates said this was incorrect because any
cancellation or mid-term adjustments would change policy year 2011 earned premium, which is not
enough of an explanation to receive full credit as it does not differentiate from calendar year premium
(need to mention when cancellation or mid-term adjustment occurs).

Questions from the 2012 exam:
5a. (0.5 point) Discuss whether or not it is appropriate to perform a classification ratemaking analysis
using premiums adjusted with aggregate on-level factors.
5b. (0.5 point) State one advantage and one disadvantage of the parallelogram method relative to the
extension of exposures method.
Exam 5 Model Solution 1 – Part a (Exam 5A Question 5a)
No. If a rate change disproportionately effects a certain class more than others, the on-level factors will
vary by class. Therefore aggregate OLF should not be used.
Exam 5 Model Solution 2 – Part a (Exam 5A Question 5a)
It would be appropriate only if all classes have had the same rate change history. If not, then we need
rate change info for each class, so that the true rate adjustment for each class can be determined.
Examiner’s Comments:
The answers to part (a) often lacked sufficient detail to demonstrate the candidates understanding of
why the aggregate on level factors may/may not be appropriate for class ratemaking.

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2012 exam:
5a. (0.5 point) Discuss whether or not it is appropriate to perform a classification ratemaking analysis
using premiums adjusted with aggregate on-level factors.
5b. (0.5 point) State one advantage and one disadvantage of the parallelogram method relative to the
extension of exposures method.
Exam 5 Model Solution 1 – Part b (Exam 5A Question 5b)
Advantage: Parallelogram method is much simpler + requires much less calculations +
computing power. It is much quicker to use.
Disadvantage: It assumes uniform premium writings throughout the year. When this assumption
does not hold, it is not accurate. Extension of exposures is more accurate.
Exam 5 Model Solution 2 – Part b (Exam 5A Question 5b)
Advantage: Easy to calculate.
Disadvantage: Not so accurate.
Exam 5 Model Solution 3 – Part b (Exam 5A Question 5b)
Parallelogram
Advantage: Does not require individual policies, only need aggregate data.
Disadvantage: If different classes have different rate changes over time, then applying aggregate on level
factors to aggregate premium will likely not produce the correct on-level premium.
Examiner’s Comments
The majority of the candidates answered part (b) of the question well.

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Section 2: Premium Aggregation – For Workers’ Compensation
Solutions to questions from the 1994 exam
(a) (1.5 points) What adjustment factor is needed to bring calendar year 1993 premiums to current level?
(Show a diagram representing the appropriate time periods.)
(b) (1.5 points) What adjustment factor is needed to bring policy year 1993 premiums to current level?
(Show a diagram representing the appropriate time periods.)
48.
+2%

+10%

+3%

+15%

1.02
1.122
1.29
1/92

1/93

7/1

1/94

7/1

(a) Calculate the numerator of the on-level factor. This is equal to (1.02)(1.10)(1.15)(1.03) = 1.329.
Calculate the average rate level factor for the calendar year. This is a weighted average of the rate level
factors in the calendar year. The weights will be relative proportions of the square. First calculate the
area of all triangles (area = .5*base*height) within a unit square and then determine the remaining
proportion of the square by subtracting the sum of the areas of the triangles from 1.0.
The average rate level factor for the calendar year = (1/2)(.5)(.5)*1.02 + (1/2)*.5*.5*1.29 +
(1.0 - .25)*1.122 = 1.130.
The on-level factor = 1.329 / 1.130 = 1.176.
(b). Calculate the numerator of the on-level factor. This is equal to (1.02)(1.10)(1.15)(1.03) = 1.329.
Calculate the average rate level factor for the policy year. This is a weighted average of the rate level
factors in the policy year. The weights will be relative proportions of the parallelogram. First calculate
the area of all triangles (area = .5*base*height) within the parallelogram and then determine the remaining
proportion of the parallelogram by subtracting the sum of the areas of the triangles from 1.0.
The average rate level factor for the policy year = (1/2)(.5)(.5)*1.290 + (1/2)(.5)(.5)*1.156 +
(1.0 - (1/4))*1.122*.50 + (1.0 - (1/4))*1.329*.50 = 1.225.
+2%

+10%

+3%
1.156

1.02
1.122
1.29
1/92

7/1

1/93

+15%

1.329
1/94

7/1

The on-level factor = 1.329 / 1.225 = 1.085.

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to the questions from the 1996 exam
Question 36.
(a). The premium adjustment factor is also known as an on-level factor. The numerator of the on-level factor
considers rate changes which impact both PY 1995, represented by the parallelogram below, and rate
changes up and through the current level. The denominator of the on-level factor considers only those
rate changes which impact PY 1995.
Calculate the numerator of the on-level factor. This is equal to (1.0)(1.15)(.90)(1.05) = . 1.08675
Calculate the average rate level factor for the policy year. This is a weighted average of the rate level factors
in the policy year. The weights will be relative proportions of the parallelogram.
First calculate the area of all triangles (area = .50 * base * height) within the parallelogram and then determine
the remaining proportion of the parallelogram by subtracting the sum of the areas of the triangles from 1.0.
Notice the area of the parallelogram at the 1.035 level.
Its area is calculated as base * height = .50*1.0 = .50.
The average rate level factor for the policy year = (1/2)(.5)(.5)*1.0 + (1/2)(.5)(.5)*1.15
+.50*1.0*1.035 + (1.0 - .125 - .125 - .50)*1.15 = 1.07375.
+15%

-10%
1.15

1.00
1.15
1.00

1/94 5/1

1/95

1.035

1.035
7/1

1/96

The on-level factor = 1.08675 / 1.07375 = 1.012.
(b) Experience rate changes are represented graphically as diagonal lines, and are computed to adjust current
rates for changes anticipated in projected experience level. These affect new and renewal policies only.
Law amendment changes are represented graphically as straight lines, and since they affect all policies inforce at a given point in time. These changes adjust premiums for statutory modifications to benefits.

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 1997 exam
Question 12. Assume that policy year 199X premium is being booked at $P per month.
Developed premium, due to final audits, is not known until 6 months after the policy expires.
At 12/31/9X+1, developed premium for only those policies issued during the 1st 6 months of PY 199X is known.
At 12/31/9X+2, developed premium for all policies issued during PY 199X is known.
Reported Premium for polices issued during the
Evaluation Date

1st 6 months of PY 199X

Last 6 months of PY 199X

Total PY 199X

12/31/9X
12/31/9X+1
12/31/9X+2

6 months * ($P/month)
6 * P * 1.07
6 * P * 1.07

6 months * ($P/month)
6*P
6 * P * 1.07

12P
12.42P
12.84P

Therefore, the PY premium development factor for 24 to 36 months is 12.84P/12.42P = 1.034

Answer D.

Solutions to questions from the 1999 exam
Question 37
Note: At 12/31/9X+1, premium for PY 199X is at 24 months of development.
At 12/31/9X+2, premium for PY 199X is at 36 months of development.
a.
Reported Premium for polices issued during the
Evaluation Date

1st 6 months of PY 199X

Last 6 months of PY 199X

Total PY 199X

12/31/9X
12/31/9X+1
12/31/9X+2

6 months * ($1M/month)
6 * ($1M/month)*.20

6 months * ($1.1M/month)

12.6M
12.6M + 1.2M = 13.8M
13.8M + 1.32M =15.12M

6 * ($1.1M/month)*.20

Therefore, the PY premium development factor for 24 to 36 months is 15.12M/13.8M = 1.096
b. CY premiums include audit premium from past policies. As long as premium volume remains steady,
next year's audit premiums associated with current exposures should approximate this year's audit
premiums due to from prior year's exposures, so the PDF is approximately = 1.00

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2001 exam
Question 15. Compute the policy year reported premium development factor from 12 to 24 months.
Assume that policy year 199X premium is being booked at $P per month.
 Final audit occurs 3 months after policy expiration.
 On average, audits result in 15% additional premium.
Developed premium, due to final audits, is not known until 3 months after the policy expires.
At 12/31/9X+1, developed premium for policies issued during the 1st 9 months of PY 199X is known.
At 12/31/9X+2, developed premium for all policies issued during PY 199X is known.
Reported Premium for polices issued during the
Evaluation Date

1st 9 months of PY 199X

Last 3 months of PY 199X

Total PY 199X

12/31/9X
12/31/9X+1
12/31/9X+2

9 months * ($P/month)
9 * P * 1.15
9 * P * 1.15

3 months * ($P/month)
3*P
3 * P * 1.15

12P
13.35P
13.80P

Therefore, the PY premium development factor for 12 to 24 months is 13.35P÷12.00P = 1.1125

Answer D.

Solutions to the questions from the 2001 exam
Question 47.
a. Describe how premiums and losses are compiled under each of the three experience periods:
1. Policy year experience compiles premiums and losses arising from policies issued in a given period
(typically a one year period). Thus, premiums and losses arising from a given block of policies can
be directly matched.
2. Calendar year experience reflects financial statement transactions for a given year. Earned
premium is defined as written premium for the year plus the unearned premium reserve at
beginning of this year minus UEP reserve at end of the year. Calendar year incurred losses are
paid losses during the year plus loss reserves at the end of the year minus loss reserves at the
beginning of the year.
3. Calendar/Accident year – Premiums are computed as calendar year earned premiums or can be
adjusted for audits or earned but not reported (EBNR) premium changes. Losses include
payments and reserves for accidents occurring in a given period.
b. (1½ points) State one advantage and one disadvantage associated with each type of experience period.
Experience period
Policy year

Advantage
It matches premiums and losses
from a given block of policies

Calendar year

It is more “mature” than similarly
aged policy year or cal/acc year
experience.

Calendar/Acc year

Accident year losses can be
matched to the corresponding
exposure year earned premium.

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Disadvantage
Policy year experience is less
“mature” than similarly aged
calendar year or cal/acc year
experience.
It is not available for individual
classifications and premium and
loss experience are not related to
a given block of policies.
Premium must be adjusted for
exposure audits or retrospective
adjustments

 2014 by All 10, Inc.

Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2002 exam
Question 27.
a. (2 points) What is the policy year 2000 earned premium after all appropriate adjustments for premium
development, current rate level, premium trend, and benefit changes?
Step 1: Draw a diagram similar to the one below which identifies periods in time in which rate changes
take place.
Benefit level change
+5%
Rate change
-6%
10%
7%

1.0

7/1/99

1/1/2000

7/1/00

1/1/2001

1.1

7/1/01

Policy year 2000 is represented by the dashed line parallelogram. Further, rate level changes are shown
separately from benefit level changes, since the problem states that although a 5% increase in benefit
levels were effective 1/1/01, no rate change to account for the benefit level change took place.
Step 2: To determine premium development, a development factor to account for premium audits
needs to be determined. At 12/31/01, policies issued between 1/1/00 – 6/30/00 have completed
their audits whereas policies issued between 7/1/00 – 12/31/00 have not. At 12/31/01, the factor
1.10
1.1
to account for future premium development is

 1.047619
.5(1.10)  .5(1.0) 1.05
Step 3: To determine the current rate level, we can ignore the -6% rate level change that was effective
7/1/99, establish a base rate level of 1.0, and determine that the current rate level is (1.0 * 1.10 *
1.07) 1.177. The average rate level for policy year 2000 is 1.05 (.50*1.0+.50*1.10) and therefore:
The on-level factor for policy year 2000 is

Current Rate Level 1.177

 1.121
Average Rate Level 1.050

Step 4: To determine the premium trend period, one must determine the time between the average
date of writing during policy year 2000 (7/1/00) and the corresponding projected date in the
forecast period. Since we are told that the effective date of the analysis is 7/1/02, and that rates
will be effective for a period of one year, average written date during the forecast period is
1/1/03. Thus, the premium trend period is 2.5 years (7/1/00 – 1/1/03), and the premium trend
factor is 1.04 2.5 = 1.103.
Step 5: Using the policy year 200 earned premium given in the problem, and the results of Steps 2 – 4,
compute on-leveled, developed and trended earned premium.
On-leveled, developed and trended policy year 2000 earned premium is
90M * 1.0476 * 1.121 * 1.103 = 116.58M

Exam 5, V1a

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2002 exam (continued)
b. (2 points) What are the policy year 2000 losses after the appropriate adjustments for loss
development, loss trend, and benefit changes?
Step 1: A development factor to account for benefit level changes needs to be determined. Since a 5%
increase in benefit levels affects all policies inforce as of its effective date (shown as the solid
vertical line at 1/1/01 in the graph above), the factor to account for this benefit level change is
1.05
 1.024
.5(1.0)  .5(1.05)

Step 2: To determine the loss trend period, one must determine the time between the average accident
during the experience period (which for policy year 2000 is 1/1/01) and the average accident
date during the effective period of the rates (which for a one year effective period beginning
7/1/02 is 7/1/03). Thus, the loss trend factor is 1.082.5 = 1.212
Therefore, losses adjusted for development, benefit changes, trend and loss adjustment expenses
are 40M * 1.80 * 1.024 * 1.212 * 1.20 = 107.28M
c. (½ point) What is the projected loss and loss adjustment expense ratio for policy year 2000?
The projected loss and LAE ratio for policy year is the ratio of the result from questions (b) to (a)
107.28
above:
 .92
116.58
d. (½ point) What is the indicated rate change based on experience from policy year 2000?
The indicated rate change based on experience from policy year 2000 is the ratio of the projected
.92
loss and LAE ratio to the garget loss and LAE ratio minus one:
 1  .278
.72
e. (1 point) What should the ratio of charged to manual premium be in order to produce the target loss
and loss adjustment expense ratio?
Since the company has been charging 25% below its manual rates for workers compensation, and
since the target loss and loss adjustment expense ratio is based on the anticipated expense costs
during the future policy period, the ratio of charged to manual premium to produce the target loss and
loss adjustment expense ratio should be 1.278 * (1.0 - .25) = .96

Exam 5, V1a

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2003 exam
33. (2 points) Calculate the factor needed to adjust policy year 2002 written premium to current level.
Show all work.
Step 1: Draw a diagram similar to the one below which identifies periods in time in which rate changes
take place.
Law amendment change +10%
Rate change
+ 5%
+7%

1.10
1.155
1.0

7/1/99

1/1/2002

7/1/02 10/1/02

1/1/03

Policy year 2002 is represented by the dashed line parallelogram. Further, rate level changes are
shown separately from law amendment changes.
Step 2: To determine the current rate level, establish a base rate level of 1.0, and determine that the
current rate level is (1.10 * 1.05 * 1.07) 1.236.
Since PY 2002 had 3 rate levels in effect, we need to determine the respective area weights to
apply to the rate levels. For the 1/1/02 level, the weight is ½ *½ *½ = 1/8. For the 10/1/02 level, the
weight is ¼ * 1.0 = ¼. Thus, the weight for the 7/1/02 level is 1.00 – 1/8 – ¼ = 5/8.
The average rate level for policy year 2002 is (1/8 * 1.0 + 5/8 * 1.10 + ¼ * 1.155) 1.101.
Current Rate Level 1.236
Therefore, the on-level factor for policy year 2002 is

 1.122
Average Rate Level 1.101

Solutions to questions from the 2004 exam
11. Given the following data, calculate the policy year 2001 premium development factor from 24 to 36 months.
• Full estimated policy year premium is booked at inception, $10 million a month in 2001.
• Premium develops upward by 5% at the final audit, three months after the policy expires.
• All policies are annual.
We are told that developed premium, due to final audits, is not known until 3 months after the policy expires.
At 12/31/02, developed premium for policies issued during the 1st 9 months of PY 2001 is known.
At 12/31/03, developed premium for all policies issued during PY 2001 is known.
This can be demonstrated mathematically as follows:
Reported Premium for polices issued during the
Evaluation Date

1st 9 months of PY 2001

Last 3 months of PY 2001

Total PY 2001

12/31/01
12/31/02
12/31/03

9 months * $10M/month
9 * $10M * 1.05
9 * $10M * 1.05

3 months * $10M/month
3 * 10M
3 * $10M * 1.05

120M
124.5M
126M

Therefore, the PY premium development factor for 24 to 36 months is $126M/$124.5M = 1.012
Answer B: > 1.010 but < 1.015

Exam 5, V1a

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2004 exam (continued):
31. (4 points) Given the following information, answer the questions below. Show all work.
• Policies are written uniformly throughout the year.
• Polices have a term of 12 months.
• The law amendment change affects all policies in force.
Assume the following rate changes:
• Experience rate change on October 1, 2001 =+7%
• Experience rate change on July 1, 2002 =+10%
• Law amendment change on July 1, 2003 = -5%
a. (2 points) Calculate the factor needed to adjust calendar year 2002 earned premium to current level.
Step 1: Draw a diagram similar to the one below which identifies periods in time in which rate changes
(both experience rate and law amendment rate) take place.
View the earning of CY 2002 EP using a unit square.
-.05
+0.10

Law amendment rate change

+0.07

Experience rate changes

1.0
1.00*1.07=1.07

1.07 * 1.10=1.177

'10/1/01

7/1/02

7/1/03

Step 2: Compute the current rate level factor, the product of the experience and law amendment rate
changes. This is the numerator of the CY 2002 on-level factor.
Current rate level factor = 1.00 * 1.07 * 1.10 * (1.00 - .05) = 1.1182.
Step 3: Calculate the denominator for the CY 2002 on-level factor. The denominator is the average rate
level factor for the CY. This is a weighted average of the varying rate levels in effect. The weights
are the relative proportions of the CY 2002 square.
First calculate the area of all triangles (area = .5 * base * height) within a unit square and then
determine the remaining proportion of the square by subtracting the sum of the areas of the triangles
from 1.0.
Since CY 2002 had 3 experience and amendment rate levels in effect, we need to determine the
respective area weights to apply to these rate levels. Prior to the 10/1/01 experience rate change
level, the relative weight associated with the 1.0 rate level during CY 2002 is .50 * .75 * .75 = .28125.
Subsequent to the 7/1/02 experience rate change, the relative weight applied to the 1.177 rate level
is .50 * .50 * .50 = .125. Therefore, the relative weight associated with the 1.07 rate level for the
remaining portion of CY 2002 is 1.00 - .28125 - .125 = .59375.
The average rate level for CY 2002 is (.28125 * 1.00 + .125 * 1.177 + .59375 * 1.07) = 1.0637
Therefore, the on-level factor for calendar year 2002 is

Exam 5, V1a

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Current Rate Level 1.1182

 1.051
Average Rate Level 1.0637

 2014 by All 10, Inc.

Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2004 exam (continued):
Question 31 (continued):
b. (2 points) Calculate the factor needed to adjust policy year 2002 earned premium to current level.
Step 1: Draw a diagram similar to the one below which identifies periods in time in which rate changes
take place.
Law amendment change
Experience rate changes

-5%
+ 7%

+10%

1.045
1.0
1.10

10/1/01

7/1/02

7/1/03

Policy year 2002 is represented by the dashed line parallelogram. Further, rate level changes are
shown separately from law amendment changes.
Step 2: To determine the current rate level, establish a base rate level of 1.0, and determine that the
current rate level is (1.00 *01.10 * .95) = 1.045.
Since PY 2002 had 3 rate levels in effect, we need to determine the respective area weights to apply
to the rate levels. Prior to the 7/1/02 experience rate change, the weight associated with the PY
2002, 1.0 rate level, is .50 (half the area of the parallelogram). The relative weight associated with
the 7/1/03 law amendment change, with a rate level of 1.10 * .95 = 1.045, is ½ *½ *½ = 1/8. Thus,
the weight for the 7/1/02,1.10 rate level, is 1.00 – 1/8 – 1/2 = 3/8.
The average rate level for policy year 2002 is (.50 * 1.00 + .375 * 1.10 + .125 * 1.045) = 1.0431.
Therefore, the on-level factor to adjust policy year 2002 earned premium to current level is
Current Rate Level 1.045

 1.002
Average Rate Level 1.0431

Exam 5, V1a

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2007 exam:
37. (2.0 points)
a. (1.0 point) Draw the diagram underlying the calculation of the current rate level factor used to adjust
policy year 2004 premium to current rate level.
b. (1.0 point) Draw the diagram underlying the calculation of the current rate level factor used to adjust
calendar year 2004 earned premium to current rate level.
Note: Policy years are represented graphically by a parallelogram. Calendar years are represented
graphically by a square.
The relative rate levels are the multiplicative product of (1.0 + rate level changes) and (1.0 + law
amendment changes).

A=1.00
B=1.00 * 1.10 =1.10
C=1.00 * 1.05 =1.05
D=1.00 * 1.10 * 1.05 =1.155

Exam 5, V1a

A=1.00
B = 1.00 * 1.10=1.10
C = 1.00 * 1.05=1.05
D = 1.00 * 1.10* 1.05=1.155

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2009 exam:
Question 19:
a. Since a rate change was effective on 10/1/07 and applies to all future policies sold, a diagonal line is
drawn at 10/1 to graphically depict the impact of the change when computing the on-level factor.
Since a law change was effective on 7/1/08 and applies to all in-force and future policies, a solid vertical
line is drawn at 7/1 to graphically depict the impact of the change when computing the on-level factor.
0.05

0.10

1.0

1.05

10/07

01/08 04/08

b. OLF 

1.155=1.05*1.10

07/08

Current Rate Level Factor
Avg Rate Level Factor

The current rate level factor equals the product of all rate changes occurring during CY 2008
CRLF = 1.0 * 1.05 * 1.10 = 1.155
The average rate level factor is a weighted average of the varying rate levels that occurred in CY 2008.
The weights will be relative proportions of the CY square. First calculate the area of all triangles (area =
.5 * base * height) or rectangles within a unit square and then determine the remaining proportion of the
square by subtracting the sum of the areas of the triangles and rectangles from 1.0.
Since all policies are semi-annual, the diagonal line is representative of a policy written 10/1/2007 and
expiring 3/31/2008.
CY 2008 Average rate level = (.50)(3/12)(6/12) * 1.0 + [(1/2) - (.50)(3/12)(6/12)] * 1.05 + (.50)*1.155
= .0625 +.459375 +.5775 = 1.099375
On-level factor for 2008 CY EP = 1.155/1.099375 = 1.05059693
c. Snowmobile insurance is not uniformly earned throughout the year. The parallelogram method
assumes uniform earnings.

Exam 5, V1a

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2010 exam:
Question 19
a. (1 point) Draw and fully label a diagram for CY 2008 earned premium reflecting the parallelogram method.
b. (1 point) Calculate the on-level factor for CY 2008 earned premium.
c. (1 point) Draw and fully label a diagram for PY 2008 earned premium reflecting the parallelogram method.
a. Since a law change was effective on 7/1/08 and applies to all in-force and future policies, a solid vertical
line is drawn at 7/1 to graphically depict the impact of the change when computing the on-level factor.
Since a rate change was effective on 10/1/08 and applies to all future policies sold, a diagonal line is
drawn at 10/1 to graphically depict the impact of the change when computing the on-level factor.
Areas A, B and C represent portions of CY 2008 that correspond to the three rate levels in effect.
Rate Change
0.10
%
of Policy
Earned

1/2008

7/1 10/1

b. OLF 

0.05

100%

Current Rate Level Factor
Avg Rate Level Factor

The current rate level factor equals the product of all rate changes occurring during CY 2008
CRLF = 1.0 * 1.10 * 1.05 = 1.155
The average rate level factor is a weighted average of the varying rate levels that occurred in CY 2008.
The weights will be relative proportions of the CY square. First calculate the area of all triangles (area =
.5 * base * height) or rectangles within a unit square and then determine the remaining proportion of the
square by subtracting the sum of the areas of the triangles and rectangles from 1.0.
Area

Rate Level

1.00

1.155

1.10

CY 08 OLF 

Weight
.50 * 1.0 =

.50

½(1/4)(1/2)=

.0625

1.0 - .50 - .0625=

.4375

1.155
1.155

 1.0964
[.50(1.0) .4375(1.10) .0625(1.155)] 1.0534375

c.
Rate Change
0.10
%
of Policy
Earned

0.05

100%
B

0%
1/2008

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Section 3: Premium Aggregation – Using the One and Two Step Procedures
Solutions to questions from the 2003 exam
11. Determine the written premium trend period.
Step 1: Determine the average written date during the experience period. For the experience period 4/1/01 –
3/31/02, and given that 6 month policies are being written, the average earned date is 10/1/01 and
the average written date is 7/1/01, or ½ the policy term earlier from the average earned date.
Step 2: Determine the average written date during the exposure period. The average written date during the
future policy period is a function of the length of time that the rates are expected to remain in effect. In
this example, since rates are reviewed every 18 months, this would make the average written date 9
months after the proposed effective date of 4/1/03, which is 1/1/04. Thus, the written premium trend
period is 2.50 years.
Answer: D.  2.4 years, but < 2.7 years

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2004 exam
Question 35.
a. (1 point) Calculate the beginning and ending dates for each of the Step 1 and Step 2 trend periods,
assuming the selected trend is based on average written premium.
Preliminary information.
The solution below includes a graphic depicting the beginning and ending dates for each of the Step 1
and Step 2 trend periods, assuming the selected trend is based on average written or average earned
premium. The graphic is included in our solution for instructional purposes only.
What are the trending periods to apply to CY/AY 2002 earned premium at current rate level using a twostep trending procedure?

2001

2002
Average
Written
Date
4/1/02

Average
Earned
Date
7/1/02

Step 1

2003

2006

Average
Date for
Latest
Trend Point
7/01/003

Future
Effective
Date
7/1/05

Average
Written
Date
1/1/06

Average
Earned
Date
4/1/06

Step 2

a. AWP

Step 1: Determine the trend period from the average written date of the experience period to the
average date for the last data point in the average written date series:
To determine the average written date, recognize that the first policies that contribute to calendar
year 2002 earned premium would be ones written on 7/2/01, since these policies would be effective
until the end of the day on 1/1/02. The last policies that would contribute to CY 2002 earned
premium would be ones written on 12/31/02. The total amount of time between the two written dates
is 18 months, so the average written date is 4/1/02.
In establishing the ending point for the first part of the trending period (step 1), it is important to
recognize that the average written premium measures in the series are 12-month averages. This
means that each figure provides a measure of the average premium at the midpoint of its 12-month
period. In other words, since the latest trend point in the series is for the year ending 12/31/03, then
the measure of the average premium for that point corresponds to 7/1/03, not 12/31/03.
Thus, the average written date of the experience period is 4/1/02 and the average date for the last
data point in the average written date series is 7/1/03.
Step 2: Determine the trend period from the average written date for the last data point in the average
written date series to the average written date under the effective period of the rates.
As stated before, the average written date for the last data point in the average written date series
under the experience period is 7/1/03. The average written date for polices effective during the
planned effective period is January 1, 2006. This is because the average written date in the future
policy period does not depend on the length of the policies. Instead, it is the length of time the rates
are assumed to be in effect before the next revision.
Therefore, the beginning and ending dates for Step 2 trend is 7/1/03 – 1/1/06.

Exam 5, V1a

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2004 exam (continued):
b. (1 point) Calculate the beginning and ending dates for each of the Step 1 and Step 2 trend periods,
assuming the selected trend is based on average earned premium.
Preliminary information.
It is important to realize that whether the selected trend is based on average written premium or average
earned premium, the two alternatives have the same length trending periods. However, these periods
are not identical. The trending period for the average earned premium approach is shifted in time so that
it is a half a policy period later than the trending period for the average written premium approach.

2001

2002
Average
Written
Date
4/1/02

2003

2006

Average
Date for
Latest
Trend Point
7/01/003

Average
Earned
Date
7/1/02

Step 1

Future
Effective
Date
7/1/05

Average
Written
Date
1/1/06

Average
Earned
Date
4/1/06

Step 2

a. AWP
Step 1

Step 2

b. AEP

Based on the discussion in part a, and the graphic above, we can determine the following:
The beginning and ending dates for Step 1 trend is 7/1/02 – 7/1/03.
The beginning and ending dates for Step 2 trend is 7/1/03 – 4/1/06.
(1 point) Describe a situation when it may be more appropriate to use a two-step trending procedure,
rather than a one-step trending procedure. Two step trending is more appropriate when there isn’t a
clear trend in the series of average written or earned premiums.

Avg WP

12-Month Moving Average Written Premium

Quarter

\
For example, if the 12 month moving average written premiums looked like the series above it would not
be appropriate to apply a single trend, since the lower average written premium at the midpoint needs
more trend applied to it than the average written premium at the beginning or end.

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2005 exam:
37a. (3.5 points) Using two-step trending, determine the total premium trend factors for each year above.
Initial comments: the two-step trending method simply divides the latest average written premium at
current by the average earned premium at current for each year in the experience period. This produces
conversion factors for adjusting the total earned premium at current rate level for each year to the latest
period’s average written premium level.
In establishing the ending point for the first part of the trending period (step 1), it is important to recognize
that the average written premium measures in the series are 12-month averages. This means that each
figure provides a measure of the average premium at the midpoint of its 12-month period. In other words,
if the latest trend point in the series is for the year ending 12/31/01, then the measure of the average
premium for that point corresponds to 7/1/01, not 12/31/01. Therefore, the first step of the two-step
trending procedure trends the premium to the midpoint of the latest trend data point in the series.
The second step of the two-step trending procedure trends the premium from the midpoint of the latest
trend data point to the average written date for the future policy period. If the target effective date were
1/1/03, then the average written date for the future policy year would be half way through, or 7/1/03, with
the standard assumption that the proposed rates will be in effect for one year. The trending period in this
example would need to extend from the midpoint of the latest average written premium measure (7/1/01)
to the average written date for the future policy period (7/1/03). Therefore, the trending period for the
second step would be two years.
Problem Specific:
First, one needs to adjust the historical premiums for the 20% rate decrease on 7/1/03.
For CAY 2004 – The average written premium does not need to be adjusted
For CAY 2003 – One half of the written premium needs to be adjusted down by 20%. Thus, the adjusted
CAY 2003 average written premium is ½(933.33) + ½(933.33)(0.8) = 840
For CAY 2002 – The entire premium needs to be adjusted downward by 20%: 1,000 × 0.80 = 800
The first step in the two-step trending is to divide the latest year’s average written premium by each year’s
average written premium. The ratios are the trend factors for step 1. They are used to trend the premiums to
7/1/04 and are computed as follows:
CAY
Trend Factor
2002 882/800 = 1.1025
2003 882/840 = 1.05
2004 882/882 = 1.0
This factor already includes the 3% trend due to shifts in limit distributions from 2002-2004.

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2005 exam (continued):
Question 37 (continued):
In step 2, project the average premiums for each year to the anticipated future level.
A prospective trend is not given, so I will use the historical trend of 1.05 reduced for the 3% trend not
continuing past 2004. Thus, the prospective trend = 1.05/1.03 = 1.019 = 1.9%
The step 2 trending period extends from 7/1/04 to the average written date of effective period. As rates are
reviewed every 18 months, and given that the planned effective date for a rate change is January 1, 2006, the
average written date will be 9 months past the effective date, or 10/1/06.
Trend factor for step 2 = (1.019)2.25 = 1.043
Thus, the total premium trend factor is calculated as follows:
CAY
Step 1
Step 2
Total
(1)
(2)
(3)=(1)*(2)
2002
1.1025
1.043
1.15
2003
1.05
1.043
1.095
2004
1.0
1.043
1.043
See page 28.
b. (0.5 point) Why is two-step trending a more suitable procedure for trending premium than for trending
loss frequency or severity?
This procedure relies on the assumption that the latest year’s average written premium is a time value. For
premiums, this assumption holds because premiums are relatively stable. Loss severity and frequency
values vary greatly over time and the assumption does not hold.
Alternatively,
“Consider the theoretical implications of two-step trending. This trending method rests on the assumption that
the last data point of the trend series is a “true” number. For loss frequency or severity, this can be a dubious
assumption because of random fluctuations around the true expected value. For average premium, on the
other hand, the individual data points are more believable because there is not as large a random element.”

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2006 exam:
Question 26
a. (1.5 points) Assume all policies have a six-month term. Use 2-step trending with average written
premium to calculate the trended premium for calendar year 2002. Show all work.
Step 1: Determine the trend period from the average written date of the experience period to the
average date for the last data point in the average written date series:
To determine the average written date, recognize that the first policies that contribute to calendar
year 2002 earned premium would be ones written on 7/2/01, since these policies would be effective
until the end of the day on 1/1/02. The last policies that would contribute to CY 2002 earned
premium would be ones written on 12/31/02. The total amount of time between the two written dates
is 18 months, so the average written date is 4/1/02.
In establishing the ending point for the first part of the trending period (step 1), it is important to
recognize that the average written premium measures in the series are 12-month averages. This
means that each figure provides a measure of the average premium at the midpoint of its 12-month
period. In other words, since the latest trend point in the series is for the year ending 12/31/04, then
the measure of the average premium for that point corresponds to 7/1/04, not 12/31/04.
Thus, the average written date of the experience period is 4/1/02 and the average date for the last
data point in the average written date series is 7/1/04. This is the period where premium will be
trended by the historic premium drift of 5%.
Step 2: Determine the trend period from the average written date for the last data point in the average
written date series to the average written date under the effective period of the rates.
As stated before, the average written date for the last data point in the average written date series
under the experience period is 7/1/04. The average written date for polices effective during the
planned effective period is December 1, 2006. This is because the average written date in the future
policy period does not depend on the length of the policies. Instead, it depends on the length of time
the rates are assumed to be in effect before the next revision.
Therefore, the beginning and ending dates for Step 2 trend is 7/1/04 – 12/1/06. This is the period
where premium will be trended by the expected future premium drift of 3%.
Thus, the trended premium for calendar year 2002 is computed as follows:

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2006 exam:
Question 26, part b:
b. (1.5 points) Assume all policies have an annual term. Use 2-step trending with average written
premium to calculate the trended premium for calendar year 2002. Show all work.
Note: The only difference in solving this problem, compared with the problem in part a, is the starting
date for the trend period. The rationale given for all other points in time in as stated in part a, for both
steps, holds.
To determine the average written date, given annual policies, recognize that the first policies that
contribute to calendar year 2002 earned premium would be ones written on 1/2/01, since these policies
would be effective until the end of the day on 1/1/02. The last policies that would contribute to CY 2002
earned premium would be ones written on 12/31/02. The total amount of time between the two written
dates is 24 months, so the average written date is 1/1/02.
Thus, the trended premium for calendar year 2002 is computed as follows:

c. (0.5 point) Explain one advantage of using 2-step trending in this example over 1-step trending.
1-step-trending assumes uniform trend from the experience period to the future policy period. This
assumption does not apply to certain situations where there are differences in trend between the past and
the future. The 2-step trending procedure solves this problem.

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2006 exam (continued):
27. (1 point)
a. (0.5 point) Explain why using average premiums is better than total premiums when analyzing
premium trend.
b. (0.5 point) Give one argument for using average earned premiums in the premium trend analysis
and one argument for using average written premiums.
CAS Model Solution
Part a.
Total premiums are affected by exposure changes, while average premiums have averaged out the exposure
effects. Thus changes in average premium are more related to the actual trend in premium.
Part b.
1 – The premiums being trended are earned premiums, thus it is better to use average earned premiums in
the premium trend analysis.
2 – Average written premiums are more responsive to recent changes.
As Jones states
“Since these trends will apply to historical earned premium at current rate level, we should evaluate trends
based on shifts in average earned premium.”
“Even though the historical premium is earned premium, we can determine the average written date for that
block of premium and then observe changes in average written premium to establish the trend. Therefore,
basing the trend analysis on average written premium is a valid approach. Furthermore, average written
premium has an important advantage in that it allows us to capture more recent data than average
earned premium. This is because of the simple fact that the premium for a given policy is not earned until
well after it is written. In fact, at any given point in time, the latest quarter’s average earned premium is based
on a group of policies that is a half a policy period older than the group of policies comprising the latest
quarter’s average written premium. Using average earned premium would unnecessarily postpone the
recognition of the effects of the most recent changes in the mix of business.”

Solutions to questions from the 2007 exam:
Question 36 - Calculate the trended premium for each year, using the two-step trending method.
Model Solution - Initial comments.
The two-step trending method requires the use of average earned premium at current rate level for each year in
the experience period. The components are total earned premium at current rate level and earned exposures. In
this problem, we are given the average earned premium at current rate level.
How the two-step trending method is used.
The two-step trending method simply divides the latest average written premium at current level by the
average earned premium at current for each year in the experience period. This produces conversion
factors for adjusting the total earned premium at current rate level for each year to the latest period’s
average written premium level.

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2007 exam:
Question 36 - Calculate the trended premium for each year, using the two-step trending method.
Model Solution
Step 1: Bring the average earned premium at current rate level to the latest level available in the series of
average written premiums at current rate level.
This accounts for shifts in the mix of business and any other factors not already accounted for with a
direct adjustment to the historical experience.

For Step 1, we don’t need to consider exposures because average written premiums at current level are used.
Step 2: Project the average written premiums at current level for each year to the anticipated future rate level.
A three percent annual trend (stated in the problem (see (3)) is applied over a two-year period.
The Step 2 trend period is 2 years (from 7/1/05 to 7/1/07) at 3%.
Latest
Total
Value of
Premium
Step 1
Step 2
Avg EP
Avg WP
Trend
@CRL
@CRL (7/05)
Trend Factor
Trend Factor
Factor
CY
(1)
(2)
(3) = (2)/(1)
(4)
(5) = (3)*(4)
2003
3,605
3,998
1.1090
1.177
1.032
2
2004
3,749
3,998
1.0664
1.131
1.03
2
1.088
2005
3,899
3,998
1.0254
1.03

CY
2003
2004
2005

Trended
Average Premium
(6) = (1)*(5)
4,242
4,242
4,242

Earned
Exposures
(7)
1,000
1,050
1,100

Trended Total
Premium
(8) = (6)*(7)
4,242,000
4,454,100
4,666,200

(4) = The selected annual trend for Step 2 (given in the problem as 3%) is applied from the midpoint
of (2) to the average written date in the future policy period (which is 7/1/2007 in this problem).
Note that the total premium trend factors in column (5) are used to compute trended average
premium in (6), and are used in place of those developed by the one-step procedure.

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2008 exam:
Model Solution – part a. – question 15
a. Question no longer applicable to the content covered in this chapter.
Model Solution – part b – question 15. - Initial comments.
The two-step trending method requires the use of average earned premium at current rate level for each year in
the experience period. This problem is based upon the example in Appendix 2 - the Two-Step Trending Method.
Keep in mind that all policies are semi-annual and thus, Jones’ comments on “What about six month policies on
pages 17 – 18 apply.
In particular “For a six-month policy term, the first step of the procedure will involve a shorter trending period than
the one used for 12-month policies. This is because the average written and average earned dates are closer
together for shorter policies. The break point between the first and second step is still the same since we use 12month moving averages of written premium in both analyses. The second step of the procedure results in the
same length trending period as was used for 12-month policies. This is because the average written date in the
future policy period does not depend on the length of the policies. Instead, it is the length of time the rates are
assumed to be in effect before the next revision.”
In step 1, bring the average earned premium at current rate level to the latest level available in the series of
average written premiums at current rate level.
In step 2, project the average premiums for each year to the anticipated future level. In this example, a 4 percent
annual trend is applied over a two-year period.
NOTE: The following is not needed to solve the problem but is provided to give you a broader understanding of
what is happening in this example.
The first policies that contribute to calendar year 2006 earned premium would be ones written on 7/2/05, since
these policies would be effective until the end of the day on 1/1/06. The last policies that would contribute to
2006 earned premium would be ones written on 12/31/06. The total amount of time between the two written
dates is 18 months, so the average written date is 4/1/06.
In establishing the ending point for the first part of the trending period (step 1), it is important to recognize that
the average written premium measures in the series are 12-month averages. This means that each figure
provides a measure of the average premium at the midpoint of its 12-month period. In other words, since the
latest trend point in the series is for the year ending 12/31/07, then the measure of the average premium for
that point corresponds to 7/1/07, not 12/31/07. Therefore, the first step of the two-step trending procedure
trends the premium to the midpoint of the latest trend data point in the series.
The second step of the two-step trending procedure trends the premium from the midpoint of the latest trend data
point to the average written date for the future policy period. Since the target effective date is 1/1/09, then the
average written date for the future policy year would be half way through, or 7/1/09, with the standard assumption
that the proposed rates will be in effect for one year. The trending period in this example would need to extend
from the midpoint of the latest average written premium measure (7/1/07) to the average written date for the
future policy period (7/1/07). Therefore, the trending period for the second step would be two years.

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2008 exam:
Model Solution – part b. – question 15
Thus, the Step 1 trend factor is 112/106 = 1.056 and Step 2 trend factor = 1.042 = 1.0816, and
The trend factor to 2006 calendar/accident year = 1.0566 x 1.0816 = 1.1428
This can also be demonstrated as shown below.

(1)

Year

Avg EP
@CRL

(2)
Latest
Value of
Avg WP
@CRL

2004
2005
2006
2007

$98
$102
$106
$110

$112
$112
$112
$112

(3)

(4)

Step 1
Trend
Factor
(3)=(2)/(1)
1.1429
1.0980
1.0566
1.0182

Step 2
Trend
Factor
1.0816
1.0816
1.0816
1.0816

(5)
Total
Premium
Trend
Factor
(5)=(3)*(4)
1.2361
1.1876
1.1428
1.1013

Solutions to questions from the 2010 exam:
Question 18
Calculate CY 2008 earned premium at prospective levels using two-step trending.
Step 1: Adjust the historical premium to the current trend level using the following adjustment factor:

Current Premium Trend Factor =

Latest Average WP at Current Rate Level
Historical Average EP at Current Rate Level

Latest Avg WP at Current Rate Level is 682,500/1,300 = 525
Historical Avg EP at Current Rate Level is 487,500/1,000 = 487.50
Thus, the current premium trend factor is 1.0769 (= 525/487.50).
The latest average WP is for CY 2009; thus, the average written date is 7/1/2009 (this will be “trend
from” date for the second step in the process).
Step 2: Compute the projected premium trend factor.
Select the amount the average premium is expected to change annually from the “trend from” date to the
projected period.
The “trend from” date is 7/01/2009.
The “trend to” date is the average written date during the period the proposed rates are to be in effect,
which is 7/01/2011.
Thus, the projected trend period is 2 years long (7/1/2009 to 7/1/2011).
Given a projected annual premium trend of 5%, the projected trend factor is 1.1025 (= (1.0 + 0.05)2).
The total premium trend factor for two-step trending is the product of the current trend factor and the
projected trend factor (i.e. 1.18728 (= 1.0769 x 1.1025)).
That number is applied to the average historical EP at current rate level to adjust it to the projected level:
CY08 EP at projected rate level = CY08 EP at current rate level x Current Trend Factor x Projected
Trend Factor.
CY 2008 earned premium at prospective levels = (487,500) (1.0769) (1.052) = 578,800.10

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2011 exam:
5. (2.25 points) Given the following information:
•
Policy term: Six months; Proposed rates in effect from 1/1/2012 to 6/30/2013
•
Selected projected premium trend: 5%
Calendar
Average Earned Premium
Average Written Premium
Year
at Current Rate Level
at Current Rate Level
2009
$375
$380
2010
$390
$395
5a. (2 points) Calculate the total premium trend factor for each of CYs 2009 and 2010 using two-step trending.
5b. (0.25 point) Briefly discuss when it is appropriate to use two-step trending.
Question 5 - Model Solution 1
a. Two-step trending = Use Step 1 and Step 2 premium trend factors
- For CY 2009
Step 1 trend = (Avg WP@CRL Latest period) / (Historical Avg EP@CRL) = 395/375 = 1.05333
AWD for CY 2010 = 7/1/10. Average written date for the period 1/1/2012 to 6/30/2013 is 10/1/2012
Step 2 trend = Starts 7/1/10, Ends 10/1/12.
Step 2 trend period from 7/1/10 - 10/1/12 = 2.25 years
Step 2 trend = (1.05)2.25 = 1.116
CY 2009 total premium trend factor = (1.0533)(1.052.25) = 1.1756
- For CY 2010
Step 1 trend = 395/390 = 1.0128 (see above formula)
Step 2 trend = trend from 7/1/10 – 10/1/12 = 2.25 years
CY 2010 Total premium trend factor = (1.0128)(1.052.25) = 1.1303
b. It is appropriate to use two step trending when the historical trend and the prospective trend are different.
Question 5 - Model Solution 2
a.
(1)
(2)
2010
CY
Avg EP Avg. WP
2009
375
395
2010
390
395

(3)
= (2)/(1)
1.0533
1.0128

(4)
Premium Trend
1.052.25
1.052.25

(5)
(5) = (3)x(4)
1.1755
1.1303

2nd step trend period is from 7/1/2010 to 9/30/2012 which is 2.25 years.
b. When the future premium trend is different from the current trend, we cannot use one-step trend, we
need to use a 2- step trend instead.

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Chapter 5 – Premium
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2012 exam:
6a. (1 point) Use the two-step trending method to calculate the projected earned premium for the
calendar year ending December 31, 2009.
6b. (1 point) After completing the analysis, the actuary determines that the assumed annual increase in
the amount of insurance to account for inflation was materially reduced post-January 1, 2012.
Discuss any necessary adjustments to the completed analysis in part a. above
Question 6 – Model Solution 1 (Exam 5A Question 6)
Step 1 factor = latest average written premium @ CRL (current rate level)
Calendar year 2009 average earned premium @ CRL = 560/(5,000,000/10,000) = 560/500 = 1.12
Step 2 => trend from = 11/15/2011 <-midpoint of latest period.
trend to = 7/1/ 2013 <-average written date in projected period
= proposed effective date + ½ the time rates are expected to be in effect.
→trend period = 1.625, and the Step 2 trend factor = (1.05) ^ 1.625
Projected Earned Premium for CY 2009
= EP @ CRL x Step 1 factor x Step 2 factor = 5,000,000 x (1.12) x (1.05) ^ 1.625 = $6,062,066.
b. The assumed annual increase in the amount of insurance to account for inflation is an ongoing and
gradual change, and is reflected in the prospective annual premium trend. So it would be necessary to
adjust the prospective annual premium trend of +5% downwards to reflect this reduction, which would
resultantly adjust the Step 2 factor. Note that since 2-step trending is used in part (a), it will be appropriate
to only adjust the Step 2 factor since this change means trend expected in the future will be different from
historical trend.
Question 6 – Model Solution 2 (Exam 5A Question 6)
Step 1: 560/ (5,000,000/10,000) =1.12
Step 2: from 11/15/2011 to 7/1/2013
From avg. of latest period (4Q11) to avg. written date of prospective period (7/1/2012 t0
6/30/2014) <-2 years. Thus, the step 2 trend factor is 1.05 ^ (1.625) = 1.0825
Total Projected EP = 5,000,000 x 1.12 .x 1.0825 = 6,062,065.69
b You would need to re-calculate your selected prospective trend in step 2. Step 1 can be left alone,
however the step 2 trend would be less than 5%, and would lower the projected premium.
Question 6 – Model Solution 3 (Exam 5A Question 6)
Average written date in 4Q 11 is Nov. 15, 2011
Average written date for 2 year effective period starting July 1, 2012 is July 1, 2013.
Thus, the Prospective Trend period is 1.625 years
Average earned premium for CY2009 is 5,000,000 ÷ 10,000 = 500
Projected Earned Premium for CY2009 is 5,000,000 (560/500) (1.05 ^ 1.625)= 6,062,065.69
b. The 5% prospective premium trend is likely too high and should be reduced in the analysis from a
Examiner’s Comments
a. The majority of candidates received full credit. Those that didn’t receive full credit typically lost
points for calculating the trend period incorrectly.
b. Most candidates either identified both or only one of the other elements needed for full credit. Some
candidates identified that the first step in two step trending would not be affected, but this was not
necessary for full credit.

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Sec
1
2
3
4
5
6
1

Description
Loss Definitions
Loss Data aggregation Methods
Common Ratios Involving Loss Statistics
Adjustments To Losses
Loss Adjustment Expenses
Key Concepts

Pages
90 - 91
91 -93
92 - 93
93 – 121
121 – 122
122 - 123

Loss Definitions

90 - 91

The text uses the term claim to mean demand for compensation and loss to refer to the amount of compensation.
Losses and LAE usually represent largest portion of premium.
This chapter discusses:
 The different types of insurance losses
 How loss data is aggregated for ratemaking analysis
 Common metrics involving losses
 Adjustments made to historical loss data to make it relevant for estimating future losses in the
ratemaking process. This includes adjusting data for:
• Extraordinary loss events
• Changes in benefit levels
• Changes in the loss estimates as immature claims become mature
• Changes in loss cost levels over time
 Treatment of LAE
Definitions
• Paid losses: Payments made to claimants.
• Case reserve: An amount expected to be paid on a claim, based on a claims adjuster’s estimate or
determined by formula.
• Reported (Case Incurred) losses: Paid Losses + Case Reserves
• Incurred but not enough reported (IBNER): Reported losses adjusted to account for any anticipated
shortfall in the case reserves
• Incurred but not reported (IBNR): Reserves for claims incurred but that have not yet been reported.
• Ultimate Losses: Reported Losses + IBNER + IBNR
Aggregated losses are based on statistics (e.g. paid or reported losses), a data aggregation method (e.g.
calendar, accident, policy, or report month/quarter/year), and a period of time.
The time period for data aggregation is defined by an accounting period and a valuation date.
The accounting period for losses should be consistent with financial statement dates (e.g. month, quarter,
or calendar year).
The valuation date (which can be different than the end of the accounting period) is the date losses are
evaluated for analysis. It is expressed as the number of months after the start of the accounting period (e.g.
AY 2010 as of 18 months implies AY 2010 as of 6/30/2011).
Valuation dates can occur prior to the end of the accounting period.

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.

Loss Data aggregation Methods

91 -93

Four ways to aggregate data are by calendar year, accident year, policy year, and report year (see Chapter 3
for comments on CY, AY and PY).
Note: Some insurers aggregate losses in twelve-month periods that do not correspond to calendar years. This is called a
fiscal accident year and the period is referred to as 12 months ending mm/dd/yy (i.e. the accounting date).

RY Loss aggregation method:
Losses are aggregated according to when the claim is reported (as opposed to when the claim occurs for AY).
 Accident dates are maintained so the lag in reporting can be determined, since report year losses can be
subdivided based on the report lag.
 This type of aggregation results in no IBNR claims, but a shortfall in case reserves (i.e. IBNER) can exist.
 RY aggregation is limited to the pricing of claims-made (CM) policies.
Claims Made policies provide coverage based on the date the claim is reported (as opposed to the date the claim
occurs).
 It is often written in lines of business for which there is often a significant lag between the date of the
occurrence and the reporting of the claim (e.g. medical malpractice).
 CM ratemaking is covered in Chapter 16.
Quantifying Reported Losses under different loss aggregation methods
Assume reserves are $0 prior to CY 2009
Claim Transaction History
Policy
Effective
Date of
Report
Transaction Incremental
Case
Date
Loss
Date
Date
Payment
Reserve
07/01/09
11/01/09
11/19/09
11/19/09
$0
$10,000
02/01/10
$1,000
$9,000
$2,500
09/01/10
$7,000
01/15/11
$3,000
$0
09/10/09
02/14/10
02/14/10
02/14/10
$5,000
$10,000
11/01/10
$8,000
$4,000
03/01/11
$1,000
$0
*Case reserve evaluated as of transaction date.
CY 2009 reported losses are $10,000: CY 2009 paid losses (i.e. the sum of the losses paid in 2009 ($0)) plus
the ending reserve at 12/312009 ($10,000) minus the beginning reserve in 2009 ($0).
CY 2010 reported losses are $17,500: CY 2010 paid losses ($1,000 + $7,000 + $5,000 + $8,000) plus the
ending reserve at 12/31/ 2010 ($2,500 + $4,000) minus the beginning reserve in 2010 ($10,000).
CY 2011 reported losses are -$2,500: CY 2011 paid losses ($3,000+$1,000) plus the ending reserve at
12/31/2011 ($0), minus the beginning reserve in 2011 ($2,500 + $4,000).

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AY 2009 reported losses as of 12/31/2011 are $11,000 (considers transactions on the first claim only):
Cumulative losses paid through 12/31/2011 on the first claim ($1,000 + $7,000 + $3,000) plus the case
reserve estimate for this claim as of 12/31/2011 ($0). (When referring to AY paid losses, the adjective cumulative is
usually implied rather than explicit.)

AY 2010 reported losses as of 12/31/2011 are $14,000 (considers transactions on the second claim only):
Losses paid on the second claim through 12/31/2011 ($5,000 + $8,000 + $1,000), plus the case reserve
estimate for this claim as of 12/31/2011 ($0).
PY 2009 reported losses as of 12/31/2011 are $25,000 (considers transactions from both policies):
The sum of the losses paid on both policies ($1,000 + $7,000 + $3,000 + $5,000 + $8,000 + $1,000) plus the
case reserve estimate as of 12/31/2011 ($0).
PY 2010 reported losses as of 12/31/2011 are $0 since neither of these policies was issued in 2010.
CY 2009, AY 2009, and PY 2009 reported losses at three different valuation dates are shown below
Reported Losses: CY09 v AY09 v PY09
Valuation Date
Aggregation Type 12/31/2009 12/31/2010 12/31/2011
Calendar Year 09 $10,000
$10,000
$10,000
Accident Year 09
$10,000
$10,500
$11,000
Policy Year 09
$10,000
$27,500
$25,000



CY reported losses are finalized at the end of the year, accident year and policy year losses are not.
PY losses undergo development during the second twelve months of the 24-month policy year period
(this longer lag time to get accurate PY data is a shortcoming of the PY aggregation method).

RY 2009 reported losses only include amounts associated with the first claim as it was reported in 2009.
 As of12/31/2009, RY 2009 reported losses are $10,000 (reflects the outstanding case reserve only)
 As of 12/31/2010, RY 2009 reported losses are $10,500: the sum of all payments made ($1,000 +
$7,000) and the $2,500 case reserve estimate as of the end of 2010.
The second claim was reported in 2010 and only contributes to RY 2010 losses.

Common Ratios Involving Loss Statistics

92 - 93

Four common ratios involving loss statistics are: frequency, severity, pure premium, and loss ratio (see
chapter 1 for more information).
Each ratio is defined by:
 a choice of statistics (e.g. paid or reported losses, or earned or written premium)
 a data aggregation method (e.g. calendar, accident, policy, or report month/quarter/year)
 an accounting period, and
 a valuation date.

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Adjustments To Losses

93 – 121

Prior to projecting losses to the cost level expected when the rates will be in effect, preliminary adjustments
may involve:
 removing individual shock losses and catastrophe losses from historical losses and replacing them
with a long-term expectations provision.
 developing immature losses to ultimate.
 restating losses to the benefit and cost levels expected during the future policy period.
Extraordinary Losses (Large Individual Losses and Catastrophe Losses)
Large losses (a.k.a. shock losses) are infrequent but are expected in insurance.
Examples: a large multi-claimant liability claim, a total loss on an exceptionally high-valued home, and a
total permanent disability of a young worker.
Historical data used to project future losses should exclude a portion of these losses above a threshold, that
corresponds to the point at which the losses are extraordinary and their inclusion causes volatility in the rates.
The threshold may be:
 based on the minimum amount of insurance offered (i.e. the “basic limit”) as it corresponds to the limit
associated with the base rate.
 a point significantly higher than the basic limit (e.g. the basic limit for personal auto liability insurance
typically equals the amount of insurance required by the financial responsibility laws, but as many
insureds select higher limits of insurance, insurers may have a significant number of losses that
exceed the basic limit).
When losses are not capped at the basic limit, the actuary must determine the threshold that best balances the
goals of: (1) including as many losses as possible and (2) minimizing the volatility in the ratemaking analysis.
Set the threshold by:
 examining the size of loss distribution and setting it at a given percentile (e.g. the 99th percentile).
Examine individual claim sizes in increasing order and choosing the claim amount for which 99% of
the claim inventory is below that amount.
 choosing a certain % losses rather than claim amounts.
In property insurance the AOI varies based on the value of the insured item, and since the expected
size of loss distribution may vary significantly from one policy to the next, it may be more appropriate
to use a threshold that is a % of the AOI rather than to use a fixed threshold.
Actual shock losses are replaced with an average expected large loss amount calculated over a longer period.
The time period may vary significantly for different lines of business and even from insurer to insurer.
Examples:
 a medium-sized homeowners insurer may derive a good estimate for expected large fire losses using
10 years of data
 a small personal umbrella insurer may need 20 years of data.
Avoid using too many years as older data becomes less relevant over time (e.g. jury awards may be much
higher today than previously).
The average should be based on the number of years to produce a reasonable estimate without including so
many years as to make the historical data irrelevant.

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BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Excess Loss Factor Calculation
 In this example, individual reported losses are capped at $1,000,000 (a.k.a. non-excess losses)
 The long-term average ratio of excess losses (the portion of each shock loss above the $1,000,000
threshold) to non-excess losses is used to determine an excess loss provision.
Excess Loss Procedure
(1)
Accident
Year
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
Total

Reported Losses
$118,369,707
$117,938,146
$119,887,865
$118,488,983
$122,329,298
$120,157,205
$123,633,881
$124,854,827
$125,492,840
$127,430,355
$123,245,269
$123,466,498
$129,241,078
$123,302,570
$123,408,837
$1,841,247,359

(2)
Number of
Excess
Claims
5
1
3
0
7
3
0
1
0
6
3
0
10
0
3
42

(3)
Ground –Up
Excess Losses
$ 6,232,939
$1,300,000
$3,923,023
$
$12,938,382
$3,824,311
$
$3,000,000
$13,466,986
$4,642,4
$
$17,038,332
$
$4,351,805
$70,718,201

(4)
Losses
Excess of
$1,000,000
$1,232,939
$300,000
$923,023
$
$5,938,382
$824,311
$
$2,000,000
$
$7,466,986
$1,642,423
$
$7,038332
$
$1,351,805
$28,718,201

(7) Excess Loss Factor

(5)

(6)

Non-Excess
Losses
$117,136,768
$117,638,146
$118,964,842
$118,488,983
$116,390,916
$119,332,894
$123,633,881
$122,854,827
$125,492,840
$119,963,369
$121,602,846
$123,466,498
$122,202,746
$123,302,570
$122,057,032
$1,812,529,158

Excess
Ratio
1.1%
0.3%
0.8%
0.0%
5.1%
0.7%
0.0%
1.6%
0.0%
6.2%
1.4%
0.0%
5.8%
0.0%
1.1%
1.6%

1.016

(4)= (3) - [$1,000,000 x (2)]
(5)= (1) - (4)
(6)= (4) / (5)
(7)= 1.0 + (Tot 6), and is applied to the non-excess losses for each year in the historical experience period.
Notes: The excess loss procedure is ideally performed on reported losses that have been trended to future
levels (i.e. excess losses are calculated by censoring trended ground-up losses).
Alternatively, some actuaries may fit statistical distributions to empirical data and simulate claim
experience in order to calculate the expected excess losses.

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BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Catastrophe Losses
Ratemaking data excludes losses arising from catastrophic events. Catastrophe losses:
 from hurricanes, tornadoes, hail storms, earthquakes, wildfires, winter storms, explosions, oil spills and
certain terrorist attacks are severe and results in a significant number of claims (unlike shock losses
from individual high severity claims)
 are defined by the Property Claims Services (PCS) unit of the Insurance Services Office (ISO) as
events that cause $25 million or more in direct insured property losses and that affect a significant
number of policyholders and insurers.
 may have alternative definitions by insurers for internal procedures.
 are removed from ratemaking data and replaced with an average expected catastrophe loss amount.
 are broken down into non-modeled catastrophe losses and modeled catastrophe losses.
Non-modeled catastrophe analysis is performed on events that occur with some regularity over decades.
Example: Hail storms (which occur with some multi-year on and off regularity) is the most common
catastrophic loss related to private passenger auto comprehensive coverage.
 Without a non-modeled cat procedure, indicated rates will increase immediately after a bad storm year
and decrease in years having few or no storms.
 The actuary can calculate the ratio of hail storm losses to non-storm losses over a longer experience
period (e.g. 10-30 years).
 The number of years used should balance stability and responsiveness.
Example: If the concentration of exposures in the most hail-prone area of a state has increased
drastically over the past 20 years, then a cat procedure based on 20 years of statewide data
may understate the expected catastrophe potential.
Once determined, the ratio can be used to adjust the non-catastrophe losses in consideration of future
expected catastrophe loss.
Alternatively, the actuary can develop a pure premium (or loss ratio) for the non-modeled cat exposure.
 Using a pp approach, compute the long-term ratio of cat losses to exposure (or amount of insurance
years) and apply that ratio to projected exposures (or projected amount of ins years). See Appendix B.
 The loss ratio indication would be similar except the denominator of the long-term ratio would be EP,
which is inflation-sensitive and the premium would need to be brought to current rate level.
Catastrophe models are used for events that are irregular and generate high severity claims (e.g. hurricanes
and earthquakes).
 30 years of data may not capture the expected damage these events can inflict.
 Stochastic models are designed by professionals from a variety of fields (e.g., insurance,
meteorologists, engineers) to estimate the likelihood that events of varying magnitudes will occur and
the damages that will likely result given the insured property characteristics.
 The modeled cat loss provision is added to the non-catastrophe loss amount to determine the
aggregate expected losses to be used for pricing.
Insures writing in cat prone areas:
 may use non-pricing actions (e.g. restrict the writing of any new business, may require higher
deductibles for catastrophe-related losses, or may purchase reinsurance) in cat prone areas to control
the concentration to minimize the financial impact any one event can have on the profitability.
 may alter the underwriting profit provision in the rates to reflect the higher cost of capital needed to
support the risk caused by the higher concentration of policies.

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BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Reinsurance
Historically, ratemaking for primary insurance was done on a direct basis (i.e. without reinsurance consideration).
Some ratemaking analyses are now performed on a net basis (i.e. with consideration of reinsurance) as
reinsurance programs have become more extensive and reinsurance costs have increased substantially for
some lines of business.
Proportional reinsurance means the same proportion of premium and losses are transferred or “ceded” to the
reinsurer (thus, proportional reinsurance may not necessarily need to be included in the pricing consideration).
With non-proportional reinsurance:
 the reinsurer agrees to assume some % of the losses (reinsurance recoverables to the insurer)
 the insurer cedes a portion of the premium (the cost of the reinsurance).
Examples of non-proportional reinsurance include:
 cat excess-of-loss reinsurance (e.g. the reinsurer covers 50% of the losses that exceed $15,000,000 up
to $30,000,000 on their entire property book of business in the event of a cat)
 per risk excess of loss reinsurance (e.g. the reinsurer will cover the portion of any large single event
that is between $1,000,000 and $5,000,000 for specified risks).
Changes in Coverage or Benefit Levels
An insurer may:
 initiate changes in coverage (e.g. expand or contract coverage with respect to the types of losses
covered) or
 opt to increase or decrease the amount of coverage offered.
Benefit levels can be impacted by a law change or court ruling (e.g. caps on punitive damages for auto liability
coverage and changes in the WC statutory benefit levels).
Benefit changes can have direct and indirect effects on losses.
 direct effects are a direct and obvious consequence of the benefit change.
 indirect effects arise from changes in claimant behavior that as a result of the benefit change (and are
more difficult to quantify than direct effects).
Example: Quantification of benefit changes.
Assume an insurer reduces the maximum amount of coverage for jewelry, watches, and furs on a standard
homeowners policy from $5,000 to $3,000. The direct effect:
 is that any claimants with jewelry, watches, and furs losses in excess of $3,000 will now only receive
$3,000 rather than at most $5,000.
 of this change can be calculated if a distribution of historical jewelry, watches, and furs losses is
available. The table below shows the how reported losses on 6 claims would be capped under the two
different thresholds.

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Direct Effect of a Coverage Limit Change
(1)
(2)
Losses
Losses
Claim
Capped
Capped
number @$5,000 @$3,000
1
$1,100
$1,100
2
$2,350
$2,350
3
$3,700
$3,000
4
$4,100
$3,000
5
$5,000
$3,000
6
$5,000
$3,000
Total
$21,250
$15,450
(1) Given
(2) = Min[(1), $3,000]
(3) = (3) / (2) - 1.0

(3)
Effect of
Change
0.0%
0.0%
-18.9%
-26.8%
-40.0%
-40.0%
-27.3%

The direct effect is -27.3%.
Example: Indirect effect
Consider an example involving a decrease in coverage.
 Insureds may feel the reduced coverage is inadequate and purchase a personal articles floater (PAF) to
cover jewelry, watches, and furs.
 If the HO is secondary to the PAF, the jewelry, watches, and furs losses from the homeowners policy
will be further reduced as they are now covered by the PAF.
 Since there is no way to know how many insureds will purchase the PAF and the amount of PAF
coverage they will purchase, it is very difficult to accurately quantify the indirect effect.
WC benefits are statutory and changes in these statutes can lead to direct and/or indirect effects on losses.
Statutes dictate the maximum/minimum benefits, the maximum duration of benefit, the types of injuries or
diseases covered treatments that are allowed, etc.
Consider the case where the WC wage replacement rate increases from 60% to 65% of pre-injury wages.
 the direct effect on wage replacement losses is easily quantified as +8.3% ( = 65% / 60% - 1.0).
 there may be an indirect effect as workers may be more inclined to file claims and claimants may have
less incentive to return to work in a timely manner.
Example: Calculation of the direct effect of a benefit level change
Suppose the WC maximum indemnity benefit for a particular state is changing. The assumptions include:
• The compensation rate is 66.7% of the worker’s pre-injury wage.
• The state average weekly wage (SAWW) is currently $1,000.
• The minimum indemnity benefit remains at 50% of the SAWW.
• The maximum indemnity benefit is decreasing from 100% of the SAWW to 83.3% of the SAWW.

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The distribution of workers (and their wages) according to how their wages compare to the SAWW is as follows:
Benefit Example
Ratio to
Total
Average
Weekly
#
Weekly
Wages
workers
Wage
<50%
7
$3,000
50-75%
24
$16,252
75-100%
27
$23,950
100-125%
19
$23,048
125-150%
12
$16,500
>150%
11
$17,250
Total
100
$100,000
Calculate the direct effect of the benefit level change.
The key is to calculate the benefits provided before and after the change.
The minimum benefit is 50% of the SAWW ($1,000) which equals $500 (= $1,000 x 50%).
The minimum benefit of $500 applies to workers who earn less than 75% of the SAWW
(i.e. $500 = 66.7% x 75% x $1,000), given the current compensation rate of 66.7%.
The aggregate benefits for 31 (= 7 + 24) employees in this category are $15,500 (= 31 x $500).
The maximum benefit is 100% of the SAWW ($1,000) and thus equals $1,000 (= $1,000 x 100%).
The maximum benefit of $1,000 applies to workers who earn more than 150% of the SAWW
(i.e. $1,000 = 66.7% x 150% x $1,000), given the current compensation rate of 66.7%.
The aggregate benefits for the 11 employees in this category are $11,000 (= 11 x $1,000).
The remaining 58 (= 27 + 19 + 12) employees fall between the minimum and maximum benefits.
This means their total benefits are 66.7% of their actual wages or $42,354 ( = ( 66.7% x 23,950 )
+ ( 66.7% x 23,048 ) + ( 66.7% x 16,500 ) ).
The sum total of benefits is $68,854 (= $15,500 + $11,000 + $42,354) under the current benefit structure.
Once the maximum benefit is reduced from 100% to 83.3% of the SAWW, more workers will be subjected
to the new maximum benefit.
Workers earning approximately >125% of the SAWW are subject to the maximum (i.e. $833.75 = (66.7%
x 125% x $1,000) > $833). These 23 (= 11 + 12) workers will receive $19,159 (= 23 x $833) in benefits.
Workers subject to the minimum benefit, 31, are not impacted by the change, and their benefits remain
$15,500.
There are now only 46 (= 27 + 19) employees that receive a benefit equal to 66.7% of their pre-injury wages or:
$31,348 (= (66.7% x 23,950) + (66.7% x 23,048)) because more workers are now impacted by the maximum.
The new sum total of benefits is $66,007 (= 19,159 + 15,500 + 31,348).
The direct effect from revising the maximum benefit is -4.1% (= 66,007 / 68,854 – 1.0).

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Benefit Example
(1)
Ratio to
Average
Wage
<50%
50-75%
75-100%
100-125%
125-150%
>150%
Total

(2)
Workers
7
24
27
19
12
11
100

(3)
(4)
Total
Current
Benefits
Weekly
Wages
$3,000
$3,500
$16,252
$12,000
$23,950
$15,975
$23,048
$15,373
$16,500
$11,006
$17,250
$11,000
$100,000
$68,854
(6) Benefit Change

(5)
Proposed
Benefits
$3,500
$12,000
$15,975
$15,373
$9,996
$9,163
$66,007
-4.1%

(4)= < Min: (2) x $500, Other (3) x 0.667 > Max: (2) x 1,000
(5)= < Min: (2) x $500 Other (3) x 0.667 >Max: (2) x $833
(6)= (Tot 5) / (Tot 4) - 1.0

There may also be an indirect effect if the max indemnity benefit is decreased.
Assuming there is no data to estimate the indirect effect, it needs to be determined judgmentally (the strength
of the indirect effect is a function of the economic environment, the nature of the insured population, etc).
Recall that a benefit change may affect:
(1) all claims on or after a certain date or
(2) claims arising from all policies written on or after the date.
The needed adjustment is different in each case and the techniques for calculating the adjustment are similar to
the parallelogram method for deriving on-level premium.
Example: Benefit Change Loss Adjustment Factor
The figure below shows a law change implemented on 8/15/2010 that only affects losses on policies written on
or after 8/15/2010. The direct effect of the change for annual policies on an AY basis is estimated at +5%.





The pre-change loss level is 1.00 and post-change loss level is 1.05.
Since scenario (1) applies, the line dividing the losses into pre- and post-change is a diagonal line
representing a policy effective on the date of the law change.
Note that the calendar accident years have been divided into accident quarters.

The benefit change loss adjustment factor is Adjustment =

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Current Loss Level
Average Loss Level of Historical Period

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Focusing on the third quarter of 2010, the portion of losses assumed to be pre- and post-change are as follows:
• 3Q 2010 Post-change: 0.0078 = 0.50 x 0.125 x 0.125
• 3Q 2010 Pre-change: 0.2422 = 0.25 - 0.0078
The adjustment factor for 3rd quarter 2010 reported losses is

Adjustment 

1.05
 1.0484
 0.2422 
 0.0078 
1.00* 
  1.05* 

 0.2500 
 0.2500 

The adjustment factors for the reported losses from all other quarters are calculated similarly.
Example: How to measure the same law change on a policy year basis.
Affect on Losses on New Annual Policies (PY Basis)

The adjustment factor applicable to the third quarter 2010 policy quarter reported losses is:

Adjustment 




1.05
 1.0244
 0.50 * 0.25 
 0.50 * 0.25 
1.00 * 
  1.05* 

 0.25 
 0.25 

Reported losses from quarters prior to the third quarter need to be adjusted by a factor of 1.05.
Reported losses from quarters after the third quarter are already being settled in accordance with the
new law, and need no adjustment.

Example: A benefit change affecting all losses occurring on or after 8/15/2010 (regardless of
the policy effective date).
Affects all New Losses (AY Basis)

i. The adjustment factor applicable to the third accident quarter 2010 losses is as follows:

Adjustment 

Exam 5, V1a

1.05
 1.0244
 0.50 * 0.25 
 0.50 * 0.25 
1.00 * 
  1.05* 

 0.25 
 0.25 

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Affects all New Losses (PY Basis)

ii. The adjustment factor applied to third policy quarter 2010 losses is

Adjustment

1.05
1.0015
 0.078 
 0.2422 

1.00*
1.05*



 0.2500 
 0.2500 

Actuaries can access industry sources to determine the effects of benefit level changes also (e.g. NCCI
publishes estimated industry effects of benefit level changes at the state level_.
Loss Development
Loss development adjusts immature losses to an estimated ultimate value.
A brief explanation of one commonly used method, the chain ladder method, is given below.
The chain ladder method assumes losses move from unpaid to paid in a consistent pattern over time (hence
historical loss development patterns can be used to predict future loss development patterns).
 The method can be performed separately on claim counts and losses to generate ultimate values of
each.
 The analysis can be done on various types of claims (e.g. reported, open, closed) and losses (e.g. paid
and reported), and to allocated loss adjustment expenses.
For most lines of business, developing reported losses including ALAE is used.
Loss development should be performed on a set of homogeneous claims.
 This can be a line of business or on a more granular level (e.g. coverages or types of losses within that
line of business).
 Liability claims and property claims are typically analyzed separately.
 Experience by geography (e.g. state) may also be analyzed separately where there is sufficient volume.
Extraordinary losses should be removed and the losses should be adjusted for any material benefit changes.
Claims data or loss data is organized in a triangle format as shown below:
In this example:
 Each row is a different AY.
 Columns represent each AYs reported losses at successive maturities (starting at 15 months and
increasing in annual increments).
 Losses are assumed to be at ultimate levels at 75 months (so no more columns are required), however
for other lines of business, ultimate may not be reached for many more years.
 Each diagonal represents a date as of which losses are evaluated (the valuation date) (e.g. the latest
diagonal represents a valuation date of 3/31/2008)

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Loss Development Triangle
Reported Losses ($000s) by AY Age (months)
Accident Year
15
27
39
51
63
75
2002
1,000
1,500
1,925
2,145
2,190
2,188
2003
1,030
1,584
2,020
2,209
2,240
2004
1,061
2,070
2,276
1,560
2005
1,093
1,651
2,125
2006
1,126
1,662
2007
1,159
The boxed value is the reported losses for accidents occurring in 2004 at 27 months of maturity (i.e. losses
paid and case reserves held as of 3/31/2006 for accidents occurring in 2004).
Prior to reviewing development patterns:
Review the magnitude of losses at first development age, 15 months, to see if loss levels at this early stage
are consistent from year to year, with consideration for loss trends and any changes in the portfolio.
i. If loss levels are different than expected, examine a similar triangle of claim counts to see if larger or
smaller than usual number of claims was reported for a particular AY.
ii. Inconsistent patterns at first development period may be expected for small portfolios or long-tailed lines
of business.
The development pattern is analyzed by taking the ratio of losses held at successive maturities (e.g. the link
ratio or the age-to-age development factor).
The following data triangle shows the link ratios for each accident year row as well as the:
 arithmetic average
 geometric average
 volume-weighted average (the ratio of total reported losses at successive maturities across all AYs)
Age-to-Age Development Factors
Accident Year
15 – 27
27 – 39
39 – 51
51 63
63 -74
2002
1.50
1.28
1.11
1.02
1.00
2003
1.54
1.28
1.09
1.01
2004
1.33
1.10
1.47
2005
1.51
1.29
2006
1.48
2007
-Arithmetic average
1.50
1.30
1.10
1.02
1.00
Geometric average 1.50
1.29
1.10
1.01
1.00
Ratio of total losses 1.50
1.29
1.10
1.02
1.00
Selected factor
1.50
1.30
1.10
1.02
1.00
The geometric average is the nth root of the product of n numbers.
The “ratio of total reported losses at successive maturities” compares the sums of an equal number of losses from each maturity (i.e.,
the most recent losses for the earlier maturity are not considered).

The boxed value shows that AY 2004 losses developed 47% (= 1.47 – 1.0) from age 15 months to age 27 months.

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Age-to-Age loss development factor (a-t-a LDF) selection:
The ratemaking actuary selects a suitable link ratio for each maturity (since the link ratios for each development
period are fairly consistent across the AYs, the all-year arithmetic average link ratios are selected).
A-t-A LDFs in practice may not be as stable as outlined above:
 If the ratemaking actuary believes patterns may be changing over time, the actuary may prefer to rely on
more recent development patterns, and select a two- or three-year average.
 If there is a desire to select based on the most recent data, but the line of business is to too volatile to
rely solely on a two- or three-year average, calculate weighted average link ratios giving more weight to
the more recent years.
 If A-t-A factors vary widely between AYs or there may be a strong anomaly in one or two AYs, consider
adjusted averages that eliminate the highest and lowest development factors from the calculation.
Loss Development:
 Reported losses develop upward as losses approach ultimate (due in part to the emergence of new
claims as well as adverse development on known claims).
 In some lines of business, development may be negative:
i. In auto physical damage coverages, an insurer may declare a vehicle a total loss (i.e. pay the total
limit for the car), take the damaged car, and sell it as scrap or for parts. The money received is
called “salvage” and is treated as a negative loss.
ii. When insurers pay losses for which another party is actually liable, it can approach the responsible
party for indemnification of those amounts (called subrogation).
Thus, when subrogation or salvage are common, or when early case reserves are set too high, age-toage development factors can be less than 1.00.
While this example assumes losses are ultimate at 75 months, for some lines of business, the historical data
triangle may not reach ultimate.
Here, actuaries may fit curves to historical development factors to extrapolate the development beyond the
patterns in the historical data.
A ‘tail factor’ accounts for additional development beyond that included in the standard chain ladder method.
Adjustments to Historical Data:
 Remove extraordinary losses from the historical data used to measure loss development patterns.
 Benefit or coverage changes may also distort loss development patterns.
i. Since benefit changes often affect policies prospectively, the effect of the change will first appear in a
new AY row.
ii. If the change impacts all claims occurring on or after a certain date, it is possible there will be a
change in the absolute amount of losses even though the development pattern is unaffected.
If it is not possible to restate the losses, any such distortions should be considered during the a-t-a ldf
selection process.
Next Step: Calculate age-to-ultimate development factors (a-t-u ldf) for each maturity.
 The a-t-u ldf is the product of each selected a-t-a ldf and the selected a-t-u ldf for subsequent maturities
(and the tail factor, if relevant).
 Example, a-t-u ldf for losses at age 51 months is the product of the selected age-to-age development
factors for 51-63 months and 63-75 months (1.02 x 1.00).

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BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Next Step: Apply the a-t-u ldfs to the reported losses at the most recent period of development (the latest diagonal in
the reported loss triangle) to yield estimated ultimate losses for each AY as shown below:
Adjusting Reported Losses to Ultimate
(1)
(2)
(3)
(4) = (2)*(3)
Accident
Reported
Age-toEstimated
Year Age
Losses
Ultimate
Ultimate
Accident (Months a/o)
($000s) Development
Losses
Year
3/31/08)
a/o 3/31/08
Factor
($000s)
2002
75
$2,188
1.00
$2,188
2003
63
$2,240
1.00
$2,240
2004
51
$2,276
1.02
$2,322
2005
39
$2,125
1.12
$2,380
2006
27
$1,662
1.46
$2,427
2007
15
$1,159
2.19
$2,538
Total
$11,650
$14,095
The chain ladder method is only one method for calculating loss development, and assumes that historical
emergence and payment patterns are indicative of patterns expected in the future.
Changes in (claims handling methodology or philosophy) or ( dramatic changes in claims staffing) may result in
claims being settled faster or slower than historical precedents, and would violate the basic assumption of the
chain ladder method.
Other methods to develop losses to ultimate:
 The Bornhuetter-Ferguson (B-F) method incorporates a priori assumptions of the expected loss ratio in
order to calculate ultimate losses and consequently the outstanding reserve at a point in time (see
Appendix C)
 The Berquist-Sherman (BS) method is used when an insurer has experienced significant changes in
claim settlement patterns or adequacy of case reserves that would distort development patterns.
The method produces adjusted development patterns estimated to be consistent with the reserve
levels and settlement rates present as of the last diagonal by restating historical development data.
 Stochastic methods (e.g. the Mack method) study variability around loss development so actuaries can
better understand the risk of adverse development.
These methods are covered in more detail in literature regarding loss reserving methodologies.
Loss Trend
It is necessary to adjust the losses for trends expected to occur between the historical experience period and
the period for which the rates will be in effect (in addition to projecting historical losses to an ultimate level).
Changes in frequency and severity are referred to as loss trends, and available data to estimate the loss
trends should be used to project historical losses.

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BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Loss Trend Selections
1. Monetary inflation, increasing medical costs, and advancements in safety technology are examples of
factors that can drive loss trends.
2. Social influences also impact loss costs.
ASOP 13, Trending Procedures in P&C Insurance Ratemaking defines social influences as “the impact on
insurance costs of societal changes such as changes in claim consciousness, court practices, and legal
precedents, as well as in other non-economic factors.”
3. Distributional changes in a book of business also affect frequencies and severities (e.g. if the proportion of
risky policies is growing, loss costs will be expected to increase).
Loss Trend Measurement
Actuaries measure loss trend by fitting curves to historical data.
Frequency and severity are analyzed separately to better understand the drivers of the trend (in addition to
analyzing pure premium data).
If an insurer heavily markets a higher deductible, the resulting shift in distribution will lower frequencies but
is likely to increase severities (which is difficult to detect in a pure premium analysis).
The years chosen to review is based on the actuary’s judgment (considering responsiveness and stability).
 Influences (e.g. the cyclical nature of insurance and random noise) may be difficult to eliminate from
the trend analysis.
 The actuary should, however, adjust the trend data for more easily quantifiable (e.g. seasonality and
the effect of benefit level changes)
Different lines of business call for different or multiple views of the losses for analyzing trend.
i. In stable, short-tailed lines of business (e.g., automobile physical damage), the actuary typically analyzes
CY paid losses for the 12 months ending each quarter.
CY data is readily available, the paid loss definition eliminates any distortion from changes in case
reserving practices, and the use of 12-month rolling data attempts to smooth out the effect of seasonality.
ii. In more volatile and long-tailed line of business (e.g. WC medical) analyze the trend in AY reported losses
that have already been developed to ultimate and adjusted for benefit changes.
Perform a trend analysis on a set of homogeneous claims:
i. Separate indemnity and medical losses within WC insurance.
ii. Analyze liability claims and property claims separately.
iii. Analyze experience by geography (e.g. state) separately.
Types of trend measurement:
Linear and exponential regression models are the most common methods used to measure the trend.
 Linear models result in a projection that increases by a constant amount for each unit change in the
ratio measured (e.g. claim severities).
A linear model will eventually project negative values when measuring decreasing trends, and since a
negative frequency or severity does not occur in insurance, this is a shortcoming of linear trend
models.
 Exponential models produce a constant rate of change in the ratio being measured.

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BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
The following shows the result of an exponential curve fit to different durations of CY paid frequency, severity,
and pure premium data for the 12 months ending each quarter.
Exponential Loss Trend Example
Year
Ending
Quarter
Mar-09
Jun-09
Sep-09
Dec-09
Mar-10
Jun-10
:::
Sep-13
Dec-13

Earned
Exposure
131,911
132,700
133,602
135,079
137,384
138,983
:::
141,800
142,986

Closed
Claim
Count
7,745
7,785
7,917
7,928
7,997
8,037
:::
7,755
7,778

Paid
Losses
$8,220,899
$8,381,016
$8,594,389
$8,705,108
$8,816,379
$8,901,163
:::
$8,702,135
$8,761,588

Annual
%
Frequency Change
0.0587
-0.0587
-0.0593
-0.0587
-0.0582
-0.9%
0.0578
-1.5%
:::
:::
0.0547
-0.7%
0.0544
-0.9%

Severity
$ ,061.45
$ 1,076.56
$ 1,085.56
$ 1,098.02
$ 1,102.46
$ 1,107.52
:::
$ 1,122.13
$ 1,126.46

Annual
%
Change
----3.9%
2.9%
:::
2.3%
3.0%

Pure
Premium
$ 62.32
$ 63.16
$ 64.33
$ 64.44
$ 64.17
$ 64.04
:::
$61.37
$ 61.28

Annual
%
Change
----3.0%
1.4%
:::
1.5%
2.1%

Number of
Frequency
Severity
Pure Premium
Points
Exponential Fit
Exponential Fit
Exponential Fit
20 point
-1.7%
0.5%
-1.2%
16 point
-1.3%
-0.1%
-1.4%
12 point
-0.7%
-0.2%
-0.9%
8 point
-1.2%
1.2%
-0.1%
6 point
-0.9%
2.5%
1.6%
4 point
-1.5%
3.3%
1.9%
As shown above, separate exponential models may be fit to the whole of the data and to more recent periods.
If separate frequency and severity trends are selected, these are used to compute a pure premium trend
(e.g. a -1% selected frequency trend and a +2% selected severity trend produce a +1%
(= (1.0 - 1%) x (1.0 + 2%) - 1.0) pure premium trend.
Exclude catastrophe losses from the loss trend analysis data.
Changes in benefit levels can affect trend analyses. Therefore, if the historical data to which loss trends will be
applied is restated to reflect the new benefit level, then either:
 data adjusted for benefit level should be used for the trend analysis, or
 the trend analysis must remove the impact of the benefit level change.
Care must be taken not to “double count” the benefit level change in the projected losses.
Is the historical data is overly volatile or inappropriate for trending purposes? For example:
 the data may be too sparse or reflect non-recurring events that cannot be appropriately adjusted.
 the statistical goodness of fit of the trending procedure may be called into question.
Circumvent the problem by:
 supplementing the loss trend data with multi-state, countrywide, or industry trend data and consider
weighting the results.
 consider non-insurance indices (e.g. the medical component of the CPI (Consumer Price Index) may
be relevant when selecting severity trends for products related to medical expense coverage.
Also, more sophisticated techniques (e.g. econometric models and generalized linear models) may be
employed for quantifying loss trends.

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Loss Trend Periods
The loss trend period is the period of time from the average loss occurrence date of each experience period
(often a calendar-accident year, CAY) to the average loss occurrence date for the period in which the rates will
be in effect (i.e. the forecast period, which is a policy year or years).
The average loss occurrence date depends on the policy term and the duration the new rates will be in effect.
Assume the following:
• The losses to be trended are from AY 2011.
• The company writes annual policies.
• The proposed effective date is January 1, 2015.
• The length of time the rates are expected to be in effect is one year.
The average loss occurrence date of CAY 2011 (called the “trend from” date) is 6/30/2011.
The average accident date for PY 2011 is 12/31/2011, as polices are in effect over a 24-month period.
The average loss occurrence date during the forecast period (called the “trend to” date) is 12/31/2015.
This is because last policy to be written will be on 12/31/2015, and losses can continue to occur until
12/31/2016, so the midpoint of that two-year time period is 12/31/2015.
Thus, the trend period for CAY 2011 is 4.5 years.

The pure premium trend (+1%) is applied to CAY Year 2011 losses by multiplying the historical losses by
(1.01)4.5 (which is the trend factor).
If the policy term were semi-annual, the “trend from” date would not change, but the “trend to” date would
be different.
Coverage for policies written between 1/1/2015 and 12/31/2015 would extend over an 18-months, of which
the midpoint would be 9 months (i.e. 9/302/015). The trend length would be 4.25 years as shown below.
Loss Trend Period for 6-month Policy Term

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
If data were aggregated by PY:
 the average loss occurrence date for an annual policy term would be one year after the start of the PY, as
policies are in effect over 24-months.
 the “trend to” date is the average loss occurrence date for the PY in which rates will be in effect.
Therefore, the trend period for PY 2011 annual term policies is 4 years (1/1/2012 to 12/31/2015), as
shown below.
Loss Trend Period for 12-month Policy Term and PY experience period

The PY2011 trend factor, applied to PY 2011 losses, is 1.0406 ( = 1.014.0).
Exhibit 6.18 (below) shows the same PY scenario but with semi-annual policies.
 Both the “trend from” and “trend to” dates are 3 months earlier than the annual policy scenario since the
average occurrence date for semi-annual policies is 9 months after the start of the PY.
 Thus, the trend length remains the same as in the annual policy scenario and is still 4 years.
Loss Trend Period for 6-month Policy Term and PY experience period

If the trend selection is based on a linear trend, the selected trend is a constant amount rather than a %.
 The projected dollar change = (the selected annual trend) * (the length of the trend period).
 Assuming the selected annual pure premium linear trend is $1.00 per year, then the dollar increase due to 4
years of trend is $4.00 (= $1.00 x 4.0).
The actuary may choose to undertake a two-step trending process.
 This is beneficial when the trend in the historical experience period and the expected trend for the
forecast period are not equal.
 For example, legislative changes in the trend data call for a 2-step trending process if the trend
exhibited in the historical period is clearly different from that expected in the future.

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BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
In the exponential trend data shown above, historical severity trend exhibits a different pattern in more recent
periods than in earlier years.
 The losses in the experience period are trended from the average accident date in the experience
period to the average accident date of the last data point in the trend data. Example:
The average loss occurrence date of CAY 2011 is 6/30/2011. If the last data point in the loss trend data
is the 12 months ending fourth quarter 2013, the average accident date of that period is 6/30/2013.
If the selected step 1 trend is -1%, the factor to adjust CAY 2011 losses to the end of the experience
period is 0.98 (= (1.0 - 1%)2 ).
 Next, these trended losses are projected from the average accident date of the last data point in the
trend data (the “project from” date of 6/30/2013) to the average loss occurrence date for the forecast
period (the “project to” date of 12/31/2015). The length of this projection period is 2.5 years.
If the trend selection is 2%, step 1 trended losses are adjusted by a factor of 1.05 (= (1.0 + 2%) 2.5).
Two-Step Trend Periods for 12-month Policy

When using CY data to measure loss trend, it is assumed that the book of business is not significantly
increasing or decreasing in size. Problems with this assumption are:
 claims (or losses) in any CY may have come from older AYs, but are matched to the most recent CY
exposures (or claims).
 a change in exposure levels causing changes in the distribution of each CY’s claims by accident year.
The solution is to match the risk with the appropriate exposure.
1. Use econometric techniques or generalized linear models to measure trend, which will absorb changes in
the size of the portfolio as well as changes in the mix of business.
2. Measure the trend using AY data (in lieu of CY data). The AY losses (or claim counts) need to be developed
to ultimate before measuring the trend, which introduces subjectivity into the trend analysis.
3. Analyze the trend in incremental CY frequencies or severities.
Assume CY 2010 has paid losses on claims from AYs 2010, 2009, and 2008.
i. CY 2010 frequency is the sum of all [paid claim counts in CY 2010/ CY 2010 exposures].
ii. Alternatively, CY 2010 frequency is the sum of the following three incremental CY 2010 frequencies:
• [CY 2010 paid claim counts from AY 2010 / CY 2010 exposures]
• [CY 2010 paid claim counts from AY 2009 / CY 2009 exposures]
• [CY 2010 paid claim counts from AY 2008 / CY 2008 exposures]
The alternative method properly matches older claim counts to older exposures and is valid whether the
portfolio is changing or not.

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Leveraged Effect of Limits on Severity Trend
When loss experience is subject to limits, consider the leveraged effect of those limits on the severity trend.
Basic limits losses are losses that have been censored at a limit referred to as a “basic limit.”
Total limits losses are losses that are uncensored
Excess limits losses are the portion of the losses that exceed the basic limit (or the difference between total
limits and basic limits losses). It is important to understand that severity trend affects each of these differently.
Consider the following simple example in which every total limits loss is subject to a 10% severity trend.
Effect of Limits on Severity Trend
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
Trended Losses
Total
Losses
Capped @ $25,000
Capped
@
Claim
Limits
Excess
Total Limits
Excess Losses
Number
Loss
$25,000
Losses
Loss
Trend
Loss
Trend
Loss
Trend
1
$10,000 $10,000
$ $11,000
10.0% $11,000
10.0% $N/A
2
$15,000 $15,000
$$16,500
10.0% $16,500
10.0% $N/A
3
$24,000 $24,000
$ $26,400
10.0% $25,000
4.2% $1,400
N/A
4
$30,000 $25,000
$ 5,000 $33,000
10.0% $25,000
0.0% $8,000
60.0%
5
$50,000 $25,000
$25,000 $55,000
10.0% $25,000
0.0% $30,000 20.0%
Total
$129,000 $99,000
$30,000 $141,900
10.0% $102,500
3.5% $39,400 31.3%
(2)=min [(1), $25,000] (3) = (1) - (2)
(4) = (1) x 1.10 (5) = (4) / (1) - 1.0
(6)=min [ (4) , $25,000]
(7)= (6) / (2) - 1.0
(8) = (4) - (6)
The 10% trend in total limits losses affects basic limits losses and excess losses differently.
Basic Limits:
The 10% total limit trend is reduced to 3.5% when considering the basic limits losses.
 The two smallest losses (Claims 1 and 2) are well below the $25,000 limit before and after the 10%
increase.
 Claim 3 was below $25,000 before trend was applied, but above the basic limit after applying trend.
 Claims 4 and 5 were already in excess of $25,000, so the amount of loss under the limit is the same
before and after trend.
Excess Limits:
The impact of positive trend on excess losses is greater than the total limits trend.
 Claims 1 and 2 are significantly below the limit and do not impact the trend in the excess layer.
 Claim 3 was below $25,000 before trend was applied, but above the basic limit after applying trend.
 Since claims 4 and 5 were already higher than the basic limit, the entire increase in losses associated
with these claims is realized in the excess losses trend.

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Effect of Limits on Severity Trend
Initial Loss Size
Basic Limits
Trend
Limit

Loss 

Total Losses
Trend

Excess Losses
Undefined

Limit
 1.0
Loss

Trend

Undefined

Trend

[ Lossx(1.0  Trend )]  Limit
Loss  Limit

1.0  Trend

Limit
 Loss  Limit
1.0  Trend
Limit  Loss

Given positive trend, then Basic Limits Trend  Total Limits Trend  _ Excess Losses Trend.
Given negative trend, then Excess Losses Trend  Total Limits Trend  Basic Limits Trend.
Final notes:
 If severity trends are analyzed on total limits loss data, the indicated trend must be adjusted
before it is applied to basic limits losses for ratemaking purposes.
 Alternatively, use basic limits data in analyzing severity trend.
 Deductibles also have a leveraging effect on severity trend. The mathematics is analogous to
excess losses except that the censoring is done below the deductible rather than above the limit.
Coordinating Exposure, Premium, and Loss Trends
It is important to make sure that all components of the formula are trended consistently.
When deriving a pure premium rate level indication, three types of trends that are considered are:
 changes in the likelihood of a claim happening,
 changes in the average cost of claims, and
 changes in the level of exposure.
When the insurer’s internal frequency and severity trend data is used as the basis of the loss trend, changes in
frequency (i.e.# of claims / exposure) account for the net effect of (1) the change in the probability of having a
claim and (2) the change in exposure. This also holds when analyzing pure premium data.
When using inflation-sensitive exposure bases, the inflation on the exposure can mask part or all of the change
in the likelihood of claims occurring.
To remove the effect of the changing exposure, examine historical frequencies (or pure premiums) that have
been adjusted for exposure trend (i.e. the denominator has been adjusted by the exposure trend).
When deriving a loss ratio indication, examine patterns in historical adjusted loss ratios.
 This is the ratio of losses adjusted for development, benefit changes, and extraordinary losses
compared to premium adjusted to current rate level. This produced a “net” trend.
 Based on the pattern in adjusted loss ratios, the actuary selects a loss ratio trend to adjust the historical
loss ratios to the projected policy period.
 One shortcoming of this approach is that trends in adjusted loss ratios over time may not be stable, and
it can be more difficult to understand what may be driving the results.

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It may be preferable to examine the individual components of the loss ratio statistic (i.e. frequency, severity, and
average premium) and adjust each component to get a better understanding of how each individual statistic is
changing and therefore how the entire loss ratio statistic is changing.
Insurers may use external indices to select loss trends (e.g. a WC insurer may use an external study as the
basis to estimate the expected increase in utilization and cost of medical procedures)
 However, the loss trend selection does not implicitly account for any expected change in the insurer’s
premium or exposure due to an inflation-sensitive exposure base.
 Thus, the exposure or premium needs to be adjusted to reflect any expected change in exposure.
Appendices A-F highlight some of the different approaches.
 The auto and homeowners examples do not have inflation-sensitive exposure bases and use internal
trend data, however, the homeowners example does include a projection of the amount of insurance
years, which is necessary for the projection of the non-modeled catastrophe loading.
 The medical malpractice loss ratio example includes a net trend approach. Trend selections are made
using internal data. Since the “frequency” is number of claims divided by premium, the frequency
selection accounts for pure frequency trend as well as premium trend.
 The WC example separately applies loss and exposure trend.
Overlap Fallacy: Loss Development and Loss Trend
Trending restates past losses to the level expected during the future period due to inflation and other factors.
Loss development brings immature losses to their expected ultimate level.
While it is true that loss development incorporates inflationary pressures that cause payments for reported
claims to increase over time, this does not prove overlap.
The timeline below shows how losses are trended and developed.

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BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Given the following:
 The historical experience period is CAY 2010.
 The average date of claim occurrence is 7/1/2010.
 Assume it is typical for claims to settle within 18 months, so this “average claim” will settle on
12/31/2011.
 The projection period is the policy year beginning 1/1/ 2012 (i.e. rates are expected to be in effect for
annual policies written from 1/1/2012 – 12/31/2012).
 The average hypothetical claim in the projected period will occur on 1/1/2013, and settle 18 months
later on 6/30/2014 (i.e. consistent with the settlement lag of 18 months).
Key comments:
Trend adjusts the average historical claim from the loss cost level that exists on 7/1/2010 to the loss cost level
expected on 1/1/2013.
Development adjusts the trended, undeveloped claim to the ultimate level, expected to occur by 6/30/2014.
This 48 month period represents 30 months of trend to adjust the cost level to that anticipated
during the forecast period and the 18 months of development to project this trended value to its
ultimate settlement value.

Loss Adjustment Expenses

121 – 122

LAE are all costs incurred by a company during the claim settlement process.
LAE have been divided into two categories:
 Allocated loss adjustment expenses (ALAE) are costs that can be related to individual claims (e.g. legal
fees to defend against a specific claim or costs incurred by a claim adjuster assigned to one claim)
 Unallocated loss adjustment expenses (ULAE) are those that are more difficult to assign to particular
claims (e.g. claim department salaries).
In 1998, the insurance industry introduced new LAE definitions; costs are now split into defense cost and
containment (DCC) expenses and adjusting and other (A&O) expenses.
 DCC expenses include costs incurred in defending claims, including expert witness fees and other
legal fees.
 A&O include all other expenses.
Despite the change in U.S. financial reporting definitions, this text will refer to the subdivisions of ALAE
and ULAE, which are more commonly used in ratemaking.

In general, ALAE or DCC vary by the dollar amount of each claim, while ULAE or A&O vary by the number of
claims reported.
 ALAE are often included with losses for ratemaking purposes (e.g. for loss development and trend).
 In commercial lines, actuaries often study development and trend patterns separately for loss and ALAE,
when ALAE are significantly high or in order to detect any changes in ALAE patterns.
 Is ALAE subject to the policy limits or not? This does not affect the treatment of ALAE in a ratemaking
context, but it emphasizes the need to understand whether the ALAE data retrieved is the entire ALAE or
only the portion included within the policy limits.

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ULAE are more difficult to incorporate into the loss projection process.
Assume ULAE expenditures track with loss dollars consistently over time, both in terms of rate of payment and
in proportion to the amount of losses paid.
Calculate the ratio of CY paid ULAE to CY paid loss plus ALAE over several years (e.g. three years or longer,
depending on the line of business).
 This ratio is applied to each year’s reported loss plus ALAE to incorporate ULAE.
 The ratio is calculated on losses that have not been adjusted for trend or development as this data is
readily available for other financial reporting.
 The resulting ratio of ULAE to loss plus ALAE is then applied to loss plus ALAE that has been adjusted
for extraordinary events, development, and trend.
ULAE Ratio
(1)
(2)
(3)
Calendar
Paid Loss
ULAE
Year
And ALAE
Paid ULAE
Ratio
2008
2009
2010
Total
(3) = (2) / (1)

$ 913,467
$1,068,918
$1,234,240
$3,216,625

$144,026
$154,170
$185,968
$484,164
(4) ULAE Factor
(4) = 1.0 + (Tot3)

15.8%
14.4%
15.1%
15.1%
1.151

Catastrophic events can cause extraordinary loss adjustment expenses (e.g. a company setting up temporary
offices in the catastrophe area).
 Since these costs are significant and irregular, the historical ratio will be distorted
 Thus cat LAE are generally excluded from the standard ULAE analysis and are determined as part of the
catastrophe provision.
The method described above is a dollar-based allocation method. Other allocation methods are:
 Count-based allocation methods that assume the same kinds of transactions cost the same amount
regardless of the dollar amount of the claim, and that there is a cost associated with a claim remaining
over time.
 Time studies showing how claim adjusters spend their time working on what types of claims, what
types of claim activities, lines of business, etc.

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BASIC RATEMAKING – WERNER, G. AND MODLIN, C.

Key Concepts

122 - 123

1. Loss definitions
a. Paid loss
b. Case reserves
c. Reported loss
d. Ultimate loss
2. Loss aggregation methods
a. CY
b. Calendar-accident year
c. Policy year
d. Report year
3. Common ratios involving losses
a. Frequency
b. Severity
c. Pure premium
d. Loss ratio
4. Extraordinary losses
5. Catastrophe losses
a. Non-modeled catastrophes
b. Modeled catastrophes
6. Reinsurance recoveries and costs
7. Changes in coverage or benefit levels
8. Loss development

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BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
The predecessor papers to the current syllabus reading “Basic Ratemaking” by Werner,
G. and Modlin, C. were numerous. While past CAS questions were drawn from prior
syllabus readings, the ones shown below remain relevant to the content covered in this
chapter.

Section 1: Loss Trending and Loss Development
Questions from the 1996 Exam:
Question 30. (4 points) You are given:
Wisconsin Personal Automobile Bodily Injury
20/40 Basic Limits
Calendar/
Accident
Year
1992
1993
1994

Ultimate
Loss &
ALAE
325,000
575,000
800,000

Written
Premium
750,000
1,000,000
1,250,000

Earned
Premium
375,000
875,000
1,125,000

Combined

1,700,000

3,000,000

2,375,000

Rate Level History
Effective
% Rate
Date
Change
1/1/91
+7.0%
10/1/93
+5.0%
7/1/94
+3.0%
1/1/95

+5.0%

• Target Loss and ALAE ratio
69.0%
• Countrywide 20/40 Indicated
+5.0%
• Proposed effective date
1/1/96
• The filed rate will remain in effect for one year.
• All policies are annual.
• Annual 20/40 severity trend
5.0%
• Annual 20/40 frequency trend -1.0%
• Statewide credibility
50.0%
Using the techniques described by McClenahan, "Ratemaking," Foundations of Casualty Actuarial Science:
(a) (2 points) Calculate the on-level earned premium for the experience period 1992-1994.
(b) (1 point) Calculate the trended on-level loss and ALAE ratio for the experience period 1992-1994.
(c) (1 point) Calculate the indicated rate level change for Wisconsin.
Question 36. (3 points)
Rate
Implementation
Change
Date
Type of Change
+8%
5/1/94
Experience
+15%
7/1/95
Law Amendment
-10%
7/1/95
Experience
+5%
4/1/96
Experience
• Policies are written uniformly throughout the year.
According to Feldblum, "Workers' Compensation Ratemaking:"
(a) (2 points) Calculate the premium adjustment factor to bring policy year 1995 premium to current rate level.
(b) (1 point) How are experience rate changes and law amendment rate changes different in their
purpose and their effect?

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 1997 Exam:
44. (4 points) You are given:
Calendar/Accident
Year

Reported Loss and ALAE

Earned Exposures

1993
1994
1995

1,800,000
2,275,000
1,975,000

2,500
2,900
3,400

Losses are evaluated as of 12/31/96
Loss (incl. ALAE) Development Factors:
12 months to ultimate
24 months to ultimate
36 months to ultimate
48 months to ultimate

LDFs
1.500
1.250
1.050
1.000

• Annual severity trend = +4.3% (trend is exponential)
• Annual frequency trend = -2.0% (trend is exponential)
• Commission = 14.0%
• Taxes = 3.0%
• Variable portion of General and Other Acquisition = 10.0%
• Total fixed expense = $30 per exposure
• Profit load = 3.0%
• All policies are annual
• Filed rates will be in effect for one year
• Proposed effective date for the rate change is 10/1/97
Using the methodology in McClenahan, "Ratemaking," of Foundations of Casualty Actuarial Science,
A. (2 points) Determine the developed and trended Loss and ALAE by accident year (chapter 6)
B. (1 point) Determine the indicated pure premium (chapter 8)
C. (1 point) Determine the indicated gross rate (chapter 8)

Questions from the 1999 exam
39. (2 points) McClenahan in "Ratemaking," chapter 2 of Foundations of Casualty Actuarial Science,
discusses the effects of limits on severity trend. Use the information shown below to determine the
one-year severity trend for the loss amounts in the following three layers of loss:
$0-$50
$50-$100
$100-$200


Losses occur in multiples of $40, with equal probability, up to $200, i.e., if a loss occurs, it has an
equal chance of being $40, $80, $120, $160, or $200.



For the next year, the severity trend will uniformly increase all losses by 10%.

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2000 exam
40. (4 points) Using the techniques described by McClenahan in "Ratemaking," chapter 2 of Foundations of
Casualty Actuarial Science, and the following data, answer the questions below.
You are given the following information for your company's homeowners business in a single state:
Calendar/
Ultimate Loss
Accident Year
and ALAE
Written Premium
Earned Premium
1997
635,000
1,000,000
975,000
1998
595,000
1,050,000
1,000,000
Effective Date
July 1, 1996
January 1, 1998
July 1, 1999

Rate Change
+4.0%
+1.8%
+3.0%

Target Loss and ALAE Ratio
Proposed effective date
Effective period for rates
Credibility
Alternative indication
Policy period
Severity trend
Frequency trend

0.670
July 1, 2000
One year
0.60
0.0%
Twelve months
+3.0%
+1.0%

Questions from the 2001 exam
Question 2. Based on McClenahan, “Ratemaking,” chapter 2, Foundations of Casualty Actuarial Science, and
the following information, answer the question below.
Assume:


Experience period is accident year 1999.



Indicated rates will become effective July 1, 2001.



The next scheduled rate increase is expected to become effective April 1, 2002.



All policies are expected to have an 18-month period.



There are no seasonal effects on the frequency of accidents.



Policies are evenly written throughout the year.

How many months are there between the midpoint of the experience period and the midpoint of the
exposure period?
A. < 22 months B. >22 months but < 28 months C. > 28 months but < 34 months
D. > 34 months but < 40 months E. > 40 months

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BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2002 exam
17. (4 points) Based on McClenahan, "Ratemaking," chapter 2 of Foundations of Casualty Actuarial
Science, and the following data, answer the questions below. Show all work.
Projected rates to be effective January 1, 2003 and in effect for 1 year.
Target loss and ALAE ratio is 65%.
Experience is from the accident period January 1, 2000 to June 30, 2001.
Developed accident period loss and ALAE is $21,500.
Annual trend factor is 3%.
All policies have one-year terms and are written uniformly throughout the year.
The rate on January 1, 1999 was $120 per exposure.
Effective Date
January 1, 2000
January 1, 2001
Year
1998
1999
2000
2001

Rate Change
+10%
-15%
Written Exposures
200
200
200
200

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BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2003 exam:
12. Given the following data and using the loss development method as described by McClenahan in
Foundations of Casualty Actuarial Science, calculate the projected ultimate accident year 2001 losses.
As of December 31, 2002
Accident Year
Paid Losses
Case Reserves
1999
$11,000
$1,000
2000
$6,000
$2,000
2001
$3,500
$4,000
2002
$1,000
$4,000


Projected ultimate accident year 2000 losses = $9,240



12-24 case-incurred link ratio = 1.71



24-36 case-incurred link ratio = 1.20

A. < $8,700
B.  $8,700, but < $9,200
D.  $9,700, but < $10,200

C.  $9,200, but < $9,700
E.  $10,200

Questions from the 2004 exam:
7. Given the following data, calculate the trended loss ratio.
Number of
Insureds
20
•
•
•
•
•
A. < 68%

Earned
Premium
$50,000

Developed
Incurred
Losses
$35,000

Years of Trend = 2.5
Annual Exposure Trend = 2.0%
Annual Premium Trend = 2.9%
Annual Frequency Trend = -1 .0%
Annual Severity Trend = 6.0%
B. > 68% but < 71%

C. > 71 % but < 74%

D. > 74%, but < 77%

E. > 77%

8. Which of the following statements are true regarding loss trends?
1. When an exponential curve is used to approximate severity, the assumption is a constant
multiplicative increase in severity.
2. This original statement no longer applies to the content in this chapter
3. Linear trends tend to underestimate future costs when inflation is increasing at a multiplicative rate.
A. 1 only
B. 3 only
C. 1 and 2 only
D. 1 and 3 only
E. 2 and 3 only

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2004 exam (continued):
37. (5 points) Given the information below, answer the following questions. Show all work.
Case-Incurred Losses
Accident Year
Age 12
Age 24 Age 36 Age 48
2000
$1,412
$1,816 $1,993 $1,993
2001
$1,624
$2,023 $2,137
2002
$1,841
$2,271
2003
$2,421


Ultimate losses are reached at age 48.

The annual frequency trend is -2%.



The annual severity trend is 8%.

Planned effective date of rate change is July 1, 2004.

 Rates are reviewed annually.
Policies have a term of 12 months.
a. (1 point) Calculate the age-to-ultimate development factor for accident year 2003 as of December 31,
2003. Explain your assumptions.
b. (0.5 point) Calculate the ultimate loss amount for accident year 2003.
c. (1 point) Calculate the trended ultimate loss amount for accident year 2003.
d. (1.5 points) Briefly describe three causes of loss development.
e. (1 point) Briefly explain why it is appropriate to both trend and develop losses (i.e. why there is no
overlap).

Questions from the 2007 exam
22. (1.5 points) The claims department of an insurance company has historically set an initial case
reserve of $10,000 for each liability claim at the time the claim is opened. If the claim is not closed
within 18 months, the case reserve is adjusted to an appropriate level based on the characteristics of
the claim. Starting with accidents occurring January 1, 2006 and later, the initial case reserve was set
at $5,000 for each liability claim. The actuarial department was not made aware of this change.
Assume incurred loss data for accident year 2006, valued as of December 31, 2006, is used to derive
rates effective July 1, 2007. Explain the impact of this change on incurred loss development and rate
adequacy for this liability line of insurance.

Questions from the 2008 exam
17. (2.0 points) Given the following payment and reserve data about 2 different claims on 2 different policies:
Policy Effective Date
Date of Loss
Transaction Date
Payment
Case Reserve
July 1, 2006
December 1, 2006 December 1, 2006
$0
$5,000
March 1, 2007
$500
$3,500
$2,000
October 1, 2007
$3,500
March 1, 2008
$3,000
$0
October 1, 2006

March 1, 2007

March 1, 2007
October 1, 2007
March 1, 2008

$5,000
$9,000
$1,000

$10,000
$1,000
$0

a. (0.5 point) Calculate the calendar-year incurred losses for 2006 and 2007.
b. (0.5 point) Calculate the accident-year incurred losses for 2006 and 2007 evaluated as of 12/31/2008.
c. (0.5 point) Calculate the policy-year incurred losses for 2006 and 2007 evaluated as of 12/31/2008.
d. (0.5 point) Identify one advantage and one disadvantage associated with using policy year incurred losses
for ratemaking.

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BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2009 exam
22. (2 points) an insurance company started writing annual policies in 2005. Given the following
information for claims associated with policies written in 2005:
Accidents Occurring in 2005
Accidents Occurring in 2006
Calendar Payments
Reserve @
Calendar Payments
Reserve @
Year
End of Year
Year
End of Year
2005 $ 1,000,000
$500,000
2005
$
$
2006 $ 300,000
$300,000
2006
$ 1,500,000 $ 1,000,000
2007 $ 250,000
$100,000
2007
$ 700,000 $ 200,000
2008 $ 50,000
$
2008
$ 100,000 $ 50,000
a. (0.5 point) Calculate the calendar year losses for 2006.
b. (0.5 point) Calculate the accident year incurred losses for 2006 evaluated as of December 31, 2007.
c. (0.5 point) Calculate the policy year incurred losses for 2005 evaluated as of December 31, 2008.
d. (0.5 point) Provide one advantage and one disadvantage associated with using calendar year
incurred losses rather than accident year incurred losses for ratemaking.
24. (1 point) Fully discuss why it may be inappropriate to apply a basic limits loss trend to total limits losses.
27. (1 point Fully discuss the "overlap fallacy" between trend and loss development.
42. (1 point) For homeowners insurance explain two reasons that hurricane rates should be priced separately
from non-hurricane rates.

Questions from the 2010 exam
20. (2 points) Given the following claim activity on an annual policy effective on December 29, 2006:
Claim
Number
1
1
1
1
1
2
2
2

Incremental
Transaction Date
Payment
December 31, 2006
December 31, 2006
October 5, 2007
July 5, 2008
January 25, 2009 $ 30,000
April 1, 2007
April 5, 2007
July 1, 2008

Case Reserve as
Of Transaction
Date
$1,000
$ 10,000
$ 25,000
$$ 25,000
$-

Transaction Description
Claim occurred
Claim reported and reserve established
Case reserve increased
Case reserve increased
Settlement made, Payment made, Claim closed
Claim occurred
Claim reported and reserve established
Claim closed without payment

a. (0.5 point) Calculate 2008 calendar year reported losses.
b. (0.5 point) Calculate 2006 accident year reported losses evaluated as of December 31, 2007.
c. (0.5 point) Calculate 2006 policy year reported losses evaluated as of December 31, 2007.
d. (0.5 point) Briefly describe one advantage and one disadvantage of using calendar year losses as
compared to accident year losses in a ratemaking application.

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2010 exam
21. (2 points) Identify four adjustments made to historical losses in projecting losses for a future policy
period for ratemaking. Briefly describe the purpose of each.
24. (1 point) Given the following countrywide calendar year information:
Calendar
Year
2006
2007
2008
2009

Earned
Premium
$696,667
$733,333
$805,673
$907,725

Paid Loss
$475,000
$500,000
$498,750
$518,700

Paid ALAE
$47,500
$50,000
$24,938
$25,935

Paid Loss
and ALAE
$522,500
$550,000
$523,688
$544,635

Paid ULAE
$26,125
$55,000
$52,369
$54,464

Select a ULAE factor to be applied to the statewide incurred losses and paid ALAE as part of
calculating statewide rate indications. Explain your selection.

Questions from the 2011 exam
6. (2.5 points) Given the following information for claims associated with annual homeowners policies
written in 2007:
Claim
Accident
Report
Transaction
Loss
Case Reserve
Number
Year
Year
Date
Payment
Balance
1
2007
2007
April 1, 2007
$100
$300
1
2007
2007
July 1, 2008
$200
$600
1
2007
2007
June 1, 2009
$500
$0
2
2007
2008
May 1, 2008
$500
$200
2
2007
2008
July 1, 2009
$200
$0
3
2008
2008
August 1, 2008
$50
$200
3
2008
2008
March 1, 2009
$100
$50
3
2008
2008
July 1, 2010
$200
$0
a. (0.5 point) Calculate the calendar year 2008 incurred losses.
b. (0.5 point) Calculate the accident year 2008 incurred losses, evaluated at December 31, 2009.
c. (0.5 point) Calculate the policy year 2007 incurred losses, evaluated at December 31, 2009.
d. (0.5 point) Calculate the report year 2008 incurred losses, evaluated at December 31, 2009.
e. (0.5 point) Briefly describe one advantage and one disadvantage associated with using policy year
losses for ratemaking.
7. (1 point) Fully explain the overlap fallacy between loss development and loss trend.

17. (1 point) Given the following data:
Claim
Number
Loss Amount
1
$10,000
2
$15,000
3
$30,000
4
$35,000
• Basic limit = $25,000
• Total limits severity trend = 10%
Calculate the excess loss trend.

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2012 exam
7. (5.75 points) An actuary is preparing a rate filing in a state that requires full supporting documentation
of the rate level indication. The actuary is given the following information:






A single trend percentage is used to trend the losses.
There are no law or benefit changes.
All policies are annual.
Rate change effective date is April 1, 2013.
Rates are reviewed annually.

AY 2010 Reported Losses and ALAE as of 12/31/2010 = $50,000

Accident Year
2004
2005
2006
2007
2008
2009

Reported Loss and ALAE Age-to-Age Development Factors
12-24
24-36
36-48
48-60
60-72
72-ult
1.58
1.35
1.05
1.06
0.98
1.00
1.75
1.31
1.05
1.01
1.01
2.63
1.20
1.08
1.04
1.82
1.23
1.02
1.46
1.18
1.66

All year Average
Average ex-hi/lo
Average last 3 years

Calendar Year
Ending
March 2008
June 2008
September 2008
December 2008
March 2009
June 2009
September 2009
December 2009
March 2010
June 2010
September 2010
December 2010

1.82
1.70
1.65

1.25
1.26
1.20

1.05
1.05
1.05

1.04
1.04

1.00

Reported Loss and ALAE
Frequency Severity
Pure
Premium
0.082
$2,410
$197.62
0.077
$3,650
$281.05
0.073
$3,700
$270.10
0.070
$3,710
$259.70
0.069
$3,685
$254.27
0.068
$2,525
$171.70
0.070
$2,580
$180.60
0.065
$2,565
$166.73
0.065
$2,605
$169.33
0.065
$2,675
$173.88
0.065
$2,715
$176.48
0.065
$2,730
$177.45

Develop the projected ultimate loss and LAE for accident year 2010 losses using the data above. In order
to satisfy the state requirements, fully describe the rationale for the selections for loss development, loss
trend, and ULAE.

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Section 2: Effects of WC Benefit Level Changes
Questions from the 1995 exam
37. (3 points) You are given:
Ratio of
Worker's Wage
to Average Wage
0.250
0.500
0.750
1.000
1.250
1.500
1.875
2.250

Cumulative
Percentage
of Workers
6%
15%
35%
60%
75%
90%
96%
99%

Cumulative
Percentage
of Wages
1%
5%
17%
38%
55%
76%
86%
92%

Current Workers' Compensation Law
• Compensation rate is one-half of worker's pre-injury wage.
• There is no maximum benefit limitation.
• Minimum benefit limit = 50% of average weekly wage.
Revised Workers' Compensation Law
• Compensation rate is two-thirds of worker's pre-injury wage.
• Maximum benefit limit = 125% of average weekly wage.
• Minimum benefit limit = 50% of average weekly wage.
Following the methodology presented by Feldblum, “Workers' Compensation Ratemaking," calculate the
direct effect of the law change.

Questions from the 1999 exam
38. (2 points) Based on Feldblum, "Workers' Compensation Ratemaking," and the information shown
below, calculate the average benefit as a percentage of the average wage.
Ratio to
Average Wage
0.00-0.50
0.50-0.75
0.75-1.00
1.00-1.50
1.50-2.00
2.00-2.50

% Of
Workers
15%
20%
25%
20%
15%
5%

Minimum benefit
Maximum benefit
Compensation rate

0.75 of average wage
1.50 of average wage
0.75 of pre-injury wage

Exam 5, V1a

% Of
Wages
6%
12%
21%
24%
26%
11%

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2001 exam:
Question 48. (2 points) Based on Feldblum, “Workers Compensation Ratemaking,” and the following
information, answer the questions below. Show all work.
Statewide Average Weekly Wage

$900

Maximum Weekly Benefit

900

Minimum Weekly Benefit

360

Compensation Rate

66.7% of pre-injury wage

Ratio to
Average Wage
0.40
0.50
0.60
0.70
0.80
1.00
1.25
1.50
1.75

Wage Distribution Table
Cumulative
Cumulative
Percentage of
Percentage of
Workers
Wages
5%
2%
15%
7%
25%
13%
35%
20%
45%
28%
65%
48%
80%
67%
90%
82%
95%
90%

a. (1 point) Calculate the average benefit as a percentage of the statewide average weekly wage.
b. (1 point) Calculate the direct effect of changing the compensation rate from 66.7% to 80.0% of the
pre-injury wage.

Questions from the 2007 exam:
40. (2.5 points) Workers compensation law changes can produce both direct and incentive (or indirect) effects.
a. (0.5 point) Explain what is meant by direct effect.
b. (0.5 point) Explain what is meant by incentive effect.
c. (0.75 point) Will implementation of cost of living adjustments have a direct effect, incentive effect,
or both? Explain your answer.
d. (0.75 point) Will changes in administrative procedures have a direct effect, incentive effect, or
both? Explain your answer.

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2008 exam
19. (3.0 points)
a. (2.0 points) You are given the following information related to workers' compensation:
Ratio to Statewide
Average
Cumulative
Cumulative
Weekly Wage
Percent
Percent
(SAWW)
of Workers
of Wages
0.50
9%
4%
0.75
35%
20%
1.00
60%
42%
1.25
81%
65%
1.50
91%
81%
 The compensation rate is 2/3 pre-injury wage subject to maximum and minimum limitations.
 Statewide average weekly wage (SAWW) = $100
 Minimum weekly benefit = $50
 Maximum weekly benefit = $67
a. Calculate the direct benefit level effect of increasing the maximum benefit to $100.
b. (0.5 point) Define incentive (or indirect) effect.
c. (0.5 point) Identify and briefly describe an incentive (or indirect) effect that may result from increasing the
maximum benefit.

Questions from the 2009 exam
26. (1 point) Given the following information regarding a change to a workers' compensation program's
indemnity benefits:
• The replacement rate for benefits is changed from 50% of gross earnings to 85% of net takehome (after-tax) pay.
• The maximum and minimum limitations do not affect the reimbursement, either before or after the
change.
• The tax rate for all participants is 30%.
a. (0.5 point) Calculate the direct effect of this benefit change.
b. (0.5 point) Briefly explain two possible indirect effects of this change.

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2010 exam
23. (2.5 points) Given the following workers compensation information:
• The compensation rate is 80% of the worker's pre-injury wage.
• The state average weekly wage (SAWW) is $1,500.
• The minimum benefit is 48% of the SAWW.
• The maximum benefit is changing from 128% of the SAWW to 112% of the SAWW.
• The distribution of workers (and their wages) according to how their wages compare to the
SAWW is as follows:
Ratio to
Average
Weekly
Wage
0 - 60%
60 - 120%
120 - 140%
140 - 160%
160 +

Number of
Workers
64
144
33
21
29

Total
Weekly
Wages
$37,550
$196,200
$64,350
$47,250
$84,000

a. (2 points) Calculate the direct effect of the change in maximum benefits on losses.
b. (0.5 point) Explain a potential indirect effect of the change in maximum benefits on losses.

Questions from the 2012 exam
7. Develop the projected ultimate loss and LAE for accident year 2010 losses using the data above. To
satisfy the state requirements, fully describe the rationale for the selections for loss development, loss
trend, and ULAE.

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
The predecessor papers to the current syllabus reading “Basic Ratemaking” by Werner, G.
and Modlin, C. were numerous. While past CAS questions were drawn from prior syllabus
readings, the ones shown below remain relevant to the content covered in this chapter.

Section 1: Loss Trending and Loss Development
Solutions to the questions from the 1996 exam
Question 30.
(b) To calculate the trend factor, one must know, the frequency and severity trend indications, the period of time
the rates will remain in effect, the proposed effective date of the rates, and the length of the policy issued.
These are given in the problem as (.99)*(1.05) = 1.0395; one year; 1/1/96; and annual policies.
Trend factors are computed based on the time between the average accident date of the experience period to
the average accident date of the effective period.

CY
1992
1993
1994
Total

Ultimate Loss
and ALAE
325,000
575,000
800,000

Average Accident Date
Experience Effective
7/1/92
1/1/97
7/1/93
1/1/97
7/1/94
1/1/97

Trend
Factor
(1.0395)4.5
(1.0395)3.5
(1.0395)2.5

Trended On-Level
Loss and ALAE
386,895
658,497
881,356
1,926,748

Thus, the trended, on-level loss and ALAE ratio = 1,926,748/2,646,299 = .728.
Question 36.
(a). The premium adjustment factor is also known as an on-level factor. The numerator of the on-level factor
considers rate changes which impact both PY 1995, represented by the parallelogram below, and rate
changes up and through the current level. The denominator of the on-level factor, considers only those rate
changes which impact PY 1995.
Calculate the numerator of the on-level factor. This is equal to (1.0)(1.15)(.90)(1.05) = . 1.08675
Calculate the average rate level factor for the policy year. This is a weighted average of the rate level factors
in the policy year. The weights will be relative proportions of the parallelogram. First calculate the area of all
triangles (area = .50 * base * height) within the parallelogram and then determine the remaining proportion of
the parallelogram by subtracting the sum of the areas of the triangles from 1.0.
Notice the area of the parallelogram at the 1.035 level. Its area is calculated as base * height = .50*1.0 = .50.
The average rate level factor for the policy year = (1/2)(.5)(.5)*1.0 + (1/2)(.5)(.5)*1.15
+.50*1.0*1.035 + (1.0 - .125 - .125 - .50)*1.15 = 1.07375.
+15%

-10%
1.15

1.00
1.00

1/94 5/1

1/95

1.15
1.035

1.035

1/96

7/1

The on-level factor = 1.08675 / 1.07375 = 1.012.

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to the questions from the 1996 exam (continued)
Question 36. (continued)
(b) Experience rate changes are represented graphically as diagonal lines, and are computed to adjust current
rates for changes anticipated in projected experience level. These affect new and renewal policies only.
Law amendment changes are represented graphically as straight lines, and since they affect all policies
inforce at a given point in time. These changes adjust premiums for statutory modifications to benefits.

Solutions to questions from the 1997 exam:
Question 44.

(a) Trend Factors:
To calculate trend factors for each year’s losses, compute:
1. The annual trend factor.
2. The midpoint of each year’s loss exposure (the average accident date for each year of the experience period).
3. The midpoint of loss occurrence during the exposure period (the period the rates are to be in effect).
On page 103, McClenahan states that “While frequency and severity trends are often analyzed separately, it
is sometimes preferable to look at trends in the pure premium, thus combining the impact of frequency and
severity”.
Using this approach, the annual trend factor is (1+.043)*(1-.020) = 1.022.
Since we are given accident year 199X losses, the midpoint of each year loss exposure is 7/1/9x.
We are told that the revised rates will be in effective for 12 months, from 10/1/97 through 9/30/98 (exposure
period), and that all policies written will be annual policies. Therefore, the average policy will run from 4/1/98
to 3/31/99, and the midpoint of loss occurrence during that policy will be 9/30/98.
(Note: Another way to remember trend period for annual policies, for which rates will be in effective for 12
months, is midpoint of experience period to one year past the effective date.)

(a) Loss Development Factors (LDFs):
The appropriate LDFs to apply to each year’s losses depends upon its age as of the loss evaluation date.
Since losses are evaluated at 12/3196, AY 1995 losses are “aged” 24 months, AY 1994 losses are “aged” 36
months, and AY 1993 losses are “aged” 48 months.
To project these losses to ultimate, the respective age to ultimate factors to be used are 1.25, 1.05, and 1.00.

With this information, we can compute developed and trended Loss and ALAE by accident year as follows:

AY
1993
1994
1995

Reported
Loss and
ALAE
(1)
1,800,000
2,275,000
1,975,000

LDF
(2)
1.00
1.05
1.25

Annual
trend
factor
(3)
1.022
1.022
1.022

Midpoint of the
experience
period
(4)
7/1/93
7/1/94
7/1/95

Midpoint of
the exposure
period
(5)
9/30/98
9/30/98
9/30/98

Trend
Factor
(6)
1.121
1.097
1.073

Developed and
trended Loss
and ALAE
(7)=(1)*(2)*(6)
2,017,800
2,620,459
2,648,969

Column (6) = Column (3)t, where t is the number of years elapsed between column 5 and column 4.

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 1999 exam
Question 39.
The severity trend rate =
Loss Amount
Before/After(x)
40/44
80/88
120/132
160/176
200/220

E[ X ']
1.0 , where X' represents losses affected by a 10% inflation rate.
E[ X ]

Probability
of loss (f(x))
.20
.20
.20
.20
.20

Distribution of Loss by Layer
0 - 50
50 - 100
40/44
0/0
50/50
30/38
50/50
50/50
50/50
50/50
50/50
50/50

100 - 200
0/0
0/0
20/32
60/76
100/100

Loss amounts before and after the impact of uniform 10% increase
Layer
0 - 50
50 - 100
100 - 200

E [ X ]  x* f ( x )
x
[.2*40 .80*50] 48

E[ X ']  x* f ( x )
x
[.2*40*(1.1) .80*50] 48.8

[.2*30 .60*50] 36

[.2*38 .60*50] 37.6

[.2*20 .20*60 .20*100] 36

[.2*32 .20*76 .20*100] 41.6

Layer
0 – 50
50 – 100
100 – 200

One year severity Trend
48.8
1.0  1.017 or 1.7%
48
37.6
1.0  1.044 or 4.4%
36
41.6
1.0  1.156 or 15.6%
36

Solutions to questions from the 2000 exam:
Question 40.
b. Calculate the trended projected ultimate on-level loss and ALAE ratio for the combined experience period
1997-1998.
With this information, we can compute developed and trended Loss and ALAE by accident year as follows:

AY
1997
1998
Total

Developed
Loss and
ALAE
(1)
635,000
595,000

Freq
trend
factor
(2)
1.01
1.01

On-level loss and ALAE ratio 

Exam 5, V1a

Sev
trend
factor
(3)
1.03
1.03

Midpoint of the
experience
period
(4)
7/1/97
7/1/98

Midpoint of
the exposure
period
(5)
7/1/2001
7/1/2001

Trend Factor
(6)
(1.01*1.03)4
(1.01*1.03)3

Developed and
trended Loss
and ALAE
(7)=(1)*(2)*(6)
743,717
669,873
1,413,590

1,413,590
Developed and Trended losses

 .684
On  Level Earned Pr emium
1,027,2831,039,290

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2001 exam
Question 2. Based on McClenahan, “Ratemaking,” chapter 2, Foundations of Casualty Actuarial Science,
and the following information, answer the question below.
Key dates given:


Experience period is accident year 1999.



Indicated rates will become effective July 1, 2001.



The next scheduled rate increase is expected to become effective April 1, 2002.



All policies are expected to have an 18-month period.



Policies are evenly written throughout the year.

How many months are there between the midpoint of the experience period and the midpoint of the
exposure period?
Step 1: Determine the midpoint of the experience period:
The midpoint of the experience period is a function of the average accident date during the experience
period. The experience period is ACCIDENT year 1999, and since all polices are written evenly
throughout the year, the average accident date during the experience period is 7/1/99.
Step 2: Determine the midpoint of the exposure period:
The midpoint of the experience period is a function of the average policy written date and the average
accident date (based on the average written date) during the exposure period. The exposure period is
from 7/1/2001 – 4/1/2002, and so the average written date during the exposure period is 11/15/2001.
Since all policies are expected to have an 18 month period, the average accident date is 9 months later,
which is 8/15/2002.
Thus, the number of months between the midpoint of the experience period (7/1/99) and the midpoint of the
exposure period (8/15/2002) is 37.5 months.
Answer D.

Solutions to questions from the 2002 exam
Question 17.
a. (1 point) Calculate the experience period trended developed loss and ALAE.
Since we are given that the developed accident period loss and ALAE is $21,500, and that the annual
trend factor is 1.03, what remains to be computed is the trend period.
The trend period is determined by the time between the average accident date of the experience
period and the average accident date associated with the effective period of the rates.
The average accident date for the eighteen month (1/1/00 – 6/30/01) accident experience period is 10/1/00.
Since the revised rates will be in effect for a one year period (1/1/2003 – 12/31/2003) and since all
polices have one year terms and written uniformly throughout the year, the average policy will run
from 7/1/2003 – 6/30/2004, and the midpoint of loss occurrence under that policy will be 1/1/2004).
The trend period is therefore 3.25 years (10/1/2000 – 1/1/2004), and the experience period trended
developed loss and ALAE is $21,500 (1.03)3.25 = 23,668

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2003 exam
12. Calculate the projected ultimate accident year 2001 losses.
Step 1: Determine AY 2001 case incurred losses at 12/31/2002 projected to 36 months.
Case incurred losses at 12/31/2002 = $3500 + $4,000 = $7,500. Note that at 12/31/02, AY 2001 case
incurred losses are at 24 months of development. The loss development factor from 24-36 months is
given as 1.20. Thus, AY 2001 case incurred losses projected to 36 months equals $9,000.
Step 2: Determine AY 2001 case incurred losses at 12/31/2002 projected to ultimate.
AY 2000 36-48 months case incurred loss development factor is $9,420/$8,000 = 1.155. Thus, at
12/31/02, AY 2001 cased incurred losses are at ultimate equals $9,000 * 1.155 = $10,395.
Answer E.  $10,200

Solutions to questions from the 2004 exam
7. Calculate the trended loss ratio.
Step 1: Based on the givens of the problem, write an equation to determine the trended loss ratio.
 Developed Incurred Losses   Freq Trend*Sev Trend 
Trended Loss Ratio = 
*

Earned Premium
Premium Trend

 


Years of Trend

Step 2: Using the equation in Step 1, and the data in the problem, solve for the trended loss ratio.
2.5
 $35,000   .99 * 1.06 
Trended Loss Ratio = 
*
  .7352 Answer C: > 71 % but < 74%
 
 $50,000   1.029 
8. Which of the following statements are true regarding loss trends?
1. When an exponential curve is used to approximate severity, the assumption is a constant multiplicative
increase in severity. True. “Since this data contains random fluctuations, the minimization of these
fluctuations will provide a better estimate of the underlying trend. This is achieved by fitting the data to a
curve. An exponential curve is selected because it assumes a constant percentage trend from year to
year.”
2. Statement no longer applicable to the content within this article
3. Linear trends tend to underestimate future costs when inflation is increasing at a multiplicative rate. True.
Note that the linear model will produce a model in which the projection will increase by a constant amount
(a) for each unit change in x. The exponential model will produce a constant rate of change of ea - 1, with
each value being ea times the prior value.
Answer: D. 1 and 3 only

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2004 exam (continued):
37.(5 points)
a. (1 point) Calculate the age-to-ultimate development factor for accident year 2003 as of December 31,
2003. Explain your assumptions.
Assumptions:
 We are told that ultimate losses are reached at age 48, and therefore our 48-ultimate loss development
factor is 1.000.
 Selected age to age development factors are set equal to age to age link ratios computed using the
given data. Age to Age link ratios are computed by dividing case-incurred losses at successive
intervals (e.g. AY 2000 12-24 link ratio = 1,816/1,412 = 1.2861)
Since accident year 2003 at 12/31/03 is at 12 months of maturity, a 12 to ultimate loss development factor
is necessary and is computed as follows:

AY
2000
2001
2002
2003
3 yr avg
Factor to Ult

12-24
1.2861
1.2457
1.2336

24-36
1.0975
1.0564

36-48
1.0000

48-ULT
1.0000
1.0000
1.0000
1.0000
1.2551
1.0769
1.0000
1.0000
1.3516
1.0769
1.0000
1.0000 , where
12 to ultimate loss development factor = 1.3516 = 1.2551 * 1.0769 * 1.0000 * 1.0000
b. (0.5 point) Calculate the ultimate loss amount for accident year 2003.
AY 2003 ultimate losses = AY 2003 case incurred losses12 months * 12 to ultimate loss development factor
= $2,421 * 1.3516 = $3,272.22
c.

(1 point) Calculate the trended ultimate loss amount for accident year 2003.
Since we have computed ultimate losses for AY 2003 as $3,272.22, what remains to be computed is
the annual trend factor and the trend period.
The annual trend factor is computed as the product of the given annual frequency and severity trend
rates. Thus, the annual trend factor equals .98 * 1.08 = 1.0548
The trend period is determined by the time between the average accident date of the experience
period and the average accident date associated with the effective period of the rates.
The average accident date for AY 2003 is 7/1/2003
Since the revised rates will be in effect for a one year period (7/1/2004 – 7/1/2005) and since all
polices have one year terms and are written uniformly throughout the year, the average policy will run
from 1/1/2005 – 12/31/2005, and the midpoint of loss occurrence under that policy will be 7/1/2005).
The trend period is therefore 2 years (7/1/2003 – 7/1/2005), and the AY 2003 trended developed loss
and ALAE is $3,272.22 (1.0548)2.00= $3,640.68

d. (1.5 points) Briefly describe three causes of loss development.
1. Development on known claims. This occurs when reserves are initially set too low, and then increase as
more loss related information becomes known.
2. Newly reported claims. These result from the late reporting of claims.
3. Re-opening of prior closed claims. This happens when additional damages, resulting from the original loss
occurrence, arise at point in time after the claim has been closed.

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2004 exam (continued):
Question 37 (continued):
e. (1 point) Briefly explain why it is appropriate to both trend and develop losses (i.e., why there is no overlap).
It is appropriate to both trend and develop losses because there is no double counting of severity trend and
loss development factors in the ratemaking process.
The trend factor reflects the severity trend from the midpoint of the experience period to the midpoint of
the exposure period.
The loss development factor reflects the underlying severity trend from the midpoint of the exposure
period to ultimate.

Solutions to questions from the 2007 exam:
22. Explain the impact of this change on incurred loss development and rate adequacy for this liability
line of insurance.
CAS Model Solution
Incurred loss development factors are based on losses prior to accident year 2006. Since the initial case
reserves were much higher, the development factors being applied to 2006 losses will be too low.
Ultimate losses for 2006 will be understated therefore indicated projected loss ratios or pure premiums
will be too low. This will result in an indication that will be too low. Ultimately, the rates based on accident
year 2006 will be inadequate.

Solutions to questions from the 2008 exam:
Model Solution - Question 17
a. (0.5 point) Calculate the calendar-year incurred losses for 2006 and 2007.
CY 2006 incurred losses = CY 2006 Paid losses + CY 2006 Ending Reserves – CY 2006 Beginning Reserves
Note: For CY 2006, we are only concerned with transactions associated with any policies effective during CY
2006 that also have losses during CY 2006. For CY 2006, the only policy meeting this criterion is the policy
effective 7/1/2006.

CY 2006 Paid losses (for policy effective 7/1/2006) = $0.
CY 2006 Ending Reserves (for policy effective 7/1/2006) = $5,000 and CY 2006 Beginning Reserves = $0.
Thus, CY 2006 incurred losses = $0 + $5,000 - $0 = $5,000
CY 2007 incurred losses = CY 2007 Paid losses + CY 2007 Ending Reserves – CY 2007 Beginning Reserves associated
with policies having CY transactions during CY 2007. Note that both the 7/1/2006 and 10/1/2006 policies have
transactions (paid and case reserve activities) during CY 2007.
i. For the policy effective 7/1/2006, total paid losses (based on 2007 transaction dates) = $500 + $3,500 = $4,000. In
addition, beginning reserves = $5,000 and ending reserves = $2,000.
Thus, CY 2007 incurred losses (for policy effective 7/1/2006) = $4,000 + $2,000 - $5,000 = $1,000.
ii. For the policy effective 10/1/2006, total paid losses (based on 2007 transaction dates) = $5,000 + $9,000 = $14,000. In
addition, beginning reserves = $0 and ending reserves = $1,000.
Thus, CY 2007 incurred losses (for policy effective 10/1/2006) = $14,000 + $1,000 - $0 = $15,000.

Thus, CY 2007 incurred losses = $1,000 + $15,000 = $16,000.

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2008 exam:
Model Solution - Question 17 (continued)
b. (0.5 point) Calculate the accident-year incurred losses for 2006 and 2007 evaluated as of 12/31/2008.
Note: Here we are concerned with final payments and reserves associated with accidents occurring during
AY 2006 and 2007 respectively.
i. For the policy effective 7/1/2006, total paid losses (on accidents occurring during 2006) as of
12/31/2008 = $500 + $3,500 + $3,000 = $7,000. Final reserves as of 12/31/2008 = $0.

Thus, AY 2006 incurred losses (for policy effective 7/1/2006) = $7,000 + $0 = $7,000.
ii. For the policy effective 10/1/2006, total paid losses (on accidents occurring during 2007) =
$5,000 + $9,000 + $1,000 = $15,000. Again, final reserves as of 12/31/2008 = $0

Thus, AY 2007 incurred losses (for policy effective 10/1/2006) = $15,000 + $0 = $15,000.
c. (0.5 point) Calculate the policy-year incurred losses for 2006 and 2007 evaluated as of 12/31/2008.
Note: Both policies are effective during 2006. No policies are effective during 2007.
Therefore, there will be no policy year 2007 incurred losses.
i. For the policy effective 7/1/2006, total paid losses (on accidents occurring during 2006) as of 12/31/2008 = $7,000
ii. For the policy effective 10/1/2006, total paid losses (on accidents occurring during 2007) as of 12/31/2008 = $15,000

Thus, PY 2006 incurred losses = $7,000 + $15,000 = $22,000.
Thus, PY 2007 incurred losses = $0
d. (0.5 point) Identify 1 advantage and 1 disadvantage associated with using PY incurred losses for ratemaking.
One advantage is that premiums and losses can be matched using policy year incurred losses.
One disadvantage is that policy year data is the least mature and least responsive compared to CY or AY data.

Solutions to questions from the 2009 exam:
Question 22
a. CY 2006 losses. The question is ambiguous with respect to whether it refers to paid or incurred losses.
Assuming Paid Losses are sought, add paid losses during CY 2006 from accidents occurring in both 2005 and
2006: 300,000 + 1,500,000 = $ 1,800,000
Assuming Incurred Losses (i.e. paid + change in reserves) are sought, use the result from above and compute
the change in reserves as the ending reserves – beginning reserves, for accidents occurring in both 2005 and
2006: $1,800,000 + (300,000 - 500,000) + (1,000,000 – 0) = $2,600,000
b. AY 2006 incurred losses @ 12/31/07 =(AY 06 paid through 12/31/07) + (AY 06 reserves @ 12/31/07)
= (1,500,000 + 700,000) + 200,000 = $2,400,000
c. PY 2005 incurred losses @ 12/31/08. Note: Question states that all claims given in the problem arise from
policies written in 2005
= (PY 05 Paid until 12/31/08) + (PY 05 reserves @ 12/31/08)
= (1,000,000 + 300,000 +250,000 + 50,000) + (0) [for accidents occurring in 2005] +
(1,500,000 + 700,000 + 100,000) + (50,000) [for accidents occurring in 2006]
= $1,600,000 + $2,350,000 = $ 3,950,000
d. CY incurred losses are more responsive than AY since loss info is known once CY is complete. AY incurred
provides a better match to premium and loss then CY basis, although not as well as PY which matches
premium and loss.

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2009 exam (continued):
Question 24 Why it may be inappropriate to apply a basic limits loss trend to total limits losses.
If loss costs are increasing, basic limit losses will trend at a lower rate than total losses, and thus a basic limit
trend will understate the actual underlying loss trend.
Basic limit losses trend at a lower rate than total losses because for losses near or at basic limits before trending,
the full trend will not be realized by limiting losses. A loss that is already at or above basic limits, in fact, will
observe no basic limit trends if losses are increasing.

Question 27 Fully discuss the "overlap fallacy" between trend and loss development.
It was believed that loss development and loss trend capture the same change in loss patterns.
Therefore, using both would be “double counting”. This belief was referred to as “overlap fallacy”.
It is incorrect, because loss trend projects losses from the midpoint of experience period to the midpoint of
exposure period, while loss development projects losses from midpoint of the exposure period to ultimates.
This can be thought graphically as possible:
Successive Evaluation Periods

Question 42: For homeowners insurance, explain two reasons that hurricane rates should be priced
separately from non-hurricane rates.
Ratemaking becomes a much easier process if premiums are split. Traditional techniques can be applied
on the non-hurricane portion without having to deduce the non-hurricane portion each time.
Allows appropriate classification. For example, it does not make sense to have a 25 % discount for fire
protection in an area where 80 % of losses are hurricane related.

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2010 exam:
Question 20
a. CY 2008 reported losses = CY 2008 Paid losses + CY 2008 Ending Reserves – CY 2008 Beginning Reserves
Note: Since two claims are given, values for each formula component above need to be aggregated. These values
are shown below as (claim 1 amount + claim 2 amount)
CY 2008 reported losses = ($0 + $0) + ($25,000 + $0) – (10,000 + $25,000) = -$10,000

b. AY 2006 Reported Loss as of 12/31/2007
Note: Here we are concerned with total payments and reserves as of 12/31/2007 associated with accidents
occurring during AY 2006. This limits transactions to claim 1 only.
i. Total paid losses (on accidents occurring during 2006) as of 12/31/2007 = $0. Final reserves as of
12/31/2007 = $10,000.

Thus, AY 2006 incurred losses $0 + $10,000 = $10,000.
c. PY 2006 reported loss as of 12/31/2007
Note: Here we are concerned with total payments and reserves at 12/31/2007 associated with both
claims because both claims arose from a single policy issued in 2006.
PY 2006 reported loss as of 12/31/2007 = ($0 + $0) + ($10,000 + $25,000) = 35,000
d. Advantage: CY losses are readily available/immediately known. No need to wait for losses to develop.
Disadvantage: AY aggregation provides a better match of premiums to losses than CY aggregation.
21. (2 points) Identify four adjustments made to historical losses in projecting losses for a future policy
period for ratemaking. Briefly describe the purpose of each.
1. Development – taking losses from an early state (e.g. 24 months) to their total ultimate state when all losses
are paid and the claims are closed.
2. Trend – taking historical losses from the midpoint of the experience period and projecting to the midpoint of
the future period (takes things such as inflation into account)
3. Benefit Level Changes – take into account anything that would change the benefits being charged to get
losses to a “current benefit level” (e.g. workers comp. change in the law affecting benefits paid)
4. Catastrophes/Shock Losses/Extraordinary Events – adjust historical losses to take out any cats and load back
in an amount to account for them. If cats were always just included, rates would increase years after cats and
decrease after years without them to volatile.
Question 24
Select a ULAE factor to be applied to the statewide incurred losses and paid ALAE as part of calculating
statewide rate indications. Explain your selection.
Calendar
Year
2006
2007
200 8
2009

Paid Loss
& ALAE
(1)
522,500
550,000
523,688
544,635

Paid
ULAE
(2)
26,125
55,000
52,369
54,464

Paid ULAE/
Paid Loss & ALAE
(3)=(2)/(1)
5%
10%
10%
10%

I would select ULAE factor =10%
Calendar Year 2006 has ULAE factor of 5 % but 2007– 2009 ULAE factors are all at 10% .
I believe there must have been a change in operation in 2007 that caused ULAE to increase to 10%.

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2011 exam:
Question 6
a. (0.5 point) Calculate the calendar year 2008 incurred losses.
b. (0.5 point) Calculate the accident year 2008 incurred losses, evaluated at December 31, 2009.
c. (0.5 point) Calculate the policy year 2007 incurred losses, evaluated at December 31, 2009.
d. (0.5 point) Calculate the report year 2008 incurred losses, evaluated at December 31, 2009.
e. (0.5 point) Briefly describe one advantage and one disadvantage associated with using policy year
losses for ratemaking.
Question 6 – Model Solution
a. CY 2008 incurred losses = CY 2008 Paid losses + CY 2008 Ending Reserves – CY 2008 Beginning Reserves
Note: Here we consider transaction date data occurring in 2008. Such data exists for claims 1, 2 and 3.
Claim 1: CY 2008 incurred losses = ($200 + $600 - $300) = $500
Claim 2: CY 2008 incurred losses = ($500 + $200 - $0) = $700
Claim 3: CY 2008 incurred losses = ($5 + $200 - $0) = $250
CY 2008 incurred losses = $500+ $700+$250=$1,450

b. AY 2008 incurred losses = AY 2008 Paid losses + AY 2008 Ending Reserves as of 12/31/2009
Note: Here we consider transaction date data occurring during AY 2008. Such data exists for claim 3 only.
Claim 3: AY 2008 paid losses = ($50 + $100) = $150. AY 2008 case reserve as of 12/31/2009 = $50
CY 2008 incurred losses = $150+ $50 = $200

c. PY 2007 incurred loss as of 12/31/2009
Note: Here we are concerned with total payments and reserves at 12/31/2009 associated with all three
claims these claims arose from policies issued in 2007.
PY 2007 paid losses as of 12/31/2009 = 100 + 200 + 500 + 500 + 200 + 50 + 100 = 1650
PY 2007 case reserves of 12/31/2009 = 0 + 0 + 50 = 50
PY 2007 incurred losses as of 12/31/2009 = 1650 + 50 = 1700
d. RY 2008 incurred loss as of 12/31/2009
Here we are concerned with total payments and reserves as of 12/31/2009 associated with accidents reported
during 2008. This limits transactions to claim 2 and claim 3.
i. Total paid losses (on accidents reported during 2008) as of 12/31/2009 = $500 + 200 + 50 + 100 = 850.
Case reserves as of 12/31/2009 for claims 2 and 3= $0 + $50 = $50

Thus, RY 2008 incurred losses as of 12/31/2009 $850 + $50 = $900.
a.
b.
c.
d.
e.

200 + 600 - 300 + 500 + 200 + 50 + 200 = 1,450
50 + 100 + 50 = 200
100 + 200 + 500 + 500 + 200 + 50 + 100 + 50 = 1700
500 + 200 + 50 + 100 + 50 = 900
Advantage: True match between premiums and losses
Disadvantage: Extended development. It takes longer to develop.

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2011 exam:
Question 7 – Model Solution 1
There is no overlap when developing loss and trending loss. Trending loss will rend loss from the
midpoint of experience period to the midpoint of the exposure period. Developing loss will develop loss
from the midpoint of the exposure period to the ultimate.
Question 7 – Model Solution 2
The overlap fallacy between loss development and trend clarifies than there actually is no overlap, or doublecounting, between the two adjustments. Trend brings historical losses to the projected cost level/ environment
of the future period, whereas development brings these losses to their ultimate settlement value.
The graph below demonstrates this:

Question 17. Given 5 claim amounts; • Basic limit = $25,000; • Total limits severity trend = 10%
Calculate the excess loss trend.
Question 17 – Model Solution

When Limit  Loss , Excess loss trend  [ Loss *(1.0  Trend )]  Limit

Loss  Limit
Excess loss trend = Excess trended losses/Excess losses
Claim #
Loss
XS Loss
Trended Loss
XS Trended Loss
= loss x (1+10%)
(1)
(2)
(3)
(4)
(5)
1
10,000
0
11,000
0
2
15,000
0
16,500
0
3
30,000
5,000
33,000
8,000
4
35,000
10,000
38,500
13,500
Total
15,000
21,500
(3) = (1) - 25,000, if (1) is greater than 25,000; otherwise (3) = 0
(5) = (4) - 25,000, if (4) is greater than 25,000; otherwise (5) = 0
Excess Loss trend = 21,500/15,000 – 1 = 43.33%

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Section 2: Effects of WC Benefit Level Changes
Solutions to questions from the 1995 exam:
Question 37.
Direct effect of a benefit change =

Average benefit (after the change)
.
Average benefit (before the change)
Current
.50

Proposed
.667

None

.667*1.875 = 1.25

.50*1.0=.50

.667*.75 = .50

Replacement (Compensation) rate =% of the preinjury wage =
Max benefit is set equal to a % state average weekly
wage (SAWW)
Min benefit is set equal to .50* (SAWW) =
Average Benefit Computed::

(R Rate)*(% SAWW)*(Cum % of workers)

The % of workers earning > (1.25 * SAWW ) receive
max benefits
The % of workers earning < (.50 * SAWW ) receive
min benefits

None

.667*1.875*(1-.96) =.05

.50*1.0*.6=.30

.667*.75*.35 = .175

Workers earning between the maximum and the
minimum receive benefits of equal to a % of their
pre-injury wage
Total

(R Rate) * (cumulative % of wages)
.50*(1-.38) = .31
.667*(.86-.17) = .46

.30 + .31 = .61

.05+.175+.46=.685

The direct effect of a benefit change = .685/.610 - 1.0 = 12.3.

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to the questions from the 1999 exam
Question 38.
To compute the average benefit, begin by re-stating the %s in the given table as cumulative %s.
Ratio to
Average Wage
50%
75
1.00
1.50
2.00
2.50

Cum % Of
Workers
15%
35%
60%
80%
95
100%

Cum % Of
Wages
6%
18%
39%
63%
89%
100%

Next, determine the % of workers receiving the maximum and minimum benefit. These values are found
by looking in the table above for the % of workers earning a certain percentage of the average wage such
that the product of (ratio to average wage ) * (compensation rate) equals 150% and 75% of the state
average wage respectively.
Maximum benefit =

1.50 of average wage

Minimum benefit =

Note: At the maximum benefit limit, the compensation rate
(.75) times the ratio to the state average wage (2.0) equals
1.50 of the state average weekly wage.
0.75 of average wage

Compensation rate =

Note: At the minimum benefit limit, the compensation rate
(.75) times the ratio to the state average wage (1.0) equals
.75 of the state average weekly wage.
0.75 of pre-injury wage

Computation of the average benefit:

Workers earning > 2.0 times the state average weekly wage receive max
benefits
Workers earning < 1.0 times the state average weekly wage receive min
benefits
Workers earning between the maximum and the minimum receive benefits
of = a % of their pre-injury wage (R Rate) * (cumulative % of wages)
Total

Benefits as a % of
wages
.75 * 2.0 * .05 = .075
.75 * 1.0 * .60 = .45

.75 * (.89 - .39) = .375
.075 + .45 + .375 = .90

Thus, the average benefit is equal to 90% of the state average weekly wage:

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to the questions from the 2001 exam
Question 48.
a. (1 point) Calculate the average benefit as a percentage of the statewide average weekly wage.
Determine the % of workers receiving the maximum and minimum benefit. These values are found by looking
in the given table for the % of workers earning a certain percentage of the average wage such that the
product of (ratio to average wage ) * (compensation rate) equals 100% (900/900) and 40% (360 / 900) of the
state average wage respectively.
Maximum benefit =

1.00 of average wage
Note: At the maximum benefit limit, the compensation rate (given as .667) times
the ratio to the state average wage (1.50) equals 1.00 of the state average
weekly wage.

Minimum benefit =

Compensation rate
=

0.40 of average wage
Note: At the minimum benefit limit, the compensation rate (.667) times the ratio
to the state average wage (.60) equals .40 of the state average weekly wage.
0.667 of pre-injury wage (given)

Computation of the average benefit:
Workers earning > 1.50 times the state average weekly wage
receive max benefits
Workers earning < 0.60 times the state average weekly wage
receive min benefits

Benefits as a % of wages
.667 * 1.5 * .10 = .10
.667 * .60 * .25 = .10

.667 * (.82 - .13) = .4602
Workers earning between the maximum and the minimum
receive benefits of = a % of their pre-injury wage (R Rate) *
(cumulative % of wages)
Total
.10 + .10 + .4602 = .6602
Thus, the average benefit is equal to 66.2% of the state average weekly wage (900) = 594.21
b. (1 point) Calculate the direct effect of changing the compensation rate from 66.7% to 80.0% of the pre-injury
wage.

Average benefit (after the change)
.
Average benefit (before the change)
Benefits as a % of wages
Workers earning > 1.25 times the state average weekly wage
.80 * 1.25 * .20 = .20
receive max benefits (.80 * 1.25 = 1.0)
.80 * .50 * .15 = .06
Workers earning < 0.50 times the state average weekly wage
receive min benefits(.80 * 50 = .40)

Direct effect of a benefit change =

.80 * (.67 - .07) = .48
Workers earning between the maximum and the minimum
receive benefits of = a % of their pre-injury wage (R Rate) *
(cumulative % of wages)
Total
.10 + .10 + .48 = .74
Thus, the average benefit is equal to 74% of the state average weekly wage (900) = 666
Direct effect of a benefit change = 666 / 594.21 = 1.121 or 12.1%

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to the questions from the 2007 exam
40. (2.5 points) Workers compensation law changes can produce both direct and incentive (or indirect) effects.
a. (0.5 point) Explain what is meant by direct effect.
b. (0.5 point) Explain what is meant by incentive effect.
c. (0.75 point) Will implementation of cost of living adjustments have a direct effect, incentive effect,
or both? Explain your answer.
d. (0.75 point) Will changes in administrative procedures have a direct effect, incentive effect, or
both? Explain your answer.
CAS Model Solution
a. A direct effect is the direct impact on premium or losses solely due to law change not taking into account
the human response to a change. For example, if the max benefit is increased, losses will automatically go
up because those already at the max will get an increase in benefits.
b. An incentive effect is the impact a change has on premium and losses because of the change in human
behavior. For example, if the duration of benefits is lengthened, more people that are ready to go back
may malinger to get benefits longer.
c. Both. Direct – Increase in indemnity payments because they will be adjusted upwards with inflation.
Indirect – More people may stay out of work longer because their benefits are keeping up with inflation.
Previously, they may have returned to work because their benefits were not a sufficient amount.
d. Incentive effect only – Administrative procedures that make it easier to file claims may cause some to file
claims they wouldn’t have in the past.

Solutions to questions from the 2008:
Model Solution - Question 19
Step 1: Write an equation to determine the direct benefit level effect of increasing the maximum
benefit to $100.
Direct effect of a benefit change = [Avg benefit (after the change)/ Avg benefit (before the change]) – 1.0
Step 2: Write an equation to determine the average benefit (effective compensation rate).
The average benefit is computed as the sum of the following:
1. Benefits, as a % of wages, for the % of workers earning the minimum % of the SAWW.
2. Benefits, as a % of wages, for the % of workers earning at least the maximum % of the SAWW.
3. Benefits, as a % of wages, for the % of workers earning between the minimum % of the SAWW and
the maximum % of the SAWW.
Step 3: Compute the % of workers earning benefits for each of the three groups of workers identified in Step 2,
before increasing the max benefit to $100.
1. The % of workers earning the minimum % of the SAWW. With a compensation rate of .667, the
minimum benefit of $50 is received by a worker making $75 ($50/.667), and $75 as a % of the SAWW
of $100 equals .75. Using this as the lookup value for table give in the problem, 35% of workers earn
the minimum benefit.
2. The % of workers earning the maximum % of the SAWW. With a compensation rate of .667, the
maximum benefit of $67 is received by workers making at least $100 ($67/.667), and $100 as a % of
the SAWW of $100 equals 1.0. Using this as the lookup value for table give in the problem, 40%
(1.0 - .60) of workers earn at least the maximum benefit.
3. The % of wages unaffected by the min and max limits for workers earning between the minimum %
and maximum % of the SAWW. Workers between the limits earn 42% - 20% = 22% of state wages.

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2008:
Model Solution - Question 19 (continued)
Step 4: Compute benefits, as a % of wages, for each of the three groups of workers indentified in Step 2.
1. % of workers * min wages as a % of the SAWW * the compensation rate = .35 * .75 * .667 = .1751
2 % of workers * max wages as a % of the SAWW * the compensation rate = .40 * 1.0 * .667 = .2668
3 % of workers * the compensation rate = .22 * .667 = .1467
Current effective compensation rate = .1751 + .2668 * .1467 = .5886
Step 5: Repeat Steps 3 and 4 to determine the % of workers earning benefits for each of the three groups of
workers identified in Step 2, after increasing the max benefit to $100.
1. Workers earning no more than 1 half of the SAWW receive the minimum benefit. [Two thirds
of 0.75 times the SAWW equals half the SAWW which equals the min benefit.]
These benefits, as a percentage of wages, are 2/3 x .75 x 35% = 17.51%
2. Workers earning at least one and a half times the SAWW receive the maximum benefit. [Two
thirds of 1.5 times the SAWW equals the revised maximum benefit].
These benefits, as a percentage of wages, are 2/3 x 1.5 x (100% - 91%) = 9%.
3. Workers earning between one half of the SAWW and one and a half times the SAWW receive
benefits equal to two thirds of their pre-injury wages.
These benefits, as a percentage of wages, are 2/3 x (81% - 20%) = 40.69%.
Revised effective compensation rate = 9% + 17.51% + 40.96% = 67.47%
Step 6: Using the equation in Step 1, and the results from Steps 3 and 5, compute the direct benefit level effect.
Direct benefit level affect = .6747/.5886 - 1.0= .1416
b. Incentive effects are the human behavioral responses to changes in the direct effects of increasing or
decreasing benefit levels, compensation rates, etc.
c. Because increasing the maximum benefit increased the effective compensation rate, we might expect to see
longer duration injuries, since injured workers are receiving more benefit, they have less incentive to return to
work. We would also expect an increase in claims, since workers will be paid more for injuries, they will report
more injuries.

Solutions to questions from the 2009 exam:
Question 26
a. (0.5 point) Calculate the direct effect of this benefit change.
b. (0.5 point) Briefly explain two possible indirect effects of this change.
a. Before the change: benefits = (.5)(pre-tax pay)
After the change: benefits = (.85)(post-tax pay)
= (.85)(1 - .30)(pre-tax pay) =(.595)(pr- tax pay)
The direct effect of the benefits change is that benefits have increased by (.595/.5 - 1= .19 = 19%
b1 .We would expect higher frequencies, since the higher benefit will provide employees with more incentive to
file claims
b2. We would expect employees to stay on disability longer, rather than returning to work, since they will receive
higher benefits.

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2010 exam:
Question 23
a. (2 points) Calculate the direct effect of the change in maximum benefits on losses.
b. (0.5 point) Explain a potential indirect effect of the change in maximum benefits on losses.
Part a.
The key is to calculate the benefits provided before and after the change to determine the direct
effect.
The minimum benefit is 48% of the SAWW ($1,500) which equals $720 (= $1,500 x 48%).
The minimum benefit of $720 applies to workers who earn less than 60% of the SAWW (i.e. $720 =
80% x 60% x $1,500), given the current compensation rate of 80%. Min compensation =

.48
=60%
.80

The aggregate benefits for 64 employees in this category are $46,080 (= 64 x $720).
The maximum benefit is 128% of the SAWW ($1,500) and thus equals $1,920 (= $1,500 x 128%).
The maximum benefit of $1,920 applies to workers who earn more than 150% of the SAWW (i.e. $1,920 =
80% x 160% x $1,500), given the current compensation rate of 80%. Max compensation=

1.28
=160%
.80

The aggregate benefits for the 29 employees in this category are $55,680 (= 29 x $1,920).
The remaining 198 (= 144 + 33 + 21) employees fall between the minimum and maximum benefits.
This means their total benefits are 80% of their actual wages or $246,240 ( = ( 80% x 196,200 ) + ( 80% x
64,350) + ( 80% x 47,250 ) ).
The sum total of benefits is $348,000 (= $46,080 + $55,680 + $246,240) under the current benefit
structure.
Once the maximum benefit is reduced from 128% to 112% of the SAWW, more workers will be subjected
to the new maximum benefit.
Workers earning approximately >140% of the SAWW are subject to the maximum (i.e. $1,680 = (80% x
140% x $1,500) > $1,680). These 50 (= 21 + 29) workers will receive $84,000 (= 50 x $1,680) in benefits.
New compensation =

1.12
=140%
.80

Workers subject to the minimum benefit, 64, are not impacted by the change, and their benefits remain
$46,080.
There are now only 177 (= 144 + 33) employees that receive a benefit equal to 80% of their pre-injury wages or:
$208,440 (= (80% x 196,200) + (80% x 64,350)) because more workers are now impacted by the maximum.
The new sum total of benefits is $338,520 (= 84,000 + 46,080 + 208,440).
The direct effect from revising the maximum benefit is -2.724 (= 338,520/348,000 – 1.0).
Part b.
An indirect effect of lowering the max benefit would be a change in claimant behavior. Higher wage earnings may
return to work faster as their benefits would not be as favorable as they had been prior. This might compound
the decrease in total compensation.

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2012 exam:
7. Develop the projected ultimate loss and LAE for accident year 2010 losses using the data above. To
satisfy the state requirements, fully describe the rationale for the selections for loss development, loss
trend, and ULAE.
Question 7 – Model Solution 1 (Exam 5A Question 7)
Loss Development
The ‘06 12-24 factor is a one-off high valve indicating a onetime event. This should be excluded from the
selection. Also, the past 3 yrs. 24-36 avg. is stable and has decreased by an absolute 0.1 value from the
‘04 and 05 levels. All other periods are stable and relatively consistent.
Based on this, I select the Avg. last 3 yrs. as my LDF.
Loss Trend:
Frequency: The frequency over the past 12 quarters has been decreasing and leveled off in the final year.
I would check w/management about any initiatives they took to decrease the frequency. I would think,
based on the data, a process was taken and was effective at bringing freq down to the 0.065 level, but we
can expect the stable value going forward.
Freq trend = 0%
Severity : The book went through a shift in Pure premium, freq, and severity after March 2009. The PP is
significantly less implying smaller risks were written which brought down severity. After the pure premium
stabilized in June ’09 we see an increasing trend in severity. To recognize this trend, but not include the
seventy values from prior ’09 June, I would use the 6pt severity trend.
Sev Trend = 5.6%
Trend period: 7/1/2010 -> 4/1/2014
3.75
ULAE: The book went through a shift after ’08 and saw a reduction in freq/sev of claims. I would consult
the claims dept about how this is effecting their operations w/the change in the type of claims going
forward. Since ’08 is considerably different than ’09 and ’10 I would take an average of the ULAE ratio for
these years as they reflect the environment going forward. Selecting only ’10 would be based on the
results of my conversations w/claim and could overstate the true ULAE ratio.
ULAE = (15+ 15.6) / 2 = 15.3%
Ult Loss & LAE = 50k x (1.65 x 1.2 x 1.05 x 1.04) Dev x (1 + 0 + .056)^3.75 trend x 1.153 ULAE

Ultimate Loss and LAE = 152.907

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Chapter 6 – Losses and LAE
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Question 7 – Model Solution 2 (Exam 5A Question 7)
Loss Development: Notice that from 36-48 and onward, the link ratio are the same. So focus on 12-24
first. Notice that the all year average is high because of Accident Year 2006 in this maturity. This is likely
an anomaly- due to a large loss. The other years in the maturity do not seem substantially different, so
select the ex-hi/lo average. Now consider the 24-36 category. There is steady decrease in age-to-age
factors here. Given this, I would select the Average 1st 3 years average.
So selected link ratios are
12-24
24-36 36-48
48-60
60-72
72-ult
1.7
1.2
1.05
1.04
1.
1
Freq
Loss Trend: Over the last year, frequency is very stable. However, it is declining in all other years. To
balance stability of selections (represent the decreasing trend) but also be responsive (recognize that the
trend has leveled off some) I would select -2% (between the 4 and 8 point fits).
Sev
Since June 2009, severity trend has been increasing at about +6%. The negative trends appear to be
the result of the June 2008 -> March 2009 year, which has much higher severity than all other years.
Therefore, adjusting or excluding the year is appropriate. Here, I choose to exclude. Since the 6-point
and 4-pt fits are so similar, I feel a 6% is well supported.
Pure prem
Our selections imply a (1.06) * .98 = 1.0388 => 3.88% pure premium trend. Looking at the pure premium
and excluding the data points from June 2008 to March 2009, we can see that a 3.88% will balance
stability and reasonableness - it falls between the 6 and 4 point fits. Thus, a 3.88% pure premium trend
is appropriate.
ULAE No compelling reason is seen in regards to differences in paid.
Loss and ALAE by year. The ULAE ratio does seem to be going, but it could be skewed by the fact that
ULAE is more responsive to claim volume growth than Paid loss is (since paid loss is often from accidents
occurring in prior years).
So, 15.6% is not appropriate, but 14.5% would not be either without more information on the claims dept.
So we select on all-year average of 15% ULAE ratio, which has the added benefit of being explainable to
regulators.
Avg. date of loss
Avg. date of future loss
Our trend paired is from
7/1/2010
->
4/1/2014, 3.75 years
Ultimate projected loss of LAE = 50,000 x 1.7 x 1.2 x 1.05 x 1.04 x 1.0388 ^ (3.75) x 1.15 = 147,745.90
Examiner’s Comments
Candidates generally justified the loss development factor selections well. Some candidates did lose
credit for not including justification. Occasionally candidates’ factors did not match the justification,
resulting in the loss of points. Most candidates were able to identify the flat frequency trend and picked a
four-point trend. The most common error was selecting a longer projection period without justification of
why a decreasing trend was reasonable given the latest points. Many candidates failed to mention either
the shock loss or the increasing pattern for severity in recent periods. Some candidates incorrectly
calculated the trend period. Some candidates failed to provide justification for the ULAE selection. Most
candidates projected ultimate loss and LAE correctly.

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Chapter 7 – Expenses and Profit
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Sec
1
2
3
4
5
6
7
8
9
10
1

Description
Simple Example
Underwriting Expense Categories
All Variable Expense Method
Premium-Based Projection Method
Exposure/Policy-based Projection Methods
Trending Expenses
Reinsurance Costs
Underwriting Profit Provision
Permissible Loss Ratios
Key Concepts

Pages
125 – 126
126 – 127
127 – 130
130 – 133
133 – 135
135 – 137
137 – 137
138 – 138
139 – 139
139 - 139

Simple Example

125 – 126

How expenses and profit are incorporated within the fundamental insurance equation in the ratemaking process.
Assume the following:
__



The average expected loss and LAE ( L  EL ) for each policy is $180.



The insurer incurs $20 in expenses ( E F ) for costs associated with printing and data entry, etc. each
time it writes a policy.
15% of each dollar of premium collected covers expenses that vary with the amount of premium, (V),
(e.g. premium taxes).




Company management has determined that the target profit provision ( QT ) should be 5% of premium.

If the rates are appropriate, the premium collected will be equivalent to the sum of the expected losses, LAE,
underwriting (UW) expenses (both fixed and variable), and the target underwriting profit.
Using the notation below, the fundamental insurance equation can be re-written.

= Exposures
__

P; P
V

= Premium; Average premium(P divided by X)
= Variable expense provision(EV divided by P)

= Target profit percentage

L; L

= Losses; Pure Premium(L divided by X)
__

EL ; EL = Loss Adjustment Expense(LAE); Average LAE per exposure(EL divided by X)
___

EF ; EF = Fixed underwriting expenses; Average underwriting expense per exposure  EF divided by X 
EV

= Variable underwriting expenses

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Chapter 7 – Expenses and Profit
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Premium = Losses + LAE + UW Expenses + UW Profit

P  L  EL  ( EF  V * P)  QT * P
P - (V  QT ) * P  L  EL  EF
P

[ L  EL  E F ]
[1.0 - V - QT ]




[ L  EL  EF ] / X [ L  E L  E F ]
P

[1.0 - V - QT ]
[1.0 - V - QT ]


Substituting the values from the example into the formula produces the following premium:




L  E L  EF
[$180  $20]
P

 $250
[1.0  V  QT ] [1.0  0.15  0.05]


The company should charge $250, composed of $180 of expected losses and LAE, $20 of fixed expenses,
$37.50 (= 15% x $250) of variable expenses, and $12.50 (= 5% x $250) for the target UW profit.
This chapter focuses on determining the fixed expense provision (i.e. $20), the variable expense provision (i.e.
15%), and the profit provision (i.e. 5%).

Underwriting Expense Categories

126 – 127

Underwriting expenses (or operational and administrative expenses) are usually classified into the
following four categories:
• Commissions and brokerage
• Other acquisition
• Taxes, licenses, and fees
• General
1. Commissions and brokerage:
 are paid as a percentage of premium written.
 may vary between new and renewal business.
Contingent commissions vary based on the quality (e.g. a loss ratio) or amount of business written
(e.g. predetermined volume goals).
2. Other acquisition costs (e.g. media advertisements, mailings to prospective insureds, and salaries of
sales employees who do not work on a commission) are expenses to acquire business other than
commissions and brokerage expenses.
3. Taxes, licenses, and fees (e.g. premium taxes and licensing fees) include all taxes and miscellaneous
fees due from the insurer excluding federal income taxes.
4. General expenses (e.g. overhead associated with the insurer’s home office (e.g. building
maintenance) and salaries of certain employees (e.g. actuaries)) include the expenses associated with
insurance operations, excluding investment income expenses.

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The u/w expense provision is further divided into two groups: fixed and variable.
Fixed expenses (e.g. overhead costs associated with the home office) are assumed to be the same for
each risk, regardless of premium size (i.e. the expense is a constant dollar amount for each risk or policy).
Variable expenses (e.g. premium taxes and commissions) vary directly with premium and thus are
constant percentage of the premium.
The magnitude and distribution of underwriting expenses vary significantly for different lines of business.
 Commissions tend to be much higher in lines that require a comprehensive inspection at the
onset of the policy (e.g. large commercial property) than for lines that do not involve such activity
(e.g. personal auto).
 Expenses can even vary significantly by company within a given line of business.
i. A national direct writer may incur significant other acquisition costs for advertising.
ii. An agency-based company may rely more heavily on the agents to generate new business; which
should lower other acquisition costs, but might be partially offset by higher commission expenses.
Three different procedures used to derive expense provisions for ratemaking:
 All Variable Expense Method
 Premium-based Projection Method
 Exposure/Policy-based Projection Method

All Variable Expense Method

127 – 130

The All Variable Expense Method treats all expenses as variable (i.e. all expenses are assumed to be a
constant percentage of premium). This method:
 assumes that expense ratios during the projected period will be consistent with the historical
expense ratios (i.e. all historical underwriting expenses divided by historical premium).
 is widely used when pricing products for which the total u/w expenses are dominated by variable
expenses (i.e. commercial lines products).
The table below shows an example of this method for deriving the other acquisition expense provision of
a commercial general liability insurer.
Other Acquisition Provisions Using All Variable Expense Method

a Countrywide Expenses
b Countrywide Written Premium
c Variable Expense % [(a)/(b)]

2013
2014
$72,009
$104,707
$1,532,091 $1,981,109
4.7%
5.3%

3-Year
2015
Average
$142,072
$2,801,416
5.1%
5.0%

Selected

5.0%

Historical CY expenses are divided by either CY written or earned premium during the same historical
experience period.

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The choice to use WP or EP depends on whether the expenses are incurred at the onset of the policy
(e.g. commissions) or throughout the policy (e.g. building maintenance).
 WP is used when expenses are incurred at policy inception (as it reflects the premium at the
onset of the policy).
 EP is used when expenses are assumed to be incurred throughout the policy (as it reflects the gradual
payment of expenses that can be proportional to the earning of premium over the policy term).
 The choice of WP or EP has little impact if an insurer’s volume of business is not changing materially
(since WP is approximately to EP).
 If the insurer is growing (or shrinking) significantly, WP will be proportionately higher (or lower) than EP.
Also, acquisition costs will be higher (or lower) during a period of stable volume.
 Use of an appropriate premium measure provides a better match to the types of expenses incurred
during the historical period.
The Annual Statement and Insurance Expense Exhibit (IEE) contain historical expense and premium data.
However, this data may not be available in the level of detail needed for ratemaking purposes (e.g.
homeowners data includes renters and mobile homes data, and as a result, may not be appropriate for
deriving expense provisions specifically for homeowners policies).
The choice to use countrywide or state data varies by type of expense.
 Other acquisition costs and general expenses are assumed to be uniform across all locations, so C/W
data from the IEE are used to calculate these ratios.
 The data used to derive commissions and brokerage expense ratios varies from carrier to carrier (e.g.
some insurers use state-specific data and some use C/W data, depending on whether the insurer’s
commission plans vary by location).
 TL&F vary by state and the expense ratios are based on state data from the Annual Statement.
Data Summarization for All Variable Expense Method
Expense
Data Used
Divided By
General Expense
Countrywide
Earned Premium
Other Acquisition
Countrywide
Written Premium
Commissions and Brokerage
Countrywide/State
Written Premium
Taxes, Licenses, and Fees
State
Written Premium
Historical expense ratios for each category and year are calculated.
The selected ratio is based on either the latest year’s ratio or a multi-year average of ratios along with
management input, prior expense loads, and judgment.
Since the ratemaking process is a projection of future costs, the actuary should select an expense ratio
consistent with what is expected in the future (examples of this are as follows):
• If the commission structure is changing, use the expected commission percentage.
• If productivity gains led to a reduction in staffing levels during the historical experience period, then the
selected ratios should be based on the expected expenses after the reduction vs. an all-year average.
• A growing portfolio can cause expense ratios to decrease (since volume will increase faster than
expenses); however, if the insurer plans to open a new call center to handle greater planned growth,
consider that fixed costs will increase in the short-term until the planned growth is achieved.
If there were non-recurring expenses during the historical period, examine the materiality and nature of the
expense to determine how to best incorporate the expense in the rates (if at all).

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A few states place restrictions on which expenses can be included when determining rates (e.g. not allowing an
insurer to include charitable contributions or lobbying expenses in its rates).
This procedure described is repeated for each of the expense categories, and the sum of the selections is the
total expense provision. This provision is used directly in the loss ratio or pure premium rate level indication
formulae (see Chapter 8).
Potential Distortions Using this Approach
By treating all expenses as variable, this understates the premium need for risks with a relatively small policy
premium and overstates the premium need for risks with relatively large policy premium.
Assume the $20 of fixed expense ( E F ) is included as a percentage with the other 15% of variable expenses (V).
The $20 as a ratio to premium is 8% (= $20 / $250).
Treating all expenses as variable, the premium calculation becomes:


P



L  EL


[1.0  (V  ( EF / P )  QT ]



$180
 $250
[1.0  (0.15  0.08)  .005]

Since the fixed dollar amount of $20 is exactly equivalent to 8% of $250 (i.e. the provision for the average risk),
this approach produces the same result (i.e. $250) as the example that had the fixed expense included in the
numerator as a fixed dollar amount.
The table below shows the results of the two methods for risks with a range of average premiums.
Results of All Variable Expense Method
Correct Premium
All Variable Expense Method
Variable
Variable
Expense
Expense
Fixed
And
Fixed
And
Loss Cost Expense Profit
Premium Expense Profit Premium
%Diff
$135
$20
20%
$193.75
$28%
$187.50
-3.2%
$180
$20
20%
$250.00
$28%
$250.00
0.0%
$225
$20
20%
$306.25
$28%
$312.50
2.0%
The All Variable Expense Method undercharges risks with premium less than the average and
overcharges the risks with premium more than the average.
Therefore, insurers that use this approach may implement a premium discount structure that reduces the
expense loadings based on the amount of policy premium charged.
 This is common for WC insurers (see Chapter 11).
 Some insurers using the All Variable Expense Method may also implement expense constants to
cover policy issuance, auditing, and handling expenses that apply uniformly to all policies.

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Chapter 7 – Expenses and Profit
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
4

Premium-Based Projection Method

130 – 133

For insurers with a significant amount of both fixed and variable u/w expenses, the premium based
projection method is used since it recognizes the two types of expenses separately.
 Like the All Variable Expense Method, it assumes expense ratios during the projected period will
be consistent with historical expense ratios
 The enhancement is that this approach calculates fixed and variable expense ratios separately
(as opposed to a single variable expense ratio) so that each can be handled more appropriately
within the indication formulae.
General Expense Provisions Premium-Based Projection Method
2013
a Countrywide Expenses
b Countrywide Earned Premium
c Ratio [(a) / (b)]

2014

2015

$26,531,974 $28,702,771 $31,195,169
$450,000,000 $490,950,000 $530,000,000
5.9%
5.8%
5.9%

d % Assumed Fixed
e Fixed Expense % [(c ) x (d)]
f Variable Expense % [(c ) x (1.0-(d))]

3-Year
Average

5.9%

Selected

5.9%
75.0%
4.4%
1.5%

Step 1: Determine the % of premium attributable to each expense type by dividing historical underwriting
expenses by EP or WP for each year during the historical experience period.
Here, general expenses are assumed to be incurred throughout the policy period, and thus are
divided by EP.
Step 2: Choose a selected ratio (e.g. if the ratios are stable over time, a 3-year average may be chosen;
if the ratios demonstrated a trend over time, the most recent year’s ratio or some other value may
be selected).
Step 3: Divide the selected expense ratio into fixed and variable ratios (using detailed expense data so
that this division can be made directly, or using activity-based cost studies that help split each
expense category appropriately).
The example assumes 75% of the general expenses are fixed, and that percentage is used to
split the selected general expense ratio of 5.9% into a fixed expense provision of 4.4% and a
variable expense provision of 1.5%.
Step 4: Sum the fixed and variable expense ratios across the different expense categories to determine
total fixed and variable expense provisions.
If the average fixed expense per exposure (required for the pure premium approach discussed in
Chapter 8) is needed, the fixed expense provision can be multiplied by the projected average premium.
Fixed Expense Per Exposure = Fixed Expense Ratio x Projected Average Premium
Potential Distortions Using this Approach
This approach assumes that historical fixed and variable expense ratios will be the same as in the projected
period. . (Note: Recall that an actuary CAN select other than the historical ratios.)
However, the fixed expense ratio will be distorted if the historical and projected premium levels are materially
different.

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Situations that can cause such a difference to exist:
1. Recent rate increases (or decreases) implemented during or after the historical period will tend to overstate (or
understate) the expected fixed expenses.
Also, using 3-year historical expense ratios increases the chances of rate changes not being fully reflected in
the historical premium.
Solution: Restate historical written or earned premium at current rate level (see Chapter 5).
2. Distributional shifts that have increased the average premium (e.g. shifts to higher amounts of insurance) or
decreased the average premium (e.g. shifts to higher deductibles) will tend to overstate or understate the
estimated fixed expense ratios, respectively.
Using 3-year historical expense ratios increases the impact of these premium changes by increasing the
amount of time between the historical and projected periods.
Solution: Trend historical premium to prospective levels (see Chapter 5).
3. Countrywide expense ratios that applied to state projected premium to determine the expected fixed expenses
can create inequitable rates for regional or nationwide carriers.
 This process allocates fixed expenses to each state based on premium.
 However, the average premium level in states varies due to overall loss cost differences (e.g. coastal
states tend to have higher overall homeowners loss costs) as well as distributional differences (e.g. some
states have a significantly higher average amount of insurance than other states).
 If significant variation exists in average rates across the states, estimated fixed expenses will be
overstated in higher-than-average premium states and understated in the lower-than-average average
premium states.
Assume the historical fixed expense ratio was calculated when the average premium level was $200 rather than
$250, then the historical expense ratio is 10% (= $20 / $200).
If the 10% is applied to the premium at current rate level, the projected dollars of fixed expense will be $25
(=$10% x $250), and the overall indicated average premium will be overstated:




[ L  EL  EF ]
[$180  $25]
P

 $256.25
[1.0  V  QT ] [1.0  0.15  0.05]


Alternatively, the actuary can use a fixed expense projection method based on exposures or number of policies.

Exposure/Policy-based Projection Methods

133 – 135

Variable expenses are treated the same way as the Premium-based Projection Method, but historical
fixed expenses are divided by historical exposures or policy count rather than premium.
If fixed expenses are assumed to be constant:
 for each exposure, historical expenses are divided by exposures.
 for each policy, historical expenses are divided by the number of policies.

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The table below shows the development of the fixed and variable expenses for the general expenses category.
(although the example uses exposures, the procedure is the same if policy counts are used instead.)
General Expense Provisions Using Exposure-Based Projection Method
2013
a Countrywide Expenses
b % Assumed Fixed
c Fixed Expense $ [(a) x (b)]
d Countrywide Earned Exposures
e Fixed Expense Per Exposure [(c) / (d)]
f Variable Expense $ [(a) x (1.0-(b))]
g Countrywide Earned Premium
h Variable Expense % [(f) / (g)]





$26,531,974

2014

2015

$28,702,771

3-Year Selected
Average

$31,195,169
75.0%

$19,898,981 $21,527,078
4,378,500
4,665,500
$4.54
$4.61
$ 6,632,994
$ 7,175,693
$450,000,000 $490,950,000
1.5%
1.5%

$23,396,377
4,872,000
$4.80 $4.65
$ 7,798,792
$545,250,000
1.4% 1.5%

$4.65

1.5%

Expenses are split into variable and fixed components (the assumption that 75% of GE are fixed is used).
Fixed expenses are then divided by the exposures for that same time period.
GEs are assumed to be incurred throughout the policy and thus are divided by earned exposures to
determine an average expense per exposure for the indicated historical period.

Data Summarization for Exposure/Policy-Based Projection Method
Divided By
Expense
General
Other Acquisition
Commissions and Brokerage
Taxes, Licenses, and Fees







Data Used

Fixed

Countrywide
Countrywide
Countrywide/State
State

Earned Exposure
Written Exposure
Written Exposure
Written Exposure

Variable
Earned Premium
Written Premium
Written Premium
Written Premium

Selected expense ratios are based on either the latest year or a multi-year average.
Similar values for the projected average expense per exposure imply expenses are increasing or
decreasing proportionately to exposures.
If the insurer is growing and the projected average expense per exposure is declining each year, then
expenses may not be increasing as quickly as exposures due to economies of scale.
Non-recurring expense items, one-time changes in expense levels, or anticipated changes in
expenses should be considered in the selection process.
If the rate level indication approach requires that the fixed expense be expressed as a percentage of
premium (i.e. when using the loss ratio approach, see Chapter 8), then the average fixed expense per
exposure should be divided by the projected average premium.

Projected Fixed Expense Ratio =

Average Projected Fixed Expense Per Exposure
Projected Average Premium

Variable expense ratios (variable expenses divided by historical premium) are treated the same way under both
the Premium-based and Exposure/Policy-based Projection Methods.
The three-year average variable expense provision is selected in the example above.

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Other Considerations/Enhancements
Shortcomings with the Exposure/Policy-based Projection Method
1. First, the method requires the actuary to split the expenses into fixed and variable portions (like the
Premium-based Projection Method and is done judgmentally).
Activity-based cost studies will more accurately segregate expenses.
Sensitivity testing shows that the overall indication not materially impacted by moderate swings in % of
expenses.

2. The method allocates countrywide fixed expenses to each state based on the exposure or policy distribution
by state (as it assumes fixed expenses do not vary by exposure or policy).
However, average fixed expense levels may vary by location (e.g. advertising costs may be higher in some
locations than others).
Note: If the insurer collects data at a finer level to make more appropriate adjustments, the cost of the data
collection should be balanced against the additional accuracy gained.
3. Some expenses considered fixed actually vary by certain characteristics (e.g. fixed expenses may vary
between new and renewal business).
 This only affects the overall statewide rate level indication if the distribution of risks for that
characteristic is either changing dramatically or varies significantly by state, or both.
 Any material fixed expense cost difference not reflected in the rates will impact the equity of the two
groups (even if there is no impact on the overall rate level indication).
 Material differences in new and renewal provisions should be reflected with consideration given to
varying persistency levels as described by Feldblum in “Personal Automobile Premiums: An Asset
Share Pricing Approach for Property/ Casualty Insurers” (Feldblum 1996). This article is part of the
2010 CAS Exam 5 Syllabus.
4. The existence of economies of scale in a changing book may lead to increasing or decreasing projected
average fixed expenses.
Internal expense trend data and actuarial judgment should suffice for incorporating the impact of economies
of scale.

Trending Expenses

135 – 137

Expenses are expected to change over time due to inflationary pressures and other factors.
 Since variable expenses automatically change as the premium changes, there is no need to trend the
variable expense ratio.
 However, average fixed expense per exposure or policy are expected to increase over time due to
inflation.
In the Premium-based Projection Method:
 If the average expenses and average premium are changing at the same rate, then the fixed expense
ratio will be consistent and no trending is needed.
 However, if average fixed expenses are changing at a different rate than average premium, then the
fixed expense ratio needs to be trended.

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BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
In the Exposure/Policy-based Projection Method:
 If an inflation-sensitive exposure base (e.g. payroll per $100) is used, no trending is needed if the
expenses and exposure base are changing at the same rate.
 If a non-inflation sensitive base (e.g. car-year or house-year) or policy counts are used, average fixed
expenses are expected to change over time and trending is appropriate.
Data used:
 Some insurers use internal expense data (examining the historical change in average expenses) to
select an appropriate trend.
 However, internal data maybe volatile and insurers may use government indices (e.g. Consumer Price
Index, Employment Cost Index, etc.) and knowledge of anticipated changes in company practices to
estimate an appropriate trend (see the procedure in Appendix B).
Trending:
The selected fixed expense ratio will be trended from the average date that expenses were incurred in the
historical expense period to the average date that expenses will be incurred in the forecast period of the rates.
 Expenses incurred at policy inception should be trended from the average date that the policies were
written in the historical period to the average written date in the projection period.
 Assume annual policies are sold, a steady book of business is maintained, and projected rates will be in
effect for one year:
Expenses Incurred at the onset of the Policy



Expenses incurred evenly throughout the policy period should be trended from the average date the
policies were earned in the historical period to the average earned date in the projection period.
Expenses Incurred Throughout Policy

Points in time:
Since the experience period is a calendar year, the average date the policies are written and earned is the same.
However, expenses incurred throughout the policy are trended 6 months longer than expenses incurred at
inception.

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To simplify, actuaries make the assumption that all expenses are incurred either a policy inception or evenly
throughout the policy period.
After trending, the expense ratio or average dollar amount of expense is called the projected (or trended) fixed
expense provision.

Reinsurance Costs

137 – 137

Some ratemaking analysis is now performed on a net basis as reinsurance programs have become more
extensive and reinsurance costs have increased substantially.
In proportional reinsurance, the same proportion of premium and losses to the reinsurer so this type of
reinsurance may not need to be explicitly considered in ratemaking analysis.
With non-proportional reinsurance, projected losses are reduced for any expected non-proportional
reinsurance recoveries. However, the cost reinsurance must be included too. This is done by:
 reducing the total premium by the amount ceded to the reinsurer, or
 the net cost of the non-proportional reinsurance (i.e. the cost of the reinsurance minus the expected
recoveries) may be included as an expense item in the overall rate level indication.

Underwriting Profit Provision

138 – 138

By writing insurance, insurers assume risk and must maintain capital (which includes a reasonable profit
provision in their rates) to support that risk.
Total profit is the sum of investment income and underwriting profit: Total Profit = II + UW Profit.
Investment Income (II)
Two sources of II are: II on capital and II on policyholder-supplied funds (PHSF).
Insurer capital funds:
 belonging to insurance company owners is known as equity.
 are also known as policy holder surplus (PHS) although the funds may be from investors rather than
policyholders.
Insurers invest these funds and earn II (although disagreement exists as to whether this source of income should be
included in ratemaking or not).
Insurers invest money from 2 types of PHS: unearned premium reserves and loss reserves.
Insurers’ invest:
 premiums paid at policy inception (i.e. unearned premium) until it is earned.
 funds to pay for claims that have occurred, but have not yet been settled (i.e. loss reserves).

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Investment time period:
 For short-tailed lines (e.g. personal auto collision coverage or HO insurance), there is a short time
between the payment of premium and the settling of claims, and II will be relatively small.
 For long-tailed lines (e.g. personal auto BI or WC) there may be years between the time the premium is
paid and all claims are settled with the opportunity for II to become much larger.
Projection of II is an advanced topic and is outside of the scope of this text.
Underwriting Profit
UW Profit = Premium - Losses - LAE - UW Expenses
The actuary determines the UW profit needed to achieve the target rate of return after consideration of II.
 For some long-tailed lines, II may be large enough that insurers can accept an UW loss and still achieve
the target rate of return.
 For short-tailed lines, II is lower and the UW profit is a larger portion of the total return.

Permissible Loss Ratios

139 – 139

The expense and profit provisions are used to calculate a variable permissible loss ratio (VPLR) and the total
permissible loss ratio (PLR).
The variable PLR is calculated as follows:
VPLR = 1.0 - Variable Expense % - Target Profit% = 1.0 – V – QT.
 This represents the % of each premium dollar to pay for the projected loss and LAE and projected
fixed expenses.
 The remaining portion of each premium dollar is intended to pay for variable expenses and for profit
The total PLR is calculated as follows:
PLR = 1.0 - Total Expense % - Target Profit% = 1.0 – F – V – QT
 This represents the % of each premium dollar to pay for the projected loss and LAE.
 The remaining portion of each premium dollar is intended to pay for all UW expenses and for profit
If all expenses are treated as variable expenses, the VPLR and PLR are the same.
These ratios are used in the calculation of the overall rate level indications (see Chapter 8).

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10

Key Concepts

139 - 139

1. Types of underwriting expenses
a. Commissions and brokerage
b. Other acquisition costs
c. Taxes, licenses, and fees
d. General expenses
2. Fixed and variable expenses
3. Expense projection methods
a. All Variable Expense Method
b. Premium-Based Projection Method
c. Exposure/Policy-Based Projection Method
4. Expense trending
5. Reinsurance costs
6. Underwriting profit provision
7. Permissible loss ratios
a. Variable permissible loss ratios
b. Total permissible loss ratios

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Chapter 7 – Expenses and Profit
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
The predecessor papers to the current syllabus reading “Basic Ratemaking” by Werner, G.
and Modlin, C. were numerous. While past CAS questions were drawn from prior syllabus
readings, the ones shown below remain relevant to the content covered in this chapter.
By relevant, we mean the concepts tested on past CAS exams relating to expenses and profits
are similar to the concepts found in this chapter relation to expenses and profits.
Questions from the 1996 exam
Question 3.

You are given:
• Rate per unit exposure
• Pure premium including loss adjustment expense
• General expense ratio
• Other acquisition expense ratio
• Commission expense ratio
• Taxes, licenses and fees ratio
• Profit and contingencies ratio

$120
$75
7.0%
3.0%
15.0%
3.0%
5.0%

• 80% of general and other acquisition expenses are considered to be fixed expense.
Using the pure premium method described by McClenahan, chapter 2, "Ratemaking," Foundations of Casualty
Actuarial Science, in what range does the fixed expense per exposure that is incorporated into the rate fall?
A. < $6 B. > $6, but < $9 C. > $9, but < $12 D. > $12, but < $15
E. > $15

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Chapter 7 – Expenses and Profit
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2004 exam:
Question 33
b. (1 point) Expenses can be related to written or earned premium. Briefly explain why other acquisition
expenses are related to written premium, while general expenses are related to earned premium.

Questions from the 2005 exam
43. (4 points) Use Werner's proposed methodology in "Incorporation of Fixed Expenses" and the
information below to answer the following questions for the projected annual policy period beginning
July 1, 2005. Show all work.
Statewide Projected Average Premium at Present Rates
$850.00
Statewide Projected Loss and LAE Ratio
68.0%
Profit and Contingencies Provision
5.0%
Annual Fixed Expense Trend
3.0%

Countrywide General Expenses
Fixed General Expense as percentage of General Expenses
Countrywide Earned Exposures
Countrywide Written Exposures
Countrywide Earned Premium
Countrywide Written Premium
Fixed
Variable
Other Acquisition
$60.00 2.5%
Taxes, Licenses, and Fees
$ 2.50
2.0%
Commissions and Brokerage
None
12.0%

Annual Policy Period
2003
2004
$25,000
$28,000
75%
75%
625
645
640
700
$435,000
$450,000
$460,000
$475,000

• Assume expenses are incurred evenly throughout the policy period.
a. (2 points) Calculate the fixed expense provision.
b. (1 point) Calculate the variable expense provision.
c. (1 point) Calculate the statewide indicated rate change.

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Chapter 7 – Expenses and Profit
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2006 exam
33. (3 points) Given the following private passenger automobile ratemaking data for the past three
calendar years, answer the following questions.

Written Premium
Earned Premium
Commissions
General Expenses
Home Office Salaries
Home Office Utilities
One-Time Expense associated
with Reduction in Staff
All Other General Expenses
Total General Expenses
Other Acquisition Expenses
Taxes, Licenses, and Fees

Calendar Year
2003
2004
$20,000,000 $25,000,000
19,000,000
24,000,000
3,000,000
3,750,000

2005
$30,000,000
28,000,000
3,000,000

798,000
209,000

1,056,000
216,000

1,008,000
280,000

0
190,000
1,197,000
1,780,000
500,000

360,000
240,000
1,872,000
2,175,000
625,000

0
280,000
1,568,000
2,640,000
750,000

a. (1 point) Beginning on January 1, 2005 all policies written and renewed had commissions changed in
order to allow the company to compete more effectively. This new commission rate is expected to
continue into the future.
As the actuary for this insurance company, briefly explain the commission provision you would
recommend for use in the next rate revision to be effective July 1, 2006. Show all work.
b. (2 points) As shown in the table above, during 2004 the company paid a one-time expense associated with
a reduction in staff. This reduction was due to increases in productivity and resulted in fewer employees
during 2005. This new level of staffing is expected to continue.
As the actuary for this insurance company, briefly explain the general expense provision you
would recommend for use in the next rate revision to be effective July 1, 2006. Show all work.

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BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2008 exam
23. (3.0 points)
a. (0.5 point) Briefly define fixed expense and variable expense.
b. (2.0 points) You are given the following information:

General Expense
Other Acquisition
Commissions & Brokerage
Taxes, Licenses & Fees

Historical Expenses
$100,000
$66,000
$110,000
$40,000

Percent
Assumed
Fixed
60%
50%
0%
25%

 Historical written premium = $1,100,000
 Historical earned premium = $1,000,000
 Projected loss & LAE ratio = 75%
 Profit provision = 5%
 General expense and taxes, licenses & fees are throughout the policy.
 Other acquisition and commissions & brokerage to occur at the onset of the policy.
Calculate the indicated rate change.
c. (0.5 point) Identify a situation that could impact the appropriateness of the historical fixed expense ratio for
projection purposes and briefly explain the impact on the estimated fixed expenses.

Questions from the 2010 exam
25. (1.5 points) Identify and explain two potential distortions with using the premium-based projection method
to determine expense ratios. In the explanation, include discussion of the direction of the distortion.

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Question 3. Calculate the fixed expense per unit of exposure, EF :
___

PI = rate per unit of exposure, and is given as $120
_________

L  EL = pure premium, and is given as $75.

 _________ ____ 
___
 L  EL  EF 
PI 
1.0  V  QT 

____

EF = fixed expense per exposure, which is what needs to be solved for.

V = variable expense factor, which requires some computation.
QT = profit and contingencies factor, and is given as .05.

The variable expense load is comprised of commissions, taxes, licenses and fees, and as stated in
the problem, 20% of the general and other acquisition expense ratio.
V = 0.15 + 0.03 + 20% (0.07) + 20% (0.03) = 0.20 (Fast solving hint: note that 20% of the sum of other
acq/gen expenses(10%) is 2%. Added to taxes of 3% is 5%, Added to commission of 15% is 20%.)

____
$75  E
____
F
. EF = 15. Answer E.
Therefore, $120 
1.0 - [.15  .03  (.07  .03) *.20)  .05]
Solutions to questions from the 2004 exam:
Question 33
b. (1 point) Expenses can be related to written or earned premium. Briefly explain why other acquisition
expenses are related to written premium, while general expenses are related to earned premium.
Other acquisition expenses are assumed to be incurred mainly at the beginning of the policy,
due to the effort/process of “acquiring” the policy, so it makes more sense to relate it to Written
Premium.
General expenses (e.g. salary/overhead) would continue to be incurred even if policies ceased to be
written, so it makes more sense to relate it to Earned Premium.

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BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2005 exam:
a. (2 points) Calculate the fixed expense provision.
This question can be answered by referencing Exhibit 2-A Sheet i and Exhibit 2B from the Werner article.
Create a table similar to the one below to compute the general fixed expense provision per exposure.

(3)=(1)*(2)
(5)=(3)/(4)

(8)=(6)(7)
(9)=(5)*(8)

(1) Total CW General Expenses (IEE)
CALCUATION: GEN FIXED EXP PROV PER EXPOSURE:
(2) Fixed General Expense as % of Total General Expense
(3) Fixed General Expense $
(4) Total CW Earned Exposures
(5) Average Fixed General Expense Per Exposure
(6) Expense Trend
(7) Trend Period from 7/1/XX to 7/1/06)
(8) Expense Trend Factor
(9) Projected Average Fixed General Expense Per Exposure

2003
$25,000

2004
$28,000

75.0%
$18,750
625
$30.00
1.03
3
1.0927
$32.78

75.0%
$21,000
645
$32.56
1.03
2
1.0609
$34.54

2-Yr Straight
Average

$33.66

Total fixed expense provision = projected average fixed general expense per exposure + other acquisition
expenses + Taxes, licenses, and fees = $33.66 + $60.00 + $2.50 = $96.16
b. (1 point) Calculate the variable expense provision.
This question can be answered by referencing Exhibit 2-A Sheet i and Exhibit 2B from the Werner article.
Create a table similar to the one below to compute the general variable expense provision
CALCULATION: GEN VARIABLE EXP PROV
(10) Variable Gen Expense as % of Total General Expense
1.0 - (2)
(11)=(1)*(10) (11) Variable General Expense $
(12) CW Earned Premium
(13)=(11)*(12) (13) Variable General Expense %

2-Yr Straight
Average
25.0%
$6,250
$435,000
1.44%

25.0%
$7,000
$450,000
1.56%

1.50%

Total variable expense provision = variable general expense % + variable other acquisition expenses + variable
Taxes, licenses, and fees + variable commission and brokerage = 1.5% + 2.5% + 2.0% + 12.0% = 18.0%
c. (1 point) Calculate the statewide indicated rate change.
This question can be answered by referencing Exhibit 2-C from the Werner article. Create a table similar
to the one below to compute the statewide indicated rate change.

Calculation of Indicated Rate Change
(1) Statewide Projected Average Premium at Present Rates
(2) Statewide Projected Loss & LAE Ratio
(3) Statewide Projected Average Loss & LAE
(3)=(1)*(2)
(4) Projected Average Fixed Expense Per Exposure
(5) Variable Expense Provision
(6) Profit and Contingencies Provision
1.0-(5)-(6)
(7) Variable Permissible Loss Ratio [100%-(5)-(6)]
(8)=[(3)+(4))]/(7)
(8) Statewide Projected Average Required Premium
(9)=(8)/(1)-1.0 (9) Indicated Rate Change

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$850.00
68.0%
$578.00
$96.16
18.0%
5.0%
77.0%
$875.49
3.0%

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Chapter 7 – Expenses and Profit
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2006 exam
Question 33.
a. (1 point) Beginning on January 1, 2005 all policies written and renewed had commissions changed in
order to allow the company to compete more effectively. This new commission rate is expected to
continue into the future.
As the actuary for this insurance company, briefly explain the commission provision you would
recommend for use in the next rate revision to be effective July 1, 2006. Show all work.
b. (2 points) As shown in the table above, during 2004 the company paid a one-time expense associated with
a reduction in staff. This reduction was due to increases in productivity and resulted in fewer employees
during 2005. This new level of staffing is expected to continue.
As the actuary for this insurance company, briefly explain the general expense provision you
would recommend for use in the next rate revision to be effective July 1, 2006. Show all work.
CAS Model Solution
a. Use the 2005 commission ratio because it is most indicative of the future. Use written premium because
commissions are generally paid at onset of policy.
3,000,000 / 30,000,000 = 10%
b. Use 3-year averages for home office utilities and all other general expense. Use the 2005 ratio for salaries
to reflect the new staffing level.
Ignore the one-time expense since it is non-recurring.
Use earned premium since general expenses are usually incurred throughout the policy period.
The general expense provision that I would recommend for use in the next rate revision to be effective
July 1, 2006 is computed as follows:
Utilities = [(209,000/19,000,000) + (216,000/24,000,000) + (280,000/28,000,000)]/3 = 1.0%
All other = {(190,000/19,000,000) + (240,000/24,000,000) + (280,000/28,000,000)]/3 = 1.0%
Salaries = 1,008,000/28,000,000 = 3.6%
Total = 1.0% + 1.0% + 3.6% = 5.6%

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Chapter 7 – Expenses and Profit
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2008 exam
Model Solution - Question 23 Initial comments: Actuaries generally divide underwriting expenses into two
groups: fixed and variable. Fixed expenses are those expenses that are assumed to be the same for each
exposure, regardless of the size of the premium (i.e., the expense is a constant dollar amount for each risk).
Typically, overhead costs associated with the home office are considered a fixed expense.
Variable expenses are those expenses that vary directly with premium; in other words, the expense is a constant
percentage of the premium. Premium taxes and commissions are two good examples of variable expenses.
a. A fixed expense is an expense that is incurred that does not vary with premium. A variable expense is an
expense that is incurred that varies with the amount of premium. A better solution is as follows:
Fixed expenses (e.g. overhead costs associated with the home office) are assumed to be the same for
each risk, regardless of premium size (i.e. the expense is a constant dollar amount for each risk or policy).
Variable expenses (e.g. premium taxes and commissions) vary directly with premium and thus are
constant percentage of the premium.
b. Calculate the indicated rate change.
Step 1: Write an equation to determine the indicated rate change.

Indicated Rate Change =

Projected L + LAE Ratio + Fixed Expense ratio
1.0 -V - Q

Step 2: Using the given expense data in the problem, compute the fixed and variable expense ratio.
Note: Since other acq. and commissions & brokerage are assumed to occur at the onset of the policy,
these expenses are related to written premiums, while all other expenses are related to E premium.

Fixed expense ratio=

.6(100k) .5(66k) .25(40k)
+
+
=.06+.03+.01=.10
1M
1.1M
1M

Variable expense ratio=

.4(100k) .5(66k) 110k .75(40k)
+
+
+
=.04+.03+.10+.03=.20
1M
1.0M 1.1M
1M

Step 3: Using the equation in Step 1, and the results from Step 2, compute the indicated rate change.

Indicated Rate Change=

.75+.10
1.0-.20-.05

-1.0=13.3% increase

c. Rate changes impact the fixed expenses as a percent of premium because the premium the ratio is applied
to is different than contemplated in the ratio itself. If there had been a large rate increase after the fixed ratio
was calculated the estimated fixed expenses would be higher than actual

Solutions to questions from the 2010 exam
Question 25 – Model Solution 1
The premium based projection method could produce distorted results if:
1. Premium is not placed at the current rate level. If rates have increased (decreased) since or throughout
the historical experience period, premium used in the expense ratios would be understated
(overstated), resulting in an overstated (understated) expense ratio.
2. Premium is not trended to reflect shifts in average premium. If average premium is trending upward
(downward) after or throughout the historical experience period, premium used in the expense
ratios would be understated (overstated), resulting in an overstated (understated) expense ratio.
Question 25 – Model Solution 2 – Acceptable Response
3. If we are using a nationwide expense ratio and apply it to a state that has significantly different
average premium but the same fixed expense, there will be a distortion. For states with higher
(lower) average premium, fixed expense will be overestimated (underestimated).

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Chapter 8 – Overall Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Sec
1
2
3
4
5

Description
Introduction and the Pure Premium Method
Loss Ratio Method
Loss Ratio Versus Pure Premium Methods
Indication Examples
Key Concepts

Pages
141 – 143
143 – 145
145 – 147
147 – 147
147 – 148

Introduction and the Pure Premium Method

141 – 143

Introduction:
This chapter explains how to determine whether current rates are appropriate (i.e. whether the profit target is
likely to be met at the current rates) in the aggregate.
Chapters 9 - 11 discuss the calculation of indications by subclasses of insureds.
Chapter 14 discusses how to calculate final rates based on the overall indications and indications by
subclasses of insureds.
Two basic approaches for determining an overall rate level need:
1. Pure premium method
2. Loss ratio method
This chapter will discuss each of these in detail, demonstrate the mathematical equivalency of the approaches,
and discuss rationale for selecting one over the other.
The Pure Premium Method:
The pure premium method:
 is the simpler and more direct of the two ratemaking formulae
 determines an indicated average rate (not an indicated change to the current average rate).
 involves projecting the average loss and loss adjustment expenses per exposure and the average
fixed expenses per exposure to the period that the rates will be in effect.
The indicated average rate per exposure is computed as follows:

Indicated Average Rate =

Exam 5, V1a

Pure Premium (including LAE) + Fixed UW Expense Per Exposure
1.0 - Variable Expense Ratio - Target Profit Percentage

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BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Recall the following notation:

= Exposures
__

P; P

= Premium; Average premium(P divided by X)

___

P1 ; PI = Indicated premium; Averageindicated premium  PI divided by X 
V

= Variable expense provision(EV divided by P)

= Target profit percentage

L; L

= Losses; Pure Premium(L divided by X)
__

EL ; EL = Loss Adjustment Expense(LAE); Average LAE per exposure(EL divided by X)
___

EF ; EF = Fixed underwriting expenses; Average underwriting expense per exposure  EF divided by X 
EV

= Variable underwriting expenses

Using the above notation, the formula can be rewritten as:

 ( L  E L )  EF 
 L  EL  EF  
X
X 

PI  
1.0  V  QT 
1.0  V  QT 


Derivation of Pure Premium Indicated Rate Formula
Begin with the fundamental insurance equation:
Premium = Losses + LAE + UW Expenses + UW Profit.

PI  L  EL  ( EF  V * PI )  (QT * PI ).
PI  V * PI  QT * PI  ( L  EL )  EF .
PI  [1.0  V  QT ]  ( L  EL )  EF ; PI 

( L  EL  EF )
[1.0  V  QT ]

Dividing by the number of exposures converts each of the component terms into averages per exposure, and
the formula becomes the pure premium indication formula:
_________ ____
 ( L  EL )  EF   L  E  E 
L
F 
___
X
X  

PI

P


X
1.0  V  QT 
1.0  V  QT  I

Given the following information:
• Projected pure premium including LAE
• Projected fixed UW expense per exposure
• Variable expense ratio
• Target profit percentage
The indicated average rate per exposure is:

= $300
= $25
= 25%
= 10%

 _________ ____ 
 L  EL  E F 
 = [$300  $25] =$500
Indicated Average Rate  
1.0  V  QT  [1.0 - 0.25 - 0.10]

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New Company
When determining rates for an insurer writing new business, no internal historical data exists. However, the
actuary can still determine the indicated rate by estimating the expected pure premium and expense provisions
and selecting a target profit provision (based on external data or determined judgmentally).

Loss Ratio Method

143 – 145

The loss ratio method:
 is the more widely used of the two rate level indication approaches.
 calculates an indicated change factor
 compares the sum of the projected loss and LAE ratio and the projected fixed expense ratio to the
variable permissible loss ratio.

Indicated Change Factor =

[Loss & LAE Ratio + Fixed Expense Ratio]
[1.0 -Variable Expense Ratio - Target UW Profit%]

When the numerator and denominator are not in-balance, the indicated change factor will be something other
than 1.0. The factor can be applied to the current premium to bring the formula back in balance.

(L + EL ) + F 
PC


The loss ratio indication formula can be rewritten as follows: Indicated Change Factor =
1.0 -V - QT 

(L + EL ) + F 
PC


The indicated change is computed by subtracting 1.0: Indicated Change =
- 1.0
1.0 -V - QT 
Derivation of Loss Ratio Indicated Rate Change Formula
Start with the fundamental insurance equation: Premium = Losses + LAE + UW Expenses + UW Profit.
Using the following notation, PC = Premium at current rates; QC = Profit percentage at current rates , the
fundamental insurance equation can be rewritten as follows:

PC  L  EL  ( EF  V * PC )  QC * PC
Rearranging the terms leads to:

QC * PC  PC - ( L  EL ) - ( EF  V * PC )
Dividing each side by the projected premium at current rate level ( PC ) yields:

QC = 1.0 -


(L + EL )+(EF +V * PC )
L  EL + E F
= 1.0 -
+V 
PC
PC  PC


Thus, Profit % at Current Rates = 1.0 – Loss Ratio – OER = 1.0 - Combined Ratio.

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Chapter 8 – Overall Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
The goal of the ratemaking: Determine whether current rates can cover the estimated losses and expenses
and produce the target profit.
 If the expected profit % at current rates (QC) is equivalent to the target profit % (QT), then the current
rates are appropriate.
 It is more likely case is that (QC) is not equivalent to (QT) and rates need to be adjusted.

QC = 1.0 -

(L + EL )+ EF
-V
PC

The objective: How much does the premium at current rates need to be increased or decreased to achieve the
target profit percentage?
Determine this by substituting:
 (QT) for (QC) and
 the indicated premium (PI) for the projected premium at current rates (PC) (indicated premium is the
projected premium at current rates times the indicated change factor):

QT = 1.0 -

Rearranging terms leads to: 1.0 -V - QT 

(L + EL )+ EF
-V
PC * Indicated Change Factor
(L + EL )+ EF
PC * Indicated Change Factor

Rearranging terms and dividing through by PC yields:

L + EL + EF
Indicated Change Factor =
=
PC * (1.0 -V - QT )

(L + EL )

E
+ F
PC
PC
, which
(1.0 -V - QT )

(L + EL ) + F 

PC

is equivalent to the loss ratio indication formula: Indicated Change Factor =
[1.0 -V - QT ]
A result greater than 1.0 means the current rates are inadequate and need to be adjusted upward (and vice versa).

(L + EL ) + F 


PC
- 1.0
Subtract 1.0 from both sides to produce an indicated change: Indicated Change =
[1.0 -V - QT ]
Example of Loss Ratio Indicated Rate Change Formula
• Projected ultimate loss and LAE ratio
= 65%
• Projected fixed expense ratio
= 6.5%
• Variable expense ratio
= 25%
• Target profit percentage
=10%

 ( L  EL )  F 
PC


[65%  6.5%]
 1.0 
 1.0  10%
Indicated Change = 
[1.0 - V  QT ]
[1.00  0.25  0.10]
Thus, the overall average rate level is inadequate and should be increased by 10%.

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Chapter 8 – Overall Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
New Company
It is not used to price rates for a new insurer since the loss ratio approach is dependent on current premium.
The LR method is only used for making rates for a company with existing rates (since the loss ratio approach is
dependent on current premium).

Loss Ratio Versus Pure Premium Methods

145 – 147

Comparison of Approaches
Two major differences between the two approaches.
1. The loss measure used in each approach: the loss ratio (i.e. projected ultimate losses and LAE divided by
projected premium at current rate level) versus the pure premium statistic (i.e. projected ultimate losses
and LAE divided by projected exposures).
 The loss ratio indication formula requires premium at current rate level and the pure premium indication
formula does not.
 The pure premium formula requires exposures whereas the loss ratio indication formula does not.
Preference:
 The pure premium approach is preferable if premium is not available or if it is difficult to calculate
premium at current rate level (e.g. the rating algorithm for personal auto includes a large number of
rating variables, and if significant changes were made to those variables during the historical period, it
may be difficult to calculate the premium at current rate level).
 The loss ratio method is preferable if exposure data is not available or if the product being priced does
not have clearly defined exposures (e.g. CGL policies have multiple sub-lines, each with different
exposure bases). Thus, it’s easier to obtain and use premium at current rate level rather than trying to
define a consistent exposure.
2. The output of the two formulae.
 The loss ratio formula produces an indicated change to rates currently charged.
 The pure premium formula produces an indicated rate (thus, the pure premium method must be used
with a new line of business for which there are no current rates to adjust).

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Chapter 8 – Overall Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Equivalency of Methods
Both formulae can be derived from the fundamental insurance equation (thus two approaches are
mathematically equivalent).

(L + EL ) + F 

PC

1. Start with the loss ratio indication formula: Indicated Change Factor =
[1.0 -V - QT ]
(L + EL ) + EF 

PC
PC 
Restate the formula as: Indicated Change Factor =
[1.0 -V - QT ]
2. The indicated adjustment factor, the ratio of the indicated premium (PI ) to the projected premium at current

(L + EL ) + EF 

PC
PC 
P
=
rates (PC), yields the following: I
PC
[1.0 -V - QT ]
3. Multiplying both sides by the projected average premium at current rates ( PC / X ) results in the pure
premium indication formula (proving the two methods are equivalent):

(L + EL ) + EF 
_________
____
X
X  [ L + EL + EF ]


=
=
X
[1.0 -V - QT ]
[1.0 -V - QT ]

Note: The equivalency depends on consistent data and assumptions used for both approaches.
Example: If the premium at current rate level is estimated using the parallelogram method rather
than the more accurate extension of exposures method, any inaccuracy introduced by
the approximation may result in inconsistency between the loss ratio and pure premium
methods.

Indication Examples

147 – 147

Chapters 1 – 8 have provided different techniques that can be used to determine an overall rate level indication.
The exact techniques used by actuaries to determine the overall rate level indication depend on various factors
(e.g. unique characteristics of the product being priced, data limitations, historical precedence, and regulatory
constraints).
Appendices A – D:
 provide overall rate level indication examples for 4 different lines of business (insurance products).
 example indications are based on several years of subject experience.
Calculating the total loss ratio (or pure premium) can be done as follows:
i. Insurers may sum projected ultimate loss and LAE across all years and divide by projected EP at
present rates (or projected exposures) across all years (i.e. equivalent to weighting each year’s loss
and LAE ratio (pure premium) by the relevant premium (or exposure).
ii. Alternatively, some insurers select weights for each AY’s experience, giving more weight to the more
recent years.

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Chapter 8 – Overall Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.

Key Concepts

147 – 148

1. Pure premium indication formula

Indicated Average Rate =

Pure Premium (including LAE) + Fixed UW Expense Per Exposure
1.0 - Variable Expense Ratio - Target Profit Percentage

(L + EL ) + EF 
_________
____
X
X  [ L + EL + EF ]

Indicated Average Rate =
=
[1.0 -V - QT ]
[1.0 -V - QT ]
2. Loss ratio indication formula

Indicated Change =

[Loss & LAE Ratio + Fixed Expense Ratio]
- 1.0
[1.0 - Variable Expense Ratio - Target Profit %]

(L + EL ) + F 

PC

Indicated Change = 
- 1.0
[1.0 -V - QT ]
3. Loss ratio versus pure premium method
a. Strengths and weaknesses of each method
b. Mathematical equivalency of methods

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Chapter 8 – Overall Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
The predecessor papers to the current syllabus reading “Basic Ratemaking” by Werner, G. and
Modlin, C. were numerous. While past CAS questions were drawn from prior syllabus readings,
the ones shown below remain relevant to the content covered in this chapter.
Questions from the 2002 exam
17. (4 points) Based on McClenahan, "Ratemaking," chapter 2 of Foundations of Casualty Actuarial Science,
and the following data, answer the questions below. Show all work.
Projected rates to be effective January 1, 2003 and in effect for 1 year.
Permissible loss and ALAE ratio (modified) is 65%.
Experience is from the accident period January 1, 2000 to June 30, 2001.
Developed accident period loss and ALAE is $21,500.
Annual trend factor is 3%.
All policies have one-year terms and are written uniformly throughout the year.
The rate on January 1, 1999 was $120 per exposure.
Effective Date
January 1, 2000
January 1, 2001
Year
1998
1999
2000
2001

Rate Change
+10%
-15%
Written Exposures
200
200
200
200

Questions from the 2003 exam:
36. (5 points) Using the following information, answer the questions below. Show all work.

a.
b.
c.
d.
e.



On-level earned premium = $500,000



Experience period losses = $400,000



Experience period earned exposure = 5,000



Premium-related expense factor = 22%



Fixed underwriting expenses (modified) = $20,000



Profit and Contingencies factor = 3%

(1 point)
(1 point)
(1 point)
(1 point)
(1 point)

Exam 5, V1a

Calculate the variable permissible loss ratio using the loss ratio method (modified).
Calculate the indicated rate level change using the loss ratio method.
Calculate the indicated rate level change using the pure premium method.
Describe a situation where the pure premium method cannot be used.
Describe a situation where the loss ratio cannot be used.

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Chapter 8 – Overall Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2004 exam:
10. Which of the following statements is false regarding the loss ratio and pure premium methods for ratemaking?
A. The loss ratio and pure premium methods are identical when using consistent assumptions.
B. The pure premium method is preferable when on-level premium is difficult to calculate.
C. The loss ratio method produces indicated rate changes.
D. The pure premium method requires well-defined, responsive exposures.
E. The loss ratio method is preferable for a new line of business.
13. Given the information below, determine the indicated rate per exposure unit.
• Frequency per exposure unit = 0.25
• Severity = $100
• Fixed expense per exposure unit = $10
• Variable expense factor = 20%
• Profit and contingencies factor = 5%
A. < $35

B. > $35 but < $40

C. > $40 but < $45

D. > $45 but < $50

E. > $50

33. (3 points) Given the following information, answer the questions below.
On-Level
Trended
Accident
Earned
Ultimate
Year
Premium Loss & ALAE
2000
$800
$512
2001
$900
$540
2002
$1,000
$550
•
•
•
•
•
•
•

Ratio of commissions to written premium = 14%
Ratio of taxes, licenses and fees to written premium = 3
Ratio of other acquisition expenses to written premium = 2%
Ratio of general expense to earned premium = 6.25%
Profit and contingency provision = 5%
Fixed U/W expense ratio (modified) = 5%
Assume each year of historical experience receives equal weighting.

a. (2 points) Determine the indicated rate change for policies to be written from January 1, 2004 to
December 31, 2004. Show all work.
b. (1 point) Expenses can be related to written premium or earned premium. Briefly explain why other
acquisition expenses are related to written premium, while general expenses are related to earned
premium. (chapter 7)

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Chapter 8 – Overall Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2005 exam:
46. (5 points) Given the following data for private passenger auto bodily injury basic limits, answer the questions
below. Show all work.
• Policies are annual.
• Proposed Effective Date = July 1, 2005
• Rates are in effect for one year.
• Current Rate = 225
Experience Period Exposures and Losses
Calendar Accident
Earned
Loss & ALAE as of
Year
Exposures
December 31, 2004
2002
450
$52,000
2003
500
$54,000
2004
530
$40,000
• Age-to-age loss development factors
12-24 months =1.50;
24-36 months =1.15;
36-48 months= 1.05;
48 - ultimate =1.06
• Frequency trend = 2%
• Severity trend = 5%
• Permissible Loss Ratio (modified) = 65%
a. (4 points) Calculate the indicated statewide rate level change using the loss ratio method.
b. (1 point) Using your results from part a. above, illustrate the equivalency of the loss ratio method and
the pure premium method.

Questions from the 2006 exam:
36. (4 points) Using the methods described by McClenahan, and the following information, answer the
questions below. Show all work.

Experience period on-level earned premium = $500,000

Experience period trended and developed losses = $300,000

Experience period earned exposure = 10,000

Premium-related expenses factor = 23%

Fixed underwriting expenses (modified) = $21,000

Profit and Contingency factor = 5%
a. (1.5 points) Calculate the indicated rate level change using the loss ratio method.
b. (1.5 points) Calculate the indicated rate level change using the pure premium method.
c. (1.0 point) Describe one situation in which it is preferable to use the loss ratio method, and one
situation in which it is preferable to use the pure premium method.

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Chapter 8 – Overall Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2007 exam:
7. You are given the following information:

Indicated base rate is $300 per unit of exposure.

Profit and contingencies provision is 3%.

Other variable expenses represent 15% of premium.
What would the revised base rate be if the company changes the profit and contingencies provision to -6%?
A. < $272.00
B. > $272.00 but < $285.00
C. > $285.00 but < $298.00
D. > $298.00 but < $311.00
E. > $311.00

8. You are given the following information:
On-level Earned Premium:
Projected Loss & ALAE:
Projected Fixed Expense Ratio (modified):
Variable Expense Ratio (modified):
Profit and Contingencies Ratio:

$100,000
$75,000
10%
25%
0%

What is the indicated rate level change?
A. < 6.5% B. > 6.5% but < 8.0% C. > 8.0% but < 9.5%

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D. > 9.5% but < 11.0%

E. > 11.0%

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Chapter 8 – Overall Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2007 exam (continued):
42. (6.0 points) You are given the following information:
Incurred
Earned
Calendar Accident Year
Losses & LAE
Premium
2004
$5,000,000
$10,000,000
2005
3,750,000
11,000,000

Weights for
Accident Year
35%
65%

Historical Rate Level Changes
July 1, 2003
5.0%
July 1, 2004
-1.0%
July 1, 2005
10.0%
July 1, 2006
0.0%
 Losses are valued as of June 30, 2006.
 Selected annual frequency trend is 4%.
 Selected annual severity trend is 1%.
 There is no premium or exposure trend.
 All policies are annual.
 Fixed expense ratio is 7%.
 Profit and contingencies provision is 5%.
 Other variable expenses are 20% of premium.
 The indication is considered to be 60% credible.
 The complement of credibility is no change.
Loss Development Factors
Age
Age to Ult.
6
3.500
12
2.500
18
2.000
24
1.700
30
1.500
36
1.400
42
1.350
Calculate the indicated rate change for rates to be effective from July 1, 2007 through June 30, 2008.
Show all work.
Note: This is a chapter 5, chapter 6 and chapter 8 question.

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Chapter 8 – Overall Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2007 exam (continued):
43. (3.0 points) Using Werner and Modlin’s notation:
a. (2.0 points) Demonstrate the equivalence of the pure premium and loss ratio approaches,
assuming identical data and consistent assumptions.
b. (0.5 point) Which approach is more appropriate when pricing a new line of business? Explain.
c. (0.5 point) Which approach is more appropriate when pricing a line of business for which the
historical rate change history is not available? Explain.

Questions from the 2008 exam:
24. (1.0 point) The indicated average rate was determined to be $300 based on the following information:
 Average fixed expense per exposure = $16
 Variable expense provision = 15%
 Profit and contingencies provision = 3%
Calculate the revised indicated average rate assuming the expected loss costs will be 10% higher than
those assumed in the original analysis.

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Chapter 8 – Overall Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2008 exam continued:
26. (5.75 points) You are given the following information:
Calendar/Accident Year
2006
2007
Earned Premium
$345,704
$396,714
Base Rate Underlying Premiums
$100
$100

Accident
Year
2002
2003
2004
2005
2006
2007

15
$164,000
$172,000
$181,000
$190,000
$200,000
$210,000

Case Incurred Loss and ALAE
Evaluation Age in Months
27
39
51
$213,200
$245,180
$262,343
$223,600
$257,140
$269,997
$235,300
$258,830
$271,772
$228,000
$250,800
$240,000

63
$262,343
$269,997







Current base rate = $110
Current rating structure is purely multiplicative.
Proposed rates will be effective January 1, 2009, and will be in effect for one year.
All policies are annual policies.
On January 1, 2005 the claims department changed case reserving practices applicable to all
outstanding claims.
 Premium trend = 3%
 Frequency trend = -1% and severity trend = 2%
 Unallocated loss adjustment provision = 10% of ultimate incurred loss & ALAE
 Fixed expense ratio = 8% and variable expense ratio = 20%
 Profit and contingencies provision = 5%
 Accident year projections should be weighted 60% to accident year 2007 and 40% to accident year 2006.
 Overall indication is assumed to be 75% credible.
 Complement of credibility should be assigned to no change.
a. (1.25 points) Calculate calendar/accident year 2006 and calendar/accident year 2007 projected premium
at present rates. (Chapter 5, but shown here)
b. (3.0 points) Calculate accident year 2006 and accident year 2007 ultimate incurred losses and loss
adjustment expenses, projected to future loss cost levels. (Chapter 6, but shown here)
c. (1.5 points) Calculate the indicated rate change. (Chapter 8)
27. (1.0 point)
a. (0.5 point) Provide an example of where a pure premium method is more appropriate than a loss ratio
method.
b. (0.5 point) Provide an example of where a loss ratio method is more appropriate than a pure premium
method.

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Chapter 8 – Overall Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2009 exam:
31. (1.5 points) For each of the following identify whether the loss ratio or pure premium ratemaking method is
preferable. Briefly explain your answer.
a. (0.5 point) Setting prices for a new line of business.
b. (0.5 point Setting prices for a product that is not written uniformly throughout the year; current systems do
not support re-rating policies.
c. (0.5 point) Setting prices for a commercial lines product that has multiple complex exposures underlying
each risk.

Questions from the 2010 exam:
26. (2 points)
a. (1.5 points) Derive the indicated pure premium rate formula starting from the fundamental insurance
equation.
b. (0.5 point) Briefly describe two instances where it is more appropriate to use the pure premium method
than the loss ratio method.

Questions from the 2011 exam:
9. (6.75 points) Given the following information for a book of business:
•
Policies have a six month term
•
Rate change history:
o -3% effective October 1, 2008
o +6% effective January 1, 2010
•
Annual premium trend = 1.5%
•
Annual loss trend = 2.2%
•
Proposed rates will be in effect for one year beginning on October 1, 2011
•
Unallocated loss adjustment expense provision = 3.2% (of loss and ALAE)
•
Fixed expense ratio = 5.6%
•
Variable expense ratio = 24.0%
•
Underwriting profit and contingencies provision = 3.5%
•
Rates developed based on calendar/accident year 2009 and 2010
Calendar
Year Ending:
December 31, 2009
December 31, 2010

Accident Year
2006
2007
2008
2009
2010

12 months
$44,860
$47,985
$51,384
$60,735
$76,094

Earned Premium (000s)
$110,865
$128,973
Incurred Losses and ALAE (000s)
24 months 36 months 48 months
$51,589
$56,748
$57,315
$54,703
$60,720
$61,327
$59,606
$64,970
$69,845

60 months
$57,315

a. (2 points) Calculate the projected calendar year earned premium at current rate level for calendar
years 2009 and 2010.
b. (4.25 points) Calculate the indicated rate change.
c. (0.5 point) Assume the 2009 incurred loss and ALAE amount includes an additional $25,000,000 in
losses attributable to a single weather event. Discuss an appropriate strategy for including this
information in the indicated rate change calculation.

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Chapter 8 – Overall Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2011 exam continued:
10. (1.5 points) Identify whether the loss ratio or pure premium ratemaking method is preferable in each
of the following scenarios. Briefly explain each answer.
a. (0.5 point) A company introduced two new rating variables within the past year.
b. (0.5 point) A company is entering a new line of business.
c. (0.5 point) A company writes a commercial product with multiple exposure bases.

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(L + EL ) 
PC 

Indicated Rate Change = PI  
- 1.0
[PLR]
___

(L + EL ) Developed and Trended losses 23,668


 .70315
33,660
On  Level Earned Premium
PC
PLR =

1.0  V  QT  = .65 (given in the problem)

Indicated Rate Change =

Exam 5, V1a

.70315
 1  0.0818
.65

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Chapter 8 – Overall Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2003 exam:
Question 36.
a. (1 point) Calculate the variable permissible loss ratio (VPLR) using the loss ratio method.

VPLR  [1.0 -V - QT ] , where V and QT are given as 0.22 and 0.03
VPLR = (1.0 - 0.22 - 0.03) = 0.75 = 75.0%
b. (1 point) Calculate the indicated rate level change using the loss ratio method (LRM).

(L + EL ) + F 
(L + EL ) + F 

PC


PC


 - 1.0 ,
Indicated Change =
- 1.0 = 
[1.0 -V - QT ]
VPLR
E
20 K
(L + EL ) 400 K
 0.04 ; VPLR = (1.0 - 0.22 - 0.03) = 0.75 = 75.0%

 0.80 , F  F 
PC
500 K
Pc 500 K
Thus, the indicated rate level change using the LRM = [0.80+0.04]/0.75 – 1.0 = .12 = 12%
c. (1 point) Calculate the indicated rate level change using the pure premium method.

 _________ ____ 
___
 L  EL  EF 
Under the pure premium method, the indicated rate (R) is computed as follows: PI 
.
1.0  V  QT 
_________

L  EL = Indicated pure premium =

____

EF = Fixed expense

Experience Period Losses
$400,000

 $80
Experience Period Exposures
5,000

Non  premium Re lated Expensess $20,000

 $4
Experience Period Exposures
5,000
___

V = Variable expense = .22;

QT = Profit load = .03; Thus,

PI 

$80 4
 $112
1 - .22 - .03

The current rate can be computed on-level earned premium/experience period earned exposures. Thus,
the current rate is computed as $500,000/5,000 = $100.
Therefore, indicated rate level change using the pure premium method = $112/$100 – 1.0 = .12 = 12%
d. (1 point) Describe a situation where the pure premium method cannot be used.
The pure premium method cannot be used if exposure information is not available.
e. (1 point) Describe a situation where the loss ratio cannot be used.
The loss ratio method cannot be used for a new line of business because the method requires existing rate.

Solutions to questions from the 2004 exam:
10. Which statements is false regarding the loss ratio and pure premium methods for ratemaking?
A. The loss ratio and pure premium methods are identical when using consistent assumptions. True.
B. The pure premium method is preferable when on-level premium is difficult to calculate. True.
C. The loss ratio method produces indicated rate changes. True.
D. The pure premium method requires well-defined, responsive exposures. True.
E. The loss ratio method is preferable for a new line of business. False. The loss ratio method cannot be
used for a new line.

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Chapter 8 – Overall Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2004 exam (continued):
13. Determine the indicated rate per exposure unit.
Step 1: Write an equation to determine the indicated rate per exposure unit, based on the given data
The given data lends itself to computing the rate per exposure unit using the pure premium
method. Under the pure premium method, the indicated rate is computed as follows:

 _________ ____ 
____
 L  EL  EF 
___
___

Freq
Sev
EF
*
 . Based on the given data, P 
PI  
I
PLR
1.0  V  QT 
Step 2: Using the equation from Step 1, and the data given in the problem, solve for the indicated rate
per exposure unit.
___



.25*$10010 $35

 $46.67
1.20.05
.75

Answer: D. > $45 but <

$50
33. (3 points)
a. (2 points) Determine the indicated rate change for policies to be written from 1/1/2004 to 12/312004.
Show all work.
Step 1: Write an equation to determine the indicated rate change (IRC).

(L + EL ) + F 

PC

Indicated Change = 
- 1.0 ,
[1.0 -V - QT ]
Step 2: Using the equation from Step 1, and the data given in the problem, solve for the experience
loss ratios and the variable expense factor.

(L + EL )  512 540 550 



 / 3  .5967 , since it is assumed that each year of historical
PC
 800 900 1,000 
experience receives equal weighting.

V  .14  .03  .02  .0625  .2525;

Q  .05;
T

F  .05

Step 3: Using the equation from Step 1, the results from Step 2, and the data given in the problem,
solve for the indicated rate change for policies to be written from 1/1/2004 to 12/31/2004.

(L + EL ) + F 
PC


(0.5967  .05)
0.6467
Indicated Change = 
- 1.0 
 1.0 
 0.0728
[1.0 -V - QT ]
(1.0  0.2525  .05)
0.6975

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Chapter 8 – Overall Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2005 exam:
46. (5 points)
a. (4 points) Calculate the indicated statewide rate level change using the loss ratio method.
Step 1: Write an equation to determine the indicated rate change (IRC).

(L + EL ) 

PC 
Indicated Rate Change = PI  
- 1.0 
[1.0 - F  V - QT ]
___

(L + EL ) 

PC 
- 1.0 .
[PLR]

Note: The problem does not mention fixed expenses, so we assume there are no fixed expenses. So the PLR
is used (which, in this case, is equal to the VPLR)
Step 2: Calculate the trended projected ultimate on-level loss and ALAE ratio for the combined experience
period 2002 - 2004. With the given information in the problem, compute the developed and trended
Loss and ALAE by accident year as follows:

AY
2002
2003
2004
Total

Loss and
ALAE at
12/31/2004
(1)
52,000
54,000
40,000

Age to
Ult
LDFS
(2)
1.113
1.280
1.920

Midpoint of the
experience
period
(3)
7/1/2002
7/1/2003
7/1/2004

Midpoint of
the exposure
period
(4)
7/1/2006
7/1/2006
7/1/2006

Trend Factor
(5)
(1.071)4
(1.071)3
(1.071)2

Developed and
Trended Loss
and ALAE
(6)=(1)*(2)*(5)
76,147.63
84,912.60
88,092.75
249,152.98

Notes:
(2) Age to ultimate LDF computations:
(4) Avg Accident date of the exposure period is one year beyond
36 – ult = (1.05)(1.06) = 1.113
the proposed effective date of the rates.
24 – ult = (1.15)(1.113) = 1.280
12 – ult = (1.50)(1.280) = 1.920
(5) A combined frequency and severity trend is computed as (1.02)(1.05) = 1.071. Thus, (5) = 1.071t,
where t is the number of years elapsed between column 3 and column 4.
Step 3: Compute the Experience Loss and ALAE ratio as

Developed and Trended losses
$249,152.98
$249,152.98


 0.748
On - Level Earned Premium
$225[450  500  530]
$333, 000
Step 4: Using the equation from Step 1, the results from Step 2, and the data given in the problem, solve
for the indicated rate change for policies to be written from July 1, 2005 to July 1, 2006.

(L + EL ) 

PC 
.748
 1  0.151
Indicated Rate Change = PI  
- 1.0 
.65
[PLR]
___

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Chapter 8 – Overall Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2005 exam (continued):
b. (1 point) Using your results from part a. above, illustrate the equivalency of the loss ratio method and the
pure premium method.

 _________ ____ 
___
 L  EL  EF 
Under the pure premium method, the indicated rate (R) is computed as follows: PI 
.
1.0  V  QT 
In this problem,
_________

L  EL = Indicated pure premium =

Experience Period Developed and Trended Losses
$249,152

 $168.35
Experience Period Exposures
(450  500  530)

____

EF = Fixed expenses per exposure, V = Variable expense, and

QT = Profit load.
___

$168.35
 $259 . Therefore, the indicated
.65
Indicated Rate  Current Rate $259  $225

 0.151
rate change using the pure premium method is IRC 
Current Rate
$225
Since F, V and QT are not given, and since (1.0 – V – QT ) = PLR,

PI 

Solutions to questions from the 2006 exam:
Question 36
a. (1.5 points) Calculate the indicated rate level change using the loss ratio method.
Step 1: Write an equation to determine the indicated rate change (IRC).

(L + EL ) + F 

PC

Indicated Change = 
- 1.0
[1.0 -V - QT ]
Step2: Using the equation from Step 1, and the data given in the problem, solve for the indicated rate
change using the loss ratio method.

.642
 300,000 
IRC  [
 21, 000 / 500, 000] / (1  .23  .05)  1.0 
 1  .108333  10.83%

.72
 500,000 

Exam 5, V1a

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Chapter 8 – Overall Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2006 exam (continued):
b. (1.5 points) Calculate the indicated rate level change using the pure premium method.

 _________ ____ 
 L  EL  EF 
Under the pure premium method, the indicated rate (R) is computed as follows: PI 
.
1.0  V  QT 
___

_________

L  EL = Indicated pure premium =

Experience Period Developed and Trended Losses $300,000

 $30.0
Experience Period Exposures
10,000

____

EF = Fixed expenses per exposure unit =

Fixed U /W Expenses
$21,000

 $2.10
Experience Period Exposures 10,000

V and QC are the premium related expense ratio and P&C load respectively, as given in the problem.

$30.0  $2.10
 $44.60 .
1.0  0.23  0.05
Experience Period On - level Earned premiums $500, 000
The current rate =

 $50.0
Experience Period Exposures
10, 000
___

Thus, PI 

Thus, the indicated rate change using the pure premium method is

IRC 

Indicated Rate  Current Rate $44.60  $50

 0.108  10.8%
Current Rate
$50

c. (1.0 point) Describe one situation in which it is preferable to use the loss ratio method, and one
situation in which it is preferable to use the pure premium method.
 The loss ratio method is preferable when the exposure unit is not available.
 The loss ratio method is preferable when the exposure unit is not reasonably consistent between risks.
 The pure premium method is preferable for a new line of business.
 The pure premium method is preferable where on-level premium is difficult to calculate.

Solutions to questions from the 2007 exam:
7. What would the revised base rate be if the company changes the profit and contingencies provision to -6%?
Step 1: Write an equation to determine the pure premium and fixed expenses associated with the current rate,
based on the given data. This will help determine what this provision is when computing the revised
based rate. The given data lends itself to computing pure premium and fixed expenses using the pure
premium method. Under the pure premium method, the base rate is computed as follows:

 _________ ____ 
___
 L  EL  EF 
.
PI 
1.0  V  QT 
Step 2: Using the equation from Step 1, and the data given in the problem, solve for the pure premium and
_________

____

L  EL  EF _________ ____
; L  EL  EF  246
fixed expenses 300 
1  .15  .03
Step 3: Using the results from Step 2, and the equation in Step 1, solve for the revised base rate.
___

PI 

Exam 5, V1a

246
 270.32
1  .15  (.06)

Answer: A

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Chapter 8 – Overall Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2007 exam (continued):
8. What is the indicated rate level change?
Step 1: Write an equation to determine the indicated rate change (IRC).

 ( L  EL )  F 

PC

Indicated Change = 
 1.0 ,
[1.0 - V  QT ]
__

L; L

= Losses; Pure Premium(L divided by X)
__

EL ; EL = Loss Adjustment Expense(LAE); Average LAE per exposure(EL divided by X)
EF ; F = Fixed underwriting expenses; Proj Fixed Exp Ratio =  EF divided by P 
EV

= Variable underwriting expenses;

= Exposures

= Premium at current rates

= Variable expense provision(EV divided by P)

= Target profit percentage

Step 2: Using the equation from Step 1, the results from Step 2, and the data given in the problem, solve
for the indicated rate change. Indicated Change =

[75,000 / 100, 000  10.0%]
 1.0  1.133%
[1.00  0.25  0.0]

Answer: E

Exam 5, V1a

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Chapter 8 – Overall Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2007 exam (continued):
42. Calculate the indicated rate change for rates to be effective from July 1, 2007 through June 30, 2008.
Step 1: Write an equation to determine the indicated rate change.

 ( L  EL )  F 

PC

Indicated Change = 
 1.0
[1.0 - V  QT ]
Note that losses will need to be adjusted by the selected annual frequency and severity trend rates, and
developed to ultimate. Premiums need to be adjusted by rate level changes only, since there is no
premium or exposure trend. Since we are given two years of premiums and losses, a weighted loss
ratio will need to be calculated. And after computing the indicated rate change, a credibility weighted
indicated rate change must be determined since the indication is considered to be 60% credible.
Step 2: Determine on-level earned premium. To do so, compute on-level factors for CYs 2004 and 2005.
This is the current rate level divided by the weighted average of the rate level factors in the experience
period. The weights will be relative proportions of each square or triangle. First calculate the area of
all triangles (area = .5 * base * height) within a unit square and then determine the remaining
proportion of the square by subtracting the sum of the areas of the triangles from 1.0.
Rate Level Factors:
Date
Rate Change
Rate Level Factor
7/1/03
5%
1.05000 = 1.05 * 1.000
7/1/04
-1%
1.03950 = 1.05 * (1-.01)
7/1/05
10%
1.14345 = 1.0395 * 1.10
7/1/06
0%
1.14345 = 1.14345 * 1.00
Current Rate Level = 1.05 * (1.0 -0.01) * 1.1 * 1.0 = 1.14345
On level Earned Premium:
2004 on level EP: 1.14345/(0.125*1.00+0.75*1.05+0.125*1.0395) * 10M = 1.097 * 10M = 10,970,000
2005 on level EP: 1.14345/(0.125*1.05+1.0395*0.75+1.14345*0.125) * 11M = 1.085 * 11M = 11,935,000
Step 3: Determine ultimate losses. As of 6/30/2006, AY 2004 losses are 30 months old while AY 2005
losses are 18 months old.
2004 ultimate losses: 5,000,000 * (30-Ult Factor) = 5,000,000 * 1.5 = 7,500,000
2005 ultimate losses: 3,750,000 * (18-Ult Factor) = 3,750,000 * 2.0 = 7,500,000
Note: Losses also need to be trended to one year beyond the effective date of the rates (i.e. 7/1/2008). For
AY 2004, the average accident date is 7/1/2004. Thus, four years of frequency/severity trend is applied.
Step 4: Determine the projected weighted loss ratio.
Ultimate
CL Earned
Loss
Trended
Loss
Premium
Trend
Loss
4
2004
7,500,000
10,970,000
[(1.04)(1 .01)]
9,130,196
3
2005
7,500,000
11,935,000
[(1.04)(1 .01)]
8,692,114
Thus, the project weighted loss ratio = 0.35(0.8323) + 0.65(0.7283) = 0.7647

Loss
Ratio
0.8323
0.7283

Indicated change = [(L+EL)/Pc +F]/[1.0 – V – QT ] – 1.0 =(0.7647+0.07)/(1 - 0.2 - 0.05) – 1.0 = .1129
Credibility weighted indicated rate change: [0.60* 1.1129 +0.4 (1.00)] - 1.0 = .0677 = +6.77%

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Chapter 8 – Overall Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2007 exam (continued):
Question 43
a. (2.0 points). Demonstrate the equivalence of the pure premium and loss ratio approaches,
assuming identical data and consistent assumptions.
b. (0.5 point) Which approach is more appropriate when pricing a new line of business? Explain.
c. (0.5 point) Which approach is more appropriate when pricing a line of business for which the
historical rate change history is not available? Explain.
Model Solution

(L + EL ) + F 


PC
1. Start with the loss ratio indication formula: Indicated Change Factor =
[1.0 -V - QT ]
(L + EL ) + EF 

PC
PC 
Restate the formula as: Indicated Change Factor =
[1.0 -V - QT ]
2. The indicated adjustment factor, the ratio of the indicated premium (PI ) to the projected premium at current

(L + EL ) + EF 
_________
____
X
X  [ L + EL + EF ]


=
=
X
[1.0 -V - QT ]
[1.0 -V - QT ]

b. The pure premium method produces an indicated rate, so no existing rate is required. The loss ratio
method produces an indicated rate change, so an existing rate is required. The pure premium method
is more appropriate for new line of business.
c. The pure premium method does not require premium at current level. The loss ratio method requires
premium at current level to calculate the indicated change. The pure premium method is more
appropriate when no historical rate changes are available.

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Chapter 8 – Overall Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2008 exam:
Model Solution - Question 24
24. (1.0 point) The indicated average rate was determined to be $300 based on the following information:
 Average fixed expense per exposure = $16
 Variable expense provision = 15%
 Profit and contingencies provision = 3%
Calculate the revised indicated average rate assuming the expected loss costs will be 10% higher than those
assumed in the original analysis.
Step 1: Write an equation to determine the revised indicated average rate.

 _________ ____ 
 L  EL  EF 
and thus the revised indicated average rate equals
Indicated Average Rate  PI 
1.0  V  QT 
___

_________ ____


1.10*
L  EL  EF 


1.0 V  QT 

Step 2: Using the equations in Step 1, solve for the revised indicated average rate.
___

____

_________

W are given that PI = $300, EF = $16, V = .15 and QT = .03, Thus, L  EL = $300(1.0-0.18)-16 = $230
___

Thus, revised

Exam 5, V1a

PI 

230(1.1)16
 328.05
1.15.03

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Chapter 8 – Overall Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2008 exam:
Model Solution - Question 26
a. (1.25 points) Calculate calendar/accident year 2006 and calendar/accident year 2007 projected premium at
present rates.
Step 1: Write an equation to determine CAY 2006 and CAY 2007 projected premium at present rates (PPPR).
PPPR = Earned Exposures * Current Base Rate * (1.0 + Premium Trend)(midpt exper period to 1 yr after proj eff date)
Step 2: Determine Earned Exposures * Current Base Rate for CAY 2006 and CAY 2007
CAY 2006 Earned Exposures * Current Base Rate = $345,704/100 * $110 = $380,274.4
CAY 2007 Earned Exposures * Current Base Rate = $396,714/100 * $110 = $436,385.4
Step 3: Compute the trend period for CAY 2006 and CAY 2007
The Trend period should extend from the midpoint of the experience period to 1 year after the projected
effective date of the rates.
For CAY 2006, the trend period is from 7/1/06 to 1/1/2010 = 3.5 years
For CAY 2007, the trend period is from 7/1/07 to 1/1/2010 = 2.5 years
Step 4: Using the equation in Step 1, and the results from Steps 2 and 3, compute PPPR
CAY 2006 PPPR = $380,274.4 * (1.03)3.5 = $421,723
CAY 2007 PPPR = $436,385.4 * (1.03)2.5 = $469,854
b. (3.0 points) Calculate accident year 2006 and accident year 2007 ultimate incurred losses and loss adjustment
expenses, projected to future loss cost levels.
Step 1: Write an equation to determine AY 2006 and AY 2007 Trended and Ultimate Incurred L+ALAE
Projected Ultimate Incurred L+ALAE+ULAE
= Case Incurred Losses * LDFULT * (1+ loss Trend)(midpt exper period to 1 yr after proj eff date) * (1+ULAE factor)
Step 2: Using the case incurred loss triangle, compute age to age factors, select age to ultimate factors, and
compute AY 2006 and AY 2007 ultimate losses.

AY
2002
2003
2004
2005
2006

15-27
1.30
1.30
1.30
1.20
1.20

Case Incurred Link Ratios
27-39
39-51
1.15
1.07
1.15
1.05
1.10
1.05
1.10

51-63
1.00
1.00

We can see the change in case reserving practices from the link ratios. We will use the link ratios below the solid
line.

Sel A-t-A
Age to Ult

1.200
1.386

1.100
1.155

1.050
1.050

1.000
1.000

AY 2006 ultimate losses = $240,000 * 1.155 = 277,200
AY 2007 ultimate losses = $210,000 * 1.386 = 291,060

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Chapter 8 – Overall Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2008 exam continued:
Model Solution - Question 26
Part b.
Step 3: Using the given frequency and severity trends, compute the loss trend and using the previously
determined trend periods, compute the loss trend factors for AY 2006 and AY 2007. Apply this facto to
compute trended and ultimate incurred losses.
Loss trend = Frequency trend * Severity trend = (1.0 - .01)*(1+.02) = 1.0098
The Trend period should extend from the midpoint of the experience period to 1 year after the projected
effective date of the rates.
 For CAY 2006, the trend period is from 7/1/06 to 1/1/2010 = 3.5 years
 For CAY 2007, the trend period is from 7/1/07 to 1/1/2010 = 2.5 years
Thus, AY 2006 trended and ultimate incurred L+ALAE = 277,200 * (1.0098)3.5 = 286,825
Thus, AY 2007 trended and ultimate incurred L+ALAE = 291,060 * (1.0098)2.5 = 298,243
Step 4: Multiply trended and ultimate incurred L+ALAE by the ULAE factor.
AY 2006 Projected Ultimate Incurred L+ALAE+ULAE = 286,825 (1.10) = 315,508
AY 2007 Projected Ultimate Incurred L+ALAE+ULAE = 298,243 (1.10) = 328,067
c. (1.5 points) Calculate the indicated rate change.
Step 1: Write an equation to determine the credibility weighted Indicated Rate change:
Credibility Weighted Indicated Rate change factor = Indicated Rate change factor * Z + (1.0 – Z)*1.0
(note that the problem states that the complement of credibility should be assigned to no change).
Step 2: Write an equation to determine the Indicated Rate change factor and solve for it:
Indicated Rate change factor =

Weighted Loss Ratio F [.40*AY 06 Loss Ratio.60*AY 07 Loss Ratio] F

,
1V QT
1V QT

since AY projections should be weighted 60% to AY 2007 and 40% to AY 2006.
AY 2006 loss ratio = 315,508/421,723 = .748. AY 2007 loss ratio = 328,067/469,854 = .698.
Thus, 

[.40*.748.60*.698].08
1.064
1.20.05

Step 3: Using the equation in Step 1, the results from Step 2, and the credibility factor to be applied to the overall
indication, compute the credibility weighted Indicated Rate change.
Credibility Weighted Indicated Rate change factor = 1.064 * Z + (1.0 – Z)*1.0 = (1.064*0.75+.25)-1=.048
Model Solution - Question 27
27. (1.0 point)
a. (0.5 point) Provide an example of where a pure premium method is more appropriate than a loss ratio method.
b. (0.5 point) Provide an example of where a loss ratio method is more appropriate than a pure premium method.
a. Pure premium method is more appropriate than loss ratio method when current rate level premiums are
difficult to calculate.
b. Loss ratio method is more appropriate than pure premium method when a well defined and responsive
exposure base is not present.

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Chapter 8 – Overall Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2009 exam:
31. (1.5 points) For each of the following identify whether the loss ratio or pure premium ratemaking
method is preferable. Briefly explain your answer.
a. (0.5 point) Setting prices for a new line of business.
b. (0.5 point Setting prices for a product that is not written uniformly throughout the year; current
systems do not support re-rating policies.
c. (0.5 point) Setting prices for a commercial lines product that has multiple complex exposures
underlying each risk.
a. Pure premium - because it produces an indicated rate, which does not require historical rates
b. Pure premium - loss ratio method requires on-level premiums which would be challenging/ not possible here
c. Loss ratio - in this situation it would be easier to use premiums and not have to deal with difficult exposures in
the pure premium method.

Solutions to questions from the 2010 exam:
Question 26
a. (1.5 points) Derive the indicated pure premium rate formula starting from the fundamental insurance equation.
b. (0.5 point) Briefly describe two instances where it is more appropriate to use the pure premium method than the
loss ratio method.

a. Begin with the fundamental insurance equation:
Premium = Losses + LAE + UW Expenses + UW Profit.
PI  L  EL  ( EF  V * PI )  (QT * PI ).
PI  V * PI  QT * PI  ( L  EL )  EF .
PI  [1.0  V  QT ]  ( L  EL )  EF ; PI 

( L  EL  EF )
[1.0  V  QT ]

Dividing by the number of exposures converts each of the component terms into
averages per exposure, and the formula becomes the pure premium indication formula:
_________ ____
 ( L  EL )  EF   L  E  E 
L
F
___
X
X  

PI
P


X
1.0  V  QT 
1.0  V  QT  I

b1. Use it for anew line of business for which you do not have a current premium level.
b2. If you are unable to get a rate change history to put historical premium on-level (which the LR method
requires).

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Cha
apter 8 – Overall Indication
n
BASIC RATTEMAKING – WERNER, G
G. AND MOD
DLIN, C.
Solution
ns to questio
ons from th
he 2011 exa
am:
9a. (2 points) Calculate
e the projecte
ed CY EP currrent rate leve l for calendarr years 2009 a
and 2010.
9b. (4.25 points) Calcu
ulate the indicated rate cha
ange.
9c. (0.5 po
oint) Assume the 2009 inc
curred loss an
nd ALAE amo unt includes a
an additional $25M in lossses
attributtable to a sing
gle weather event.
e
Discuss
s an appropri ate strategy ffor including tthis informatio
on in the IRC
calcula
ation.
Question
n 9 – Model Solution
S
1
a. Projec
cted calendarr year earned premium at current
c
rate le
evel = EP * O
OLF * Premium
m trend factorr
Curre
ent rate level is 1.0 * (1.0 - 0.03) * (1.0 + .06) = 1.028
82

CY 09 at 1.0 level: Are
ea = 1/2 *b*h.. b = 3mos/12
2mos. h is a function of w
when a rate ch
hange occurs and
the length
h of the policie
es being writte
en. h = 1/2 as
a it intersectss CY 09 three
e months afterr the 10/1/08 rate
change im
mpacting the six
s month policies being written.
2009 on le
evel factor = 1.0282 / [1/16
6*(1) + (15/16
6)*.97] = 1.058
8; 1/16 = 1//2*(1/4)*(1/2)
2010 on le
evel factor = 1.0282 / [1/4**(.97) + 3/4*(1
1.0282)] = 1.0
014; 1/4 = 1/2
2*(1/2)*(1)
3
2009 prem
mium = 11086
65 * 1.058 * 1.015 = 122,6
653 = EP * OL
LF * Premium
m trend factor
2010 prem
mium = 12897
73 * 1.014 * 1.0152 = 134,7
731
2009 prremium trend period from avg
a written da
ate of 4/1/09 tto average wrritten date 4/1
1/12 or 3 yearrs
2010 prremium trend period from avg
a written da
ate of 4/1/10 tto average wrritten date 4/1
1/12 or 2 yearrs
[L
o
s
s
&
L
A
E
R
a
tio
+
F
ix
e
d
E
x
p
en
n
se
R
a
tio
]
b. In d ic a te d C h a n g e F a c to r =
[ 1 .0 - V a ria b lee E x p e n se R a tio - T a r g ett U W P r o fit % ]

Selected
AT
TU

12-2
24
1.15
5
1.14
4
1.16
6
1.15
5
1.15
5
1.27
78

24-36
1.1
1.1
1.09

36-48
1.01
1.01

48-60
1

1.1
1.111

1.01
1.01

1
1

2009 loss
ses: 69845 x 1.111
1
x 1.022
23 (1.032) = 85
5483 = Latestt Losses * LD
DF to Ult * Losss trend facto
or * ULAE
2009 loss
ses: 69845 x 1.111
1
x 1.022
23 (1.032) = 85
5483
Loss ratio
o = 85,483/12
22,653 = .697
7
2010 loss
ses: 76094 x 1.278
1
x 1.022
22 (1.032) = 10
04824.5
Loss ratio
o = 104,824.5
5/134,731 = .7
778
2010 Trend: fro
om 7/1/2010 to
o 7/1/2010 orr 2 years; UL
LAE factor = 1
1.032
U
Loss and LAE Ratio = 190,279//257,426 = .739
Overall Trrended and Ultimate
Indicate ra
ate change = [LR + F / (1 - V - Q)] - 1.0 = [.739 + .05
56] / (1 - .24 - .035) = 1.096
655 - 1 = 9.66
6%
c. Given that 25m is a large proporttion of the inc
curred to date
e losses of $6
69,845,000, I w
would exclude this loss
and inc
clude a CAT load
l
based on
n a cat model or longer terrm historical a
average of ca
at losses inste
ead.

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Chapter 8 – Overall Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2011 exam continued:
Question 9 – Model Solution 2
a.
OLF09 = 1.0282 / [1000 * (½ * ½ * ¼) + 0.97 * (1 - 0.0625)] = 1.05795; 1/2*1/2*1/4 = 0.0625
OLF10 = 1.0282 / [0.97 * (½ * 1 * ½) + 1.0282 * (1 - .25)] = 1.01435
(1)
(2)
(3)
(4)
(5)
(6) = (1)*(2)*(5)
CY
EP
OLF
Trend From
Trend To
Trend Factor Trended on-level EP
2009 110,865 1.05795
4/1/09
4/1/12
1.0153
122,648
2010 128,973 1.01435
4/1/10
4/1/12
1.0152
134,778
257,426
(3) = avg. written date of policies earned in calendar year
(4) = avg. written date of projection period
b.
Weighted avg
LDF
To Ultimate

CY
2009
2010

(1)
Loss &
ALAE
69,845
76,094

12-24
1.150
1.27765

24-36
1.100
1.111

(2)
LDF

(3)
ULAE
Load
1.032
1.032

1.111
1.27765

36-48
1.010
1.010

48-60
1.000
1.000

(4)
Trend
From
7/1/09
7/1/10

(5)
Trend
To
7/1/12
7/1/12

(6)
Trend
Factor
1.0223
1.0222

(7) = (1)*(2)*(3)*(6)
Trended Ultimate
Loss & LAE

85,483
104,796
18,279
Indicated change = [LR + F / (1 - V - Q)] - 1 = [0.7352 + 0.056 / (1 - 0.24 - 0.035)] - 1 = +9.677%

LR
0.69699
0.7775
0.7392

c. This amount is a catastrophic loss and will distort indications. It should be excluded from the analysis
and an appropriate catastrophe load should be incorporated based on separate analysis.
Question 10
10. (1.5 points) Identify whether the loss ratio or pure premium ratemaking method is preferable in each
of the following scenarios. Briefly explain each answer.
a. (0.5 point) A company introduced two new rating variables within the past year.
b. (0.5 point) A company is entering a new line of business.
c. (0.5 point) A company writes a commercial product with multiple exposure bases.
Question 10 – Model Solution
a. Pure premium because bringing historical premium to CRL with the new variables may be difficult.
b. Pure premium because there is no existing rate to which an indicated change can be applied.
c. Loss ratio because an accurate and consistent exposure measure will be difficult to calculate.

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Statement of Principles Regarding P & C Insurance Ratemaking
CAS COMMITTEE ON RATEMAKING PRINCIPLES
Section 1
Section 2
Section 3
Section 4

Background
Definitions
The Statement of Principles
Considerations

Section 1

Background

A. Background regarding the Principles:
1. The principles are limited to the portion of the ratemaking process involving the estimation of costs
associated with the transfer of risk.
2. Provides the foundation for the development of actuarial procedures and standards of practice.
3. Applies to other risk transfer mechanisms.
The ratemaking process considers marketing goals, competition, legal restrictions, etc., to the extent
they affect the estimation of future costs associated with the transfer of risk
B. The costs associated with transfer of risk include:
1. Claims 2. Settlement expenses 3. Operational and administrative

Section 2

Definitions

Select Definitions:
Other acquisition
expense
U/W P&C provision
TL&F

Section 3
Principle 1
Principle 2
Principle 3

Principle 4

4. Cost of Capital.

All costs, except commission and brokerage, associated with the acquisition of
business.
Amounts that, when considered with net investment income and other income,
provide an appropriate total after-tax return.
Taxes, licenses and fees except federal income taxes.

The Statement of Principles
A rate is an estimate of the expected value of future costs.
A rate provides for all costs associated with the transfer of risk.
A rate provides for the costs associated with an individual risk transfer.
(When an individual risk's experience does not provide a credible basis for estimating costs,
it is appropriate to consider the aggregate experience of similar risks).
A rate is reasonable and NOT excessive, inadequate, or unfairly discriminatory if it is an
actuarially sound estimate of the expected value of all future costs associated with an
individual risk transfer.

Notes:


Ratemaking produces cost estimates that are actuarially sound if it is based on
principles 1, 2 and 3.The actuary need not be completely bound by these precedents.
Material assumptions should be documented and available for disclosure.

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Statement of Principles Regarding P & C Insurance Ratemaking
CAS COMMITTEE ON RATEMAKING PRINCIPLES

Section 4

Considerations

Data

Consider historical premium, exposure, and loss data (external and internal).

Exposure Unit

Should vary with the hazard, and be practical and verifiable.

Mix of Business

Changes in deductibles, coverage limits affecting frequency and severity.

Credibility

Homogeneity. A group should be large enough to be statistically reliable.

Actuarial Judgment

Can be used effectively. It should be documented and available.

Policy Provisions

Review subrogation and salvage, coinsurance, deductibles, 2nd injury fund
recoveries.

Reinsurance

Examine the effects of various arrangements.

Individual Risk Rating

Examine the impact of individual risk rating plans on overall experience.

Trends

Consider past and prospective changes in frequency, severity, exposure,
expenses.

Organization of Data

CY, AY, RY, PY. Availability, clarity, and simplicity dictate the choice.

Catastrophe

Consider including an allowance for the catastrophe exposure in the rate.

Operational changes

Review U/W, Claims, Reserving, Marketing.

Other Influences

Regulatory, Residual Markets, Economic Variables need to be considered.

Loss Development

Expected development is subject to CAS Statement of Reserving Principles.

Risk

Risk of random variation from expected costs; It should be consistent with
the cost of capital, and therefore influences the U/W profit provision.
Risk of systematic variation of estimated costs from expected costs. This
charge should be reflected when determining the Contingency provision.

Investment and other
income
Class Plans

Properly defined, it enables the development of actuarially sound rates.

Homogeneity

Subdivide or combine to minimize effects of procedural changes.

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Statement of Principles Regarding P & C Insurance Ratemaking
CAS COMMITTEE ON RATEMAKING PRINCIPLES
Question from the 1989 exam
4. According to the Statement of Principles Regarding Property and Casualty Insurance
Ratemaking, which of the following are true?
1. A rate is an estimate of the expected value of future costs.
2. Informed actuarial judgment should not be used in ratemaking, unless there is a lack of credible data.
3. Consideration should be given in ratemaking to the effects of subrogation and salvage.
A. 1

B. 2

C. 1, 3

D. 2, 3

E. 1, 2, 3

Question from the 1990 exam
1. (1 point) According to the "Statement of Principles Regarding Property and Casualty Insurance
Ratemaking," which of the following are true?
1. Marketing, underwriting, legal and other business considerations should NOT be a factor when
applying the principles set forth in the above statement.
2. Historical premium, exposure, loss and expense experience is usually the starting point of
ratemaking.
3. Accident year is the best acceptable method of organizing data to be used in ratemaking.
A. 1

B. 2

C. 3

D. 1, 2

E. None of the above.

Question from the 1991 exam
18. (1 point) According to the CAS Committee on Ratemaking Principles, "Statement of Principles Regarding
Property and Casualty Insurance Ratemaking," which of the following are stated principles?
1. A rate provides for all costs associated with the transfer of risk.
2. A rate is an estimate of the expected value of future costs.
3. A rate provides for the costs associated with an individual risk transfer.
A. 1

B. 1, 2

C. 1, 3

D. 2, 3

E. 1, 2, 3

Question from the 1992 exam
There were no questions from this article tested on the above referenced exam.

Question from the 1993 exam
23. According to Statement of Principles Regarding Property and Casualty Insurance Ratemaking, which
of the following are true?
1. The charge for any systematic variation of the estimated costs from the expected cost should
be reflected in the determination of the contingency provision.
2. Experience should be organized on an accident year basis whenever possible.
3. A rate provides for the costs associated with an individual risk transfer.
A. 2 only

Exam 5, V1a

B. 3 only

C. 1, 3 only

Page 265

D. 2, 3 only E. 1, 2, 3.

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Statement of Principles Regarding P & C Insurance Ratemaking
CAS COMMITTEE ON RATEMAKING PRINCIPLES
Question from the 1994 exam
39. (3 points) You are an actuary analyzing recommended rates for a line of business for which you only
write two classes. The company has a monopoly, and all insureds must buy insurance. There are
no legal restrictions on the rates charged. Below is a summary of the current rate situation.
Class
A
B
Average

Current
$100
$200
$150

Indicated
$ 75
$225
$150

Recommended
$100
$200
$150

Are the recommended rates consistent with the Principles set forth in the '“Statement of Principles
Regarding Property and Casualty Insurance Ratemaking"? Be specific and explain why or why not.

Questions from the 1995 exam
1. (1 point) According to the “Statement of Principles Regarding Property and Casualty Insurance Ratemaking”,
which of the following are true?
1. Affordability is specifically stated as an important factor that should be considered in the ratemaking
process.
2. The cost of reinsurance should be considered in the ratemaking process
3. Changes in the underwriting process should be considered in the ratemaking process.
A. 1 only

B. 2 only

C. 3 only

D. 2, 3 only

E. 1, 2, 3.

28. (2 points) Your company wants to start writing Automobile Insurance in State X. You have developed
rates and have filed them with the insurance department. The insurance department accuses your
company of filing excessive rates because they are significantly higher than your rates for identical
insureds in neighboring State Y.
Using the “Statement of Principles Regarding Property and Casualty Insurance Ratemaking," list and
briefly describe four external influences that you could cite that justify higher rates in State X.

Question from the 1996 exam
1. According to the "Statement of Principles Regarding Property and Casualty Insurance Ratemaking,"
which of the following are true of ratemaking?
1. Consideration should be given to the effect of reinsurance arrangements.
2. Consideration should be given to the quality of company management.
3. Consideration should be given to changes in claims handling practices.
A.

1 only

2 only

3 only

1, 3 only

1, 2, 3

Question from the 1997 exam
25.
A. (1 point) According to the "Statement of Principles Regarding Property and Casualty Ratemaking," what
are three desirable features for exposure units to have?

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Statement of Principles Regarding P & C Insurance Ratemaking
CAS COMMITTEE ON RATEMAKING PRINCIPLES
Question from the 1998 exam
46. Assume that a state has a monopoly on a line of insurance, and it mandates that each insured pays
the same fixed rate, based upon what it believes the average insured can afford. Any deficit is made
up from the state's general revenues, and any surplus goes into other state funds.
Based on the "Statement of Principles Regarding Property and Casualty Insurance Ratemaking,"
answer the following questions.
a. (1.5 points) Identify principles 1, 2, and 3 and state whether the system described above satisfies
each principle. Briefly explain why or why not.
b. (.50 point) If the state changes the system so that if there is a deficit, there is an equal surcharge on all
policyholders, and if there is a surplus there is an equal rebate, how would your answer to part (a)
change?

Question from the 1999 exam
Question 41. As the ratemaking actuary for your company, you have proposed to change the exposure base for
automobile coverage to "actual miles the vehicle is driven."
Based on the "Statement of Principles Regarding Property and Casualty Insurance Ratemaking," state
three criteria for a desirable exposure base and briefly discuss whether your proposal satisfies (or
does not satisfy) each criteria.

Question from the 2000 exam
22. According to the Statement of Principles Regarding Property and Casualty Insurance Ratemaking, which of
the following statements is true?
A. Subdividing the data to minimize the effects of operational or procedural changes may increase credibility.
B. Creating homogeneous groupings of data will tend to decrease the credibility of the data.
C. Data should not be organized by calendar year for purposes of producing rates.
D. When considering the trade-off between partitioning of data into homogeneous groups versus increasing the
volume of ratemaking data in each grouping, preference should be given to creating the most homogeneous
groupings.
E. None of A, B, C, or D is true.

Question from the 2000 exam
42. (2 points)
According to the Statement of Principles Regarding Property and Casualty Insurance Ratemaking, ratemaking
produces actuarially sound cost estimates if rates are based on three principles.
a. (1 point) State these three principles.
b. (1 point) If a rate is actuarially sound, it complies with four criteria commonly used by actuaries. Name these
four criteria.

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Statement of Principles Regarding P & C Insurance Ratemaking
CAS COMMITTEE ON RATEMAKING PRINCIPLES
Questions from the 2001 exam
Question 3. According to the Statement of Principles Regarding Property and Casualty Insurance
Ratemaking, which of the following statements is true?
A. Unallocated loss adjustment expenses are the claim settlement costs directly assignable to specific claims.
B. Taxes, licenses, and fees exclude federal income taxes.
C . Policyholder dividends are a return of premium not assigned as an expense.
D. Allocated loss adjustment expenses include all costs associated with the settlement of claims.
E. General administrative expenses are all costs, except commission and brokerage costs, associated
with the acquisition of business.
Question 4. According to the Statement of Principles Regarding Property and Casualty Insurance
Ratemaking, which of the following statements is true?
A. Consideration should be given to changes in case reserving that affect the continuity of the experience.
B. Consideration should be given to the determination of an appropriate exposure unit or premium basis,
although it is not essential.
C. Ratemaking is retrospective because the property and casualty insurance rate must be developed
after the transfer of risk.
D. Credibility is generally increased by making groupings more heterogeneous due to the diversification
benefit from combining uncorrelated items.
E. Changes in policy provisions, such as coordination of benefits and second injury fund recoveries, are
outside the scope of ratemaking data and thus need not be considered in ratemaking methodologies.

Questions from the 2002 exam
1. Based on the Statement of Principles Regarding Property and Casualty Insurance Ratemaking, which
of the following statements is false?
A.
B.
C.
D.

A rate is an estimate of the expected value of current costs.
A rate provides for all costs associated with the transfer of risk.
A rate provides for the costs associated with an individual risk transfer.
Rates that are actuarially sound comply with the following criteria: reasonable, not excessive, not
inadequate, and not unfairly discriminatory.
E. Ratemaking is prospective because the property and casualty insurance rate must be developed
prior to the transfer of risk.

Questions from the 2003 exam
30. (3 points) The Statement of Principles Regarding Property and Casualty Insurance Ratemaking lists
numerous considerations involved in the ratemaking process. State and briefly discuss three of these
considerations that have been impacted by the recent rise in worldwide terrorist activity.

Questions from the 2004 exam
9. Which of the following is true regarding ratemaking expense provisions?
1. Taxes, licenses and fees do not include federal income tax.
2. Other acquisition expenses include commission and brokerage expenses.
3. General administrative expenses represent all costs associated with the claim settlement process not directly
assignable to specific claims.
A. 1 only
B. 2 only
C. 3 only
D. 1 and 2 only
E. 1 and 3 only

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Statement of Principles Regarding P & C Insurance Ratemaking
CAS COMMITTEE ON RATEMAKING PRINCIPLES
Questions from the 2004 exam (continued):
38. (1.5 points) Credibility is an important consideration in ratemaking methodology.
a. (0.5 point) Define credibility.
b. (0.5 point) One method of increasing credibility is by increasing the size of the groupings analyzed.
Briefly describe another method to increase credibility.
c. (0.5 point) Explain a potential weakness in increasing credibility by the method you provided in part
b. above.

Questions from the 2005 exam
35. (2 points) State the four ratemaking principles of the Casualty Actuarial Society.

Questions from the 2006 exam
25. (1.5 points) The ratemaking actuary for ABC Insurance Company is proposing to change the
exposure base for Homeowners Insurance from number of homes to amount of Coverage A.
a. (0.5 point) According to the Statement of Principles regarding P&C Insurance Ratemaking, state
two desirable characteristics of an exposure base.
b. (1.0 point) Determine which exposure base better satisfies each of the characteristics stated in
part a. above. Explain.

Questions from the 2007 exam
11. Which of the following is true based on the Statement of Principles Regarding Property and Casualty
Insurance Ratemaking?
A. Unallocated loss adjustment expenses are the claim settlement costs directly assignable to
specific claims.
B. Taxes, licenses, and fees exclude federal income taxes.
C. Policyholder dividends are a return of premium not assigned as an expense.
D. Allocated loss adjustment expenses include all costs associated with the settlement of claims.
E. General administrative expenses are all costs, except commission and brokerage costs,
associated with the acquisition of business.

Questions from the 2009 exam
39. (1.75 points)
a. (1 point) Identify two considerations from the "Statement of Principles Regarding Property & Casualty
Ratemaking" that could apply to the concept of insurance to value. Briefly explain the relevance of each to
insurance to value.
b. (0.75 point) An insurance company increases the insurance to value of its book of business.
Briefly describe the impact on each of the following:
• Premium
• Losses
• Expenses

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Statement of Principles Regarding P & C Insurance Ratemaking
CAS COMMITTEE ON RATEMAKING PRINCIPLES
Solution to the question from the 1989 exam
Question 4.
1. T.
2. F.
3. T.

Answer C.

Solution to the question from the 1990 exam
Question 1.
1. F.
2. T.
3. F.

Answer B.

Solution to the question from the 1991 exam
Question 18.
1. T.
2. T.
3. T.

Answer E.

Solution to the question from the 1993 exam
Question 23.
1. T. Risk
2. F. Organization of Data.
3. T.

Answer C.

Solution to the question from the 1994 exam
Question 39.
Principle 1: A rate is an estimate of the expected value of future costs. The recommended average rate of
$150 is consistent with the indicated estimate of the expected value of future costs.
Principle 2: A rate provides for all costs associated with the transfer of risk. By recommending an average
rate, which provides for the costs associated with the transfer of risk, equal to the indicated
average rate, equity among insureds is maintained.
Principle 3: A rate provides for the costs associated with an individual risk transfer. The recommended rate
of $200 for class B does not provide for the costs associated with an individual risk transfer, as
it is $25 below that which is indicated.

Solutions to questions from the 1995 exam
Question 1.
1. F. Affordability is not one of the considerations.
2. T. Reinsurance.
3. T. Operation Changes

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Answer D.

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Statement of Principles Regarding P & C Insurance Ratemaking
CAS COMMITTEE ON RATEMAKING PRINCIPLES
Solutions to questions from the 1995 exam
Question 28.
1. Other Influences: The judicial environment, residual markets, guaranty fund assessment all vary by state.
2. Trends: Consideration of past and prospective changes in frequency, severity, exposure, expenses, which
can vary by state.
3. Economic variables: Costs associated with repair and replacement all vary by state.
4. Catastrophe: The types of natural catastrophe’s vary by state, and degree of frequency and severity.

Solution to the question from the 1996 exam
Question 1.
The "Statement of Principles Regarding Property and Casualty Insurance Ratemaking," identifies 18
considerations.
1. Reinsurance is specifically listed.
2. Quality of company management is not listed.
3. Changes in claims handling practices is just one of the items mentioned under the category "Operational
Changes".
Answer D.

Solution to the question from the 1997 exam
Question 25.
A. Exposure units should vary with the hazard, and be practical and be verifiable.

Solution to the question from the 1998 exam
Question 46.
a.
Principle 1: A rate is an estimate of the expected value of future costs. The recommended rate, based on
affordability, and not on expected future costs, is not consistent with this principle.
Principle 2: A rate provides for all costs associated with the transfer of risk. Since any deficit is made up by
the state's general fund, this principle is not satisfied.
Principle 3: A rate provides for the costs associated with an individual risk transfer. Since the
recommended rate is fixed, this principle is not satisfied, as the costs associated with individual
risk transfer are not recognized.
b. Principle 2 is now satisfied since offering a rebate or imposing a surcharge provides a mechanism to
target all costs associated with the transfer of risk.

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Statement of Principles Regarding P & C Insurance Ratemaking
CAS COMMITTEE ON RATEMAKING PRINCIPLES
Solutions to questions from the 1999 exam
Question 41.
The statement of principles state that "it is desirable that exposure unit:
1. be Practical
2. be Verifiable
3. vary with the level of risk
The proposed exposure base is "actual miles the vehicle is driven."
1. The proposed exposure base is not practical from a number of aspects, including:
Accuracy - asking insureds to provide exposure base information makes the exposure base easy to
manipulate, and thus, gives rise to a moral hazard.
Expense - the expense of having the odometer read by company personnel may outweigh the benefits
gained from using this exposure base.
2. The proposed exposure base is verifiable (odometers can be read), but is subject to the following types of
manipulation:
a. odometers can malfunction
b. odometers can be adjusted by individuals and automobile shops.
3. For auto liability and collision, actual miles driven (as an exposure unit) clearly varies with the level of risk.

Solutions to questions from the 2000 exam
Question 22. Which of the following statements is true?
A. T. Subdividing the data to minimize the effects of operational or procedural changes may increase credibility.
Credibility is increased by making groupings more homogeneous or by increasing the size of the group
analyzed. Homogenous groups require refinement and portioning of the data. See page 3.
B. F. Creating homogeneous groupings of data will tend to decrease the credibility of the data.
Credibility is increased by making groupings more homogeneous or by increasing the
size of the group analyzed. See page 3.
C. F. Data should not be organized by calendar year for purposes of producing rates. Acceptable methods of
organizing data include calendar year, accident year, report year and policy year. See page 3.
D. F. When considering the trade-off between partitioning of data into homogeneous groups versus
increasing the volume of ratemaking data in each grouping, preference should be given to
creating the most homogeneous groupings. Each situation requires balancing homogeneity
and the volume of data. See page 3.
Answer A.

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Statement of Principles Regarding P & C Insurance Ratemaking
CAS COMMITTEE ON RATEMAKING PRINCIPLES
Solutions to questions from the 2000 exam
Question 42.
a. State the three principles in which ratemaking produces actuarially sound cost estimates
Principle 1
Principle 2
Principle 3

A rate is an estimate of the expected value of future costs.
A rate provides for all costs associated with the transfer of risk.
A rate provides for the costs associated with an individual risk transfer.
(When an individual risk's experience does not provide a credible basis for estimating costs,
it is appropriate to consider the aggregate experience of similar risks).

b. If a rate is actuarially sound, name the four criteria commonly used by actuaries.
Principle 4: A rate is actuarially sound if it is:
1. Reasonable
2. NOT excessive
3. NOT inadequate
4. NOT or unfairly discriminatory if it is an actuarially sound estimate of the expected value of all future
costs associated with an individual risk transfer.

Solutions to questions from the 2001 exam
Question 3. Which of the following statements is true?
A. Unallocated loss adjustment expenses are the claim settlement costs directly assignable to specific claims.
False. Allocated loss adjustment expenses are claim settlement costs directly assignable to specific claims.
B. Taxes, licenses, and fees exclude federal income taxes. True. Answer B.
C . Policyholder dividends are a return of premium not assigned as an expense. False. Policyholder
dividends are a non-guaranteed return of premium charged to operations as an expenses.
D. Allocated loss adjustment expenses include all costs associated with the settlement of claims. False.
Allocated loss adjustment expenses are the claim settlement costs directly assignable to specific claims.
E. General administrative expenses are all costs, except commission and brokerage costs, associated
with the acquisition of business. False. General administrative expenses are all other operational and
administrative costs.

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Statement of Principles Regarding P & C Insurance Ratemaking
CAS COMMITTEE ON RATEMAKING PRINCIPLES
Solutions to questions from the 2001 exam
Question 4. According to the Statement of Principles Regarding Property and Casualty Insurance
Ratemaking, which of the following statements is true?
A. Consideration should be given to changes in case reserving that affect the continuity of the
experience. True.
Answer A.
B. Consideration should be given to the determination of an appropriate exposure unit or premium basis,
although it is not essential. False. The determination of an appropriate exposure unit or premium
basis it is essential.
C. Ratemaking is retrospective because the property and casualty insurance rate must be developed
after the transfer of risk. False. Ratemaking is prospective because the property and casualty
insurance rate must be developed prior to the transfer of risk.
D. Credibility is generally increased by making groupings more heterogeneous due to the diversification
benefit from combining uncorrelated items. False. Credibility is generally increased by making
groupings more homogeneous or by increasing the size of the group analyzed.
E. Changes in policy provisions, such as coordination of benefits and second injury fund recoveries, are
outside the scope of ratemaking data and thus need not be considered in ratemaking methodology. False.
Changes in policy provisions, such as coordination of benefits and second injury fund recoveries, need to
be considered in ratemaking methodology

Solutions to questions from the 2002 exam
1. Based on the Statement of Principles Regarding Property and Casualty Insurance Ratemaking, which
of the following statements is false?
A. A rate is an estimate of the expected value of current costs.
False. A rate is an estimate of the expected value of future costs.
B. A rate provides for all costs associated with the transfer of risk. True.
C. A rate provides for the costs associated with an individual risk transfer. True.
D. Rates that are actuarially sound comply with the following criteria: reasonable, not excessive, not
inadequate, and not unfairly discriminatory. True.
E. Ratemaking is prospective because the property and casualty insurance rate must be developed
prior to the transfer of risk. True.

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Statement of Principles Regarding P & C Insurance Ratemaking
CAS COMMITTEE ON RATEMAKING PRINCIPLES
Solutions to questions from the 2003 exam
30. (3 points) The Statement of Principles Regarding Property and Casualty Insurance Ratemaking lists
numerous considerations involved in the ratemaking process. State and briefly discuss three of these
considerations that have been impacted by the recent rise in worldwide terrorist activity.
1. Reinsurance. Reinsurance has become more expensive because of the major losses on Sept 11. In
addition, many reinsurers have become insolvent, making recoveries uncertain. Both the cost of
reinsurance and the solvency of the reinsurer must be considered.
2. Catastrophe losses. Terrorist attacks were considered a catastrophe. The potential for future
catastrophic losses from terrorist attacks needs to be considered in any allowance for the catastrophe
exposure in the rates.
3. Legislation. There is a bill that has or is about to be passed about government involvement in losses
sustained in terrorist attacks. When this bill is passed, the effect on net losses for insurers will need to
be considered in ratemaking process.

Solutions to questions from the 2004 exam
9. Which of the following is true regarding ratemaking expense provisions?
1. Taxes, licenses and fees do not include federal income tax. True. See Section 1: Definitions.
2. Other acquisition expenses include commission and brokerage expenses. False. Other acquisition
expenses are all costs, except commission and brokerage, associated with the acquisition of business.
3. General administrative expenses represent all costs associated with the claim settlement process not
directly assignable to specific claims. False. General administrative expenses are all other
operational and administrative costs.
Answer A. 1 only
38. (1.5 points) Credibility is an important consideration in ratemaking methodology.
a. (0.5 point) Define credibility.
According to the CAS Statement of Principles regarding P&C ratemaking, “credibility is a measure of the
predictive value that the actuary attaches to a particular body of data.”
Note: The CAS model solution from the 2004 exam reads as follows: “Credibility is determined by how
much experience is expected to be a good predictor of future experience.”
b. (0.5 point) One method of increasing credibility is by increasing the size of the groupings analyzed.
Briefly describe another method to increase credibility.
Another method would be to increase the homogeneity of groupings analyzed. The more stable and
homogeneous a group, the larger the credibility. Obtaining homogeneous groupings requires refinement
and partitioning of the data. See the CAS Statement of Principles regarding P&C ratemaking.
c.

(0.5 point) Explain a potential weakness in increasing credibility by the method you provided in part b.
above.
There needs to be a balance between the size of the groupings and how homogeneous you make the
groupings. If groups are segregated too much in an attempt to increase homogeneity, the groups will be
too small to be credible. According to the CAS statement of principles, there is a point at which partitioning
divides data into groups too small to provide credible patterns. Each situation requires balancing
homogeneity and the volume of data.”

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Statement of Principles Regarding P & C Insurance Ratemaking
CAS COMMITTEE ON RATEMAKING PRINCIPLES
Solutions to questions from the 2005 exam
35. (2 points) State the four ratemaking principles of the Casualty Actuarial Society.
1. A rate is an estimate of the expected value of future costs.
2. A rate provides for all costs associated with the transfer of risk.
3. A rate provides for the cost associated with an individual risk transfer.
4. A rate is reasonable, not inadequate, excessive, or unfairly discriminatory if it is an actuarially
sound estimate of the expected value of future costs associated with an individual transfer of risk.

Solutions to questions from the 2006 exam
25. (1.5 points) The ratemaking actuary for ABC Insurance Company is proposing to change the
exposure base for Homeowners Insurance from number of homes to amount of Coverage A.
a. (0.5 point) According to the Statement of Principles regarding P&C Insurance Ratemaking, state
two desirable characteristics of an exposure base.
b. (1.0 point) Determine which exposure base better satisfies each of the characteristics stated in
part a. above. Explain.
Initial comments:
Exposure Unit—The determination of an appropriate exposure unit or premium basis is essential. It is
desirable that the exposure unit vary with the hazard and be practical and verifiable.
CAS Model Solution:
Part a.
1 – Verifiable.
2 – Vary with hazard.
- OR 3 – Be practical

Part b.
1 – It is easier to verify that there is a home (# homes) rather than the value of home. Thus number of
homes is better for verifiability.
2 – Coverage A amount is a better exposure base for varying with hazard. The amount of damage and
loss depends on the value of the home.
- OR 3 – The number of homes is more practical since Coverage A amount is subject to some judgment.

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Statement of Principles Regarding P & C Insurance Ratemaking
CAS COMMITTEE ON RATEMAKING PRINCIPLES
Solutions to questions from the 2007 exam
11. Which of the following is true based on the Statement of Principles Regarding Property and Casualty
Insurance Ratemaking?
A. Unallocated loss adjustment expenses are the claim settlement costs directly assignable to specific
claims False. Unallocated loss adjustment expenses are all costs associated with the claim settlement
function not directly assignable to specific claims. See Definitions.
B. Taxes, licenses, and fees exclude federal income taxes. True. See Definitions.
C. Policyholder dividends are a return of premium not assigned as an expense. False. Policyholder
dividends are a non-guaranteed return of premium charged to operations as an expense. See Definitions.
D. Allocated loss adjustment expenses include all costs associated with the settlement of claims.
False. Allocated loss adjustment expenses are claims settlement costs directly assignable to specific
claims. See Definitions.
E. General administrative expenses are all costs, except commission and brokerage costs, associated
with the acquisition of business. False. Statement E. is the definition of other acquisition expenses.
General administrative expenses are all other operational and administrative costs. See Definitions.

Solutions to questions from the 2009 exam
Question 39 – Model Solution
a. Mix of business - changing mix of ITV in the book will influence premium and loss trends.
Economic/Social
Social trends = if there is a movement towards lower insurance to value because people are purchasing
lower amounts of coverage to save money on premium due to hard economic times, the actuary may want to
evaluate the insurance to value contemplated on the current rates.
b. Premium - could see higher prem. as a result of larger exposure amounts written
could see lower premium if there are higher cancel/non-renews
Losses – expect to see larger total and near total claim amts. from larger exposures
Losses may decrease from higher cancel/non-renew
Losses may decrease if reinspection also leads to loss control measures implemented by homeowners.
Expenses – increased inspection/reinspection may create additional expenses, however increase relative to
premium change is unclear.

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ASOP 13
Trending Procedures in Property/Casualty Insurance
Sec
1
2
3
4
1

Description
Purpose, Scope, Cross References, and Effective Date
Section 2. Definitions
Section 3. Analysis of Issues and Recommended Practices
Section 4. Communications and Disclosures

Pages
1-1
1-2
2-3
3-4

Purpose, Scope, Cross References, and Effective Date

1-1

1.1 Purpose—To provide guidance to actuaries when performing trending procedures to estimate future values.
1.2 Scope—This standard applies to actuaries when performing work for insurance or reinsurance
companies, as well as self insurers.
A trending procedure does not encompass “development,” which estimates changes over time in
losses (or other items) within a given exposure period (e.g. accident year or underwriting year).
If the actuary departs from the guidance in this standard to comply with applicable law (statutes,
regulations, and other legally binding authority) or for any other reason the actuary deems
appropriate, refer to section 4.3.
1.3 Cross References—When referring to the provisions of other documents, the reference includes the
referenced documents as they may be amended or restated in the future, and any successor to them,
by whatever name called.
If any amended or restated document differs materially from the originally referenced document,
consider the guidance in this standard to the extent it is applicable and appropriate.

Section 2. Definitions

1-2

2.1 Coverage—The terms and conditions of a plan or contract, or the requirements of applicable law, that
create an obligation for claim payment associated with contingent events.
2.2 Experience Period—The period of time to which historical data used for actuarial analysis pertain.
2.3 Forecast Period—The future time period to which the historical data are projected.
2.4 Social Influences—The impact on insurance costs of societal changes (e.g. changes in claim
consciousness, court practices, and legal precedents, as well as in other noneconomic factors).
2.5 Trending Period—The time over which trend is applied in projecting from the experience period to the
forecast period.
2.6 Trending Procedure—A process by which the actuary evaluates how changes over time affect items such as
claim costs, claim frequencies, expenses, exposures, premiums, retention rates, marketing/solicitation
response rates, and economic indices. Trending procedures estimate future values by analyzing changes
between exposure periods (e.g. accident years or underwriting years).

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ASOP 13
Trending Procedures in Property/Casualty Insurance
3

Section 3. Analysis of Issues and Recommended Practices

2-3

3.1 Purpose or Use of Trending Procedures—Trending is an important component in ratemaking,
reserving, valuations, underwriting, and marketing.
Where multiple purposes or uses are intended, the actuary should consider the potential conflicts
arising from those multiple purposes or uses and should consider adjustments to accommodate the
multiple purposes or uses to the extent that, in the actuary’s professional judgment, it is appropriate
and practical to make such adjustments.
The actuary may present the trend estimate resulting from the trending procedure in a variety of ways
(e.g. a point estimate, a range of estimates, a point estimate with a margin for adverse deviation, or a
probability distribution of the trend estimate).
3.2 Historical Insurance and Non-Insurance Data
The actuary should select data (historical insurance or non-insurance information) appropriate for the
trends being analyzed.
When selecting data, the actuary should consider the following:
1. the credibility assigned to the data by the actuary;
2. the time period for which the data is available;
3. the relationship to the items being trended; and
4. the effect of known biases or distortions on the data relied upon (e.g. the impact of
catastrophic influences, seasonality, coverage changes, nonrecurring events, claim practices,
and distributional changes in deductibles, types of risks, and policy limits).
3.3 Economic and Social Influences
Consider economic and social influences that can have a significant impact on trends in selecting the
appropriate data to review, the trending calculation, and the trending procedure.
Consider the timing of the various influences.
3.4 Selection of Trending Procedures
In selecting trending procedures, the actuary may consider relevant information as follows:
a. procedures established by precedent or common usage in the actuarial profession;
b. procedures used in previous analyses;
c. procedures that predict insurance trends based on insurance, econometric, and other noninsurance data; and
d. the context in which the trend estimate is used in the overall analysis.
3.5 Criteria for Determining Trending Period
The actuary should consider the following when determining the trending period:
 the lengths of the experience and forecast periods
 changes in the mix of data between the experience and forecast periods when determining
the trending period.
When incorporating non-insurance data in the trending procedure, the actuary should consider the
timing relationships among the non-insurance data, historical insurance data, and the future values
being estimated.

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ASOP 13
Trending Procedures in Property/Casualty Insurance
3.6 Evaluation of Trending Procedures—The actuary should evaluate the results produced by each
selected trending procedure for reasonableness and revise the procedure where appropriate.
3.7 Reliance on Data or Other Information Supplied by Others—When relying on data or other information
supplied by others, the actuary should refer to ASOP No. 23, Data Quality, for guidance.
3.8 Documentation —The actuary should prepare and retain appropriate documentation regarding the
methods, assumptions, procedures, and the sources of the data used.
The documentation should be in a form such that another actuary qualified in the same practice area
could assess the reasonableness of the actuary’s work, and should be sufficient to comply with the
disclosure requirements in section 4.

Section 4. Communications and Disclosures

3-4

4.1 Actuarial Communication—When issuing an actuarial communication subject to this standard, the
actuary should refer to ASOP Nos. 23 and 41, Actuarial Communications.
In addition, the actuary should disclose the following, as applicable, in an actuarial communication:
a. the intended purpose(s) or use(s) of the trending procedure, including adjustments that the
actuary considered appropriate in order to produce a single work product for multiple purposes
or uses, if any, as described in section 3.1; and
b. significant adjustments to the data or assumptions in the trend procedure, that may have a
material impact on the result or conclusions of the actuary’s overall analysis.
4.2 Additional Disclosures—The actuary may need to make the following disclosures in addition to those in 4.1:
a. When the actuary specifies a range of trend estimates, disclose the basis of the range provided.
b. Disclose changes to assumptions, procedures, methods or models that the actuary believes
might materially affect the actuary’s results or conclusions as compared to those used in a prior
analysis, if any, performed for the same purpose.
4.3 Deviation—If the actuary departs from the guidance set forth in this standard, the actuary should
include the following where applicable:
4.3.1 the disclosure in ASOP No. 41, section 4.2, if any material assumption or method was
prescribed by applicable law (statutes, regulations, and other legally binding authority)
4.3.2 the disclosure in ASOP No. 41, section 4.3.1, if any material assumption or method was
selected under applicable law by a party other than the actuary, and the actuary disclaims
responsibility for the assumption or method;
4.3.3 the disclosure in ASOP No. 41, section 4.3.2, if the actuary disclaims responsibility for any
material assumption or method in any situation not covered under section 4.3.1 or 4.3.2; and
4.3.4 the disclosure in ASOP No. 41, section 4.4, if the actuary deviated from the guidance of this ASOP.

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ASOP 13
Trending Procedures in Property/Casualty Insurance
Question from the 1993 exam
21. Based on the "Actuarial Standard of Practice No. 13, Trending Procedures in Property/Casualty
Insurance Ratemaking," which of the following are examples of biases or distortions which should be
considered when examining historical insurance data for trending purposes?
1. Hurricane Andrew which struck Florida in 1992.
2. The increase in the Massachusetts automobile Personal Injury Protection coverage from
$2,000 to $8,000.
3. The impact of school vacations on automobile miles driven.
A. 1 only

B. 2 only

C. 1, 3 only

D. 2, 3 only

E. 1, 2, 3

Question from the 1994 exam
19. Based on "Actuarial Standard of Practice No. 13, Trending Procedures in Property/Casualty Insurance
Ratemaking," which of the following items should be considered in the trending procedure used in ratemaking
for Workers Compensation insurance?
1. An enacted reform that restricts the use of lump sum settlements.
2. Annual revisions in the hourly rate of compensation for union employees.
3. A decrease in attorney representation as Workers Compensation returns to a true "first party"
coverage.
A. 1 only

B. 2 only

C. 1, 2 only

D. 2, 3 only

E. 1, 2, 3

Question from the 1995 exam
There were no questions associated with this article appearing on the 1995 exam.

Question from the 1997 exam
2. Based on the "Actuarial Standard of Practice No. 13, Trending Procedures in Property/Casualty Insurance
Ratemaking," which of the following are biases or distortions that could affect the selection of trending
procedures?
1. Revising Homeowners policy coverage from actual cash value to replacement cost value.
2. A new underwriting requirement for percentage hurricane deductibles.
3. An automatic insurance to value program at policy renewal.
A. 1

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B. 2

C. 3

D. 1, 2

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E. 1, 2, 3

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ASOP 13
Trending Procedures in Property/Casualty Insurance
Questions from the 2001 exam
Question 14. According to “Actuarial Standard of Practice No. 13: Trending Procedures in Property/Casualty
Insurance Ratemaking,” which of the following items should be considered in the trending
procedure used in ratemaking for private passenger automobile insurance?
A. A decrease in automobile usage due to rising gas prices
B. The introduction of higher policy limits
C. A recently enacted tort reform that strengthens the verbal threshold for lawsuits
D. Changes in price levels in the economy as measured by external indices such as the Consumer
Price Index
E. All of the above should be considered.

Questions from the 2007 exam
6. According to ASOP No. 13, Trending Procedures in Property/Casualty Insurance Ratemaking, which of the
following should be considered when selecting trending procedures?
1. Known biases (e.g., seasonality)
2. The impact on the overall indication
3. The credibility of the data
A. 1 only

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B. 1 and 2 only

C. 1 and 3 only

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D. 2 and 3 only

E. 1, 2, and 3

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ASOP 13
Trending Procedures in Property/Casualty Insurance
Solutions to questions from the 1993 exam
Question 21.
Analysis of Historical Insurance Data
Select trending procedures with considerations to: The effect of known biases or distortions (Cats,
Seasonality, Deductible changes, Coverage changes , Type of Risks, and Policy Limits).
1. T. CATS
2. T. Coverage changes
3. T. Seasonality
Answer E.

Solutions to questions from the 1994 exam
Question 19.
1. T. Non-recurring changes (tort reform
2. T. Economic Influences
3. T. Coverages changes

Answer E.

Solutions to questions from the 1997 exam
Question 2.
Select trending procedures with considerations to:
a. Those established by precedent or common usage in the actuarial profession.
b. Those used in previous analyses.
c. The choice of the data base and methodology, with emphasis given to the credibility of the data.
d. The effect of known biases or distortions (e.g. Cats, Nonrecurring events, Seasonality, Deductible
changes, Coverage changes, Type of Risks, and Policy Limits).
Thus, 1, 2, and 3 are true.

Answer E.

Solutions to questions from the 2001 exam
Question 14. Which of the following items should be considered in the trending procedure used in
ratemaking for private passenger automobile insurance?
A. A decrease in automobile usage due to rising gas prices. True. Economic influences (such as
rising gas prices) impact trend.
B. The introduction of higher policy limits. True. Trending procedures should consider the effect of
known biases or distortions when using historical data (Cats, Seasonality, Deductible changes,
Coverage changes, Type of Risks, and Policy Limits).
C. A recently enacted tort reform that strengthens the verbal threshold for lawsuits. True. Social
inflation (the impact on insurance costs from changes in claim consciousness, court practices,
judicial attitudes) impacts trend.
D. Changes in price levels in the economy as measured by external indices such as the Consumer
Price Index. True. Consideration should be given to non-insurance data that supplements
insurance data.
E. All of the above should be considered. True.
Answer E.

Solutions to questions from the 2007 exam
6. According to ASOP No. 13, Trending Procedures in Property/Casualty Insurance Ratemaking, which of the
following should be considered when selecting trending procedures?
1. Known biases (e.g., seasonality).

True.

2. The impact on the overall indication.

False.

3. The credibility of the data.

True

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Risk Classification Statement of Principles
AMERICAN ACADEMY OF ACTUARIES COMMITTEE ON RISK CLASSIFICATION
Section
1
2
3
4
5
1

Description
Summary
Economic Security and Insurance
The Need for Risk Classification
Considerations in Designing a Risk Classification System
Conclusion
Summary

3 elements associated with the economic uncertainty of losses:
1. Occurrence.
2. Timing.
3. Financial impact.
Risk classification:
a. is necessary to maintain a financially sound and equitable system.
b. enables the development of equitable insurance prices, which in turn assures the availability of
needed coverage to the public.
c. is achieved through the grouping of risks to determine averages and the application of these
averages to individuals.
Risk classification is:
the grouping of risks with similar risk
characteristics for the purpose of setting prices.

Risk classification is not:
a. the prediction of experience for individual risks
(it is both impossible and unnecessary to do so).
b. to identify good or bad risks OR to reward or penalize
certain groups of risks at the expense of others.

3 primary purposes of risk classification:
1. Protect the insurance system's financial soundness.
2. Be fair.
3. Encourage availability of coverage through economic incentives.
Note: Achieving an appropriate balance among these purposes is not easy. However, they are in the
public interest and are not incompatible.
5 basic principles to achieve the primary purposes:
A risk classification system should:
1. Reflect expected cost differences.
2. Distinguish among risks based on relevant cost-related factors.
3. Be applied objectively.
4. Be practical and cost-effective.
5 Be acceptable to the public.
Marketing, underwriting and administration combine with risk classification to provide an entire
system of insurance.

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Risk Classification Statement of Principles
AMERICAN ACADEMY OF ACTUARIES COMMITTEE ON RISK CLASSIFICATION
2

Economic Security and Insurance

3 mechanisms for coping with the financial impact of chance occurrences (both natural and societal):
1. Hazard avoidance and reduction.
a. Some hazards may be avoided or exposure to them reduced. Choose not to engage in a hazardous
activity or implement safety precautions to reduce the incidence and severity of other hazards.
However, the practical application of hazard avoidance and hazard reduction is limited.
b. While some financially insignificant hazards may be retained and funded through savings or reserves,
retention of major financial uncertainties may be undesirable and unwise.
2. Transfer of financial uncertainty (governmental assistance, self-insured group pension, private ins, etc).
Programs for transferring financial uncertainty include charitable activities by individuals and
organizations; governmental assistance and insurance programs; self-insured group pension and
welfare plans; and private insurance programs.
3. Public vs. Private insurance programs:
Similarities
1. The transfer of financial uncertainty
and the subsequent pooling of risks.
2. The exposure to loss is (should be)
broad enough to assure reasonable
predictability of total losses.

Differences
1. Gov't plans are usually compulsory while Private
programs are usually voluntary.
2. Gov't plans are provided by law while Private plans
are subject to contractual agreement.
3. Competition plays an important role in Private but not
public plans.
4. Gov't plans often provide coverage for risks which
are "uninsurable" privately.
5. In Gov't programs, the benefits received by, or paid
on behalf of a class, are not necessarily related to
the amount paid into the plan by that class.
6. Private insurance programs are highly diverse.

The Need for Risk Classification

Although the exchange of uncertainty for a fixed price does not alter the uncertainty, the firm should find a
way of establishing a fair price for assuming the uncertainty.
3 Means of Establishing a Fair Price:
1. Reliance on wisdom, insight, and good judgment.
2. Observation of the risk's actual losses over an extended period of time.
(Not appropriate for life insurance applications. Also, a gradual change in the hazard may render
past information useless).
3. Observation of losses from groups of individual risks with similar characteristics.
This is the most frequently used method.
Its major problem: identification of similar risk characteristics (determined by fact and informed
judgment) and related classes before the observation period.

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Risk Classification Statement of Principles
AMERICAN ACADEMY OF ACTUARIES COMMITTEE ON RISK CLASSIFICATION
3

The Need for Risk Classification

3 Primary Purposes of Risk Classification
1. Protect the insurance program's financial soundness.
This is threatened by adverse selection (in markets where buyers are free to select, with a
motivation to minimize the price for the coverage sought, adverse selection is possible).
Risk classification minimizes the effects of adverse selection.
Regulation can control adverse selection by restricting the buyer's freedom (e.g. participation can be
made mandatory).
2. Enhanced fairness


Produce prices that are not unfairly discriminatory.



Price differentiation should reflect differences in expected costs with no redistribution or subsidy
among classes.



Prices and expected costs should also match within each class.

3. Economic incentive


Risk classification will help ensure adequate prices for the assumed uncertainty.



Selling to higher cost risks will increase market penetration which provides economies of scale.



Competition will motivate an insurer to refine its risk classification system so that it can better
serve both lower and higher cost risks.



A risk classification system should be efficient. It should not cost more to refine than the
reduction in expected costs.

Finally, while there is a close, and reinforcing, relationship among the 3 distinct primary purposes of risk
classifications, a system which serves any one tends to serve the other two as well.

Considerations in Designing a Risk Classification System

1. Underwriting is the process of determining the acceptability of a risk based on its own merits.


is in contrast to the assignment of a risk to a classed based on general criteria.

 controls the practical impact of the classification system.
2. Marketing influences the insurer’s mix of business and restrictions on / adjustments to a risk
classification system may produce unintended changes in the mix of business.
3. Program Design elements related directly to risk classification include:


degree of choice available to the buyer (compulsory programs use broad classes while voluntary
programs are more refined).



experience based pricing (when purchased by or through an organization, the price adjustment is
referred to as an experience rating adjustment; when purchased by an individual, it is recognized
by a dividend or in the premium paid).



classes used for experience rating (may be different than those used for the original pricing). The
need for less refined classes exists when experience rating is used.
premium payer. Use a broad class system to reduce the chance of adverse selection if the
premium payer is not the individual insured (i.e. group insurance).



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4

Considerations in Designing a Risk Classification System

4. Statistical Considerations may be conflicting. An increase in the number of classes may improve
homogeneity at expense of credibility.


Homogeneity. The overlap phenomenon (actual claim experience of some risks in one class
being the same as those in another class) is both anticipated and a statistically inevitable
ramification.



Credibility. Each class in the risk classification should be large enough to permit credible
predictions.



Predictive stability requires the risk classification system to be:
(a) responsive to changes in the nature of insurance losses, yet
(b) stable in avoiding unwarranted abrupt changes in prices.

5. Operational Considerations
 expense - costs to obtain and maintain data, assigning risks to a class, and determining fair prices by
class.


constancy - the lack of constancy in the characteristics used increases expense and reduces its utility.



maximize coverage availability. Properly matching expected costs and price will enhance availability.



extreme discontinuity avoidance. Attention is needed in defining classes at the extreme ends of a
range. There should be enough classes to establish a reasonable continuum of expected losses but
few enough to allow significant differences between classes



absence of ambiguity - classes should be collectively exhaustive and mutually exclusive.



minimize abilities to manipulate the system.



measurability - class variables (age, sex, occupation, location) should be reliably measurable.

6. Hazard Reduction Incentives (e.g. recognizing sprinklers for risk classification) are desirable but not
necessary features of a risk classification system.
7. Public Acceptability Considerations:
Are difficult to apply in practice because social values:


are difficult to ascertain.



vary among segments of society.



change over time.

Public acceptability considerations should:


not differentiate unfairly among risks.



be based on clearly relevant data.



respect personal privacy.



be structured so that risks tend to identify naturally with their classification.

Regulatory and legislative restrictions on the risk classification system must balance the desire of
public acceptability with the potential economic side effects of adverse selection or market
dislocation.

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4

Considerations in Designing a Risk Classification System

8. Causality:


Class characteristics may be more publicly acceptable if there is a demonstrable cause and effect
relationship between the risk characteristic and expected costs, since such relationships tend to
boost confidence that such information is useful in predicting the future.



It is often impossible to prove statistically any postulated cause and effect relationship.

Thus, causality cannot be made a requirement of a risk classification system.
Causality may be used in a general sense, implying the existence of plausible relationships between
characteristics of a class and the insured hazard.
9. Controllability:
Refers to the ability of an insured to control its own characteristics as used in the classification
system.
Controllability as a
Desirable risk characteristic:
Undesirable risk characteristic:
1. Its close association with an effort to reduce hazards. 1. Susceptibility to manipulation.
2. Its general acceptability by the public.
2. Its irrelevance to predictability of future costs.

Conclusion



Classification of risks is fundamental to any true insurance system.



Risk classification is done to determine average claim costs and to apply those averages to
individual risks.



Any risk classification is only part of an entire insurance structure and does not operate in a vacuum.

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AMERICAN ACADEMY OF ACTUARIES COMMITTEE ON RISK CLASSIFICATION
Questions from the 1991 Exam:
2. According to the ”Risk Classification Statement of Principles," by the American Academy of Actuaries,
which of the following statistical considerations are involved in designing a sound risk classification
system?
1. Creation of classes large enough to allow credible statistical predictions regarding the class.
2. Creation of classes small enough to be homogenous.
3. Creation of classes that are publicly acceptable.
A. 1 only
B. 3 only
C. 1 and 2
D. 2 and 3
E. 1, 2 and 3.
3. According to the ”Risk Classification Statement of Principles, " by the American Academy of
Actuaries, which of the following statements is true?
A) In insurance programs that are largely or entirely compulsory, with broad classifications and no
voluntary choice among competing institutions, adverse selection will likely occur.
B) Risk classification reduces adverse selection by balancing the economic forces governing buyers
and sellers.
C) Causality is a necessary requirement for risk classification systems.
D) Controllability is always a desired characteristic in a risk classification system.
E) None of the above statements is true.
20. (2 points) According to the "Risk Classification Statement of Principles" by the American Academy of
Actuaries, briefly discuss how and why individual risk rating affects the needed level of refinement in
a classification system.

Questions from the 1992 Exam:
1. Based on the American Academy of Actuaries' paper Risk Classification Statement of Principles,
which of the following are true:
1. The application of experience based pricing, based on the risk's actual losses, increases the
need for a refined classification system.
2. The presence of strong competition decreases the need for an insurer to have a refined
classification system.
3. Homogeneity and credibility are somewhat conflicting considerations for a risk classification
system.
A. 1 only
B. 3 only
C. 1 and 3
D. 2 and 3
E. All of the Above

Questions from the 1994 Exam:
5. According to the American Academy of Actuaries' "Risk Classification Statement of Principles",
which of the following are considered primary purposes of risk classification?
1. To protect the insurance program's financial soundness.
2. To enhance fairness.
3. To permit economic incentives to operate.
A. 2 only
B. 1 and 2
C. 1 and 3
D. 2 and 3
E. 1, 2 and 3

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Questions from the 1994 Exam (continued):
26. (2 points) In the American Academy of Actuaries' monograph "Risk Classification Statement of
Principles", several operational considerations in designing a successful classification system are
cited. List four of these considerations, and briefly explain how each contributes to the success of a
classification system. (Only the first four considerations listed will be graded.)

Questions from the 1995 Exam:
4. According to the American Academy of Actuaries' "Risk Classification Statement of Principles," which
of the following are true?
1. In contrast to the assignment of a risk to a class based on individual and possibly unique
characteristics of each risk, the underwriting process involves the evaluation of the risk based
on general criteria.
2. To the extent that prices are adjusted based on a risk's emerging actual experience after the
insurance and its initial price have been established, less refined initial risk classification
systems are needed.
3. As the proportion of the total premium paid by the insured increases, the use of a broader
classification system becomes more appropriate.
A. 1 only
B. 2 only
C. 3 only
D. 2 and 3
E. 1, 2 and 3
5. According to the American Academy of Actuaries' "Risk Classification Statement of Principles," which
of the following are true?
1. Operational expenses for a risk classification system include those expenses associated with
determining a price for each class.
2. Particular attention often is required in defining classes at the extreme ends of the expected claim
cost range, in order to reduce large differences in anticipated average claim costs between the
extreme class and the adjacent class.
3. Hazard reduction incentives are desirable and necessary features of a risk classification system.
A. 1 only
B. 3 only
C. 1 and 2
D. 2 and 3
E. 1, 2, and 3

Questions from the 1996 Exam:
17. According to "Risk Classification Statement of Principles" by the American Academy of Actuaries,
which of the following are the primary purposes of risk classification?
1. To protect the financial soundness of the insurance program.
2. To permit economic incentives to operate and thus encourage widespread coverage availability.
3. To identify unusually high and low quality risks.
A. 2
B. 3
C. 1, 2
D. 1, 3
E. 1,2,3
47. a. (1.25 points) According to the American Academy of Actuaries' "Risk Classification Statement of
Principles" promulgated in 1980, what are the five basic principles that should be present in any
sound risk classification system?
b. (0.5 point) The Actuarial Standards Board's "Actuarial Standard of Practice No. 12 Concerning
Risk Classification" was promulgated in 1989. Which of the five principles from part (a) did this
Standard explicitly omit?
c. (0.75 point) List three reasons given by the American Academy of Actuaries in "Risk Classification
Statement of Principles" on why the principle identified in part (b) is difficult to apply in practice.

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Questions from the 1996 Exam (continued):
48. A property insurance company is considering adding a new classification rating variable to its
homeowners insurance program based on individual risk's actual loss experience over the past five
year period as follows:
Class A - No claims
Class B - One or two claims
Class C - Three or more claims
a. (1.5 points) Evaluate this new classification rating variable based on the following considerations
as described in the American Academy of Actuaries' "Risk Classification Statement of Principles":
1. Controllability
2. Operational Expense
3. Hazard Reduction Incentives
b. (1.5 points) Considering the basic principles that should be present in any sound risk classification
system, would you recommend the addition of this new classification? Why or why not?

Questions from the 1997 Exam:
48. (3 points) As the personal lines actuary for the department of insurance in the state of Crazyfornia,
you have been asked by the state’s insurance commissioner to comment on Proposition 99.
Proposition 99- The ratemaking for personal automobile insurance should be based on a new
classification system using the following 6 criteria:
1. Insureds are to be classified based on nationality.
2. Insureds are to be classified based on the ability to pass an annual random drug test
3. Insureds are to be classified based on whether they can pass a comprehensive, individually
administered 8 hour driving test every year.
4. Insureds are to be classified based on their weights.
5. Insureds are to be classified as either ‘good eyesight’ or ‘bad eyesight’. Each eye doctor can
have his/her own definition of good/bad eyesight.
6. Insureds are to be classified as ‘right handed’ or ‘left handed’.
For each criterion, identify which one of the five basic principles of a sound risk classification system
(as mentioned in “Risk Classification Statement of Principles” by the American Academy of Actuaries
Committee on Risk Classification) is violated. You may not use the same principle for more than 2
criteria.

Questions from the 1999 Exam:
43. You are the actuary for Aggressive Mutual Insurance Company. The marketing department has
approached you with a plan to increase business by liberalizing protection class definitions. The
new definition would allow you to classify any risk within eight miles of the nearest fire department
using the protection class of that town, without any verification of its ability to respond to the location
of that risk.
a. (0.75 point) According to the American Academy of Actuaries Committee on Risk Classification's
"Risk Classification Statement of Principles," what are the three primary purposes of risk
classification?
b. (1.5 points) Based on these principles, what would you tell the marketing director about the
appropriateness of the proposed class definitions? Include a discussion of all three
principles from part a.

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Questions from the 1999 Exam (continued):
46. Based upon the American Academy of Actuaries Committee on Risk Classification's "Risk
Classification Statement of Principles," answer the following questions.
In an insurance program, an individual buying insurance exchanges the uncertainty of occurrence,
timing, and magnitude of a particular event for the certainty of a fixed price.
a. (1 point) List three methods for determining this price.
b. (1 point) List one deficiency for each method described in part a.

Questions from the 2000 Exam:
16. According to the American Academy of Actuaries Committee on Risk Classification’s ‘Risk
Classification Statement of Principles,” which of the following are not operational considerations
relating to classification plans?
A. Availability of Coverage
B. Avoidance of Extreme Discontinuities
C. Absence of Ambiguity
D. Measurability
E. All of the above are operational considerations.
35. Adverse selection is a financial threat to an insurance program’s solvency. Based on the American
Academy of Actuaries Committee on Risk Classification’s “Risk Classification Statement of
Principles,” answer the following.
a. (0.5 point) Briefly describe adverse selection.
b. (1.5 points) Briefly explain the two methods described for controlling adverse selection.

Questions from the 2001 Exam:
3. According to the American Academy of Actuaries Committee on Risk Classification’s “Risk
Classification Statement of Principles,” in which of the following situations would a refined risk
classification program be most appropriate?
A. Insurance premiums are determined prior to the policy period and are not adjusted on the basis of
actual experience.
B. Participation in the insurance program is entirely compulsory.
C. Dividends are paid after the initial insurance premium has been established and are based on the
risk’s actual experience.
D. The insurance premium is paid by someone other than the individual insured.
E. None of A, B, C, or D are appropriate situations for a refined risk classification program.
23. (1.5 points) List and briefly describe the three primary purposes of risk classification according
to the American Academy of Actuaries Committee on Risk Classification’s “Risk Classification
Statement of Principles.”

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Questions from the 2002 Exam:
20. Which of the following best describes a basic principle of a sound risk classification system?
A.
B.
C.
D.
E.

The system should be applied subjectively.
The system should produce prices based on the observed actual losses of each risk.
The system should reflect expected cost differences.
The system should be based solely on public acceptability.
The system should be the same for all competitors.

46. (2 points) Your company is planning to implement a new classification system. List and describe two
statistical and two operational considerations in designing this new classification system.
48. (4 points) Your company is planning to purchase a block of boat owner’s insurance business from
Zeron. Zeron has raised overall rates on this block of business for three consecutive years, but does
not classify risks by age or size. Despite the rate increases, loss ratios continue to worsen and
growth remains high.
a. (1 point) Explain how adverse selection could be impacting the seller's poor results.
b. (3 points) Using the information below, calculate rates to address the adverse selection problem.
Briefly justify your methods in light of risk classification principles.
Age
Group

Boat
Size

Ethnicity
Group

Exposures

Premium

Losses

1
1
1
1
1
1
2
2
2
2
2

Large
Medium
Small
Large
Medium
Small
Large
Medium
Small
Large
Medium

A
A
A
B
B
B
A
A
A
B
B

75
35
5
15
20
45
100
60
20
25
25

15,000
7,000
1,000
3,000
4,000
9,000
20,000
12,000
4,000
5,000
5,000

4,600
3,200
350
1,100
1,800
6,500
11,100
8,500
2,500
2,600
2,800

Small

10,000

7,200

Questions from the 2003 Exam:
1. According to the American Academy of Actuaries Committee on Risk Classification's "Risk
Classification Statement of Principles," which of the following statements are intentions of risk
classification?
1. to identify good and bad risks
2. to predict the experience for an individual risk
3. to group individual risks having reasonably similar expectations of loss
A. 1 only

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C. 3 only

D. 1 and 3 only

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Questions from the 2004 Exam:
23. (3 points)
a. (1.5 points)

Given the following information:

Type of
Vehicle
Cars
Trucks

Earned
Exposures
100,000
75,000

Number of
Claims per year
5,000
4,000

Pure
Premium
$200
$300

Would a classification plan that assigns cars and trucks to different classes be statistically sound?
Explain why or why not.
b. (1.5 points)

Given the following information:

Type of
Vehicle
Type A
Type B

Earned
Exposures
99,950
50

Number of
Claims per year
4,950
5

Pure
Premium
$199
$2,199

Would a classification plan that assigns Type A and Type B cars to different classes be statistically
sound? Explain why or why not.
24. (4 points)
a. (2 points)
b. (2 points)

List and describe four operational considerations in designing a risk classification
plan.
Compare the use of miles driven and the use of accident and violation history for
auto insurance based on the following risk classification considerations:
i.
ii.

Hazard Reduction Incentives
Availability of Coverage

Questions from the 2005 Exam:
1. (3 points)
a. (1.5 points) Describe three statistical considerations in designing a risk classification system.
b. (1.5 points) Discuss one advantage and two disadvantages of using controllability as a consideration for
identifying rating variables.

Questions from the 2006 Exam:
1. (1.5 points) Describe three primary purposes of risk classification.

Questions from the 2007 Exam:
1. (2 points) The American Academy of Actuaries, "Risk Classification Statement of Principles", discusses
three statistical considerations that an actuary must contemplate when designing a risk
classification system.
a..(1.5 points) Identify and briefly explain these three statistical considerations.
b..(0.5 point) Explain how two of these considerations may be in conflict with one another.

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Questions from the 2008 Exam:
1. (1.5 points) According to "Risk Classification Statement of Principles" the process of risk classification
should serve three primary purposes.
a. (0.75 point) State these three primary purposes of risk classification.
b.
(0.75 point) Briefly describe how each of these purposes helps to establish and
maintain a viable insurance system.
2. (3 points) A company is considering changing its "Age of Home" rating system, which has been in use
for five years, and has compiled the following data:
Age
Current
2005 — 2007 Combined
2007
Of
Age
Loss
Earned
Earned
Loss
Home
Discount
Ratio
Exposures
Premium ($)*
Ratio
0
5%
40,000
28,000,000
54%
27%
1
5%
35,000
23,625,000
65%
62%
2
5%
35,000
23,100,000
65%
50%
3
3%
25,000
16,125,000
60%
48%
4
3%
20,000
12,600,000
45%
40%
5
3%
25,000
15,375,000
60%
53%
6+
0%
30,000
18,000,000
60%
59%
Total
210,000
136,825,000
63%
50%
*At current discounts
Provide a recommendation whether the company should adopt each of the three changes below.
Defend the recommendation on the basis of at least one of the Statistical and one of the Operational
considerations presented in the AAA publication "Risk Classification Statement of Principles".
a.
b.
c.

(1 point) Set the discount for Age 0 (new homes) to 15%, leaving other discounts unchanged.
(1 point) Set the discount for Age 4 to 25%, leaving other discounts unchanged.
(1 point) Disaggregate the Age 6+ group and implement discounts of 2% for Age 6 and Age
7 and 1% for Age 8 and Age 9, leaving discounts for Age 10+ at 0%.

Questions from the 2009 Exam:
1. (2 points) With respect to a private, voluntary insurance program, discuss the extent to which each of
the following assumptions is or is not important for defining a risk classification system.
a. (0.5 point) The system should contemplate the level of competition in the market place.
b. (0.5 point) The characteristics of the system should be based on causality.
c. (0.5 point) The system should provide incentives for risks to reduce their expected losses.
d. (0.5 point) The system should balance between providing a reasonable continuum of expected
claim costs and maintaining significant differences in prices between classes.

Questions from the 2011 Exam:
12. (1 point) Describe two primary purposes of risk classification.

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Solutions to questions from the 1991 Exam:
Question 2. Which statistical considerations are involved in designing a sound risk classification system?
1. T. This is one of 3 statistical considerations (homogeneity, credibility, and predictive stability).
See page 14.
2. T. "There should be no clearly identifiable subclasses with significantly different potential for
losses". See page 14.
3. F. This is a consideration, Public Acceptability, (see page 19), but not a statistical one.
Answer C.
Question 3. Which statements listed in the problem are true?
1. F. Adverse selection occurs when prices are not reflective of expected costs. Broad
classifications and having no voluntary choice among competing institutions leads to pricing
on an expected cost basis.
Adverse selection is controlled by restricting the buyers' freedom, and risk classification is the
primary means to control the instability caused by adverse selection. See page 8.
2. T. Based on the above.
3. F. It is often impossible to prove statistically any postulated cause and effect relationship. Thus,
causality cannot be made a requirement of a risk classification system. See page 21.
4. F. Controllability has two undesirable risk characteristics:
(a) its susceptibility to manipulation.
(b) its irrelevance to predictability of future costs. See page 21.
Answer B.
Question 20. Briefly discuss how and why individual risk rating affects the needed level of refinement in a
classification system.
To the extent that prices are adjusted based on a risk's actual experience, after the insurance and its
initial price have been established, less refined initial risk classification systems are needed.
Experience rating refunds, premium adjustments, or dividends, ultimately produce a refined risk
classification system. See page 13.

Solutions to questions from the 1992 Exam:
Question 1. Which statements listed in the problem are true?
1. F. Experience rating refunds, premium adjustments, or dividends, ultimately produce a refined
risk classification system. See page 13.
2. F. Competition will motivate an insurer to refine its risk classification system so that it can better
serve both lower and higher cost risks. See page 10.
3. T. The statistical considerations of Homogeneity, Credibility, and Predictive stability are
somewhat conflicting. Increasing the number of classes may improve homogeneity but at the
expense of credibility. See page 16.
Answer B.

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Solutions to questions from the 1994 Exam:
Question 5. Which statements are considered primary purposes of risk classification?
These are the 3 Primary Purposes of Risk Classification. See page 2.

Answer E.

Question 26. List four considerations and briefly explain how each contributes to the success of a
classification system.
Four of the seven operational considerations are as follows (See pages 16 - 18):
1. Expense - The costs to obtain and maintain data, assigning risks to a class, & determining fair
prices by class.
2. Absence of ambiguity - classes should be collectively exhaustive and mutually exclusive.
3. Minimize abilities to manipulate the system.
4. Measurability - class variables (age, sex, occupation, location) should be reliably measurable.

Solutions to questions from the 1995 Exam:
Question 4. Which statements listed in the problem are true?
1. F. underwriting is the process of determining the acceptability of a risk based on its own merits.
See page 11.
2. T. the need for less refined classes when experience rating is used. See page 13.
3. F. As the more of the price is paid by other than the individual insured, the individual becomes
more indifferent to the classification structure. It is possible that broad classification systems
may be appropriate, since the distinction between payer and insured can operate to reduce
the likelihood of adverse selection. See page 13.
Answer B.
Question 5. Which statements listed in the problem are true?
1. T. expense includes costs to obtain and maintain data, assigning risks to a class, & determining
fair prices by class. See page 16.
2. T. extreme discontinuity avoidance. Attention is needed in defining classes at the extreme ends
of a range.


There should be enough classes to establish a reasonable continuum of expected losses
but few enough to allow significant differences between classes



Particular attention often is required in defining classes at the extreme ends of the
expected claim cost range, in order to reduce large differences in anticipated average
claim costs between the extreme class and the adjacent class. See page 18.
3. F. Hazard Reduction Incentives (i.e recognizing sprinklers for risk classification) are desirable but
not necessary features of a risk classification system. See page 19.
Answer C.

Solutions to questions from the 1996 Exam:
Question 17. Which statements are considered primary purposes of risk classification?
The 3 primary purposes of risk classification:
1. Protect the insurance system's financial soundness.
2. Be fair.
3. Encourage availability of coverage through economic incentives.
Thus, 1 is true, 2 is true and 3 is False.
Answer C.

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Solutions to questions from the 1996 Exam: (continued)
Question 47. Answer the statements listed in the question.
A. 5 basic principles to achieve the primary purposes:
A risk classification system should:
1. Reflect expected cost differences.
2. Distinguish among risks based on relevant cost-related factors.
3. Be applied objectively.
4. Be practical and cost-effective.
5 Be acceptable to the public.
B. ASB 12 omitted the principle of being acceptable to the public.
C. Public Acceptability Considerations:
Are difficult to apply in practice because social values


are difficult to ascertain.



vary among segments of society.



change over time.

Question 48. Answer the statements listed in the question.
A. Controllability:
Refers to the ability of an insured to control its own characteristics as used in the classification
system.
Controllability as a
Desirable risk characteristic:
Undesirable risk characteristic:
1. Its close association with an effort to reduce hazards.
1. Susceptibility to manipulation.
2. Its general acceptability by the public.
2. Its irrelevance to predictability of future costs.
The use of a individual risk's actual loss experience over the past five year period as a rating
variable certainly has both desirable risk characteristics as noted above.
The operational cost of utilizing this rating variable should be less than the benefits received by
using it.
Hazard Reduction Incentives (e.g. reduced prices for better experience) are desirable but not
necessary features of a risk classification system.
B. The 5 basic principles to achieve the primary purposes:
A risk classification system should:
1. Reflect expected cost differences.
2. Distinguish among risks based on relevant cost-related factors.
3. Be applied objectively.
4. Be practical and cost-effective.
5. Be acceptable to the public.
I would recommend implementation of the new rating variable, since its use will comply with most of
the basic principles, especially principles 1, 2, and 5.

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Solutions to questions from the 1997 Exam:
Question 48. For each criterion, identify which one of the five basic principles of a sound risk
classification system is violated.
The 5 basic principles of a sound risk classification system are to:
1. Reflect expected cost differences.
2. Distinguish among risks based on relevant cost-related factors.
3. Be applied objectively.
4. Be practical and cost-effective.
5 Be acceptable to the public.
Proposition 99 Criteria
1. Insureds are to be classified based on
nationality.
2. Insureds are to be classified based on the ability
to pass an annual random drug test
3. Insureds are to be classified based on whether
they can pass a comprehensive, individually
administered 8 hour driving test every year.
4. Insureds are to be classified based on their
weights.
5. Insureds are to be classified as either ‘good
eyesight’ or ‘bad eyesight’. Each eye doctor can
have his/her own definition of good/bad
eyesight.
6. Insureds are to be classified as ‘right handed’ or
‘left handed’.

Statement of principle violated
Principle 1: Reflect expected cost differences.
Principle 5: Be acceptable to the public.
Principle 4: Be practical and cost-effective.
Principle 5: Be acceptable to the public.
Principle 3: Be applied objectively.

Principle 2: Distinguish among risks based on
relevant cost-related factors.

Solutions to questions from the 1999 Exam:
Question 43.
a. (0.75 point) what are the three primary purposes of risk classification?
b. (1.5 points) Based on these principles, what would you tell the marketing director about the
appropriateness of the proposed class definitions? Include a discussion of all three
principles from part a.
a 3 primary purposes of risk classification:
1. Protect the insurance system's financial soundness.
2. Be fair.
3. Encourage availability of coverage through economic incentives.
b. 1. The financial soundness of Aggressive Mutual's new plan is threatened by adverse selection,
since equitable rates are not being charged. A deterioration in its overall profitability is likely to
materialize over time. Risk classification minimizes the effects of adverse selection.
2. A plan is fair if its prices are not unfairly discriminatory, and reflect differences in expected costs
with no redistribution or subsidy among classes. By liberalizing the protection class definitions,
there are fewer opportunities for justifiable price discrimination.
3. Economic incentives (profitability through justifiable price discrimination) motivate insurers to
refine their risk classification, to better serve low and high cost risk. Liberalizing the protection
class definitions works against these incentives.

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AMERICAN ACADEMY OF ACTUARIES COMMITTEE ON RISK CLASSIFICATION
Solutions to questions from the 1999 Exam (continued):
Question 46.
a. (1 point) List three methods for determining this price.
b. (1 point) List one deficiency for each method described in part a.
a. 1. Reliance on wisdom, insight, and good judgment.
2. Observation of the risk's actual losses over an extended period of time.
3. Observation of losses from groups of individual risks with similar characteristics. This is the most
frequently used method.
b. 1. Valuable information about expected future loss experience is lost when a risk's actual loss
experience is not reviewed.
2. Gradual changes in the hazard may render past information useless.
3. Identification of similar risk characteristics (commonly determined by fact and informed judgment)
and related classes before the observation period is problematic.

Solutions to questions from the 2000 Exam:
Question 16. Which are not operational considerations relating to classification plans?
All of the operational considerations listed relate to classification plans. See pages 11 – 13. Answer E.
Question 35
a. (0.5 point) Briefly describe adverse selection.
Adverse selection arises when buyers (looking to secure the minimum price) are free to select among
different sellers, and when sellers react by offering a similar product in order to incite the movement of
buyers in an attempt to gain an economic advantage, often at a price where the seller has not matched
price to cost. See page 7.
b. (1.5 points) Briefly explain the two methods described for controlling adverse selection.
1. Risk classification in a voluntary market - charges each risk the appropriate rate through proper
risk identification and balances the economic forces governing buyer and seller actions. This is
the primary means to control instability caused by adverse selection.
2. Compulsory insurance with limited choices (e.g. group insurance) reduces the voluntary choice
among competing institutions. Restriction of buyer freedom prevents movement or reduces the
price incentive. See pages 8 and 12-13.

Solutions to questions from the 2001 Exam:
3. In which of the following situations would a refined risk classification program be most appropriate?
A. Insurance premiums are determined prior to the policy period and are not adjusted on the basis of
actual experience. True. To the extent that prices are NOT adjusted based on a risk’s actual
experience, MORE refined risk classifications systems are needed.
B. Participation in the insurance program is entirely compulsory. In government programs,
participation is usually compulsory and the benefits received by, or paid on behalf of a class, are
not necessarily related to the amount paid into the plan by that class.
C. Dividends are paid after the initial insurance premium has been established and are based on the
risk’s actual experience. To the extent that prices are adjusted based on a risk’s actual
experience, less refined risk classifications systems are needed.
D. The insurance premium is paid by someone other than the individual insured. Here, the individual insured
is indifferent to the classification system, and thus, broad classification systems may be appropriate.
E. None of A, B, C, or D are appropriate situations for a refined risk classification program. False. A is true.

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AMERICAN ACADEMY OF ACTUARIES COMMITTEE ON RISK CLASSIFICATION
Solutions to questions from the 2001 Exam (continued):
23. (1.5 points) List and briefly describe the three primary purposes of risk classification according to the
American Academy of Actuaries Committee on Risk Classification’s “Risk Classification Statement of
Principles.”
1. Protect the insurance system's financial soundness. Risk classification is the primary means to
control instability caused by adverse selection.
2. Be fair. A proper risk classification system produces prices which are reflective of expected costs.
3. Encourage availability of coverage through economic incentives. A proper risk classification
system will allow an insurer to write and better serve both higher and lower cost risks.
See pages 2 – 9.

Solutions to questions from the 2002 Exam
20. Which of the following best describes a basic principle of a sound risk classification system?
A. The system should be applied subjectively. False. The system should be applied
objectively. See page 2.
B. The system should produce prices based on the observed actual losses of each risk. False. A
system that produces prices based on observed actual losses of each risk is an example of
experience based pricing. Further, to the extent that prices are adjusted based on a risk’s
emerging actual experience after the insurance and its initial price have been established, less
refined initial risk classification systems are needed. See pages 12 and 13.
C. The system should reflect expected cost differences. True. See page 2.
D. The system should be based solely on public acceptability. False. Although the system should
be acceptable to the public, it should not be based solely on public acceptability. See page 2.
E. The system should be the same for all competitors. False. Insurers should refine their risk
classification systems and thus their pricing structures to be more successful than their
competitors, so that it could serve both lower cost and higher cost risks in the marketplace. See
pages 9 and 10.
46. (2 points) Your company is planning to implement a new classification system. List and describe two
statistical and two operational considerations in designing this new classification system.
Statistical:
1. Homogeneity. Individual risks within a class should have reasonably similar expected costs. Within a
class there should be no clearly identifiable subgroups with significantly different loss potential.
2. Credibility. The larger the numbers of observations, the more accurate are the statistical predictions
that can be made. Each class does not have to be large enough to stand on its own, since accurate
predictions can be made based on statistical analysis of the experience of broader grouping of
correlative classes.
Note: Candidates would also receive credit for listing and defining Predictive Stability.
Operational:
1. Manipulation. The ability to manipulate or misrepresent a risk’s characteristics to affect its class
assignment should be minimized.
2. Measurability – Risk characteristics should lend themselves to reliable and convenient measurement,
such as age, sex, occupation or location.
Note: Candidates would also receive credit for listing and defining Expense, Constancy, Availability of
Coverage, Avoidance of Extreme Discontinuities, Absence of Ambiguity, and Hazardous
Reduction Incentives.

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AMERICAN ACADEMY OF ACTUARIES COMMITTEE ON RISK CLASSIFICATION
Solutions to questions from the 2002 Exam (continued):
48. (4 points) Your company is planning to purchase a block of boatowner's insurance business from
Zeron. Zeron has raised overall rates on this block of business for three consecutive years, but does
not classify risks by age or size. Despite the rate increases, loss ratios continue to worsen and
growth remains high.
General information. Based on the given information, we know that Zeron does not classify risks by age
or size, that their loss ratios are worsening and that their growth remains high (presumably due to writing
a large proportion of poor risks). This implies that that Zeron’s competitors do classify by age and size,
which impacts the types of risks they underwrite, and the rates they charge.
a. (1 point) Explain how adverse selection could be impacting the seller's poor results.
Apparently, Zeron’s worsening loss ratios and high growth rate are the result of writing a large proportion
of poor risks at inadequate rates. A review of the given premium and exposure data indicates that Zeron
charges an average rate for all risks. Assuming that Zeron’s competitors classify risks by age and size,
better risks will purchase from Zeron’s competitors at an actuarially fair rate while poorer risks will
purchase from Zeron. Zeron’s pricing is causing a significant shift in the types of risks it underwrites.
b. (3 points) Using the information below, calculate rates to address the adverse selection problem.
Briefly justify your methods in light of risk classification principles.
Rates should be based on measurable risk characteristics (e.g. age and size) and not on ethnicity
group (since this is not a publicly acceptable classification criteria). Therefore, the data should be
configured as follows:

Group

Age

Boat
Size

Premium

Exposures

Rates

(1)

(2)

1
1
1

L
M
S

18,000
11,000
10,000

90
55
50

2
2
2

L
M
S

25,000
17,000
14,000

125
85
70

Total

Current

Loss
Losses

Ratio

(3)=(1)/(2)

(4)

(5)=(4)/(1)

200
200
200

5,700
5,000
6,850

0.3167
0.4545
0.6850

200
200
200

13,700
11,300
9,700

0.5480
0.6647
0.6929

52,250

0.5500

95,000

Given the significant variability in the loss ratios, rates should be based on differences in expected
costs. This can be reflected by adjusting current rates by loss ratio relativities.
Age
Group
1
1
1

Proposed
Rates
(6)
115.15
165.29
249.09

2
199.27
2
241.71
2
251.95
(6) = (3) * [(5) ÷ (5)total]

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AMERICAN ACADEMY OF ACTUARIES COMMITTEE ON RISK CLASSIFICATION
Solutions to questions from the 2003 Exam:
1. Which of the following statements are intentions of risk classification?
1. to identify good and bad risks. False. This is not mentioned.
2. to predict the experience for an individual risk. False. This is not mentioned.
3. to group individual risks having reasonably similar expectations of loss. True. See page 121.
Answer C. 3 Only.

Solutions to questions from the 2004 Exam:
Question 23- Model Solution 1
a. (1.5 points) Given the following information:

Type of
Vehicle
Cars
Trucks

Earned
Exposures
100,000
75,000

Number of
Claims per year
5,000
4,000

Pure
Premium
$200
$300

Would a classification plan that assigns cars and trucks to different classes be statistically sound?
Explain why or why not.
Yes, assigning cars and trucks to different classes would be statistically sound. Both cars and trucks
have large volumes of data (100,000 earned exposures for cars; 75,000 for trucks). Also, the pure
premiums of cars and trucks are significantly different ($200 for cars versus $300 for trucks).
b. (1.5 points)

Given the following information:

Type of
Earned
Number of
Pure
Vehicle
Exposures
Claims per year
Premium
Type A
99,950
4,950
$199
Type B
50
5
$2,199
Would a classification plan that assigns Type A and Type B cars to different classes be statistically sound?
Explain why or why not.
No, assigning Type A and Type B to different classes would not be statistically sound. Even though
Type B has much higher pure premium than Type A, there are only 50 exposures for Type B, which is
too small to derive statistical conclusions. The high cost of Type B may only be random loss fluctuation.
Question 23 – Model Solution 2
a. There would be homogeneity within the class. There are enough exposures in each to have
statistical credibility. These are mutually exclusive classes that could not be manipulated by the
insureds. There are differences in severity. Yes, assigning cars and trucks to different classes
would be o.k.
b. No, there are not enough exposures in Type B to have statistical credibility.
Question 23, part b only- Model Solution 3
b. I would say yes. While Type B has very small volume, by examining the credibility-weighted
differences between the types would still bring value. Type B is significantly worse in the three types of
characteristics identified in A above (frequency, severity and pure premium).

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AMERICAN ACADEMY OF ACTUARIES COMMITTEE ON RISK CLASSIFICATION
Solutions to questions from the 2004 Exam (continued):
Question 24 - Model Solution 1
a. (2 points)
List and describe four operational considerations in designing a risk classification
plan.

b. (2 points)
i.
ii.

Compare the use of miles driven and the use of accident and violation history for
auto insurance based on the following risk classification considerations:

Hazard Reduction Incentives
Availability of Coverage

a. 1. Measurability – the variables should be easy to defined & measure.
2. Manipulation – the plan should not allow for insureds to manipulate their classifications.
3. Expense – the expenses of the classification plan should be as low as possible while maximizing
company value.
4. Absence of ambiguity – the classifications should be all encompassing and mutually exclusive;
each insured should fit into one and only one class.
b. i. It would be difficult to significantly alter the number of miles driven since most are of necessity (work,
etc.). It doesn’t provide much hazard reduction incentive. Some drivers may avoid long trips.
Hazard reduction incentives would work for accident and violation history because drivers
would be more cautious in order to avoid higher rates.
ii. Miles driven would allow for more availability of coverage because miles driven have an impact on
loss exposure. Using this as a classification would improve rate accuracy and thus encourage
widespread availability.
Use of accident and violation history may have the same impact as described for miles driven.
However, insurers may use this information to deny coverage to drivers with more than a certain
number of accidents. This would reduce availability.
Question 24 - Model Solution 2
a. 1. Measurability – it should be easy to measure or quantify the value of the classification (e.g., age or
sex).
2. Expense – the value added by having the classification should be greater than the expense of
having it in the plan.
3. Avoidance of extreme discontinuity – we should avoid a large jump in rates between a class and
the one next to it.
4. Maximize coverage availability – the plan should accurate price risks so that the availability of
coverage is maximized.
b. i. Hazard reduction incentive
a. Use of miles driven – to the extent that an insured will avoid unnecessary road trips, this may reduce
the hazard. But this does not seem like an effective way to reduce hazard because people still need to
drive.
b. Accident / violation – this will create an incentive for insureds to drive safely and avoid accidents.
ii. Availability of coverage
a. Use of miles driven – to the extent that costs are correlated with miles drive, this may more
accurately price risks and thus result in more availability of coverage.
b. Accident / violation – since accident / violation history is correlated with costs, having this variable
will promote more accurate rates, leading to better availability.

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Risk Classification Statement of Principles
AMERICAN ACADEMY OF ACTUARIES COMMITTEE ON RISK CLASSIFICATION
Solutions to questions from the 2005 Exam:
1. (3 points)
a. (1.5 points) Describe three statistical considerations in designing a risk classification system.
b. (1.5 points) Discuss one advantage and two disadvantages of using controllability as a consideration for
identifying rating variables.
Question 1 – Model Solution 1
a. Statistical Considerations
1. Homogeneity: risks in the same class should have reasonably similar loss potential.
2. Credibility: the number of claims should be voluminous to warrant credibility.
3. Predictive Stability: responsive to changes in the nature of insurance yet stable in avoiding
unwarranted abrupt changes.
b. Advantage: If the rating variable is closely associated with the efforts to reduce hazard, then the
classification will help reduce the potential loss.

Disadvantages:
1. If the variable is susceptible to manipulation then the insured may misuse it.
2. If the variable is irrelevant to the predictability of the losses, then the variable may not be useful in
predicting future losses and this may not be acceptable to the public.
Question 1 – Model Solution 2
a. Statistical Considerations
1. Homogeneity: Risks are grouped according to their traits as homogeneously as allowed (but not
forgoing credibility).
2. Credibility: Risks are grouped in volumes that are adequate for the group to be credible.
3. Predictive Stability: Risks are grouped according to traits that are responsive enough to changes;
but stable enough to not allow abrupt changes.

b. Advantage: It is a good way to encourage reduction in hazard; insureds will want to control how much
they pay in premium.
Disadvantages:
1. Manipulation: Risks may tend to manipulate their exposure to reduce premiums.
2. Impractical: Some traits may not be practical to implement in a classification system.

Solutions to questions from the 2006 Exam:
1. (1.5 points) Describe three primary purposes of risk classification.
Question 1 – Model Solution 1
1. Protect the insurance system’s financial soundness. This is threatened by adverse selection which
can occur if insurance companies are not allowed to classify.
2. Enhance fairness. Charge insureds appropriately for their potential for loss, do not punish or reward
insureds at the expense of others.
3. Provide economic incentive to make coverage available. With classification, companies will be able to
charge appropriately and will be able to serve higher and lower risk insureds and will be incented to provide
coverage.

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AMERICAN ACADEMY OF ACTUARIES COMMITTEE ON RISK CLASSIFICATION
Solutions to questions from the 2006 Exam:
Question 1 – Model Solution 2
1. To protect insurance system’s financial soundness. Minimize potential for adverse selection by
matching expected costs with price.
2. To enhance fairness. By ensuring prices valid and equitable with no subsidization between classes.
Each risk is charged appropriate rate through proper risk identification.
3. To permit economic incentives to operate and thus encourage widespread availability of coverage
 By charging higher premiums for higher risks and lower premiums for lower risks
 Economies of scale by offering coverage to all at appropriate rates
 Financial incentive to be a better risk and thus reduce one’s premium
See pages 2 – 3.

Solutions to questions from the 2007 Exam:
Question 1 – Model Solution 1
Credibility -> enough risks in the class to allow reasonable and credible inferences to be drawn
Homogeneity -> risks in the class should be similar (i.e. no subgroups identifiable)
Predictive stability -> use of the classes should be responsive to changing conditions, but avoid large
swings in rates from year to year
Credibility and homogeneity may be in conflict. We want the risks to be very similar, but we also want
enough experience so that they are credible.
Question 1 – Model Solution 2
Homogeneity ->the risks within the class should be similar (i.e. there should be little variation within the
class)
Credibility ->there must be enough data in the class to be able [to] rely on
Predictive Stability-> should be responsive to the nature of insurance losses yet stable enough to avoid
abrupt price changes
Homogeneity and credibility are in conflict since making a class more homogeneous by eliminating risks
comes at the expense of credibility, since there may not end up being enough risks in the class to make it
credible.

Solutions to questions from the 2008 Exam:
Question 1 – Model Solution 1
a. State these three primary purposes of risk classification.
1. Enhance insurance system financial soundness
2. Enhance fairness
3. Permit economic incentives to operate and increase availability of insurance
b. Briefly describe how each of these purposes helps to establish and maintain a viable insurance
system.
1. Risk classification minimizes adverse selection which will exist when buyers are free to select who
they purchase insurance from
2. Rate should be in line with their expected loss costs and there shouldn’t be any subsidy between
risk classes
3. Each risk class should be priced to their expected losses so that insurers have same profit potential
on all risks and are willing to write high risks and low risks, rather than just going after low risks. This
increases availability.

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AMERICAN ACADEMY OF ACTUARIES COMMITTEE ON RISK CLASSIFICATION
Solutions to questions from the 2008 Exam:
Question 1 – Model Solution 2
a.
1. To protect the financial soundness of the insurance system
2. Enhance fairness
3. Economic incentives to make coverage available
b.
1. Risk classification protects insurers from adverse selection which could impair an insurance company
2. It would provide rates that are reflective of insured’s expected cost making them fair and not unfairly
discriminatory
3. Encourages insurer to refine system to better serve both high and low risk insureds because of
competition.

Solutions to questions from the 2008 Exam:
a. (1 point) Set the discount for Age 0 (new homes) to 15%, leaving other discounts unchanged.
b. (1 point) Set the discount for Age 4 to 25%, leaving other discounts unchanged.
c. (1 point) Disaggregate the Age 6+ group and implement discounts of 2% for Age 6 and Age 7 and 1%
for Age 8 and Age 9, leaving discounts for Age 10+ at 0%.
Question 2 – Model Solution 1
a. Yes
Stat
Credibility
There seems to be enough data to provide a reasonable prediction.
Oper
Manipulation
The age of the home would not be subject to manipulation since it should
be well documented.
b. No
Stat
Predictive Stability
This is probably random loss fluctuation and should not be too
responsive.
Oper
Discontinuity
There would be a discontinuity of coverage changing discount from 3% to
25% back to 3%.
c. No
Stat
Homogeneity
These risks should be similar and therefore can be grouped
Oper
Expense
Expensive to implement and change system when there is not an
apparent need.
Question 2 – Model Solution 2
a. Agree with making the change
i. From the statistical consideration, this age group has the most exposures and thus the most
credibility and their loss ratios would support this change in discount.
ii. From an operational consideration, this is one that could not be manipulated by the insured.
b. Disagree with making the change
i. Statistical – although the discount may be supported by loss ratios, this is smallest age group
category so has the least credibility.
ii. Operational – This would result in Age group 3 with a 3% discount, age group 4 with a 25% discount,
and then age group 5 with a 3% discount again. This is an extreme discontinuity which we want to
avoid.
c. Disagree
i. Statistical – The credibility for making this change might be in question.
ii. Operational – The expense of making this change would likely outweigh the benefits.

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AMERICAN ACADEMY OF ACTUARIES COMMITTEE ON RISK CLASSIFICATION
Solutions to questions from the 2009 Exam:
1. (2 points) With respect to a private, voluntary insurance program, discuss the extent to which each of
the following assumptions is or is not important for defining a risk classification system.
a. (0.5 point) The system should contemplate the level of competition in the market place.
b. (0.5 point) The characteristics of the system should be based on causality.
c. (0.5 point) The system should provide incentives for risks to reduce their expected losses.
d. (0.5 point) The system should balance between providing a reasonable continuum of expected
claim costs and maintaining significant differences in prices between classes.
Question 1 – Model Solution 1
a. This is important, the less competition the less refined classification system is required.
b. Causality is not necessary and is impossible to prove so it is not important, nice though.
c. Incentives to reduce loss are good, but not a requirement for a risk classification system.
d. This is an important operational consideration. They should aim to avoid extreme discontinuities in the
price, but differences should still be significant.
Question 1 – Model Solution 2
a. Important – in a competitive market risk classification is important to avoid adverse selection.
b. Not important – may help with public acceptance, but difficult to prove; can use plausibility instead.
c. Not important – thought hazard reduction incentives are beneficial to society, the utility is limited.
d. Important – system should avoid extreme discontinuities, but should have significant enough
differences to justify different class.

Solutions to questions from the 2011 Exam:
Question 12 – Model Solution 1
1. To ensure the insurance system’s financial soundness by protecting it against adverse selection, which
happens in a competitive environment when others are using risk classification.
2. To be fair. Risk classification allows the insurer to better match expected costs and premiums for the
policy holders based on how they classify with respect to exposure to risk.
[These purposes come from AAA Risk Classification Principles]
Question 12 – Model Solution 2
1. Protect financial soundness of the insurance system. If buyers are free to purchase insurance in a
competitive market, adverse selection could result if appropriate risk classification is not used. This
could put the solvency of insurers at risk.
2. Encourage availability of coverage through economic incentives. Equitable pricing ensures that prices
reflect expected differences in cost. In the long run, this allows insurers to better serve both low and
high cost insureds.
[These purposes come from AAA Risk Classification Principles]

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Personal Vehicle Manual
ISO – EDITION 6-98 – GENERAL RULES 1 - 6
Section
1
2
3
4
5
6

Description
Select Definitions
Personal Auto Policy – Eligibility
Premium Determination
Classifications
Safe Driver Insurance Plan (SDIP)
Model Year/Age Groups for Comprehensive and Collision

Excerpts from the ISO Personal Vehicle Manual, included in the CAS Exam 5 Study Kit, is copyrighted.
Copyright, Insurance Services Office, Inc., 1998
SELECT information from each of the 6 sections will be provided in this review. For additional
information, consult the syllabus reading.

Select Definitions

A. Private Passenger Auto:
1. is a four wheel motor vehicle, owned or leased under contract for a continuous period of at least 6
months, and
2. can also be considered a pickup or van, and
3. can also be considered a farm family owned or a farm family co-partnership, or farm family
corporation motor vehicle.
B. AUTO refers to a private passenger auto or a vehicle considered as a private passenger auto.
C. LIABILITY refers only to Bodily Injury and Property Damage Coverages.
D. OWNED includes an auto leased under contract for a continuous period of at least 6 months.

Personal Auto Policy – Eligibility

A Personal Auto Policy shall be used to afford coverage to:
A. private passenger autos and motor vehicles considered as private passenger autos in Rule 1., if:
1. They are written on a specified auto basis, and
2. They are owned by an individual or by a husband and wife who are residents in the same
household.
B. private passenger autos, and pickups and vans as defined in Rule 1., that are owned jointly by two
or more:
1. Resident relatives other than husband and wife;
2. Resident individuals; or
3. Non-resident relatives, including a non-resident husband and wife; If:


They are written on a specified auto basis, and



The Joint Ownership Coverage endorsement is attached.

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ISO – EDITION 6-98 – GENERAL RULES 1 - 6
C. motorcycles, motor homes, golf carts or other similar type vehicles and snowmobiles if:
1. They are written on a specified vehicle basis,
2. They are owned by:
a. An individual;
b. A husband and wife;
c. Two or more relatives other than husband and wife; or
d. Two or more resident individuals; and
3. Coverage is limited in accordance with the miscellaneous type vehicle or snowmobile
endorsement.
D. a named individual who does not own an auto. The named non-owner coverage endorsement
must be attached.
Note: Exposures in A. B. or C. above may be written under a commercial auto policy when combined
with a commercial risk.

Premium Determination

Single Limit Liability, or BI and PD Liability; Medical Payments; Comprehensive and Collision premiums
are determined as follows:
A. Refer to the Classification Rule to determine the applicable classification, rating factors and statistical
Code.
B. Refer to the Model Year/Age Group Rule and the Symbol and Identification section to determine the
model year/age of the auto and the appropriate symbol of the auto.
NOTES:


When a model year is used in rating and the rates for a model year are not displayed in the
Rate Pages, use the rates shown for the latest model year.



If no Rating Symbol is shown in the Symbol and Identification (S&I) Section, use the following
procedure to determine an interim rating symbol.
a. If the S&I section displays a rating symbol for the PRIOR MODEL YEAR version of the
same vehicle, use the prior model year’s Rating Symbol for the new model year vehicle.
b. If the S&I Section does NOT display a rating symbol for the PRIOR MODEL YEAR version
of the same vehicle, assign a symbol based on the cost new of the auto, using the
Price/Symbol Chart located in the reference pages of the S&I Section.
C. Refer to Territory Definitions to determine the territory code for the location where the auto is
principally garaged.
D. Refer to the Rate pages to determine base rates for the desired coverage for the appropriate territory.
E. Expense Fees
The premium for each coverage is determined by multiplying the base rate by the appropriate rating
factor and adding the appropriate Expense Fees (see page 2 for more details).
Notes:


Expense Fees are added separately to the premium for the Single Limit Liability or BI and PD
Liability, Comprehensive, Collision and No-Fault Coverages applying to each auto.



Expense Fees are not subject to modification by the provisions of any rating plans or other
rating rules (e.g. Classifications, Safe Driver Insurance Plan



Expense Fees are subject to the Cancellation and Suspension provisions of this manual.

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ISO – EDITION 6-98 – GENERAL RULES 1 - 6
4

Classifications

A. Classifications:
Autos owned by an individual, or owned jointly by two or more relatives or resident individuals are
classified as follows:
1. Primary Classification
Classify the auto according to the sex and marital status of the operators, the use of the auto and
the eligibility of youthful operators for the Driver Training and/or Good Student classes.
2. Secondary Classification
Refer to the Symbol and Identification section to determine if the auto is:
a. 1. Standard performance.
2. Intermediate performance
3. High performance.
4. Sports, or
5. Sports premium.
b. 1. A single car, or
2. Part of a multi-car risk.
3. Classification Changes
Premium adjustments are made on a pro-rata basis when changes in Primary and Secondary
Rating Classifications are made.
Exceptions.
A policy may not be changed mid-term:
a. because of the attained age of an operator of the auto.
b. to effect a change in the Driving Record Sub Classification.
c. due to a change in symbol assignment based on a review of loss experience.
B. Definitions.
1. Use Classifications:
a. BUSINESS USE (other than going to or from the principal place of occupation, profession or
business)
b. FARM USE
c. PLEASURE USE means:
1. No Business use.
2. includes driving to and from work or school
a. less than 3 road miles one way
b. 3 or more, but less than 15, road miles one way for not more than 2 days per week, or
more than 2 weeks per 5 week period.

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ISO – EDITION 6-98 – GENERAL RULES 1 - 6
d. WORK LESS THAN 15 MILES means:
e. WORK MORE THAN 15 MILES means:
1) No Business use.
1. No Business use.
2) includes driving to and from work or school:
2. includes driving to and from work or school:
a. 3 or more, but less than 15, road miles one
way for not more than 2 days per week, or
more than 2 weeks per 5 week period.
b. 15 or more road miles one way for not more
15 or more road miles one way more than 2
than 2 days per week, or more than 2 weeks
days per week, or more than 2 weeks per 5
per 5 week period.
week period.
Note: An auto driven part way to or from work or school (e.g. to a railroad or bus depot) shall be
considered as driving to or from work a school.
2. Age, Sex and Marital Status Classifications
YOUTHFUL OPERATOR means any operator resident in the same household who customarily
operates the auto, and is one of the following:
a. YOUTHFUL UNMARRIED FEMALE OPERATOR - unmarried female under 25 years of age.
b. YOUTHFUL MARRIED MALE OPERATOR - married male under 25 years of age.
c. YOUTHFUL UNMARRIED MALE OPERATOR - unmarried male under 25 years of age who is
not an owner or principal operator.
d. YOUTHFUL UNMARRIED MALE OWNER OR PRINCIPAL OPERATOR unmarried male
under 30 years of age who is an owner or principal operator.
3. Driver Training
Driver Training Classification applies to each Youthful Operator under 21 years of age where
“Satisfactory Evidence” is presented that such operator has successfully completed a driver
education course meeting the following standards:
a. The course included a minimum of 30 clock hours of classroom instruction plus a minimum of
6 clock hours of actual driving experience per student.
b. The course was conducted by instructors certified by the State Department of Education or
other responsible educational agency.
"Satisfactory Evidence" is a certificate signed by a school official certifying to the fulfillment of the
requirements.
4. Good Student
The Good Student Classification applies provided the owner or operator is 1) At least 16 years of age, and
2) A full time high school, college or university student.
A certified statement from a school official is presented to the Company on each anniversary date
of the policy indicating that the student has met one of the following requirements during the
immediately preceding school semester.
1) Is in the upper 20% of his/her class scholastically, or
2) Maintains a "B" average, or its equivalent.
3) When in a school maintaining a numerical grade, must have at last a 3 in a 4, 3. 2. 1 point
system
4) Student is included in a "Dean's List " 'Honor Roll" or comparable list indicating scholastic
achievement.
Note: A classification change resulting from a change in the scholastic standing of the student
cannot be effected between anniversary dates of the policy.

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ISO – EDITION 6-98 – GENERAL RULES 1 - 6
5. a. Youthful Operators
1) Single Car Risks
The youthful operator with the highest Primary Rating Factor shall apply.
2) Multi-Car Risks
(a) Assign any youthful principal operators to the autos they principally operate.
(b) Assign other youthful operators to remaining autos (see page 5 for details)
b. Operators Age 50 and Over
1) The Principal Operator Age 50-64 Class shall apply if the principal operator of the auto is
age 50 to 64.
2) The Principal Operator Age 65-74 or 75 or Over Classes shall apply if the principal
operator of the auto is age 65 or over.
c. Multi-Car Discount
The Multi-Car Rating Factor applies if:
1) more than one private passenger auto is owned by an individual or owned jointly by two or
more relatives or resident individuals, and
2) two or more autos are insured in the same company for any of the following coverages:
single limit liability (or BI and PD liability,) medical payments, no-fault, comprehensive or
collision.
d. TOTAL BASE PREMIUM is the sum of the base premium for single limit liability or BI and PD
liability, medical payments, no-fault, comprehensive and collision coverages that apply to the auto.
6. Vehicles Equipped With Anti-Theft Devices
These discounts apply to comprehensive coverage only.
7. Safety Equipment Discounts
a. Passive Restraint Discount
The following discounts apply to Medical Payments and/or any No-Fault Coverage only.
1) 20% discount shall be afforded when the restraint is installed in the driver-side only position.
2) 30% discount shall be afforded when the restraints are true in both front outboard seat positions.
b. Anti-Lock Braking System Discount
A 5% for BI and PD Liability (or Single Limit Liability) coverages shall be afforded for those
private passenger autos equipped with a factory installed four wheel Anti-Lock Braking
System (ABS).

Safe Driver Insurance Plan (SDIP)

SECTION I.
The SDIP applies to policies written in Companies authorizing its use. For companies electing not to
use the Plan see Section II of this Rule. When SDIP is used it is to be applied to all eligible autos.
A. Eligibility:
An auto is eligible for rating under this Plan if it is:
1. Owned by an individual, or owned jointly by two or more relatives or resident individuals.
2. Owned by a family partnership or family corporation, provided the vehicle is:
a. Garaged on a farm a ranch; and
b. Not rated as part of a fleet; and
c. Not used in any occupation other than farming or a ranching.

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ISO – EDITION 6-98 – GENERAL RULES 1 - 6
B. Definitions:
1. Driving Record Points
a. Convictions
Points shall be assigned for convictions during the experience period for motor vehicle
violations of the applicant or any other currently resident operator as follows:
(1) 3 points are assigned for conviction of:
(a) Driving while intoxicated or under the influence of drugs; or
(b) Failure to stop and report when involved in an accident; or
(c) Homicide or assault arising out of the operation of a motor vehicle; or
(d) Driving while license is suspended or moving traffic violation in connection with
revoked.
(2) 2 points are assigned for the accumulation of points under a State Point System or a
series of convictions requiring the filing of evidence of Financial Responsibility under
any Financial Responsibility Law as of the effective date of the policy.
(3) 1 point is assigned for conviction of any other moving traffic violation resulting in:
(a) Suspension a revocation of an operators license, or
(b) The filing of evidence of financial responsibility under any Financial Responsibility
Law as of the effective date of the policy.
b. Accidents
Points shall be assigned for each accident
1 point is assigned for each auto accident that results in:
(a) Bodily injury, or death; or
(b) Total damage to all property, including his or her own, in excess of $500.
c. Inexperienced Operator
(1) If the principal operator of the auto has no point assigned for an accident or conviction
but has been licensed less than 2 years, 1 point is assigned. Sub-Classification 1B
applies.
(2) Sub-Classification 1A applies only when the policy has total of 1 point assigned based on
any operator's accident or conviction record.
d. Refund of Surcharged Premium
If a point has been assigned for an accident and it is later determined that the accident falls
under one of the exceptions in this rule, the company shall refund to the Insured the
increased portion of the premium generated by the accident.
C. Driving Record Sub-Classification
The driving record sub-classification shall be determined from the number of Driving Record
Points accumulated during the experience period as follows:
Number of Driving
Record Points
0
1
2
3
4 or more

Exam 5, V1a

Driving Record
Sub-classification
0
1
2
3
4

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ISO – EDITION 6-98 – GENERAL RULES 1 - 6
SECTION II
For companies electing not to use SDIP, rate eligible private passenger autos by adding 0.20 to the
Rating Factor otherwise applicable.
Use the following Secondary Rating Factors and Codes:
1971 and Later Model Autos
Single Car
Code
Factor
Standard Performance
19
+0.00
Intermediate Performance
39
+0.15
High Performance
59
+0.30
Sports
79
+.015
Sports Premium
99
+0.15
Note: Factors also apply to Multi-Car and to 1970 and Prior Model Autos

Model Year/Age Groups for Comprehensive and Collision

A. Where Model Year Is Used in Rating:
1. The model year of the auto is the year assigned by the auto manufacturer.
2. Rebuilt or Structurally Altered Autos - the model year of the chassis determines the model
year of the auto.
3. If the rates for a model year are not displayed in the Rate Pages, use the rates shown for the
latest model year.
B. Where Age Is Used in Rating:
1. Age is determined as follows:
Age Group
Definition
1
Autos of “current model year”
2
Autos of first preceding year
3
Autos of 2nd preceding year
“”
“”
Note: The “current model year" changes effective October 1 of each calendar year
regardless of the actual introduction of the makes and models.
2. Rebuilt or Structurally Altered Autos - the age of the chassis determines the age of the autos.
C. Coding applicable whether Model Year or Age is used in rating:
1. Policies effective July 1, 1980 and subsequent:
Code the last two digits of the model year, e.g. code 1980 vehicles as 80, 1981 as 81, etc.
2. Policies effective prior to July 1, 1980:
Description
Current Model Year
First Preceding Model Year
Second Preceding Model
Year
Third Preceding Model Year
Fourth Preceding Model Year

Exam 5, V1a

Code
1
2
3
4
5

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ISO – EDITION 6-98 – GENERAL RULES 1 - 6
Questions from the 2002 exam
8. Based on Insurance Services Office, Inc., Personal Automobile Manual (Effective 6-98), which of the
following is false?
A. The Manual describes the types of vehicles eligible for coverage.
B. The Manual specifies that all Liability and Physical Damage policies must have a policy period of
no longer than 12 months.
C. The Manual specifies which drivers must be categorized as "Youthful Operators".
D. The Manual sets forth rating factor adjustments for companies electing not to use the Safe Driver
Insurance Plan.
E. The Manual describes the primary and secondary classifications applicable.
Questions from the 2004 exam
21. (2 points) Using Rule 4 of the Insurance Services Office, Inc. Personal Auto Manual and the following
information, determine the appropriate primary classification factor. Explain how you arrived at your
selection.
The insured:
• Is a 28 year-old unmarried male.
• Owns the insured vehicle.
• Drives 25 miles one way to work twice a week.

Primary Classification
Description
Pleasure
Youthful Unmarried Male 2.0
Operator
Youthful Unmarried Male 2.5
Owner or Principal Operator
All Other
1.5

Work Less
Than 15
Miles
2.1

Work 15
or More
Miles
2.3

Business
2.4

2.6

2.8

3.0

1.6

1.7

1.8

Questions from the 2005 exam
6. A driver's insurance premium, before discounts and without expense fees, is as follows:
• Bodily Injury and Property Damage Liability= $210
• Comprehensive (Other than Collision) = $100
• Collision = $320
• Medical Payments = $20
The driver's vehicle has a qualifying alarm, dual-side passive restraints and anti-lock brakes. If the
premium is calculated using the ISO Personal Automobile Manual, how much does the driver save by
having these safety features?
A. < $21.60
B. > $21.60, but < $24.60
C. > $24.60, but < $27.60
D. > $27.60, but < $30.60
E. > $30.60

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ISO – EDITION 6-98 – GENERAL RULES 1 - 6
Questions from the 2006 exam:
3. According to the ISO Personal Automobile Manual, which of the following mid-term changes to an
annual policy can result in a mid-term premium adjustment?
A. The use of a vehicle on the policy is changed from "Business Use" to "Pleasure Use."
B. An operator on the policy attains a certain age that results in a Classification change.
C. An operator is involved in an accident that results in a change in the Driving Record Sub-Classification.
D. A review of loss experience results in a change in symbol assignment of a vehicle that is on the
current policy.
E. An operator on the policy now qualifies for the Good Student Classification.

Questions from the 2007 exam
5. A driver's insurance premium, before discounts and without expense fees, is as follows:
 Single Limit Liability = $250
 Comprehensive (other than Collision) = $125
 Collision = $325
 Medical Payments = $30
The driver's vehicle has an alarm and a fuel system disabling device which is manually activated using a
switch under the dashboard. It also has driver-side passive restraints and anti-lock brakes. If the premium
is calculated using the ISO Personal Automobile Manual, how much does the driver save by having these
safety features?
A. < $22.50
E. > $30

Exam 5, V1a

B. > $22.50 but < $25.00

C. > $25.00 but < $27.50

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D. > $27.50 but < $30.00

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Personal Vehicle Manual
ISO – EDITION 6-98 – GENERAL RULES 1 - 6
Questions from the 2011 exam
1. (2 points) Given the following information for a semi-annual ISO Personal Automobile Policy:
•
Principal operator is a 16-year-old single male
•
Auto is driven to school every day, 10 miles from operator's residence
•
Operator is full-time student
o
3.2 grade point average on a 4-point scale
o
Not in the top 20% of students at his school
•
Good student discount is 20%
•
Bodily injury and property damage base rate is $200
Age/Sex/Marriage Status Classification
Youthful Unmarried Female Operator
Youthful Married Male Operator
Youthful Unmarried Male Operator
Youthful Unmarried Male Owner or Principal Operator

Multiplicative Rate Factor
1.4
1.2
1.85
2.1

Use Classification
Business Use
Pleasure Use
Work Less Than 15 Miles Use
Work 15 or More Miles Use

Multiplicative Rate Factor
1.4
0.9
1.1
1.3

•
No other rating factors apply
a. (1 point) Calculate the premium for bodily injury and property damage liability coverage.
b. (0.5 point) Exactly three months after the policy is sold, the driver moves to a new home that is two miles
from school. Assuming all other policy characteristics remain consistent with part a above, determine the
impact of the mid-term adjustment for the remaining three months.
c. (0.5 point) Exactly four months after the policy is sold, the driver has an accident that results in a change
to the driving record sub-classification. Assuming all other policy characteristics remain consistent with
part a above, determine the impact of the mid-term adjustment for the remaining two months.

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ISO – EDITION 6-98 – GENERAL RULES 1 - 6
Questions from the 2012
1. (2.5 points) Given the following information for personal automobile policy:


Principal operator is a 35-year-old male.



Operator just obtained his driver's license, and has no prior driving experience or accidents.



The only vehicle is a 2011 Honda Accord sedan.
o Vehicle is equipped with anti-theft passive disabling device and anti-lock braking system.
o The physical damage rating symbol for this car is 13.



The current model year is 2012.



Operator drives 10 miles to work every weekday.



The policy expense fee is $60.



Selected coverage:
o The bodily Injury limits are $100,000/300,000.
o The property damage limit is $100,000.
o $1,000 deductible for both Collision and Comprehensive.
Primary Classification
Factor
Secondary Classification

Factor

Pleasure Use

1.00

0.00

Less Than 15 Miles

1.05

0.40

15 or More Miles

1.15

0.50

Business Use

1.20

0.90

Farm Use

0.85

1.50

2.20

Collision Relativities

Comprehensive Relativities

Symbol

2012

2011

2012

2011

1.11

1.05

1.06

1.00

Bodily Injury Limit

Factor

Property Damage Limit

Factor

Coverage

Base Rate

$25,000/$50,000

1.00

$25,000

1.00

Bodily Injury

$88

$50,000/$100,000

1.25

$50,000

1.06

Property Damage

$109

$100,000/$300,000

1.54

$100,000

1.12

Collision

$231

Collision Deductible

Factor

Comprehensive Deductible

Factor

$100

118%

Full Coverage

157%

$500

100%

$500

100%

$1,000

83%

$1,000

73%

Calculate the premium for this policy using the ISO Personal Automobile Manual.

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ISO – EDITION 6-98 – GENERAL RULES 1 - 6
Solutions to Questions from the 2002 Exam.
8. Based on Insurance Services Office, Inc., Personal Automobile Manual (Effective 6-98), which of the
following is false?
A. True. See page G-1.
B. False. "No policy may be written for a period longer than 12 months for Liability Coverage or 36
months for Physical Damage."
C. True. See section 4: Classifications, page G-5.
D. True. See section 5: Safe Driver Insurance Plan, section 2 page G-8.
E. True. See section 4: Classifications, page G-2.

Solutions to questions from the 2004 Exam:
21. (2 points) Using Rule 4 of the Insurance Services Office, Inc. Personal Auto Manual and the following
information, determine the appropriate primary classification factor. Explain how you arrived at your selection.
To determine the appropriate primary classification factor, candidates must use the information found within
the ISO Personal Auto Manual excerpt that accompanied the exam (which can also be obtained from the CAS
exam 5 Study Kit).
Based on the given data, find the information within Rule 4 which answers the following questions:
1. What Driving Category (Pleasure, Work, Business) does the insured fall into?
Under Rule 4: 4.C. Definitions
1.d. (2) (b) states - 15 or more road miles one way, for not more than 2 days per week or not more than 2
weeks in any 5-week period, shall be classified as WORK LESS THAN 15 MILES.
2. What Primary Classification Description does the insured belong to?
Under Rule 4: 4.C. Definitions
Under 2.a. (4) states - unmarried male under 30 years of age who is an owner or principal operator, shall
be classified as Youthful Unmarried Male -Owner or Principal Operator.
Therefore, the primary class factor = 2.6

Solutions to questions from the 2005 exam
6. If the premium is calculated using the ISO Personal Automobile Manual, how much does the driver
save by having these safety features?
Initial comments: On page G-6 of the ISO Personal Automobile Manual, it states that a 5% discount on
comprehensive coverage (premium) shall be afforded on vehicles equipped with alarm only devices
which sound an audible alarm that can be heard at a distance of at least 300 feet for a minimum of three
minutes; a 30% discount applicable to medical payments (premium) shall be afforded with restraints are
installed in both front outboard seats; a 5% for BI and PD (premium) shall be afforded for those autos
equipped with a factory installed four wheel anti-lock braking system.
In light of the above, the amount saved resulting from these safety features is .05 ($100) + .30 ($0.20)
+.05 ($210) = $5 + $6 + $10.50 = $21.50
Answer: A < $21.60

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ISO – EDITION 6-98 – GENERAL RULES 1 - 6
Solutions to questions from the 2006 exam
3. According to the ISO Personal Automobile Manual, which of the following mid-term changes to an
annual policy can result in a mid-term premium adjustment?
A. The use of a vehicle on the policy is changed from "Business Use" to "Pleasure Use."
B. An operator on the policy attains a certain age that results in a Classification change.
C. An operator is involved in an accident that results in a change in the Driving Record Sub-Classification.
D. A review of loss experience results in a change in symbol assignment of a vehicle that is on the
current policy.
E. An operator on the policy now qualifies for the Good Student Classification.
Answer: A – See page G-3 – Section 4: Classifications

Solutions to questions from the 2007 exam
5. If the premium is calculated using the ISO Personal Automobile Manual, how much does the driver
save by having these safety features?
Initial comments: On page G-6 of the ISO Personal Automobile Manual, it states that a 5% discount on
comprehensive coverage (premium) shall be afforded on vehicles equipped with alarm only devices which
sound an audible alarm that can be heard at a distance of at least 300 feet for a minimum of three minutes; a
20% discount applicable to medical payments (premium) shall be afforded with restraints are installed in the
driver side only position ;and a 5% for BI and PD / Single Limit (premium) shall be afforded for those autos
equipped with a factory installed four wheel anti-lock braking system.
In light of the above, the amount saved resulting from these safety features is
.05 ($125) + .20 ($30) +.05 ($250) = $6.25 + $6.00 + $12.50 = $24.75
Answer B. > $22.50 but < $25.00

Solutions to questions from the 2011 exam
Question 1
a. (1 point) Calculate the premium for bodily injury and property damage liability coverage.
b. (0.5 point) Exactly three months after the policy is sold, the driver moves to a new home that is two miles
from school. Assuming all other policy characteristics remain consistent with part a above, determine the
impact of the mid-term adjustment for the remaining three months.
c. (0.5 point) Exactly four months after the policy is sold, the driver has an accident that results in a change to
the driving record sub-classification. Assuming all other policy characteristics remain consistent with part a
above, determine the impact of the mid-term adjustment for the remaining two months.
Note: Access to ISO PAM (effective 6-98) is needed to answer the question. Section 4. Classifications
a. Base rate = $200
Youth, unnamed, male, principal op: multiplier = 2.1
10 mi/day: work <15 mi: multiplier = 1.1
Full-time, 16y/o, 3.2 GPA: disc = 20%
Premium = $200 * 2.1 * 1.1 * (1-.2) = $369.6
b. per ISO PAM part 4.Cc. (page G-3) use class = pleasure use, thus the multiplier = 0.9
Prem = $200 * 2.1 * 0.9 * (1-.2) = $302.4. Policy is semi-annual so Total prem = ½ (369.6 + 302.4) = $336
Thus, the impact of the mid-term adjustment is a decrease is premium of $369.6 - $336 = $33.6
c. According to ISO PAM part 4.A3 (page G-3), a policy shall not be changed mid-term to effect a change
in driving record sub-class, so there is no impact from part a.

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ISO – EDITION 6-98 – GENERAL RULES 1 - 6
Questions from the 2012 exam
12a. (1 point) Calculate the premium for bodily injury and property damage liability coverage.
12b. (0.5 point) Exactly three months after the policy is sold, the driver moves to a new home that is two miles
from school. Assuming all other policy characteristics remain consistent with part a above, determine the
impact of the mid-term adjustment for the remaining three months.
12c. (0.5 point) Exactly four months after the policy is sold, the driver has an accident that results in a change to
the driving record sub-classification. Assuming all other policy characteristics remain consistent with part a
above, determine the impact of the mid-term adjustment for the remaining two months.
Question 1 – Model Solution
Based on the given data in the problem, key rating manual classifications to identify prior to solving this
problem are as follows:
 Inexperienced operator = subclass 1B
 10mi commute everyday = work less than 15mi
 Passive disabling device = 15% discount on comp
 Anti lock braking = 5% discount on BI PD
 Vehicle is a 2011 model => use 2011 relativities
BI
88 x 1.54 x (1.05 + 0.5) x 0.95 = 199.55
Property
109 x 1.12 x (1.05 + 0.5) x 0.95 = 179.76
Collision
231 x 0.83 x 1.05 x (1.05 + 0.5) = 312.04
Comprehensive
60 x 0.73 x 1.00 x (1.05 + 0.5) x (0.85) = 57.71
Total Prem
(57.71 +312.04 + 179.76 +199.55) + 60 expense fee = $809
Examiner’s Comments
A very small number of candidates received full credit.
Most candidates did sum the 4 components and add the expense fee correctly.
Most candidates made mistakes in calculating and applying the primary and secondary classification factor.
Many multiplied the primary and secondary classification factors, instead of adding them together.
Some candidates did not correctly calculate other components (beyond the primary and secondary
classification factor) of the premium (base rate, ILF and other factors and discounts).

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Actuarial Notes for
Spring 2014 CAS Exam5

Syllabus Section A
Ratemaking, Classification Analysis,
Miscellaneous Ratemaking Topics

Volume 1b

Table of Contents
Exam 5 – Volume 1b: Ratemaking, Classification Analysis and
Miscellaneous Ratemaking Topics – Part 2
Syllabus Section/Title

Author

Page

A. Chapter 9: Traditional Risk Classification .................. Modlin, Werner ......................................................................... 1
A. Chapter 10: Multivariate Classification ....................... Modlin, Werner ....................................................................... 45
A. Chapter 11: Special Classification ................................ Modlin, Werner ....................................................................... 68
A. Chapter 12: Credibility .................................................. Modlin, Werner ...................................................................... 138
A. Chapter 13: Other Considerations ............................. Modlin, Werner ...................................................................... 171
A. Chapter 14: Implementation ....................................... Modlin, Werner ...................................................................... 190
A. Chapter 15: Commercial Lines Rating Mech ............. Modlin, Werner ...................................................................... 218
A. Chapter 16: Claims Made Ratemaking ........................ Modlin, Werner ...................................................................... 258

Appendix A: Auto Indication ............................................. Modlin, Werner ..................................................................... 290
Appendix B: Homeowners Indication. ............................. Modlin, Werner ...................................................................... 302
Appendix C: Medical Malpractice Indication. .................. Modlin, Werner ...................................................................... 311
Appendix D: Workers Compensation Indication. ........... Modlin, Werner ...................................................................... 320
Personal Auto Premiums: Asset Share Pricing ................ Feldblum ................................................................................. 328

Spring 2013 – Exam 5 – SS A and B ................................. CAS .......................................................................................... 368
Including Solutions and Examiner’s Comments.

Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Sec
1
2
3
4
5
6

Description
Introduction and Importance of Equitable Rates
Criteria For Evaluating Rating Variables
Typical Rating (or Underwriting) Variables
Determination of Indicated Rate Differentials
Appendix E - Univariate Classification Examples
Key Concepts

Pages
150 – 154
154 – 159
159 – 159
159 - 168
168 -168
169 - 169

Introduction and Importance of Equitable Rates

150 – 154

INTRODUCTION
The fundamental insurance equation is in balance in the aggregate when total premium covers the total costs
and allows for the target underwriting profit.
 It is also important to develop a balanced indication for individual risks or risk segments as well.
 Other considerations (e.g. marketing, operational, and regulatory) may require implementing a rating
algorithm other than what is indicated by the actuary’s analysis.
Very large risks (e.g. a multi-billion dollar manufacturing corporation with property, commercial liability, and WC
exposures) may have enough historical experience to estimate the amount of premium required for a future
policy term (see rating techniques covered in Chapter 15).
For smaller risks with not enough individual historical experience, classification ratemaking (i.e. grouping risks
with similar loss potential and charging different manual rates to reflect differences in loss potential among the
groups) is used.
First, class ratemaking requires risk criteria to segment risks into groups with similar expected loss experience
(e.g. a homeowners insurer may recognize that the expected loss for a homeowners policy varies based on
the age of the home).
 The characteristic examined is a rating variable (which refers to any variable used to vary rates, even if
it is based on a characteristic considered as an UW characteristic).
 The different values of the rating variable are known as levels (e.g. age of the home is the rating
variable, and the different ages or age ranges are the levels).
The insured population is then subdivided into appropriate levels for each rating variable.
Next, the actuary calculates indicated rate differentials relative to the base level for each level priced.
 A rate differential applied multiplicatively is known as a rate relativity.
 A rate differential applied additively is known as an additive.
 The term class refers to a group of insureds belonging to the same level for each of several rating
variables (e.g. in personal lines auto, class refers to a group of insureds with the same age, gender,
and marital status).

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Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
This chapter discusses:
• The importance of charging equitable rates
• Criteria for evaluating potential rating variables
• Traditional univariate (one-way) techniques used to estimate rate differentials for various levels of a
given rating variable.
To eliminate distortions inherent in univariate techniques, multivariate classification ratemaking
techniques (discussed in Chapter 10) are used.
Chapter 11 outlines special classification ratemaking techniques used for certain rating variables.
IMPORTANCE OF EQUITABLE RATES
An insurer that fails to charge the right rate for individual risks (when others are doing so) is subject to adverse
selection (and thus, deteriorating financial results).
An insurer that differentiates risks using a valid risk characteristic (when others are not) may achieve favorable
selection, and gain a competitive advantage.
Adverse Selection - Example
The goal of class ratemaking: Determine a rate commensurate with the individual risk.
Assume Simple Insurer charges an average rate for all risks (and others have implemented a rating variable
that varies rates to recognize the differences in expected costs).
 Simple will attract and retain higher-risk insureds and lose lower-risk insureds to those offering lower rates).
 A distributional shift toward higher-risk insureds makes Simple’s previously “average” rate inadequate
and causes the insurer to be unprofitable.
 Thus, Simple must raise the average rate.
 The increase in the average rate will encourage more lower-risk insureds to switch to competing
insurers, causing the revised average rate to be unprofitable.
 This downward spiral will continue until Simple:
i. improves their rate segmentation, or
ii. becomes insolvent, or
iii. decides to focus solely to higher-risk insureds and raises rates.
When Simple receives a disproportionate number of higher cost insureds, relative to its classification plan, it
is being adversely selected against.
As stated above, if adverse selection continues, Simple must either lose money, change its underwriting
criteria, or increase its premiums.
Example - The Adverse Selection Cycle









___

The average loss ( L ) and LAE ( E L ) is $180. Therefore, assuming no UW expenses or profit,
average total cost is $180.
The insured population consists of 50,000 high-risk insureds (Level H) and 50,000 low-risk insureds
(Level L).
The market consists of two insurers (Simple and Refined) each insuring 25,000 of each class of risk.
H risks have a cost of $230, and L risks have a cost of $130.
Simple charges H and L risks the same rate, $180. Refined implements a rating variable to vary the
rates according to the cost and charges H and L risks $230 and $130, respectively.
1 out of every 10 insureds shops at renewal and bases the purchasing decision on price.

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Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
The risks are distributed evenly amongst the two companies and the rates are set as follows:
Original Distribution, Loss Cost, and Rates
(1)
(2)
(3)
(4)
(5)
True
Refined Insurer
Simple Insurer
Expected
Insured
Charged
Insured
Charged
Risk
Cost
Risks
Rate
Risks
Rate
H
$230.00
25,000
$230.00
25,000
$180.00
L
$130.00
25,000
$130.00
25,000
$180.00
Total
$180.00
50,000
$180.00
50,000
$180.00
As shown below, if there is no movement of risks between the insurers, aggregate premium collected by both
insurers is the same.
 For Refined, the premium charged varies by level of the rating variable and is equitable.
 For Simple, H risks are not charged enough premium (the $1,250,000, shortfall is completely offset by the
excess premium collected from L risks).
Thus, L risks are subsidizing the H risks at Simple Insurer.
Static Distribution With Results
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Refined
Simple
True
Total
Total
Expected
Insured
Charged
$Excess/
Insured
Charged
$Excess/
Risk
Cost
Risks
Rate
($Shortfall)
Risks
Rate
($Shortfall)
H
$230.00
25,000
$230.00
$25,000
$180.00
$(1,250,000)
L
$130.00
25,000
$130.00
$25,000
$180.00
$1,250,000
Total
$180.00
50,000
$180.00
$50,000
$180.00
$(4)= [(3)-(1)] x (2)
(7)= [(6)-(1)] x (5)
Since 1 out of 10 insureds shops at renewal and makes their purchase based on price, the distribution of
insureds will not remain static.
 2,500 =[.10 * (25,000)] Refined H risks will buy from Simple and 2,500 Simple L risks buy from Refined.
 This movement results in the following distribution of risks for policy year one:
Policy Year One Distribution With Results
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Refined Company
Simple Company
True
Total
Total
Expected
Insured
Charged
$Excess/
Insured
Charged
$Excess/
Risk
Cost
Risks
Rate
($Shortfall)
Risks
Rate
($Shortfall)
H
$230.00
$230.00
$$180.00
$(1,375,000)
22,500
27,500
L
$130.00
$130.00
$$180.00
$1,125,000
27,500
22,500
Total
$180.00
50,000
$175.00
$50,000
$180.00
$(250,000)
[(22,500 * $230) + (27,500 * $130)]/50,000 = 175.00
(4)= [(3)-(1)] x (2)
(7)= [(6)-(3)] x (5)

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Because Simple’s distribution has shifted toward more H risks, the excess premium from the L risks fails to
make up for the shortfall from the H risks. It is forced to increase the rate from $180 to $185, the new average
cost based on the new distribution to make up for the $250,000 = [ ($185.00 - $180.00) * 50,000] shortfall.
Until Simple changes its price by risk level, this cycle will continue each year.
Policy Year Five Distribution With Results
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Refined Company
Simple Company
True
Total
Total
Expected
Insured
Charged
$Excess/
Insured
Charged
$Excess/
Risk
Cost
Risks
Rate
($Shortfall)
Risks
Rate
($Shortfall)
H
$230.00
14,762
$230.00
$35,238
$197.20
$(1,155,798)
L
$130.00
35,238
$130.00
$14,762
$197.20
$992,023
Total
$180.00
50,000
$159.52
$50,000
$197.20
$(163,775)
(4)= [(3)-(1)] x (2)
(7)= [(6)-(1)] x (5); (7tot)=(7H)+(7L)
This trend will continue until such time that Simple:
 segments its portfolio in a more refined manner
 loses too much money to continue
 only insures H risks at the rate of $230.
There are many factors that affect the adverse selection cycle (e.g. raising rates to the new true average cost
each year may not be feasible, and many jurisdictions require a company to obtain approval to change rates).
Favorable Selection
When an insurer identifies a characteristic that differentiates risk that other companies are not using, the insurer
has two options for making use of this information:
1. Implement a new rating variable.
2. Use the characteristic for purposes outside of ratemaking (e.g. for risk selection, marketing, agency
management).
If the insurer implements a new rating variable and prices it appropriately:
 its’ new rates will be more equitable.
 it may write a segment of risks that were previously considered uninsurable.
 it will attract more lower-risk insureds at a profit.
 some of the higher-risk insureds will remain and will be written at a profit
Over the long run, the insurer will be better positioned to profitably write a broader range of risks.
The motorcycle insurance market is a good example of favorable selection.
 Initially, motorcycle insurers rating algorithms did not include variation based on age of operator.
 Insurers recognizing that age of operator is an important predictor of risk charged higher rates for
youthful operators.
To keep overall premium revenue neutral, they lowered rates for non-youthful operators and were able to
attract a large portion of the profitable adult risks from their competitors.
Also, youthful operators who chose to insure with them were written profitably.

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Chapter 9 – Traditional Risk Classification
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At times, insurers may not be able to (or may choose not to) implement a new or refined rating variable.
 If allowed by law, the insurer may continue to charge the average rate but use the characteristic to
identify, attract, and select the lower-risk insureds (a.k.a. “skimming the cream).”
 This will allow the insurer to lower the average rate to reflect the better overall quality of the risks
insured.

Criteria For Evaluating Rating Variables

154 – 159

The first step in class ratemaking is to identify rating variables to segment insureds into different groups of similar
risks for rating purposes (e.g. the number, type, and skill level of employees are risk characteristics that may be
used as rating variables for WC insurance).
Criteria to evaluate the appropriateness of rating variables can be grouped into the following categories:
 Statistical
 Operational
 Social
 Legal
Statistical Criteria
The following statistical criterion helps to ensure the accuracy and reliability of a potential rating variable:
 Statistical significance
 Homogeneity
 Credibility
The rating variable should be a statistically significant risk differentiator:
 Expected cost estimates should vary for the different levels of the rating variable
 Estimated differences should be within an acceptable level of statistical confidence
 Estimated differences should be relatively stable from one year to the next.
Risk potential should be homogeneous within groups and heterogeneous between groups.
Identify and group risks for which the magnitude and variability of expected costs are similar (since by doing so
more accurate and equitable rates will be developed).
The number of risks in each group should either be large enough or stable enough or both to accurately
estimate costs (a.k.a. having sufficient credibility as discussed in Chapter 12).
Thus, group risks into a sufficient number of levels to ensure the risks within each group are homogeneous while
being careful not to create too many defined groups that may lead to instability in the estimated costs.

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Operational Criteria
For a rating variable to be practical, it should be
 Objective
 Inexpensive to administer
 Verifiable
Examples:
1. Levels within a rating variable should have objective definitions.
 Estimated costs for medical malpractice insurance vary by the skill level of a surgeon. Example:
However, the skill level of a surgeon is difficult to determine and subjective (thus, it is not a practical
choice for a rating variable).
 More objective rating variables like board certification, years of experience, and prior medical
malpractice claims can serve as proxies for skill level.
2. The cost to obtain information to properly classify a risk should not be high. Example:
 Building techniques and features that improve the ability of a home to withstand high winds can
significantly reduce expected losses, and should be implemented as a rating variable to recognize
differences, but cannot be easily identified without a very thorough inspection of the home performed by
a trained professional.
 Thus, if the cost of the inspection outweighs the benefit, do not use that risk characteristic as a rating
variable.
3. The levels of a rating variable should not be easily manipulated by the insured and should be easy for the
insurer to verify. Example:
 Number of miles driven is a risk differentiator for personal auto insurance. However:
 Many car owners cannot accurately estimate how many miles their car will be driven in the upcoming
policy period, and
 Insurers may not have a cost-effective way to verify the accuracy of the amount estimated by the
insured.
Since insureds may not report accurate data, insurers may not use annual miles driven as a rating variable.
Note: As technology (e.g. on-board diagnostic devices) become standard equipment in cars, this rating variable
may become more verifiable and how it is used in rating may make it miles driven a viable rating variable.
Social Criteria
The following affect social acceptability of using a risk characteristic as a rating variable:
 Affordability
 Causality
 Controllability
 Privacy concerns
1. Affordability: It is desirable for insurance to be affordable for all risks. This is true when:
 it is required by law (e.g. states require “proof of financial responsibility” from owners of vehicles)
 it is required by a third party (e.g. lenders require homeowners insurance)
 it facilitates ongoing operation (e.g. stores purchase commercial general liability insurance).

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Social Criteria (continued)
2. Causality: It is preferable if rating variables are based on characteristics that are causal in nature.
Examples:
 A sump pump in a house has a direct effect on water damage losses to the house, and a corresponding
reduction in premium for the presence of a sump pump is socially acceptable.
 While insurance credit scores (a measure of the insured’s financial responsibility) have been
incorporated into rating algorithms (given its strong statistical power in predicting losses), use of this
variable has resulted in a consumer backlash from a belief of a lack of obvious causality to losses.
3. Controllability: It is preferable for an insured to have some control as to the class they belong to (affecting
the premium charged). For example:
 The type and quality of a company’s loss control programs affects WC expected losses, since approved
loss control programs can reduce expected losses and thus the charged premium.
 In contrast, insureds cannot control their age or gender. Although age and gender have been shown to
impact personal lines loss costs, some jurisdictions do not allow them as rating variables.
4. Privacy: There are privacy concerns associated with the use of particular rating variables. Examples:
 When technology to determine how safely a car is being driven is standard in all vehicles, this can
greatly improve an insurer’s ability to accurately price a given risk. To address the privacy concern, the
data is deemed to be protected and the insurer is only able to use it with the consent of the insured.
 Some insurers have implemented usage-based insurance programs on a voluntary basis.
However, any such usage-based programs will be most effective if they can be used on all risks rather
than just the ones who volunteer.
Legal Criteria
Most jurisdictions worldwide have laws and regulations related to P&C insurance products.
In the U.S. P&C insurance products are regulated by the states.
 Most states have statutes that require insurance rates to be “not excessive, not inadequate, and not
unfairly discriminatory.”
 Some states’ statutes may require certain rates to be “actuarially sound.”
 Some states have regulations about what is allowed and not allowed in risk classification rating for
various P&C insurance products.
 Some states statutes prohibit the use of gender in rating while others permit it as a rating variable.
 Some states may allow the use of a rating variable, but may place restrictions on its use (e.g. allosing a
credit score to be used for rating personal insurance for new business, but not allowing insurers to raise
rates for renewal risks should the insured’s credit worsen (although they may allow companies to reduce
rates if the insured’s credit score improves).
 Some states prohibit variables from use in the rating algorithm but allow their use in U/W (which may be
used to guide risk selection decisions and or guide risk placement decisions).
To be familiar with the laws and regulations of each jurisdiction the insurer writes in, the actuary should work with
lawyers or regulatory compliance experts in determining what is acceptable and what is not.

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Chapter 9 – Traditional Risk Classification
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3

Typical Rating (or Underwriting) Variables

159 – 159

Examples of rating variables by line of business are as follows:
Type of Insurance
Rating Variables
Personal Automobile
Driver Age and Gender, Model Year, Accident History
Homeowners
Amount of Insurance, Age of Home, Construction Type
Workers Compensation
Occupation Class Code
Commercial General Liability
Classification, Territory, Limit of Liability
Medical Malpractice
Specialty, Territory, Limit of Liability
Commercial Automobile
Driver Class, Territory, Limit of Liability
Note: Some risk characteristics may be used as both rating variables and underwriting variables.

Determination of Indicated Rate Differentials

159 - 168

The actuary must identify the amount of rate variation among the levels of each rating variable. The rate for all
non-base levels is expressed relative to the base level (see chapter 2) as prescribed in the rating algorithm.
This chapter discusses traditional univariate methods that use the historical experience for each level of a rating
variable to determine the differentials.
Each of the approaches described below assume that the rating algorithm is multiplicative, so differentials
are called relativities.
Differentials could be derived in an additive/subtractive manner (but this is not addressed in the examples).
The following approaches are discussed:
1. Pure Premium
2. Loss Ratio
3. Adjusted Pure Premium
The output of these approaches is a set of indicated rate relativities.
 If relativities are changed for some or all of the levels of the rating variables, more or less premium
being collected overall can result, and the base rate can be altered to compensate for the expected
increase or decrease in premium.
 This topic (base rate offsetting) is discussed in Chapter 14.

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Assumptions for Simple Example
The assumptions:
 All UW expenses are variable. The variable expense provision (V) is 30% of premium, the target profit
percentage ( QT ) is 5% of premium, so the PLR is 65% (= 1 – 30% - 5%).


There are only 2 rating variables: amount of insurance (AOI) and territory. Exposures are distributed
across the two rating variables as follows:
Exposure Distribution (in number and in percentage)
Territory
Territory
AOI
1
2
3 Total
1
2
3
Total
7
130 143
280
1% 13% 14% 28%
Low
360
11% 13% 13% 37%
Medium 108 126 126
179 129
40
348
18% 13% 4% 35%
High
294 385 309
988
30% 39% 31% 100%
Total


The “true” underlying loss cost relativities (which the actuary is attempting to estimate) as well as
the relativities currently used in the insurer’s rating structure are as follows:
True and Charged Relativities for AOI and for Territory
True
Charged
True
Charged
AOI
Relativity
Relativity
Terr
Relativity
Relativity
Low
0.7300
0.8000
1
0.6312
0.6000
Medium
1.0000
1.0000
2
1.0000
1.0000
High
1.4300
1.3500
3
1.2365
1.3000
Note: The base levels are Medium AOI and Territory 2:



The exposure, premium, and loss information needed for the analysis is summarized as follows:
Simple Example Data
Premium @
Current Rate
AOI
Terr Exposure
Loss & LAE
Level
Low
1
7
$210.93
$335.99
Medium
1
108
$4,458.05
$6,479.87
High
1
179
$10,565.98
$14,498.71
Low
2
130
$6,206.12
$10,399.79
Medium
2
126
$8,239.95
$12,599.75
High
2
129
$12,063.68
$17,414.65
Low
3
143
$8,441.25
$14,871.70
Medium
3
126
$10,188.70
$16,379.68
High
3
40
$4,625.34
$7,019.86
TOTAL
988
$65,000.00
$100,000.00

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Pure Premium Approach
Given a rating variable R1 with a rate differential for each level i given by R1i, then the rate for each level of
rating variable R1 (Ratei) is the product of the base rate (B) and the rate differential (R1i): Ratei = R1i x B.
The indicated differential is calculated as follows:

R1I,i =

RateI,i
, where subscript I denotes indicated.
BI
_________

____

[ L  EL  EF ]
The formula for the indicated rate using the pure premium method is Indicated Rate 
.
[1.0 - V - QT ]


If all UW are considered to be variable or if fixed expenses are handled through a separate fee, then the
fixed expense component (F) is set equal to zero and the formula simplifies to the following:

Indicated Rate 


[ L  EL ]
[1.0 - V - QT ]

If fixed expenses are material and a separate expense fee is not used (i.e. the base rate includes a
provision for fixed expenses), include the fixed expense loading in the formula.
This will “flatten” the otherwise indicated relativities to account for the fact that the fixed expenses
represent a smaller portion of the risks with higher average premium.

Assuming the fixed component is not necessary and substituting the formula for the indicated rate and base rate,
________

the indicated differential for level i is calculated as follows: R1I ,i 

[ L  EL ]i
[1.0 - V - QT ]i

________

[ L  E L ]B
[1.0 - V - QT ]B
_________

Assuming all policies have the same UW expenses and profit provisions, then R1I ,i 

[ L  EL ]i
_________

[ L  E L ]B

Pure Premium Approach in Practice
 It is not always feasible to allocate ULAE to different classes of business, so the pure premiums used in
class analysis generally only include L + ALAE.
 If the actuary chooses to incorporate U/W expense provisions and target profit provisions that vary by
type of risk, the indicated PP for each level can be adjusted by the applicable provisions prior to
calculating the indicated relativities.
Depending on the portfolio, it may not always be necessary to trend and develop the loss and (A)LAE.
 In stable portfolios for short-tailed lines of business (e.g. HO), it is acceptable to ignore these
adjustments for class analysis.
 If the portfolio is growing or shrinking, or the distribution of loss and (A)LAE by class is changing over
time, a multi-year PP analysis would be improved by applying aggregate trend and development factors
to the individual year’s loss and (A)LAE before summing.
 In long-tailed lines (e.g. WC), it is possible that classes of risk undergo trend and development at
materially different rates. For example:
i. WC risks with return-to-work programs may experience less development over time than risks without
such a program.
ii. If trend and development are materially different by level or claim type (e.g. WC indemnity and
medical), consider developing and/or trending individual risks or levels prior to classification analysis.

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It is common to adjust losses for extraordinary and catastrophic events in classification data as they can have
a disproportionate impact on a level or levels for the rating variable being analyzed. For example:
 a catastrophic event may only affect one territory.
 one extraordinary loss only impacts one level.
Thus, the actuary should consider replacing these actual losses with an average expected figure for each level
(if such data is available).
The following shows the Pure Premium Method calculations for the simple example:
(1)
(2)
(3)
(4)
(5)
(6)
Indicated
Indicated
Pure
Indicated Relativity to
Terr
Exposures Loss & LAE
Premium
Relativity
Base
1
294
$15,234.96
$51.82
0.7877
0.7526
2
385
$26,509.75
$68.86
1.0467
1.0000
3
309
$23,255.29
$75.26
1.1439
1.0929
Total
988
$65,000.00
$65.79
1.0000
0.9554
(4)= (3)/(2);
(5)= (4)/(Tot4);
(6)= (5)/(Base5)
In this example, loss and LAE in (3) is not developed or trended, and implicitly assumes that all levels of the
rating variable experience development and trend at the same rate.
 In many short-tailed lines of business (e.g. HO), the assumption may be reasonable.
 In long-tailed lines (e.g. WC), risks may undergo trend and development at different rates (e.g. WC risks
with return-to-work programs may experience less development than risks without such a program).
 If trend and development are materially different by level, consider developing and/or trending individual
risks or levels prior to class analysis.



Adjust class data for extraordinary and catastrophe losses as they can have a disproportionate impact
on a level or levels for the rating variable being analyzed (e.g. a cat event may only affect one territory).
While column (6) can be calculated directly from column (4), column (5) was included as insurers
typically compare current, indicated, and competitors’ relativities all normalized so that the total average
exposure-weighted relativity is 1.00 for each (thus relativities can be compared on a consistent basis).

Distortion (in the true vs. indicated relativities)
Compare the true underlying pure premium relativities and the relativities indicated by the pure premium analysis:
Pure
True
Premium
Terr
Relativity
Indication
1
0.6312
0.7526
2
1.0000
1.0000
3
1.2365
1.0929
Key! The indicated and true territorial relativities do not match due to a shortcoming of the univariate
pure premium approach.

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Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
The pure premium for each level is based on the experience of each level and assumes a uniform
distribution of exposures across all other rating variables.
 If one territory has a disproportionate number of exposures of high or low AOI homes, this
assumption is invalid.
 By ignoring the exposure correlation between territory and AOI, the loss experience of high or low
AOI homes can distort the indicated territorial relativities resulting in a “double counting” effect.
i. Territory 1 indicated PP relativity is higher than the true relativity due to a disproportionate share of
high-value homes in Territory 1.
ii. Territory 3 indicated PP relativity is lower than the true relativity due to a disproportionate share of
low-value homes in Territory 3.
If AOI were distributed in the same way within each territory, the indicated relativities would not have
been affected. This does not mean that each of the three AOI levels needs to be 1/3rd of the exposures
within each territory, but that the distribution of AOI must be the same within every territory.
Note: Since in reality there are many characteristics that affect an insured’s risk potential, to the extent there is
a distributional bias in some or all of the other characteristics, the resulting pure premiums can be biased.
The Adjusted Pure Premium, discussed later, minimizes the impact of the distributional bias resulting from
the AOI relativities.
Loss Ratio Approach
The major difference between the PP and LR approaches is that the LR approach uses premium (vs. exposure).
The LR approach compares LRs for each of the levels to the total LR to determine the appropriate
adjustment to the current relativities.

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Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Loss Ratio Approach Calculations:
Step 1: Start with the PP indicated differential formula (assumes all policies have the same UW expenses and
profit provisions):

( L  EL )i
[ L  EL ]i
Xi
R1I ,i  _________

( L  EL ) B
[ L  E L ]B
XB
_________

Step 2: Multiply both sides of the equation by the ratio of the avg. premium at current rates for the base level
______

_____

( PC , B ) to the avg. premium at current rates for level i of the rating variable being reviewed PC ,i
______

PC , B

R1I ,i  ______
PC ,i

___________

_____

[ L  EL ]i PC ,B


[ L  EL ]B _____
PC ,i

Step 3: Average premium equals total premium divided by total exposures and average PP equals total losses
__

and LAE divided by total exposures: P 

_________
L  EL
P
and L  EL 
X
X

Step 4: The current differential for level i ( R1C ,i ) equals the ratio of the current average premium for level i
_____

divided by the current average premium at the base level:

R1C ,i 

C ,i
_____

PC , B
Step 5: Transform the Step 4 formula as follows:

Indicated Differential Change 

R1I ,i
R1C ,i

( L  EL )i
PC ,i
Loss & LAE Ratio for i
=

( L  EL ) B Loss & LAE Ratio for B
PC , B

Loss Ratio Approach in Practice
Similar to the PP premium approach, many of the same data limitations and assumptions regarding losses
apply (e.g. ULAE cannot be allocated by class).
 In the LR approach, however, it is important to bring earned premium to the current rate level of each
class.
 This is most accurately done via extension of exposures, though the parallelogram method can be
performed at the class level if data limitations preclude use of extension of exposures.

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Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Calculations for the Loss Ratio Method:
(1)
(2)
(3)

Terr
1
2
3

Premium @
Current Rate
Level
$21,314.57
$40,414.19
$38,271.24
$100,000.00
(4)= (3)/(2);

Loss & LAE
$ 15,234.96
$ 26,509.75
$ 23,255.29
$ 65,000.00
(5)= (4)/(Tot4) ;

(4)
Loss &
LAE
Ratio
71.5%
65.6%
60.8%
65.0%

(5)
(6)
(7)
Indicated
Relativity
Change
Current
Indicated
Factor
Relativity
Relativity
1.1000
0.6000
0.6600
1.0092
1.0000
1.0092
0.9354
1.3000
1.2160
1.0000
(7)= (5)x(6);
(8)= (7)/(Base7)

(8)
Indicated
Relativity
Base
0.6540
1.0000
1.2049

Noteworthy comments:
 Column 4 should be adjusted for any extraordinary or catastrophic losses.
 The validity of the assumption that trend and development apply uniformly to all risks applies should be
challenged.
 Column 5 represents the amount the territory relativities should be changed to make the loss and LAE
ratios for every territory equivalent.
 Column 7 relativities have the same overall weighted average as the current relativities.
Since it is useful to compare the current, indicated, and competitors’ relativities for a variable, each set of
relativities should be adjusted so that the overall weighted-average relativity is the same.
The proper way to make such an adjustment is shown in column 8, which adjusts the relativities to the
base level by dividing the indicated relativity for each level by the indicated relativity at the base level.
Distortion (in the true vs. indicated relativities)
Compare the true underlying pure premium relativities and the relativities indicated by the pure premium analysis:
Pure
Loss
True
Premium
Ratio
Terr
Relativity
Indication Indication
1
0.6312
0.7526
0.6540
2
1.0000
1.0000
1.0000
3
1.2365
1.0929
1.2049
The indicated LR territorial relativities are closer to the true relativities than those computed using the PP
approach.
 Since the PP approach relies on exposures (i.e. one exposure for each house year), the risks in each
territory are treated the same regardless of the AOI.
 In contrast, LR approach relies on premium (in the denominator of the loss ratio) which reflects the fact
that the insurer collects more premium for homes with higher AOI.
Using the current premium helps adjust for the distributional bias.
 Regardless, the LR method did not produce the correct relativities (the distortion coming from the
variation in AOI relativities being charged rather than the true variation).
If the current AOI relativities equaled the true AOI relativities, then the LR method will produce the true
territorial relativities.

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Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Indicated relativities (using the LR method) “adjust” for the inequity present in the other rating variables.
 The rate relativity for Territory 1 is higher than the true relativity because the process by which it takes
into account the high proportion of high-valued homes relies on the current AOI relativities that are
under-priced.
 The downside to this adjustment is that all homes in Territory 1, not just the high-value homes, are
being charged an extra amount to correct for the inequity in AOI relativities.
Adjusted Pure Premium Approach
It is possible to make an adjustment to the PP approach to minimize the impact of any distributional bias.
The PP approach can be performed using exposures adjusted by the exposure-weighted average relativity of
all other variables.
Calculation of the current exposure-weighted average AOI relativities by territory is shown below:
Charged
AOI
Exposures by Territory
AOI
Factor
1
2
3
Low
0.8000
7
130
143
Medium
1.0000
108
126
126
High
1.3500
179
129
40
Total
294
385
309
Wtd Avg AOI Relativity by Terr
1.2083
1.0497
0.9528



If there are more than two rating variables, the above table needs to be expanded so that the exposureweighted average relativity is based on all rating variables.
If this is not practical, the actuary may focus only on rating variables suspected to have a distributional
bias across the levels of the rating variable being analyzed.

Adjusted Pure Premium Method
(1)
(2)
(3)
Wtd Avg
Earned
AOI
Terr
Exposures
Relativity
1
294
1.2083
2
385
1.0497
3
309
0.9528
Total
988
(4)= (2)*(3)
(6)= (5)/(4);

(4)

(5)

(6)
Indicated
Adjusted
Pure
Exposures
Loss & LAE
Premium
355.24
$15,234.96
$42.89
404.13
$26,509.75
$65.60
294.42
$23,255.29
$78.99
1,053.79
$65,000.00
$61.68
(7)= (6)/(Tot6);
(8)= (7)/(Base7)

(7)
Indicated
Relativity
0.6954
1.0636
1.2806
1.0000

(8)
Indicated
Relativity
@Base
0.6538
1.0000
1.2040
0.9402

Distortion
 Since the current AOI relativities were used for the adjustment, the resulting indicated relativities are
equivalent to those calculated using the LR approach (except for rounding).
 The same comments made about the distortion associated with the LR approach apply.
Since univariate techniques cause distortion, many insurers have moved to multivariate techniques, which are
possible to perform with today’s technology, and are covered in the next chapter.

Exam 5, V1b

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Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
5

Appendix E - Univariate Classification Examples

168 -168

The following show examples of classification analysis using a pure premium and loss ratio analysis.
Pure Premium Approach
Wicked Good Auto Insurance Company
Classification Relativities Using the Pure Premium Approach
(1)

Class
J
K
L
M
N
P
TOTAL

(2)

(3)

Reported
Earned
Loss &
Pure
Exposures
ALAE
Premium
16,520
$878,200
$53.16
11,328
$740,940
$65.41
1,266
$136,830
$108.08
12,836
$888,582
$69.23
4,200
$753,156
$179.32
11,538
$518,146
$44.91
57,688 $3,915,854 $67.88

(4)

Indicated
Relativity
0.7831
0.9636
1.5922
1.0198
2.6418
0.6616
1.0000

(5)

(6)

(7)

(8)

(9)
CredibilityWeighted
Credibility- Indicated
Normalized
Weighted Relativity
Current
Current
Indicated
@ Base
Relativity Relativity Credibility Relativity
Class
1.00
0.7811
1.00
0.7831
1.0000
1.15
0.8983
1.00
0.9636
1.2304
1.95
1.5232
0.34
1.5466
1.9748
1.35
1.0545
1.00
1.0198
1.3022
3.50
2.7340
0.62
2.6771
3.4184
0.85
0.6640
1.00
0.6616
0.8448
1.2802
1.0000
1.0016

(10)

Selected
Relativity
1.00
1.23
1.98
1.30
3.42
0.84
1.2776

(11)

(12)

Relativity
Change
0.0%
7.0%
1.5%
-3.7%
-2.3%
-1.2%
-0.2%

Relativity
Change
with OffBalance
0.2%
7.2%
1.7%
-3.5%
-2.1%
-1.0%
0.0%

(3) = (2) / (1)
(4) = (3) / (Tot3)
(Tot5) = (5) Weighted by (1)
(6) = (5) / (Tot5)
(7) = [ (1) / 11,050 ] ^ 0.5 limited to 1.0
(8) = (4) * (7) + [ 1.0 - (7) ] * (6)
(Tot8) = (8) Weighted by (1)
(9) = (8) / (Base8)
(Tot10) = (10) Weighted by (1)
(11) = (10) / (5) - 1.0
(12) = [ 1.0 + (11) ] / [ 1.0 + (Tot11) ] - 1.0

Column 1: Earned exposures are the best match to reported losses to produce pure premiums
Column 2: Calendar accident year reported loss and ALAE. These amounts have been adjusted to
convert historical losses and ALAE to projected loss and LAE (e.g. development, trend, ULAE
adjustment) at the aggregate level.
Column 4: Note that the total exposure-weighted average relativity is 1.00, which is important for comparing
indicated pure premium relativities to those currently used by the insurer or competitors (assuming those are
normalized to 1.00 also).
Column 5: The current class relativities found in the rating manual having base class J (with a relativity of 1.0)
Column 6: Current class relativities normalized so that the total exposure-weighted average relativity is 1.00.
 Weight the relativities using premium adjusted to the base class, but exposures are used as a proxy.
 By normalizing these relativities, they can be compared to the indicated relativities in Column 4.
Column 7: Full credibility standard is 11,050 exposures, and partial credibility is computed using the square
root rule (11,050 is based on a 663 claim standard and an expected frequency of 6%).
As discussed in Chapter 12, the 663 standard assumes no variation in the size of loss and that there is a 99% chance that the
observed value will be within 10% of the true value.

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Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Column 8: Credibility-weights the indicated relativities with the current normalized relativities.
The all class pure premium is another common complement of credibility, but it was ruled out due to the
significant variation between the classes.
Column 11: Shows the expected change in premium for each class due to the change between the
current and selected manual relativities.
 A total exposure-weighted average relativity of -0.2% change (= 1.2776 / 1.2802 -1.0) means that if
the selected class relativities are implemented without any other changes, the overall premium will
change by -0.2%.
 This is the amount the base rate needs to be offset by if no overall premium change is desired (i.e.
to make the rate change revenue neutral).
Column 12: Displays the relativity change assuming the base rate will be offset so that there is no overall
increase or decrease due solely to the implementation of the selected relativities.
Loss Ratio Approach – Part 1
Wicked Good Auto Insurance Company
Classification Relativities - Using the Loss Ratio Approach

Class
J
K
L
M
N
P
TOTAL

(1)

(2)

Premium
at
Current
Rate
Level
$1,114,932
$917,284
$166,314
$1,162,236
$1,056,318
$666,978
$5,084,062

Reported
Loss &
ALAE
$878,200
$740,940
$136,830
$888,582
$753,156
$518,146
$3,915,854

(3)

Loss
Ratio
78.8%
80.8%
82.3%
76.5%
71.3%
77.7%
77.0%

(4)

(5)

Indicated
Change
2.3%
4.9%
6.8%
-0.7%
-7.4%
0.9%
0.0%

Number
of
Claims
826
652
124
866
736
490
3,694

(6)

(7)

CredibilityWeighted
Indicated
Credibility
Change
1.00
2.3%
0.99
4.8%
0.43
2.9%
1.00
-0.7%
1.00
-7.4%
0.86
0.7%

(8)

(9)

Current
Relativity
1.00
1.15
1.95
1.35
3.50
0.85

CredibilityWeighted
Indicated
Relativity
1.0227
1.2056
2.0075
1.3401
3.2400
0.8563

(3) = (2) / (1)
(4) = (3) / (Tot3) - 1.0
(Tot5) = (5) Weighted by (1)
(6) = [ (1) / 663 ] ^ 0.5 limited to 1.0
(7) = (4) * (6) + 0.0% * [ 1.0 - (6) ]
(9) = [ 1.0 + (7) ] * (8)

Column 1: It is critical that the premium is adjusted at the granular level rather than at the aggregate level
(i.e. it is not sufficient to use the parallelogram method at the aggregate level if the rate changes varied by the
classes being examined).
Column 2: The same comments about aggregate adjustments made in the pure premium approach apply.
Column 3: Indicated change is the % the current class relativities (column 8) need to be increased or
decreased so that the expected loss ratio will be the same for every class.
Columns 5 through 7: The full credibility standard is 663 claims, partial credibility is calculated using the
square root rule, and the complement of credibility is no change.
Column 9: Credibility-weighted indicated relativities are adjusted to the base class level in Column 10.

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Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Loss Ratio Approach – Part 2
(9)

(10)
CredibilityWeighted
Credibility- Indicated
Weighted Relativity
Indicated
@ Base
Relativity
Class
1.0227
1.0000
1.2056
1.1789
2.0075
1.9630
1.3401
1.3104
3.2400
3.1682
0.8563
0.8373

(11)

Selected
Relativity
1.00
1.18
1.96
1.31
3.17
0.84

(12)

(13)

Relativity
Change
0.0%
2.6%
0.5%
-3.0%
-9.4%
-1.2%
-2.3%

Relativity
Change
with OffBalance
2.4%
5.0%
2.9%
-0.7%
-7.3%
1.2%
0.0%

(10) = (9) / (Base9)
(12) = (11) / (8) - 1.0
(Tot12) = (12) Weighted by (1)
(13) = [ 1.0 + (12) ] / [ 1.0 + (Tot12) ] - 1.0
Column 10: Uses column (9) credibility-weighted indicated relativities to adjust to the base class level
Column 11: Selected relativities, and
Column 12: The total change (-2.3%):
 is the weighted average of the class changes using premium at current rate level as the weight.
 represents the expected change in premium due to the selected class relativity changes, and is the
amount the base rate needs to be offset if these relativity changes are to be implemented on a
revenue-neutral basis.
Column 13: The relativity change for each class if the base rates are offset.

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Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
6

Key Concepts

169 - 169

1. Definitions used in classification ratemaking
a. Rating variable
b. Level of a rating variable
c. Rate differentials
2. Importance of equitable rates
a. Adverse selection
b. Favorable selection
3. Considerations for evaluating rating variables
a. Statistical criteria
b. Operational criteria
c. Social criteria
d. Legal criteria
4. Calculating indicated rate differentials
a. Pure premium approach
b. Loss ratio approach
c. Adjusted pure premium approach

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Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
The predecessor papers to the current syllabus reading “Basic Ratemaking” by Werner, G. and Modlin, C.
were numerous. While past CAS questions were drawn from prior syllabus readings, the ones shown
below remain relevant to the content covered in this chapter.
By relevant, we mean concepts tested on past CAS exams relate to similar to the concepts found in this
chapter.

Section 1: Criteria Used In Traditional Risk Classification
Questions from the 1991 exam
3. According to Werner and Modlin, "Basic Ratemaking", statistical criteria are used to achieve which of the
following goals when establishing a classification system?
1. Homogeneity
A. 1

B. 2

2. Credibility
C. 3

D. 1, 2

3. Causality

E. 1, 3.

Questions from the 1993 exam
31. a. (1 point) Identify the three statistical criteria for selecting rating variables mentioned in Werner
and Modlin, "Basic Ratemaking".

Questions from the 1997 exam
31. (3 points) According to Werner and Modlin, "Basic Ratemaking",
a. (2 points) Identify and explain three statistical criteria that should be considered when selecting rating
variables for a classification plan.
b. (1 point) Question no longer applicable to the content covered in this chapter.

Questions from the 1998 exam
43. Werner and Modlin, "Basic Ratemaking" list a number of social criteria that any rating plan should satisfy.
a. (1 point) List and briefly describe four of these social criteria.

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Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2004 exam
29. (4 points) ABC Insurance Company writes standard auto business in State X and uses driver classification to
rate policies. Based on the most recent analysis, a 5% rate level increase is needed in order to maintain rate
adequacy. This rate level need varies by driver classification, as detailed in the table below.
Driver
Classification
A
B
C
D
State Total

Indicated
Rate Change
-40%
-20%
+20%
+40%
+5%

a. (1 point) Other than an overall rate level increase, describe an action the insurance company could
undertake to restore overall rate adequacy. Assume that the indicated rate need by driver classification
does not change when the proposed action is taken.
b. (1 point) Suppose that ABC Insurance Company's chief competitor in State X has the same
underwriting rules and writes a similar distribution of business as ABC Insurance Company. The
competitor is rate adequate by driver classification as well as on a statewide basis. Describe the
situation that could result if ABC Insurance Company fails to reflect the indicated changes by driver
classification.
c. (1 point) Suppose regulation was enacted abolishing the use of the driver classification rating variable
for State X. Briefly describe the impact on ABC Insurance Company's profitability.
d. (1 point) Briefly describe the social consequences of the abolishment of the driver classification rating
variable.
40. (2 points) Finger, in "Classification Ratemaking," discusses several criteria for rating variables. Some
companies use information from credit reports as a rating variable. State four criteria for rating variables
and explain whether or not they are fulfilled by information from credit reports.

Questions from the 2005 exam:
45. (2 points) Finger, in "Risk Classification," discusses the effect of market forces on the refinement of
insurance classification plans.
a. (1 point) Describe how the behavior of policyholders creates pressure on insurers to refine
classification plans.
b. (1 point) Explain why classification plans may also become more refined as insurance coverage
becomes more expensive. Discuss the perspective of both the insurer and the policyholder.

Questions from the 2006 exam
8. Which of the following changes might cause an insurer to develop a more refined classification plan?
1. The market becomes more competitive.
2. Coverage becomes more expensive.
3. The market becomes larger.
A. 1 only
B. 2 only
C. 1 and 3 only
D. 2 and 3 only
E. 1, 2, and 3

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Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2006 exam
38. (3 points) Werner and Modlin, "Basic Ratemaking" discuss various criteria for selecting rating variables.
As the actuary for an insurance company, you are developing an auto class plan in which one of the
proposed rating variables is estimated miles driven during the coverage period.
a. (1.5 points) Identify and briefly describe two statistical criteria, and explain whether mileage
defined this way satisfies these criteria.
b. (1.5 points) Identify and briefly describe two operational criteria, and explain whether mileage
defined this way satisfies these criteria.

Questions from the 2008 exam:
28. (2.0 points) An insurance company wants to use color of car as a rating variable within its risk
classification system.
a. (1.0 point) Identify two operational risk classification criteria and evaluate the variable "color of car" with
respect to each criterion.
b. (1.0 point) Identify two social risk classification criteria and evaluate the variable "color of car" with respect
to each criterion.

Questions from the 2009 exam:
33. (1 point) Fully discuss how an insurance company can "skim the cream" to gain a competitive advantage.
34. (1.5 points) An insurance company is considering using a rating factor based on a detailed
psychological profile.
a. (1 point) Identify and briefly explain two of the criteria for desirable classification rating factors.
b. (0.5 point) Evaluate if the rating factor based on the new psychological profile meets each of the
criteria identified in part a. above.

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Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Section 2: Traditional classification analysis using PP and LR analyses.
Questions from the 1991 exam
41. (2 points) This question should be answered using Chapter 5, "Risk Classification" from the CAS
textbook Foundations of Casualty Actuarial Science.
Using the loss ratio method and the data that follows, calculate the revised territorial relativities.
Territory A is the base class. Show all work.
Territory
A
B
C
Total

EP @ Present
Rates
2,000,000
1,500,000
500,000
4,000,000

Incurred
Losses
1,400,000
900,000
400,000
2,700,000

Credibility
.85
.50
.40

Existing
Relativity
1.000
.900
1.200

Questions from the 1994 exam
42. (4 points) Use the methodologies described by Finger in chapter 5, "Risk Classification,"
Foundations of Casualty Actuarial Science, and the information below:
Territory
A
B
C

Earned
Exposures
800
1,800
400

Base
Exposure
1,000
1,500
500

Earned
Premium
$200,000
300,000
100,000

Incurred
Losses
$108,000
180,000
72,000

Claim
Count
530
1,200
271

Current
Relativity
1.000
0.900
0.800

The full credibility standard is 1,082 claims.
(a) (2 points) What are the territory relativities using the loss ratio approach?
(b) (2 points) What are the first iteration territory relativities using the pure premium approach?

Questions from the 1996 exam
Question 32. (4 points) You are given:
Current
Incurred
Class
Class
Losses
Relativity
1
500,000
1.000
2
400,000
1.100
3
360,000
0.900
Total
1,260,000
Current Territory Relativity:

A
2,000
1,500
2,000
5,500
1.000

Historical Earned Exposure
Territory
B
Total
3,000
5,000
1,500
3,000
2,000
4,000
6,500
12,000
0.600

Using the pure premium method described by Finger, chapter 5, "Risk Classification,"
Foundations of Casualty Actuarial Science:
(a) (2 points) Determine the first iteration classification relativities.
(b) (1 point) Determine the first iteration territory base exposures.
(c) (1 point) Explain your selection of exposures for weighting classification relativities in (a) above.

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Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 1997 exam
43. (3 points) You are given:
Territory

Prior Year
Base
Rates

Prior Year
Earned
Premium

Current
Year Base
Rates

Current Year
Earned
Premium

A
100
250,000
110
300,000
B
60
400,000
55
350,000
C
120
200,000
100
250,000
D
150
100,000
160
150,000
• Full credibility is 1,082 claims
• Territory A is the base territory
• Incurred losses and claim counts are developed and trended
• No weighting is used to combine the two years of data

Combined
Years
Earned
Premium @
Current
Rates
575,000
716,667
416,667
256,667

Combined
Years
Incurred
Losses

Combined
Years
Claim
Counts

330,000
525,000
290,000
135,000

435
800
390
275

Based on Finger, "Risk Classification," chapter 5 of Foundations of Casualty Actuarial Science, calculate the
indicated territorial relativities using the loss ratio approach.

Questions from the 1999 exam
13.

Based on Finger, "Risk Classification" chapter 5 of Foundations of Casualty Actuarial Science, use
the loss ratio approach for setting classification relativities and the data below to determine the
adjustment to class B's relativity after balancing to no overall rate change.

Class
A
B
Total
A. < -10%

Earned
Premium
$100
$200
$300

Incurred
Loss
$60
$90
$150

B. > -10% but < -8%

Credibility
0.50
1.00

C. > -8% but < -6%

D. > -6% but < -4%

E. > -4%

Exam 5, V1b

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Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2000 exam
21.

Using the loss ratio approach described by Finger in "Risk Classification," chapter 5 of Foundations of
Casualty Actuarial Science, and the following data, calculate the indicated balanced adjustment to
territory 3's relativity.
Territory
Earned Premium
Incurred Losses
Credibility
1
$1,200,000
$600,000
1.00
2
800,000
500,000
0.80
3
500,000
300,000
0.60

A. < 1.010

B. > 1.010 but < 1.030 C. > 1.030 but < 1.050

D. > 1.050 but < 1.070 E. > 1.070

Questions from the 2005 exam
49. (3 points) Using a loss ratio approach, calculate the territorial relativities indicated by the following
information. Show all work.
• Territory A is the base class.
• 2005 earned premium is an accurate estimate of next year's writings.
• Incurred losses are for the experience period 2003-2004 and are fully trended and developed.
• The full credibility standard is 1,082 claims. Partial credibility is determined using the square
root rule.

Territory
A
B

Current
Relativity
1.00
0.40

Earned Premium
2003
2004
2005
$500,000
$100,000

$600,000
$200,000

Base Rates
2003 2004 2005
$50
$40

$55
$40

$55
$60

Incurred
Losses
$500,000
$300,000

Claim
Count
1,500
300

Questions from the 2008 exam
30. (3.0 points) You are given the following information:
Incurred Loss
Claim
Current
Territory
Premium
& ALAE
Count
Relativity
1
$520,000
$420,000
600
0.60
2
$1,680,000
$1,250,000
1,320
1.00
3
$450,000
$360,000
390
0.52
$2,650,000
$2,030,000
2,310
• Full credibility standard is 1,082 claims and partial credibility is calculated using the square root rule.
• The complement of credibility is no change.
Calculate indicated territorial relativities using this most recent experience. Assume that Territory 2 remains
the base territory.

Exam 5, V1b

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Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2009 exam:
37. (3 points) Given the following information:

Territory
1
2
3

Historical
Earned
Exposures
4,000
16,000
3,750

Average
Current Relativity
Territorial for Other
Relativity Factors*
0.60
1.30
1.00
1.05
0.52
1.20

Reported
Losses
$ 420,000
$1,250,000
$ 360,000

Reported
Claim Count
600
1,320
390

*Weighted-average rate relativity for all factors except territory.
• Territory 2 will remain the base territory.
• Full credibility standard is 1,082 claims.
• Complement of credibility is no change.
Calculate the indicated territorial relativities.

Questions from the 2010 exam:
29. (3 points) A private passenger auto insurance company uses only two rating variables: territory and marital
status.
The distribution of exposures is:
Marital
Status
Married
Single

1
123
74

Territory
2
79
123

3
87
33

The rating factors for each variable are:
Marital
Status

Current
Relativity

Territory
Territory

Current
Relativity

Married
Single

1.00
1.15

1
2

0.60
1.00

0.90

Losses/LAE for each category during the experience period are:
Territory

Marital
Status

Loss &
LAE

Married

$7,760

Single

$5,789

Married

$8,307

Single

$16,038

Married

$8,233

Single

$3,873

• No fixed expense adjustment is necessary.
• All policies have the same underwriting expense and target profit.
a. (2.5 points) Using the adjusted pure premium approach and maintaining the same base classes,
develop the indicated relativity for policyholders who are single.
b. (0.5 point) Explain why the adjusted pure premium approach is preferable to the pure premium method.

Exam 5, V1b

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Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2011 exam:
11. (2.75 points) Given the following information for State X:
• Only two insurance companies write automobile policies
• Total expected costs (including expenses) per policy are the same for 2010 and 2011
• All policies are annual policies effective January 1
• 10% of class 1 risks shop for new insurance every year
• 20% of class 2 risks shop for new insurance every year
• All insureds who shop always select the carrier with the lowest rate
2010 Policy Year
Total
Class
Insureds
1
10,000
2
10,000
Total
20,000

#
Insureds
5,000
5,000
10,000

Company A
Expected
2010
Costs
Rates
100
150
200
150
150
150

2011
Rates
100
200

Company B
#
Expected
2010
Insureds
Costs
Rates
5,000
100
150
5,000
200
150
10,000
150
150

2011
Rates
150
150

Company A will introduce a new rating variable effective January 1, 2011, that segments the market into
two 2 classes.
The 2011 rate levels will be consistent with the expected costs associated with each class of business.
Company B will not be changing rates on January 1, 2011. Company B uses one rate level for all insureds.
a. (1.5 points) Calculate the total profit for Company A and Company B for Policy Year 2011.
b. (0.5 point) Company A's goals were to improve profitability and increase market share. Briefly explain whether
the goals were achieved.
c. (0.25 point) Provide one recommendation to Company A to help achieve its goals of improved profitability and
increased market share.
d. (0.5 point) Describe the impact of Company A's action on Company B.

Exam 5, V1b

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Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2011 exam:
15. (3 points) Given the following information:
Developed Incurred Loss and
Earned
ALAE Total for Accident Years
Territory
Exposures
2009 and 2010
A
20,000
$500,000
B
5,000
$125,000
C
15,000
$250,000
Total
40,000
$875,000

Current
Relativity
1.00
0.95
1.25

• The effective date for the proposed rate change is January 1, 2012 and rates will be in effect for one year.
• Average date of loss is January 1, 2010.
• All policies are annual.
• Full credibility standard 11,050 exposures
On a statewide basis, annual pure premium trends have been holding steady at 0%.
However, due to fraudulent claim behavior, pure premiums are expected to trend at different rates
throughout the state as follows:

Territory
A
B
C
Total

Annual Pure
Premium Trend
-5%
0%
10%
0%

This fraudulent behavior is expected to continue into the foreseeable future.
a. (2.75 points) Assuming Territory A is the base territory, calculate the credibility-weighted indicated relativities
to the base territory.
b. (0.25 point) Briefly describe a reason multivariate classification techniques are preferred over univariate
classification techniques when performing territorial relativity analyses.

Exam 5, V1b

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Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2012 exam:
13. (1.75 points) Given the following information:
As of January 1, 2011
Base Rate
$200
Good Driver Discount Factor
0.85
Territory 1 Factor
1.00
Territory 2 Factor
1.10

Exposures
Territory 1
Territory 2

Loss and ALAE
Territory 1
Territory 2



As of July 1, 2011
$250
0.75
1.00
1.10

Good Driver Discount
Yes
No
750
250
600
150
Good Driver Discount
Yes
No
$90,000
$40,000
$80,000
$20,000

The rating algorithm is base rate x good driver discount factor x territory factor.
Territory 1 and No Good Driver Discount remain the base classification.

Use the loss ratio method to calculate indicated territorial relativities.

Exam 5, V1b

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Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
The predecessor papers to the current syllabus reading “Basic Ratemaking” by Werner, G. and
Modlin, C. were numerous. While past CAS questions were drawn from prior syllabus readings,
the ones shown below remain relevant to the content covered in this chapter.
By relevant, we mean concepts tested on past CAS exams relate to similar to the concepts
found in this chapter.

Section 1: Criteria Used In Traditional Risk Classification
Solutions to questions from the 1991 exam
Question 3.
1. T.
2. T.
3. F. This is one of the social criteria.

Answer D.

Solutions to questions from the 1993 exam
Question 31. The three statistical criteria are: Credibility, Homogeneity, and Statistical
Significance.

Solution to questions from the 1997 exam
Question 31.
a Credibility: A rating group should be large enough so that costs can be measured with sufficient accuracy.
Homogeneity: If all are charged the same rate, then all members should have the same expected costs.
Statistical Significance: The rating variable should be a statistically significant risk differentiator, meaning:
 Expected cost estimates should vary for the different levels of the rating variable
 Estimated differences should be within an acceptable level of statistical confidence
 Estimated differences should be relatively stable from one year to the next.
b. Question no longer applicable to the content covered in this chapter.

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Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solution to questions from the 1998 exam
Question 43.
1. Privacy. People in general are reluctant to provide any information than what is normally justifiable for
securing insurance. Although some insureds may choose to pay more in order to avoid disclosing personal
information, others might secure insurance from carriers that do not require this information for rating
purposes. Therefore, introducing this rating element into the plan does not satisfy one of the social criteria
that should be a part of any sound rating plan.
2. Affordability. High rates, and higher rates for lower income groups cause affordability problems. If there was
a tendency for lower income households to have a greater than average number of children, then the
proposal would not satisfy this social criterion.
3. Causality. Causality implies that an intuitive relationship exits between the rating variable and the cost of
insurance. The proposal satisfies this criteria, since the greater the number of children in a household, the
more likely it is that liability losses may ensue from careless or reckless behavior. However, additional
studies should be conducted to determine whether this is truly a causal relationship and not a highly
correlated one.
4. Controllability. When insureds have some control over a rating variable, they can implement accident
prevention measures. Therefore, the proposal fails this criterion since the insured realistically cannot
control this exposure.

Solutions to questions from the 2004 exam:
29. a. (1 point) Other than an overall rate level increase, describe an action the insurance company could
undertake to restore overall rate adequacy. Assume that the indicated rate need by driver classification
does not change when the proposed action is taken.
The insurer should try to retain its lower cost insureds within a classification by adjusting its
underwriting practices. In this case, it should try to retain more insureds in driver classifications
A and B.
b. (1 point) Suppose that ABC Insurance Company's chief competitor in State X has the same
underwriting rules and writes a similar distribution of business as ABC Insurance Company. The
competitor is rate adequate by driver classification as well as on a statewide basis. Describe the
situation that could result if ABC Insurance Company fails to reflect the indicated changes by driver
classification.
If ABC fails to reflect indicated changes by driver classification, ABC will receive a disproportionate
number of higher cost insureds, relative to its classification plan. ABC will be adversely selected
against. “If the adverse selection continues, ABC must either lose money, change its underwriting
criteria, or increase its premiums. Premium increases may induce ABC’s lower-cost insureds to move
to another insurer, creating more adverse selection and producing a need for further premium
increases.”
c. (1 point) Suppose regulation was enacted abolishing the use of the driver classification rating variable
for State X. Briefly describe the impact on ABC Insurance Company's profitability.
If drivers were equally distributed among A, B, C and D, then there would be no impact. However, the
state total indicated rate change is positive (+5) which implies that there are more C and D drivers who
need an increased rate for ABC to be profitable. Thus, if the driver classification rating variable was
abolished, ABC would be less profitable.
d. (1 point) Briefly describe the social consequences of the abolishment of the driver classification rating
variable. “Abolition will create subsidies. Insurers may voluntarily insure underpriced groups.
Otherwise, residual markets will expand; since most residual markets are subsidized by the voluntary
market, subsidies will be created.”

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Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2004 exam:
40. (2 points) State four criteria for rating variables and explain whether or not they are fulfilled by information
from credit reports.
1. Privacy: not fulfilled, since people are reluctant to have personal information disclosed to others, and
consider credit report data a very private issue.
2. Causality: not fulfilled, since a bad credit report has no causal connection to an individual’s propensity
to have more claims, or more severe claims.
3. Controllability: is fulfilled. Since insureds have control over managing their finances and paying off
their debts, the use of credit reports as a rating variable allows insureds to reduce their
premiums through fiscal responsibility.
4. Availability: fulfilled, since companies have access to and can run credit reports easily to determine an
insured’s fiscal responsibility.

Solutions to questions from the 2005 exam
45. (2 points) Finger, in "Risk Classification," discusses the effect of market forces on the refinement of insurance
classification plans.
a. (1 point) Describe how the behavior of policyholders creates pressure on insurers to refine classification plans.
Policyholders shop around for the most affordable coverage. Therefore, insurers who can identify lower
cost risks can make greater profits by offering discounts to lower cost insureds. This process is known as
“skimming the cream”.
Conversely, insurers who don’t recognize high-cost characteristics will be adversely selected against.
In either case, this puts pressure on insurers to refine their classification plans.
b. (1 point) Explain why classification plans may also become more refined as insurance coverage becomes more
expensive. Discuss the perspective of both the insurer and the policyholder.
Insurer:
• has more “expense” dollars on more expensive coverages with which to refine the classification system.
• has incentive to keep large premium accounts that are profitable.
Insured:
• has more incentive to shop around as coverage becomes more expensive since he/she is paying the
premium. Thus, the more insureds shop, the more incentive an insurer has to refine its class plan.

Solutions to questions from the 2006 exam:
8. Which of the following changes might cause an insurer to develop a more refined classification plan?
1. The market becomes more competitive. True. A competitive market tends to produce more refined
classifications and accurate premiums.
2. Coverage becomes more expensive. True. Classification systems may also become more refined
as coverage becomes more expensive. From the buyer’s side, shopping for favorable prices is
encouraged when coverage is more expensive. From the insurer’s side, more expense dollars may
be available to classify and underwrite; in addition, the cost of making mistakes, or of not having as
refined a system, is higher when premiums are higher.
3. The market becomes larger. True. Classification systems usually are more refined for larger markets.
A. 1 only

B. 2 only

C. 1 and 3 only

D. 2 and 3 only

E. 1, 2, and 3

Answer: E.

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Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2006 exam:
38. (3 points) Werner and Modlin, "Basic Ratemaking" discuss various criteria for selecting rating variables.
As the actuary for an insurance company, you are developing an auto class plan in which one of the
proposed rating variables is estimated miles driven during the coverage period.
a. (1.5 points) Identify and briefly describe two statistical criteria, and explain whether mileage
defined this way satisfies these criteria.
b. (1.5 points) Identify and briefly describe two operational criteria, and explain whether mileage
defined this way satisfies these criteria.
CAS Model Solutions
Part a.
1 – Homogeneity (relates to similar insureds being grouped together) – If you group insured by miles driven,
you are in fact putting similar exposures to loss together, so their average loss cost should be similar.
2 – Credibility (having enough data to estimate future costs) – If you segment miles driven into large enough
discrete ranges, you should have enough data to accurately estimate future loss costs.
Part b.
1 - Verifiable/Available (the rating variable is easily available for rating purposes) – “Estimated” miles would
need to be audited at end of year and therefore not easily available/verifiable.
2 – Cost Effective (the increase in accuracy should be balanced by the cost of getting data) – Since audits
would be required, this variable may not be cost effective.
- OR 3 – Objective (should have little ambiguity, mutually exclusive and exhaustive classes) – Classes which are
mutually exclusive and exhaustive should be easy to derive, and mileage is an objective measure, so
mileage is objective.

Solutions to questions from the 2008 exam:
Question 28.
a. 1. Verifiable - color would be easy to verify
2. Objective Definition - color would also satisfy this criteria
b. 1. Privacy - color would satisfy this criteria since color is not a very private issue
2. Controllability -the insured can choose the color of their car, so it is controllable

Solutions to questions from the 2009 exam:
Question 33
If an insurer notices a positive characteristic that is not used in their rating structures (or competitors), the
insurer can market to those with the positive characteristic and try to write more of them (skimming the
cream). The insurer will then benefit from lower loss ratios and better profitability.

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Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2009 exam:
Question: 34
a. Cost effective‐ the cost of obtaining the information should not exceed the benefit of additional
accuracy.
Privacy – insured may rather pay more to avoid disclosing certain information
b. For cost effectiveness, detailed psychological profile may cost a lot to obtain. This is most likely not
cost effective.
For privacy, many people will not want to take the psychological test for the profile or may not wish to
disclose their profile to insurance company.
Alternate Solution:
a. 1. Social criteria: privacy, affordability, causality and controllability
2. Operational: Low administrative expense, objective definition, verification intuitively related,
underlying losses
b. 1. Social: privacy not met, insured may not want to disclose that information and it’s not something
that’s easily controllable, although it may be good from causality standpoint.
2. Operational: increased administrative expense, but it is objectively defined, verifiable, and likely
intuitively related.

Exam 5, V1b

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Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Section 2: Traditional classification analysis using PP and LR analyses.
Solutions to questions from the 1991 exam
Question 41.

Territory
A
B
C
Total

Loss
Ratio

LR
relativity

(1)

2= 1/1 tot

.700
.600
.800
.675

1.037
0.888
1.185
1.000

Cred

Credibility wtd
LR relativity

Premium
Extension

Balanced
LR relativity

Existing
Relativity

Territory
Relativity

(3)

4 = (2-1.0)*3 + 1.0

5 = EP * 4

6 = 4/4tot

(7)

8=6*7/6base

.85
.50
.40

1.0315
.944
1.074
1.004
4017K/4000K

2,063,000
1,417,000
537,000
4,017,000

1.027
.940
1.070

1
.90
1.20

1.00
.824
1.25

Solution to questions from the 1994 exam
Question 42. Note: The values shown above are identical to those asked in question 38, on the 1992 exam.
a. Territory relativities using the loss ratio approach.

Terr

Loss Ratio
IL / EP
relativity

A
B
C
Total

Cred

Credibility wtd
LR relativity

Premium
Extension

Balanced
LR relativity

Existing
Relativity

Territory
Relativity

(1)

2= 1/1 tot

(3)

4 = (2-1.0)*3 + 1.0

5=EP*4

6=4/4tot

(7)

8=6*7/6base

.54
.60
.72
.60

.90
1.0
1.2

.70
1.0
.50

.93
1.0
1.1
.9933
596K/600K

186,000
300,000
110,000
596,000

.937
1.007
1.108

1.000
.900
.800

1.000
.967
.945

Note: Credibility = Min ( claim count / 1082 , 1.0)
b. Territory relativities using the pure premium approach.

Terr
A
B
C
Total

Pure Premium
IL/B.Exp
relativity

Cred

Credibility wtd
PP relativity

Premium
Extension

Balanced
PP relativity

Existing
Relativity

Territory
Relativity

(1)

2= 1/1 tot

(3)

4 = (2-1.0)*3 + 1.0

5=EP*4

6=4/4tot

(7)

8=6*7/6base

108
120
144
120

.90
1.0
1.2

.70
1.0
.50

.93
1.0
1.1
.9933
596K/600K

186,000
300,000
110,000
596,000

.937
1.007
1.108

1.000
.900
.800

1.000
.967
.945

Note: 1. Credibility = Min ( claim count / 1082 , 1.0)
2. The suggested solution accompanying the 1994 CAS exam does not follow the procedure in the 1995
errata to this syllabus reading.

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Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solution to questions from the 1996 exam
(a) (2 points) Determine the first iteration classification relativities.
Question 32: An approach to calculating class relativities using the pure premium method:
New class relativity = Current class relativity * Indicated adjustment.
3

The indicated adjustment, for class (i) = Class

pure premium/  Class

pure premium .

i =1

The class (i) pure premium is computed using "base exposures"
Base exposures in this example are earned exposures adjusted for current territorial relativities.

Class
1
2
3
Total

Current Class
& Territory A
Relativity
1.000
1.100
0.900

Incurred
Losses
500,000
400,000
360,000
1,260,000

Current Class
& Territory B
Relativity
.600
.660 = .6*1.10
.540 = .6*.900

Total
Base
Exposures
3,800
2,640
2,880
9,320

Historical Earned Exposures
A
B
2,000
3,000
1,500
1,500
2,000
2,000
5,500
6,500

Pure
Premium
131.58
151.52
125.00
135.19

Indicated Adj.
(Pure premium
relativity
0.973
1.121
0.925

Base Exposures
A
B
2,000
1,800
1,650
990
1,800
1,080
5,450
3,870

Current
Class
Relativity
1.000
1.100
0.900

First Iteration
Class
Relativity
1.000
1.267 = 1.121/.973*1.10

.855

(b) Using the first iteration class relativities, compute the first iteration territory base exposures.

Class
1
2
3
Total

Indicated Class
& Territory A
Relativity
1.000
1.267
0.855

Terr B
Relativity

Indicated Class
& Territory B
Relativity
.600
.760 = 1.267*.600

.513
.600

Historical Earned Exposures
A
B
2,000
3,000
1,500
1,500
2,000
2,000
5,500
6,500

Base Exposures
A
B
2,000
1,800
1,900
1,140
1,710
1,026
5,610
3,966

(c) "the reason for using base exposures instead of actual exposures is to correct for varying exposure levels
in the non-reviewed relativities. For example, Territory A and B may differ in the distribution of insureds by
class".

Exam 5, V1b

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 2014 by All 10, Inc.

Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solution to questions from the 1997 exam
Question 43: Based on Finger, "Risk Classification," chapter 5 of Foundations of Casualty Actuarial Science,
calculate the indicated territorial relativities using the loss ratio approach.
1. Replace unclear column headings with more meaningful ones.
Column 12 in exhibit II is labeled “Preliminary adjustment”. Its counterpart in the exhibit below is labeled
“Combined years loss ratio relativity”.
2. Compute only those values necessary to calculate the territorial relativities.
Combined years Experience

Territory

Loss Ratio Relativity
(1)
(2)
(1)

A
B
C
D
Total

0.574
0.733
0.696
0.526
0.651

Credibility
(3)

2= 1/(1 tot)

0.881
1.125
1.068
0.807

Credibility wtd Current EP Balanced Current
LR relativity
* (4)
Crd LR rel Relativity
(4)
(5)
(6)
(7)

Territory
Relativity
(8)

4 = (2-1.0)*3 + 1.0

5=EP*4

8=6*7/6base

0.925
1.108
1.041
0.903
1.010

277,500
387,800
260,250
135,450
1,061,000

0.634
0.860
0.600
0.504

6=4/4tot

0.915
1.097
1.030
0.894

1.000
0.500
0.909
1.455

1.000
0.599
1.023
1.420

Note: Column (3) Credibility = Min ( claim count / 1082 , 1.0) .
Column (4) total, 1.010 = Column (5) total  Current year earned premium total (1,050,000), which is given.
Column (7) relativities are based on the Current year base rates in each territory relative to the base
territory (a).

Solutions to questions from the 1999 exam
Question 13.
1. Replace unclear column headings with more meaningful ones.
Column 12 in exhibit 2 is labeled “Preliminary adjustment”. Its counterpart in the exhibit below is labeled
“Combined years loss ratio relativity”.
2. Compute only those values necessary to calculate the territorial relativities.

Class

Combined years
Loss Ratio
IL / EP
relativity

Credibility wtd

Premium

Balanced

Credibility

LR relativity

Extension

LR relativity

(1)

2= 1/1 tot

(3)

4 = (2-1.0)*3 + 1.0

5=EP*4

6=4/4tot - 1

A
B

.60
.45

1.20
.90

.50
1.00

1.10
.90

110
180

-.069

Total

.50

.966

290

290/300

Note: Column (3) credibility is given
Column (4) total, .966 = Column (5) total  Current year earned premium total, which is given.
Thus, the adjustment to class B's relativity after balancing to no overall rate change is -.069.
Answer C.

Exam 5, V1b

Page 37

 2014 by All 10, Inc.

Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2000 exam
Question 21.

Terr
1
2
3
Total

Earned

Incurred

Loss

Premium

Losses

1.2M
800K
500K
2.5M

600K
500K
300K
1.4M

Ratio

Loss
ratio
relativity

Credibility wtd

Premium

Balanced

Cred

LR relativity

Extension

Adjustment

(1)

2= 1/1 tot

(3)

4 = (2-1.0)*3 + 1.0

5 = EP * 4

6 = 4/4tot

.500
.625
.600
.560

.893
1.116
1.071

1.00
.80
.60

.893
1.093
1.043
.987
2.468M/2.5M

1.072M
874.4K
521.5K
2.468M

1.0567

Answer D.

Exam 5, V1b

Page 38

 2014 by All 10, Inc.

Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2005 exam
49. (3 points)
Using a loss ratio approach, calculate the territorial relativities indicated by the given information.
Step 1: Compute on-level earned premium for 2003 and 2004. Create a table similar to the one below to
compute on-level earned premium to be used in Step 2 below.

Territory
A
B
Total

2003
2004
2003
Premium Base Rate
Premium
(1)
(2)
(3)
500,000
600,000
50
200,000
40
100,000
600,000
800,000
(5) = [(1)/(3)+(2)/(4)]*2005 base rates

2004
Base Rate
(4)
55
40

Onlevel
2003-2004
Earned
Premium
(5)
1,150,000
450,000
1,600,000

Trend&Dev
2003-2004
2005
Incurred
Premium
Losses
(6)
$600,000
500,000
$200,000
300,000
800,000
800,000

Claim
Count
1,500
300
1,800

Step 2: Compute the indicated territorial relativities ((8) below) by creating a table similar to the one
below and performing the notated computations.
Territory relativities using the Loss Ratio Approach.
Experience (2003-2004)
Credibility wtd
Relativity
Credibility LR relativity
Territory Loss Ratio
(1)
(2)
(3)
(4)
A
0.435
0.870
1.000
0.870
1.333
0.527
1.176
B
0.667
Total
0.500
0.946
Notes

Exam 5, V1b

Curr EP
* (4)
(5)
521,739
235,104
756,843

Balanced
Crd LR rel
(6)
0.919
1.243

Current
Relativity
(7)
1.000
0.400

Territory
Relativity
(8)
1.000
0.541

See page 321
(2)= 1/1 tot. (3) = Sqrt[Claim Count / 1082] Full Cred = 1.0 if CC > 1,082
(4) = [(2)-1.0]*3 + 1.0. (4) Total = 756,843/800,000
(6) = (4) / (4,Total)
(8) = [(7)*(6)] / (6,A)

Page 39

 2014 by All 10, Inc.

Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2008 exam:
Model Solution 1 - Question 30

1
2
3

Preliminary

Premium

Incurred Loss
& ALAE

Loss Ratio

(1)

(2)

(3)=(2)/(1)

$520,000
$1,680,000
$450,000
$2,650,000

$420,000
$1,250,000
$360,000
$2,030,000

0.8077
0.7440
0.8000
0.7660

Territory

Adjustment
4=(3)/(3)total

Current

Credibile
Credibility

Adjustment
Relativities
6=[(4)-1]*(5)+1
(7)
1.040
0.600
0.971
1.000
1.027
0.520

(5)
0.745
1.000
0.600

1.054
0.971
1.044

Indicated
Relativities
(8)=(6)/(6)2*(7)
0.643
1.000
0.550

(1), (2) and (7) are given
(4) 1.054=.8077/.7666
Min ( claim count / 1082 , 1.0)
(8) = [(6)/.971] * (7), since territory 2 remains the base territory.

Column (5) Credibility =

Model Solution 2 - Question 30
Initial comments.
In this model solution, premiums are adjusted to the territory 2 level, as shown in (2) below, prior to computing
loss ratios in (4) below. By doing so, this allows us to compute indicate relativities to territory 2, since the latter will
remain as the base territory. Indicated relativities are generally credibility weighted with existing relativities hence
the need to compute (6) and (7).

Prem

Prem at
Ter 2 Level

(1)
520,000
1,680,000
450,000

Territory
1
2
3

Credibility
Weighted
Credibility Relativities

Loss & ALAE

Loss
Ratio

Indicated
Relativities

(2)

(3)

(4)=(3)/(2)

(5)=(4)/(4)2

(6)

(7)

866,667
1,680,000
865,385

420,000
1,250,000
360,000

0.4846
0.7440
0.4160

0.6513
1.0000
0.5591

0.745
1.000
0.600

0.638
1.000
0.543

(1) and (3) are given
(2) = (1)*[Territory 2 Current Relativity/Territory Relativity]
(6) Credibility = Min ( claim count / 1082 , 1.0)
(7) = (5)(6) + [1.0-(6)](CurRel)

Solutions to questions from the 2009 exam:
Question: 37

Terr.
1
2
3
Total

Exam 5, V1b

(1)
(Historical
x all
relativities)
Base
Exposures
3,120
16,800
2,340
22,260

(2)

(3)=
(2)/(1)

(4)=
(3)/91.1

(5)

(6)=
(5)x((4)-1) +1

(7)
=( 6 )/.816xCur.
Rel.

Rep.
Losses
420,000
1,250,000
360,000
2,030,000

Base
Premium
134.615
74.405
153.846
91.19

Prelim.
Adjustment
1.4761
0.8159
1.6870

Credibility

Cred.
Adjustment
1.355
0.816
1.412

New
Relativity
0.996
1.000
0.900

Page 40

0.74467
1
0.6004

 2014 by All 10, Inc.

Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2010 exam:
Question 29 – Model Solution - Part a.
The Adjusted PP approach can be performed using exposures adjusted by the exposure-weighted average
relativity of all other variables (see (2) below).
The calculation of the current exposure-weighted average Marital Status relativities by territory is shown below:
Exposure Weighted Marital Status Relativity
Married: [123 (.60) + 79(1.0) + 87(.90)]/[123+79+ 87] = 231.1/289 = .7997
Single: [74(. 60) + 123 (1.0) + 33 (.90)]/[74+123+33] = 197.1/230 = .8570
Adjusted Pure Premium Method
Marital
Status

Exposures

Married
Single

(1)
289
230

Married
Single

Adjusted
PP Rel
(6)=(5)/(5 tot)
0.9005
1.1167

Exposure
Adjustment
(2)
0.7997
0.857

Adjusted
Exposures
(3)=(1)*(2)
231.11
197.11
428.22

Loss and
ALAE
(4)
24,300
25,700
50,000

Adjusted
Pure Prem
(5)=(4)/(3)
105.143
130.384
116.762

Ind Rel
To Base
(7)=(6)/(6 married)
1.2401

(1) and (4) are given

Question 29 – Model Solution - Part b
The pure premium method gets distorted since it assumes uniform distribution of exposures across all other
variables, thus ignoring the correlation between variables.
The adjusted pure premium method minimizes the impact of any distributional bias.

Solutions to questions from the 2011 exam:
a. (1.5 points) Calculate the total profit for Company A and Company B for Policy Year 2011.
b. (0.5 point) Co. A's goals were to improve profitability and increase market share. Did it achieve its goals?
c. (0.25 point) Provide one recommendation to Company A to help achieve its goals.
d. (0.5 point) Describe the impact of Company A's action on Company B.
Question 11 – Model Solution
[Co. A class 1 rate = 100; Co. B class 1 rate = 150]; [Co. A class 2 rate = 200; Co. B class 2 rate = 150]
Profitability = Sum[# of policies * (2011 rate – expected costs)]
• 10% of class 1 risks (from Co. B) shop for new insurance (due to a lower rate) = 10% * 5,000 = 500
• 20% of class 2 risks (from Co. A) shop for new insurance (due to a lower rate) = 20% * 5,000 = 1000
a. Class 1: 10% switch from B to A (500 new policies to A); Class 2: 20% switch from A to B (1000 policies)
A: 5500(=5000+500) * (100-100) + 4000(=5000-1000) * (200-200) = 0
B: 4500(=5000-500) * (150-100) + 6000(=5000+1000) * (150-200) = 225,000 – 300,000= -75,000
b. No. profit will always be zero as long as rates are equal to costs. Market share decreased. They lost more
customers than they gained.
c. It should increase rates on Class 1, but not to 150 or more. It will attract business AND be profitable!
d. Company B will lose its Class1 customers, who are over-priced in that company. Company A will continue to
send Class 2 customers to Company B, who ruin B’s profit margin. Company A can “skim the cream” while B
is adversely selected against.

Exam 5, V1b

Page 41

 2014 by All 10, Inc.

Chapter
C
9 – Traditiional Risk Classiffication
BASIC RATTEMAKING – WERNER, G
G. AND MOD
DLIN, C.
Solution
ns to questio
ons from th
he 2011 exa
am:
• The
e effective datte for the prop
posed rate ch
hange is Janu
uary 1, 2012 a
and rates will be in effect fo
or one year.
• Ave
erage date of loss is Janua
ary 1, 2010.
• All policies
p
are annual.
• Fulll credibility sta
andard 11,050 exposures
15a. (2.75
5 points) Assu
uming Territorry A is the base territory, ccalculate the ccredibility-weiighted indicatted
relativities to the base territory.
15b. (0.25
5 point) Briefly
y describe a reason
r
multiv
variate classifiication techniques are prefferred over un
nivariate
classifiication techniques when pe
erforming terrritorial relativiity analyses.
Question
n 15 – Model Solution 1 – part a.

Terr
A
B
C

(1
1)
PP = L&A
ALAE/EE
25
25
16.6
667
21.8
875

(5)
Curr
rel

(6)
EE

1.00
0.95
1.25
1.0875
5

20k
5k
15k

(2)
Annual trend
d
0.953
13
1.13

(3) = (1) x (2)
Trended
d PP
21.4343
375
25
22.18
83
21.87
75

(7
7) = (5)/(5) To
ot
Curr
Cred((z) =
Rel
Min( EE/1
11050 ,1.0)
0.919540
0.873563
1.103448

1
0.672671794
1

(4) = (3)/(3) tot
In
nd. Chg.
0
0.979857
1.142857
1.014095

(8) = (4)*z +(1-z)*(7)
Cred weightted ind. chg

0.979
9857
1.0547097
1.014
4095

(9)=(8)/(8
8a)
Cred
weighted iind
Chg to ba
ase
1
1.076391
1.03494
4

(5) total is exposure weighted; Trrend from the avg. date of loss in the exxperience perriod to avg. da
ate of
loss in the
t exposure period (1 yea
ar after the efffective date o
of the rates, since 1 year policies are isssued)

Question
n 15 – Model Solution 1 – part b.
Because territorial
t
relativities are ge
enerally highly
y dependent o
of other varia
ables in the model. Thus, itt is better to
use a mulltivariate class
sification tech
hnique because it considerr the exposurre correlationss between variables.

Exam 5, V1b

Page 42

 2014 by Alll 10, Inc.

Chapter
C
9 – Traditiional Risk Classiffication
BASIC RATTEMAKING – WERNER, G
G. AND MOD
DLIN, C.
Solution
ns to questio
ons from th
he 2011 exa
am (cont’d):
Question
n 15 – Model Solution 2 – part a.
Note: the difference be
etween modell solution 1 an
nd model solu
ution 2 lies in how the trended pure prem
mium for all
territories are calculate
ed. In this solution it is calc
culated as Su
um [losses * p
pp trend]/Sum
m[exposures]=
=22.161
Terr
A
B
C

Pure Prem
25
25
16.67

Pure Prem
m Trend
0.953
1
1.13

Tre
ended Pure P
Prem
21.43
25
22.18
22.161

Ind R
Rel
0.967
72
1.128
81
1.001
10

Trend period
p
is show
wn below

Terr
A
B
C

Credibility
1.00
0.673
1.00

Curr Rel
1.00
0.95
1.25
1.0875

Adj. Curr Rel
R
0.9195
0.8739
1.1494
1.00

Cred w
weight Rel.
0
0.9672
1
1.0448
1.001

2nd Rel @ Base
1.00
1.08
1.03

Question
n 15 – Model Solution 2 – part b.
Territories
s are generallly heavily corrrelated with other
o
variabless. Multivariate
e techniques take into acccount
the effects
s of other varriables, where
eas univariate
e techniques d
do not.

Exam 5, V1b

Page 43

 2014 by Alll 10, Inc.

Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2012 exam:
13. Use the loss ratio method to calculate indicated territorial relativities.
Question 13 – Model Solution 1 (Exam 5A Question 13)
First, calculate current premium for both territories.
→Territory 1 = 250(1)(.75)(750) [prem for good drivers]+ 250(1)(1.00)(250) [prem for remaining drivers]
= $203,125
→Territory 2 = 250 (1.1)(.75)(600) + 250 (1.1)(1.00)(150)= $165,000

Question 13 – Model Solution 2 (Exam 5A Question 13)

Terr
1
2

Curr Var Prem
750 x 250 x 0.75 + 250 x 250 = 203,125
600 x 250 x 0.75 x 1.1 + 150 x 250 x 1.1 = 165,000
Indic Rd to Base

Terr
1
2

OLEP
203,125
165,000
368,125

L+ALAE
90k + 40k = 130
80k + 20k = 100
230k

LR
0.640
0.606

Indic Rd to
1 (base) 0.60606 x 1.1 = 1.04167
1.0417
0.64

Examiner’s Comments
Candidates in general performed well on this question. Most frequently candidates failed to use current rate
level premium, which in this question is calculated via the extension of exposures method.
Candidates also frequently calculated only the indicated change factors to the current relativities, as opposed
to calculating the final indicated relativity.
A subset of candidates misinterpreted the class plan and used the loss ratio method to solve for 4 different
relativities concurrently (each combination of territory/good driver), as opposed to solving for the requested
indicated territorial relativities.
A small group of candidates solved for indicated territory relativities by using a pure premium approach as
opposed to the requested loss ratio approach. Some candidates made adjustments to the exposure bases to
reflect the class plan relativities.

Exam 5, V1b

Page 44

 2014 by All 10, Inc.

Chapter 10 – Multivariate Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Sec
1
2
3
4
5
6
7
8
9
10
11
12
1

Description
Shortcomings Of Univariate Methods
Minimum Bias Procedures
The Adoption of Multivariate Methods
The Benefits Of Multivariate Methods
GLM’S
Sample GLM Output
A Sample Of GLM Diagnostics
Practical Considerations
Data Mining Techniques
Augmenting Multivariate Analysis With External Data
Key Concepts
Appendix F – A Multivariate Classification Example

Pages
170 - 171
171 - 174
174 -174
174 -175
176 - 177
177 - 179
179 - 182
183 - 183
183 - 185
185 - 185
187 - 187

Shortcomings Of Univariate Methods

170 - 171

Class ratemaking:
 produces more equitable individual risk pricing by analyzing loss experience of groups of similar risks.
 protects the insurer against adverse selection.
 may provide insurers with a competitive advantage and help expand the types of risks the insurer is
willing and able to write profitably.
Univariate class ratemaking approaches (pure premium or loss ratio) use loss experience of the levels within
each rating variable to establish rate differentials to the base level.
The major shortcoming of univariate approaches:
Its failure to accurately account for the effect of other rating variables.
 The PP approach does not consider exposure correlations with other rating variables.
If a rating algorithm contained several rating variables, this shortcoming could be mitigated using a
two-way analysis or by making some manual adjustments.
To illustrate the distortion created when using univariate methods, consider the following:
Assume a one-way PP analysis for a personal auto book of business shows that older cars have
high claims experience relative to newer cars.
However, in reality it can be shown that this analysis is distorted by the fact that older cars tend to
be driven by younger drivers (who have higher claims experience).
Therefore, although the experience for both young drivers and old cars looks unfavorable, it does so
primarily because of the youthful driver effect.
 The LR approach uses current premium to adjust for an uneven mix of business to the extent the
premium varies with risk, but premium is only an approximation since it deviates from true loss cost
differentials.
The adjusted pure premium approach multiples exposures by the exposure-weighted average of all other
rating variables’ relativities to standardize data for the uneven mix of business before calculating the oneway relativities. But, this is an approximation to reflect all exposure correlations.

Exam 5, V1b

Page 45

 2014 by All 10, Inc.

Chapter 10 – Multivariate Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
2

Minimum Bias Procedures

171 - 174

Minimum bias procedures are iterative univariate approaches. Each procedure involves the:
 selection of a rating structure (e.g. additive, multiplicative or combined) and
selection of a bias function (e.g. balance principle, least squares,  , and maximum likelihood bias
functions).
The bias function compares the procedure’s observed loss statistics (e.g. loss costs) to indicated loss
statistics and measures the mismatch.
Both sides of this equation are weighted by the exposures in each cell to adjust for an uneven mix of
business.
“Minimum bias” refers to the balance principle that requires that the sum of the indicated weighted pure
premiums to equal the sum of the weighted observed loss costs for every level of every rating variable (a.k.a.
“minimizing the bias” along the dimensions of the class system).
2



The balance principle applied to a multiplicative personal auto rating structure is shown below.
 There are only two rating variables: gender and territory.
 Gender has values male (with a rate relativity g1) and female (g2).
 Territory has values urban (t1) and rural (t2).
 The base levels relative to multiplicative indications are female and rural (hence g2 = 1.00 and t2 = 1.00).
 The base rate is $100.
The actual loss costs (pure premiums) are as follows:
Urban
Rural
Total
Male
$650
$300
$528
Female
$250
$240
$244
Total
$497
$267
$400
The exposure distribution is as follows:
Urban
Rural
Male
170
90
Female
105
110
Total
275
200

Total
260
215
475

Step 1: Write four equations with observed weighted loss costs on the left and indicated weighted loss costs
(the base rate, the exposure, and the indicated relativities) on the right.
Males
170 x $650 +90 x $300 = ($100 x 170 xg1 x t1 ) +( $100 x 90 x g1 x t2 )
Females
105 x $250 + 110 x $240 = $100 x 105 x g2 x t1 + $100 x 110 x g2 x t2
Urban
170 x $650+ 105 x $250 = $100 x 170 x g1 x t1 + $100 x 105 x g2 x t1
Rural
90 x $300+ 110 x $240 = $100 x 90 x g1 x xt2 + $100 x 110 x g2 x t2
Step 2: Choose initial (or seed) relativities for the levels of one of the rating variables.
A sensible seed is the univariate PP relativities.
The urban relativity is the total urban loss costs divided by the total rural loss costs:
t1 = 1.86 = ($497.27/$267.00)
t2 = 1.00.

Exam 5, V1b

Page 46

 2014 by All 10, Inc.

Chapter 10 – Multivariate Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Step 3: Substituting these seed values into the first two equations, solve for the first values of g1 and g2:
170 x $650 + 90 x $300 = ($100 x 170 x g1 x 1.86) + ($100 x 90 x g1 x 1.00)
$137,500 = ($31,620 x g1) + ($9,000 x g1)
$137,500 = $40,620 x g1
g1 = 3.39.
105 x $250 + 110 x $240 = ($100 x 105 x g2 x 1.86) + ($100 x 110 x g2 x 1.00)
$52,650 = ($19,530 x g2) + ($11,000 x g2)
$52,650 = $30,530 x g2
g2 = 1.72.
Step 4: Using these seed values for gender, g1 and g2, set up equations to solve for the new intermediate
values of t1 and t2:
170 x $650 + 105 x $250 = ($100 x 170 x 3.39 x t1) + ($100 x 105 x 1.72 x t1)
$136,750 = ($57,630 x t1) + (18,060 x t1)
$136,750 = $75,690 x t1
t1 = 1.81.
90 x $300 + 110 x $240 = ($100 x 90 x 3.39 x t2) + ($100 x 110 x 1.72 x t2)
$53,400 = ($30,510 x t2) + ($18,920 x t2)
$53,400 =$49,430 x t2
t2 = 1.08.
This procedure is repeated (each time discarding the previous relativities and solving for new ones)
until there is no material change in the values of g1, g2, t1, and t2.
Step 5: Upon convergence, normalize the base class relativities to 1.00.
Assuming the relativities derived above represent the final iteration, then normalizing the base class
relativities to 1.00 would result in:
g1 = 3.39 / 1.72 = 1.97
g2 = 1.72 /1.72 = 1.00
t1 = 1.81 /1.08 = 1.68
t2 = 1.08 /1.08 = 1.00.
The initial univariate relativity for t1 was 1.86, but after one iteration, the replacement value for t1 is 1.68,
(reflecting the fact that the cell for urban males has considerably more exposure than the other cells, and
thus the experience in that cell is given more weight).
Step 6: Adjust the base loss cost (to a normalized basis):
Since the base levels are female and rural (g2 and t2), and since the base loss cost = $100, then the
Adjusted base loss cost = $100 x 1.72 x 1.08 = $185.76.

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Chapter 10 – Multivariate Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
The example above only considers one minimum bias method (multiplicative structure with balance principle)
using the pure premium statistic. In addition, it considers only two rating variables each with two levels.
The computation required to incorporate several rating variables requires at least spreadsheet programming.
Sequential analysis:
 is related to minimum bias analysis.
 is mandated as the only class ratemaking method for pricing private passenger auto insurance CA.
 uses an adjusted one-way PP approach on the first variable to determine the indicated relativities.
exposures are adjusted using the adjusted one-way PP approach and indicated relativities are
calculated for the second variable; this continues until indicated relativities for every variable have been
calculated.
 involves making only one pass through the sequence of chosen rating variables (rather than iterating
until convergence is achieved).
The main criticism of the non-iterative sequential approach: since it does not have a closed form solution; the
results vary depending on the order of the rating variables in the sequence.

The Adoption of Multivariate Methods

174 -174

Minimum bias procedures are a subset of generalized linear models (GLMs).
Iterating the minimum bias procedure a sufficient number of times may result in convergence with GLM
results (however GLMs are more computationally efficient).
Reasons for the adoption of GLMs for class ratemaking in the late 20th century/early 21st century:
1. Computing power increased.
2. New data warehousing improved the granularity and accessibility of data for ratemaking purposes (enhanced
computing power and better data enabled its use in class ratemaking).
3. Competitive pressure called for adoption of multivariate methods (putting the rest of the industry in a position
of adverse selection and decreased profitability).

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BASIC RATEMAKING – WERNER, G. AND MODLIN, C.

The Benefits Of Multivariate Methods

174 -175

1. The main benefit: consideration of all rating variables simultaneously and automatically adjust for
exposure correlations between rating variables
2. The methods attempt to remove unsystematic effects in the data (a.k.a. noise) and capture only the
systematic effects (a.k.a. signal) as much as possible.
This is not the case with univariate methods (which include both signal and noise in the results).
3. The methods produce model diagnostics (additional information about the certainty of results and the
appropriateness of the model fitted).
4. They allow interaction between two or more rating variables.
Interactions occur when the effect of one variable varies according to the levels of another (e.g. the effect of
square footage varies across different levels of AOI).
Clarifying interaction with exposure correlation:
 Interaction (a.k.a. response correlation); Exposure correlation (describes a relationship between the
exposures of one rating variable and another).
 Examples:
i. Gender exposures may be uniformly distributed across age (i.e. at any age there is an identical
distribution of men and women and no exposure correlation exists), but the two variables may interact
if the loss experience for men relative to women is distinctly different at the youthful ages than at the
middle and senior ages.
ii. A variable’s exposures may be unevenly distributed across the levels of another rating variable (i.e.
exposure correlation exists), yet no interaction is present.
5. Benefits vary among different types of multivariate methods.
GLMs are transparent; the model output includes parameter estimates for each level of each explanatory
variable in the model, as well as a range of statistical diagnostics.
In contrast, neural networks are criticized for a lack of transparency.
How the methods mentioned before stack up to this list of benefits/disadvantages:
Univariate methods:
 are distorted by distributional biases.
 heavily distorted by unsystemic effects (noise).
 require no assumptions about the nature of the underlying experience.
 produce a set of answers with no additional information about the certainty of the results.
 can incorporate interactions but only by expanding the analysis into two-way or three-way tables.
 scores high in terms of transparency (but is plagued by the inaccuracies of the method).
Minimum bias methods:
 account for an uneven mix of business but iterative calculations are computationally inefficient.
 require no assumptions about the structure of the model and the bias function.
 do not produce diagnostics
 scores high on transparency and outperforms univariate analysis in terms of accuracy (but does not
provide all of the benefits of full multivariate methods).

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Chapter 10 – Multivariate Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
GLMs are the standard for class ratemaking.
The iterations of a GLM can be tracked, and the output is a series of multipliers that can be used in rating
algorithms and rating manuals.
A Mathematical Foundation for GLMs: Linear Models
A good way to understand GLMs is to first review linear models (LMs).
 Both LMs and GLMs express the relationship between an observed response variable (Y) and a number
of explanatory variables (a.k.a. predictor variables). Example:
 The response variable may be claim frequency for homeowners insurance, and the predictor variables
may include AOI, age of home, and deductible.
Observations in the data (e.g. claims on individual exposures) are realizations of the response variable.
Linear models:
 express the response variable (Y) as the sum of its mean (µ) and a random variable (  ) (a.k.a. error
term): Y    


assume that the mean can be written as a linear combination of the predictor variables. Example:

Y  ( 1 X 1   2 X 2   3 X 3   4 X 4 )   where X 1 , X 2 , X 3 , and X 4 are each predictor variables,
and

1 ,  2 , 3 , and  4 are the parameter estimates to be derived by the LM.



2.



assume that the random variable,



attempt to find the parameter estimates, which, when applied to the chosen model form, produce the
observed data with the highest probability.
This is achieved using the likelihood function (or the log-likelihood), as maximum likelihood relies on
linear algebra to solve a system of equations.
Due to the high volume of observations in class ratemaking datasets, numerical techniques such as
multi-dimensional Newton-Raphson algorithms are used. These techniques find the maximum of a
function by finding a zero in the function’s first derivative.
The likelihood function is equivalent to minimizing the sum of squared error between actual and indicated.

, is normally distributed with a mean of 0 and constant variance,

Generalized Linear Models: Loosening the Restrictions
GLMs:
 are LMs that remove the restrictions of the normality assumption and a constant variance.
 use a link function to define the relationship between the expected response variable (e.g. claim severity)
and the linear combination of the predictor variables (e.g. age of home, amount of insurance, etc.).
Choice of link functions means predictor variables do not have to relate strictly in an additive fashion (as
they do with LMs). Example: GLMs fit to claims experience for ratemaking often specify a log link
function which assumes the rating variables relate multiplicatively to one another.

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Chapter 10 – Multivariate Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
To solve a GLM, the modeler must:
• have a dataset with a sufficient number of observations of the response variable and associated predictor
variables.
• select a link function defining the relationship between the systematic and random components.
• specify the distribution of the underlying random process (e.g. a member of the exponential family such
as normal, Poisson, gamma, binomial, inverse Gaussian); this is done by specifying the mean and the
variance of the distribution, the latter being a function of the mean.
The maximum likelihood approach:
 maximizes the logarithm of the likelihood function and
 computes the predicted values for each variable.

Sample GLM Output

177 - 179

GLMs are often performed on loss cost data (usually frequency and severity separately).
Statistical and practical reasons for doing so include:
 Modeling loss ratios requires premiums at a current granular rate level (which is difficult to obtain).
 An a priori expectation of frequency and severity patterns (e.g. youthful drivers have higher
frequencies) are needed.
 LRMs are obsolete when rates and rating structures are changed.
 There is no commonly accepted distribution for modeling loss ratios.
Graphing GLM output is useful to strengthen an understanding of GLMs.
 The rating variable (vehicle symbol) has 17 discrete levels and each level’s exposure count is shown as
yellow bars (on the right y-axis).
 Each symbol groups vehicles having common characteristics (e.g. weight, number of cylinders,
horsepower, and cost).
 Discrete variables (a.k.a. categorical factors), and continuous variables (a.k.a. variates) can be
incorporated into GLMs. Variates can take the form of polynomials or splines (a series of polynomial
functions with each function defined over a short interval) within GLMs.

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Chapter 10 – Multivariate Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Effect of Vehicle Symbol on Automobile Collision Frequency

The output is from a multiplicative model.
The base level (to which all other levels’ parameter estimates are expressed relatively) is vehicle symbol 4.
Its multiplicative differential is 1.00, and is chosen as one with the largest volume of exposure (so that
statistical diagnostics are relative to a large and stable base).
Notice that the GLM indicates that vehicle symbol 10 has a 25% higher indicated collision frequency than
vehicle symbol 4, all other variables being considered.
The pink line with square markers represents the results of a univariate analysis.
The disparity b/t the GLM and univariate lines suggest vehicle symbol is strongly correlated with another
variable in the model (e.g. age of driver, prior accident experience, etc).
It is important to understand the phrase “all other variables being considered.”
GLM results of one variable are only meaningful if the results for all other variables are considered at the
same time (a.k.a. “all other variables being constant” or “all other variables at the base level.”)
Chapter 13 discusses how the insurer’s final rate relativities often deviate from the actuary’s indicated
relativities for business reasons.

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Chapter 10 – Multivariate Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
7

A Sample Of GLM Diagnostics

179 - 182

Statistical significance is an important criterion for evaluating rating variables, and statistical diagnostics are a
major byproduct of GLMs. Statistical diagnostics:
 aid the modeler in understanding the certainty of the results and the appropriateness of the model.
 can determine if a predictive variable has a systematic effect on losses (and be retained in the model).
 assess the modeler’s assumptions around the link function and error term.
A common statistical diagnostic for deciding whether a variable has a systematic effect on losses is the
standard errors calculation.
 “standard errors are an indicator of the speed with which the log-likelihood falls from the maximum
given a change in parameter.”
 2 standard errors from the parameter estimates are akin to a 95% confidence interval.
i. the GLM parameter estimate is a point estimate
ii. standard errors show the range in which the modeler can be 95% confident the true answer lies within.
The following graph is identical to the graph shown previously but now includes standard error lines for
the non-base levels (i.e., +/- two standard errors from the differentials indicated by the GLM).
Standard Errors for Effect of Vehicle Symbol on Automobile Collision Frequency

Results:
 The upward pattern and narrow standard errors suggest this variable is statistically significant.
 Wide standard errors may suggest the factor is detecting mostly noise and be eliminated from the model.
Symbol 17 shows wide standard errors, but that is a function of the small volume present in that level
(and thus does not invalidate the strong results for symbols 1- 16, where most of the business lies).
Deviance measures (an additional diagnostic) assess the statistical significance of a predictor variable.
 Deviance measures of how much fitted values differ from the observations.
 Deviance tests are used when comparing nested models (one is a subset of the other) to assess
whether the additional variable(s) in the broader model are worth including.
i. The deviance of each model is scaled so that the results can be compared.
ii. Chi-Square or F-test gauge the theoretical trade-off between the gain in accuracy by adding the
variables versus the loss of parsimony in adding more parameter estimates to be solved.

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Chapter 10 – Multivariate Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
A practical diagnostic in modeling is to compare GLM results for individual years to gauge consistency of results
from one year to the next.
Consistency over time of vehicle symbol on auto collision frequency separately for the two years

The two lines show some random differences but in general the patterns are the same.
Model validation techniques compare the expected outcome with historical results on a hold-out sample of data
(i.e. data not used in developing the model so that it could be used to test the effectiveness of the model).
The following output is a validation of a frequency model.
 The bands of expected frequencies from the GLM (from lowest to highest) track closely to the actual
weighted frequency of each band in the hold-out sample of data (for most of the sample)
 The volatile results for the high expected frequency bands are a result of low volume of data.

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Chapter 10 – Multivariate Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Model Validation

Over-fitting and Under-fitting Models:
 If the modeler retains variables that reflect a non-systematic effect on the response variable (i.e.
noise) or over-specifies the model with high order polynomials, the result is over-fitting.
The model will replicate historical data very well (including the noise) but will not predict future
outcomes reliably (the future experience will not have the same noise).
 If the model is missing important statistical effects (containing few explanatory variables and fits to
the overall mean), the result is under-fitting.
This model will hardly help the modeler explain what is driving the result.
See Appendix F includes for additional examples and more details.

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Chapter 10 – Multivariate Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
When using GLMs, the actuary should focus on:
• ensuring data is adequate for the level of detail of the class ratemaking analysis (avoiding the GIGO
principle: Garbage In, Garbage Out)
• identifying when anomalous results call for additional exploratory analysis
• reviewing model results as it relates to both statistical theory and business application
• developing methods to communicate model results in light of an insurer’s ratemaking objectives
(e.g. policyholder dislocation, competitive position)
More work can be done.
 Retrieving of data requires careful consideration of needed volume of data; definition of homogeneous
claim types; method of organization (e.g. PY vs. CAY); treatment of midterm policy changes, large
losses, U/W changes during the experience period, and the effect of inflation and loss development.
 Balance stability and responsiveness as it relates to experience period as well as to geographies to be
included in the analysis (e.g. countrywide versus individual state analysis).
 Commercial considerations (e.g. IT constraints, marketing objectives, and regulatory requirements) have
to be carefully incorporated into the statistical analysis before any results are implemented in practice.

Data Mining Techniques

183 - 185

Data mining techniques are used to enhance classification analysis in the following five ways:
1. Factor Analysis
Factor analysis is a technique to reduce the number of parameter estimates in a class analysis (e.g. a GLM).
This can be a reduction in the number of variables or a reduction in the levels within a variable.
Example:
 Summarize the exposure correlation between two variables in a scatter plot,
 Fit a regression line that summarizes the linear relationship between the two variables.
 A variable can then be defined that approximates this regression line.
 This combined variable replaces the original variables and thus reduces the parameter estimates of the
model.
This technique can be used to compress a long list of highly correlated variables into a score variable that
represents linear combinations of the original variables.
Examples:
 The vehicle symbols discussed earlier may have been derived as a linear combination of correlated
variables (e.g. vehicle weight, vehicle height, number of cylinders, horsepower, cost when new, etc.).
 Combining geo-demographic variables which describe average characteristics of an area (e.g.
population density, average proportion of home-ownership, average age of home, median number of
rooms in the home, etc.)

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Chapter 10 – Multivariate Classification
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2. Cluster Analysis
 combines small groups of similar risks into larger homogeneous categories or “clusters.”
 minimizes differences within a category and maximizes differences between categories.
 is used in rating for geography, with actuaries starting with small geographic units (e.g. zip code)
 applies different algorithms to group these units into clusters based on historical experience, modeled
experience, etc.
3. CART (Classification and Regression Trees)
CART is used to develop tree-building algorithms to determine a set of if-then logical conditions that help
improve classification.
In personal auto, a tree may start with an if-then condition around gender.
 If the risk is male, the tree then continues to another if-then condition around age.
 If the risk is male and youthful, the tree may continue to an if-then condition involving prior accident
experience.
Examining the tree may help actuaries identify the strongest list of initial variables and determine how to
categorize each variable.
CART can also help detect interactions between variables.
4. MARS (Multivariate Adaptive Regression Spline)
MARS algorithm:
 operates as a piecewise linear regression where breakpoints define a region for a particular linear
regression equation.
 is used to select breakpoints for categorizing continuous variables. Example:
In HO insurance, AOI may be treated as a categorical factor despite being continuous in nature, and
can help select the breakpoints used to categorize the AOI factor before using it in a GLM.
 can help detect interactions between variables.
5. Neural Networks
Neural networks are sophisticated modeling techniques but are criticized for their lack of transparency.
Test data is gathered and training algorithms are invoked to automatically learn the structure of the data
(a.k.a. a recursion applied to a GLM).
The results of a neural network can be fed into a GLM (or vice versa), which helps highlight areas of
improvement in the GLM (e.g. a missing interaction).
The data mining techniques listed above can enhance a ratemaking exercise by:
• whittling down a long list of explanatory variables to a more manageable list for use within a GLM;
• providing guidance in how to categorize discrete variables;
• reducing the dimension of multi-level discrete variables (i.e. condensing 100 levels, many of which have
few or no claims, into 20 homogenous levels);
• identifying candidates for interaction variables within GLMs by detecting patterns of interdependency
between variables.

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Chapter 10 – Multivariate Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
10

Augmenting Multivariate Analysis With External Data

185 - 185

Insurers using GLMs seek to augment data that has already been collected and analyzed about their own
policies with external data. This includes but is not limited to information about:
• geo-demographics (e.g. population density of an area, average length of home ownership of an area);
• weather (e.g. average rainfall or number of days below freezing of a given area);
• property characteristics (e.g. square footage of a home or business, quality of the responding fire
department);
• information about insured individuals or business (e.g. credit information, occupation).

Key Concepts

187 - 187

1. Shortcomings of univariate approach
2. Minimum bias techniques
3. Circumstances that led to the adoption of multivariate techniques
a. Computing power
b. Data warehouse initiatives
c. Early adopters attaining competitive advantage
4. Overall benefits of multivariate methods
a. Adjust for exposure correlations
b. Allow for nature of random process
c. Provide diagnostics
d. Allow interaction variables
e. Considered transparent
5. Mathematical foundation of generalized linear models (GLMs)
6. Sample GLM output
7. Statistical diagnostics, practical tests, and validation techniques
a. Standard errors
b. Deviance tests
c. Consistency with time
d. Comparison of model results and historical results on hold-out sample
8. Practical considerations
9. Data mining techniques
a. Factor analysis
b. Cluster analysis
c. CART
d. MARS
e. Neural networks
10. Incorporation of external data in multivariate classification analysis

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Chapter 10 – Multivariate Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
12

Appendix F – A Multivariate Classification Example

The appendix includes output from a GLM analysis. It includes:
 several tests used to evaluate the predictive power of a potential rating variable
 hold-out sample testing used to evaluate the overall effectiveness of a particular model.
EXAMPLE - PREDICTIVE VARIABLE (a multiplicative GLM fit to homeowners water damage frequency data)
 The graphical output isolates the effect of the prior claim history variable as a significant predictor of
water damage frequency, however
 The model contains other explanatory variables that must be considered in conjunction with the prior
claims history effect.
Parameters and Standard Errors
The graph shows indicated frequency relativities for prior claims history (all other variables considered).
 The x-axis represents the levels of the variable (0, 1, or 2 claims), with the level for zero claims being the
base level, and all other levels expressed relative to it.
 The bars relate to the right y-axis, which show the number of policies in each level. The line with the circle
marker shows the indicated relativities, and the lines with the triangle markers represent two standard
errors on either side of the indicated relativities.
Main Effect Test for Prior Claim History

Conclusions:
 The upward sloping indicated relativity line with relatively tight standard errors suggests that the expected
frequency is higher for risks with prior claims.
 Risks with 1 or 2 prior claims have a frequency about 35% and 65% higher than risks with no prior claims.

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Chapter 10 – Multivariate Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Consistency Test
The prior graph shows the indicated relativities for the whole dataset.
 The following shows the pattern of relativities for each of the individual years included in the analysis.
 The lines represent the indicated frequency relativities for prior claims history, separately for each year.
Consistency Test for Prior Claim History

Each year’s indicated line slopes upward with roughly the same shape suggests that the pattern is consistent
over time, and provides the actuary with a test supporting the stability of this variable’s predictive power.
Statistical Test
The actuary can test the predictive power of a variable using deviance diagnostics
 Using the Chi-Square test, the actuary fits models with and without the variable being studied and analyzes
the trade-off between the increased accuracy of the model with the variable versus the additional complexity
in having additional parameters to estimate.
 The null hypothesis is that the two models are approximately the same.
 Calculate a Chi-Square percentage based on the results of the two models (a percentage of less than 5%
suggests the actuary should reject the null hypothesis that the models are the same and should use the
model with the greater number of parameters).
Here, the Chi-Square percentage is 0%; the actuary rejects the null hypothesis and selects the model with the
greater number of parameters (e.g. select the model with prior claims history variable in it).

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Chapter 10 – Multivariate Classification
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Judgment
 Evaluate the reasonableness of the model and diagnostic results based on knowledge of the claims
experience being modeled.
 In this case, the statistical results are consistent with what is intuitively expected (i.e. that frequency is
higher given the presence of prior claims).
Decision
All four tests suggest the rating variable is predictive, should be included in the model, and ultimately the
rating algorithm.
EXAMPLE UNPREDICTIVE VARIABLE (from a multiplicative GLM fit to HO wind damage frequency data).
 The output isolates the effect of fire safety devices as an insignificant predictor of wind damage
frequency, though
 The model contains other explanatory variables that must be considered in conjunction with this variable.
Parameters and Standard Errors
The graph shows indicated frequency relativities for the fire safety device variable (all other variables considered).
 The x-axis represents the different levels of fire safety devices (the base being the level “none”)
 The bars are the number of policies in each level.
 The lines represent the indicated wind damage frequency relativities and two standard errors on either side
of the indicated relativities.
Main Effect Test for Fire Safety Device



The indicated line is flat (i.e. indicated relativities are close to 1.00) for the levels that have a significant
number of policies. The sprinkler system has very wide standard errors around the indicated relativity, which
is due to the small number of policies in that category.
There is little variable predictive power, and should be removed from the wind damage frequency model.

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Chapter 10 – Multivariate Classification
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Consistency Test
 The pattern for each of the individual years included in the analysis is shown below.
 The categories on the x-axis represent different fire safety devices, the bars are the number of policies in
each level, and the lines represent the indicated relativities for each year.
Consistency Test for Fire Safety Device Claim

These results confirm the conclusions derived from the parameter results and standard errors.
 The patterns are consistent across the years for all categories but the sprinkler system.
 That sprinkler has little data, and the predictions are very volatile.
Statistical Test
The Chi-Square percentage for this variable is 74%.
 Percentages above 30% indicate that the null hypothesis that the models are the same should be accepted.
 If the models are “the same,” the actuary should select the simpler model that does not include the additional
variable (%s between 5% and 30% are often thought to be inconclusive based on this test alone).
Judgment
The existence of smoke detectors, sprinklers, and fire alarms does not seem to have any statistical effect on the
frequency of wind damage losses (and consistent with intuition)
Decision
All four tests suggest the rating variable is not predictive (exclude it from the wind damage frequency model).

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Chapter 10 – Multivariate Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
OVERALL MODEL VALIDATION
The most common test to analyze the overall effectiveness of a given model is one which compares predictions
made by the model to actual results on a hold-out dataset (i.e. data not used to develop the model).
This test requires that insurers set aside a portion of the data for testing (although this may not be possible for
smaller insurers).
Validation Test Segmented by Variable
The following shows observed and predicted frequencies for various levels of AOI.
 If the model is predictive, the frequencies should be close for any level with enough volume to produce
stable results.
 The insurance process is random and will create small differences between the lines; however, either large or
systematic differences or both should be investigated as possible indicators of an ineffective model.
Example: A model may contain too much noise from retaining statistically insignificant variables or not having
enough explanatory power because statistically significant variables are omitted.
Actual Results v Modeled Results for AOI

The amount of insurance is a variable for which there is a natural order to view for the different levels.
 The modeled results for the first four levels appear to be higher than the actual results (i.e. the model may be
over-predicting the frequency for homes with low AOI)
 Similar-sized discrepancies can be seen for medium AOI (actual results appear higher than the modeled
results) and for high AOI (actual results appear lower than modeled results but with considerable volatility).

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Validation Test Segmented by Fitted Value
1. Use the frequency and severity models to determine a modeled pure premium for each observation in a
hold-out dataset.
2. Order each observation according to the modeled pure premium result from lowest to highest expected value.
3. Group the observations into 10 groups and compare actual and modeled results for each group on the chart.
** If the model is predictive, the actual result will be close to the modeled result for each group.
Special attention should be paid to the lowest and highest groups (where results are likely to deviate as
models are generally less able to predict extreme observations).
Actual Results v Modeled Results

Conclusions:
 Actual results are very close to the modeled results for the first seven groups.
 There appears to be a lot of difference between actual and modeled results for the last few groups
(because the low volume in those groups suggests the results may be distorted by noise and
therefore less valid).

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Chapter 10 – Mu
ultivariate
e Classifiication
BASIC RATEMAKING
A
– WERNER, G. AND MO
ODLIN, C.
Question
ns from the
e 2011 exam
m
13. (1 point) A compan
ny applied gen
neralized linear modeling tto its homeow
wners data.
A graph of indicate
ed relativities and their stan
ndard errors ffor a fire safetty device ratin
ng variable iss shown
below
w.
Evaluate the effecttiveness of the variable in the
t model.

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Chapter 10 – Multivariate Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
The predecessor papers to the syllabus reading “Basic Ratemaking” by Werner, G. and Modlin,
C. were numerous, but none covered the topics that are presented in this chapter. Thus, there
are no past CAS questions that are relevant to the content covered in this chapter.
Solutions to questions from the 2010 exam
36. (1 point) Company XYZ applied generalized linear modeling to its personal auto data. Graphs of the
actual and modeled pure premiums by the driver groupings were produced by the analysis.
The first graph is a plot of the values using the modeling dataset.
The second graph is a plot of the values using a hold-out dataset. The modeling dataset and the holdout dataset have the same number of exposures.
Explain whether or not the model appears to be appropriate.
Question 36 – Model Solution
The model appears to be over fitted in that it’s fitting the data’s “noise” in addition to its “signal”. This is why it
fits the original data so well.
In the hold-out data, however, the model is projecting the same data fluctuations as in the original modeling
dataset (in age ranges without many exposures, where experience is likely to be volatile).

Solutions to questions from the 2011 exam
13. A company applied generalized linear modeling to its homeowners data. A graph of indicated relativities
and their standard errors for a fire safety device rating variable is shown below.
Evaluate the effectiveness of the variable in the model.
Question 13 – Model Solution
This is not a good variable. “None,” “Smoke Detector,” and “Fire Alarm” all receive the same rate relatively.
“Sprinkler system” receives a different relativity than the others, but it is a class with low volume.
The error bars are also very wide. Probably reject this rating variable.

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Sec
1
2
3
4
5
6
1

Description
Territorial Ratemaking
Increased Limits Ratemaking
Deductible Pricing
Size Of Risk For Workers Compensation
Insurance To Value (ITV)
Key Concepts

Pages
188 - 192
192 - 198
199 - 204
204 - 206
206 - 213
215 - 215

Territorial Ratemaking

188 - 192

Certain rating variables and risk characteristics call for special ratemaking procedures.
Geography is a primary driver of claims experience and is a widely used rating variable.
Insurers define territories as small geographic units (e.g. postal/zip codes, counties, census blocks) and
establish rate relativities for each territory.
Territorial ratemaking challenges.
1. Location is heavily correlated with other rating variables (e.g. high-value homes tend to be located in the
same area) making univariate analysis of location susceptible to distortions.
2. Data in each individual territory is often sparse.
Territorial ratemaking generally involves two phases:
I. Establishing territorial boundaries
II. Determining rate relativities for the territories

I. Establishing Territorial Boundaries
In the past, most companies used the same or very similar boundaries, which were developed by a third-party
(e.g. ISO or NCCI). Insurers subdivide/modify territories to gain a competitive advantage, using operational
knowledge and judgment.
Recently actuaries
 apply more advanced methods (e.g. geo-spatial techniques) to develop or refine territorial boundaries.
 use both internal and external data in their analyses.
Step 1: Determining Geographic Unit
Typical units:
 should be homogenous with respect to geographic differences while still having observations in most units.
 are postal/zip codes, census blocks, counties, etc.
i. zip codes have the advantage of being readily available but the disadvantage of changing over time.
ii. counties have the advantage of being static and readily available, but due to their large size, tend to
contain very heterogeneous risks.
iii. census blocks are static over time, but require a process to map insurance policies to the census blocks.

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Next: Estimate the geographic risk associated with each unit.
 Actual experience contains both signal and random noise. The signal is driven by non-geographic
elements (e.g. age, amount of insurance, number of employees) and geographic elements (e.g. density,
weather indices, crime rates).
 The key to accurately estimating the geographic risk is isolating the geographic signal in the data.
Components of Actual Experience

Step 2: Calculating the Geographic Estimator
Historically, actuaries used univariate techniques (e.g. pure premium approach) to develop an estimator for
each geographic unit. Two major issues with this approach.
1. The geographic estimator reflects both the signal and the noise.
Since geographic units tend to be small, the data is sparse and the resulting loss ratios or pure
premiums or both will be too volatile to distinguish the noise from the signal.
2. Since location is highly correlated with other non-geographic factors, the resulting estimator is biased.
A better approach involves using a multivariate model (e.g. a GLM) on loss cost data using a variety of nongeographic and geographic explanatory variables.
 Non-geographic variables include rating variables (e.g. age of insured, claim history) as well as other
explanatory variables not used in rating.
 Geographic variables include geo-demographic variables (e.g. population density) and geo-physical
variables (e.g. average rainfall).
Components of Actual Experience Further Refined



By including geographic and non-geographic predictors in the Signal model, the actuary controls for nongeographic effects and isolates the signal stemming from the geographic predictors.
If the actuary cannot fully explain the geographic effect via the geographic predictors, there will be some
systematic variation not captured by the geographic variables (a.k.a. geographic residual variation).
The parameters from each geographic predictor, including the geographic residual variation, can be
combined to form one composite risk index or score that represents the geographic signal for each unit.

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Step 3: Spatial Smoothing
Geographic risk close in proximity tend to be similar.
Spatial smoothing techniques improve the estimate of any unit by using information from nearby units.
Two basic types of spatial smoothing: distance-based and adjacency-based.
1. The distance-based approach:
 smoothes by weighting information from one unit with information from nearby geographic units based
on the distance from the primary unit and some measure of credibility.
 The influence of nearby areas diminishes with increasing distance.
Advantage: Easy to understand and implement.
Disadvantages:
i. The assumption that a certain distance (e.g. a mile) has the same impact on similarity of risk regardless
of whether it is an urban or rural area.
ii. The presence of a natural or artificial boundary (e.g. river or highway) between two geographic units is
not taken into consideration when determining distance.
2. Adjacency-based approach:
 weights information from one geographic unit with information estimators of rings of adjacent units (i.e.
immediately adjacent units get more weight than the units adjacent to adjacent units, etc).
 handles urban/rural differences appropriately.
 accounts for natural or artificial boundaries better than the distance-based smoothing.
 is most appropriate for perils driven heavily by socio-demographic characteristics (e.g. theft).
Balance over and under-smoothing:
 Using too much smoothing (e.g. using data from dissimilar units in another part of the state) may mask
the real spatial variation among the risks.
 Using not enough smoothing may leave noise in the estimator.
The mechanics of spatial smoothing techniques are beyond the scope of this text.
Smoothing techniques are applied in one of two ways.
1. Applied to the geographic estimators themselves (done when the geographic estimator is based on the
univariate approaches as the estimators still contain a significant amount of noise).
2. Applied within a more sophisticated framework to improve the predictive power of a multivariate analysis of
geographical effects.
Smoothing techniques are applied to geographic residuals to see if there are any patterns in the residuals
(i.e. to detect any systematic geographic patterns that are not explained by the geographical factors in the
multivariate model).
Any pattern in the residuals (i.e. all positive or negative in a certain region) indicates the existence of
geographic residual variation. Spatially smoothed residuals can be used to adjust the geographic
estimators to improve the overall predictive power of the model.

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Step 4: Clustering
Units are grouped into territories to minimize within group heterogeneity and maximize between group
heterogeneity.
Basic types of clustering routines include:
• Quantile methods: Create clusters based on equal numbers of observations (e.g. geographic units) or
equal weights (e.g. exposure).
• Similarity methods: Create clusters based on how close the estimators are to one another. Closeness
can be based on a different statistics:
i. The average linkage similarity method creates boundaries based on the overall average difference
between the estimators from one cluster to the next (tends to join clusters with smaller variances).
ii. The centroid similarity method creates boundaries based on the overall average difference in
estimators squared (tends to be more responsive to outliers).
iii. Ward’s clustering method creates boundaries that lead to the smallest within cluster sum of
squares difference (tends to produce clusters that have the same number of observations).
These types of clustering routines do not produce contiguous groupings (i.e. groupings that only include
geographic units that are adjacent to each other). If contiguous territorial boundaries are desired, then a
contiguity constraint needs to be added to the clustering routine.
Since geographic risk changes gradually, a discontinuity at self created boundaries will occur.
Thus, the actuary should select the number of clusters that minimizes noise without creating significant
discontinuities.
Many insurers have eliminated grouping units into territories and simply derive rate relativities for each
geographic unit (i.e. no different than creating a large number of small territories).
Rather than rating territories, insurers can geo-code every risk, and the latitude and longitude of the insured
item creates a unique rate relativity that changes gradually from one location to the next.

II. Calculating Territorial Relativities
Rate relativities or differentials can be accomplished using the techniques described in chapters 9 and 10.
Since location tends to be highly correlated with other variables (e.g. low or high-valued homes tend to be
concentrated in certain areas), perform this analysis using multivariate classification techniques (e.g. a new
territorial boundary could be modeled along with other explanatory variables in a GLM).

Increased Limits Ratemaking

192 - 198

Insurance providing protection against third-party liability claims are offered at different limits of insurance.
The lowest limit offered is the basic limit (BL) and higher limits are referred to as increased limits (IL).
Reasons to establish rate relativities (i.e. to use increased limits ratemaking) for various limits:
1. As personal wealth grows, individuals have more assets to protect and need more insurance coverage.
2. Inflation drives up costs and trends in costs have a greater impact on IL losses than on BL losses.
3. The propensity for lawsuits and the amount of jury awards have increased significantly (i.e. social
inflation) and this has a disproportionate impact on IL losses.

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Lines of business in which IL ratemaking is used include private passenger and commercial auto liability,
umbrella, any commercial product offering liability coverage (e.g. contractor’s liability, professional liability, etc).
Two types of policy limits offered:
1. Single limits: Refers to the total amount the insurer will pay for a single claim (e.g. if an umbrella policy has a
limit of $1,000,000, then the policy will only pay up to $1,000,000 for any one claim).
2. Compound limits: Applies two or more limits to the covered losses. Examples:
i. A split limit: includes a per claimant and a per occurrence limit (e.g. in personal auto insurance, a split
limit for bodily injury liability of $15,000/$30,000 means that if the insured causes an accident, the policy
will pay each injured party up to $15,000 with total payment to all injured parties not to exceed
$30,000).
ii. An occurrence/aggregate limit: limits the amount payable for any one occurrence and for all
occurrences incurred during the policy period (e.g. if an annual professional liability policy has a limit of
$1,000,000/$3,000,000, the policy will not pay more than $1,000,000 for any single occurrence and will
not pay more than $3,000,000 for all occurrences incurred during the policy period).
The text will focus determining indicated increased limit factors (ILFs) for a single limit (compound and split
limits are more complex).
Standard Approach to Computing LAS and ILFs
The ILF is used to modify the base rate (B, which assumes the basic limit) if the insured selects a limit of
liability (H) that is different than the basic limit: Rate at Limit H = ILF for Limit H x B.
Assuming all UW expenses are variable and variable expense and profit provisions do not vary by limit, the
_________

Indicated ILF ( H ) 

( L  EL ) H
_____________

(derived in the same way as Chapter 9).

( L  EL ) B

Actuaries may vary the profit provision by limit:
 because higher limits offer coverage for claims that are less frequent and very severe, and this variability
adds uncertainty which makes it difficult to price and risky for insurers.
 to reflect the higher cost of capital needed to support the additional risk.
Assume frequency and severity are independent: Indicated ILF ( H ) 

FrequencyH x SeverityH
FrequencyB x SeverityB

Assume frequency is the same regardless of the limit chosen: Indicated ILF ( H ) 

SeverityH
SeverityB

For some lines of business, frequency may vary by the limit chosen.
Personal auto insureds who select a very high limit tend to have lower accident frequencies than insureds
who select low limits. Selecting higher limit tends to be a sign of risk aversion and a higher degree of
overall responsibility that also applies to driving behavior.
A severity limited at H is referred to as the limited average severity at H or LAS (H).

Indicated ILF ( H ) 



LAS ( H )
LAS ( B)

LAS (H) is the severity assuming every loss is capped at limit H (regardless of actual policy limit), and
LAS (B) is the severity assuming every loss is capped at the basic limit.

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Example: Given the following 5,000 reported uncensored claims categorized by the size of the loss
(i.e. a $150,000 loss is slotted in the $100,000 100K.
This is equivalent to dividing the losses in the layer by the total claim count for those policies:

1,579 $132,876,901
=
2,981
2,981
Thus, LAS($250K) = $77,046 + $44,575 = $121,621 . ILF (250K) = 121,621/77,046 = 1.5785
$44,575  $84,153 *

Calculating LAS ($500,000) using the same techniques:
For losses in the $250K to $500K layer, only policies with a $500K limit or greater can be used:

$81,092,725 - 232 * $250,000
1,518
Thus, LAS($500K) = $77,046 + $44,575 + $15,213= $136,834
$15,213 =

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Other Considerations
Historical losses used in ILF analysis should be adjusted for any expected trend and for loss development.
Recall that loss trends have a leveraged effect on increased limits losses.
Assuming a constant positive percentage trend in total losses, the following relationship holds:
Basic Limits Trend < Total Limits Trend < Increased Limits Trend.
(See Chapter 6 for a numeric example that demonstrates this relationship).
Fitted Data Approach
Actuaries may fit curves to empirical data to smooth out the random fluctuations in the data.
Let f(x) represent a continuous distribution of losses of size x, and H be the limit being priced.
H



LAS ( H )   xf ( x)dx  H  f ( x)dx

The ILF for the limit H is represented as follows: ILF ( H ) 



0
B

H


 xf ( x)dx  H  f ( x)dx
 xf ( x)dx  B  f ( x)dx

The challenge with this approach is determining a distribution that is representative of the expected losses.
ISO Mixed Exponential Methodology
 is designed to address some of the issues with the empirical data (trend, censoring by policy limits, etc.).
 is outside the scope of this text.
Multivariate Approach
Actuaries may analyze ILFs using GLMs which can more effectively deal with sparse data.
A major difference between a GLM approach and the univariate approaches using LAS is that the GLM does
not assume the frequency is the same for all risks. Thus,
 GLM results are influenced by both the limiting of losses and the behavioral differences among
insureds at different limits.
 This may produce counter-intuitive results (e.g., expected losses decrease as limit increases)
Therefore, actuaries may use both approaches to guide the selection of increased limit factors.

Deductible Pricing

199 - 204

Two basic types of deductibles: flat dollar deductibles and percentage deductibles.
Flat dollar deductibles are the most common.
i. A flat dollar deductible (e.g. $250 deductible) specifies a dollar amount below which losses are not
covered by the policy.
ii. Flat dollar deductibles may range from small amounts (e.g. $100 or $250) on personal lines policies to
large deductibles (e.g. $100,000 or more) on large commercial policies.
Percentage deductibles state the deductible as a % of the coverage amount (e.g. a 5% deductible on a
home insured for $500,000 is equivalent to a flat dollar deductible of $25,000).
% deductibles are common property policies, and are applied specifically to perils that are susceptible to
catastrophic losses (e.g. earthquake or hurricane).

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Several reasons why deductibles are popular among both insureds and insurers:
• Premium reduction: A deductible reduces the rate as the insured pays a portion of the losses.
• Eliminates small nuisance claims: Deductibles minimize the filing of small claims (and the expense
associated with investigating and handling small claims, which is often greater than the claim amount).
• Provides incentive for loss control: Since the insured is responsible for the first layer of loss, the
insured has a financial incentive to avoid losses.
• Controls catastrophic exposure: For insurers writing a large number of policies in cat prone areas,
the use of large cat deductibles can reduce its exposure to loss.
Loss Elimination Ratio (LER) Approach
Deductible relativities can be determined using a LER approach.
Assuming all expenses are variable and that variable expenses and profit are a constant % of premium, the
indicated deductible relativity for deductible D is given by the following formula (where the base level in this
_________

example assumes no deductible): Indicated Deductible Relativity 

( L  EL ) D
_________

( L  EL ) B
The indicated deductible relativity is the ratio of ultimate losses and LAE after application of the deductible to
ground-up ultimate losses and LAE.
In the LER approach, calculate the amount of losses that are eliminated going from full coverage to a
deductible or by going from one deductible to a higher deductible:

LER ( D ) 

Losses and LAE Eliminated by Deductible ( L  EL ) B  ( L  EL ) D

Total Ground - up Losses and LAE
( L  EL ) B

The formula is re-written as follows: ( L  EL ) D  ( L  EL ) B  (1.0 - LER ( D )).
The indicated deductible relativity can be restated as:
_______

Indicated Deductible Relativity 

( L  EL ) B x(1.0 - LER( D))
_________

 (1.0 - LER( D)).

( L  EL ) B
Empirical Distribution (Discrete Case)
The LER can be calculated as follows: LER ( D )  [1 



AllLosses

Maximum[0, ( Loss Amount  D )]



Loss Amount

]

AllLosses

(assuming the ground-up loss is known for every claim)

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Consider the size of loss distribution of ground-up homeowners losses:

Size of Loss Distribution
(1)

Size of Loss
X <= $ 100
$ 100 < X <= $ 250
$ 250< X <= $ 500
$ 500 < X <= $ 1,000
$ 1,000 < X
Total

(2)
Reported
Claims
3,200
1,225
1,187
1,845
2,543
10,000

(3)
Ground-Up
Reported
Losses
$240,365
$207,588
$463,954
$1,551,938
$11,140,545
$13,604,390

To calculate LER ($250), compute the amount of losses in each layer that will be eliminated by the deductible.
 The first two rows contain losses less than $250 and are completely eliminated by the deductible.
 The remaining rows contain individual losses that are at least $250; thus $250 will be eliminated for each
of the 5,575 claims (=1,187+1,845+2,543).
The LER = losses eliminated/ total losses:

LER($250) 

($240,365  $207,588)  $250  (1,187  1,845  2,543)
 0.135
$13, 604,390

The rate credit for going from full coverage to a $250 deductible is 13.5%; the deductible relativity is 0.865.
The following table shows the calculations discussed above:
(1)

(2)

Size of Loss
X <= $ 100
$ 100 < X <= $ 250
$ 250< X <= $ 500
$ 500 < X <= $ 1,000
$ 1,000 < X
Total
(4) Losses < 250
(4) Losses>=250
(5) LER

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=
=
=

(3)

Reported
Claims
3,200
1,225
1,187
1,845
2,543
10,000

Ground-Up
Reported
Losses
$240,365
$207,588
$463,954
$1,551,938
$11,140,545
$13,604,390
(5) LER =

(4)
Losses
Eliminated By
$250
Deductible
$240,365
$207,588
$296,750
$461,250
$635,750
$1,841,703
0.135

(3)
(2) x $250
(Tot4) / (Tot3)

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Other Considerations
Insurers may not know the ground-up losses for every claim (e.g. insureds may not report claims that are less
than the deductible on their policy).
When this is the case, data from policies with deductibles greater than the deductible being priced cannot be
used to calculate the LER. For example:
 data from policies with a $500 deductible cannot be used to determine LERs for a $250 or $100
deductible, however
 data from policies with deductibles less than the deductible being priced can be used to determine
LERs (e.g. data from policies with a $500 deductible can be used to determine the LER associated
with moving from a $750 deductible to a $1,000 deductible).
Calculating the credit to change from a $250 to a $500 deductible.
LER Calculation to Move from a $250 to $500 Deductible
(1)
(2)
(3)
(4)
(5)
Net Reported Net Reported
Losses
Losses
Reported
Net Reported
Assuming
Assuming
Deductible
Claims
Losses
$500 Ded
$250 Ded
Full Cov
500
$680,220
$524,924
$588,134
$100
680
$1,268,403
$1,049,848
$1,176,269
$250
1,394
$2,940,672
$2,624,621
$2,940,672
$500
2,194
$5,249,242
$5,249,242
Unknown
Unknown
Unknown
$1,000
254
$859,755
Total
5022
$10,998,292
(7) Net Reported Losses for Ded <=$250
(8) Losses Eliminated <=$250 Ded
(9)LER

(3)= Net of the deductible
(4) =(3) Adjusted to a $500 deductible
(6)= (5) - (4) (7)= Sum of (5) for $0, $100, $250 Deductibles
(8)=Sum of (6) for $0, $100, $250 Deductibles (9)=(8)/(7)




(6)
Losses
Eliminated
Moving from
$250 to $500
$63,210
$126,421
$316,051
Unknown
Unknown
$4,705,075
$505,682
0.107

(5)=(3) Adjusted to a $250 deductible

Each row contains data for policies with different deductible amounts.
The analysis can only use policies with deductibles of $250 or less (since the goal is to determine the
losses eliminated when changing from a $250 to a $500 deductible)
Columns 4 and 5 contain the net reported losses in Column 3 restated to $500 and $250 deductible
levels, respectively.
Columns 4 and 5 are not Column 3 minus the product of Column 2 and the assumed deductible.
This is because not every reported loss exceeds the assumed deductible.
The losses in Columns 4 and 5 are based on an assumed distribution of losses by deductible and size of loss, and
cannot be recreated given the data shown.

The comments made earlier with respect to trend and development in the ILF section apply to deductible
pricing, too.

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Fitted Data Approach
Let f(x) represent a continuous distribution of losses of size x, and D be the size of the deductible.
 This formula is very similar to the formula used in the ILF section




The expected loss eliminated through the use of a deductible, D:

LER( D) 





xf ( x)dx  D  f ( x)dx
D



xf ( x)dx  D  f ( x)dx





xf ( x)dx

Practical Considerations
Like the ILF pricing, the LER approach assumes claim behavior is the same for each deductible.
This may not be the case (e.g. an insured with a $250 deductible and an insured with a $1,000 deductible
both having a $1,100 loss are both not likely to report such a loss since the insured with the $1,000 deductible
may choose not to report the claim for fear of an increase in premium from the insurer applying a claim
surcharge).
Also, lower-risk insureds tend to choose higher deductibles, since they are unlikely to have a claim and are
willing to accept the risk associated with a higher deductible.
Since the LER approach does not recognize these behavioral differences, higher deductible policies may end
up being more profitable than lower deductible policies.
The LER approach determines an average % credit applied to all policies with a certain deductible amount.
 In the prior example, the credit for a $250 deductible is 13.5%.
 But, if the total policy premium is $3,000, then the credit for moving from full coverage to a $250
deductible is $405, and since premium savings exceeds the amount of the deductible, the insured will be
better off to select the deductible.
 An insurer may handle this circumstance in different ways.
i. A cap on the amount of dollar credit from the deductible may be used (e.g. the maximum dollar credit
for moving from full coverage to a $250 deductible might be $200)
ii. Calculate different set of credits for different policies (e.g. a homeowners insurer may have different
deductible credits for low, medium, and high-valued homes)
Note: % deductibles do not have this issue since the deductible increases with the amount of insurance.

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
4

Size Of Risk For Workers Compensation

204 - 206

To account for differences in expense and loss levels for larger insureds, some WC insurers vary the expense
component for large risks, incorporate premium discounts or loss constants, or all of these.
Expense Component
Commercial lines insurers use the All Variable Approach to determine the applicable expense provisions.
The assumption is that UW expenses are a constant % of the premium charged.
Since some expenses are fixed, using the all variable approach will cause policies with small average
premium (i.e. small risks) to be undercharged and policies with large average premium (i.e. large risks) to be
overcharged.
Insurers may adjust for this in a few different ways.
1. WC insurers may calculate a variable expense provision that only applies to the first $5,000 of standard
premium (generally defined as premium before application of premium discounts and expense constants).
2. Insurers may charge an expense constant to all risks, which accounts for costs that are the same
regardless of policy size (e.g. UW and administrative expenses). Since the expense constant is a flat dollar
amount, it is a decreasing % of written premium as the size of the policy increases.
3. WC insurers apply a premium discount to policies with premium above a specified amount. The following
shows the calculation of the premium discount for a policy with standard premium of $400,000.
Workers Compensation Premium Discount Example
(1)
(2)
(3)
(4)
(5)
Premium
Premium Range
in Range
Prod
General
$0
$5,000
$5,000
15.0%
10.0%
$5,000
$100,000 $95,000
12.0%
8.0%
$100,000 $500,000 $300,000
9.0%
6.0%
$500,000
above
6.0%
4.0%
Standard Premium
$400,000

(6)

(7)

(8)

Taxes
3.0%
3.0%
3.0%
3.0%

Profit
5.0%
5.0%
5.0%
5.0%

Total
33.0%
28.0%
23.0%
18.0%

(3)= Min of [(2) - (1), Standard Premium - Sum Prior(3)]
(9)= (8Row 1)-(8)
(10)= (9)/[1.0 -(6) - (7)]

(9)
Expense
Reduction
0.0%
5.0%
10.0%
15.0%

(10)
Discount
%
0.0%
5.4%
10.9%
16.3%

(11)=

(11)
Premium
Discount
$0
$5,130
$32,700
$0
$37,830

(3) x (10)

Loss Constants
Small WC risks tend to have less favorable loss experience (as a % of premium) than large risks for several
reasons. Small companies:
 have less sophisticated safety programs because of the large amount of capital to implement and
maintain.
 may lack programs to help injured workers return to work.
 premiums are unaffected or slightly impacted by experience rating; small insureds may not be eligible
for ER and may have less incentive to prevent or control injuries than large insureds.

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
When WC insurers charge the same rate per exposure for small and large insureds, the premium will be
inadequate for small companies and excessive for large companies.
A loss constant added to the premium for small risks equalizes the final expected loss ratios between small
and large insureds.
WC Loss Constant calculation example:
(1)

(2)

Premium Range
$1
$2,500
$2,501
above

(3)

(4)

Policies
1,000
1,000

(6)= (5) / (4) (7) = Given

(5)

Reported
Premium
Loss
$1,000,000 $750,000
$5,000,000 $3,500,000

(6)
Initial
Loss
Ratio
75.0%
70.0%

(7)
Target
Loss
Ratio
70.0%
70.0%

(8)

(9)

Premium
Shortfall
$71,429
$0

Loss
Constant
$71.43
$0.00

(8)= [(5)/(7)] -(4)(9)= (8) / (3)

The unadjusted expected loss ratios for small (premium less than or equal to $2,500) and large (premium
greater than $2,500) risks are 75% and 70% (see (6))
To achieve an expected loss ratio of 70% for both types of risks, the computations in (8) and (9) are performed.
With sophisticated multivariate techniques, insurers add a rating variable to account for the size of the risk,
making the loss constant no longer necessary.

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
5

Insurance To Value (ITV)

206 - 213

Insurance to value (ITV) indicates how the level of insurance chosen relates to the overall value or replacement
cost of the insured item, and thus how rates vary based on the policy limit chosen (e.g. if an item is insured to
full value, then the AOI equals the total value or replacement cost).
Consider the following example:
• Two homes worth $250,000 and $200,000 are each insured for the full amount.
• Expected claim frequency is assumed to be 1% for both homes.
• Expected losses are uniformly distributed.
That information yields the following expected size of loss distributions and rates for each home:
Rate calculations for a $250,000 Home
(1)
(2)
Reported
Loss
Loss
Size of Loss ($000s)
Distribution ($000s)
$ - < X <= $ 25
10.0%
$13
$ 25 < X <= $50
10.0%
$38
$ 50 < X <= $75
10.0%
$63
$ 75 < X <= $100
10.0%
$88
$ 100 < X <= $125
10.0%
$113
$ 125 < X <= $150
10.0%
$138
$ 150 0).
2. The face amount of insurance is less than the coinsurance requirement (i.e. F < cV).
3. The loss is less than the coinsurance requirement (i.e. L < cV).
The amount of the penalty is as follows:

 L - I , if L  F

e =  F - I , if F  L  cV
0,
if cV  L

Example 1:
Assume a home valued at $500,000 is insured only for $300,000 despite a coinsurance requirement of 80%
(or $400,000 in this case).
Since F is $300,000 a coinsurance deficiency exists and a = 0.75 (=$300,000 / $400,000).
The indemnity payments and coinsurance penalties for a $200,000 loss are:

F
$300, 000
 $200, 000 
 $150, 000
cV
$400, 000
e  L - I  $200, 000 - $150, 000  $50, 000
IL 

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Example 2:
The indemnity payments and coinsurance penalties for a $300,000 loss:

F
$300, 000
 $300, 000 
 $225,000
cV
$400, 000
e  L - I  $300, 000 - $225,000  $75, 000
I  L

Example 3:
The following are the indemnity payments and coinsurance penalties for a $350,000 loss:

$300, 000
F
 $350, 000 
 $262,500
$400, 000
cV
e  F - I  $300, 000 - $262,500  $37,500
I L 

Example 4:
The following are the indemnity payments and coinsurance penalties for a $450,000 loss:

F
$300, 000
 $450,000 
 $337,500, but $337,500  F , so I  F  $300, 000
cV
$400, 000
e  F  I  $300,000 - $300, 000  $0.
I  L

The coinsurance penalty for loss values between $0 and $500,000 (i.e. the full value of the home):

The magnitude of the co-insurance penalty:
 the dollar coinsurance penalty increases linearly between $0 and F (where the penalty is the largest).
 the penalty decreases for loss sizes between F and cV.
 there is no penalty for losses larger than the cV, but the insured suffers a penalty in that the payment
does not cover the total loss.

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Achieving Rate Equity by Varying Rates Based on ITV Level
I. A coinsurance penalty corrects for inequity caused by similar homes insured to different ITV levels by
adjusting the indemnity payment in the event of a loss.
II. Another way to achieve equity is to calculate and use rates based on the level of ITV.
Recall that the indicated rate per $1,000 of insurance was the same for the two homes insured to full value
(i.e. $50 per $1,000 of insurance) and higher for the underinsured home (i.e. $60 per $1,000 of insurance).
If those indicated rates were used, the premium would have been equitable and no coinsurance penalty
would have been necessary.
A rate can be calculated given the expected frequency, the size of loss distribution, and the full value of the
property. Using the following notation:
f
= frequency of loss
s(L) = probability of loss of a given size
V
= maximum possible loss (which may be unlimited for some insurance)
F
= face value of policy
The rate is the expected indemnity payment/policy face value (AOI is often shown in $100 or $1,000 increments).
Given an empirical distribution of losses, the rate is as follows:
F
F

f    Ls ( L)  F  (1.0   s ( L)) 
L 1
 L1

Rate 
F

Given a continuous distribution of losses, the rate is as follows:
F
F

f    Ls ( L)dL  F  (1.0   s ( L)dL) 
0
0

Rate 
F

If partial losses are possible, the rate per AOI decreases as F gets closer to the value of the insured item.
The rate of change of the decrease varies depending on the shape of the loss distribution:
• Left-skewed distribution (i.e. small losses predominate): the rate will decrease at a decreasing rate
as F increases.
• Uniform distribution (i.e. all losses equally likely): the rate will decrease at a constant rate as
F increases.
• Right-skewed distribution (i.e. large losses predominate): the rate will decrease at an increasing rate as
F increases.
Under the rate (versus the co-insurance penalty) approach:
 the coinsurance is any portion of the loss that exceeds F should the insured choose F less than V.
 the major difficulty is determining the loss distribution.

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Insurance to Value Initiatives
The HO policy settles losses based on replacement cost, subject to the policy limit.
 One policy feature encouraging insurance to full value is guaranteed replacement cost (GRC), allowing
replacement cost to exceed F if the property is 100% insured to value and subject to annual indexation.
 Insurers are now using more sophisticated property estimation tools, with component indicator tools
considering customized features of the home (e.g. granite countertops, hardwood floors, age of
plumbing and electricity).
By increasing the AOI on underinsured homes to ITV level assumed in the rates, insurers generate additional
premium without increasing rates.
 Since homeowners loss distributions are left-skewed (i.e. small losses predominate), the increased
premium is more than the additional losses generated from this action.
 As the insureds receive increased coverage, they are more accepting of the increased premium than if
rate increases were implemented.
Also, the industry has made better use of property inspections, indexation clauses, and education of insureds.

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.

Key Concepts

215 - 215

1. Territorial ratemaking
a. Establishing territorial boundaries
i. Defining basic geographic units
ii. Creating geographic estimators
iii. Smoothing geographic estimators
iv. Combining units based on clustering techniques
b. Calculating territorial rate relativities
2. Increased limit factors
a. Limited Average Severity
i. Uncensored losses
ii. Censored losses
b. Fitted data approach
c. Other considerations
d. Multivariate approach
e. ISO mixed exponential approach
3. Deductible LER approach
a. Discrete approach
b. Fitted data approach
c. Practical considerations
4. Workers compensation size of risk
a. Expense component
b. Loss constants
5. Insurance to Value (ITV)
a. Importance of ITV
b. Coinsurance
i. Penalty
ii. Varying rates based on ITV level
c. ITV initiatives

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
The predecessor papers to the current syllabus reading “Basic Ratemaking” by Werner, G.
and Modlin, C. were numerous. While past CAS questions were drawn from prior syllabus
readings, the ones shown below remain relevant to the content covered in this chapter.

Section 1: Increased Limits Ratemaking
Questions from the 2004 exam
45. (2 points) Given the following data, calculate the annual claims inflation rate in the layer $50,000 excess
of $50,000. Assume aground-up annual claims inflation rate of 15%. Show all work.
Date of Loss
Ground-up Loss
February 1, 2003
$37,000
July 15, 2003
$47,000
October 1, 2003
$64,000
December 1, 2003
$93,000
Note: This is more of a chapter 6 question

Questions from the 2005 exam
50. (1 point) Explain two reasons why claim inflation produces larger cost trends on increased limits
coverage than on basic limits coverage.
Note: This is more of a chapter 6 question

Questions from the 2006 exam
31. (3.25 points)
a. (2 points) Given the following claim information for accident year 2005, calculate the annual
inflation rate for claims in the layer $50,000 excess of $100,000 for 2006. Assume a ground-up
annual claims inflation rate of 10%. Show all work.
Claim
Ground-up Loss
1
$75,000
2
100,000
3
125,000
4
150,000
b. (1.25 points) How would you expect the inflation rate in the layer $50,000 excess of $100,000
to differ from the inflation rate for claims limited to $100,000?
Explain two reasons for the difference between the inflation rates.
Note: This is more of a chapter 6 question

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2007 exam
46. ( 2.0 points) You are given the following information:

Claim
A
B
C
D
E

Ground-up
Uncensored
Loss Amount
$35,000
125,000
180,000
206,000
97,000

If all claims experience an annual ground-up severity trend of 8.0%, calculate the effective trend in
the layer $100,000 in excess of $100,000. Show all work.
Note: This is more of a chapter 6 question
47. (2.0 points) You are given the following information:
Ground-up
Uncensored
Claim
Loss Amount
A
$250,000
B
300,000
C
450,000
D
750,000
E
1,200,000
F
2,500,000
G
4,000,000
H
7,500,000
I
9,000,000
J
15,000,000
Basic limit is $1,000,000.
Using the methods described by Palmer in Increased Limits Ratemaking for Liability Ratemaking,
calculate the following:
a. (1.25 points) The $5,000,000 increased limit factor.
b. (0.75 point) The limited average severity in the layer $4,000,000 in excess of $1,000,000. Show all work.

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2008 exam
18. (1.25 points) You are given the following information:

Claim
A
B
C
D
E
Total

Loss
Amount
$50,000
$70,000
$90,000
$110,000
$20,000
$340,000

• Total limit trend = 10%
• Basic limit = $50,000
a. (0.5 point) Calculate the basic limit trend.
b. (0.5 point) Calculate the excess limit trend.
c. (0.25 point) Identify a situation in which the excess limit trend will be less than the basic limit trend.
Note: This is more of a chapter 6 question
34. (2.0 points)
a. (1.0 point) You are given the following distribution of losses.
Layer of Loss
Lower Limit ($)
Upper Limit ($)
Total $ Loss
$1
$10,000
$500,000
$10,001
$250,000
$16,000,000
$250,001
$500,000
$17,500,000
$500,001
$1,000,000
$11,500,000

Occurrences
100
80
50
20

Calculate the $500,000 increased limit factor assuming the basic limit is $250,000.
b. (1.0 point) Identify and briefly explain two issues that arise when using empirical data to construct
increased limit factor tables.

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2009 exam
36. (2 points) Given the following information:
• Basic Limit = $1,000,000
• ULAE Provision as % of Loss (Basic Limit) = 10.0%
• ULAE Provision as % of Loss (Increased Limit) = 20.0%
• Expected Frequency (Basic Limit) = 0.15
• Expected Frequency (Increased Limit) = 0.10
• Assume no risk load
Ground-Up
Claim
Uncensored Loss
1
$300,000
2
$600,000
3
$750,000
4
$1,250,000
5
$4,500,000
6
$10,000,000
Calculate the increased limit factor at $5,000,000, assuming there is no ALAE.

Questions from the 2010 exam
31. (3 points) Given the following information:
Censored Loss Distribution by Policy Limit
Policy Limit
$300,000

Size of Loss

$100,000

X <= $100,000
$100,000 < X <= $300,000
$300,000 < X <= $500,000
Total

$97,000,000

$46,000,000
$150,000,000

$97,000,000

$196,000,000

$500,000
$11,000,000
$107,000,000
$160,000,000
$278,000,000

Censored Claim Distribution by Policy Limit
$100,000

Policy Limit
$300,000

$500,000

1,573

753
637

1,573

1,390

168
561
407
1,136

Size of Loss
X <=
$100,000 < X <=
$300,000 < X <=
Total

$100,000
$300,000
$500,000

Calculate the increased limit factor for the $300,000 policy limit, assuming a basic limit of $100,000.

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2012 exam
12. (1.25 points) Given the following information:
Paid Losses
$50,000
$100,000
$300,000
$500,000
Total

Claim Counts by Policy Limit
$100,000 $300,000 $500,000
30
25
80
150
60
120
35
50
30
180
120
280

Calculate an indicated increased limit factor for the $300,000 policy limit, assuming a basic limit of
$100,000.

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.

Section 2: Deductible Pricing
Questions from the 2003 exam:
38. (3 points) Given the information below, calculate the loss elimination ratio for ABC Company's
collision coverage in State X at a $250 deductible. Show all work.
• ABC insures 5,000 cars at a $250 deductible with the following fully credible data on the
collision claims:
o Paid losses are $1,000,000 per year.
o The average number of claims per year is 500.
• A fully credible study found that in State X:
o The average number of car accidents per year involving collision damage was
10,000.
o The average number of vehicles was 67,000.
• Assume ABC Company's expected ground-up claims frequency is equal to that of State X.
• Assume the average size of accidents that fall below the deductible is $150.

Questions from the 2004 exam:
39. (3 points) Given the information below, calculate the premium for a policy with a $5,000 deductible.
Show all work.
Loss Distribution
Frequency
Loss Amount
0.45
$500
0.35
$2,500
0.15
$10,000
0.05
$25,000
•
•
•
•
•
•
•

First dollar premium is $500,000.
Ground-up expected loss ratio is 60%.
Allocated Loss Adjustment Expenses (as a percentage of loss) is 10%.
Fixed expense is $95,000.
Variable expense is 12%.
Profit and contingency provision is 3%.
Assume the deductible applies to loss and ALAE.

Questions from the 2005 exam:
19. Given the following information, calculate the loss elimination ratio at a $500 deductible.
Loss Amount
Below $500
$500
Over $500
A. < 0.4

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Claim Count
150
6
16

B. > 0.4, but < 0.5

Total Loss
$15,000
$3,000
$22,000

C. > 0.5, but < 0.6

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D. > 0.6, but < 0.7

E. > 0.7

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2008 exam:
32. (2.5 points) Given the following information:
Ground-up Severity
$100
$250
$500
$1000
$3000
$8000







Probability
20%
10%
15%
30%
20%
5%

Premium for a policy with no deductible = $350
Ground-up expected loss ratio = 60.9%
Fixed expenses = $31.70
Variable underwriting expense provision = 22%
Profit provision = 2%
Allocated loss adjustment expenses (ALAE) are 10% of loss and are the responsibility of the
insurer.

a. (1.0 point) Calculate the loss elimination ratio (LER) for a $500 deductible.
b. (1.5 points) Calculate the premium for a policy with a $500 deductible

Questions from the 2010 exam
30. (1 point) Given the following information:

Net Reported
Policy Deductible
Full Coverage
$250
$500

Losses
$680,000
$2,900,000
$5,200,000

Net Reported
Losses
Assuming a

$250 Deductible
$590,000
$2,900,000
N/A

$500 Deductible
$525,000
$2,600,000
$5,200,000

Calculate the loss elimination ratio associated with moving from a $250 deductible to a $500
deductible.

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2011 exam
14. (1.5 points) Given the following information:

Deductible
Full coverage
$250
$500
$750
$1,000
Total

Reported
Claim Counts
990
2,770
4,360
1,350
500
9,970

Net Reported
Losses
$1,347,000
$5,167,000
$9,198,000
$3,230,000
$1,692,000
$20,634,000

Net Reported
Losses
Assuming
$750 Deductible
$772,000
$4,024,000
$8,244,000
$3,230,000
Unknown

Net Reported
Losses
Assuming
$1000 Deductible
$605,000
$3,505,000
$7,345,000
$2,926,000
$1,692,000

a. (1 point) Use the loss elimination ratio approach to deductible pricing to calculate the credit associated
with moving from a $750 deductible to a $1,000 deductible.
b. (0.5 point) An assumption of the loss elimination ratio approach is that claim behavior will be the same
for each deductible. Describe why this assumption may not hold in practice.

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.

Section 3: Size of Risk for Workers Compensation
3a. Premium Discounts
Questions from the 2000 exam
52. (2 points) Based on Schofield, "Going from a Pure Premium to a Rate," and the following information, use
the Workers' Compensation Method to calculate the dollar amount of Premium Discount.
 Standard premium = $ 475,000
 Expense Table:
Expense Provisions
Profit and
Premium Range ($)
Production
General
Taxes
Contingencies
1 – 5,000
12.0%
10.0%
4.0%
2.5%
5,001 - 100,000
9.0%
7.5%
4.0%
2.5%
100,001 - 500,000
7.0%
5.0%
4.0%
2.5%
500,001 +
6.0%
2.5%
4.0%
2.5%

Questions from the 2002 exam
29. (3 points) Based on Schofield, "Going From a Pure Premium to a Rate," and the information below,
use the Worker's Compensation Method to calculate the discounted premium. Show all work.
 Standard Premium of 500,000
 For each premium gradation of 200,000 above 10,000, commissions and general expenses
decrease by 25%.
 For the first 10,000 of Standard Premium commissions are 15% and general expenses are 10%.
 All other expenses total 8% of the discounted premium.

Questions from the 2011 exam
16. (1.75 points) Workers compensation insurers often offer a premium discount for large premium dollar
accounts. Given the following expense information for workers compensation policies:

Premium Range
$0 - $7,500
$7,500 - $75,000
$75,000 - $200,000
$200,000 & above

Expense Percentage by Type:
Production
General
Taxes
14%
10%
3%
10%
8%
3%
7%
6%
3%
5%
4%
3%

Profit
5%
5%
5%
5%

Calculate the total amount of premium discount for a policy with premium of $180,000.

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
3b. Loss Constants
Questions from the 1995 exam
35. Feldblum, “Workers' Compensation Ratemaking,” states that loss experience for large risks tends to be better
than for small risks.
(a) (1 point) Give two explanations that support this observation.
(b) (2 points) In 1990 the NCCI recommended application of loss constants to all risks, rather than to small
risks only. Using Feldblum's methodology and the information below, calculate the appropriate loss
constant to be applied to all risks.
Premium Size
Small Risk $0 - $2,000
Large Risk $2,001 or more

Number
of Risks
100
50

Earned
Premium
$75,000
$200,000

Incurred
Losses
$63,000
$144,000

Loss
Ratio
84.0%
72.0%

Questions from the 1998 exam
34. Based on Feldblum, "Workers' Compensation Ratemaking," answer the following.
a. (1 point) Give two reasons why small risks generally show higher loss ratios than larger risks.
b. (1 point) Using the information below, calculate the loss constant necessary to bring the experience
of the smaller risks in line with the experience of the larger risks.
Premium
Range
$0-1,000
>1,000

Number of
Risks
1,000
2,000

Earned
Premium
1,200,000
13,000,000

Incurred
Loss
1,100,000
10,000,000

Questions from the 2000 exam
48. (3 points) Based on Feldblum, "Workers' Compensation Ratemaking," answer the following questions.
a. (1/2 point) What is the purpose of an Expense Constant?
b. (1/2 point) Why is an Expense Constant important for small policies?
c. (1/2 point) What is the purpose of a Loss Constant?
d. (1 1/2 points) Given the following data, calculate the loss constant. Assume loss constants are to be
used for risks with annual premium of $1,000 or less.
Premium Range
# of Risks
Earned Premium
Incurred Loss
$ 0 - 1,000
200
$130,000
$104,000
> $1,000
200
$960,000
$720,000

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.

Section 4: Insurance to Value (ITV)
Questions from the 1990 exam
49. (3 Points)
a. (2 Points) A building is insured for $150,000 under an agreed value policy. Assume a 12.5% loss
frequency and the following size of loss distribution. Using the methods discussed by
Head “Insurance to Value," calculate the pure premium rate per $100 for the building.

Size of Loss (L)
0 < L <$ 50,000
$50,000 < L < 100,000
$100,000 < L < $150,000
$150,000 < L < $200,000
$200,000 < L < $250,000
$250,000 < L
TOTAL

Number Of
Losses
340
75
50
25
10
0
500

Dollars Of
Loss
$3,762,000
5 625,000
6,375,000
4,463,000
2,275,000
0
$22,500,000

b. (1 Point) Is this rate higher or lower than the rate for a comparable building insured for $200,000? Why?

Question from the 1992 exam
5. According to the Study Note Reading: Head, G.L.; Insurance to Value, if losses less than the policy
face are possible, which of the following are true concerning the pure premium rate as the coinsurance
percentage increases?
1. If small losses outnumber large ones, pure premium rates should decrease at a decreasing rate.
2. If large losses outnumber small ones, pure premium rates should decrease at a decreasing rate.
3. If losses of all sizes are equally numerous, pure premium rates should decrease at a constant rate.
A. 1

B. 3

C. 1, 3

D. 2, 3

E. 1, 2, 3

Questions from the 1994 exam
43.
(a) (2 points) Using the methods described by Head in the Study Note Reading Insurance to Value,
calculate the pure premium rate per $100 for 20%, 50%, and 80% coinsurance. You
have the following data:
 The value of property insured is $200,000.
 Loss frequency is 2%.

Coinsurance
Percentage
(Cn)
20%
50%
80%

Conditional
Probability
of Losses in Interval
(Cn-1, Cn]
.50
.20
.05

Arithmetic Mean Loss
of Losses in Interval
as % of Total Value
5%
35%
60%

(b) (1 point) This question no longer applies to the content covered in this chapter

Exam 5, V1b

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 1995 exam
46. (4 points) You are given:

Replacement
Cost
$100,000

Loss
Frequency
10%

$200,000

10%

Size of Losses
in Interval
($000)
$ 0- 20
21- 50
51- 80
81-100
$ 0- 20
21- 50
51- 80
81-100
101-160
161-200

Conditional
Probability
of Losses in
the Interval
.80
.10
.08
.02
.70
.15
.09
.04
.01
.01

Arithmetic Mean
Loss of Losses
in Interval
($000)
$2
3
60
95
$3
35
65
95
150
190

A client has asked you to determine the pure premium cost of insuring his house with a $200,000 replacement
cost.
(a) (1 point) As described in the study note reading by Head, “ Insurance To Value," determine the pure
premium rate per $100 for insuring this house for $100,000.
(b) (1.5 points) How does this pure premium per $100 compare to the rate for this house if it were insured for
$200,000? Explain.
(c) (1.5 points) Would the pure premium rate per $100 derived in (a) match that of a house with a
replacement cost of $100,000 and insured for $100,000? Why or why not?

Questions from the 1996 exam
44. (3 points) You are given:

Coinsurance
Percentage (Cn)
40%
60%
80%
100%

Conditional
Probability of
Losses in Interval
[Cn-1 ,Cn]
65%
20%
10%
5%

Arithmetic Mean
Loss in Interval
[Cn-1 ,Cn]
$100,000
$250,000
$350,000
$500,000

• Value of Property: $500,000
• Loss Frequency:
5%
(a) (2 points) Using the methods described by Head, "Insurance to Value," calculate the pure premium rate
per $100 for 60% coinsurance.
(b) (1 point) The property is actually insured for $200,000, with a 60% coinsurance clause. A loss of
$80,000 occurs. What is the total indemnity amount payable to the insured?

Exam 5, V1b

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 1998 exam
5. Based on Head, Insurance to Value, calculate the pure premium rate per $100 of insurance for a
$100,000 risk and a 50% coinsurance percentage.

Losses
At Least
Less Than
0%
10%
10%
20%
20%
30%
30%
40%
40%
50%
50%
60%
60%
70%
70%
80%
80%
90%
90%
100%
A. < $1.00

Unconditional
Probability
Of Loss
.0100
.0075
.0050
.0035
.0020
.0010
.0005
.0003
.0002
.0005

Arithmetic
Mean
Loss
4%
14%
23%
33%
43%
53%
62%
72%
82%
98%

B. > $1.00 but < $1.05 C. > $1.05 but < $1.10 D. > $1.10 but < $1.15 E. > $1.15

Questions from the 1999 exam
15. (1 point) Based on Head, "Insurance to Value," and given the information below, what is the coinsurance
penalty applicable to the insured?
Coinsurance Requirement:
Full Value of Structure:
Amount of Insurance on Structure:
Amount of Loss:

80%
$1,000,000
$700,000
$600,000

A. < $20,000
B. > $20,000 but < $40,000
D. > $60,000 but < $80,000
E. > $80,000

C. > $40,000 but < $60,000

Questions from the 2000 exam
24. Based on Head, Insurance to Value, and the following information, calculate the absolute difference
between the pure premium rate per $100 for a 50% coinsurance clause and a 75% coinsurance clause.
• The value of the insured property is $100,000.
• The loss frequency is 5%.
Arithmetic Mean Loss
Loss, as Percentage of
Conditional Probability
in Interval, as a
Total Property Value
of a Loss in Interval
Percent of Total Value
Less than or equal to 10%
0.50
4%
11 % to 25%
0.25
18%
26% to 50%
0.15
40%
51 % to 75%
0.07
70%
A. < 0.10
E. > 0.40

Exam 5, V1b

B. > 0.10 but < 0.20

C. > 0.20 but < 0.30

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D. > 0.30 but < 0.40

 2014 by All 10, Inc.

Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2001 exam
Question 7. Based on Head, Insurance to Value, and the following information, calculate the ratio of the pure
premium rate per $100 for a 60% coinsurance clause to the pure premium rate per $100 for a
40% coinsurance clause.
Loss, as a Percentage of
Total Property Value

Unconditional Probability
of a Loss in Interval

Arithmetic Mean Loss in Interval
as a Percent of Total Value

Less than or equal to 20%

0.050

12%

21% to 40%

0.025

30%

41% to 60%

0.015

52%

61% to 80%

0.007

75%

80% to 100

0.003

95%

A. < 0.65

B. > 0.65 but < 0.75

C. > 0.75 but < 0.85

D. > 0.85 but < 0.95

E. > 0.95

Questions from the 2002 exam
42. (2 points) Based on Head, Insurance to Value, and the following information, calculate the pure
premium rate per $100 for a 50% coinsurance clause.
The value of the insured property is $200,000. The loss frequency is 3%.
Loss, as Percentage of
Total Property Value
Less than or equal to 25%
26% to 50%
51% to 75%

Conditional Probability
of a Loss in Interval
0.75
0.12
0.08

Arithmetic Mean Loss in Interval
as a Percent of Total Value
9%
40%
70%

Questions from the 2003 exam
40. (2.25 points) An insurer writing fire insurance uses coinsurance in its rating structure by means of
an "average clause." A coinsurance percentage of 80% applies to all policies. Based on the
following information, answer the questions below. Show all work
Policy

Amount of Loss

Property Value

Face Amount of
Insurance

1
2
3

$50,000
$155,000
$375,000

$200,000
$160,000
$480,000

$150,000
$120,000
$400,000

a. (1.5 points) For each of the policies above, calculate the indemnity payment made by the insurer.
b. (0.75 points) For each of the policies above, calculate the additional insurance, if any, that would
have been required for the insurance company to indemnify the full amount of the loss.

Exam 5, V1b

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2004 exam
41. (4 points) Given the following information on an individual property policy, answer the questions below.
Show all work.
• The property value is $200,000.
• Assume no deductible applies.
• The frequency of non-zero loss is 10%.
• The severity of loss distribution is as follows:
• 70% at 10% of value
• 20% at 50% of value
• 8% at 80% of value
• 2% at 90% of value
• Coinsurance to 80% underlies the expected rate.
• Permissible loss ratio is 65%.
a. (2 points) The insured purchases a policy insuring the property to 80% of value. Determine the
premium charged for the policy.
b. (1 point) The insured instead purchases a policy insuring the property to 70% of value. Assuming the
same rate per $100 of insured value as in part a. above, determine the expected loss ratio for this
policy.
c. (1 point) Assume the insurer incorporates a coinsurance clause into the policy. The insured continues
to insure the property to 70% of value. What is the expected loss ratio for this policy? Briefly explain
your answer.

Questions from the 2005 exam
51. (2 points) Using the following information, answer the questions below. Show all work.
• All properties are valued at $500,000.
• The company writes 1,000 policies.
• Each policy has a face value equal to the value of the insured property.
• Assume only one loss per policy per period is possible, and exactly 20 insureds will incur a loss
of some size during any one policy period.
• Assume no coinsurance clause or deductible applies
Assume losses are distributed as shown:
50% at $50,000
20% at $250,000
30% at $500,000
a. (1 point) Calculate the pure premium rate per $100 of insurance for a policy face equaling
$300,000.
b. (1 point) Does the pure premium rate per $100 of insurance for a $500,000 policy face differ
from the rate for the $300,000 policy face? Briefly explain your answer.

Exam 5, V1b

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2006 exam
44. (2.5 points) You are given the following assumptions for an insured book of property business:


A company writes 1,000 property policies.




Each property is valued at $500,000.
Exactly 20 of these properties will experience a loss during one policy period.



The losses are distributed as shown in the table below:
S(L)
50%
20%
10%
5%
15%

L
$100,000
200,000
300,000
400,000
500,000

Find the premium rate per $100 of insurance for a policy face equaling $400,000. Show all work.

Questions from the 2007 exam
49. (1.0 point) A property is valued at $300,000. The coinsurance requirement for the policy is 80% of
the property value. The insured chooses a $200,000 face value. Assume there is no deductible.
Calculate each of the following:
a.
(0.25 point) Coinsurance requirement.
b.
(0.25 point) Coinsurance apportionment ratio.
c.
(0.25 point) Coinsurance deficiency.
d.
(0.25 point) Maximum coinsurance penalty.
Show all work.

Questions from the 2008 exam
36. (2.0 points) You are given the following information:
 Home is valued at $350,000.
 Coinsurance requirement = 80% of the property value
 Face value of policy = $275,000
a. Calculate the coinsurance deficiency.
b. Calculate the coinsurance apportionment ratio.
c. Calculate the maximum coinsurance penalty possible.
d. Calculate the coinsurance penalty for a $300,000 loss.

Questions from the 2009 exam
40. (2 points) Given the following:
• Property is valued at $500,000.
• Coinsurance requirement is 88% of the property value.
• Policy face value is $300,000.
Graph and label the coinsurance penalty function.

Exam 5, V1b

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2010 exam
32. (2 points) Given the following information:
• Amount of loss = $200,000
• Amount of coverage = $350,000
• Replacement cost of property = $450,000
• Minimum insurance-to-value requirement = 80%
a. (1 point) Calculate the coinsurance penalty.
b. (0.5 point) Identify the problem with underinsurance from the insurer's perspective.
c. (0.5 point) Identify the problem with underinsurance from the insured's perspective.

Questions from the 2012 exam
15. (2.25 points) You are given the following information on expected claim payment distribution for
properties with a replacement cost of $350,000.
Claim Payment Probability
$0
97.0%
$10,000
1.5%
$50,000
0.8%
$200,000
0.5%
$350,000
0.2%


Assume no expenses or profit.

a. (0.5 point) Assuming all homeowners purchase full coverage, calculate the pure premium per
$1,000 of insurance.
b. (0.75 point) Demonstrate with an example that the use of a fixed rate per $1,000 of insurance is
inequitable if a subset of the insured group purchases only partial coverage.
c. (1 point) Describe two insurer initiatives that would reduce the inequity from part b. above,
including an explanation of how the inequity would be reduced.

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Section 1: Increased Limits Ratemaking
Solutions to questions from the 2004 exam:
45. (2 points) Calculate the annual claims inflation rate in the layer $50,000 excess of $50,000. Assume
aground-up annual claims inflation rate of 15%. Show all work.
Date of Loss
Ground-up Loss
February 1, 2003
$37,000
July 15, 2003
$47,000
October 1, 2003
$64,000
December 1, 2003
$93,000
Note: This is more of a chapter 6 question
To determine the annual claims inflation rate in the layer $50,000 excess of $50,000, compare losses in the layer
50,000 excess of $50,000 prior to inflation with losses in the layer $50,000 excess of $50,000 after inflation. Be
sure to trend ground up claims by the annual claims inflation rate of 15% prior to computing losses in the layer.
Then ratio the losses in the layer prior to, and post the application of inflation.

Date of
Loss
2/1/03
7/15/03
10/1/03
12/1/03

Ground-up
Loss
(1)
37,000
47,000
64,000
93,000
241,000

Ground-up
Annual
Losses
Claims inflation
50K xs 50K
Rate
(2)
(3)
0
1.15
0
1.15
14,000
1.15
43,000
1.15
57,000

Trended
Ground-up
Loss
(4)=(1)*(3)
42,550
54,050
73,600
106,950
277,150

Trended
Losses
50K xs 50K
(5)
0
4,050
23,600
50,000
77,650

Annual Layer
Claims
Inflation
Rate
(6)=(5)/(2)-1.0

0.3623

Col (2) and Col (5) are capped at 50,0000

Solutions to questions from the 2005 exam
50. (1 point) Explain two reasons why claim inflation produces larger cost trends on increased limits
coverage than on basic limits coverage.
Note: This is more of a chapter 6 question
1. For losses above the basic limit, inflation will impact the increased limits portion of the loss only.
2. For losses near the basic limit, inflation may cause the loss to pierce the increased limit layer, resulting in
increased frequency of increased limit losses.
Alternatively:
“First, the whole effect of the trend is in the excess portion of the increased limits claim while the effect
on the basic limits portion is zero. Second, although uniform frequency trends affect equally basic and
increased limits, a rising cost trend causes a rise in increased limits claim frequency since additional
claims (previously only basic limits losses) break through the lower boundary of the increased limits layer
of losses becoming new excess claims.”

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2006 exam
31. (3.25 points)
a. (2 points) Calculate the annual inflation rate for claims in the layer $50,000 excess of $100,000 for 2006.

2005
Ground-up
Loss
(1)
75,000
100,000
125,000
150,000
450,000

Ground-up
2005
Annual
Losses
Claims inflation
50K xs100K
Rate
(2)
(3)
0
1.10
0
1.10
25,000
1.10
50,000
1.10
75,000

2006
Trended
Ground-up
Loss
(4)=(1)*(3)
82,500
110,000
137,500
165,000
495,000

Trended
2006
Losses
50K xs100K
(5)
0
10,000
37,500
50,000
97,500

Annual Layer
Claims
Inflation
Rate
(6)=(5)/(2)-1.0

0.3000

Col (2) and Col (5) are capped at 50,0000
b. (1.25 points) How would you expect the inflation rate in the layer $50,000 excess of $100,000 to
differ from the inflation rate for claims limited to $100,000? Explain two reasons for the difference
between the inflation rates.
Note: This is more of a chapter 6 question
The excess layer inflation rates are greater than the basic limit inflation rates for two reasons:
1. For losses already in the excess layer, inflation impacts only the portion of the loss in the excess
layer. The basic limits portion does not change.
2. For losses near the basic limit, inflation causes the losses to pierce the increased limits layer, resulting
in increased frequency of increased limits losses.

Exam 5, V1b

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2007 exam
46. If all claims experience an annual ground-up severity trend of 8.0%, calculate the effective trend in
the layer $100,000 in excess of $100,000. Show all work.
Note: This is more of a chapter 6 question
Initial comments: Analysis of trend on excess loss layers.
Two factors need to be considered.
1. The portions of losses below the layer are removed from both the pre-trend and post-trend loss amounts.
See columns (2) and (5) below.
This is a smaller % of the post-trend loss, which produces a "leveraging" effect.
Compare [1.0 - (2)/(1)] to [1.0 - (5)/(4)].
2. However, some losses may be capped by the upper limit of the layer, mitigating the effect (See claim D
below).

Claim
A
B
C
D
E
Total

Ground-up
Loss
(1)
35,000
125,000
180,000
206,000
97,000
643,000

Ground-up
Annual
Losses
Claims inflation
100K xs100K
Rate
(2)
(3)
0
1.08
25,000
1.08
80,000
1.08
100,000
1.08
0
1.08
205,000

Trended
Ground-up
Loss
(4)=(1)*(3)
37,800
135,000
194,400
222,480
104,760
694,440

Effective Trend
Trended
Rate in the
Losses
100K XS 100K
100K xs100K
Layer
(5)
(6)=(5)/(2)-1.0
0
35,000
94,400
100,000
4,760
234,160
0.1422

Col (1) and Col (3) are given
Col (2) equals Col (1) - 100,000, capped at 100,000, if (1) is greater than 100,000
Col (5) equals Col (4) - 100,000, capped at 100,000, if (4) is greater than 100,000
Thus the effective trend in the 100K xs 100K layer is 234,160/205,000 - 1.0 = 0.1422 = 14.22%

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2007 exam (continued):
47. (2.0 points) Using the methods described by Palmer in Increased Limits Ratemaking for Liability
Ratemaking, calculate the following:
a. (1.25 points) The $5,000,000 increased limit factor.
b. (0.75 point) The limited average severity in the layer $4,000,000 in excess of $1,000,000. Show all work.
Initial comments:
An Increased Limit Factor (ILF) at limit L relative to basic limit B can be defined as:
ILF ( L ) 

Expected Indemnity Cost(L)
Expected Indemnity Cost(B)

ILFs are developed on a per-claim or per-occurrence basis:
 A per-claim limit is a limit on the amount that will be paid to a single plaintiff for losses arising
from a single incident.
 A per-occurrence limit is a limit on the total amount that will be paid to all plaintiffs for losses
arising from a single incident.
To evaluate an appropriate provision for indemnity costs at various limits of liability, we develop (LAS) at
various limits of liability. LAS is the average size of loss when all losses have been capped at the given limit.
Part A

Claim
A
B
C
D
E
F
G
H
I
J

Ground-up
Loss Amount
(1)
250,000
300,000
450,000
750,000
1,200,000
2,500,000
4,000,000
7,500,000
9,000,000
15,000,000

Loss at
$1,000,000 Limit
(2)
250,000
300,000
450,000
750,000
1,000,000
1,000,000
1,000,000
1,000,000
1,000,000
1,000,000

Loss at
$5,000,000 Limit
(3)
250,000
300,000
450,000
750,000
1,200,000
2,500,000
4,000,000
5,000,000
5,000,000
5,000,000

Limited Average

775,000

2,445,000

Losses in the Part B
4M x/s 1M
Layer
(4)
0
0
0
0
200,000
1,500,000
3,000,000
4,000,000
4,000,000
4,000,000
2,783,333

Col (2) equals Col (1) capped at 1,000,000; Col (3) equals Col (1) capped at 5,000,000
Col (4) equals Col (1) - 1,000,000, capped at 4,000,000, if (1) is greater than 1,000,000

a. The indemnity-only ILF at 5,000,00 given a basic limit of 1,000,000 equals 2,445,000/775,000 = 3.1548
b. LAS (4M xs 1M) = (200,000 + 1,500,000 + 3,000,000 + [3 x 4,000,000])/6 = 2,783,333, or
LAS (4M xs 1M) = (2,445,000 – 775,000)/0.6 = 2,783,333, where .60 is equal to the probability that a
loss is greater than 1M, given that a loss has occurred, or
[(3 * 5,000,000 + 4,000,000 + 2,500,000 + 1,200,000]/6 - [(6 * 1,000,000]/6 = 3,783,333 -1,000,000 = 2,783,333

Exam 5, V1b

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2008:
Model Solution – Question 18
a. Calculate the basic limit trend and b. Calculate the excess limit trend
Note: This is more of a chapter 6 question
a. Basic limits trend: All losses except the $20K loss are at or exceed the basic limit of $50,000.
So the BL trend is simply [($50+$50+$50+$50+$20*1.1)/($50+$50+$50+$50+$20)] -1.0 = 1%
b. Excess limits trend is computed as [($50+$70+$90+$110]*1.1 -$ 50*4)]/[0+0+20+40+60] – 1.0 = 26.7%
This can also be computed as follows:

 110,000
 110,000
Excess Limits Trend    ( x *1.1  50, 000)  /  ( x  50,000)
 x50,000
 x50,000
Alternatively, the basic limits trend and excess limits trend can be computed as follows:
Effects of +10% Trend on Basic (50,000) and Excess Loss Limits
Loss
Amount
$50,000 Limit
Excess Limit
($)
Pre Trend($)
Post Trend($)
Pre Trend($)
Post Trend($)
20,000
20,000
22,000
0
0
50,000
50,000
50,000
0
5,000
70,000
50,000
50,000
20,000
27,000
90,000
50,000
50,000
40,000
49,000
110,000
50,000
50,000
60,000
71,000
Total
220,000
222,000
120,000
152,000
Trend [Post ($)/Pre ($)]

1.00%
0.009

27.00%
0.267

Note: 22,000 = 20,000 * 1.1; 27,000=70,000 * 1.1 - 50,000

c. When loss trends are negative.

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2008 (continued):
Model Solution – Question 34
a. Initial comments:
An Increased Limit Factor (ILF) at limit (L) relative to basic limit (B) can be defined as:
ILF ( L ) 

Expected Indemnity Cost(L)
Expected Indemnity Cost(B)

Step 1: Write an equation to determine the $500,000 ILF given a $250,000 basic limit
ILF (500,000) 

Expected Indemnity Cost(500,000) LAS (500)

Expected Indemnity Cost(250,000) LAS (250)

Step 2: Recall that to evaluate LAS at $5000,000, include all loss dollars from losses of:
i. $500,000 or less, plus
ii. the first $500,000 of each loss that is in excess of $500,000.
The same holds true when computing LAS at $250,000, except that $250,000 is used in i. and ii. above.
Finally, recognize that since LAS is the average size of loss when all losses have been capped at a
given limit, we must divide the loss amounts describe above by the total number of loss occurrences.
Step 3: Using the guidance in Step 2, and the data given in the problem, compute LAS (500K) and LAS (250K).

$500,000  $16,000,000  $17,500,000  20 *$500,000 $44,000,000

 $176,000
100  80  50  20
250
$500,000  $16,000,000  70 *$250,000 $34,000,000
LAS (250k ) 

 $136,000
100  80  50  20
250

LAS (500k ) 

Notes:
i. The losses given in this problem are assumed to be the total losses that actually occurred. None of the
losses were limited, or "censored," by the insured’s policy limit. For more information on working with
losses that are limited, or "censored," by the insured’s policy limit, see Section 4 in your manual.
ii. There are only 20 losses in excess of $500,000, while there are 70 losses in excess of $250,000.
Step 4: Using the equation in Step 1, and the results from Step 3, solve for the $500,000 ILF

ILF (500k ) 

LAS (500k ) $176,000

 1.294
LAS (250k ) $136,000

b. Two issues with using empirical data are:
1. Credibility - Data could be sparse for large losses, which makes ILFs susceptible to random
fluctuations and therefore unreliable (or less credible).
2. Ground-up loss data may not be available, especially for first party coverages where small losses
under the policy deductible are not reported.

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2009 exam
Question: 36
Initial comments: In the predecessor paper to WM, Palmer states that:
An Increased Limit Factor (ILF) at limit L relative to basic limit B can be defined as:
ILF ( L ) 

Expected Indemnity Cost(L)+ALAE (L) +ULAE(L) +RL(L)
, where
Expected Indemnity Cost(B) +ALAE(B) +ULAE(B) +RL(B)

ALAE(X) = the Allocated Loss Adjustment Expense provision at each limit,
ULAE(X) = the Unallocated Loss Adjustment Expense provision at each limit, and
RL(X) = the Risk Load provision at each limit.
In addition, for illustrative purposes, examine the "indemnity-only" ILF:
ILF ( L ) 

Expected Indemnity Cost(L)
Expected Indemnity Cost(B)

Assumptions:
**Key: When working with ILFs, it’s often assumed that frequency is independent of severity. **
The above formula can then be expressed as:
ILF ( L ) 

Expected Frequency (L)  Expected Severity (L)
Expected Frequency (B)  Expected Severity (L)

However, it is generally assumed that the frequency is independent of the policy limit
purchased (i.e. Expected Frequency (L) = Expected Frequency (B))
Problem specific solution
ILF = [LAS (5,000,000) + ULAE (5M)] x Freq(5M)/ [LAS (1,000,000) + ULAE (1M)]x Freq(1M)
Compute the following:

LAS (1, 000, 000)  [300, 000  600, 000  750, 000  3(1, 000, 000)] / 6  775, 000
LAS (5, 000, 000  [300, 000  600, 000  750, 000  1, 250, 000  4,500, 000  5M ] / 6  2, 066, 667
Thus, ILF = [LAS (5,000,000) + ULAE (5M)] x Freq(5M)/ [LAS (1,000,000) + ULAE (1M)]x Freq(1M)
= [2,066,667 x 1.2 x .10]/ [775,000 x 1.1 x .15] = 1.9394

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2010 exam
Question: 31
Calculate the increased limit factor for the $300,000 policy limit, assuming a basic limit of $100,000.

Indicated ILF($300 K)=

LAS($300 K)
LAS($100 K)

To calculate LAS by limit, calculate a LAS for each layer of loss and combine the estimates for each
layer taking into consideration the probability of a claim occurring in the layer.
The LAS of each layer is based solely on loss data from policies with limits as high as or higher than
the upper limit of the layer.
Example: When calculating the LAS ($100K), use the experience from all policies limits censored at $100,000:

LAS ($100 K ) 

$97 M  $46 M  637($100 K )  $11M  (561  407)$100 K
$314,500, 000

 $76, 726
(1,573  1,390  1,136)
4, 099

Note: When calculating LAS ($300,000), the actuary cannot use the policies that have a $100,000 limit as
there is no way to know what the claim amounts would be if each of those policies had a limit of
$300,000.
Calculating LAS ($300,000):
Combine LAS ($100K) with LAS for the layer ($100,000 to $300,000).
Step 1: Determine the losses in the $100K - $300 K layer.
i. Policies with a limit of $100,000 cannot contribute any losses to that layer and the data is not used.
ii. Of the 1,390 claims with policies having a $300K limit, 637 claims have losses in the $100K to $300K layer.
Total censored losses for those 637 claims are $150,000,000.
Eliminating the first $100K of each of those losses results in losses in the $100K to $300K layer.
$150,000,000 - 637 x $100,000 = $86,300
iii. Policies with a limit of $500K also contribute loss dollars to the $100K to $300K layer.
Of the 1,136 claims associated with a limit of $500K limit, 561 have losses in the $100K to $300K layer.
These claims contribute $50,900,000 (=$107,000,000 – 561 x $100,000) of losses to the layer.
Another 407 claims exceed $300,000, and each contributes $200,000 to the $100K to $300K layer.
$81,400,000 = 407x ($300,000- $100,000)
The sum of the above values are the losses in the $100K to $300 layer:
$86,300,000+ $50,900,000+ $81,400,000 = $218,600,000.
These loss dollars were derived from 1,605 (=637 + 561 + 407) claims.
LAS(100K-300K) =

$136,199 =

$218, 600, 000
1, 605

Thus, LAS(100K-300K)*Pr(100 0.6, but < 0.7

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2008 exam
Model Solution - Question 32
Part a. Calculate the loss elimination ratio (LER) for a $500 deductible.
Step 1: Write an equation to determine the LER for a $500 deductible.
x D

E[ X ; D]

The loss elimination ratio (LER) 
E[ X ]

 x * f ( x)dx  D[1 F (d )]
x 1

E[ X ]

Step 2: Using the equation in Step 1, and the data given in the problem, solve for the LER for a $500 deductible.

E[ X ;500]  100(0.2)  250(0.1)  500(1  0.2  0.1)  395
E[ X ]  100(0.2)  250(0.1)  500(0.15)  1000(0.30)  3000(0.20)  8000(0.05)  1, 420
Thus, the LER 

395
 0.278169  27.82%
1, 420

Part b. Calculate the premium for a policy with a $500 deductible
Step 1: Write an equation to determine the premium for a $500 deductible policy

Prem500Ded =

Losses above ded + ALAE + Fixed Exp
1.0 - %Comm Exp- %Other Var Exp-%P&C

Step 2: Compute losses excess of the deductible and ALAE
Expected losses X/S of the deductible = Expected losses * X/S ratio
= SP * ELR * X/S ratio = $350 * .609 * (1 - .2782) = $153.8583
Note: the X/S ratio = 1 - LER
ALAE = Expected losses * ALAE % of loss = SP * ELR * ALAE % = $350 * .609 * .10 = $21.315
Step 3: Using the equation in Step 1, the results from Step 2 and the givens in the problem, solve for $100,000
deductible policy premium.

Prem500Ded =

$153.8583+$21.315+31.70
 $272.20
1.0 - .22 - .02

Solutions to questions from the 2010 exam
Question 30 Calculate the LER associated with moving from a $250 deductible to a $500 deductible.
In the LER approach, calculate the amount of losses that are eliminated going from full coverage to a
deductible or by going from one deductible to a higher deductible:

LER ( D ) 

Losses and LAE Eliminated by Deductible ( L  EL ) B  ( L  EL ) D

Total Ground - up Losses and LAE
( L  EL ) B

Ignore $500 data due to censoring of data.
Losses eliminated = (2,900,000 + 590,000) – (2,600,000 + 525,000) = 365,000
LER (500) = 365,000/(2,900,000+590,000) = 0.10458

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2011 exam
14a. (1 point) Use the loss elimination ratio approach to deductible pricing to calculate the credit
associated with moving from a $750 deductible to a $1,000 deductible.
14b. (0.5 point) An assumption of the loss elimination ratio approach is that claim behavior will be the
same for each deductible. Describe why this assumption may not hold in practice.
Initial comments
Insurers may not know the ground-up losses for every claim (e.g. insureds may not report claims that are less
than the deductible on their policy).
When this is the case, data from policies with deductibles greater than the deductible being priced cannot be
used to calculate the LER. For example:
 data from policies with a $500 deductible cannot be used to determine LERs for a $250 or $100
deductible, however
 data from policies with deductibles less than the deductible being priced can be used to determine
LERs (e.g. data from policies with a $500 deductible can be used to determine the LER associated
with moving from a $750 deductible to a $1,000 deductible).
LER Calculation to Move from a $750 to $1000 Deductible
(1)
(2)
(3)
(4)
(5)
Net Reported
Net Reported
Losses
Losses
Reported
Net Reported
Assuming
Assuming
Deductible
Claims
Losses
$1000 Ded
$750 Ded
Full Cov
990
$1,347,000
$605,000
$772,000
$250
2770
$5,167,000
$3,505,000
$4,024,000
$500
4360
$9,198,000
$7,345,000
$8,244,000
$750
1350
$3,230,000
$2,926,000
$3,230,000
$1,000
500
$1,692,000
$1,692,000
Unknown
$20,634,000
Total
9970
(7) Net Reported Losses for Ded <=$750
(8) Losses Eliminated <=$750 Ded
(9)LER

(6)
Losses
Eliminated
Moving from
$750 to $1000
$167,000
$519,000
$899,000
$304,000
Unknown
$16,270,000
$1,889,000
0.1161

(3)= Net of the deductible
(4) =(3) Adjusted to a $1000 deductible (5)=(3) Adjusted to a $750 deductible
(6)= (5) - (4) (7)= Sum of (5) for $0, $250, $500, 750 Ded
(8)=Sum of (6) for $0, $250, $500, $750 Deductibles (9)=(8)/(7)
 Each row contains data for policies with different deductible amounts.
 The analysis can only use policies with deductibles of $750 or less (since the goal is to determine the
losses eliminated when changing from a $750 to a $100 deductible)
 Columns 4 and 5 contain the net reported losses in Column 3 restated to $1000 and $750 deductible
Columns 4 and 5 are not Column 3 minus the product of Column 2 and the assumed deductible.
This is because not every reported loss exceeds the assumed deductible. The losses in Columns 4
and 5 are based on an assumed distribution of losses by deductible and size of loss, and cannot be
recreated given the data shown.
Question 14 – Model solution
a. LER = [(772 - 605) + (4024 - 3505) + (8244 - 7345) + (3230 - 2926)] / (772 + 4024 + 8244 + 3230)
= [16,270 - 14381] / 16,270 = 0.1161 Credit
b. Low risk drivers more likely to purchase higher deductibles

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.

Section 3: Size of Risk for Workers Compensation
3a. Premium Discounts
Solutions to questions from the 2000 exam
Question 52.
Calculate the dollar amount of Premium Discount.
• Given Standard premium = $ 475,000
1. Partition the $475,000 into "gradations" (the first $5,000 of premium; the next $95,000 of premium, etc.)
2. Compute Premium in Range:
Gradation of Premium in
Premium Range ($) Premium
the range
Production
(1)
1 - 5,000
5,000
5,000
12.0%
5,001 - 100,000
95,000
95,000
9.0%
100,001 - 500,000 400,000
375,000
7.0%
500,001 +
500,000+
0
6.0%

General
(2)
10.0%
7.5%
5.0%
2.5%

(3)
(1)+(2)
22%
16.5%
12%
8.5%

Taxes
(4)
4.0%
4.0%
4.0%
4.0%

Profit and
Contingencies
(5)
2.5%
2.5%
2.5%
2.5%

3. Compute the Expense reduction
The expense reduction in expenses is simply the difference between the expenses in a particular
Premium Range and those expenses in the Premium Range of $1 - $5,000.
Note: Each gradation of premium has a set of expense percentages associated with it.
The Production and General Expenses percentages vary with the premium gradation and
represent percentages of Standard Premium (taxes and P&C contingencies are fixed %s).
4. Compute the Discount Percent is calculated as:
Discount Percent =

Expense Reduction
Expense Reduction
=
1-"all other expenses" as a % of discounted premium 1.0-Taxes-Profit & Cont.

Premium Range ($)
1 - 5,000
5,001 - 100,000
100,001 - 500,000
500,001 +

5. Total Discount =

Premium in
the range
(6)
5,000
95,000
375,000
0



Expense
Reduction
(7)
0%
22%-16.5%= 5.5%
22%-12%= 10%
22%-8.5%= 13.5%

Discount
Percent
(8) = (7)/[1.0-[(4)+(5)]
0
5.882%
10.695%
14.439%

Premium
Discount
(9)=(6)*(8)
0
5,588
40,106
0
45,694

(Discount Percent)*(Premium in range) = 45,694.

premium range

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2002 exam
29. Use the Worker's Compensation Method to calculate the discounted premium. Show all work.
Given Standard premium = $500,000
Step 1. Partition the $500,000 into "gradations" as stated in the problem (the first $10,000 of premium; the
next $200,000 of premium, etc.)
Step 2. Compute Premium in Range and the reduction of commissions and general expenses by gradation.
Step 3. Compute the Expense reduction (the difference between the expenses in a particular
Premium Range and those expenses in the Premium Range of $0 - $10,000).
Step 4. Compute the Discount Percent, which is calculated as:
Discount Percent =

Expense Reduction
1-"all other expenses" as a % of discounted premium

Premium in
Premium Range ($) the range Commissions
(1)
(2)
0 – 10,000
10,000
15.0%
10,001 - 210,000
200,000
11.25%
210,001 - 410,000 200,000
8.44%
410,001 – 610,000 90,000
6.33%
(2i+1) = (2i) * .75.
(3i+1) = (3i) * .75.
(6) = [(.15+.10) - (4)].
(7) = (6)/[1.0 - (5)]

Gen Exp
(3)
10.0%
7.5%
5.63%
4.22%

All Other
Expenses
(4)=(2)+(3) (5)
25%
8.0%
18.75%
8.0%
14.07%
8.0%
10.55%
8.0%

Expense
Reduction
(6)
0.00%
6.25%
10.93%
14.45%

Discount
Percent
(7)
0.00%
6.79%
11.88%
15.71%

Step 5: Compute the premium discount and the discounted premium.
Premium discount = Sumproduct[(1)*(7)] = [200,000 * .0679 + 200,000 * .1188 + 90,000 * .1571] = 51,483
Discounted premium = 500,000 – 51,483 = 448,516.

Solutions to questions from the 2011 exam
16. Calculate the total amount of premium discount for a policy with premium of $180,000.
Question 16 – Model Solution
Prem Range

(1)
Prem in Range

(2)
Prod + Gen

(3)
Diff. From 1st Range

(4) = (3) / (1-.08)
Discount

(5) = (4) * (1)
$Discount

0-7500
7500-75000
75000-200000
200000+

7500
67500
105000
0

.24
.18
.13
.09

0
.06
.11
.15

0
.06522
.1196
.163

0
4402.17
12554.35
0
16956.52

(1)= 7,500 – 0; 75,000-7,500; 180,000-75,000;

(3)= (2Row 1)-(2);

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(4) = (3)/[1.0 –taxes - profit)]

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
3b. Loss Constants
Solutions to the questions from the 1995 exam
Question 35.
(a) 1. The experience of large firms receives greater credibility than that of small firms, and thus
large firms have greater incentives to reduce losses.
2. Safety programs require large fixed costs, which may be more cost effective for larger firms.
(b) Chosen such that loss ratio for small risks (with premium < 2,000) = loss ratio for large risks
(with premium > 2,000).
Let X = the loss constant per risk. Solve the equation for $X. ,
$63,000
$144,000
=
. $X = 160.
$75,000 + 100 * $X
$200,000 + 50 * $X
(c) This question is no longer applicable to the content covered in this chapter

Solutions to questions from the 1998 exam
Question 34.
a. Explanations to why loss experience tends to be better for large risks than for small risks.
1. Good loss experience reduces the cost of future insurance. Since experience rating gives more weight
(more credibility) to a larger risk's experience, it gives them more incentive to reduce losses.
2. The large expenditures required to implement safety programs are more cost effective for larger risks than
for smaller risks.
3. Post injury and back-to-work programs may not be offered by smaller risks, since severe injuries do not
occur with great frequency.
b. Loss constants are flat dollar premium additions designed to flatten loss ratios by size of risk.
The loss constant can be calculated in two ways.
Method 1. Loss Constants Applied to Small Risks Only.
The loss constant is chosen such that loss ratio for small risks (with premium < $1000) is equal to
the loss ratio for large risks (with premium > 1,000).
Based on the given information, compute the loss ratios for small risks and large risks:
Number of Risks
Premium Range
Earned Premium Incurred Losses Loss Ratio
Small Risks
1,000
$ 0 - 1,000
1,200,000
1,100,000
.917
Large Risks
2,000
> $1,000
13,000,000
10,000,000
.769
Let X = the total loss constant premium. Solve for X such that the loss ratio for small risks will equal the
loss ratio produced by large risks.

1,100,000
.769 . X = 230,429. Since there are 1,000 small risks, the loss constant equals $230.43
1,200,000  X
Method 2. Loss Constants Applied to All Risks.
The use of a loss constant for all risks flattens the loss ratio for small risks.

1100
, ,000
10,000,000

. X = 294, 871.
1,200,000  X 13,000,000  2 X
Given 1,000 small risks, the loss constant equals $294.87

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to the questions from the 2000 exam
Question 48.
a. The purpose of an expense constant is to charge for expenses which do not vary by policy size
(e.g. setting up files), and is uniform for all risks.
b. An expense constant is important for small policies since it ensures that an adequate premium is being
charged. Without an expense constant, the premium computed for small insureds may be so low that it
would be inadequate to cover the expenses of writing the policy.
c. Loss constants (flat dollar premium additions either for all or small insureds) are a means of flattening the
loss ratios by size of risk.
d. Given the following data, calculate the loss constant. Assume loss constants are to be used for risks with
annual premium of $1,000 or less.
The loss constant is chosen such that loss ratio for small risks (with premium < $1000) is equal to
the loss ratio for large risks (with premium > 1,000).
Based on the given information, compute the loss ratios for small risks and large risks:
Let X = the total loss constant premium. Solve for X such that the loss ratio for small risks will equal the
loss ratio produced by large risks.
Premium Range
$ 0 - 1,000
> $1,000

# of Risks
200
200

Earned Premium
$130,000
$960,000

Incurred Loss
$104,000
$720,000

Loss ratio
.80
.75

Method 1. Loss Constants Applied to Small Risks Only.
104,000
 .75 . X = 8,666.66. Since there are 200 small risks, the loss constant equals $43.33
130,000  X

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.

Section 4: Insurance to Value (ITV)
Solutions to questions from the 1990 exam
Question 49.

Size of Loss (L)
(1)
0 < L <$ 50,000
$50,000 < L < 100, 000
$100,000 < L < $150,000
$150,000 < L < $200,000
$200,000 < L < $250,000
$250,000 < L
TOTAL

Number Of
Losses
(2)
340
75
50
25
10
0
500

Conditional
Pr[of Loss]
(3)=(2) / 2(tot)
.68
.15
.10
.05
.02
0

Unconditional
Pr[of Loss]
(4) = (3)*.125
.085
.01875
.0125
.00625
.0025
0

Dollars Of
Loss
(5)
$ 3,762,000
5,625,000
6,375,000
4,463,000
2,275,000
0
$22,500,000

Pure premium
(6)=[(5)/(2)]*(4)
941
1,406
1,594
938
375
0
5,254

Note: For L > 150,000, column (6) pure premium = $150,000 * (4)
The pure premium rate per $100 for the $150,000 building = 5,254 / [150,000/100] = 3.502.
(b). This rate is higher.
Whenever losses < F are possible, the PP rate should decrease as F increases.

Solutions to questions from the 1992 exam
Question 5.
1. T.
2. F.
3. T.

Answer C.

Solutions to questions from the 1994 exam
C
C

 Ls(L)dL + F[1- s(L)dL] 


0

R = f 0


F / 100







Question 43.



(a).

Co-Ins
%
.20
.50
.80

General Pure premium rate
Equation
.02*

[.50(10,000) + (1-.50) * (40,000)]
40,000 / 100

02*

[.50(10,000)+.2 * 70,000 + (1-.70) * (100,000)]
100,000 / 100

02*

[.50 * 10,000+.2 * 70,000+.05 * 120,000 + (1-.75) * (160,000)]
160,000 / 100



Pure prem
rate per $100
1.25

.98
.8125

(b). This question no longer applies to the content covered in this chapter

Exam 5, V1b

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 1995 exam
Question 46.
C

C
 Ls(L)dL + F[1- s(L)dL] 


0
0
.

R= f


F / 100






Note the mistake in the example. For a replacement cost of $100,000 and a size of loss interval
between 21,000 and 50,000, the arithmetic mean loss cannot be 3,000, but is more likely to be 30,000.



See (b) below.
(a). $200,000 replacement cost, at 50% co-insurance . C = cV = .50 * 200,000 = 100,000.
.10[.70*(3,000) + .15*(35,000) + .09*(65,000) + .04*(95,000) + .02*(100,000)] / [100,000 / 100] = $1.90.
(b). The pure premium per $100 computed in (a) of $1.90 is higher than the computed pure premium rate for
the house if it were insured for 200,000 ( which is equivalent to a 100 % co-insurance rate).
Whenever losses < F are possible, the PP rate should decrease as F increases, even if large losses
predominate.
200,000 replacement cost, at 100% co-insurance . C = cV = 1.0 * 200,000 = 200,000.
.10[.70*(3,000) + .15*(35,000) + .09*(65,000) + .04*(95,000) + .01*(150,000) + .01(190,000)] / [200,000 / 100] =

$1.02.
(c). $100,000 replacement cost, at 100% co-insurance . C = cV = 1.0*100,000 = 100,000.
.10[.80*(2,000) + .10*(30,000) + .08*(60,000) + .02*(95,000)] / [100,000 / 100] = $1.13.
Since there is a probability of a loss > 100,000 associated with a $200,000 replacement cost policy, and
since the policy limit of $100,000 caps the indemnity at $100,000 on a $100,000 policy, the pure
premium rate associated with the latter (1.13) is < the pure premium rate associated with the former
(1.90).

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 1996 exam
Question 44.
(a) The computation of % coinsurance rates. Begin with the pure premium rate equation.
C
C

 Ls(L)dL + F[1- s(L)dL] 
Symbol
Description


f
frequency of loss
0

R = f 0


c
coinsurance %
F / 100


V
property value




F
policy face (expressed in $'s)
C
cV
L
Loss amount



Assume that L is a continuous variable, "because this assumption clarifies some relationships which might be
nearly unintelligible in discrete notation."
"Pure premium coinsurance rates are computed on the assumption of a policy face equal to the
coinsurance requirement."
Since the assumed policy face, F, = C = cV = .60 * $500,000, and using the information in the table
below, we can compute the pure premium rate per $100 for 60% coinsurance as follows:
Coinsurance
Percentage
(Cn)
.40
.60
.80
1.00

Conditional
Probability of
Losses in Interval
[Cn-1 ,Cn]
65%
20%
10%
5%

Cumulative
Conditional
Probability of Loss > Cn
.35
.15
.05
0

Arithmetic Mean
Loss in Interval
[Cn-1 ,Cn]
$100,000
$250,000
$350,000
$500,000

C = cV
$200,000
$300,000
$400,000
$500,000

 $100,000*.65 + $250,000*.20 + $300,000 * (1.0 - .85) 
Therefore, R = .05* 
 = 2.67.
$300,000 / 100


(b) "If a policy should be less that its agreed amount, coinsurance reduces every indemnity payment
proportionately."
The proportion is based on the ratio of the amount of insurance purchased to the amount of insurance
assumed in the pure premium coinsurance rate calculation.
We are given that the insured purchased a $200,000 policy. The 60% coinsurance requirement
called for the purchase of a $300,000 ($500,000 * .60) policy.

 $200,000 
Therefore, the indemnity paid to the insured = $80,000* 
 = $53,333.33.
 $300,000 

Exam 5, V1b

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 1998 exam
Question 5.
The formula to calculate the pure premium rate per $100 of insurance:
C
C

 Ls(L)dL + F[1- s(L)dL] 


0
0


R= f


F / 100









At Least
0%
10%
20%
30%
40%
50%

Losses
Less Than
10%
20%
30%
40%
50%

Unconditional
Probability
Of Loss
.0100
.0075
.0050
.0035
.0020
.0025

(%)
4%
14%
23%
33%
43%
50%

Arithmetic Mean Loss
100,000 risk
4,000
14,000
23,000
33,000
43,000
50,000

Note that the unconditional probability of a loss exceeding 50% of its value is
.0010+.0005+.0003+.0002+.0005 = .0025. In addition, the policy face equals the co-insurance
requirement (C = cV = .50 (100,000) = 50,000).
Co-Insurance %
.50

Pure premium rate per $100

[.01*4,000.0075*14,000.005*23,000.0035*33,000.002*43000.0025*50,000]
= $1.17
50,000 /100

Answer E.

Solutions to questions from the 1999 exam
Question 15.
Given:
Coinsurance Requirement:
Full Value of Structure:
Amount of Insurance on Structure:
Amount of Loss:

80%
$1,000,000
$700,000
$600,000

c
V
F
L

 700,000 
 F 
Since I  L 
  525, 000
 , then I  $600, 000 * 
 cV 
 .80*1,000,000 
The coinsurance penalty equals loss amount - the indemnity payment = 600,000- 525,000 = 75,000.
Answer D.

Exam 5, V1b

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2000 exam
Question 24.

The general pure premium rate equation for percentage co-insurance
is :

Co-Ins
%
.50
.75

C
C

 Ls(L)dL + F[1- s(L)dL] 


0
0


R= f


F / 100









General Pure premium rate
Equation

Pure prem
rate per $100

[.50(.04*100,000) .25*(.18*100,000) .15*(.40*100,000)  (1-.90)*(.50*100,000)]
.05*
50,000 /100

1.75

.05 *

[.50(.04*100,000) .25*(.18*100,000) .15*(.40*100,000) .07*(.70*100,000)]

75,000 /100

.05 *

[(1-.97)*(.75*100,000)]
75,000 /100

1.31

the absolute difference between the pure premium rate per $100 for a 50% coinsurance clause and a 75%
coinsurance clause is 1.75 – 1.31 = .44.
Answer E.

Solutions to questions from the 2001 exam
Question 7.
The general pure premium rate equation for percentage co-insurance
is :

C
C

 Ls(L)dL + F[1- s(L)dL] 


0

R = f 0


F / 100









Unlike problem 24 from the 2000 exam, we are not given in this particular problem the value of the insured
property, nor the loss frequency (f). However, this information is not necessary to compute the ratio of the pure
premium rate per $100 for a 60% coinsurance clause to the pure premium rate per $100 for a 40% coinsurance
clause.
Co-Ins
%
.40

General Pure premium rate
Equation

.60

[.05*.12 .025*.30 .015*.52  (007 .003)*.60]
.60

[.05*.12 .025*.30  (.015 .007 .003)*.40]
.40

Pure prem
rate per $100
.0588
.0455

The ratio of the pure premium rate per $100 for a 60% coinsurance clause to the pure premium rate per $100 for
a 40% coinsurance clause is .0455 ÷ .0588 = .77381
Answer E.

Exam 5, V1b

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2002 exam
Question 42
C
C

  Ls(L)dL+F[1- s(L)dL] 
0
:
The general pure premium rate equation for percentage co-insurance is R=f  0


F/100




Using the data given in the problem, and the discrete counterpart to the continuous function above, the pure
premium rate per $100 for a 50% coinsurance clause is computed as follows:

General Pure premium rate
Equation
.03 *

[.75(.09*$200,000)  .12*(.40*$200,000) .08*(.50*$200,000)  (1-.95)*(.50*$200,000)]

(.50*$200,000) /100

Exam 5, V1b

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Pure prem
rate per $100
1.083

 2014 by All 10, Inc.

Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2003 exam
40. (2.25 points) An insurer writing fire insurance uses coinsurance in its rating structure by means of
an "average clause." A coinsurance percentage of 80% applies to all policies. Based on the
following information, answer the questions below. Show all work
Policy

Amount of Loss

Property Value

Face Amount of
Insurance

1
2
3

$50,000
$155,000
$375,000

$200,000
$160,000
$480,000

$150,000
$120,000
$400,000

a. (1.5 points) For each of the policies above, calculate the indemnity payment made by the insurer.
Note: “Insurance to value" (ITV) exists only if property is insured to the exact extent ($ amount or % value)
assumed in the rate calculation. To evaluate coinsurance applications, the following formulas are
given: the coinsurance requirement C = cV the coinsurance deficiency d = [cV – F] CAR = a =
[F/cV] < 1.00.
Compute ITV for each policy:
For policy 1, ITV = $150,000/$200,000 = .75. This policy does not meet the coinsurance requirement.
For policy 2, ITV = $120,000/$160,000 = .75. This policy does not meet the coinsurance requirement.
For policy 3, ITV = $400,000/$480,000 = .833. This policy does meet the coinsurance requirement.
Note: A standard coinsurance clause may be represented algebraically as follows:
I = L*[F/cV], subject to two constraints:
1.
I < L The indemnity payment cannot exceed the loss. This constraint is in concert
with the principle of indemnity, which states that no insured should profit from any loss.
2.
I < F The indemnity payment cannot exceed the policy face. This sets the overall
limit on the amount insurance payable from a single occurrence.
For policy 1, I  L*

FV
$120,000
FV
$150,000
$50,000*
$46,875 . For policy 2, I  L *
 $155, 000 *
 $145, 312 ,
.80*$160,000
cV
.80*$200,000
cV

but is capped at policy limits of $120,000. For policy 3, since the coinsurance requirement was met and the
loss was less than policy face, indemnity equals loss amount $375,000.
b. (0.75 points) For each of the policies above, calculate the additional insurance, if any, that would
have been required for the insurance company to indemnify the full amount of the loss.
For policy 1, the coinsurance requirement is $160,000, so an additional $10,000 is needed. For
policy 2, an additional $35,000 is needed ($155,000 - $120,000). For policy 3, no additional amount
is needed, since the policy limits purchased meet the coinsurance requirement and the loss is less
than the policy limit.

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2004 exam
41. (4 points) Given the following information on an individual property policy, answer the questions below.
Show all work.
a. (2 points) The insured purchases a policy insuring the property to 80% of value. Determine the
premium charged for the policy
Step 1: Write an equation to compute the premium charged for a policy insuring the property to 80% of value:
L F

 L F

E  I   f *   L*s ( L ) dL  F [1  s ( L )]  

 Expected Losses  
 L 1

L 1
Premium= 
=
PLR
PLR

 






Step 2: Using the equation in Step 1, and the data given in the problem, solve for the expected losses under
the policy and then for the premium.

E  I   $200, 000*.10*[.70*.10  .20*.50  (1  .70  .20)*.80]  $5, 000
Premium = $5,000/.65 = $7,692
b. (1 point) The insured instead purchases a policy insuring the property to 70% of value. Assuming the
same rate per $100 of insured value as in part a. above, determine the expected loss ratio for this
policy.
Step 1: Determine the rate per $100 charged under the policy insuring the property to 80% of value, and
then compute the premium charged for a policy insuring the property to 70% of value.
The rate per $100 charged under the policy insuring the property to 80% of value is
Premium/[AOI/100]. In this problem, the rate per $100 is $7,692/[.80 * 200,000/100] = $4.81
Thus, the premium charged for a policy insuring the property to 70% of
value is $4.81 * [200,000/100 * .70] = $6,734.
Step 2: Determine the Expected Losses under the policy:

E  I   $200, 000*.10*[.70*.10  .20*.50  (1  .70  .20) *.70]  $4,800
Step 3: Compute the loss ratio as the ratio of the results from Step 2 and Step 3:
Loss Ratio = $4,800/$6,734 = .7131= 71.3%
c. (1 point) Assume the insurer incorporates a coinsurance clause into the policy. The insured continues
to insure the property to 70% of value. What is the expected loss ratio for this policy? Briefly explain
your answer.
Once the insurer incorporates a coinsurance clause into the policy, the expected loss ratio for the policy
will equal the permissible loss ratio underlying the expected rate, which in this case is 65%. This is due
to the fact that indemnification for losses under the policy will be reduced by the amount of coinsurance
the insured maintains relative to the amount the insured is required to maintain (80% in this problem).
This can be demonstrated numerically as follows:

E  I   $200, 000*.10*[.70*.10*.7 / .80)  (.20*.50*.7 / .8)  (1  .70  .20)*.70]  $4,375
Loss Ratio = $4,375/$6,731 = 65.0%

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2005 exam
51. (2 points)
a. (1 point) Calculate the pure premium rate per $100 of insurance for a policy face equaling $300,000.
Step 1: Write an equation to determine the insured’s pure premium rate for each unit of face amount.
L F
 L F

R  f *   L * s ( L)dL  F [1   s ( L)]  / [ F / 100] , where f is the frequency of loss (i.e. the
L 1
 L1


number of insureds divided by the number of policies).
Step 2: Using the equation in Step 1, and the data given in the problem, compute the pure premium rate
per $100 of insurance for a policy face equaling $300,000.
f = 20/1,000 = .02

R  2% *

[.50($50,000) +(.20)*($250,000)+(1-.70)*($300,000)]
 $1.10
$300,000/100

b. (1 point) Does the pure premium rate per $100 of insurance for a $500,000 policy face differ from the
rate for the $300,000 policy face? Briefly explain your answer.
As the policy face (F) increases, the pure premium rate decreases at a decreasing rate, if small losses
F

outnumber large ones. Here, the second derivative is negative

dR

dF

 f *  L * s ( L)dL
0

Since small losses predominate in this example, we show the pure premium rate per $100 of insurance
for a $500,000 policy is smaller than that for a $300,000 policy face as follows:

R  2%*

[.50($50,000) +(.20)*($250,000)+(.3)*($500,000)] $4,500

 $0.90
$500,000/100
$5, 000

Solutions to questions from the 2006 exam
44. (2.5 points) Find the premium rate per $100 of insurance for a policy face equaling $400,000.
Show all work.
Step 1: Write an equation to determine the insured’s pure premium rate per $100 of insurance for a
policy face equaling $400,000.
L F
 L F

R  f *   L * s ( L)dL  F [1   s( L)]  / F , where f is the frequency of loss (i.e. the number of
L 1
 L 1

losses divided by the number of exposures), and s( L) represents the percentage of losses exactly

equaling L, or the conditional probability of a loss of L, given some loss greater than zero.
Step 2: Using the equation in Step 1, and the data given in the problem, compute the pure premium rate
per $100 of insurance for a policy face equaling $400,000.
f = 20/1,000 = .02

R  2%*

Exam 5, V1b

[.50($100,000) +(.20)*($200,000)+(.10)*($300,000)+(1-.80)*($400,000)]
 $1.00
$400,000/100

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2007 exam
49. (1.0 point) Calculate each of the following:
a.
(0.25 point) Coinsurance requirement.
b.
(0.25 point) Coinsurance apportionment ratio.
c.
(0.25 point) Coinsurance deficiency.
d.
(0.25 point) Maximum coinsurance penalty.
Model Solution
a. The coinsurance requirement may be in the form of a stated sum or a specified % of the value of the
insured property. Thus, the coinsurance requirement equals $300,000 * 0.80 = $240,000
b. The coinsurance apportionment ratio (CAR) is the ratio of the amount of insurance purchased to
either a (i) stated sum, or (ii) a specified % of the value of the insured property. The maximum
coinsurance apportionment ratio is 1.00. Thus, the $200,000/$240,000 = 0.83333
c. The coinsurance deficiency is the amount by which a coinsurance requirement exceeds the amount
of the carried insurance. Thus, the coinsurance deficiency equals $240,000 - $200,000 = $40,000
d. A coinsurance penalty is the amount by which the indemnity payment resulting from a loss is reduced
due to the coinsurance clause. The face amount that should have been purchased (given the
coinsurance requirement) equals $240,000. Since $200,000 was purchased instead, the maximum
penalty = $200,000 * (1 - $200,000/$240,000) =$33,333.33. Due to underinsurance, the maximum
penalty occurs when the loss equals the face value of policy.

Solutions to questions from the 2008 exam
Model Solution – Question 36
a. Calculate the coinsurance deficiency.
b. Calculate the coinsurance apportionment ratio.
c. Calculate the maximum coinsurance penalty possible.
d. Calculate the coinsurance penalty for a $300,000 loss.
a. The coinsurance deficiency is the amount by which a coinsurance requirement exceeds the amount
of the carried insurance. Algebraically, this is computed as cV – F, where c is the co-insurance
requirement as a % of the insured property, V = the value of the insured property and F = Face value
of the property.
Based on the givens in the problem, the coinsurance requirement equals 0.80 * $350,000 =
$280,000, F = $275,000 and thus, the coinsurance deficiency equals $280,000 - $275,000 = $5,000
b. The coinsurance apportionment ratio (CAR) is the ratio of the amount of insurance purchased to
either a (i) stated sum, or (ii) a specified % of the value of the insured property. The maximum
coinsurance apportionment ratio is 1.00. Thus, $275,000/$280,000 = 0.9821.
c. The maximum coinsurance penalty occurs when the Loss = F. Since CAR = 0.9821, the maximum
indemnity payment is 0.9821 * $275,000 = $270,089.28. Therefore, if L equaled F, then the maximum
coinsurance penalty would equal $275,000 - $270,089.28 = $4,910.72
d. The coinsurance penalty = e = L – I if L < F
e = F – I if F < L < cV
e = 0 if L > cV
First compute I. I = L * CAR = $300,000 * 0.98211 = 294,633
But since L = 300,000 > cV = $280,000 (the 3rd condition shown above), there is no co-insurance penalty.

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Chaptter 11 – Special
S
Classificattion
BASIC RATTEMAKING – WERNER, G
G. AND MOD
DLIN, C.
Solutions to questtions from the
t 2009 exa
am
Question: 40
Property value = 500,000
Coins. Req.
R
= 500
0,000 x 0.88 = 440,000
Face value
= 300
0,000
Coinsura
ance apportionment ratio = 300/440 = 68.18% (which
h is applied to
o the loss to d
determine the indemnity).
Max co-in penalty occ
curs when los
ss is = 300,000 (the face va
alue of the po
olicy)
pe
enalty = 300,0
000 (1– 0.681
18)= 95,454.5
50

Losss amount

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All 10, Inc.

Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2010 exam
Question 32.
a. (1 point) Calculate the coinsurance penalty.
We are given the following: L = 200,000 = amount of loss, V = value of property = 450,000,
F= face amount = 350,000 C = Co-ins req = 80%
The home is valued at $450,000 and is insured only for $350,000 despite a coinsurance requirement of 80%
(or $360,000 in this case).
Since F is $350,000 a coinsurance deficiency exists and a = 0.9722 (=$350,000 / $360,000), where a =
apportionment ratio.
The indemnity payments and coinsurance penalties for a $200,000 loss are:

F
$350, 000
 $200, 000 
 $194, 444.44
cV
$360, 000
e  L - I  $200, 000 - $194,144.44  $5,555.55
IL 

b. (0.5 point) Identify the problem with underinsurance from the insurer's perspective.
If policyholders are underinsured this is a problem from insurer’s perspective because if rates are
calculated assuming all properties are insured to value, the premium charged will not be adequate to
cover expected losses arising from those policies not insured to value.
c. (0.5 point) Identify the problem with underinsurance from the insured's perspective.
The insured may pay a lower premium if home is underinsured but in the case of a total loss, insured
won’t get payment for full value of home. If there is a co-ins penalty partial losses will be subject to
that penalty, so insured is still not compensated for full value of loss.

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2012 exam (cont’d)
15a. (0.5 point) Assuming all homeowners purchase full coverage, calculate the pure premium per
$1,000 of insurance.
15b. (0.75 point) Demonstrate with an example that the use of a fixed rate per $1,000 of insurance is
inequitable if a subset of the insured group purchases only partial coverage.
15c. (1 point) Describe two insurer initiatives that would reduce the inequity from part b. above,
including an explanation of how the inequity would be reduced.
Question 15 – Model Solution 1 (Exam 5A Question 15)
a. Expected loss = (0) (97%) + 10k(1.5%) + 50 k (.8%) + 200k (.5%) + 350k (.2%) = 2,250
PP rate = 2,250/ (350k /1,000) = $6.43
b. Assume the purchase of 10k coverage
expected loss = 0 (97%) + 10k (1-97%) = 300
if used fixed rate, the premium = 6.43 

10k
 64.3
1k

Thus the premium is inequitable 64.3 vs. 300
c. –Offer incentive for higher ITV (guaranteed replacement cost @ 100% ITV)


More insureds purchase high ITV reducing inequity
-Coinsurance clause


Reduces amount of loss paid (by ratio of face/requirement) and keeps the premium to loss adequate

Question 15 – Model Solution 2 (Exam 5A Question 15)
a. PP = .015 x 1 0k + .008 x 50k + .005 x 200k + .002 x 350k = 2,250
PP rate = 2,250/ (350k /1,000) = $6.429
b. example: insured w/ 80% ITV. Face Value is 80% x 350K = 280k
PP = .015 x 10k +.008 x 50k + .005 x 200k + .002 x 280k = 2,110
PP rate = 2,110/(280k/1,000) = $7.536
If charge the rate from (a) assuming insured to full value, the home will be undercharged by
7.536 - 6.429 = $1.107 per $1000 of coverage.
c(1). a coinsurance clause would reduce the indemnity payments by the proportion of selected coverage out of
the required coverage. This would reduce the loss ratios for underinsured homes to the same loss ratio as
fully insured homes.
c(2). could begin initiatives to increase ITV through home inspections, etc, forcing underinsured homes to
purchase the right amount. This would increase premiums for underinsured homes and equalize loss ratios.

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Chapter 11 – Special Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Question 15: Examiner’s Comments
a. This question was generally well-answered by candidates. A common mistake was to forget to divide
by the amount of insurance. Another common mistake was to divide by 1000s of premium instead of
amount of insurance.
b. Many amounts of insurance were commonly used by candidates and were deemed acceptable.
A common demonstration by candidates was to calculate the premium that would be charged with the
rate in A) and compare this with the expected loss of underinsured risk to demonstrate the
inadequacy.
Some candidates calculated loss ratios or compared the fixed rates that should be charged in a) with
b) to demonstrate an inequity. All those solutions were accepted and received full marks.
Many candidates demonstrated poorly the inequity created by the situation in b). Some only
calculated the rate per $1000 of insurance for underinsured risks and did not explain why there was
an inequity.
c. A common mistake for candidates was to simply list and describe initiatives to increase insurance to
value. However, the question clearly asked for an explanation of how the measure reduces inequity.
Another common mistake was to identify an ITV initiative that would have no impact on the example in b).
For example, the indexing of amounts of insurance at each renewal for all risks would not reduce inequity
over time caused by a subset buying partial coverage.

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Chapter 12 – Credibility
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Sec
1
2
3
4
5
6
1

Description
Necessary Criteria For Measures Of Credibility
Methods For Determining Credibility Of An Estimate
Desirable Qualities Of A Complement Of Credibility
Methods For Developing Complements Of Credibility
Credibility When Using Statistical Methods
Key Concepts

Pages
216 - 216
216 - 223
223 - 224
224 - 236
236 –236
238 - 238

Necessary Criteria For Measures Of Credibility

216 - 216

The credibility (Z) given to observed experience, assuming homogenous risks, is based on three criteria:
1. 0 < Z < 1 (i.e. no negative credibility and capped at fully credible).
2. Z should increase as the number of risks increases (all else being equal).
3. Z should increase at a non-increasing rate.

Methods For Determining Credibility Of An Estimate

216 - 223

As defined in Actuarial Standard of Practice (ASOP) No. 25, credibility is “a measure of the predictive
value in a given application that the actuary attaches to a particular body of data.”
Two common credibility methods are classical credibility and Bühlmann credibility.
Both methods calculate a measure of credibility to blend subject experience and related experience.
A third method, Bayesian analysis, introduces related experience into the actuarial estimate in a probabilistic
measure (it does not explicitly calculate a measure of credibility).
1. Classical Credibility Approach
The classical credibility approach (a.k.a. limited fluctuation credibility) is the most frequently used method in
insurance ratemaking. The goal is to limit the effect that random fluctuations in the observations have on the
risk estimate.
Z is the weight assigned to the observed experience (a.k.a. subject experience or base statistic) and the
complement of Z is assigned to some related experience (as shown in the following linear expression):
Estimate = Z x Observed Experience + (1.0 - Z) x Related Experience.
First, determine the expected number of claims, (E(Y), for the observed experience to be fully credible (Z=1.00).
The observed experience is fully credible when the probability (p) that the observed experience will not differ
significantly from the expected experience by more than some arbitrary amount (k).
Stated in probabilistic terms:

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Pr[(1- k)E(Y)  Y  (1+k)E(Y)] = p

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Chapter 12 – Credibility
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
According to the Central Limit Theorem,

S  E (S )
~ N (0,1).
Var ( S )

Therefore, the probabilistic expression can be transformed as follows:

 (1  k ) E ( S ) - E ( S ) S  E ( S ) (1  k ) E ( S ) - E ( S ) 
Pr 


 p
Var ( S )
Var ( S )
Var ( S )


Since the normal distribution is symmetric about its mean, this is equivalent to:

 (1  k ) E ( S )  E ( S ) 

  z( p 1) , where z( p 1) is the value in the Standard Normal (SN) table for
2
2
Var ( S )


values (p+1)/2.
Make simplifying assumptions about the observed experience:
• Exposures are homogeneous (i.e. each exposure has the same expected number of claims).
• Claim occurrence follows a Poisson distribution; thus E(Y) = Var(Y).
• There is no variation in the size of loss (i.e. constant severity).

 kE (Y ) 
  z( p 1)
2
 E (Y ) 

Based on those assumptions, the expression above can be simplified to: 

 z( p 1)
2
Thus, the expected number of claims needed for full credibility can be expressed as: E (Y )  
 k







Example: Full and Partial Credibility Calculations
Assume an actuary regards the loss experience fully credible if there is a 90% probability that the observed
experience is within 5% of its expected value.
 This is equivalent to a 95% probability that observed losses are no more than 5% above the mean.
In the SN table, the 95th percentile is 1.645 standard deviations above the mean; therefore, the expected
2

 1.645 
number of claims needed for full credibility is: E (Y )  
  1, 082
 0.05 


If the number of observed claims > the standard for full credibility (1,082 in the example), the measure of
credibility (Z) is 1.00: Z  1.00 where Y  E (Y )



If the number of observed claims is < the standard for full credibility, the square root rule is applied to
calculate Z: Z 

Y
, where Y  E (Y ).
E (Y )

In the example, if the observed number of claims is 100, Z 

100
 0.30.
1, 082

The square root formula, with a maximum of 1.0, meets the three criteria for Z.

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Chapter 12 – Credibility
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Example: A full credibility standard based on the number of exposures (rather than the number of claims).
The exposure standard is calculated by [number of claims needed for full credibility/ expected frequency].
The number of claims and exposures needed for full credibility using example values for k and p:
(1)
(2)
(3)
(4)
(5)
(6)
Number of
Number of
Claims for
Projected Exposures for
k
p
Zp/2
Full Credibility Frequency Full Credibility
5%
90%
1.645
1,082
5.0%
21,640
10%
90%
1.645
271
5.0%
5,420
5%
95%
1.960
1,537
5.0%
30,740
10%
95%
1.960
384
5.0%
7,680
5%
99%
2.575
2,652
5.0%
53,040
10%
99%
2.575
663
5.0%
13,260
(3)= From Normal Distribution Table
(4)= [(3) / (1)]^2
(6)= (4) / (5)
Assuming there is variation in the size of losses, the number of claims needed for observed data to be
considered fully credible is as follows:

 zp
E (Y )   2
 k


  2 
2
  1  s  , where  s is the coefficient of variation squared.
   s2 
s2


Example - Calculating the credibility-weighted pure premium estimate
Assume:
• Full credibility is set so that the observed value is to be within +/-5% of the true value 90% of the time.
• Exposures are homogeneous, claim occurrence follows a Poisson distribution, and no variation in claim
costs exists.
• The observed pure premium of $200 is based on 100 claims.
• The pure premium of the related experience is $300.
Based on values of k and p above, the corresponding value on the SN table is 1.645.
2



 1.645 
The standard for full credibility is therefore: E (Y )  
  1, 082
 0.05 



Since observed claims are < 1,082, compute Z using square root rule: Z  Min 




100
,1.00   0.30
 1, 082


The credibility-weighted estimate is $270 (=0.30 x $200 + (1-0.30) x $300).

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Chapter
C
12
1 – Cred
dibility
BASIC RATTEMAKING – WERNER, G
G. AND MOD
DLIN, C.
Comme
ents on Classical Credibility Approac
ch
3 Advan
ntages:
1. It is the most co
ommonly used
d and thus ge
enerally accep
pted.
2. Th
he data requirred is readily available.
a
3. Th
he computatio
ons are straigh
htforward.
Disadva
antage: Simp
plifying assum
mptions may not
n be true in practice (e.g.. no variation in the size off losses).
2. Bühlm
mann Credibility
The goal of
o Bühlmann credibility (a.k.a. least squ
uares credibiliity): minimize
e the square o
of the error be
etween the
estimate and
a the true expected
e
valu
ue of the quan
ntity being esttimated.
The cre
edibility-weighted estimate is defined as: Estimate = Z x Observed
d Experience + (1.0 - Z) x P
Prior Mean.
This forrmula conside
ers a prior me
ean, the actua
ary’s a priori a
assumption off the risk estim
mate (wherea
as classical
credibility considered
d related expe
erience).
Z is define
ed as follows::




Z

N
NK

A comparison of Z for d
different value
es of K is sho
own below.

N re
epresents the
e number of observations
K is
s the ratio of the expected value
v
of the process
p
varian
nce (EVPV) tto the variancce of the hypo
othetical
mea
ans (VHM) (i.e. the ratio off the average risk variance
e to the varian
nce between rrisks).
i. K can be diffic
cult to calculatte and the me
ethod of calcu
ulation is beyo
ond the scope
e of this text.
ii. Since
S
K is a constant
c
(for a given situatiion), Z meets the criteria lissted earlier.

es this visually
y:
The chart demonstrate
Z appro
oaches 1.0 as
symptotically as
a N gets larg
ger (the class ical credibilityy measure eq
quals 1.0 at th
he point the
numberr of claims or exposures eq
quals the full credibility
c
sta ndard (Nf))

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Chapter
C
12
1 – Cred
dibility
BASIC RATTEMAKING – WERNER, G
G. AND MOD
DLIN, C.
The chart below shows
s a compariso
on of Z at diffe
erent numberrs of observattions (N) unde
er classical an
nd
Bühlmann
n approaches
s.

Commentts:
 Bühlmann cred
dibility estima
ate is closest to
t the classica
al credibility e
estimate when K equals 5,000 (i.e.
th
he line with da
ashes and dots is close to the solid line)), for these sp
pecific valuess of Nf and K a
and for a
re
elatively small number of observations.
o
 As
A N gets large
er, the Bühlm
mann credibilitty estimate is closest to the
e classical cre
edibility estim
mate when
K equals 1,500
0 (i.e. the dottted line).
 Practitioners using classical credibility as
ssume there iis no variation
n in the size o
of losses and that the
ris
sks in the sub
bject experien
nce are homo
ogeneous. If th
hese assump
ptions are mad
de with least squares
crredibility, then
n
i. VHM = 0 (be
ecause all exp
posures have
e exactly the ssame claim distribution).
ii. when VHM = 0, then Z = 0 (no credib
bility is assign
ned to the obsserved experie
ence).
The assum
mptions unde
er the Bühlma
ann credibility formula are a
as follows:
* (1.0
0 - Z) is applie
ed to the priorr mean.
* Risk parameters
s and risk proc
cess do not shift over time
e.
* The
e EVPV of the
e sum of N ob
bservations in
ncreases with N.
* The
e VHM of the sum of N obs
servations inc
creases with N
N.
Simple Ex
xample
Calculate the Bühlman
nn credibility-w
weighted estim
mate assumin
ng the followin
ng:
• The observed value is $200
0 based on 21
1 observation s.
• EVPV = 2.00, VHM
V
= 0.50 and
a the prior mean is $225
5.

Thus,

K

2
21
1
EVPV 2.00
 4.000, Z 
 0.84; and
=
0.50
4
VHM
21  4.00

Bühlmann
n Credibility-w
weighted Estim
mate = 0.84 x $200 + (1- 0
0.84) x $225 = $204.

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Chapter 12 – Credibility
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Comments on Least Squares Credibility (LSC)
 It is used and is generally accepted.
 The major challenge is determining EVPV and VHM.
 It is based on assumptions that needs to be evaluated for suitability purposes (like classical credibility).
Bayesian Analysis
 There is no calculation of Z, but a distributional assumption must be made.
 Is based on a prior estimate to be adjusted to reflect the new information (introduced into the prior
estimate in a probabilistic manner, via Bayes Theorem).
This differs from LSC where new information is introduced into the prior estimate via credibility weighting.
 Bayesian analysis is not used as commonly as Bühlmann credibility (due to the greater complexities of its
probabilistic nature).
Notes:
 Bühlmann credibility is the weighted least squares line associated with the Bayesian estimate.
 The Bayesian estimate is equivalent to the LSC estimate (in certain mathematical situations).

Desirable Qualities Of A Complement Of Credibility

223 - 224

The credibility-weighted actuarial estimate using classical credibility is:
Estimate = Z x Observed Experience + (1- Z) x Related Experience.
Note: Theoretically when credibility is based on the Bühlmann approach, the complement of credibility should be the
prior mean (however, actuaries have used other related experience when Bühlmann credibility is used).

Once Z is determined, the next step is to select the related experience (the “complement of credibility”).
According to ASOP 25, the related experience:
i. should have frequency, severity, or other characteristics to be similar to the subject experience.
ii. should not be used (if it does not or cannot be adjusted to meet such criteria).
The complement of credibility (CC) can be more important than the observed data (e.g. if the observed
experience varies around the true experience with a standard deviation equal to its mean, it will probably
receive a very low credibility. Therefore, the majority of the rate (in this context, expected loss estimate) will be
driven by the complement of credibility.
In “Complement of Credibility” Boor states desirable qualities for a complement of credibility:
1. Accurate: A CC that causes rates to have a low error variance around the future expected losses being
estimated is considered accurate.
2. Unbiased: Differences between the complement and the observed experience should average to 0 over time.
Accurate vs. Unbiased:
 An accurate statistic may be consistently higher or lower than the following year’s losses, but it is
always close.
 An unbiased statistic varies randomly around the following year’s losses over many successive years,
but it may not be close.
3. Independent: The complement should also be statistically independent from the base statistic (otherwise,
any error in the base statistic can be compounded).
4 and 5. Available and Easy to Compute: If not, the CC is not practical and justification to a third party (e.g.
regulator) for approval is needed.
6. Logical relationship (to the observed experience): is easier to support to any third party reviewing the
actuarial justification.

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Chapter 12 – Credibility
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
4

Methods For Developing Complements Of Credibility

224 - 236

A variety of complements are used in practice.
 First dollar ratemaking is performed on products that cover claims from the first dollar of loss (or after
some small deductible) up to some limit (e.g. personal auto, HO, WC, and professional liability insurance)
 Excess ratemaking is performed on insurance products covering claims that exceed some high
attachment point (e.g. personal umbrella policies, large deductible commercial policies, and excess
reinsurance).

I. First Dollar Ratemaking
Boor describes six commonly used methods for developing complements for first dollar ratemaking:
• Loss costs of a larger group that includes the group being rated
• Loss costs of a larger related group
• Rate change from the larger group applied to present rates
• Harwayne’s method
• Trended present rates
• Competitor’s rates
The complements are discussed in terms of pure premium ratemaking (although some methods can be used
with loss ratio methods by replacing the exposure units with earned premium).

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Chapter 12 – Credibility
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
1. Loss Costs of a Larger Group that Include the Group being Rated
This complement considers a larger group’s experience to which the subject experience belongs.
Examples that may apply:
* A multi-state insurer using data from regional states to supplement the state experience being reviewed.
* A medical malpractice insurer using experience of all primary care physicians to supplement the experience
of primary care pediatricians.
* An auto insurer using data of all 16-19 year old insureds to supplement the experience of 16- year-olds.
* An insurer using data from a longer-term period to credibility-weight experience that is short-term.
Consider the following data and possibilities for a complement of credibility to the observed experience, the
latest year pure premium from Rate Group A, Class 1 ( = $50).

Candidates for complement of credibility are:
 the 3-year pure premium for Rate Group A, Class 1;
 the 1 or 3-year pure premium for Rate Group A;
 the 1 or 3-year pure premium for the total of all experience.
Another option is the total of all Class 1 experience across all rate groups (not shown).
Advantages and disadvantages of complement of credibility candidates.
* The 3-year pure premium of Rate Group A, Class 1 experience (i.e., $64) is problematic.
i. Lack of independence (the 1-year experience comprises over 1/3rd the exposures of the 3-year experience).
ii. Bias. The huge difference between the 1-year pure premium ($50) and the3-year pure premium ($64)
indicates the 3-year data may be biased (i.e. changes in loss costs makes older data less relevant).
* Using the total of all experience combined is:
i. Better with respect to independence (Rate Group A, Class 1 is a small portion of the total experience (100
out of 4,000 exposures)).
ii. Biased. The difference between the 1-year Rate Group A, Class 1 pure premium ($50) and the 1-year total
pure premium ($74) implies a bias may be present.
* The 1-year Rate Group A experience appears to be the best.
i. The Rate Group A data should reflect risks that are more similar to Class 1.
ii. The 1-year pure premium ($55) and 3-year pure premium ($57) suggests it has a low process variance.
iii. The 1-year result is not too different than the 1-year Rate Group A, Class 1 result, which suggests little bias.
* If the Class 1 data from all rate groups combined were available, it may be a reasonable option.

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Chapter 12 – Credibility
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Complement Evaluation
1. It has a lower process variance (because the complement is based on a greater volume of data than the
subject experience).
2. The subject experience has been split out of the larger group suggests that the actuary believes the subject
experience is different than the larger group.
i. If so, the larger group is a biased estimator of the subject experience.
ii. The actuary may be able to make an adjustment to reduce this bias.
The complement can include or exclude the subject experience.
i. If it excludes the subject experience, it is likely to be independent.
ii. If it includes the subject experience, ensure it does not dominate the group.
3. Loss cost data of the larger group is typically available and the loss cost is easy to compute.
4. There is a logical connection between the complement and the subject experience (as long as all the risks in
the larger group have something in common).
2. Loss Costs of a Larger Related Group
Use loss costs of a separate but similar large group (e.g. a HO insurer may use the contents loss experience
from the owners forms to supplement the contents experience for the condos form).
Complement Evaluation
1. It is biased (though the magnitude and direction of bias are unknown)
i. If the related experience can be adjusted to match the exposure to loss in the subject experience, the bias
can be reduced.
ii. In the example, consider how the exposure to loss for condos differs from owned homes and adjust the
experience accordingly.
2. Independent (since the complement does not contain the subject experience)
3. The data is readily available and the loss cost is easy to compute
4. It may be difficult to explain adjustments made to the related experience to correct for bias
5. The complement will have a logical relationship to the base statistic (if the groups are closely related)
3. Rate Change from the Larger Group Applied to Present Rates
This approach mitigates bias by using the rate change indicated for a larger group and applying it to the current
loss cost of the subject experience (rather than using the larger group’s loss costs directly)
The complement (C) can be expressed as:



Larger Group Indicated Loss Cost
C = Current Loss Cost of Subject Experience × 

 Larger Group Current Average Loss Cost 
Assume the following:
• Current loss cost of subject experience is $200.
• Indicated loss cost of larger group is $330.
• Current average loss cost of larger group is $300.
Then the complement of credibility is calculated as follows: C = $200 x $330/300 = $220.

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Chapter 12 – Credibility
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Complement Evaluation
1. This complement is largely unbiased (even when the overall loss costs for the subject experience and the
larger group are different).
2. It is likely to be accurate (assuming the rate changes are relatively small).
3. The level of independence depends on the size of the subject experience relative to the larger group.
4. The data is readily available and the calculations are very straightforward.
4. It is logical that the rate change indicated for a larger related group is indicative of the rate change for the
subject experience.
4. Harwayne’s Method
 Is used when the subject experience and related experience have different distributions (the related
experience requires adjustment before it can be blended with the subject experience).
 can be applied to the subject experience within a geographical area (e.g., a state), and the desired
complement of credibility considers related experience in other geographical areas (e.g., other states).
Other states may have distinctly different cost levels than the subject experience due to legal environment
and population density.
Example:
The complement of credibility is determined using countrywide data (excluding the base state being reviewed),
but the countrywide data is adjusted to remove overall differences between states.
Steps to calculate the complement for class 1 of state A.

State
A

All

Class
1
2
Subtotal
1
2
Subtotal
1
2
Subtotal
1
2
Total

Exposure
100
125
225
190
325
515
180
450
630
470
900
1,370

Losses
250
500
750
600
1,500
2,100
500
1,800
2,300
1,350
3,800
5,150

$
$
$
$
$
$
$
$
$
$
$
$
___

Step 1: Calculate the average pure premium for state A:

LA 

Pure
Premium
2.50
4.00
3.33
3.16
4.62
4.08
2.78
4.00
3.65
2.87
4.22
3.76

100  2.50  125  4.00
 3.33.
100  125

Step 2: Calculate the average pure premium for states B and C based on the state A exposure
distribution by class:

100  3.16  125  4.62
100  2.78  125  4.00
 3.46,
LˆB 
 3.97, LˆC 
100  125
100  125

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Chapter 12 – Credibility
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Step 3: Compute adjustment factors by dividing the average pure premium for state A by the reweighted
average pure premium for B and C:

FB =

Lˆ A 3.33
L 3.33

 0.84, FC = A =
 0.96
LˆB 3.97
LˆC 3.46

Step 4: Apply the adjustment factors to the class 1 pure premium in states B and C, to adjust for the difference
in loss costs by state A. The adjusted loss costs for class 1 in states B and C, respectively, are:
____

_____

Lˆ1, B  L1,B  FB  3.16  0.84  2.65, Lˆ1,C  L1,C  FC  2.78  0.96  2.67
Step 5: Compute (C) by combining the adjusted Class 1 loss costs by state into a single Class 1 loss cost
according to the proportion of class 1 risks in each state:

C

Lˆ1,B  X 1,B  Lˆ1,C  X 1,C 2.65  190  2.67  180

 2.66
190  180
X 1, B  X 1,C

Complement Evaluation
1. It is unbiased as it adjusts for the distributional differences.
2. It is accurate as long as there is sufficient countrywide data to minimize the process variance.
3. It is independent since the subject experience and related experience consider data from different states.
4. The data for the complement is available but the computations can be time-consuming and complicated.
5. The complement has a logical relationship to the subject experience.
6. The complement may be harder to explain because of the computational complexity.
5. Trended Present Rates
Actuaries may rely on the current rates as the best available proxy for the indicated rate (when there is no larger
group to use for the complement).
Two adjustments are made before using the current rates:
1. Adjust current rates to what was previously indicated rather than what was implemented (since insurers do
not always implement the rate that is indicated, see reasons for this in chapter 13).
2. Adjust for changes in trends due to changes in loss cost level may have occurred between the time the
current rates were implemented and the time of the review. (e.g. due to changes in monetary inflation,
distributional shifts, safety advances, etc).
Trend from the original target effective date of the current rates to the target effective date of the new rates.

C = Present Rate × Loss Trend Factor ×

Prior Indicated Loss Cost
Loss Cost Implemented with Last Review

Example: Assume the following:
• Present average rate is $200.
• The selected annual loss trend is 5%.
• The rate change indicated in the last review was 10%, and the target effective date was 1/1/2011.
• The rate change implemented with the last review was 6%, and the actual effective date was 2/1/2011.
• The proposed effective date of the next rate change is 1/1/2013.

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Chapter
C
12
1 – Cred
dibility
BASIC RATTEMAKING – WERNER, G
G. AND MOD
DLIN, C.
Before ca
alculating the complement
c
of
o credibility, the loss trend
d length mustt be measured
d.
This is the
t length from the target effective
e
date
e of the last ra
ate review (1/1
1/2011) to the
e target effecttive
date of the next rate change (1/1/2
2013), or two
o years.
Then the complemen
nt of credibility
y is calculated
d as follows:

C = $200
$
* (1.05)) 2 *

1.10
 $2
229
1.06

This proce
edure can als
so be used to calculate a complement
c
fo
or an indicate
ed rate change factor when
n
using the loss ratio app
proach:

Loss Trend
d Factor
(1.0 + Prioor % Indicattion)

Premium
P
Tren
nd Factor (1.0 + Priorr % Rate Chaange)

Complem
ment Evaluation
1. Accura
acy depends largely on the
e process variance of the h
historical loss costs (that iss why it is use
ed primarily
for indiications with voluminous
v
data)
2. It is un
nbiased since pure trended
d loss costs (i.e. no updatin
ng for more re
ecent experie
ence) are unbiased.
3. It may or may not be independen
nt depending on the historiical experiencce used to de
etermine the ssubject
experie
ence and com
mplement (e.g
g. if the complement comess from a revie
ew that used data from 200
07 through
2010, and
a the subje
ect experience
e is based on data from 20
008 through 2
2011, the two are not indep
pendent).
4. The da
ata required is
s readily availlable, the calc
culations are very straightfforward, and tthe approach
h is easily
explain
nable.
es
6. Competitors’ Rate
 New
w or small com
mpanies with small volume
es of data find
d their own da
ata too unrelia
able for ratem
making.
 The
e rationale forr using compe
etitors’ rates as
a a complem
ment is that if ccompetitors h
have a much larger
num
mber of expos
sures, the com
mpetitors’ stattistics have le
ess process e
error.
Evaluatio
on
1. Compe
etitors’ manua
al rates are ba
ased on theirr marketing co
onsiderations, judgment, a
and the effectss of the
regulattory process—
—all of which can introduce
e inaccuracy to the rates.
2. Bias from competito
ors having diffferent underw
writing and cla
aim practices may be difficcult to quantifyy.
3. The co
ompetitors’ rates will be ind
dependent of the companyy data.
4. The ca
alculations ma
ay be straighttforward, but the
t data need
ded may be d
difficult or time
e-consuming to obtain.
5. Rates of a similar co
ompetitor hav
ve a logical re
elationship an
nd are accepte
ed as a comp
plement by regulators.
6. This co
omplement is often the onlly viable alternative.

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Chapter 12 – Credibility
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
II. Excess Ratemaking




Deals with volatile and low volumes of data so the complement is more important than the subject
experience.
Actuaries try to predict the volume of excess loss costs below the attachment point (since there are very
few claims in the excess layers).
Losses for liability lines of business are slow to develop, and inflation inherent in excess layers is higher
than that of the total limits experience.

Four methods that can be used to determine the complement of credibility for excess ratemaking analyses:
• Increased limits analysis
• Lower limits analysis
• Limits analysis
• Fitted curves
The first 3 methods use loss data and ILFS to calculate the complement of credibility.
The last method relies on historical data to fit curves, and the complement is calculated from the distribution.
1. Increased Limits Factors (ILFs) Methods
 are used when data is available for ground-up loss costs through the attachment point (i.e., losses have
not been truncated at any point below the bottom of the excess layer being priced).
 are used to adjust losses capped at the attachment point to produce an estimate of loss costs in the
specific excess layer.
The complement is defined as follows: C =

__
 ILFA L  ILFA  __  ILFA L

LA  
 1.0  , where
  L A 
ILFA


 ILFA


* L A is the loss cost capped at the attachment point A;
* ILFA is the increased limits factor for the attachment point A;
* ILFA+L is the ILF for the sum of the attachment point A and the excess insurer’s limit of liability L.
Example: Calculate the complement of credibility for the excess layer between $500,000 and $750,000 (i.e.
$250,000 of coverage in excess of $500,000).
Assume losses capped at $500,000 are $2,000,000 and the following ILFs apply:
Increased
Limit of
Limits
Liability
Factor
$100,000
1.00
$250,000
1.75
$500,000
2.50
$750,000
3.00
$ 1,000,000
3.40

 3.00

C  $2, 000, 000  
 1.0   400, 000.
 2.50


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Chapter 12 – Credibility
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Complement Evaluation
1. Biased results will occur if the subject experience has a different size of loss distribution than that used to
develop the ILFs (i.e. if the ILFs are based on industry data rather than the insurer’s own data). Despite the
issues with accuracy, this is often the best available estimate.
2. The error is parameter error associated with the selected ILFs (the error associated with this estimate tends
to be independent of the error associated with the base statistic).
3. To the extent that ILFs (preferably industry factors) and ground-up losses that have not been truncated
below the attachment point is available, the procedure is practical.
4. In terms of acceptability, the estimate is more logically related to the data below the attachment point (which
is used for the projection) than to the data in the layer (and this may be controversial).
2. Lower Limits Analysis
Losses capped at the attachment point are used to estimate the losses in the excess layer being priced.
If losses are too sparse use losses capped at a limit lower than the attachment point (i.e. the basic limit).
___ 
ILFA+L  ILFA 
C  Ld  
 where
ILFd



•

Ld is the loss cost capped at the lower limit, d;

•

ILFA is the ILF for the attachment point A;

•

ILFd is the ILF for the lower limit, d;

•

ILFA+L is the ILF for the sum of the attachment point A and the excess insurer’s limit of liability L (i.e.
this sum is the top of the excess layer being priced).

Note the first excess procedure is a special case of this procedure where d = the attachment point.
Example: Calculate the complement of credibility for the layer between $500,000 and $750,000.
Assume losses capped at $250,000 are $1,500,000, and the ILFs from the prior Table apply.

 3.00 - 2.50 
C  $1,500, 000  
  $428,571.
 1.75

Evaluation
1. It is difficult to determine whether this is more or less accurate than the previously complement.
2. It is more biased (as the differences in size of loss distributions will be exacerbated when using losses
truncated at lower levels).
3. Stability of the estimate is increased when using losses capped at lower limits.
4. The error is generally independent of the error of the base statistic.
5. The data may not be available if some other lower limit is chosen, and the calculations are simple.
6. The complement is more logically related to the lower limits losses that to the losses in the layer being priced.

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Chapter 12 – Credibility
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
3. Limits Analysis
Insurers sell policies with a wide variety of policy limits.
 Some policy limits fall below the attachment point and some extend beyond the top of the excess layer.
 Thus, each policy’s limit and ILF needs to be considered in the calculation of the complement.
i. Policies at each limit of coverage are analyzed separately.
ii. Estimated losses in a layer are computed using the premium and expected loss ratio in that layer.
iii. An ILF analysis on each first dollar limit’s loss costs is performed.

C  LR   Pd 

( ILFmin( d , A L ) - ILFA )

ILFd

d A

, where

LR = Total loss ratio,
Pd= Total premium for policies with limit d.
Calculate expected loss for the layer between $500,000 and $750,000 assuming a total limits loss ratio of 60%.
(1)

(2)

(3)

(4) = (2)*(3)

(5)

(6)

(7)

(8)

(9) = (4)*(8)
Expected

Limit of
Liability (d)

Premium

Expected
Loss Ratio

Expected
Capped
Losses

ILF @
d

ILF @
A

ILF @
A+L

% Loss
In Layer

Loss in
Layer

100,000

$1,000,000

60.0%

$ 600,000

1.00

2.50

3.00

0.0%

$
$
$

250,000
500,000
750,000

$ 500,000
$ 200,000
$ 200,000

60.0%
60.0%
60.0%

$ 300,000
$ 120,000
$ 120,000

1.75
2.50
3.00

2.50
2.50
2.50

3.00
3.00
3.00

0.0%
0.0%
16.7%

$ 1,000,000
Total

$ 75,000
$1,975,000

60.0%

$ 45,000
1,185,000

3.40

2.50

3.00

14.7%

$20,040
$6,615
$26,655

(8): if d< =A then 0.0%; if A < d < A +L then [(5)- (6)]/(5); if d >A+L then [(7)- (6)]/(5)
Complement Evaluation
1. It is biased and inaccurate to the same extent as the prior two complements, and it assumes that ELR
does not vary by limit.
2. It may be the only method available for reinsurers that use this method and do not have access to the
full loss distribution
3. It is more time-consuming to compute, but the calculations are straightforward.
4. It is not based on actual data from the layer being priced.

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Chapter 12 – Credibility
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
4. Fitted Curves
Curves are fit curves to smooth out the volatility of the data and to extrapolate the distribution to higher limits.
The techniques described in Chapter 11 can determine the expected losses in the layer being priced.
The percentage of the curve’s total losses expected in the excess layer is:
A L

% of Losses in Layer ( A, A  L) 





( x  A) f ( x)dx 





( A  L) f ( x)dx

A L

 xf ( x)dx



This % is applied to total limits loss costs to determine the expected losses in the layer.
Evaluation
1. It is less biased and more stable than the other excess methods (assuming the fitted curve replicates the
shape of the actual data).
2. It is more accurate than the others when there are few claims in the higher layers.
3. It is dependent on the existence or non-existence of larger claims because of the curve-fitting process.
4. The error is less independent than complements determined from the other approaches.
5. It is the most computationally complex and requires data that may not be readily available.
6. It is the most logically related to the losses in the layer than the others (as the data is more fully used).
7. Its computational complexity may make it difficult to communicate.

Credibility When Using Statistical Methods

236 –236

When performing a multivariate classification analysis (e.g. a GLM), diagnostics from the model results
gauge to what extent the model results are meaningful given the data provided.
 Statistical diagnostics include standard errors of the parameter estimates and standardized deviance
tests (e.g. Chi-Square or F-test), as well as practical tests such as consistency of model results over
time.
 Statistical methods also provide diagnostics (deviance residual plots and leverage plots) that inform the
modeler of the appropriateness of the model assumptions (e.g. the link function or error term selected).
Typically, the results of a multivariate classification analysis are not credibility-weighted with traditional
(univariate) actuarial estimates.

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Chapter 12 – Credibility
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
6

Key Concepts

238 - 238

1. Criteria for measures of credibility
2. Methods for determining credibility
a. Classical credibility
b. Bühlmann credibility
c. Bayesian analysis
3. Desirable qualities for the complement of credibility
a. Accurate
b. Unbiased
c. Independent
d. Available
e. Easy to calculate
f. Logical relationship to the base statistic
4. Methods for determining the complement of credibility
a. First dollar ratemaking
i. Loss costs of a larger group that includes the group being rated
ii. Loss costs of a larger related group
iii. Rate change from the larger group applied to present rates
iv. Harwayne’s method
v. Trended present rates
vi. Competitors’ rates
b. Excess ratemaking
i. Increased limits analysis
ii. Lower limits analysis
iii. Limits analysis
iv. Fitted curves
5. Credibility when using statistical modeling methods

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Chapter 12 – Credibility
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
The predecessor papers to the current syllabus reading “Basic Ratemaking” by Werner, G.
and Modlin, C. were numerous. While past CAS questions were drawn from prior syllabus
readings, the ones shown below remain relevant to the content covered in this chapter.
Questions from the 1996 exam
Question 42. (3 points) You are given:
Great Northeast Insurance Company
State
Vermont

Maine

New Hamp.

Countrywide

Class
1
2
Subtotal
1
2
Subtotal
1
2
Subtotal
1
2
Total

Exposures
200
500
700
150
600
750
100
400
500
450
1,500
1,950

Losses
600
2,000
2,600
600
2,700
3,300
350
1,800
2,150
1,550
6 500
8,050

Pure
Premium
3.00
4.00
3.71
4.00
4.50
4.40
3.50
4.50
4.30
3.44
4.33
4.13

In his article "The Complement of Credibility," Boor discusses a method used by Harwayne to determine a
complement of credibility that involves a separate adjustment to each state's data. You are reviewing Class 1
rates for Vermont. Using Harwayne's method:
(a) (2 points) Calculate the adjusted Class 1 pure premiums for Maine and New Hampshire.
(b) (1 point) To the extent that Vermont Class 1 experience is not fully credible, calculate the pure premium
to be used for the complement of credibility.

Questions from the 1997 exam
7. You are given:
Limit of Liability
Increased Limit Factors
Historical Losses Capped at Limit
$50,000
1.00
$350,000
$100,000
1.65
$650,000
$250,000
2.00
$800,000
$500,000
2.75
$1,050,000
$1,000,000
3.30
$1,200,000
Based on methodology described by Boor, "The Complement of Credibility," and using losses capped at
$100,000, in what range does the complement fall for losses in the layer $500,000 to $1,000,000?
A. < $150,000 B. > $150,000, but < $200,000 C. > $200,000, but < $250,000
D. > $250,000, but < $300,000 E. > $300,000
21. (3 points) Boor, "The Complement of Credibility," discusses using competitor's rates as the
complement when one is faced with ratemaking data that is unreliable.
According to Boor, what are three desirable characteristics and three undesirable characteristics of using
competitor's rates as complements?

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Chapter 12 – Credibility
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 1998 exam
4. In Boor, "The Complement of Credibility," a First Dollar Ratemaking procedure is discussed in which
the rate change from the larger group is applied to present rates.
Use this procedure and the following data to determine in which range the complement of credibility for
Class 1 loss costs falls.
Class 1 present loss cost
Class 1 indicated loss cost
All Class present average loss cost
All Class indicated loss cost
A. < $110

B. > $110 but < $120

$125
$115
$150
$165
C. > $120 but < $130

D. > $130 but < $140

E. > $140

53. (2 points) Using the Limits Analysis as described in Boor, "The Complement of Credibility," and the
following data, calculate the complement of credibility that could be used to estimate the losses in the
layer of insurance between $250,000 and $500,000.
Limit
of
Liability
$250,000
$500,000
$1,000,000

Premium
$1,000,000
$700,000
$500,000

Increased
Limits
Factor
1.80
2.60
3.20

Estimated All Limits Loss Ratio = 65.0%

Questions from the 1999 exam
50. (2 points) Boor in 'The Complement of Credibility," suggests that trended present rates may provide an
appropriate complement of credibility for use in ratemaking. Using the trended present rates method
outlined in the reading and the information shown below, calculate the complement of credibility.
Information from the Previous rate chance which established the current rates:
Target effective date:
4/1/97
Actual effective date:
6/15/97
Requested change:
+19.6%
Approved and implemented change:
+ 4.0%
Information on current filing being Prepared:
Present pure premium rate:
Target effective date:
Expected regulatory delay (not contemplated in target effective date):
Annual frequency trend:
Annual severity trend:

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$325
10/1/99
6 months
+ 3.5%
+11.4%

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Chapter 12 – Credibility
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 1999 exam
53. (3 points) The Actuarial Standard of Practice (SOP) No. 25, "Credibility Procedures Applicable to
Accident and Health, Group Term Life, and Property/Casualty Coverages " lists the following as
criteria for selecting credibility procedures:
 The procedure does not tend to bias the results in a material way.
 The procedure is practical to implement.
Boor, in 'The Complement of Credibility' lists a number of credibility complements used in (first dollar)
ratemaking for a given class. For each of the credibility complements given below, explain how they
fit (or do not fit) the SOP No. 25 criteria listed above.
1. Loss Costs of a larger group including the class - Classic Bayesian
2. Trended Present Rates
3. Competitors Rates

Questions from the 2000 exam
23. Based on Boor, "The Complement of Credibility," and the following data, calculate the complement of
credibility "C" using the "trended present rates" method.
•
•
•
•
•
•
•

Proposed rate change effective date = January 1, 2000
Present pure premium rate = $200
Annual inflation (trend) = 3.0%
Amount requested (indicated) in last rate change = +10.0%
Effective date requested for last rate change = January 1, 1998
Rate request approved by regulator = +5.0%
Effective implementation date of last rate change = July 1, 1998

A. < $210
E. > $240

B. > $210 but < $220

C. > $220 but < $230

D. > $230 but < $240

Questions from the 2001 exam
Questions 17. Based on Boor, “The Complement of Credibility,” and the following information, calculate
the complement of credibility for class 2.
Class

Indicated Loss
Cost Rate

Current Loss
Cost Rate

Complement of
Credibility

150

120

140

160

150

A. < 155

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B. > 155 but < 170

C. > 170 but < 185

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D. > 185 but < 200

E. > 200

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Chapter 12 – Credibility
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2002 exam
10. Boor, "Complement of Credibility," discusses using competitor's rates as the
complement of credibility when using ratemaking data that is unreliable. Derive the pure
premium complement of credibility for Small Company, Class 1, pure premium using the
data below.

Small Company, Class 1

$80

Present manual rate
Permissible loss ratio

60%

Competitor Company, Class 1
Present manual rate
Permissible loss ratio
Projected Loss Ratio from Schedule P Analysis
Average frequency of loss per exposure

$70
62%
75%
0.040

Due to the assumed growth of Small Company, 10% more losses are expected for Small Company
than Competitor Company.
A. < $45 B. > $45, but < $50 C. > $50, but < $55 D. > $55, but < $60 E. > $60

Questions from the 2003 exam
39. (3 points) In "The Complement of Credibility," Boor discusses several methods for calculating
complements of credibility in first dollar ratemaking. Briefly discuss three of these methods and
comment on the effectiveness of each method as a complement of credibility.

Questions from the 2004 exam
46. (2 points) Boor, in "The Complement of Credibility," discusses using the trended present rates as the
complement of credibility when using ratemaking data that is not fully credible.
a. (1 point) Derive the pure premium complement of credibility using the data below. Show all work.
• Present pure premium rate is $150.
• Annual inflation rate is 4%.
• Original target effective date of the current rates was October 1, 2002.
• Amount indicated and requested in last rate change was 18%.
• Actual effective date was February 1, 2003.
• Amount approved in last rate change was 10%.
• Target effective date of the new rates is December 1, 2004.
b. (1 point) State and briefly describe one advantage and one disadvantage of using this
complement of credibility.

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Chapter 12 – Credibility
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2005 exam
18. Using the following data, calculate the complement of credibility for class 3 based on the rate changefor the larger group applied to the present rate.
Class
1
2
3
Total
A. < 400

Exposures
200
300
500
1,000
B. > 400, but < 425

Losses
$100,000
$135,000
$215,000
$450,000

Present Pure
Premium
$550.00
$500.00
$455.00
$487.50

C. < 425, but < 450

D. > 450, but < 475

E. > 475

Questions from the 2006 exam
There were no questions drawn from the content within this article appearing on the above referenced exam.

Questions from the 2007 exam
32. (3.0 points) Using the following data, calculate the complement of credibility for the pure premium of
Class 1 in State A, using Harwayne's full method. Show all work.
State
A
A
B
B
C
C
D
D

Class
1
2
1
2
1
2
1
2

Exposure
$130
160
150
200
130
180
140
250

Losses
$180
450
330
600
180
500
320
500

Questions from the 2008 exam
22. (1.5 points)
a. (1.0 point) You are given the following information:
 Present average rate = $200
 Annual loss trend = 10%
 20% rate change requested in last filing.
 15% rate change approved with last filing.
 Effective date requested in last filing was January 1, 2006.
 Actual effective date of last change was June 1, 2006.
 Proposed effective date of next change is January 1, 2008.
Calculate the complement of credibility using the trended present rate approach.
b. (0.5 point) Identify one advantage and one disadvantage to using the trended present rate as the
complement of credibility.

Exam 5, V1b

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Chapter 12 – Credibility
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Questions from the 2009 exam
32. (2 points) Given the following information:
• Current State X loss cost = $1,600
• Current countrywide loss cost = $1,800
• Indicated countrywide loss cost = $1,710
• State X losses and LAE = $1,000,000
• State X exposures = 1,000
• State X fixed expenses = $200,000
• Variable expense factor = 25%
• Profit and contingency factor = 5%
• Full credibility standard is 16,000 exposures.
• Partial credibility is assigned using the square root rule.
• Complement of credibility is determined using the "Rate Change from a Larger Group" method.
Calculate the credibility-weighted indicated rate for State X.

Questions from the 2012 exam
9. (1.75 points) Given the following information:


Projected Loss and LAE Ratio = 58.5%.



Projected Fixed Expense Provision = 11.5%.



Variable Expense Provision =15%.



Underwriting Profit Provision = 5%.



Credibility of the indicated rate change = 0.7.



Last rate change was taken January 1, 2012, the entire indicated change was implemented.



Proposed effective date of next rate change is July 1, 2013.



Annual Loss Ratio Trend = +2.5%.

Calculate the credibility-weighted indicated rate change.

Exam 5, V1b

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Chapter 12 – Credibility
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
The predecessor papers to the current syllabus reading “Basic Ratemaking” by Werner, G.
and Modlin, C. were numerous. While past CAS questions were drawn from prior syllabus
readings, the ones shown below remain relevant to the content covered in this chapter.
Solutions to questions from the 1996 exam
42(a) The adjusted Class 1 pure premiums for each state is computed as follows:
P’c,j = Fj Pc,j = (Ps / P j ) *Pc,j = (Ps /

 Em,s * Pm, j  Em,s ) *P

c,j.

Step 1: Compute the base state average pure premium, Ps

Ps 

L

i, s

(total state losses divided by total state exposure units)

E

i, s

This is given in the problem as 3.71.

Step 2: Compute the state average pure premium, P j .
Pj 

 Em,s * Pm,j  Em,s

m
m
(combine the state j class pure premiums using the base state exposure distribution).

[200 * 4.00 + 500 * 4.50]
= 4.36.
(200 + 500)
[200 * 3.50 + 500 * 4.50]
= 4.21.
For New Hampshire, P j 
(200 + 500)

For Maine, P j 

Step 3: Compute the individual state adjustment factors, Fj .

Fj  Ps

For Maine, Fj = 3.71 / 4.36 = .851
For New Hampshire, Fj = 3.71 / 4.21 = .881
Step 4: Compute the class 1 adjusted pure premium, P’c,j .
P’c,j = Fj Pc,j
For Maine, P’c,j = .851* 4.00 = 3.40.
For New Hampshire, P’c,j = .881* 3.50 = 3.08.
(b) The pure premium to be used for the complement of credibility is computed as follows:

C

E

c , j P' c , j

C

E

c, j

[150 * 3.40 + 100 * 3.08]
= 3.27
(150 + 100)

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Chapter 12 – Credibility
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 1997 exam
7. Using losses capped at $100,000, compute the complement of credibility for losses in the layer $500,000 to
$1,000,000.
Step 1: Write an equation to determine the complement for losses in the layer $500,000 to $1,000,000.
 ILFA  L  ILFA 
C  Pd 
 , where
Symbol
Description
ILFd


Pd
Historical losses capped at limit 100,000.
A
Attachment Point = $500,000.
L
Layer Limit = $1,000,000.
Step 2: Using the equation in Step 1, and the data given in the problem, solve for the complement of credibility.
 3.30 2.75 

= $216,667.
C = 650,0000* 
.
165
. 
 165

Answer C.

Question 21.
Boor discuses the advantages and disadvantages when using competitor’s rates on pages 23 and 24.

Statistic’s Quality
Independence
Availability
Process Error

Statistic’s Quality
Explainable
relationship
Bias
Computation

Exam 5, V1b

Statistic’s Desireable Characteristic:
Prediction errors in the competitor’s rates are independent of the subject loss
costs. (Errors stem mostly from inter-company differences)
Competitor’s rates are generally available through a regulatory agency
A competitor may write more exposures and thus have have less process error

Statistic’s UnDesireable Characteristic:
It can be difficult to explain since the competitor’s rates may be unrelated to the
subject loss costs.
Might be biased due to different underwriting and claim practices.
This data does not exist in any other part of the rate filing and will have to be
posted manually.

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Chapter 12 – Credibility
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 1998 exam
4. Determine the range the complement of credibility for Class 1 loss costs falls in using the present rates
adjusted for rates changes in a larger group method.
Step 1: Write an equation to determine the complement using the present rates adjusted for rates changes in a
larger group method.

F
GH

The formula to compute this statistic is C  Rc 1 
Notation

Rc
Pg

Pg  Rg
Rg

I
JK

Description
Class 1 present loss cost
All class indicated loss cost
All class present average loss cost

Step 2: Using the equation in Step 1, and the data given in the problem, solve for C.
165150
C  125 * 1
 137.50
Answer D.
150

FG
H

IJ
K

53. Calculate the complement of credibility that could be used to estimate the losses in the layer of insurance
between $250,000 and $500,000.
Step 1: Use a limits analysis when losses limited to a single capping point are not available.
This method assumes that all the limits will experience the same loss ratio (in this case, .65).
ILFs can be used to determine the percentage of losses in the layer. The sum of losses within
a layer can be used as the complement of credibility.
The formula to compute the complement of credibility is: C  LRT *  Wd
dA
Symbol
LRT
Wd
A
L

FG ILF
H

min( d , A  L )  ILFA

ILFd

IJ , where
K

Description
Estimated total limits loss ratio
The premium with policy limits of d
Attachment Point (in this case $250,000).
Layer Limit (in this case, $250,000).

Step 2: Using the equation in Step 1, and the data given in the problem, solve for the complement of
credibility to estimate the losses in the layer of insurance between $250,000 and $500,000.
Expected
Losses at a
% of Expected
Expected Losses
65%
Losses
in
the
layer
in the layer
Policy Limit Premium
ILF
ELR
Loss ratio
250K - 500K
250K - 500K
(1)
(2)
(3)
(4)
(5) = (2)*.65
(6)
(7) = (5) * (6)
250,000
1,000,000
1.80 .650
650,000
(1.8-1.8)/1.8= 0.0
0
$500,000
$700,000
2.60 .65
455,000
(2.6-1.8)/2.6 = .308 140,140
$1,000,000 $500,000
3.20 .65
325,000
(2.6-1.8)/3.2 = .25
81,250
221,390
Thus, C = 221,390

Exam 5, V1b

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Chapter 12 – Credibility
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 1999 exam
Question 50

The formula for the complement of credibility using trended present rates.

Symbol

T
t

RL
PL

Definition
the annual trend factor, expressed as (1+ the inflation
rate). In the problem, it is the frequency and severity
trend.
the number of years between the target effective date of
the current rates and that of the new rates
Loss cost presently in the rate manual.
The last indicated (requested) pure premium (rate
change).
The pure premiums actually being charged (rate change
approved) in the current manual. This may differ from RL

because

The factor

P 
C  RL * T t *  L 
 PC 
As Given in the Problem
(1.035)(1.114) = 1.15299

4/1/97 - 10/1/99 = 2.5 years
$325
1.196
1.04

PL and PC may be taken from a broader group.

 PL   last indicated pure premium 
    actual pure premium in present rates  adjusts the loss cost in the present rates, RL, for

 PC  

inadequacies which stem from the current rate being less than the indicated rate at the last rate filing.
2.5 1.196 
Based on the above, C  $325 * (1.15299) 
 $533.51
 1.04 
Question 53

Complement
Loss Costs of a
larger group

Does not tend to bias the results in a material way.
Since the true class expected losses are not equal
to the group expected losses, this statistic is
biased.

Trended Present
Rates

The pure trended loss costs are unbiased since
they are based on present rates which are
presumed to be unbiased.
Might be biased due to different underwriting and
claim practices.

Competitors
Rates

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Page 164

Practical to implement.
Very practical, as long as all the
classes in the group have something
in common. Using national or
statewide averages in the ratemaking
process is common.
Same rationale as above

They are often available from
regulators, although the process
takes some work. It is also a
manually intensive and time
consuming process.

 2014 by All 10, Inc.

Chapter 12 – Credibility
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2000 exam
23. Calculate the complement of credibility "C" using the "trended present rates" method.
Step 1: Write an equation to determine the complement of credibility "C" using the "trended present rates"
P 
method: C T t * RL * L  .
 Pc 




P  

last indicated pure premium
The factor  L   
 adjusts the loss cost in the present rates, RL, for
 Pc   actual pure premium in present rates 
inadequacies which stem from the current rate being less than the indicated rate at the last rate filing.

t is the number of year between the original target effective date of the current rates (not
necessarily the date they actually went into effect), and the target effective date of the new rates.

Step 2: Using the equation in Step 1, and the data given in the problem, solve for C
2 1.10 
C  $200 *1.03 
 $222.28
1.05 

Answer C.

Solutions to Questions from the 2001 exam
17. Calculate the complement of credibility for class 2 using a “Rate Change from the larger Group Applied to
Present Rates”
Step 1: Write an equation to determine the complement of credibility "C" using the “Rate Change from
 Pg  R g 
.
the larger Group Applied to Present Rates”: C  Rc 1 
Rg 





Step 2: Compute 1 

Pg  Rg 
 Pg  Rg 
 Pg  Rg  140
. 140  120* 1 
; Thus 1 

 1.1666


Rg 
Rg 
Rg  120



Step 3: Using the equation in Step 1, the results from Step 2, and the data given in the problem, solve for
the complement of credibility for class 2. C2  150*1.1667  175 . Answer C.

Solutions to questions from the 2002 exam
Question 10. Derive the pure premium complement of credibility for Small Company, Class 1, pure premium.
Since new companies and companies with small volumes of data often find their own data too unreliable
for ratemaking, actuaries use competitor’s rates for the complement of credibility. In this problem, we are
also told that due to the assumed growth of Small Company, 10% more losses are expected for Small
Company than Competitor Company.
Compute the pure premium complement of credibility as follows:
Pure premium complement of credibility = Competitor present manual rate * Competitor Projected loss ratio *
Company expected % increase loss per exposure = $70 * .75 * 1.10 = $57.75.
Answer D.

Exam 5, V1b

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Chapter 12 – Credibility
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2003 exam
39. (3 points) In "The Complement of Credibility," Boor discusses several methods for calculating
complements of credibility in first dollar ratemaking. Briefly discuss three of these methods and
comment on the effectiveness of each method as a complement of credibility.
1. Classic Bayesian credibility uses a larger group mean (including the base class) to compute the complement.
This complement is biased and inaccurate, independent if the base class doesn’t predominate the data, is readily
available, easy to compute, and has an explainable relationship to the base class.
2. The trended present rates method uses the present rate, which is adjusted for the residual indication and
trended from the last filing’s target effective date, as the complement. It is unbiased, accurate,
independent, available, easy to compute, and easy to explain since it is using the rates of the base class.
3. The rate change from a larger group is applied to present rates. A rate change from a larger group is
applied to present rates. This complement is unbiased, accurate, independent, available, easy to
compute, and easily explainable.

Solutions to Questions from the 2004 exam
46. (2 points)
a. (1 point) Derive the pure premium complement of credibility using the trended present rates method.
• Present pure premium rate is $150.
• Annual inflation rate is 4%.
• Original target effective date of the current rates was October 1, 2002.
• Amount indicated and requested in last rate change was 18%.
• Actual effective date was February 1, 2003.
• Amount approved in last rate change was 10%.
• Target effective date of the new rates is December 1, 2004.
The formula for the complement of credibility using trended present rates.

The factor

P 
C  RL * T t *  L  .
 PC 

 PL   last indicated pure premium 
    actual pure premium in present rates  adjusts the loss cost in the present rates, RL, for

 PC  

inadequacies which stem from the current rate being less than the indicated rate at the last rate filing.
Symbol

T
t

RL
PL
PC

Definition
the annual trend factor, expressed as (1+ the inflation rate).

As Given in the Problem
1.04

the number of years between the target effective date of the
current rates and that of the new rates
Loss cost presently in the rate manual.

10/1/02 - 12/1/04 = 2.167 years

The last indicated (requested) pure premium (rate change).

1.18

The pure premiums actually being charged (rate change
approved) in the current manual.

1.10

$150

1.18 
1.10   $175.183
b. (1 point) State and briefly describe one advantage and one disadvantage of using this COC.
Advantage: It is unbiased in the sense that pure trended loss costs (e.g. with no updating for more current
loss costs) are unbiased.
Disadvantage: It is less accurate for loss costs with high process variance.

Based on the above, C  $150 * (1.04)

Exam 5, V1b

2.167

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Chapter 12 – Credibility
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2005 exam
18. Using the following data, calculate the complement of credibility for class 3 based on the rate changefor the larger group applied to the present rate.
Class
1
2
3
Total

Exposures
200
300
500
1,000

Present Pure
Premium
$550.00
$500.00
$455.00
$487.50

Losses
$100,000
$135,000
$215,000
$450,000

The complement of credibility approach using a “Rate Change from the larger Group Applied to Present
Rates”. The formula for the complement of credibility is as follows:
 Pg  R g 
, where
C  Rc 1 
Rg 


C is the compliment of credibility
RC is the present pure premium (present manual loss cost) for the class under consideration
PG is the indicated loss cost for the entire group of classes
RG is the average loss cost for the entire group of classes
Using the above equation, and the data given in the problem, compute the compliment
 ($450,000 /1,000) $487.50) 
C $455.00 1
  420
$487.50


Answer B. > 400, but < 425

Solutions to questions from the 2006 exam
There were no questions drawn from the content within this article appearing on the above referenced exam.

Exam 5, V1b

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Chapter 12 – Credibility
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2007 exam
32. (3.0 points) Using the following data, calculate the complement of credibility for the pure premium of
Class 1 in State A, using Harwayne's full method. Show all work.
Step 1: Write an equation to determine the complement of credibility "C" for the pure premium of Class 1 in
State A, using Harwayne's full method

C

E

c , j P' c , j

E

c, j .

The numerator is the sum product of the adjusted class 1 pure premiums

and their class 1 exposures, summed over all states other than state A.
Also, compute the following pure premiums:
Class Pure Premium=Loss/Exposure
State
A
1
180/130=1.38
A
2
2.81
B
1
2.20
B
2
3.00
C
1
1.38
C
2
2.78
D
1
2.29
D
2
2.00
The adjusted Class 1 pure premiums for each state are computed as follows:
Step 2: Compute the base state average pure premium, Ps

Ps 

L

i, s

E

i, s

Total state losses divided by total state exposure units: PA = (180+450)/(130+160) = 2.172.
Step 3: Compute the state average pure premium, P j .
Pj 

 Em,s * Pm,j  Em,s

m
m
Combine state j class pure premiums using the base state exposure distribution
PB = (2.20* 130 + 3.00* 160)/(130 + 160) = 2.641
PC = (1.38*130 + 2.78*160)/(130 + 160) = 2.153
PD = (2.29*130 + 2.00*160)/(130 + 160) = 2.130

Step 4: Compute the individual state adjustment factors, Fj and then compute the class 1 adjusted
pure premium, P ' c j

Fj  Ps

P ' c j  Fj Pc j

P'B1 = PA/ PB * PB1 = 2.172/2.641 *2.20 = 1.809
P'C1 = PA/ PC * PC1 = 2.172/2.153 *1.38 = 1.392
P'D1 = PA/ PD * PD1 = 2.172/2.130 * 2.29 = 2.335
Step 5: The pure premium to be used for the complement of credibility is computed as follows:

C

E

c , j P' c , j

Exam 5, V1b

E

c, j

C = (1.809*150 + 1.392*130 + 2.335*140)/(150 + 130 + 140) = 1.855

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Chapter 12 – Credibility
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2008 exam
22. Calculate the complement of credibility using the trended present rate approach.
Step 1: Write an equation to determine the COC of credibility using trended present rates:

The factor

P 
C  RL * T t *  L  .
 PC 

 PL   last indicated pure premium 
    actual pure premium in present rates  adjusts the loss cost in the present rates, RL, for

 PC  

inadequacies which stem from the current rate being less than the indicated rate at the last rate filing.
Symbol

T
t

RL
PL
PC

Definition
the annual trend factor, expressed as (1+ the inflation rate).

As Given in the Problem
1.10

the number of years between the target effective date(not
necessarily the date they actually went into effect) of the
current rates and that of the new rates
Loss cost presently in the rate manual.

1/1/06 - 1/1/08 = 2 years

$200

The last indicated (requested) pure premium (rate change).

1.20

The pure premiums actually being charged (rate change
approved) in the current manual.

1.15

Step 2: Using the equation in Step 1, and the data given in the problem, solve for C.
2 1.20 
C  $200 * (1.10) 
 $252.52
1.15 
b. (1 point) State and briefly describe 1 advantage and 1 disadvantage of using this complement of credibility.
Advantage: It is unbiased in the sense that pure trended loss costs (e.g. with no updating for more current
loss costs) are unbiased.
Disadvantage: It is less accurate for loss costs with high process variance.

Solutions to questions from the 2009 exam
Question 32



Larger Group Indicated Loss Cost
C = Current Loss Cost of Subject Experience × 

 Larger Group Current Average Loss Cost 
Complement = 1,600 x (1,710/1,800) =1,520
Credibility= Z 

Y
1, 000
, where Y  E (Y ); Z=
 .25
E (Y )
16, 000

Indicated loss lost =

1, 000, 000
 1000
1000

Credibility weighed indicated loss costs = .25 (1,000) + .75 (1520) = 1,390

 ( L  EL )  EF 
 L  EL  EF  
X
X 


Indicated Rate 

1.0  V  QT 
1.0  V  QT 
Credibility weighed indicated rate (1390 + 200,000/1,000)/(1.0 - .25 -.05) = 2,271.43

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Chapter 12 – Credibility
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Solutions to questions from the 2012 exam
9. Calculate the credibility-weighted indicated rate change.
Question 9 – Model Solution (Exam 5A Question 9)
An equation to determine the COC of credibility using trended present rates:

The factor

P 
C  RL * T t *  L  .
 PC 

 PL   last indicated pure premium 
    actual pure premium in present rates  adjusts the loss cost in the present rates, RL, for

 PC  

inadequacies which stem from the current rate being less than the indicated rate at the last rate filing.
Complement of credibility = Trended present rate = (indicated/approved) (loss trend)^ t -1
t = from 1/1/12 last date to 7/1/13 next date
COC = (1) (1.025)^1.5 -1 = 3.7733%
Ind Rate Change -> LR Method = (0.585 + .115) / (1.0 - .15 - .05 ) -1 = -12.5%
(.70) (-12.5%) + (1.0 - .70)(3.7733%) = -7.618%
Examiner’s Comments
Candidates typically lost points on the compliment of credibility.
Given the information in the question, using 0 was determined to not be worth full credit.
Candidates lost varying amount of points for using 0 as a compliment depending on the completeness
of the explanation. Other candidates trended a projected loss ratio that was already trended.

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Chapter 13 Other Considerations
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Sec
1
2
3
4
1

Description
Regulatory Constraints
Operational Constraints
Marketing Considerations
Key Concepts

Pages
239 - 241
241 - 244
244 - 260
262 - 262

Regulatory Constraints

239 - 241

This chapter outlines some reasons why a company might implement rates and/or rating differentials other than
those calculated using techniques from prior chapters (that balance the fundamental insurance equation).
Those reasons are:

Regulatory constraints

Operational constraints

Marketing considerations
The U.S. P&C insurance industry is highly regulated through state law and state regulatory agencies.
Regulatory scrutiny varies by jurisdiction and by insurance product. Examples:
 Scrutiny is high for personal auto insurance (since car owners have to meet state-mandated financial
responsibility requirements by purchasing this coverage)
 However, oversight is lower for other types of commercial insurance (e.g. directors and officers
insurance), which may not be compulsory and are purchased by more sophisticated buyers.
U.S regulation often requires insurers to file proposed manual rates with the state insurance department.
Filing requirements vary considerably by jurisdiction and product.
 Some regulation requires regulator’s approval of the new rates before the company can use them.
 Other regulation requires a copy of the manual rates to be on file with the regulator.
 Regulators may promulgate rates to be used but allow a specified range of deviation from these rates (in
some extreme cases).
In Canada, insurance rate regulation is executed by the individual provinces. For the personal auto product:
i. some provinces require approval of filed rates; others operate more on open competition.
ii. a few provinces have a government insurer for compulsory liability coverages, but allow open competition
for other coverages.
The United Kingdom has less rigid rate regulation than in the U.S. (and relies on competitive pressures to
“regulate” the market).
In Latin American markets:
 regulation is focused more on rate adequacy (i.e. ensuring that insurers collect the minimum premium to
meet their obligations) than equity among classifications.
 rating plans are unsophisticated.
One exception is Brazil; carriers use a wider range of rating variables on some products (e.g. personal
auto) and rates are required to be filed with the regulators for approval.
In many developing markets (e.g. India) rate regulation is heavier on compulsory coverages (e.g. personal auto
liability), but other insurance products are deregulated and operate on open competition.

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Examples of U.S. Regulatory Constraints
Regulatory constraints, causing insurers to implement rates different from those indicated by its ratemaking
analyses, follow.
1. Regulations that limit the amount of an insurer’s rate change (to either the overall average rate change for
the jurisdiction or to the change in premium for any individual or group of customers, or both)
Example: A jurisdiction may prohibit a rate change that generates an overall premium increase greater
than 25% and/or a rate change that results in a significant number of existing customers getting
an increase greater than 30%.
2. Regulatory requirements regarding the magnitude of the requested change.
Example: An insurer may be required to provide written notice to all insureds or hold a public hearing in
the event a proposed rate change exceeds some specified threshold (but may decide to
implement a rate change that is less than the threshold to avoid the extra requirements).
3. Regulations prohibiting the use of a characteristic for rating (even if it can be demonstrated to be
statistically strong predictors of risk).
Example: The use of insurance credit score for underwriting or rating personal lines insurance (e.g.
personal automobile or homeowners).
i. An individual’s insurance credit score is a strong predictor of risk in personal lines.
ii. Where allowed, insurers charge higher premium for individuals with poor credit scores
than for individuals with good credit scores.
iii. Because credit score is perceived to be correlated with certain socio-demographic
variables, some jurisdictions have placed limitations on the use of credit and some have
banned the use of credit
4. Regulations prescribing the use of certain ratemaking techniques.
Examples: The state of Washington requires that multivariate classification analysis be used to develop
rate relativities if insurance credit score is used to differentiate premium in personal auto
insurance.
Other states mandate the use of a certain method for incorporating investment income in the
derivation of the target underwriting provision.
5. Regulators disagreeing with actuarial ratemaking assumptions (e.g. a regulator may disagree with the
method the actuary used to calculate loss trend, or may disagree with the trend selected).
There may be a cost (e.g. delayed implementation of new rates, requirement of specialized staff
resources) associated with negotiating with the regulator to resolve such differences.
Insurer actions that can be taken with respect to regulatory restrictions:
• An insurer can take legal action to challenge the regulation.
• An insurer may revise its U/W guidelines to limit business written at what it considers to be inadequate
rate levels (although some locations require insurers to “take all comers” for personal lines).
• An insurer may change marketing directives to minimize new applicants whose rates are thought to be
inadequate (e.g. concentrate its advertising on areas in which it believes the rate levels to be
adequate).
•
In the case of banned or restricted usage of a variable (e.g. insurance credit scores), an insurer can
use a different allowable rating variable (e.g. payment history with the company) it believes can explain some
or all of the effect associated with the restricted variable.

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2

Operational Constraints

241 - 244

Operational constraints include items like systems limitations and resource constraints. For example:
Modifying rating algorithms can require significant systems changes, and the complexity of the change
depends on:
 The extent of the changes (e.g. the number of rating variables, the number of levels within each rating
variable, how the rating variables are applied in the rating algorithm)
 The number of systems (e.g. quotation, claims, monitoring, etc.) impacted by the rate change
Also, implementing a new rating variable may require data that has not been previously captured.
 It is often necessary to get this data directly, either through a questionnaire sent to insureds or by
visually inspecting the insured item.
 These approaches can call for additional staff with unique skills.



When an operational constraint arises, a cost-benefit analysis can determine the appropriate course of
action. The cost of implementing the change is the cost associated with modifying the system.
The benefit is the incremental profit that can be generated by charging more accurate rates, and
attracting more appropriately priced customers.

Cost-benefit analysis example:
Assume that:
 a risk characteristic accounts for a 10% difference in projected ultimate losses and expenses between
Class A and Class B.
 the characteristic is not currently reflected in the rates (both classes are charged a rate of $1,050, and
that this average rate reflects a target profit provision of 5.2 %.)
The table below depicts the number of risks for each class, as well as the projected costs, current rates, and
actual profit for each class.
Calculation of Profit (Current Rate)
(1)
(2)

Class
A
B
Total




# Risks
50,000
1,000,000
1,050,000

Projected
Losses &
Expenses
$45,000,000
$1,000,000,000
$1,045,000,000

(3)
Projected
Losses &
Expenses
per Risk
$900
$1,000
$995

(4)

(5)

Current
Rate per
Risk
$1,050
$1,050
$1,050

Target
Profit
%

5.2%

(6)

(7)

Actual Profit
$
%
14.3%
$7,500,000
4.8%
$50,000,000
5.2%
$57,500,000

(3) = (2)/(1)
(6) = [(4)-(3)]x(1)
(7) = (6)/[(4)x(1)]
using the current average rate, Class A risks will be more profitable than Class B risks.
if the rating variable is implemented, the company can decrease the rate for Class A and increase the
rate for Class B in line with the difference in expected costs.

Instead of charging $1,050 for all risks, charge Class A risks $950 and Class B risks $1,055.
Assuming no change in the risks insured, there will be no change in the total profit but the cross-subsidy will be
eliminated (as shown in the table below).

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Chapter 13 Other Considerations
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Calculation of Profit (After Rate Change)
(1)
(2)

Class
A
B
Total

# Risks
50,000
1,000,000
1,050,000

Projected
Losses &
Expenses
$45,000,000
$1,000,000,000
$1,045,000,000

(3)
Projected
Losses &
Expenses
per Risk
$900
$1,000
$995

(4)

(5)

Current
Rate per
Risk
$950
$1,055
$1,050

Target
Profit
%

5.2%

(6)

(7)

Actual Profit
$
%
5.3%
$2,500,000
5.2%
$55,000,000
5.2%
$57,500,000

(3) = (2)/ (1)
(6) = [(4) - (3)] x (1)
(7) = (6)/ [(4) x (1)]
If rate changes are made, the insurer will write more Class A risks and possibly fewer Class B risks.
Assuming the change results in 25% more Class A business and no change in Class B business, the profit
projections are as follows:
Calculation of Profit (After Rate Change and Distributional Shift)
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Projected
Projected
Losses &
Current
Target
Losses &
Expenses
Rate per
Profit
Actual Profit
Class
# Risks
Expenses
per Risk
Risk
%
$
%
A
$900
$950
$3,125,000
5.3%
62,500
$56,250,000
B
1,000,000
$1,000,000,000
$1,000
$1,055
$55,000,000
5.2%
Total
1,062,500
$1,056,250,000
$994
$1,049
5.2%
$58,125,000
5.2%
(3) = (2)/ (1)
(6) = [(4) - (3)] x (1)
(7) = (6)/ [(4) x (1)]
Conclusion/Course of Action:
 Implementing the rating variable will generate an additional $625,000 (= $58,125,000 - $57,500,000)
in profits.
 Compare the profit to the cost of making the change to determine the appropriate course of action.
 There may also be other costs associated with this change (e.g. changes in staffing for the UW
department to handle the increased number of Class A insureds).

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Cha
apter 13 Other
O
Co
onsideratiions
BASIC RATEMAKING
G – WERNER
R, G. AND MODLIN, C
3

Marketing Considera
ations

24
44 - 260

The relatio
onship betwe
een price and profit (assum
ming the numb
ber of policiess is fixed) is shown below:
Profit As
ssuming Fixe
ed Volume

d curve shows
s that that the
e demand for a product de creases as th
he price increases.
A demand
Sample
e Demand Curve

To determ
mine true expe
ected profitab
bility, the two curves
c
should
d be considerred simultane
eously.
Expecte
ed profit as a function of prrice is an arc--shaped curve
e.
Expecte
ed Profit Cons
sidering Demand

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Chapter 13 Other Considerations
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Total profit increases to the price at which lost business outweighs the benefit associated with higher prices on the
business that remains.
This does not mean that the actuarial rate indication is incorrect (since the latter is determined without regard to
whether or not the product will be purchased).
Thus, the insurer should consider both the cost-based rate indication and marketing conditions.
Marketing considerations:
Insurers often categorize insureds into new and renewal business.
 These groups are analyzed separately since purchasing behavior and expected profitability of each group
can be quite different.
 Factors that affect an insured’s propensity to renew an existing product or purchase a new product are:
1. Price of competing products: If the same product is offered at a lower price, they are likely to
purchase the competing product.
2. Overall cost of the product: If the product is costly, insureds are likely to compare prices to determine
any potential savings (and vice versa).
3. Rate changes: Significant increases (or decreases) in premium for an existing policy can cause
existing insureds to look for better options.
4. Characteristics of the insured:
i. A large established business may be less sensitive to the price of its commercial package policy than
a sole practitioner.
ii. A young policyholder may shop (and change insurers) more frequently than an older policyholder.
5. Customer satisfaction and brand loyalty: Poor claims handling or a bad customer service experience
may cause existing insureds to explore other options.
Notes:
 The above are more relevant for personal lines insureds than for larger commercial lines purchasers.
 Commercial entities have less access to competitive price information and stay with an existing carrier
based on service.
Techniques for Incorporating Marketing Considerations
The decision-maker considers the traditional actuarial rate indication along with marketing information
(incorporated judgmentally) to determine the set of rates to be implemented.
Marketing information includes:
* Competitive comparisons
* Close ratios, retention ratios, growth
* Distributional analysis
* Dislocation analysis
1. Competitive Comparisons via Premiums Charged
All information needed to accurately determine the premium charged by competitors can be difficult to obtain.
* U.S. commercial lines insurers adjust the manual rate via schedule and experience rating (see Chapter 15).
* For U.S. personal lines, estimating a competitor’s premium is difficult if the competitor makes extensive use
of risk placement to vary the rate charged (e.g. insurers use U/W tiers that function as a rating variable, but
the guidelines or algorithms that allocate risks into tiers are not always publicly available).

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In sophisticated, less regulated markets (e.g. the U.K.), rate manuals may not be available, and rates may
change (as frequently as daily).
Insures may rely on obtaining competitive price quotes from brokers, questioning potential or existing
customers about price information, or surveying Web-based quoting engines.
Though data is hard to obtain, it is valuable to compare premium to competitors’. Insurers are interested in 2
levels of competitiveness:
1. how competitive their rates are on average (i.e. for all risks combined, a.k.a. a base rate advantage).
2. how competitive their rates are for individual risks or groups of risks (e.g. for new homes or claims-free
drivers).
Overall competitive position compares premiums for a set of sample risks, for all quoted risks (for new business),
or for all existing insureds (for renewal competitiveness).
When doing so, companies typically focus on one or more of the following metrics:

Competitor Premium
(or the reciprocal) - 1.0
Company Premium



% Competitive Position =



$ Competitive Position = Competitor Premium - Company Premium (or the reverse)



%Win =



Rank = Rank of Company Premium when compared to the premium from several competitors

Number of Risks Meeting Criteria (e.g. Premium Lower than Competitor )
Total Number of Risks

The chart below shows a distribution of policies for different ranges of the percentage competitive measure:
Policy Count by Percentage Competitive

The x-axis represents different ranges of the % competitive position.
i. if 2 insurers charge the same premium, then all policies are in the range containing 0% (i.e. -5% to 5%).
ii. if the competitor has a different premium structure, the bars will be dispersed across the different
ranges.
a. the overall average competitive position is -7% ( on average, the competitor’s premium is 7% lower
than the insurer’s premium), but the competitiveness ranges from -60% to over 100%.
b. this variation highlights significant differences in the rating algorithms/relativities between the 2
insurers.

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Cha
apter 13 Other
O
Co
onsideratiions
BASIC RATEMAKING
G – WERNER
R, G. AND MODLIN, C
Competittive Analysis
s using Rate Relativities
The chart below shows
s a compariso
on of age rela
ativities for personal auto ccoverage.
 the x-axis shows
s the different age levels off the variable being studied
d (i.e. ages)
 the bars represent the numbe
er of vehicles for each leve
el of age (rightt y-axis),
 the lines represe
ent the rate re
elativities by company
c
(left y-axis).

This type
e of competitiv
ve analysis is
s effective when rating algo
orithms are siimilar betwee
en companies.
Howeverr, rating algorrithms have be
ecome much more comple
ex (and includ
de many more
e risk charactteristics, thus
individua
al rate relativitty comparison
ns may be les
ss meaningfu l). Exampless:
i. Com
mparing age re
elativities may
y not be usefu
ul if one insurrer includes other age-relatted factors in its rating
algorrithm (e.g. rettiree discountts, inexperienced operator surcharges) while the other insurer doe
es not.
ii. Ratin
ng variables may
m be additiv
ve for one ins
surer and mulltiplicative for another insurer.
e total premium comparis
sons for groups of risks sharing the rating chara
acteristic of iinterest.
Thus, use
The chart on the next page
p
shows the average premium
p
by ag
ge rather than
n the rate rela
ativities by ag
ge.

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Chapter 13 Other Considerations
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Average Premium Comparisons

This shows where competitive threats and opportunities exist for the company’s existing rating variables.
When using this type of analysis, note that a change in one variable’s rate relativities can have an unintended
impact on the average premium of a certain level of another variable. Example:
If square footage introduced as a rating variable in HO insurance, it may significantly change the average
premium of certain territories or AOI levels (since those are highly correlated with square footage).
2 Close Ratios, Retention Ratios, Growth
The Close ratio (a.k.a. hit ratio, quote-to-close ratio, or conversion rate) measures rate at which
prospective insureds accept new business quotes: Close Ratio 



Number of Accepted Quotes
Total Number of Quotes

If an insurer issues 25,000 quotes in a month and generates 6,000 new policies then the close ratio is
24% (= 6,000 / 25,000).
Understand the data used to calculate the denominator of the ratio. Example:
i. Insurer A may include all quotes issued, while insurer B may only include one quote per applicant.
ii. Insurer A will have a lower close ratio if applicants request more than one quote before making a
decision (e.g. if an applicant gets several quotes with different limits).

The Retention ratio (a.k.a. persistency ratio) measures the rate at which existing insureds renew their
policies upon expiration: Retention Ratio =




Number of Policies Renewed
Total Number of Potential Renewal Policies

If 30,000 policies are up for renewal in a month and 24,000 renew, then the retention ratio is 80%
(= 24,000 / 30,000).
Renewal customers are less expensive to service and generate fewer losses than new customers.
Understand the data used to calculate the denominator of retention ratio.
If insurer A excludes all policies that were non-renewed (because they no longer met the eligibility criteria),
and insurer B includes them, then insurer A will have a better retention ratio than insurer B.

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Cha
apter 13 Other
O
Co
onsideratiions
BASIC RATEMAKING
G – WERNER
R, G. AND MODLIN, C
Both abso
olute ratios an
nd changes in
n the close an
nd retention ra
atios are anallyzed.
 Insu
urers rely on close
c
ratios and
a retention ratios
r
as prim
mary signals o
of the competiitiveness of ra
ates for
new
w business an
nd renewal cu
ustomers, resp
pectively.
 Cha
anges in ratios are used to
o gauge chang
ges in compe
etitiveness.
 Close ratios and retention ratiios are review
wed when rate
e changes are
e implemente
ed. Rate chan
nges:
i. affect
a
renewal business dire
ectly (since any change ca
an motivate existing custom
mers to shop
ellsewhere).
ii. influence the insurer’s com
mpetitive posittion (e.g. If an
n insurer takess a rate decre
ease, the exp
pectation is
that
t
the close and retention
n ratios will im
mprove, and vvice versa)
The follow
wing are chartts comparing close ratios and
a retention by month (x--axis).
 The
e bars represe
ent the numbe
er of applican
nts or renewa ls (right y-axiss) for each m
month.
 The
e line represents the close or retention ratio
r
(left y-axxis) for each m
month.
 The
e increase in each
e
ratio ove
er the last cou
uple months ccoincides with
h a rate decre
ease impleme
ented in July.
Close Ratios
R
by Mo
onth

Retention Ratios by
y Month

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Cha
apter 13 Other
O
Co
onsideratiions
BASIC RATEMAKING
G – WERNER
R, G. AND MODLIN, C
Growth captures
c
new business obta
ained and existing custom
mers retained.
Policy gro
owth rate is de
efined as:

%PoliccyGrowth =

(New Policiees Written - Lost
L Policiess) Policiees at End of Period
=
- 1..0 , where
Policiees at Onset off Period
Policiess at Onset off Period

a “lost policy”
p
can eitther be a canc
celled or non--renewed pol icy.
Example:
 Assume
A
there were 360,000
0 policies at the beginning of the month
h.
 If 9,600 new po
olicies were added
a
and 6,0
000 policies w
were lost durin
ng the month, then the mo
onthly policy
grrowth is 1.0%
% (= [9,600 - 6,000]
6
/ 360,000).
Growth pe
ercentages arre tracked ove
er time.
i. Low or
o negative grrowth can ind
dicate uncomp
petitive rates and vice verssa.
ii. Chan
nges in growth can also be
e significantly impacted by items other tthan price. Exxample:
If an insurer tighten
ns or loosens
s the underwriiting standard
ds, growth can
n be affected.
Polic
cy Growth by
y Month

The close
e, retention, and growth rattios may be trracked for spe
ecific groups of insureds.
 If any of the ratios look wors
se for a segm
ment despite h
having similarr competitiven
ness as otherr segments,
th
hen it may ind
dicate that:
i. the segmentt is more price
e sensitive
ii. competitive rate compariisons are not valid, or
iii. something other
o
than pric
ce is driving the
t purchasin
ng decision.


Consider
C
the char
c
below of close ratios by
b age of nam
med insured.
i. The bars rep
present the nu
umber of app
plicants (right y-axis)
i. the line represents the clo
ose ratio (left y-axis) by ag
ge of applican
nt (x-axis).

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Cha
apter 13 Other
O
Co
onsideratiions
BASIC RATEMAKING
G – WERNER
R, G. AND MODLIN, C
Close Ra
ates by Age of
o Named Ins
sured




Eve
en if the comp
petitive positio
on is similar across
a
all age groups, the cclose rate is tthe lowest forr the younger
insu
ureds (since younger
y
insurreds tend to be
b more price -sensitive).
Sim
milar analysis can be perforrmed for reten
ntion and grow
wth.

3. Distrib
butional Ana
alysis
A distrib
butional analy
ysis includes both
b
the distribution by seg
gment at a givven point of time and chan
nges in
distributtions over tim
me. For example,
An insurer may wish to review its distribution of HO policies by amounts of insurance (AOI)
Policies
s and Averag
ge Premium by AOI Rang
ge

 the distributional
d
analysis
a
may uncover that while 15% off homes in a market are va
alued under $
$200,000,
only 5% of the homes in the insurers portfollio have an AO
OL in that ran
nge. Reasons for this inclu
ude:
i. ins
surer rates for homes in this range are uncompetitive
u
e.
ii. po
oor marketing
g or inadequa
ate agent plac
cement.
 a co
omparison of distributions over time can
n reveal whetther this low p
penetration ha
as been consistent or if it
is a recent development (if the
e latter, it cou
uld indicate th at a competittor began targ
geting homess valued less
than
n $200,000 viia marketing strategy,
s
price
e strategy, ettc).

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Cha
apter 13 Other
O
Co
onsideratiions
BASIC RATEMAKING
G – WERNER
R, G. AND MODLIN, C
4. Policy
yholder Dislo
ocation Analy
ysis
Quantifies
s the number of existing cu
ustomers thatt will receive sspecific amou
unts of rate ch
hange.
 It is used to extra
apolate how the
t rate chang
ge may affectt retention.
believes will p
 Thresholds defin
ne the magnitu
ude and dispe
ersion of rate
e changes tha
at the insurer b
produce an
una
acceptable efffect on retentiion (in total orr by customerr segment).
If th
he effects are outside the to
olerance leve
el, the insurer could revise the proposed
d rate change
e.
 Kno
owledge of the
e expected diislocation can
n be shared w
with the sales and custome
er support uniits (e.g. call
cen
nters) prior to implementatio
on to prepare
e them for cusstomer respon
nse (e.g. a cu
ustomer callin
ng an agent
abo
out a large pre
emium increase).
 When a base ratte change is made,
m
the am
mount of disloccation is unifo
orm across all insureds.
 If ra
ate relativities also change, the amount of dislocation
n can vary sig
gnificantly for different insureds or
clas
sses of insure
eds.
Ratte Change Diistribution

Assimilatting the Inforrmation
One mustt weigh all info
ormation and select rates that
t
best mee
et the insurer’’s goals (done
e judgmentallly).
Assume the following about
a
a class of business:
C
averag
ge premium
0
• Current
= $1,000
• In
ndicated avera
age premium
= $1,200
0 (or 20% inccrease)
• Competitor’s
C
average
a
premium = $1,000
0
• Close
C
ratio, rettention ratio, and
a growth are all significa
antly below ta
arget
mpacts:
Options/Im
1. Implem
menting a 20%
% increase wiill cause signiificant loss off renewal custtomers and prohibit busine
ess growth.
2. If the in
nsurer decide
es not to imple
ement the fulll increase, it ccan consider other non-priicing solutionss to improve
profitab
bility (e.g. rev
vise UW guide
elines or mark
keting strateg
gies).

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Chapter 13 Other Considerations
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Systematic Techniques for Incorporating Marketing Considerations
These techniques incorporate both marketing information and actuarial indications when proposing rates.
1. Lifetime Value Analysis
Examines the profitability of an insured over a long period of time noting that not all insureds will renew.
To do this, assumptions are made regarding the:
i. propensity of the insured to renew (see (6) below)
ii. expected profitability of the insured over the time period being projected (see (3) and (4) below).
The following is a personal auto lifetime value calculation analyzing the long term profitability of a 22-year-old
and a 70-year-old.
Four-Year Time Horizon for 22-Year-Old
(1)
(2)
(3)
(4)
Year
1
2
3
4
Total

Age
22
23
24
25

Prem
$810
$800
$790
$780
$3,180

Losses
$800
$750
$700
$650
$2,900

Expense
$35
$15
$15
$15
$80

Four-Year Time Horizon for 70-Year-Old
(1)
(2)
(3)
(4)
Year
1
2
3
4
Total

Age
70
71
72
73

Prem
$600
$600
$600
$600
$2,400

Losses
$550
$578
$606
$640
$2,374

Expense
$35
$15
$15
$15
$80

(5)= (2) - (3) - (4);
(7)= (6) x (Prior7);
(10)= (2) x (7)discounted by 5% per annum;

(5)
Profit
($25)
$35
$75
$115
$200

(5)
Profit
$15
$7
($21)
($55)
($54)

(6)
Renewal
Prob
100.00%
75.00%
75.00%
80.00%

(7)
Cumulative
Persistency
100.0%
75.0%
56.3%
45.0%

(6)
Renewal
Prob
100.00%
95.00%
96.00%
97.00%

(7)
Cumulative
Persistency
100.0%
95.0%
91.2%
88.5%

(8)= (5) x (7);

(8)
Adj Profit
($25.00)
$26.25
$42.19
$51.75
$95.19

(8)
Adj Profit
$15.00
$6.65
($19.15)
($48.66)
($46.16)

(9)
PV of
Adj Profit
($25.00)
$25.00
$38.27
$44.70
$82.97

(10)
PV of
Premium
$810.00
$571.43
$403.06
$303.21
$2,087.70

(9)
PV of
Adj Profit
$15.00
$6.33
($17.37)
($42.03)
($38.07)

(10)
PV of
Premium
$600.00
$542.86
$496.33
$458.51
$2,097.69

(11)
Profit%
-3.1%
4.4%
9.5%
14.7%
4.0%

(11)
Profit%
2.5%
1.2%
-3.5%
-9.2%
-1.8%

(9)= (8) discounted by 5% per annum;
(11)= (9) / (10)

Conclusions:
 The % profit over a one-year time horizon (i.e. the first row in each table) show that a 70-year-old is more
profitable to insure than a 22-year-old.
 When persistency is considered over a four-year time horizon, the 22-year-old (age 25 at the end of the time
period) is more profitable than the 70-year-old (age 73 by the end of the time period).
Improvements to this type of analysis include:
 refining the assumptions
 increasing the time horizon
 incorporating results from other products the customer may purchase.
For related information on lifetime value analysis, see “Personal Automobile Premiums: An Asset Share
Pricing Approach for Property/Casualty Insurance” (Feldblum 1996), which is also part of the CAS Exam 5
Syllabus of Readings.

Exam 5, V1b

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Cha
apter 13 Other
O
Co
onsideratiions
BASIC RATEMAKING
G – WERNER
R, G. AND MODLIN, C
2. Optimized Pricing
Multivaria
ate statistical modeling
m
tech
hniques are being
b
applied to develop re
enewal and co
onversion mo
odels (i.e.
customer demand mod
dels). These models
m
are us
sed to estima
ate the probab
bility that:
 an applicant
a
will accept a quo
ote (i.e. conve
ersion model) or
 an existing
e
custo
omer will acce
ept the renew
wal offer (i.e. re
etention model).
Historical data used to develop thes
se models inc
cludes:
 a se
eries of obserrvations and
 a co
orresponding response forr each observ
vation.
Examples
s of model dattasets:
i. a co
onversion mo
odel dataset contains
c
a serries of new bu
usiness quote
es and whether each quote
e was
acc
cepted or rejec
cted.
ii. a retention
r
model dataset contains a serie
es of renewal offers and whether each o
offer was accepted or not.
Each data
observation ((e.g. risk charracteristics su
aset should in
nclude relevan
nt information
n about each o
uch as
amount off premium quoted, rate cha
ange information (for reten
ntion models), and an indiccator of the
competitiv
veness of the premium).
The mode
els help predic
ct the change
e in close rate
e or retention rate in respon
nse to a prop
posed rate cha
ange.
The chart below is an output
o
from a retention mo
odel.
 The
e bars represe
ent the % of policies
p
(right y-axis) gettin
ng different % change in prremium (x-axiis).
 The
e lines illustratte the insured
d’s propensity
y to renew (lefft y-axis) depending on wh
hether it is the
e first or
sub
bsequent rene
ewal for the in
nsured.
As premiu
um changes increase, the blue (bottom)) line drops m
more steeply th
han the red (ttop) line, sugg
gesting
that the lo
onger the insu
ured is with th
he carrier, the less sensitive
e he or she iss to premium increases.
Retention
n Model Outp
put

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Chapter 13 Other Considerations
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Scenario testing rate changes (a precursor to full price optimization):
A loss cost model and a customer demand model can be jointly used to estimate expected premium volume,
losses, and total profits for a given rate proposal.
 For renewal business, the loss cost and retention models project the expected profitability and
probability of renewal for each risk at a given price.
 Using these models, an insurer can test several rate change scenarios on the in-force distribution to
determine the expected volume, premium, losses, and profit of each scenario.
i. The objective: Identify the rate change that best achieves the company’s profitability and volume
goals on the renewal portfolio.
ii. Added benefit: This same process can test multiple rate scenarios on new business by applying the
results of loss cost models and conversion models on a portfolio of quotes.
Optimization algorithms:
 incorporate loss cost models, demand models, and other assumptions as inputs, and generate
hundreds of thousands of scenarios to determine the premium for each individual risk that optimizes
overall profit while achieving an insurer’s overall volume goals (or optimize volume while achieving an
insurer’s overall profitability goals).
 require the actuary to translate individually optimized premium into a manual rate structure, depending
on the product being priced.
In summary, optimized pricing systematically combines knowledge of loss costs and customer demand to
develop rates that meet volume and profitability objectives of the insurer.
3. Underwriting Cycles
The industry undergoes cyclical results (i.e. overall industry profitability oscillates systematically), and
understanding which phase of the cycle one is in is important when determining which rates to implement.
The terms “hard market” and “soft market” refer to the highs and lows of the cycle.
 The hard market refers to periods of higher price levels and increased profitability.
i. insurers respond to this profitability by trying to expand their market share.
ii. insurers become more aggressive in their pricing (deviating from actuarial indications), which puts
pressure on other insurers to reduce prices.
 This leads to a soft market, during which profits are lower. In response to the low profits, insurers focus
more on the actuarial indications and take appropriate rate increases.
Thus, competitive pressures ease and the cycle begins again. The U/W cycle is shown below.

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Cha
apter 13 Other
O
Co
onsideratiions
BASIC RATEMAKING
G – WERNER
R, G. AND MODLIN, C
The U/W
W cycle

When ma
aking pricing decisions,
d
the
e actuary mus
st understand the existence
e of U/W cycles and consid
der the
current cy
ycle stage of the
t industry.
Refer to “T
The Impact of
o the Insuranc
ce Economic Cycle on Ins urance Pricin
ng” (Boor 2004
4) for more detailed
informatio
on on U/W cyc
cles.

Key Conce
epts

26
62 - 262

1. Regula
atory constrain
nts
2. Operational constraiints
a. Typ
pes of operatiional constraints
b. Cost-benefit ana
alysis
3. Market consideration
ns
a. Tra
aditional analy
ysis
i. Competitive
C
comparisons
c
ii. Close
C
ratios
iii. Retention rattios
iv. Distributional analysis
v. Policyholder dislocation an
nalysis
b. Sys
stematic analysis
i. Lifetime custom
mer value
ii. Optimized
O
pric
cing
derwriting cyc
cles
c. Und

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Cha
apter 13 Other
O
Co
onsideratiions
BASIC RATEMAKING
G – WERNER
R, G. AND MODLIN, C
The pred
decessor pa
apers to the
e syllabus reading
r
“Ba
asic Ratema
aking” by W
Werner, G. a
and Modlin, C.
were numerous, bu
ut none cove
ered the top
pics that are
e presented
d in this cha
apter. Thus
s, there are no
past CAS
S questions
s that are re
elevant to th
he content c
covered in this chapte
er.
ns from the 2012 exam:
Question
11. (1.5 points) An insu
urer uses sev
veral rating va
ariables, includ
ding vehicle w
weight, to dettermine premium
charges fo
or commercia
al automobiles
s. Your mana
ager has requ ested a revie
ew of the vehiccle weight ratting
relativities
s. The followin
ng diagnostic chart display
ys the results for vehicle weight from a g
generalized linear
model.

y managemen
nt plans to exp
pand its comm
mercial auto m
market-share with an emphasis on writing more
Company
businesse
es that operatte with extra-h
heavy weight vehicles. Ma
anagement wa
ants to charge
e the same ra
ates for
both heav
vy and extra-h
heavy weight vehicles.
Based on the model re
esults, provide
e your recomm
mendation to management and explain the considerrations
supporting
g your positio
on. Include a discussion
d
of any potentiall risks associa
ated with it

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Chapter 13 Other Considerations
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
The predecessor papers to the syllabus reading “Basic Ratemaking” by Werner, G. and Modlin, C.
were numerous, but none covered the topics that are presented in this chapter. Thus, there are no
past CAS questions that are relevant to the content covered in this chapter.
Solutions to Questions from the 2012 exam:
Question 11 – Model Solution (Exam 5A Question 11)


The error bars are fairly wide around the relativity for the extra-heavy vehicles due to low volume of data
for this level.



The relativity for heavy vehicles, of about 1.55 is found in the 95% confidence interval for extra
heavy vehicles.



Finally, since management wants to expand its comm. auto market share, and given the two facts
above, I suggest we charge the same relativities for heavy and extra-heavy.



The risk is that when we gather enough data over time, we may realize that the rate for extra-heavy
vehicles turns out to be insufficient. At this point we can adjust the rate accordingly.

Examiner’s Comments
Many candidates lost points for not including any discussion of potential risks or for incomplete
considerations supporting the recommendation.

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Chapter 14 - Implementation
BASIC RATEMAKING – WERNER, G. AND MODLIN, C

Sec
1
2
3
4
5
6
1

Description
Example Imbalance
Non-Pricing Solutions
Pricing Solutions
Calculating New Rates Based On Bureau Or Competitor Rates
Communicating and Monitoring
Key Concepts

Pages
263 – 263
263 - 264
264 -285
285-286
286 -287
288 - 288

Example Imbalance

263 – 263

This chapter discusses actions an insurer can take if its current rates do not produce an average premium
equal to the sum of the expected costs and target underwriting profit.
This chapter uses the notation found in the Foreword to this text, and uses the same pricing example and
assumptions described in prior chapters (a.k.a. the “simple example”):
__

* The average expected loss and LAE ( ( L  EL ) ) for each policy is $180.
__

* For each policy written, the insurer incurs $20 in fixed expenses ( E F ) for costs associated with
printing and data entry, etc.
* 15% of each dollar of premium collected covers expenses (V) that vary with the amount of
premium, such as premium taxes.
* The target profit provision (QT ) is 5% of premium (determined by management)
The indicated average premium per exposure is $250 (= ($180 + $20) / (1.0 - 0.15 - 0.05)).
If the projected average premium per exposure is $235, the fundamental insurance equation is not in balance.
The insurer can bring the equation into balance by reducing its costs (non-pricing solutions), increasing its
rates, or both.

Non-Pricing Solutions

263 - 264

1. Balance can be achieved through expense reductions (i.e. reduction in UW or LAE expenses, by reducing the
marketing budget or staffing levels).
i. if fixed expenses are reduced from $20 to $8, or variable expenses are reduced from 15% to 10%, the
equation will be brought into balance.
ii. if the actuary projects a reduction in expenses, recalculate the overall rate level indication.

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Chapter 14 - Implementation
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
2. Balance can be achieved by reducing the average expected loss as follows:
a. Change the make-up of the portfolio of insureds.
i. An insurer may tighten the U/W criteria or non-renew policies with grossly inadequate premium.
Note: When the portfolio changes, expected losses and expected premium change; but if the loss
reduction is greater than the premium reduction, the UW action could move the fundamental
equation to the balanced position.
ii. If the insurer does this, adjust the premium and loss projections and recalculate the overall rate level
indication.
b. Reduce the coverage provided by the policy (a.k.a. a coverage level change).
Example: A HO insurer may adjust the policy to exclude coverage for mold losses.
i. If this eliminates previously covered losses and rates are not decreased accordingly, then this coverage
level change is equivalent to a rate level increase.
In the simple example, the insurer needs to reduce the average expected loss and LAE from $180 to
$168 to bring the fundamental insurance equation into equilibrium.
ii. If an insurer does this, adjust the premium and loss projections and recalculate the overall rate level
indication.
c. Institute better loss control procedures.
Example: A WC insurer may reduce average severity by applying proactive medical management
procedures and return-to-work programs for disability claims likely to escalate.

Pricing Solutions

264 -285

Most insurers choose to change current rates (i.e. implement a rate change) to get closer to the desired
equilibrium (since achieving the target U/W profit is important). However:
 Chapter 13 addressed reasons why an insurer may implement rates different from those indicated.
 If the insurer decides that $235 is the most that can be charged in the short run, it is forced to accept a
target U/W profit provision of -0.1% [ = ($235 - $180 - $20 – (0.15 x $235)) / $235 ] until rates can be
increased.
To calculate a final set of rates for an existing product, the insurer must:
1. Select an overall average premium target for the future policy period (see chapter 8).
2. Finalize the structure of the rating algorithm (see chapter 2, and the example below).
3. Select the final rate differentials for each of the rating variables (see chapters 9 - 11, 13).
4. Calculate proposed fixed expense fees, if applicable (see example below).
5. Derive the base rate necessary to achieve the overall average premium target (see example below).
Example Rating Algorithm
Assume a simple multiplicative rating algorithm includes:
 a base rate (B),
 an additive fixed policy fee (A),
 two multiplicative rating variables (R1 and R2),
 [1.0 - two discounts (D1 and D2)]
P and X are used to denote premium and exposures, respectively.
Subscript P refers to “proposed” and subscripts i, j, k, m refer to different levels for the different rating
variables/discounts.

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Chapter 14 - Implementation
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
The proposed rating algorithm for a given risk is defined as follows:

PP , ijkm  [ BP  R1P , i  R 2 P , j  (1.0  D1P , k  D 2 P ,m )  AP ]  X ijkm
The portion of the premium:
 calculated prior to adding the fixed policy fee and other dollar additives is the variable premium
 derived from the additive fixed policy fee and other dollar additives is the flat or additive premium.
Example Rating Variables
Assume the insurer relies on the following data to select proposed rate differentials for each rating variable:
Current
Indicated Competitor Proposed
Current Indicated Competitor Proposed
R1

Differential

Differentia

Differential

1
2
3

0.8000
1.0000
1.2000

0.9000
1.0000
1.2500

0.9200
1.0000
1.2500

0.9000
1.0000
1.2500

A
B
C

Current

Indicated

Competitor

Proposed

Discount

Y
N

5.0%
0.0%

4.0%
0.0%

5.0%
0.0%

Differential Differential Differential Differential
1.0000
1.0500
1.2000

Current

1.0000
0.9000
1.3000

1.0000
0.9500
1.3500

Indicated Competitor

1.0000
0.9500
1.3000
Proposed

Discount

5.0%
0.0%

Y
N

10.0%
0.0%

2.5%
0.0%

7.5%
0.0%

5.0%
0.0%

Calculation of Fixed Expense Fees and Other Additive Premium
Scenario 1: When a rating algorithm incorporates fixed expenses through an additive per exposure expense
___

fee, the fee is based on the average fixed expense per exposure ( E F ).
Also, the fee must be adjusted to account for V and QT in the same way that losses and LAE per exposure
are adjusted for these items in the rate level indication formulae (e.g. the insurer incurs variable expenses and
expects target profit on all premium, including that which comes from fixed expense fees).
 The adjustment is the average fixed UW expense divided by the variable permissible loss ratio:
___

EF
AP 
(1.0  V  QT )
Calculation of the proposed expense fee:
(1) Average Fixed Expense
$20.00
(2) Variable Expense %
15.0%
(3) Target Profit %
5.0%
(4) Variable Permissible Loss Ratio
80.0%
(5) Proposed Fee
$25.00
(4)= 1.0 - (2) - (3)
(5) = (1)/(4)
Here, the proposed $25 additive fee includes $20 to cover the fixed expenses and $5 to cover the
variable expense (e.g. premium tax) and profit associated with the $20.
Some insurers use a fixed per policy expense fee rather than a fixed per exposure expense fee in the
rating algorithm.
It is important that base rate derivation formulae (see next section) combine average variable premium and
average flat premium on a consistent basis (i.e. per policy or per exposure). A per policy expense fee can be
converted to a per exposure expense fee by dividing by the average number of exposures per policy.


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Chapter 14 - Implementation
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Scenario 2: The (V) used to adjust the flat fee may differ from the (V) used in calculating the overall rate level
indication.
Insurers may elect not to apply certain aspects of the variable expenses to the flat fee (i.e. some
insurers may not make the flat fee subject to agent commissions).
If the premium-based expense projection method is used (see chapter 7), a fixed expense ratio is calculated
(rather than a fixed expense dollar amount).
 The ratio can be converted to a dollar amount by multiplying it by the projected premium per exposure,
as shown in the following table.
Calculation of $Fee (Using the Fixed Expense Ratio)
(1) Fixed Expense Ratio
8.0%
(2) Projected Average Premium per Exposure
$250.00
(3) Average Fixed Expense
$20.00
(4) Variable Expense %
15.0%
(5) Target Profit %
5.0%
(6) Variable Permissible Loss Ratio
80.0%
(7) Proposed Fee
$25.00
(3) = (1) x (2); (6) = 1.0 – (4) – (5);
(7) = (3) / (6)
Some rating algorithms have other additive premium components (in addition to fixed expense fees):
 In HO insurance, endorsements that add or extend coverage are priced separately and are added to the
variable premium of the standard policy.
 The same adjustment (described above for fixed expense fees) applies to other additive premium.
Derivation of Base Rate: No Rate Differential Changes
The base rate is derived:
 so that proposed average premium (or change in average premium) is expected to be achieved.
 after the actuary selects:
i. the proposed average premium per exposure (or change in proposed average premium),
ii. the proposed rate differentials,
iii. the proposed fixed expense fees, and other additive premium
 to achieve the target UW profit.
Consider the simple scenario when there is only variable premium and rate differentials are not changing.
Here, the proposed base rate (PBR) = current base rate (CBR) times the ratio of the proposed average
___

premium to current average premium: B p  BC 

PP
PC

If there are flat premium components (and rate differentials are still not changing), the PBR equals
___

(P  A )
B p  BC  P P
( PC  Ac )
Note: The when a 5.0% overall average premium change is targeted, it can be achieved by:
i. increasing the base rate 5.0% and increasing the flat premium 5.0%, or
ii. increasing the base rate by 5.56% in order to achieve the 5.0% overall change (i.e. 5.0% = 90%
(5.56%) + 10% (0.0%)), assuming the flat premium is 10% of the total average premium.

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Chapter 14 - Implementation
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Derivation of Base Rate: Rate Differential Changes
Three approaches for deriving the proposed base rate:
1. Extension of exposures
2. Approximated average rate differential
3. Approximated change in average rate differential



The extension of exposures method is the most direct and most accurate.
The approximated methods are used when the extension of exposures is not practical for the product
being priced.

1. Extension of Exposures Method
The same technique (see chapter 5) is applied when deriving a proposed base rate.
Proposed rate differentials (R1P, R2P, D1P, D2P), a proposed fixed expense fee per policy (AP), and a starting
value for the proposed base rate (BS) is used to rerate individual policies (a.k.a. a “seed” base rate).
The proposed premium per policy is:

PS ,ijkm = [[ BS  R1P ,i  R 2 Pj  (1.0 - D1P ,k - D 2 P ,m )  AP ]  X ijkm
The proposed average premium (assuming the seed base rate) is:
___

PS 

 [[ B

___

P S  BS 

Exam 5, V1b

 R1P ,i  R 2 Pj  (1.0  D1P ,k  D 2 P ,m )  AP ]  X ijkm ]
, which can be simplified as:

[ R1

P ,i

 R 2 Pj  (1.0  D1P ,k  D 2 P ,m )  X ijkm ]
 AP

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Chapter 14 - Implementation
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Using the extension of exposures method, the resulting proposed average premium assuming a base rate seed
of $215 is $246.83.
Extension of Exposures (Assuming Seed Base Rate)
(1)

(2)

(3)

(4)

(5)

Exposures

A
Y
10,000
1
Y
A
Y
7,500
2
Y
A
Y
3,000
3
Y
9,000
1
B
Y
Y
20,000
2
B
Y
Y
5,000
3
B
Y
Y
1,875
1
C
Y
Y
5,000
2
C
Y
Y
2,000
3
C
Y
Y
3,500
1
A
N
Y
7,500
2
A
N
Y
3,500
3
A
N
Y
15,000
1
B
N
Y
36,000
2
B
N
Y
9,000
3
B
N
Y
3,750
1
C
N
Y
10,000
2
C
N
Y
2,000
3
C
N
Y
N
A
3,500
1
Y
N
A
7,500
2
Y
N
A
3,500
3
Y
15,000
1
B
Y
N
36,000
2
B
Y
N
9,000
3
B
Y
N
N
3,750
1
C
Y
N
10,000
2
C
Y
N
5,000
3
C
Y
48,000
1
A
N
N
112,500
2
A
N
N
25,000
3
A
N
N
11,000
1
B
N
N
250,000
2
B
N
N
65,000
3
B
N
N
28,125
1
C
N
N
68,000
2
C
N
N
15,000
3
C
N
N
869,500
(7) Avg Prop Prem (Base Seed = $215)

(6)
Proposed
Premium
(assuming Seed
Base Rate =$215)
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$

1,991,500.00
1,638,750.00
800,625.00
1,713,982.50
4,176,500.00
1,273,960.00
471,365.63
1,382,750.00
678,875.00
730,887.50
1,719,375.00
981,093.75
2,994,667.50
7,885,350.00
2,407,873.50
989,896.88
2,905,250.00
713,834.00
730,887.50
1,719,375.00
981,093.75
2,994,667.50
7,885,350.00
2,407,873.50
989,896.88
2,905,250.00
1,784,585.00
10,488,000.00
27,000,000.00
7,343,750.00
2,297,075.00
57,312,500.00
18,220,312.50
7,777,968.75
20,706,000.00
5,615,625.00
214,616,746.63
246.83

(6)= Calculated via extension of exposures with BS =$215; (7)= (Tot6) / (Tot 1)

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The proposed average premium assuming a seed base rate is lower than the target average premium of $250
so the seed base rate needs to be increased.
1A. Pure Premium Method (PPM):
___

If the pure premium method has been used for the overall rate level indication, ( P S ) is compared to the
target average premium, and if these values are not equal, the seed base rate needs to be adjusted.
___




The actuary can derive P S via trial and error (i.e. testing various base rates until the target average
premium is achieved).
Alternatively, the actuary can calculate the amount the (BS) needs to be adjusted via formula.
The formula for proposed average premium (assuming a seed base rate) is:
___

PS  BS 

 [ R1

P ,i

 R 2 Pj  (1.0  D1P ,k  D 2 P ,m )  X ijkm ]
 AP

The formula for proposed average premium (assuming a proposed base rate) is:
___

PP  BP 

 [ R1

P ,i

 R 2 Pj  (1.0  D1P ,k  D 2 P ,m )  X ijkm ]
 AP

Rearranging the terms and dividing one formula by the other yields:
___

( PP  AP )
___

( PS  AP )



BP
BS
___

Thus, the PBR via extension of exposures is given by the following: B p  BS 

( PP  AP )
___

( PS  AP )

___

If no fixed expense fee or other additive premium applies, B p  BS 

PP
___

PS
The table summarizes the calculation of the proposed base rate
Proposed Base Rate Calculation (Extension of Exposures)
(1) Seed Base Rate
$
215.00
(2) Average Premium assuming Seed Base Rate $
246.83
(3) Proposed Fixed Fee per Policy
$
25.00
(4) Proposed Average Premium
$
250.00
(5) Proposed Base Rate
$
218.07
(2)= from table above.
(5)= (1) x [(4) - (3)] /[(2) - (3)]

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1B. Loss Ratio Method (LRM):
If the LRM is used to calculate an overall rate level indication, the change in average premium is computed.
____

____

The proposed average premium based on the selected change (  ) is PP  (1   %)  PC
___

This can then be used in the base rate derivation formula: BP  BS 

PP  AP
___

PS  AP

___

 BS 

(1  %)  PC  AP
___

PS  AP

Assume that current average premium (using extension of exposures on current rates) is $242.13.
If the indicated % change in average premium is 3.25%, the resulting proposed average premium is $250.
Proposed Base Rate (Extension of Exposures, Loss Ratio Method)
(1) Target % Change in Average Premium
3.25%
(2) Current Average Premium
$242.13
(3) Proposed Average Premium
$250.00
(4) Seed Base Rate
$215.00
(5) Average Premium assuming Seed Base Rate
$246.83
(6) Proposed Fixed Fee per Policy
$25.00
(7) Proposed Base Rate
$218.07
(3)= (1.0 + (1)) x (2).
(7)= (4) x [(3) - (6)] /[(5) - (6)]

2a. Approximated Average Rate Differential Method
An insurer may not be able to retrieve the detailed data to perform the extension of exposures method for deriving
the PBR. One alternative method involves estimating the weighted average proposed rate differential across all
___

rating variables (a.k.a. S P ). Using the extension of exposures technique the proposed average premium is:

 [[ R1

P ,i

___

PP  BP 

 R 2 Pj  (1.0  D1P ,k  D 2 P ,m )  X ijkm ]
 AP

___

S P is substituted for the weighted average proposed rate differential across all rating variables:

 [[ R1

P ,i

___

SP 

 R 2 Pj  (1.0  D1P ,k  D 2 P , m )  X ijkm ]

X
___

Solve for the PBR: BP 

PP  AP
___

SP
___

When a rating algorithm is purely multiplicative, S P is typically approximated as the product of the exposureweighted average differentials for each of the rating variables.
In our example rating algorithm, which has discounts that are additive in nature:
___

SP 

X

 R1P ,i

Exam 5, V1b



X

 R 2 Pj


  X k  D1P ,k  X m  D 2 P ,m  


 1.0   k
 m


X
X


 

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Chapter 14 - Implementation
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
___

The following tables show the approximation of S P for the example, using exposures as weights:
Proposed Differentials Wtd by Exposures
(1)
(2)
R1
1
2
3
Total

Exposures
152,500
570,000
147,000
869,500

(1)

(2)

R2
A
B
C
Total

Exposures
235,000
480,000
154,500
869,500

(1)

(2)

D1
Y
N
Total

Exposures
156,625
712,875
869,500

(1)

(2)

D2
Y
N
Total

Exposures
153,625
715,875
869,500

(3)
Proposed
Differential
0.9000
1.0000
1.2500
1.0247
(3)
Proposed
Differential
1.0000
0.9500
1.3000
1.0257
(3)
Proposed
Discount
0.0500
0.0000
0.0090
(3)
Proposed
Discount
0.0500
0.0000
0.0088

___

(Tot3) = (3) weighted by (2).

(4)

S p = 1.0323 (4) = (Tot3R1) x (Tot3R2) x (1.0 - Tot3D1 - Tot3D2)

The proposed base rate, assuming the exposure-weighted average proposed rate differential across all rating
___

variables from the table above, is: BP 

PP  AP
___



$250  $25
 $217.96
1.0323

This proposed base rate ($217.96) is different than that which was calculated using the extension of exposures
method ($218.07).
 Exposure-weighting each variable’s differentials independently and then combining those averages
according to the rating algorithm ignores the dependence of the exposure distribution by level of one
rating variable on the level of another rating variable (i.e. the distributional bias between variables,
discussed in Chapters 9 and 10).
 The example data was not largely biased, but in practice bias can drive larger discrepancies in the PBR.
To mitigate this bias, use variable premium at current rate level and at base level instead of exposures for
weights in the approximation.
i. Variable premium is the premium before addition of any fixed expense fees or other additive premium.
ii. The current rate level adjustment for the premium in this analysis should be done at the class level (i.e.
applying the parallelogram method to fully aggregated data would not be suitable).

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iii. The phrase “at base level” means that the variable premium for non-base levels is adjusted to remove
the effect of the current rate differential.
* For multiplicative factors, divide the variable premium for each non-base level by the current rate
differential for the given variable. Assuming the rating algorithm is entirely multiplicative, calculating
variable premium at base level may be a feasible improvement.
* When the rating algorithm has both multiplicative and additive components, deriving variable premium
at current rate level and at base level becomes so challenging that the effort to improve the
approximation would be better spent compiling data to use the extension of exposures technique.

2. Approximated Change in Average Rate Differential Method
One issue with this method is that the actuary still needs to calculate the average proposed and current rate
relativities for each rating variable.
For rating algorithms that are complex, the actuary may prefer using the following approach which:
 is used when overall rate level indications are performed with the LRM.
 calculates the change in the average rate differential.
The proposed average premium is the current average premium multiplied by the proposed overall change in
___

___

average premium: PP  (1.0   %)  Pc

The proposed overall change in average premium is comprised of a change to the variable and additive
premium components (  v % and

___

 A % ) . Thus, PP  (1.0   v %)  ( Pc  AC )  (1.0   A %)  ( AC )
___

* The last term on the right side of the equation is

___

AP ; Thus, ( PP  AP )  (1.0  V %)  ( PC  AC )

* The proposed change in variable premium given the overall change, the current average premium, and the
___

current and proposed additive premium is (1.0   v %) 

PP  AP

___



PC  AC

(1.0  %) PC  AC
___

Pc  AC

* The change in variable premium is comprised of the change in base rate and the change in the average
___

B S
rate differential: (1.0  V %)  P  P
BC SC
___

___

B
(1.0  %) PC  AP SC
* The base rate adjustment is P 

___
BC
SP
Pc  AC
Finally, using the

 B % and  S % as the % base rate change and the % change in average rate differential,
___

the equation becomes: 1.0   B % 

(1.0   %) PC  AP
___



Pc  AC

1.0
(1.0   S %)

Comments:
 The final term of the equation (the reciprocal of the adjustment to the rate differentials)
is a.k.a. the off-balance factor (OBF)
 OBF is the amount the base rate needs to be adjusted to balance the change in the rate differentials.

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Calculation of the change in the average rate differentials (  S ).



If current and proposed average differentials are available, use them as they are an exact calculation (as
described earlier in the extension of exposures section).
If the data is not available, the change in average rate differentials is approximated as the product of the
change in the average rate differential for each of the rating variables (w) that are changing with the
review: 1.0   S % 

 (1.0  

S ,w

The formula for the change in average rate differential for R1 is given as:

R1P ,i

 R1

(1.0   S , R1 %) 

 ( PC ,i  AC )

C ,i

 (P

C ,i

 AC )

This formula is the change in the current variable premium due to the change in the rate differentials
for the given rating variable.
The use of variable premium as weights may be difficult for various reasons.
1. It may be difficult to obtain the current variable premium data (particularly at current rate level).
2. Weighting by variable premium is challenging when a rating algorithm has additive components.
Therefore, one may choose to measure the average change in rating differentials using exposures as weights.
 This method of weighting introduces the same distributional bias as discussed in the previous section,
but it may be the most feasible alternative.
 In the example rating algorithm, the additive discounts can be combined and restated as a single
multiplicative variable (i.e. 1 - D1 - D2). The formula for the average rate differential across all variables
in the example is as follows:
1.0 + ∆S% ≈ (1.0 + ∆S,R1%) x (1.0 + ∆S,R2%) x (1.0 + ∆S,(1 − D1 − D2 )%)
Actuaries approximate the average rate differential changes for multiplicative variables (e.g. R1) as follows:
___

(1.0   S , R1 %) 

R1P

, where the current and proposed average differentials are determined using exposures

___

R1C
___

as weights:

R1P 

 R1

P ,i

 Xi

___

and R1C 

 R1

C ,i

 Xi

The change in (1 - D1 - D2) can be approximated as follows: (1.0   S ,(1 D1 D 2) R1 %) 

1  D1p  D 2 P
1  D1C  D 2C

where the current and proposed average discounts are determined using exposures as weights, as
___

shown below for D1:

Exam 5, V1b

D1P 

 D1

P ,i

 Xi

___

and D1C 

Page 200

 D1

C ,i

 Xi

 2014 by All 10, Inc.

Chapter 14 - Implementation
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
The following table shows the approximation of the average change in differentials ((1.0 + ∆S%) for the
example using exposures as weights.
Proposed Average Change in Differentials (Using Exposures)
(1)
(2)
(3)
(4)
Current
Proposed
D1
Exposures
Discount
Discount
Y
156,625
0.0500
0.0500
N
712,875
0.0000
0.0000
Total
869,500
0.0090
0.0090
(1)

(2)

D2
Exposures
Y
153,625
N
715,875
Total
869,500
(Tot3) = (3) Weighted by (2)
(5)

(6)

R1
1
2
3
Total

Exposures
152,500
570,000
147,000
869,500

(5)

(6)

R2
A
B
C
Total

Exposures
235,000
480,000
154,500
869,500

(10)

(11)

1-D1-D2
Total

Exposures
235,000

(3)
(4)
Current
Proposed
Discount
Discount
0.1000
0.0500
0.0000
0.0000
0.0177
0.0088
(Tot4) = (4) Weighted by (2)
(7)
Current
Differential
0.8000
1.0000
1.2000
0.9987

(8)
Proposed
Differential
0.9000
1.0000
1.2500
1.0247

(9)
Proposed /
Current
1.1250
1.0000
1.0417
1.0260

(7)
Current
Differential
1.0000
1.0500
1.2000
1.0631

(8)
Proposed
Differential
1.0000
0.9500
1.3000
1.0257

(9)
Proposed /
Current
1.0000
0.9048
1.0833
0.9648

(12)
Current
Differential
0.9733

(13)
Proposed
Differential
0.9822

(14)
Proposed /
Current
1.0091

(15) Average Change in Differential

0.9989

(9)= (8) / (7)
(Tot9)= (9) Weighted by (6)
(12)= 1 - (Tot3D1) - (Tot3D2)
(13)= 1 - (Tot4D1) - (Tot4D2)
(14)= (13) / (12)
(15) = (Tot9R1) x (Tot9R2) x (Tot14)

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___

Using the results from the prior table and (1.0   B %) 

(1.0  %)  PC  AP
___

PC  AC



1.0
, the
(1.0   S %)

proposed base rate can be calculated as shown in the following table.
Proposed Base Rate (Approximated Method)
(1) Current Base Rate
$210.00
(2) Current Average Premium
$242.13
(3) Target Change in Average Premium
3.25%
(4) Proposed Average Premium
$250.00
(5) Proposed Additive Premium per Policy (same as Current)
$ 25.00
(6) Average Rating Differential Adjustment
0.9989
(7) Proposed Base Rate Adjustment
1.0374
(8) Proposed Base Rate
$217.85
(4)= (1.0 + (3)) x (2)
(7)= [ (4) - (5) ] / [ (2) - (5) ] x [ 1.0 /(6) ]

(8)=(1) x (7)

Other Considerations
Minimum Premium
A minimum premium ensures that, on an individual risk basis, premium covers the expected fixed
expenses plus some minimum expected loss
 Insurers that use a minimum premium requirement do not have additive fixed expense fees in their
rating algorithms.
 Implementing a minimum premium can increase total premium. The effect is calculated as follows:

Effect =


Premium With Minimum
- 1.0
Premium Without Minimum

To offset this increase in premium, the base rate should be multiplied by the following factor:

Offset Factor 

1.0
1.0  Effect

Limiting the Premium Effect of a Single Variable
Actuaries may decide to limit the premium impact caused by the change in rate differentials for a rating variable.
Example: A territorial analysis may be performed to determine a set of proposed relativities.
i. After taking into account business considerations (e.g. marketing) the actuary may decide to limit or “cap”
the premium impact on any one territory by adjusting the proposed relativities.
ii. If a proposed relativity for any one territory is capped, this reduces the proposed average rate differential,
which will necessitate an offsetting increase in the proposed base rate to achieve the target average
premium for all territories combined.

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The following outlines a rate change scenario with the insurer is targeting an overall rate level change of 15.0%.
 Also, the insurer is revising relativities for a particular rating variable, and management requires that the
premium increase for any level of this variable not exceed 20%.
 The table below shows the current and selected relativities (prior to capping) in Columns (3) and (4).
i. these relativity changes would result in an off-balance factor of 0.9749 (= 1/(1+2.57%)).
ii. The total change to each level is shown in Column (8).
Rate Change Before Capping
(1)
(2)
(3)

Level
1
2
3
Total

Premium
$138,000
$659,000
$203,000
$1,000,000

Current
0.8000
1.0000
1.2000

(4)

(5)

(6)

(7)

(8)

(9)

Selected
0.9000
1.0000
1.2500

Relativity
Change
12.50%
0.00%
4.17%
2.57%

OffBalance
Factor
0.9749
0.9749
0.9749
0.9749

Selected
Overall
Change
15.0%
15.0%
15.0%
15.0%

Total
Change
26.13%
12.12%
16.79%
15.00%

Premium on
Proposed
Rates
174,063
738,855
237,082
1,150,000

(5)= (4) / (3) - 1.0; (Tot5) = (5) weighted by (2)
(6)= 1.0 / (1.0 + (Tot5))
(8)= [1.0 + (5)] x (6) x [1.0 + (7)] - 1.0
(9)= (2) x (1.0 + (8))
Interpreting the Results:
 The total change for Level 1 is 26.13%, which exceeds the desired maximum change of 20.0%.
 The new capped relativity for Level 1 (a.k.a. X) equals the product of the relativity change factor (new
capped relativity for Level 1 / current relativity for Level 1 = X / 0.8000), the off-balance factor (0.9749),
and the overall change factor (1.1500) that results in a 20% total change.
The new capped relativity for Level 1 (X) that satisfies this equation is 0.8563.
If the total change for Level 1 were limited to 20.0%, the premium would be $165,600 (=$138,000 x 1.20).
 This results in a shortfall of $8,459 (=$174,059 - $165,600) which needs to be made up by charging the
other levels (Levels 2 and 3) higher premium.
 The premium proposed for Levels 2 and 3 is $975,889 ( = $738,805 + $237,084). This premium must be
increased to cover the $8,459 shortfall.
i. One way to achieve this is to increase the base rate by 0.87% (=$8,459 / $975,889).
ii. However, since all levels are affected by any base rate change, the premium for capped Level 1 will
increase beyond the desired 20% limit. Therefore, the capped relativity for Level 1 must be further
reduced by 0.87% to undo the effect of the base rate increase on this level.
This adjustment results in a relativity for Level 1 of 0.8489 (= 0.8563 / 1.0087).

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The table below summarizes these calculations
Rate Change after Capping Non-Base Level at 20%
(1)

Level
1
2
3
Total

(2)

Premium
$138,000
$659,000
$203,000
$1,000,000

(3)

Current
0.8000
1.0000
1.2000

(10)
(11)
(12)
(13)

(4)

(5)

(6)

OffDifferential Balance
Selected Change
Factor
0.9000
12.50%
0.9749
1.0000
0.00%
0.9749
1.2500
4.17%
0.9749
2.57%
0.9749

(7)

(8)

(9)

Selected
Overall
Change
15.0%
15.0%
15.0%
15.0%

Total
Change
26.13%
12.11%
16.79%
15.00%

Premium
Above 20%
Cap
8,459
0
0
8,459

Proposed Premium from Non-capped Levels (2, 3)
Proposed Level 1 Relativity to Comply with Cap
Base Rate Adjustment to cover Shortfall
Proposed Lev 1 relativity adjusted for base rate offset

$975,889
0.8563
1.0087
0.8489

(5)= (4) / (3) - 1.0
(Tot5)= (5) weighted by (2)
(6)= 1.0 / (1.0 + (Tot5))
(8)= [1.0 + (5)] x (6) x [1.0 + (7)] - 1.0
(9)= max of [(2) x ((1.0 + (8))] - [ (2) x (1.0 + 20%)] and 0
(10)= (2) x (1+(8)) summed over Levels 2 and 3
(11)= [(1.0 + 20%) / ((6Row 1) x (1.0 + (7Row 1))] x (3Row 1)
(12)= 1.0 + (Tot9) / (10)
(13)= (11) / (12)
The final base rate offset factor equals the original off-balance factor (0.9749) times the base rate adjustment to
cover the premium shortfall from capping (1.0087).
 The revision to the Level 1 relativity achieves the 20% desired cap, and the adjustment to the base rate
ensures the overall change is still 15.0%.
 The calculations are slightly different if capping is necessary for the base class.
The table below shows a rate change scenario (with the same selected overall change and same premium
capping requirement) in which the base class exceeds the premium cap.
Rate Change Before Capping Base Level Impact
(1)
(2)
(3)
(4)
(5)

Level
1
2
3
Total

Premium
$138,000
$659,000
$203,000
$1,000,000

Current
0.8000
1.0000
1.2000

Selected
0.6500
1.0000
1.0500

Relativity
Change
-18.75%
0.00%
-12.50%
-5.13%

(6)
OffBalance
Factor
1.0541
1.0541
1.0541
1.0541

(7)
Selected
Overall
Change
15.0%
15.0%
15.0%
15.0%

(8)
Total
Change
-1.51%
21.22%
6.07%
15.01%

(9)
Premium on
Proposed
Rates
135,916
798,840
215,322
1,150,078

(5)= (4) / (3) - 1.0;
(Tot5) = (5) weighted by (2)
(6)= 1.0 / (1.0 + (Tot5))
(8)= [1.0 + (5)] x (6) x [1.0 + (7)] - 1.0
(9)= (2) x (1.0 + (8))

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Interpreting the Results:
 The base rate is adjusted downward to cap the change for the base level.
 The non-base relativities are adjusted upward to cover the amount of premium shortfall due to the cap
and to offset the effect of the base rate change in the non-base levels.
i. To limit the total change for Level 2 to 20.0%, the base rate is decreased by a factor of 0.9899 (=
1.2000/1.2122). This results in a shortfall in Level 2 premium of $8,040 (= (21.22% - 20.00%) x $659,000).
The premium collected from the non-base levels need to make up for that shortfall.
ii. Prior to capping, the premiums from Levels 1 and 3 was $351,238 (=135,916 + 215,322).
The relativities for these levels need to increase by 2.29% (=$8,040/$351,238).
Also, the relativities for Level 1 and Level 3 need to be adjusted to negate the effect of the base rate offset.
Thus, the final adjustment factor for these levels’ relativities is 1.0333 (=1.0229 / 0.9899).
The following table summarizes these calculations.
Rate Change After Capping Non-Base Level at 20%
(1)
(2)
(3)
(4)
(5)

Level
1
2
3
Total

Premium
$138,000
$659,000
$203,000
$1,000,000

Current
0.8000
1.0000
1.2000

(10)
(11)
(12)
(13)

(6)

OffDifferential Balance
Selected Change
Factor
0.6500
-18.75%
1.0541
1.0000
0.00%
1.0541
1.0500
-12.50%
1.0541
-5.13%
1.0541

(7)

(8)

Selected
Overall
Change
15.0%
15.0%
15.0%
15.0%

Total
Change
-1.51%
21.22%
6.07%
15.00%

Base Rate Adjustment to Comply with Cap
Premiuim from Non-capped Levels (1,3)
Adjustment to Level 1,3 Relativities due to Cap
Total Adjustment to Level 1,3 Relativities

(9)
Premium
Shortfall if
Total Change
capped to 20%
0
8,040
0
8,040
0.9899
$351,238
1.0229
1.0333

(5)= (4) / (3) - 1.0; (Tot5)= (5) weighted by (2)
(6)= 1.0 / (1.0 + (Tot5))
(8)= [1.0 + (5)] x (6) x [1.0 + (7)] - 1.0
(9)= max of [(2) x ((1.0 + (8))] - [ (2) x (1.0 + 20%)] and 0
(10)= (1.0 + 20.0%) / (1.0 + (8Row 2))
(11)= (2) x (1+(8)) summed over Levels 1 and 3
(12)= 1.0 + (9) / (11)
(13)= (12) / (10)
Interpreting the Results:
 The revised Level 1 selected differential is 0.6716 (= 0.6500 (selected) x 1.0333) and the Level 3 selected
differential is 1.0850 (=1.0500 (selected) x 1.0333).
 The final base rate offset factor (1.0435) = the original off-balance factor (1.0541) * the base rate
adjustment to comply with the cap (0.9899).
These changes result in a 15.0% overall change with no level’s premium exceeding the 20.0% limit.

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Premium Transition Rules
A premium transition rule dictates the maximum and/or minimum amount of change in premium that an
insured can receive at a single renewal. Example:
 An insurer may cap the renewal premium increase for each individual insured to 15%.
 If the insurer’s rate change results in an insured receiving a 20% premium increase, the insured will
receive a:
i. 15% rate change at the first renewal following the implementation of the rate change, and
ii. the remaining 4.3% ( = 1.20 / 1.15 - 1.0 ) at the second renewal.
Some key considerations when using a premium transition rule:
* The insurer needs to determine the maximum and minimum premium change amounts (test various
scenarios of min and max amounts, to determine the optimal selections)
* Often premium transition rules apply only to premium changes resulting from insurer initiated rate changes.
i. If premium change is affected by a change in risk characteristics (e.g. the insured buys a newer car),
the transition rule algorithm must be adjusted to neutralize the effect of the risk characteristic change.
ii. The premium change may be calculated as the ratio of [new premium on new risk characteristics/ old
premium on new risk characteristics].
* The time needed to fully transition the renewal portfolio to the manual rates depends on the proposed rate
change and the premium transition rule implemented.
* The effect on the average premium level should also be considered and the base rate altered.
i. Decide whether the base rate should be set so that the equilibrium is achieved over the whole time the
proposed rates are in effect, or by the expected end of the transition period.
ii. Example: if the insurer is targeting an average premium of $250 and using a premium transition rule that
is expected to span 2 years, then the insurer needs to decide whether the base rate should be set so that
average premium will equal $250 over the two years combined or at the end of the two-year period.
This is important since the cap does not apply equally to premium increases and decreases, and the rate
changes are not uniformly distributed.
Expected Distribution
Actuaries often use the latest in-force distribution of policies as the best estimate of the expected future
distribution.
 By doing so, the actuary assumes the rate change will not alter the existing portfolio.
 The validity of that assumption can vary significantly based on product, market conditions, and the extent
of the proposed changes.
Example: Assume a non-standard auto insurer implements a significant rate change that varies widely by
age of insured.
i. the insurer is likely to see a significant change in the overall volume and distribution of business (i.e.
insureds receiving large rate changes may non-renew their policies).
ii. the actual average premium change realized is likely to differ than what is proposed using the latest inforce distribution.
* if all risks are equally profitable, then loss of premium will be offset by a loss in expected costs, and
the overall rate level adequacy will be unaffected.
* If the risks are not equally profitable, then the distributional shift can affect the adequacy of the
overall rates.
This is a shortcoming of standard actuarial techniques.
Price optimization techniques (Chapter 13), address this by taking into consideration how the rate change
is expected to affect demand (i.e. volume).

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4

Calculating New Rates Based On Bureau Or Competitor Rates

285-286

When writing a new insurance product, insurers often do not have the data to generate rates, and often rely on
similar products sold by competitors (if the data is publicly available), or data from rating bureaus, and make
adjustments.
If the insurer has rates from a related product or rating bureau, the insurer still needs:
 a copy of the relevant rating manual or rating bureau filing.
 a copy of the competitor’s underwriting guidelines (which may not be available).
 to obtain information regarding the relative expense levels and profitability of the target competitor (which
can be obtained from recent rate filings or from annual statement data).
The insurer may use the competitor’s manual as a starting point and make adjustments (the following are a few
examples):
1. Estimate whether its fixed expenses will be higher or lower than those of the target competitor.
i. the insurer can increase or decrease the competitor’s expense fee by the appropriate percentage.
ii. if the insurer estimates its fixed expenses will be 10% lower than the competitor’s, and the competitor has
an expense fee of $25.00, then the insurer should implement an expense fee of $22.50 (= $25 multiplied
by a factor of 0.90 ( = 1.0 - 0.10 ).
2. Estimate whether its variable expenses will be higher or lower than those of the target competitor.
i. the insurer can adjust the base rate and the expense fee by the ratio of [the target competitor’s variable
permissible loss ratio/ the expected variable permissible loss ratio].
ii. if the insurer plans to use a commission % that is 5 percentage points higher than the competitor’s, and
that the competitor’s variable expense ratio is 15% and the target profit % is 5%, then the insurer should
adjust the competitor’s base rate and expense fees by 1.067 [ = ( 1.0- 0.15 - 0.05 ) / (1.0 - 0.20 - 0.05 ) ].
3. Estimate whether its expected loss costs will be different than the target competitor’s due to
operational differences or a lack of experience with the product, and change the base rate.
If the insurer feels its lack of experience in settling claims for the new product will result in expected costs
that are 5% to 10% higher than those of the target competitor’s, it should increase the base rates by 5% to
10% to account for this.
4. Target a certain segment of the market that the competitor does not seem to be targeting.
If the insurer chooses to reduce the rate differential in that territory, it can adjust the base rate to offset the
change in the average territorial differential.

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5

Communicating and Monitoring

286 -287

Prior to implementing a final set of rates, the actuary will communicate the expected rate change effect to
regulators and insurer management.
If the proposed rates apply to a brand new product, then communication to regulators may be limited to the
source of the derivation of rates (e.g. competitor or bureau rates) and some justification for any judgmental
adjustments made.
If the insurer is implementing rate changes that will impact existing policies, then the communications will be
more extensive.
 Internal management may want to understand the assumptions and selections involved in the overall
rate level indication or rate differential changes, and will want to understand the impact on competitive
position, expected volume, and expected profitability.
 The actuary will often prepare competitive comparisons (e.g. % wins) under the current and final
proposed rates, as well as policyholder dislocation analysis for insurer management (in total as well as
by key segments as discussed in Chapter 13).
This is useful for marketing, sales, and customer service to prepare for any potential repercussions of
large policyholder premium impacts or to focus advertising on customer segments that will be priced
more competitively.
 Some insurers use models to estimate the conversion and retention rates (per individual risk and in
aggregate) expected after implementation of a rate change.
i. These can be used to estimate future expected loss costs, premium, and expenses on these risks.
ii. This allows calculation of expected profitability after the rate change.
 Regulators may require considerable detail about the methods and assumptions underlying the overall
rate level and rate differential indications and selections, and may want to understand the expected
policyholder dislocation.
It is important to monitor the actual effect of the rate change against the expected effect (e.g. comparing actual
and expected close rates, retention rates, distributions, and claim frequencies against those expected).

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6

Key Concepts

288 - 288

1. Non-pricing solutions to an imbalanced fundamental insurance equation
a. Reduce expenses
b. Reduce loss costs
2. Pricing solutions for an existing product
a. Calculation of additive fixed expense fee and other additive premium
b. Derivation of base rate
i. Extension of exposures method
ii. Approximated average rate differentials method
iii. Approximated change in average rate differentials method
c. Other considerations
i. No fixed expense fees or additive premium
ii. Minimum premium
iii. Limit on the premium effect of a single variable
iv. Premium transition rules
v. Expected distribution
3. Pricing solutions for a new product
a. Use of related data, competitor’s rates, or bureau rates
b. Consideration of differences in expected loss, expense, and target segments
4. Communicating rate change effect to key stakeholders
a. New product
b. Existing product

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Chapter 14 - Implementation
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
The predecessor papers to the current syllabus reading “Basic Ratemaking” by Werner, G. and Modlin,
C. were numerous. While past CAS questions were drawn from prior syllabus readings, the ones
shown below remain relevant to the content covered in this chapter.
By relevant, we mean the concepts tested on past CAS exams relating to Expense Fee Ratios and
Expense Fees are similar to the concepts found in this chapter.

Questions from the 2000 exam
29. Based on Schofield, "Going from a Pure Premium to a Rate," and the following information concerning
Private Passenger Auto Bodily Injury Liability Coverage, calculate the Expense Fee Ratio.
Expense Type
Total Expense
% Fixed
Commission
0.180
0%
Other Acquisition
0.030
75%
General
0.050
80%
Taxes, Licenses and Fees
0.025
15%
Profit and Contingencies
0.035
0%
A. < 0.085
E. > 0.115

B. > 0.085 but < 0.095

C. > 0.095 but < 0.105

D. > 0.105 but < 0.115

Questions from the 2002 exam
4. Based on Schofield, "Going From a Pure Premium to a Rate," and the following data,
use the Expense Fee Methodology to calculate the expense fee.

Earned Premium at Current Rate Level

325,000

Earned Exposures

1,100

Total Fixed Expense Ratio

0.13

Total Variable Expense Ratio including Profit
and Contingency Provision

0.23

A. < $40

B. > $40, but < $60

C > $60, but < $80

D. > $80, but < $100

E. > $100

Questions from the 2007 exam
41. (3.5 points) You are given the following information about an automobile book of business:
Expense Category
Countrywide
% of Expenses
Total Expenses
Assumed to Be Fixed
Commissions
$1,400,000
0%
General Expenses
1,200,000
50%
Other Acquisition
400.000
100%
Premium Tax
300.000
0%
Licenses & Fees
100.000
100%
 Countrywide total premium volume:
$10,000,000
 Profit and contingencies provision:
5%

(2.0 points) Calculate the countrywide expense fee ratio.

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Questions from the 2009 exam
38. (3.5 points) Given the following information:

Class
1
2
3
Total

On-Level
Premium
$500,000
$100,000
$400,000
$1,000,000

Current Proposed
Relativity Relativity
1.00
1.00
1.25
1.15
1.60
1.40

• Class 1 will remain the base class.
• The current base rate is $100.
• The proposed overall change is 15%.
a. (1 point) Calculate the revised base rate.
b. (2.5 points) Assume the actuary wants to cap all class changes at 20% while still achieving the
overall change of 15%. Calculate the revised base rate and class relativities.

Questions from the 2010 exam
28. (2.25 points) Given the following information:
• Current base rate $90
• Current average premium per exposure = $110
• Loss and LAE ratio = 75%
• Fixed expense ratio = 10%
• Variable expense ratio = 15%
• Target profit provision = 5%
• Rating algorithm = {Base Rate x Factor x (1.0 - Discount)} + Expense Fee•
• Selected rate change = indicated rate change

Factor

Current
Differential

Proposed
Differential

1
2

0.95
1.00

Factor
1
1
2
2

Discount
Yes
No
Yes
No

Discount
Yes
No

Current
Discount

Proposed
Discount

5.0%
0.0%

10.0%
0.0%

Exposures
100
200
300
400

a. (0.75 point) Calculate the indicated rate change.
b. (0.5 point) Calculate the proposed expense fee.
c. (1 point) Calculate the proposed base rate.

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Questions from the 2011 exam
18. (1.25 points) An insurance company is planning to implement new rates and expects the following:
Premium
Range
$0-$50
$51-$100
$101-$200
$201-$500

Policy
Count
26
34
45
150

Avg Premium
at Proposed
Rates
$35
$80
$150
$300

The resulting base rate from the proposal is $100.
a. (0.75 point) Calculate the new base rate that achieves a revenue-neutral impact if the company
were to implement a minimum premium of $100.
b. (0.5 point) Explain the purpose of a minimum premium.

Questions from the 2012 exam
14. (4 points) An insurance company sells auto insurance where the premium for each car is the same,
equal to the current statewide average pure premium. The company is considering developing a more
sophisticated rating structure to better compete in the marketplace and has determined the following
information about three potential rating variables for its existing book of business.






Garaging Location
Urban
Rural

Exposures
800
200

Losses
$430,000
$70,000

Base Class
Yes
No

Driver Skill
High
Low

Exposures
950
50

Losses
$476,000
$24,000

Base Class
Yes
No

Marital Status
Married
Single

Exposures
500
500

Losses
$210,000
$290,000

Base Class
No
Yes

The garaging location is determined by the garaging zip code of the vehicle being insured.
Driver skill is determined by self-assessment when the policy is originally issued.
Marital status of the principal operator determined as of the policy's effective date.
Assume that each variable is independent.

a. (2.25 points) For each potential rating variable, recommend whether the variable should be used
and justify the recommendation.
b. (1.75 points) Develop a base rate and rating factors for the rating plan structure recommended in
part a. above.

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Solutions to questions from the 2000 exam
Question 29.
The Expense Fee Ratio is the proportion of the total rate that is needed to cover the fixed expenses (i.e.
the fixed expense ratio loaded for variable expenses).
F = fixed expenses per exposure
H = fixed expense ratio (fixed expenses as a proportion of the rate)
V = variable expense ratio (variable expenses as a proportion of the rate)
Using the given information from the problem (columns (1) and (2)) below, Compute columns
(3) and (4).
Expense Type
Total Expense
% Fixed
Variable
Fixed
(1)
(2)
(3)=(1)*(2)
(4)=(1)*(1.0 (2))
Commission
0.180
0%
0.0000
.1800
Other Acquisition
0.030
75%
0.0225
.0075
General
0.050
80%
0.04
.01
Taxes, Licenses and Fees 0.025
15%
0.00375
.02125
Profit and Contingencies
0.035
0%
0
.035
.06625
.25375
H = .06625 V+Q = 0.25375

Expense Fee Ratio 

H
.06625

 .0887
1V Q 1.25375

Answer B.

Solutions to questions from the 2002 exam
Question 4. Use the Expense Fee Methodology to calculate the expense fee.
Step 1: Write an equation to compute the expense fee.
F
H * R , where H = fixed expense ratio (fixed expenses as a % of the rate).
E xp en se F ee 

1V  Q 1V  Q
Step 2: Assign symbols to the given data in the problem and solve for any unknown terms:
Description

Amount

Earned Premium at Current Rate Level

325,000

Earned Exposures

1,100

Total Fixed Expense Ratio

0.13

Total Variable Expense Ratio including Profit
and Contingency Provision

0.23

Symbol

H
V+Q

Note: R = rate per unit of exposure = 325,000 ÷ 1,100 = 295.45
Step 3: Using the equation in Step 1, and the data from Step 2, solve for the expense fee.
E xpense F ee 

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Chapter 14 - Implementation
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Solutions to questions from the 2007 exam
41. a. (2.0 points) Calculate the countrywide expense fee ratio.
Step 1: Write an equation to determine the expense fee ratio.
Expense Fee Ratio = H/(1.0 - V - Q), where H is the proportion of countrywide total expenses
that are assumed to be fixed, V is the percentage of expenses assume to be variable, and Q is
the profit and contingencies provision.
Step 2: Construct a table similar to the one below to compute H and V.

Commissions
General Expenses
Other Acquisition
Premium Tax
Licenses & Fees

Total
Expense
(1)
1,400,000
1,200,000
400,000
300,000
100,000

Countrywide total premium

10,000,000

Expense Type

% Fixed
(2)
0.00%
50.00%
100.00%
0.00%
100.00%

Fixed
Expense
(3)=(1)*(2)
0
600,000
400,000
0
100,000
1,100,000
H = 0.11

Variable
Expense
(4)=(1)*(1.0 - (2))
1,400,000
600,000
0
300,000
0
2,300,000
V = 0.23

Thus, the Expense Fee Ratio = H/(1.0 - V - Q) = 0.11/(1.00-0.23-0.05) = 0.1528

Solutions to questions from the 2009 exam
Question 38
a. Revised Base Rate = Current Base Rate * Proposed Overall Change * Off-balance factor

Off balance factor 

 On - Level EP
indicated relativity
 On - Level EP 
current relativity

Premium weighted ratio of proposed relativity to current relativity
= 500/1 x 1 + 100/1.25 x 1.15 + 400/1.6 x 1.4= 942
Off-balance factor = 1,000/942 = 1.062
Revised Base = 1.062 x 1.15 x 100 = $ 122.08
b. Determine if any class experiences changes greater than 20% (the cap)
Class
Base rate * Cur Rel Rev Base rate * Prop Rel
% change in rate
1
100
122.08
22.08% .0208 (500K) =10,400 must be spread to 2 & 3
2
125
140.392
12.31%
3
160
170.91
6.08%
Class 2 and 3 proposed relativities need to be increased because the class 1 base rate is being reduced to 20%
Revised Base Rate= 100 x 1.2 = $120
Factor to apply to relativities

1.2208
10, 400
 1.037
 .0193 , and 1.0193 
(100, 000  1.1231  400, 000  1.068)
1.2

Revised proposed class relativities
Class
Revised Relativities
1
1
2
1.15 x 1.037 = 1.193
3
1.40 x 1.037 = 1.452

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Solutions to questions from the 2010 exam
Question 28

(L + EL ) + F 


PC
0.75  0.10
a. Indicated Rate Change =
 1.0  .0625 ;
- 1.0; IRC 
1  0.15  0.05
[1.0 -V - QT ]
b. Calculation of $Fee (Using the Fixed Expense Ratio)
(1) Fixed Expense Ratio
10.0%
(2) Proj Av Premium per Exposure (110 * 1.0625) $116.88
(3) Average Fixed Expense
$11.68
(4) Variable Expense %
15.0%
(5) Target Profit %
5.0%
(6) Variable Permissible Loss Ratio
80.0%
(7) Proposed Expense Fee
$14.60
(3) = (1) x (2); (6) = 1.0 – (4) – (5);
(7) = (3) / (6)

(given)
(given Current Avg Prem per Exp * 1.0625)
(given)
(given)

Alternatively, Expense Fee = 110  (1.0625)  (0.1)  110  (1.0625)  (0.1) = 14.609
1- V - Q

1- 0.15 - 0.05

c. Calculation of the change in the average rate differentials (  S ). Under this method,
Proposed Base Rate = Current Base Rate * Proposed Base Rate Adjustment
___

Proposed Base Rate Adjustment = (1.0   B %) 

(1.0  %)  PC  AP
___

PC  AC



1.0
,
(1.0   S %)

Proposed Base Rate Adjustment = (Proposed Avg Prem - Proposed Expense Fee)/(Current Avg Prem Current Expense Fee)*(1/Average Change in Differential)
Computation of the proposed average change in differentials (using exposures)
Current
Factor

Discount

Exposure Differential
(1)

1
1
2
2

Y
N
Y
N

100
200
300
400

1.0 - Current
Discount
(2)

(3) = (1)*(2)

0.95
95.00%
0.95
100.00%
1.00
95.00%
1.00
100.00%
Exposure weighted =

0.9025
0.9500
0.9500
1.0000
0.9653

Proposed

1.0 - Proposed

Differential

Discount

(4)

(5)

(6) = (4)*(5)

0.950
90.00%
0.950
100.00%
1.000
90.00%
1.000
100.00%
Exposure weighted =

0.8550
0.9500
0.9000
1.0000
0.9455

Average Change in Differential = 0.9455/0.9653 = 0.9795
Ac = (110 * .10)/(1-.20) = 13.75
Proposed BR = 90 * (110 * 1.0625 - 14.61)/(110 - 13.75)*(1.0/0.9795) = 97.62
Note: In the text, the proposed additive premium Ap = Ac.

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Chapter 14 - Implementation
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Solutions to questions from the 2011 exam
18a. (0.75 point) Calculate the new base rate that achieves a revenue-neutral impact if the company were
to implement a minimum premium of $100.
18b. (0.5 point) Explain the purpose of a minimum premium.
Initial comments:
A minimum premium ensures that, on an individual risk basis, premium covers the expected fixed
expenses plus some minimum expected loss.
 Insurers that use a minimum premium requirement do not have additive fixed expense fees in their
rating algorithms.
 Implementing a minimum premium can increase total premium. The effect is calculated as follows:
P rem iu m W ith M in im u m
E ffect =
- 1 .0
P rem iu m W ith o u t M in im u m
 To offset this increase in premium, the base rate should be multiplied by the following factor:

Offset Factor 

1.0
1.0  Effect

Question 18 – Model Solution 1
a.
Avg Premium
Policy at Proposed
Total Prem w/o
Total Prem
minimum
Count
Rates
with minimum
(1)
(2)
(3)=(1)*(2)
(4)
(5)=(1)*(4)
26
$35
910
$100
2600
34
$80
2720
$100
3400
45
$150
6750
$150
6750
150
$300
45000
$300
45000
55,380
57,750
Effect = 57,750/55,380 = 1.0428 = 4.28% . Base Rate = 100*1/1.0428 = $95.90

b. The purpose of a minimum premium is to cover expected fixed expenses and some expected losses.
Question 18 – Model Solution 2
a. Total Prem required = 26 * 35 + 34 * 80 + 45 * 150 + 150 * 300 = 55,380
Avg. w/ min prem
0-50
100
51-100
100
101-200
150x
201-500
300x
Total w/ Min = 26 * 100 + 34 * 100 + x[45 * 150 + 150 * 300] = 6,000 + 51,750x
Set = 55380 →x = .9542

New base rate = 100*.9542 = 95.42
b. Minimum premium ensures that the company collects an amount of premium that is enough to cover
fixed expenses and a minimum risk provision for small-premium policies.

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Chapter 14 - Implementation
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Solutions to questions from the 2012 exam
14a. (2.25 points) For each potential rating variable, recommend whether the variable should be used
and justify the recommendation.
14b. (1.75 points) Develop a base rate and rating factors for the rating plan structure recommended in
part a. above.
Question 14 – Model Solution – part a (Exam 5A Question 14a)
Compute pure premiums for each of the rating variables given in the problem since the company sells auto
insurance where the premium for each car is the same, equal to the current statewide average pure premium.
pure premium = losses/exposures
Urban 537.5
High 501
Married 420
Rural 350
Low 480
Single 580

- Garaging location should be used because loss costs differ significantly, the variable is easy to identify
and “measureable” based on zip code, and it is also easy to verify.
- Driver skill should not be used. The fact that it is self-identified by the insured and very open to
interpretation means it is not measureable and open to moral hazard. Further, it obviously does not work
based on experience. It makes no sense that loss costs for highly skilled drivers would be higher.
- Marital status should be used. It can be verified by public records and is straight-forward categorization.
Loss costs also differ significantly.
Examiner’s comments
A variety of reasons whether a characteristic should be included were accepted. However, we didn’t
expect the candidate to identify all critical pieces of evidence as long as there was sufficient justification
for including or excluding a variable.
Question 14 – Model Solution – part b (Exam 5A Question 14b)

Garaging
Urban
Rural
TOTAL

Pure Prem=loss/ exp
537.5
350.0
500.0

Marital Status
Married
Single
TOTAL

Pure Prem
420
580
500

PP Rel
1.00
0.65

ARF=wtd avg of exp+factor

0.93
PP rel
0.72
1.00

ARF

0.86

Assume this data is representative of SW Avg PP => $500

Bp 

500
0.7998

 625.16

Examiner’s comments
Candidates lost points for various reasons like: used the wrong characteristic as the base class,
calculating separate base rates for marital status and garaging location, and calculating the base rate as
a simple average of the pure premiums for single policyholders and urban policyholders.

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Chapter 15 – Commercial Lines Rating Mechanisms
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Sec
1
2
3
1

Description
Manual Rate Modification Techniques
Rating Mechanisms For Large Commercial Risks
Key Concepts

Pages
289 - 298
298 -309
311 - 311

Manual Rate Modification Techniques

289 - 298

Commercial Risks manual rates are modified to adjust for past experience and/or risk characteristics not
adequately reflected in the manual rates.
There are two types of manual rate modification techniques: Experience Rating and Schedule rating.
Experience Rating (ER)
ER is used when an individual insured’s past experience, with adjustments, can be predictive of the future
experience. Eligibility for ER is often based on size of manual premium.
The ER adjustment for the future policy period manual premium equals the credibility weighting of:
i. adjusted past experience (a.k.a. the “experience” component) and
ii. expected results (a.k.a. the “expected” component).
Recall: Techniques to derive credibility measures and ways to develop the complement of credibility are
discussed in Chapter 12.
The experience component and the expected component should be consistent (e.g. ALAE should be included in
the experience component if it is included in the expected component).
Comparison of experience and expected components can be performed in different ways:
1. Actual paid loss (and ALAE) compared to expected paid loss (and ALAE) for the experience period as of
a particular date.
2. Actual reported loss (and ALAE) compared to expected reported loss (and ALAE) for the experience
period as of a particular date.
3. Projected ultimate loss (and ALAE) compared to expected ultimate loss (and ALAE) for the experience
period.
4. Projected ultimate loss (and ALAE) for the experience period that has been adjusted to current exposure
and dollar levels compared to expected ultimate loss (and ALAE) based upon the current exposure and
dollar levels
Key components of the ER formula, including necessary adjustments to each, follows.
Experience Component
1. Determine the length of the historical experience period to be used in the ER formula.
The experience period usually ranges from 2-5 policy years, ending with the last complete year.
i. a shorter experience period is more responsive to changes, but more subject to large fluctuations (due
to its relative loss immaturity and reduced aggregate exposure of the shorter period).
ii. a longer experience period is less subject to large fluctuations in the experience, but is less responsive to
changes.

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Chapter 15 – Commercial Lines Rating Mechanisms
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2. Adjust the historical experience for extraordinary losses.
 Many ER plans apply per occurrence caps on the losses to exclude unusual or catastrophic losses.
i. This is referred to as the maximum single limit per occurrence or MSL.
ii. The caps could apply to losses only (or loss and ALAE).
iii. If the actual losses are subject to a per occurrence cap, then the expected losses need to be as well.
 Caps may also be applied to the aggregate of all losses in the policy period.
3. If the experience modification (emod) is based on projected ultimate losses, then historical losses and ALAE
(assuming ALAE is included) need to be developed to an ultimate level (see chapter 6).
 Expected losses also need to be at an ultimate level.
 If capped losses are used, the LDFs should be developed from data that has also been capped.
4. If the ER formula is based on projected ultimate losses at current exposure and dollar levels (i.e. the 4th
comparison listed above), adjustments for economic and social inflation (e.g. changes in judicial decisions or
litigiousness) as well as changes in risk characteristics (e.g. size and type of entity) and changes in policy
limits are needed.
i. Historical losses are developed to ultimate, trended to current cost levels, and totaled.
ii. If the exposure base is sensitive to inflation (e.g. payroll), trend and sum historical exposures.
iii. The ratio of [i./ii.] is then multiplied by a current exposure measure.
Trended Projected Ultimate Losses & ALAE at Current
(1)

(2)

(3)

(4)

Policy
Year

Trended
Ultimate
Losses &
ALAE

Exposures

Pure
Premium

Current
Exposures

(5)
Projected
Ultimate Losses
& ALAE @
Current
Exposures

2006
2007
2008
Total

$2,568,325
$1,954,725
$1,465,741
$5,988,791

688
564
414
1,666

$3,594.71

400

$1,437,885

(3) = (Tot1) / (Tot2) (4) = Number of Vehicles Currently Insured (5) = (Tot3) x (Tot4)

Expected Component
The expected component should relate to the experience component.
For the comparisons listed above, the first three use past exposure and the fourth uses current exposure.
Expected losses are the product of an expected loss rate and an exposure measure.
 The expected loss rate is the expected loss cost in the manual rates.
 If the loss rates are needed for a prior period, the expected loss rate can be based on:
i. the manual rates for the prior period or
ii. the manual rates for the current period, adjusted to the appropriate level (i.e. de-trended).
Other Considerations
The e-mod may be subjected to maximum or minimum changes.
When the total premium under the ER plan does not equal the total expected premium, an off-balance
correction is needed (see Chapter 14.)

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Chapter 15 – Commercial Lines Rating Mechanisms
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Example ER Plan – Commercial General Liability
The following is a simplified version of the ER portion of the 1997 ISO CGL ER and SR Plan (“company” refers
to the insurer using the ER plan).
The ER debit/credit is:

CD 

( AER - EER)
 Z , where
EER

CD = Credit/debit percentage
AER = Actual experience ratio (i.e. the experience component)
EER = Expected experience ratio (i.e. the expected or exposure component)
Z = Credibility
Assume the following:
• The policy being experience rated is an occurrence policy with an annual term, and the effective date is
7/1/2010.
• The experience period consists of the last three completed policies effective 7/1 to 6/30 (i.e. annual
policies originating in July 2006, 2007, and 2008), evaluated at 3/31/2010.
• Losses are capped at basic limits, and ALAE are unlimited.
• A MSL is applied to the basic limits losses and unlimited ALAE combined.
• The Z of the company is 0.44.
• The expected experience ratio (EER) is 0.888.
Table 1 shows the calculation of the ER debit/credit. Table 2 supports the derivation of certain inputs to Table 1.
Table 1 and Table 2 are shown on the next page.
 Actual experience is the ultimate losses and ALAE for the 3-year experience period, consisting of:
i. reported losses and ALAE as of 3/31/2010 [given as 1(a) in Table 1] and
ii. expected unreported losses and ALAE at 3/31/2010 (derived in column 8 of Table 2).
For both the reported and unreported losses and ALAE, losses are capped at basic limits and a MSL is
applied to the basic limited losses and ALAE combined.
 Company subject basic limit loss and ALAE costs [1(d) in Table 1] represent expected loss and ALAE
underlying the current rating manual rates adjusted to the dollar level of the experience period (see Table 2).
 The actual experience ratio (AER) is [ultimate losses and ALAE (at basic limits and limited by the MSL)
divided by company subject basic limits loss and unlimited ALAE costs].
 The expected experience ratio (EER) is the complement of an expected deviation of the company’s loss
costs from the loss costs underlying the manual rate (here, the deviation is caused by applying the MSL).
The ER credit/debit is calculated as a credibility weighting of the AER and the EER
 An experience credit reduces premium and an experience debit increases premium.
 This plan does not have any minimums, maximums, or an off-balance correction.

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Chapter 15 – Commercial Lines Rating Mechanisms
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Table 1: Experience Credit/Debit Calculation
(1) Experience Components
(a)
Reported Losses and ALAE at 3/31/10
Limited by Basic Limits and MSL
(b)
Expected Unreported Losses and ALAE at
3/31/10 Limited by Basic Limits and MSL
(c)
Projected Ultimate Losses and ALAE
Limited by Basic Limits and MSL
(d)
Company Subject Basic Limit Loss and
ALAE Costs
(e)
Actual Experience Ratio
(2)
Expected Experience Ratio
(3)
Credibility
(4)
Experience (Credit)/Debit
(1a)=
(1b)
(1c)=
(1d)=
(1e)=
(2),(3)=
(4)=

$141,500.00
$ 58,760.24
$200,260.24
$181,365.61
1.104
0.888
.044
10.7%

Given
Table 2
(1a) + (1b)
Table 2
(1c)/(1d)
Given
[((1e) - (2)) / (2)] x (3)

Table 2 shows the derivation of two elements in Table 1:
(d) company subject basic limits loss and unlimited ALAE costs and
(b) expected unreported losses and ALAE.
Table 2 - Calculation of Company Subject Loss Costs and Expected Unreported Losses
(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

Expected

Policy

Current

Company

Percentage B/L

Expected B/L

Company B/L

Subject B/L

Expected

Losses & ALAE

Losses & ALAE
Unreported at

Loss & ALAE

Detrend

Loss & ALAE

Experience

Unreported at

Costs

Factors

Costs

Ratio

3/31/10

41,546.70

0.888

0.192

0.839

15,815.15

0.888

0.4256

0.849

43,872.08

0.888

0.300

0.876

16,512.60

0.888

0.545

Period

Coverage

7/1/06-07

Prem/Ops

51,675.00

0.804

Products

18,851.00

7/1/07-08

Prem/Ops

51,675.00

Products

18,850.00

7/1/08-09
Total

3/31/10
$

Prem/Ops

51,675.00

.0897

46,352.48

0.888

0.394

Products

18,850.00

0.916

17,266.60

0.888

0.639

211,575.00

181,365.61

(4)= the reciprocal of the loss and ALAE trend factor;

(5)= (3) x (4)

(6),(7) = given

7,083.55
5,982.68
11,687.52
7,991.44
16,217.43
9,797.62
58,760.24

(8)= (5) x (6) x (7)

Company subject basic limits losses and unlimited ALAE costs (column 5 above) are the product of:
• the current company basic limits loss and ALAE costs (i.e. the loss costs underlying the current
manual rates) and
• the detrend factors, which bring current company basic limits loss and ALAE to the average accident
date of each of the policy periods in the experience period, using the loss and ALAE trend underlying
the current rates.

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Chapter 15 – Commercial Lines Rating Mechanisms
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Detrend Factors:
The detrend factor:
 for each policy period in the experience period is the reciprocal of the loss and ALAE trend factor.
 is used to adjust the current loss costs to a historical experience period. For example:
i. the average accident date of the prospective policy period is 1/1/2011.
ii. for the policy period beginning 7/1/2008, the length of the detrend period is two years (the length of
time between 1/1/2011 and 1/1/2009).
iii. given a loss trend of 4.5%, the detrend factor for the 2008 policy period is the reciprocal of the
trend plus 1.0, raised to the length of the detrend period [=0.916 = (1/1.045)2].
Expected basic limits losses and ALAE unreported at 3/31/2010 (column 8) are the product of:
• The company subject basic limits losses and ALAE
• The expected experience ratio (EER)
• The expected percentage basic limits losses and ALAE unreported at 3/31/2010 (these %s are
derived from a separate analysis).
Example ER Plan – Workers Compensation (WC)
The National Council on Compensation Insurance (NCCI) has been designated by the majority of states as the
licensed rating and statistical organization of WC insurance.

The NCCI ER Plan divides losses into primary and excess components.

The mod formula credibility weights primary and excess losses separately:

M

Z P  AP  (1.0  Z P )  EP  Z e  Ae  (1.0  Z e )  Ee
, where
E

M = Experience Modification Factor
AP = Actual Primary Losses,
EP = Expected Primary Losses
E = Ep + Ee
ZP = Primary Credibility

Ae = Actual Excess Losses
Ee = Expected Excess Losses
Ze = Excess Credibility

NCCI uses an alternative (algebraically equivalent) formula by substituting some terms.

M

AP  w  Ae  (1.0  w)  Ee  B
, where
EB

B = Ballast Value, which is based on: ZP = E/(E +B); w = Excess Loss Weighting Value = Ze/Zp.
How primary and excess credibility factors are expressed in NCCI’s formula:
 The primary credibility factor is a function of the ballast value (B).
 The excess credibility factor is a function of both (B) and (w).
 The ballast value and weighting value:
i. are obtained from a table based upon the policy’s expected losses
ii. both increase as expected losses increase.

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Chapter 15 – Commercial Lines Rating Mechanisms
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
WC ER Plan:
 The experience period consists of the 3 most recent complete policy years.
 Actual losses are reported losses evaluated at 18 months, 30 months, and 42 months from the beginning
of the most recent, second most recent and third most recent policy years, respectively.
 Actual primary losses are capped at $5,000 per loss.
 Expected losses are the actual payroll (in hundreds) by class for the experience period multiplied by the
expected loss rates by class for the prospective period.
i. Expected loss rates reflect the losses expected to be reported at the respective evaluations of the
experience period policies (18, 30, and 42 months).
ii. Expected primary losses are the expected losses multiplied by a D-ratio (the loss elimination ratio at
the primary loss limit determined using the LER techniques described in Chapter 11).
NCCI emod calculation.
 The effective date of the policy being rated is 9/1/2010
 The policy is comprised of only one class code.
 Table 1 below lists the actual losses from the last three complete policy years.
i. The losses are separated into primary and excess components.
ii. Primary losses are capped at $5,000; Excess losses are the portion of each individual loss above $5,000.
 Table 2 expected loss costs reflects expected losses as of policy evaluation date.
Expected losses are separated into the primary and excess components based upon a D-ratio.
Table 1 - Actual Losses as of 3/31/10
(1)
(2)
(3)
Reported
Primary
Excess
Policy Year
Claim #
Losses
Losses
Losses
9/1/06-07
1
$15,000
$5,000
$10,000
2
$100,000
$5,000
$95,000
3
$25,000
$5,000
$20,000
9/1/07-08
1
$45,000
$5,000
$40,000
2
$50,000
$5,000
$45,000
3
$10,000
$5,000
$5,000
9/1/08-09
1
$20,000
$5,000
$15,000
$5,000
$50,000
2
$55,000
Total
$320,000
$40,000
$280,000
Table 2 - Expected Losses
(1)
Policy
Year
9/1/06-07
9/1/07-08
9/1/08-09
Total

Payroll
$1,956,000
$2,128,000
$2,317,000
$6,401,000

(2)

(3)

(4)

Expected
Loss Cost
4.10
3.52
2.37

Expected
Losses
$80,196.00
$74,905.60
$54,912.90
$210,014.50

D-Ratio
0.24
0.24
0.24

(3) = [ (1) / $100 ] x (2)

(5)
Expected
Primary
Losses
$19,247.04
$17,977.34
$13,179.10
$50,403.48

(6)
Expected
Excess
Losses
$60,948.96
$56,928.26
$41,733.80
$159,611.02

(5) = (3) x (4)

(6) = (3) - (5)

Assuming a ballast value (B) of $30,000 and a weighting value (w) of 0.25, the ER Mod factor is

M

AP  w  Ae  (1.0  w)  Ee  B
EB

M 

4 0 , 0 0 0  [0 .2 5  $ 2 8 0 , 0 0 0 ]  [(1 .0 - 0 .2 5 )  $ 1 5 9 , 6 1 1 .0 2 ]  $ 3 0 , 0 0 0
$ 5 0 , 4 0 3 .4 8  $ 1 5 9 , 6 1 1 .0 2  $ 3 0 , 0 0 0

 1 .0 8 2

The e-mod factor of 1.082 is applied multiplicatively to policy standard premium.

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Chapter 15 – Commercial Lines Rating Mechanisms
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Schedule Rating (SR)
SR:
 is used to modify the manual rate, in commercial lines pricing, to reflect characteristics that are:
i. expected to have a material effect on the insured’s future loss experience but that are not actually
reflected in the manual rate, or
ii. not adequately reflected in the prior experience (if ER applies).
Example: if an insured implements a new loss control program, it is expected that losses will be lower
than that indicated by the actual historical experience (hence an underwriter can use SR to
reflect this).
 is applied as % credits (reductions) and debits (increases) to the manual rate.
Characteristics can be objective (e.g. the number of years a physician has been licensed) or subjective
(e.g. quality of company management).
i. Objective characteristics are generally easier to quantify and validate.
ii. SR requires significant underwriting judgment (and documentation is required to support application of
each credit and debit).

if used in addition to ER (e.g. a newly implemented safety program), then the latter will eventually be
reflected in the loss experience, so the key for the underwriter is to avoid double-counting the risk
characteristic effect in both the e-mod and SR.
Schedule credits and debits are often subject to an overall maximum modification.
SR Plan - Example
The following is a SR plan for WC and EL. In this plan:
 the underwriter has discretion in applying the credits or debits.
 there are five categories for which an insured can be eligible for a schedule credit or debit with minimums
and maximums specific to each category.
 An overall maximum credit or debit also applies.

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Chapter 15 – Commercial Lines Rating Mechanisms
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Schedule Rating Worksheet
Category

Available Range

Credit

Debit

of Modification

Applied

Reason / Basis

(Credit to Debit)

Premises
- General Housekeeping
- Preventative Maintenance
- Workplace Design
- Physical Condition

-10% to +10%

Classification
- Exposures not contemplated in class
- Hazards peculiar to a classification
have been eliminated
- Exposure variation due to technology

-15% to +15%

Medical Facilities
- First Aid
- Medical Assistance on Site

-5% to +5%

Safety Organization
- Written Safety Program
- Emergency and Disaster Plans
- Loss Control Programs
- Ergonomics

-15% to +15%

Employees
- Pre-employment Physicals
- Drug-Free Workplace
- New Hire Training
- Job-Specific Training

-15% to +15%

Total

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Max = 25% (Credit) / Debit

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Chapter 15 – Commercial Lines Rating Mechanisms
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2

Rating Mechanisms For Large Commercial Risks

298 -309

Rating mechanisms (loss-rated composite risks, large deductible policies, and retrospective rating plans) in this
section develop premium for the large commercial entities based on their experience.
1. Composite Rating
In composite rating:
 all coverages are rated using a single composite exposure base.
 an initial deposit premium is based on a composite rate and estimated composite exposures at the
beginning of the policy period.
 the final premium is based on an audit of final composite exposures after the end of the policy period.
The composite rate:
 may be based on manual rates adjusted by SR and/or ER modification.
 can also be based entirely on a large insured’s prior experience (a.k.a. loss-rated risks)
Example Composite Rating Plan for Loss-Rated Risks
In ISO’s Composite Rating Plan, an insured is eligible for being classified as “loss-rated” if its historical reported
losses and ALAE over a defined period exceed a specified aggregate dollar amount.
 the threshold varies based on coverage and limits.
 if eligible, the insured’s historical experience is 100% credible for determining the composite rate.
Step 1: Compute Trended Ultimate Loss & ALAE by coverage by year =
(Reported Loss & ALAE) x (Development Factor) x (Loss & ALAE Trend Factor).
do so for each type of coverage and for each of the past five completed years of experience
Step 2: Compute Trended Composite Exposure = Composite Exposure x Exposure Trend Factor.
i. select a composite exposure base
ii. trend the composite historical exposures to the average accident date of the proposed experience
period (do so for sales and payroll which are inflation-sensitive, but not for number of vehicle years
used in commercial auto)
Step 3: Compute

Adjusted Premium =

Trended Ultimate Loss & ALAE
Expected Loss & ALAE Ratio

i. the expected loss and ALAE ratio is the same as the PLR discussed in Chapter 7
(1.0 minus the sum of the provisions for expenses and profit).
ii. dividing the loss and ALAE by the expected loss and ALAE ratio incorporates a provision for other
expenses and profit.
Step 4: Compute

Composite Rate =

Adjusted Premium
(for coverage to be written)
Trended Composite Exposure

For loss-rated risks:
i. the composite rate is not adjusted by any ER plan (since the insured’s own experience has already been
reflected in the rate).
ii. SR (however) may apply.

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Chapter 15 – Commercial Lines Rating Mechanisms
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Composite Rate Calculation Example
 Bob’s Rentals sells new and used equipment, operates a repair and service shop, and offers leases and
rentals on equipment it owns.
 In calculating the commercial general liability (CGL) policy premium, each of the three operations is rated
separately, and the exposure base for each operation is different.
i. the exposure for sales on new and used equipment is receipts (in $000s) related to the latter.
ii. the exposure for the repair and service shop is payroll (in $00s) relating to the latter.
iii. the exposure for leases and rentals is receipts (in $000s) attributable only to leases and rentals.
 Bob’s Rentals is large enough to meet ISO’s Composite Rating Plan eligibility requirements for loss rating
and desires coverage up to $250,000 per occurrence with $500,000 general aggregate.
 The last five years of reported losses and ALAE over all 3 operations, separated into BI and PD is shown
below. Amounts are capped at $250,000 per occurrence.
 The selected composite exposure base is total receipts.
 Assume the following:
* Loss and ALAE annual trend (for bodily injury and property damage) is 6%.
* Exposure annual trend rate is 4%.
* Expected loss & ALAE ratio is 72%.
Reported Loss & ALAE a/o 12/31/08

Policy
Year

Bodily Injury

Property
Damage

7/1/03-04
7/1/04-05
7/1/05-06
7/1/06-07
7/1/07-08

$1,842,705
$1,406,353
$1,356,511
$1,355,545
$1,193,012

$626,162
$591,899
$517,616
$623,184
$568,669

Total

$7,154,126

$2,927,530

Policy
Year

New/Used
Equipment

Repair and
Service

Lease and
Rentals

Total

7/1/03-04
7/1/04-05
7/1/05-06
7/1/06-07
7/1/07-08

$56,498,756
$58,564,822
$61,193,878
$63,245,228
$65,721,869

$22,599,503
$23,425,929
$24,477,551
$25,298,091
$26,288,748

$33,899,254
$35,138,893
$36,716,327
$37,947,137
$39,433,121

$112,997,513
$117,129,644
$122,387,756
$126,490,456
$131,443,738

Total

$305,224,553

$122,089,822

$183,134,732

$610,449,107

Age to
Ultimate

Bodily
Injury

Property
Damage

66-Ult
54-Ult
42-Ult
30-Ult
18-Ult

1.10
1.25
1.45
1.70
1.95

1.03
1.10
1.20
1.35
1.50

Receipts

Development Factors

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Chapter 15 – Commercial Lines Rating Mechanisms
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Calculate: the loss-rated composite rate for Bob’s Rentals for its upcoming annual policy effective 7/1/2009.
Step 1: Develop trend factors to be applied to the loss and ALAE and the exposure base.
 The AAD of the proposed policy period is 12/31/2009, and the AAD of each policy year from the
experience period is 12/31.
 Based on the assumed trend rates, the trend factors are calculated as follows:
Trend Factors
(1)
(2)
(3)
(4)
(5)
Annual
Loss &
Loss &
ALAE
Annual
Exposure
Policy
Trend
ALAE
Trend
Exposure
Trend
Period
Trend
Factor
Trend
Factor
Year
7/1/03-04
7/1/04-05
7/1/05-06
7/1/06-07
7/1/07-08

(3) =[1.0 + (2)]^(1)

6
5
4
3
2

(5)

6.00%
6.00%
6.00%
6.00%
6.00%

1.4185
1.3382
1.2625
1.1910
1.1236

4.00%
4.00%
4.00%
4.00%
4.00%

1.2653
1.2167
1.1699
1.1249
1.0816

=[1.0+ (4)]^(1)

Step 2: Estimate the trended ultimate loss and ALAE.
Trended Ultimate Loss & ALAE
(1)
Policy
Year

(2)

Incurred Loss and ALAE
BI
PD

7/1/03-04
7/1/04-05
7/1/05-06
7/1/06-07
7/1/07-08

1,842,705
1,406,353
1,356,511
1,355,545
1,193,012

626,162
591,899
517,616
623,184
568,669

Total

7,154,126

2,927,530

(3)

(4)

Development Factors
BI
PD
1.10
1.25
1.45
1.70
1.95

1.03
1.10
1.20
1.35
1.50

(5)
Loss &
ALAE
Trend Factor

(6)
Trended
Ultimate Loss &
ALAE

1.4185
1.3382
1.2625
1.1910
1.1236

3,790,122
3,223,764
3,267,451
3,746,558
3,572,348

17,600,243

(6) = [ (1) x (3) + (2) x (4) ] x (5)

Step 3: Compute trended composite exposures.
Trended Composite Exposure

(1)
Policy
Year

Total Receipts
($000's)

7/1/03-04
7/1/04-05
7/1/05-06
7/1/06-07
7/1/07-08
Total

112,998
117,130
122,388
126,490
131,444
610,449

(1) = Sum of receipts from table on prior page

Exam 5, V1b

(2)
Exposure
Trend
Factor

(3)
Trended
Exposure

1.2653
1.2167
1.1699
1.1249
1.0816

142,976
142,512
143,181
142,289
142,170

713,127
(3) = (1) x (2)

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Step 4: Compute the composite rate:
Composite Rate
(1) Trended Ultimate Loss & ALAE
(2) Expected Loss & ALAE Ratio
(3) Adjusted Premium
(4) Trended Composite Exposure
(5) Composite Rate

(3) = (1) / (2)

$17,600,243
72.0%
$24,444,782
$713,129
$34.28

(5) = (3) / (4)

Step 5: Compute the Deposit premium:
Assuming total receipts for the proposed policy period are estimated to be $142,500, then the deposit
premium is $4,884,900 (= $142,500 x 34.28).
Step 6: Final premium: Is calculated according to the audited exposure (and any difference from the deposit
premium can be charged or credited to the insured).
Large (and Small) Deductible Policies
The purpose of small deductibles is for the insurer to keep premium low by avoiding expenses associated with
processing and investigating small nuisance (frivolous) claims.
Under a large deductible policy, the insured is bearing significant risk (either from a large number of small claims
or a small number of large claims).
Thus, the following pricing considerations must be addressed (in addition to those associated with small
deductible pricing):
* Claims handling: Will the insured or insurer handle claims that fall within the deductible?
i. If it is the insurer, the premium must cover the cost for all claim handling expenses (even those expenses
associated with claims that do not pierce the deductible).
ii. If it is the insured, the insurer should evaluate the insured’s claim handling expertise to determine the
likelihood of claims leakage above the deductible (as any increase in expected costs as a result of the
insured’s inexperience should be reflected in the pricing).
* Application of the deductible:
i. The deductible may apply to losses or to losses and ALAE.
ii. LER calculation should be based on data consistent with the treatment of ALAE in the policy terms.
* Deductible processing:
i. When the insurer is responsible for paying the entire claim and seeks reimbursement for amounts below
the deductible from the insured, the premium should reflect the cost of invoicing and monitoring deductible
activity as well as a provision for the risk that the insured may become bankrupt and be unable to pay for
any future deductible invoices (i.e. credit risk).
ii. Even if collateral is received to cover potentially uncollectible deductible amounts, it is rare that this credit
risk is fully collateralized.
* Risk margin: Since losses above a large deductible are more uncertain than losses below the deductible,
the profit margin may need to be adjusted to reflect the increased risk assumed by the insurer.
With the exception of these considerations, pricing for a large deductible policy is the same as pricing a standard
deductible (see Chapter 11)

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Chapter 15 – Commercial Lines Rating Mechanisms
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Large Deductible Premium Calculation Example
Consider a large deductible CGL policy being priced based on the following provisions and assumptions:
* The deductible is $500,000 per occurrence.
* The insurer will handle all claims (including those that fall below the deductible)
* The deductible is not expected to reduce ALAE costs. ALAE costs are estimated to be 11% of total losses.
* The deductible applies to losses only.
* Total ground-up losses without recognition of a deductible are estimated to be $1,000,000.
* Fixed expenses are assumed to be $50,000.
* Variable expenses are assumed to be 13% of premium.
* The insurer will make the payments on all claims and seek reimbursement for amounts below the deductible
from the insured. The cost to process deductibles is estimated to be 4% of the losses below the deductible.
* Deductible recoveries are not fully collateralized, and the credit risk is estimated to be 1% of the expected
deductible payments.
* The desired UW profit for full-coverage (i.e. no deductible) premium is 2%.
* An additional risk margin of 10% of excess losses for policies with a deductible of $500,000 is charged.
* The % of total losses below the deductible (i.e. the LER) and the % of total losses above the deductible (i.e.
excess ratio) are summarized below.
Loss Elimination Ratios

Premium =

Loss Limit

LER

Excess
Ratio
[1.0-LER]

$100,000
$250,000
$500,000

60%
80%
95%

40%
20%
5%

Losses above Deductible+ ALAE + Fixed Expense+ Credit Risk + Risk Margin
(1.0 -Variable Expense Provision - Profit Provision)

Step 1: Estimate losses above the $500,000 deductible.
(1) Expected total ground-up losses
(2) Excess ratio
(3) Estimated losses above deductible (1) x (2)

$1,000,000
5%
$ 50,000

Step 2: Compute the premium as follows:
(1) Estimated Losses Above the Deductible
(2) ALAE
(3) Fixed Expenses
(a) Standard
(b) Deductible Processing
(4) Credit Risk
(5) Risk Margin
(6) Variable Expenses and Profit (.13 + .02)
(7) Premium
(1) = prior table,row (3);
(2) = 11% x prior table, Row (1)
(4) = 1% x prior table, Row (1) x LER
(5) = 10% x (1)

Exam 5, V1b

$50,000
$110,000
$50,000
$38,000
$9,500
$5,000
15%
$308,824

(3a) = Provided (3b) = 4% x prior table, Row (1) x LER
(7) = [(1) + (2) + (3a) + (3b) + (4) + (5)] / [1.0 - (6)]

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Chapter 15 – Commercial Lines Rating Mechanisms
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Retrospective Rating
A retrospective rating plan uses the insured’s actual experience during the policy period to determine the
premium for that same period.
 actual losses used to determine the final retrospective premium may be limited to reduce the effect of any
single unusual or catastrophic event.
 total premium charged may be subject to a minimum and maximum amount, to stabilize the year-to-year
cost and to protect the insured from exceeding an aggregate cost due to a large number of claims
incurred in any one year.
Premium for a retro rated policy consists of an initial premium and periodic premium adjustments made after the
policy period to reflect actual claims experience for a pre-determined number of adjustments or until the insurer
and insured agree no more adjustments are needed.
Three ways in which initial premium and premium adjustments can be structured are as follows:
1. Initial premium is based on total expected expenses, profit, and costs associated with any caps.
At the end of the policy period, the insured is billed annually for all losses incurred under the policy after
capping rules apply.
Annual premium adjustments continue each year for a pre-determined length of time.
2. Initial premium is based on expenses, profit, and costs associated with any caps (excluding LAE
associated with the policy).
Annual premium adjustments associated with reported losses during the policy period will include a
provision for LAE costs (i.e. a pre-determined percentage chosen to reflect LAE costs).
3. Initial premium is based on an estimate of the final premium under the policy (including provision for
total expected ultimate losses and expenses).
Periodic premium adjustments are due to changes in the revised estimate of final premium based on the
latest loss data.
All 3 examples above should produce the same total premium for a retro rated policy.
Retrospective Rating Plan Premium Calculation – WC
Basic Formula
The basic formula for retrospective premium is as follows:
Retro Premium = [Basic Premium + Converted Losses] x Tax Multiplier, where the retro premium is subject
to a maximum and minimum.
Basic Premium = [Expense Allowance - Expense Provided Through LCF + Net Ins Charge] x Standard Premium
where:
LCF = Loss Conversion Factor
Expense Provided Through LCF = Expected Loss Ratio x (LCF -1.0)
Net Insurance Charge = [Insurance Charge - Insurance Savings] x Expected Loss Ratio x LCF.
The Basic Premium provides for:
1. The insurer’s target UW profit and expenses (excluding expenses provided for by the LCF and the tax
multiplier), and
2. The cost of limiting the retrospective premium (to be between the minimum and maximum premium
negotiated under the policy), and

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Converted Losses: Converted Losses = Reported Losses x LCF.
 Converted losses are reported losses limited by the selected limit (and then multiplied by the LCF).
 The LCF is a factor to include the ALAE and ULAE not already included in the losses.
Expenses
Are introduced into the formula through 3 components:
1. the tax multiplier (to account for the cost of premium taxes)
2. the expense allowance (to account for all other underwriting expenses).
3. the LCF (to account of expenses that varies with losses, e.g. ALAE, and is negotiated between the insurer
and insured).
Standard Premium
 is the premium before consideration of the retro rated plan and any premium discount.
 is determined based on the exposure, the insurer’s rates, the experience modification, and any premium
charges (excluding premium discount).
Minimum/Maximum Retrospective Premium
Minimum Retro Premium = Standard Premium x Minimum Retro Premium Ratio.
Maximum Retro Premium= Standard Premium x Maximum Retro Premium Ratio.
Minimum and maximum retrospective premium ratios are negotiated between the insured and insurer.
Insurance Charge and Insurance Savings
Applying a minimum and maximum will affect the total premium collected by the insurer and therefore the cost of
doing so needs to be considered as part of the final premium.
 The insurance charge: the cost associated with limiting the retrospective premium to be no higher than
the maximum retrospective premium.
 The insurance savings: the savings by requiring the retrospective premium to be no lower than the
minimum retrospective premium.
 The insurance charge and insurance savings:
i. are contained in a table of values.
ii. are expressed as a % of expected unlimited losses.
Notes:
* In the following example, the impact of the per occurrence loss limitation is incorporated into the values
contained within this table;
* There are table that represent only the effect of the maximum and minimum premiums, and the effect of
the per occurrence loss limitation is computed as an additional charge.
Retro Rated Premium Calculation - Example
Assume the following:
 The 1st computation of the retrospective premium occurs 6 months after the end of the policy period and
annually thereafter until the insurer and insured agree that the latest computation will be the final one.
 The policy is an annual policy and limited reported losses valued as of 18 months are $153,000.

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Chapter 15 – Commercial Lines Rating Mechanisms
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Retrospective premium calculation:
Note: The givens in computing the retrospective premium are shown in the first 10 rows of Table below.
(1) Minimum retrospective premium ratio (negotiated)
(2) Maximum retrospective premium ratio (negotiated)
(3) Loss Conversion Factor (negotiated)
(4) Per Accident Loss Limitation (negotiated)
(5) Expense Allowance (excludes tax multiplier)
(6) Expected Loss Ratio
(7) Tax Multiplier
(8) Standard Premium
(9) Insurance Charge for Maximum Premium
(10) Insurance Savings for Minimum Premium
(11) Basic Premium
(12) Converted Losses
(13) Preliminary Retrospective Premium
(14) Minimum Retrospective Premium
(15) Maximum Retrospective Premium

60.0%
140.0%
1.10
$100,000
20%
65%
1.03
$769,231
0.42
0.03
$318,346
$168,300
$501,245
$461,539
$1,076,923
$501,245

(16) Retrospective Premium
(11) = [(5)-(6) x [ (3)-1.0 ]+[ (9)-(10)] x (6) x (3)] x (8)
(12) = $153,000 x (3)
(13) = [(11)+(12) ] x (7)
(14) = (1) x (8)
(15) = (2) x (8)
(16) = Min [Max[(13),(14)] , (15) ]

Key Concepts

311 - 311

1. Manual rate modification plans
a. ER
i. Actual experience
ii. Expected experience
iii. Other considerations
iv. Examples for CGL and workers compensation
b. Schedule rating (with example plan for workers compensation and employer’s liability)
2. Rating techniques for large commercial risks
a. ISO loss-rated composite risks (with example for a CGL policy)
b. Large deductible policies
c. Retrospective rating plans (with example for a WC policy)

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Chapter 15 – Commercial Lines Rating Mechanisms
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
The predecessor papers to the current syllabus reading “Basic Ratemaking” by Werner, G. and
Modlin, C. were numerous. While past CAS questions were drawn from prior syllabus readings,
the ones shown below remain relevant to the content covered in this chapter.
Questions from the 1992 exam
33. (2 Points) The Acme Widget Company has a retrospectively rated workers' compensation policy with a
$100,000 limitation on individual losses.
Using the information below and methods outlined in the Foundations of Casualty Actuarial Science,
Chapter 3 -"Individual Risk Rating", compute Widget's retrospective premium.
Standard Premium
Net (after discount) Premium
Incurred Losses Limited to $100,000
Retrospective Premium Development Factor
Excess Loss Premium Factor
Basic Minimum Premium Factor
Maximum Premium Factor
Loss Conversion Factor
Tax Multiplier

$100,000
90,500
40,000
0.15
0.12
0.20
1.00
1.10
1.00

A. < $87,000 B > $87,000 but < $89,000 C. > $89,000 but < $91,000 D. > 91,000 but< $93,000
E. > $93,000

Questions from the 1994 exam
26. The XYZ Construction Company has a Workers Compensation policy that is rated under the National
Council on Compensation Insurance (NCCI) Experience Rating Plan. Given the following information,
the Experience Modification Factor for XYZ falls into which range?
Actual Primary Losses
$ 50,000
Actual Excess Losses
10,000
Expected Primary Losses
40,000
Expected Excess Losses
20,000
Weighted Loss Factor (w)
0.10
Ballast (B)
6,000
A. < 0.98
B. > 0.98 but <1.02
C. >1.02 but < 1.06
D. > 1.06 but < 1.10 E > 1.10

Questions from the 1995 exam
32. (2 points) You are given:
• Workers' Compensation Manual Premium
• Experience Modification
• Premium Discount Factor
• Basic Premium Factor
• Converted Losses
• Tax Multiplier
• Minimum Retrospective Factor
• Maximum Retrospective Factor

$100,000
10.0% Credit
9.0%
30.0%
$80,000
1.05
80.0%
120.0%

According to Tiller, “Individual Risk Rating," calculate the Workers' Compensation retrospective premium.

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Chapter 15 – Commercial Lines Rating Mechanisms
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Questions from the 1997 exam
13. According to Werner/Modlin, the basic premium in the NCCI retrospective rating plan provides for which of
the following costs?
1. Risk control services
2. Premium taxes
3. An allowance for profit and contingencies
A. 1

B. 3

C. 1, 2

D. 1, 3

E. 1, 2, 3

36. (3 points) You are given:

Claim 1
Claim 2
Claim 3
Claim 4

Total Reported
Loss Amount

Allocated Loss
Adjustment Expense

80,000
145,000
110,000
125,000

50,000
120,000
80,000
250,000

Basic Limits Earned Premium (Subject Premium)
Basic Loss Limit
Maximum Single Loss
Expected Loss and ALAE Ratio (Not Limited by MSL)
Expected Unreported Basic Limits Loss and ALAE (Limited by MSL)
D-Ratio
Credibility

1,500,000
100,000
200,000
0.700
250,000
0.80
0.50

Based on Tiller, "Individual Risk Rating," chapter 3 of Foundations of Casualty Actuarial Science,
a. (2 points) What is the actual loss ratio that will be used in calculating the experience modification?
b. (1 point) Determine the experience modification.

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Chapter 15 – Commercial Lines Rating Mechanisms
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Questions from the 1998 exam
19. Based on Tiller, "Individual Risk Rating," chapter 3 of Foundations of Casualty Actuarial Science, calculate
the NCCI experience modification factor.
Actual primary loss
$50,000
Actual excess loss
$100,000
Expected primary lose
$55,000
Expected excess loss
$25,000
Stabilizing value
$20,000
Excess loss weighting factor
0.75
32. (2 points) A commercial risk is being rated based on a one year experience period. The actual experience is
as follows:
Actual basic limit loss & ALAE
$30,000
Current basic limit premium
$50,000
Loss & ALAE development factor
1.25
Credibility
.80
Detrend factor
.85
Expected loss & ALAE ratio
.70
Note: There is no maximum single loss limitation.
Based on Tiller, "'Individual Risk Rating," chapter 3 of Foundations of Casualty Actuarial Science, calculate
the experience modification.

Questions from the 1999 exam:
36. (2 points) Using the ISO experience rating plan described in Tiller, "Individual Risk Rating," chapter 3 of
Foundations of Casualty Actuarial Science and the information shown below, answer the following.
Loss experience for this risk:

Claim #1
Claim #2

Loss
15,000
35,000

Allocated Loss Adjustment
Expense (ALAE)
15,000
10,000

Basic limit
Current basic limit premium
Detrend factor
Expected Percentage of Basic Limits Loss
and ALAE Unreported
Policy Adjustment Factor
Credibility
Expected loss and ALAE ratio

$25,000
$100,000
.85
20%
1.00
.80
.70

Assume there is no maximum single loss
a. (1 1/2points) Calculate the experience modification.
b. (1/2 point) According to Tiller, state one advantage and one disadvantage of using a one-year
experience period as compared to a longer period.

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BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Questions from the 2000 exam
28. Based on Tiller, "Individual Risk Rating," chapter 3 of Foundations of Casualty Actuarial Science, and the
following information, calculate the experience modification factor following the ISO CGL Experience
Rating Plan.
 Subject Premium = $100,000
 Adjusted Expected Loss and ALAE Ratio = 65.0%
 Actual Loss and ALAE Ratio = 68.0%
 Actual Losses and ALAE Limited by Basic Limits and MSL = $32,917
 Credibility = .35
A. < 0.80
B. > 0.80 but < 1.10
C. > 1.10 but < 1.40
D. > 1.40 but < 1.70
E. > 1.70

Questions from the 2001 exam
Question 13. Based on Tiller, “Individual Risk Rating Study Note,” and the following data, calculate the
Adjusted Expected Loss & ALAE Ratio.


D-ratio

0.624



Off-balance factor

1.050



Subject Premium



Total Limits Earned Premium

$100,000



Expected Basic Limits Losses & Unlimited ALAE

$ 60,000



Expected Total Limits Losses & Unlimited ALAE

$ 74,500

A. < 42.0%

B. > 42.0% but < 44.0%

$80,000

C. > 44.0% but < 46.0%

D. > 46.0% but < 48.0%

E. > 48.0%

Questions from the 2002 exam
2. Based on Tiller, "Individual Risk Rating - Study Note," and the following data, calculate
the experience modification factor using NCCI's "Revised Experience Rating Plan".
Expected total loss
210,000
Expected primary loss
50,000
Actual total loss
320,000
Actual primary Loss
40,000
Ballast factor
30,000
Excess loss weighting factor
0.25
A. < 1.070 B. > 1.070 but < 1.080 C. > 1.080 but < 1.090 D. > 1.090 but < 1.110 E > 1.110

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BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Questions from the 2003 exam
17. Which of the following statements are true regarding individual risk rating?
1. Schedule rating directly reflects an entity's claim experience.
2. Experience rating is used when the past, with appropriate adjustments, is predictive of the future.
3. Individual risk rating is appropriate when entities in a rating group are homogeneous.
A. 1 only

B. 2 only

C. 3 only

D. 1 and 2 only

E. 2 and 3 only

42. (3 points)
a. (1.5 points) Using the following information, calculate the final retrospective premium. Show all work.


Standard premium = $300,000



Basic premium factor = 0.18



Loss conversion factor = 1.20



Excess loss premium factor = 0.25



Tax multiplier = 1.04



Loss limit per accident = $50,000
Reported losses
$70,000
$15,000
$25,000

b. (1 point) Explain why the retrospective rating process tends to produce back-and-forth payments
between the insured and insurer.
c. (0.5 point) Briefly describe a mechanism that can be used to smooth these back-and-forth payments.

Questions from the 2004 exam:
43. (3 points) Using the ISO experience rating plan for a policy with premises/operations coverage and the following
information, calculate the experience debit or credit. Show all work.
Expected Percent of
Basic Limits
Detrend
Loss & ALAE Unreported
Policy Period
Factors
as of September 30, 2003
1999
0.78
15%
2000
0.85
25%
2001
0.94
40%
•
•
•
•
•
•

Policy being rated is a January 1, 2004 - December 31, 2004 occurrence policy.
Premises/operations premium is $240,000.
Reported loss and ALAE for experience period as of September 30, 2003 (limited by basic limits losses
and MSL) is $300,000.
Expected experience ratio is 0.90. Expected loss and ALAE ratio is 0.62.
Maximum single limit per occurrence is $100,000. Credibility is 0.35.
All policies in experience period are occurrence policies.

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BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Questions from the 2005 exam:
54. (2 points) Given the following information for an insured, determine the General Liability premium after
adjustments for experience and schedule rating. Show all work.
• Manual premium = $75,800
Experience rating information:
• Reported Limited Losses and ALAE = $93,500
• Expected Unreported Limited Losses and ALAE = $25,200
• Company Subject Basic Limits Loss and ALAE Costs = $153,900
• Credibility = 0.35
• Expected Experience Ratio = 0.92
The underwriter has determined that the following schedule rating modifications are appropriate:
• Premises - Condition, Care = +4%
• Equipment - Type, Condition, Care = -7%
• Classification Peculiarities = -8%
• Employees - Selection, Training, etc. = +3%

Questions from the 2006 exam:
10. John's Car Wash is a new single-location business. It is purchasing commercial general liability
insurance. Which of the following rating methods might be used in calculating the premium?
1. Schedule Rating
2. Experience Rating
3. Composite Rating
A. 1 only

B. 2 only

C. 3 only

D. 1 and 2 only

E. 1 and 3 only

49. (2 points) Given the following information for a commercial general liability risk, calculate the experience
(Credit)/Debit based on the ISO CGL Experience Rating Plan. Show all work.
Actual Losses in the experience period valued as of March 31, 2006:
Loss
ALAE
Claim
1
$1,000
$200
2
1,500
200
3
5,000
800
4
6,000
1,000
5
12,000
1,800
6
23,000
2,200
7
120,000
40,000
Expected Unreported Losses and ALAE @ March 31, 2006 = $45,000
Company Subject Basic Limits Loss and ALAE costs = $250,000
Basic Limit = $100,000
MSL = $150,000
Expected Experience Ratio = 0.9 Credibility = 0.6

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Chapter 15 – Commercial Lines Rating Mechanisms
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Questions from the 2007 exam:
52. (1.5 points) Given the following information for a General Liability risk:
Company subject basic limit loss and ALAE costs:
Expected experience ratio at MSL of $200,000:
Projected ultimate loss and ALAE limited by basic limits and MSL:
Credibility factor (Z):
Credit/Debit limit:

$150,000
0.9
$250,000
0.4
+/-25%

Calculate the experience rating credit/debit using the ISO CGL experience rating plan. Show all work.

Questions from the 2008 exam:
39. (1.0 point) Identify and briefly explain two of the three types of prospective individual risk rating systems.
41. (2.0 points) You are given the following information:
 Premises/Operations Manual Premium = $200,000
 Expected Basic Limits Loss and ALAE Ratio = 70%
 Policy Adjustment Factor = .9
 Detrend Factor = .95
 Reported Loss and ALAE Limited by Basic Limits and MSL = $115,000
 Expected Unreported Loss and ALAE Limited by Basic Limits and MSL = $35,000
 Expected Experience Ratio= .85
 Credibility = 45%
 Maximum Credit or Debit = +-50%
a. Calculate the Actual Experience Ratio using the ISO Commercial GL Experience Rating Plan.
b. Calculate the Experience Credit or Debit using the ISO Commercial GL Experience Rating Plan.

Exam 5, V1b

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Chapter 15 – Commercial Lines Rating Mechanisms
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Questions from the 2009 exam:
35. (3.25 points) Given the following information:
Frequency of
Claims
45%
20%
15%
15%
5%

Loss Amount
per Claim
$22,000
$35,000
$150,000
$250,000
$1,000,000

• Full Coverage Premium = $24,793
• Expected Ground-up Loss Ratio = 59%
• ALAE = 10% of losses (assume the deductible does not apply to ALAE)
• Incremental Fixed Expenses for processing a deductible = 4% of losses in deductible layer
• Load for uncollectible deductible payments = 1% of losses in deductible layer
• Profit = 8%
• Additional Risk Load = 5%
• Commission = 10%
• Other Variable Expenses = 5%
Calculate the final premium for a policy with a $100,000 deductible.

43. (1 point) An insurance company uses experience rating and schedule rating to calculate Commercial
General Liability (CGL) premium for bowling ball manufacturers.
• A schedule rating credit of up to 10% can be judgmentally given for loss control programs.
• There are no caps on the experience modification factors.
• The insured is a bowling ball manufacturer whose loss control program has reduced fosses by an
estimated 5% each year for the last 10 years.
Determine the appropriate schedule rating credit, assuming no changes to the insured's loss control
program. Briefly explain your answer.
44. (1.5 points) Contrast experience rating and retrospective rating with respect to the following concepts:
a. (0.75 point) Providing incentive to the insured to control losses during the policy period.
b. (0.75 point) Providing stability in the premium charged to the insured.

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Chapter 15 – Commercial Lines Rating Mechanisms
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Questions from the 2010 exam:
33. (3 points) Given the following information:
• Payrolls during the experience period were $10,920,000.
• Primary losses are capped at $10,000.
• The standard for full credibility is 1,082 claims.
• Weighting value = w = 0.36.
• The expected loss cost is 2.90 per $100 payroll.
• The D-ratio at $10,000 is 0.82.
Claim Size
$4,000
$8,000
$15,000
$16,000
$23,000
$42,000

Number of
Claims
32
15
3
1
2
1

Calculate the NCCI experience modification factor.
34. (1 point) An insurer has been tracking the claims experience of a very large construction company for
the three years the construction company has been insured by this insurer. The construction company
will implement a new safety program starting in the upcoming year.
a. (0.5 point) Determine whether the insurer should use experience rating, schedule rating, or both to
rate the construction company for the upcoming policy period. Briefly explain your answer.
b. (0.5 point) Assuming no additional changes, determine whether the insurer should use experience
rating, schedule rating, or both to rate the construction company five years from now. Briefly explain
your answer.

Exam 5, V1b

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Chapter 15 – Commercial Lines Rating Mechanisms
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Questions from the 2011 exam:
19. (2.5 points) The owner of a parking lot is looking to purchase workers compensation insurance for
herself and her employees. The insured has enough prior experience to be eligible for an NCCI
experience rating modification factor. Given the following characteristics of the policy:
• The insured has a dedicated return-to-work program that makes it eligible for a 15% premium
discount.
• The expense constant is $250.
• Actual primary losses in the experience period = $47,000
• Actual excess losses in the experience period = $10,000
• Expected primary losses = $75,000
• Expected excess losses = $15,000
• Primary credibility = 0.5
• Excess credibility = 0.1
• Exposures and applicable rates for the insured are as follows:

Class Code
8392 - Auto Parking Lot
8742 - Salespersons
8810 - Clerical Office Employees

Payroll
$2,500,000
$500,000
$1,000,000

Rate per $100 of
Payroll
4.1
0.5
0.3

a. (1 point) Calculate the experience rating modification factor.
b. (0.75 point) Calculate the standard premium.
c. (0.75 point) Calculate the final premium for the insured.

Exam 5, V1b

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Chapter 15 – Commercial Lines Rating Mechanisms
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
The predecessor papers to the current syllabus reading “Basic Ratemaking” by Werner, G. and
Modlin, C. were numerous. While past CAS questions were drawn from prior syllabus readings,
the ones shown below remain relevant to the content covered in this chapter.
Solutions to questions from the 1992 exam
Question 33.
Retro premium = [Basic prem + Converted losses + Excess Loss prem + Retro Dev prem] * Tax Multiplier
Basic premium = Standard Premium * Basic Premium Factor
Converted losses = Reported limited losses at Evaluation Date * Loss conversion factor.
Retro development premium = Standard premium * Retro Development Factor * Loss conversion factor.
Retro premium = [(100,000)*(.20) + {40,000 + (100,000)*(.12) + (100,000)*(.15)} * 1.10] * 1.00 = 93,700.
This premium is less than the maximum premium of [SP * maximum premium factor] and greater than the
minimum premium of [SP * minimum premium factor] and is thus the correct premium.
Answer E.

Solutions to questions from the 1994 exam
Question 26.

Ap +[w *Ae ] +[(1- w) * Ee ] + B 50,000 + .10 * 10,000 + .90 * 20,000 + 6,000
=
= 1.136. Answer E.
E+B
(40,000 + 20,000) + 6,000

Solutions to questions from the 1995 exam
Question 32.
Retro premium = [Basic prem + Converted Losses + Excess Loss prem + Retro Devel.prem] * tax multiplier

H < R = (B + cL)T < G
B = Basic premium = Standard premium * Basic premium factor .
cL = Converted losses = Reported limited losses at the evaluation date * Loss conversion factor
H = Minimum premium = Standard premium * Minimum premium factor.
G = Maximum premium = Standard premium * Maximum premium factor.
Standard premium = Manual premium modified for experience rating, loss constants, and minimum premium
excluding premium discount and expense constant. SP = 100,000 * (1-.10) = 90,000.
Retro premium = [(90,000)*(.30) +80,000] * 1.05 = 112,350.
However, the retro premium calculation is subject to a maximum of
SP * maximum premium factor = 90,000 * 1.20 = 108,000.

Exam 5, V1b

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Chapter 15 – Commercial Lines Rating Mechanisms
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Solutions to questions from the 1997 exam
Question 13
13. According to Werner/Modlin, the basic premium in the NCCI retrospective rating plan provides for which of the
following costs?
1. Risk control services
2. Premium taxes
3. An allowance for profit and contingencies
According to Werner/Modlin, the following elements are included in the basic premium.
1. The insurer’s target UW profit and expenses (excluding expenses provided for by the LCF and the
tax multiplier), and
2. The cost of limiting the retrospective premium (to be between the minimum and maximum premium
negotiated under the policy), and
3. The cost of limiting each occurrence to a negotiated loss limitation (if applicable).
Thus, 1 is False, 2 is False, and 3 is True

Exam 5, V1b

Answer B.

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Chapter 15 – Commercial Lines Rating Mechanisms
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Solutions to questions from the 1997 exam
Question 36
Write formulas to compute the
1. Actual Loss and ALAE ratio (ALR)
2. Adjusted Expected Loss and ALAE ratio (AELR) and
3. The Experience mod (M).

ALR =

(Reported L + ALAE Limited by BL & MSL) + E[Unreported L + ALAE Limited by BL & MSL]
.
Subject Premium

AELR = Expected Loss and ALAE ratio.

ALR - AELR
*Z.
AELR
(i) Compute actual basic limits losses to be included in the experience rating calculation:
Step 1: Define basic limit losses. This is given in the problem as $100,000.
Step 2: Define and calculate actual basic limits loss and ALAE included in Mod Calculation:
Paid and O/S losses (including ALAE) with
(a) indemnity limited to basic per occurrence limits and
(b) (indemnity + ALAE) limited by the MSL (Given as 200,000).
M=

Losses

Limited Losses +

Actual Loss + ALAE

Unlimited

Limited

ALAE

Unlimited ALAE

limited by the MSL

(1)
80,000
145,000
110,000
125,000

(2)
80,000
100,000
100,000
100,000

(3)
50,000
120,000
80,000
250,000

(4) = (2)+(3)
130,000
220,000
180,000
350,000

(5)
130,000
200,000
180,000
200,000
710,000

Thus, the ALR =

[710 ,000  250 ,000 ]
= .640
1,500 ,000

(ii) Compute Adjusted Expected Loss and ALAE ratio (AELR): AELR = .70 * .80 = .560.

 .640.560 
Mod = 
 .50 = .071.
 .560 

Exam 5, V1b

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Chapter 15 – Commercial Lines Rating Mechanisms
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Solutions to questions from the 1998 exam
Question 19.

Ap +[w *Ae ] +[(1- w) * Ee ] + B 50,000 + .75 * 100,000 + .25 * 25,000 + 20,000
=
= 1.512.
E+B
(80,000 + 20,000)
Answer D.

Question 32.
Write formulas to compute the
1. Actual Loss and ALAE ratio (ALR)
2. The Experience mod (M).
(Reported L + ALAE Limited by BL & MSL) + E[Unreported L + ALAE Limited by BL & MSL]
.
ALR=
Subject Premium
=

30,000 + E[Unreport ed L + ALAE Limited by BL & MSL]
Subject Premium

ELR = Expected Loss and ALAE ratio.
M=1+

ALR - ELR
* Z.
ELR

Subject Premium = Current Basic Limits Premium * PAF1 * PAF2 * Detrend Factor
= 50,000 * 1.00 * 1.00 * .85 = 42,500
Expected Unreported Losses = Subject Premium * ELR * Expected % Unreported

FG
H

= Subject Premium * ELR * 1

F
H

= 42,500 * .70 * 1
Thus, ALR =
Mod = 1 +

Exam 5, V1b

[30,000  5,950]
= .846
42,500

1
1.25

I
K

1
LDFULT

IJ
K

= 5,950

ELR = .70

FG .846.70IJ *.80 = 1.17
H .70 K

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Chapter 15 – Commercial Lines Rating Mechanisms
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Solutions to questions from the 1999 exam
Question 36.
Write formulas to compute the
1. Actual Loss and ALAE ratio (ALR)
2. The Experience mod (M).

(Reported L + ALAE Limited by BL & MSL) + E[Unreported L + ALAE Limited by BL & MSL]
.
Subject Premium

ALR=
=

30,000 + E[Unreport ed L + ALAE Limited by BL & MSL]
Subject Premium

ELR = Expected Loss and ALAE ratio.

ALR - ELR
* Z.
ELR
Subject Premium = Current Basic Limits Premium * PAF1 * PAF2 * Detrend Factor
= 100,000 * 1.00 * 1.00 * .85 = 85,000
Expected Unreported Losses = Subject Premium * ELR * Expected % Unreported
= Subject Premium * ELR * .20
= 85,000 * .70 * .20 = 11,900
Note: Since there is no maximum single loss, loss limitation to basic limits is all that is necessary.
Unlimited Basic Limits Loss
Allocated Loss Adjustment
Loss
(Limited to
Expense (ALAE)
Basic Limits Loss + ALAE
$25,000)
(1)
(2)
(3)
(4) = (2) + (3)
Claim #1
15,000
15,000
15,000
30,000
Claim #2
35,000
25,000
10,000
35,000
M=1+

Thus, ALR =

[ 65,000 11,900]
= .905
85,000

b. advantage: more responsive.

ELR = .70

Mod = 1 +

FG .905.70IJ *.80 = 1.234
H .70 K

disadvantage: less stable

Solutions to questions from the 2000 exam
Question 28.
The formula for the experience modification factor is: M  1 

ALR - AELR
* Z , where
AELR

ALR = Actual Loss and ALAE ratio =.68
AELR = Adjusted Expected Loss and ALAE Ratio = 65.0%
Z = Credibility = .35
Thus, M  1 

Exam 5, V1b

.68 - .65
*.35  1.016
.65

Answer B.

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Chapter 15 – Commercial Lines Rating Mechanisms
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Solutions to questions from the 2001 exam
Question 13. Calculate the Adjusted Expected Loss & ALAE Ratio.
On page 1 and 2 of the “Individual Risk Rating Study Note”, Tiller states that the adjusted expected loss
and ALAE ratio (AELR) is computed as follows:

AELR 
Thus,

Expected basic limits losses and (unlimited ) ALAE * D  ratio
Subject Premium

AELR 

60, 000*.624
 .468
80, 000

Answer D.

Solutions to questions from the 2002 exam
Question 2. Calculate the experience modification factor using NCCI's "Revised Experience Rating Plan".
Step 1: Write an equation to compute the experience modification factor:

M 

Ap  [ w * Ae ]  [(1- w) * Ee ]  B
EB

Step 2: Assign symbols to the given data in the problem and solve for any unknown terms:
Description
Amount
Symbol
Expected total loss
210,000
E
Expected primary loss
50,000
Ep
Actual total loss
320,000
A
Actual primary Loss
40,000
Ap
Ballast factor
30,000
B
Excess loss weighting factor
0.25
W
Note: Ae = A - Ap = 320,000 – 40,000 = 280,000. Ee = E - Ep = 210,000 – 50,000 = 160,000.
Step 3: Using the equation in Step 1, and the data from Step 2, solve for the experience modification factor.

M

40,000 + .25*280,000 + .75*160,000 + 30,000
 1.08333 .
210,000 + 30,000

Answer C.

Solutions to questions from the 2003 exam
17. Which of the following statements are true regarding individual risk rating?
1. Schedule rating directly reflects an entity's claim experience. False. Schedule rating takes into
consideration characteristics that are expected to affect losses and ALAE but that are not reflected in past
experience. Schedule rating is the only individual risk rating system that does not directly reflect an entity’s
claim experience.
2. Experience rating is used when the past, with appropriate adjustments, is predictive of the future. True.
3. Individual risk rating is appropriate when entities in a rating group are homogeneous. False. Individual
risk rating is appropriate when there is a combination of non-homogeneous rating groups and entities with
credible experience.
Answer: B. 2 only

Exam 5, V1b

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Chapter 15 – Commercial Lines Rating Mechanisms
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Solutions to questions from the 2003 exam
42. (3 points)
a. (1.5 points) Calculate the final retrospective premium. Show all work.
Step 1: Write an equation to determine the final retrospective premium:
H < R = (B + cL)T < G, where
R  Retro premium=[Basic prem+Converted Losses+Excess Loss prem+Retro Devel.prem]*tax mult

B = Basic premium = Standard premium* Basic premium factor
cL = Converted losses = Reported limited losses at the evaluation date * Loss conversion factor
H = Minimum premium = Standard premium * Minimum premium factor.
G = Maximum premium = Standard premium * Maximum premium factor.
Step 2: Using the formulas in Step 1, and the data given in the problem, solve for the basic premium, converted
losses, and excess losses.
a. Standard premium = Manual premium modified for experience rating, loss constants, and minimum premium
excluding premium discount and expense constant. In this problem, SP is given as $ 300,000.
b.

B = Basic premium = Standard premium* Basic premium factor = $300,000* .18 = $54,000 .

c. cL = Converted losses = Reported limited losses at the evaluation date * Loss conversion factor
= ($50,000 (limited) + $15,000 + $25,000) * 1.20 = $108,000
d. Excess Losses = SP * ELPF * LCF = $300,000 * .25 * 1.20 = $90,000
Retrospective Premium Development premium is to be ignored (since this elective option was not referenced)
Step 3: Using the equation in Step 1, the results from Step 2, and the data given in the problem, solve for the
retrospective premium = [($54,000) + $108,000 + $90,000] * 1.04 = $262,080.
b. (1 point) Explain why the retrospective rating process tends to produce back-and-forth payments
between the insured and insurer.
The back and forth premium payments are due to the retrospective premium adjustments that modify the
premium based on loss experience incurred. The 1st adjustment (typically at 18 months after inception) is
usually a return premium (i.e. a refund) because minimal loss experience is reported. Subsequent
adjustments typically require additional premium from the insured, as losses develop over time.
c. (0.5 point) Briefly describe a mechanism that can be used to smooth these back-and-forth payments.
The retrospective development premium can be used to offset and smooth out some of the uneven cash flows.

Exam 5, V1b

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Chapter 15 – Commercial Lines Rating Mechanisms
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Solutions to questions from the 2004 exam:
43. (3 points) Using the ISO experience rating plan for a policy with premises/operations coverage and the
following information, calculate the experience debit or credit. Show all work.
Step 1: Write an equation to determine the experience modification:

M 

AER - EER
*Z , where AER is the actual experience ratio, EER is the expected experience ratio,
EER

and Z is the credibility applied to this ratio.
Step 2: Write an equation to determine the AER

AER=

Projected Ultimate Loss and ALAE (Limited by BL and MSL)
Company Subject Basic Limits Loss and ALAE

Projected Ultimate Losses and ALAE are comprised of the following two components:
a. Reported Losses and ALAE limited by Basic Limits and the MSL (given in the problem)
b. Expected Unreported Losses and ALAE Basic Limits and the MSL
Note: The experience used in the computation of the experience debit or credit, given in the problem is
three policy periods completed at least 6 months prior to the 1/1/04 rating date.
Step 3: Using the data given in the problem, compute the values for the numerator and denominator of the
equation shown in Step 2:
Subject
Policy
Subject
Detrend
Loss
%
Unreported
Premium
ELR
Factor
Cost
EER
Unreported
Losses
Period
(2)
(3)
(4)=(1)*(2)*(3)
(5)
(6)
(7)=(4)*(5)*(6)
(1)
1999
240,000
0.62
0.78
116,064
0.90
0.15
15,669
2000
240,000
0.62
0.85
126,480
0.90
0.25
28,458
0.90
0.40
50,354
2001
240,000
0.62
0.94
139,872
382,416
94,481
Note: The numerator is computed as the sum of the reported loss and ALAE for experience period as of
9/30/03 (limited by basic limits losses and MSL) of $300,000 and the Unreported losses shown in col
(7) above of 94,481 = 394,481. The denominator, computed in col (4) above, is 382,416

 394, 481 
1.0315 - .90
 1.0315 , and M 
*.35=.051 (experience debit)

.90
 382, 416 

Thus, AER  

M = +5.1% experience debit.

Exam 5, V1b

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Chapter 15 – Commercial Lines Rating Mechanisms
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Solutions to questions from the 2005 exam:
54. (2 points) Given the following information for an insured, determine the General Liability premium after
adjustments for experience and schedule rating. Show all work.
Step 1: Write an equation to determine the GL premium after adjusting for experience and schedule rating:
Adjusted premium = Manual premium * Experience Modification Factor * Schedule Rating factors
Note: The experience modification factor (M) = 1 + [AER – EER]/AER * Z
Step 2: Using the formula in Step 1, and the data given in the problem, solve for the adjusted premium.
I. Experience Components
A. Reported Losses and ALAE Limited by Basic Limits and MSL
93,500
B. Expected Unreported Losses and ALAE Limited by Basic Limits and MSL 25,200
C. Projected Ultimate Losses and ALAE Limited by Basic Limits and MSL (A)+(B)
118,700
D. Company Subject Basic Limits Loss and ALAE Costs
153,900
E. Actual Experience Ratio (C)/(D):
AER = .7713
II. Exposure Component: Expected Experience Ratio:
EER = .9200
III. Credibility
.35
(M) = 1 + [AER – EER]/AER * Z
1 + [(0.7713 – 0.92)/0.92]*.35 = .943
Cumulative additive impact of schedule
+.04 - .07 - .08 + .03 = -.08
Adjusted premium = $75,800 * .943 * (1 - .08) = $65,761

Solutions to questions from the 2006 exam:
10. John's Car Wash is a new single-location business. It is purchasing commercial general liability
insurance. Which of the following rating methods might be used in calculating the premium?
1. Schedule Rating
2. Experience Rating. ER is not appropriate to use, since this is a new business, it has no actual experience
to modify application of manual rates.
3. Composite Rating. CR is not appropriate to use, since the business is small, its exposures are not
complex, and it has no experience to modify application of manual rates.
A. 1 only

B. 2 only

C. 3 only

D. 1 and 2 only

E. 1 and 3 only

“Schedule rating takes into consideration characteristics that are expected to affect losses and ALAE but that are
not reflected in past experience.
Experience rating uses an entity’s actual experience to modify manual rates (determined by the entity’s rating
group).
Composite rating simplifies the premium calculation for large, complex entities and, in some instances, allows the
entities’ experience to affect the premium developed from manual rates or to determine the rates regardless of
rating group.”
Answer A:

Exam 5, V1b

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Chapter 15 – Commercial Lines Rating Mechanisms
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Solutions to questions from the 2006 exam:
Question 49:
Calculate the experience (Credit)/Debit based on the ISO CGL Experience Rating Plan
Step 1: Write an equation to determine experience Credit/Debit: [AER – EER]/AER * Z, where:

AER = Actual Experience Ratio
= Projected Ultimate Losses and ALAE Limited by Basic Limits and MSL
/ Company Subject BL Loss and ALAE
EER = Expected Experience Ratio and Z = Credibility.
Step 2: Using the formula in Step 1, and the data given in the problem, determine what components need to be
solved for.
I. Experience Components
A. Reported Losses and ALAE Limited by Basic Limits and MSL
?
B. Expected Unreported Losses and ALAE Limited by Basic Limits and MSL 45,000
C. Projected Ultimate Losses and ALAE Limited by Basic Limits and MSL (A)+(B)
? +45,000
D. Company Subject Basic Limits Loss and ALAE Costs
250,000
E. Actual Experience Ratio (C)/(D):
AER = ?
II. Exposure Component: Expected Experience Ratio:
EER = 0.900
III. Credibility
0.60
Step 3: Compute the Reported Losses and ALAE Limited by Basic Limits and MSL.
Unlimited losses are first capped by the basic limit. ALAE is then added to these resulting losses, and then
capped by the MSL.
Losses
Unlimited
Limited to
Claim
Losses
100,000
ALAE
(1)
(2)
(3)
1
1,000
1,000
200
2
1,500
1,500
200
3
5,000
5,000
800
4
6,000
6,000
1,000
5
12,000
12,000
1,800
6
23,000
23,000
2,200
7
120,000
100,000
40,000
Losses & ALAE Limited by Basic Limits and MSL

Limited Losses and
ALAE capped
by the MLS
(4)=(2)+(3)
1,200
1,700
5,800
7,000
13,800
25,200
140,000
194,700

Step 4: Using the equation in Step 1, the result of Step 3 and the data given in the problem, solve for experience
(Credit)/Debit
AER = (Reported + Unreported) / Company Subject BL Loss and ALAE
= (194,700 + 45,000) / 250,000 = 0.9588
EER = 0.90
Experience (Credit)/Debit = (AER – EER)/EER x Z = (0.9588 – 0.9)/0.9 x 0.6 = 0.0392

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Solutions to questions from the 2007 exam:
52. (1.5 points) Calculate the experience rating credit/debit using the ISO CGL experience rating plan.
Show all work.
Step 1: Write an equation to determine the experience rating credit/debit
CD = [(AER-EER)/EER](Z), subject to the CD Limit, where CD = Experience Credit or Debit, AER
= Actual Experience Ratio, EER = Expected Experience Ratio and Z = Credibility.
Step 2: Compute the AER.
AER = (Projected Ult Loss and ALAE limited by BL and MSL)/(Company Subject BL Loss and ALAE)
= 250,000/150,000 = 1.6667
Step 3: Using the equation in Step 1, and the givens in the problem, solve for the experience rating credit/debit
CD = [(AER-EER)/EER](Z) =[(1.6667 - 0.90)/0.90] (0.40); CD = 0.3407
This implies a debit of +34.07%. But CD is limited by ± 25%, so min (34.07%, 25%) = 25%

Solutions to questions from the 2008 exam:
Model Solution – Question 39
1. Schedule Rating – Based on the characteristics of loss exposures of insured, underwriters assign debit or
credit for the policy. Actual experience is not considered.
2. Experience Rating – Based on insured’s experience, underwriters adjust the premium to be charged. Large
risks that have more credible experience get more credibility towards their experience whereas small risks get
less credibility towards their experience.
Model Solution – Question 41
a. Calculate the Actual Experience Ratio using the ISO Commercial GL Experience Rating Plan.
Step 1: Write an equation to determine the Actual Experience Ratio (AER)
 (Re ported L  ALAE Limited by BL & MSL )  E[Unreported L  ALAE Limited by BL & MSL ] 
AER 

Company Subject BL Losses and ALAE



Step 2: Write an equation and compute the Company Subject BL loss and ALAE
On page 166 of the 4th edition of the "Foundations of Casualty Actuarial Science", Tiller provides an example
of how to compute Company Subject BL Loss and ALAE
Company Subject BL L+ALAE = Prem/Ops Manual Premium * ELR * PAF1 * PAF2 * Detrend Factor
= 200,000 * 0.70 * 0.90 *.95 = 119,700
Note: Although we are only given one PAF, know that
 PAF1 adjusts current company basic limits loss and ALAE up to an occurrence level.
 PAF2 adjusts for the experience period being CM, reflecting the CM year.
Step 3: Using the equation in Step 1, the result from Step 2 and the data given in the problem, solve for the AER.
 $115,000  $35,000 
AER  
  1.253
$119,700



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Chapter 15 – Commercial Lines Rating Mechanisms
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Solutions to questions from the 2008 exam (continued):
Model Solution – Question 41 (continued):
b. Calculate the Experience Credit or Debit using the ISO Commercial GL Experience Rating Plan.
Step 1: Write an equation to determine the experience rating credit/debit
CD = [(AER-EER)/EER](Z), subject to the CD Limit, where CD = Experience Credit or Debit, AER
= Actual Experience Ratio, EER = Expected Experience Ratio and Z = Credibility.
Step 2: Using the equation in Step 1, the results from part a, and the data given in the problem, solve for the
experience rating credit/debit
CD = [(AER-EER)/EER](Z) =[(1.253 - 0.85)/0.85] (0.45); CD = 0.213
This implies a debit of +21.3%. But the CD is limited by ± 50%, so min (21.3%, 50%) = 21.3%

Solutions to questions from the 2009 exam:
Question 35. Calculate the final premium for a policy with a $100,000 deductible.

Premium =

Losses above Deductible + ALAE + Fixed Expense + Credit Risk + Risk Margin
(1.0 -Variable Expense Provision - Profit Provision)

Compute the following:

LER(100 K ) 

.45 (22, 000)  .20(35,000)  (.15  .15  .05)(100, 000)
51,900

 .4089
.45 (22, 000)  .2(35, 000)  .15(150, 000)  .15(250, 000)  .05(1, 000, 000) 126,900

Excess ratio  1.0  LER100 K  0.591
Losses = Full coverage premium * Expected ground up LER = 24,793 (.59) = 14,627.87
Thus, Excess loss = .591 (14,627.87) = 8,645.31, and
Losses in the deductible layer = 14,627.87 – 8,645.31= 5,982.56
Since the problem does not state, assume ALAE is not reduced by ded: Thus, .10(14,627.87) = 1,462.787
Incremental Fixed Expenses for processing a deductible = .04 * (5,982.56) = 239.302
Load for Uncollected Deductible payments = .01 * (5,982.56) = 59.826
Risk Load (assume it applies to losses from excess layer) = .05(8,645.31) = 432.27

Pr emium 

8, 645.31  1, 4 6 2.787  239.302  59.826  432.27
 14, 077.27
1- .08 - .10 - .05

Question: 43
No schedule rating credit should be given. The reduced loses has already been measured and would be
reflected in the experience rating. If the insured were to also be given a schedule credit then there would be
a double counting of credits.
Question 44
a. In retrospective rating, insurer will try to control losses incoming period because their loss experience will
be used to calculate their rate. In experience rating, they have less motivation to control losses, because
rate is based on past experience.
b. Experience rating is more stable because it uses experience over several periods and retrospective rating is
very likely to fluctuate because it is based on loss experience during a single policy period only.

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Chapter 15 – Commercial Lines Rating Mechanisms
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Solutions to questions from the 2010 exam:
Question 33

The NCCI ER Plan divides losses into primary and excess components.

The mod formula credibility weights primary and excess losses separately:

M 






Z P * AP  (1.0  Z P )  EP  Z e  Ae  (1.0  Z e )  Ee
, where
E

M = Experience Modification Factor
AP = Actual Primary Losses,
Ae = Actual Excess Losses
Ee = Expected Excess Losses
EP = Expected Primary Losses
E = Ep + Ee
ZP = Primary Credibility
Ze = Excess Credibility
Primary losses are capped at $10,000; Excess losses are the portion of each individual loss above $10,000.
Expected losses are separated into the primary and excess components based upon a D-ratio of .82
Ep = Payroll/$100 * expected loss cost per $100 payroll * D-ratio
w = Excess Loss Weighting Value = Ze/Zp.

Calculate Ap
Calculate Ae
Calculate Ep
Calculate Ee

= 4,000 * 32 + 8,000 * 15 + 10,000 * (3+1+2+1) = 318,000
= 15,000 * 3 + 16,000 * 1 + 23,000 * 2 + 42,000 * 1 -10,000 * (3+1+2+1) = 79,000
=10,920,000/100 * 2.9 * 0.82 = 259,678
= 10,920,000/100 * 2.9 * (1 – 0.82) = 57,002

There are 32 + 15 + 3 + 1 + 2 + 1 = 54 primary claims in the experience period. By the square root rule:
Zp = √(54/1082) = 0.223
Calculate Ze. Since w = Ze/Zp: Ze = Zp * w = 0.223 * .36 = 0.080
M
= [Zp * Ap + (1.0 – Zp) * Ep + Ze * Ae + (1.0 – Ze) * Ee] / E
= [0.223 * 318,000 + (1 – 0.223) * 259,678 + 0.080 * 79,000 + (1 – 0.080)*57,002] / (259,678 + 57,002)
= 1.0467
34. (1 point) An insurer has been tracking the claims experience of a very large construction company for the three
years the construction company has been insured by this insurer. The construction company will implement a
new safety program starting in the upcoming year.
a. (0.5 point) Determine whether the insurer should use experience rating, schedule rating, or both to rate the
construction company for the upcoming policy period. Briefly explain your answer.
b. (0.5 point) Assuming no additional changes, determine whether the insurer should use experience rating,
schedule rating, or both to rate the construction company five years from now. Briefly explain your answer.
Question 34
A. Both. Experience rating should be used to reflect the claims experience over the previous 3 years and
schedule rating to reflect the new safety program and the expected reduction in losses it will create.
B. Just experience rating. 5 years after the safety program has been implemented, the effects of the
program should be seen as experience and taken into account through experience rating.

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BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Solutions to questions from the 2011 exam:
a. (1 point) Calculate the experience rating modification factor.
b. (0.75 point) Calculate the standard premium.
c. (0.75 point) Calculate the final premium for the insured.
Initial comments
To account for differences in expense and loss levels for larger insureds, some WC insurers vary the expense
component for large risks, incorporate premium discounts or loss constants, or all of these.
Standard premium is a term defined by the National Council of Compensation Insurers (NCCI). In general, it is
premium before application of premium discounts and expense constants.
• The insured has a dedicated return-to-work program that makes it eligible for a 15% premium discount.
• The expense constant is $250.

M 

Z P * AP  (1.0  Z P )  EP  Z e  Ae  (1.0  Z e )  Ee
, where
E
M = Experience Modification Factor
AP = Actual Primary Losses,
EP = Expected Primary Losses
E = Ep + Ee
ZP = Primary Credibility

M

Ae = Actual Excess Losses
Ee = Expected Excess Losses
Ze = Excess Credibility

AP  w  Ae  (1.0  w)  Ee  B
, where
EB

B = Ballast Value, which is based on: ZP = E/(E +B); w = Excess Loss Weighting Value = Ze/Zp.
Question 19 – Model Solution 1
a. M = [ZpAp + (1 - Zp)Ep + ZeAe + (1 - Ze)Ee] / (Ep + Ee)
= [0.5(47,000) + 0.5(75,000) + 0.1(10,000) + 0.9(15,000)] / (75,000 + 15,000) = 0.8388 →Mod factor
b. Manual premium = 2,500,000 / 100 * 4.1+ 500,000 / 100 * 0.5 + 1,000,000 / 100 * 0.3 = 108,000
Standard premium = 108,000 * 0.8388 = 90,590
c. Final premium = 90,590 * (1 - 0.15) + 250 = 77,252; where 0.15 = discount and 250 = exp. constant
Question 19 – Model Solution 2 – part a
a. M = [Ap + w * Ae + (1 - w)Ee + B] / (E + B) = [47 + .2(10) + .8(15) + 90] / (90 + 90) = .8389
w = Ze/Zp,

• Excess credibility = 0.1; •Primary credibility = 0.5;

W= .1/.5 = .2

Zp = E/(E + B)→ .5 = 90,000 / (90,000 + B) →B = 90,000
E= Ep + Ee = 75,000 + 15,000 = 90,000

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Chapter 16 – Claims Made Ratemaking
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Sec
1
2
3
4
5
1

Description
Report Year Aggregation
Claims Made Policy Principles
Determining Rates
Coordinating Policies
Key Concepts

Pages
312 –314
314 – 317
317 - 317
317 - 320
321 - 321

Report Year Aggregation

312 –314

To understand how claims-made (CM) coverage and occurrence coverage differ, review the following
diagram that categorizes claims by the year reported and the report lag:
Note: Report lag refers to the time between the occurrence date and report date of a claim.
Report Year Aggregation

Report Year

0
2010
2011
2012
2013
2014
2015

L(2010,0)
L(2011,0)
L(2012,0)
L(2013,0)
L(2014,0)
L(2015,0)

Report year Lag
1
2
L(2010,1)
L(2011,1)
L(2012,1)
L(2013,1)
L(2014,1)
L(2015,1)

L(2010,2)
L(2011,2)
L(2012,2)
L(2013,2)
L(2014,2)
L(2015,2)

L(2010,3)
L(2011,3)
L(2012,3)
L(2013,3)
L(2014,3)
L(2015,3)

L(2010,4)
L(2011,4)
L(2012,4)
L(2013,4)
L(2014,4)
L(2015,4)

Examples:
 L(2010,0) represents a claim that occurs in 2010 and is reported in year 2010 (i.e. there is 0 time lag
between when the claim occurred and when it was reported).
 L(2012,2) represents a claim that is reported in 2012 after a report lag of two years (i.e. the claim
occurred in 2010).
In general, each:
 row corresponds to claims reported in a given year (i.e. the report year)
 column corresponds to claims that share the same reporting lag
 diagonal (top left to bottom right) represents claims that occurred in the same year (i.e. the same AY).
Occurrence policies
Occurrence policies cover claims that occur during the policy period regardless of when the claim is reported,
and are aggregated by accident year (i.e. each diagonal in the table). Example:
 An annual occurrence policy written on 1/1/2010 covers claims incurred during the policy period and
reported either during or after the policy period.
 This policy covers claims reported in 2010 with no report lag, claims reported in 2011 with a one-year
report lag, claims reported in 2012 with a two-year report lag, etc.
Thus, Occurrence Policy (2010) = L(2010,0)+ L(2011,1)+ L(2012,2)+ L(2013,3)+ L(2014,4).
Given a maximum report lag of N, the occurrence policy for year Y can be written as follows:
Occurrence Policy (Y ) 

 L(Y  i, i)
i 0

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Chapter 16 – Claims Made Ratemaking
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Claims-Made policies
The coverage trigger for a CM policy is the report date. A CM policy is represented by the entries in a row.
A CM policy written on 1/1/2010 covers all claims reported in 2010 (regardless of the report lag):
CM Policy (2010) = L(2010,0)+ L(2010,1)+ L(2010,2)+ L(2010,3)+ L(2010,4).
This can be written as: CM Policy (Y ) 

 L(Y , i)
i 0

Compare a 2010 CM policy (within the dotted box) to a 2010 occurrence policy (within the solid diagonal box).
Comparison of 2010 Claims-Made and Occurrence Policies

Report Year

Report Year Lag
2

2010

L(2010,0)

L(2010,1)

L(2010,2)

L(2010,3)

L(2010,4)

2011

L(2011,0 )

L(2011,1 )

L(2011,2 )

L(2011,3 )

L(2011,4 )

2012

L(2012,0)

L(2012,1)

L(2012,2)

L(2012,3)

L(2012,4)

2013

L(2013,0)

L(2013,1)

L(2013,2)

L(2013,3)

L(2013,4)

2014

L(2014,0)

L(2014,1)

L(2014,2

L(2014,3)

L(2014,4)

2015

L(2015,0)

L(2015,1)

L(2015,2

L(2015,3)

L(2015,4)

Claims Made Policy Principles

314 – 317

In “Rating Claims-Made Insurance Policies” (Marker and Mohl 1980), the authors list five principles of claims-made
policies that detail how pricing risk is reduced when compared to pricing occurrence policies.
1. A claims-made policy should always cost less than an occurrence policy as long as claim costs are
increasing.
2. If there is a sudden, unexpected change in the underlying trends, a claims-made policy priced based on the
prior trend will be closer to the correct price than an occurrence policy based on the prior trend.
3. If there is a sudden, unexpected shift in the reporting pattern, the cost of a mature claims-made policy (i.e. a
policy that covers claims reported during the policy period regardless of accident date) will be affected
relatively little, if at all, relative to the occurrence policy.
4. Claims-made policies incur no liability for IBNR, so the risk of reserve inadequacy is greatly reduced.
5. Investment income earned from claims-made policies is substantially less than under occurrence policies.

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To demonstrate these principles, assume the following:
 Exposure levels are constant.
 The average loss cost for RY 2010 is $1,000 (see below).
 Loss costs increase by 5% each report year (see below).
 An equal number of incurred claims are reported each year and all claims are reported within 5 years of
occurrence (i.e. 20% reported each year).
 Loss costs do not vary by report year lag. Any trends affecting settlement lag have been ignored.
 The data underlying these assumptions is shown in the table below:
Report
Year
2010
2011
2012
2013
2014
2015
2016
2017
2018

0
$200.00
$210.00
$220.50
$231.53
$243.10
$255.26
$268.02
$281.42
$295.49

Accident
Year
2010
2011
2012
2013
2014

Occurrence
Loss Costs
$1,105.13
$1,160.39
$1,218.41
$1,279.33
$1,343.29

Loss Costs by Report Year Lag
1
2
3
$200.00
$200.00
$200.00
$210.00
$210.00
$210.00
$220.50
$220.50
$220.50
$231.53
$231.53
$231.53
$243.10
$243.10
$243.10
$255.26
$255.26
$255.26
$268.02
$268.02
$268.02
$281.42
$281.42
$281.42
$295.49
$295.49
$295.49

4
$200.00
$210.00
$220.50
$231.53
$243.10
$255.26
$268.02
$281.42
$295.49

Claims Made
Loss Costs
$1,000.00
$1,050.00
$1,102.50
$1,157.65
$1,215.50
$1,276.30
$1,340.10
$1,407.10
$1,477.45

Using Loss Costs by Report Year Lag from above
=200 + 210 + 220.50 + 231.53 + 243.10

Principle 1
“A claims-made policy should always cost less than an occurrence policy as long as claim costs are increasing.”
Since there is a shorter period of time between coverage trigger and settlement date for CM policies, and since
short-term projections are more accurate than long-term ones, a CM policy should always cost less.
Example: An actuary pricing a 2011 CM policy only needs to project the ultimate cost of claims that will be
reported in that year.
An actuary pricing a 2011 occurrence policy has to project the ultimate value of claims that occur in
2011 and may not even be reported until 2015.

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Chapter 16 – Claims Made Ratemaking
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Principle 2
“If there is a sudden, unpredictable change in the underlying trends, the claims-made policy priced based on the
prior trend will be closer to the correct price than an occurrence policy based on the prior trend.”
The following table assumes actual loss cost trend by report year is 7% instead of 5%:
Unexpected Trend
Report
Year
0
2010
$200.00
2011
$214.00
2012
$228.98
2013
$245.01
2014
$262.16
2015
$280.51
2016
$300.15
2017
$321.16
2018
$343.64
Accident
Year
2010
2011
2012
2013
2014

Occurrence
Loss Costs
$1,150.15
$1,230.66
$1,316.81
$1,408.99
$1,507.62

Loss Costs by Report Year Lag
1
2
3
$200.00
$200.00
$200.00
$214.00
$214.00
$214.00
$228.98
$228.98
$228.98
$245.01
$245.01
$245.01
$262.16
$262.16
$262.16
$280.51
$280.51
$280.51
$300.15
$300.15
$300.15
$321.16
$321.16
$321.16
$343.64
$343.64
$343.64

4
$200.00
$214.00
$228.98
$245.01
$262.16
$280.51
$300.15
$321.16
$343.64

Claims Made
Loss Costs
$1,000.00
$1,070.00
$1,144.90
$1,225.05
$1,310.80
$1,402.55
$1,500.75
$1,605.80
$1,718.20

Using Loss Costs by Report Year Lag from above
=214 + 228.98 + 245.01 + 262.16 + 280.51



The unexpected increase in trend resulted in RY 2011 loss cost for the CM policy to be 1.9%
(=$1,070.00 / $1,050.00 – 1.0) higher than the original estimate in the prior Table.
 The unexpected trend increase resulted in an AY 2011 loss cost for the occurrence policy that is 6.1%
(=$1,230.66/1,160.39 -1.0) higher than the original estimate.
Since occurrence policies cover claims reported in the future and are more significantly affected by trend, an
error made in the trend selection has more of an impact on occurrence policies.
Principle 3
“If there is a sudden, unexpected shift in the reporting pattern, the cost of a mature CM policy will be affected
relatively little, if at all, relative to the occurrence policy.”
Example: Assume that 5% of the claims are reported one year later than expected, but all claims are reported
within five years (e.g. in 2010, $50 of the loss cost shifts from lag 0 to lag 1, $50 of the loss costs from
lag 1 shift to lag 2, and so on).
Since an equal amount of loss costs are shifting in and out of lag periods 1, 2, and 3, the only impact
is on the first and last lag periods.

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Unexpected Reporting Shift
Report
Year
0
2010
$150.00
2011
$157.50
2012
$165.38
2013
$173.64
2014
$182.33
2015
$191.44
2016
$201.02
2017
$211.07
2018
$221.62
Accident
Year
2010
2011
2012
2013
2014

Occurrence
Loss Costs
$1,115.91
$1,171.70
$1,230.30
$1,291.80
$1,356.40

4
$250.00
$262.50
$275.63
$289.41
$303.88
$319.07
$335.03
$351.78
$369.37

Total
All Lags
$1,000.00
$1,050.00
$1,102.51
$1,157.64
$1,215.51
$1,276.29
$1,340.11
$1,407.11
$1,477.46

Using Loss Costs by Report Year Lag from above
=150 + 210 + 220.50 + 231.53 + 303.88
=157.50 + 220.50 + 231.53 + 243.10 + 319.07

Conclusions:
 There is no impact on the loss cost estimates for the CM policies
 Estimates for the occurrence policies have changed (e.g. for AY 2011 loss cost estimate for the
occurrence policies has changed by 1% (= ($1,171.70 / $1,160.39) – 1.0).
Principle 4
“Claims-made policies incur no liability for IBNR, so the risk of reserve inadequacy is greatly reduced.”
 When pricing occurrence policies, reserves for incurred but not reported (pure IBNR) claims and incurred
but not enough reported (IBNER) must be established.
 CM policies have no pure IBNR component. Only the IBNER reserve has to be determined and so the risk
of reserve inadequacy is greatly reduced.
Principle 5
“The investment income earned from claims-made policies is substantially less than under occurrence policies.”
Insurers are required to hold unearned premium reserves, case reserves, IBNR reserves, and IBNER reserves
which are invested over a period of time.
Since the CM policy has a shortened period of time between collection of premium and payment of claim, funds
are invested for a shorter time and less investment income is earned relative to an occurrence policy.
This principle has pricing risk implications for CM policies (e.g. when determining the target UW profit provision,
the actuary should take into account both reduced investment income as well as reduced pricing risk).

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Chapter 16 – Claims Made Ratemaking
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
3

Determining Rates

317 - 317

Once expected loss costs are determined, rates are derived using techniques previously discussed.

Coordinating Policies

317 - 320

Insureds converting from one policy type to the other should be aware of coverage overlaps or gaps, since
occurrence and CM policies have different coverage triggers.
 Consider an insured that had an occurrence policy in 2010 and switches to a CM policy starting in
2011.
 Notice the overlapping coverage between the occurrence policy and the claims-made policy.
Comparison of Several Claims-Made and Occurrence Policies
Report
Report Year Lag
Year
0
1
2
3
L(2010,0)
L(2010,1)
L(2010,2)
L(2010,3)
2010

4
L(2010,4)

2011

L(2011,0)

L(2011,1)

L(2011,2)

L(2011,3)

L(2011,4)

2012

L(2012,0)

L(2012,1)

L(2012,2)

L(2012,3)

L(2012,4)

2013

L(2013,0)

L(2013,1)

L(2013,2)

L(2013,3)

L(2013,4)

2014

L(2014,0)

L(2014,1)

L(2014,2)

L(2014,3)

L(2014,4)

2015

L(2015,0)

L(2015,1)

L(2015,2)

L(2015,3)

L(2015,4)

Claims-made = within dotted rectangle Occurrence Policy = shaded

Retroactive Date
CM policies have a retroactive date (only claims that occur on or after the retroactive date are covered).
To obtain complete coverage without overlap, the retroactive date should coordinate with the expiration of the last
occurrence policy.
By applying the retroactive date to the table above, the results are shown in the table below.
 The insured can purchase a 1st year CM policy in 2011 with a retroactive date of 1/1/2011.
The 1st year CM policy will only provide coverage for claims that occurred on or after 1/1/2011, and were
reported in 2011 (i.e. L(2011,0)).
 A 2nd year CM policy with a retroactive date of 1/1/2011 will cover L(2012,0) and L(2012,1).
 This continues until a mature CM policy is issued in 2015.
Coordinating
Report
Year
2010
2011
2012
2013
2014
2015

the Switch from Occurrence to Claims-Made Policy
Report Year Lag
0
1
2
3
L(2010,0)
L(2010,1)
L(2010,2)
L(2010,3)
L(2011,0)
L(2011,1)
L(2011,2)
L(2011,3)
L(2012,0)
L(2012,1)
L(2012,2)
L(2012,3)
L(2013,0)
L(2013,1)
L(2013,2)
L(2013,3)
L(2014,0)
L(2014,1)
L(2014,2)
L(2014,3)
L(2015,0)
L(2015,1)
L(2015,2)
L(2015,3)

Claims-made = within dotted rectangle

Exam 5, V1b

4
L(2010,4)
L(2011,4)
L(2012,4)
L(2013,4)
L(2014,4)
L(2015,4)

Occurrence Policy = shaded

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Chapter 16 – Claims Made Ratemaking
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Rating CM policies includes using a step factor to recognize the growth in exposure for each successive CM
policy during the transition.
 The step factor is a % of the mature claims-made rate.
 Computing step factors requires evaluating the expected reporting lag and factors affecting claim costs
during the lag time and leads to a distribution of costs to each of the lags of a mature claims-made policy.
Example: Consider the 2015 mature claims-made policy from 2015
 Loss estimates for L(2015,0), L(2015,1), L(2015,2), L(2015,3) and L(2015,4) expressed as a ratio to the
total losses for RY 2015 can be used to determine the step factors.
 The cumulative values of these ratios, by year of lag, are used to determine the step structure.
 The table below shows a potential step factor structure for a CM policy.
Claims-Made Year

Step Factor

First
Second
Third
Fourth
Fifth or More

40%
70%
85%
95%
100%

i. 40% of the of the costs of a mature CM policy come from claims that occurred and were reported
during that year.
ii. 70% of the costs come from claims that occurred during that year and one year prior (and the
progression continues until the mature stage is reached).
Example: An insured switching from a CM policy to an occurrence policy in 2011.

Report Year

Report Year Lag
2

2010

L(2010,0) L(2010,1 ) L(2010,2)

L(2010,3)

2011
2012
2013
2014
2015

L(2011,0)
L(2012,0)
L(2013,0)
L(2014,0)
L(2015,0)

L(2011,3)
L(2012,3)
L(2013,3)
L(2014,3)
L(2015,3)

L(2011,4)
L(2012,4)
L(2013,4)
L(2014,4)
L(2015,4)

L(2011,1)
L(2012,1)
L(2013,1)
L(2014,1)
L(2015,1)

Claims-made = within dotted rectangle

L(2011,2)
L(2012,2)
L(2013,2)
L(2014,2)
L(2015,2)

Occurrence Policy Coverage = shaded

This causes a coverage gap, since there is no coverage for claims that occurred before 2011, but were
not reported until after the expiration of the last CM policy.
Thus, insurers offer an extended reporting endorsement (or tail coverage) that covers claims that
occurred but were not reported before the expiration of the last CM policy.
Switching from Claims-Made to Occurrence Policy with Tail Coverage

Report Year

Report Year Lag
2

2010

L(2010,0) L(2010,1 ) L(2010,2)

L(2010,3)

2011
2012
2013
2014
2015

L(2011,0)
L(2012,0)
L(2013,0)
L(2014,0)
L(2015,0)

L(2011,3)
L(2012,3)
L(2013,3)
L(2014,3)
L(2015,3)

L(2011,4)
L(2012,4)
L(2013,4)
L(2014,4)
L(2015,4)

L(2011,1)
L(2012,1)
L(2013,1)
L(2014,1)
L(2015,1)

L(2011,2)
L(2012,2)
L(2013,2)
L(2014,2)
L(2015,2)

CM = within dotted rectangle
Tail Coverage = within the dotted triangle
Occurrence Policy Coverage = shaded

Exam 5, V1b

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Chapter 16 – Claims Made Ratemaking
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
A gap in coverage can also occur in the case of retirement.
 If physicians with CM policies retire, they need protection against claims that are reported after the
expiration of the last CM policy.
 This protection is given by a tail policy that covers losses occurring during the period for which CM
coverage was in force and that are reported after the insured’s last CM policy expires.

Key Concepts

321 - 321

1. Rationale for claims-made coverage
2. Aggregating losses by report year and report lag
3. Coverage triggers for claims-made coverage
5. Coordinating coverage
a. Retroactive date
b. First- and second-year claims-made policies
c. Mature claims-made policies
d. Extended reporting endorsement or tail coverage

Exam 5, V1b

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Chapter 16 – Claims Made Ratemaking
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Questions from the 1988 exam
26. (1 point) According to Werner and Modlin, "Basic Ratemaking, which of the following is not one of the
principles of claims-made (C-M) ratemaking?
A.
B.
C.
D.
E.

Substantially less investment income is earned on C-M policies than under occurrence policies.
Sudden unexpected shifts in the reporting pattern will have less of an impact on the cost of mature C-M
coverage than on the cost of occurrence coverage.
C-M policies have less risk of case reserve inadequacies than do occurrence policies.
A C-M policy should always cost less than an occurrence policy as long as pure premiums are
increasing.
Whenever there is a sudden, unpredictable increase or decrease in the underlying trend, C-M policies
priced on the basis of the prior trend will be closer to the correct price than occurrence policies priced the
same way.

58. (a) (2 points)
An insurance company will give policyholders a choice of purchasing an occurrence policy or a claims-made
policy beginning 1/1/88. From reviewing the company's experience, you know that all losses are reported
within 4 years of occurrence, that the losses reported in 1986 totaled $400 and that these losses were
produced in equal proportions from accidents that occurred between 1983 and 1986.
(a) (1 point) Assume that there will be no change in the reporting pattern of claims and that inflation will be
10% per year. Ignoring investment income and risk, determine the multiplier that should be
applied to the adequate rate for an occurrence policy to get the rate for:
1) a first year claims-made policy.
2) a mature claims-made policy.
(b) (1 point) As a current occurrence coverage policyholder, you must decide whether to purchase the
claims-made coverage policy. You plan to retire in 2 years and the company assures you that
they will sell you tail coverage at that time.
Which coverage should you purchase? Assume that the conditions outlined above apply and that your
decision will be based solely on the cost of the total coverage (i.e. 2 years of occurrence policies versus a first
year claims made policy, a second year claims made policy, and tail coverage).
Assume all prices quoted are based on the same expected loss ratio. Explain the reasons for your decision.

59.
Using Werner and Modlin, "Basic Ratemaking,” Made Insurance Policy" and given the fact that
L0,0 = L1,0=L2,0 = L3,0 =L4,0 available for ratemaking (where Li j represents the pure premium
for accident year lag i and report year j)
(a) (1 point) Demonstrate and identify the first principle of claims made ratemaking by pricing both occurrence
and claims made policies (ignore expenses) effective at the beginning of year 1 assuming losses
will increase $50 for each report year for each lag.
(b) (1 point) Demonstrate and identify the second principle of claims made ratemaking by pricing both
occurrence and claims made policies (ignore expenses) effective at the beginning of year 1
assuming the increase in losses was underestimated by $15 per year per lag (i.e. losses are
actually increasing at $65 per reported year).

Exam 5, V1b

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Chapter 16 – Claims Made Ratemaking
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Questions from the 1989 exam
20. From Werner and Modlin, "Basic Ratemaking”, which of the following are true?
1. A claims-made policy should always cost less than an occurrence policy.
2. The investment income earned from claims-made policies is about the same as is earned from occurrence
policies.
3. A sudden unexpected shift in the reporting pattern will have relatively little effect on the cost of a mature
claims-made policy relative to the effect on the cost of an occurrence policy.
A. 1

B. 2

C. 3

D. 2, 3

E. 1, 2, 3.

55. (3 points)
You are acting as a consultant for a doctor beginning a private practice in an obscure specialty. The doctor wants
the lowest-cost malpractice coverage available for the first two years of practice, in order to pay off a substantial
loan debt. Beginning in the third year, the doctor would prefer to pay a higher cost for a policy that would cover
any claims that may emerge from that year's practice.
After some thought, you explain to the doctor that at today's price levels you can recommend policies that would
have a relatively low first year cost, a higher second year cost, and a much higher third year costs. Fourth and
subsequent years costs would be lower than year three.
a. (1 point) What type of policies do you recommend for years 1, 2, and 3?
b. (2 points) The doctor is confused by your comment on the third year premiums. Use the report year/lag
diagram approach outlined in Werner and Modlin, "Basic Ratemaking” , to illustrate your
recommendations by labeling the sections corresponding to the policies you recommended in
part (a). Assume that all claims would be settled by the end of the third year following their
occurrence.

Questions from the 1990 exam
23. You are pricing a claims-made policy for the 1991 year using occurrence year 1990 data. Under existing
conditions, you estimate that 1990 occurrence year losses will be reported in the following manner:
Report
Year
1990
1991
1992
1993
1994

1990 Occurrence Year
Percentage of Losses Reported
40%
20%
20%
10%
10%

If loss costs are increasing at an annual rate of 10%, what is the 1991 mature claims-made multiple of the 1991
occurrence pure premium?
A. < .88

Exam 5, V1b

B. > 88 but <.92

C. > .92 but <.96

D.> 96 but < 1.00

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Chapter 16 – Claims Made Ratemaking
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Questions from the 1990 exam
24. According to Werner and Modlin, "Basic Ratemaking”, which of the following statements about the use of
occurrence data to price claims-made coverage is FALSE?
A. In order to distribute the mature pure premium to lags, exponential regression is preferred to linear regression
through the origin.
B. An adjustment to claim reporting patterns is needed since it is assumed that claims will be reported sooner
under claims-made coverage.
C. Under claims-made coverage there are assumed to be additional incidents reported that would not have been
reported under an occurrence policy.
D. Expenses should be separated into their fixed and variable portions and the final rate calculated accordingly.
E. None of the above.

Questions from the 1991 exam
For the next two questions use the techniques described by Werner and Modlin, "Basic Ratemaking” and
using the following data:

L
A
G

0
1
2
3
4

1
L0,1
L1,1
L2,1
L3,1
L4,1

2
L0,2
L1,2
L2,2
L3,2
L4,2

3
L0,3
L1,3
L2,3
L3,3
L4,3

Report Year
4
L0,4
L1,4
L2,4
L3,4
L4,4

5
L0,5
L1,5
L2,5
L3,5
L4,5

6
L0,6
L1,6
L2,6
L3,6
L4,6

7
L0,7
L1,7
L2,7
L3,7
L4,7

27. Which of the following expressions defines a second-year claims made policy written at the beginning
of year 4?
A.
B.
C.
D.
E.

L0,3 + L1,3
L0,4 + L1,4
L1,4 + L2,5 + L3,6 + L4,7
L0,3 + L1,4 + L2,5 + L3,6 +L4,7
None of A, B, C, or D.

28. Which of the following expressions defines a tail policy for an insured at the end of year 3 who had previously
purchased three consecutive claims-made policies, the first in year 1?
A. L0,4 + L0,5 + L0,6 + L0,7
B. L1,3 + L2,3 + L3,3 + L4,3
C. L0,3 + L1,4 + L2,5 + L3,6 +L4,7
D. L1,4 + L2,4 + L3,4 + L2,5 + L3,5 +L4,5
E. None of A, B, C, or D.

Questions from the 1992 exam
1. According to Werner and Modlin, "Basic Ratemaking”, which of the following are true?
1. The cost of mature claims-made coverage is less susceptible to changes in the reporting pattern than
occurrence coverage.
2. Occurrence pricing is less affected by sudden, unpredictable changes in trend than claims-made pricing.
3. While claims costs are increasing, occurrence policies should always cost more than claims-made.
A. 1

Exam 5, V1b

B. 2

C. 1, 2

D. 1, 3

E. 1, 2, 3

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Chapter 16 – Claims Made Ratemaking
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Questions from the 1992 exam (continued):
The following information should be used to answer questions #29 and #30.
Questionable Insurance Company specializes in claims-made accountant's malpractice insurance. Questionable
has been writing this line for ten years and has never experienced a claim with a report lag greater than four
years. Fully developed and credible 1990 reporting and exposures show the following pure premiums:
1990 Pure Premium
Lag
Per Accountant
0
$1,000
1
$2,000
2
$3,000
3
$2,000
4
$1,000
The overall trend is 10% annually.
29. Compute the pure premium for an individual accountant with a policy written 1/1/92 with a 1/1/91
retroactive date.
A. < $3,700 B. > $3,700 but < $3,800 C. > $3,800 but < $3,900 D. > $3,900 but < $4,000 E. > $4,000
30. Addem and Up is a firm of 10 accountants, each of whom has been with the firm at least five years. Compute
the pure premium for Addem and Up for a mature claims-made policy written 1/1/92.
A. < $100,000 B. > $100,000 but < $105,000 C. > $105, 000 but <$110,000 D. > $110,000 but < $115, 000
E. > $115, 000
60.

(2 points)
You are given the following incurred loss information evaluated at 12/31/87 and presented in a manner
consistent with the Werner and Modlin, "Basic Ratemaking”.
Report Year
1982
1983
1984
1985 1986 1987
1981
0
10
30
25
20
40
50
30
1
10
10
30
25
20
40
50
L
2
5
20
15
30
25
20
40
A
3
5
10
20
14
36
30
24
G
4
5
10
20
15
0
0
0
5
0
10
15
10
20
0
0
Create the corresponding cumulative incurred loss triangle by Accident Year and Report Period. Put your
answer in the following format:

Reporting Period (through months)
Accident
24
36
48
60
Year
12
1981
1982
1983
1984
1985
1986
1987

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Chapter 16 – Claims Made Ratemaking
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Question from the 1993 exam
47. (3 points)
When Charlie Frye entered the actuarial profession January 1, 1962 he purchased a professional liability
policy providing coverage on an occurrence basis. Charlie renewed this policy until 1989. In 1989 he
switched to a "claims made policy" which he renewed until he retired at the end of 1992. His loss history is
as follows:

L
A
G

0
1
2
3
4

1988
2,000
950
450
215
0

1989
2,100
1,000
475
225
0

Report Year
1990
1991
2,200
2,300
1,050
1,100
500
525
238
250
0
o

1992
2,400
1,150
550
263
0

a. (2 points) What is Charlie's pure premium for each of his policies carried in the years 1988 through 1992?
b. (1 point) Assuming Charlie's historical reporting patterns continue, what is his anticipated pure premium for
a tail policy purchased January 1, 1993, covering his entire tail exposure?

Questions from the 1994 exam
None

Question from the 1995 exam
39. Professional Services, Inc., has the following expected General Liability loss experience over seven report
years:

Lag
0
1
2
3
4

1991
$25,000
$20,000
$15,000
$10,000
$0

General Liability Expected Losses
Report Years
1992
1993
1994
$30,000
$35,000
$40,000
$25,000
$28,000
$35,000
$20,000
$25,000
$32,000
$15,000
$20,000
$21,000
$0
$0
$0

1995
$45,000
$42,000
$40,000
$25,000
$0

1996
$50,000
$46,000
$44,000
$28,000
$0

1997
$55,000
$50,000
$48,000
$30,000
$0

Professional Services purchased an occurrence policy at the beginning of 1991 and then switched to a
claims-made policy for 1992 and 1993. At the beginning of 1994 they reverted to an occurrence policy.
Using methods described in Werner and Modlin, "Basic Ratemaking”.
(a) (1 point) Calculate the expected losses for the occurrence policy purchased in 1991.
(b) (1 point) Calculate the expected losses for the 1993 claims-made policy.
(c) (1 point) Calculate the expected losses for a tail policy purchased at the end of the 1993 CM policy.
(d) (1 point) Briefly explain why claims-made rates are both more accurate and more responsive to
changing conditions than are rates for an occurrence policy.

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Chapter 16 – Claims Made Ratemaking
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Question from the 1996 exam
Question 12. You are given the following data:
General Liability Expected Pure Premiums
Report Year
Lag
1995
1996
19970
1998
1999
2000
2001
0
100
102
104
106
108
1101
113
1
500
600
720
864
1037
1244
1493
2
50
55
61
67
74
811
89
3
30
33
36
40
44
48
53
4
20
22
24
27
29
32
35
5
0
0
0
0
0
0
Using the approach described by Werner and Modlin, "Basic Ratemaking”, calculate the difference in
expected annual pure premiums between an occurrence policy purchased 1/1/96 and a mature claimsmade policy purchased 1/1/96. In what range does the difference fall?
A.

< $80

B. > $80, but < $100

C. > $100, but < $120

> $120, but < $140

E. > $140

Question from the 1997 exam
5. Based on Werner and Modlin, "Basic Ratemaking”, which of the following are true?
1. Occurrence policies have less risk of reserve inadequacy than do claims-made policies.
2. Occurrence policies will generate more investment income than will claims-made policies.
3. An occurrence policy should cost more than a claims-made policy, if claim costs are increasing at a rate
greater than investment returns.
A. 1

B. 3

C. 1, 2

D. 2, 3

E. 1, 2, 3

33. (3 points)You are given: Medical Malpractice data
Estimated
Estimated
Estimated
Estimated
Estimated
Actual 1994
1995 Loss
1996 Loss
1997 Loss
1998 Loss
1999 Loss
Loss Costs
Costs
Costs
Costs
Costs
Costs
0
1,000
1,050
1,103
1,158
1,216
1,276
1
1,000
1,050
1,103
1,158
1,216
1,276
2
1,000
1,050
1,103
1,158
1,216
1,276
3
500
525
551
579
608
638
4
0
0
0
0
0
0
Your latest rate changes for both occurrence and claims-made policies, effective 1/1/96, were based on
1994 experience and followed the methodology described by Werner and Modlin, "Basic Ratemaking”. You
assumed that loss costs would increase 5% annually. You have now learned that inflation in loss costs has
been 10% annually, since 1994, and you expect this pattern to continue.
Lag

A. (1 point)
B. (1 point)
C. (1 point)

Exam 5, V1b

Determine the loss cost inadequacy, as a percentage of the loss costs assumed in the
rates, for a second-year claims-made policy effective 1/1/96.
Determine the loss cost inadequacy, as a percentage of the loss costs assumed in the
rates, for an occurrence policy effective 1/1/96.
Determine the loss cost inadequacy, as a percentage of the loss costs assumed in the
rates, for a claims-made tail policy effective 1/1/97 following a second-year claims-made
policy. Assume that an occurrence policy was purchased 1/1/94 and that claims-made
policies were purchased 1/1/95 and 1/1/96.

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Chapter 16 – Claims Made Ratemaking
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Questions from the 1998 exam
15. According to Werner and Modlin, "Basic Ratemaking”, which of the following are true?
1. The confidence interval about the projected losses for a claims-made policy is generally narrower than
for an occurrence policy priced at the same time.
2. The longer the settlement lag, the greater will be the difference in investment income between claimsmade and occurrence policies.
3. A claims-made policy should always cost less than or equal to an occurrence policy.
A. 1

B. 2

C. 3

D. 1, 2

E. 2, 3

35. (2 points) You are given the following incurred loss experience for the Leaning Tower Consulting Firm.

Lag
0
1
2
3

1992
1,000
600
500
0

1993
1,300
800
700
0

Report Year
1994
1995
1,400
1,100
900
1,000
800
400
0
0

1996
1,800
1,200
500
0

1997
1,900
1,300
700
0

Leaning Tower Consulting purchased the following varying types of policies to cover their liability exposure:
 Up to and including 1992 they purchased occurrence policies.
 In 1993 and 1994 they purchased claims-made coverage with a
 1/1/93 retroactive date.
 In 1995 they switched back to occurrence coverage.
 In 1995 they also bought tail coverage in the form of a single payment reporting endorsement.
Calculate the loss incurred under each of the following policies:
a.
b.
c.
e.

1992
1993
1994
1995

Occurrence
Claims-Made
Claims-Made
Tail Coverage

Questions from the 1999 exam
Question 31. (2 points) Based on Werner and Modlin, "Basic Ratemaking”, and the information shown below,
determine the total premium for a third-year claims-made policy.
Mature claims-made pure premium
Commission
Profit
Taxes
Variable general expense
Fixed general expense
Unallocated loss adjustment expense

$1,000
12%
-3%
4%
6%
$75
10% of loss

Annual Lag Factors
Lag
Factor
0
.30
1
.25
2
.20
3
.15
4+
.10

Exam 5, V1b

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Chapter 16 – Claims Made Ratemaking
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Questions from the 1999 exam
Question 32. (3 points) Based on Werner and Modlin, "Basic Ratemaking”, and the information shown below,
determine the undiscounted pure premium for a mature claims-made policy effective 1/1/98.
 60% of all claims are lag 0
30% of all claims are lag 1
10% of all claims are lag 2


For each report year, lag 1 claims settle for twice the value of lag 0 claims, and lag 2 claims
settle for three times the value of lag 1 claims.



Report year severity is increasing 10% per year across all lags.



An occurrence policy effective 1/1/96 has an undiscounted pure premium of $1,000.



Claims are uniformly distributed throughout the year and frequency has been constant during
the experience period.

Questions from the 2001 exam
Question 46. (2 points) Werner and Modlin, "Basic Ratemaking”, discusses five principles of claimsmade ratemaking. In each of the subparts of this question, one of these five principles is
listed. For each of the stated principles, briefly describe why it is true.
a. (½ point) A claims-made policy should always cost less than an occurrence policy, as long as
claims costs are increasing.
b. (½ point) Whenever there is a sudden, unpredictable change in underlying trend, claims-made
policies priced on the basis of the prior trend will be closer to the correct price than occurrence
policies priced in the same manner.
c. (½ point) Whenever there is a sudden unexpected shift in the reporting pattern, the cost of mature
claims-made coverage will be affected very little, if at all, relative to occurrence coverage.
d. (½ point) The investment income earned from claims-made policies is substantially less than
under occurrence policies.

Questions from the 2002 exam
43. (3 points) Based on Werner and Modlin, "Basic Ratemaking”, and the following information, calculate
the dollars of "pure" IBNR reserve inadequacy for a company writing occurrence policies for five years.
Show all work.
Losses of $1,500 reported in the last year were produced in equal proportions from occurrences
in the last five years.
Losses are forecast to increase at a rate of $10 per year.
Actual results show an unexpected shift of $5/per year/per lag towards later reportings.

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Chapter 16 – Claims Made Ratemaking
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Questions from the 2003 exam
22. (3 points)
a. (1 point) Define a coverage trigger.
b. (1 point) State how the coverage trigger for claims-made forms differs from the coverage trigger
for occurrence forms.
c. (0.5 point) A dentist begins his practice on January 1, 2003 and retires three years later. He buys
the following professional liability insurance policies:
• An occurrence policy to cover his first year of practice
• A 1st - year claims-made policy for 2004
• A 2nd - year claims-made policy for 2005
• A tail policy at the end of 2005
A loss that occurred in 2004 was not reported until 2006. State which policy, if any, covers the
loss and explain why.
d. (0.5 point) Assume that the dentist in part c. above instead purchased three occurrence policies,
one for each of his first three years of practice. State which policy, if any, would cover the loss
described in part c. above and explain why.
29. (3 points) Given the information below, calculate the premium for an occurrence policy written in Year 1.
Show all work.
Loss Reporting Pattern
Year
Percent Reported
1
50%
2
80%
3
95%
4
100%


Mature claims-made pure premium for Year 1 = $600



Loss trend = 5%



Fixed expense per policy = $150



Commissions = 12%



Premium taxes = 5%



Loss adjustment expense as percent of loss = 8%



Profit provision = 3%

Questions from the 2004 exam
12. Which of the following statements are true regarding claims-made ratemaking?
1. The investment income earned under claims-made policies is substantially less than the investment
income earned under occurrence policies.
2. An occurrence policy will generally cost less than a claims-made policy.
3. Claims-made policies incur no liability for IBNR claims.
A. 1 only

Exam 5, V1b

B. 3 only

C. 1 and 2 only

D. 1 and 3 only

Page 274

E. 2 and 3 only

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Chapter 16 – Claims Made Ratemaking
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Questions from the 2005 exam
15. A claim occurred in May 2001 and was reported in September 2003. Which of the following would cover
this claim?
1. A one-year occurrence policy effective January 1, 2003
2. A second-year claims-made policy effective January 1, 2003
3. Tail coverage effective January 1, 2003 for a physician retiring after 10 years of practice covered by
claims-made coverage
A. 2 only
B. 3 only
C. 1 and 3 only
D. 2 and 3 only
E. None of 1, 2, or 3

Questions from the 2007 exam
13. Which of the following statements are true regarding claims-made ratemaking?
1. The investment income earned under claims-made policies is substantially less than the
investment income earned under occurrence policies.
2. A claims-made policy should always cost less than an occurrence policy, as long as claim costs
are increasing.
3. Claims-made policies incur no liability for IBNR claims.
A. 1 only
B. 3 only
C. 1 and 2 only
D. 2 and 3 only
E. 1, 2, and 3

Questions from the 2008 exam
20. (2.0 points) Using the techniques contained in Werner and Modlin, "Basic Ratemaking”, draw and label a
diagram representing the following five different types of policies written in 2000 through 2004. Assume all
claims are reported within four years.
a. Occurrence policy written in year 2000
b. 1st year claims-made policy written in 2001
c. 2nd year claims-made policy written in 2002
d. 3rd year claims-made policy written in 2003
e. Tail policy written in 2004
21. (1.0 point) Explain the following terms as they pertain to a claims-made policy.
a. Retroactive Date
b. Extended Reporting Period

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Chapter 16 – Claims Made Ratemaking
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Questions from the 2009 exam
28. (2 points) An insured has purchased the following policies:
Policy
Effective Date
Term
Policy Type
Retroactive Date
January 1, 2004
1 Year
Occurrence
N/A
January 1, 2005
1 Year
Occurrence
N/A
January 1, 2006
1 Year
First-Year Claims Made
January 1, 2006
January 1, 2007
1 Year
Second-Year Claims Made
January 1, 2006
January 1, 2008
1 Year
Third-Year Claims Made
January 1, 2006
A tail policy is also purchased on January 1, 2009 to cover any losses that occurred while the claimsmade policies were in effect but had not been reported as of December 31, 2008.
Draw and label a diagram that shows what losses each policy covers, based on when the losses
occurred and when they were reported, assuming all claims are reported within 3 years of
occurrence.

Questions from the 2010 exam
22. (2 points)
a. (0.5 point) Explain the major difference between claims-made and occurrence policies.
b. (0.5 point) Explain how claims-made coverage reduces pricing risk.
c. (0.5 point) Explain how claims-made coverage reduces reserving risk.
d. (0.5 point) Explain the purpose of an extended reporting endorsement (or tail policy).

Questions from the 2012 exam
8. (2 points) A physician maintained medical malpractice coverage with occurrence policies through 2011.
Effective January 1, 2012, the physician switched to claims-made coverage. The physician will retire
on December 31, 2014. The last claims-made policy to be issued prior to the physician's retirement
date will be effective from January 1, 2014 to December 31, 2014.
The following table contains anticipated loss costs used to evaluate pricing for the physician's policy.
All claims are reported within 3 years.
Report Year Lag
Report Year
0
1
2
3
2011
$350 $300 $250 $100
2012
$368 $315 $263 $105
2013
$386 $331 $276 $110
2014
$405 $347 $290 $116
2015
$426 $365 $304 $122
2016
$447 $383 $319 $128
2017
$469 $402 $335 $134
a. (0.5 point) Briefly describe two advantages that claims-made coverage has over occurrence
coverage for a medical malpractice insurer.
b. (1 point) Calculate the loss costs associated with a 2011 occurrence policy and the loss costs associated
with a mature 2012 claims-made policy. Briefly describe the overlap in loss costs between the two and the
mechanism used to prevent it.
c. (0.5 point) Identify the loss costs in the table above for which the physician would still have exposure at the
time of retirement, and the coverage that the physician would need to purchase to transfer that exposure to
the insurer.

Exam 5, V1b

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Chapter 16 – Claims Made Ratemaking
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Solutions to questions from the 1988 exam
Question 26.
Answer C.

Question 58.
Report Year
Lag
0
1
2
3

1986
100
100
100
100

1987

1988

1989

100*1.1
100*1.1
100*1.1

100*(1.1)2
100*(1.1)2

100*(1.1)3

Occ. Price
Mature CM

464.1
400

1990

1991

1992

510.51

561.56

617.71

400*(1.1) = 484

(a)
A 1st year claims made policy, purchased at the beginning of 1988, costs $100 * (1.10)2 = 121.
An occurrence policy, purchased at the beginning of 1988, costs
$100 [(1.1)2 + (1.1)3 + (1.1)4 + (1.1)5 = $561.56
The multiplier that should be applied to the adequate rate for an occurrence policy to get a rate for a first year
claims made policy is $121 / $561.56= .215.
A mature claims made policy costs $100 * 4 * (1.10)2 = 484.
The multiplier that should be applied to the adequate rate for an occurrence policy to get a rate
for a mature year claims made policy is 484 / 561.56 = .861.
(b)
The cost of 2 occurrence policies (after 1/1/88) = 561.56 + 617.71 = 1179.28.
The cost of a 1st year claims made, a 2nd year CM policy and tail coverage
= 100(1.1)2 + 2*100*(1.1)3 + 2*100*(1.1)4 + 2*100*(1.1)5 + 100*(1.1)6 = 1179.28.
Therefore, each form of coverage is equal in price.

Question 59.
See the example provided in the summary of this article.

Exam 5, V1b

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Chapter 16 – Claims Made Ratemaking
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Solutions to questions from the 1989 exam
Question 20.
1. F. The statement is true only when claims costs are rising.
2. F.
3. T..

Answer C.

Question 55.
a. The recommended coverage would consist of the purchase of a 1st year CM policy, a 2nd year CM
policy and a combined tail policy (to cover the remaining exposure under the 1st 2 policies) and an
occurrence policy, thereafter.
b.

Lag

/
0
1
2
3
Occ. Price
Mature CM

1
L0,1

2
L0,2
L1,2

Report Year
4

3
L0,3
L1,3
L2,3

L1,4
L2,4
L3,4

L2,5
L3,5

L3,6

L0,1 are the losses generated by a 1st year CM policy.
L0,2 and L1,2 are the losses generated by a 2nd year CM policy.
The sum of the remaining cells comprise the combined tail and occurrence coverage purchased in the third
year.
Fourth and subsequent years costs would be lower than year three since only occurrence coverage would
be purchased.

Solutions to questions from the 1990 exam
Question 23.
Solution: Assume that the losses reported in 1990 totaled $1 and that these losses emerged according to the
reporting pattern above from accidents that occurred between 1987 and 1990.

Lag

/
0
1
2
3
4
Occ. Price
Mature CM

1990
.4
.2
.2
.1
.1

1991
.4*(1.1)1
.2
.2*(1.1)-1
.1*(1.1)-2
.1*(1.1)-3

1992

Report Year
1993
1994

1995

.2
.1
.1
1.0

1.1

.979

To determine the mature claims-made multiple of the 1991 occurrence pure premium, solve for X:
X*(1.1) = .979. X = .89.
Answer B.

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Chapter 16 – Claims Made Ratemaking
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Solutions to questions from the 1990 exam
Question 24.
Answer A.

Solutions to questions from the 1991 exam
Question 27.
Answer B.

Question 28.

0
1
2
3
4

L
A
G

Answer E.

1
L0,1
L1,1
L2,1
L3,1
L4,1

2
L0,2
L1,2
L2,2
L3,2
L4,2

Report Year
4
5
L0,4
L0,5
L1,4
L1,5
L2,4
L2,5
L3,4
L3,5
L4,4
L4,5

3
L0,3
L1,3
L2,3
L3,3
L4,3

6
L0,6
L1,6
L2,6
L3,6
L4,6

7
L0,7
L1,7
L2,7
L3,7
L4,7

The tail policy should include L1,4 + L2,4 + L3,4 + L2,5 + L3,5 + L4,5 + L3,6 + L4,6 + L4,7.

Solutions to questions from the 1992 exam
Question 1.
1. T.
2. F.
3. T.

Answer D.

Question 29.
The coverage required is provided by a 2nd year claims made policy.
The pure premium =$1,000*(1.1)2 + $2,000*(1.1)2 = 3630.

Answer A.

Question 30.

Lag
0
1
2
3
4
Mature CM e

1990 Pure Premium
Per Accountant
$1,000
$2,000
$3,000
$2,000
$1,000

1991 Pure Premium
Per Accountant
$1,000*(1.1)1
$2,000(1.1)1
$3,000(1.1)1
$2,000(1.1)1
$1,000(1.1)1

The pure premium = 10 * 10,890 = 108,900.

Exam 5, V1b

Page 279

1992 Pure Premium
Per Accountant
$1,000*(1.1)2
$2,000*(1.1)2
$3,000*(1.1)2
$2,000*(1.1)2
$1,000*(1.1)2
10,890
Answer C.

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Chapter 16 – Claims Made Ratemaking
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Solutions to questions from the 1992 exam
Question 62.
To complete the AY by RP matrix, it is important to realize that losses along the same diagonal in a report year by
Lag matrix are associated with the same AY. First, complete an AY by incremental report period matrix by filling
out the following grid (rows by Lag above become columns. Next, using the results of the losses by AY reported
within the interval, create a cumulative loss matrix:
Losses by AY reported within the interval
Accident
Year
0 - 12
13 - 24
25 - 36
36-48
1981
10
10
15
14
1982
30
30
30
36
1983
25
25
25
30
1984
20
20
20
24
1985
40
40
40
1986
50
50
1987
30
Reporting Period (through months)
Accident
Year
1981
1982
1983
1984
1985
1986
1987

12
10
30
25
20
40
50
30

24
20
60
50
40
80
100

36
35
90
75
60
120

48
49
126
105
84

60
49
126
105

72
49
126

Solutions to questions from the 1993 exam
Question 47.
a.
In 1988, Charlie purchased a occurrence policy. The pure premium = 2,000+1,000+500+250+0 = 3750.
In 1989, Charlie purchased a 1st year CM policy. The pure premium = 2,100.
In 1990, Charlie purchased a 2nd year CM policy. The pure premium = 2,200 + 1050 = 3250.
In 1991, Charlie purchased a 3rd year CM policy. The pure premium = 2,300 + 1100 + 525 = 3925.
In 1992, Charlie purchased a 4th year CM policy. The pure premium = 2,400 + 1150 + 550 + 263= 4363.
b. The tail policy would cover all losses from the three AY’s subsequent to the last CM policy. By extending
the matrix above to account for losses reported during the next three years of tail coverage, the
anticipated pure premium for a tail policy = 1200 + 575 + 600 + 275 + 288+ 300 = 3238.

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Chapter 16 – Claims Made Ratemaking
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Solutions to questions from the 1995 exam
Question 39.

Lag
0
1
2
3
4
Occ Price

1991
$25,000
$20,000
$15,000
$10,000
$0

1992
$30,000
$25,000
$20,000
$15,000
$0

1993
$35,000
$28,000
$25,000
$20,000
$0

Report Years
1994
$40,000
$35,000
$32,000
$21,000
$0

1995
$45,000
$42,000
$40,000
$25,000
$0
$96,000

1996
$50,000
$46,000
$44,000
$28,000
$0

1997
$55,000
$50,000
$48,000
$30,000
$0

(a) Since an occurrence policy provides for all losses arising from the same AY, expected losses for
AY 1991 = the sum of the losses along the AY 1991 diagonal = $96,000.
(b) Since the insured purchased CM coverage beginning in 1992, the CM policy purchased in 1993
is a 2nd year CM. The expected losses for a 2nd year 1993 CM policy are those from RY 1993,
lag 0 and lag 1 = 35,000 + 28,000 = 63,000.
(c) The expected losses for a tail policy purchased at the end of the 1993 CM policy are the losses
associated with LAGs 1 and 2 for RY 1994, LAGs 2 and 3 for RY 1995, LAGs 3 and 4 for RY 1996, and
LAG 4 for RY 1997= 35,000 + 32,000 + 40,000 + 25,000 + 28,000 + 0 = 160,000 .
(d) CM rates are more accurate since the coverage period to which they apply are shorter in duration than
the coverage period associated with an occurrence policy. Any changes in external conditions, such
as changes in trend or reporting patterns, are more apparent as losses are reported. Therefore, CM
rates are more responsive to changing conditions.

Solutions to questions from the 1996 exam
Question 12.
The expected annual pure premium for an occurrence policy purchased 1/1/96
= $102 + 720 + 67 + 44 + 32 = 965.
The expected annual pure premium for a mature claims made policy purchased 1/1/96
= $102 + 600 + 55 + 33 + 22 = 812.
Thus, the difference = $965 - $812 = $153.

Answer E.

Solutions to questions from the 1997 exam
Question 5.
The answer to each of these questions can be found by reviewing the 5 principles of claims made ratemaking.
1. F. CM policies incur no liability for IBNR claims so the risk of reserve inadequacy is greatly reduced.
If the claim is not reported, it is not covered. The IBNR need for a CM policy is always 0.
2. T. Investment income (II) earned from a CM policy is substantially less than under an occurrence policy.
3. T. A CM policy should always cost less than an occurrence policy, as long as claim costs are
rising.
Answer D

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Chapter 16 – Claims Made Ratemaking
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Solutions to questions from the 1997 exam
Question 33.

Lag
0
1
2
3
4

Based on an assumption of 5% loss cost inflation
Estimated
Estimated
Estimated
Estimated
Actual 1994
1995 Loss
1996 Loss
1997 Loss
1998 Loss
Loss Costs
Costs
Costs
Costs
Costs
1,000
1,050
1,103
1,158
1,216
1,000
1,050
1,103
1,158
1,216
1,000
1,050
1,103
1,158
1,216
500
525
551
579
608
0
0
0
0
0

Estimated
1999 Loss
Costs
1,276
1,276
1,276
638
0

Based on an assumption of 10% loss cost inflation
Lag
0
1
2
3
4

Actual 1994
Loss Costs
1,000
1,000
1,000
500
0

Estimated
1995 Loss
Costs
1,100
1,100
1,100
550
0

Estimated
1996 Loss
Costs
1,210
1,210
1,210
605
0

Estimated
1997 Loss
Costs
1,331
1,331
1,331
666
0

Estimated
1998 Loss
Costs
1,464
1,464
1,464
732
0

Estimated
1999 Loss
Costs
1,611
1,611
1,611
805
0

(a) Since the above RY by Lag table was constructed based on 1994 experience, a 2nd year CM policy would be
based on losses associated with RY 1996.
The expected losses for a 2nd year CM policy, are from RY 1996, lag 0 and lag 1.
Expected losses, assuming 5% inflation = 1,103 + 1,103 = 2,206.
Expected losses, assuming 10% inflation = 1,210 + 1,210 = 2,420.
The loss cost inadequacy, as a percentage of the loss costs assumed in the rates, for a second-year claims 2,206  2,420 
made policy effective 1/1/96 = 
 = -9.7%.
2,206


(b) Since an occurrence policy provides for all losses arising from the same AY, expected losses for
AY 1996 = the sum of the losses along the AY 1996 diagonal = L0,3 + L1,4 + L2,5 + L3,6
Expected losses, assuming 5% inflation = 1,103 + 1,158 + 1,216 +638 = 4,115.
Expected losses, assuming 10% inflation = 1,210 + 1,331 + 1,464 + 805 = 4,810.
The loss cost inadequacy, as a percentage of the loss costs assumed in the rates, for an occurrence policy
 4 ,115  4,810 
effective 1/1/96 = 
 = -16.9%.
4 ,115


(c) The losses under a claims-made tail policy effective 1/1/97 following a second-year claims-made policy, is
shown below:

L
A
G
(i)

0
1
2
3
4

Report Year (j)
3
4

1
L(0,1)

1st
L(1,2)

2nd year
CM
L(2,3)

L(1,4)
L(2,4)
L(3,4)

L(2,5)
L(3,5)
L(4,5)
1 st y ear oc c . pol

L(3,6)
L(4,6)

Tail
Policy

Expected losses, assuming 5% inflation = 1,158 + 1,158 + 1,216 + 608 + 638 = 4,778
Expected losses, assuming 10% inflation = 1,331 + 1,331 + 1,464 + 732+ 805 = 5,663.
The loss cost inadequacy, as a percentage of the loss costs assumed in the rates, = [4,778-5,663]/4,778=-18.5%

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Chapter 16 – Claims Made Ratemaking
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Solutions to questions from the 1998 exam
Question 15. The answer to each question can be found by reviewing the 5 principles of CM ratemaking.
1. T.
2. F. The longer the reporting lag or the shorter the settlement lag, the greater the difference will be.
3. F. A CM policy should always cost less than an occurrence policy, as long as claim costs are rising.
Answer A

Solutions to questions from the 1998 exam
Question 35.
Lag
0
1
2
3
Occ Price

1992
1,000
600
500
0

1993
1,300
800
700
0

Report Years
1994
1995
1,400
1,100
900
1,000
800
400
0
0
2,600

1996
1,800
1,200
500
0

1997
1,900
1,300
700
0

(a) Since an occurrence policy provides for all losses arising from the same AY, expected losses for AY 1992 =
the sum of the losses along the AY 1992 diagonal = $1,000 + 800 + 800 + 0 = 2,600.
(b) Since the insured purchased CM coverage beginning in 1993, the CM policy purchased in 1993 is a 1st
year CM. The expected losses for a 1st year 1993 CM policy are those from RY 1993, lag 0 = 1,300.
(c) Since the insured purchased CM coverage beginning in 1993, the CM policy purchased in 1994 is a 2nd
year CM. The expected losses for a 2nd year 1994 CM policy are those from RY 1994, lag 0 and lag 1 =
1,400 + 900 = 2,300.
(d) The expected losses for a tail policy purchased in 1995 are the losses associated with LAGs 1 and 2 for RY
1995, LAGs 2 and 3 for RY 1996 = 1,000 + 400 + 500+ 0 +0 = 1,900 .

Solutions to questions from the 1999 exam
Question 31.

We are given:

Report Year
/
1
2
3
4
5
6
0
L0,1
L0,2
L0,3
1
L1,2
L1,3
L1,4
2
L2,3
L2,4
L2,5
3
L3,4
L3,5
L3,6
Mature CM
1,000
 L0,3 and L1,3 and L2,3 are the losses generated by a 3rd year CM policy.
 The losses are determined by applying annual lag factors (LF) to a mature claims made premium.
Write equations for:
1. The pure premium for a 3rd year claims made policy:
Lag

PP3  PPMCM *

 LF  1,000 * (.30.25.20)  750 , where
i

PP3 and PPMCM represent the pure premium

i 0

for a 3rd year and mature claims made policy respectively.

PP3 ULAE  FE
.
1.0 V  Q
V = Total variable expenses = commission + variable gen. expense + taxes + profit = .12+.06+.04=.22
Q = Profit and Contingencies = -.03.
FE = Fixed expense = $75
$750  $750*.10  $75
. .
 1,11111
Thus, the total premium for a third-year claims-made policy is
1.0.22  ( .03)

2. The formula for the total premium for a third-year claims-made policy is

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Chapter 16 – Claims Made Ratemaking
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Solutions to questions from the 1999 exam
Question 32. We are given that:
 For each report year, lag 1 claims settle for twice the value of lag 0 claims, and lag 2 claims settle for three
times the value of lag 1 claims. Let X = the severity of lag 0 claims in 1996, 2X = the severity of lag 1 claims
and 3 * 2X = the severity of lag 2 claims.
 Report year severity is increasing 10% per year across all lags.
These givens can be translated diagrammatically as shown below:
Report Year
Lag /
1996
1997
1998
0
.6*X
.6*(1.1)X
.6*(1.1)2 *X
1
.3*2X
.3(1.1)2X
.3*(1.1)2 *2X
2
.1*3*2X
.1*(1.1) *3*2X
.1*(1.1)2 *3*2X
Occ. Price
$1,000
Mature CM
??
Set up an equation to determine the premium for an occurrence policy, effective 1/1/96, with an
undiscounted pure premium of $1,000, and solve for X:
.6X + .3*(1.1)*2X + .1*(1.1)2 *3*2X = 1,000;
.6X + .66x + .726X = 1.986X = 1,000. X = 503.52
The mature claims made pure premium for policy effective 1/1/98 is calculated as follows:

98 MCM Pr emium  Sev0,96 *

 RYLF

i , 98

 503.5 *[.6 * (11
. ) .3 * (11
. ) * ( 2 ) .1 * (11
. ) * ( 6)]  1098

i 0

Solutions to questions from the 2001 exam
Question 46. For each of the stated principles, briefly describe why it is true.
a. A claims-made policy should always cost less than an occurrence policy, as long as claims costs are
increasing.
Under a claims-made policy, we are always pricing next year’s claims. This reduces the amount of time
inflation has to impact losses. Under an occurrence policy, we must take into account claims to be reported
many years in the future. This increases the amount of time inflation has to act upon losses, and thus
increases the cost of occurrence policies.
b. Whenever there is a sudden, unpredictable change in underlying trend, claims-made policies priced on the
basis of the prior trend will be closer to the correct price than occurrence policies priced in the same
manner.
For claims-made policies, failing to incorporate the true change in the underlying trend results in a small
change to the proper rate level, since the period for trending losses is shorter. However, when pricing an
occurrence policy, the error in not incorporating the true change in the underlying trend is compounded over
a longer period. Stated another way, the confidence interval about the projected losses for a claims-made
policy is narrower than for an occurrence policy priced at the same time.
c. Whenever there is a sudden unexpected shift in the reporting pattern, the cost of mature claims-made
coverage will be affected very little, if at all, relative to occurrence coverage.
Given an unexpected shift in the reporting pattern, only the first and last lags are affected since the other
lags have the same dollars shifting in and out, leaving the same total dollars reported. This results in a
mature claims-made policy still being correctly price.
d. The investment income earned from claims-made policies is substantially less than under occurrence
policies.
Claims-made policies incur no liability for IBNR claims. Because there is no need for IBNR, the time lapsed
between the collection of premium and the payment of claims is reduced. This reduces the time in which
premiums may be invested to generate investment income.

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Chapter 16 – Claims Made Ratemaking
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Solutions to questions from the 2002 exam
Question 43. Calculate the dollars of "pure" IBNR reserve inadequacy for a company writing occurrence
policies for five years.
Step 1: Recognize that the dollars of "pure" IBNR reserve inadequacy result from the difference in what actually
occurs and what is estimated to occur.
Step 2: Create a table which shows the results of losses of $1500 being reported in the last year produced
in equal proportions from occurrences in the last 5 years (column (1) below) and that losses are
forecast to increase at a rate of $10 per year (columns (2) – (5)). This is what is estimated to
occur.

Lag
0
1
2
3
4+
IBNR =

Report Year
0
1
2
300
310
320
300
310
320
300
310
320
300
310
320
300
310
320
3200
3200 = 1240+960+660+340

3
330
330
330
330
330

4
340
340
340
340
340

Step 3: Create a table which displays actual results showing an unexpected shift of $5/per year/per lag towards
later reportings.
Note: The impact of an unexpected shift of $5/per year/per lag affects the 1st and last lags differently.
Begin with Lag 0. Lag 0 shows the shift of $5/per year across report years 2 – 5.
For report year 2, lags 1, 2 and 3, show a $5/per lag shift.
Report year 2, lag 4+ shows the cumulative effect of a $5/per year/per lag shift.
Similar results are shown for report years 3 – 5.

Shift

$15

$20

$10
Report Year
3

Lag

0
1

300

305
$310

310
320

$315
330

$320
340

$310

320

330

340

$310

320

330

340

$315

$330

$345

$360

IBNR =

3250
3250 = 1245+970+675+360

Step 4: Using the information from Step 1, and the results from Steps 2 and 3, calculate the dollars of "pure"
IBNR reserve inadequacy for a company writing occurrence policies for five years.
IBNR reserve inadequacy = 3,250 – 3200 = 50.

Exam 5, V1b

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Chapter 16 – Claims Made Ratemaking
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Solutions to questions from the 2003 exam
22. (3 points)
a. (1 point) Define a coverage trigger.
A coverage trigger is an event that must occur, subject to requirements in the policy, before the policy
will respond to a claim.
b. (1 point) State how the coverage trigger for claims-made forms differs from the coverage trigger
for occurrence forms.
For claims-made forms, coverage is triggered for a loss that is reported to the insurer during the
effective period of the policy. The claim may be subject to a retroactive date. For occurrence forms,
coverage is triggered when a loss occurs during the effective period of the policy.
c. (0.5 point) A dentist begins his practice on January 1, 2003 and retires three years later. He buys
the following professional liability insurance policies:
• An occurrence policy to cover his first year of practice
• A 1st - year claims-made policy for 2004
• A 2nd - year claims-made policy for 2005
• A tail policy at the end of 2005
A loss that occurred in 2004 was not reported until 2006. State which policy, if any, covers the
loss and explain why.
The tail policy covers the claim because the tail policy covers all claims reported 1/1/2006 and
afterwards for losses that occurred between 1/1/2004 and 12/31/2005.
d. (0.5 point) Assume that the dentist in part c. above instead purchased three occurrence policies,
one for each of his first three years of practice. State which policy, if any, would cover the loss
described in part c. above and explain why.
The 2004 occurrence policy covers the loss because the 2004 occurrence policy covers all losses that
occurred in 2004 regardless of when the loss is reported.
29. (3 points) Calculate the premium for an occurrence policy written in Year 1.
Using the loss reporting % pattern and the fact that a mature claims-made pure premium for year 1 is $600,
compute 1st the RY by Lag distribution for the mature claims-made pure premium and then the year 1
occurrence pure premium.
Report Year
Lag
1
2
3
4
0
$300.00
1
$180.00 $189.00
2
$90.00
$99.22
$34.73
3
$30.00
$600.00
$622.95
Note: The calculations supporting the computation of the occurrence year pure premium are as follows:
300 * 1.0 = $300 $180 * (1.05) = $189 $90 * (1.05)2 = $99.23 $30 * (1.05)3 = $34.73
Pure premium = $300 + $189 + $99.23 + $34.73 = $622.96
To compute the premium for an occurrence policy written in year 1, the rate calculation is as follows:
R= (PP + FE)/(1.0-VE-P), where R is the rate, PP is the pure premium, P is the profit allowance and E = FE + VE
is the expense, broken down into its fixed and variable components. Thus, the premium is calculated as:
Premium = [$622.96 * (1.08) + $150] / (1 – 0.12 – 0.05 – 0.03) = $1,028.50

Exam 5, V1b

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Chapter 16 – Claims Made Ratemaking
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Solutions to questions from the 2004 exam
12. Which of the following statements are true regarding claims-made ratemaking?
1. The investment income earned under claims-made policies is substantially less than the
investment income earned under occurrence policies. True. This is principle number 5.
2. An occurrence policy will generally cost less than a claims-made policy. False. This is a
misstatement of principle number 1. A claims-made policy should always cost less than an
occurrence policy, as lone as claim costs are increasing.
3. Claims-made policies incur no liability for IBNR claims. True. This is principle number 4. Claimsmade policies incur no liability for IBNR claims so the risk of reserve inadequacy is greatly
reduced.
Answer D: 1 and 3 only

Solutions to questions from the 2005 exam
15. A claim occurred in May 2001 and was reported in September 2003. Which of the following would cover
this claim?
1. A one-year occurrence policy effective January 1, 2003. False. Occurrence policies cover claims
occurring during the policy period. An accident occurring on 5/1/2001 would not be covered by a
policy covering the period 1/1/2003 – 12/31/2003
2. A second-year claims-made policy effective January 1, 2003. False. This is due to the retroactive date.
The retroactive date restricts coverage to accidents occurring on or after that date. Normally, this would be
the date on which an insured's first claims-made policy commences. Thus, a second-year claims-made
policy effective January 1, 2003 would cover claims occurring anytime after 1/1/2002.
3. Tail coverage effective 1/1/2003 for a physician retiring after 10 years of practice covered by claimsmade coverage. True. A claims made policy covers claims reported (made) (in this example, 9/1/2003)
during the policy period (i.e. 1/1/2003 – 12/31/2003), regardless of when the accident date occurred.
Answer B. 3 only

Solutions to questions from the 2007 exam
13. Which of the following statements are true regarding claims-made ratemaking?
1. The investment income earned under claims-made policies is substantially less than the
investment income earned under occurrence policies. True. This is principle number 5.
2. A claims-made policy should always cost less than an occurrence policy, as long as claim costs
are increasing. True. This is principle number 1.
3. Claims-made policies incur no liability for IBNR claims. True. This is principle number 4.
Answer: E. 1, 2, and 3

Exam 5, V1b

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Chapter 16 – Claims Made Ratemaking
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Solutions to questions from the 2008 exam
Model Solution - Question 20
Lag
0
1
2
3
4
5
6

2000
A

2001
B
A

Report Year
2002
2003
2004
C
D
C
D
E
A
D
E
A
E
A

2005

E
E
E
A

2006

E...
E...
E...
A...

a. Occurrence policy in $2000 = All A’s
b. 1st year claims made = B
c. 2nd year claims made = All C’s
d. 3rd year claims made = All D’s
e. 2004 tail policy = All E’s
Model Solution - Question 21
a. Retroactive date is a date which activates the claims made policy. Claims occurred on or after that date and
reported during the policy period will be covered by the CM policy.
b. Extended reporting period extends the periods for the claims to be reported under CM policy after the policy
period ends. Claims occurred during the policy period and reported before the extended reporting period
ends will be covered by CM policy.

Solutions to questions from the 2009 exam
Question#: 28
Report Year
LAG 2004 2005 2006 2007 2008 2009 2010 2011
0 A
B
C
D
E
1
A
B
D
E
F
2
A
B
E
F
F
A
B
F
F
F
3
A: Occurrence Policy Effective 1/1/2004
B: Occurrence Policy Effective 1/1/2005
C: 1st year CM Policy 1/1/2006
D: 2nd year CM Policy Effective 1/1/2007
E: 3rd year CM Policy Effective 1/1/2008
F: Tail Policy Effective 1/1/2009

Exam 5, V1b

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Chapter 16 – Claims Made Ratemaking
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Solutions to questions from the 2010 exam
Question 22
a. Coverage Trigger
Claims made coverage is triggered when claim is first reported to the insurer, (given that it occurred
on or after the retroactive date and is reported in policy period)
Occurrence coverage is triggered when the loss occurs during the policy period.
b. Claims made policies have a much shorter tail than occurrence policies, so they are not affected as
severely by inflation, trend, etc. This makes pricing the future easier.
c. There is no IBNR (incurred but not reported) on a claims‐made policy since all claims are reported at
the end of policy period. So, it makes it easier to set reserves.
d. To provide coverage for claims that maybe reported after the claims‐made policy expires
For example, a doctor may retire, so no more new claims occur, but he/she needs coverage for
claims that might be reported after he/she is done practicing medicine.

Solutions to questions from the 2012 exam
8a. (0.5 point) Briefly describe two advantages that claims-made coverage has over occurrence coverage
for a medical malpractice insurer.
8b. (1 point) Calculate the loss costs associated with a 2011 occurrence policy and the loss costs
associated with a mature 2012 claims-made policy. Briefly describe the overlap in loss costs between
the two and the mechanism used to prevent it.
8c. (0.5 point) Identify the loss costs in the table above for which the physician would still have exposure
at the time of retirement, and the coverage that the physician would need to purchase to transfer that
exposure to the insurer.
Question 8 – Model Solution (Exam 5A Question 8)
a. 1. Med Mal has a very long tail, so for occurrence policies it takes a very long time to develop. For
claims made policies losses are known at the end of the year. No Pure IBNR component needed.
2. Since CM policies have a shorter time frame, they would be less subject to changes in trend
or inflation than occurrence policies.
b. 2011 Occurrence = 350 + 315 + 276 + 116 = 1057 (sum of the loss costs along the diagonal)
2012 Mature CM = 368 + 315 +263 + 105 = 1051 (sum of the loss costs across the RY 2012 row)
Overlap would be for RY2012 Lag 1- the 315 would be covered by both. To prevent this, CM
policies have retro dates which signal the beginning of coverage (losses that occurred before retro
date would not be covered, only losses that occurred after). So after the occurrence policy in 2011,
the CM policy in 2012 should have a retro date of 1/1/2012 + be a first year CM policy.
c. Loss costs after would be 365 + 304 + 122 + 319 +128 +134 = 1372
L(2015,1) + L(2015,2) + L(2015,3) + L(2016,2) + L(2016,3) + L(2017,3)
There would be losses that occurred while the CM policies were still in place but were reported after
the physician retired. Physician would need to purchase a tail coverage to cover these losses.
Examiner’s Comments
a. Many candidates received full credit on this part. Some candidates did not provide enough detail to
receive credit, using statements like “less pricing risk” and “less reserving risk”. Other candidates
provided slightly different variations on the same item.
b. Most candidates did identify the correct loss costs. Full credit was given for identifying the
overlap graphically. Some candidates lost credit for not addressing the specific overlap for this
question.
c. Many candidates were able to identify the loss costs associated with the tail exposure. Some
responses were not able to identify tail coverage, instead listing some combination of claims made
or occurrence policies that did not match the exposure.

Exam 5, V1b

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Appendix A – Auto Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Since the appendices use real examples from various rate filings, some of the procedures may vary from those
discussed within the actual text.
APPENDIX A: AUTO INDICATION
The following show an example of an overall rate level indication using the loss ratio approach.
 This is for the property damage liability coverage of personal automobile insurance in State XX.
 All policies are semi-annual
 The proposed effective date for the revised rates is 1/1/2017.
The individual exhibits are as follows:
 LR Indication: The overall indicated premium change using the LR method on 5 AYs of State XX data
evaluated as of 3/31/2016.
 Credibility: To be applied to the experience period using the classical credibility approach and the
square-root rule.
 Current Rate Level: The calculation of the current rate level factors using the parallelogram method.
 Premium Trend: Premium trend factors are computed using the two-step trending approach.
 Loss Development: Computation and selection of the loss development factors using the chain ladder
method.
 Loss Trend: Selection of the loss trend factors based on the pattern of historical changes of frequency,
severity, and pure premium.
 ULAE Ratio: Computing the ULAE factor based on the historical relationship of ULAE to losses and ALAE.
 Expense: Computing fixed and variable expense provisions using the premium-based projection method.

Exam 5, V1b

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Appendix A – Auto Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
LR (LOSS RATIO) INDICATION EXHIBIT
The overall indication process:
 A projected loss and LAE ratio is selected and added to the fixed expense provision.
This ratio is compared to the variable PLR to obtain the overall indicated rate change, which is credibilityweighted with the trended present rates indication from the prior rate change analysis.
State XX
Wicked Good Insurance Company
Private Passenger Auto: Property Damage Liability
Indicated Rate Chance- Loss Ratio Method
(1)

Calendar
Accident
Year
2001
2012
2013
2014
2015
Total

Earned
Premium
$1,122,372
$1,154,508
$1,280,545
$1,369,976
$1,397,750
$6,325,151

(2)

(3)

Current
Rate Level
Factor
1.2161
1.2176
1.1311
1.0892
1.0991

Premium
Trend
Factor
1.1342
1.1116
1.0879
1.0663
1.0452

(4)
Projected
Earned
Premium at
Current
Rate Level
$1,548,088
$1,562,608
$1,575,741
$1,591,109
$1,605,706
$7,883,253

(5)

(6)

Reported
Losses
Loss
and
Development
Paid ALAE
Factor
$856,495
1.0000
$867,184
0.9799
$835,120
1.0003
$821,509
1.0282
$797,866
1.0966
$4,178,174

(2) From Current Rate Level Exhibit - 2
(3) From Premium Trend Exhibit - 3
(4) = (1)*(2)*(3)
(5) Case Incurred Losses and ALAE Evaluated As Of 03/31/2016
(6) From Loss Development Exhibit
(7) From Loss Trend Exhibit
(8) From ULAE Ration Exhibit
(9) = (5)*(6)*(7)*(8)
(10) = (9)/(4)
(12) From Expense Exhibit
(13) From Expense Exhibit
(14) Selected Profit Provision
(15) = 100% - (13) - (14)
(16)= { [ (11) + (12) ] / (15) } - 1.0
(17) From Credibility Exhibit
(18) From Credibility Exhibit
(19) = (16) * (17) + (18) * [ 1.0 - (17) ]

(7)

Loss
Trend
Factor
0.9912
0.9962
1.0012
1.0062
1.0113

(8)

ULAE
Factor
1.143
1.143
1.143
1.143
1.143

(9)

(10)

Projected
Ultimate
Losses
and LAE
$970,359
$967,578
$955,974
$971,450
$1,011,357
$4,876,718

Projected
Loss and
LAE Ratio
62.7%
61.9%
60.7%
61.1%
63.0%
61.9%

(11) Selected Projected Loss and LAE Ratio
(12) Fixed Expense Provision
(13) Variable Expense Provision
(14) UW Profit Provision
(15) Variable Permissible Loss Ratio
(16) Indicated Rate Change
(17) Credibility
(18) Trended Present Rates Indication
(19) Credibility- Weighted Indicated Rate Change
(20) Selected Rate Change

61.9%
11.3%
17.0%
5.0%
78.0%
-6.2%
100.0%
6.2%
-6.2%
-6.2%

Noteworthy Commentary:
Projected Premium at Current Rate Level: Columns 1 – 4.
Projected ultimate loss and LAE: Columns 5 – 9.
Row 11: The (selected) 5-year average projected loss and LAE ratio
Rows 12 – 15: U/W expense and Profit items.
 Row 12 is the projected fixed expense ratio (as a % of premium).
 Rows 13 – 15: the calculation of the VPLR, where rows 13 and 14 are %s of premium
Row 15: VPLR is the % of each premium dollar that is available to pay for losses, LAE, and fixed expenses.
Row 16 is the calculation of the indicated rate change using the formula:
Indicated Change =

Loss & LAE Ratio + Fixed Expense Ratio
[Row 11 + Row 12]
-1.0 =
 1.0
Variable Permissible Loss Ratio
[Row 15]

Row 17: The credibility to be applied to the indicated rate change.
Row 18: The trended present rates indication (from the prior review) and used as the complement of credibility.
Row 20 is the (selected) credibility-weighted indicated rate change.

Exam 5, V1b

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Appendix A – Auto Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
CREDIBILITY EXHIBIT
The credibility value is calculated based on a full credibility standard of 1,082 claims,
The complement of credibility is the residual indication based on the latest rate change and indication (i.e. the
“trended present rates” approach to derive complement of credibility, as discussed in Chapter 12).
State XX
Wicked Good Insurance Company
Private Passenger Auto: Property Damage Liability
Credibility Calculations
(1) Total Number of Claims in Historical Period
(2) Number of Claims for Full Credibility
(3) Credibility
Min { [ (1)/(2) ] ^ 0.5, 1.0 }
(4) Latest Indicated Rate Change
(5) Last Rate Change Taken
From Current Rate Level Exhibit - 2
(6) Residual Loss Trend
{ [ 1.0 + (4) ] / [ 1.0 + (5) ] } - 1.0
(7) Projected Loss Trend
From Loss Trend Exhibit - 1
(8) Projected Premium Trend
From Premium Trend Exhibit - 1
(9) Net Trend
{ [ 1.0 + (7) ] / [ 1.0 + (8) ] } - 1.0
(10) Trend Period
From Last Rate Change Effective Date (01/01/2016) to Proposed Effective Date (01/01/2017)
(11) Trended Present Rates Indication
{ [ 1.0 + (6) ] * [ 1.0 + (9) ] ^ (10) } - 1.0

3,612
1,082
100.0%
13.2%
5.0%
7.8%
0.5%
2.0%
-1.5%
1
6.2%

Row 3: Since the number of claims (3,612) exceeds the number of claims needed for full credibility (1,082), the
credibility is 100%.
Rows 4 – 11: Derivation of the complement of credibility.
The trended present rates indication in Row 11 and is used as the complement of credibility.

Exam 5, V1b

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Appendix A – Auto Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
CURRENT RATE LEVEL EXHIBIT
Shows the calculation of the current rate level factors using the parallelogram method for each year.
Sheet 1 - Cumulative rate level indices for each rate level group during or after the historical period.
Columns (1) and (2): Rate change history
Columns 3 and 4: The rate factor in Column 3 and the cumulative rate level in Column 4
State XX
Wicked Good Insurance Company
Private Passenger Auto: Property Damage Liability
Rate Change History
(1)
Rate
Level
Effective
Group
Date
A
B
04/01/2011
C
07/01/2012
D
10/01/2013
E
07/01/2014
F
10/01/2015
G
01/01/2016
(3) = 1.0 + (2)
(4) = Cumulative Product of (3)

(2)
Rate
Change
-5.0%
10.0%
5.0%
-2.0%
5.0%
5.0%

(3)
Rate
Level
Index
1.0000
0.9500
1.1000
1.0500
0.9800
1.0500
1.0500

(4)
Cumulative
Rate Level
Index
1.0000
0.9500
1.0450
1.0973
1.0753
1.1291
1.1855

Sheet 2 -Calculation of current rate level factors.
State XX
Wicked Good Insurance Company
Private Passenger Auto: Property Damage Liability
Calculation of Current Rate Level Factors
(1a)

Calendar Year
2011
2012
2013
2014
2015
(1b) Cumulative Rate Level

A
50.00%
0.00%
0.00%
0.00%
0.00%
1.0000

Portion of Earned Premium Assumed in Each Rate Level Group
B
C
D
E
F
50.00%
0.00%
0.00%
0.00%
0.00%
75.00%
25.00%
0.00%
0.00%
0.00%
0.00%
93.75%
6.25%
0.00%
0.00%
0.00%
6.25%
68.75%
25.00%
0.00%
0.00%
0.00%
0.00%
93.75%
6.25%
0.9500
1.0450
1.0973
1.0753
1.1291

G
0.00%
0.00%
0.00%
0.00%
0.00%
1.1855

(2)
Average
Cumulative
Rate Level
0.9750
0.9738
1.0483
1.0885
1.0787

(3)
Current
Rate Level
Index
1.1855
1.1855
1.1855
1.1855
1.1855

(4)

CRL Factor
1.2159
1.2175
1.1309
1.0891
1.0991

(1a) Portion of Each Calendar Year's Earned Premium by Rate Level Group
(1b) Cumulative Rate Level for Each Rate Level Group
(2) (1b) Weighted by (1a) Within Each Calendar Year
(4) = (3) / (2)

Column 1a %s are calculated based on the assumption that the six-month policies are written uniformly
throughout the year.
Column 2 shows the average rate level for each CY (i.e. the cumulative rate level associated with each rate level
group weighted by the portion of the CY premium represented by the rate level group).
Column 4 is the factor to be applied to earned premium in each CY to bring it to current rate level.

Exam 5, V1b

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Appendix A – Auto Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
PREMIUM TREND EXHIBIT
Historical premium needs to be adjusted to account for the change in average premium level due to
distributional changes in the book of business. Shown is the calculation of premium trend factors using a twostep trending approach.
Sheets 1 - 2
Sheet 1: Historical annual changes in average written premium at current rate level.
State XX
Wicked Good Insurance Company
Private Passenger Auto: Property Damage Liability
Premium Trend Selection

Year Ending
Quarter - X
2010 - 2
2010 - 3
2010 - 4
2011 - 1
2011 - 2
2011 - 3
2011 - 4
2012 - 1
2012 - 2
2012 - 3
2012 - 4
2013 - 1
2013 - 2
2013 - 3
2013 - 4
2014 - 1
2014 - 2
2014 - 3
2014 - 4
2015 - 1
2015 - 2
2015 - 3
2015 - 4

(1)
Written Premium
at CRL
$1,314,117
$1,323,381
$1,333,726
$1,343,014
$1,354,391
$1,364,644
$1,374,283
$1,384,951
$1,393,570
$1,403,987
$1,415,881
$1,428,087
$1,438,647
$1,448,311
$1,458,540
$1,468,617
$1,479,666
$1,492,537
$1,503,294
$1,514,903
$1,524,242
$1,536,215
$1,547,368

(2)
Written
Exposure
12,752
12,776
12,806
12,825
12,863
12,893
12,917
12,953
12,973
13,005
13,044
13,082
13,108
13,128
13,155
13,183
13,217
13,262
13,292
13,325
13,341
13,383
13,414

(3)
Average Written
Premium at CRL
$103.05
$103.58
$104.15
$104.72
$105.29
$105.84
$106.39
$106.92
$107.42
$107.96
$108.55
$109.16
$109.75
$110.32
$110.87
$111.40
$111.95
$112.54
$113.10
$113.69
$114.25
$114.79
$115.35

(4)
Annual
Trend

2.2%
2.2%
2.2%
2.1%
2.0%
2.0%
2.0%
2.1%
2.2%
2.2%
2.1%
2.1%
2.0%
2.0%
2.0%
2.1%
2.1%
2.0%
2.0%

Exponential Trend
20 pt
2.1%
16 pt
2.1%
12 pt
2.0%
8 pt
2.0%
6 pt
2.0%
4 pt
2.0%
Selected Projected Premium Trend

2.0%

(3) = (1) / (2)
(4) Percent Change in Avg WP at CRL From Prior Year

Column 3: Average written premium at current rate level for the 12-month period ending each quarter.
Average written premium at current rate level for each quarter (rather than the 12-month rolling quarter) is
preferable to use, but that data was not readily available.
Column 4 calculates an annual trend of average written premium at current rate level (i.e. the percentage
change from the prior year).

Exam 5, V1b

Page 294

 2014 by All 10, Inc.

Appendix A – Auto Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Sheet 2 displays Sheet 1 data in graphical format, and shows the selected projected premium trend (which is
based on the more recent data because this trend is to be applied to historical premium already trended to the
most recent period).
State XX
Wicked Good Insurance Company
Private Passenger Auto: Property Damage Liability
Premium Trend

Average Premium Trend
A 116
v
e 114
r
a 112
g
e 110
Actual

P
108
r
e
m 106
i
u 104
m

20 pt fit
12 pt fit
6 pt fit

2015 ‐ 4

2015 ‐ 3

2015 ‐ 2

2015 ‐ 1

2014 ‐ 4

2014 ‐ 3

2014 ‐ 2

2014 ‐ 1

2013 ‐ 4

2013 ‐ 3

2013 ‐ 2

2013 ‐ 1

2012 ‐ 4

2012 ‐ 3

2012 ‐ 2

2012 ‐ 1

2011 ‐ 4

2011 ‐ 3

2011 ‐ 2

2011 ‐ 1

2010 ‐ 4

2010 ‐ 3

2010 ‐ 2

102

Year Ending Quarter

Exponential Trend
20 pt
2.1%
12 pt
2.0%
6 pt
2.0%

Selection
2.0%

Sheet 3 - Derivation of the premium trend factors.
State XX
Wicked Good Insurance Company
Private Passenger Auto: Property Damage Liability
Premium Trend Calculation

Calendar
Year
2011
2012
2013
2014
2015

(1)

(2)

Earned
Premium at
at CRL
$1,364,916.59
$1,405,728.94
$1,448,424.45
$1,492,177.86
$1,536,267.03

Earned
Exposure
12,900
13,020
13,130
13,258
13,380

(3)
Average
Earned
Premium
at CRL
$105.81
$107.97
$110.31
$112.55
$114.82

(4)
Most Recent
Average Written
Premium
at CRL
115.354704
115.354704
115.354704
115.354704
115.354704

(1) = [ LR Indication Exhibit (1) ] * ( Current Rate Level Exhibit -1 (4) ]
(3) = (1) * (2)
(4) = Average Written Premium for Year Ending 2015, Quarter 4
(5) = (4) / (3)
(6) From Premium Trend Exhibit - 1
(7) From 06/30/2015 to 06/30/2017
(8) = [ 1.0 + (6) ] ^ (7)
(9) = (5) * (8)

Exam 5, V1b

(5)
Current
Trend
Factor
1.0902
1.0684
1.0457
1.0249
1.0047

(6)
Selected
Projected
Premium
Trend
2.0%
2.0%
2.0%
2.0%
2.0%

(7)

(8)

(9)

Projected
Trend
Period
2.0000
2.0000
2.0000
2.0000
2.0000

Projected
Trend
Factor
1.0403
1.0403
1.0403
1.0403
1.0403

Total
Trend
Factor
1.1342
1.1115
1.0878
1.0662
1.0452

[ From Premium Trend Exhibit -1 ]

Page 295

 2014 by All 10, Inc.

Appendix A – Auto Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
LOSS DEVELOPMENT EXHIBIT
Since historical Losses and ALAE are not fully mature, they need to be developed, and the Loss Development
Exhibit shows the calculation of the LDFs using the chain ladder technique.
State XX
Wicked Good Insurance Company
Private Passenger Auto: Property Damage Liability
Loss Development

Accident Year
2009
2010
2011
2012
2013
2014
2015
Age-to-Age Factors
2009
2010
2011
2012
2013
2014

Reported Losses and Paid ALAE Evaluated As of
15 Months
27 Months
39 Months
705,088
725,592
738,686
712,475
753,295
782,248
714,196
763,913
855,150
764,101
861,114
884,498
774,384
846,167
835,120
785,068
821,509
797,866

51 Months
753,027
800,258
874,106
867,184

63 Months
732,239
813,949
856,495

63-Ult

15-27
1.0291
1.0573
1.0696
1.1270
1.0927
1.0464

27-39
1.0180
1.0384
1.1194
1.0272
0.9869

39-51
1.0194
1.0230
1.0222
0.9804

51-63
0.9724
1.0171
0.9799

(1) All-Year Average
(2) 3-Year Average
(3) 4-Year Average
(4) Average Excluding Hi-Lo
(5) Geometric Average

1.0703
1.0887
1.0839
1.0665
1.0699

1.0380
1.0445
1.0430
1.0279
1.0371

1.0113
1.0085
1.0113
1.0208
1.0111

0.9898
0.9898
0.9799
0.9896

(6) Selected Age-to-Age
(7) Age-to-Ultimate

1.0665
1.0965

1.0279
1.0281

1.0208
1.0002

0.9799
0.9799

1.0000
1.0000

(1) Straight Average
(2) Straight Average
(3) Straight Average
(4) Sraigtht Average Excluding Highest and Lowest Values
(5) = (Product of Age-to-Age Factors) ^ (1.0 / Number of Age-to-Age Factors)
(7) = Cumulative Product of (6)

The age-to-age factors (i.e. link ratios) are calculated for each AY by dividing the reported loss and paid ALAE
at one valuation point by the value at the previous valuation point.
Rows 1 - 5 show various averages used as guides for selections.
Row 6 shows the selected age-to-age factors.
Row 7 converts the selected age-to-age factors to age-to-ultimate factors by multiplying each age-to-age factor
by all of the subsequent age-to-age factors (e.g. the 39-ultimate factor is the product of the selected 39-51, 5163, and 63-ultimate age-to-age factors).

Exam 5, V1b

Page 296

 2014 by All 10, Inc.

Appendix A – Auto Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
LOSS TREND EXHIBIT
The proposed rates will be in effect in a period later than the historical period, and loss and ALAE need to be
adjusted to account for expected trends in the frequency and severity of claims between the two periods.
A two-step loss trending approach is used, and regional data is used to determine appropriate trends.
Sheets 1-4
Sheet 1: Historical frequencies, severities, and pure premiums.
State XX
Wicked Good Insurance Company
Private Passenger Auto: Property Damage Liability
Loss Trend Selections - Regional Data
(1)
Year
Ending
Quarter - X
2011 - 1
2011 - 2
2011 - 3
2011 - 4
2012 - 1
2012 - 2
2012 - 3
2012 - 4
2013 - 1
2013 - 2
2013 - 3
2013 - 4
2014 - 1
2014 - 2
2014 - 3
2014 - 4
2015 - 1
2015 - 2
2015 - 3
2015 - 4

Earned
Exposure
131,911
132,700
133,602
135,079
137,384
138,983
140,396
140,997
140,378
139,682
138,982
138,984
139,155
139,618
139,996
140,141
140,754
141,534
141,800
142,986

(2)
Closed
Claim
Count
7,745
7,785
7,917
7,928
7,997
8,037
7,939
7,831
7,748
7,719
7730
7,790
7,782
7,741
7,720
7,691
7,735
7,769
7,755
7,778

(1) Shown on a 4-Quarter Rolling Total Basis
(2) Shown on a 4-Quarter Rolling Total Basis
(3) Shown on a 4-Quarter Rolling Total Basis
(4) = (2) / (1)
(5) = (3) / (2)
(6) = (3) / (1)

(3)

(4)

(5)

(6)

Paid
Losses
$8,220,899
$8,381,016
$8,594,389
$8,705,108
$8,816,379
$8,901,163
$8,873,491
$8,799,730
$8,736,859
$8,676,220
$8,629,925
$8,642,835
$8,602,105
$8,535,327
$8,466,272
$8,412,159
$8,513,679
$8,614,224
$8,702,135
$8,761,588

Frequency
0.0587
0.0587
0.0593
0.0587
0.0582
0.0578
0.0565
0.0555
0.0552
0.0553
0.0556
0.0560
0.0559
0.0554
0.0551
0.0549
0.0550
0.0549
0.0547
0.0544

Severity
$1,061.45
$1,076.56
$1,085.56
$1,098.02
$1,102.46
$1,107.52
$1,117.71
$1,123.70
$1,127.63
$1,124.01
$1,116.42
$1,109.48
$1,105.38
$1,102.61
$1,096.67
$1,093.77
$1,100.67
$1,108.79
$1,122.13
$1,126.46

Pure
Premium
$62.32
$63.16
$64.33
$64.44
$64.17
$64.04
$63.20
$62.41
$62.24
$62.11
$62.09
$62.19
$61.82
$61.13
$60.48
$60.03
$60.49
$60.86
$61.37
$61.28

Exponential
Trend
20 pt
16 pt
12 pt
8 pt
6 pt
4 pt

Frequency
-1.7%
-1.3%
-0.7%
-1.3%
-0.9%
-1.4%

Severity
0.5%
-0.1%
-0.2%
1.2%
2.5%
3.3%

Pure
Premium
-1.2%
-1.4%
-0.9%
-0.1%
1.6%
1.9%

Selections
Current
Projected

-1.0%
-1.0%

0.5%
1.5%

-0.5%
0.5%

Columns 1 - 3 are the earned exposures, closed claim counts, and paid losses on a rolling 12-month basis.
 Changes in paid losses are used as the best estimate of the trend since using paid losses eliminates
any distortions caused by changes in overall reserve adequacy.
 LAE are not included with the losses in the trend data, and are therefore affected by the same trend.
Exponential trends are fit to the frequency, severity, and pure premiums columns for various durations. While
not displayed, actuaries may view the R-squared statistic to gauge the goodness of fit of the exponential trends.

Exam 5, V1b

Page 297

 2014 by All 10, Inc.

Appendix A – Auto Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Sheets 2 through 4: Graphical representation of the data and the selected trends.
Sheet 5: Shows the derivation of the total loss trend factor.
State XX
Wicked Good Insurance Company
Private Passenger Auto: Property Damage Liability
Loss Trend

Accident
Year
2011
2012
2013
2014
2015

(1)
Selected
Current
Trend
-0.5%
-0.5%
-0.5%
-0.5%
-0.5%

(2)
Current
Cost Trend
Period
4.00
3.00
2.00
1.00
0.00

(3)
Current
Trend
Factor
0.9801
0.9851
0.9900
0.9950
1.0000

(4)
Selected
Projected
Trend
0.5%
0.5%
0.5%
0.5%
0.5%

(5)
Projected
Cost Trend
Period
2.25
2.25
2.25
2.25
2.25

(6)
Projected
Projected
Trend
1.0113
1.0113
1.0113
1.0113
1.0113

(7)
Loss Trend
Factor
0.9912
0.9962
1.0012
1.0062
1.0113

(1) From Loss Trend Exhibit - 1
(2) From 07/01/20XX to 06/30/215
(3) = [ (1.0 + (1) ] ^ (2)
(4) From Loss Trend Exhibit - 1
(5) From 07/01/2015 to 09/30/2017
(6) = [ (1.0 + (4) ] ^ (5)
(7) = (3) * (6)

Column 2: The current cost trend period (for each AY) is the number of years between the average date of loss
in the accident year (6/30/20XX) to the average date of loss for the most recent period used to select the loss
trends (6/30/2015).
Column 5: The selected projected pure premium trend is used to trend losses and ALAE from 6/30/2015, to the
average date of loss for the projected period.
ULAE RATIO EXHIBIT
3 CYs of countrywide data are used to determine the factor needed to adjust the State XX reported loss and
paid ALAE to include ULAE.
State XX
Wicked Good Insurance Company
Private Passenger Auto: Property Damage Liability
ULAE Ratio

Calendar
Year
2013
2014
2015
Total
(3) = (2) / (1)
(5) = 1.0 + (4)

(1)
Countrywide
Paid Losses
and ALAE
$283,299,252
$290,213,410
$293,934,810
$867,447,472

(2)
Countrywide
Paid
ULAE
$41,170,520
$41,262,210
$41,959,671
$124,392,401

(3)
ULAE
Ratio
14.5%
14.2%
14.3%
14.3%

(4) Selected Ratio
(5) ULAE Factor

14.3%
1.143

CY paid information is used as it is readily available accounting data and is not susceptible to changes in
reserving practices. The selection in Row 4 is based on the historical ratios.
The selected percentage is converted into a factor in Row 5 by adding 1.0.

Exam 5, V1b

Page 298

 2014 by All 10, Inc.

Appendix A – Auto Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
EXPENSE EXHIBIT
U/W expense ratios are determined using the premium-based projection method.
Assume that the historical relationship between expenses and premium will continue during the projected period.
Expenses are divided into five categories: general, other acquisition, license and fees, commissions and
brokerage, and taxes (calculations and selections are performed separately for each category).
State XX
Wicked Good Insurance Company
Private Passenger Auto: Property Damage Liability
Expense Calculation
3-year
2013
2014
2015 Weighted Avg
(1) General Expenses
a Countrywide Expenses
b Countrywide Earned Premium
c Ration [(a)/(b)]
d % Assumed Fixed
e Fixed Expense % [(c)*(d)]
f Variable Expense % [(c)*(1.0-(d))]
(2) Other Acquisition
a Countrywide Expenses
b Countrywide Earned Premium
c Ration [(a)/(b)]
d % Assumed Fixed
e Fixed Expense % [(c)*(d)]
f Variable Expense % [(c)*(1.0-(d))]
(3) Licenses and Fees
a State Expenses
b State Written Premium
c Ration [(a)/(b)]
d % Assumed Fixed
e Fixed Expense % [(c)*(d)]
f Variable Expense % [(c)*(1.0-(d))]
(4) Commission and Brokerage
a State Expenses
b State Earned Premium
c Ration [(a)/(b)]
d % Assumed Fixed
e Fixed Expense % [(c)*(d)]
f Variable Expense % [(c)*(1.0-(d))]
(5) Taxes
a State Expenses
b State Written Premium
c Ration [(a)/(b)]
d % Assumed Fixed
e Fixed Expense % [(c)*(d)]
f Variable Expense % [(c)*(1.0-(d))]
(6) Fixed Expense Provision
(7) Variable Expense Provision

Selected

$29,143,368
$466,001,205
6.3%

$29,940,978
$478,971,842
6.3%

$30,763,160
$491,904,082
6.3%

6.3%

6.3%
75.0%
4.7%
1.6%

$40,158,296
$468,850,020
8.6%

$40,912,479
$482,345,783
8.5%

$41,652,543
$495,356,701
8.4%

8.5%

8.5%
75.0%
6.4%
2.1%

$3,124
$1,289,484
0.2%

$3,190
$1,380,129
0.2%

$3,229
$1,407,811
0.2%

0.2%

0.2%
100.0%
0.2%
0.0%

$145,073
$1,289,484
11.3%

$154,235
$1,380,129
11.2%

$158,172
$1,407,811
11.2%

11.2%

11.2%
0.0%
0.0%
11.2%

$27,338
$1,289,484
2.1%

$27,549
$1,380,129
2.0%

$29,853
$1,407,811
2.1%

2.1%

2.1%
0.0%
0.0%
2.1%
11.3%
17.0%

(1e) + (2e) + (3e) + (4e) + (5e)
(1f) + (2f) + (3f) + (4f) + (5f)

Row “a” shows the expense associated with each category for each of the three years (and the expense is
aggregated either at the state or countrywide level, depending on the category).
Row “b” displays the corresponding premium. The premium used in this calculation is either state or
countrywide and either written or earned depending on the nature of the expense category.
Row “c” is the calculation of the expense ratio for each expense category for each year as well as the premiumweighted average of the three years; the selected percentage is displayed in the last column.
No expense trend is applied to the fixed expense ratio (assumes the expenses and premium will trend at the
same rate and the ratio will remain constant).

Exam 5, V1b

Page 299

 2014 by All 10, Inc.

Appendix A – Auto Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Questions from the 2009 exam
30. (6 points) Given the following information:
• All policies have annual terms.
• Proposed effective date is October 1, 2009 and rates will be in effect for 12 months.
• Rate change history:
• - 4% effective July 1, 2007
• +5% effective January 1, 2009
• Selected premium trend = 1%
Calendar
Accident Year
2007
2008
•

Case Incurred
Losses and ALAE
$250,000
$350,000

Historical Accident Year Case Incurred Loss and ALAE Link Ratios:
Accident
Year
2001
2002
2003
2004
2005
2006
2007

•
•
•
•
•
•
•

Earned
Premium
$600,000
$650,000

12-24
Months
1.40
1.40
1.40
1.40
1.30
1.30
1.30

24-36
Months
1.07
1.07
1.07
1.07
1.15
1.15

36-48
Months
1.05
1.05
1.05
1.05
1.05

48-60
Months
1.03
1.03
1.03
1.03

60-72
Months
1.02
1.02
1.02

A tail development factor of 1.01 is needed to account for development beyond 72 months.
Selected annual frequency trend = -2%
Selected annual severity trend = 5%
ULAE is consistently 4% of ultimate losses and ALAE.
Projected fixed expense provision = 10% of premium
Variable expense provision = 20% of premium
Profit and contingencies provision = 3% of premium

a. (2 points) Calculate 2007 and 2008 projected calendar year earned premium at current rate level.
b. (1 point) Select 12-month and 24-month age to ultimate factors. Briefly explain your selection.
c. (1.5 points) Calculate the 2007 and 2008 projected calendar accident year losses and LAE.
d. (1.5 points) Calculate the indicated rate change, giving 40% weight to calendar accident year 2007
and 60% weight to calendar accident year 2008.

Exam 5, V1b

Page 300

 2014 by All 10, Inc.

App
pendix A – Auto In
ndication
n
BASIC RATTEMAKING – WERNER, G
G. AND MOD
DLIN, C
Solutions
s to question
ns from the 2009
2
exam
Question
n: 30
Current Le
evel Factor = 1.0 * (1-.04) * 1.05 = 1.00
08

2007 on-le
evel factor =

1.008
 1.0131
[(.5)(.5)(.5)](.9
[
96)  (.875)((1)

2008 on-le
evel factor =

1.008
 1.00446
(.125)(1)
(
 (.875) .96

2007 trend factor goes
s from 7/1/07 to
t 10/1/10 = 3.25
3
years
2008 trend factor goes
s from 7/1/08 to
t 10/1/10 = 2.25
2
years
2007 proje
ected CY earrned premium
m = (600,000)((1.0131)(1.01 )3.25 = 627,8
839
2008 proje
ected CY earrned premium
m = (650,000)((1.0446 )(1.01
1)2.25 = 694,3
363
b.
Average factors
f
chosen

12-24
1.3

24-3
36
1.1
15

36-48
1.05

48- 60
1.03

60-72
1.02

01)=1.6657
12-month age-to-ulttimate =(1.3)((1.15)(1.05)(1.03)(1.02)(1.0
24-month age-to-ulttimate =

(1.15)(1.05)(1.03)(1.02)(1.0
01)= 1.2813

There was
s probably a change
c
in res
serving metho
ods underlying
g the abrupt cchanges in hiistorical 12-24
4 and
24-3 6 link
k ratios. The new method reflecting the recent link ra
atios should b
be used.
c. 2007 AY projected lo
oss + LAE = (250,000)(1.2
(
2813)(1.04) [(..98 )(1.05)]3.25 =
2.25
5

2008 AY
A projected lo
oss + LAE = (350,000)(1.6
6657)(1.04) [((.98)(1.05)]

365,5
573
646,5
596

= (.4)(365,573
3/627,839) + (.6)(646,596/
(
694,363) =.7
7916
d. Weighted loss ratio=
Indicatted rate chang
ge= (.7916 + .1)/(1 - .2 - .0
03) - 1= 15.8%
%

Exam 5, V1b

Page 301

 2014 by Alll 10, Inc.

Appendix B – Homeowners Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
This is an example of a homeowners (HO) rate level indication using the pure premium approach.
All policies are annual, and the proposed effective date for new rates in State XX is 1/1/2017.
The individual exhibits are as follows:
• PP Indication: Calculation of the overall indicated rate per exposure using the pure premium method on
5 AYs of experience as of 3/31/2016.
• Non-Modeled Cat: Calculation of the cat provision for non-modeled catastrophes.
• AIY Projection: Selection of the projected average amount of insurance years (AIY) in the effective
period, used in the derivation of the non-modeled cat pure premium.
• Reinsurance: Projected net reinsurance cost per exposure.
• Loss Development: Derivation and selection of the LDFs using the chain ladder method.
• Loss Trend: Selection of the loss trend factors based on the historical changes of frequency, severity,
and pure premium.
• ULAE Ratio: Computation of the ULAE factor based on the historical relationship of ULAE to losses and
ALAE.
• Expense: Fixed and variable expense provisions using the exposure-based projection method.

Exam 5, V1b

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Appendix B – Homeowners Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
PP (PURE PREMIUM) INDICATION EXHIBIT
The overall rate level indication is based on the latest 5 AYs as of 3/31/2016.
 The projected non-cat PP for State XX is credibility-weighted with a regional non-cat PP, and then added
to the sum of the non-modeled cat PP and modeled cat PP.
 The total projected PP, projected fixed expense per exposure and the projected net reinsurance cost per
exposure are combined and divided by the VPLR to obtain the overall indicated rate.
State XX
Wicked Good Insurance Company
Homeowners
Pure Premium Indication
(1)
Calendar
Accident
Year
2011
2012
2013
2014
2015
Total

Earned
Exposure
12,760
12,766
12,805
12,834
13,411
64,576

(2)

(3)
(4)
Non-Cat
Reported
Loss
Loss
Losses and Development
Trend
Paid ALAE
Factor
Factor
$5,161,624
1.0000
1.1939
$4,820,968
1.0012
1.1705
$4,112,172
1.0054
1.1476
$5,052,052
1.0185
1.1251
$6,559,224
1.0553
1.1030
$25,706,040

(2) Reported Losses and Paid ALAE As of 03/31/2016
(3) From Loss Development Exhibit
(4) From Loss Trend Exhibit - 1
(5) From ULAE Ratio Exhibit
(6) = (2) * (3) * (4) * (5)
(7) = (6) / (1)
(11) = Min { [ (9) / (10) ] ^ 0.5 , 1.0 }
(13) = (8) * (11) + (12) * [ 1.0 - (11) ]
(14) From Non-Modeled Cat Exhibit
(15) From Hurricane Catastrophe Model
(16) = (13) + (14) + (15)
(17) From Cost of Reinsurance Exhibit
(18) From Expense Exhibit - 1
(19) From Expense Exhibit - 1
(21) = 100% - (19) - (20)
(22) = [ (16) + (17) + (18) ] / (21)

(5)

(6)

ULAE
Factor
1.0118
1.0118
1.0118
1.0118
1.0118

Projected
Ultimate NonCat Losses
and LAE
$6,235,484
$5,716,355
$4,800,456
$5,857,361
$7,725,146
$30,334,802

(8) Selected Projected Non-Cat Pure Premium
(9) Number of Claims
(10) Claims Required for Full Credibility
(11) Credibility
(12) Regional Non-Cat Pure Premium
(13) Credibility-Weighted Non-Cat Pure Premium
(14) Non-Modeled Cat Pure Premium
(15) Modeled Cat Pure Premium
(16) Total Pure Premium
(17) Projected Net Reinsurance Cost Per Exposure
(18) Projected Fixed Expense Per Exposure
(19) Variable Expense Provision
(20) Profit and Contingency Provision
(21) Variable Permissible Loss Ratio
(22) Indicated Rate
(23) Selected Rate

(7)
Projected
Non-Cat
Pure
Premium
$488.67
$447.78
$374.89
$456.39
$576.03
$469.75
$469.75
683
1082
79.5%
$585.75
$493.59
$29.11
$74.57
$597.27
$15.68
$77.74
13.8%
5.0%
81.2%
$850.30
$850.30

Columns 2 - 7 show the calculation of the projected non-cat pure premium (including LAE).
Rows 9 - 3 show the derivation of the credibility-weighted non-cat PP.
Row 11 full credibility standard: 1,082 claims based on the classical credibility approach; partial credibility is
calculated using the square root rule.
Row 22 indicated rate per exposure: Sum of the total PP (Row 16), the projected fixed expense per exposure
(Row 18), and the projected net reinsurance cost per exposure (Row 17), divided by the VPLR (Row 21).

Exam 5, V1b

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Appendix B – Homeowners Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
NON-MODELED CAT EXHIBIT
This exhibit outlines the calculation of the non-modeled catastrophe provision.
State XX
Wicked Good Insurance Company
Homeowners
Calculation of Non-Modeled Cat Loading

Calendar
Year
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015

(1)
Amount of
Insurance
Years
($000s)
$1,752,020
$1,911,500
$2,110,710
$2,333,580
$2,494,580
$2,545,420
$2,631,470
$2,738,710
$2,858,230
$2,927,850
$2,936,440
$2,923,330
$2,910,500
$2,944,090
$2,916,440
$2,665,300
$2,771,912
$2,882,788
$2,998,100
$3,208,151

(3) = (2) / (1)
(4) = Average of (3)
(5) From ULAE Ratio Exhibit
(6) = (4) * (5)
(7) From AIY Projection Exhibit
(8) = (6) * (7)

(2)
Reported
Cat Losses
and
Paid ALAE
$4,412
$26,236
$155,872
$38,689
$145,490
$227,118
$222,464
$833,316
$173,649
$2,668,809
$96,981
$256,753
$54,333
$475,524
$1,230
$70,299
$485,029
$29,025
$69,868
$178,200

(3)

Cat-to-AIY
Ratio
0.003
0.014
0.074
0.017
0.058
0.089
0.085
0.304
0.061
0.912
0.033
0.088
0.019
0.162
0.000
0.026
0.175
0.010
0.023
0.056

(4) All-Year Arithmetic Average
(5) ULAE Factor
(6) Non-Modeled Cat Provision Per AIY
(7) Selected Average AIY Per Exposure
(8) Non-Modeled Cat Pure Premium

0.110
1.012
0.112
$262.20
$29.28

Column 1:
 AIY (in $000s) represents the sum total of amount of insurance for all policies in-force during the CY.
 If the non-modeled cat provision was based on the ratio of non-modeled cat losses and ALAE to house
years, the ratio would increase over time due to the influence of inflation and other factors on the
numerator during the twenty year period.
 Using AIY in the denominator adjust the ratio for inflation. .

Exam 5, V1b

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Appendix B – Homeowners Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
AIY PROJECTION EXHIBIT
The projected average AIY is used to calculate the expected non-modeled catastrophe pure premium.
The AIY Projection Exhibit details how the projected average AIY is calculated.
State XX
Wicked Good Insurance Company
Homeowners
Calculation of Projected Average AIY

Calendar
Year
2011
2012
2013
2014
2015
2016
2017
2018

(1)
Amount of
Insurance
Years
($000s)
$2,665,300
$2,771,912
$2,882,788
$2,998,100
$3,208,151

(2)

(3)

Earned
Exposure
12,760
12,766
12,805
12,834
13,411

AIY-to-Earned
Exposure
Ratio
$208.88
$217.13
$225.13
$233.61
$239.22

(4)

Annual
Change
4.0%
3.7%
3.8%
2.4%

(6) Projected Average AIY in Effective Period
(3) = (1) / (2)
(7) Selected AIY in Effective Period
(4) = Current Year (3) / Prior Year (3) - 1.0
(5) Exponential Fit of (3) Using Data From Calendar Years 2011 Through 2015
(6) Average of (5) For Latest 2 Years

(5)
AIY-to-Earned
Exposure
Exponential
Fit
$209.58
$216.92
$224.52
$232.39
$240.53
$248.96
$257.68
$266.71
$262.20
$262.20

Row 6: Average AIY for the effective period (PY 2017), or the average of Column 5 for 2017 and 2018.
Row 7 shows the selected projected average AIY.

Exam 5, V1b

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Appendix B – Homeowners Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
REINSURANCE EXHIBIT
The net reinsurance cost per exposure considers both the reinsurance recoveries and the cost of the
reinsurance contract.
State XX
Wicked Good Insurance Company
Homeowners
Cost of Reinsurance
(1) Expected Reinsurance Recoveries
(2) Cost of Reinsurance (Expected Ceded Premium)
(3) Net Cost of Reinsurance
(4) Latest Year Exposures
(5) Expected Annual Exposure Increase
(6) Projection Period
(7) Projected Exposures
(8) Projected Net Reinsurance Cost Per Exposures

$458,672
$673,248
$214,576
13,411
1.0%
2.0
13,681
$15.68

(3) = (2) - (1)
(4) From Pure Premium Indication Exhibit
(5) Based on Company Goals
(6) From Midpoint of Latest Year to Midpoint of Reinsurance
Contract [ (07/01/2015) to (07/01/2017) ]
(7) = (4) * [ 1.00 + (5) ] ^ (6)
(8) = (3) / (7)

Row 1: Expected reinsurance recoveries from the reinsurance contract (obtained from the output of
catastrophe models and is the expected recoveries in an “average year”).

LOSS DEVELOPMENT EXHIBIT
This is the same procedure used for the personal automobile example in Appendix A. Thus, the same
comments apply.
LOSS TREND EXHIBIT
This is the same procedure used for the personal automobile example, except that the data is at the pure
premium level rather than at the frequency and severity level. Thus, the same comments apply.
ULAE RATIO EXHIBIT
This is the same procedure used for the personal automobile example. Thus, the same comments apply.
EXPENSE EXHIBIT
The U/W expense provisions are determined using the exposure-based projection method.
Assumes the historical relationships of variable expenses to premium and fixed expenses to exposures are
expected to continue during the projected period.

Exam 5, V1b

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Appendix B – Homeowners Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Sheet 1
Expenses are divided into: GE; Other Acq; TL&F; and Comm and brokerage.
The calculations and selections are performed for each category independently.
State XX
Wicked Good Insurance Company
Homeowners
Expense Calculation
2013

2014

2015

(1) General
a Countrywide Expenses
b % Assumed Fixed
c Fixed Expense $ [(a)*(b)]
d Countrywide Earned Exposures
e Fixed Expense Per Exposure [(c)/(d)]
f Variable Expense % [(a)*(1.0-(b))]
g Countrywide Earned Premium
h Variable Expense % [(f)/(g)]
(2) Other Acquisition
a Countrywide Expenses
b % Assumed Fixed
c Fixed Expense $ [(a)*(b)]
d Countrywide Written Exposures
e Fixed Expense Per Exposure [(c)/(d)]
f Variable Expense % [(a)*(1.0-(b))]
g Countrywide Written Premium
h Variable Expense % [(f)/(g)]
(3) Taxes, Licenses and Fees
a State Expenses
b % Assumed Fixed
c Fixed Expense $ [(a)*(b)]
d State Written Exposures
e Fixed Expense Per Exposure [(c)/(d)]
f Variable Expense % [(a)*(1.0-(b))]
g State Written Premium
h Variable Expense % [(f)/(g)]
(4) Commission and Brokerage
a State Expenses
b % Assumed Fixed
c Fixed Expense $ [(a)*(b)]
d State Written Exposures
e Fixed Expense Per Exposure [(c)/(d)]
f Variable Expense % [(a)*(1.0-(b))]
g State Written Premium
h Variable Expense % [(f)/(g)]

$0
$0
$0
12,820
13,123
13,478
$0.00
$0.00
$0.00
$1,115,970 $1,207,693 $1,244,644
$11,217,062 $11,810,250 $12,332,420
9.9%
10.2%
10.1%

(5) Total Fixed Expenses
(6) Fixed Expense Trend
(7) Trend Period
(8) Fixed Expense Trend Factor
(9) Projected Fixed Expense
(10) Variable Expense Provision

(1e) + (2e) + (3e) + (4e)
From Expense Exhibit - 2
From 07/01/2015 to 07/01/2017
[1.0 + (6)]^ (7)
(5) * (8)
(1h) + (2h) + (3h) + (4h)

$2,238,241

$2,301,402

3-year
Weighted Avg

Selected

$2,432,343
75.0%

$1,678,681 $1,726,052 $1,824,257
56,884
57,452
58,027
$29.51
$30.04
$31.44
$559,560
$575,351
$608,086
$51,764,213 $53,143,516 $53,965,296
1.1%
1.1%
1.1%
$2,582,786

$2,715,731

$30.33

$31.44

1.1%

$2,912,054
75.0%

$1,937,090 $2,036,798 $2,184,041
56,602
57,740
58,317
$34.22
$35.28
$37.45
$645,697
$678,933
$728,014
$51,907,954 $53,554,406 $55,235,122
1.2%
1.3%
1.3%
$200,879

$205,363

$35.65

$37.45

1.3%

$210,002
25.0%

$50,220
$51,341
$52,501
12,820
13,123
13,478
$3.92
$3.91
$3.90
$150,659
$154,022
$157,502
$11,217,062 $11,810,250 $12,332,420
1.3%
1.3%
1.3%
$1,115,970

$1,207,693

$3.91

$3.90

1.3%

$1,244,644
0.0%

$0.00

10.1%

10.1%
$72.78
3.4%
2.00
1.0681
$77.74
13.8%

Row “a” shows the expense associated with each category for each of the 3 CYs. The expense is either at the
state or countrywide level, depending on the category.
Row “d” displays the exposure per year; the exposures are state or countrywide and written or earned
depending on the expense category.
Row 7 is the length of the trend period (from the average written date of the latest year to the average written
date for the time period the rates are to be in effect).

Exam 5, V1b

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Appendix B – Homeowners Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Sheet 2: Outlines the procedure for selecting the fixed expense trend.
State XX
Wicked Good Insurance Company
Homeowners
Calculation of Annual Expense Trend
(1) Employment Cost Index - Finance, Insurance & Real Estate, excluding Sales Opportunity (annual change over latest 2 years)
U.S. Department of Labor

4.8%

(2) % of Other Acquisition and General Expense used for Salaries and Employee Relations & Welfare Insurance Expense Exhibit, 2015
(3) Consumer Price Index, All Items (annual change over latest 2 years)
(4) Annual Expense Trend [ (1) * (2) ] + [ (3) * { 100% - (2) } ]
Selected Annual Expense

Exam 5, V1b

50.0%

1.9%
3.4%
3.4%

Page 308

 2014 by All 10, Inc.

Appendix B – Homeowners Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Questions from the 2010 exam
27. (7 points) Given the following information for a Homeowners book of business in State X:
• All policies are annual.
• Proposed effective date is January 1, 2011 and rates will be in effect for twelve months.
• Selected current loss trend is 3%.
• Selected projected loss trend is 2%.

Calendar
Accident
Year
2005
2006
2007
2008
2009
2010
2011
2012

•
•
•
•
•
•
•
•
•
•

Earned
Exposures
5,400
8,600
9,600
10,000
11,000

Non-Cat
Reported
Losses and
Paid ALAE
$2,025,000
$3,440,000
$3,408,000
$5,400,000
$5,500,000

Loss
Development
Factor to
Ultimate
1.0000
1.0500
1.1000
1.1500
1.2000

ULAE
Factor
1.050
1.050
1.050
1.050
1.050

•

Amount of Insurance Years
(AIY)
to
Earned Exposure
Exponential Fit
$270.00
$283.50
$297.68
$312.56
$328.19
$344.60
$361.83
$379.92

Selected fixed expense using expense data through 2009 = $47 per exposure
Variable expenses = 14.4% of premium
Fixed expense trend = 3.0%
Profit and contingency provision = 9% of premium
State X experience is fully credible
Modeled catastrophe pure premium = $35.15
Arithmetic average of last 20 years' non-modeled cat to AIY ratio = 0.370
Expected reinsurance recoveries from a reinsurance contract with coverage from June 1, 2011 to
June 1, 2012 = $350,000
Cost of reinsurance = $680,000
Assume no projected growth in exposures

a. (3 points) Calculate the projected ultimate non-catastrophe pure premium.
b. (1 point) Calculate the projected non-modeled catastrophe pure premium.
c. (0.75 point) Calculate projected net reinsurance cost per exposure.
d. (0.75 point) Calculate the projected fixed expense per exposure.
e. (1.5 points) Calculate the indicated rate.

Exam 5, V1b

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Appendix B – Homeowners Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Solutions to questions from the 2010 exam
Question 27 Initial comments: The following is the Model CAS solution to question 27. Rather than
expound upon the solution, its best for candidates to see ‘exactly’ what it takes to solve the problem in the
most efficient manner (which this model solution does). Our notes appear in << >> below.
a. (3 points) Calculate the projected ultimate non-catastrophe pure premium.
1

CAY

2
Non-Cat
Rept Loss
+Pd ALAE

2005
200 6
2007
200 8
2009

5400
8600
9600
10000
11000

2025
3440
3408
5400
5500

LDF

ULAE
Factor

Trend
Factor

1.00
1.05
1.10
1.15
1.20

1.05

1.1826
1.1482
1.1147
1.0823
1.0508

44,600

6
Proj ult
Non-Cat
Loss & LAE

7
Proj
Non-Cat
PP

2514.503
4354.663
4387.727
7057.137
7282.044

465.65
506.36
457.55
705.71
662.00

25,596.074

573.90

Notes: (1) – (4) Given; (2) & ( 6 ) in 000’s
(5) ’09 trended from 6/30/09 to 12/31/11 @ 2%, ’08 & prior @ 3 % to‘09
(6) = (2)x(3)x(4) x(5)
(7)= (6)/ (1) x 1000
<< See PP indication exhibit, Appendix B, Page 5>>

b. (1 point) Calculate the projected non-modeled catastrophe pure premium.
(1) 20 yr non‐modeled CAT – to AIY Ratio

0.370

(2) ULAE Factor
(3) Aug AIY – to –EE ratio in projected period

1.050
370,875

(4) Projected non‐modeled CAT PP

144.08

Notes: (1), (2) Given
(3) = avg fitted CAY 2011 & CAY 2012 Ratio
(4) = (1) x(2) x(3)
<< See Non-Modeled Cat Exhibit, Appendix B, Page 6>>

c. (0.75 point) Calculate projected net reinsurance cost per exposure.
Projected Net Reins cost per exposure = ( 680,000 – 350,000)/11,000 = 30
<< See Reinsurance Exhibit, Appendix B, Page 8>>

d. (0.75 point) Calculate the projected fixed expense per exposure.
Trend from 7/1/09 to 7/1/11 (Avg written date) = 47*1.032 = 49.86

e. (1.5 points) Calculate the indicated rate.
(1) Non‐CAT PP (part a.) =

573.90

(2) CAT PP = modeled (given) + non modeled (part b.) = 35.15+144.08 =
(3) Net Reins per EE (part c.) + fixed expense per EE (part d.)

179.23
79.86

(4) Permissible LR = 1.0 ‐ .144 ‐ .09

0.766

(5) Indicated Rate= [(1)+(2)+(3)]/(4)
Notes (4) = 1.0 – Var. Exp provision – Target & cont. provision
<>

Exam 5, V1b

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Appendix C – Medical Malpractice Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
The overall rate level indication for a medical malpractice (MM) insurance program using the loss ratio
indication approach is shown.
 While MM insurance can be written on an occurrence or CM; the data used in this example is based on
occurrence policies.
 Due to the longer-tailed nature and higher frequency of large losses, the data is more volatile and
ratemaking techniques are slightly different than those used for personal automobile and homeowners.
All policies are annual and the proposed effective date of the rate change in State XX is 5/1/2016.
The individual exhibits are as follows:
• LR Indication: The overall indicated rate change using the loss ratio method based on 5 years of State
XX calendar-accident year experience evaluated as of 9/30/2015.
• Current Rate Level: Calculation of the current rate level factors using the parallelogram method.
• Loss Development: Selected ultimate loss and ALAE using a combination of the chain ladder and
Bornhuetter-Ferguson methods.
• Net Trend: Selection of net trend factors based on historical changes of frequency, severity, and
premium.
• Expense and ULAE Ratio: Derives the expense provision using all ULAE and underwriting expenses.
LR (LOSS RATIO) INDICATION EXHIBIT
The overall rate level indication is calculated on the LR (Loss Ratio) Indication Exhibit.
The projected loss and ALAE ratio is calculated and compared to the permissible loss ratio to obtain the
indicated statewide rate change (which is credibility-weighted with the countrywide rate indication)
Note: Certain factors in the exhibits below are displayed to a certain number of decimal places. However, certain calculations shown
in these exhibits may be based on unrounded factors. Thus, these values do not match those in the corresponding exhibit in the text.
However, the formulas, which are shown correctly, are what matter most when preparing for the exam.

Exam 5, V1b

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Appendix C – Medical Malpractice Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
State XX
Wicked Good Insurance Company
Medical Malpractice
Indicated Rate Change
(1)

(2)

(3)

(4)

(5)

(6)

(7)

Projected

Current

Earned

Ultimate

Net

Ultimate

Accident

Earned

Rate Level

Premium

Loss

Trend

Loss

Loss and

Year

Premium

Factor

@ CRL

and ALAE

Factor

and ALAE

ALAE Ratio

2010

$14,904,664

1.2029

$17,928,820 $11,673,500

1.7902

$20,897,845

116.6%

2011

$14,494,543

1.2058

$17,476,994 $11,199,932

1.6439

$18,411,446

105.3%

2012

$14,442,449

1.2724

$18,376,646

1.5095

$9,492,557

51.7%

2013

$14,834,605

1.3018

$19,312,280 $18,257,633

1.3862

$25,308,200

131.0%

2014

$18,265,093

1.2390

$22,631,001 $23,362,271

1.2729

$29,737,466

131.4%

Total

$76,941,354

$103,847,514

108.5%

Calendar

$6,288,376

$95,725,741 $70,781,712

(1) From Net Trend Exhibit - 1

(8) Selected Loss and ALAE Ratio

(2) From Current Rate Level Exhibit - 2

(9) Expense and ULAE Ratio

(3) = (1)*(2)

(10) Profit and Contingency Provision

(4) From Loss Development Exhibit - 6

(11) Permissible Loss Ratio

(5) From Net Trend Exhibit - 3

(12) Statewide Indicated Rate Change

(6) = (4)*(5)

(13) Number of Reported Claims

(7) = (6)/(3)

(14) Claims Required for Full Credibility Standard

(9) From Expense & ULAE Ratio Exhibit - 2

(15) Credibility

(11) = 100% - (9) - (10)

(16) Countrywide Indicated Rate Change

(12)= [(8)/ (15)] - 1.0

(17) Credibility - Weighted Indicated Rate Change

(13) Derived From Net Trend Exhibit - 2

(18) Selected Rate Change

108.5%
34.8%
‐5.0%
70.2%
54.6%
283
683
64.4%
18.5%
41.7%
41.7%

(14) = Min { [ (13) / (14) ] ^ 0.5, 1.0 }
(17) = (12) * (15) + (16) * [ 1.0 - (15) ]

(12)= [(8)/ (11)] - 1.0; Also
Column 4: ultimate losses and ALAE selected for each AY.
Companies cap losses at the basic limit to minimize the impact of extraordinary losses, but since basic limits
losses were not available, total limit losses were used.
Row 6 selected loss and ALAE ratio is the projected loss and ALAE ratio across all accident years.
Row 10 shows the target UW profit provision.
Note that the UW profit provision is negative; Recall that the insurer’s total profit is UW profit plus investment
income.; II is expected to be high in this long-tailed line of business, so WU profit can be negative.
Row 12 statewide rate indication is calculated by comparing the selected projected loss and ALAE ratio (Row
8) to the permissible loss ratio (Row 11).
Rows 13 - 15 show the calculation of the credibility factor.
Row 13 shows the number of reported claims for the five most recent accident years as of 9/30/ 2015.
The number of claims for full credibility, 683, is based on a 95% probability that the observed experience will
be within 7.5% of the expected experience.
Row 15, the credibility measure, is calculated using the square root rule.
The countrywide indication is displayed in Row 16. Row 17 shows the credibility-weighted rate indication of the
statewide and countrywide results. A rate change is then selected in Row 18.

Exam 5, V1b

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Appendix C – Medical Malpractice Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
CURRENT RATE LEVEL EXHIBIT
These two sheets use the same parallelogram method that was used to adjust earned premium to current rate
level in the personal automobile rating example.
Sheet 1: cumulative rate level indices for each rate level group during or after the historical period.
Sheet 2: current rate level factors computation.
LOSS DEVELOPMENT EXHIBIT
The calculation of ultimate loss and ALAE uses three loss development techniques, since it is common to use
multiple methods in long-tailed lines of business.
The results of the three techniques are used to judgmentally select ultimate loss & ALAE by AY
Sheets 1-3
Sheets 1 and 2: Calculation and selection of age-to-ultimate LDFs using the chain ladder approach.
This is the same approach as used in the personal automobile and homeowners rating examples.
Sheet 1: The chain ladder approach applied to paid losses and paid ALAE.
Sheet 2: The chain ladder approach applied to reported losses and paid ALAE.
The losses are total limit losses; if capped losses were available, the loss development analysis would have
been conducted on that basis as well.
Sheet 3: Calculation of claim count development factors based on historical reported claim counts. The
resulting ultimate claim counts are used in deriving the net loss ratio trend (discussed later).
State XX
Wicked Good Insurance Company
Medical Malpractice
Claim Count Development Factors
Accident
Year
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014

Reported Losses & Paid ALAE Evaluated As of
21 Months 33 Months 45 Months 57 Months
33
41
52
59
15
33
48
48
26
52
74
85
37
59
70
85
44
81
85
107
19
44
59
67
15
44
63
63
48
59
67
33
56
30

69 Months
63
48
85
85
107
67

81 Months
63
48
89
85
107

93 Months
63
48
93
85

105 Months 117 Months 129 Months
63
63
63
48
48
96

21-33
1.2424
2.2000
2.0000
1.5946
1.8409
2.3158
2.9333
1.2292
1.6970

33-45
1.2683
1.4545
1.4231
1.1864
1.0494
1.3409
1.4318
1.1356

45-57
1.1346
1.0000
1.1486
1.2143
1.2588
1.1356
1.0000

57-69
1.0678
1.0000
1.0000
1.0000
1.0000
1.0000

69-81
1.0000
1.0000
1.0471
1.0000
1.0000

81-93
1.0000
1.0000
1.0449
1.0000

93-105
1.0000
1.0000
1.0323

105-117
1.0000
1.0000

117-129
1.0000

(1) All-Year Average
(2) 3-Year Average
(3) 4-Year Average
(4) Average Excluding Hi-Lo
(5) Geometric Average

1.8948
1.9532
2.0438
1.8415
1.7370

1.2863
1.3028
1.2394
1.2977
1.2542

1.1274
1.1315
1.1522
1.1266
1.1397

1.0113
1.0000
1.0000
1.0000
1.0089

1.0094
1.0157
1.0118
1.0000
1.0103

1.0112
1.0150
1.0112
1.0000
1.0140

1.0108
1.0108
1.0108
1.0000
1.0147

1.0000
1.0000
1.0000

1.0000

(6) Selected Age-to-Age
(7) Age-to-Ultimate

1.7370
2.6041

1.2542
1.4992

1.1397
1.1953

1.0089
1.0488

1.0103
1.0396

1.0140
1.0289

1.0147
1.0147

1.0000
1.0000

Age-to-Age Factors
2005
2006
2007
2008
2009
2010
2011
2012
2013

129 to Ult

1.0000
1.0000

(1) Straight Average
(2) Straight Average
(3) Straight Average
(4) Sraigtht Average Excluding Highest and Lowest Values
(5) = Average Weighted by Loss
(7) = Cumulative Product of (6)

Exam 5, V1b

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 2014 by All 10, Inc.

Appendix C – Medical Malpractice Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Sheets 4-5
Since MM has relatively more large losses than other lines of business, the link ratio patterns are less stable,
and is especially true for the more recent evaluation points; thus the reported Bornhuetter-Ferguson (BF)
method (Sheets 4 and 5) is used to develop losses and ALAE to ultimate for the 3 most recent AYs.
 A two-year (2010-2011) average expected loss and ALAE ratio is calculated and adjusted to the rate and
cost level of each of the three most recent years (i.e. 2012, 2013, and 2014).
 This ratio is multiplied by EP to compute expected losses and ALAE for each of the three years.
 Age-to-ultimate factors from the reported chain ladder method are used to calculate the portion of these
losses that are unreported as of 9/30/2015.
 Add these estimated unreported losses to actual reported losses as of the same valuation date to derive
the ultimate losses and ALAE for each year.
Sheet 4: Calculation of the two-year (2010-2011) average ultimate loss and ALAE ratio forecasted to the rate
level and cost level of 2011.
Loss Development - 4
State XX
Wicked Good Insurance Company
Medical Malpractice
Bornhuetter-Ferguson Developed Losses

Accident
Year
2010
2011

(1)

(2)

(3)

Earned
Premium
14,904,664
14,494,543

Ultimate
Loss and
ALAE
$11,673,500
$11,199,932

Ultimate
Loss and
ALAE Ratio
78.3%
77.3%

(4)
Adjustment
to Avg
Rate Level
in 2011
0.9976
1.0000

(1) From Net Trend - 1
(2) From Loss Development Exhibit - 6
(3) = (2) / (1)
(4) From (2) in Current Rate Level - 2
(5) From (14) in Net Trend - 1
(6) From 07/01/20XX to 07/01/2011
(7) = [ 1 + (5) ] ^ (6)
(8) = (3) / (4) * (7)
(9) Straight Average of (8)

(5)

(6)

Selected
BF Net
Trend
13.3%
13.3%

Trend
Length
1.00
0.00

(7)
Net
Trend
Adjustment
to 2011
1.1330
1.0000

2- Year Avg
Ultimate Loss and
ALAE Ratio
(9) (2010-2011)

(8)
Ultimate
Loss and
ALAE Ratio
as of 2011
88.9%
77.3%

83.1%

Column 2:
Column 3:
Column 4:
Column 5:

Is a straight average of ultimate loss and ALAE from the reported and paid chain ladder methods.
Is the ratio of Column 2 to Column 1.
Is the ratio of the 2011 average rate level to the average rate level of each respective year.
Is based on a review of the trend in severity and adjusted frequency from 2005-2011 (see Net Trend
– 1 exhibit). As ultimate losses have not yet been derived for the most recent years, this trend
analysis (for the purpose of applying the BF method) does not consider the most recent years.
Column 6: The number of years from the midpoint of each accident year (7/1/20xx) until the midpoint of AY
2011 (7/1/2011).

Exam 5, V1b

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Appendix C – Medical Malpractice Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Sheet 5: Calculation of the ultimate loss and ALAE ratio for AY 2012-2014, using the BF method.
State XX
Wicked Good Insurance Company
Medical Malpractice
Bornhuetter-Ferguson Developed Losses
(1)
(2)
2- Year Avg
Ultimate Loss
and ALAE
Accident
Ratio
Earned
Year
(2010-2011) Premium
2012
83.1% 14,442,449
2013
83.1% 14,834,605
2014
83.1% 18,265,093

(3)

(4)

(5)

Average
Rate
Level
0.9454
0.9240
0.9708

Rate
Level
2011
0.9976
0.9976
0.9976

(6)

Average
Selected
Rate
BF
Level
Net
Adjustment
Trend
0.9476
13.3%
0.9262
13.3%
0.9732
13.3%

(7)
Trend
Length
from
2011
1.00
2.00
3.00

(8)

(9)

Expected
Net
Losses
Trend
and ALAE
Adjustment
Ratio
1.1330
99.4%
1.2837
115.2%
1.4544
124.2%

(1) From Loss Development Exhibit - 4
(2) From Net Trend - 2
(3) From Current Rate Level - 2
(4) From Current Rate Level - 2
(5) = (3) / (4)
(6) From Net Trend - 1
(7) From 07/01/2011 to 07/01/20XX
(8) = [ (1) + (6) ] ^ (7)
(9) = (1) / (5) * (8)
(10)

(11)

(12)

(13)

Expected
Reported
Losses
Reported
Losses
and
Age-to-Ult Percent
and ALAE
ALAE
Factor
Unreported a/0 9/30/15
14,351,088 1.9190
47.9%
$1,954,200
17,087,637 4.9128
79.6%
$3,873,900
22,687,265 36.3756
97.3%
$1,298,700

(14)
(15)
Expected Losses
B-F
and ALAE
Ultimate
Not Yet
Losses
Reported
and
a/o 9/30/15
ALAE
$6,872,518
$8,826,718
$13,609,484
$17,483,384
$22,063,571
$23,362,271

(10) = (2) * (9)
(11) From Loss Development Exhibit - 2
(12) = 1 - 1 / (11)
(13) From Loss Development Exhibit - 6
(14) = (10) * (12)
(15) = (13) + (14)

Exam 5, V1b

Page 315

 2014 by All 10, Inc.

Appendix C – Medical Malpractice Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Sheet 6: Shows the derivation of the selected ultimate loss and ALAE for each AY.

(1)

Acident
Year
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014

Paid
Losses
& ALAE
a/o 9/30/15
$5,735,000
$2,701,000
$4,591,700
$8,524,800
$7,377,800
$7,770,000
$7,895,800
$1,029,200
$170,200
$873,200

State XX
Wicked Good Insurance Company
Medical Malpractice
Developed Loss Selection
(2)
(3)
(4)
Reported
Losses
& ALAE
a/o 9/30/15
$5,735,000
$2,701,000
$4,739,700
$8,543,300
$7,414,800
$11,673,500
$8,191,800
$1,954,200
$3,873,900
$1,298,700

Paid
Age-toUltimate
Factor
1.0000
1.0000
1.0000
1.0040
1.0426
1.1596
1.6452
6.3690
23.6441
140.4900

Reported
Age-toUltimate
Factor
1.0000
1.0000
1.0000
1.0125
1.0284
1.0837
1.1487
1.9190
4.9128
36.3756

(1) From Loss Development Exhibit - 1
(2) From Loss Development Exhibit - 2
(3) From Loss Development Exhibit - 1
(4) From Loss Development Exhibit - 2
(5) = (1) * (3)
(6) = (2) * (4)
(7) From Loss Development Exhibit - 5

(5)
Ultimate
Losses & ALAE
Using Paid
Age-to-Ultimate
Factors
$5,735,000
$2,701,000
$4,591,700
$8,558,831
$7,691,841
$9,010,050
$12,990,063
$6,555,025
$4,024,229
$122,675,851

(6)
Ultimate
Losses & ALAE
Using Reported
Age-to-Ultimate
Factors
$5,735,000
$2,701,000
$4,739,700
$8,650,182
$7,625,361
$12,650,602
$9,409,801
$3,750,035
$19,031,882
$47,241,015

(7)
Ultimate
Losses
Using
B-F Method

$8,826,718
$17,483,384
$23,362,271

(8)

Selected
Losses
& ALAE
$5,735,000
$2,701,000
$4,739,700
$8,604,507
$7,658,601
$11,673,500
$11,199,932
$6,288,376
$18,257,633
$23,362,271

(8) Judgementally Selected Based On Combinations of (5), (6) and (7)
2005-2011: max [ (2), average of (5) and (6) ]
2012-2013: max [ (2), average of (6) and (7) ]
2014 uses (7) only

Because of the volatility in the more recent years:
 An average of the reported chain ladder and BF results is used for AYs 2012 and 2013
 The BF result is used for AY 2014.
For all AYs, an additional criterion is applied to the selected ultimate loss and ALAE: each year’s selected ultimate
loss and ALAE must be equal to or greater than that year’s reported losses and paid ALAE as of 9/30/2015.
NET TREND EXHIBIT
In the personal auto: premium trend and loss trend components are analyzed and selected separately.
In MM: premium trend is considered within the loss trend.
 Adjusted frequency trend is based on ratios of ultimate claim counts to earned premium at current rate
level; changes in this ratio represent the net effect of changes in frequency and average premium.
 The severity trend is based [ultimate loss and ALAE/ultimate claim counts] (both derived using the chain
ladder method).
 The selected net trend is based on the combined severity trend and the adjusted frequency trend.
Due to the long-tailed nature of MM, loss trends are based on ultimate losses and ultimate claim counts rather
than paid losses and reported claim counts (common in short-tailed lines).
The BF method considers trended losses in deriving ultimate loss estimates for the 3 most recent years, but
the trend used within this method does not consider the 3 most recent years; thus, there are separate trends
selected for the BF method and for the overall LR indication.
Sheet 1: Trend analysis conducted for the BF method.
Sheet 2: Trend analysis for the LR indication.

Exam 5, V1b

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 2014 by All 10, Inc.

Appendix C – Medical Malpractice Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Sheet 1
 Severity and adjusted frequency trends are analyzed separately for AYs 2005- 2011.
 Exponential trends are fit to the data; Trend selections are made based on the results.
State XX
Wicked Good Insurance Company
Medical Malpractice
Net Trend Calculation for Borhuetter-Ferguson Method

Accident
Year
2005
2006
2007
2008
2009
2010
2011

(1)
Selected
Ultimate
Loss &
ALAE
$5,735,000
$2,701,000
$4,739,700
$8,604,507
$7,658,601
$11,673,500
$11,199,932

(2)
Reported
Claim
Count
63
48
96
85
107
67
63

(3)
Reported
Age-toUltimate
Factor
1.0000
1.0000
1.0000
1.0147
1.0289
1.0396
1.0488

(4)

(5)

(6)

Developed
Claim
Count
63
48
96
86
110
70
66

Severity
$91,032
$56,271
$49,372
$99,762
$69,562
$167,602
$169,509

Earned
Premium
$17,944,254
$17,942,995
$18,532,758
$18,265,093
$15,590,108
$14,904,664
$14,494,543

(7)
Current
Rate
Level
Factor
1.2029
1.2029
1.2029
1.2029
1.2029
1.2029
1.2058

(1) From Loss Development Exhibit - 6
Exponential
(2) From Loss Development Exhibit - 3
Trend
(3) From Loss Development Exhibit - 3
(10)
2005-2011
17.0%
(11)
(4) = (2) * (3)
(5) = (1) / (4)
(7) From Current Rate Level Exhibit - 2
Selected
(8) = (6) * (7)
Severity
(9) = [ (4) / (8) ] * 1,000,000
(12)
Trend for BF
10.0%
(13)
(10) Exponential Fit to Severity (2005-2011)
(11) Exponential Fit to Adjusted Frequency (2005-2011)
(12) Forecasted Severity Trend based on (10) and judgment, for use in BF loss development method
(13) Forecasted Adj Freq Trend based on (11) and jdgmnt in BF loss development method
(14)

(8)
Earned
Premium
at Current
Rate Level
$21,585,143
$21,583,629
$22,293,055
$21,971,080
$18,753,341
$17,928,820
$17,476,994

(9)

Adjusted
Prequency
2.92
2.22
4.31
3.93
5.87
3.88
3.78

2005-2011

Exponential
Trend
8.2%

Selected
Adjusted
Frequency
Trend for BF

3.0%

Selected
Total Net
Trend for BF

13.3%

Column 9: By dividing developed claim counts by premium instead of exposures, adjusted frequency reflects
frequency and premium trends within one measure.
Rows 12 and 13 selected trends are made in consideration of the exponential trends and judgment with
respect to the volatility of the data.
Row 14: The selected severity and adjusted frequency trends are combined to form the net trend
Sheet 2: Same format as Sheet 1 except that the most recent accident years (2012-2014) are considered.
 Exponential trends are fit to 2005-2014 and to 2010-2014.
 Row 16 selected net trend relies more heavily on the recent period.

Exam 5, V1b

Page 317

 2014 by All 10, Inc.

Appendix C – Medical Malpractice Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Sheet 3: The calculation of each AYs net trend factors for use in the LR Indication.
State XX
Wicked Good Insurance Company
Medical Malpractice
Net Trend Factors

Accident
Year
2010
2011
2012
2013
2014

(1)
Selected
Net Trend
8.9%
8.9%
8.9%
8.9%
8.9%

(2)
Trend
Period
6.83
5.83
4.83
3.83
2.83

(3)
Net Trend
Factor
1.7902
1.6439
1.5095
1.3862
1.2729

(1) From Net Trend Exhibit - 2
(2) From 07/01/20XX to 05/01/2017
(3) = [ 1.0 + (1) ] ^ (2)

Column 2: the number of years between the midpoint of the historical period (7/1/20XX) and the average
expected loss date for when the rates will be in effect (5/1/2017).
EXPENSE AND ULAE RATIO EXHIBIT
Unlike the personal automobile and homeowners, all U/W expenses are treated as variable expense and
ULAE are included within this analysis.
Due to the volatility of this line of business, the expense ratios are calculated using countrywide data.
Sheet 1: Computation of the selected ULAE ratio.
State XX
Wicked Good Insurance Company
Medical Malpractice
ULAE Ratio

Calendar Year
Year
2010
2011
2012
2013
2014
Total

(1)
Countrywide
Earned Premium
($000s)
$455,119
$724,423
$870,129
$596,311
$548,096
$3,194,078
(4) Selected Ratio

(2)
Countrywide
Paid ULAE
($000s)
$16,310
$34,010
$4,799
$10,086
$12,573
$77,778

(3)
ULAE
Ratio
3.6%
4.7%
0.6%
1.7%
2.3%
2.4%
2.4%

(3) = (2) / (1)

Column 3 selected ULAE ratio is based on the five-year ratio in Column 3, and while it is more intuitive to study
the relationship between ULAE and losses, ULAE are a small portion of the total expenses so comparing
ULAE to earned premium is acceptable.

Exam 5, V1b

Page 318

 2014 by All 10, Inc.

Appendix C – Medical Malpractice Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
Sheet 2: Expense ratio for each category of expense using the three most recent CY of countrywide data.
State XX
Wicked Good Insurance Company
Medical Malpractice
Expense and ULAE Ratio Calculation

(1) General Expenses
a Countrywide Expenses
b Countrywide Earned Premium
c Ratio [(a)/(b)]
(2) Other Acquisition
a Countrywide Expenses
b Countrywide Written Premium
c Ratio [(a)/(b)]
(3) Taxes, Licenses, and Fees
a Countrywide Expenses
b Countrywide Written Premium
c Ratio [(a)/(b)]
(4) Commission and Brokerage
a Countrywide Expenses
b Countrywide Written Premium
c Ratio [(a)/(b)]
(5) UW Expense Ratio
(6) ULAE Ratio
(7) UW Expense and ULAE Ratio

3-year
Weighted
2015 Average Selected

2013

2014

$67,766
$870,129
7.8%

$41,658
$596,311
7.0%

$35,243
$548,096
6.4%

7.1%

6.4%

$29,041
$768,631
3.8%

$17,853
$579,383
3.1%

$15,103
$576,253
2.6%

3.2%

2.6%

$21,678
$768,631
2.8%

$14,800
$579,383
2.6%

$12,225
$576,253
2.1%

2.5%

2.1%

$159,751
$768,631
20.8%

$123,221
$579,383
21.3%

$122,211
$576,253
21.2%

21.1%

21.2%

(1c) + (2c) + (3c) + (4c)
From Expense and ULAE Ratio Exhibit - 1
(5) + (6)

32.3%
2.4%
34.7%

(1b) from Expense and ULAE Ratio - 1
(3b) from (2b)
(4b) from (2b)

Row “a” shows expenses paid for that CY and Row “b” shows premium.
 EP is used to calculate the expense ratio for GE since these expenses are incurred throughout the life of
the policy.
 All other expense ratios use WP since these expenses are assumed to be incurred at policy inception
(when written).
All expenses are assumed to be variable (i.e. vary by premium).
The latest year in Row “c” historical variable expense ratios (Row “c”) is selected due to the downward trend
exhibited.
Row 7 is not trended which assumes that expenses and premium will increase/decrease at the same rate.

Exam 5, V1b

Page 319

 2014 by All 10, Inc.

Appendix D – Workers Compensation Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
The overall rate level indication using the loss ratio approach is shown.
 The example uses WC industry data to determine advisory loss costs, including LAE
 Individual WC insurers that intend to use these loss costs as a basis for rates must include their own U/W
expense and profit assumptions (described later in the appendix)
 Five AYs of experience evaluated as of 12/31/2016 are used.
Since it is industry data, the experience is more stable than that of an individual WC insurer.
An insurer may wish to use more years of data to increase the stability of the results.
 The experience is from annual policies and the proposed effective date for the revised loss costs is
7/1/2017.
The exhibits included in this appendix are as follows:
• Premium: Calculates projected loss cost premium.
• Indemnity: Determines the indemnity loss ratio for each AY.
• Medical: Determines the medical loss ratio for each AY.
• LAE: Determines the ALAE and ULAE factors.
• Indication: Combines medical and indemnity loss ratios with the ALAE and ULAE ratios to develop an
indicated change to advisory loss costs.
• Company: Computes the adjustment necessary to account for individual company UW expenses and
profit (as well as deviations to expected losses).
Note: Certain factors in the exhibits below are displayed to a certain number of decimal places. However, certain calculations shown in
these exhibits may be based on unrounded factors. Thus, these values do not match those in the corresponding exhibit in the text.
However, the formulas, which are shown correctly, are what matter most when preparing for the exam.

PREMIUM EXHIBIT
Historical loss cost premium needs to be adjusted for current rate level, exposure trend, and expected
experience modification factors.
Workers Compensation
Calculation of Projected Premium

Accident
Year
2012
2013
2014
2015
2016
Total

(1)
Industry
Loss
Cost
Premium
$3,900,972,841
$4,148,612,420
$4,334,300,493
$4,659,789,168
$4,795,461,580
$21,839,136,502

(2)
Annual
Payroll
Level
Change
2.5%
3.0%
3.7%
4.2%
3.5%

(3)
(4)
Exposure Trend
Factor to
Expected
Current Wage
Future Wage
Level
Level Change
1.152
6.1%
1.118
6.1%
1.078
6.1%
1.035
6.1%
1.000
6.1%

(5)
Factor to
Adjust to
Future Wage
Level
1.222
1.187
1.144
1.098
1.061

(6)
Historical
Average
Experience
Modification
0.991
0.985
0.981
0.982
0.957

(7)
Expected
Average
Experience
Modification
0.970
0.970
0.970
0.970
0.970

(8)
Projected
Loss
Cost
Premium
$4,666,705,987
$4,847,754,029
$4,903,940,552
$5,054,547,098
$5,157,100,516
$24,630,048,184

(1) Industry loss costs at current rate level (assuming no company derivations and no provision for expense and profit)
(2) Determined in separate study
(3) = [ 1.0 + (2NextRow) ] * (3NextRow)
(4) Based on 3% trend projected for 2 years
(5) = (3) * [1.0 + (4) ]
(6) Determined in a separate analysis
(7) Selected
(8) = (1) * (5) * (7) / (6)

Column 1: Loss cost premium:
 represents the hypothetical portion of the premium charged by insurers assuming the current advisory
loss costs and historical experience modification factors were used.
 does not reflect any company deviations from advisory loss costs or any provision for expense and profit.
 has been adjusted for subsequent changes in advisory loss costs (i.e. brought to current level) using the
extension of exposures technique.

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Appendix D – Workers Compensation Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Column 2:
 The exposure base for WC insurance is payroll, which is inflation-sensitive, so premium changes as
payroll changes.
 shows the historical changes in payroll by AY, assuming a constant number of workers.
Column 3: Converts the annual changes into cumulative factors such that the factor for the most recent AY
period (2016) is indexed to 1.00.
Column 4: is the wage increase expected between the most recent historical period and the time the rates are
to be in effect (i.e. the selected trend of 6.1% is based on an assumed trend of 3.0% for two years (=
(1.032) -1.0).
Column 5: combines the current and projected future wage changes into a composite exposure trend factor.
Per Chapter 15, insurers use ER to modify the manual rate for larger risks based on their actual experience.
Column 6: The average e-mod factor for each historical accident year
Column 7: The e-mod expected during the projected period
INDEMNITY EXHIBITS
Sheet 1: Indemnity Loss Development
Workers Compensation
Reported Indemnity Loss Development
Accident
Year
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015

12 to
24

24 to
36

1.861
1.910
1.931
1.873
1.952
1.782
1.448
1.503
1.684

3-Year
Average
Average
xHi Lo

1.792 1.247 1.100 1.055 1.035 1.022 1.013 1.009 1.006 1.004 1.003 1.002 1.001 1.001 1.001 1.000 1.001 1.001 1.001 1.001 1.001 1.001 1.001 1.001 1.001 1.001

Selected

1.792 1.247 1.100 1.055 1.035 1.022 1.013 1.009 1.006 1.004 1.003 1.002 1.001 1.001 1.001 1.000 1.001 1.001 1.001 1.001 1.001 1.001 1.001 1.001 1.001 1.001

1.230
1.260
1.291
1.276
1.325
1.263
1.187
1.158
1.221

36 to
48

1.092
1.109
1.117
1.118
1.123
1.106
1.069
1.069
1.087

48 to
60

1.048
1.062
1.071
1.068
1.068
1.052
1.035
1.033
1.055

60 to
72

1.031
1.038
1.047
1.042
1.045
1.034
1.021
1.023
1.032

72 to
84

1.016
1.022
1.031
1.030
1.026
1.021
1.014
1.015
1.021

84 to
96

1.009
1.013
1.020
1.016
1.022
1.013
1.007
1.011
1.012

96 to 108 to 120 to 132 to 144 to 156 to 168 to 180 to 192 to 204 to 216 to 228 to 240 to
108
120
132
144
156
168
180
192
204
216
228
240
252
1.000
1.002 0.998
1.001 1.001 1.002
1.000 1.003 1.000 1.002
1.000 1.001 1.000 1.001 1.001
0.999 1.001 1.000 1.001 1.001 1.001
0.999 1.002 1.000 1.000 1.001 1.001 1.000
0.999 1.001 1.001 1.001 1.000 1.001 1.000 1.000
1.001 1.001 1.001 1.000 1.000 1.002 0.999 1.000 1.000
1.001 1.004 1.001 1.000 1.001 1.000 1.001 1.001 1.000
1.002 1.003 1.001 1.002 1.001 1.002 1.000 1.003 1.000
1.002 1.003 1.004 1.003 1.001 1.000 1.001 1.001 1.000
1.006 1.008 1.004 1.003 1.002 1.001 1.000 1.003 1.000
1.007 1.008 1.005 1.003 1.006 1.002 1.002 1.001
1.015 1.006 1.005 1.004 1.001 1.002 1.001
1.013 1.009 1.007 1.000 1.002 1.002
1.017 1.007 0.998 1.003 1.003
1.011 1.003 1.001 1.004
1.002 1.007 1.005
1.008 1.003
1.006

252 to
264
1.000
1.002
1.001
1.000
1.002
1.000
1.001
1.000

264 to
276
1.000
1.001
1.002
1.002
1.002
1.001
1.001

276 to
288
1.000
1.001
1.003
1.000
1.001
1.000

288 to
300
1.000
1.001
1.002
1.001
1.001

300 to
312
1.000
1.001
1.001
1.000

312 to
324
1.000
1.001
1.001

1.545 1.189 1.075 1.041 1.025 1.017 1.010 1.005 1.004 1.001 1.002 1.002 1.002 1.001 1.002 1.000 1.001 1.000 1.000 1.000 1.000 1.001 1.000 1.001 1.001 1.001

Selected
Tail Factor 1.000
Cumulative 2.878 1.606 1.288 1.171 1.110 1.072 1.050 1.036 1.026 1.020 1.016 1.013 1.011 1.010 1.009 1.008 1.008 1.007 1.006 1.006 1.005 1.004 1.003 1.003 1.002 1.001

The selected link ratios are based on the average excluding the highest and lowest link ratios.
A tail factor, selected based on a separate study, represents the development expected beyond 348 months,
and since reported losses are expected to reach their ultimate level by 348 months, the tail factor is set to 1.00.

Exam 5, V1b

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Appendix D – Workers Compensation Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Sheet 2: Indemnity Benefit Cost Level Factors
Indemnity loss costs are impacted by changes in the legislative benefits, changes in utilization of indemnity
benefits for each AY, and inflationary pressures.
Workers Compensation
Indemnity Benefit Cost Level Factors
(1)

Accident
Year
2012
2013
2014
2015
2016

Benefit
Level
Change
0.0%
0.0%
-30.0%
0.0%
0.0%

(2)
Annual Impact
on Benefit
Due to
Wage
Inflation
1.0%
2.0%
2.0%
1.5%
0.9%

(3)
Combined
Impact
on
Benefits
1.0%
2.0%
-28.6%
1.5%
0.9%

(4)
Factor to Adjust
Indemnity
Benefits to
Projected
Cost Level
0.761
0.746
1.045
1.029
1.020

Projected

0.0%

2.0%

1.000

(1) Based on average impact of legislative changes
(1 Proj) Selected
(2) Based on the weekly wages of injured workers
(2 Proj) Selected (1% annual trend)
(3) = [ 1.0 + (1) ] * [ 1.0 + (2) ]-1.0
(4) = [ 1.0 + (3NextRow) ] * (4NextRow)

Column 1:
 displays the estimated average annual impact of changes in the applicable indemnity benefit levels,
considering both direct and indirect effects.
 AY 2014 effect of -30% is due to a law change (the impact was calculated in a separate study).
 The last row includes any known changes in benefits that occur after the experience period.
Column 2:
 displays the annual impact of wage inflation on benefits
 %s were calculated in a separate study
 %s reflects the impact of any maximum and minimum benefit level restrictions
 last row is the expected increase in benefits due to wage increases that will occur between the historical
period and the projected period; the selection is based on an estimated 1% trend for two years (i.e. from
the average loss date of the latest accident year, 7/1/2016, to the average loss date of the policy
projection period, 7/1/2018).
 figures in Column 2 are significantly lower than the factors used to adjust loss cost premium to future
wage level (in Sheet 1) due to the impact of maximum benefit level restrictions.
Column 4: the factor needed to adjust each historical Ay’s reported losses to the projected level.

Exam 5, V1b

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Appendix D – Workers Compensation Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Sheet 3: Indemnity Loss Ratios
Workers Compensation
Loss Ratios-Indemnity Losses Only

Year
2012
2013
2014
2015
2016
Total

(1)
Projected
Loss
Cost
Premium
$4,666,705,987
$4,847,754,029
$4,903,940,552
$5,054,547,098
$5,157,100,516
$24,630,048,184

(2)
Reported
Indemnity
Losses
$1,678,705,592
$1,982,528,857
$1,345,482,170
$931,871,212
$668,971,913
$6,607,559,744

(3)
Indemnity
Loss
Development
Factor
1.110
1.171
1.288
1.606
2.878

(4)
Factor to Adjust
Indemnity
Benefits to
Projected Cost
0.761
0.746
1.045
1.029
1.020

(5)
Projected
Ultimate
Indemnity
Losses
$1,417,388,212
$1,732,058,164
$1,810,516,788
$1,540,391,665
$1,963,948,014
$8,464,302,843

(6)
Expected
Indemnity
Loss
Ratio
30.4%
35.7%
36.9%
30.5%
38.1%
34.4%

(1) From Premium Exhibit
(2) Input
(3) From Indemnity Sheet 1 (Development)
(4) From Indemnity Sheet 2 (Cost Change)
(5) = (2) * (3) * (4)
(6) = (5) / (1)

MEDICAL EXHIBITS
Sheet 1: Medical Loss Development
 represents the development triangle for the reported medical losses by accident year.
 is organized in the same way as in the Indemnity Loss Development section.
 Unlike indemnity losses, reported medical losses (in this example) are expected to develop beyond 348
months, and a tail factor greater than 1.00 is selected.
Sheet 2: Medical Benefit Cost Level Factors
Legislative and regulatory changes impact the cost of medical benefits.
 The fees for many medical services in WC are subject to a fee schedule.
 Thus, medical loss costs are impacted by changes in the medical fee schedules and changes due to
general utilization and inflation.

Exam 5, V1b

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Appendix D – Workers Compensation Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Workers Compensation
Medical Benefit Cost Level Factors

Accident
Year
2012
2013
2014
2015
2016

(1)
Medical
Fee
Schedule
Change
0.0%
0.0%
-20.0%
0.0%
10.0%

(2)
Annual
"Other Medical"
Level
Change
2.5%
2.0%
4.0%
4.1%
3.9%

(3)
Protion of
Medical Losses
Subject to Fee
Schedules
75.0%
75.0%
70.0%
70.0%
70.0%

(4)

Combined
Effect
0.6%
0.5%
-12.8%
1.2%
8.2%

(5)
Factor to Adjust
Medical Benefits
to Projected
Cost Level
0.983
0.978
1.122
1.108
1.025

Projected

0.0%

8.2%

70.0%

2.5%

1.000

(1) Based on evaluations of the cost impact of changes to the Fee Schedule
(1 Proj) Selected
(2) Based on medical component of the Consumer Price Index
(2 Proj) Selected (4% annual trend)
(3) Selected Based on separate study
(4) = (1) * (3) + [ (2) * ( 1 - (3) ]
(5) = [ 1.0 + (4NextRow) ] * (5NextRow)

Column 1:
 Shows the estimated average changes in the applicable medical fee schedule (considering both direct
and indirect effects).
 The medical fee schedule is not expected to change from the most recent period through the projected
time period.
Column 2:
 Shows the annual average change in medical benefits not subject to the medical fee schedule.
 The %s are based on the medical component of the Consumer Price Index (CPI).
 The projected “other medical” change is based on an expected annual change of 4% for two years (and
considers any expected changes between the most recent period and the projected period).
Column 5: Converts the changes in Column 4 to adjust historical accident year reported medical losses to the
projected loss cost levels.
Sheet 3: Medical Loss Ratios
 Calculates expected medical loss ratios for each accident year in the experience period.
 The calculations are the same as in the indemnity loss ratio section.
LAE EXHIBITS
Sheet 1: ALAE Loss Development
This sheet represents the development triangle for paid ALAE by AY (organized in the same way as described
in the Indemnity Loss Development section).
 The selected factors are based on the all-year average excluding the highest and lowest
 Paid ALAE are expected to develop beyond 348 months, so a tail factor greater than 1.00 is selected.

Exam 5, V1b

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Appendix D – Workers Compensation Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
Sheet 2: ALAE Ratio: Calculates the ratio of ultimate ALAE to ultimate projected losses.
Workers Compensation
ALAE Ratio

Year
2012
2013
2014
2015
2016
Total

(1)
(2)
Projected Ultimate
Indemnity and
Paid
Medical Losses
ALAE
$4,339,828,939 $350,034,124
$4,423,762,673 $336,178,599
$4,602,457,877 $201,330,551
$4,525,988,662 $155,896,057
$4,711,677,739
$93,338,368
$22,603,715,890 $1,136,777,699

(3)
ALAE
Development
Factor
1.469
1.676
2.075
3.102
6.992

(6)

(4)

(5)

Ultimate
ALAE
$514,051,816
$563,316,173
$417,746,386
$483,638,473
$652,596,546
$2,631,349,393

ALAE
Ratio
11.8%
12.7%
9.1%
10.7%
13.9%
11.6%

Selected Ratio

11.6%

(1) Derived from Indemnity Sheet 3 and Medical Sheet 3
(2) Input
(3) From LAE Sheet 1 (Development)
(4) = (2) * (3)
(5) = (4) / (1)
(6) Selected

Column 5 is the ratio of the ultimate ALAE to ultimate losses, and since it is expressed as a % of losses, is
different from the ratios computed for indemnity and medical (which are expressed as a % of premium).
This ratio is used as ALAE are more directly related to the amount of losses than the amount of premium.
Sheet 3: ULAE Ratio
Calculates the ULAE ratio based on the historical relationship of CY paid ULAE and paid losses.
Workers Compensation
ULAE Ratio

Calendar
Year
2012
2013
2014
2015
2016
Total
(1) Input
(2) Input
(3) = (2) / (1)

(1)
Calendar Year Paid
Indemnity and
Medical Losses
$4,306,514,977
$4,007,631,598
$3,641,833,560
$3,203,661,824
$3,034,498,823
$18,194,140,782

(2)

(3)

Calendar Year Paid
ULAE
$288,536,503
$272,518,949
$320,481,353
$288,329,564
$273,104,894
$1,442,971,263

ULAE as % of
Losses
6.7%
6.8%
8.8%
9.0%
9.0%
7.9%

(4) Selected Ratio

9.0%

Row 4 selection is based on the latest year because the actuary expects those years to be more representative
of the future.

Exam 5, V1b

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Appendix D – Workers Compensation Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
INDICATION EXHIBIT
This exhibit brings together the results from the previous exhibits and calculates the indicated loss cost
premium change.
Workers Compensation
Overall Indication

Accident
Year
2012
2013
2014
2015
2016
Total

(1)
Expected
Indemnity
Loss Ratio
30.4%
35.7%
36.9%
30.5%
38.1%
34.4%

(2)
Expected
Medical
Loss Ratio
62.6%
55.5%
56.9%
59.1%
53.3%
57.4%

(1) From Indemnity Sheet 3
(2) From Medical Sheet 3
(3) From LAE Sheet 2
(4) From LAE Sheet 3
(5) = [ (1) + (2) [ * [ 1.0 + (3) + (4) ]
(6) Selected
(7) = (6) - 1.0






(3)
Expected
ALAE
Ratio
11.6%
11.6%
11.6%
11.6%
11.6%
11.6%

(4)
Expected
ULAE
Ratio
9.0%
9.0%
9.0%
9.0%
9.0%
9.0%
(6) Selected
(7) Indication

(5)
Expected
Loss & LAE
Ratio
112.2%
110.1%
113.2%
108.0%
110.2%
110.7%
110.7%
10.7%

The objective is to determine the advisory loss costs
The premium does not include any UW expenses or profit; therefore, the target loss ratio is 100%.
The [selected loss ratio - 1.0] produces the overall indicated change to the current advisory loss cost
premium.
Conduct a separate analysis to determine whether the change should be applied uniformly to all risks or
whether it should vary by type of risk.

Exam 5, V1b

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Appendix D – Workers Compensation Indication
BASIC RATEMAKING – WERNER, G. AND MODLIN, C
COMPANY EXHIBIT
Calculates the adjustment an individual company should make to the advisory loss costs to account for U/W
expenses, profit targets, and operational differences that would affect loss cost levels.
Workers Compensation
Company Adjustment
(1) General Expenses
(2) Other Acquisition Costs
(3) Taxes, License and Fees
(4) Commissions and Brokerage Fees
(5) Target Profit Provision
(6) Total Expense and Profit
(7) Expense and Profit Adjustment

10.0%
8.0%
2.5%
8.0%
1.5%
30.0%
1.429

(8) Expected Loss Cost Difference
(9) Operational Adjustment
(10) Proposed Deviation

-5.0%
0.950
1.358

(11) Current Deviation
(12) Industry Loss Cost Change
(13) Company Change

1.400
10.7%
7.4%

(1) - (5) Inputs
(10) = (7) * (9)
(6) = (1) + (2) + (3) + (4) + (5)
(11) Given
(7) = 1.0 / [ 1.0 - (6) ]
(12) From Indication Sheet
(8) Selection
s/b 10.8% (rounding)
(9) = 1.0 - (8) = 1.0 - .05
(13) = (10) / (11) * [ 1.0 + (12) ] - 1.0
s/b 7.5% (rounding)

Rows 1 - 4: Expected U/W expense (for GE, Other Acq., TL&F and Com & Brkg) as a % of total premium
Row 5: Target profit as a % of total premium.
Row 7 adjustment applies multiplicatively to advisory loss costs to include a provision for U/W expenses and profit.
(Equivalently, this adjustment is expressed as the [advisory loss costs/1.0-total expense and profit percentages].)

Row 8:
 is the expected difference in loss costs due to any known operational differences between the individual
company and the industry.
 an overall average adjustment of -5% was selected to reflect an expectation of lower losses attributable to
the company’s more stringent underwriting and claims handling practices.
Row 10: Combines the adjustment for expenses and profit with the adjustment for operational differences, and
represents the deviation factor that the company should apply to the industry advisory loss costs.
Row 11 (the current company deviation factor); Row 12 (the industry loss cost change).
Row 13 (Company Change):
 assumes that the company’s distribution of risks is similar to the industry distribution, and that the
industry loss cost change applies uniformly to all risks.
 otherwise, the industry loss cost change may be different from the actual impact for the company.

Exam 5, V1b

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Personal Automobile Premiums: An Asset Share Pricing Approach
For Property-Casualty Insurance – Feldblum, S.
Section
1
2
3
4
5
6

Topic Covered
Introduction
Asset-Share Components
Asset Share Modeling – Four Illustrations
Illustration 1 – Business Expansion
Illustration 2 – Classification Relativities
Illustration 3 – Competitive Strategy

Introduction

General characteristics of Asset Share Pricing models:


Used for life and health insurance premium determination.



Examines the profitability from inception to termination (including renewals) of the policy.

 Its importance is highlighted when cash flows and reported income vary by policy year.
Predominant Property and Casualty insurance ratemaking methods: Loss Ratio and Pure Premium
Financial pricing models are used to set underwriting profit provisions.


These models presume the contract is in effect for a single policy period.

 Most examine the duration of loss payments, and not the duration of the insurance contract.
A. Life versus Casualty Rate Making:
Factors affecting the differing rate making philosophies:
Factor
1. Cancellation

Life and Health
Few, except for non-payment of premium.

2. Claim Costs

Vary by duration due to:
a. Policyholder age (mortality rises with age).
b. Underwriting selection (but “wears off”
over time).
WL commission rates are high in the 1st year
but low for renewals.

3. Expenses

4. Level premiums.

Much life insurance is provided by level
premium contracts.

Property and Casualty
Carrier has the right to cancel at
renewal and often during the term.
Relationship between expected
losses and duration since policy
inception is less apparent.
Commission rates do not differ
over time for independent agency
system.
Rates may be revised each year.

B. Developments in Casualty Insurance
Attributes that motivate asset share pricing.
1. Commissions: Direct writers of personal lines policies charge higher commission rates in the first
year than in renewal years.
2. Cancellations: Insurers rarely cancel or non-renew the contract, since profitability depends on the
stability of the book of business.
3. Loss costs: Expected loss costs are greater for new business than for renewal business.
The question faced by all insurers: "Is it profitable to write the insurance policy?"
Financially strong carriers examine the stream of future profits during the original policy year and from renewal
years. “Asset share pricing enables the actuary to provide quantitative estimates of long-term profitability”

Exam 5, V1b

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Personal Automobile Premiums: An Asset Share Pricing Approach
For Property-Casualty Insurance – Feldblum, S.
2

Asset-Share Components

In property and casualty insurance, asset share pricing is not common since:


The data needed are not always available.



Casualty pricing techniques are still somewhat undeveloped.



The casualty insurance policy allows great flexibility in premiums and benefit levels.



Liability claim costs are uncertain, both in magnitude and in timing.

Factors influencing asset share pricing techniques include:
A. Premiums
Life and Health
Premiums for whole life policies remain constant
until the termination or forfeiture of the contract.
Premiums for renewable term life policies are
generally guaranteed for the first several years.
Life insurance benefits are fixed in nominal terms.

Factor
Premiums

1. Inflation
2. Underwriting
cycles
3. Classification

Not found in individual life insurance.
Generally is not subject to change after policy
inception.

Property and Casualty
Fluctuate widely from year to
year, for a variety of reasons.*

Raises loss costs, which impact
premiums.
Raise and lower the premiums
charged.
Class and or exposure may
change each year (i.e. single vs.
marital status in personal auto
insurance ).

Level premiums associated with whole life policies have lead life actuaries to place greater reliance on
asset-share pricing models than P&C actuaries (which work with premiums that fluctuate widely).
B. Claims
LOB

Auto:
WC:

O. Liability

Life and Health
Mortality rates are stable over
time and their influences (age,
sex, etc.) are well documented.

Property and Casualty
Claim rates are more variable and less well understood.

Rural vs Urban: Traffic density, road conditions,
number of attorneys, medical treatment.
Recessions: increased filing of minor, non-disabling
injuries.
Prosperous times: Accidental injuries among young,
inexperienced workers are more common.
Statutory enactments and judicial precedents affect the
frequency of claims.

1. Policy Duration and Claim Frequency
a. Policy duration has a strong influence on claim frequency, particularly in Personal Automobile.
b. New insureds have higher average loss ratios than renewal policyholders.
c. Older drivers have lower average claim frequencies and loss ratios.

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Personal Automobile Premiums: An Asset Share Pricing Approach
For Property-Casualty Insurance – Feldblum, S.
2. Inexperience, Youth, Transience, and Vehicle Acquisition
Factors impacting the relationship between duration of the policy and expected claim frequency:
a. Experience: Good driving habits are acquired over time. Inexperienced drivers have high
claim frequencies.
b. Youth: Young male and female drivers have higher than average claim frequencies, even after
adjusting for driving experience.
Note: adolescent drivers living at home and insured on their parents' policies, may cause a
temporary reversal in the inverse relationship of frequency with policy duration.
c. Transience: Many high risk drivers (young males), are "transient" insureds (they often drop
coverage with one carrier and purchase a policy from another). .
Reasons for high (20-30%) termination rates for young male drivers:
i. Young male drivers are more likely to voluntarily cancel their policies (move, get married,
switch to their wives' insurers, drop coverage after an accident).
ii. Company underwriters are more likely to cancel their coverage (more likely to have caused
an accident or are considered too risky).
iii. More likely to experience financial difficulties and or fail to pay premiums.
iv. Have more incentive to shop around for cheaper coverage.
d. Acquisition of the Vehicle: Policy duration is correlated with the time since acquisition.
Accident frequency often decreases with time since acquisition, as the insured becomes
familiar with the operation of the vehicle.
The vehicle’s age is a classification factor for physical damage coverages (the value of the car
declines over time) Time since acquisition of the vehicle, not its age, is important for liability
coverages. The two factors are the same only when the insured purchases a new vehicle.
3. Reunderwriting:
a. Affects the relationship between loss ratios and the policy duration. D'Arcy and Doherty state
that private information collected by the insurer causes declining loss ratios as the policy ages.
b. In WC, loss engineering services and the encouragement of a safe work environment reduce
claim frequency.
C. Expenses
Insurance expenses are greater when the policy is first issued than in renewal years since:
1. Underwriting and acquisition expenses are incurred predominantly at policy inception.
2. This is true for both "per policy" expenses (costs of underwriting and setting up files), and
"percentage of premium" expenses (commissions and premium taxes).
Premiums derived for Life insurance policies incorporate these expense differences by policy year.
Premiums derived using the loss ratio and pure premium methods for P&C policies do not account
for these expense differences by policy year
a. An ELR, derived from company budgets, agency contracts, state statutes or Insurance Expense
Exhibit data is compared to the experience loss ratio, after trending, development, and other
adjustments to determine the indicated rate change
b. This treatment does not recognize their actual incidence.

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Personal Automobile Premiums: An Asset Share Pricing Approach
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Policy Duration and Insurance Expenses
The similarity in Life insurance and P&C expense costs are that they are greater in the year of policy
issuance than in renewal years.
1. Underwriting expenses incurred predominantly in the first year (salaries, policy issuance and
underwriting reports costs, overhead, etc).
2. Loss control expenses incurred either at or before policy issuance (technical inspections, Landfill
inspections, loss engineering services, financial analyses, building inspections). Few inspections
are repeated at renewal dates.
3. Acquisition expenses for direct writers: Three types of commission schedules are used in propertycasualty insurance:
a. Independent agency companies pay level commissions ( needed because the agent "owns the
renewals". A lower commission in renewal years would encourage agents to move the policy to
another insurer to obtain a "first year" commission). Level commission structures do not reflect
actual incidence of acquisition expenses (agents spend more effort writing new policies). Thus,
the independent agency system is inefficient.
b. Direct writers pay commissions that vary by policy year: high in the first year and low renewal
commissions (the insurer owns the renewals which prevents the agent from moving the
policyholder to a competing carrier.
c. Direct writers have either (i) a salaried sales force or (ii) a combined salary - commission based
sales force.
4. Most "other acquisition expenses," (advertising, development costs for expanding or automating
distributions systems) are expended at or before the policy inception date.
State Farm has high retention rates
because:
1. it targets a suburban and rural insured
population.

Many independent agency co.’s have low retention
rates:
1. because the agents can move the insured to
whichever company offers the lowest rates.

2. it offers low premium rates.

2. because these carriers use little consumer advertising.

3. it provides renewal discounts.
D. Persistency
Persistency rates (retention rates) are the key to asset share pricing models and vary widely by company.
They are most important when the net insurance income varies by duration since policy inception.
For Long-Term Ordinary Life, persistency improves with duration since policy inception (termination
rates, or “lapse rates”, decline over time).
An intuitive relationship between duration and persistency exists for both life and casualty insurance.
1. Initially, policyholders are undecided about the value of the policy and the required premiums.
2. Some feel the insurance is not worthwhile; the carrier's service is not acceptable; the premium is
too high or it is unaffordable.
3. Thus, voluntary termination rates during the first year coupled with carriers' reevaluation of newly
acquired risks that have had recent accidents impact persistency.
However, after several renewals, continued renewals are more likely.
Termination Rates and Probabilities of Termination
Persistency may be analyzed either by termination rates or by probabilities of termination.
1. The termination rate = the number of terminations  [number of terminations + policies persisting].
2. The probability of termination = the number of terminations  the number of originally issued
policies in that cohort.

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Personal Automobile Premiums: An Asset Share Pricing Approach
For Property-Casualty Insurance – Feldblum, S.
Example: 100 auto policies are written in 1990, 20 risks lapse the 1st year, 10 lapse the 2nd year,
5 lapse the 3rd year.
The termination rates: 20% [=20  100] the 1st year, 12.5% [=10  80] the 2nd year, and 7.1%
[= 5  70] the 3rd year. The probabilities of termination: 20% [=20  100] the 1st year, 10%
[=10  100] the 2nd year, and 5% [=5 100] the 3rd year.
Conclusions: Termination rates more clearly distinguish persistency patterns by classification.
Probabilities of termination, in certain analyses, better portray the insurer's profitability.
Life insurance persistency patterns are analyzed by issue age, duration, interest rates, sex, rating,
policy face amount, premium payment pattern, policy form, distribution system, etc.
As described above, it is the relationship between the distribution system and persistency patterns that
is particularly important for casualty insurance.
E. Discount Rates
Cash flows over the policy’s lifetime for each PY are discounted to the issue date to determine PVs.
Claim payments:
Reserves:
Discount:

Matching experience:
U/W and investment

Life and Health
Paid soon after death. No
settlement lag.
Policy reserves are a known
quantity.
The discount rate to determine
the above is limited by the
State’s standard valuation law.
Is essential for asset share
pricing.

Property and Casualty
Not settled immediately due to determination of
liability, claim investigations, tort system, etc.
Loss reserves are affected by inflation rates, court
decisions, jury awards, and social expectations.
Property - Liability insurance accounting records
incurred losses on an undiscounted basis,
regardless of the basis (Statutory or GAAP).
Both Statutory and GAAP accounting do not match
the experience for the same block of policies.

Methods of matching underwriting and investment experience:
1. Record undiscounted incurred claims, but include an offsetting investment income account tied to
the assets supporting the unpaid losses.
2. Record cash transactions, not the accounting statement incurred losses. The asset share model
looks like an expanded (multi-period) internal rate of return model.
3. Record discounted loss reserves, using market interest rates, risk-free rates, or "risk adjusted" rates.
Feldblum uses the third method. He states that:
a. the discount rate used to determine the present value of unpaid losses at the accident date need
not equal
b. the discount rate used to determine the present value of future earnings at the issue date.

Asset Share Modeling – Four Illustrations

A. Business Expansion: Most risks from new business have high loss and expense ratios, and although
generally "unprofitable," the "loss may be offset by the future profits in a stable renewal book. Asset
share modeling helps determine true profitability.
B. Classification Relativities: Traditional rate making methods determine classification relativities. If
persistency is ignored, then rate relativities are too low for the poorly persisting classes and too high
for the long-persisting classes. Thus, pricing using class relativities for young drivers is shown.
C. Competitive Strategy: Traditional rate making procedures ignore:
1. the future profits and losses from renewals, and
2. the effects of rate revisions on policyholder retention and new business production
Competitive pricing strategy maximizes long term income by determining the change in policyholder
retention, and new business production from raising or lowering rates.

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Personal Automobile Premiums: An Asset Share Pricing Approach
For Property-Casualty Insurance – Feldblum, S.
D. Underwriting Cycles: It is unclear whether market share gains during the soft cycle combined with
profits gained during the hard cycle will lead to satisfactory long-term income. Asset share modeling
helps to determine the effects of different pricing strategies on overall returns.
Rate Revisions and Rates


Casualty pricing methods determine rate revisions and rate relativities, not actual rates.



Asset share pricing determines rates, not rate revisions. The actuary determines an actual rate
for a selected a “pivotal" classifications, and relies on interpolation and relativity analyses other
(non-pivotal) classifications.

Illustration 1 – Business Expansion

Company expansion or contraction distorts reported financial results:


Expansion raises the statutory combined ratio (loss reserves are held at undiscounted values and
acquisition costs are written off when incurred)



A “GAAP operating ratio” (derived after deferring acquisition expenses and adding investment
income) does not resolve the problem, since the investment income received in any calendar
year is not derived solely from those policies issued during that year.
Aspects of the model to circumvent the above problems:


Use of all figures on a fully discounted basis.



Use of a policy year model, not a calendar year model, (hence no "property-casualty type"
deferred acquisition cost).
A. Growth in a New Territory
Example:
1. A Personal Automobile direct writer expands into a new geographic area in 1992
2. "Fixed" costs peculiar to the expansion (subsidies for new agents, construction costs for a new
branch office, and extra advertising expenses during the first year), are charged to a corporate
account, and not included).
3. 10,000 policies are written in 1992, 1993, and 1994. Losses of 5.6 million are incurred over this time.
4. The asset share model shows that the company is earning a 19% return on surplus.
Question: How can a 19% return on surplus be consistent with losses of $5.6 million in three years?
B. Asset Share Assumptions
1. Premiums: Average rate increases of 9% per annum are expected.
2. Losses: The fully discounted loss ratio on new business is 82% in 1992.
Loss costs are increasing at 10% per annum, and average loss costs on any policy are expected to
improve by 3% a year since policy inception, after adjusting for inflation.
3. Expenses: Variable expenses, (commissions and premium taxes), increase at the same rate as
premium. “Fixed" expenses, (salaries and rent), increase at 5% per annum.
4. Persistency: The termination rates chosen begin at 15% and decline to 8% after 15 years.
5. Present Values: The discount rate is set equal the company’s cost of capital (12%) and is used to
determine the present value of future earnings.
C. The Model: The asset share model is shown on the next page. The PV of current and future profits
and premium (column 12 and 13) is $480 and $5,012 respectively. Their ratio suggests a return on sales
of 9.6%, and assuming a 2 to 1 premium to surplus ratio, the return on surplus is 2 * 9.6% = 19.2%.

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Personal Automobile Premiums: An Asset Share Pricing Approach
For Property-Casualty Insurance – Feldblum, S.
Exhibit 1: Asset share model for Company Growth (page 289)
Policy
Year
(1)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Total:

PV of
Variable Expense
Fixed Expense
Persistency Cumulative
Discount Present Value of
Premium Loss
Year 1
Renewal Year 1
Renewal Rate
Persistency Profit
Factor
Profit
Premium
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
800
656
242
0
142
0
1.000
1.000
-240
1.00
-240
800
872
701
0
54
0
32
0.850
0.850
72
1.12
65
662
950
748
0
59
0
34
0.860
0.731
80
1.25
64
554
1036
799
0
64
0
35
0.870
0.636
87
1.40
62
469
1129
853
0
70
0
37
0.880
0.560
95
1.57
60
402
1231
911
0
76
0
39
0.890
0.498
102
1.76
58
348
1342
973
0
83
0
41
0.900
0.448
110
1.97
56
305
1462
1039
0
91
0
43
0.900
0.403
117
2.21
53
267
1594
1110
0
99
0
45
0.910
0.367
125
2.48
50
236
1738
1186
0
108
0
47
0.910
0.334
133
2.77
48
209
1894
1266
0
117
0
50
0.920
0.307
142
3.11
46
187
2064
1352
0
128
0
52
0.920
0.283
150
3.48
43
168
2250
1444
0
140
0
55
0.920
0.260
159
3.90
41
150
2453
1542
0
152
0
57
0.920
0.239
168
4.36
38
135
2673
1647
0
166
0
60
0.920
0.220
176
4.89
36
120
480
5,012

Column (1) = year since policy inception. Figures in exhibit pertain to this policy only.
Column (2) = an average premium per car of $800 increasing a 9% per annum.
Column (3) is the present value at the beginning of that policy year.
Columns (4) through (7): Variable expenses are 30.2% of premium in the 1st year and 6.2% in renewal
years. Fixed expenses are 17.8% of premium in the 1st year, are .038 * $800 * 1.05 in the
1st renewal year, and then increase 5% per year thereafter.
Column (8) shows the expected persistency rate.
Column (9) = the downward product of column (8).
Column (10) = Column (9) *{Column (2) -  of Columns (3, 4, 5, 6, and 7)}.
Column (11) uses a rate of 12% per year compounded annually.
Column (12) = column (10)  column (11).
Column (13) = column (2) * column (9)  column (11).
Accounting Results and Long-Term Profitability
The reported earnings of a negative $5.6 million for the first three policy years, even after full
discounting of losses, is the result that traditional actuarial pricing techniques. Calendar year
statutory financial statements use undiscounted loss reserves and write off all underwriting and
acquisition expenses when incurred would show worse results.
The dependence of loss and expense ratios on the year since the policy was first issued explains the
difference between the $5.6 million loss shown by traditional pricing analyses and the 19% return on
surplus shown by the asset share model.
D. Federal Income Taxes
1. Federal income taxes are not considered in these illustrations.
2. The simplest way of computing income taxes is to multiply the "profit” column by the marginal tax
rate (the discount rate used for losses should be pre-tax, while the discount rate used for profits
should be after-tax).
3. Alternatively, if after tax premiums, losses and expenses are used, then after tax discount rates
should be applied.

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Personal Automobile Premiums: An Asset Share Pricing Approach
For Property-Casualty Insurance – Feldblum, S.
E. Profitability Measures
Different measures of profitability can be incorporated in an asset share model:
1. In the example, future earning are discounted at the company's cost of capital, which implies that
profits should be measured using a return on equity (the examples assumes that surplus = equity).
2. Determine the "break-even” point by determining when writing policies is more profitable than
investing the equity in financial securities of similar risk.
3. Combine cash transactions from the insurance operations with assumed equity flows to determine
the internal rate of return to the equity providers.

Illustration 2 – Classification Relativities

Classification relativities are determined by comparing relative loss ratios or pure premiums among
groups of insureds. For example:
Driver Category
Adult (base class)
Young male

Average losses
500
1500

Class relativity
1.00
3.00

A. Expense Flattening and Persistency
Expense flattening procedures are used to separate expenses into fixed and variable. These
procedures fail to incorporate differences in persistency among insureds which impact class
relativities.
1. Fixed expenses, as a % of total premium, are lower for young male drivers than for adult drivers.
2. Variable expenses, as a % of total premium, are equal for the two classes.
3. Young male drivers have higher termination rates than adult drivers and so the ratio of total
expenses to total premium over the lifetime of the policy is greater for young male drivers.
B. Determinants of Rate Relativities
The correct relativity depends on the:
1. Classification system.
2. Average losses and persistency rates by classification, and
3. Strength of loss ratio improvement by policy year.
The following example compares young male drivers with adult drivers to determine classification relativity
factors. The information listed below is needed, the 2nd and 3rd items are essential for the model.
1. The dimensions of the classification system.
2. The relative average loss costs of these two groups of insureds.
3. The relative average persistency rates of these two groups of insureds.
4. The strength of loss ratio improvement by policy year for these insureds.
C. The Classification System
Renewal discounts and age boundaries between driver classes affect future years' premium.
Example: A asset share model is being used for an 18 year old unmarried male driver.
Given that the insurer differentiates between “males aged 25 and under" and "adult drivers," then
1. The driver will spend 8 years in the "young male" classification.
2. The premium is probably too low for the next 3 or 4 years and too high for the subsequent 4 or 5
years, since average losses decline between ages 17 and 25.

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Personal Automobile Premiums: An Asset Share Pricing Approach
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3. Expected termination rates will start high but decline over the next 8 years since termination
decrease with duration of the policy.
4. A renewal discount will improve persistency but reduce renewal premiums.
The results of an asset share model should be used to design the classification system. This is
the case for the "competitive strategy" illustration (shown in the following section).
D. Coverage Mix
2 types of differences affect classification relativities:
1. Average losses for any coverage vary by classification (e.g. young male drivers have higher
expected BI losses than adult drivers).
2. The coverage mix varies by classification (e.g. young male drivers are less likely to buy physical
damage or excess limits for liability coverages than adult drivers).
Classification relativities and Loss cost relativities would be similar if the ratio of expenses to premium
did not vary with the above mentioned items.
E. Policy Basis versus Coverage Basis Rate Relativities
The asset share pricing model can be used to develop rate relativities on either a policy or coverage
basis.
1. The policy basis model compares experience for all coverages combined among classes of
insureds, and the resultant rate relativities must be allocated to coverages.
2. The coverage basis model compares experience for an individual coverage among classes of
insureds.
a. “Fixed" expenses must be allocated to coverage before the asset share pricing model is used.
b. Premiums rates are not additive and there should be a “multiple coverages" discount.
F. Policy Basis Loss Cost Relativities
3 factors account for the policy basis loss cost differences between young male and adult drivers:
1. Young male driver rate relativities by coverage: Rate relativities vary among insurers, depending on:
a. the definition of young male drivers and
b. the other classification dimensions( years of driving experience, past accident history).
2. Physical damage coverage by classification: (young male drivers are less likely to buy physical
damage coverage, due to high premiums, the relative value of their auto, etc.).
3. Average liability increased limits and physical damage deductibles:
G. Persistency by Classification
Whole life policies, guaranteed issue policies, and Personal Automobile policies typically show an
accounting loss during the first policy year, since either expenses or loss costs or both are higher that
year. In any case, the loss turns into a profit as the policyholder persists.
Classification differences may be based on either current classification or original classification.
Classification does not change in most lines with the exception of Personal Automobile, since age,
geographic domicile and the value of the auto change over time.

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Personal Automobile Premiums: An Asset Share Pricing Approach
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H. Young Male Drivers
1. Traditional rate making considers current classification.
2. Asset share pricing models consider original classification and expected future changes.
Although persistency rates by duration are easily determined for current classifications (the % of
young male drivers in their 5th policy year who persist into their 6th year), it is persistency rates by
original classification, not current classification, that it is needed. Notice the difference: the
persistency of young male drivers in their 5th policy year does not tell us the expected 5th year
persistency of young male drivers.
I. Model Assumptions
Given: Adult pleasure use is the base class. Unmarried males aged 21 and 22 who drive to work will
be compared.
3 differences by classification are needed to form rate relativities:
1. average loss costs,
2. average fixed expense costs, and
3. persistency rates.
Assume the following differences
1. Average liability loss costs are $400 and $1,000 per annum for adults and young male drivers
respectively.
2. Average premium for all drivers is $550, average 1st year fixed expenses (F.E.) are 17.8% of this ($98).
a. Adult drivers are less expensive to underwrite, etc, so average fixed expenses per coverage is
10% less, or $88 per policy. 2nd year F.E = the ratio of renewal to 1st year F.E. Subsequent
years F.E. increase at 5% per annum.
b. Young male drivers are more expensive to underwrite, so average 1st year fixed expenses per
coverage are 20% higher, or $117 per policy.
3. Adult drivers have higher retention rates than young male drivers.
Givens: the classification plan, average loss costs, average fixed expenses, and persistency rates.
Assume: Writing at a 2:1 premium to equity ratio and desire for a 14% return on equity.
Approach: Step 1: Use the asset share pricing model to determine a 7.0% return on premium.
Step 2: Derive the rate relativities from the resulting premiums.
For each class:
1. Select a starting gross premium and increase it 9% per annum, which determines the variable
expenses in all future years.
2. Loss costs are discounted to the beginning of the policy year.
3. A 12% cost of capital rate is used to determine the present values of future profits and premiums at
the original policy issuance date.
The Goal: Determine the original premium such that the ratio of the present value of all future profits to
the present value of all future premiums is 7.0% for both classes.
Asset Share Results
1. The loss cost relativity is 2.50, or $1,000  $400.
2. The fixed expense cost relativity is 1.33, or (= $117  $88).
3. The rate relativity is 2.68, or $1,272  $475.
A premium rate relativity of 2.68 is needed to equalize the returns between these two classes.

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Personal Automobile Premiums: An Asset Share Pricing Approach
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Exhibit 3: Young Male Drivers.
Policy
Year
(1)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Total:

PV of
Variable Expense
Fixed Expense
Persistency Cumulative
Discount Present Value of
Premium Loss
Year 1
Renewal Year 1
Renewal Rate
Persistency Profit
Factor
Profit
Premium
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
1272
1000
384
0
117
0
100%
100%
-230
1.00
-230
1,272
1386
1068
0
86
0
26
60%
60%
124
1.12
110
743
1511
1141
0
94
0
28
65%
39%
97
1.25
78
470
1647
1218
0
102
0
29
70%
27%
81
1.40
58
320
1796
1301
0
111
0
30
73%
20%
70
1.57
45
227
1957
1389
0
121
0
32
76%
15%
63
1.76
36
168
2133
1484
0
132
0
34
79%
12%
58
1.97
29
129
2325
1584
0
144
0
35
82%
10%
55
2.21
25
103
2535
1692
0
157
0
37
85%
8%
54
2.48
22
85
2763
1807
0
171
0
39
88%
7%
55
2.77
20
73
3011
1930
0
187
0
41
90%
7%
56
3.11
18
64
3282
2061
0
204
0
43
90%
6%
58
3.48
17
56
3578
2201
0
222
0
45
90%
5%
59
3.90
15
49
3900
2351
0
242
0
47
90%
5%
61
4.36
14
43
4251
2511
0
264
0
50
90%
4%
62
4.89
13
38
269
3,841

Column (1) = year since policy inception. Figures in exhibit pertain to this policy only.
Column (2) = chosen such that the PV of profits = 7.0% of the PV of premiums.
Column (3): 1st year average losses = 1,000. loss cost trend = 10% per annum. Losses decrease 3%
per year.
Columns (4) through (7): Variable expenses are 30.2% of premium in the 1st year and 6.2% in renewal
years. Fixed expenses are 20% higher than the average $98 per policy in the 1st year, are
550 * 1.20 * 1.05 * .038 in the 1st renewal year, and then increase 5% per year.
Column (8) shows the expected persistency rate.
Column (9) = the downward product of column (8).
Column (10) = Column (9) *{Column (2) -  of Columns (3, 4, 5, 6, and 7)}.
Column (11) uses a rate of 12% (cost of capital) per year compounded annually.
Column (12) = column (10)  column (11).
Column (13) = column (2) * column (9)  column (11).

Illustration 3 – Competitive Strategy
Example
Business expansion.
Classification relativities
Traditional ratemaking

Competitive strategy

Given:
The environment.
Insured population
Insured population

Question:
Is the growth strategy profitable?
What prices are equitable?
What are the anticipated losses and expenses that
determine premiums such that economic profits are
eliminated?
How can the pricing structure create a more
profitable consumer base?

2 Considerations when seeking to change the insured population:


Recognize that any strategy affects new business growth or retention rates.



Traditional ratemaking procedures are cost-based. Premium rates and relativities impact
consumer demand and the mix of insureds, which impact profitability.

Exam 5, V1b

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Cars and Courage
Physical limitations:
Awareness of the above:
Exposure to road hazards:
Cost of Insurance:

Older/Retired Drivers
Makes them less capable of escaping
dangerous situations
Makes them more likely to avoid
dangerous situations
After retirement, spend less time behind
the wheel.
Have less impetus to price shop at
renewal due to lower premiums and less
information about competing carriers

Young Drivers
N/A

Drive to work. attend late
parties, etc
Have higher premiums, and
are more informed about
competing carrier’s rates.

Considerations for the asset-share model for the pricing of a older driver


Expected loss costs by policyholder age.



Persistency rates by policyholder age and policy duration.

 Price elasticity of demand: (the effects of price on retention rates).
First, we will discuss the relevance of these elements to retired drivers.
A. Retired Drivers
Average loss costs:
Price elasticity of
demand:

Older Drivers
Decrease with age. Still drives to work and
is exposed to road hazards.
More likely to switch carriers for a better
rate.

Retired Drivers
Drives less often than older
drivers.
“Consumer loyalty” is more
likely.

Optimal Pricing Strategy: Requires underpricing older drivers (50’s) to gain market share and
eventually reap greater profits as insureds age and persist. Requires offering a discount before the
data seems to justify it. This requires determination of the Age and Optimal Magnitude.
Age:

Magnitude:

Before any substantial decline in losses.
Depends on relationship between age and
persistency, discounts offered, E[loss
costs].
Depends on price elasticity of demand and
peer co. rates structure , and E[loss costs].

Before retirement.

B. Model Assumptions:
Determinations of the optimal age and magnitude for the retired driver discount:
1. Loss Costs by Age of Policyholder
Shown below is the loss ratio relativities by policyholder age, for new and renewal business. The
relativity equals the ratio of the loss ratio in that row to the average loss ratio for all rows combined.

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2. Persistency Rates for Older Drivers
Table 6: Loss Ratio Relativities by Policyholder Age
Policy holder
New Business
Renewal Business
Age
LR Relativity
LR Relativity
20 - 49
1.02
1.03
50 - 54
1.00
0.98
55 59
0 94
0.83
60 - 64
0.84
0.72
65 - 69
0.82
0.65
70 - 74
0.98
0.76
75 & older
1.10
0.98
Total:
1.00
1.00
Retention rates improve as both the policy and the policyholder ages.
Table 7: Persistency Rates by Policyholder Age
Policyholder Age
50
54
58
62
66
70
Persistency Rate (%)
96
95
94
92
90
88

74
85

78
80

2 differences in the persistency rates shown above compared with those for adult and young male
drivers:
a. Most insureds aged 50 and over represent mature renewal business.
b. Persistency drops as policyholders advance into their 70's due to death or illness.
Since persistency rates depend upon the premium discount that is offered, replace the
"persistency rates" in Exhibit 9 with a set of rows, showing persistency rates with no discount, a 5%
discount, a 10% discount, etc. Since these persistency rates depend on the discounts offered by
other carriers, there are no "absolute” expected rates, as expected rates also depend on other
carriers' discounts.
Persistency rate assumptions are subjective, but are essential for determining optimal prices.
2 sets of persistency rates for the asset share model are used:
a. One set, with lower rates, and no premium discount offered to older or retired drivers.
b. The other set, with higher rates, and a 7.5% “market” discount rate.
Table 8: Persistency Rates by Policyholder Age
Policyholder Age
50
54
58
62
66
70
Persistency: with discount
98
97
96
94
92
90
Persistency: without
90
85
80
75
80
80
discount

74
85
85

78
80
80

To determine the optimal premium discount, the asset-share pricing model is run 3 times, after
considering each of the following:
a. No carrier offers a retired driver discount (Exhibit 4).
b. Peer companies offer the discount, but your company does not (Exhibit 5).
c. Your company offers a 7.5% “market” discount rate (Exhibit 6).

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In each case, we use a 15 year asset-share model for insureds aged 52. We assume:
a. that persistency rates depend on the premium discount offered, but
b. average loss costs do not.

Consideration
Persistency:

Profitability:

No discounts offered.
High quality insureds with
high persistency rates
exist with declining loss
costs.
Profitability is good, since
the insurer has already
paid the high cost of new
business production and
is reaping the benefits of
the renewal book.

Peers offer discount, but
your company does not.

Your company offers
the market discount.

Persistency drops.
Retention rates are lower
as more insureds leave
each year.
Loss and expense ratios
remain unchanged, so the
full profit margin is
maintained. However, the
impact of lower persistency
reduces the PV of future
profits.

Persistency is increased
as the market reacts
favorably to the market
discount offered.
Although the 7.5%
discount cannot be
justified on a short term
basis, persistency
increases to the highest
level of the three
scenarios and the PV of
future profits has
increased compared
with the adjacent
scenario.

Note: A return on
premium is relevant when
market shares remain
constant.

Exam 5, V1b

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Personal Automobile Premiums: An Asset Share Pricing Approach
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Questions from the 1996 exam
Question 8. (1 point) According to Feldblum, "Personal Automobile Premium: An Asset Share Pricing Approach
For Property-Casualty Insurance," which of the following are evidence that property-casualty
insurance is taking on attributes that motivate asset share pricing?
1. A greater emphasis is being placed on the investment income component of rates.
2.

Insurers rarely cancel or non-renew policies.

Expected loss costs are greater for new business than for renewal business.

1 only

2 only

1, 3 only

2, 3 only

1, 2, 3

Question 38. (3 points) You are given the following for an average policy:

Policy
Year
1
2
3
4
5

Premium
1,000
1,000
1,000
1,000
1,000

Rate
1.000
0.850
0.850
0.850
0.850

PV Loss
800.00
776.00
752.72
730.14
708.23

Persistency
Cumulative
1.000
0.850
0.723
0.614
0.522

Variable Expense
Year 1
Renewal
250
0
0
50
0
50
0
50
0
50

Profit
(200.00)
113.90
113.63
110.46
105.33

Total

Discount
Factor
1.000
1.100
1.210
1.331
1.464

Fixed Expense
Year 1
Renewal
150
0
0
40
0
40
0
40
0
40

Present Value
Profit
Premium
(200.00)
1,000.00
103.55
772.73
93.91
597.11
82.99
461.40
71.94
356.56
152.39

3,187.80

• PV Loss is the present value at the beginning of each policy year.
• Assume all policies are annual and have January 1 effective dates.
• The policy count at year 0 is 1,000.
Using the asset share pricing model described by Feldblum, "Personal Automobile Premiums:
An Asset Share Pricing Approach for Property-Casualty Insurance:"
(a) (2 points) If you increase rates 10% on January 1 of year 1 and then keep rates constant
throughout the five-year period, you project a 20% policy count decrease in year 1 and all
other patterns will remain the same. Calculate the revised present value 5-year
aggregate profit.
(b) (1 point) If you increase rates 10% on January 1 of year 1 and then keep rates constant throughout
the five-year period, what decrease in year 1 policy counts would result in the original
estimated present value 5-year aggregate profit of $152,390, assuming all other patterns
will remain the same?

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Questions from the 1997 exam
3. According to Feldblum, "Personal Automobile Premiums: An Asset Share Pricing Approach for PropertyCasualty Insurance," which of the following are true?
1. Analysis of persistency rates is a key part of asset share pricing models.
2. Agency ownership of policy renewals affects persistency rates.
3. It is preferable to review persistency rates by current driver classification rather than by the original
driver classification.
A. 1

B. 2

C. 1, 2 D. 1, 3 E. 1, 2, 3

30. (4 points)
The Innovative Insurance Company is considering offering a 5% renewal discount to its Personal Auto
policyholders to improve the company's retention. It has asked for a four-year study of profitability using the
information below.
Year
New
2
3
4

Retention Without
Discount
100%
80%
80%
80%

Retention With
Discount
100%
98%
98%
98%

Commission
15%
3%
3%
3%

Fixed Expense
$10
$5
$5
$5

Premium Taxes
3%
3%
3%
3%

• First Year Average Premium
:
$1,000
• First Year Average Loss + LAE:
$800
• Annual Cost of Capital:
10%
• Loss + LAE Trend:
5%
• Fixed Expense Trend:
4%
• Average Annual Premium Growth due to Rate Changes
6%
• Assume annual renewals
• Assume there are no new policies written
• Present Value Profit without a renewal discount over the four-year study is:$282.09
Based on Feldblum, "Personal Automobile Premiums: An Asset Share Pricing Approach for PropertyCasualty Insurance," which alternative is more profitable (5% discount or with no discount) over the fouryear study?

Exam 5, V1b

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Personal Automobile Premiums: An Asset Share Pricing Approach
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Questions from the 1998 exam
52. You are given the following information for a group of new business auto liability policies:
Average year 1 premium
$500
Annual rate increases
10.0%
New business loss ratio
80%
Renewal loss ratio
60%
New business commission
20%
Renewal commission
5%
Other expenses (all variable)
15%
Cost of equity capital
15%
Premium to surplus ratio
2.5
Policy year
1
2
3
4
Probability of termination
30%
30%
20%
20%
Assume all policies are annual and cancel or lapse on their anniversary.
Determine the following using the method described by Feldblum, "Personal Automobile Premium , An
Asset Share Pricing Approach for Property-Casualty Insurance."
a. (1 point) The persistency rate by policy year.
b. (3 points) The four year underwriting return on premium and the four year underwriting return on surplus.

Questions from the 1999 exam
20. According to Feldblum, "Personal Automobile Premiums: An Asset Share Pricing Approach for
Property-Casualty Insurance,' which of the following are true?
1. Variability of premiums and losses in property and casualty insurance is a reason why property and
casualty actuaries have not relied on asset share pricing.
2. The principal benefit to asset share pricing is the determination of profitability over the entire time a
policyholder stays with the company.
3. The asset share pricing model is inappropriate to use for high risk drivers, such as young males,
because they do not tend to remain with one company long enough to permit completion of a long
term analysis.
A. 3

B. 1, 2

C. 1, 3

D. 2, 3

E. 1,2,3

34. (4 points) Based on Feldblum, "Personal Automobile Premiums: An Asset Share Pricing Approach for
Property-Casualty Insurance," and the information shown below, calculate the present value of expected
profits as a percentage of the present value of premium. Assume that the cost of capital is 12%.
Policy
Present Value
Variable
Fixed
Persistency
Year
Premium
of Loss*
Expense
Expense
Rate
1
900
810
135
72
1.00
2
990
826
149
30
.75
3
1,089
843
163
30
.80
4
1,198
860
180
30
.85
* Present Value of Loss is the present value at the start of each respective policy year.
Assume there are no policies in policy year five.

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Personal Automobile Premiums: An Asset Share Pricing Approach
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Questions from the 2000 exam
7. T/F According to Feldblum in "Personal Automobile Premiums: An Asset Share Pricing Approach for
Property-Casualty Insurance," the fundamental issue in asset share pricing methods is the
predictability of losses.
8. T/F According to Feldblum in "Personal Automobile Premiums: An Asset Share Pricing Approach for
Property-Casualty Insurance," asset share modeling is considered particularly valuable when
differences in termination rates influence expected profits.
47. (3 points) Using the procedure described by Feldblum in "Personal Automobile Premiums: An
Asset Share Pricing Approach for Property-Casualty Insurance" and the following information,
complete the asset share model and compute the return on premium for policies written during
1999. Use a 3-year time horizon.


Average policy premium in 1999 = $1,000



Loss cost trend = 8% per annum



"Fixed" expense trend = 4% per annum



Expected rate increases = 6% per annum



Discounted loss ratio on new business written in 1999 = 75%



Loss costs improve by 3% per year since policy inception, after adjusting for inflation.



Termination rates are 10% each year after the year of policy issuance.



Cost of capital = 9%

Acquisition and underwriting expenses for Policy Year 1999:
New Policies
Renewal Policies
Fixed
Variable
Fixed
Variable
15%
25%
5%
10%

Questions from the 2001 exam
Question 11. Based on Feldblum, “Personal Automobile Premiums: An Asset Share Pricing Approach for
Property-Casualty Insurance,” and the following information, calculate the termination rate
for the third year.


Number of policies originally issued = 1,000



Number of first-year lapses = 350



Number of second-year lapses = 200



Number of third-year lapses = 100

A. < 12%

Exam 5, V1b

B. >12% but < 16%

C. > 16% but < 20%

Page 345

D. > 20% but < 24%

E. > 24%

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Personal Automobile Premiums: An Asset Share Pricing Approach
For Property-Casualty Insurance – Feldblum, S.
Questions from the 2001 exam (continued):
Questions 12. According to Feldblum, “Personal Automobile Premiums: An Asset Share Pricing
Approach for Property-Casualty Insurance,” which of the following is false ?
A. Asset share pricing determines rates, not rate revisions.
B. Life insurance policy claim rates are more certain than property-casualty policy claim rates.
C. It is appropriate to assume the same pattern of persistency ratios for both direct writers and
independent agency companies.
D. A level commission structure is inappropriate for the persisting and profitable risks.
E. The dominant market share of the direct writers makes asset share pricing a more appropriate
model for personal automobile insurance.
23. (2 points) In his paper “Personal Automobile Premiums: An Asset Share Pricing Approach,”
Feldblum gives four reasons for the relationship between the duration of an auto policy and the claim
frequency for that policy. State and explain these four reasons.

Questions from the 2002 exam
36. (3 points) Based on Feldblum, "Personal Automobile Premiums: An Asset Share Pricing Approach for
Property-Casualty Insurance," and the following information, answer the questions below. Show all work.
30 policies terminate in the third year
The probability of termination in year 2 is 0.0816
The termination rate in year 2 is 0.1000
The termination rate in year 3 is 0.0750
a. (1½ points) Calculate the number of policies terminated in year 1 and year 2.
b. (1 point) Calculate the original number of policies in the cohort.
c. (½ point) Calculate the termination rate and the probability of termination in year 1.

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Questions from the 2003 exam
43. (5 points)
a. (4 points) You are given the following information:
• Indicated adult class new business policy premium = $1,000
• Adult class discounted new business loss ratio = 80%
• Youthful class discounted loss cost during first policy year = $1,500
• Premiums increase 10% annually
• Losses increase 5% annually
• Variable expense ratio = 20% for all business
• All expenses are variable
• Annual adult class lapse rate = 10%
• Annual youthful class lapse rate = 25%
• The company's cost of capital is 10%
• 3-year present value of premium for adult class = $2,710
• 3-year present value of profit for adult class = $90.30
• Assume same return is earned for all classes
Using the procedure described by Feldblum in "Personal Automobile Premiums: An Asset Share
Pricing Approach for Property/Casualty Insurance," calculate the indicated premium relativity for
youthful drivers. Use a three year time horizon to determine your answer. Show all work.
b. (1 point) How might traditional ratemaking methods be misleading in determining classification
relativities?

Questions from the 2004 exam
44. (4 points)
a. (3 points) You are given the following information about a group of policies:
• The first year average policy premium is $1,000 and increases by 12% annually.
• Premiums are collected at the beginning of each year.
• The discounted first year loss ratio is 75%.
• Loss cost trend is 10% per annum.
• Loss costs improve by 4% per year, after adjusting for loss costs trends.
• Expenses are $400 in year 1 and $100 in all subsequent years.
• 90% of first year policyholders persist into the second year.
• 90% of second year policyholders persist into the third year.
• The company's cost of capital is 15%.
• The premium to surplus ratio is 3 to 1.
Using the asset share pricing model, determine the return on equity over the three-year period.
Show all work.
b. (1 point) Explain how asset share pricing models and property/casualty insurance ratemaking methods
differ in their consideration of the profitability of an insurance policy.

Exam 5, V1b

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Personal Automobile Premiums: An Asset Share Pricing Approach
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Questions from the 2005 exam
56. (3 points)
Given the information below, use an asset share pricing approach to determine whether a company
should write this business. Show all work and explain your answer.
Policy Year

Premium

1
2
3

$500
$550
$605

Present Value of
Losses
$415
$440
$460

Expenses
$100
$110
$121

Annual
Persistency
100%
85%
85%

• Equities of similar risk are yielding 10% per year.

Questions from the 2006 exam
50. (5.75 points) You are the actuary for an insurance company that is considering offering a
5% discount to retired drivers in order to improve retention.
Using the Asset Share Pricing approach described by Feldblum, and the information provided below,
determine which alternative is more profitable for a cohort of 65 year-old existing insureds over a threeyear time period. Show all work.

Year
1
2
3










Persistency
With
Discount
100%
98%
95%

Persistency
Without
Discount
100%
90%
85%

Fixed
Expense
$40
42
44

First-year average premium (with no discount) $800
First-year average losses
$500
Average annual premium trend
5%
Loss cost trend per annum
5%
New Business Variable Expenses
30% of premium
Renewal Business Variable Expenses 20% of premium
Annual Cost of Capital
10%
Assume there are no taxes.
For this cohort of business, average loss costs in any policy year are 1% lower than in the
preceding policy year after adjustment for loss cost trend.

Exam 5, V1b

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Personal Automobile Premiums: An Asset Share Pricing Approach
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Questions from the 2007 exam
28. (4.0 points) A personal automobile carrier marketing its business through direct writers is planning
expansion into a new territory. You are given the following information:
First-year territory average premium
Termination rate in year 1
Termination rate in year 2
First-year average fixed expenses
Second-year average fixed expenses
First-year discounted loss ratio
Average annual premium trend
Average annual loss cost trend
Average annual fixed expense trend
New business variable expenses
Renewal business variable expenses
Annual cost of capital
Premium to surplus ratio

$900
15%
12%
$60
$40
75%
5%
7%
5%
27%
5%
5%
2:1

of premium
of premium

For this cohort of business, average loss costs in any policy year are 3% lower than in the preceding
policy year after adjustment for loss cost trend.
Using the Asset Share Pricing approach described by Feldblum, determine whether this opportunity
exceeds the company's target return on surplus of 15% over a three-year time period. Show all work.

Questions from the 2008 exam
43. (1.0 point) Contrast the asset share pricing model to traditional techniques for calculating rate relativities.
44. (3.0 points) You are given the following information:
Class A
Class B
Premium-First Year
$633.80
X
Loss Cost-First Year
$500
$1,000
Fixed Expense
First Year
$90
$120
Subsequent Years
$80
$110
Variable Expense
First Year
10%
10%
Subsequent Years
5%
5%
Persistency Rate
80%
60%
 Loss costs for renewal business are 10% lower than-for new business.
 There is no premium trend and there are no rate changes.
 There is no expense trend.
 Interest rate for discount is 10%.
 Premium-to-Surplus ratio is 2:1.
 Target pre-tax return on equity is 6%.
Calculate the indicated rate relativity for Class B as compared to the base class (Class A) using the asset
share pricing model and a two-year time horizon.

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Personal Automobile Premiums: An Asset Share Pricing Approach
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Questions from the 2009 exam
45. (2.25 points)
a. (0.75 point) Define persistency rates. Briefly explain why persistency rates are important for
classification ratemaking.
b. (0.75 point) For private passenger auto, identify whether a direct writer or an independent agency
company is expected to have higher persistency rates. Explain your answer.
c. (0.75 point) There are two private passenger auto insurers. One targets non-standard insureds and
the other targets preferred or standard insureds. Identify which insurer is likely to have higher
persistency rates and explain your answer.

Questions from the 2010 exam
35. (3 points) Given the following information:
• Year 1 premium is $1000.
• Premium rates increase 10% per annum.
• Loss ratio for new business is 65%.
• Loss trend is 10% per annum.
• Loss costs improve 5% each renewal period.
• Variable expenses are 30% in Year 1 and 10% in subsequent years.
• Fixed expenses are $200 in Year 1 and $50 in subsequent years.
• Persistency is 75%.
• Cost of capital is 10%.
• Premium to surplus ratio is 2.0.
• Target return on surplus is 4%.
Determine whether this business will achieve the target return on surplus based on a two-year time horizon.

Questions from the 2011 exam
20. (3.5 points) Given the following information:
• Company cost of capital is 8%
• No premium or loss trend
Annual premium
Present value of losses (1st year)
Present value of losses (2nd year)
Variable expense ratio (1st year)
Variable expense ratio (2nd year)
Fixed expense (1st year)
Fixed expense (2nd year)
Probability of annual termination

Class A
$800
$550
$550
20%
12%
$42
$20
15%

Class B
Unknown
$650
$650
24%
20%
$50
$30
40%

a. (0.5 point) Using a one-year period, calculate the premium for class B if the same profit loading is
targeted for all classes.
b. (1.25 points) Using a two-year period, calculate the return on sales for class A.
c. (1.75 points) Using a two-year period, calculate the premium for class B that would achieve the
same return on sales as calculated in part b above.

Exam 5, V1b

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Personal Automobile Premiums: An Asset Share Pricing Approach
For Property-Casualty Insurance – Feldblum, S.
Solutions to questions from the 1996 exam
Question 8.
Comment: This question is just one of the numerous list type questions that can be asked from this paper.
Answers to this question are found on page 195, Proceedings, November 1996.
Feldblum lists 3 attributes about P&C insurance that motivate the use of asset share pricing:
1. Commissions. Commission rates tend to be higher in the 1st year than in renewal years.
2. Cancellations. Insurers rarely cancel or non-renew policies, since profitability depends on the
stability of the book of business.
3. Loss costs.
Feldblum states that this phenomenon is valid for personal auto insurance as well
as for other lines of business.
Therefore, 1 is False, 2 is True and 3 is True.

Answer D.

Related topic. On page 196, Feldblum lists 4 reasons why asset share pricing is not yet common in P&C
insurance.

Solutions to questions from the 1996 exam
Question 38.
(a) On page 289, Feldblum shows in Exhibit 1, an example of how to use the asset share model for company
growth. The revised present value 5-year aggregate profit is calculated as follows:
5-year aggregate profit = (Policy count at time 0)*(policy count impact)*(aggregate present value profit).

The policy count at time zero, and expected policy count impact are given as 1,000 and .80 (1.0 - .20)
respectively. Therefore, the only element that must be computed is 5 year aggregate present value profit.
The following is given:
Policy
Year Premium

(1)
1
2
3
4
5
Total

(2)
1,000
1,000
1,000
1,000
1,000

PV of
Loss

(3)
800.00
776.00
752.72
730.14
708.23

Variable Expense
Year 1 Renewal

(4)
250

(5)
0
50
50
50
50

Fixed Expense
Persistency
Year 1 Renewal
Rate

(6)
150

(7)
0
40
40
40
40

(8)
1.000
0.850
0.850
0.850
0.850

Cumulative
Persistency

(9)
1.000
0.850
0.723
0.614
0.522

Profit

(10)
-200.00
113.90
113.63
110.46
105.33

Discount
Factor

Present Value of
Profit
Premium

(11)
(12)
(13)
1.000 -200.00 1,000.00
1.100 103.55
772.73
1.210 93.91
597.11
1.331 82.99
461.40
356.54
1.464 71.94
152.38 3,187.77

Only the values in bold need to be adjusted in accordance with the 10% rate increase.
This produces the following impact:
Policy
Year Premium
(1)
(2)
1
1,100
2
1,100
3
1,100
4
1,100
5
1,100
Total

PV of Variable Expense Fixed Expense Persistency Cumulative
Discount PV of
Loss
Year 1 Renewal Year 1 Renewal
Rate
Persistency Profit Factor Profit
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
800.00
0
150
0
1.000
1.000 -125.00
1.000 -125.00
275
776.00
55
40
0.850
0.850 194.65
1.100 176.95
752.72
55
40
0.850
0.723 182.27
1.210 150.64
730.14
55
40
0.850
0.614 168.80
1.331 126.82
708.23
55
40
0.850
0.522 154.92
1.464 105.81
435.22

Therefore, the revised present value 5-year aggregate profit = 1,000 * .80 * $435.22 = $348,176
(b) To answer this , use the same equation as shown in part (a) and solve for the policy count impact.
5-year aggregate profit = (Policy count at time 0)*(policy count impact)*(aggregate present value profit).
$152,380 = 1,000 * (1 - x) * $435.22. x = .6498.

Exam 5, V1b

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Personal Automobile Premiums: An Asset Share Pricing Approach
For Property-Casualty Insurance – Feldblum, S.
Solutions to questions from the 1997 exam
Question 3.
1. T. Feldblum states “Persistency rates (retention rates) are the crux of asset share pricing models.
They are most important when the net insurance income varies by duration since policy inception.”
page 207.
2. T. On page 208, Feldblum compares persistency rates among direct writers and independent agency
companies:
State Farm has high retention rates because:
(a) it targets a suburban and rural insured
population.
(b) it offers low premium rates.
(c) it provides renewal discounts.

Many independent agency co.’s have low retention rates
(a) because the agents can move the insured to whichever
company offers the lowest rates.
(b) because these carriers use little consumer advertising.

3. F. page 239.
Although persistency rates by duration are easily determined for current classifications (the % of young
male drivers in their 5th policy year who persist into their 6th year), it is persistency rates by original
classification, not current classification, that it is needed.
Notice the difference: the persistency of young male drivers in their 5th policy year does not tell us the
expected 5th year persistency of young male drivers.
Answer C.

Exam 5, V1b

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Personal Automobile Premiums: An Asset Share Pricing Approach
For Property-Casualty Insurance – Feldblum, S.
Solutions to questions from the 1997 exam
Question 30.
On pages 292-294, Feldblum shows in Exhibits 4-6, an example of how to use the asset share model when
discounts are offered.
The easiest way to solve the problem:
1. Prepare an exhibit like the one below, without any of the numbers filled in.
2. Fill in the “givens” in the problem. These appear in italics.
3. Adjust your initial year premiums, losses, and expenses according to the growth / trend rates given
in the problem.
4. Memorize the formulas for calculating columns 9 through 12. These are relatively easy and apply
to a number of exhibits in the article.
5. Key. Recognize that the only computation which differs in the tables below is column 2.
Not offering a 5% renewal discount to its Personal Auto policyholders
Policy
Year

(1)
1
2
3
4
Total

Premium

(2)
1,000.00
1,060.00
1,123.60
1,191.02

Loss

(3)
800.00
840.00
882.00
926.10

Variable Expense
Year 1 Renewal

(4)
180

(5)
0
63.60
67.42
71.46

Fixed Expense
Persistency
Year 1 Renewal
Rate

(6)
10

(7)
0
5
5
5

(8)
1.000
0.800
0.800
0.800

Cumulative
Persistency

Discount
Factor

Profit

(9)
(10)
1.000 10.00
0.800 121.12
0.640 108.28
0.512 96.49

PV of
Profit

(11)
(12)
1.000 10.00
1.100 110.11
1.210 89.49
1.331 72.49
282.09

Column (2) = an average premium per car of $1,000, with 6% annual growth due to rate changes.
Column (3) shows the initial year Loss and LAE of $800 increased by 6% trend.
Columns (4) through (7): Variable expenses are 18% of premium in the 1st year and 6.0% in renewal years.
Fixed expenses are $10 of premium in the 1st year, and $5 in the following years.
Column (8) shows the expected persistency rate.
Column (9) = the downward product of column (8).
Column (10) = Column (9) *{Column (2) - of Columns (3, 4, 5, 6, and 7)}.
Column (11) uses the 10% annual cost of capital.
Column (12) = column (10) / column (11).
Offering a 5% renewal discount to its Personal Auto policyholders
Policy
Variable Expense Fixed Expense Persistency Cumulative
Discount PV of
Year Premium
Loss
Year 1 Renewal Year 1 Renewal
Rate
Persistency Profit
Factor Profit
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
1,000.00 800.00
10
1.000
1
180
0
0
1.000 10.00
1.000 10.00
5
1.100 90.50
2
60.42
0.980 99.55
1,007.00 840.00
0.980
5
3
64.05
0.960 111.77
1.210 92.37
1,067.42 882.00
0.980
5
4
67.89
0.941 124.69
1.331 93.68
1,131.47 926.10
0.980
Total
286.55
Column (2) = an average premium per car of $1,000, with 6% annual growth due to rate changes, times (1-.05 credit)

Exam 5, V1b

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Personal Automobile Premiums: An Asset Share Pricing Approach
For Property-Casualty Insurance – Feldblum, S.
Solutions to questions from the 1998 exam
Question 52
a.
On page 210. Feldblum discusses the relationship among three terms: persistency rates (a.k.a. retention
rates), termination rates, and the probabilities of termination.
Persistency may be computed using termination rates or probabilities of termination.
 The termination rate equals the number of terminations  [number of terminations + policies
persisting].
 The probability of termination equals the number of terminations  the number of originally issued
policies in that cohort.
Feldblum concludes that termination rates more clearly distinguish persistency patterns by classification.
Thus, the persistency rate is 1.0 - termination rate.
Policy Year
0
1
2
3
Total

Probability of
Termination
(1)
.30
.30
.20
.20
1.00

Number of
terminations
(2)
30
30
20
20
100

Termination
Rate
(3)
0
30/(30+70) = .30
30/(30+40) = .43
20/(20+20) = .50

Persistency
Rate
(4) = 1.0 - (3)
1.00
.70
.57
.50

Cumulative
Persistency rate
(5)
1.00
1.00 *.70 =.70
.70* .57 = .40
.40 * .50 = .20

(2) = (1)/(1 total) * 100
(5) = downward product of column (4). These values will be used in part b.

Exam 5, V1b

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Personal Automobile Premiums: An Asset Share Pricing Approach
For Property-Casualty Insurance – Feldblum, S.
Solutions to questions from the 1998 exam
Question 52
b. On page 289, Feldblum shows in Exhibit 1, an example of how to use the asset share model under a
company growth scenario.
The easiest way to solve problems like this one is to:
1. Prepare an exhibit like the one below, without any of the numbers filled in.
2. Fill in the “givens” in the problem. These appear in bold.
3. Adjust your initial year premiums, losses, and expenses according to the growth / trend rates given
in the problem.
4. Memorize the formulas for calculating columns 11 through 14. Note: columns 9 and 10 were
computed in part a above. These are relatively easy to calculate and apply to a number of
exhibits in the article.
5. Write formulas to compute what is asked for in the problem:
4 year underwriting return on premium =
4 year underwriting return on surplus =

Policy
Year

Premium

(1)
1
2
3
4
Total

(2)
500.00
550.00
605.00
665.50
2,320.50

Policy
Year

Profit

(1)
1
2
3
4
Total

(11)
-75.00
77.00
48.28
26.55

Annual Losses
Year 1
Renewal

(3)
400

Discount
Factor

(12)
1.000
1.150
1.323
1.521

(4)
0
330
363
399

 PV of profit
 PV of premium

Pr emium
 PV of profit
*
 PV of premium Surplus

Commission Expense
Year 1
Renewal

(5)
100

(6)
0
28
30
33

O. Variable Expense Persistency Cumulative
Year 1
Renewal
Rate
Persistency

(7)
75

(8)
0
83
91
100

(9)
1.00
0.70
0.57
0.50

(10)
1.00
0.70
0.40
0.20

Present Value of
Profit
Premium

(13)
(14)
-75.00
500.00
66.96
334.78
36.51
182.53
87.30
17.46
45.92 1,104.61

Column (2) = an average premium per car of $500, with10% annual growth due to annual rate increases.
Column (3) is column (2) * .80 new business loss ratio. Column (4) is column (2) * .60 renewal loss ratio.
Columns (5) through (8): Commissions are 20% of premium in the 1st year and 5.0% in renewal
Variable expenses are 15% of premium in the 1st year, are 15% in the following years.
Column (9) is the persistency rate calculated in part a of the question.
Column (10) = the downward product of column (9).
Column (11) = Column (10) *{Column (2)-  -of Columns (3, 4, 5, 6, 7 and 8)}.
Column (12) uses a rate of 15% per year compounded annually.
Column (13) = column (11) / column (12).
Column (14) = column (2) * column (10) / column (12) .

4 year underwriting return on premium = 46 / 1105 = 4.2%
4 year underwriting return on surplus = .042 * 2.5 = 10.5%

Exam 5, V1b

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Personal Automobile Premiums: An Asset Share Pricing Approach
For Property-Casualty Insurance – Feldblum, S.
Solutions to questions from the 1999 exam
Question 20
1. T. Level premiums associated with whole life policies have lead life actuaries to place greater reliance
on asset-share pricing models than P&C actuaries (which work with premiums that fluctuate
widely). page 197.
2. T. It examines the profitability from inception to termination (including renewals) of the policy. page 192
3. F. Feldblum demonstrates how asset share pricing is used to determine class relativities for young
drivers. page 217.
Answer B.
Question 34.
Calculate the present value of expected profits as a percentage of the present value of premium.
Assume that the cost of capital is 12%.
Policy
Year
1
2
3
4
Policy
Year
1
2
3
4

Premium
(1)
900
990
1,089
1,198

Present Value
of Loss
(2)
810
826
843
860

Cumulative
Persistency
(6)
1.00
0.75
0.60
0.51

Profit
(7)
-117.00
-11.25
31.80
65.28

Variable
Expense
(3)
135
149
163
180
Discount
Factor
(8)
1.00
(1.12)-1 = .893
(1.12)-2 = .797
(1.12)-3 = .712

Fixed
Expense
(4)
72
30
30
30

Persistency
Rate
(5)
1.00
.75
.80
.85

Present Value
Profit
Premium
(9) = (7)*(8)
(10)
-117.00
900
-10.04
663.05
25.35
520.75
46.47
435.01
-55.22
2,518.81

(6) is the downward product of column (5)
(7) is [(1) - {(2) + (3) + (4)}] * (6)
(10) is [(1) * (6) * (8)

Exam 5, V1b

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Personal Automobile Premiums: An Asset Share Pricing Approach
For Property-Casualty Insurance – Feldblum, S.
Solutions to questions from the 2000 exam
7. F. Persistency rates (the term "retention rates" are used interchangeably in this paper) are the crux
(and hence the fundamental issue) of asset share pricing models. See page 207.
8. T. Termination rates more clearly distinguish persistency patterns by classification. Probabilities of
termination, in certain analyses, provide a better portrayal of the insurer's profitability. See pages
210 - 211.
Question 47
On page 289, Feldblum shows in Exhibit 1, an example of how to use the asset share model under a
company growth scenario.
The easiest way to solve problems like this one is to:
1. Prepare an exhibit like the one below, without any of the numbers filled in.
2. Fill in the “givens” in the problem. These appear in bold (in this case, only the premium is a given input).
3. Adjust your initial year premiums, losses, and expenses according to the growth / trend rates given
in the problem.
4. Memorize the formulas for calculating columns 11 through 14.
5. Write formulas to compute what is asked for in the problem:
3 year underwriting return on premium =
Policy
Year

Premium

(1)
1999
2000
2001

(2)
1,000.00
1,060.00
1,123.60

Total

(3)
750.00

(4)
0.00
786.41
824.58

Fixed Expense
New
Renewal

(5)
150

Variable Expense
Year 1
Renewal

(6)
0
52.00
54.08

(7)
250

(8)
0
106.00
112.36

Persistency
Rate

(9)
1.00
0.90
0.90

Cumulative
Persistency

(10)
1.00
0.90
0.81

3,183.60

Policy
Year

Profit

(1)
1999
2000
2001

(11)
-150.00
104.03
107.39

Total

Annual Losses
Year 1
subsequent

 PV of profit
 PV of premium

Discount
Factor

(12)
1.000
1.090
1.188

Present Value of
Profit
Premium

(13)
-150.00
95.44
90.39

(14)
1,000.00
875.23
766.03

35.83

2,641.26

Column 2 is an average premium per car of 1000 with a 6% annual growth due to annual rate increases
Column (3) is column (2) * 0.75 new business loss ratio. Column (4) is column (3) * 1.049 net trend.
Column (5). First year fixed expenses are .15*1,000. Fixed renewal expenses in renewal year 1.
equal fixed renewal expenses in policy year 1 times fixed expense trend: 52 = 1,000*.05*1.04
Variable expenses are 25% of 'premium' in the 1st 'year', and 10% in the following years
Column (9) is 1.0 - termination rate of 10%
Column (10) = the downward product of column (9).
Column (11) = Column (10) *{Column (2) - Sum of Columns (3, 4, 5, 6, 7 and 8)}.
Column (12) uses a rate of 9% per year compounded annually.
Column (13) = column (11) / column (12).
Column (14) = column (2) * column (10) / column (12) .

3 year underwriting return on premium = 35.83 / 2,641.26 = 1.36%

Exam 5, V1b

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Personal Automobile Premiums: An Asset Share Pricing Approach
For Property-Casualty Insurance – Feldblum, S.
Solutions to questions from the 2001 exam
Question 11. Calculate the termination rate for the third year.
General information:
“Persistency may be analyzed either by termination rates or by probabilities of termination.
The termination rate is the number of terminations during a given renewal period divided by the sum of
terminations during that period plus policies persisting through that period.
Termination rates more clearly distinguish persistency patterns by classification.”
Solution:
The termination rates by year are:
35% [ = 350  1000] the 1st year
30.79% [=200  (1,000 – 350) = 650] the 2nd year, and
22.22% [=100  (1,000 – 350 – 200) = 450] the 3rd year.
See page 210.

Answer D.

Questions 12. Which of the following is false ?
A. Asset share pricing determines rates, not rate revisions. True. See page 215.
B. Life insurance policy claim rates are more certain than property-casualty policy claim rates. True.
Claim rates in casualty insurance are more variable and less well understood. See page198.
C. It is appropriate to assume the same pattern of persistency ratios for both direct writers and
independent agency companies. False. Direct writers, like Sate Farm, have high retention rates
because they offer low premium rates and provide renewal discounts. Many independent agency
companies have low retention rates because they can move the insured to whichever company
offers the lowest rate. See page 208.
Answer C.
D. A level commission structure is inappropriate for the persisting and profitable risks. True. A level
commission structure works wells for risks that terminate quickly. It works poorly for risks that
endure with the carrier. See page 206.
E. The dominant market share of the direct writers makes asset share pricing a more appropriate
model for personal automobile insurance. True. In the personal lines of business, direct writers
are steadily gaining market share, See page 206.
23. (2 points) 4 factors which help to explain the relationship between the duration of an auto policy and
the claim frequency for that policy include:
1. Experience: Good driving habits and safety precautions exercised by experienced drivers
contribute to lower claim frequency. Inexperienced drivers are more careless and tend to
have high claim frequencies.
2. Youth: Young male and female drivers have relatively new policies and have higher than
average claim frequencies, even after adjusting for driving experience.
3. Transience: Many high risk drivers (young males), are "transient" insureds. Young male
drivers tend to cancel policies as they shop for cheaper coverage, move often, and after
causing accidents, either voluntarily drop coverage or tend to be non-renewed by
underwriters.
4. Acquisition of the Vehicle: Accident frequency often decreases with time since acquisition, as
the insured becomes familiar with the operation of the vehicle. See pages 200 – 203.

Exam 5, V1b

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Personal Automobile Premiums: An Asset Share Pricing Approach
For Property-Casualty Insurance – Feldblum, S.
Question 36.
30 policies terminate in the third year
The probability of termination in year 2 is 0.0816
The termination rate in year 2 is 0.1000
The termination rate in year 3 is 0.0750
a. (1½ points) Calculate the number of policies terminated in year 1 and year 2.
b. (1 point) Calculate the original number of policies in the cohort.
c. (½ point) Calculate the termination rate and the probability of termination in year 1.
Step 1: Write equations to determine the termination rate and the probability of termination.
1. The termination rate =

the number of terminations
number of terminations + policies persisting

2. Probability of termination =

the number of terminations
number of policies in cohort

Step 2: Set up a table similar to the one below, and enter the given data to develop a structured approach
to answering questions a, b, and c.
Number of policies
a. in cohort
b. Terminating
c. Persisting
Rates/Probabilities
d. Termination rate
e. Prob of termination

Year
2

3
30

0.1000
0.0816

0.0750

Step 3: Fill in the table above by working backwards from time 3 to time 0.

The termination rate3 =.075=

30-(30*.075)
30
 370
. No. of polices persisting (3) =
.075
30+ policies persisting

No. of policies persisting (2) = No. of polices terminating (3) + No. of polices persisting (3) = 30 + 370 = 400

The termination rate2 =.100=

No of policies terminating
.100(400)
 44
. No. of polices terminating (2) =
No of policies terminating+ 400
(1.0 - .100)

No. of policies persisting (1) = No. of polices terminating (2) + No. of polices persisting (2) = 44 + 400 = 444

Probability of termination 2 =

the number of terminations 2
. No. of polices in cohort (1) = 44 ÷ .0816 = 539
number of policies in cohort1

No. of policies terminating (1) = No. of policies in cohort - No. of polices persisting (2) = 539 - 444 = 95
Step 4: By examining the formulas in Step 1, recognize that the termination rate and the probability of
termination in year 1 are equal.

The termination rate1 =Prob of termination=
Answers:

Exam 5, V1b

95
 .176
539

a. The number of policies terminated in year 1 and 2 are 95 and 44 respectively.
b. The original number of policies in the cohort equals 539.
c. The termination rate and the probability of termination in year 1 equal .176.

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Personal Automobile Premiums: An Asset Share Pricing Approach
For Property-Casualty Insurance – Feldblum, S.
Solutions to questions from the 2003 exam
Question 43.
a. Using the procedure described by Feldblum in "Personal Automobile Premiums: An Asset Share Pricing
Approach for Property/Casualty Insurance," calculate the indicated premium relativity for youthful drivers.
Use a three year time horizon to determine your answer. Show all work.
• Indicated adult class new business policy premium = $1,000
• Youthful class discounted loss cost during first policy year = $1,500
• Premiums increase 10% annually
• Losses increase 5% annually
• Variable expense ratio = 20% for all business
• All expenses are variable
• Annual youthful class lapse rate = 25%
• The company's cost of capital is 10%
• 3-year PV of premium for adult class = $2,710
• 3-year PV of profit for adult class = $90.30
• Assume same return is earned for all classes
Step 1: Write a generic formula to compute the indicated premium relativity for youthful drivers:

Premium Relativity youthful 

Youthful Premiumyr1
Adult Premiumyr1



?
$1,000

Step 2: It is assumed that the same rate of return is earned for all classes. We can compute that return as
the PV of the profit/PV of the premium. Since these values are only unknown for the youthful class,
create a table similar to the one below to compute these values:
Policy
Year

Premium

(1)
1
2
3

(2)
P
1.1*P
1.21*P

Policy
Year

Profit

(1)
1
2
3
Total

(11)

Annual Losses
Year 1
subsequent

(3)
1,500

Discount
Factor

(12)
1.000
1.100
1.210

(4)
0
1,575
1,654

Variable Expense
Year 1
Renewal

(7)
.20P

(8)
0
.22P
.242P

Persistency
Rate

Cumulative
Persistency

(9)
1.0000
0.7500
0.7500

(10)
1.0000
0.7500
0.5625

Present Value of
Profit
Premium

(13)

(14)

Compute Compute

Step 3: Compute the 3 year PV of profit and PV of premium for the youthful class:
3-year present value of profit
= [(P - 0.2P - 1,500) * 1/1] + [(1.1P - 0.22P – 1,575) * 0.75/1.10] + [(1.21P - 0.242P – 1,653.75) * 0.5625/1.21]
= [(0.8P – 1,500)] + [(0.88P – 1,575) * 0.682] + [(0.968P – 1,653.75) * 0.465]= 1.85P – 3,342.65
3-year present value of premium = P * 1/1 + 1.1P * 0.75/1.1 + 1.21P * 0.5625/1.21 = 2.3125P
Step 4: Equate the 3 year PV of profit and PV of premium for the adult and youthful class:
Adult return $90.30 / $2,710 = 3.33%. Thus, 3.33% = (1.85P – 3,342.65) / 2.3125P. P = 1,885 for youthful class
Therefore, the indicated premium relativity for youthful drivers = $1,885 / $1,000 = 1.885
b. How might traditional ratemaking methods be misleading in determining classification relativities?

Exam 5, V1b

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Personal Automobile Premiums: An Asset Share Pricing Approach
For Property-Casualty Insurance – Feldblum, S.
Since traditional ratemaking methods don’t take into account persistency rates, they tend to
underestimate relativities for classifications with poor persistency and overestimate relativities for
classifications with good persistency.

Solutions to questions from the 2004 exam
44. (4 points)
Using the asset share pricing model, determine the return on equity over the three-year period. Show all work.
Preliminary comments:
The easiest way to solve problems like this one is to:
1. Prepare an exhibit like the one below, without any of the numbers filled in.
2. Fill in the “givens” in the problem. These appear in bold.
3. Adjust initial year premiums, losses, and expenses according to the growth / trend rates given in the
problem.
4. Memorize the formulas for calculating columns 8 through 14.
5. Write formulas to compute what is asked for in the problem:
Policy
Year

(1)
1
2
3
Policy
Year

(1)
1
2
3
Total

Premium

(2)
1,000.00
1,120.00
1,254.40
3,374.40

Annual Losses
subsequent
Year 1

(3)
750

Profit

Discount
Factor

(9)
-150.00
204.06
255.45

(10)
1.000
0.870
0.756

(4)
0
793.27
839.03

Fixed Expenses
Year 1
Renewal

(5)
400.00

Present Value of
Profit
Premium

(11)
-150.00
177.44
193.15
220.60

(12)
1,000.00
876.52
768.29
2,644.81

(6)
0
100.00
100.00

Persistency
Rate

Cumulative
Persistency

(7)
1.0000
0.9000
0.9000

(8)
1.0000
0.9000
0.8100

Return on
Premium
Equity

(13)

8.34%

(14)

25.02%

Column (2) is computed based on the givens in the problem
Column (3) is col (2) * .75 (1st year loss ratio). Col (4) is col (2) * 1.10/1.04 net loss cost trend
Columns (5) - (7) are based on the givens in the problem
Column (8) = the downward product of column (7).
Column (9) = Column (8) *{Column (2) - sum of Columns (3, 4, 5, 6)}.
Column (10) uses a discount rate of 15% per year compounded annually.
Column (11) = column (9) * column (10).
Column (12) = column (2) * column (8) * column (10) .
Column (13) = column (11) Total / column (12) Total
Column (14) = column (13) * Premium to Surplus ratio (given in the problem)

b. (1 point) Explain how asset share pricing models and property/casualty insurance ratemaking methods differ
in their consideration of the profitability of an insurance policy.
Traditional P&C ratemaking methods consider only the profitability of the future policy period to determine
whether there will be enough premium to cover losses and expenses during the forecast period.
Asset share pricing models look at profitability over the life of the policy, taking into account policyholder
persistency patterns.

Exam 5, V1b

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Personal Automobile Premiums: An Asset Share Pricing Approach
For Property-Casualty Insurance – Feldblum, S.
Solutions to questions from the 2005 exam
56. (3 points) Use an asset share pricing approach to determine whether a company should write this
business. Show all work and explain your answer.
Initial comments: “Profits may be measured as the net present value of premiums minus the net present
value of expenditures (losses, expenses, and taxes). Thus, Anderson, recommends that “the profit
objective be defined by the criterion that the present value of the profits which will be received in the
future be equal to the present value of the surplus depletion, with both present values based on a yield
rate or yield rates which represent adequate return to the stockholders for the degree of risk incurred in
expending surplus in the expectation of receiving future profits. That is, the present value of the entire
series of profits and losses is zero.”
Based on the above, setup a table similar to the one below to determine the present value of the profits.
Policy
Year

(1)
1
2
3

PV of the
Losses
(2)
(3)
$500
$415
$550
$440
$460
$605
1,655.00 1,315.00
Premium

Expenses
(4)
$100
$110
$121

Persistency
Rate

(5)
100%
85%
85%

Cumulative
Persistency

(6)
100.000%
85.000%
72.250%

Discount
Factor

PV of the
Profits

(7)
1.0
1.1-1
1.1-2

(8)
-$15.00
$0.00
$14.33
-$0.67

Column (6) = the downward product of column (5).
Column (8) = (6) *{ (2) - (3) - (4) )} * (7).
Using a 10% cost of capital (the yield from equities of similar risk), this business should not be written
since the present value of the profits are negative. A better return could be obtained by simply investing
in the equities of similar risk that are yielding 10% per year.
See pages 265 – 266.

Solutions to questions from the 2006 exam
50. (5.75 points) Determine which alternative is more profitable for a cohort of 65 year-old existing insureds
over a three-year time period. Show all work.
Preliminary comments:
The easiest way to solve problems like this one is to:
1. Prepare an exhibit like the one below, without any of the numbers filled in.
2. Fill in the “givens” in the problem. These appear in bold.
3. Adjust initial year premiums, losses, and expenses according to the growth / trend rates given in the
problem.
4. Memorize the formulas for calculating columns 8 through 13.
5. Write formulas to compute what is asked for in the problem:
Note: It is assumed that the persistency rates given are for that year only, and are not cumulative
persistency rates.

Exam 5, V1b

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Personal Automobile Premiums: An Asset Share Pricing Approach
For Property-Casualty Insurance – Feldblum, S.
Solutions to questions from the 2006 exam
Question 50
Profitability analysis for 65 year-old existing insureds - Without Offering a Discount
Policy
Year

(1)
1
2
3
Policy
Year

(1)
1
2
3
Total

Premium

(2)
800.00
840.00
882.00
2,522.00

Annual Losses
Year 1
subsequent

(3)
500

Profit

Discount
Factor

(9)
100.00
99.18
92.73

(10)
1.000
0.909
0.826

(4)
0
519.80
540.39

Expenses
Variable
Fixed

(5)
160.00
168.00
176.40

(6)
40.00
42.00
44.00

Persistency Cumulative
Rate
Persistency

(7)
1.0000
0.9000
0.8500

(8)
1.0000
0.9000
0.7650

Present Value of
Return on
Profit
Premium
Premium

(11)
100.00
90.16
76.63
266.80

(12)
800.00
687.27
557.63
2,044.90

(13)

13.05%

Column (2) is computed based on the givens in the problem
Column (3) is given; Col (4) is col (2) * 1.05/1.01 net loss cost trend
Columns (5) = .20 * (2)
Columns (6) - (7) are based on the givens in the problem
Column (8) = the downward product of column (7).
Column (9) = Column (8) *{Column (2) - sum of Columns (3, 4, 5, 6)}.
Column (10) uses a discount rate of 10% per year compounded annually.
Column (11) = column (9) * column (10).
Column (12) = column (2) * column (8) * column (10) .
Column (13) = column (11) Total / column (12) Total
Profitability analysis for 65 year-old existing insureds - With Offering a Discount
Policy
Year

(1)
1
2
3
Policy
Year

Premium

(2)
760.00
798.00
837.90
2,395.90
Profit

Annual Losses
subsequent
Year 1

(3)
500

Discount
Factor

(4)
0
519.80
540.39

Expenses
Variable
Fixed

(5)
152.00
159.60
167.58

(6)
40.00
42.00
44.00

Persistency Cumulative
Rate
Persistency

(7)
1.0000
0.9800
0.9500

(8)
1.0000
0.9800
0.9310

Present Value of
Return on
Profit
Premium
Premium

(1)
(9)
(10)
(11)
(12)
(13)
1
68.00
1.000
68.00
760.00
2
75.07
0.909
68.24
710.95
644.70
3
80.00
0.826
66.12
Total
202.36
2,115.64
9.56%
Note: $760 = $800 * (1-.05)
The more profitable solution is to not offer a discount. For this cohort over a three-year period, the return
on premium without the discount is 13.0%, and the return on premium with the discount is only 9.6%.

Exam 5, V1b

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Personal Automobile Premiums: An Asset Share Pricing Approach
For Property-Casualty Insurance – Feldblum, S.
Solutions to questions from the 2007 exam
28. (4.0 points) Using the Asset Share Pricing approach described by Feldblum, determine
whether this opportunity exceeds the company's target return on surplus of 15% over a threeyear time period. Show all work.
Preliminary comments:
The easiest way to solve problems like this one is to:
1. Prepare an exhibit like the one below, without any of the numbers filled in.
2. Adjust initial year premiums, losses, and expenses according to the growth / trend rates given in the
problem.
3. Memorize the formulas for calculating columns 10 through 16.
4. Write formulas to compute what is asked for in the problem:

Policy
Year

Premium

(1)
1
2
3

(2)
900.00
945.00
992.25

Policy
Year

Profit

Discount
Factor

(11)
-78.00
133.06
128.80

(12)
1.0000
1.0500
1.1025

(1)
1
2
3
Total

Annual Losses
Year 1
subsequent

(3)
675

(4)
0
701.21
728.45

Fixed Expenses
Year 1
Renewal

Variable Expenses
Year 1
Renewal

(5)
60.00

(7)
243.00

(6)
0
40.00
42.00

Present Value of
Profit
Premium

(13)
-78.00
126.72
116.83
165.55

(14)
900.00
765.00
673.20
2,338.20

(8)
0
47.25
49.61

Persistency
Rate

Cumulative
Persistency

(9)
1.0000
0.8500
0.8800

(10)
1.0000
0.8500
0.7480

Return on
Premium
Equity

(15)

7.08%

(16)

14.16%

Column (2) is computed based on the givens in the problem
Column (3) is col (2) * .75 (1st year loss ratio). Col (4) is col (2) * 1.07/1.03 net loss cost trend
Columns (5) and (6) are based on the givens in the problem
Columns (7) and (8) are based (2) and the givens in the problem
Column (9) is 1.0 - the given termination rates in the problem
Column (10) = the downward product of column (9).
Column (11) = Column (10) *{Column (2) - sum of Columns (3, 4, 5, 6,7,8)}.
Column (12) uses a discount rate of 5% per year compounded annually.
Column (13) = column (11) / column (12).
Column (14) = column (2) * column (10) / column (12) .
Column (15) = column (13) Total / column (14) Total
Column (16) = column (15) * Premium to Surplus ratio (given in the problem)

Since the computed return on surplus does not exceed the target return on surplus, the company should
not pursue this opportunity.

Exam 5, V1b

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Personal Automobile Premiums: An Asset Share Pricing Approach
For Property-Casualty Insurance – Feldblum, S.
Solutions to questions from the 2008 exam
Model Solution - Question 43
43. (1.0 point) Contrast the asset share pricing model to traditional techniques for calculating rate relativities.
Traditional techniques only consider a single period. By doing so, they fail to consider differences in persistency
amongst different risks. Persistency can have a significant impact due to loss and expense differences between
new and renewal business. The asset share pricing model accounts for this by introducing multiple periods,
persistency, and different assumptions for new and renewal business.
Model Solution - Question 44
Calculate the indicated rate relativity for Class B as compared to the base class (Class A) using the asset share
pricing model and a two-year time horizon.
Initial comments: The easiest way to solve problems like this one is to:
1. Prepare an exhibit like the one below, without any of the numbers filled in.
2. Fill in the “givens” in the problem. These appear in bold (in this case, only the premium is a given input).
3. Adjust your initial year premiums, losses, and expenses according to the growth / trend rates given
in the problem.
4. Memorize the formulas for calculating columns 10 through 16.
5. Write formulas to compute what is asked for in the problem:
Step 1: Write a generic formula to compute the indicated premium relativity for Class B:

Class B Premium Relativity 

Class B Premiumyr1
Class A Premiumyr1



?
$633.80

Step 2: It is assumed that the same rate of return is earned for all classes. We can compute that return as
the PV of the profit/PV of the premium. Since these values are only unknown for the Class B, create
a table similar to the one below to compute these values:
Policy
Year

Premium

(1)
1
2

(2)
X
X

Policy
Year

(1)
1
2
Total

Profit

(11)

Annual Losses
Year 1
subsequent

(3)
1,000

Discount
Factor

(12)
1.000
1.100
`

(4)
0
900

Variable
Expense

Fixed
Expense

Persistency
Rate

Cumulative
Persistency

(7)
.10X
.05X

(8)
120
110

(9)
1.0000
0.6000

(10)
1.0000
0.6000

Present Value of
Profit
Premium

(13)

(14)

Compute Compute

Return on
Premium
Equity

(15)

(16)

Compute

Step 3: Compute the 2 year PV of profit and PV of premium for the Class B:
2-year present value of profit:
Year 1: [(X – 1,000 – 120 – 0.1X) * 1.0]/1.0 = (0.90X – 1,120)/1.0
Year 2: [(X – 900 – 110 – 0.05X) * 0.60]/1.1 = (0.95X – 1,010)* 0.6/1.1
Year 1 + Year 2: 1.418X – 1671
2-year present value of premium = X + .60X/1.1= 1.545X

Exam 5, V1b

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Personal Automobile Premiums: An Asset Share Pricing Approach
For Property-Casualty Insurance – Feldblum, S.
Step 4: Equate the 2 year PV of profit and PV of premium for Class B to the targeted pre-tax return on
equity of 6% and the given P/S ratio of 2:1, to solve for Class B year 1 premium.
PV of the profit/PV of premium = 1.418X – 1671/1.545X = .06 * [1.0/2.0]. X = 1,218.24
Therefore, the indicated premium relativity Class B = $1,218.24/$633.8 = 1.92

Solutions to questions from the 2009 exam
Question 45
a. Persistency rates are the proportion of business that remains in force from one period to the next. They are
important in classification ratemaking because certain classes may have higher persistency than others,
making them more profitable to write when you consider the complete expected lifetime of the policy.
b. A direct writer is expected to have higher persistency rates since the company owns the renewals. In
the independent agency systems, the agent owns the renewals and may put the insured with a
different company in order to get a better rate.
c. Standard insured has higher persistency rate because non-standard insureds are high risk, which means they
are either likely to shop around for cheap coverage or more likely to be cancelled by the company.

Solutions to questions from the 2010 exam
Question 35
1

Prem

PV Loss

Var Exp

Fixed Exp

Cumulative

1
2

1000
1100

650
680.95

300
110

200
50

Persistency
Rate
1
.75

Profit

Discount
1

PV profit

PV Prem
1000

PY
1

‐150

194.29

(1) Use prem trend of 10 %
(2) Use 1st yr LR of 65 %
For yr 2 take 650*1.1/1.05
(3) For yr 1= 30 % (1000)
For yr 2 = 1100 (.1)
(6) is downward product of (5)
(7) = [(1) - (2) - (3) - (4)]x6
(9) = (7)/(8)
(10) = (1)x(6 )/(8)

Exam 5, V1b

1.1

‐150
176.63

750

26.63

1750

1
.75

ROP = 26.63 = 1.522 %
1750
ROS = Profit/Premium * Premium/Surplus = ROP *P/S = 1.522 % (2) = 3.04%
↓
Doesn’t meet company’s goals of 4 % ROS

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Personal Automobile Premiums: An Asset Share Pricing Approach
For Property-Casualty Insurance – Feldblum, S.
Solutions to questions from the 2011 exam
20a. (0.5 point) Using a one-year period, calculate the premium for class B if the same profit loading is
targeted for all classes.
20b. (1.25 points) Using a two-year period, calculate the return on sales for class A.
20c. (1.75 points) Using a two-year period, calculate the premium for class B that would achieve the
same return on sales as calculated in part b above.
Initial comments:
Year 1 Premium = PV Losses + Variable Expenses + Fixed Expenses + Profit.
Year 2 Profit = [Premium - PV Losses – Variable Expenses – Fixed Expenses]*Cumulative Persistency
Return on Sales = PV Profit/PV Premium
The probability of termination is the number of terminations during a given renewal period divided by the
number of originally issued policies in that cohort. A cohort is a group of policies written in a given issue period.
Persistency may be computed using termination rates or probabilities of termination.
 The termination rate = the number of terminations  [number of terminations + policies persisting].
 The probability of termination = the number of terminations  the number of originally issued policies
in that cohort.
Feldblum concludes that termination rates more clearly distinguish persistency patterns by classification.
Thus, the persistency rate is 1.0 - termination rate.
Question 20 – Model Solution
a. Profit for class A = 800 - 550 - (800 * .2) - 42 = 48
Profit loading for class A = .06 = 48/800
Premium for class B
.06PB = PB - 650 - .24PB – 50; .07PB = 700; PB = 1,000
b.
Year
Prem PV Loss VarExp
Fixed exp Persis Discount Factor
(1)
(2)
(3)
(4)
(5)
(6)
1
800
550
160
42
1.0
1.0
2
800
550
96
20
.85
1.08

PV Prem
(7)
800
630
1,430

PV Profit
(8)
48
105.463
153.463

(7) = [(1) * (5)]/(6)
(8) = {[(1) - (2) - (3) - (4)]*(5)}/(6)

Return on sales = 153.463/1,430 = .1073
c.
Year
1
2

Prem
(1)
P
P

PV Loss
(2)
650
650

VarExp
(3)
.24P
.20P

Fixed exp
(4)
50
30

Persis
(5)
1.0
.60

Disc Factor
(6)
1.0
1.08

PV Prem
(7)
P
.5556P
1.5556P

PV Profit
(8)
.76P - 700
.444P - 377.78
1.204P - 1,077.78

(7) = [(1) * (5)]/(6)
(8) = {[(1) - (2) - (3) - (4)]*(5)}/(6)

(1.204P - 1,077.78)/1.5556P = .1073
.167P = 1.204P - 1,077.78; 1.037P = 1,077.78; P = 1,039.32

Exam 5, V1b

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CAS Exam 5 – Spring 2013 - Questions, Solutions and Commentary
1. (2 points) Given the following information for an insurance company that writes 24-month term policies:
Policy

Effective

Group

Date

Expiration
Date

Number of
Vehicles

January 1, 2010

December 31, 2011

July 1, 2010

June 30, 2012

100

All policies within each group have the same effective date.
a. (0.5 point) Calculate the earned car-years for calendar year 2011.
b. (0.5 point) Calculate the earned car-years for policy year 2010 evaluated as of December 31,
2010 and as of December 31, 2011.
c. (0.5 point) Assume Policy Group B cancels on January 1, 2011. Calculate the 2010 policy year written
car-years evaluated as of December 31, 2010 and as of December 31, 2011 for Policy Group B.
d. (0.5 point) Assume Policy Group B cancels on July 1, 2011. Calculate the 2010 and 2011 calendar
year written car-years for Policy Group B.
S13 - Exam 5 - Question #1
Preliminary comments:


Since we are asked to compute CY and PY earned car years, and CY and PY written car years for
Policy Group B that cancels its policy on different dates, it is helpful to set up a timeline to illustrate
effective, expiration, and cancellation dates.



For auto policies, one typically works with exposures (car years) written on annual policies. Since the
given policies are 24-month policies, the number of car years are multiplied by two.
Typically, the concepts of written and earned exposures are based on the assumption of annual
policies being issued.
Per the text “If the policy term is shorter or longer than a year, then the aggregation for each type of
exposure is calculated differently. For example, if we are given six-month policies, each policy would
represent one-half of a written exposure.” In this problem, 24-month policies are issued; thus each
policy covers twice the number of exposures (car years) that a 12 month policy would cover.
Further, since we are given multiple vehicles on policies within a policy group, we compute earned
and written exposures by multiplying by the number of vehicles within a policy group.



Written exposures arise from policies issued (i.e. underwritten or written) during a specified period
of time (e.g. a calendar quarter or a CY). CY 2011 written exposures are the sum of the exposures
for all policies that had an effective date in 2011.
If a policy cancels midterm, the policy will contribute a written exposure to two different CYs if the
policy cancellation date is in a different CY year than the original policy effective date.
In part d., Policy Group B is cancelled on 7/1/2011, one half way through its policy period. Each
policy in policy group B will contribute 1 written exposure to CY 2010 and -(1/2)(1)=-1/2 written
exposure to CY 2011.



Earned exposures are the portion of written exposures for which coverage has already been
provided as of a certain point in time. Note: Unlike CY earned exposure, exposure for one policy
cannot be earned in two different PYs.



Since CY captures transactions occurring on or after the first day of the year, and on or before the last
day of the year, CY is represented graphically as a square



PY is represented graphically using a parallelogram starting with a policy written on the first day of
the PY and ending with a policy written on the last day of the PY.

Exam 5, V1b

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CAS Exam 5 – Spring 2013 - Questions, Solutions and Commentary
7/1

CY 2010

CY 2011

CY 2012

7/1

 mos. exposed to loss in a specific CY  Policy term 
CY ECYs =  No. of vehicles  


 Total mos. exposed to loss over policy term   in years 
1a. PGA CY 2011 ECYs = (50) * (12/24) * 2 = 50
1b. PGB CY 2011 ECYs = (100) * (12/24) * 2 = 100
Total CY 2011 Earned Car Years = 50 + 100 = 150
Policy Year – General Comments
PY 2010

1/1

12/31

For PYs, all earned exposure is assigned to the year the policy was written and increases in relation to time. By
the time the policy year is complete (24 months after the beginning of the policy year for annual policies), the
policy year earned and written exposures are equivalent. Unlike CY EE, exposure for one policy cannot be
earned in two different PYs.
b1. Evaluated as of 12/31/2010

 mos. exposed to loss as of a point in time  Policy term 
PY ECYs =  No. of vehicles  


 Total mos. exposed to loss over policy term   in years 
PGA PY 2010 ECYs as of 12/31/2010 = (50) * (12/24) * 2 = 50
PGB PY 2010 ECYs as of 12/31/2010 = (100) * (6/24) * 2 = 50
Total earned car years = 50 + 50 = 100
b2. Evaluated as of 12/31/2011
PGA PY 2010 ECYs as of 12/31/2011 = (50) * (24/24) * 2 = 100
PGB PY 2010 ECYs as of 12/31/2011 = (100) * (18/24) * 2 = 150
Total earned car years = 100 + 150 = 250
Policy year written exposures – Policy Cancellation– General Comments
Since policy year written exposure is aggregated by policy effective dates, the original written exposure
and the written exposure due to the cancellation are all booked in the same policy year.
c. Evaluated as of 12/31/2010, but cancelling on 1/1/2011
PGB PY 2010 WCYs as of 12/31/2010 = (100) * 2 = 200
Evaluated as of 12/31/2011
= PGB PY 2010 WCYs as of 12/31/2010 - PY 2010 WCY cancellations on 1/1/2011 as of
12/31/2011
= (100) * 2 – [(100) * (18/24) * 2] = 200 – 150 = 50

Exam 5, V1b

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CAS Exam 5 – Spring 2013 - Questions, Solutions and Commentary

Calendar year written exposures – Policy Cancellation– General Comments
If a policy cancels midterm, the policy will contribute a written exposure to two different CYs if the policy
cancellation date is in a different CY year than the original policy effective date.
In part d., Policy Group B is cancelled on 7/1/2011, one half way through its policy period. Each policy in
policy group B will contribute 1 written exposure to CY 2010 and -(1/2)(1)=-1/2 written exposure to CY
2011.
d.

PGB CY 2010 WCYs = (100) * 2 = 200
PGB CY 2011 WCYs = - (100) * (12/24) * 2 = -100

Examiner’s Report
a. Most candidates answered this question correctly. A small number of candidates misread the problem and
assumed that the provided vehicle counts were actually the exposures over the two year period, which
caused the answer to be halved.
b. Most candidates answered this question correctly. A small number of candidates misread the problem and
assumed that the provided vehicle counts were actually the exposures over the two year period, which
caused the answer to be halved. A few others calculated only the earned car-years for one of the evaluation
dates requested.
c. Candidates generally answered this answer correctly. A small number of candidates misread the problem
and assumed that the provided vehicle counts were actually the exposures over the two year period, which
caused the answer to be halved. Some candidates also provided the combined values for both Policy A & B
instead of just policy B. Full credit was given to candidates that clearly identified the portion attributable to
Policy B. A few others calculated only the written car-years for one of the evaluation dates requested.
d. Candidates generally answered this answer correctly. A small number of candidates misread the problem
and assumed that the provided vehicle counts were actually the exposures over the two year period, which
caused the answer to be halved. Some candidates also provided the combined values for both Policy A & B
instead of just policy B. Full credit was given to candidates that clearly identified the portion attributable to
Policy B. A few others calculated only the written car-years for one of the calendar years requested.
There were also some candidates who weren’t familiar with the concept of having negative calendar year
counts in cases where a multiple-year policy was cancelled in a subsequent year. These candidates often got
the 2010 value correct, but would either answer the 2011 value as 0 or 100.

Exam 5, V1b

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CAS Exam 5 – Spring 2013 - Questions, Solutions and Commentary
2. (2 points) Given the following information for an insurance company:


Proposed effective date of the next rate change is January 1, 2014.



Rates will be in effect for 1 year.



All policies have 12-month terms and are written uniformly throughout the year.



Calendar year 2012 earned premium at current rate level is $114,208,050.
12 Month Period

Written Premium at

Ending

Current Rate Level

Written Exposures

December 31, 2011

$104,500,000

110,000

June 30, 2012

$113,800,500

121,000

December 31, 2012

$123,916,100

133,100

a. (1 point) Utilizing one-step trending, calculate the calendar year 2012 projected earned premium
at current rate level for use in calculating the rate change.
b. (0.25 point) Briefly discuss why a premium trend should be utilized in a rate level indication.
c. (0.25 point) Briefly discuss why it is inappropriate to use written premium at historical rate levels
to determine premium trends.
d. (0.5 point) The insurance company decides to move all existing business with a $100 deductible to a
$500 deductible upon renewal during calendar year 2013. Given this new information, discuss whether
the true projected earned premium will be higher, lower, or unchanged from that in part a. above.
S13 - Exam 5 - Question #2
Preliminary comments:
Data to use for premium trend. A decision to use earned or written premium must be made.
Written premium is a leading indicator of trends that will emerge in earned premium and the trends
observed in written premium are appropriate to apply to historical earned premium.
Assuming adequate data is available, the actuary will use quarterly average written premium at
current rate levels (as opposed to annual average written premium) to make the statistic as responsive
as possible.
Changes in the quarterly average WP are used to determine the amount historical premium needs to
be adjusted for premium trend.
One-Step Trending
The trend factor adjusts historical premium to account for expected premium levels from distributional
shifts in premium writings.
The Process: Using the changes discussed previously, the actuary selects a trend factor.
Next:

Determine the trend period.

Assume:

WP is used as the basis of the trend selection and EP for the overall rate level indications

Compute: The trend period as the length of time from the average written date of policies with
premium earned during the historical period to the average written date for policies that will
be in effect during the time the rates will be in effect. *

Exam 5, V1b

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CAS Exam 5 – Spring 2013 - Questions, Solutions and Commentary
a.

Proj CY 2012 EP @ CRL = Current CY 2012 EP @ CRL * Selected trend factor * Trend Period

Since we are given 12 month period ending data at semiannual evaluation dates, select a semiannual
trend factor. Do so by first computing average written premium at current rate levels at 12/31/2011,
6/30/2012 and at 12/31/2012.
Dec 31 2011
950 գ
-1%
104,500/110 = 950
June 30 2012

940.5 գ

Dec 31 2012

931

-1%
selected semiannual trend factor = -1%

Next, determine the trend period using the average written date of policies with premium earned
during the historical period to the average written date for policies that will be in effect during the time
the rates will be in effect. *
Trend period: 1/1/2012- 7/1/2014

based on avg. written dates.
Thus, the trend period is 2.5 years (5 half years)

Projected CY 2012 EP @ CRL = 114,208,050 ((1 - .01))5 =108,610,779
OR
Trend period 1/1/12 to 7/1/14
2.5 years.

2.5 yrs. Here an annual trend will be selected and the trend period is

Projected EP = CY 2012 Earned from @CRL * Trend 2.5
AVG WP @ CRL
Semi Ann
12 mo ending AWP at CCRL

12/31/2011

Chng

950
-0.01

6/30/2012

940.5

-2% annual trend
-0.01

12/31/2012

931

Projected 2012 EP @ CRL = 114,208,050 * (1-.02)2.5 = 108,583,017.3
b. It takes into account changes in exposure distributions, for what is expected to occur when rates are in effect.
OR
Premium trend accounts for the gradual shift in the book of business for things such as inflation or mix of
business
c. Using historical rates would cause a double-counting effect in the trend calculation
OR
Using written premium at historical rate levels to determine premium trend would include rate changes in the
selected trend number, when we don’t necessarily expect those rate changes to continue into the future.
d. This change would cause premiums to go lower because fewer losses would be paid. The true projected
premium is lower than that calculated above.
OR
The true projected earned premium will be lower because a higher deductible gives the insured a discount on
premium.

Exam 5, V1b

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CAS Exam 5 – Spring 2013 - Questions, Solutions and Commentary
Question 2 - Examiner’s Report
a. In general candidates scored well. Some of the common errors were:
• -1% trend (not annual)
• Wrong trend period
• 8.5% or 8.9% trend (using total WP or WP over EP)
• Apply trend to WP
• Calculating EP from WP instead of projecting the given EP
b. A common error was to say the premium trend is used to bring historical premium to expected future cost level
which is stating what the premium trend does but not why you’d do it. The other common mistake was to
mention rate changes as part of the premium trend.
c. Candidates often compared average premium to total premium instead of historical premium to current level
premium. The other common mistake was to compare written premium to earned premium instead of
historical premium to current level premium.
d. Candidates scored very well on this part. When candidates missed points it was due to not responding to the
actual question asked but instead describing how the issue could be addressed.

Exam 5, V1b

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CAS Exam 5 – Spring 2013 - Questions, Solutions and Commentary
3. (2.5 points) An actuary has submitted the following analysis for a rate level indication:
Accident
Year

Accident Year

Calendar/

Calendar

Reported

Reported Loss

Accident

Year Earned

Losses and

and Paid ALAE

Year
2010

Premium
$1,023,549

Paid ALAE
$703,902

Ratio
68.8%

2011

$1,086,756

$773,430

71.2%

2012

$1,222,930

$749,249

61.3%

Three Year Average Reported Loss and Paid ALAE Ratio

67.1%

Fixed Expense Provision

11.0%

Variable Expense Provision

15.0%

Underwriting Profit Provision

8.0%

Variable Permissible Loss Ratio

77.0%

Indicated Rate Change

1.4%

Recommend five improvements to the analysis and briefly explain the purpose of each.
S13 - Exam 5 - Question #3
1. Adjust the earned premium to current rate level. This will avoid an indication that ignores past rate
changes and provides a better projection of future loss ratios.
2. Determine a loss trend and apply to the Loss + ALAE. This will created a better projection of future
losses if there is an ongoing or past change in frequency or severity of losses
3. Develop losses to ultimate. The rate must account for all losses from the policies, not just the ones
that have been reported thus far. Ignoring IBNR will create an inadequate rate.
4. Include a ULAE load. The rate must provide for all costs associated with the transfer of risk so it must
include adjustment expenses that are not allocated to specific claims
5. Use a volume-weighted average of loss ratios. 2012 has significantly more premium than past years
and will be more responsive to changes in the book so it should be given more weight.

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CAS Exam 5 – Spring 2013 - Questions, Solutions and Commentary
Question 3 - Examiner’s Report
The question presented an analysis for a rate indication. The candidate was requested to provide 5
improvements for the analysis and briefly explain the purpose of each. Suggesting improvements to the
company's operation did not address the question asked and did not receive credit.
The majority of candidates recommended and received full credit for at least four enhancements to the
analysis. Many recommended and received full credit for five. Those that did not receive credit for all 5
recommendations didn't attempt an answer or suggested enhancements that did not improve the analysis.
Additionally, some candidates confused various concepts (for example, "trend losses to ultimate"), provided a
response that summarized prior enhancements, were too general in their recommended improvement, or
simply identified a shortcoming in the analysis without offering an enhancement, and did not receive credit.

Candidates generally struggled to receive credit for briefly explaining the purpose of each recommendation;
most candidates received less than full credit on four of the five explanations requested.
Most candidates did not provide an explanation or attempted to give further explanation of the enhancement
without explaining its purpose -- these did not receive credit.
Many candidates restated a version of the original recommended improvement to the analysis in their
explanation of the purpose (i.e. "Earned premium can be adjusted to the current rate level.
This makes sure that all premiums are on-level."), which did not get credit for explaining the purpose of the
bringing the premium to current rate levels.

Exam 5, V1b

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CAS Exam 5 – Spring 2013 - Questions, Solutions and Commentary
4. (3 points) Given the following information:


Annual loss trend rate = +4%.



Rate change history:
o

+3% effective April 1, 2009.

+2% effective July 1, 2010.



All policies have annual terms.



Calendar year 2012 earned premium = $50,000.



Accident year 2012 reported losses at December 31, 2012 = $4,200.
Percentage of Loss
Reported at:
12 months

10%

24 months

35%

36 months

65%

Selected Ultimate Loss Ratio
Accident Year 2009

66%

Accident Year 2010

67%

Accident Year 2011

70%

Use the reported Bornhuetter-Ferguson technique to estimate ultimate losses for accident year 2012.
S13 - Exam 5 - Question #4
Step 1:

AY 2012 BF Ultimate Losses = CY 2012 EP * AY 2012 Selected Ultimate ELR * AY
2012 % unreported at 12/31/2012 + AY reported losses at 12/31/2012

Step 2:

Compute on-level factors to use in the selection of the ELR

1.03  1.02
 1.0418
9 / 32  1.03  23 / 32  1
1.03  1.02
For 2010: On-level factor:
 1.0184
1 / 32  1  1 / 8  1.03  1.02  27 / 32  1.03
1.03  1.02
For 2011: On- level factor:
 1.00246
1/ 8  1.03  7 / 8  1.03  1.02
For 2009: On- level factor:

Step 3:

Compute the selected ELR = Avg (2009 – 2011) selected ultimate loss ratio
adjusted for loss trend and on-level premium factors

1
1.042
1.042
1.04 
 67% 
 70%
 66% 
  71.68%
3
1.0418
1.0184
1.00246 
Step 4:

AY 2012 BF Ultimate Losses = 50,000 x 71.68% x (1 – 10%) + 4,200 = 36,456

Exam 5, V1b

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CAS Exam 5 – Spring 2013 - Questions, Solutions and Commentary
OR
BF ULT. Losses = 4,200 + [% unreported @ 12/31/12 x 2012 Selected Ult. LR x CY 2012 EP]
AY 2011 ULT loss ratio =

2011 Loss + LAE
2011 EP

On level factor for 2011 EP =

1.02
 1.002
11/ 8   1.02  7 / 8 

AY 2011 Selected Ultimate LR adj for AY 2012 = .7 *

1.04
 .727
1.002

BF ULT Loss for AY 2012 = 4,200 + 0.90 * 0.727 * 50,000 = 36,915
Examiner’s Report Question 4
Many candidates did not identify the need to adjust historical loss ratios for the future 2012 level.
Some did not develop on-level-factors or apply them appropriately to the historical loss ratios, while others
did not apply loss trend to the historical loss ratios.
Some thought that the 2012 on-level earned premium was the only on-level adjustment needed, but this
number was provided and the historical loss ratios still need adjustment for future levels.
We also frequently saw misidentified loss trend periods (2 years instead of 3, 1.5 years instead of 1, etc.).

Exam 5, V1b

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CAS Exam 5 – Spring 2013 - Questions, Solutions and Commentary
5. (4 points) A company is reviewing the rate level adequacy. Given the following information for a
book of business:












All policies are annual.
Current rates have been in effect for three years.
New rates will be in effect for 18 months beginning on July 1, 2013.
Annual premium trend = -1%.
Annual loss trend = +3%.
Loss adjustment expense provision = 2.5% of loss.
Historical expense ratios:
o

Fixed = 6%.

Variable = 30%.

Underwriting profit and contingencies provision = 5%.
Ultimate losses are estimated using the reported development technique.
On January 1, 2014, the company will reduce agency commissions by 3% of premium.

Calendar Year Ending

Earned Premium ($000s)

December 31, 2011

$2,163

December 31, 2012

$2,120
Reported Losses ($000s)

Accident Year

12 months

24 months

36 months

48 months

60 months

2008

$780

$928

$1,030

$1,083

$1,094

2009

$765

$921

$1,004

$1,053

$1,012

2010

$760

$920

2011

$805

$966

2012

$890

Calculate the indicated rate change.

Exam 5, V1b

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CAS Exam 5 – Spring 2013 - Questions, Solutions and Commentary
S13 – Exam 5 - Question #5

Indicated Rate Change 

LR  F
1
1  V  QT

(F = 6%, v = 30%, Q = 5%, V will have to be adjusted due to the reduction in agency commissions.
Step 1:
a.

Compute Proj Ult Claims = Reported Losses * LDF to Ult * Loss Trend * LAE Loading
Compute volume weighted age to age factors; 60 to ultimate factor (1.0) judgmentally selected.

Selected ATAF reported Losses

12-24

24-36

36-48

48-60

60+

1.200965

1.100036

1.050147

1.010157

1.0

(1030  1004  1012) / (928  921  920)
b Compute the loss trend period for AY 2011 and AY 2012
Trend Period = Average accident date during the experience period to the average accident
date during the period the rates are in effect.
AAD during the experience period (AY) is 7/1/20XX.
The period the rates will be in effect is 7/1//2013 – 12/31/2014; the AWD during that period
is 3/1/2014 and the AAD for a policy written on 3/1/2014 is 10/1/2014.
Trend Period for AY 2011 is 7/1/2011 – 10/1/2014 = 3.25 years, and for AY 2012 is 2.25 years
Reported Losses
2011 (24 mos)
2012 (12 mos)

966,000
890,000

CDF-ULT

Loss Trend

LAE loading

Projected ult claims

1.166933

3.25

1.025

1,271,943.715

2.25

1.025

1,366,387.864

1.03

1.401446

1.03

=1.1200965 * 1.100036 *1.0500147*1.010157
Step 2:

Compute Projected Trended Premium = EP * On-level Factor * Premium Trend Factor

Avg written date of CY 20XX EP - Avg written date of  07/01/2013-12/31/2014 PY 









01 / 01 / 20 XX

04/01/2014

For CY Ending 12/31/2011, the trend period is 1/1/2011 – 4/1/2014 = 3.25 years
For CY Ending 12/31/2012, the trend period is 1/1/2012 – 4/1/2014 = 2.25 years
The given premium trend is -1%
EP
2011
2012

On-level Factor*

2163000

Premium Trend

1.00

2120000

Projected Trended Premium

3.25

2,093,490.054

2.25

2,072,597.876

0.99

1.00

0.99

*Already on-level as no rate change in past 3 years
Step 3:

Compute the projected LR
1, 271, 943.715  1, 366, 387.864

LR 

Step 4:

2, 093, 490.054  2, 072, 597.876

 .603328

Compute V during the effective period of the rates.
1/3 period

V approx in forecast period =

2/3 period

07.01.2013  31.12.2013  0.30  01.01.2014  31.12.2014  0.27
07.01.2013  31.12.2014

= 1/3(0.3)+2/3(0.27)= 0.28
Step 5:

Compute the indicated rate change

Indicated Rate Change 

Exam 5, V1b

LR  F
1  V  QT

On 1/1/2014, V = .30-.03=.27

1 

0.633288  0.06
1  0.28  0.05

 1  3.476%

Page 379

CAS Exam 5 – Spring 2013 - Questions, Solutions and Commentary
Question #5 (continued)
OR
Compute Trended and Developed Losses
12-24

24-36

36-48

Rpt

Los Dev

1.19

1.11

1.05

1.20

1.09

1.049

1.01

1.21.

1.1

1.2

Sel

1.2

1.1

1.05

1.01

1.167

1.0605

1.01

To ULT 1.400
CY
2011
2012

Loss
966,000
890,000

Prem

LDF
1.167
1.400

48-60

Trend Fact

LAE

Trended Dev Losses’

3.25

1.025

1,272,017

2.25

1.025

1,364,978

1.03

(1/1/12 – r4/1/14)

Compute Trended EP and Trended and Developed On-Level Loss Ratios (Selection = Avg)
CY
2011
2012

EP
2,163,000
2,120,000

Trend

Trended Ep

3.25

2,093,490

.6076

2.25

2,072,598

.99

.6586
Avg: .6331

Compute the Weighted Average PLR during the effective period of the rates:
PLR from 7/1/13 -1/1/13 =1.0 -.3 -.05 = .65
PLR from 1/1/14 – 12/31/14 = 1 - .27 - .05 = .68
WTD PLR = 1/3 (.65) + 2/3 (.68) = .67
Compute the Indicate Change Factor

Ind Change

.6331  .06
 1.0345
0.67

+3.45%

Examiner’s Report - Question 5
In general, this question was completed well although there were a couple common errors on this question.
1. Most candidates recognized an adjustment needed to be made for the commission change, but the
adjustment wasn’t consistently done correctly.
2. The trend period for losses and premium was often determined incorrectly. Although rates were in effect
for 18 months, candidates are expected to know how to properly determine trend periods.

Exam 5, V1b

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CAS Exam 5 – Spring 2013 - Questions, Solutions and Commentary
6. (2.5 points)
a. (0.5 point) Contrast the components of IBNR for a claims-made policy and an occurrence policy.
b. (0.5 point) Explain why a claims-made policy should cost less than an occurrence policy, provided claim
costs are increasing.
c. (0.5 point) Explain why a change in underlying trends will impact the estimated premium for an occurrence
policy more than for a claims-made policy.
d. (0.5 point) Briefly describe the provision that exists to eliminate coverage overlap if an insured switches
from an occurrence policy to a claims-made policy, and why an overlap would exist without it.
e. (0.5 point) Explain why there would be a coverage gap if an insured switches from a claims-made policy to
an occurrence policy and what an insurer can do to provide coverage.
S13 – Exam 5 - Question #6
a. Occurrence Policy has both pure IBNR + IBNER, CM policy only has IBNER
OR
CM has no pure IBNR @ report year end because all claims in the report have be reported (by def.), development
is limited to IBNER. Occurrence policies will see development due to both pure IBNR + IBNER, since polices can
be reported long after they occur.
b. A claims made policy has a much shorter period of time between the coverage trigger and the settlement
date- it is not impacted by loss cost increases as much as an occurrence policy is impacted.
OR
Occurrence policies incur liability for claims that occur now but are reported much later so inflation/loss
trend accumulates on these costs whereas CM policies incur liability for claims reported @ today’s cost
levels.
c. Under an occurrence policy, claims are covered that are reported much further out into the future.
These loss trends will therefore have a greater impact on the losses covered by an occurrence policy
- more of an impact due to inflation/loss trends
OR
An occurrence policy can have losses reported much later, trends have more leverage on future costs
than on current costs → a ∆ in trend affects occurrence more than CM.

Report Year

d. Retroactive date means that losses are only covered by a CM policy if they occur after the retro date.
Lag
0

L(10,0)

L(10,1)

L(10,2)

L(11,0)

L(11,1)

L(11,2)

L(12,0)

L(12,1)

L(12,2)

Occurrence policy in 10 would cover losses on shaded diagonal.
CM policy in 11, without a retro date would cover entire row=overlap on L(11,1)

Exam 5, V1b

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CAS Exam 5 – Spring 2013 - Questions, Solutions and Commentary
Question #6 (continued)
OR
Apply retroactive date to the new CM policy to limit coverage to losses that occur after such a date.
A=occ. Policy covg
B= CM covg w/o adj
LAG
year

13
“

A/B
↑

(Over Lap)

(previous years as well if occ coverage was provided before 2011)
e. Use Extended reported period endorsement = provides coverage for losses that occurred when CM
coverage was effective, but reported after expiration of last CM policy.
CM policy in 10 covers entire row. Occurrence policy in 11 covers diagonal = L(11,0) and L (12,1).
No coverage for L(11,1) or L(11,2) or L(12,2).
LAG
Report Year

L(10,0)

L(10,1)

L(10,2)

L(11,0).

L(11,1).

L(11,2)

L(12,0)

L(12,1)

L(12,2)

OR
Year

Covg
Gap

A
A

Purchase tail coverage to cover during gap

Exam 5, V1b

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CAS Exam 5 – Spring 2013 - Questions, Solutions and Commentary
Examiner’s Report Question 6
a. More than half of the candidates provided enough components of IBNR for both claims-made and
occurrence to get full credit. Many candidates named only the pure IBNR component but did not state that
it was the only difference between the policies. No credit was granted for candidates stating that
Occurrence has IBNR and Claims-Made does not, because Claims-Made has IBNER, a component of IBNR
Other candidates named additional components of IBNR, such as claims in transit or reopened claims. No
credit was granted or deducted for these additional components, unless they were assigned incorrectly.
In general the majority of candidates seemed to understand the question and what was being asked. The
most common mistakes were not including both Pure IBNR and IBNER in their contrast or simply stating that
Claims-made has no IBNR.
b. About half of the candidates received full credit for either some reference to Occurrence policies having
claims reported further in the future at a higher cost level, or additional pricing risk associated with having
to make a longer projection for Occurrence policies.
Several candidates received partial credit for showing a specific numeric example of lower costs, but without
a full explanation of the cause.
Some candidates received no credit for simply stating that Claims-Made lack pure IBNR, or have no claims
reported after the policy expiration, so the overall cost is less. However, these claims are balanced by claims
reported from earlier accident years, such that it is the higher future cost levels (& additional pricing risk), not
additional claims, that result in Claims-Made policies costing less than Occurrence policies.
Many candidates stated that Claims-Made policies have only one year of trend, or are fully settled &/or paid
at the end of the year, while Occurrence policies have many years of trend. These responses received no
credit, as it is the report lag that is shorter for the Claims-Made policies, not the settlement lag. Just like for
Occurrence policies, inflation will act on Claims-Made policies for as long as the settlement lag lasts, which will
likely be several years for a long-tailed line.
In general, a large number of candidates spent far too much time on this part. A simple statement with one
or two sentences would have garnered full credit, but candidates seemed to misunderstand the intent and
provided much lengthier responses – which cost them time and also increased the risk that they would
misstate something resulting in only partial credit.
c. About half of the candidates received full credit for some reference to Occurrence policies having claims
reported further in the future.
Several candidates received partial credit for showing a specific numeric example of the higher impact, but
without a full explanation of the cause.
Many candidates stated that Claims-Made policies have only one year of trend, or are fully settled &/or paid
at the end of the year, while Occurrence policies have many years of trend. These responses received no
credit, as it is the report lag that is shorter for the Claims-Made policies, not the settlement lag. Just like for
Occurrence policies, inflation will act on Claims-Made policies for as long as the settlement lag lasts, which will
likely be several years for a long-tailed line.
Similar to part B, we found that candidates provided much lengthier responses than was necessary for
full credit.

Exam 5, V1b

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CAS Exam 5 – Spring 2013 - Questions, Solutions and Commentary
Examiner’s Report Question 6
d. More than half the candidates received credit for stating any of the following for the provision:
retroactive date, first-year claims-made policy (or second-year, etc.), or for describing the provision as
a date restricting the mature claims-made policy to cover only claims occurring on or after that date.
Several candidates did not get credit for the provision because they incorrectly described it as the date
on or after which claims must be reported for the claims-made policy, which is simply the effective date
of the claims-made policy.
About half of the candidates received partial credit for the overlap description using either a written
description or a diagram showing at least one occurrence & claims-made policies, and where the
policies intersected as the overlap.
Several candidates did not get credit for the written overlap description because they did not mention
both the reporting & occurring situation for the overlap to happen, or they did not assign them
correctly.
Several candidates did not get credit for the diagram overlap description because they labeled one axis
as AY with the Occurrence policy on the diagonal, which is incorrect. Other candidates did not get credit
for the diagram because they did not identify the following: the axis labels, the occurrence and
claims-made policies & the overlap.
In both the written response and diagram, several candidates received no credit for describing the
overlap as happening when both the claims-made and occurrence policies were effective at the same
time (rather than in a subsequent year), which would cause an overlap regardless of the type of policy.
Based on the responses of the candidates, it does seem that they understood the question part and
formulated appropriate responses. Some candidates did spend more effort than necessary elaborating
on the provision and overlap rather than ‘briefly describing’ them as requested.
e. Most candidates received at least partial credit for stating either of the following for the provision: tail
policy or extended reporting endorsement. Similar responses were also accepted, as long as either
the tail or extended reporting period for the claims-made policy was included in the response.
About half of the candidates received credit for the gap description using either a written description or
a diagram showing at least one occurrence & claims-made policies, and the area between the policies
where the gap would be.
Several candidates did not get credit for the written gap description because they did not mention both
the reporting & occurring situation for the gap to happen, or they did not assign them correctly.
Several candidates did not get credit for the diagram gap description because they labeled one axis as
AY with the Occurrence policy on the diagonal, which is incorrect. Other candidates did not get credit
for the diagram because they did not identify the following: the axis labels, the occurrence and
claims-made policies & the gap (or alternatively, the area where the tail coverage would fill in).
In both the written response and diagram, several candidates received no credit for describing the gap
as happening when both the claims-made and occurrence policies were effective at the same time,
rather than in a subsequent year.
As with part D, candidates did demonstrate a strong understanding of what was being asked, but some
provided responses that were more involved than needed.

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7. (3 points) An actuary is reviewing workers compensation indemnity loss experience for a rate level
indication analysis. Given the following information:


A benefit change having an impact of +5.0% applies to all indemnity losses for accidents occurring
after July 1, 2011.



A benefit change having an impact of +2.0% applies to indemnity losses on policies written after
October 1, 2012.



No other benefit changes are expected within the next few years.



The annual impact on benefits due to wage inflation has been +2.0% and is expected to continue.



The proposed effective date for revised loss costs is July 1, 2013.



Policies are annual.



Revised loss costs would be in effect for one year.



Losses occur uniformly throughout the year.
Estimated
Ultimate Losses at
Pre-July 1, 2011
Accident

Benefit Levels

Year
2010

($000s)
$1,875

2011

$1,875

2012

$2,000

Calculate the 2010, 2011, and 2012 accident year projected ultimate losses to be used in the rate level
indication.
S13 – Exam 5 - Question #7
Since the proposed effective date is 7/1/2013, and since annual pols will be issued and in effect for 1
year, trend losses to the avg loss date of 7/1/2014.
AY

ULT Loss (000s)

Trend

Benefit Changes*

ULT Losses (000s)

2010

1,875

(1.02)4

(1.05)(1.02) = 1.071

2,173.7

1,875

(1.05)(1.02)

2,131.0

(1.05)(1.02)

2,228.5

2011
2012

2,000

(1.02)
(1.02)

*since all losses are reported at pre July 2011 benefit levels, all years need both the 2% and 5% adjustment.
Examiner’s Report
This question was a straightforward calculation. The most challenging part for candidates was the part of the
question where it stated that losses given were prior to the 7/1/11 benefit change, and that all accident years
needed to adjusted by the both benefit changes (the full amounts) for full credit.
The majority of candidates missed this subtlety and approached the question by adjusting each accident year
by a different amount. A common mistake among these candidates was to treat the 7/1/11 benefit change as
applying to policies written on or after 7/1/11 (question stated that it applied to losses on or after) and/or treat
the 10/1/12 benefit change as applying to losses on or after 10/1/12 (question stated that it was applied to
policies written on or after).
Several candidates correctly calculated the average benefit level for losses in each of the given accident years,
but then multiplied the given losses by the average benefit level (rather than using the average benefit level
to calculate a benefit level adjustment factor before applying).

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8. (3 points) Given the following information:
All policies are annual and written on January 1.
Rate change effective date is January 1, 2013.
Rate level is reviewed annually.
Underwriting guidelines were revised on January 1, 2011, substantially changing the composition
of the book of business.






Reported Loss
Accident Year

& ALAE as of
June 30, 2012

2010

$10,000,000

2011

$6,000,000

2012

$1,500,000

Selected Reported Loss & ALAE Age-to-Ultimate Factors
Month

Factor

6.50

2.00

1.55

1.20

1.12

1.08

1.05

1.02

1.01

1.00

Reported Loss & ALAE
Calendar
Year

Annual

Annual Pure

Frequency

Severity

Premium

Pure

# of

Exponential

Fit

Ending

Frequency

Severity

Premium

Points

Sep 2009

0.058

$20,355

$1,181

15.9%

-1.7%

13.9%

Dec 2009

0.059

$20,125

$1,187

16.0%

-1.7%

14.0%

Mar 2010

0.062

$20,500

$1,271

4.7%

2.9%

7.7%

Jun 2010

0.063

$21,575

$1,359

4.1%

2.5%

6.7%

Sep 2010

0.063

$21,388

$1,347

Dec 2010

0.065

$19,903

$1,294

Mar 2011

0.078

$19,567

$1,526

Jun 2011

0.078

$19,238

$1,501

Sep 2011

0.079

$19,538

$1,543

Dec 2011

0.082

$20,063

$1,645

Mar 2012

0.081

$20,050

$1,624

Jun 2012

0.082

$19,950

$1,636

Calculate the 2010 accident year trended ultimate loss & ALAE to be used in a rate change analysis.
Justify any trend selections.

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S13 – Exam 5 - Question #8
Preliminary comments
“In some circumstances, the actuary may choose to undertake a two-step trending process. This technique is
beneficial when the actuary believes that the loss trend in the historical experience period and the expected
trend for the forecast period are not identical.” This is the case in this problem since underwriting guidelines
were revised on January 1, 2011, substantially changing the composition of the book of business.
While we believe the point of this problem was for candidates to recognize that two trending periods were
necessary to use due to the u/w guidelines change on 1/1/11, we do not believe that the 4/1/2012 ending date
of the 1st step trend period is the only date that can be used. We believe examiners were testing candidates’
ability to recognize that two step trending was appropriate and then looking for judgment needed to determine
the ending date of the 1st step trend period.
First, the losses in the experience period are trended from the average accident date in the experience period
to the average accident date of the last data point in the trend data. For example, the average loss occurrence
date of Calendar-Accident Year 2010 (the “trend from” date) is assumed to be July 1, 2010.
Note the assumption being made here:
**Assume that 6-month reporting periods for trend period selection is appropriate**
Since the last data point in the loss trend data is the 6 months ending second quarter 2012, the average
accident date of that period (the “trend to” date) is March 31, 2012.
Second, these trended losses are projected from the average accident date of the latest data point (the
“project from” date of April 1, 2012) to the average loss occurrence date for the forecast period (assuming
annual policies, the “project to” date of July 1, 2013). Note that the problem states that all policies are annual
and are written on 1/1, and thus the AAD is 7/1. This differs from the AAD if the problem stated that all
policies are uniformly written over the year.
Historical vs. Projected Trending Periods
Historical Period:
Use 2-part trend since historical trend is different due to changing book of business.

Assume 6-month reporting periods for trend period selection.
Historical trend period = 7/1/2010 - 4/1/1012 = 1.75
Projected trend period = 4/1/2012 - 7/1/2013 = 1.25
Historical trend selection: freq = 16% sev = -1.7%
Use 8 point trends tor both frequency and severity; this will account for the change in the book of business.

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Projection Period:
“Underwriting guidelines were revised on January 1, 2011, substantially changing the composition of the
book of business.”
Use 4 point trends for frequency and severity since this includes the period after the mix of business changed
and should be indicative of future patterns.
Future trend selection: freq = 4.1% sev = 2.5%
2010 AY trended Ult Loss + ALAE
= AY 2010 Reported Loss & ALAE as of 6/30/2012 * 30-Ult LDF * (Hist Freq * Sev)^Hist
Trend Period * (Future Freq * Sev)^Projected Trend Period
= 10,000,000 x 1.12 x (1.16 x .983)1.75 x (1.041 x 1.025)1.25 = $15,282,922
Loss Development
Used the 30 month CDF-ULT factor of 1.12 since AY 2010 is 30 months old when evaluated at 6/30/2012.
Examiner’s Report Question 8
Only a very small number of candidates received the full credit.
One of the most popular mistakes is the incorrect trending periods. Very few candidates got it right.
A significant portion of candidates missed the assumption that "All policies are annual and written on January
1" and therefore calculated the total trending period as incorrect 3.5 years.
Another common mistake is the application of one step trending without any adjustment.
Most candidates did not use two step trending or one step trending plus onetime adjustment to account for the
underwriting guidelines change. Regarding the loss development part, most candidates got it correct.
A small percentage of candidates misread the ultimate LDFs provided in the question as age-to-age factors.
Almost all candidates understood the correct trend factor calculation (freq*sev) ^ trend period.
They also understood that projected ultimate loss is calculated by multiply the incurred loss by the loss
development factor to ultimate and trend factor.
About 10% of all candidates did not attempt the question (having a blank or almost blank answer sheet).

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9. (2 points) An actuary develops an overall indicated rate increase of 4.5% using the following assumptions:




All expenses are variable.
Total permissible loss ratio = 65%.
Profit and contingency provision = 5%.

The actuary's manager asks that the expenses be split into fixed and variable components as follows:



Fixed = 75% of total expenses.
Variable = 25% of total expenses.

a. (1.25 points) Calculate the revised overall rate indication with the new expense split suggested by the
actuary's manager.
b. (0.25 point) Briefly explain why splitting the expenses as described above results in a different indication.
c. (0.5 point) Identify two reasons an actuary may want to split expenses into fixed and variable components.
S13 – Exam 5 - Question #9
a. Since the PLR = .65 =1 – V – Q, and Q = .05, then V = .3 represents total current expenses
When splitting expenses into fixed and variable, the respective %’s are:
Fixed % = .75(.3) = .225
Variable % = .25 (.3) = .075
To compute the revised overall rate indication, one needs to determine the experience period loss
ratio in in initial rate indication:

Loss Ratio
, therefore, the experience period loss ratio =.67925
.65
.67925  .225
Revised Indication =
 1.0334 , which is a 3.34% Increase
1  .075  .05
1.045=

b. Splitting expenses into fixed + variable accounts for the fact that certain expenses are a set amount
for each risk, regardless of premium size. Depending on ratio of fixed vs. variable, indication will
differ due to fixed included on top off equation added to loss ratio.
OR
Allows fixed expenses to be added in with the loss of ratio and the revised permissible loss ratio to be
higher which lowers indication.
OR
Because fixed expenses are not changing with premium they are a set in stone percentage. That’s why
we add them to the LR rather than include it in the permissible ratio.

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c.
1. Assuming all variable expenses when some are truly fixed will over charge high premium risks and
under charge low premium risks.
2. Fixed expenses may be affected by trend, so separating allows us to apply trend factors to get more
accurate expense load.
OR
1. including fixed and variable expenses together could distort your indication
2. Including them together could cause you to undercharge small premium policies and overcharge large
premium policies.
OR
1. because some expenses do not vary with premium and in order to correctly account for it, it should be fixed.
2. Also it helps better track expenses and understand expenses
Examiner’s Report
a. Many candidates received full credit for this question. When there was an error committed,
candidates either used the permissible loss ratio as the experience loss ratio or flipped the variable
and fixed expense percentages.
b. Many candidates had trouble with this question. The answer was a verbalization of part a of this
question. Many didn’t realize this and tried to define fixed and variable expenses rather than stating
how reflecting fixed expenses impacted the indication.
c. The most common mistakes on this part was providing the similar responses twice, only defining
fixed and variable expenses.

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10. (2.25 points) Given the following information for a policy:


Annual earned premium = $1,000.



New business expected loss ratio = 60%.



Losses expected to decrease $25 per year.



New business expenses = $420.



Renewal business expenses = $350.



Probability of first renewal = 85%.



Probability of second renewal = 90%.



Probability of third renewal = 0%.



Assume an annual discount rate of 3%.

a. (1.75 points) Calculate the lifetime value of the expected total profit as a percentage of premium.
b. (0.5 point) Identify two considerations used in the analysis in part a. above that differ from
standard actuarial ratemaking techniques.
S13 – Exam 5 - Question #10
a.
Duration

(1)

(2)

(3)

(4)

(5)

(6)

7)=[ (1) - (2) -(3) ]
x (5) / (6)

Premium

Loss

Expense

Persistency

Cumulative

Discount

Persistency

Factor

PV of
Premium

PV of Profit

$1,000

$600

420

100%

1.000

-20

1,000

575

350

85%

1.030

61.89

825.24

1,000

550

350

90%

76.5%

1.0609

72.11

721.09

114

2,546.33

Profit/premium = $114/$2,546.33 = 4.477%

b.
i. Standard actuarial ratemaking techniques typically do not consider persistency, the likelihood of and
insured renewing his policy.
ii. Standard actuarial ratemaking techniques only consider premium and losses for the period in which
rates will be in effect, not over the lifetime of the insured with the insurer.
Examiner’s Report
Generally speaking, the candidate pool did very well on both parts of this question.
a. When candidates did make mistakes, the most common ones were:
1. Only calculated the lifetime value of the expected total profit but did not calculate the expected
premium (the denominator for the final ratio)
2. Didn’t apply cumulative persistency to the expected premium
3. Incorrect discounting (for example, multiplying by 0.97 in year 2 instead of dividing by 1.03)
4. Mathematical error (with credit given for the remainder of Part A in situations where the correct
answer would have been calculated without the math error)
b. Candidates scored well on this part too, with credit was typically given for the following themes:
1. The use of multiple policy years (i.e. “lifetime” of the policy)
2. The use of persistency (i.e. “retention”)
3. Reflection of discounting
4. Differences in expenses/losses for new business versus renewal business

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CAS Exam 5 – Spring 2013 - Questions, Solutions and Commentary
11. (3.5 points) An insurance company is researching three new rating variables to include in its
homeowners risk classification system. The insurer has determined the following information about
the existing book of business:
Competitor's Rating
Credit

Exposures

Pure Premium

Plan Factor

Base Class

Excellent

1,500

$116.67

0.85

Good

2,500

$128.00

Yes

Fair

1,000

$155.00

1.3

Total

5,000

$130.00

Age of

Competitor's Rating

Homeowner

Exposures

Pure Premium

Plan Factor

Base Class

Under 30 years

800

$150.00

0.7

30 to 40 years

1,200

$116.67

Yes

1.2

Competitor's Rating

Base Class

Over 40 years

3,000

$130.00

Total

5,000

$130.00

Loss Prevention

Exposures

Pure Premium

Plan Factor

Fire extinguisher

100

$100.00

0.9

Smoke detector

4,700

$128.72

Yes

None

200

$175.00

1.5

Total

5,000

$130.00



Credit is determined using the credit score for the primary homeowner.



Age of homeowner is determined using the age of the primary homeowner.



A homeowner with both a fire extinguisher and smoke detector would be classified with a
smoke detector.



Full credibility claim standard = 400.



The square root rule is used to determine partial credibility.



A competitor's rating relativities are used as the credibility complement.



Frequency for every risk classification = 10%.



Assume that the insurer can implement only one new rating variable at this time.



Assume that each variable is independent.

a. (1.5 points) For each potential rating variable, briefly describe two possible concerns of adding it
to a risk classification system.
b. (0.75 point) Without performing any calculations, recommend and justify which rating variable the
insurer should implement within a risk classification system.
c. (1.25 points) Develop the indicated credibility weighted rating factors for the variable
recommended in part b. above.

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S13 – Exam 5 - Question #11
Credit: -Lacks causality as is correlated with loss exposure; however, difficult to show causality
-Invades privacy of insureds
Age: -Lacks controllability since insured can’t control their age
-The indicated relativities from the insurer’s data differ significantly from competitor relativities.
(e.g. Ind Under 30 Rel > 1.00)
Loss Prevention:
-Difficult and expensive to verify as it is subject to manipulation from the insureds
-Non-sensical definition. Why would someone with both a fire extinguisher and a smoke detector
be rated higher than someone with just a fire extinguisher
b. I would recommend credit score as score as a variable.
-significant loss cost differentiation
-objective definition
-Easy and inexpensive to verify and administer
-Social concerns are not sufficient to prevent using this variable (assuming it is legal to do so)
c. (1.25 points) Develop the indicated credibility weighted rating factors for the variable recommended
in part b. above.
Credibility



Full credibility claim standard = 400.
The square root rule is used to determine partial credibility.
Credit

Excellent
Good-Base
Fair
Total

PP
(1)
116.67
128.00
155.00
130.00

PP Ind Rel Comp Rel /1.015
(2)=(1)/Tot(1)
(3)
0.8975
0.8374
0.9846
0.9852
1.1923
1.2808
1.0000
1.0000

Z
(4)
61.237%
79.057%
50.000%

Z-wtd Rel Z-wtd Rel/Base (=good)
(5)
(6)
0.8741949
0.8877
0.9847424
1.0000
1.2365479
1.2557

(3): 1.015 = 5075/5000 = Sum[Comp Rel * Exposures]/Sum[Exposures]
(4) = Sqrt [(Exposures * Freq)/Full Cred Standard]
(5) =

 2    4    3  1   4  
 2    4   3  1   4  

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Examiner’s Report Question 11
a. Candidates needed to provide a brief description along with the characteristic they listed.
Most candidates lost points for either no, or an insufficient, description of the characteristic listed.
For example, a common insufficient answer is that “credit is discriminatory”. Such an answer is not
quite accurate, since all classification plan factors discriminate among insureds. Thus, a
clarification of the nature of discrimination that causes concern is warranted.
Some candidates mentioned concern that the age of homeowners relativities curve does not trend
monotonically.
Candidates who received credit typically mentioned lack of credibility in the youngest age group or
the dissimilar direction compared to competitor relativities. However, the lack of monotonic
relationship in and of itself was not accepted as a valid concern.
b. Many candidates did not provide a description commensurate with the point value assigned.
In order to receive full credit, candidates needed to briefly describe at least three reasons to support
their choice.
Some candidates provided reasons for choosing a variable that contradicted the concerns listed in
Part A, which lost them points.
Often, candidates described reasons why they wouldn’t choose other variables.
Points were awarded when the reason a variable wasn’t selected for one variable was a valid reason
to select the chosen variable.
For example, if the candidate didn’t select loss prevention because it is difficult to verify and they
were choosing credit score (which is not difficult to verify), points were awarded. However, if a
candidate said they didn’t select age of homeowner because of lack of credibility and they chose loss
prevention (which has an issue with credibility), points were not awarded.
Many candidates who chose credit score lost points for saying the levels were “fully credible”, as
opposed to “good credibility” which leads to a different discussion and also lead to candidates losing
points in Part C.
c. To receive full credit, candidates needed to correctly calculate:






the full credibility standard,
the credibility using the square root rule,
the company indicated relativities,
credibility weight the company relativities with the competitor relativities, and finally
re-base the credibility weighted relativities.

The most common mistake here was claiming full credibility, not recognizing that the 400 full
credibility standard refers to claim count and not exposure.
For candidates who calculated the indicated company relativities relative to the total pure premium, a
common mistake was not calculating the revenue neutral competitor relativities as well.
Additionally, some candidates missed the instruction to use the competitor’s relativities as the
complement of credibility.

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12. (3 points) An insurer is planning to revise burglar alarm and deductible rating plan factors for its
Homeowners program. Given the following generalized linear model output:

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12 (continued)
Burglar Alarm

GLM

-2 Standard

+2 Standard

Prediction

Errors

Policies

None

1.00

Local Alarm

0.98

0.950

1.010

320,000
27,500

Central Reporting

0.86

0.730

0.990

2,500

Deductible

GLM

-2 Standard

+2 Standard

Policies

Prediction

Errors

$250

1.75

1.60

1.90

2,700

$500

1.10

1.05

1.15

87,000

$1,000

1.00

$2,500

0.95

150,000
0.90

1.00

60,000

$5,000

0.85

0.80

0.90

50,100

$7,500

1.25

0.90

1.60

150

$10,000

0.40

0.00

0.80

Propose revised burglar alarm and deductible rating plan factors. Document the relevant analysis and
rationale to support the proposal.
S13 – Exam 5 - Question #12
Burglar Alarm: Relatively low volume and wide confidence interval for both Local Alarm and Central
Reporting groups.
The Local Alarm std errors suggest it’s not significantly different than the None category (the confidence
interval encompass the relativity for none).
Central reporting has very for few exposures and large standard errors. I would recommend this variable not
be used (1.00 factor for all groups.
Deductible:
250

500

2500

5000

7500

10000

1.50

1.000

0.95

0.85

0.75

0.65

1. 250 not enough data
2. 500, 1000, 2500, and 2000: fit very well and sufficient data factor directionally also make sense.
Use indicated factors.
3. 7500: reversal should be lower than 5,000
10,000: indicated factors are too small, may be due to sparse data judgmentally select 0.65.
7500: Select the average factors of 5,000 (.85) and 10,000 (.65)
Examiner’s Report Question 12
In general, the response to this question was poor. Many candidates recognized the small data volume but
incorrectly went about combining alarm types or deductibles into one category. This was often accompanied
by a calculation of a proposed factor by weighted the GLM output. Time was unnecessarily lost by this
calculation. Another common error was candidate’s often recognized unintuitive output that seemed to be the
result of sparse data but yet still proposed to select the predicted factor.

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13. (2 points) Given the following for a large deductible commercial general liability policy:
Per occurrence deductible

$250,000

Loss elimination ratio for a $250,000 deductible

80%

ALAE/ground up loss ratio

10%

Ground up loss estimate

$2,000,000

Fixed expenses

$100,000

Variable expenses as % of premium

12%

Underwriting profit as a % of premium

Deductible processing cost as a % of losses below the deductible

Credit risk as a % of losses below the deductible

Additional risk margin as a % of excess losses



The insurer will handle all claims, including those that fall below the deductible.



The insurer will make the payments on all claims and will seek reimbursement for amounts
below the deductible from the insured.



The deductible is for loss only.



All ALAE is paid by the insurer.

Calculate the premium for the large deductible policy.
S13 – Exam 5 - Question #13
Write an equation to compute the premium for a large deductible policy. Assign symbols to the given
data, compute values for various terms used in the equation and written.
LER =. 80

ALAE = .10

ALAE$ = .1 x (2,000,000) = 200,000;

L = 2M
Loss = (1 - .8) x (2,000,000) =400,000

Fee for handling the ded: 80 x (2,000,000) x .05 = 80,000
Credit Risk = .8 x (2,000,000) x .02 = 32,000
Risk Margin = (1 - .8) (2,000,000) (.08) =32,000
L  E L  Ded Fee  Credit Risk  Risk Margin  F
1V  Q
400, 000  200, 000  80, 000  32, 000  100, 000
1  .12  .03

 992, 941.176

Examiner’s Report
Many candidates received full credit on this question. Some common mistakes that were made on this
problem:


Forgetting fixed expense is in the numerator.



Treating the loss elimination ratio as the excess loss ratio. If the candidate used the incorrect LER
“correctly” (applied the deductible processing and credit risk loads to the losses under the
deductible, the excess risk margin to the losses above the deductible, and used the losses above
the deductible in the numerator) candidates still received some partial credit.



Applying the ALAE % to excess losses.

Exam 5, V1b

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CAS Exam 5 – Spring 2013 - Questions, Solutions and Commentary
14. (1.25 points) An insurer proposes to increase rates by 6.0% where many individual policy impacts
will be above 10%. The insurer proposes a capping rule that will restrict premium changes at the
policy level to plus or minus 10.0%.
a. (0.5 point) Identify two problems that a capping rule may cause for an insurer.
b. (0.75 point) Explain why an insurer would propose a capping rule in light of the problems
identified in part a. above.
S13 – Exam 5 - Question #14
a. 1. Insurer will not be charging what they should be to keep the fundamental insurance equation in balance
and earn their target underwriting profit.
2. Systems limitations-need to program this rule into computer systems. Can get complicated as to what
gets capped and what doesn’t and how this changes the rating algorithm
OR
1. May cause need for premium transition
2. Insurer may not get all the rate needed
OR
1. Can cause rates to be inadequate
2. Can be subject to adverse selection
b. May have a concern that they will not retain policyholders if they raise rates substantially at renewal-may
cause insureds to shop- Also might be regulation reasons-restrictions on the amount of rate increase a
policyholders can see at each renewal
OR
Keep customers from getting shocked at renewal and shopping.
OR
An insurer would propose a capping rule in light of the problems in (a) to maximize the retention. An
insurer might be able to get an increase in rate in the future which will make rates adequate again. The
more profitable business they retain the more profits they will enjoy in the long run.
Examiner’s Report
a. Candidates not receiving partial credit on often restated the same item twice or two sides of the
same item. To receive full credit, 2 separate ideas were necessary.
b. On part b, very few candidates only received partial credit. Examples of full credit statements include:


“An insurer’s retention may decline if a rate cap is not adopted.”



“State laws may require a maximum rate change be followed for all policies.”

Exam 5, V1b

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CAS Exam 5 – Spring 2013 - Questions, Solutions and Commentary
15. (2.5 points) An employer negotiated a workers compensation retrospective policy with an insurer,
effective from January 1, 2011 to December 31, 2011. The first adjustment of the retrospective premium
occurs six months after the end of the policy period and annually thereafter until the tenth adjustment.
The reported losses during the policy period evaluated as of June 30, 2012 are as follows:
Claim

Reported Losses

$300,000

$200,000

$100,000

The provisions for this retrospective rating plan are as follows:
Minimum retrospective premium ratio

50%

Maximum retrospective premium ratio

150%

Loss Conversion Factor
Per Accident Loss Limitation

1.2
$150,000

Expense Allowance Excluding Tax Multiplier

25%

Expected Loss Ratio

60%

Tax Multiplier

1.05

Net Insurance Charge
Standard Premium

44.6%
$540,000

a. (2 points) Calculate the retrospective premium as of June 30, 2012.
b. (0.5 point) Discuss what could cause the retrospective premium in part a. above to change for
the insured between June 30, 2012 and the tenth adjustment.
S13 – Exam 5 - Question #15
a. Basic Premium = [e - (c – 1.0)(E)+ cl] *SP
Basic Premium = [0.25 - (1.2 – 1.0)(0.60)+ 0.446] * 540,000 = 311,040
Retro Premium = [Basic Premium + Converted Losses] x Tax Multiplier, where the retro premium is
subject to a maximum and minimum.
Limited Reported Losses = 100,000 + Min(200,000,150,000) + Min(300,000,150,000) = 400,000
Retro Premium = [311,040 + 400,000(1.2)]1.05 = 830,592 before min/max
Maximum Retro Premium= Standard Premium x Maximum Retro Premium Ratio.
Max Retro Premium is 1.5(590,000) = 810,000
So the final retrospective premium is 810,000
b. The retro premium could decrease from the max cap if reports losses develop downward or if claims
are closed with no payment.

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CAS Exam 5 – Spring 2013 - Questions, Solutions and Commentary
Examiner’s Report Question 15
This question was answered poorly with few candidates receiving full credit.
a. To get full credit, candidates would need to calculate the basic premium and retrospective premium
correctly, and calculate and apply the maximum/ minimum premium. The common errors
included:


incorrectly calculating the capped losses



when calculating the basic premium, applying factors to adjust the net insurance charge that
was provided in the question



incorrect basic premium formula



not applying the max/ min premium

b. Candidates did better on this part. The most common error was to provide reasons that the
premium could increase, as it was already at the maximum level. However, if candidates incorrectly
calculated the retrospective premium in part a, and produced a number that was in between the
min and max, we did award them full credit in part b if they stated that premium could rise or fall.

Exam 5, V1b

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CAS Exam 5 – Spring 2013 - Questions, Solutions and Commentary
16. (1.75 points) Given the following information:
Accident

Cumulative Closed Claim Counts

Half-Year

6 Months

12 Months

18 Months

24 Months

30 Months

36 Months

2010-1

4,898

7,349

7,571

7,647

2010-2

5,576

6,786

7,487

7,569

2011-1

6,580

10,215

10,618

10,724

2011-2

7,514

9,564

10,953

2012-1

8,894

13,807

2012-2

10,265

Accident

Age-to-Age factors

Half-Year

6-12

12-18

18-24

24-30

30-36

2010-1

1.500

1.030

1.010

1.000

2010-2

1.217

1.103

1.011

1.000

2011-1

1.552

1.039

1.010

2011-2

1.273

1.145

2012-1

1.552

Assume no closed claim count development after 36 months.
a. (1.25 point) Estimate the ultimate claim count for accident year 2012.
b. (0.5 point) Briefly discuss two advantages for analyzing this data using accident half-years as
opposed to full accident years.
S13 – Exam 5 - Question #16
a. There appears to be a seasonal pattern in the age-to-age factors that causes differences between
XXXX-1 and XXXX-2 half years.
I would select a separate pattern for each half year (-1 and -2) using simple all year averages.
6-12

12-18

18-24

24-30

30-36

36-ult

Sel (-1)

1.535

1.035

1.010

1.000

Sel (-2)

1.245

1.124

1.011

1.000

ULT count for AY 2012 = 13,807(1.035)(1.01) + 10,265(1.245)(1.124)(1.011)= 28,956
b. Allows for recognition of seasonal patterns in claims development
Allow for better recognition of growing portfolio as average accident date shifts.
OR
ADV 1: Since there is a pretty clear seasonality effect based on the ATA values that vary significantly by period,
using this type of analysis captures these differences to produce a more accurate development projection.
ADV 2: Using shorter time frames such as half year can help the accuracy of projection during times of greatly
increasing exposure (due to higher granularity). This could be useful here, since the claims closed down the 6
and 12 month columns are increasing noticeably, which may be due in part to an exposure increase.
OR
1. Because of the developmental seasonality it helps to pick different patterns for the different half years’
2. The counts appear to be increasing at a decent rate. When counts are increasing like this it could mean an
increase in exposures. Splitting the years into half-years better deals with the changing average date of loss
that accompanies rapidly increasing exposures.

Exam 5, V1b

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CAS Exam 5 – Spring 2013 - Questions, Solutions and Commentary
Examiner’s Report
a. Most candidates were able to properly apply development factors, while not everyone reflected the
seasonality in the data. Some of the common mistakes were as follows:


Developing the 6 month closed claims for the first half of the year instead of the 12 month
closed claims.



Failing to reflect seasonality.



Applying 1st half factors to the 2nd half closed claims and vice-versa



Only calculating the ultimate claims for one half of the year

b. Most candidates were able to recognize the seasonality. A significant number also recognized the
exposure growth and shifting of average accident date. A common mistake was to misinterpret the
question as referring to development age (6, 12, 18, etc vs 12, 24, 36, etc). This resulted in many
responses along the lines of making the LDFs less leveraged.

Exam 5, V1b

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CAS Exam 5 – Spring 2013 - Questions, Solutions and Commentary
17. (1.25 points) The following information is available for a self-insured entity:
Industry

Industry

Accident

Case

Reported CDF

Paid CDF

Year

Outstanding

to Ultimate

2010

$30

1.005

1.105

2011

$60

1.035

1.235

2012

$110

1.120

1.560

a. (0.5 point) Using a case outstanding development technique, estimate the unpaid claims for
accident year 2012 as of December 31, 2012.
b. (0.5 point) Identify two limitations to the technique used in part a. above.
c. (0.25 point) Briefly describe a situation when this technique is particularly useful.
S13 – Exam 5 - Question #17
Preliminary Information
Exhibit III

Chapter 12 - Case Outstanding Development Technique
Self-Insurer Case Outstanding Only - General Liability
Development of Unpaid Claim Ratio ($000)

Accident
Year
(1)
1998
1999
2000
2001
2002
2003
Total

Case
Outstanding
at 12/31/08
(2)
500,000
650,000
800,000
850,000
975,000
1,000,000
4,775,000

CDF to Ultimate
Reported
(3)
1.015
1.020
1.030
1.051
1.077
1.131

Paid
(4)
1.046
1.067
1.109
1.187
1.306
1.489

Case
Outstanding
(5)
1.506
1.454
1.421
1.445
1.439
1.545

Unpaid
Claim
Estimate
(6)
753,065
945,128
1,136,911
1,228,356
1,403,157
1,544,858
7,011,474

Column Notes:
(2) Based on data from Self-Insurer Case Outstanding Only.
(3) and (4) From Exhibit I, Sheet 2 in Chapter 8.
(5) = { [(3) - (1) * (4) ]/ ((4) -(3))} +1
(6) = [(2) * (5)].
The following formula is used to develop the case O/S development factor:

Exam 5, V1b

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CAS Exam 5 – Spring 2013 - Questions, Solutions and Commentary
S13 – Exam 5 - Question #17
17a. (0.5 point) Using a case outstanding development technique, estimate the unpaid claims for
accident year 2012 as of December 31, 2012.
17b. (0.5 point) Identify two limitations to the technique used in part a. above.
17c. (0.25 point) Briefly describe a situation when this technique is particularly useful.
Problem Specific Solutions

a. Case O/S development factors for AY 2012



1.12  1 1.56  1  1.425
1.56  1.12

AY 2012 unpaid claims as of 12/31/2012 = 110 x 1.425 = 156.75
b. 1. Industry benchmark CDF often prove to be inaccurate for a particular insurer
2. Analysis can be distorted by large losses in case outstanding
OR
- Industry benchmarks aren’t accurate or don’t apply to this self-insured entity
- Paid CDFs might be highly leveraged→ subject to inaccurate estimates
c. This technique is useful when no other technique is available because the only information the
self-insured has is case O/S.
Examiner’s Report
a. About ½ the candidates received full credit on this question. The most common error was providing
IBNR instead of total unpaid claims.
b. Many candidates got partial credit on this question for only listing the “industry development/mix
might not be like carrier development/mix” limitation. The other two limitations (large loss and
leveraged) were not very common. There were several common limitations that did not receive
credit, such as “this method only produces unpaid claims” or answers that made reference to the
other case outstanding method (references to claims made policies).
c. Many candidates got this question completely correct. A wide variety of answers were accepted,
but did not give credit for candidates who said that the insurer had “limited” or “thin” data. Credit
was not given for candidates that referenced the other case outstanding method (references to
claims made policies).

Exam 5, V1b

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CAS Exam 5 – Spring 2013 - Questions, Solutions and Commentary
18. (2 points)
a. (0.25 point) Briefly explain the key assumption of the Bornhuetter-Ferguson method.
b. (0.5 point) Briefly explain how the Bornhuetter-Ferguson method can be considered a
credibility-weighted method and how the credibility is calculated.
c. (0.25 point) Briefly describe one situation where the credibility-weighted assumption underlying
the Bornhuetter-Ferguson method may not apply.
d. (0.5 point) Explain whether the paid or reported Bornhuetter-Ferguson method is more
responsive in a situation where claim ratios are increasing.
e. (0.5 point) Compare and contrast the Cape Cod method and Bornhuetter-Ferguson method by
providing one similarity and one difference.
S13 – Exam 5 - Question #18
a. Key assumption: Losses reported (paid) to date do not tell you anything about the losses that are yet
to be reported (paid)
(Unpaid) Unreported losses are better estimated based on an a priori initial expected ultimate.
OR
Assumes the actuary’s a priori estimate is a better indicator of unpaid/unreported claims than
experience to date
b. The method is considered a cred weighted method of the Development Method and Initial Expected.
Z (Dev Method) + (1-Z) Initial Expected Ultimate
Z = The percent reported to date=

1
1
=
cumulative dev.factor
CDF
from development method

OR
Cred weighting of Development and Expected Claim techniques,
The weight is based on % paid (or % reptd.), i.e. B-F Ult = % paid * Dev Ult + (1 - % paid) x Exp Clm. Ult
c. On a pattern that goes above 100% reported or paid, you’ll see this on lines with salvage +
subrogation or short tailed lines with strong case reserves. The % reported amount (2) cannot go
above 1 in credibility theory. Therefore, in this situation, in theory, the method shouldn’t be used.
OR
Would not apply if % paid is greater than 100% (violates credibility definition)
d. The reported method would be more responsive because the development method is responsive to
increasing claim ratios, and the reported BF method will give more weight to the development
method early on since % Rpt is often greater than % paid.
OR
Reptd is more responsive, since % reptd is usually greater than % paid, thereby putting more
weight on the developed emerging exp. And less on the a priori estimate

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S13 – Exam 5 - Question #18
e. Similarity: CC (Cape Cod) and BF methods both assume the unreported amount should be based off
of another estimate and not developed as in the development technique. In other words, they both
assume that experience to date in an AY doesn’t tell you everything about future development.
Difference: The two methods calculated the “initial expected” ultimate differently. The BF method
relies on an a priori selected loss ratio and the CC method calculates the LR (or PP) using the losses
to date divided by the “used up” premium. Therefore the CC method is more responsive.
OR
Both methods are cred weighting of Dev &Exp Claims but B-F initial exp loss ratio is an a priori
estimate, while Cape Cod determines IELR using reported losses & used-up premium
Examiner’s Report
a. The majority of candidates received full credit. Those that didn’t receive full credit typically lost
points because they didn't differentiate between total claim versus unreported/unpaid claim.
b. The majority of candidates received full credit. Those that didn’t receive full credit were often mentioning
the credibility calculation but were not mentioning to which method this factor would apply.
Another common mistake was to weight Z with [Actual loss / reported/ paid] instead of
[Development Method Ultimate Loss/ reported / paid]
c. The majority of candidates did not receive full credit. A common mistake for candidates was that they
were mentioning situation where BF method was not appropriate instead of referring to a situation
where credibility weighting assumption itself of BF method was not appropriate.
d. The majority of candidates did not receive full credit. Most of the candidate identified the right
method, but only a few had a clear explanation on why the reported method was more appropriate.
e. Most candidates received full credit on this part.

Exam 5, V1b

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CAS Exam 5 – Spring 2013 - Questions, Solutions and Commentary
19. (3.25 points) Given the following information:
Reported as of
Accident

December 31, 2012

Payroll

Year

Claim Counts

Seventies

($000)

2010

1,549

$22,418

$63,438

2011

1,455

$18,730

$62,893

2012

1,023

$12,501

$67,005

Reporting Patterns
As of

(Reported %)

Month

Claim Count

Seventies

85.0%

43.0%

95.0%

67.0%

98.0%

83.0%



The reported claim counts for accident year 2012 are unusually low due to a temporary
slowdown of claims being opened.



Annual frequency trend = -2%.



Annual severity trend = +5%.



Annual payroll trend = +4%.

Use an appropriate frequency-severity technique to estimate the IBNR for accident year 2012 at
December 31, 2012 and justify all selections.
S13 – Exam 5 - Question #19
AY 2012 ULT Losses = Sel PP x payroll ($100)
Sel PP = Trended and Developed Claim Counts * Trended and Developed Severity/Trended Exposures
AY 2010 IBNR = AY 2012 ULT Losses – AY 2012 Reported Losses
= AY 2012 ULT Losses – AY 2012 (Reported Claim Counts * Reported Severities)
Because 2012 frequency is off, severity is probably also impacted (smaller claims open faster), so 2012
will not be used in the calculation.
As of 12/31/2012, AY
Counts

CDF

Trend

Trend + Dev counts (a)

2010

1,549

1/.98

.982

1518.02

2011

1,455

1/.95

.98

1500.95

Sev

CDF

Trend

Trend + Dev sev (b)

2010

22,418

1/.83

1.052

29778.13

2011

18,730

1/.67

1.05

29352.99

Exposure

Trend
2

Trended Exp (c)

2010

63,438

1.04

= 68,614.54

2011

62,893

1.04

= 65,408.72

Exam 5, V1b

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Trended PP 

 a  b  ;
c

2010 = 658.81;

2011 = 673.57; Sel = Avg = 666.19

AY 2012 ULT Losses = Sel PP x payroll($100) = 666.19 x 67,005=44,638,060.95
AY 2010 IBNR = AY 2012 ULT Losses – AY 2012 Reported Losses
= AY 2012 ULT Losses – AY 2012 (Reported Claim Counts * Reported Severities)
= 44,638,060.95 – (1023) x 12501 = $31,849,537.95
OR
AY 2012 ULT Losses = Ultimate Trended Frequency * Ultimate and Trended Severity * payroll ($100)
ULT and Trended Frequency = Trended Ultimate Counts/Trended Payroll
ULT Trended Severity = Reported Severity Cumulative Reported Severity % * Severity Trend
AY 2010 IBNR = AY 2012 ULT Losses – AY 2012 Reported Losses
= AY 2012 ULT Losses – AY 2012 (Reported Claim Counts * Reported Severities)
Because 2012 frequency is off, severity is probably also impacted (smaller claims open faster), so
ULT claims

Claim Trend Trended Ult Claims

Trended Payroll = Payroll * Payroll Trend

1549/0.98

1.0192

= 1,642

63,438 x 1.042 = 68,615

1455/0.95

1.01921

= 1,561

62,893 x 1.041 = 65,409

1023/0.85

= 1,204

67,005 x 1.040 = 67,005

1.0192

Freq trend = Claim Trend / Payroll Trend = 0.98 = Claim Trend/1.04; Claim Trend = 1.0192
ULT and Trended Frequency = Trended Ultimate Counts / Trended Payroll
2010 Freq = 1,642/68,615= 0.0239
2011 Freq= 1,561/65,409 = 0.0239
→ Selected frequency trend = 0.0239
ULT Trended Severity = Reported Severity / Cumulative Reported Severity % * Severity
Trend
22,418/0.83 x 1.052 = 29,778
18,730/0.67 x 1.05 = 29,353
12,501/0.43 x 1.00 = 29,072
→All Average Sel= 29,401
AY 2012 Ultimate = 0.0239 x $29,401 x $67,005 = $47,083,335
AY 2012 IBNR = $47,803,335 – 1,023 x $12,501 = $34,294,812
Selected Frequency based on 2010 + 2011 because 2012 had a slowdown in claim counts, making it
project an inaccurately low ULT claim count.
Severity is still reliable because it is an average number i.e. average is based on counts and dollars.
Used an all years average for stability.

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S13 – Exam 5 - Question #19
OR
Ultimate Claims

Trended Exposure
2

Frequency

2010

1,549 /.98 = 1580

63,438 x 1.04

2.30%

2011

1,455 /.95 = 1532

62,893 x 1.04

2.34%

Trended Frequencies
2010 .023 (.98)2 = .0221
2011 .0234(.98) = .0229
Selected Frequency = Simple Average = .0225
Ultimate Severity
2010

22, 418
.83

2011

18, 730
.47

2012

12, 501
.43

 27, 010
 27, 955
 29, 072

Trended Ultimate Severity
29,779
29,353
29,072

Selected Severity = Simple Average = 29,401
Ultimate Claims= 29,401 x .0225 x 67,005 = 44,325,315
IBNR= 44,325,315 - 1,023 x 12,501= 31, 536,792
Since AY 2012 claim counts were subject to a temporary slowdown they were removed from the
calculation of the ultimate frequency because using the current report patterns would severely
underestimate ultimate freq. for that year.
Severity was assumed to be unaffected since there was no mention of a change in claim department
methodology, just a slowdown in opening all claims.

Exam 5, V1b

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CAS Exam 5 – Spring 2013 - Questions, Solutions and Commentary
Examiner’s Report
Candidates generally performed well on the calculation portion of this question.
Some candidates did not calculate frequency (claim counts / payroll) and simply multiplied the average
of 2010 and 2011 claim counts by a severity selection to determine 2012 ultimate claims. This does not
account for the 2012 exposure levels and was not awarded full credit.
Some candidates calculated the ultimate loss indication correctly and subsequently lost points by failing
to calculate the indicated IBNR associated with the ultimate loss. A small portion of candidates
calculated the IBNR for all 3 accident years rather than just 2012.
Some candidates did not justify their selections, as specified in the question. Additionally, a portion of
candidates simply wrote out their selection in words; for example, writing "select average of 2010 and
2011" does not constitute a justification and did not receive credit.
There were some candidates that spent time converting the percentage reported factors to loss
development factors and subsequently multiplying by the claim counts and severities. The
mathematical equivalent of dividing by the percentage reported could have saved the candidates time.
A smaller portion of candidates used the percentage reported figures to create triangles of counts and
severities that were unnecessary and subsequently not used in their solution.
Common mistakes included:


Not using trend factors



Not using loss development factors



Applying loss development factors or trend factors to the incorrect year (for example, applying
the 36-month factor to 2012 rather than 2010)



Assuming that the inverse of the given percentage reported factors were age-to-age factors
rather than age-to-ultimate factors

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20. (3 points) Given the following information for a line of business:


Assume no reported claims development past 36 months.



Annual claim severity trend = +5%.



Paid claim development method ultimate loss for accident year 2012 = $10,275,000.



Reported claim development method ultimate loss for accident year 2012 = $9,650,000,
Cumulative Paid Claims ($000s)

Cumulative Closed Claim Count

Accident

Year

12 Months

24 Months 36 Months

2010

$2,100

$6,410

2011

$2,210

$7,000

2012

$2,550

Year

$8,300

Cumulative Reported Claims ($000s)

12 Months 24 Months 36 Months

2010

2011

2012

Cumulative Reported Claim Count

Accident

Year

12 Months

24 Months 36 Months

2010

$5,300

$7,810

2011

$5,500

$8,130

2012

$6,000

$8,500

Year

12 Months 24 Months 36 Months

2010

2011

2012

Outstanding Claims ($000s)

100

Outstanding Claim Count

Accident

Year

12 Months

24 Months 36 Months

2010

$3,200

$1,400

2011

$3,290

$1,130

2012

$3,450

$200

Year

12 Months 24 Months 36 Months

2010

2011

2012

Fully discuss the considerations in deciding between using the paid or the reported claim development
method to estimate ultimate claims for this line of business, and recommend an ultimate loss estimate
for accident year 2012.

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CAS Exam 5 – Spring 2013 - Questions, Solutions and Commentary
S13 – Exam 5 - Question #20
Based on the given data, one should create triangles of various claim averages (i.e. avg. paid severities,
avg. case o/s, avg. closed to reported counts, avg. reported severities) and analyze each for anomalies
or trends in the data.
Check avg. paid severities:
AY

85.47

83.84

1.05
11

1.024
63.14

87.5

1.01
12

63.75

=7,000/80

Avg paid severities appears to be trending at rate less than 5% for most recent AY.
This could indicate a change in settlement practices; Insurer could be closing more small claims.
Check Avg Case Outstanding:
AY

Avg Case out =

71.11
1.05

O/S $Claim
 O/SClaim Count 

60.87

200

1.09
74.77

66.47

1.02
12

76.19

Avg. case outstanding increased by less than 5% per year at 12 months and greater than 5% per year
at 24 months. This could indicate a change in type of claim being closed at the pd.
Look at closed to reported ratios: Closed Ct/Rep Ct
AY

.4375

.7653

.99

.4430

.8247

.4878

=99/100

Closed to report count ratio appears to be increasingly, indicating a speed up in claim settlement.
Since there is a speed up in settlement and avg. paid severity is trending at rate lower than 5%, it
appears the insurer is closing more small claims quickly.
Look at avg rep clm
AY

66.25
1.05

79.69

1.05
69.62

83.81

1.05
12

73.17

Avg. Rep. CLM increasing at steady rate of 5%.
Due to the diagnostics and explanations above, I would select the reported dev method ultimate of $9.65 mil.

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Examiner’s Report Question 20
Candidates were supposed to evaluate Average Paid (and/or Outstanding) and Average Reported
trends and compare them to the known severity of 5%.
They should have noticed the increase in paid settlement and that reported trends matched the 5%
severity.
From there they were to conclude to use the reported method and not the paid. This conclusion should
have been reached by evaluating changes (or lack of change) in both case adequacy and settlement rates.
Many candidates calculated Average Paid and Average Case severities, but did not calculate the
Average Reported severities. Most candidates did calculate trend from year to year.
Many of those lost credit by not making any statement on the stability or instability of the resulting
trends.
Also, comparisons of the observed paid severity to the outstanding severity, or the observed severities
along the diagonal rather than down the columns of the triangle did not receive full credit.
Many candidates that only looked at average paid and case and decided the change in trend of the case
outstanding disproved using the reported method.
But case alone is inconclusive in determining reported stability.
Many of those candidates did not test for settlement rate changes, likely with the thought that they had
identified the relevant piece of information to make their choice.
Some candidates further went on to test the settlement rate but did not see how an apparent case
adequacy change is influenced by a real settlement rate change.
Those that did calculate Average Reported often noticed that the year to year trend was stable and
some of those mentioned that the trend was consistent with the 5% severity.
A large number of candidates went off onto a Berquist-Sherman technique or an “adjusted” reported
methodology which was incorrect as the reported method without adjustment is the preferred method.
Full credit for the selection of the reported method was given if the correct choice was made or even if
the words “select the reported method” and no numerical choice was made.
If the candidate mistook the reported ultimate for incurred and then applied an LDF, or created their
own LDF instead of using the ultimate given, full credit was still awarded.
If they adjusted the reported triangle using a BS or other methodology and then developed to ultimate,
no credit was given for selecting the reported method.
The question asked the candidates to choose between the paid and reported methods.
Some candidates choose an average of them and got a number “Between.” Since the reported was
accurate and the paid was not candidates did not receive full credit.

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21. (2 points) Given the following information as of the December 31, 2011 actuarial valuation:
Accident

Ultimate

Reported

Paid

Year

Claims

2010

$1,200

$280

$125

2011

$1,300

$125

$75

Total

$2,500

$405

$200

Cumulative

Age in

Percent

Months

Reported

Paid

40%

12%

25%

10%

Given the following information as of December 31, 2012:
Accident

Reported

Paid

Year

Claims

2010

$470

$200

2011

$320

$175

Total

$790

$375

a. (0.5 point) Based on the 2011 actuarial valuation, calculate expected paid claims for each
accident year during calendar year 2012.
b. (0.5 point) Based on the 2011 actuarial valuation, calculate expected reported claims for each
accident year during calendar year 2012.
c. (0.5 point) Discuss a scenario that explains any differences between actual and expected paid
and reported claims as of December 31, 2012.
d. (0.5 point) Using the scenario discussed in part c. above, justify the selection of a reserving
technique for estimating ultimate claims as of December 31, 2012.
S13 – Exam 5 - Question #21
Expected paid during CY 2012 = (Ult – Paid) / % Unpaid × (% Paid at 2012 - % Paid at 2011)
a.

Ultimate-Paid

% unpaid

developed in CY 2012

2010

1075

90%

1075 / .9  * 12%  10%   23.89

2011

1225

95%

 1225 

 *  .1  .05   64.47
 .95 

OR
Expected paid during CY 2012 = Ultimate Paid * x (% Paid at 2012 - % Paid at 2011)
Yr

Ult Paid

% pd

%pd age+12

% pd in age

Exp paid in 2012

(1)

(2)

(3)

(4)=(3)-(2)

(1)*(4)

2010

1200

.10

.12

.02

2011

1300

.05

.10

.05

65
89

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S13 – Exam 5 - Question #21
OR
Exp. Emergence ($) = Paid Claims x [Age to Age factor - 1.0]
Expected paid claims in CY 2012= Paid Claims x [Age to Ult x+12/ Age to Ult x – 1.0]



1
 1


 1   25
 .10 .12 
 1 1 
AY 2011 = 75 
  1  75
 .05 .1 

AY 2010 = 125 

b.
Expected reported during CY2012= (Ult – Rptd) / % Unrptd × (% Rptd at 2012 - % Rptd at 2011)
Ultimate-Reported
2010

920

%
unreported
.75

2011

1175

 920  .4  .25  184



 .75 

 1175 

  .25  .1  195.83
 .9 

OR
Expected reported during CY2012 = Ultimate Reported * x (% Paid at 2012 - % Paid at 2011)
Yr

Ult
% rptd %rptd age+12 % rptd in age
Reported

Exp reported in
2012

(1)

(2)

(3)

(4)=(3) – (2)

(5)=(1)*(4)

2010

1200

.25

.15

180

2011

1300

.25

.15

195
375

OR
Exp. Emergence ($) = Reported Claims x [Age to Age factor - 1.0]
Expected reported claims in CY 2012= Reported Claims x [Age to Ult x+12/ Age to Ult x – 1.0]




 1 1 
  1   168
 .25 .1 
1 1

AY 2011 = 125  
 1   187.5
 .1 .25 

AY 2010 = 280 

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Question 21 - continued
c. Using reported and paid claims as of 12/31/2011 and using the expected reported and expected
paid from solution 1:
Reported at 12/31/2011 + Expected Reported

Paid at 12/31/2011 + Expected Paid

2010: 280 + 184 = 464

125 + 23.89 = 148.89

2011: 125 +195.83 = 320.83

75 + 64.47 = 139.47

Actual Reported as of 12/31/2012

Actual Paid as of 12/31/2012

2010 = $470

2010 = $200

2011 = $320

2011 = $175

Expected reported is close to actual

Expected paid is much less than actual

The higher actual paid can be a result of speed up in the claim settlement.
OR
Increase in rate of claim settlement. The reported losses tracked quite close to expected, while the paid
losses were much larger than expected.
OR
Reported claims expected are less than actual, so are paid claims. They could be understated due to
change in the mix of business towards business with worse claim experience.
d. The actuary can use the reported development technique because the projected vs. actual
development was very close, and it is not affected by the speed up in claim settlement as the paid claim
dev. method.
OR
I would use a reported dev. technique as it is not affected by decrease in settlement lag.
OR
I would suggest using the expected claims technique because you can judgmentally adjust the
expected claims ration up due to the shift.
Examiner’ Report
a. Most candidates performed well , either applying the formula from the Friedland text or another reasonable
estimation technique of expected loss emergence.
b. Most candidates performed well , either applying the formula from the Friedland text or another reasonable
estimation technique of expected loss emergence.
c. Many candidates skipped this part. Some candidates focused on explaining the relatively minor difference
in emerging reported losses while overlooking the more drastic difference in paid loss emergence. Other
candidates described a scenario that would only partially explain the results derived in part a. and part b.
Other candidates described scenarios that would result in the opposite results from those seen in part a.
and part b., reversing the actual and expected losses. These responses generally received partial credit.
d. Many candidates skipped part d. No credit was given for simply stating a reserve technique, as the question
required the candidate to justify the technique. Some responses failed to link the response back to the
scenario described in part c. as the question required.

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22. (3 points) An actuary is assisting a manufacturing company in reserving its self-insured workers
compensation program as of December 31, 2012. The program began on January 1, 1998 and has
undergone the following changes in recent years:


On January 1, 2007, the per-occurrence retention was increased from $300,000 to $750,000.



On January 1, 2010, the company automated some of its production process. As a result, the
company replaced a significant portion of its assembly-line staff with sales staff.

The actuary would like to use the following methods and data to estimate ultimate claims as of
December 31, 2012:


Development method using company-specific claim development triangles.



Expected claims method using payroll as exposure base and the average of the reported and
paid claim development projections as initial estimates of ultimate claims.



Frequency-severity method using company-specific claim count development triangles.

a. (1 point) Discuss necessary adjustments the actuary should make to the company-specific data
to use the development method.
b. (1 point) Briefly describe four adjustments the actuary should consider making to historical
claims and exposures to put them on current levels in the expected claims method.
c. (1 point) Describe two diagnostic tests the actuary should perform before using the
frequency-severity method.
S13 – Exam 5 - Question #22
a. If possible, the actuary should restate the historical triangles to a $300k retention (one triangle) and
to a $750K retention (a separate triangle) in order to remove the distortion that the change in
retention would otherwise create. The actuary should then review these triangles separately and
select LDFs to be applied to the appropriate retention by year.
OR
The actuary should adjust the claims data to be used in development method since the retention
was increased from $300,000 to $750,000. The increase in retention will increase the claims
reported and paid. Therefore, claims data before 2007 should be adjusted to current level before
applying the development method. In addition, the change from assembly-line to sales will have an
impact to the claims. Less injury will be expected when the company automated some of its
production process. Hence, claims data before 2010 should be adjusted.
b. -Adjust the losses so they are on the 750,000 retention level by using ILFS.
-Adjust losses to account for the change in workers. Sales staff will have fewer losses (injuries) than
assembly staff
-Adjust the exposures to account for inflation.
-Adjust the losses to account for benefit changes related to inflation. As the workers get raises, the
losses will increase.
OR
1. Cap the historical claims, select large loss load
2. Apply loss trend
3. Apply benefit level change adjustment
4. Apply exposure trend

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S13 – Exam 5 - Question #22
22c. (1 point) Describe two diagnostic tests the actuary should perform before using the
frequency-severity method.
c. Look at the avg severity amount → claims/closed counts. The change in per occurrence retention
could have an effect on severity.
Look at frequency triangle → claims/exposures. Change in production could have significant
increases on frequency.
OR
1. Paid to reported claim counts to determine if there were any changes in claim settlement rate.
2. Average case outstanding per open claim to see if there were any changes in case outstanding adequacy.

Examiner’s Comment
a. Many candidates did not include a detailed discussion of how the changes in retention and/ or risk
profile would affect the data.
Some candidates did not recognize that the actuary was working for a self-insured client and not an
insurance company; in these cases, some candidates said premium should be adjusted to current rate
level, but the actuary would not have premium to use as an exposure base for the self-insured layer.
b. Again, some candidates said premium should be adjusted to current rate level; however the actuary
in the question would not have access to premium information for the self-insured layer.
c. Some candidates discussed the need to review the data for changes in frequency and severity, but
failed to identify diagnostics that could be used to test for changes.

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23. (2 points) Given the following information:
Unadjusted Case Outstanding Claims ($000s) Accident
Year

12 Months

24 Months

36 Months

2010

$10,300

$21,300

$37,500

2011

$11,400

$29,400

2012

$15,600

Accident

Open Claim Counts

Year

12 Months

24 Months

36 Months

2010

1,030

1,420

1,500

2011

1,140

1,470

2012

1,200

Unadjusted Cumulative Paid Claims ($000s)
Accident
Year

12 Months

24 Months

36 Months

2010

$2,575

$15,975

$30,000

2011

$2,850

$18,200

2012

$3,900

Selected annual severity trend = +5%
a. (1.5 points) Calculate the adjusted cumulative reported claim triangle using the
Berquist-Sherman case outstanding adjustment technique.
b. (0.5 point) Discuss whether IBNR estimated using the Berquist-Sherman case outstanding
adjustment technique should be higher or lower than IBNR estimated using an unadjusted
reported claim development technique.

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S13 – Exam 5 - Question #23
($000) Adj Rept = (Adj Avg Case x open) + Paid
a. Avg case = Case/Open

Example AY 2011 at 12 mos = 13/1.05=12.38

Adj Avg Case ($000)
12

11.791

19.048

2011

12.381

2012

13(=15.6/1.2)

2010

($000) Adj Rept = (Adj Avg Case x open) + Paid
12

2010

14,720.12

43,022.62

67,500

2011

16,964.29

47,600

2012

19,500

b. Original Avg Case
12

2010

2011

2012

Adj Avg Case amounts are higher than original avg case amounts so adjusted case will result in higher
reported amounts in earlier years, and lower LDFS, thus less IBNR.
Unadjusted reported claim development technique would overstate IBNR so adjusted technique will
produce lower IBNR than the unadjusted technique.
OR
Whether the B/S case OS method produces higher or lower IBNR depends on how the trend in case
reserves relates to the selected severity trends.
If the case trend is higher, the adjusted amount will be higher in the B/S than development method.
This will lead to lower CDFs, and lower IBNR amounts.
Vice Versa if the trend in case OS is lower than the select severity trend.

Examiner’s Report
a. A majority of the candidates received full credit on this part. When there were errors, the most
common was calculation errors in the Acc Year 2010 at 24 months despite correct answers elsewhere
in the final triangle.
b. Many candidate provided answers that were factually correct but did not fully explain the issue at
hand and/or the mechanics of the adjustment.

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24. (2.5 points) Given the following information:
Paid Claims Gross of Salvage & Subrogation
Accident
Year

12 Months

24 Months

36 Months

48 Months

2009

$2,000

$2,400

$2,500

2010

$2,100

$2,300

$2,400

2011

$2,100

$2,400

2012

$2,500
Paid Salvage & Subrogation

Accident
Year

12 Months

24 Months

36 Months

48 Months

2009

$98

$166

$250

2010

$105

$163

$240

2011

$107

$170

2012

$75



Assume no development after age 48.



Ultimate claims for accident year 2012 = $2,985.

a. (0.75 point) Using a development approach, estimate the ultimate salvage and subrogation for
accident year 2012.
b. (1.5 points) Using a ratio approach, estimate the ultimate salvage and subrogation for accident
year 2012.
c. (0.25 points) Briefly discuss which approach, the development or ratio approach, to select in
recommending an ultimate salvage and subrogation estimate for accident year 2012.
S13 – Exam 5 - Question #24
a. Compute Paid S&S ATA factors
Select all year weighted avg.
12-24

24-36

36-48

48-ULT

1.6097

1.4894

1.000

e.g. 1.6097 = (166 + 163 + 170)/(98 + 105 + 107)
AY 2012 Ult S&S = AY 2012 Paid S&S * 12-Ult LDF = (75) (1.6097) (1.4894) = 179.81
b. Compute the ratio of paid S&S/ to paid claims
12

ULT ratio (using the selected all year avg ratios)

0.049

0.069

0.1

0.10 = .10 * 1.0

0.05

0.07

0.1

0.051

0.071

0.03

0.10= .10 * 1.0
0.071(1.429)(1.0) = 0.10
0.03(1.4701)(1.429)= 0.06; however selected = .10

Select all yr weighted avg of ratios:
12-24

24-36

36-48

1.407
1.429
1.0
AY 2012 S&S Ult = AY 2012 Ultimate Claims * Selected Ult Ratio of S&S/Paid Claims = (2,985)(0.1)=298.5
c. Ratio approach provides more stability, less subject to leveraging at early maturities

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S13 – Exam 5 - Question #24
Examiner’s Report
a. Most candidates received full credit. In limited cases, there were mathematical errors or no final
calculation of the ultimate paid S&S.
b. Most candidates received high partial credit. Very few candidates selected an ultimate ratio for
accident year 2012 that considered ultimate ratios from prior years.
c. Many candidates received full credit. Some of the common mistakes were not selecting a method by
saying it does not matter and therefore not having a reason, or not giving a valid reason.

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25. (2.25 points) Given the following information for a portfolio written on claims-made policy form:
Year-End

Year-End

Calendar

Paid

Outstanding

Year
2009

ULAE
$409

Claims
$3,625

Case Reserve
$7,575

IBNR
$6,250

2010

$476

$5,875

$10,450

$7,500

2011

$614

$7,950

$13,750

$8,750

2012

$761

$10,375

$16,500

$10,625

Claim amounts include ALAE.
a. (1.5 points) Calculate a ULAE provision as of December 31, 2012 using the Kittel adjustment.
b. (0.5 point) Explain the purpose of the Kittel adjustment.
c. (0.25 point) Briefly explain a shortcoming of the classical method that is not addressed by the
Kittel adjustment.
S13 – Exam 5 - Question #25
a.
Kittel ULAE Ratio = (CY paid ULAE)/ ½ ×(CY paid + CY reported)
ULAE = Ratio * [(.50) * (case o/s + IBNR)]
Note the ULAE provision is being made for a portfolio written on claims-made forms
PD ULAE

Pd claims

Reported claims

Ratio

(1)

(2)

(3)

(4)=(1)/[(2)+(3)]

409

3,625

17,450

.0388

476

5,875

23,825

.0320

614

7,950

30,450

.0320

761

10,375

37,500

.0318

2,260

27,825

109,225

.0330

Selected CY 09-12 Avg = .0330
(3) = Pd claims + case ols +IBNER
(assuming “year-end O/S IBNR” = IBNER)
(4) = Pd ULAE/Avg (Pd claims and reported claims)
Unpaid ULAE= .0330 * [50% * (16,500 +10,625)] = 447.6
Note: This provision is for a portfolio of claims made policies and thus CY 2012 Case O/S and Year End IBNR O/S
only is used.

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S13 – Exam 5 - Question #25
OR
ULAE Reserve = Paid to Paid Ratio * % at closing * (Case Reserves) + Paid to Paid Ratio*(IBNR Reserves)
Paid to Paid ratio = paid ULAE/(paid loss + % at opening (change in total reserves))
Change in reserves = (Case o/s at 2010 - Case o/s at 2009) + (IBNR at 2010 – IBNR at 2009)

Pd ULAE

Reported = Paid + ∆ case + ∆ IBNR

476

5,875

10,000 = 5,875+(10,450 – 7,575) + (7,500 –
6,250)

614

7,950

12,500

761

10,375

15,000

ULAE / Avg(paid, reported)
10

476/((5,875+10,000)/2)

=.05997

=.06000

Selected ratio (.600) is based on a straight average of the ratios above
ULAE Reserves = .50 * Paid to Paid Ratio * (Case Reserves) + Paid to Paid Ratio*(IBNR Reserves)
.06 x .5 x 16,500 + .06 x 10,625 = 1,132.5
b. It accounts for ULAE on reported but not yet paid claims. It is a adjustment to the classical technique.
It is useful for cases like this where there is growing business + it is not steady state.
c. A short coming of the classical method is the assumption that 50% of the ULAE is incurred when
claims are opened and 50% of the ULAE is closed. This is not addressed by the Kittel method.
The problem is that the 50%-50% assumption is inflexible and doesn’t distinguish between the cost
of closing a claim and maintaining a claim.
OR
When inflation affects paid ULAE and claims differently
OR
Both assume 50% of ULAE is paid on opening and 50% on closing. This assumption is not always true.

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Examiner’s Report
a. Candidates generally did not score well on this part.
Many candidates received partial credit for:


using the average of paid and incurred losses in the denominator of the ULAE ratio



selecting a ULAE ratio that was appropriate given the ratios calculated by year



calculating the ULAE provision

Most candidates failed to properly calculate incurred losses as the sum of paid losses, the change in case
reserves, and the change in IBNR.
Errors made in the incurred loss calculation included simply adding paid losses to the year-end reserve
values or not including IBNR.
Some candidates did not properly use the average of paid and incurred losses in the denominator of the ratio.
Additionally, many candidates calculated a ULAE ratio based on the sum of all years (a weighted average)
instead of calculating the ratio by year to identify potential trends.
Some candidates determined a ULAE ratio but did not calculate the ULAE provision.
Finally, of candidates that did calculate the ULAE provision, almost all candidates failed to properly calculate
the ULAE provision.
The most common errors in this final step of the calculation included applying the ratio to the sum of year-end
case reserves and IBNR for all years, or applying the ratio to 50% of case reserves and 100% of IBNR, despite
the question clearly identifying the policy as being claims-made.
b. Most candidates received either no credit or partial credit on this part. Many candidates failed to
describe the purpose of the Kittel adjustment, and simply mentioned that the adjustment used the
average of paid and reported losses in the denominator of the ratio. Candidates receiving partial credit
failed to mention that the adjustment is intended to improve upon the classical method in the case of
growing lines of business.
c. The majority of candidates who attempted this part provided an acceptable response.

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26. (2 points) An actuary is conducting a reserve review for a line of business and calculates the
following:
Claims as of

Projected Ultimate Claims

December 31, 2012 Development Method

BF Method

Frequency-Severity
Claim Count

Accident

Disposal

and Severity

Rate

Technique

$77,758 $78,022

$77,474

$77,817

2010

$104,000 $98,100 $113,782 $113,828 $113,374 $113,165

$112,669

$106,363

2011

$107,200 $55,100 $130,379

$94,770 $127,393 $102,646

$132,743

$107,447

2012

$58,100 $20,400 $120,014

$89,600 $121,397 $115,159

$123,383

$93,012

Year

Reported

Paid

2009

$76,700 $75,800

Reported

Paid

$77,501

$77,483

Reported

Paid

a. (1.5 points) Suggest a reason for the disparity between the estimates of ultimate claims for
accident year 2011 and propose diagnostic tests that would verify the assumption.
b. (0.5 point) Determine what steps the actuary should take to determine the most appropriate
methodology to project ultimate claims for accident year 2011.

S13 – Exam 5 - Question #26
a. Perhaps case outstanding adequacy was strengthened for AY 2011, with no change in payment pattern.
Thus the DFM (reported) is applying too-high DFs to reported losses and coming up with too high estimate
of ultimate. If severity in the F-S technique includes reported losses’ severity, then this will similarly
produce a high result.
To verify produce triangles of average paid and average case OS. Look for a jump between 2010 and 2011
at 24 months that is larger than the average increase in pd avg down the columns.
OR
A slowdown in the settlement pattern could have caused the differences as it would have applied the
historic CDF’s to a lower paid amount at early maturities.
-This could be tested by looking at the paid-to-reported claims ratios and the closed count-to reported
count if these ratios decrease for a given maturity for new accident years, this would support the reason.
b. Discuss these questions with claims dept manager, and examine payment patterns to make sure they are
consistent. If so, use a paid DFM or BF.
OR
The actuary should confirm there was a change to the settlement pattern and check if there were changes
to the case strength. If there were changes the data could be adjusted using the Berquist Sherman
technique the actuary should talk to the claims department to get insight into the process.

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S13 – Exam 5 - Question #26
Examiner’s Report
a. There were many potential causes to the discrepancy in the data – the most common responses
were case reserve strengthening, claim payment slowdown, and the presence of an unpaid large
loss. Credit was given to any explanation that made sense given the data.
In addition to stating a reason for the discrepancy between paid and reported methods, candidates
received credit for explaining how the ultimates for some of the methods were impacted instead of
merely stating the result of reported method is overstated or paid method is understated.
A more complete answer would be giving case reserve strengthening as a reason and explaining
how the same historical cdfs are applied to higher reported losses resulting in a possible
overestimate.
The question asked the candidate to propose “diagnostic tests” to verify the assumption. In order to
receive full credit, candidates had to provide more than one test (some candidates only provided
one test). In addition, some indication of how the diagnostic tests would be used to verify the
assumption was required for full credit. Candidates did not receive full credit for simply listing tests
without further explanation.
Other errors:


Candidates assume a speed up in claim settlement when it should be a slowdown (candidates
were able to receive points on the rest of the question with this answer).



Merely stating there was a change in claim settlement

b. Some candidates listed diagnostic tests in part b but not in part a. For these candidates, credit was
given in part a. for diagnostic tests listed in part b.
Many of the students gave only half the answer. They either explained what they would do to
confirm their reason for the discrepancy without following-up with a solution or they would only give
a solution.
Full credit was awarded if the candidate indicated how their findings or confirmation steps will lead
to a solution.

END OF EXAM

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Actuarial Notes for
Spring 2014 CAS Exam5

Syllabus Section B
Estimating Claim Liabilities

Volume 2

Table of Contents
Exam 5– Volume 2: Estimating Unpaid Claims Liabilities
Title .................................................................................. Author ..........................................................Page

Part 1 – Introduction – Estimating Unpaid Claims Using Basic Techniques
Chapter 1 – Overview ....................................................... Friedland .......................................................... 1
Chapter 2 – The Claims Process ...................................... Friedland ........................................................ 12

Part 2 – Information Gathering– Estimating Unpaid Claims Using Basic Techniques
Chapter 3 – Understanding the Types of Data Used ........ Friedland ........................................................ 19
Chapter 4 – Meeting with Management ............................ Friedland ........................................................ 38
Chapter 5 – The Development Triangle ............................ Friedland ........................................................ 42
Chapter 6 – Development Triangle as a Diagnostic Tool . Friedland ........................................................ 58

Part 3 - Basic Techniques for Estimating Unpaid Claims
Chapter 7 – Development Technique ............................... Friedland ........................................................ 75
Chapter 8 – Expected Claims Technique ......................... Friedland ...................................................... 113
Chapter 9 – Bornhuetter-Ferguson Technique ................. Friedland ...................................................... 135
Chapter 10 – Cape Cod Technique .................................. Friedland ...................................................... 174
Chapter 11 – Frequency-Severity Techniques ................. Friedland ...................................................... 198
Chapter 12 – Case Outstanding Development TechniqueFriedland ...................................................... 247
Chapter 13 – Berquist-Sherman Techniques.................... Friedland ...................................................... 275
Chapter 14 – Recoveries: Salvage & Subro and Reins .... Friedland ...................................................... 344
Chapter 15 – Evaluation of Techniques ............................ Friedland ...................................................... 360

Part 4 – Estimating Unpaid Claim Adj Expenses– Estimating Unpaid Claims Using Basic Techniques
Chapter 16 — Estimating Unpaid Claim Adj Expenses .... Friedland ...................................................... 403
Chapter 17 – Estimating Unpaid ULAE ............................ Friedland ...................................................... 427
Unpaid Claim Estimates .................................................... ASOP 43 ...................................................... 458
Statement of Principles: Loss and LAE Reserves ............ CAS .............................................................. 464
This volume (including C11 Exhibits that can be downloaded from our website) includes numerous
sample and past CAS questions and solutions associated with the following articles that are no longer
on the syllabus but were used extensively by Friedland in authoring of her paper.
Adler, M.; and Kline, C.D. Jr., "Evaluating Bodily Injury Liabilities Using a Claims Closure Model"
Berquist, J.R.; and Sherman, R.E., "Loss Reserve Adequacy Testing: A Comprehensive, Systematic Approach"
Bornhuetter, R.L.; and Ferguson, R.E., "The Actuary and IBNR"
Fisher, W.H.; and Lange, J.T., "Loss Reserve Testing: A Report Year Approach"
Fisher, W.H.; and Lester, E.P., "Loss Reserve Testing in a Changing Environment"
Wiser, R.F.; Cockley, J.E; and Gardner A., "Loss Reserving," Foundations of CAS (Fourth Edition)

Chapter 1 – Overview: ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES
FRIEDLAND

Sec
1
2
3
4
5
6

Description
Importance of Accurately Estimating Unpaid Claims
Further Requirements for Accurate Reserves
Organization of This Book
Ranges of Unpaid Claim Estimates
Background Regarding the Examples
Key Terminology

Pages
1-5
5-7
7 -10
10
10 - 12
12 - 16

Importance of Accurately Estimating Unpaid Claims

1-5

Accurately estimating unpaid claims is critical to an insurer since it must report financial results on a regular
basis. However, an insurer may not be able to quantify the exact costs of covered claims for years due to
lengthy settlement periods.
Three viewpoints of the importance of accurately estimating unpaid:
1. Internal management; 2. Investors; 3. Regulators
1. Internal Management
Accurately estimating unpaid claims is essential for pricing, underwriting, strategic, and financial decisions.
It is very important in pricing since inaccurate estimates could ruin an insurer’s financial condition.
Scenario 1: An inadequate estimate of unpaid claims could cause an insurer to reduce its rates not
realizing that the estimated unpaid claims were insufficient to cover historical claims.
a. the new lower rates would be insufficient to pay the claims arising from the new policies.
b. if the insurer gains market share as a result of the lower rates, the premiums collected
would prove to be inadequate to cover future claims, and could lead to a situation where
the future solvency of the insurer is at risk.
Scenario 2: An excessive estimate of unpaid claims could cause the insurer to increase rates
unnecessarily, resulting in a loss of market share and a loss of premium revenue to the
insurer, negatively impacting the insurer’s financial strength.
Scenario 3: An inaccurate estimate of unpaid claims could lead to poor underwriting, strategic, and
financial decisions, because financial results influence an insurer decisions (e.g. where to
increase business and whether to exit an underperforming market).
An inaccurate estimate can have a negative impact on the insurer's decisions regarding its
reinsurance needs and claims management procedures and policies.
Unpaid claims estimates impact financial decision-making such as capital management (i.e. which lines of
business get a larger proportion of allocated capital).

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2. Investors
Inaccurate reserves may lead to misstated balance sheets and income statements and misleading key
financial metrics used by investors.
Investor decisions about an insurer could be affected by an insurer with:
 insufficient reserves presenting itself in a stronger position than it truly is, and
 excessive reserves showing itself in a weaker position than its true state.
3. Regulators
Regulators rely on accurate financial statements to perform supervisory duties (e.g. assisting insurers that
mask their true financial position because of inadequate reserves to regain strength).

Further Requirements for Accurate Reserves

5-7

1. State Law
Accurate estimation of unpaid claims is required by law, and many jurisdictions tie legal requirements to do so
to the actuary (e.g. the role of the Appointed Actuary has been created by insurance legislation in countries
around the world).
2. National Association of Insurance Commissioners (NAIC)
In 1990, the NAIC required that most P&C insurers in the U.S. obtain a Statement of Actuarial Opinion signed
by a qualified actuary regarding the reasonableness of the carried statutory loss and loss adjustment expense
(LAE) reserves as shown in the statutory annual statement.
In 1993, qualified actuaries signing statements of opinion used the title of Appointed Actuary because the
NAIC required that they be appointed by the Board of Directors.
3. Other U.S.-Regulated Entities
Many state insurance departments require opinions for captive insurers, self-insurers, self-insurance pools and
some underwriting pools and associations.
4. Canada
The Insurance Companies Act requires federally regulated insurers to have an Appointed Actuary to value the
actuarial and other policy liabilities of the company at their financial year end.
The Appointed Actuary's:
 valuation must be in accordance with the rules and the standards set by the Canadian Institute of
Actuaries (CIA).
 responsibilities are set forth by the Office of the Superintendent of Financial Institutions Canada
(OSFI).
Most provinces have adopted similar legislation defining the responsibilities of the Appointed Actuary.

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5. Other Examples — Australia and Slovenia
Australia: Insurance legislation requires insurance companies to have an Appointed Actuary. The signed
actuary's report must contain a statement of the actuary's opinion about each of the following:
* The adequacy of all or part of the amount specified in the general insurer's accounts in respect of its
liabilities, and the amount that the actuary considers would be adequate in the circumstances
* The accuracy of any relevant valuations made by the actuary
* The assumptions used by the actuary in making those valuations
* The relevance, appropriateness, and accuracy of the information on which those valuations were based
Slovenia: Every company with insurance operations is obliged to appoint a certified actuary. The insurance
legislation defines the tasks of the certified actuary as follows:
A certified actuary shall be obliged to examine whether premiums are calculated and technical provisions set
aside in accordance with the regulations, and whether they are calculated or set aside so as to ensure the
long-term meeting of all the insurance under writing’s obligations arising from the insurance contracts. ...
A certified actuary shall be obliged to submit to the supervisory boards and boards of directors, together with
the opinion on the annual report, a report on the findings of the certified actuary with regard to the
supervision carried out in the preceding year pursuant to the first paragraph hereunder.
The said report must, in particular, include the reasons for issuing a favorable opinion, an opinion with a
reservation or an unfavorable opinion of a certified actuary on the annual statements.

Organization of This Book

7 -10

This text focuses on estimating unpaid claims for P&C insurers, reinsurers, and self-insured entities.
Actuaries wanting to expand their knowledge beyond the scope of this text should look to:
* Casualty Actuarial Society (CAS) seminars (e.g. the Reserve Variability Limited Attendance Seminar and
the Casualty Loss Reserve Seminar)
* CAS publications (including the Proceedings of the CAS(PCAS), Forum, Discussion Paper Program, and
Variance)
* International actuarial organizations (e.g. The Institute of Actuaries of Australia and The Institute of
Actuaries / The Faculty of Actuaries (UK))
Organization of the book:
* Part 1 — Introduction
* Part 2 — Information Gathering
* Part 3 — Basic Techniques for Estimating Unpaid Claims
* Part 4 — Estimating Unpaid Claim Adjustment Expenses
Part 1: Estimating unpaid claims from the perspective of the claims department.
A claim is traced from its first report to the insurer, to the establishment of an initial case outstanding
(case O/S), to partial payments and changes in the case O/S, to ultimate claim settlement.

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Part 2: Information gathering; Types of data to analyze; and Development triangles.
Information gathering includes:
 summarizing historical claims and exposure experience
 understanding the insurer's internal and external environment, which involves:
conducting meetings with those involved in the claims and underwriting processes and obtaining
detailed information the actuary should seek from such meetings.
Types of data actuaries use and methods for organizing the data are discussed.
Since the development triangle is used to evaluate the performance of an insurer and to determine estimates
of unpaid claims, Part 2, Chapter 5, describes how to create and use development triangles.
Part 3: A review of basic techniques for estimating unpaid claims.
 Examples of actual experience of U.S. and Canadian insurers are shown
 Similar portfolios of insurance in successive chapters allow a comparison of the results from different
techniques.
 Detailed examples of the impact of various changes (e.g. an increase in claim ratios, a shift in the
strength of case outstanding, and a change in product mix) on each method for estimating unpaid
claims is demonstrated.
 An evaluation of all the methods presented is given as well as a discussion of on-going monitoring of
unpaid claim estimates.
Part 4: Techniques to estimate unpaid claim adjustment expenses.
Claim adjustment expenses:
 are the costs of administering, determining coverage for, settling, or defending claims
 may be small (e.g. when a claim is a house fire that is settled with only a few phone calls).
 may be large (e.g. when an asbestos claim involving complex legal and medical issues, results in high
defense costs and expert fees and thus, very high expenses)
 in some cases (e.g. asbestos claims) may be significantly greater than the indemnity payment.
Claim adjustment expenses are categorized as allocated loss adjustment expenses (ALAE) and unallocated
loss adjustment expenses (ULAE).
 ALAE are costs the insurer is able to assign/allocate to a claim (e.g. legal and expert witness
expenses)
 ULAE are costs not easily allocated to a specific claim (e.g. payroll, rent, and computer expenses for
the claims department).
In Canada, actuaries still separate claim adjustment expenses into ALAE and ULAE
In 1998, the NAIC promulgated two new categories of adjustment expenses for U.S. insurers reporting on
Schedule P of the P&C Annual Statement: defense and cost containment (DCC) and adjusting and other (A&O).
 DCC expenses include defense litigation and medical cost containment expenses regardless of
whether internal or external to the insurer;
 A&O expenses include all claims adjusting expenses, whether internal or external to the insurer.

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The Actuarial Standards Board (ASB) is a U.S. actuarial organization associated with the Academy that
promulgates the standards of practice for the U.S. actuarial profession.
 CAS members are required to observe the Academy's standard if they practice in the U.S.
 CAS members who do not practice in the U.S are required to observe the standards set by other
recognized actuarial organization for the jurisdiction in which they practice (e.g. the CIA in Canada or
the Institute/Faculty of Actuaries in the United Kingdom).
 These organizations provide standards of practice, educational notes, statements of principles, and
other professional guidelines.
 Selected CAS and Academy documents related to the estimating unpaid claims are in the appendices.

Ranges of Unpaid Claim Estimates

This text focuses on obtaining point estimates for unpaid claims.
 However, several methods applied to the same line of business produce different unpaid claims
estimates.
 Since each method produces a different value of the unpaid claim estimate, we recognize that we are
dealing with the estimation of the mean of a stochastic process, since actual unpaid claims almost
always differ from the estimate.
While a range of estimates of the unpaid and a statement of confidence that the actual unpaid claims is
valuable to management, regulators, policyholders, investors, and the public, the insurer's balance sheet
requires the insurer to record a point estimate of the unpaid claims.
Actuarial Standard of Practice No. 43 (ASOP 43) defines the actuarial central estimate as an expected value
over the range of reasonably possible outcomes.
This text does not address ranges of unpaid claim estimates.

Background Regarding the Examples

10 - 12

1. Differences in Coverages and Lines of Business Around the World
Differences in the types of P&C insurance offered and in the names used for similar coverages include:
 in the U.S. and Canada, insurers use the name "automobile insurance" to refer to the P&C coverage
for automobiles and trucks;
 in the U.K., coverage is called "motor insurance";
 in India, coverage is called "car insurance";
 in the U.S., coverage protecting personal homes and possessions is "homeowners insurance"
 in Canada, "home insurance"
 in South Africa, some insurers differentiate between "household content" and "household building"
insurance.
Some major coverages for U.S. P&C insurers (e.g. workers compensation (WC) or medical malpractice (MM),
may not exist at all in other countries.
In Canada, WC insurance is not categorized as a P&C insurance coverage and is not sold by insurers (it is
provided by monopolistic provincial funds; pension and life (not P&C) actuaries typically provide actuarial
services to the provincial WC funds).

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The authors relied on claim development data from Best's Aggregates & Averages Property/Casualty United
States & Canada — 2008 Edition for many of the examples presented (as well as their actuarial colleagues at
Canadian insurers who volunteered data from their organizations).
2. Description of Coverages Referred to in This Book
To assist the reader in understanding types of coverage, a brief description of each P&C coverage is given.
o Accident benefits - Canadian no-fault automobile (Auto) coverage that provides numerous benefits
following a covered accident including: medical and rehabilitation expenses, funeral benefits, death
benefits, and loss of income benefits
Since this is a no-fault coverage, it is payable by the insured's insurer regardless of fault for the accident
o Auto property damage - A sub-coverage of auto liability insurance
Provides protection to the insured against a claim or suit for damage to the property of a third-party arising
from the operation of an auto
o Collision - A sub-coverage of auto physical damage coverage providing protection against claims
resulting from any damages to the insured's vehicle caused by collision with another vehicle or object
It’s a first-party coverage and responds to the claims of the insured when he or she is at fault.
o Commercial auto liability - A coverage that provides protection from the liability that can arise from the
business use of owned, hired, or borrowed autos or from the operation of an employee's autos on behalf of
the business
o Crime insurance - Protects individuals and organizations from loss of money, securities, or inventory
resulting from crime
Including but not limited to: employee dishonesty, embezzlement, forgery, robbery, safe burglary,
computer fraud, wire transfer fraud, and counterfeiting
o Direct compensation – A Canadian auto coverage that provides for damage to, or loss of use of, an auto
or its contents, to the extent that the driver of another vehicle was at fault for the accident
It is called direct compensation because, even though someone else caused the damage, the insured
person collects from his or her insurer instead of from the person who caused the accident
o General liability (GL) – In the U.S. and Canada covers a wide array of insurance products
The principal exposures covered by GL insurance are: premises liability, operations liability, products
liability, completed operations liability, and professional (i.e., errors and omissions) liability
o Medical malpractice – (medical professional liability insurance)
* is often separated into hospital professional and physician/surgeon professional liability insurance
* responds to the unique GL exposures present for insureds (both individuals and organizations) offering
medical care and related professional services
* an example from "Loss Reserve Adequacy Testing: A Comprehensive, Systematic Approach" by James
R. Berquist and Richard E. Sherman (PCAS, 1977) is used by the authors (see chapter 13)
While the data for the MM example is very dated, the methodology, approach, and conclusions remain
applicable today
o Personal auto insurance – (private passenger auto insurance)
Auto insurance (either personal or commercial) provide a variety of coverages, including first-party and
third-party coverages, and are dependent upon the jurisdiction in which the insurance is written
o Primary insurance - Refers to the first layer of insurance coverage
It pays compensation in the event of claims arising out of an insured event ahead (first) of any other
insurance coverage that the policyholder may have
o Private passenger auto liability – Provides third-party liability protection to the insured against a claim or
suit for bodily injury or property damage arising out of the operation of a private passenger auto
o Private passenger auto physical damage - A personal lines coverage providing protection against
damage to or theft of a covered private passenger auto

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o Property insurance - Provides protection against most risks to property, such as fire, theft, and some
weather damages
Specialized forms of property insurance include fire insurance, flood insurance, earthquake insurance,
home insurance, and boiler and machinery insurance
o Umbrella and excess insurance - Refers to liability types of coverage available to individuals and
companies protecting them against claims above and beyond the amounts covered by primary insurance
policies or in some circumstances for claims not covered by the primary policies
o U.S. workers compensation - Provides coverage for the benefits the insured (i.e. the employer)
becomes legally responsible for due to workplace injury, illness, and/or disease
The complete name for this U.S. coverage - workers compensation and employers liability insurance
U.S. WC also covers the cost to defend against, and pay, liability claims made against the employer (i.e.
the insured) on account of bodily injury to an employee.

Key Terminology

12 - 16

Definitions from the Standards of Practice and Statements of Principles are used.
Insurer: any risk bearer for P&C exposures, whether an insurance company, self-insured entity, or other.
A. Reserves
U.S. and Canada financial statements contain different types of reserves including:
1. case reserves,
2. loss reserves, bulk and IBNR reserves, case LAE reserves,
3. unearned premium reserves,
4. reserves for bad debts,
5. reserves for rate credits and retrospective adjustments,
6. general and contingency reserves, and
7. earthquake reserves.
The focus of the text is estimating unpaid claims and claim adjustment expenses.
ASOP 43 limits the term reserve to its strict definition as an amount booked in a financial statement. It defines
the term unpaid claim estimate to be the actuary's estimate for future payment resulting from claims due to past
events.
This text uses terminology consistent with ASOP 43.
Unpaid claim estimate vs. carried reserve for unpaid claims:
1. The unpaid claim estimate results from an estimation technique.
For the same line of business and the same experience period:
 different estimation techniques will often produce different unpaid claim estimates.
 unpaid claims estimate will often change from one valuation date to another (for the same portfolio).
2. The carried reserve is the amount reported in an external/internal statement of financial condition.

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Key: The unpaid claims estimate includes 5 components:
1. case outstanding on known claims,
2. provision for future development on known claims,
3. estimate for reopened claims,
4. provision for claims incurred but not reported, and
5. provision for claims in transit (i.e., claims reported but not recorded).
Terminology:
 Case O/S or unpaid case refers to estimates of unpaid claims set by the claims department, third-party
adjusters, or independent adjusters for known and reported claims only (it does not include future
development on reported claims).
 The sum of the remaining 4 components is a broad definition of incurred but not reported (IBNR).
IBNR claims are often separated into 2 components:
1. Incurred but not yet reported claims (pure IBNR or narrow definition of IBNR)
2. Incurred but not enough reported (IBNER, a.k.a. development on known claims)
An important reason for separating IBNR into its components is to test the adequacy of case O/S over time,
since it can be a useful when determining which methods are most appropriate for estimating unpaid claims.
In Chapter 3, a discussion of the different types of data provided for estimating unpaid claims is provided (e.g.
does the data include or exclude: IBNR, estimates of unpaid claim adjustment expenses, recoverables from
salvage and/or subrogation, reinsurance recoveries, and policyholder deductibles?).
B. Claims, Losses, and Claim Counts
“Claims” and “losses” are used interchangeably.
 Claims rather than losses are used more frequently in the U.S. and Canadian actuarial organizations
 Claims are more frequently used for financial reporting purposes of insurers.
 While “losses” often used to refer to ultimate losses, expected losses, loss ratios, and LAE, the authors
have chosen to select the term ”claims”.
Thus, ultimate claims, expected claims, claim ratios, and claim adjustment expenses are used.
Note: Claims (dollar values) and Claim Counts (or number of claims) are differentiated.
C. Reported Claims
“Reported claims” instead of incurred claims (or incurred losses) are used.
 Incurred losses can be misunderstood as to whether or not it includes IBNR.
 Actuaries use the labels case incurred or incurred on reported claims to specifically note that the losses
do not include IBNR.
 Reported claims refer to the sum of cumulative paid claims and case outstanding estimates at a
particular point in time.

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D. Ultimate Claims
Ultimate claims are at their total dollar value after they are settled and closed.
 For short-tail lines of insurance (e.g. property insurance and automobile physical damage), ultimate
claims are known within a short time period (e.g. 1-2 years after the end of the accident period).
 For long-tail lines of insurance (e.g. GL and WC) it may take many years before the value of ultimate
claims are known.
Projecting ultimate claims:
 allows calculating the estimate of unpaid claims for IBNR and the total unpaid claim estimate (i.e. the
sum of IBNR and case outstanding).
 is valuable for evaluating and selecting the final unpaid claim estimate and for determining the accuracy
of the prior estimate of unpaid claims.
Evaluation of numerous estimation techniques are discussed in chapter 15.
E. Claim-Related Expenses
Claim adjustment expenses and claim-related expenses refer to total claim adjustment expenses (i.e. the sum
of ALAE and ULAE, or the sum of DCC and A&O).
Because the terms ALAE and ULAE are widely used and accepted, claims include ALAE and exclude ULAE
in the examples in the text.
F. Experience Period
Refers to the years included in a specific technique for estimating unpaid claims.
F. Emergence
 Refers to the reporting or development of claims and claim counts over time.
 in Canada, it refers to the rate of payment of ultimate claims, particularly when calculating estimates of
discounted claim liabilities.

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Sample Questions:
1. Friedland differentiates between “Carried Reserves” and “Unpaid Claim Estimates.” Define each.
2. Friedland discusses terminology surrounding “Reserves” and prefers the term “Unpaid Claim
Estimate.” One component of the Unpaid Claims Estimate is the “Case Outstanding” which is made
up of the estimated future dollar amounts that the claims/adjusting departments predict will be
required to settle/close existing claims (that are known and reported).
There are four other components that are often grouped together under the broad definition of IBNR.
a. Identify the four components that are included under the broad definition of IBNR.
b. Friedland also notes two subdivisions of this broad definition: Pure IBNR and IBNER.
Describe each, and note what Friedland generally uses in the text as IBNR.
1994 Exam Questions (modified):
3. True/False: Accident year approaches to reserve estimation produce reserve indications consistent
with the broad definition of IBNR
1995 Exam Questions (modified):
37. (1 point) The CAS Statement of Principles on Loss Reserving lists five elements that comprise the total
loss reserve. Which of these may be alternatively classified as either reported reserves or IBNR?

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Solutions to Sample Questions:
1. Friedland differentiates between “Carried Reserves” and “Unpaid Claim Estimates.” Define each.
An actuary may come up with several “Unpaid Claim Estimates” using different methods (as
discussed in Friedland Chapter 7 – 13 for example). The “Carried Reserve” is the amount the
company actually selects and reports in its published financial statements. Both reflect amounts to
represent the case outstanding and IBNR, broadly defined.
2. Friedland discusses terminology surrounding “Reserves” and prefers the term “Unpaid Claim
Estimate.” One component of the Unpaid Claims Estimate is the “Case Outstanding” which is made
up of the estimated future dollar amounts that the claims/adjusting departments predict will be
required to settle/close existing claims (that are known and reported). There are four other
components that are often grouped together under the broad definition of IBNR.
a. Identify the four components that are included under the broad definition of IBNR.
1) Provision for claims incurred but not reported ("pure" IBNR)
2) Provision for future development on known claims
3) Reopened claims reserve
4) Provision for claims in transit (incurred and reported, but not recorded).
b. Friedland also notes two subdivisions of this broad definition: Pure IBNR and IBNER.
Describe each, and note what Friedland generally uses in the text as IBNR.
Pure IBNR is for claims which are exactly that: “incurred, but not reported” (1 above)
IBNER is “incurred, but NOT ENOUGH reported” (2,3 and 4 above)
Friedland uses the broad definition of IBNR (including pure and IBNER).
Note: See Conger for more discussion of IBNR.
Solutions to 1994 Exam Questions (modified):
3. True/False: Accident year approaches to reserve estimation produce reserve indications consistent with the
broad definition of IBNR.
True.
Solutions to 1995 Exam Questions (modified):
37. The CAS Statement of Principles on Loss Reserving lists five elements that comprise the total loss reserve.
Which of these may be alternatively classified as either reported reserves or IBNR?
The five elements:
1) Case reserves (for known/reported claims)
2) Provision for future development on known claims
3) Reopened claims reserve
4) Provision for claims incurred but not reported ("pure" IBNR)
5) Provision for claims in transit (incurred and reported, but not recorded).
The following elements can be categorized as either reported or IBNR losses:
2) Provision for future development on known claims
3) Reopened claims reserve
5) Provision for claims in transit (incurred and reported, but not recorded).

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Chapter 2 – The Claims Process
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND.
Sec
1
2
3
4
5
1

Description
Overview
Claims Professionals
A Claim is Reported
The Life of a Claim
Further Claim Examples

Pages
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17 - 18
18 – 19
19 - 24
24 - 25

Overview

5 elements comprise the total unpaid claim estimate:
1. Case O/S
2. Provision for future development on known claims
3. Provision for reopened claims
4. Provision for claims incurred but not reported (IBNR)
5. Provision for claims in transit (incurred and reported but not recorded)
Claims professionals estimate case O/S on claims (a.k.a. "unpaid case" or "case estimates”)
 According U.S. insurance industry data, unpaid case (net for reinsurance) represents less than 50%
of total unpaid claims and claim expenses.
 Unpaid case to total unpaid claims ratio varies greatly by type of business and insurer
Actuaries estimate the other four components of total unpaid claims.
Chapter 2 focuses on how claims professionals estimate the unpaid claim.
It is important for the actuary to understand why the estimated value of a reported claim varies over time and
how changes in case O/S are processed by insurers.

Claims Professionals

17 - 18

The claims professional (a.k.a. claims examiner or claims adjuster) can be an employee of the insurer or an
employee of an outside organization.
 Large commercial insurers have internal claim adjusters
 Small to mid-sized commercial insurers hire claim administrators (TPAs) outside the company
Outside claim administrators (TPAs)
 handle a specific book of claims.
 handle claims from the initial report to the final payment.
 report details of the claims to insurers on a predetermined basis (e.g. monthly or quarterly).
 manage all the claims of an insurer, largely in an unsupervised manner.
 compensation is based on work done for the entire book as a whole (not on a claim by claim basis).

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Chapter 2 – The Claims Process
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND.
An insurer may hire an independent adjuster (IA) paid on a fee per claim basis to handle:
 an individual claim or a group of claims.
 a specific type of claim or a claim in a region of unknown expertise
 large volume of claims after a natural disaster such as a tornado or hurricane.

A Claim is Reported

18 – 19

The estimation process for unpaid claims begins when an insured first reports a claim to the insurer and a
claims professional then reviews the report.
A claims adjuster must decide whether or not the reported claim is covered under the terms of a policy. Claims
professionals review the following to determine if the incident represents a covered claim and to establish a
case O/S estimate:
*
*
*
*
*
*
*
*
*
*
*
*
*
*

Effective dates of the policy
Date of occurrence
Terms and conditions of the policy
Policy exclusions
Policy endorsements
Policy limits
Deductibles
Reinsurance or excess coverage
Reporting requirements
Mitigation of loss requirements
Extent of injury and damages
Extent of fault
Potential other parties at fault
Potential other sources of recovery

If a liability exists for a covered incident, the claims professional establishes an initial case O/S.
 Insurers use a formula or tabular value as the basis of the initial case O/S (e.g. an insurer may initially
set all automobile physical damage glass claims at $500)
Note: Tabular estimates are set based on predetermined formula, which takes into account characteristics
of the injured party and the insurance benefits.



For WC claims, an insurer may use a tabular system where injury type dictates the initial case O/S value.

Case O/S is estimated based on the information known at that time, and the value of a claim changes as more
information is uncovered.

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Chapter 2 – The Claims Process
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND.
Approaches used by insurers to set case O/S:
Example: A claim is reported under a medical malpractice policy with a policy limit of $1 million.
Approach 1: Establish case O/S based on the best estimate of the ultimate settlement value of such a claim
including inflation.
Approach 2: Set case O/S equal to the maximum value (i.e. the $1 million policy limit)
Approach 3: Seek the advice of legal counsel.
Assume that legal counsel estimates that there is an 80% chance that the claim will settle without payment
and a 20% chance of a full policy limit claim.
1. Set the case O/S based on the mode ($0 in this case).
2. Set the case O/S based on the expected value calculation or $200,000 = [(80% x $0) + (20% x $1 million)].
Approach 4: Establish case O/S for the estimated claim amount only.
Approach 5: Establish case O/S for the estimated claim amount and all claim-related expenses.
Approach 6: Establish case O/S for ALAE (or DCC) only, Establish case O/S for ULAE (or A&O) only.
Practices for the establishing case O/S for salvage and subrogation recoveries include:
•

setting case O/S based on an estimate of the salvage or subrogation recovery that the insurer expects
to receive (i.e. case O/S is net of expected salvage and subrogation recoveries).

• not setting case O/S but tracking actual salvage and subrogation recoveries as they arise.
Case O/S for reinsurance recoveries is easily determined:
•

for proportional (i.e. quota share) reinsurance, ceded case O/S is based on the reinsurers share of the
total case O/S.

•

for excess of loss reinsurance, ceded case O/S for a claim that exceeds the insurer's retention is the
total case O/S estimate less the insurer's retention.

The Life of a Claim

19 - 24

A single insurance claim may extend over a number of years.
Example: An automobile insurer issues a 1 year policy effective 12/1/2007 – 11/30/2008.
 An accident occurred on 11/15/2008, but the insurer does not receive notice of the claim until
2/20/2009.
 On 2/20/2009 (the report date of the claim), a claims professional records a number of transactions
related to this claim which could include:
* Establishment of the initial case O/S estimate
* Notification to the reinsurer if the claim is expected to exceed the insurer's retention
* A partial claim payment to injured party
* Expense payment for independent adjuster
* Change in case O/S estimate
* Claim payment (assumed to be final payment)
* Takedown of case O/S and closure of claim
* Re-opening of the claim and establishment of a new case O/S estimate
* Partial payment for defense litigation
* Final claim payment
* Final payment for defense litigation
* Closure of claim

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Chapter 2 – The Claims Process
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND.
The transactions details for a sample claim are shown in the following table.
Date
February 20, 2009
April 1, 2009
May 1, 2009
September 1, 2009
March 1, 2010

January 25, 2011
April 15, 2011

September 1, 2011

March 1, 2012

Transaction
Case 0/S of $15,000 established for claim only
Claim payment of $1,500 - case 0/S reduced to
$13,500 (case 0/S change of -$1,500)
Expense payment to IA of $500; no change in case O/S
Case 0/S for claim increased to $30,000
(case 0/S change of +$16,500)
Claim thought to be settled with additional
payment of $24,000 – case 0/S reduced to $0
and claim closed (case 0/S change of -$30,000)
Claim reopened with case 0/S of $10,000 for
claim and $10,000 for defense costs
Partial payment of $5,000 for defense litigation
and case 0/S for defense costs reduced to
$5,000 – no change in case 0/S for claim
Final claim payment for an additional $12,000
case 0/S for claim reduced to $0 (case 0/S
change of -$10,000)
Final defense cost payment for an additional
$6,000 – case 0/S for defense costs reduced to
$0 and claim closed (case 0/S change of -$5,000)

Reported Value
of Claim to Date
$15,000
$15,000

Cumulative
Paid to Date
$0
$1,500

$15,500

$2,000

$13,500

$32,000

$2,000

$30,000

$26,000

$46,000

$26,000

$20,000

$46,000

$31,000

$15,000

$48,000

$43,000

$5,000

$49,000

Case O/S
$15,000
$13,500

Key characteristics of insured claims found in the above example:
 Claim activity extends over time (i.e. 3 years for this claim)
 Its estimated value is not ultimately established until the claim finally closes; changes over time (e.g.
the claim is closed on 3/1/2010, but then reopens on 1/25/2011, with an increase to the case O/S)
 The estimated case O/S value is reasonable at the time of the claim professionals estimate but can
later turn out to be too high or low
 An insured claim can have many different types of payments associated with it
Example - the insurer makes an initial claim payment to the injured party on 4/1/2009.
This claim payment provides for out-of-pocket medical expenses reported by the claimant.
o Since the insurer questioned the validity of the claim, they hired an IA; as a result, there was a
payment of $500 for the IA's services on 5/1/2009 (in the U.S. it would be classified as A&O; in
Canada would be categorized as ALAE.)
o On 3/1/2010, the insurer makes another payment of $24,000 to the claimant for lost wages and
additional medical expenses.
o Roughly one year later, a claims professional reopens the claim.
o Over the course of the following year, the insurer makes further payments for defense litigation,
additional lost wages, and medical expenses.
 There are many dates associated with each claim:
* Policy effective date is the date the insurer issues the policy (12/1/2007)
* Accident date, or date of loss, is the date the covered injury occurs (11/15/2008)
* Report date is the date the insurer receives notice of the claim (2/20/2009)
* Transaction date is the date on which either a case O/S transaction takes place or a payment is
made (see all the dates in the preceding table)
* Closing dates are the dates the claim is initially closed (3/1/2010) and finally closed (3/1/2012)
* Reopening date is the date the insurer reopens the claim (1/25/2011)

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Chapter 2 – The Claims Process
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND.
This example does not cover every combination of transactions possible. Some claims open and close on the
same day with a single payment (one transaction and no case O/S value).
As an insurer makes a specific payment, it may:
 reduce the case O/S more than the payment
 reduce the case O/S less than the payment
 not reduce the case O/S at all
 increase the case O/S
When referring to paid claims, it is important to know whether the claims are cumulative or incremental
 Cumulative paid claims are sum of all claim payments through the valuation date.
 Incremental paid claims are the sum of all claim payments made during a specific period of time
In the above example, the cumulative paid claims including claim-related expenses are:
* $1,500 at April 1, 2009
* $2,000 at May 1, 2009
* $26,000 at March 1, 2010
* $31,000 at April 15, 2011
* $43,000 at September 1, 2011
* $49,000 at March 1, 2012
The incremental paid claims from
 1/1/2009 to 12/31/2009 are $2,000
 2010, 2011, and 2012 are $24,000, $17,000, and $6,000, respectively
The case O/S is the estimated amount of future payments on a specific claim at any given point in time.
Example: The initial case O/S on the report date of the claim is $15,000.
 just before the claim initially closes in March 2010, the case O/S is $30,000.
 when the claim is reopened on 1/25/2011 a new case O/S is established for both claim amount and
defense costs.
 it settles for a greater amount than the case O/S for both claim amount and defense costs.
“Reported claims" (or case incurred) are the sum of cumulative claim payments and the case O/S at the same
point in time. Using the example above, the reported claims are:
* $15,000 at the time of first report (i.e. 2/20/2009)
* $15,500 at 5/1/2009 after a payment of $500 to an IA
* $32,000 at 9/1/2009, when the insurer increases the case O/S to $30,000 ($2,000 cumulative paid
claims + $30,000 case O/S)
* $26,000 upon initial closing on 3/1/2010 ($26,000 cumulative paid claims + $0 case O/S)
* $46,000 upon reopening on 1/25/2011 ($26,000 cumulative paid claims + $10,000 claims and
$10,000 defense costs case O/S)
* $48,000 at 9/1/2011 after final claim payment ($43,000 cumulative paid claims and LAE +
$5,000 case O/S for defense costs)
* $49,000 at 3/1/2012 after final defense costs payment ($49,000 cumulative paid claims and LAE
+ $0 case O/S)

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Chapter 2 – The Claims Process
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND.
Calculate reported claims over a given period of time as:
 reported claims at the end of the period minus the reported claims at the beginning of the period, or
 incremental paid claims + ending case O/S minus beginning case O/S.
Example: Reported claims for the period 1/1/2009 - 12/31/2009 are $32,000.
 As of 1/1/2009, the claim was not yet reported and thus there are $0 reported claims for the claim.
 Incremental claim payments during 2009 are $2,000 and the change in case O/S is $30,000 ($30,000
ending case O/S minus $0 beginning case O/S).
 Reported claims over the period 1/1/2010 to 12/31/2010 are -$6,000.
 Incremental claim payments in 2010 are $24,000 and the change in case O/S is -$30,000 (ending
case O/S of $0 minus beginning case O/S of $30,000).
The term "reported claims" are used under two contexts: incremental and cumulative and time periods involved
to differentiate them between these two contexts.
 Reported claims equal the sum of cumulative paid claims through a specific date and case O/S as of
that same date (for a given claim or an aggregate of a group of claims).
 Reported claims can refer to claim activity over an interval of time (e.g. the insurer's income
statement).
Thus, the formulae for reported claims over a given period of time are as follows:
 Reported claims = (reported claims at end of period) – (reported claims at beginning of period)
 Reported claims = paid claims during period + case O/S at end of period - case O/S at beginning of period

Exam 5, V2

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Chapter 2 – The Claims Process
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND.

Further Claim Examples

24 - 25

Table 2 shows additional examples of how claim transactions can affect reported claims.
Table 2 - Examples of Changes in Reported Values
At December 31, 2007
Transactions During 2008
At December 31, 2008
Cumulative
Change
Cumulative
Example
Paid
Case
Reported
Paid
in
Reported
Paid
Case
Reported
Number
Claims
O/S
Claims
Claims Case O/S Claims
Claims
O/S
Claims
‘(1)
‘(2)
‘(3)
‘(4)
‘(5)
‘(6)
‘(7)
‘(8)
‘(9)
‘(10)
1
100
100
100
100
2
200
200
50
50
250
250
(Making payments where there had been no previous case outstanding increases reported claim.)
3
1000
1000
1000
1000
(Establishing a case outstanding increases reported claim by the amount of the case outstanding.)
4
1000
1000
100
(100)
100
900
1000
(Payment with offsetting case outstanding reduction has no effect on reported claim.)
5
500
5,000
5,500
200
(1,000)
(800)
700
4,000
4,700
(If case O/S is reduced by a larger amount than the claim payment, the impact is a reduction to reported claim.)
6
5,000
10,000
15,000
12,000
(10,000)
2,000
17,000
17,000
(If payment on closing exceeds case outstanding, reported claim transaction is positive.)
7
5,000
10,000
15,000
6000,
(10,000)
(4,000)
11,000
11,000
(If payment on closing is less than case outstanding estimate, reported claim transaction is negative)
8
5,000
15,000
20,000
4,500
4,500
9,500
15,000
24,500
(Claim payment with no change in case outstanding increases the reported claim.)
9
3,000
10,000
13,000
(4,000)
(4,000)
3,000
6,000
9,000
(No payment and decrease in case outstanding decreases the reported claim.)
10
2,000
10,000
12,000
1,000
5,000
6,000
3,000
15,000
18,000
(Payment and increase in case outstanding result in increase in reported claim.)






Columns (4) and (10) show reported claims as of year-end 2007 and 2008, respectively.
Reported claims at a point in time (i.e. year-end 2007 and 2008) equal to the cumulative claim
payments plus the case outstanding at that point in time.
Reported claims shown in Column (7) represent the incremental reported value during the period of
time running from 1/1/2008 to 12/31/2008.
Reported claims over the year are equal to sum of the payments during the year (Column (5)) and the
changes in case outstanding (Column (6)).

The transactions shown in Table 2 vary with respect to their impact on total reported claims.
 There are payments in the first two examples made in 2008 on claims where there was no prior existing
case O/S at 12/31/2007; thus total reported claims for both of these claims increase.
These payments could occur when the insurer reopens a claim.
 Ex. 4 – There is no change to reported claims if the payment made equals the reduction in case O/S.
 Ex. 8 – Reported claims increase when the payment made is larger than the reduction in case O/S.
 Ex. 5 and 7 – Reported claims decrease when the payment is smaller than the reduction in case O/S.
 Ex. 3 and 9 - A change in case O/S without any associated payment impacts reported claims.

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Chapter 3 – Understanding the Types of Data Used
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Sec
1
2
3
4
1

Description
Sources of Data
Homogeneity and Credibility of Data
Types of Data Used by Actuaries
Organizing Data

Pages
28 - 29
29 - 31
31 - 38
38 - 43

Sources of Data

28 - 29

Actuaries rely on data from an insurer’s management information systems to generate claims and exposure
data for the unpaid claims’ estimation.
The Need for External Data:
 Smaller insurers may have less internal data because of a limited volume of business written or
because the organizations’ system does not provide such data. Thus, actuaries must turn to external
sources of data.
 Large insurers who have entered a new line of insurance or have focused on a new geographical
region may also need external sources of information when developing estimates of unpaid claims.
Available external data varies (by jurisdiction and by product) in the:
United States
* Insurance Services Office, Inc. (ISO)
* National Council on Compensation Insurance (NCCI)
* Reinsurance Association of America (RAA)
* The Surety & Fidelity Association of America (SFAA)
* A.M. Best Company (Best)
* NAIC Annual Statement data
Canada
* Best
* General Insurance Statistical Agency (GISA)
* Insurance Bureau of Canada (IBC)
* Reinsurance Research Council (RRC)
* Market-Security Analysis & Research Inc. (MSA)
Insurers use internally generated data and external industry benchmarks.
External information is needed for selecting:
 tail development factors
 trend rates
 expected claim ratios (i.e., expected loss ratios).
External information is beneficial when an actuary evaluates or resolves the results of different estimations and
makes final selections of claims and unpaid claim estimates.

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Chapter 3 – Understanding the Types of Data Used
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Shortcomings of External Data:
The International Actuarial Association (IAA) feels that entity-specific data is far better than external data.
External data may be misleading/irrelevant due to differences relating to:

insurance products,

case outstanding and settlement practices,

insurers' operations,

coding,

geographic areas, and

mix of business and product types

Homogeneity and Credibility of Data

29 - 31

Different lines of insurance have different claim behaviors.
 Even though the insurance coverages may be identical, claims from personal insurance policies
differ from those generated from business insurance.
 Claims for umbrella and excess insurance differ from claims for primary insurance. Subcoverages under a single line of insurance differ greatly.
•

Property damage claims for automobile liability policies are reported and paid quickly and have a
low severity (i.e. settlement value).

•

Claims from auto accidents involving catastrophic spinal injuries can take years to settle and cost
millions of dollars.

Estimating unpaid claims can be made more accurate by subdividing experience into groups exhibiting
similar characteristics, such as
 comparable claim experience patterns,
 settlement patterns,
 size of claim distributions.
When separating data into groups for analyzing unpaid claims, actuaries focus on key characteristics:
* Consistency of coverage triggered by the claims in the group (i.e. group claims subject to the same or
similar laws, policy terms, claims handling, etc.)
* Volume of claim counts
* Length of time to report the claim once an insured event has occurred (i.e. reporting patterns)
* Ability to develop a case outstanding estimate from earliest report through the life of the claim
* Length of time to settle the claim (i.e. settlement, or payment, patterns)
* Likelihood of claim to reopen once it is settled
* Average settlement value (i.e. severity)
Claims are grouped by lines and sublines of business with similar traits based on the characteristics listed
above or by policy limits to achieve similar claims attributes within a block of business.
The goal: Divide data into homogeneous groupings without dividing the data into small groups which do not
provide enough information to the actuaries.

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Chapter 3 – Understanding the Types of Data Used
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Credibility is:
 the predictive value given to a group of data
 increased by increasing the homogeneity of the data or by increasing the amount of data in the group.
Changes in the portfolio also need to be considered when grouping data. Despite different claims
development, it may be appropriate to combine personal automobile and commercial automobile data.
•

Groupings do not work as well if the volume of business is changing between these two lines of
insurance.

•

As described in Part 3, in a portfolio where the volume of personal automobile is increasing at 5% per
year while the commercial automobile volume is increasing at 30%, the changing proportion on the
different estimation techniques can be significant.

Types of Data Used by Actuaries

31 - 38

1. Claims and Claim Count Data
Common types of data used by actuaries to establish and test unpaid claim estimates include:
* Incremental paid claims
* Cumulative paid claims
* Paid claims on closed claims
* Paid claims on open claims
* Case outstanding
* Reported claims (i.e., sum of cumulative paid claims plus case outstanding)
* Incremental reported claims
* Reported claim counts
* Claim counts on closed with payment
* Claim counts on closed with no payment
* Open claim counts
* Reopened claim counts
The data types can be used for claims only (i.e. losses only), claim-related expenses, or claims and claimrelated expenses combined.
2. Claim-Related Expenses
The actuary uses claim data based on how the insurer handles expenses.
 If the claim data and policy limits include claim adjustment expenses, combine ALAE experience
and historical claims when determining unpaid claims (here, claims refers to both claims and ALAE
combined).
 If the claim analysis includes only ALAE and not ULAE, a separate analysis is used to evaluate the
unpaid ULAE estimate.
Claim-related expenses can be classified many ways.
Insurers categorize LAE by the function of the expenses as either defense and cost containment (DCC) or
as adjusting and other (A&O).
A&O includes all claim adjuster costs regardless of whether or not they are attributable to:
i. internal adjusters (viewed as overhead and difficult to attribute to an individual claim) or
ii. external independent adjusters (which are easily attributable to an individual claim).

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Chapter 3 – Understanding the Types of Data Used
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Reporting requirements and insurer claim management processes determine how insurers categorize claims
expenses.
An actuary must determine which claim expenses are included in the data and how expenses are defined.
Example: Different people working for the same insurer may define the term ALAE based on:
o financial reporting systems (to meet external reporting requirements) or
o meeting internal claim management needs.
3. Multiple Currencies
If claims data exist in information systems in different currencies, an adjustment needs to be made to the data
prior to analysis. Separate the data, translate the currencies using exchange rates to single currency, and then
combine the resulting amount.
Example: If the claims data are in Euros, pounds sterling, and U.S. dollars, and a final unpaid claim estimate
is needed in Euros, convert all amounts to Euros using the current exchange rates.
4. Large Claims
Large claims in the data distorts the results from traditional methods used for estimating unpaid claims. To
circumvent the problem,
 exclude large claims from the initial projection, and then
 add a case specific provision for the reported portion of large claims, and then
 add a smoothed provision for the IBNR portion of large claims.
The size criterion of a large claim varies by:
 line of business
 geographic region
It may even vary between analyses of unpaid claims.
Actuaries consider the following when establishing a large claim threshold:
* Size of claim relative to policy limits
* Size of claim relative to reinsurance limits
* Number of claims over the threshold each year
* Credibility of internal data regarding large claims
* Availability of relevant external data
Actuaries look at large claims reports from an insurer's claims departments that track individual experience of
claims exceeding a certain threshold.
5. Recoveries
Numerous types of recoveries affect an insurer's net claims experience.
Deductibles are common.
 For auto physical damage, deductibles reduce claim payments to policyholders, and the insurer applies
the deductible before issuing payment to the insured.
 For general liability, the injured party is not the insured party and the insurer usually makes claim
payments to the injured party first, and then seeks a recovery of the deductible from the insured.

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Chapter 3 – Understanding the Types of Data Used
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Insurers differ on how they set case outstanding given that a deductible exists:
 Some set case outstanding net of the deductible.
 Others do not consider the deductible when setting case outstanding.
Salvage and Subrogation:
When an insurer pays an insured for a claim considered to be a total loss, the insurer acquires the rights to
the damaged property.
 Salvage is what the insurer collects from the sale of such damaged property.
 Subrogation is the insurer's right to recover the amount of claim payment to a covered insured from a
third-party responsible for the injury or damage.
An actuary must know whether or not the insurer records paid claims as net or gross of these recoveries.
Questions to ask include:
* are salvage and subrogation recoveries tracked separately from claim payments?
* are claim payments only recorded net of salvage or subrogation recoveries?
* is data for salvage and subrogation recoveries available to the actuary?
6. Reinsurance
When conducting an analysis of ceded or net unpaid claims, it is important to understand the reinsurance
program of the insurer and the affect of reinsurance on claims.
Because current and previous reinsurance plans and retentions affect an insurer's estimates of unpaid claims,
actuaries analyze claims both gross and net of reinsurance recoveries.
Some actuaries:
 separately analyze gross claims and ceded claims (claims ceded to reinsurers), and then
 determine the estimate of net (estimated gross unpaid claims minus estimated ceded unpaid claims)
Other actuaries:
 separately analyze gross claims and net claims (gross claims minus ceded claims), and then
 determine the estimate of ceded unpaid claims (estimated gross unpaid claims minus estimated net
unpaid claims)
The implied net or ceded unpaid claim estimate is reviewed for reasonableness.
3 possible treatments of ALAE in excess of loss reinsurance which the actuaries must focus on are:
1. Included with the claim amount in determining excess of loss coverage (most common)
2. Not included in the coverage
3. Included on a pro rata basis; the ratio of the excess portion of the claim to the total claim amount times
the ALAE amount determines coverage for ALAE
How ALAE is treated will have an effect on data requirements and possibly the method selected for estimating
unpaid claims.

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Chapter 3 – Understanding the Types of Data Used
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
7. Exposure Data
Some methods for estimating unpaid claims require a measure of the insurer's exposure to claims.
 Earned premium is the most common type of exposure and is used for estimation by insurers and
reinsurers.
 Other types of exposures used by insurers include:
o written premium
o policies in force
o policy limits by region (the early estimation of unpaid claims related to a catastrophe)
o the number of vehicles insured (personal automobile insurance)
o payroll (workers compensation).
Actuaries often adjust historical premiums to current rate levels (on-level premiums).
2 ways in which this is done include:
1. A re-rating of historical exposures at current rates (very computer-intensive and does not work in all
situations)
2. Computing rate level changes over the experience period and adjusting the premiums in the aggregate
for historical rate changes.
Note: The actuary might not always be able to collect accurate rate changes data (therefore use premium
data from insurer on unadjusted basis)
Self-insurers and insurers collect premiums in different ways.
Actuaries working with self-insurers use other observable/available exposure bases that are more closely
related to the risk and therefore claims potential.
The following table summarizes, by line of business, types of exposures used for analyzing self-insurers' unpaid
claims.

Table 1 - Examples of Exposures for Self-Insurers
Line of Insurance
Exposure
U.S. workers compensation
Payroll
Automobile liability
Number of vehicles or miles driven
General liability for public entities
Population or operating expenditures
General liability for corporations
Sales or square footage
Hospital professional liability
Average occupied beds and outpatient visits
Property
Property values
Crime
Number of employees
Exposures are important:
 as input to certain techniques for estimating unpaid claims.
 for evaluating and reconciling the results of the various techniques.
8. Insurer Reporting and Understanding the Data
It is important to know what types of claims data are contained in the insurer's claims reports and information
systems, since different insurers, TPAs, IAs, and different departments in an organization may have different
definitions for the same terms.

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"Incurred loss" is a term with a universal meaning but actually means different things to different people. For:
 the finance department, it means transactional losses incurred during a defined period (e.g. a calendar
(or fiscal) quarter or year.)
Incurred loss = sum of payments made + the change in total unpaid claims + IBNR.
 the actuary working on an incurred claim development triangle, incurred losses = cumulative claim
payments through a valuation date + case outstanding at the same valuation date (a.k.a case incurred
or incurred on reported claims).
 TPA loss reporting, incurred losses refer to case outstanding only.
The authors use the term “reported claims” to refer to case incurred losses.
“Unpaid claims" and "reserves" are terms that also have many different meanings.
In a report from:
 the finance department, unpaid claims (or reserves) means the estimate of total unpaid claims including
both case outstanding and IBNR.
 the claims department, unpaid claims (or reserves) refers to case outstanding only.
 a TPA, unpaid claims (or reserves) represent the total reported value of the claims (cumulative
payments + current case outstanding estimates).
The actuary subtracts cumulative paid claims from the reserves to determine unpaid claims.
The actuary must know whether unpaid claims is net or gross of deductibles or other types of recoveries,
including:
 salvage,
 subrogation,
 and reinsurance recoveries (also where in the claims process those recoveries are included).
The actuary needs to know whether or not case outstanding include claim-related expenses.
 Some insurers record case outstanding and payments for claim-related expenses separately from claim
only case outstanding and payments.
 Other insurers record expense payments separately (from claim payments) but do not carry case
outstanding for expense.
"Reserves" can be used differently in the actuarial and accounting professions in South Africa and the U.K.
 South African and British accountants distinguish between provisions (unpaid claim estimates) and
reserves.
 Actuaries use "reserves" to refer to unpaid claim estimates and do not distinguish between different
types of reserves.
Paid claims can be:
 cumulative or incremental, including or excluding claim-related expenses (and based on what kind of
claims expenses)
 net or gross of recoveries.
Actuaries need to know how the insurer's system tracks claim counts, which are critical to diagnostic analyses
(after analyzing unpaid claims) as well as being an important data piece for several estimation techniques for
unpaid claims.
Actuaries use claim counts to evaluate and select a final value for the unpaid claim estimate.

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Actuaries should be able to answer questions such as:
 Does the insurer counts an automobile accident with payments for multiple coverages (bodily injury
liability and physical damage) or to multiple parties (claimants) as one claim or multiple claims?
 How are reopened claims (especially in U.S. WC and accident benefits coverages) treated and are they
considered a new claim?
For a proper estimate of unpaid claims, actuaries must:
 identify the specific data that exists and
 identify the data they are requesting from the insurer
 understand the data that they receive
9. Verification of the Data
Actuaries must have ways to review data other than relying on a formal audit of the data.
The data review may include the following components:
* Consistency with financial statement data – Can the actuary reconcile the data with financial statement data
(that may be subject to some form of external audit)?
* Consistency with prior data – Is the current data consistent with data used in the prior analysis? If not, why?
* Data reasonableness – Are there certain values that appear questionable (e.g. large negative paid claims
or apparent inconsistencies between data elements? Questionable values are not always incorrect values,
but the actuary should investigate them anyway).
* Data definitions – Does the actuary know how each of the data items is defined? An actuary should
determine the proper definition of a data piece instead of just assuming the definition (similar labels do not
always imply similar definitions).
Proper documentation of the verification process, findings, and data verification are essential to any actuarial
analysis which include:
 discussions with external auditors, and
 reliance on their work regarding data verification.

Organizing Data

38 - 43

1. Key Dates
Several key dates for organizing claim data include:
* Policy effective dates
* Accident date
* Report date
* Accounting date
* Valuation date
1. Policy effective dates are the beginning and ending dates of the policy term (i.e. the period for which
the policy triggered by the claim was effective).
Some systems only capture the policy year (the year that the policy became effective).
Reinsurers refer to it as the underwriting date (or year).

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2. The accident date is
 the date that the accident or event occurred that triggered the potential policy coverage.
Some systems only capture the accident year (the year that the triggering event occurred).
For claims-made policies, the accident date:
 is the date the claim was reported (date of the event that triggered coverage)
 may be defined as the date that an injury occurred with the injury not covered by the policy unless
the resulting claim was reported during the policy period.
3. The report date
 is the date when the claim was reported to the insurer and recorded in its claims system.
Some databases split the report date into:
o report date, and
o record date, and
o (possibly) a notification date (the date the insurer is put on notice that an event occurred that may
result in a claim).
Example: An insured motorist notifies their insurer that they got in an accident (not filing a claim);
this is the notification date.
 A week later, the insurer receives claim from the other party in the accident - this is
the report date (the date on which the claim was reported).
 The following day, the claims department records the claim into their system - this is
the record date.
 Notification dates are not commonly used in many actuarial analyses.
4. The accounting date
 is the date that defines the group of claims for which liability may exist (i.e. all insured claims
incurred on or before the accounting date).
 may be any date selected for a statistical or financial reporting purpose.
 must follow a date for which the history is frozen in time (e.g. month, quarter, or year-end with
quarter, and year-end dates as the most common).
Claims Activities and Accounting Dates Example:
 Given an accounting date for an occurrence-based policy of 12/31/2008, the total unpaid claim
estimate as of this accounting date must provide for all incurred claims, whether reported or not,
as of 12/31/2008.
 An insured loss that occurred on 12/31/2008, for a policy written on 12/15/2008, would be
included in the estimate of unpaid claims for the accounting date 12/31/2008, regardless of when
the claim is reported to the insurer.
 An insured loss that occurred on 1/5/2009, for the same policy that was written on 12/31/2008,
would not be included in the unpaid claim estimate for the accounting date 12/31/2008, because
this accident occurred after the accounting date.
5. The valuation date
 is the date through which transactions are included in the database used in the evaluation of the liability.
 does not depend on when the actuary does his/her analysis.
 may happen before, coincident with, or after the accounting date.
 may be at month-end, quarter-end, half-year-end, or year-end.

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Example: To determine total unpaid claims at 12/31/2008, actuaries use data valued as of 12/31/2008.
 Here, the valuation date and the accounting date are the same.
 In some situations, the actuary does not have time to wait for the 12/31/2008 data to be available
because of internal financial reporting requirements at year-end for some insurers.
 Actuaries often use data at an earlier valuation date to estimate the requirement for unpaid claims at
the accounting date of 12/31/2008 will be.
 Some insurers use data as of 9/30/2008 to estimate unpaid claims as of 12/31/2008; in this situation
the valuation date is 9/30/2008 and the accounting date is 12/31/2008.
The valuation date can be later than the accounting date.
Example – If the actuary wants to re-estimate what claim liabilities were at 12/31/2006, he takes into
account the actual experience of 2007 and 2008.
 The actuary can use a 12/31/2008 valuation date and thus include actual paid and reported claims
experience through 2007 and 2008.
 The estimation of unpaid claims at 12/31/2008 (the accounting date) is the valuation date (the
projected ultimate claims that he or she derives using data through 12/31/2008) minus the actual
payments at 12/31/2006.
Aggregation by Calendar Year (CY)
Calendar year data is transactional data.
Examples:
 CY 2008 paid claims are claim payments made by the insurer between 1/1/2008 and 12/31/2008
 CY 2008 reported claims = CY 2008 paid claims + case O/S at 12/31/2008 – case O/S at 1/1/2008
 CY 2008 reported claim counts claim counts reported from 1/1/2008 to 12/31/2008
 Closed claim counts are the number of claims closed during the year.
CY data is used for:
 aggregation of exposures and
 diagnostic testing when analyzing AY claims data.
CY 2008 written premium (WP) is the sum of all written premium reported/recorded in the accounting systems
during 2008.
CY earned premium (EP) is:
WP + Beginning Unearned Premium Reserve (UEPR) - Ending UEPR
Advantages of using CY data:
 no future development as the value remains fixed as time goes unlike claims and exposures
aggregated based on accident year, policy year, and even report year bases.
 readily available because most insurers conduct financial reporting on a CY basis.
Disadvantage to using CY data:
 it cannot be used for loss development purposes.
 very few techniques for estimating unpaid claims are based on CY claims.
Note: CY exposures and AY claims are frequently used in estimation techniques.

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Aggregation by Accident Year (AY)
Aggregation by AY:
 is the most common grouping of claims data for an actuarial analysis of unpaid claims.
 groups claims according to the date of occurrence (the accident date or coverage triggering event).
Example: AY 2008 consists of all claims with an occurrence date in 2008.
Self-insurers' AY data may have their fiscal year ends that do not coincide with calendar year-end.
Example - AY 2008 may coincide with a self-insurer's 8/1/2007 to 7/31/2008 fiscal year or include claims
occurring during the 1/1/2008 to 12/31/2008 CY period.
Insurers compile claims data according to a variety of accident periods including accident month, accident
quarter, accident half-year and accident year.
Financial reporting schedules and statistical organizations for insurers in the U.S. and Canada require claim
information by AY. In some areas (e.g. Lloyds of London), financial reporting by underwriting year is more
common than AY.
Actuaries use CY exposures with accident year claims.
 CY EP match the claims that occur during the year with the insurance premiums earned by an insurer
during the year.
 Claims and exposures aggregated by policy year (PY) provide an exact match.
 For self insurers, CY exposures represent an exact match with AY claims.
Advantages to using AY Aggregation
 AY aggregation is the norm for P&C insurers in the U.S. and Canada.
 AY grouping is easy to achieve and easy to understand
 Since AY includes claims occurring over a shorter time frame than for PY or underwriting (U/W) year
aggregation, ultimate AY claims should be reliably estimable sooner than those for PY or U/W year.
 Many industry benchmarks are based on AY experience.
 Tracking claims by AY is valuable when there is change due to economic or regulatory forces (e.g.
inflation or law amendments) or major claim events (e.g. atypical weather or a major catastrophe)
which can influence claims experience.

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Aggregation by Policy Year (PY) or Underwriting Year (U/W Y)
Aggregation by PY:
 Claims are grouped according to the year in which the policy was written.
 Matches the premiums and claims arising from a group of policies.
U/W Y data is often used by reinsurers and refers to claims grouped by the year in which the
reinsurance policy became effective.
Claims arising from a PY or U/W Y can extend over 24 calendar months if the policy term is 12 months.
Example: PY 2010 refers to policies with effective dates between 1/1/2010 – 12/31/2010.
Claims for annual policies effective 12/31/2010 will have occurrence dates between
12/31/2010 – 12/31/2011
Advantages of PY Aggregation:
The key advantage is the true matching of claims and premiums.
 PY experience is very important when underwriting or pricing changes occur (e.g. a shift from full
coverage to large deductible policies, a change in emphasis on certain classes of business, or an
increase/decrease in the price charged leading to a change in expected claim ratios and possibly a
change in the type of insured).
 PY aggregation is useful for self-insureds, who often issue a single policy.
Disadvantages of PY Aggregation
 The primary disadvantage is the extended time to gather complete data (i.e. it can take up to 24
months to gather all reported claims) and to reliably estimate ultimate claims.
 PY data can make it difficult to understand and isolate the affect of a single large event (e.g. a
major catastrophe or court ruling), which changes how insurance contracts are interpreted.
Aggregation by Report Year (RY)
RY data is used for lines of insurance in which coverage depends on the date the claims is reported (i.e.
claims made (CM) coverage).
CM coverage is often used for medical malpractice, products liability, errors and omission, and
directors' and officers' liability.
For these lines of business, RY data is used for developing estimates of unpaid claims.
RY aggregation groups claims by the date they are reported to the insurer, regardless of the claim’s accident
date. Aggregating claims by RY can be used to test the adequacy of case O/S on known claims over time.
Note: If CM policies have extended reporting endorsements that are not coded as a new policy,
development beyond 12 months may be possible even for annual policies.
Advantages of Report Year Aggregation
The number of claims is fixed at the close of the year (other than for claims reported but not recorded).
The RY approach substitutes a known quantity (i.e. the number of reported claim counts) for an estimate.
Thus, a RY approach will often result in more stable data and more readily determinable development patterns
than an AY approach, since the number of AY claims is subject to change at each successive valuation.
Disadvantage of Report Year Aggregation
RY estimation techniques only measure development on known claims (and not pure IBNR)

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Sample Questions:
1.
List and describe 5 key dates regarding data organization for loss reserving, per Friedland Ch 3.
2.

Which of the following statements is/are true?
 The valuation date can be on, after, or before the accounting date.
 Case Outstanding represents estimated settlement values assigned to specific known claims.
 A liability provision for reopened claims is generally needed only for the medical malpractice
line of business.
1. 
2.  3. , 
4. , , 
5. Neither 1,2,3 or 4

3. Friedland discusses how techniques to estimate unpaid losses may be used with data arranged into different
time intervals. List the 6 groupings discussed (See Chapters 5 and 9).
For the next three questions, assume data is organized and analyzed by Accident Year:
4. Assume you have applied one of the techniques, on paid claims, described in Friedland to estimate unpaid
claims of $X. Let paid claims at the evaluation date = P and case outstanding estimates = C.
a. Define the Ultimate Claims in terms of X, P, and C.
b. Define the IBNR (broad definition) in terms of X, P, and C.
c. Define the Total Unpaid Claims (Total Reserves) in terms of X, P, and C.
d. Define Reported Claims in terms of X, P and C.
5. Assume you have applied one of the techniques, on reported claims, described in Friedland to estimate
IBNR of $Y. Let paid claims at the evaluation date = P and case outstanding estimates = C.
a. Define the Ultimate Claims in terms of Y, P, and C.
b. Define the IBNR (broad definition) in terms of Y, P, and C.
c. Define the Total Unpaid Claims (Total Reserves) in terms of Y, P, and C.
6. Throughout the Study Manual we see that most often past questions have provided reported claims
data, and asked us to solve for IBNR. However, some questions will ask for total unpaid claims or
ultimate claims.
a. What do we need to add to “IBNR” (broadly defined) to get total unpaid claims?
b. What do we need to add to the “paid” claims to get ultimate claims?
c. What do we need to add to the “reported” claims to get ultimate claims?
7. The previous three questions apply to Accident Year data (by far, the most common).
a. What is the key fact about development, when working with Calendar Year data?
b. What is the key fact about development, when working with Report Year data?
1995 Exam Questions (modified):
1. True/False. According to the CAS Statement of Principles on Reserves, policy effective date is one
of the key dates in the organization of a reserving database.

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1997 Exam Questions (modified):
48. You are given the following information:
Cumulative Report Year Claims ($000’s) at:
Report Year Ending
12/31/93
12/31/94
12/31/95
12/31/96

12 mos.
60
50
70
80

24 mos.
66
55
77

36 mos.
69
58

48 mos.
69

Selected Age-to-Age:
Cumulative CDF:

1.10
1.16

1.05
1.05

1.00
1.00

Estimated
Ultimate
($000’s)

Required
Reserve
($000’s)

69
58
81
92
Total:

0
3
11
0
14

“Required reserve” = Estimated ultimate claims - Reported Claims = $14,000 for all insured
claims incurred on or before 12/31/95 (for financial reporting purposes).
a. (0.25 point) What is the accounting date of the evaluation of the required reserve (unpaid
claims estimate)?
b. (0.25 point) What is the valuation date of the evaluation of the required reserve (unpaid
claims estimate)?
c. (0.5 point) Which of the following components of a total loss reserve (unpaid claims estimate)
are considered in the “required reserve” for this report year analysis? Explain your answer.
Case outstanding estimates at 12/31/06
Future development on known claims
Incurred but not reported claims
53. You have recently been hired as the chief actuary of a small multi-line property and casualty
company. The company has the following written premium by line during the past three years:

Line of Business
Commercial Auto Bodily Injury Liability
Personal Auto Bodily Injury Liability
Commercial Auto Property Damage Liability
Personal Auto Property Damage Liability
Commercial Multi-Peril
Workers Compensation

1994
$100,000
200,000
20,000
40,000
250,000
5,000

1995
$150,000
220,000
30,000
44,000
275,000
5,000

1996
$225,000
242,000
45,000
48,400
300,000
5,000

a. (1 point) Your first task as chief actuary is to determine how the data will be segregated by line for
reserving purposes. Based on the discussion in Friedland, list three considerations you would
use in making this decision.
b. (2 points)
Based on your answers from (a), list the different lines of business that you would
combine for reserving purposes, explaining your rationale for each grouping.
2011 Exam Questions:
21. (2 points) A large insurer is considering combining data from a long-tailed, low-frequency line of
business with data from a short-tailed, high-frequency line of business to estimate unpaid claims.
Identify and briefly discuss four characteristics of the data that should be considered before
combining these lines of business.

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2012 Exam Questions:
16. (2 points) Below are four independent scenarios for estimating ultimate losses as of December 31,
2011. For each, briefly explain why the actuary should not use the data described and identify a more
appropriate alternative.
a. (0.5 point) Prior to 2010, all policies for a property insurer had a $1,000 deductible. Effective
January 1, 2010, all policies were written with a $5,000 deductible. The actuary intends to use
accident year data.
b. (0.5 point) The insurer writes general liability coverage in one state only. The average severity of
litigated claims reported after January 1, 2010 is twice the value of claims reported prior to
January 1, 2010. The actuary intends to use accident year data.
c.

(0.5 point) A general liability claim with both bodily injury and property damage case estimates
would have counted as one claim count prior to 2010 and is now recorded as two. The actuary
intends to use claim counts as an exposure base.

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Solutions to Sample Questions:
1.
List and describe five key dates regarding data organization for loss reserving, per Friedland Ch 3.
Policy Effective Date, Accident Date, Report Date, Accounting Date, Valuation Date
2.

Which of the following statements is/are true?
 The valuation date can be on, after, or before the accounting date
True.
 Case Outstanding represents estimated settlement values assigned to known claims.
True.
 A liability provision for reopened claims is generally needed only for the medical malpractice line of
business
False. Generally only for the workers compensation line of business
1. 
2.  3. , 
4. , , 
5. Neither 1,2,3 or 4

3. Friedland discusses how techniques to estimate unpaid losses may be used with data arranged into different
time intervals. List the 6 groupings discussed (See also Chapters 5 and 9).
Accident Year, Policy Year, Treaty Year, Underwriting Year, Report Year, Fiscal Year
For the next three questions, assume data is organized and analyzed by Accident Year:
4. Assume you have applied one of the techniques, on paid claims, described in Friedland to estimate unpaid
claims of $X. Let paid claims at the evaluation date = P and case outstanding estimates = C.
a. Ultimate Claims = Paid Claims + Unpaid Claims = P + X
b. IBNR = Total Unpaid Claims – Case Outstanding = X - C
c. Total Unpaid Claims = Total Reserve = Case Outstanding “C” + IBNR = X
d. Reported Claims = Paid Claims + Case Outstanding = P + C
Note: Friedland comments that actuaries often use “case incurred” or even just “incurred” losses to
describe “reported” claims. Also note, it generally refers to a cumulative total amount.
5. Assume you have applied one of the techniques, on reported claims, described in Friedland to estimate
IBNR of $Y. Let paid claims at the evaluation date = P and case outstanding estimates = C.
a. Ultimate Claims = Paid Claims + Case Outstanding + IBNR (broadly defined) = P + C + Y
b. IBNR = Y
c. Total Unpaid Claims = Total Reserve = Case Outstanding “C” + IBNR = C + Y
6. Throughout the Study Manual we see that most often past questions have provided reported claims
data, and asked us to solve for IBNR. However, some questions will ask for total unpaid claims or
ultimate claims.
a. We need to add Case Reserves to “IBNR” (broadly defined) to get total unpaid claims.
b. We need to add Total Unpaid Claims (Reserves) to the “paid” claims to get ultimate claims.
c. We need to add IBNR (broadly defined) to the “reported” claims to get ultimate claims.
7. The previous three questions apply to Accident Year data (by far, the most common).
a. On Calendar Year data, there is no future development.
b. On Report Year, Friedland comments: “Estimation techniques based on claims aggregated by
report year only measure development on known claims and not pure IBNR …”

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Solutions to 1995 Exam Questions (modified):
1. True/False. According to the CAS Statement of Principles on Reserves, policy effective date is one of the
key dates in the organization of a reserving database.
False, their key dates are: accident date, report date, recorded date, accounting date, and valuation date.
Friedland, however, does discuss Policy Effective date.
Solutions to 1997 Exam Questions (modified):
48. (0.25 point) NOTICE REPORT YEAR (NOT ACCIDENT YEAR) Friedland comments: “Estimation
techniques based on claims aggregated by report year only measure development on known claims and not
pure IBNR …”
a. (0.25 point) What is the accounting date of the evaluation of the required reserve?
Accounting date = 12/31/95; this is the date at which we are interested in estimating our liability for
financial reporting purposes.
b. (0.25 point) What is the valuation date of the evaluation of the required reserve?
Valuation date = 12/31/96; this is the date through which actual claims transactions are included in
the data we are using for the analysis.
c.

(0.5 point) Which of the following components of a total loss reserve (unpaid claims estimate) are
considered in the “required reserve” for this report year analysis? Explain your answer.
Case reserves: Not included since they’re a part of incurred loss as of December 31, 1995
Future development on known claims: Included
Incurred but not reported claims:
Claims that are truly incurred but not reported are excluded
from the analysis because the data is accumulated on a report year basis. Claims that have been
reported but not yet recorded would be included.

53. a. (1 point) Your first task as chief actuary is to determine how the data will be segregated by line for
reserving purposes. Based on the discussion in Friedland, list three considerations you would use in
making this decision.
Segregate data for reserving purposes based on expected differences in
•

Volume of data needed for credibility

•

Homogeneity of data (loss ratios and development patterns)

• Growth rates/change in exposure levels
Friedland comments in Chapter 3: “It is often possible to improve the accuracy of estimating unpaid
claims by subdividing experience into groups exhibiting similar characteristics, such as comparable
claim experience patterns, settlement patterns, or size of claim distributions.”

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Solutions to 1997 Exam Questions (modified - continued):
b. (2 points)
Based on your answers from (a), list the different lines of business that you would
combine for reserving purposes, explaining your rationale for each grouping.
•

Combine commercial auto liability BI and PD because:

1. Mix of business is constant:
1994
$120,000
83.3%
16.7%

Commercial Auto, Total
% BI
% PD

1995
$180,000
83.3%
16.7%

1996
$270,000
83.3%
16.7%

2. Development characteristics are comparable (both 3 party commercial liability lines)
•

Combine commercial auto liability BI and PD because:

1. Mix of business is constant:
1994
240,000
83.3%
16.7%

Personal Auto, Total
% BI
% PD

1995
264,000
83.3%
16.7%

1996
290,400
83.3%
16.7%

2. Development characteristics are comparable (both 3rd party personal liability lines)
•

Keep commercial multi-peril and workers compensation separate because: different development
characteristics and different growth rates of these lines of business. Find another source of data to
supplement the limited available workers compensation data.

Solutions to 2011 Exam Questions
21. Identify and briefly discuss four characteristics of the data that should be considered before
combining these lines of business.
Question 21 – Model Solution
1. Credibility of the Data
Consider if each line has a large volume of data to be credible. If not, combining the data may be the
alternative.
2. Severity (average claim size)
Average claim size for each line should be considered since using combined data can distort the
results from various estimation techniques, causing the unpaid claim estimate to be inaccurate.
Usually, long-tailed-low-frequency lines have higher severity (e.g. medmal) than short-tailed high
frequency (e.g. Auto PD)
3. Case reserve adequacy
Review the case reserving philosophy used on each line of business. Different reserving practices may
affect the results of estimated unpaid claims.
4. Claim settling rate
Consider the differences between the claim settlement rates of the lines being reviewed. This is crucial
because long-tailed low frequency lines usually have longer reporting patterns. Applying the resulting
development factors from this data to the combined data may distort the true unpaid claim estimate.

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Chapter 3 – Understanding the Types of Data Used
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2012 Exam Questions
16. (2 points) Below are four independent scenarios for estimating ultimate losses as of December 31,
2011. For each, briefly explain why the actuary should not use the data described and identify a more
appropriate alternative.
a. (0.5 point) Prior to 2010, all policies for a property insurer had a $1,000 deductible. Effective
January 1, 2010, all policies were written with a $5,000 deductible. The actuary intends to use
accident year data.
b. (0.5 point) The insurer writes general liability coverage in one state only. The average severity of
litigated claims reported after January 1, 2010 is twice the value of claims reported prior to
January 1, 2010. The actuary intends to use accident year data.
c.

d. (0.5 point) The rate of growth of earned exposures has increased dramatically over the past two
years, which has changed the average accident date significantly. The actuary intends to use
accident year data.
Question 16 – Model Solution 1 (Exam 5B Question 1)
a. Using AY data is not appropriate because of the shift in the mix of business (changing deductibles). An
analysis using Policy year data is more appropriate.
b. Use of Report year data is better than AY data because of the shift in severity. The change in
severity will likely cause the occurrence data to better be correlated with the report data so RY data is
best.
c. Using earned exposure instead of the claim counts would be better to use because of the change in the
definition of a claim count. Using claim counts would distort the analysis because of the changed)
d. Use of accident quarter would be better used then AY data because of the shift in growth over the
past two years. AY data will be distorted because of the growth distribution change.
Question 16 – Model Solution 2 (Exam 5B Question 1)
a. Because there is a change in deductible, policy year data should be used.
b. Average severity is more correlated to when the claim was reported so report year data would be more
appropriate.
c. There is a change in claim count definition so the actuary should use earned exposures instead.
d. Because the average accident data has changed, the actuary should use accident quarter data.
Examiner’s comments
Overall, the candidates did well on this question. Many candidates have no problem stating the alternative to
use.
Some had trouble explaining the inappropriateness of using accident year data (for 3 of the 4 parts in the
question).
Sometimes candidates provided explanation that either would have rendered accident year data inappropriate
even before the change, or would have continued to be a problem even with their suggested alternative.
Candidates with the better answers were able to point out the essence of the change described in the question
and explain how accident year data/claim count fails to continue to be appropriate.

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Chapter 4 – Meeting with Management
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Sec
1
2
3
1

Description
Understanding the Environment
Sample Questions for Department Executives
Additional Questions

Pages
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45 - 49
49 - 50

Understanding the Environment

44 - 45

Before developing estimates of unpaid claims, the actuary must first understand:
 circumstances within the insurer's organization as well as
 the economic, social, legal, and regulatory environments that affect the insurer’s liabilities.
A sound understanding of the insurer’s internal and external environment is needed to correctly interpret
patterns and changes in the data.
Claims reporting and payment patterns, frequency, and severity can be altered by changes in:
* Classes of business written or geographical writings
* Policy provisions (e.g. policy limits and deductibles)
* Reinsurance arrangements (including limits and attachment points)
* Claims management philosophy that often occur when managerial changes occur
* Claims processing lags (e.g. when a new technology is implemented) or when department staffing is
disrupted (e.g. in the event of a merger or a major catastrophe) overwhelming the claim department's
capacity
* Legal and social environment (e.g. introduction of no-fault auto insurance, court system back-logs, new
court rulings, and implementation of tort reform)
Note: Tort reform is legislation designed to reduce liability costs through limits on various kinds of damages and/or
through modification of liability rules.

* Economic environment (e.g. an increase in the inflation rate or a decrease in the interest rate).
Information gathering requires a great deal of back-and-forth dialogue between the actuary and management.
To collect data and information, the process must include both a review of quantitative data and discussions
with members of the insurer's claim and underwriting departments.
Based on the Berquist/ Sherman paper "Loss Reserve Adequacy Testing: A Comprehensive, Systematic
Approach”, the appendix contains a list of possible interview questions for the various departments of an insurer.
By asking such questions, the actuary gains a better understanding of the specific circumstances of
particular books of business, and thus guides the actuary to choose the most appropriate methods for
determining unpaid claim estimates.
The following questions are presented from the perspective of a consultant interviewing insurance company
management.

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Chapter 4 – Meeting with Management
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND

Sample Questions for Department Executives

45 - 49

A. Questions for a Claims Executive
1. What specific objectives and guidelines does the department have in setting unpaid case?
Are unpaid case established on what it would cost today to settle the case, or has a provision for inflation
between now and the time of settlement been included in the case outstanding?
2. Have there been any significant changes in setting and reviewing unpaid case during the last 5 years?
3. Have there been any changes in the definitions of or rules for setting bulk or formula reserves for reported
claims in the last 5 years?
4. Are any special procedures or guidelines used in the reserving of large or catastrophic claims?
5. Has the adjuster’s caseload size changed significantly in the past several years?
6. When is a claim file established?
7. Are claims files setup for each claimant or for each accident? What procedures are used when there are
multiple claimants from the same accident? Is a claim file setup for each coverage or for all coverages
combined?
8. What procedures are used in recording reopened claims? Are such claims coded to the report date of the
original claim or to the date of reopening? How will the reopening of a claim affect aggregate data for
paid, open or reported claims and paid, outstanding or incurred losses?
9. Have there been shifts in the reporting or non-reporting of small/trivial claims? In the procedures for the
recording of such?
10. Has there been any shift in emphasis in settling large versus small claims? In the relative % of such
claims? In attitudes in adjusting such claims?
11. Have there been any changes in the guidelines on when to close a claim?
For example, is a P.D. (property damage) claim kept open until the associated B.I. (bodily injury) claim is
closed, or only until the P.D. portion is settled?
12. Have there been changes in the rate of settlement of claims recently?
13. Has there been any shift from the use of company adjusters to independent adjusters? Or vice versa? If
so, how has this affected the operations of the claims department?
14. Has there been any change in the timing of the payment of ALAE? Are such payments made as the
expenses are accrued (or incurred) or when the claim is closed?
15. Has there been any change in the definition and limit for one-shot or fast-track claims in recent years?
What is that limit?
16. What safeguards against fraudulent claims are used? Are any special procedures followed in the event
of the filing of questionable or non-meritorious claims? Have these safeguards changed in recent years?
17. Have there been any shifts toward (or away from) more vigorous defense of suits in recent years?
18. Could you provide copies of all bulletins to the field issued in the last 5 years in which details of the
changes in claims procedures are provided?
19. Could you provide copies of recent claim audits?
20. For WC, what mortality table was used (year and general population or disabled lives table) to set the
unpaid case for permanently disabled claimants?
21. For large open claims, has there been any revision in the reserve since the latest evaluation date of the
claims experience?
22. Are unpaid cases set at an expected level, the most likely settlement amount, or the minimum possible
amount (or some other standard)?

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Chapter 4 – Meeting with Management
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
B. Questions for an Underwriting Executive
1. What changes have occurred in your company's book of business and mix of business in the past 5-7
years? How are the risks insured today different from those of the past?
2. Do you underwrite any large risks which are not characteristic of your general book of business?
3. Have any significant changes occurred in your underwriting guidelines in recent years?
4. Has the proportion of business attributable to excess coverages for self-insurers changed in recent years?
Can a distribution of such business be obtained by line, retention limit, class, etc.?
Is a record of self-insured losses and claims available?
5. How many different programs or types of risk are premium and claims experience tracked and compiled
into claim ratio runs?
6. Are any details of excess policies (e.g. attachment points, exclusions, per occurrence, sunset clauses,
aggregate caps, etc.) available?
7. How frequent are experience summaries run? How far back are these available?
8. How are new programs priced? If you are relying on another insurer's filings, how similar are the
underlying books of business?
C. Questions for a Data Processing or Accounting Executive
1. Has there been any date change as to when books are closed for the quarter? The year?
2. How are claim payments handled for claims which have already been paid, but which have not yet been
processed to the point where they can be allocated to accident quarter? Are they excluded from the loss
history until they are allocated to accident quarter or are they loaded into an arbitrary quarter?
3. Have new data processing systems been implemented in recent years? Have they had a significant
impact on the rate of processing claims or on the length of time required from the reporting to the
recording of a claim?
4. Are data sources crosschecked and audited for accuracy and for balancing to overall company statistics?
Comment on the degree of accuracy with which each kind of statistic has been properly allocated to
accident quarter, to line of business, to size of loss, etc.
5. Have there been any changes in coding procedures which would affect the data supplied?
6. Can partial payments exceed the case outstanding on a claim? If so, what adjustments are made?
Are unpaid case taken down by the amount of partial payments?
7. How far back can the claims data be actively re-compiled by various key criteria?
8. What data elements are available for each claim? For each risk?
9. By what key criteria could the historical claims data be freshly compiled? Examples of criteria: size of loss
breakdowns, type of claim breakdowns (e.g., liability vs. property for commercial multi-peril or homeowner
multi-peril), separate compilations by policy limit, or deductible, or type of claim, or state.
10. Can data be compiled either by claimant or occurrence, if multiple claims are set for one occurrence?
D. Questions for Actuaries Specializing in Ratemaking
1. Have there been any changes in company operations or procedures which have caused you to depart
from standard ratemaking procedures? If so, please describe those changes and how they were treated.
2. What data used for ratemaking purposes could also be used in testing unpaid claims?
3. Has there been any significant shifts in the business by type of risk or type of claim within the past several
years?

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Chapter 4 – Meeting with Management
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
D. Questions for Actuaries Specializing in Ratemaking (continued):
4. Do you have any of the following sources of information which may be of value in reserve testing:
a. External economic indices,
b. Combined claims data for several companies (e.g., data obtainable from bureau rate filings),
c. Special rating bureau studies,
d. Changes in state laws or regulations, and
e. Size of loss or cause of loss studies?
5. Could we obtain copies of recent rate filings?
6. Were there any changes in statues, court decisions, extent of coverage that necessitated some reflection
in the rate analysis?
7. How are new programs priced? If you are relying on another insurer's filing, how similar are the
underlying books of business?
E. Questions for In-House Actuaries
1. Could we obtain copies of any and all actuarial studies done by consultants, auditors or internal
actuaries?
2. What areas of disagreement are there between these different studies?
3. What specific background information did you take into account in making your selections?

Additional Questions

49 - 50

The authors recommend the following questions be added for meetings with senior management of the insurer.
F. Questions for Those Managing Reinsurance
* Please provide details of reinsurance treaties for both assumed and ceded business.
* Please provide details of all reinsurance ceded treaties including:
i. Retention level or Q.S. %
ii. Reinsurers involved (including participation)
iii. Details of any sliding scale premium, commission, or profit commission (including currently booked
amounts)
iv. Any problems or delays encountered in collecting reinsurance
* Please provide details of any internal or sister company reinsurance agreements (cover notes, relevant
amounts, and by-line breakdowns).
* Have the reinsurance programs for next year been secured? If so, under what terms?
G. Questions for Senior Management
Please provide a brief description of the company's operations including:
* An organization chart (with recent changes highlighted)
* Details of ownership
* Description of types of business written (including all special programs)
* Description of marketing (i.e., direct writer, independent agent, etc.)

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Chapter 5 – The Development Triangle
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Sec
1
2
3
4
5

Description
Rows, Diagonals, and Columns
Alternative Format of Development Triangles
Detailed Example of Claim Development Triangles
Other Types of Development Triangles
Naming Convention for Examples

Pages
52 - 54
54
54 - 60
60 - 62
62

INTRODUCTION
A development triangle shows changes in the value of various cohorts (groups of claims) over time.
The table below shows how cumulative paid claims by insurers arising out of auto accidents that occurred
during 2006, 2007, and 2008 (the cohorts) increased from year-end 2006 to year-end 2007 to year-end 2008

Table 1 - Paid Claims and Expenses ($US Billions)
by Year End Accounting Date
Accident Year Year-end 2006
Year-end 2007
Year-end 2008
2006
100
150
170
2007
110
161
2008
115
Development for any of these AY cohorts is the change in the value for the cohort over time.
Paid claims and expense for AY 2006 experienced development of $50 billion (due to the change from
$100 billion to $150 billion)
It is easier to observe development by looking at the age (or maturity) of the cohort rather than the
accounting date for the cohort.
The above triangle reformatted to reflect this approach is shown below:

Table 2 – Paid Claims and Expenses ($US Billions)
by Age
Accident Year
12 Months
24 Months 36 Months
2006
100
150
170
2007
110
161
2008
115
Age (or maturity) is measured from the start of the cohort period. For example, the:
 age of AY 2006 (valued at year-end 2006) is 12 months from the start of the AY.
 age of AY 2006 (valued at year-end 2007) is 24 months from the start of the AY.
Both approaches result in data in a triangle shape (hence the term development triangle).
However, in the second triangle it is easier to see how the:
 volume (or scale) of the AY cohort changes from one AY to the next (vertically) and
 value of cumulative paid claims for an AY changes from age to age (horizontally).

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Chapter 5 – The Development Triangle
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Development can be positive or negative. For example:
 the number of claims occurring in an AY will increase from one valuation point to another until all
claims are reported. The number of claims decreases at successive valuations.
however, the number of claims can decrease from one valuation point to another (see Chapter 11,
private passenger auto collision coverage)
 reported claim development can show downward patterns if the insurer:
i. settles claims for a lower value than the case O/S estimate or
ii. includes recoveries with the claims data.
Development patterns are critical inputs to many techniques used to estimate unpaid claims.
In this chapter, we demonstrate:
 how to build development triangles for paid claims, case O/S, reported claims, and reported claim counts.
 the use of payment and case O/S for a sample of 15 claims over a 4-year time horizon.

Rows, Diagonals, and Columns

52 - 54

There are three important dimensions in a development triangle:
1. Rows
2. Diagonals 3. Columns

Table 3 — Reported Claim Triangle
Accident
Reported Claims as of (months)
12
24
36
48
Year
2005
1,500
2,420
2,720
3,020
2006
1,150
1,840
2,070
2007
1,650
2,640
2008
1,740
Each row in the triangle represents one AY.
Data organized by AY groups claims according to the date of occurrence (i.e. the accident date), and each
row consists of a fixed group of claims.
Rows:
The first row of the triangle contains claims occurring in 2005; the second row, claims occurring in 2006; the
third row, claims occurring in 2007; and the final row, claims occurring in 2008.
Diagonals:
Each diagonal represents a successive valuation date.
* The first diagonal (a single point) is the 12/31/2005 valuation
* The next diagonal is the 12/31/2006 valuation for AYs 2005 and 2006
* The next diagonal is the 12/31/2007 valuation for AYs 2005 through 2007
* The last diagonal is the 12/31/2008 valuation for AYs 2005 through 2008

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Chapter 5 – The Development Triangle
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
The diagonals and corresponding valuation dates are shown below.
CY in the diagram below refers to calendar year.

The first diagonal (in the upper left corner of the triangle) is at the 12/31/2005 valuation date and represents
AY 2005 at 12 months of maturity.
AY 2005 begins on 1/1/2005 and is 12 months old at 12/31/2005.
The second diagonal at the 12/31/2006 valuation date and consists of AY 2005 ay 24 months old and AY
2006 at 12 months old.
The last diagonal of the triangle at a valuation date of 12/31/2008 represent claims for:
* AY 2005 as of 48 months (counting from the start of the AY, 1/1/ 2005, to the valuation date of 12/31/2008)
* AY 2006 as of 36 months (counting from 1/1/2006 to 12/31/2008)
* AY 2007 as of 24 months
* AY 2008 as of 12 months
Column:
Each column represents an age (or maturity) related to the combination of AY and valuation date used to
create the triangle.
The data is AY using annual valuations, and thus the ages in the columns are 12 months, 24 months, 36
months, and 48 months.
Different valuations can be used (e.g. 6 months, 12 months, 18 months, etc.).

Alternative Format of Development Triangles

Development triangles with the rows corresponding to the experience period (e.g. AY in the prior
example) and the columns representing the maturity ages are the most common presentation of
development triangles.
Some insurers reverse this orientation and present AYs (or policy or underwriting years) as the columns
and the maturity ages as the rows.

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Chapter 5 – The Development Triangle
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
3

Detailed Example of Claim Development Triangles

54 - 60

Understanding the Data
To better understand how to create a claim development triangle, individual claims data underlying the
reported claim triangle shown in Table 3 is used.
How claims amounts in the claims listing below are integrated into the cells of a claim development triangle is
shown below. Case O/S means case outstanding.

Claim
ID
1
2
3
4

Accident
Date
Jan-5-05
May-4-05
Aug-20-05
Oct-28-05

Table 5 – Detailed Example – Claims Transaction Data
2007 Transactions
2005 Transactions
2006 Transactions
Ending
Ending
Ending
Report
Total
Case
Total
Case
Total
Case
Date
Payments
0/S
Payments
0/S
Payments
0/S
Feb-1-05
400
200
220
0
0
0
May-15-05
200
300
200
0
0
0
Dec-15-05
0
400
200
200
300
0
May-15-06
0
1,000
0
1,200

5
6
7

Mar-3-06
Sep-18-06
Dec-1-06

Jul-1-06
Oct-2-06
Feb-15-07

8
9
10
11

Mar-1-07
Jun-15-07
Sep-30-07
Dec-I2-07

Apr-1-07
Sep-9-07
Oct-20-07
Mar-10-08

12
13
14
15

Apr-12-08
May-28-08
Nov-12-08
Oct-15-08

Jun-18-08
Jul-23-08
Dec-5-08
Feb-2-09

260
200

190
500

190
0
270
200
460
0

2008 Transactions
Ending
Total
Case
Payments
0/S
0
0
0
0
0
0
300
1,200

0
500
420

0
230
0

0
270
650

200
390
400

200
0
400
60

0
390
400
530

400
300
0

200
300
540

The table above contains detailed information for 15 claims that occurred in AYs 2005 - 2008.
The first column shows the claim ID number; the next two columns are the accident date and the report date.
The accident date is needed to determine the appropriate row of the triangle.
The report date determines when the claim first enters the triangle.
The table includes claim payments made in the year and the ending case O/S value.
Claim payments are not cumulative paid values, but are transactional payments made during the year.
The case O/S values are the ending case O/S values.

Step-by-Step Example
A step by step demonstration on how to create the paid claims, case O/S, reported claims, and reported claim
count triangles will be given.
To build the cumulative paid triangle, begin with incremental paid claim development triangle (see Table 6
below, an excerpt of Table 5, which shows payment transactions).

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Using the data below, an incremental payments triangle showing the amounts paid in each 12-month
calendar period for the fixed group of claims will be constructed.
Table 6 - Detailed Example – Claims Transaction Paid Claims Data
Incremental Payments in Calendar Year
Claim
Accident
Report
ID
Date
Date
2005
2006
2007
2008
1
Jan-5-05
Feb-1-05
400
220
0
0
2
May-4-05 May-15-05
200
200
0
0
3
Aug-20-05 Dec-15-05
0
200
300
0
4
Oct-28-05 May-15-06
0
0
300
5
6
7

Mar-3-06
Sep-18-06
Dec-1-06

Jul-1-06
Oct-2-06
Feb-15-07

8
9
10
11

Mar-1-07
Jun-15-07
Sep-30-07
Dec-12-07

Apr-1-07
Sep-9-07
Oct-20-07
Mar-10-08

12
13
14
15

Apr-12-08
May-28-08
Nov-12-08
Oct-15-08

Jun-18-08
Jul-23-08
Dec-5-08
Feb-2-09

260
200

190
0
270

0
230
0

200
460
0

200
0
400
60
400
300
0

For claims that occurred during 2005, the insurer paid a total of:
 $600 (400 +200) during the first 12-month period (2005),
 $620 (220+200+200) during the second 12-month period (2006), and
 $300 in each of the following two 12-month periods (2007 and 2008).
For claims that occurred during 2006, the insurer paid
 $460 (260+200) during 2006,
 $460 (190+270) during 2007 and

$230 during 2008.
Use the same approach for each AY grouping of claims to derive the following triangle of incremental paid claims.

Table 7 — Incremental Paid Claim Triangle
Accident
Incremental Paid Claims as of (months)
Year
12
24
36
48
2005
600
620
300
300
2006
460
460
230
2007
660
660
2008
700
The incremental paid claim triangle is used for diagnostic purposes and for some frequency-severity techniques.

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Actuaries use cumulative paid claim triangles more often, created using the incremental paid claim triangle.
1. The first column in the incremental and cumulative triangles at age 12 months is the same (i.e.
incremental paid claims equal cumulative paid claims at the first maturity age).
2. The second column of the cumulative paid claim triangle equals the second column (i.e. age 24 months)
of the incremental paid claim triangle added to the first column of either triangle.
3. The third column of the cumulative paid claims at 36 months is equal to the cumulative paid claims at 24
months plus the incremental paid claims at 36 months.

Table 8 — Cumulative Paid Claim Triangle
Accident
Cumulative Paid Claims as of (months)
Year
12
24
36
48
2005
600
1,220
1,520
1,820
2006
460
920
1,150
2007
660
1,320
2008
700
An alternative way to computing the cumulative claim triangle rather than simply cumulating the incremental
paid triangle follows:
The first cell of the cumulative paid claim development triangle is AY 2005 at a valuation date of 12/31/2005
(a.k.a. AY 2005 at 12 months).
4 claims occurred in 2005 (Claim IDs 1, 2, 3, and 4).
 The first 3 claims occurred and were reported to the insurer during 2005.
 Claim ID 4 was not yet reported as of the 12/31/2005 valuation date.
 Claim ID 3 did not have any payments as of 12/31/2005.
Thus, the $600 paid claims appearing in the first cell of the triangle are the payments for Claim IDs 1 and 2
during the year 2005.
Constructing the second diagonal of the cumulative paid claim triangle (i.e. the 12/31/2006 valuation).
It contains two points: AY 2005 at 24 months and AY 2006 at 12 months.
 First calculate the value of paid claims at 24 months for AY 2005.
Total payments made during 2006 for Claim IDs 1, 2, 3, and 4 are $620 ($220 + $200 + $200 + $0).
Cumulative claim payments for AY 2005 through 12/31/2006 equal the sum of the payments made
during 2005 and the payments made during 2006 = 600+ 620 = $1,220.
 Next calculate the payments for AY 2006 at 12 months.
Claim IDs 5 and 6 were reported in 2006, and Claim ID 7 is not included in the calculation for the
12/31/2006 valuation since it was not reported as of the 12/31/2006 valuation.
Thus, paid claims for AY 2006 as of 12/31/2006 equal to the sum of claim payments ($260 + $200)
for Claim IDs 5 and 6.
Constructing the third and fourth diagonals of the cumulative paid claim triangle (i.e. the 2007 and 2008 diagonals).
The third diagonal consists of three points:
* AY 2005 at 36 months; * AY 2006 at 24 months; * AY 2007 at 12 months
The fourth diagonal consists of AY 2005 at 48 months, AY 2006 at 36 months, AY 2007 at 24 months and
AY 2008 at 12 months.
A similar procedure to the example above is used in cumulating claim payments made through 12/31/2007
and 12/31/2008.

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Chapter 5 – The Development Triangle
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Case O/S Triangle
Table 9 is an excerpt from Table 5 presented earlier in the chapter.
Table 9 – Detailed Example – Claims Transaction Ending Case Outstanding Data
Ending Case Outstanding
Claim
Accident
Report
ID
Date
Date
2005
2006
2007
2008
1
Jan-5-05
Feb-1-05
200
0
0
0
2
May-4-05
May-15-05
300
0
0
0
3
Aug-20-05
Dec-15-05
400
200
0
0
4
Oct-28-05
May-15-06
1,000
1,200
1,200
5
6
7

Mar-3-06
Sep-18-06
Dec-1-06

Jul-1-06
Oct-2-06
Feb-15-07

8
9
10
11

Mar-1-07
Jun-15-07
Sep-30-07
Dec-12-07

Apr-1-07
Sep-9-07
Oct-20-07
Mar-10-08

12
13
14
15

Apr-12-08
May-28-08
Nov-12-08
Oct-15-08

Jun-18-08
Jul-23-08
Dec-5-08
Feb-2-09

190
500

0
500
420

0
270
650

200
390
400

0
390
400
530
200
300
540

Use the table above to create the case O/S development triangle below.

Accident
Year
2005
2006
2007
2008

Table 10 – Case Outstanding Triangle
Case Outstanding as of (months)
12
24
36
48
900
1,200
1,200
1,200
690
920
920
990
1,320
1,040

Case O/S for AY 2005 at 12 months is computed by adding the ending case O/S values for Claim IDs 1, 2,
and 3 (200+300+400) to derive the case O/S value of $900.
Claim ID 4 case O/S is not included since it is not reported until 5/15/2006.
Case O/S for AY 2005 at 24 months equal case O/S values for Claim IDs 3 and 4 or $1,200 ($200 + $1,000).
Note that case O/S for Claim IDs 1 and 2 are both $0 at December 31, 2006.
Case O/S for AY 2005 at 36 months and 48 months equal the ending case O/S for Claim ID 4 of $1,200.
Case O/S for AY 2006 at 12 months (i.e., valuation date December 31, 2006) equals $690 which is equal to
the sum of the ending case O/S for Claim IDs 5 and 6 ($190 + $500).
Case O/S at 24 months equals the sum of case O/S on all three AY 2006 claims ($0 + $500 + 420).
Case O/S for AY 2006 at 36 months equals $920 which is equal to the case O/S for Claim IDs 6 and 7
Continue in a similar manner to build the remainder of the case O/S development triangle.

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Chapter 5 – The Development Triangle
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Reported Claim Development Triangle
Definition: Reported claims equal cumulative paid claims plus case O/S at the valuation date.
Thus, reported claim development triangle equal cumulative paid claim triangle plus the case O/S triangle.

Table 11 — Reported Claim Development Triangle
Accident
Reported Claims as of (months)
Year
12
24
36
48
2005
1,500
2,420
2,720
3,020
2006
1,150
1,840
2,070
2007
1,650
2,640
2008
1,740
Commentary: What happened to AY 2005 claims over time?
Claim ID 1 occurred and was reported 2005.
 By 12/31/2005 (the first year of development), there were $400 in payments and case O/S of $200.
 During 2006, a claim payment of $220 ($20 more than the case O/S) and the case O/S was reduced to $0.
 There was no further activity on this claim through year-end 2008.
Claim ID 2 occurred and was reported in 2005
 By 12/31/2006, there was a claim payment of $200 and case O/S of $300
 During 2006, the claims was settled $200 ($100 less than the $300 case O/S).
Claim ID 3 occurred and was reported in 2005.
 By 12/31/2005 an initial case O/S of $400 as set.
 During 2006, a $200 payment was made and case O/S was reduced to $200.
 During 2007, a final payment was made for $300, causing the final incurred value to be $500 ($100
more than the reported claim estimates at year-ends 2005 and 2006).
Claim ID 4 occurred in 2005 and was reported 2006.
 By 12/31/2006, the case O/S was $1,000 for this claim.
 By 12/31/2007, the case O/S had increased to $1,200.
 There were no payments in either 2006 or 2007.
 In 2008, claim payments were $300 but there was no change in the ending case O/S.
Thus, the reported claim increased by $300 during 2008 from
$1,200 (cumulative claim payments through 12/31/2007 of $0 plus ending unpaid case of $1,200) to
$1,500 (cumulative claim payments through 12/31/2008 of $300 plus ending unpaid case of $1,200).

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Chapter 5 – The Development Triangle
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Reported Claim Count Development Triangle
The data in Table 5 can be used to build a reported claim count triangle. A description of how to build the
claim count development triangle by using AYs 2005 and 2008 follows.

Table 12 — Reported Claim Count Development Triangle
Accident
Reported Claim Counts as of (months)
Year
12
24
36
48
2005
3
4
4
4
2006
2
3
3
2007
3
4
2008
3
There are 4 claims for 2005, but only 3 of them were reported as of 12/31/2005.
Thus, the first cell in the reported claim count triangle which represents AY2005 as of 12/31/2005 shows 3
claims reported.
By 12/31/2006, all 4 claims were reported.
No further claims were reported for AY 2005, and thus the number of reported claims remains unchanged at
4 for ages 36 months and 48 months.
There are 4 claims for AY2008, but as of 12 months, only 3 claims were reported for AY 2008 (claim ID 15
was not reported until 2009 and thus is not included in the triangle).

Other Types of Development Triangles

60 - 62

Actuaries use a wide variety of data when creating development triangles.
First, determine the time interval (i.e. the rows of the triangles) data will be organizing into.
Other that AY, common intervals include:
* Report year
* Underwriting year
* Treaty year (i.e. a period of 12 months covered by a reinsurance contract or treaty)
* Policy year
* Fiscal year
AY is most common used by actuaries in the U.S. and Canada use when creating development triangles.
RY is used for analyzing claims-made coverages (e.g. medical malpractice and errors and omissions liability).
U/W year is used by reinsurers and PY is a similar to underwriting year.
For self-insurers:
 PY, FY, and AY are often the same. For example, a self-insured public entity may:
have a FY 4/1 to 3/31, and
issue documents of coverage to departments and agencies with an 4/1 to 3/31 coverage period; and
arrange excess insurance with a PY of 4/1 to 3/31.
 Finally, this public entity may aggregate development triangles using AY periods of 4/1 to 3/31.

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Chapter 5 – The Development Triangle
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Time intervals (for organizing claims) other than annual intervals:
 include monthly, quarterly, and semi-annual data for developing estimates of unpaid claims.
 are selected after considering the credibility of the experience or the stability of development or both.
Types of claims data commonly appearing in development triangles include:
* Reported claims
* Case O/S
* Cumulative total paid claims
* Cumulative paid claims on closed claim counts
* Incremental paid claims Reported claim counts
* Claim counts on closed with payment
* Claim counts on closed with no payment
* Total closed claim counts
* O/S claim counts
Actuaries use the data types previously listed to create triangles of ratios and average claim values, which include:
* Ratio of paid-to-reported claims
* Ratio of total closed claim counts-to-reported claim counts
* Ratio of claim counts on closed with payment-to-total closed claim counts
* Ratio of claim counts on closed without payment-to-total closed claim counts
* Average case O/S (case O/S divided by O/S claim counts)
* Average paid on closed claims (cumulative paid claims on closed claims divided by claim counts closed
with payment)
Cumulative paid claims on closed claim counts may be difficult to obtain; Actuaries may determine that interim or pre-closing payments
are immaterial enough to justify the inexact match from including all payments, even those from open claims/closed claim counts.

* Average paid (cumulative total paid claims divided by total closed claim counts)
* Average reported (reported claims divided by reported claim counts)
Triangles of ratios and average values provide useful insight into the relationships between the various types of
data at different points in time during the experience period (see Chapter 6 as to how actuaries use these types
of triangles as diagnostic tools).
LAE data may be analyzed independently of claims only. The actuary may create development triangles with:
 ratios of paid LAE-to-paid claims only and
 ratios of reported LAE-to-reported claims only.
How many development periods are needed to be evaluated?
rd
th
 Should development be analyzed through the 3 , 5 , 10th or the 20th maturity year?
The actuary should analyze development out to the point at which the development ceases (i.e., until
the selected development factors = 1.000).
 The number of development periods required varies by line, jurisdiction, and by data type. For example:
Paid claims often require more development periods than reported claims.
Reported claims often require more development periods than reported claim counts.
Auto physical damage claims settle more quickly than general liability claims, and thus an analysis of
unpaid claims for auto physical damage requires fewer development periods than for general liability.

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Chapter 5 – The Development Triangle
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
In the following chapters, the development triangle is used as a diagnostic tool for numerous estimation
techniques for unpaid claims.

Naming Convention for Examples

Naming conventions include the terms:
* "reported claims" to refer to cumulative reported claims and
* "paid claims" to refer to cumulative paid claims.
* "reported claim counts" and "closed claim counts" to refer to cumulative reported and closed claim counts.
For some examples in Chapters 11 - 13, incremental claims and claim counts are used

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Chapter 5 – The Development Triangle
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Questions from the 2010 Exam:
10. (1.75 points) Given the following information:
Cumulative Paid Loss ($000)
Accident
Year
12 Months
24 Months
36 Months
48 Months
2006
75,000
212,500
288,000
337,000
2007
50,000
165,000
310,000
2008
115,000
238,000
2009
85,000
Case Outstanding ($000)
Accident
Year
2006
2007
2008
2009

12 Months
188,000
175,000
115,000
208,000

24 Months
115,000
94,000
238,000

36 Months
74,000
45,000

48 Months
35,000

a. (0.25 point) Calculate reported claims for accident year 2007 as of December 31, 2009.
b. (0.5 point) Calculate paid claims for calendar year 2009.
c. (0.5 point) Calculate the change in case reserves for calendar year 2009.
d. (0.5 point) Briefly describe two benefits of organizing data for reserving on an accident year basis.

15. (1.75 points) Given the following claim detail ($000):

Claim

Accident Date

2007

12/31/07

2008

Case
Reserve
at
12/31/08

1
2
3
4

January 1, 2007
July 1, 2007
January 1, 2008
July 1, 2008
January 1, 2009

75
25

250
250

50
50
0
100

250
200
500
50

Paid
During

Case
Reserve at

Paid
During

Paid
During
2009

Case
Reserve
at
12/31/09

300
200
50
100

0
0
600
0

105

645

a. (0.5 point) Construct an accident year cumulative paid loss triangle.
b. (0.5 point) Construct an accident year cumulative reported loss triangle.
c. (0.75 point) Perform a diagnostic test to determine whether the data suggests a speed-up in claim
payments.

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Chapter 5 – The Development Triangle
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Questions from the 2011 Exam:
22. (2.25 points) Given the following claim detail (000s):

Claim
1
2
3
4
5
6
7
8

Accident Date
March 3, 2008
July 18, 2008
December 1, 2008
March 1, 2009
October 3, 2009
November 3, 2009
April 12, 2010
June 28, 2010

Paid
During

Case
Reserve
at Year

2008
$225
$150

End 2008
$190
$500

Paid
During

Case
Reserve
at Year

2009
$250
$0
$105
$200
$320
$0

End 2009
$0
$500
$75
$200
$280
$100

Paid
During

Case
Reserve
at Year

2010
$0
$230
$75
$150
$200
$50
$45
$500

End 2010
$0
$270
$25
$100
$0
$100
$55
$500

a. (1.25 points) Construct the cumulative accident year reported loss development triangle as of
December 31, 2010.
b. (0.5 point) Describe a situation in which it is preferable to use accident year data for estimating
unpaid claims rather than report year data.
c. (0.5 point) Describe a situation in which it is preferable to use report year data for estimating unpaid
claims rather than policy year data.

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Chapter 5 – The Development Triangle
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to questions from the 2010 Exam
10a. (0.25 point) Calculate reported claims for accident year 2007 as of December 31, 2009.
10b. (0.5 point) Calculate paid claims for calendar year 2009.
10c. (0.5 point) Calculate the change in case reserves for calendar year 2009.
10d. (0.5 point) Briefly describe two benefits of organizing data for reserving on an accident year basis.
Initial comments
AY Reported Claims = Cumulative AY Paid claims (as of a given evaluation date)
+ Latest AY case reserves (as of a given evaluation date)
CY Paid Claims = Sum (of the diagonal for a given evaluation date) of Cumulative Paid
- Sum (of the diagonal 1 year prior to the given evaluation date) of Cumulative Paid
CY Chg in Case Res = Sum (of the diagonal for a given evaluation date) of Case Reserves
- Sum (of the diagonal 1 year prior to the given evaluation date) of Case Reserves
For AY reported claims data, cumulative paid claims (for a given AY) + latest case reserves (for the same
AY) are needed.
For CY paid claims data, cumulative paid claims (for all AYs as of a given evaluation date) + cumulative
paid claims (for AYs 1 year prior to a given evaluation date) are needed.
Question 10 – Model Solution 1
a. AY 2007 reported claims at 12/31/2009 = $310M + $45M = $355M.
b. ($337M + $310M + $238M + $85M) - $288M - $165M - $115M = $402M
c. ($35M + $45M + $238M + $208M) - $74M - $94M - $115M = $243M
d. Numerous benchmarks are tracked by accident year. Tracking claims by accident year can be very useful
when economic/regulatory changes have recently occurred, or if a significant large loss has occurred.
Question 10 – Model Solution 2
a. AY 2007 Reported Claims = Cumulative Paid + Cumulative Outstanding = $310,000 + $45,000 = $355,000.
b. Paid claims in 2009 = $85,000 + ($238,000 - $115,000) + ($310,000 - $165,000) + ($337,000 - $288,000)
= $402,000
c. 2008 case outstanding = $115,000 + $94,000 + $74,000 = $283,000.
2009 case outstanding = $208,000 + $238,000 + $45,000 + $35,000 = $526,000.
Change in case = $526,000 - $283,000 = $243,000.
d. It’s easier to understand. It’s a shorter time period than policy year.

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Chapter 5 – The Development Triangle
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to questions from the 2010 Exam
Question 15 – Model Solution 1
15a. (0.5 point) Construct an accident year cumulative paid loss triangle.
15b. (0.5 point) Construct an accident year cumulative reported loss triangle.
15c. (0.75 point) Perform a diagnostic test to determine whether the data suggests a speed-up in claim payments.
a. Cumulative Paid Triangle
12
2007

100

2008

100

2009

105

24
200

700=75+25+50+50+300+200

250=0+100+50+100

b. Cumulative Reported = Cumulative Paid + Outstanding Case at Valuation
2007

600

650

700=700+0

2008

650

2009

750=105+645

850=250+600

c. Cumulative Paid/Cumulative Reported
12

2007

0.167

0.308

1.000

2008

0.154

2009

0.14=105/750

0.294=250/850

There seems to be a decrease in the paid/reported ratio at 12 months. This could be caused by either a
slow-down in claim payments or increase in reserve adequacy. However, it is difficult to draw conclusions
with this small amount of data. There is little credibility.
Question 15 - Model Solution 2 – Part c
a. Cumulative Paid Triangle (same as Model Solution 1)
b. Cumulative Reported (same as Model Solution 1)
c. Closed Claim Count/Reported Claim Count
12

2007

2008

2009

36
100%=2closed/2reported

50%=1closed/2reported

It seems that we closed the claim in the 12-24 period faster for AY 2008 than for AY 2007. However, the
experience is too thin, so it might be just random change but not intentional speeding up.
Question 15 - Model Solution 3 – Part c
Average Cumulative Payment
12

100

350=700/2

2008

125=250/2

2009

105=105/1

2007

Again, it seems that we increase the average cumulative payment over years, but experience is too thin
to draw meaningful conclusions.

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Chapter 5 – The Development Triangle
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to questions from the 2011 Exam
Question 22 – Model Solution 1
22a. (1.25 points) Construct the cumulative accident year reported loss development triangle as of 12/31/2010.
22b. (0.5 point) Describe a situation in which it is preferable to use accident year data for estimating unpaid
claims rather than report year data.
22c. (0.5 point) Describe a situation in which it is preferable to use report year data for estimating unpaid claims
rather than policy year data.
Incremental Paid Loss (as of months)
AY
12
24
36
8
375
355
305=230+75
9
520
400=150+200+50
10
545=45+500
Case reserve (as of months)
AY
12
24
36
8
690
575
295=270+25
9
580
200=100+100
10
555=55+500
Cumulative Rpt Loss (as of mths) = Cum. Paid loss + Case reserve
AY
12
24
36
8
1065
1305
1330 = 375+355+305+295
9
1100
1120=520+400+200
10
1100=545+555

b. AY data is widely used to estimate unpaid claims so there are many industry benchmarks and data
based on AY aggregation. When the actuary wants to use such benchmarks or industry data, then it is
preferable to use internal AY data as well.
c. When there is a severe change in legal or social climate that the average severity tracks more with the
report date rather than the occurrence/accident date of the policy.
Question 22 – Model Solution 2
Initial comment on part a. The cumulative AY reported loss development triangle as of 12/31/2010 is
correct, but no supporting calculations were given in this model solution. As a general rule, it’s important to
shown CAS examiners some of the calculations you performed to arrive at a triangle of values shown below.
a.
AY
12
24
36
2008
2009
2010

1065
1100
1100

1305
1120

1330

b. When you want to estimate IBNR – this is impossible (or at least very difficult) with report year data,
since unpaid claims for a report year are IBNER, not pure IBNR.
c. When you are modeling claims made policies, it makes sense to use report year, since CM policies
operate on report years and have no IBNR at year end.

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Chapter 6 – Development Triangle as a Diagnostic Tool
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Sec
1
2
3
4
5
6
7
8
9
10
11
1

Description
Introduction
Detailed Example – Background Information
Premium History
The Reported and Paid Claim Triangles
The Ratio of Paid to Reported Claims
The Ratio of Paid Claims to On-Level Earned Premiums
Claim Count Triangles
The Ratio of Closed to Reported Claim Counts
Average Claims
Summary Comments for XYZ Insurer
Conclusion

Pages
63
63 – 64
64 – 65
65 - 67
68 - 69
69
69 - 71
71 - 72
72 - 76
76
76 - 77

Introduction

The main topics discussed in previous chapters:
 Chapter 3 described types of data and how data is organized.
 Chapter 4 discussed the importance of meeting with “management” and understanding the
insurer’s internal and external environments.
 Chapter 5 demonstrated the construction of development triangles.
In this chapter, development triangles are used to better understand how changes in an insurer's
operations and the external environment can influence the claims data.

Detailed Example – Background Information

63 – 64

Company: XYZ Insurer
Experience: Private passenger automobile bodily injury liability experience in a single state over the 2002
to 2008 experience period.
Purpose: To demonstrate how to use development triangles for diagnostic review.
Goal: To teach you how to look at relationships and how to develop your own observations and questions.
Management Disclosures:
 The strength of current case outstanding is much greater than in prior years.
 New information systems have been implemented in the past three years for the purpose of
speeding up the claims reporting and settlement processes.
 Significant changes to the automobile insurance product in this geographic region.
1. Major tort reforms were implemented in 2006 resulting in caps on awards as well as pricing
restrictions and mandated rate level changes for all insurers operating in the region.
2. Management decided to reduce its presence in this market as a result.
Review Goals and Expectations:
 To determine if the effect of the changes implemented by management in the claims data can be observed.
 Expect that the review will lead to further questions and discussions with management.
To determine what types of data and which techniques will be most appropriate to estimate unpaid claims
for XYZ Insurer.

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Chapter 6 – Development Triangle as a Diagnostic Tool
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
3

Premium History

64 – 65

Given: XYZ Insurer provided the earned premium and rate level changes by year.
Compute: Cumulative average rate level and annual change in exposures from year to year.

Table 1 – Summary of Earned Premium and Rate Changes
Calendar
Year
2002
2003
2004
2005
2006
2007
2008

Earned
Rate
Premiums Changes
($000)
61,183
69,175
5.00%
99,322
7.50%
138,151
15.00%
107,578
10.00%
62,438 -20.00%
47,797 -20.00%

Cumulative
Average
Rate Level
0.00%
5.00%
12.90%
29.80%
42.80%
14.20%
-8.60%

Annual
Exposure
Change
7.70%
33.60%
21.00%
-29.20%
-27.50%
-4.30%

The average rate level is calculated by successive multiplication of annual rate changes.
For 2004, the cumulative average rate level is 12.9% = {[(1.00 + 5.0%) x (1.00 + 7.5%)] – 1.001
The annual exposure change is equal to the annual change in earned premiums divided by the rate change in the year.
For 2008, the annual exposure change is -4.3% = {[(47,797 / 62,438) / (1 –20.0%)] – 1.001
Assume that the rate changes in the above table represent the average earned rate level for the year.

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Chapter 6 – Development Triangle as a Diagnostic Tool
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Table 2 - Reported Claim Development Triangle
Accident
Reported Claims ($000) as of (months)
24
36
48
60
Year
12
2002
12,811
20,370
26,656
37,667
44,414
2003
9,651
16,995
30,354
40,594
44,231
2004
16,995
40,180
58,866
71,707
70,288
2005
28,674
47,432
70,340
70,655
2006
27,066
46,783
48,804
2007
19,477
31,732
2008
18,632

72
48,701
44,373

Table 3 - Paid Claim Development Triangle
Accident
Paid Claims ($000) as of (months)
24
36
48
60
72
Year
12
2002
2,318
7,932
13,822
22,095
31,945
40,629
2003
1,743
6,240
12,683
22,892
34,505
39,320
2004
2,221
9,898
25,950
43,439
52,811
2005
3,043
12,219
27,073
40,026
2006
3,531
11,778
22,819
2007
3,529
11,865
2008
3,409

84
48,169

84
44,437

Analysis:
 Look down the columns at the experience of different AYs at the same age of development.
 In a stable environment, one will see stability in the claim experience down each column.
Two diagnostic triangles:
1. The ratio of reported claims to earned premium (a.k.a. the reported claim ratio) and
2. The ratio of reported claims to on-level earned premium.
Calculate the on-level premium using the average rate level changes by year and restating the
earned premium for each year as if it was written at the 2008 rate level.

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Chapter 6 – Development Triangle as a Diagnostic Tool
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Accident
Year
200
200
200
200
200
200
200

Table 4 - Ratio of Reported Claims to Earned Premium
Ratio of Reported Claims to Earned Premium as of (months)
12
24
36
48
60
72
0.20
0.33
0.43
0.61
0.72
0.79
0.14
0.24
0.43
0.58
0.63
0.64
0.17
0.40
0.59
0.72
0.70
0.20
0.34
0.50
0.51
0.25
0.43
0.45
0.31
0.50
0.39

84
0.78

Table 5 - Ratio of Reported Claims to On-Level Earned Premium
Accident Ratio of Reported Claims to On-Level Earned Premium as of (months)
12
24
36
48
60
72
84
Year
200
0.22
0.36
0.47
0.67
0.79
0.87
0.86
200
0.16
0.28
0.50
0.67
0.73
0.73
200
0.21
0.50
0.73
0.89
0.87
200
0.29
0.48
0.72
0.72
200
0.39
0.67
0.70
200
0.39
0.63
200
0.39

Questions/Observations:
1a. For AY 2003, why are reported claims so low after 12 and 24 months of development?
 Per Table 1, the insurer had a 5% higher rate level in 2003 than 2002.
 It appears that the insurer experienced an exposure growth of approximately 8% in 2003
([(($69,175 / 1.05) / $61,183) - 1.00]).
 Knowing the insurer actually increased its exposure base, a 25% drop in reported claims for
2003 after 12 months is surprising.
1b. What led to the lower level of reported claims for the first 24 months?
Was there a change in systems?
Were paid claims or case outstanding driving the decrease in reported claims?
The paid claim triangle for AY 2003 shows that paid claims are also down at 12 and 24 months of
development (roughly of the same magnitude as for the reported claims).
2. What happened in AY 2004, at and after the 24-month valuation?
 While EP are up 44% over 2002 and 34% over 2003 (after adjustment for rate changes), reported
claims for 2004 after 24 months of development are up by 97% [($40,180 / $20,370) - 1.00] over
2002 and 136% [($40,180 / $16,995) - 1.00] over 2003.
 Are large claims or more claim counts or both driving the increase?
 Was there a change in case outstanding adequacy that had an effect on the 12/31/05 valuation?

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Chapter 6 – Development Triangle as a Diagnostic Tool
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Questions/Observations (continued):
3. What happened in AYs 2005 and 2006 to drive reported claims up so much at 12 months of development?
 One possible answer: Higher EP volume for these two years
 However at the 12-month valuation, reported claims are again increasing at a rate that is greater than the
increase in exposures and our knowledge of the inflationary environment.
Compare reported claims between AYs 2004 and 2005:
[(AY2005 / AY2004) - 1.00] = [($28,674 / $16,995) - 1.00] = 69% (greater than the increase in
exposures between these years, which is 21%)
Compare reported claims between accident years 2004 and 2006:
[(AY2006/ AY2004) - 1.00] = [($27,066 / $16,995) - 1.00] = 59% (is greater than the change in exposures
between these years, which is actually a decrease of 14%).
4. Looking down the 24-month column, there are:
 Large volumes of reported claims for AYs 2004 - 2006 and larger volumes of paid claims for
2004, 2005 and 2006.
 At 24 months, AY 2007 reported claims are lower than the preceding three accident years.
Could the lower claims in 2007 be a result of the tort reforms introduced during 2006?
5. For AY 2006, the insurer experienced a significant reduction in exposures during the year.
 EP dropped from $138,151 in 2005 to $107,578 even with a 10% rate increase (indicating a drop
in exposures of almost 30%). However, reported claims:
 After 12 months of development differ from 2005 by less than 6% [($27,066 / $28,674) — 1.00] and
 At 24 months of development by less than 2% [($46,783 / $47,432) — 1.00].
6. For AYs 2007 and 2008, reported claims are significantly lower than for 2005 and 2006 though the
claim ratios are not.
Determine the change in exposures based on the given premium information:
 While exposures were reduced 30% during 2007 (from 2006), the change in earned premiums
between 2007 and 2008 was primarily due to the rate change and not due to changes in
exposure volume.
 Reported claim volume at 12 months for AYs 2007 and 2008 is consistent with EP.
At this point, analyze additional development triangles to look for answers to some of the questions raised
in this initial review of the claims data.

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Chapter 6 – Development Triangle as a Diagnostic Tool
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
5

The Ratio of Paid to Reported Claims

68 - 69

Often, reported and paid claim development triangles are the only triangles available to the actuary.
 The ratio of paid to reported helps determine whether there might have been changes in case
outstanding adequacy or in settlement patterns.
 Determine whether there are changes in paid claims (i.e. the numerator) occurring or whether
changes in case outstanding, which impact reported claims (i.e. the denominator), taking place
Note: Changes in the ratio of paid-to-reported claims may be taking place, but cannot be
observed, because offsetting changes in both claim settlement practices and the adequacy
of case outstanding can result in no change to the ratio of paid-to-reported claims.
Recall that claims management believes that:
 New claims settlement practices resulted in a speed-up in claims closure.
Thus, expect paid claims to be increasing along the latest diagonals relative to prior years.
 New policies related to case outstanding are resulting in stronger unpaid case than in prior years.
Thus, reported claims should also be increasing along the latest diagonals of the triangle.
Therefore, the ratio of paid-to-reported claims may be unchanged along the latest diagonals when compared
with prior years' diagonals.

Accident
Year
2002
2003
2004
2005
2006
2007
2008

Table 6 - Ratio of Paid Claims-to-Reported Claims
Ratio of Paid Claims-to-Reported Claims as of (months)
12
24
36
48
60
72
0.181
0.389
0.519
0.587
0.719
0.834
0.181
0.367
0.418
0.564
0.780
0.886
0.131
0.246
0.441
0.606
0.751
0.106
0.258
0.385
0.567
0.130
0.252
0.468
0.181
0.374
0.183

84
0.923

Look down each column to compare the experience from AY to AY.
 It is difficult to discern changes in this ratio.
 Recall that a downward trend in the ratio of paid-to-reported claims could be the result of
decreasing paid claims or of increasing case outstanding adequacy.
However, management states that the rate of claims settlement has increased. Thus:
 Is the change in case outstanding adequacy masking the changes in the settlement process?
 Is the type of claims being reported changing (since different types of claims have different
settlement and reporting characteristics, and this could affect on both paid and reported claims)?

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6

The Ratio of Paid Claims to On-Level Earned Premiums

This diagnostic triangle can help to determine whether there was a speedup in claims payment or
possibly deterioration in underwriting results.

Table 7 - Ratio of Cumulative Paid Claims to On-Level Earned Premium
Accident

Ratio of Cumulative Paid Claims to On-Level Earned Premium as of (months)

Year
2002
2003
2004
2005
2006
2007
2008

12
0.041
0.029
0.028
0.031
0.051
0.071
0.071

24
0.142
0.104
0.123
0.126
0.171
0.238

36
0.247
0.211
0.323
0.278
0.331

48
0.395
0.38
0.54
0.412

60
0.571
0.573
0.657

72
0.727
0.653

84
0.795

Observations/Questions:
 There seems to be evidence of a possible speed-up in payments, particularly at 12 and 24 months.
 Has there been a shift in the type of claim settled at each age?
Additional data (reported and closed claim counts development diagnostic triangles) are needed for further review.

Claim Count Triangles

69 - 71

Reported and Closed claim counts triangles are reviewed next.

Accident
Year
2002
2003
2004
2005
2006
2007
2008

Table 8--Reported Claim Count Development Triangle
Reported Claim Counts as of
12
24
36
48
60
72
1,342
1,514
1,548
1,557
1,549
1,552
1,373
1,616
1,630
1,626
1,629
1,629
1,932
2,168
2,234
2,249
2,258
2,067
2,293
2,367
2,390
1,473
1,645
1,657
1,192
1,264
1,036

Accident
Year
2002
2003
2004
2005
2006
2007
2008

Table 9 — Closed Claim Count Development Triangle
Closed Claim Counts as of
12
24
36
48
60
72
203
607
841
1,089
1,327
1,464
181
614
941
1,263
1,507
1,568
235
848
1,442
1,852
2,029
295
1,119
1,664
1,946
307
906
1,201
329
791
276

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1,554

84
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Chapter 6 – Development Triangle as a Diagnostic Tool
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
The importance in understanding claim count development triangles (i.e. what types of data are contained
in such triangles?).
1. How does the insurer treat reopened claims?
Are they coded as a new claim or is a previously closed claim re-opened?
If treated as a reopened claim, there could potentially be a decrease across a row in the closed claim
count development triangle.
2. Does the insurer include claims closed with no payment (CNP) in the reported and closed claim count
triangles?
3. How are claims classified that have only expense payments and no claim payment?
XYZ Insurer indicated that:
 Closed claim count development data excludes CNP claim counts.
 Reported claim count development triangles also excludes CNP counts.
Observations and Questions:
1. At 12 months:
 Reported claim counts experienced an increase of 40% [(1,932/1,373) - 1.00] and
 Closed claim counts had an increase of 30% [(235/181) - 1.00] between AYs 2003 and 2004.
 However, a 76% increase in reported claims is observed.
At 24 months for AY 2005, increase in claim counts [(2,293/2,168) - 1.00 = 5.8%] are not as
significant as the increases in reported claims [($47,432/$40,180) - 1.00 = 18.0%].
Why are claims increasing so much more than the number of claims?
Could large claims be driving the increases?
2. Reported claim counts for AYs 2004 and 2005 area at the highest values at all ages (and this is
consistent with the experience shown in the reported claim triangle).
However, we do not observe a similar increase in the closed claim count triangle where 2006 and
2007 are highest at 12 months.
At 24 months, the highest closed claim count values are for AY 2005 and 2006. Are the higher closed
claim counts due to the new systems implemented at the insurer?
3. The decrease in reported claim counts for 2006 and 2007 is consistent with the decrease in exposures
for these years.
However, there is not a similar decrease in closed claim counts. Is this due to the speed-up in claims
settlement processes that management discussed?
4. Finally, for AY 2008, reported and closed claim counts are lower than we would expect given reported
claims, paid claims, and the steady-state of exposures between 2007 and 2008.
Therefore, why are the number of claims down for the latest year?

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8

The Ratio of Closed to Reported Claim Counts

71 - 72

If changes in the settlement rate of claims are suspected, reviewing the ratio of paid-to-reported claims, is
an important diagnostic tool to use.
Factors affecting the reporting and closing of claims include:
* A change in guidelines on how claims are established
* A decrease in the statute of limitations duration (from a major tort reform action)
* Delegating higher claim settlement limits to a TPA
* Restructuring of the claim field offices (e.g. merging or adding of new offices).
* Introducing a new claim call center
 A change resulting in a temporary increase in closing patterns occurs when a claim department makes
an extra effort to get the backlog as low as possible before making a transition to a new system.
 A speed-up due to faster processing occurs when the new system leads to a slowdown in closing, due
to a learning curve necessary before the new system is fully operational.
Management at XYZ Insurer stated that claims are now settling much more quickly, and the new system
is having an affect on the entire portfolio of outstanding claims (not just claims from the latest AY).
As a result, we would then expect to see greater closed-to-reported claim counts ratios for the latest
diagonals than for prior years.

Accident
Year
2002
2003
2004
2005
2006
2007
2008

Table 10 - Ratio of Closed-to-Reported Claim Counts
Ratio of Closed-to-Reported Claim Counts as of
12
24
36
48
60
72
0.151
0.401
0.543
0.699
0.857
0.943
0.132
0.38
0.577
0.777
0.925
0.963
0.122
0.391
0.645
0.823
0.899
0.143
0.488
0.703
0.814
0.208
0.551
0.725
0.276
0.626
0.266

84
0.98

Change is clearly evident in this diagnostic triangle.
 For 2002 - 2005 at 12 months of development, closed-to-reported claim counts was roughly 0.14.
 For 2006 – 2008 at 12 months, the ratio is in excess of 0.20; and for the latest year it is 0.266.
 The same type of increases for the 24-month through 48-month development periods are evident.
The experience of closed and reported claim counts is consistent with management's emphasis on
settling claims faster.
Now, the actuary must consider the consequences of such a change.
 Less complicated and less expensive claims close the quickest. More complicated claims
(involving litigation and expert witnesses), take longer to close.
 If emphasis is on closing small claims quickly, there will likely be a shift in the type of claims
closed or open at any particular age in the claim development triangle.
This is discussed in the next section on average claims.

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Chapter 6 – Development Triangle as a Diagnostic Tool
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
9

Average Claims

72 - 76

Reported and paid claim development triangles as well as the reported and closed claim count triangles
are used to calculate various average values.

Average Value
Average reported claim
Average paid claim
Average case outstanding

Table 11 - Definitions of Average Values
Definition
Reported claim triangle / reported claim count triangle
Paid claim triangle / closed claim count triangle
Reported claim triangle - paid claim triangle
Reported claim count triangle - closed claim count triangle

Two important issues related to average values:
1. Have a clear understanding of the definition of closed and reported claim counts.
Some insurers include claims with no payment (CNP) in the definition of closed claim counts and some
include claims with no case outstanding and no payments in the definition of reported claim counts.
Including CNPs in closed claim count statistics or claims with no case outstanding or payments in
reported claim counts produces a much lower average value.
A change in the definition of claim counts can impact the results of diagnostic analyses using claim
counts and on estimation techniques that rely on the number of claims.
2. Large claims. Both the presence and absence of such claims can distort average claims. Methods to
deal with large claims include:
a. Removing large claims from the database before conducting both ratio and average value calculations
and handling the unpaid large claim estimate separately.
b. Use development triangles using limited claims (e.g. claims can be limited to $500,000 or $1 million
per occurrence in the reported and paid claim development triangles). See previous discussion of
determining a large claims threshold in Chapter 3.
Two other aspects affecting average values are policy deductibles and retentions.
For XYZ Insurer:
 Closed claim counts exclude claims closed without any payment.
 Reported claim counts exclude claims in which there are no case outstanding and no payments.
 Paid claims include partial payments as well as payments on closed claims. Thus, the average
paid claim triangle will be a combination of payments on settled claims as well as payments on
claims that are still open.
The average reported claim triangle is often used to detect changes in case outstanding adequacy.
It is not quite as valuable as the average case outstanding triangle since reported claims include both paid
claims and unpaid case reserves, and changes in paid claims can mask changes in case outstanding
adequacy.
The average reported claim triangle may be all that is available for diagnostic purposes.

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Chapter 6 – Development Triangle as a Diagnostic Tool
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We expect to see changes down the columns limited to inflationary forces only, however the changes
observed below are greater than the annual inflation (assumed to be 5% for this region's auto BI liability).
Are the increases are due to greater levels of payments or stronger case outstanding?

Accident
Year
2002
2003
2004
2005
2006
2007
2008

Table 12 - Average Reported Claim Development Triangle
Average Reported Claims as of (months)
12
24
36
48
60
72
9,546
13,455
17,219
24,192
28,673
31,379
7,029
10,517
18,622
24,966
27,152
27,239
8,796
18,533
26,350
31,884
31,129
13,872
20,686
29,717
29,563
18,375
28,440
29,453
16,340
25,104
17,985

84
30,997

The average paid claim triangle:
A mismatch exists in the average paid claim triangle since the numerator (cumulative paid claims) includes
partial claim payments and the denominator (closed claim counts) represents only claims with final settlement.
Consider this limitation when drawing any conclusions from this particular diagnostic triangle.
Notice that in the average paid triangle below, the average values along the latest diagonal are generally
the highest value in each column (particularly at 12 to 36 months). The average paid claim triangle appears
relatively stable for ages 48 and older.
The next important question to ask is whether or not there has been a change in the type of claim that is
being closed at these particular ages (since this can affect the actuary's selection of estimation techniques
and claim projection factors).

Accident
Year
2002
2003
2004
2005
2006
2007
2008

Table 13 - Average Paid Claim Development Triangle
Average Paid Claims as of
12
24
36
48
60
72
11,417
13,067
16,436
20,290
24,073
27,752
9,631
10,163
13,478
18,125
22,896
25,077
9,452
11,673
17,996
23,455
26,028
10,315
10,920
16,270
20,569
11,502
13,000
19,000
10,726
15,000
12,351

84
29,178

The change in average paid claims only at 12, 24, and 36 months is consistent with insurers having the
greatest control on closure rates of the less complicated and less expensive claims.
Finally, we review the average case outstanding (a.k.a. average open claim amount) triangle, since it is
one of the most important diagnostic tools for testing changes in case outstanding adequacy.
A decreasing pattern down the column is an indicator of potential weakening in the case outstanding,
An increasing pattern down the column is an indicator of possible strengthening in the case outstanding.
Questions regarding case outstanding adequacy:
 Has there been a change in case outstanding practices, policies, philosophy, staff, or senior
management of the claims department?
 Has there been changes in the mix of business in the portfolio that have nothing to do with
changes in case outstanding strength?

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Looking down the columns, we see that the average case outstanding is generally increasing by more
than the 5% inflation we expect.

Accident
Year
2002
2003
2004
2005
2006
2007
2008

Table 14 — Average Case Outstanding Development Triangle
Average Case Outstanding as of (months)
12
24
36
48
60
72
84
9,213
13,714
18,151
33,273
56,167
91,729
120,366
6,634
10,733
25,647
48,766
79,718
82,826
8,706
22,941
41,561
71,204
76,320
14,464
29,994
61,547
68,983
20,185
47,368
56,984
18,480
42,002
20,031

For 2002 through 2004, the average case outstanding at 12 months of development was less than $10,000.
For 2006 and 2008 at 12 months, the average case outstanding is greater than $20,000.
We see similar increases at 24 and 36 months.
We also observe increasing values of average case outstanding at 48 and 60 months.
We know that management increasing case outstanding strength is a priority, and a review of the average case
outstanding shows increasing average values for outstanding claims.
However, what affect, if any, is the change in claims settlement having on the average case outstanding?
 If smaller claims are settling more quickly, only the more complex/expensive claims are left.
 This, in and of itself, would lead to an increase down the columns in the average case outstanding.
Thus, it is very important for the actuary to determine how much of the increase in the average case
outstanding is truly due to a:
 systemic change in the overall level of case outstanding adequacy
 different mix of claims.

Summary Comments for XYZ Insurer

Every claim development diagnostic that was reviewed shows evidence of the changes noted by
management.
Now the actuary must determine how to incorporate all this information in the development of an unpaid
claim estimate to be carried on XYZ Insurer's financial statements.
The changing environment will have an effect on the actuary's choice of estimation techniques, types of
data, and actuarial factors within the techniques.

Conclusion

76 - 77

The development triangle is an excellent tool for exploring the data. It is important for the actuary to take the
information obtained during meetings with management and then seek confirmation in the actual claims
experience behavior.
Discussions the actuary has with those involved with the insurer's operations (especially claims operations)
must be ongoing, since understand data is a complex process that requires input of many people.

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Chapter 6 – Development Triangle as a Diagnostic Tool
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Questions from the 2009 Exam
8. (2 points) Given the following information for a single line of business for an insurance carrier:
Cumulative Paid Loss
24 Months
36 Months

Accident Year

12 Months

2005
2006
2007
2008

$4,531,950
4,871,246
5,294,951
6,675,164

Accident
Year

Premium
(On-Level)

Exposures

Average
Premium

2005
2006
2007
2008

$11,641,265
12,726,119
13,538,710
14,905,384

18,384
19,333
20,871
22,391

$633
658
649
666

5,919,356
6,312,582
6,962,001

6,511,844
6,894,515

48 Months
6,768,106

Using statistics drawn from the above data, discuss one reason why it is not appropriate to use the paid
development method to estimate the ultimate losses for accident year 2008.
Questions from the 2011 Exam
23. (1 point) Given the following data as of December 31, 2010:
Reported Claims (000s)
Accident
Year
2007
2008
2009
2010

12 Months
$500
$448
$312
$426

24 Months
$554
$470
$346

36 Months
$586
$512

48 Months
$592

Paid Claims (000s)
Accident
Year
2007
2008
2009
2010

12 Months
$85
$81
$59
$85

24 Months
$200
$225
$175

36 Months
$500
$472

48 Months
$570

Fully discuss whether the data indicates a speed-up in claim closure.

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Chapter 6 – Development Triangle as a Diagnostic Tool
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Questions from the 2012 Exam
17. (1.5 points) Given the following:
Age in
Incremental
Months
% Paid Claims
0-12
50.0%
12-24
25.0%
24-36
15.0%
36-48
2.5%
•

Incremental
% Reported Claims
40.0%
30.0%
20.0%
10.0%

Assume all outstanding claims are reported and paid by the 60th month.

a. (1 point) Calculate the paid age-to-age factors for ages 12-24, 24-36, 36-48, and 48-60.
b. (0.5 point) Provide two observations that may indicate a problem with the data.

23. (1.5 points) An insurance company faces the following scenarios:
•
•

For property claims, a new claims processing system is implemented that will result in claims
closing faster.
For liability claims, a tort reform change is passed that will reduce the statute of limitations on
reporting a claim.

a. (1 point) For each scenario above, explain the effect on the average case outstanding triangle.
b. (0.5 point) Briefly describe two additional scenarios that could cause a change in the ratio of
closed to reported claim counts.

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Chapter 6 – Development Triangle as a Diagnostic Tool
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to questions from the 2009 Exam
8. Using statistics drawn from the above data, discuss one reason why it is not appropriate to use the
paid development method to estimate the ultimate losses for accident year 2008.
Question 8 - Model Solution 1
Ratio of Cumulative Paid Loss to On-level Premium:
AY
12
24
2005
0.3893
0.5085
2006
0.3828
0.496
2007
0.3911
0.514
2008
0.4478
.4478 / .3911 = +14.5%
When we look at the ratio of cumulative paid claims to on-level premium at 12 months of development, we
see that there is a large increase (+14.5%) in the ratio from 2007 to 2008.
This may be caused by a speed-up in the settlement of the claims and thus, the paid development method
will overstate the ultimate losses for AY 2008 because it will overstate the LDF.
Paid claim development method assumes that past development is indicative of the future development and
this will not be the case because there is an increase in the settlement rate from 2007 to 2008.
Question 8 - Model Solution 2
Exposure Trend
18,384
19,333
5.16%
20,871
8%
22,391
7.3%

Paid @ 12 mth
4,531,950
4,871,246 7.5%
5,294,951 8.7%
6,675,164 26%

We can see that in AY 2008, paid claim increased tremendously (26%) compared to the exposure growth
increase of (7.3%). We also know that the Avg prem doesn’t change a lot, so the big paid increase is caused by
settlement rate speed up.
If we use LDFs derived from past years and applied it to the most recent year (AY 2008 paid losses are
too high relative to past paid losses at 12 months), this will overestimate AY 2008 ultimate losses.

Solutions to questions from the 2011 Exam
23. Fully discuss whether the data indicates a speed-up in claim closure.
Question 23 – Model Solution
Look at disposal rate or ratio of paid to reported.
Cumulative paid losses /cumulative reported losses.
AY
07
08
09
10

12
0.170
0.181
0.189
0.200

24
0.361
0.479
0.506

36
0.853
0.922

48
0.963

Looking down each column, the ratios indicate that there may be a speed up in claims closure rate,
however the increasing ratio may also be due to a decrease in case reserving adequacy.

Exam 5, V2

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Chapter 6 – Development Triangle as a Diagnostic Tool
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to questions from the 2012 Exam
17a. (1 point) Calculate the paid age-to-age factors for ages 12-24, 24-36, 36-48, and 48-60.
17b. (0.5 point) Provide two observations that may indicate a problem with the data.
Question 17 – Model Solution 1 (Exam 5B Question 2)
Age
Cum. Paid %
Cum. Reported %
0-12
50%
40%
12-24
75%
70%
24-36
90%
90%
36-48
92.5%
100%
48-60
100%
100%
Example: 92.5% = 50% + 25% + 15% + 2.5%
Age

paid CDF

Reported CDF

Incr. paid
LDF

LDF

12-ult.
24-ult.
36-ult.
48-ult.
60-ult.

2
2.5
1.5
1.749
1.333
1.429
1.1998
1.286
1.111
1.111
1.028
1.111
1.081
1
1.081
1
1
1
1
1
2=1/0.5;
1.5 = 2/1.333
b) (i) Reported CDFs are usually less than paid CDF. Here, at ages 12 and 24, Reported CDF are higher.
(ii) There should be a smooth decrease of incremental LDF across dev. period.
Here, paid LDF 36-48 is 1.028 and 48-60 is 1.081.
Question 17 – Model Solution 2 (Exam 5B Question 2)
Age
12
24
36
48
60

% paid
50%
75%
90%
92.5%
100%

1 /% paid = CDF
2
1.333
1.111
1.081
1.000

Age to Age
12-24
24-36
36-48
48-60

Age to Age factors
2/1.333=1.5
1.333/1.111=1.2
1.111/1.081=1.028
1.081/1.00=1.081
60-ult= 1.00
b. 1. After 1 year we see that half of claims are paid, but only 40% are reported. This implies
negative case outstanding, which doesn’t make much sense.
2. The 48-60 age-to-age factor is larger than the 36-48 age-to-age factor. Generally, age-toage factors should steadily decrease as the experience matures.
Examiner’s Comments
a. Most candidates received full credit on this part.
b. Most candidates were able to identify one observation but not both. Some candidates
restated the same observation in a slightly different manner.

Exam 5, V2

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Chapter 6 – Development Triangle as a Diagnostic Tool
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to questions from the 2012 Exam
23a. (1 point) For each scenario above, explain the effect on the average case outstanding triangle.
23b. (0.5 point) Briefly describe two additional scenarios that could cause a change in the ratio of closed
to reported claim counts.
Question 23 – Model Solution 1 (Exam 5B Question 7)
a. Property- We will likely see an increase in average case outstanding. Often an increase in settlement
rate means small claims are being closed quicker. A higher percentage of open claims will likely be
large claims.
Liability- This will result in a speed up in reporting rate as people need to file claims sooner. Its effect
on average case is difficult to tell. It could lower average case at early maturities if we see a lot of
claims filed that we believe will result in no payment. When statute of limitations decreases, we may
see more filing claim first ask questions later behavior.
b. 1. Change in claims department strategy to fight more claims in court will result is a decrease of
closed to reported claim counts.
2. Increase in average case load per claims adjuster due to staff cuts could also result in decrease of
closed to reported ratio.

Question 23 – Model Solution 2 (Exam 5B Question 7)
a. Claims closing faster: both case reserves and open counts should be lower at each age, since as
payments are made, claims close and case is reduced. As such it is unclear how the ratio of these two
will react to the denominator and numerator changing. For example, if it is small claims being closed
more quickly, then average case will go up, and vice versa.
Tort Reform: we would see an influx of claims reported as people try to get their claims in before the
new cap on reporting date. This would increase open counts and case O/S. If these new claims have
higher severity than the old average claim, we would see average case rise as the reserves put up
would outpace the number of new open counts in the denominator.
b1. CAT hits an insurer creating a backlog of reported claims -> ratio goes down
b2. Focus on closing small claims quickly -> ratio goes up.
Examiner’s Comments
a. Any reasonable explanation was accepted, including explanations of an increase, decrease, or no
change to average case outstanding for either scenario.
Many candidates did not “explain” the effect to average case outstanding, and instead limited their
answer to either stating an effect or only explaining what would happen to case outstanding (not
average case outstanding). These candidates received no credit.
Candidates often confused tort reform vs. statue of limitations and assumed there would be a
reduction in severity rather than the claim reporting impact due to the change in statute of limitations.
b. Most candidates offered reasonable scenarios which were accepted for full credit. Explanations were
not required for full credit.
Common mistakes not receiving credit include: stating “changes in settlement rates” without identifying
a scenario, offering scenarios that affect “case reserve adequacy” instead of claim reporting
or settlement rates, and identifying the same scenario from Part a) and not offering a new scenario.

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Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Sec
1
2
3
4
5
6
7
8
9
10

Description
Introduction
Key Assumptions
Common Uses of the Development Technique
Mechanics of the Development Technique
Unpaid Claim Estimate Based on the Development Technique
Reporting and Payment Patterns
Observations and Common Relationships
When the Development Technique Works and When It Does Not
XYZ Insurer
Influence of a Changing Environment on the Claim Development
Technique

Pages
84
84
84 - 85
85 - 92
92 - 93
93 - 94
95
95 - 97
97 - 98
98 -104

Introduction

In this chapter, estimates of ultimate claims and unpaid claims based on the reported and paid claim
development methods (a.k.a. the chain ladder technique) are developed.

Key Assumptions

The underlying assumption in the development technique is that:
 claims recorded to date will continue to develop in a similar manner in the future (i.e. the past is
indicative of the future).
 the relative change in a given year's claims from one evaluation point to the next is similar to the
relative change in prior years' claims at similar evaluation points.
Other key assumptions of the development method include:
 consistent claim processing,
 a stable mix of types of claims,
 stable policy limits, and
 stable reinsurance (or excess insurance) retention limits throughout the experience period.

Common Uses of the Development Technique

84 - 85

The development technique can be applied to:
 paid and reported claims as well as number of claims.
 all lines of insurance including short-tail lines and long-tail lines.
To use the development method, data is organized into different time intervals, including:
* Accident year; * Policy year; * Underwriting year; * Report year;
* Fiscal year (e.g. for a self-insured public entity with a fiscal year ending March 31, the actuary will likely
organize the claim development data by April 1 to March 31 fiscal year).
This technique can be applied to monthly, quarterly, and semiannual and annual data.

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Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
4

Mechanics of the Development Technique

85 - 92

The development method consists of seven steps:
Step 1 — Compile claims data in a development triangle
Step 2 — Calculate age-to-age factors
Step 3 — Calculate averages of the age-to-age factors
Step 4 — Select claim development factors
Step 5 — Select tail factor
Step 6 — Calculate cumulative claim development factors
Step 7 — Project ultimate claims
To demonstrate the seven steps, industry-aggregated accident year claim development data for U.S. private
passenger automobile insurance (labeled "U.S. Industry Auto") is used.
Step 1 — Compile Claims Data in a Development Triangle
Exhibit I, Sheets 1 and 2: consists of:
 cumulative reported and paid claim development triangles, respectively.
Part 1 of each exhibit is the data triangle for AYs 1998 - 2007.
The 10 diagonals in each triangle have annual valuation dates of 12/31/1998 – 12/31/2007
 data net of reinsurance and includes the defense cost portion of claim adjustment expenses (a.k.a. DCC
for U.S. statutory accounting).
U.S. Industry Auto
Reported Claims($000)
PART 1 - Data Triangle
Accident
Year
12
1998
37,017,487
1999
38,954,484
2000
41,155,776
2001
42,394,069
2002
44,755,243
2003
45,163,102
2004
45,417,309
2005
46,360,869
2006
46,582,684
2007
48,853,563

Exhibit I
Sheet 1

24
43,169,009
46,045,718
49,371,478
50,584,112
52,971,643
52,497,731
52,640,322
53,790,061
54,641,339

PART 2 - Age-to-Age Factors
Accident
Year
12-24
24 - 36
1998
1.166
1.056
1999
1.182
1.062
2000
1.200
1.061
2001
1.193
1.062
2002
1.184
1.059

Exam 5, V2

Reported Claims as of (months)
36
48
45,568,919
46,784,558
48,882,924
50,219,672
52,358,476
53,780,322
53,704,296
55,150,118
56,102,312
57,703,851
55,468,551
57,015,411
55,553,673
56,976,657
56,786,410

36 - 48
1.027
1.027
1.027
1.027
1.029

60
47,337,318
50,729,292
54,303,086
55,895,583
58,363,564
57,565,344

Age-to-Age Factors
48 - 60
60 - 72
1.012
1.004
1.010
1.004
1.010
1.005
1.014
1.005
1.011
1.004

Page 76

72
47,533,264
50,926,779
54,582,950
56,156,727
58,592,712

84
47,634,419
51,069,285
54,742,188
56,299,562

96
47,689,655
51,163,540
54,837,929

72 - 84
1.002
1.003
1.003
1.003

84 - 96
1.001
1.002
1.002

96 - 108
1.001
1.000

108
120
47,724,678 47,742,304
51,185,767

108 - 120
1.000

To Ult

 2014 by All 10, Inc.

Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Step 2 — Calculate Age-to-Age Factors (a.k.a. report-to-report factors or link ratios)
To calculate the age-to-age factors for the 12-month-to-24-month period, divide the claims as of 24 months by the
claims as of 12 months.
Using the reported claims presented in Exhibit I, Sheet 1, calculate the following:
12-24 factor for accident year 1998:

reported claims at 24 months for accident year 1998
$43,169, 009
= = 1.166
reported claims at 12 months for accident year 1998 $37, 017, 487
36-48 factor for accident year 2002

reported claims at 48 months for accident year 2002 $57, 703,581
= = 1.029
reported claims at 36 months for accident year 2002 $56,102,312
Continue in the same manner down the columns and across the rows of the triangles.
Step 3 — Calculate Averages of the Age-to-Age Factors
The most common averages include:
* Simple (or arithmetic) average
* Medial average (average excluding high and low values)
* Volume-weighted average
* Geometric average (the nth root of the product of n historical age-to-age factors)
Shown In Part 3 of Exhibit I, Sheets 1 and 2, are:
* Simple averages for the latest five years and the latest three years
* Medial average for the latest five years excluding one high and one low value (medial latest 5x1)
* Volume-weighted averages for the latest five years and the latest three years
* Geometric average for the latest four years
PART 3 - Average Age-to-Age Factors
12-24
Simple Average
Latest 5
1.168
Latest 3
1.164
Medial Average.
Latest 5x1
1.165
Volume-weighted Average
Latest 5
1.168
Latest 3
1.164
Geometric Average
Latest 4
1.164

24 - 36

36 - 48

1.058
1.056

1.027
1.027

1.057

Averages
48 - 60

60 - 72

72 - 84

84 - 96

96 - 108

108 - 120

1.011
1.012

1.004
1.005

1.003
1.003

1.002
1.002

1.001
1.001

1.000
1.000

1.027

1.010

1.004

1.003

1.002

1.001

1.000

1.058
1.056

1.027
1.027

1.011
1.012

1.004
1.005

1.003
1.003

1.002
1.002

1.001
1.001

1.000
1.000

1.057

1.027

1.011

1.004

1.003

1.002

1.001

1.000

To Ult

Examples (simple average and medial average):
For reported claims, the 12-24 month simple average of the latest five factors is based on the average of the
12-24 month factors for AYs 2002 – 2006 = 1.168 = (1.184 + 1.162 + 1.159 + 1.160 + 1.173) / 5.
To calculate the 24-36 month medial average development factor of the latest 5x1, consider the 24-36 month
factors for AYs 2001 - 2005; we exclude the highest value (1.062 for accident year 2001) and the lowest value
(1.055 for accident year 2004) and take an average of the remaining three values.
The 24-36 month medial average of the latest 5x1 = 1.057 = (1.059 + 1.057 + 1.056) / 3).

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Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Examples (volume weighted and geometric):
The formula for this type of average uses the sum of the claims for the specific number of years divided by the sum
of the claims for the same years at the previous age.
The 36-48 month volume-weighted average of the latest three years = the sum of the reported claims for AYs
2002 - 2004 at 48 months ($57,703,851 + $57,015,411 + $56,976,657 = $171,695,919) divided by the sum of the
reported claims for AYs 2002 - 2004 as of 36 months ($56,102,312 + $55,468,551 + $55,553,673 = $167,124,536),
or 1.027.
The geometric average (a.k.a. geometric mean) for the latest four years is equal to the fourth root of the product of
the last four age-to-age factors.
.25
The geometric average for the latest four years at 12-24 months = (1.162 x 1.159 x 1.160 x 1.173) = 1.164.
.25
The geometric average for the latest four years at 48-60 months = (1.010 x 1.014 x 1.011 x 1.010) = 1.011.
Actuaries often rely on the most recent experience as this data reflects the effect of the latest changes in the
insurer's internal and external environments.
There is often a trade-off between stability (the number of experience periods included in the average values) and
responsiveness (where only the most recent experience periods are considered).
Step 4 — Select Claim Development Factors
The selected age-to-age factor (a.k.a. claim development factor or loss development factor) represents the growth
anticipated in the subsequent development interval.
Selections are based on a review of the historical claim development data, the age-to-age factors, the various
averages of the age-to-age factors, and a review of the prior year's claim development factor selections.
Benchmarks:
When the credibility of the insurer's historical experience is limited, there may be a need to supplement the
experience with benchmark data. Possible benchmark includes:
 experience from similar lines with similar claims handling practices within the insurer.
 claim development patterns from the insurance industry when comparable.
When using benchmarks, there may be significant differences between the line of business being analyzed and
the benchmark with regard to claims practices, policy coverages, underwriting, geographic mix, claim coding,
policyholder deductibles and/or limits, legal precedents, etc.
When selecting claim development factors, consider the following characteristics:
1. Smooth progression of individual age-to-age factors and average factors across development periods.
A steadily decreasing incremental development from valuation to valuation
2. Stability of age-to-age factors for the same development period.
A relatively small range of factors (small variance) within each development interval (i.e. down the columns).
3. Credibility of the experience.
Credibility is based on the volume and the homogeneity of the experience for a given AY and age.
Benchmark development factors from the insurance industry may be needed when credibility is lacking.
4. Changes in patterns.
May suggest changes in the internal operations or external environment.
5. Applicability of the historical experience.
Has the insurer’s book of business and insurer operations changed over time?
Have the effects of changes in external factors manifested themselves in the reported claims experience?

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Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
In Part 4 of Exhibit I, Sheets 1 and 2:
Actuarial judgment is used to choose selected factors after reviewing all of the age-to-age factors, the various
averages, and the prior year's selected factors.
The term "To Ult" (i.e. To Ultimate) is used to designate the tail factor (e.g. 120 months-to-ultimate).
Selections of development factors are subjective, differ from one actuary to another, and there is more than one
reasonable selection of age-to-age and tail factors.

Reported
Paid

12-24
1.164
1.702

24-36
1.056
1.186

36-48
1.027
1.091

Table 1 - Selected Age-to-Age Factors
48-60
60-72
72-84
84-96
1.012
1.005
1.003
1.002
1.044
1.019
1.009
1.005

96-108
1.001
1.002

108-120
1.000
1.002

120-ultimate
1.000
1.002

Step 5 — Select Tail Factor
If data is available, analyze development out to the point at which the development ceases (i.e. until the selected
development factors are equal to 1.000).
When development factors for the most mature development periods are still greater than 1.000, a tail factor is
needed to bring the claims from the latest observable development period to an ultimate value.
The tail factor is crucial as it:
 influences the unpaid claim estimate for all accident years (in the experience period) and
 can create a disproportionate leverage on the total estimated unpaid claims.
Approaches to select the tail factor:
1. Use industry benchmark development factors
2. Fit a curve to the selected or observed development factors to extrapolate the tail factors (exponential decay is a
common for curve fitting).
3. For paid development, when reported development is at ultimate, use reported-to-paid ratios at the latest
observed paid development period.
Step 6 – Calculate Cumulative Claim Development Factors (CDF)
Cumulative claim development factors (a.k.a. age-to-ultimate factors and claim development factors to ultimate):
 are calculated by successive multiplications beginning with the tail factor and the oldest age-to-age factor.
 projects the total growth over the remaining valuations.
Using the selected age-to-age factors from Step 4 and the tail factor in Step 5, calculate the following:
Reported CDF at 120 months = selected tail (120-ultimate) factor = 1.000
Reported CDF at 108 months = (selected tail factor) x (selected development factor 108-120 months)
= 1.000 x 1.000 = 1.000
Reported CDF at 96 months = (selected tail factor) x (selected development factor 108-120 months) x (selected
development factor 96-108 months)
= (CDF at 108 months) x (selected development factor 96-108 months) = 1.000 x 1.001 = 1.001
Continue in this manner until computing the Reported CDF at 12 months
= (CDF at 24 months) x (selected development factor 12-24 months) = 1.110 x 1.164 = 1.292
Table 2 summarizes the cumulative claim development factors based on the selected age-to-age factors, from
Exhibit I, Sheets 1 and 2.

Reported
Paid

Exam 5, V2

12
1.292
2.39

24
1.11
1.404

Table 2 — Cumulative Claim Development Factors
72
36
48
60
84
1.051
1.023
1.011
1.006
1.003
1.184
1.085
1.04
1.02
1.011

Page 79

96
1.001
1.006

108
1.000
1.004

120
1.000
1.002

 2014 by All 10, Inc.

Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Step 7 — Project Ultimate Claims
Ultimate claims equal the product of the latest valuation of claims (the amounts on the last diagonal of the claim
triangles) and the cumulative claim development factors.
Calculations are shown in Exhibit I, Sheet 3, an excerpt of which is shown below:
 Column (3) is the last diagonal of the reported claim development triangle in Exhibit I, Sheet 1, and Column
(4) is the last diagonal of the paid claim development triangle in Exhibit I, Sheet 2.
 Columns (5) and (6) are the cumulative claim development factors that are calculated in Step 5.
 Each cumulative claim development factor refers to a specific age.
Chapter 7 - Development Technique
U.S. Industry Auto
Projection of Ultimate Claims Using Reported and Paid Claims($000)

Accident
Year
(1)
1998
2007

Age of
Year
(2)
120
12

Claims at 12/31/07
CDF to Ultimate
Reported
Paid
Reported Paid
(3)
(4)
(5)
(6)
47,742,304 47,644,187
1.000
1.002
48,853,563 27,229,969
1.292
2.390

Exhibit I
Sheet 3

Projected Ultimate Claims
Using Dev. Method with
Reported
Paid
(7)= [(3) x (5)] (8)= [(4) x (6)]
47,742,304
47,739,475
63,118,803
65,079,626

Projected ultimate claims for accident year 1998
= (reported claims for 1998 as of 12/31/07) x (reported CDF at 120 months) = $47,742,304 x 1.000 = $47,742,304

Projected ultimate claims for accident year 2007
= (reported claims for 2007 as of 12/31/07) x (reported CDF at 12 months) = $48,853,563 x 1.292 = $63,118,803

Perform similar calculations for the projection of ultimate claims using the paid claim development technique.
Projected ultimate claims for accident year 2007
= (paid claims for 2007 as of 12/31/07) x (paid CDF at 12 months) = $27,229,969 x 2.390 = $65,079,626

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Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
5

Unpaid Claim Estimate Based on the Development Technique

92 - 93

Using the development technique, unpaid claim estimates = projected ultimate claims - actual paid claims.
 Because AY data is used, the unpaid claim estimate includes both case outstanding and the broad definition
of IBNR.
 To compute estimated IBNR based on the development technique:
i. IBNR = projected ultimate claims - reported claims
ii. IBNR = estimated total unpaid claims - case O/S
Exhibit I, Sheet 4, summarizes the calculations for the unpaid claim estimate based for U.S. Industry Auto.
Columns (2) and (3) contain reported and paid claims data as of 12/31/2007 (the latest diagonals in our claim
development triangles).
Columns (4) and (5) are the projected ultimate claims (developed in Exhibit I, Sheet 3).
The equations to compute Columns (6) – (10) are shown below in the excerpt from Exhibit I, Sheet 4.
Chapter 7 - Development Technique
U.S. Industry Auto
Projection of Ultimate Claims Using Reported and Paid Claims($000)

Projected Ultimate Claims
Claims at 12/31/07
Using Dev. Method with
Reported
Paid
Reported
Paid
(2)
(3)
(4)
(5)
47,742,304 47,644,187 47,742,304 47,739,475
51,185,767 51,000,534 51,185,767 51,204,536

Accident
Year
(1)
1998
1999

Exhibit I
Sheet 4

Unpaid Claim Estimate at 12/31/07
Case
IBNR - Based on
Total - Based on
Outstanding Dev. Method with
Dev. Method with
at 12/31/07
Reported
Paid
Reported
Paid
(6) =[(2) - (3)]

(7)=[(4) - (2)]

98,117
185,233

0
0

Reporting and Payment Patterns

(8)=[(5) - (2)] (9)=[(6) + (7)] (10)=[(6) + (8)]

-2,829
18,769

98,117
185,233

95,288
204,002

93 - 94

A reporting pattern of claims is the % of ultimate claims that are reported in each year.
Reporting patterns are derived from cumulative reported claim development factors (CDFs).
The following table shows the reporting pattern from the cumulative reported CDFs for U.S. Industry

Table 3 — Reporting Pattern
Cumulative
Age Reported Claim
(Months) Development
12
1.292
24
1.110
36
1.051
48
1.023
60
1.011
72
1.006
84
1.003
96
1.001
108
1.000
120
1.000

Cumulative%
Reported
77.40%
90.10%
95.10%
97.80%
98.90%
99.40%
99.70%
99.90%
100.00%
100.00%

Incremental %
Reported
77.4%
12.7%
5.0%
2.7%
1.1%
0.5%
0.3%
0.2%
0.1%
0.0%

The % reported = 1/CDF

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Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
At 12 months, the percentage reported = 1.000/1.292 = 77.4% (i.e. 77.4% of ultimate claims are reported
through 12 months).
The incremental percentage reported for the 12-24 month period = 90.1% - 77.4%, or 12.7%.
An implied payment pattern based on the cumulative paid claim development factors can also be determined.

Age
(Months)
12
24
36
48
60
72
84
96
108
120

Table 4 — Payment Pattern
Cumulative
Cumulative Incremental %
Paid Claim
Paid
Paid
Development
2.390
41.8%
41.8%
1.404
71.2%
29.4%
1.184
84.5%
13.3%
1.085
92.2%
7.7%
1.040
96.2%
4.0%
1.020
98.0%
1.8%
1.011
98.9%
0.9%
1.006
99.4%
0.5%
1.004
99.6%
0.2%
1.002
99.8%
0.2%

Note: The incremental %s reported and paid in each successive interval are less than or equal to that of the
previous age interval. These patterns are consistent with reasonable expectations for the underlying
process of settling a portfolio of claims. When underlying development patterns are erratic, actuarial
judgment is needed in the selection process to achieve claim development patterns that exhibit such a
steady, decreasing pattern.
The reporting and payment patterns can be used in other techniques for estimating unpaid claims and in
monitoring the development of claims during the year.
The payment pattern is also often used for present value (i.e. discounting) calculations.

Observations and Common Relationships

Cumulative CDFs are often greatest for the most recent AYs and the smallest for the oldest accident years.
Actuaries refer to the most recent, less-developed AYs as immature and the oldest, most-developed AYs as
mature.
Therefore, the highest values of estimated IBNR are for the most recent accident years (the less mature
years).
As AYs mature and more claims are reported and settled, the estimate of total unpaid claims will go to zero.

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Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Also, development factors tend to increase as the retention increases.
In E. Pinto and D.F. Gogol’s paper titled "An Analysis of Excess Loss Development" they observed that:
 excess business exhibits much slower reporting than primary business,
 there is a relationship between the layer for which business is written and the resulting development pattern.
 development is not only caused by late reported claims and increases in the average reported loss per
claim but also by changes at successive maturities in the proportion of claims with losses which are
large multiples of the average.
Thus, the shape of the size of loss distribution changes at successive valuations.
Pinto and Gogol developed a model which illustrates the two influences underlying claim development:
1. the reporting pattern of claims over time and
2. the changing characteristics of the size of claims distribution at successive maturities.
Pinto and Gogol conclusions:
 Loss and ALAE development varies significantly by retention.
 Pricing and reserving estimates using development factors may produce large errors if this is not taken
into account.
 As this applies to paid as well as reported loss development, recognizing the retention is a major factor in
estimating discounted losses using paid development factors.

When the Development Technique Works and When It Does Not

95 - 97

The primary assumption of the development technique is that the reporting and payment of future claims will be
similar to the patterns observed in the past.
 When using reported claims, it is assumed that there have been no significant changes in the adequacy of
case outstanding during the experience period;
 When using paid claims, it is assumed that there have been no significant changes during the experience
period in the speed of claims closure and payment.
The development method is appropriate for insurers in a stable environment.
 If there are changes to the insurer's operations (e.g. new claims processing systems; revisions to tabular
formulae for case outstanding; or changes in claims philosophy, policyholder deductibles, or the insurer's
reinsurance limits), the past may not be predictive of the future.
 Environmental changes, such as a major tort reform occurring (e.g. a cap on claim settlements or a
restriction in the statute of limitations), may cause historical claim development experience to be less
predictive of future claims experience.
The development technique requires a large volume of historical claims experience.
 It works best when the presence or absence of large claims does not greatly distort the data.
 If the volume of data is not sufficient, large claims could greatly distort the age-to-age factors, the projection
of ultimate claims, and finally the estimate of unpaid claims using a development method.
The development technique may not be suitable when there is not a sufficient volume of credible data, as in the
following situations:
 When entering a new line of business or new territory
 For smaller insurers with limited portfolios.
While the development technique may be used in such situations, relying on benchmark patterns (e.g. from
comparable lines of business or available industry data) to select claim development factors, may be warranted.

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Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
The development technique is used for high-frequency, low-severity lines with stable and timely reporting of
claims (evenly spread throughout the AY, PY, RY, etc.)
For long-tail lines of insurance (e.g. WC or GL), cumulative CDFs can become very large for the most recent AYs
when using the paid claim development technique.
These highly leveraged factors can result in unreasonable projections of ultimate claims for the most recent
accident years.
In these situations, alternative techniques for estimating unpaid claims are often used.

XYZ Insurer

97 - 98

Chapter 6 Recap: After discussions with XYZ insurer claims department management, we know that:
 both a speed up in the rate of claims settlement and a strengthening in case reserves have been
implemented.
 during the experience period, a major tort reform modifying the liability covered by the insurance product
resulted in a change in the insurance product and in the insurer's market presence.
Q: Given the above, is the development technique appropriate for XYZ Insurer to use?
 A primary assumption of the reported claim development method is that there have been no significant
changes in the adequacy of case outstanding over the experience period.
 A primary assumption of the paid claim development method is that there have been no significant
changes in the rate of settlement over the experience period.
A: The underlying assumptions do not hold true, and we conclude that an adjustment for these changes is
necessary for the development technique to be appropriate for XYZ Insurer.
However, for demonstration and comparison purposes to other methods presented in later chapters, the
development technique is shown in Exhibit II, Sheets 1 - 4, for XYZ Insurer.
Exhibit II, Sheets 1 and 2 contain the reported and paid claim development triangles.
There is significant variability in the age-to-age factors down each column of the triangle, which we expect
given our knowledge of the changing environment.
Selected age-to-age factors are based on the volume-weighted average of the latest two years (although in
reality a higher degree of judgment would be needed in selecting the age-to-age factors).

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Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
XYZ Insurer - Auto BI
Reported Claims($000)

Exhibit II
Sheet 1

PART 1 - Data Triangle
Accident
Reported Claims as of (months)
Year
12
24
36
48
60
11,171
12,380
13,216
1998
13,255
16,405
19,639
22,473
1999
:::
:::
:::
:::
:::
19,477
31,732
2007
2008
18,632

72
14,067
23,764
:::

84
14,688
25,094
:::

96
16,366
24,795
:::

PART 2 - Age-to-Age Factors
Accident
Year
12-24
24 - 36
1998
1999
1.238
:::
:::
:::
2007
1.629
2008

Age-to-Age Factors
48 - 60
60 - 72
1.068
1.064
1.144
1.057
:::
:::

72 - 84
1.044
1.056
:::

84 - 96
1.114
0.988
:::

96 - 108
0.988
1.011
:::

108 - 120 120-132
0.980
0.999
1.001

To Ult

Averages
48 - 60

60 - 72

72 - 84

84 - 96

96 - 108

108 - 120 120-132

To Ult

36 - 48
1.108
1.197
:::

108
16,163
25,071
:::

120
15,835
25,107

132
15,822

PART 3 - Average Age-to-Age Factors
12-24
24 - 36
Simple Average
Latest 5
1.827
1.417
Latest 3
1.671
1.330
Latest 2
1.679
1.263
Medial Average.
Latest 5x1
1.715
1.419
Volume-weighted Average
Latest 4
1.802
1.376
Latest 3
1.674
1.325
Latest 2
1.687
1.265
Geometric Average
Latest 3
1.670
1.314

36 - 48
1.247
1.187
1.111

1.124
1.083
1.035

1.082
1.062
1.050

1.040
1.033
1.013

1.031
1.003
1.011

0.997
0.997
1.002

0.991
0.991
0.991

0.999
0.999
0.999

1.273

1.118

1.080

1.046

1.011

0.993

0.991

0.999

1.185
1.147
1.102

1.094
1.060
1.020

1.081
1.060
1.050

1.033
1.028
1.010

1.019
1.005
1.011

0.998
0.998
1.000

0.993
0.993
0.993

0.999
0.999
0.999

1.178

1.080

1.061

1.033

1.003

0.997

0.991

0.999

PART 4 - Selected Age-to-Age Factors
Development Factor Selection
12-24
24 - 36
36 - 48
48 - 60
60 - 72
Selected
1.687
1.265
1.102
1.020
1.050
CDF to Ultima 2.551
1.512
1.196
1.085
1.064
Percent Repo 39.2%
66.1%
83.6%
92.2%
94.0%

72 - 84
1.010
1.013
98.7%

84 - 96
1.011
1.003
99.7%

96 - 108
1.000
0.992
100.8%

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108 - 120 120-132
0.993
0.999
0.992
0.999
100.8%
100.1%

To Ult
1.000
1.000
100.0%

 2014 by All 10, Inc.

Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Projected ultimate claims based on the development technique applied to reported and paid claims are shown in
Exhibit II, Sheet 3.
XYZ Insurer - Auto BI
Projection of Ultimate Claims Using Reported and Paid Claims($000)

Accident
Year
(1)
1998
1999
:::
2007
2008
Total

Age of
Year
(2)
132
120
:::
24
12

Claims at 12/31/08
Reported
Paid
(3)
(4)
15,822
15,822
25,107
24,817
:::
:::
31,732
11,865
18,632
3,409
449,626
330,627

CDF to Ultimate
Reported Paid
(5)
(6)
1.000
1.010
0.999
1.014
:::
:::
1.512
6.569
2.551
21.999

Exhibit II
Sheet 3
Projected Ultimate Claims
Using Dev. Method with
Reported
Paid
(7)
(8)
15,822
15,980
25,082
25,164
:::
:::
47,979
77,941
47,530
74,995
514,929
605,028

Column Notes:
(2)
Age of accident year in (1) at December 31, 2008.
(3) and (4) Based on data from XYX insurer.
(5) and (6) Based on CDF from Exhibit 2, Sheets 1 and 2.
(7)
= [(3) x (5)].
(8)
= [(4) x (6)].

Estimated IBNR and the total unpaid claim estimate for the two development projections are shown in Exhibit II,
Sheet 4.
XYZ Insurer - Auto BI
Projection of Ultimate Claims Using Reported and Paid Claims($000)

Accident
Year
(1)
1998
1999
:::
2007
2008
Total

Claims at 12/31/08
Reported
Paid
(2)
(3)
15,822
15,822
25,107
24,817
:::
:::
31,732
11,865
18,632
3,409
449,626
330,627

Case
Projected Ultimate Claims
Using Dev. Method with Outstanding
Reported
Paid
at 12/31/08
(4)
(5)
(6)
15,822
15,980
0
25,082
25,164
290
:::
:::
:::
47,979
77,941
19,867
47,530
74,995
15,223
514,929
605,028
118,999

Exhibit II
Sheet 4
Unpaid Claim Estimate at 12/31/08
IBNR - Based on
Total - Based on
Dev. Method with
Dev. Method with
Reported
Paid
Reported
Paid
(7)
(8)
(9)
(10)
0
158
0
158
-25
57
265
347
:::
:::
:::
:::
16,247
46,209
36,114
66,076
28,898
56,363
44,121
71,586
65,303
155,402
184,302
274,401

Column Notes:
(2) and (3) Based on data from XYZ Insurer.
(3) and (4) Developed in Exhibit 2, Sheet 3
(6)
=[(2) - (3)].
(7)
=[(4) - (2)].
(8)
=[(5) - (2)].
(9)
=[(6) + (7)].
(10)
=[(6) + (8)].

Comparison of the estimated IBNR for the U.S. Industry Auto and for XYZ Insurer:
 For U.S. Industry Auto, the estimated IBNR generated by the reported and paid claim development
methods differs by approximately 10% and the estimate of total unpaid claims differs by only 4%.
 For XYZ Insurer, the estimated IBNR using the paid claim development technique differs by 138% from
the reported claims indication; the total unpaid claim estimate differs by almost 50%.
Thus, alternative projection methods should be reviewed.

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Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
10

Influence of a Changing Environment on the Claim Development
Technique

98 -104

Changes in Claim Ratios (i.e. loss ratios) and Case Outstanding Adequacy
To examine the effect of a changing environment on the estimates produced by the development technique, the
U.S. private passenger automobile is used as an example.
Similar reporting and payment patterns as well as a similar ultimate claim ratio are used.
Compare estimated IBNR from the development technique to the "actual IBNR" under the following 4 scenarios:
* Scenario 1 is a steady-state environment: Claim ratios are stable; there are no changes from historical levels of
case outstanding strength (U.S. PP Auto Steady-State)
* Scenario 2 environment: Increasing claim ratios; no change in case outstanding strength (U.S. PP Auto
Increasing Claim Ratios)
* Scenario 3 environment: Sable claim ratios; an increase in case outstanding strength (U.S. PP Auto Increasing
Case Outstanding Strength)
* Scenario 4 environment: Increasing claim ratios and increasing case outstanding strength (U.S. PP Auto
Increasing Claim Ratios and Case Outstanding Strength)
This example with its four scenarios are used in Chapters 8, 9, and 10.
Key Assumptions
Computation of Actual IBNR (not known in real life)
For the purpose of demonstrating the affect of a changing environment, we calculate the "actual" or "true" IBNR
requirement . In this example:
 A ten-year experience period is used (AYs 1999 – 2008).
 Assume EP is $1M for the first year (i.e. 1999), and increases 5% annually.
Actual IBNR is calculated in Exhibit III, Sheet 1, equals ultimate claims projection (based on the given ultimate
claim ratio for each AY) minus the reported claims as of 12/31/2008.

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Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Impact of Changing Conditions
Summary of Earned Premium and Claim Ratio Assumptions and Actual IBNR
Reported
Claims at
12/31/08
(5)

Actual
IBNR
(6)

700,000
::
884,463
919,306
935,722
930,797
836,166
8,365,888

0
::
8,934
18,761
49,248
103,422
249,764
438,636

Increasing Case Outstanding Strength
70.0%
700,000
700,000
::
::
::
70.0%
893,397
884,463
70.0%
938,067
933,377
70.0%
984,970
962,808
70.0%
1,034,219
979,922
931,185
70.0%
1,085,930
8,804,524
8,551,189

0
::
8,934
4,690
22,162
54,296
154,745
253,335

Accident
Year
(1)

Earned
Premium
(2)

Ultimate
Claim Ratio
(3)

1999
::
2004
2005
2006
2007
2008
Total

1,000,000
::
1,276,282
1,340,096
1,407,100
1,477,455
1,551,328
12,577,892

70.0%
::
70.0%
70.0%
70.0%
70.0%
70.0%

1999
::
2004
2005
2006
2007
2008
Total

1,000,000
::
1,276,282
1,340,096
1,407,100
1,477,455
1,551,328
12,577,892

Ultimate
Claims
(4)
Steady-State
700,000
::
893,397
938,067
984,970
1,034,219
1,085,930
8,804,524

Exhibit III
Sheet 1
Reported
Ultimate
Ultimate
Claim at
Claim Ratio
Claim
12/31/08
(7)
(8)
(9)
Increasing Claim Ratios
70.0%
700,000
700,000
::
::
::
80.0%
1,021,026 1,010,815
85.0%
1,139,082 1,116,300
90.0%
1,266,390 1,203,071
95.0%
1,403,582 1,263,224
1,551,328 1,194,523
100.0%
10,249,349 9,647,367

Actual
IBNR
(10)
0
::
10,211
22,782
63,319
140,358
356,805
601,982

Increasing Claim Ratios and Case Outstanding Strength

70.0%
::
80.0%
85.0%
90.0%
95.0%
100.0%

700,000
::
1,021,026
1,139,082
1,266,390
1,403,582
1,551,328
10,249,349

700,000
::
1,010,815
1,133,386
1,237,897
1,329,895
1,330,264
9,901,691

0
::
10,211
5,696
28,493
73,687
221,064
347,658

Column Notes:
(2)
Assume 51,000,000 for first year in experience period (1999) and 5% annual increased thereafter.
(3) and (7) Ultimate claim ratios assumed to be known for purpose of example.
(4)
=[(2) * (3)].
(5)
Latest diagonal of reported claim triangles in Exhibit III, Sheet 2 and 6
(6)
=[(4) - (5)].
(8)
=[(2) + (7)].
(9)
Latest diagonal of reported claim triangles in Exhibit III, Sheet 4 and 8
(10)
=[(8) - (9)].

In the steady-state environment, assume an ultimate claim ratio of 70% for all ten accident years
Table 5 — Key Assumptions
Steady-State Environment
Reporting and Payment Patterns
%
0/0
Reported
Paid
As of Month
12
77%
42%
24
90%
71%
36
95%
84%
48
98%
92%
60
99%
96%
72
99%
98%
84
100%
99%
96
100%
99%
108
100%
100%
120
100%
100%

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Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
In the increasing claim ratio scenarios, assume the following claim ratios by accident year:
Table 6 - Key Assumptions
Increasing Claim Ratio Scenarios
Accident Year
Ultimate Claim Ratio
1999-2003
70%
2004
80%
2005
85%
2006
90%
2007
95%
2008
100%

EP, ultimate claim ratios, and the above reporting and payment patterns are used to create reported and paid
claim development triangles for each of the 4 scenarios (shown in Exhibit III, Sheets 2 – 9).
To simplify, select reported and paid age-to-age factors based on a five-year volume-weighted average.
By not incorporating judgmental adjustments to the examples showing changes in the environment, we
demonstrate how the development technique reacts to a changing situation.
Scenario 1 — U.S. PP Auto Steady-State
As expected, the projected ultimate claims are the same for both the reported and paid claim development
methods. Both methods produce estimated IBNR equal to actual IBNR (see the top section of Exhibit III,
Sheet 10).
Impact of Changing Conditions
U.S. PP Auto - Development of Unpaid Claim Estimate

Exhibit III
Sheet 10

Age of
Claims at 12/31/08
CDF to Ultimate
Accident
Accident
Case
Year
at 12/31/08 Reported
Paid
Outstanding Reported
Paid
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Steady-State
120
1999
700,000
700,000
0
1.000
1.000
::
::
::
::
::
::
::
24
1.408
930,797
734,295
196,502
1.111
2007
836,166
2008
456,090
380,076
1.299
12
2.381
Total
8,365,888 7,573,547
792,341

Projected Ultimate Claims
Using Dev. Method with
Reported
Paid
(8)
(9)
700,000
::
1,034,219
1,085,930
8,804,527

700,000
::
1,034,218
1,085,928
8,804,522

Estimate IBNR
Using Dev. Method with
Reported
Paid
(10)
(11)
0
::
103,422
249,764
438,639

0
::
103,421
249,762
438,634

Actual
IBNR
(12)

Difference from Actual
IBNR
Reported
Paid
(13)
(14)

0
::
103,422
249,764
438,636

0
::
-1
0

0
::
1
2

0
::
140,358
356,805
601,982

0
::
0
0

0
::
-1
0

Scenario 2 — U.S. PP Auto Increasing Claim Ratios (and no case reserve strengthening)
See the bottom section of Exhibit III, Sheet 10.
Increasing Claim Ratios
120
1999
700,000
::
::
::
24
1,263,224
2007
1,194,523
2008
12
Total
9,647,367

700,000
::
996,544
651,558
8,575,113

0
::
266,680
542,965
1,072,254

1.000
::
1.111
1.299

1.000
::
1.408
2.381

700,000
::
1,403,582
1,551,328
10,249,351

700,000
::
1,403,583
1,551,328
10,249,350

0
::
140,358
356,805
601,984

0
::
140,359
356,805
601,983

Observations:
When comparing the top and bottom sections of Sheet 10, there are differences between reported and paid
claims in Columns (3) and (4), as well as differences in the claim development triangles.
The claim development triangles in Sheets 4 and 5 (increasing claim ratio scenario) are the same as the
triangles in Sheets 2 and 3 (steady-state) for AYs 1999 - 2003.
However, beginning in AY 2004, the reported and paid claims for all remaining years are higher for the
increasing claim ratio scenario than the steady-state scenario (consistent with our assumption of increasing
claim ratios for AYs 2004 - 2008).

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Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Key: Since we assume no change in the adequacy of case outstanding, there are no changes in the
age-to-age factors, and thus no changes in the cumulative claim development factors between the
increasing claim ratio scenario and the steady-state environment.
The higher value of projected ultimate claims is solely due to higher values of claims reported and paid as
of 12/31/2008.
The estimated IBNR is the same for both the reported and paid claim development methods, and is equal
to the actual IBNR.
Thus, conclude that the development technique is responsive to changes in the underlying claim ratios
assuming no changes in the underlying claims reporting or payment pattern.
Scenario 3 — U.S. PP Auto Increasing Case Outstanding Strength (and stable claim ratios)
Exhibit III, Sheets 6 and 7 contain the claim development triangles for this scenario.
Sheet 11 shows the calculations for projected ultimate claims and estimated IBNR in the top section.
Assume that case outstanding adequacy increased by 6% in 2007 and 25% in 2008 over the steady-state
case outstanding (for the latest 4 AYs only) in the reported triangle.
This means that the next to last diagonal is 6% greater in this scenario than the steady-state scenario, and
that the last diagonal is 25% greater in this scenario than the steady-state scenario.
What is expected to be seen?
 The true ultimate claims have not changed from the steady-state environment (since ultimate claims
equal 70% of EP for each year in the experience period).
 We expect higher reported claims since case outstanding strength has increased.
 Given the same value of ultimate claims with higher values of reported claims at December 31, 2008, the
IBNR should decrease.
 However, actual IBNR for this scenario of stable claim ratios and increases in case outstanding strength
 are $253,336, which is lower than the actual IBNR of the steady-state, which are $438,638.
See the top section of Exhibit III, Sheet 11.
Impact of Changing Conditions
U.S. PP Auto - Development of Unpaid Claim Estimate
Age of
Claims at 12/31/08
Accident
Accident
Year
at 12/31/08
Reported
Paid
(1)
(2)
(3)
(4)
Increasing Case Outstanding Strength
120
1999
700,000
700,000
:::
:::
:::
:::
24
979,922
734,295
2007
931,185
2008
456,090
12
Total
8,551,189
7,573,547

Exhibit III
Sheet 11

CDF to Ultimate
Case
Outstanding Reported
Paid
(5)
(6)
(7)
0
:::
245,627
475,095
977,642

1.000
:::
1.119
1.318

1.000
:::
1.408
2.381

Projected Ultimate Claims
Using Dev. Method with
Reported
Paid
(8)
(9)

Estimate IBNR
Using Dev. Method with
Reported
Paid
(10)
(11)

700,000
:::
1,096,235
1,227,589
9,052,121

0
:::
116,313
296,404
500,932

700,000
:::
1,034,218
1,085,928
8,804,522

0
:::
54,296
154,743
253,333

Actual
IBNR
(12)
0
:::
54,296
154,745
253,335

Difference from Actual
IBNR
Reported
Paid
(13)
(14)
0
:::
-62,017
-141,659
-247,597

0
:::
1
2
2

Comparing the projections of Scenario 3 with those of the steady-state environment, we notice:
 For AYs 2005 – 2008, reported claims in Column (3) are greater than those in the steady-state.
 Reported CDFs (Column (6)) are higher for the latest three AYs as well.
 Projected ultimate claims based on the reported claim development technique are greater in Scenario 3
than the steady-state projection due to higher reported claims and higher CDFs.
Conclusion: Without adjustment, the reported claim development method overstates the projected ultimate claims
and thus the IBNR in times of increasing case outstanding strength.
 An increase in case outstanding adequacy leads to higher CDFs.
 Multiplying a higher value of reported claims by a higher CDFs leads to an overstated the estimate of total
unpaid claims.

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Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Paid claim development triangles:
 There are no differences between the paid claim development triangles of Scenario 3 and the steadystate environment (because only the case outstanding are affected).
 Thus, the age-to-age factors, CDFs, and projected ultimate claims remain the same as the steady-state.
 Since there has been no change in the settlement of claims, the primary assumption of the development
technique still holds true for paid claims.
 In times of changing case outstanding adequacy, the paid claim development method is an alternative to
the reported claim development method.
One problem with the paid claim development method: The highly leveraged nature of the CDF for the most
recent years in the experience period (especially for long-tail lines of insurance).
Scenario 4 — U.S. PP Auto Increasing Claim Ratios and Case Outstanding Strength
See the bottom section of Exhibit III, Sheet 11.
 The claim ratios are the same as those of the second scenario
 Assume changes in case outstanding strength that is similar to the third scenario.
Impact of Changing Conditions
U.S. PP Auto - Development of Unpaid Claim Estimate

Exhibit III
Sheet 11

Age of
Claims at 12/31/08
CDF to Ultimate
Accident
Accident
Case
Year
at 12/31/08
Reported
Paid
Outstanding Reported
Paid
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Increasing Claim Ratios and Case Outstanding Strength
120
1999
700,000
700,000
0
1.000
1.000
:::
:::
:::
:::
:::
:::
:::
24
1.408
1,329,895
996,544
333,351
1.120
2007
1,330,264
2008
651,558
678,706
1.320
12
2.381
Total
9,901,691
8,575,113
1,326,578

Projected Ultimate Claims
Using Dev. Method with
Reported
Paid
(8)
(9)

Estimate IBNR
Using Dev. Method with
Reported
Paid
(10)
(11)

700,000
:::
1,488,875
1,756,504
10,595,469

0
:::
158,980
426,240
693,778

700,000
:::
1,403,583
1,551,328
10,249,350

0
:::
73,688
221,064
347,659

Actual
IBNR
(12)
0
:::
73,687
221,064
347,658

Difference from Actual
IBNR
Reported
Paid
(13)
(14)
0
:::
-85,293
-205,176
-346,119

0
:::
-1
0
-1

Column Notes:
(2)
Age of accident year at December 31, 2008
(3) and (4) From last diagonal of reported and paid claim triangles in Exhibit III, Sheets 6 through 9.
(5)
=[(3) - (4)].
(6) and (7) CDF based on 5-year volume-weighted average age-to-age factors presented in Exhibit III, Sheets 6 throught 9.
(8)
=[(3) * (6)].
(9)
=[(4) * (7)].
(10)
=[(8) - (3)].
(11)
=[(9) - (3)].
(12)
Developed in Exhibit III, Sheet 1.
(13)
=[(12) - (10)].
(14)
=[(12) - (11)].

Again, the paid claim development method produces the actual value for IBNR.
The reported claim development method, while responsive to the increasing claim ratios, overstates the estimate
of unpaid claims due to the changing case outstanding adequacy.
Effects of Changes in Product Mix on the Development Technique
A portfolio of contains both private passenger and commercial automobile insurance for the purpose of estimating
unpaid claims.
While these types of business have different underlying claim development patterns and ultimate claim ratios, the
development technique is an acceptable method for determining estimates of unpaid claims for the combined
portfolio as long as there are no changes in the mix of business (i.e., one line of business is not significantly
increasing or decreasing in volume relative to the other line of business).

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Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Key Assumptions
We compare a steady-state environment that has no change in product mix (called U.S. Auto Steady-State) with a
changing product mix (called U.S. Auto Changing Product Mix).
Assume:
 For U.S. Auto Changing Product Mix, the portfolio includes the same private passenger premiums as the
steady-state, but commercial automobile insurance premiums increase at 30% instead of 5% per year
starting in 2005.
 The ultimate claim ratio is 70% for private passenger automobile and 80% for commercial automobile.
 The following table shows the reporting and payment patterns for the two categories of business.

Table 7 — Key Assumptions — Product Mix Scenarios
Reporting and Payment Patterns
Private Passenger
Commercial Automobile
As of
Month
12
24
36
48
60
72
84
96
108
120

Reported
77%
90%
95%
98%
99%
99%
100%
100%
100%
100%

Paid
42%
71%
84%
92%
96%
98%
99%
99%
100%
100%

Reported
59%
78%
89%
96%
98%
100%
100%
100%
100%
100%

Paid
22%
46%
67%
82%
91%
95%
97%
98%
99%
100%

The claim development triangles are created using the EP and ultimate claim ratios by AY as well as the given
reporting and payment patterns.
Exhibit IV, Sheets 2 and 3 show reported and paid development triangles assuming no change in product mix;
Exhibit IV, Sheets 4 and 5 show the claim development triangles based on a changing product mix
Exhibit IV, Sheet 6 shows the calculation of actual IBNR.
U.S. Auto Steady-State (No Change in Product Mix) – See the top section of Exhibit IV, Sheet 6
 Both the reported and paid development techniques produce estimated IBNR equal to the actual IBNR.
 As long as the distribution between the different categories of business remains consistent (and
there are no other operational or environmental changes), the claim development method should
produce an accurate estimate of unpaid claims.
Impact of Change in Product Mix Example
U.S. Auto - Development of Unpaid Claim Estimate

Exhibit IV
Sheet 6

Age of
Claims at 12/31/08
CDF to Ultimate
Accident
Accident
Case
Year
at 12/31/08 Reported
Paid
Outstanding Reported
Paid
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Steady-State ( No Change in Product Mix)
120
1999
1,500,000 1,500,000
0
1.000
1.000
:::
:::
:::
:::
:::
:::
:::
24
1.734
1,852,729 1,277,999
574,730
1.196
2007
1,568,393
1.484
2008
729,124
839,269
12
3.191
Total
17,472,205 15,270,788 2,201,417

Exam 5, V2

Projected Ultimate Claims
Using Dev. Method with
Reported
Paid
(8)
(9)

Estimate IBNR
Using Dev. Method with
Reported
Paid
(10)
(11)

1,500,000
:::
2,216,183
2,326,992
18,866,839

0
:::
363,454
758,599
1,394,634

Page 92

1,500,000
:::
2,216,183
2,326,992
18,866,837

0
:::
363,454
758,599
1,394,632

Actual
IBNR
(12)
0
:::
363,454
758,599
1,394,634

Difference from Actual
IBNR
Reported
Paid
(13)
(14)
0
:::
0
0
0

 2014 by All 10, Inc.

0
:::
0
1
1

Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
U.S. Auto Changing Product Mix – See the bottom section of Exhibit IV, Sheet 6
 There are no differences between the two examples until AY 2005, in which commercial auto began to
increase at a 30% annual rate.
 We expect higher reported and paid claims for 2005 through 2008.
 We expect higher CDFs for both paid and reported claims for AYs 2006, 2007 and 2008.
However, even with larger claims and CDFs, the development technique falls short of the actual IBNR.
Impact of Change in Product Mix Example
U.S. Auto - Development of Unpaid Claim Estimate

Exhibit IV
Sheet 6

Age of
Accident
Accident
Case
Claims at 12/31/08
CDF to Ultimate
Year
at 12/31/08 Reported
Paid
Outstanding Reported
Paid
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Changing Product Mix
120
1999
1,500,000 1,500,000
0
1.000
1.000
:::
:::
:::
:::
:::
:::
:::
24
1.750
2,680,487 1,766,164
914,323
1.200
2007
2,556,695 1,097,644 1,459,051
2008
1.503
12
3.273
Total
20,067,180 16,738,685 3,328,495

Projected Ultimate Claims
Using Dev. Method with
Reported
Paid
(8)
(9)

Estimate IBNR
Using Dev. Method with
Reported
Paid
(10)
(11)

1,500,000
:::
3,217,775
3,842,646
22,219,968

0
:::
537,288
1,285,951
2,152,788

1,500,000
:::
3,091,665
3,592,941
21,789,881

0
:::
411,178
1,036,246
1,722,701

Actual
IBNR
(12)
0
:::
596,924
1,445,385
2,391,083

Difference from Actual
IBNR
Reported
Paid
(13)
(14)
0
:::
59,637
159,434
238,296

0
:::
185,746
409,139
668,382

Column Notes:
(2)
Age of accident year at December 31, 2008
(3) and (4) From last diagonal of reported and paid claim triangles in Exhibit IV, Sheets 2 through 5.
(5)
=[(3) - (4)].
(6) and (7) CDF based on 5-year volume-weighted average age-to-age factors presented in Exhibit IV, Sheets 2 throught 5.
(8)
=[(3) * (6)].
(9)
=[(4) * (7)].
(10)
=[(8) - (3)].
(11)
=[(9) - (3)].
(12)
Developed in Exhibit IV, Sheet 1.
(13)
=[(12) - (10)].
(14)
=[(12) - (11)].

What is the correct age-to-age factor when a portfolio is changing its composition (see Exhibit IV, Sheet 1)?
 We know that commercial auto has a longer reporting pattern than private passenger automobile (and
thus requires higher selected age-to-age factors).
 Since commercial auto claims are increasing in the portfolio, increasing age-to-age factors appear.
 Changing from a 5-year to 3-year volume-weighted average for selecting age-to-age factors helps move
the estimated IBNR closer to the actual IBNR, but its still falls short by a significant amount.
Conclusions
 the reported development method is more responsive than the paid claim development method due to
the shorter time frame in which claims are reported versus paid.
 both methods result in estimated IBNR that are significantly lower than the actual IBNR.

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Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Sample Questions:
1. When analyzing data triangles of claims by accident year (AY), using Development Techniques:
a. Explain how a cumulative reported CDF is calculated.
b. Explain how a cumulative reported CDF is applied to calculate ultimate claim estimates, for one AY.
c. What is the term used in Friedland to describe the ultimate claims minus the reported claims?
d. Explain how a cumulative paid CDF is calculated.
e. Explain how a cumulative paid CDF is applied to calculate ultimate claim estimates, for one AY.
f. What is the term used in Friedland to describe the ultimate claims minus the paid claims?
(“CDF” = claim development factor)
2. Describe a typical relationship between reporting patterns and payment patterns for many lines
of P&C insurance.
3. What name does Brosius give to the method described in Friedland as the “Development” technique?
What name does Patrik use for this method?
4. Summarize Friedland’s key points re: “When the Development Technique Works and When it Does Not.”
List the two limitations mentioned.
5. List 5 characteristics Friedland suggests that actuaries may reference when reviewing claim development
experience.
6. Based on the following data as of 12/31/08:

Earned
Premium

Accident
Year

2,000
2,200
2,500
2,650
3,000
3,150

2003
2004
2005
2006
2007
2008

Reported Claims including ALAE ($000's omitted)
1st
2nd
3rd
4th
5th
Report
Report
Report
Report
Report
940
1,200
1,250
1,400
1,500
2,250

1,620
1,690
1,725
1,550
1,900

1,700
1,710
1,800
1,900

1,750
1,800
1,950

1,750
1,800

6th
Report
1,750

a. Estimate the IBNR as of 12/31/08 using the following method: Development Technique
To select claim development factors, use the volume-weighted averages for the latest three years.
See also Friedland Chapter 8 and 9 for other methods.
b. Using the data above and based on the discussion by Friedland, what is the 12-24 month age-to-age factor
using:
(i) Simple (arithmetic) average of the last three years
(ii) Geometric average of the last four years
(iii) Medial average for the latest five years excluding one high and low value, “Medial latest 5x1”

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Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
1995 Exam Questions (modified):
38. (1 point) Friedland states that the selection of a tail factor can be difficult. Describe two complicating factors.
44.

You are given the following:

Earned
Premium

Accident
Year

4,500
5,000
5,200
5,300
5,700

1990
1991
1992
1993
1994

Reported Claims including ALAE ($000's omitted) as of
12 mo,
24 mo,
36 mo,
48 mo,
60 mo,
Report 1
Report 2 Report 3 Report 4 Report 5
2,000
2,102
2,234
2,339
2,482

2,600
2,638
2,938
2,985

2,990
3,086
3,408

3,283
3,343

3,283

a. (1.5 points) See Friedland Chapter 9.
b. (0.5 points) Using the Development Technique described in Friedland, determine the IBNR as of
12/31/94. Select development factors using latest 3 years, volume-weighted. Show all work.
c. (1.5 points) See Friedland Chapter 15.
2002 Exam Questions (modified):
22. (4 points) You are given the following information:
Accident
Year

Earned
Premium

Reported Claims
at 12-31-01

Expected
Claim Ratio

1998
1999
2000
2001

200
1,000
1,500
1,500

100
1,000
900
600

80%
80%
80%
80%

Selected age-to-age reported claim development factors:
1.25
12 - 24 months
24 - 36 months
1.10
36 - 48 months
1.05
48 - 60 months
1.08

No further development after 60 months.
a.
b.
c.
e.

(1 point) Calculate the IBNR reserve as of December 31, 2001 using the Development technique.
(1 point) See Friedland Chapter 9
(0.5 points) See Friedland Chapters 9 and 15
(1 point) See Friedland Chapters 9 and 15

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Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2003 Exam Questions (modified):
23. (3 points) You are given the following information:
Earned
Premium

Accident
Year

1,000
1,000
1,500
1,800

1999
2000
2001
2002

Reported Claims including ALAE ($000's omitted)
at age
at age
at age
at age
12 mo
24 mo
36 mo
48 mo
250
500
750
825
200
350
490
300
450
400

•

Claim development factors should be calculated using an all-years simple average.

•

The tail factor is 1.05 for development from 48 months to ultimate.

a. (1 point) Using the Development method, calculate the total IBNR reserve. Show all work.
b. (1 point)

See Friedland Chapter 9.

c. (1 point) See Friedland Chapter 9.
2005 Exam Questions (modified):
10. (4 points) You are given the following information:
Earned
Premium

Accident
Year

19,000
20,000
21,000
22,000

2001
2002
2003
2004

at age
12 mo
4,850
5,150
5,400
7,200

Reported Claims by Development Age
at age
at age
at age
24 mo
36 mo
48 mo
9,700
14,100
16,200
10,300
14,900
10,800

Assume an expected Claim Ratio = 0.90 for all years.
Choose selected factors using a straight average of the age to age factors.
Assume no development past 48 months.
a. (1 point) Using the Development method, calculate the indicated IBNR for accident year 2004
as of December 31, 2004.
b. (0.5 point) See Friedland Chapter 9
c. (1 point) See Friedland Chapter 15.
d. (0.5 point) See Friedland Chapter 15.
e. (1 point) See Friedland Chapter 15.

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Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2008 Exam Questions (modified):
2. (1.5 points) Given the following for policy year 2006 for a line of business:
Premium
1,600,000
Expected loss emerged at 24 months
68%
Expected loss emerged at 36 months
82%
Reported loss as of December 31, 2007
800,000
Bornhuetter-Ferguson estimate of ultimate loss
1,133,000
a. (0.5 point) See chapter 9.
b. (0.5 point) Calculate the ultimate loss estimate for policy year 2006 using the chain ladder method
(Note Friedland terminology: ultimate claims estimate using Reported Loss Development Method)
c. (0.5 point) See chapters 9 and 15.
2008 Exam Questions (modified):
Question 10.
Given the following for an accident year:
- Earned Premium:
- Reported Losses as of 12 months:
- Expected loss ratio:
- Expected reporting pattern:

$20,000,000
$10,000,000
70%

Age (months) % Reported
12
40%
24
60%
36
80%
48
90%
60
100%
a. (1.5 points) This portion of the problem is associated with the Brosius article that is now on Exam 7.
b. (1 point) Estimate the ultimate value of the claims currently aged at 12 months. Use the Development
Method on reported claims, as described in Friedland.
c. (.75 points) See Mack/Benktander and Friedland Ch 9.

Exam 5, V2

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Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2009 Exam Questions
12. (2 points) Given the following information:
Accident
Incremental
Valuation Date
Year
Paid Loss
Dec. 31, 2008
Dec. 31, 2008
Dec. 31, 2008
Dec. 31, 2008
Dec. 31, 2007
Dec. 31, 2007
Dec. 31, 2007
Dec. 31, 2006
Dec. 31, 2006
Dec. 31, 2005

2008
2007
2006
2005
2007
2006
2005
2006
2005
2005

Accident
Year

12 Months

2005
2006

$1,100
1,500

$1,500
2,100

2007

1,500

2008

1,000

Age-to-age factors

$1,000
500
100
50
1,500
600
150
1,500
400
1,100

Cumulative Paid Loss
24 Months
36 Months
$1,650
2,200

48 Months
$1,700

12-24 Mos

24-36 Mos

36-48 Mos

48-Ultimate

1.069

1.030

1.000

a. (1 point) Using the volume-weighted average for B, calculate the values for A and B.
b. (1 point) Use the development technique to estimate the unpaid claim liability for accident year 2008 as
of December 31, 2008.

Exam 5, V2

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Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2010 Exam Questions
13. (1.5 points) Given the following information about two lines of insurance:
Commercial Auto Property Damage Reported Claims ($000)
Accident
Year
12 Months
24 Months
36 Months
48 Months
2006
10,000
14,000
16,800
18,480
2007
15,000
21,000
25,200
2008
20,000
28,000
2009
25,000
Personal Auto Property Damage Reported Claims ($000)
Accident
Year
12 Months
24 Months
36 Months
48 Months
2006
10,000
12,000
13,200
13,332
2007
11,000
13,200
14,520
2008
12,000
14,400
2009
13,000
a. (1 point) Based on the data, provide two reasons why it would be inappropriate to combine these two
lines of business for estimating unpaid claims.
b. (0.5 point) Briefly describe two additional factors that generally should be considered when deciding
whether to combine lines of business for estimating unpaid claims.
2011 Exam Questions
24. (3.5 points) Given the following claim data as of December 31, 2010:
Accident
Year
2006
2007
2008
2009
2010

12 Months
$105
$100
$116
$122
$128

Cumulative Reported Claims (000s)
24 Months 36 Months
48 Months
$265
$340
$375
$275
$360
$390
$285
$375
$310

60 Months
$380

• No development is expected after 60 months.
• Use an all-year straight average for all factor selections.
• Accident year 2009 paid claims as of December 31, 2010 = $250,000
a. (1.5 points) Use the reported development technique to estimate the unpaid claims for AY 2009.
b. (0.5 point) Calculate the expected reported claims for accident year 2010 during the next 12 months.
c. (0.75 point) State three assumptions underlying the reported development technique.
d. (0.25 point) Briefly describe when a tail factor may be needed to estimate unpaid claims under the
reported development technique.
e. (0.5 point) Briefly describe two approaches to determine a tail factor.

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Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2011 Exam Questions
36. (3 points) An insurance company uses both the reported claim development method and the paid claim
development method to estimate unpaid claims for its automobile liability business.
a. (0.5 point) A state enacts legislation creating a court that specializes in hearing insurance liability
claims to combat a backlog of liability cases in the regular court system. Briefly describe the expected
impact of the legislation on the estimated unpaid claims in this state for each method.
b. (0.5 point) A state enacts tort reform legislation that places caps on non-economic damages awarded
in automobile liability lawsuits. Briefly describe the expected impact of the legislation on the estimated
unpaid claims in this state for each method.
c. (0.5 point) To gain market share, company management is focusing on writing $1,000,000 policy limits
whereas previously policies were written with $500,000 policy limits. Briefly describe the expected
impact of this strategic change on the estimated unpaid claims for each method.
d. (1.5 points) For each scenario described in parts a, b, and c above, discuss a diagnostic test that
would indicate whether the expected impact of the change is present in the data.

Exam 5, V2

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Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to Sample Questions:
1. When analyzing data triangles of claims by accident year (AY), using Development Techniques:
a. A cumulative reported CDF is calculated by “successive multiplications beginning with the tail factor and
the oldest age-to-age factor” as calculated and selected using triangles of reported claims.
b. For the Reported Claim Development method, ultimate claims for each accident year are estimated as
the product of the cumulative reported CDF at the valuation age, and reported claims through the
valuation date. Note: (cumulative) Reported Claims = (cumulative) Paid Claims + Case Outstanding at
valuation date
c. Term used in Friedland to describe the ultimate claims minus the reported claims = IBNR (broadly
defined)
d. A cumulative paid CDF is calculated by “successive multiplications beginning with the tail factor and the
oldest age-to-age factor” as calculated and selected using triangles of paid claims.
e. For the Paid Claim Development method, ultimate claims for each accident year are estimated as the
product of the cumulative paid CDF at the valuation age, and paid claims through the valuation date.
f. Term used in Friedland to describe the ultimate claims minus the paid claims = Unpaid Claim Estimate
Note: Unpaid Claim Estimate includes all Case Outstanding and IBNR (pure IBNR and IBNER)

2. Typical relationship between reporting and payment patterns: Cumulative paid CDFs are usually greater
than cumulative reported CDFs at the same maturity factor.
3. What does Brosius and Patrick call the method described in Friedland as the “Development” technique?
The “Link Ratio” (for ultimate loss estimates with full credibility to actual experience) in Brosius,
and Patrik’s “Chainladder” methods are analogous to the Development technique in Friedland.
4. Summarize Friedland’s key points re: “When the Development Technique Works and When it Does Not.”
Friedland lists limitations:
The development technique may not be suitable when there is not a sufficient volume of credible data,
when entering a new line of business or new territory, or for smaller insurers with limited portfolios.
For long-tail lines of insurance (e.g. WC or GL), the cumulative claim development factors can become
very large for the most recent AYs when using the paid claim development technique. These highly
leveraged factors can result in unreasonable projections of ultimate claims for the most recent accident
years. See Friedland Chapter 7.
5. List 5 characteristics Friedland suggests that actuaries may reference when reviewing claim development
experience:
-Smooth progression of individual ATA factors and average factors across development periods
-Stability of ATA factors for the same development period
-Credibility of the experience
-Changes in patterns
-Applicability of the historical experience

Exam 5, V2

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Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
6. a. Estimate the IBNR as of 12/31/08 using the following method: Development Technique
To select claim development factors, use the volume-weighted averages for the latest three years.

Earned
Premium

Accident
Year

2,000
2,200
2,500
2,650
3,000
3,150

2003
2004
2005
2006
2007
2008

Reported Claims including ALAE ($000's omitted)
1st
2nd
3rd
4th
5th
Report
Report
Report
Report
Report
940
1,200
1,250
1,400
1,500
2,250

Selected CDF calculations
ATA: 3-yr Volume-weighted average
Note: 1st report at 12 months
Reported CDF to Ultimate

1,620
1,690
1,725
1,550
1,900

1,700
1,710
1,800
1,900

1st to 2nd
Report

2nd to 3rd
Report

1.2470
at 12 mo
1.4344

1.0896
at 24 mo
1.1503

1,750
1,800
1,950

1,750
1,800

3rd to 4th 4th to 5th
Report
Report
1.0557
at 36 mo
1.0557

1.0000
at 48 mo
1.0000

6th
Report
1,750

5th to 6th
Report
1.0000
at 60 mo
1.0000

* Example of Age-to-Age calculation for 2nd to 3rd report, using 3-year volume-weighted average:
(1900+1800+1710)/(1550+1725+1690) = 1.0897 or 1.09 as shown
** Example of Ultimate CDF calculation for claims at 24 months of development:
(1.0896 for 2nd-to-3rd) * (1.0557 for 3rd-to-4th) * (1.00 for 4th-to-5th) * (1.0 tail) = 1.1503

Accident
Year
2003
2004
2005
2006
2007
2008
Total

Age of
Data at
12/31/08

Reported
Claims at
12/31/08

Reported
CDF to
Ultimate

Expected
Ultimate
Claims

IBNR
(broadly
defined)

(1)
72 months
60 months
48 months
36 months
24 months
12 months

(2)
1,750
1,800
1,950
1,900
1,900
2,250

(3) above
1.0000
1.0000
1.0000
1.0557
1.1503
1.4344

(4)=(2)*(3)
1,750
1,800
1,950
2,006
2,186
3,227

(5)=(4)-(2)
0
0
0
106
286
977
1,369

OR:
Shortcut

IBNR
(broadly
defined)

(5)=(2)*[(3) - 1.0]
0
0
0
106
286
977
1,369

b. What is the 12-24 month age-to-age factor using:
(i) 12-24 month age-to-age factor using Simple (arithmetic) average of the last three years =1.25
(ii) 12-24 month age-to-age factor using Geometric average of the last four years =1.28
(iii) 12-24 month age-to-age factor using Medial latest 5x1 =1.35

Exam 5, V2

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Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Age-to-Age Development Factors by Accident Year
Note: Did not need these "Link Ratios" to calculate the volume-weighted ATA selections:
ATA factors by AY:

Accident
Year

1st to 2nd
Report
12:24 mo

2nd to 3rd
Report
24:36 mo

2003
2004
2005
2006
2007

1.7234
1.4083
1.3800
1.1071
1.2667

1.049
1.012
1.043
1.226

Alternative ATA Selections
3-year Simple (Arithmetic) Average
4-year Geometric Average
"Medial latest 5x1"

12:24 mo
1.2513
1.2849
1.3517

Example:
12:24 month ATA
for AY 2005 =
(1725)/(1250) = 1.38
between 1st and 2nd
annual valuation dates

3rd to 4th 4th to 5th
Report
Report
36:48 mo 48:60 mo
1.029
1.053
1.083

1.000
1.000

5th to 6th
Report
50:72 mo
1.000

Calculation Details
= (1.3800+1.1071+1.2667)/3
= (1.4083*1.38*1.1071*1.2667)^(1/4)
= (1.4083+1.38+1.2667)/3

Note: "Medial latest 5x1" excludes the highest and lowest values (1.7234 and 1.1071) in 5-yr period

Solutions to 1995 Exam questions (modified):
38. (1 point) Friedland states that the selection of a tail factor can be difficult. Describe two complicating
factors.
1. Lack of data on which to base the estimate of the tail factor.
2. The tail factor affects all accident years reserve needs, thus has a disproportionate leverage on the
total reserve need.
In chapter 7, Friedland Comments:
“Sometimes the data does not provide for enough development periods … [w]hen this occurs, the actuary
will need to determine a tail factor … For some lines of insurance and some types of claims data, the tail
factor can be especially difficult to select due to the limited availability of relevant data.”
“The tail factor is crucial as it influences the unpaid claim estimate for all accident years (in the experience
period) and can create a disproportionate leverage on the total unpaid claims.”

Exam 5, V2

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Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 1995 Exam questions (modified):
44. a. See Friedland Chapter 9.
b. Using the Development Technique described in Friedland, determine the IBNR as of 12/31/94.
c. See Friedland Chapter 15.

Earned
Premium

Accident
Year

4,500
5,000
5,200
5,300
5,700

1990
1991
1992
1993
1994

Reported Claims including ALAE ($000's omitted) as of
12 mo,
24 mo,
36 mo,
48 mo,
60 mo,
Report 1
Report 2
Report 3
Report 4 Report 5
2,000
2,102
2,234
2,339
2,482

2,600
2,638
2,938
2,985

2,990
3,086
3,408

3,283
3,343

3,283

Selected CDF calculations
ATA: 3-yr Volume-weighted average

12: 24 mo
1.2825
at 12 mo
1.6224

24: 36 mo
1.1600*
at 24 mo
1.2650**

36:48 mo
1.0905
at 36 mo
1.0905

48:60 mo
1.0000
at 48 mo
1.0000

Reported CDF to Ultimate

tail
1.00

* Example of Age-to-Age calculation for 24-to-36 months, using 3-year volume-weighted average:
(2990+3086+3408)/(2938+2638+2600) = 1.1600
** Example of Ultimate CDF calculation for claims at 24 months of development:
(1.1600 for 24:36 mo) * (1.0905 for 36:48 mo) * (1.0000 for 48:60 mo) * (1.0000 tail) = 1.2650

Accident
Year
1990
1991
1992
1993
1994
Total

Exam 5, V2

Age of
Data at
12/31/94

Reported
Claims at
12/31/94

Reported
CDF to
Ultimate

Expected
Ultimate
Claims

IBNR
(broadly
defined)

(1)
60 months
48 months
36 months
24 months
12 months

(2)
3,283
3,343
3,408
2,985
2,482

(3) above
1.0000
1.0000
1.0905
1.2650
1.6224

(4)=(2)*(3)
3,283
3,343
3,716
3,776
4,027

(5)=(4)-(2)
0
0
308
791
1,545
2,644

Page 104

OR:
Shortcut

IBNR
(broadly
defined)

(5)=(2)*[(3) - 1.0]
0
0
308
791
1,545
2,644

 2014 by All 10, Inc.

Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2002 Exam Questions (modified):
Question 22.
22 a. (1 point) Calculate the IBNR reserve as of December 31, 2001 using the Development technique.
b. (1 point) See Friedland Chapter 9.
c. (0.5 point) See Friedland Chapters 9 and 15.
d. (0.5 point) See Friedland Chapters 9 and 15.
e. (1 point) See Friedland Chapters 9 and 15.

Earned
Premium

Accident
Year

200
1,000
1,500
1,500

1998
1999
2000
2001

12 mo,
Report 1

100
1,000
900
600

ATA Factors
Reported CDF to Ultimate

Accident
Year
1998
1999
2000
2001
Total

Exam 5, V2

Reported Claims including ALAE ($000's omitted) as of
24 mo,
36 mo,
48 mo,
60 mo,
Report 2
Report 3
Report 4 Report 5

12: 24 mo
1.25
at 12 mo
1.5593

24: 36 mo
1.10
at 24 mo
1.2474

36:48 mo
1.05
at 36 mo
1.1340

48:60 mo
1.08
at 48 mo
1.0800
OR:
Shortcut

Age of
Data at
12/31/01

Reported
Claims at
12/31/01

Reported
CDF to
Ultimate

Expected
Ultimate
Claims

IBNR
(broadly
defined)

(1)
48 months
36 months
24 months
12 months

(2)
100
1,000
900
600

(3) above
1.0800
1.1340
1.2474
1.5593

(4)=(2)*(3)
108
1,134
1,123
936

(5)=(4)-(2)
8
134
223
336
700

Page 105

given
tail
1.00
IBNR
(broadly
defined)

(5)=(2)*[(3) - 1.0]
8
134
223
336
700

 2014 by All 10, Inc.

Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2003 Exam Questions (modified):
23. (3 points)
a. (1 point) Using the Development method, calculate the total IBNR reserve. Show all work.
b. (1 point) See Friedland Chapter 9.
c. (1 point) See Friedland Chapter 9.
Earned
Premium
1,000
1,000
1,500
1,800

Reported Claims including ALAE ($000's omitted)
Accident
at age
at age
at age
Year
12 mo
24 mo
36 mo
1999
250
500
750
2000
200
350
490
2001
300
450
2002
400

ATA factors by AY:
Example:
12:24 for AY 2000
1.75 = 350/200

at age
48 mo
825

12:24 mo

24:36 mo

36:48 mo

1999
2000
2001

2.0000
1.7500
1.5000

1.500
1.400

1.100

12: 24 mo
1.75*
at 12 mo
2.9309**

24: 36 mo
1.45
at 24 mo
1.6748

36:48 mo
1.10
at 36 mo
1.1550

ATA: Simple Average (all yr)
Reported CDF to Ultimate

See tail factor

at 48 mo given
1.05 tail

* Example of Age-to-Age calculation for 12:24 months report, using all-year simple average:
(2.00 + 1.75 + 1.50) / 3 = 1.75
** Example of Ultimate CDF calculation for claims at 12 months of development:
(1.75 for 12:24 mo) * (1.45 for 24:36 mo) * (1.10 for 36:48 mo) * (1.05 for tail at 48 mo) = 2.9309

Accident
Year
1999
2000
2001
2002
Total

Exam 5, V2

Age of
Data at
12/31/02

Reported
Claims at
12/31/02

Reported
CDF to
Ultimate

Expected
Ultimate
Claims

IBNR
(broadly
defined)

(1)
48 months
36 months
24 months
12 months

(2)
825
490
450
400

(3) above
1.0500
1.1550
1.6748
2.9309

(4)=(2)*(3)
866
566
754
1,172

(5)=(4)-(2)
41
76
304
772
1,193

Page 106

OR:
Shortcut

IBNR
(broadly
defined)

(5)=(2)*[(3) - 1.0]
41
76
304
772
1,193

 2014 by All 10, Inc.

Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2005 Exam Questions (modified):
10. (4 points) You are given the following information:
Note: Expected Claim Ratio is not used in the Development technique.
Choose selected factors using a straight average of the age to age factors.
Assume no development past 48 months.
a.
b.
c.
d.
e.

(1 point) Using the Development method, calculate the indicated IBNR for accident year 2004.
(0.5 point) See Friedland Chapter 9
(1 point) See Friedland Chapter 9
(0.5 point) See Friedland Chapters 9 and 15
(1 point) See Friedland Chapters 9 and 15

Earned
Premium

Accident
Year

19,000
20,000
21,000
22,000

2001
2002
2003
2004

ATA factors by AY:

Reported Claims by Development Age
at age
at age
at age
at age
12 mo
24 mo
36 mo
48 mo
4,850
9,700
14,100
16,200
5,150
10,300
14,900
5,400
10,800
7,200
AY

12:24 mo

24:36 mo

36:48 mo

2001
2002
2003

2.000
2.000
2.000

1.4536
1.4466

1.1489

12: 24 mo
2.0000
at 12 mo
3.3320

24: 36 mo
1.4501
at 24 mo
1.6660

36:48 mo
1.1489
at 36 mo
1.1489

ATA: Simple Average (all yr)
Reported CDF to Ultimate

Accident
Year
2001
2002
2003
2004
Total

See tail factor

at 48 mo given
1.00 tail

Age of

Reported

Expected

IBNR

OR:

IBNR

Data at
12/31/04

Claims at
12/31/04

CDF to
Ultimate

Ultimate
Claims

(broadly
defined)

Shortcut

(broadly
defined)

(1)
48 months
36 months
24 months
12 months

(2)
16,200
14,900
10,800
7,200

(3) above
1.0000
1.1489
1.6660
3.3320

(4)=(2)*(3)
16,200
17,119
17,993
23,990

(5)=(4)-(2)
0
2,219
7,193
16,790
26,202

(5)=(2)*[(3) - 1.0]
0
2,219
7,193
16,790
26,202

Note: Only the calculations for Accident Year 2004 are required:
7200 * (3.3320 - 1) =

Exam 5, V2

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16,790

 2014 by All 10, Inc.

Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2008 Exam Questions
Question 2

a) See chapter 9.
b) Calculate the ultimate claim estimate for policy year 2006 using the Development Method.
(applied to reported claims)
Recall, claim development factor = the inverse of the percent emerged
CDF at 24 months: 1/.68 =
1.4706(& at 36 mo: 1.2195)
For the development method , Ultimate Claims = Reported Claims * CDF
Dev. Method Ultimate Claims = 800,000 * (1.4706) =
1,176,480
OR using an extra step:
Ultimate Claims = Reported Claims + IBNR
where Development Method est. IBNR = (Reported Loss)*(CDF - 1)
so expected IBNR = (800,000)*[1.4706 - 1]
376,480
Ultimate Claims = 800,000 + 376,480 =
1,176,480
c) See chapters 9 and 15.
Question 10
b. See also solution to 2008 #36 - See also Brosius, Mack, Friedland Ch 8 & 9
Ultimate $ = [Reported Losses] * [Development Factor to ult]
= $10M* 1/.4 since development factor = inverse of % reported
= 25,000,000
* Note: we used cumulative development factor of 2.5 ( = 1/.4) The detail of % reported at other ages was not used.

Solutions to 2009 Exam Questions
Question 12 – Model Solution
a. (1 point) Using the volume-weighted average for B, calculate the values for A and B.
b. (1 point) Use the development technique to estimate the unpaid claim liability for accident year 2008 as of
December 31, 2008.
a. A = AY 2007 Incremental paid loss valued as of 12/31/2008 = 500 + 1,500 = 2,000
B = 12-24 month vol wtd avg = (1,500 + 2,100 + 2,000) / (1,100 + 1,500 + 1,500) = 5,600/4,100 = 1.366
b. 2008 Ult = AY 2008 paid loss at 12 months * LDF to ult = (1,000)(1.366)(1.069)(1.03)(1.0) = 1,504.06
2008 Unpaid = Ult - Paid = 1,504.06 – 1,000 = 504.06
Also, 1,000 × (1.366 × 1.069 × 1.03 × 1 – 1) = 504.06

Exam 5, V2

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Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2010 Exam Questions
13a. (1 point) Based on the data, provide two reasons why it would be inappropriate to combine these two
lines of business for estimating unpaid claims.
13b. (0.5 point) Briefly describe two additional factors that generally should be considered when deciding
whether to combine lines of business for estimating unpaid claims.
Question 13 - Solution 1
a1. Commercial is growing at a faster pace than personal.
a2. Commercial has a longer reporting pattern (or at least a different reporting pattern)
b1. Severity may differ between the two lines
b2. Credibility of each line – may want to combine to improve credibility
Question 13 - Solution 2
a1. All year weighted Average LDF’s
12-24

24-36

36-48

Commercial APD

1.4

1.1

Personal APD

1.2

1.2 = [25.2+16.8]/[21+14]
1.1

1.01

The two lines of business have different reporting patterns as seen above
a2. AY trend in C-APD
12

2006-2007

50%

50%=21/14 -1.0

50%

2007-2008

33%

2008-2009

25%

AY trend in P-APD
12

36
10%

2006-2007

10%

2007-2008

9.1%

9.1%=14.4/13.2 -1.0

2008-2009

8.3%

It appears that the CAPD book is growing much faster than the PAPD based on the AY trends
b1. Credibility of data – want block of data to be large enough and homogenous enough
b2. Coverage trigger – don’t want to group claims made policies with occurrence

Exam 5, V2

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Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2011 Exam Questions
a. (1.5 points) Use the reported development technique to estimate the unpaid claims for accident year 2009.
b. (0.5 point) Calculate the expected reported claims for accident year 2010 during the next 12 months.
c. (0.75 point) State three assumptions underlying the reported development technique.
d. (0.25 point) Briefly describe when a tail factor may be needed to estimate unpaid claims under the reported
development technique.
e. (0.5 point) Briefly describe two approaches to determine a tail factor.
Question 24 – Model Solution 1
Age-age LDFs
2006
2007
2008
2009

12-24
2.524
2.750
2.457
2.541

24-36
1.283
1.309
1.316

36-48
1.103
1.083

48-60
1.013

60-ult
1.0

All Year Straight Average

12-24
2.568
12-ult

24-36
1.303
24-ult

36-48
1.093
36-ult

48-60
1.013
48-ult

60-ult
1.0
60-ult

Cumulative LDFs

3.705

1.443

1.107209=1.013*1.093

1.013

1.0

Note: This model solution ignored rounding issues beyond thee decimal places, yet still received full credit.
Examiners will focus more on determining if the technique is applied correctly than on rounding. For example,
see model solution 2.
a. AY 2009 Reported Losses * 24-ult LDF – AY 2009 Paid losses at 24 months
[310 x 1.443 -250] x 1000 = 197,330
b. [2010 reported claims @ 12 mo] x [ 12-24 factor – 1 ]
128,000 x (2.568 – 1) = 200,704
c. 1. Future development will be similar to prior dev.
2. Implied assumption that losses to immature accident year tell you something about losses not reported yet.
3. Stable claims practices (i.e., no change in case reserve adequacy)
d. When you don’t have enough loss history in the triangles such that development ceases.
e. 1. Extrapolate based on selected development pattern
2. Use industry benchmarks that are appropriate for line of business being reviewed.

Exam 5, V2

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Chapter 7 – Development Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2011 Exam Questions
Question 24 – Model Solution 2
a. Age-to-age factors as of
AY
2006
2007
2008
2009

12-24
2.5238
2.75
2.4569
2.5410

24-36
1.2830
1.3091
1.3158

Selected link ratio
CDF to ultimate

36-48
1.1029
1.0833

12-24
2.5679
3.7050

48-60
1.0133

24-36
1.3026
1.4428

36-48
1.0931
1.1076

48-60
1.0133
1.0133

60-ult
1.00
1.00

AY 2009 unpaid claims = 310,000 x 1.4428 – 250,000 = 197,268
b. Initial comments
Expected reported claims in the calendar year are equal to:
[(ultimate claims selected at 12/31/2010 - actual reported claims at 12/31/2010) / (%
unreported at 12/31/2010)] x (% reported at 12/31/2011 - % reported at 12/31/2010)]
The % unreported is computed as [1.00 - (1.00 / cumulative claim development factor)]. See Chapt 15.
AY 2010 Expected Reported Claims in the next 12 months
= [(128 * 3.7050 - 128)/(1 - (1/3.7050))] x (1/1.4428 – 1/3.7050) x 1000 = 200,694
c. 1. Reported claims will continue to development in a similar manner in the future.
2. Consistent claims process. no change in case reserve adequacy.
3. Consistent policy limits, retention limits, mix of claim types.
d. When the age-to age factor is still greater than 1.00 in the last development period.
e. 1. Use insurance industry benchmark data.
2. Fit a curve using average or selected LDF exponential decay model.
Question 36 – Model Solution 1
a. Will cause speed-up in claim settlement
Paid method – will overstate ult. Because dev. Factors are selected based on old pattern.
Reported method- accurate assuming reserves were set correctly and unaffected by this change
b. Will cause lower severity
Both methods will produce inaccurate estimates (overstated) if unadjusted for tort reform impact.
c. Increase in losses due to writing higher limits
Both methods will produce inaccurate estimates (Understated) – old data will have smaller dev factors
because pol. limits were reached quicker.
d. i) look at ratios of paid-to-reported claims. They will show increase if there is speedup.
ii) Look at avg. paid and avg reported – they will show decrease
iii) Look at avg. paid, avg. reported, ult loss ratios – they all should show increase.

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ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2011 Exam Questions
Question 36 – Model Solution 2
a. There will be a speed up of claims being settled so if the two developments are not adjusted for this
change, they will overstate the unpaid claim liabilities.
b. The average losses should decrease with this tort reform, so the unadjusted methods (both paid and
reported) will overstate the unpaid claim liabilities.
c. The losses should increase as the policy limits increase, so both methods will understate the unpaid claims
if they are not adjusted.
d. i. Look at paid claims to reported and see if there is an increasing ratio or look at closed counts to reported
counts ratios and see if it is increasing.
ii. Analyze the average reported and average paid values and look for downward trend between accident
years.
iii. Analyze average reported and average paid amount and look for a positive trend between subsequent
accident years.

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Chapter 8 – Expected Claims Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Sec
1
2
3
4
5
6
7
8
9

Pages
131
131
131
131 - 132
133 - 134
134
135
136 - 137
137

11
12

Description
Introduction
Key Assumption
Common Uses of the Expected Claims Method
Mechanics of the Expected Claims Method
Step-by Step Example – Auto BI Insurer
Step-by Step Example – GL Self Insurer
Step-by Step Example – U.S. Industry Auto
XYZ Insurer
When the Expected Claims Technique Works and When It Does
Not
Influence of a Changing Environment on the Expected Claim
Technique
U.S. Auto Steady-State (No Change in Product Mix)
U.S. Auto Changing Product Mix

Introduction

131

137 - 139
139
139

Expected claims are a critical component of other methods including the Bornhuetter-Ferguson and Cape
Cod techniques (discussed in Chapters 9 and 10)
 The expected claims method can be used with all lines of insurance.
 The method can be used with data organized by AY, PY, U/W Y, and CY data.

Key Assumption

131

A better estimate of total unpaid claims can be made based on an a priori (or initial) estimate than from claims
experience observed to date. At times, the claims experience reported to date may provide little information
about ultimate claims (compared to the a priori estimate).

Common Uses of the Expected Claims Method

131

This method is used in lines of business with longer emergence patterns and settlement patterns. The
expected claims method is often used:
* when an insurer enters a new line of business or a new territory.
* when operational or environmental changes make recent historical data irrelevant for projecting future
claims activity for that cohort of claims.
* for the most recent years in the experience period, since cumulative CDFs are highly leveraged.
* when data is unavailable for other methods.
* for the latest year in the experience period after major changes in the legal environment take place.
Examples: an increase in the statute of limitations for filing claims or expanded coverage due to recent
court decisions are changes in the legal environment that can affect insurers' claims liabilities.

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4

Mechanics of the Expected Claims Method

131 - 132

Ways to determine the a priori expected claims (from mathematically simple to complex statistical modeling):
1. Commercial insurers apply a claim ratio method. Ultimate claims = a selected expected claim ratio * EP
This approach implicitly relies on accurate underwriting and policy pricing.
2. A complex simulation model may require variables such as the opinions of an experts, lawyers, and
various practitioners as well as a detailed analyses of the frequency rate and severity of claims.
This chapter focuses on exposure-based methods for determining expected claims.
Expected claims = A predetermined exposure base * claims per unit of exposure (a.k.a. the pure premium
or the loss rate).
The unpaid claim estimate = projected expected claims - paid claims.
Two challenges when using the expected claims method:
1. Determining the appropriate exposure base (often EP).
2. Estimating the measurement of claims relative to that exposure base (often the claim ratio).
Since self-insureds do not collect premiums in the same way that an insurer does, the exposure base that is
chosen needs to be one that is closely related to the risk and thus the potential for claims and is readily
observable and available.
The following table shows types of exposures often used for the analysis of self-insurers' unpaid claims.

Line of Insurance
U.S. workers
Automobile liability
General liability for public
General liability for
Hospital professional
Property
Crime

Exposure
Payroll
Number of vehicles or miles driven
Population or operating expenditures
Sales or square footage
Average occupied beds and outpatient visits
Property values
Number of employees

Computing the claim ratio or pure premium:
 Begin with a review of the historical claims and exposure experience.
 Two examples of the expected claims method are shown in Exhibit I, Sheets 1 and 2.
 The expected claims method is used to estimate unpaid claims for AY 2008 only.
 Historical reported and paid claims data as well as exposure data from each organization is also used.

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5

Step-by Step Example – Auto BI Insurer

133 - 134

Step 1: Compute initial selected ultimate claims as the average of the reported and paid claim
development projections. See Column (8).
Step 2: Develop an expected claim ratio for AY 2008. See Column (13) and Line (14)
Trended adjusted claim ratios = trended adjusted claims/ on-level earned premiums.
Trended adjusted claims = initial selected ultimate claims * trend factor * tort reform factor.
Trend factor uses a 14.5% annual claim trend rate for auto BI, incorporating both frequency and severity
trends. The trend period is from the midpoint of the AY to 7/1/2008.
Tort reform factor. To account for the significant reform during 2004, historical claims are multiplied by a
reform adjustment factor of 0.67 (i.e. removing 33% of the claims for the oldest years in the experience
period). The adjustment rationale is that if the same type of claims that occurred in 2000 - 2003 were to
occur in 2008, they would cost 33% less.
Since the reform was introduced during 2004, the pro rata adjustment factor for 2004 is only 0.75, a 25%
reduction.
Development of Unpaid Claim Estimate for Accident Year 2008
Auto BI Insurer

Accident
Year
(1)
2000
:::
2004
2005
2006
2007
2008

Exhibit I
Sheet 1

Projected Ultiamate
Claims at 12/31/098
CDF to Ultimate
Claims Based On
Reported
Paid
Reported Paid
Reported
Paid
(2)
(3)
(4)
(5)
(6) = [(2) x (4)] (7) = [(3) x(5)]
10,000,000 9,500,000
1.005
1.050
10,050,000
9,975,000
:::
:::
:::
:::
:::
:::
16,500,000 11,200,000 1.200
1.750
19,800,000
19,600,000
18,500,000 10,200,000 1.400
2.500
25,900,000
25,500,000
16,500,000 6,000,000
1.800
5.000
29,700,000
30,000,000
14,000,000 3,000,000
2.900
15.000
40,600,000
45,000,000
8,700,000
750,000
4.000
90.000
34,800,000
67,500,000

Initial Selected
On-Level
Trend at Adjusted
Trended
Trended
Ultimate
Earned
14.50%
for Tort
Adj. Ultimate
Adjusted
Claims
Premium
to 7/1/08
Reform
Claims
Claim Ratio
(12) = [(8) x(10) x(11)] (13) = [(12) / (9))]
(8) =[(6)+(7)]/2
(9)
(10)
(11)
10,012,500
24,000,000
2.954
0.670
19,816,540
82.6%
:::
:::
:::
:::
:::
:::
19,700,000
32,000,000
1.719
0.750
25,398,225
79.4%
25,700,000
47,000,000
1.501
1.000
38,575,700
82.1%
29,850,000
50,000,000
1.311
1.000
39,133,350
78.3%
42,800,000
57,000,000
1.145
1.000
49,006,000
86.0%
51,150,000
62,000,000
1.000
1.000
51,150,000
82.5%
(14) Average Claim Ratio at 7/1/2008 Cost Level
Average 2000 to 2005
79.8%
Average 2000 to 2005 Excluding High and Low
79.9%
Average 2001 to 2006
79.0%
Average 2001 to 2006 Excluding High and Low
79.0%
(15) Selected Claim Ratio at 7/1/2008 Cost Level
80.0%
(16) Expected Claims for 2008 Accident Year
49,600,000
(17) Unpaid Claim Estimate for 2008 Accident Year
Total
48,850,000
IBNR
40,900,000

Column and Line Notes:
(2) and (3) Based on data provided by commercial insurer.
(4) and (5) Based on commercial insurer historical claim development experience.
(9)Based on data provi-ded by commercial insurer.
(10) Assume 14.5% annual trend iii private passenger auto bodily injury liability claims. Trend front midpoint of accident year to 7/1/08.
(11) Adjusts for law reforms-in private passenger auto implemented during experience period.
(14) Various averages of claim ratios in (13).
(15) Selected based on claim ratios by year in (13) and various averages in (14).
(16) Based on selected claim ratio at 2008 cost level and accident year 2008 earned premiums. (16) = [ (15) x (9) for 2008].
(17) Total unpaid claim estimate is equal to expected claims in (16) less paid claims for 2008. IBNR is equal to expected claims in (16) less reported claims for 2008.

Step 3: Determine the selected claim ratio at 7/1/2008 cost level. See Line (15)
Review various averages of individual AYs claim ratios, excluding claims ratios from the most recent years,
because the paid and reported development factors from those years are highly leveraged.
Select a claim ratio based on a review of the individual AY projected claim ratios and the various averages.
Step 4: Determine expected claims for AY 2008 See Line (16)
Expected claims Line (16) equal to selected claim ratio of 80% * CY 2008 EP
Step 5: Compute estimated unpaid claims and IBNR for AY 2008 See Line (17)
Estimated unpaid claims = AY 2008 Expected claims – AY 2008 Paid claims
Estimated IBNR = AY 2008 Expected claims – AY 2008 Reported claims

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6

Step-by Step Example – GL Self Insurer

134

Exhibit 1, Sheet 2: Calculation for a public entity self-insurer's general liability program (GL Self-Insurer).
Step 1: Compute initial selected ultimate claims as the average of the reported and paid claim development
projections. See Column (8).
Step 2: Develop trended pure premiums for AY 2008. See Column (12) and Line (13)
Trended pure premiums = trended ultimate claims/ population
Trended ultimate claims = initial selected ultimate claims * trend factor
Trend factor uses a 7.5% annual claim trend rate, incorporating both frequency and severity
trends. The trend period is from the midpoint of the AY to 7/1/2008.
An alternative to trending claims and exposures separately when the exposures are inflationsensitive is to use a residual pure premium trend rate. For WC a residual pure premium trend is
used that represents the trend in claims that is in excess of the trend in payroll.
Development of Unpaid Claim Estimate for Accident Year 2008
Auto BI Insurer

Accident
Year
(1)
1998
:::
2007
2008

Claims at 12/31/08
CDF to Ultimate
Reported
Paid
Reported
Paid
(2)
(3)
(4)
(5)
900,000
890,000
1.015
1.046
:::
:::
:::
:::
1,200,000 750,000
1.940
5.093
600,000
170,000
3.104
20.373

Exhibit I
Sheet 2

Projected Ultiamate
Claims Based On
Reported
Paid

Initial Selected
Ultimate
Claims
(6)=[(2)x(4)] (7)=[(3)x(5)]. (8) =[(6)+(7)]/2
913,500
930,940
922,220
:::
:::
:::
2,328,000 3,819,750
3,073,875
1,862,400 3,463,410
2,662,905

Population
(9)
709,000
:::
785,000
790,000

Trend at
Trended
7.50%
Ultimate
Trended
to 7/1/08
Claims
Pure Premium
(10)
(11)=[(8)x(10) (12)=[(11)/(9)]
2.061
1,900,695
2.68
:::
:::
:::
1.075
3,304,416
4.21
1.000
2,662,905
3.37

(13) Average Pure Premium at 7/1/2008 Cost Level
Average 2000 to 2005
Average 2000 to 2005 Excluding High and Low
Average 2001 to 2006
Average 2001 to 2006 Excluding High and Low
(14) Selected Pure Premium at 7/1/2008 Cost Level
(15) Expected Claims for 2008 Accident Year
(16) Unpaid Claim Estimate for 2008 Accident Year
Total
IBNR

3.55
3.52
3.50
3.45
3.50
2,765,000
2,595,000
2,165,000

Column and Line Notes:
(2) and (3) Based on data provided by public entity.
(4) and (5) Based on insurance industry benchmark claim development patterns.
(9) Based on data provided by public entity.
(10) Assume 7.5% annual trend in general liability claims. Trend from midpoint of accident year to 7/1/08.
(13) Various averages of pure premium in (12).
(14) Selected based on pure premium by year in (12) and various averages in (13).
(15) Based on selected pure premium at 2008 cost level and accident year 2008 population. (15) = [(14) x (9) for 2008].
(16) Total unpaid claim estimate = expected claims in (15) - paid claims for 2008. IBNR = expected claims in (15) - reported claims for 2008

Step 3: Determine the selected pure premium at 7/1/2008 cost level. See Line (14)
Review averages of individual AYs trended pure premiums, excluding claims ratios from the most recent
years, because the paid and reported development factors from those years are highly leveraged.
Select a pure premium based on a review of individual AY projected pure premiums and the various
averages.
Step 4: Determine expected claims for AY 2008 See Line (15)
Expected claims Line (16) = selected pure premium * the 2008 population
Step 5: Compute estimated unpaid claims and IBNR for AY 2008 See Line (16)
Estimated unpaid claims = AY 2008 Expected claims – AY 2008 Paid claims
Estimated IBNR = AY 2008 Expected claims – AY 2008 Reported claims

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7

Step-by Step Example – U.S. Industry Auto

135

Exhibits II through V continue with the examples presented in Chapter 7.
Exhibit II – Expected Claims Technique for the U.S. Industry Auto.
Step 1: Compute initial selected ultimate claims as the average of the reported and paid claim development
projections. See Column (8).
Step 2: Compute estimated claims ratios equal to Step 1 result / EP. Since the EP represents consolidated
results for the entire U.S. insurance industry, there is no detailed information regarding rate changes
and thus premiums cannot be adjusted an on-level basis.
Chapter 8 - Expected Claims Technique
U. S. Industry Auto
Projection of Expected Claims ($000)

Accident
Year
(1)
1998
:::
2002
2003
2004
2005
2006
2007

Claims at 12/31/098
CDF to Ultimate
Reported
Paid
Reported Paid
(2)
(3)
(4)
(5)
47,742,304 47,644,187 1.000
1.002
:::
:::
:::
:::
58,592,712 57,807,215 1.006
1.020
57,565,344 55,930,654 1.011
1.040
56,976,657 53,774,672 1.023
1.085
56,786,410 50,644,994 1.051
1.184
54,641,339 43,606,497 1.110
1.404
48,853,563 27,229,969 1.292
2.390

Exhibit II
Sheet 1
Projected Ultimate Claims
Initial Selected
Using Dev. Method with
Ultimate
Reported
Paid
Claims
(6)= [(2) x (4)] (7)= [(3) x (5)] (8)=[((6)+(7))/2]
47,742,304
47,739,475
47,740,890
:::
:::
:::
58,944,913
58,971,536
58,958,225
58,200,926
58,141,265
58,171,096
58,297,009
58,359,672
58,328,341
59,671,116
59,964,795
59,817,955
60,632,434
61,234,435
60,933,434
63,100,513
65,080,550
64,090,532

Earned
Premium
(9)
68,574,209
:::
79,228,887
86,643,542
91,763,523
94,115,312
95,272,279
95,176,240

Claim Ratio
Expected
Estimated Selected
Claims
(10)
(11) (12)=[(9)x(11)]
69.6%
75.0% 51,430,657
:::
:::
:::
74.4%
75.0% 59,421,665
67.1%
65.0% 56,318,302
63.6%
65.0% 59,646,290
63.6%
65.0% 61,174,953
64.0%
65.0% 61,926,981
67.3%
65.0% 61,864,556

Column and Line Notes:
(2) and (3) Based on Best's Aggregates & Averages U.S. private passenger automobile experience.
(4) and (5) Developed in Chapter 7, Exhibit I, Sheets 1 and 2.
(8) Based on average of paid and reported claim projections. (8) = [((6)+(7))/2].
(9) Based on Best's Aggregates & Averages U.S. private passenger automobile experience.
(10) = [(8) / (9)]
(11) Selected judgmentally based on experience in (10).

Note: (6) and (7) based on unrounded (4) and (5) CDFs. Thus, these values do not match those in the
corresponding exhibit in the Friedland text. However, the formulas, which are shown correctly, are what
matter most when preparing for the exam.
Exhibit II differs somewhat from the prior two examples in this chapter in the time period for which the
expected claims method is used.
 In the first two examples, we use historical experience to select an expected claim ratio and an
expected pure premium for the 2008 AY only. Experience period exposures and claims are adjusted
to the 2008 cost level.
 In U.S. Industry Auto example, ultimate claims for each year in the experience period are projected
based on the expected claims technique. This requires a claim ratio at the expected cost level for
each year in the experience period.
For the most recent years, either review estimated claim ratios from prior years on a trended and
adjusted basis, or use significant judgment when selecting expected claim ratios.
See Column (11) of Exhibit II, Sheet 1.
Selected expected claim ratios are 75% for AYs 1998 - 2002 and 65% for AYs 2003 - 2007.
Actuarial judgment is used by selecting two different claim ratios to reflect the change in experience
that is apparent between the older accident years and the more recent accident years.

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ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
In Exhibit II, Sheet 2
Estimated IBNR Column (6) = expected claims Column (4) - reported claims in Column (2).
Estimated total unpaid claims = expected claims - paid claims = (sum of case outstanding + IBNR).
U. S. Industry Auto
Exhibit II
Development of Unpaid Claim Estimate($000)
Sheet 2

Accident
Year
(1)
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
Total

Claims at 12/31/07
Reported
Paid
(2)
(3)
47,742,304
47,644,187
51,185,767
51,000,534
54,837,929
54,533,225
56,299,562
55,878,421
58,592,712
57,807,215
57,565,344
55,930,654
56,976,657
53,774,672
56,786,410
50,644,994
54,641,339
43,606,497
27,229,969
48,853,563
543,481,587 498,050,368

Expected
Claims
(4)
51,430,657
51,408,736
51,680,983
54,408,716
59,421,665
56,318,302
59,646,290
61,174,953
61,926,981
61,864,556
569,281,839

Case
Outstanding
at 12/31/07
(5) = [(2) - (3)]
98,117
185,233
304,704
421,141
785,497
1,634,690
3,201,985
6,141,416
11,034,842
21,623,594
45,431,219

Unpaid Claim Estimate Based
on Expected Claims Method
IBNR
Total
(6) = [(4) - (2)]
(7) = [(4) - (3)]
3,688,353
3,786,470
222,969
408,202
-3,156,946
-2,852,242
-1,890,846
-1,469,705
828,953
1,614,450
-1,247,042
387,648
2,669,633
5,871,618
4,388,543
10,529,959
7,285,642
18,320,484
34,634,587
13,010,993
25,800,252
71,231,471

Column Notes:
(2) and (3) Based on Best's Aggregates & Averages U.S. private passenger automobile experience.
(4) Developed in Exhibit II, Sheet 1.

Negative IBNR for AYS 2000, 2001, and 2003:
 While negative IBNR is possible (e.g. for first-party lines subject to salvage and subrogation (S&S)
recoveries, it is not likely for U.S. Industry Auto.
 Use of a priori estimate to determine expected claims is at times a strength of the expected claims
method and at times (as in this example)a weakness of the method.
 The negative IBNR is a result of the selected a priori claim ratio being too low for certain AYs years.
An approach to correct negative IBNR:
 Use a 65% claim ratio assumption for AYs 2005 - 2007 and rely on the estimated claim ratios in
Column (10) for all prior years (i.e. AYs 1998 - 2004).
Why this approach is sound:
Since expected claims unreported and unpaid for the older years are low, the claim development
methods produce more reasonable results (Note, for AY 2004, the % of claim unreported at 12/31/2007 is
only 2% and the % unpaid is 8%).

XYZ Insurer

136 - 137

See Exhibit III
Q. Should the claim development method be used for XYZ Insurer?
A. Due to the various changes experienced by XYZ Insurer, the primary claim development assumptions do
not hold.

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Alternatives for selecting expected claim ratios for XYZ Insurer include:
1. Use insurance industry experience for benchmark claim ratios.
Ultimate claim ratios for the aggregated insurance industry experience are approximately 50%.
However, since XYZ Insurer's undeveloped reported claim ratios (i.e. current value of reported claims/EP)
are greater than 70% for 6 of the 7 earliest AYs in the experience period, using industry claim ratios does
not appear reasonable.
2. Use the unadjusted reported and paid claim development methods as a starting point.
XYZ Insurer - Auto BI
Projection of Expected Claims ($000)

Accident
Year
(1)
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008

Claims at 12/31/08
Reported
Paid
(2)
(3)
15,822
15,822
25,107
24,817
37,246
36,782
38,798
38,519
48,169
44,437
44,373
39,320
70,288
52,811
70,655
40,026
48,804
22,819
31,732
11,865
18,632
3,409

Exhibit III
Sheet 1

Projected Ultimate Claims Initial Selected
CDF to Ultimate
Using Dev. Method with
Ultimate
Reported
Paid
Reported
Paid
Claims
(6)= [(2) x (4)] (7) = I(3) x (5)] (8)=[((6)+(7))/2]
(4)
(5)
1.000
1.010
15,822
15,980
15,901
0.999
1.014
25,082
25,165
25,124
0.992
1.031
36,948
37,932
37,440
0.992
1.054
38,488
40,598
39,543
1.003
1.116
48,310
49,598
48,954
1.013
1.268
44,948
49,856
47,402
1.064
1.525
74,758
80,555
77,656
1.085
2.007
76,651
80,346
78,499
1.196
3.160
58,346
72,098
65,222
1.512
6.569
47,990
77,938
62,964
2.551
21.999
47,536
74,994
61,265

Earned
Premium
(9)
20,000
31,500
45,000
50,000
61,183
69,175
99,322
138,151
107,578
62,438
47,797

Claim Ratio
Estimated
Selected
(10)=[(8)/(9)]
(11)
79.5%
78.3%
79.8%
78.3%
83.2%
78.3%
79.1%
78.3%
80.0%
78.3%
68.5%
78.3%
78.2%
87.1%
56.8%
78.3%
60.6%
65.8%
100.8%
63.8%
128.2%
82.5%

Expected
Claims
(12) =[(9)x(11)]

15,660
24,665
35,235
39,150
47,906
54,164
86,509
108,172
70,786
39,835
39,433

Column and Line Notes:
(2) and (3) Based on data from XYZ Insurer.
(4) and (5) Developed in Chapter 7, Exhibit II, Sheets 1 and 2.
(8) Based on average of paid and reported claim projections. (8) = [((6) + (7)) / 2].
(9) Based on data from insurer.
(11) Selected for 1998 through 2003, based on average of estimated claim ratios in (10) for these years. For 2004 through 2008, selected in Exhibit III, Sheet 2.

Exhibit III calculations:
i. Use the reported and paid development methods to determine an initial estimate of ultimate claims for
AYs 1998 – 2003 (the most mature years in the experience period) and select the expected claim ratio
based on the average of the estimated claim ratios in Column (10).
ii. Use selected expected claim ratios in Exhibit III, Sheet 2.

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Chapter 8 – Expected Claims Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
XYZ Insurer - Auto BI
Selection of Expected Claim Ratios ($000)

Exhibit III
Sheet 2

Accident
Year
(1)
2002
:::
2008

Initial Selected
Ultimate
Claims
(2)
48,954
:::
61,265

2004
(3)
1.070
:::
0.874

Trend Adjustment
2005
2006
2007
(4)
(5)
(6)
1.106
1.144
1.183
:::
:::
:::
0.904
0.935
0.967

2008
(7)
1.224
:::
1.000

Accident
Year
(1)
2002
:::
2008

Earned
Premium
(2)
61,183
:::
47,797

2004
(14)
1.129
:::
0.205

Real Level Adjustment
2005
2006
2007
(15)
(16)
(17)
1.298
1.428
1.142
:::
:::
:::
1.420
1.563
1.250

2008
(18)
0.914
:::
1.000

2004
(8)
1.000
:::
1.493

Tort Reform Adjustmant
2005
2006
2007
(9)
(10)
(11)
1.000
0.893
0.670
:::
:::
:::
1.493
1.333
1.000

2008
(12)
0.670
:::
1.000

Trended Adjusted On-Level Claim Ratios
2004
2005
2006
2007
2008
(19)
(20)
(21)
(22)
(23)
75.8%
68.2%
57.3%
55.5%
71.8%
:::
:::
:::
:::
:::
135.4% 121.7% 102.3%
99.2%
128.2%

(24) Average Claim Ratios
All Years
90.9%
All Years excluding High and Low 87.1%
Latest 5 Years
98.9%
Latest 3 Years
117.8%

81.8%
78.3%
89.0%
105.9%

68.7%
65.8%
74.7%
89.0%

66.6%
63.8%
72.5%
86.3%

86.1%
82.5%
93.7%
111.5%

(25) Selected Expected Claim Ratio

78.3%

65.8%

63.8%

82.5%

87.1%

Column and Line-Notes:
(2) Developed in Exhibit III, Sheet 1
(3) through (7) Assume annual pure premium trend rate of 3.425%. Adjust claims to average cost level of particular AY
(8) through (12)Based on independent analysis of tort reform. Adjust claims to tort environment of particular AY
(13) Based on data from XYZ Insurer.
(14) through(18) Based on rate level changes in Chapter 6. Adjusts earned premium to rate level in effect for particular AY
Students should refer to ratemaking papers for the on-level factors calculation procedure
(19) through (23) Equal to [(initial selected ultimate claims x trend adjustment x tort- reform adjustment)/(Epx rate level adj)].
(24) Averages based on claim ratios in (19) through (23).
(25) Selected based on review of claim ratios by year in (19) through (23) and average claim ratios in (24).

For the most recent AYs, 2004 - 2008, Columns (3) through (7) contain trend factors adjusted for inflation.
The annual claim trend rate is 3.425% (based on an annual frequency trend of -1.50% and an annual severity
trend of 5.00%).
Loss and Premium Adjustments:
Next, adjust the initial ultimate claims for each year in the experience period using these factors to the
cost level for each particular year under examination (i.e. 2004 - 2008).
Adjusting initial ultimate claim examples:
For AY 2008 adjustment to the inflation level expected in accident year 2004, compute
(2004-2008)
= .874 (appearing at the bottom of column (3)).
1.03425
For AY 2002 adjustment to the inflation level expected in accident year 2004, compute
(2004-2002)
= 1.070 (appearing at the top of column (3)).
1.03425
A second adjustment to ultimate claims is for tort reform, shown in Columns (8) through (12).
A third adjustment is to bring EP to current rate level changes.
In Chapter 6, we summarized EP and the historical rate level changes for XYZ Insurer.
Columns (14) - (18) show on-level factors that adjust the EP in Column (13) to the rate level for the
particular AY (i.e. this adjustment restates the premium as if the exposures were written at the rate level
that was in effect for each particular year).

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Chapter 8 – Expected Claims Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Next Step: Computing trended and adjusted on-level claim ratios (see Columns (19) - (23))
These claim ratios equal:
The initial estimate of ultimate claims * the trend factors and the tort reform adjustment factors/ EP
adjusted to the appropriate rate level for each year.
Next Step: Select expected claim ratios (see Line (25) of Exhibit III, Sheet 2) after examining various
averages of the claim ratios by year.
Final Step: Compute expected claims in Column (12) in Exhibit III, Sheet 1.
Expected claims in Column (12), for AYs 2004 – 2008, are calculated selected expected claim ratios in
Column (11) (from Line (25) above) time EP in Column (9)
Estimated IBNR and estimated total unpaid claims are calculated in Exhibit III, Sheet 3.

XYZ Insurer - Auto BI
Development of Unpaid Claim Estimate ($000)

Accident
Year
(1)
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
Total

Claims at 12/31/08
Reported
Paid
(2)
(3)
15,822
15,822
25,107
24,817
37,246
36,782
38,798
38,519
48,169
44,437
44,373
39,320
70,288
52,811
70,655
40,026
48,804
22,819
31,732
11,865
3,409
18,632
449,626
330,627

Exhibit III
Sheet 3

Case
Unpaid Claim Estimate Based
Expected Outstanding on Expected Claims Method
Claims
at 12/31/08
IBNR
Total
(4)
(5)=[(2)-(3)] (6) = [(4)-(2)) (7) = [(4) - (3)]
15,660
0
-162
-162
24,665
290
-443
-153
35,235
464
-2,011
-1,547
39,150
279
352
631
47,906
3,732
-263
3,469
54,164
5,053
9,791
14,844
86,509
17,477
16,221
33,698
108,172
30,629
37,517
68,146
70,786
25,985
21,982
47,967
39,835
19,867
8,103
27,970
39,433
15,223
20,801
36,024
561,516
118,999
111,890
230,889

Column Notes:
(2) and (3) Based on data from XYZ Insurer.
(4) Developed in Exhibit III, Sheet 1.
Finally, we compare the results of the expected claims method with the claim development method in:
 Exhibit III, Sheet 4 (projected ultimate claims) and in
 Exhibit III, Sheet 5 (estimated IBNR).

Exam 5, V2

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Chapter 8 – Expected Claims Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
XYZ Insurer - Auto BI
Summary of Ultimate Claims ($000)

Accident
Year
(1)
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
Total

Exhibit III
Sheet 4

Projected Ultimate Claims
Development Method
Expected
Reported
Paid
Claims
(4)
(5)
(6)
15,822
15,980
15,660
25,082
25,164
24,665
36,948
37,922
35,235
38,488
40,599
39,150
48,314
49,592
47,906
44,950
49,858
54,164
74,786
80,537
86,509
76,661
80,332
108,172
58,370
72,108
70,786
47,979
77,941
39,835
47,530
74,995
39,433
514,929
605,028
561,516

Column Notes:
(2) and (3) Based on data from XYZ Insurer.
(4) and (5) Developed in Chapter 7, Exhibit II, Sheet 3.
(6) Developed in Exhibit III, Sheet 1.

XYZ Insurer - Auto BI
Summary of IBNR ($000)
Case
Accident Outstanding
Year
at 12/31/08
(1)
(2)
1998
0
1999
290
2000
465
2001
278
2002
3,731
2003
5,052
2004
17,477
2005
30,629
2006
25,985
2007
19,867
2008
15,223
Total
118,997

Exhibit III
Sheet 5

Estimated IBNR
Development Method
Expected
Reported
Paid
Claims
(3)
(4)
(5)
0
158
-162
-25
57
-443
-298
676
-2,011
-310
1,801
352
145
1,423
-263
577
5,485
9,791
4,498
10,249
16,221
6,006
9,677
37,517
9,566
23,304
21,982
16,247
46,209
8,103
28,898
56,363
20,801
65,303
155,402
111,890

Column Notes:
(2) Based on data from XYZ Insurer.
(3) and (4) Estimated in Chapter 7, Exhibit II, Sheet 4
(5) Estimated in Exhibit III, Sheet 3

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Chapter 8 – Expected Claims Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
9

When the Expected Claims Technique Works and When It Does
Not

137

An important assumption:
A reliable value of the expected claim ratio can be made that takes into account a changing legal
environment for the insurance coverage.
Advantage to using the expected claims technique:
The technique maintains stability over time since actual claims do not enter into the calculations.
The claim ratios can be judgmentally adjusted based on historical experience due to a belief that either the
pricing or underwriting or both are changing.
Disadvantage:
It is not responsive when actual claims experience differs from the initial expectations.
This was evident in the U.S. Industry Auto example discussed in this chapter.

Influence of a Changing Environment on the Expected Claim
Technique

137 - 139

In Chapter 7, the performance of the development method during times of change was discussed. Below,
the same examples are shown but now using the expected claims technique.
Scenario 1 — U.S. PP Auto Steady-State Environment
Exhibit IV, Sheet 1, top section.
Assume the expected claim ratio equals the ultimate claim ratios which equals 70%.
 Thus, the expected claims technique generates the correct estimate of IBNR in a steady-state
environment.
 This is also true of the development technique in a steady-state environment.
Impact of Changing Conditions
U. S. PP Auto - Development of Unpaid Claim Estimate

Accident
Year
(1)
Steady-State
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
Total

Earned
Premium
(2)
1,000,000
1,050,000
1,102,500
1,157,625
1,215,506
1,276,282
1,340,096
1,407,100
1,477,455
1,551,328
12,577,892

Earned
Claim
Ratio
(3)
70.0%
70.0%
70.0%
70.0%
70.0%
70.0%
70.0%
70.0%
70.0%
70.0%

Exhibit IV
Sheet 1

Expected
Claims
(4)

Reported
Claims at
12/31/2008
(5)

Estimated
IBNR
(6)

Actual
IBNR
(7)

Difference
from
Actual IBNR
(8)

700,000
735,000
771,750
810,338
850,854
893,397
938,067
984,970
1,034,219
1,085,930
8,804,524

700,000
735,000
771,750
810,338
842,346
884,463
919,306
935,722
930,797
836,166
8,365,888

0
0
0
-1
8,508
8,934
18,761
49,248
103,422
249,764
438,636

0
0
0
0
0
0
0
0
0
0
0

Column Notes:
(2) Assume 51,000,000 for first year in experience period (1999) and 5% annual increase thereafter.
(3) Assumed equal to 70% for all years.
(4) = [(2) x (3)].
(5) From last diagonal of reported claim triangles presented in Chapter 7, Exhibit III, Sheets 2 and 4.
(6) = [(4) - (5)].
(7) Developed in Chapter 7, Exhibit III, Sheet 1.
(8)= [(7) - (6)]

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Chapter 8 – Expected Claims Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Scenario 2 — U.S. PP Auto Increasing Claim Ratios
Exhibit IV, Sheet 1, bottom section.
Exhibit IV
Sheet 1

Impact of Changing Conditions
U. S. PP Auto - Development of Unpaid Claim Estimate

Accident
Earned
Year
Premium
(1)
(2)
Increasing Claim Ratios
1,000,000
1999
2000
1,050,000
2001
1,102,500
2002
1,157,625
2003
1,215,506
2004
1,276,282
2005
1,340,096
2006
1,407,100
2007
1,477,455
2008
1,551,328
Total
12,577,892

Earned
Claim
Ratio
(3)
70.0%
70.0%
70.0%
70.0%
70.0%
70.0%
70.0%
70.0%
70.0%
70.0%

Expected
Claims
(4)

Reported
Claims at
12/31/2008
(5)

Estimated
IBNR
(6)

Actual
IBNR
(7)

Difference
from
Actual IBNR
(8)

700,000
735,000
771,750
810,338
850,854
893,397
938,067
984,970
1,034,219
1,085,930
8,804,524

700,000
735,000
771,750
810,338
842,346
1,010,815
1,116,300
1,203,071
1,263,224
1,194,523
9,647,367

0
0
0
-1
8,508
-117,418
-178,233
-218,101
-229,006
-108,593
-842,843

0
0
0
-1
8,508
10,211
22,782
63,319
140,358
356,805
601,982

0
0
0
0
0
127,628
201,014
281,420
369,364
465,398
1,444,825

Unless the 70% expected claim ratio assumption is changed, the projected ultimate claims will be
unchanged from Scenario 1.
 Since claims are increasing in Scenario 2, the estimated IBNR will be lower than the actual IBNR.
 One test to assess the adequacy of the expected claim ratio is to compare the reported claim ratio
to date to the selected claim ratio. This test would:
i. have alerted the actuary to the fact that for AYs 2004 through 2008, the reported claim ratios
are already greater than the expected claim ratio.
ii. suggest a higher expected claim ratio for more recent accident years and avoid the negative
values for IBNR seen in Column (6) of Exhibit IV, Sheet 1 (bottom section).

Exam 5, V2

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Chapter 8 – Expected Claims Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Scenario 3 — U.S. PP Auto Increasing Case Outstanding Strength
Exhibit IV, Sheet 2, top section.
Impact of Changing Conditions
U. S. PP Auto - Development of Unpaid Claim Estimate
Earned
Accident
Earned
Claim
Year
Premium
Ratio
(1)
(2)
(3)
Increasing Case Outstanding Strength
1999
1,000,000
70.0%
2000
1,050,000
70.0%
2001
1,102,500
70.0%
2002
1,157,625
70.0%
2003
1,215,506
70.0%
2004
1,276,282
70.0%
2005
1,340,096
70.0%
2006
1,407,100
70.0%
2007
1,477,455
70.0%
70.0%
2008
1,551,328
Total
12,577,892

Exhibit IV
Sheet 2

Reported
Expected
Claims at
Estimated
Claims
12/31/2008
IBNR
(6) = [(4) - (5)].
(4)= [(2) x (3)].
(5)
700,000
735,000
771,750
810,338
850,854
893,397
938,067
984,970
1,034,219
1,085,930
8,804,524

700,000
735,000
771,750
810,338
842,346
884,463
933,377
962,808
979,922
931,185
8,551,189

0
0
0
-1
8,508
8,934
4,690
22,162
54,296
154,745
253,335

Actual
IBNR
(7)

Difference
from
Actual IBNR
(8) =[(7) - (6)]

0
0
0
-1
8,508
8,934
4,690
22,162
54,296
154,745
253,335

0
0
0
0
0
0
0
0
0
0
0

Column Notes:
(2) Assume $1,000,000 for first year in experience period (1999) and 5% annual increase thereafter.
(3) Assumed equal to 70% for all years.
(5) From last diagonal of reported claim triangles presented in Chapter 7, Exhibit III, Sheets 6 and 8.
(7) Developed in Chapter 7, Exhibit III, Sheet I.




The expected claims method produces an accurate estimate of IBNR
Changes in the adequacy of case outstanding have no effect on the expected claim ratio method
since actual claims experience does not enter the calculation.
Scenario 4 — U.S. PP Auto Increasing Claim Ratios and Case Outstanding Strength
Exhibit IV, Sheet 2, bottom section.
Impact of Changing Conditions
U. S. PP Auto - Development of Unpaid Claim Estimate
Earned
Reported
Accident
Earned
Claim
Expected
Claims at
Estimated
Year
Premium
Ratio
Claims
12/31/2008
IBNR
(6) = [(4) - (5)].
(1)
(2)
(3)
(4)= [(2) x (3)].
(5)
Increasing Claim Ratios and Case Outstanding Strength
1999
1,000,000
70.0%
700,000
700,000
0
2000
1,050,000
70.0%
735,000
735,000
0
2001
1,102,500
70.0%
771,750
771,750
0
2002
1,157,625
70.0%
810,338
810,338
-1
2003
1,215,506
70.0%
850,854
842,346
8,508
2004
1,276,282
70.0%
893,397
1,010,815
-117,418
2005
1,340,096
70.0%
938,067
1,133,386
-195,319
2006
1,407,100
70.0%
984,970
1,237,897
-252,927
2007
1,477,455
70.0%
1,034,219
1,329,895
-295,677
-244,334
70.0%
1,085,930
1,330,264
2008
1,551,328
Total
12,577,892
8,804,524
9,901,691
-1,097,167

Exhibit IV
Sheet 2

Actual
IBNR
(7)

Difference
from
Actual IBNR
(8) =[(7) - (6)]

0
0
0
-1
8,508
10,211
5,696
28,493
73,687
221,064
347,658

0
0
0
0
0
127,628
201,014
281,420
369,364
465,398
1,444,825



IBNR falls short of the actual IBNR requirements (similar to the situation in Scenario 2), and
actual IBNR and estimated IBNR differ by the same amount for Scenarios 2 and 4.
Without a change in the expected claim ratio assumption, the expected claims method will not react
appropriately to an environment of changing claim ratios.

Exam 5, V2

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Chapter 8 – Expected Claims Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
11

U.S. Auto Steady-State (No Change in Product Mix)

139

U.S. Auto Steady-State (No Change in Product Mix)
Exhibit V, top section.
Assume that the expected claim ratio can be estimated appropriately for the combined portfolio (easier
when the proportion of each of the two categories remains consistent over time).
Impact of Change in Product Mix Example
U. S. PP Auto - Development of Unpaid Claim Estimate
Earned
Accident
Earned
Claim
Year
Premium
Ratio
(1)
(2)
(3)
Steady-State (No Change in Product Mix)
1999
2,000,000
75.0%
2000
2,100,000
75.0%
2001
2,205,000
75.0%
2002
2,315,250
75.0%
2003
2,431,013
75.0%
2004
2,552,563
75.0%
2005
2,680,191
75.0%
2006
2,814,201
75.0%
2007
2,954,911
75.0%
75.0%
2008
3,102,656
Total
25,155,785

Exhibit V

Expected
Claims
(4)= [(2) x (3)]

Reported
Claims at
12/31/2008
(5)

Estimated
IBNR
(6) = [(4) - (5)]

Actual
IBNR
(7)

Difference
from
Actual IBNR
(8) = [(7) - (6)]

1,500,000
1,575,000
1,653,750
1,736,438
1,823,260
1,914,422
2,010,143
2,110,651
2,216,183
2,326,992
18,866,839

1,500,000
1,575,000
1,653,750
1,736,438
1,814,751
1,885,068
1,948,499
1,937,577
1,852,729
1,568,393
17,472,205

0
0
0
-1
8,509
29,354
61,644
173,074
363,454
758,599
1,394,634

0
0
0
-1
8,508
29,354
61,644
173,074
363,454
758,599
1,394,634

0
0
0
0
0
0
0
0
0
0
0

Column Notes:
(2) For no change scenario, assume 52,000,000 for first year in experience period (1999) and 5% annual
increase thereafter. For change scenario, assume annual increase of 30% for commercial auto beginning in 2005.
(3)Assumed equal to 75% for all years.
(5) From last diagonal of reported claim triangles presented in Chapter 7, Exhibit IV, Sheets 2 and 4.
(7) Developed in Chapter 7, Exhibit IV, Sheet I.

If so, the expected claims technique generates the correct IBNR requirement in times of no change.

Exam 5, V2

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Chapter 8 – Expected Claims Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
12

U.S. Auto Changing Product Mix

139

U.S. Auto Changing Product Mix
Exhibit V, bottom section.
 Assume that the volume of commercial auto insurance is increasing at a greater rate than that of
private passenger automobile insurance.
 Since commercial auto has higher ultimate claim ratios, the expected claim ratio assumption will
need to be modified (critical to the expected claims technique).
 Without a change in the expected claim ratio, the expected claims technique produces an
inadequate IBNR estimate.
Impact of Change in Product Mix Example
U. S. PP Auto - Development of Unpaid Claim Estimate

Accident
Earned
Year
Premium
(1)
(2)
Changing Product Mix
1999
2,000,000
2000
2,100,000
2001
2,205,000
2002
2,315,250
2003
2,431,013
2004
2,552,563
2005
2,999,262
2006
3,564,016
2007
4,281,446
2008
5,196,516
Total
29,645,066

Earned
Claim
Ratio
(3)
75.0%
75.0%
75.0%
75.0%
75.0%
75.0%
75.0%
75.0%
75.0%
75.0%

Exhibit V

Expected
Claims
(4)= [(2) x (3)]

Reported
Claims at
12/31/2008
(5)

Estimated
IBNR
(6) = [(4) - (5)]

Actual
IBNR
(7)

Difference
from
Actual IBNR
(8) = [(7) - (6)]

1,500,000
1,575,000
1,653,750
1,736,438
1,823,260
1,914,422
2,249,447
2,673,012
3,211,085
3,897,387
22,233,800

1,500,000
1,575,000
1,653,750
1,736,438
1,814,751
1,885,068
2,193,545
2,471,446
2,680,487
2,556,695
20,067,180

0
0
0
-1
8,509
29,354
55,902
201,566
530,598
1,340,692
2,166,620

0
0
0
-1
8,508
29,354
71,855
239,057
596,924
1,445,385
2,391,083

0
0
0
0
0
0
15953
37491
66327
104693
224,464

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Chapter 8 – Expected Claims Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Sample Questions:
1. For the “Reported Claim Development” method, ultimate claims for each accident year were estimated as the
product of the cumulative reported CDF for the valuation age & reported claims through the valuation date.
For the “Paid Claim Development” method, ultimate claims for each accident year were estimated as the
product of the cumulative paid CDF for the valuation age & paid claims through the valuation date.
How are ultimate claims estimated using the “Expected Claims” technique, a given accident year?
2. Describe 2 ways an insurer may select an Expected Claim Ratio for use in Expected Claim methods.
3. List 2 challenges of the Expected Claims method, according to Friedland.
4. What name does Brosius give to the method described in Friedland as the “Expected Claims” technique?
Note: This question applies to the Brosius article, now on exam 7.
5. Summarize Friedland’s key points re: “When the Expected Claims Technique Works and When it Does Not.”
Include 3 cases where the method may be appropriate, and potential disadvantage/advantage.
6. Based on the following data:
Reported Claims including ALAE ($000's omitted)
Earned
Premium

Accident
Year

1st
Report

2nd
Report

3rd
Report

4th
Report

5th
Report

6th
Report

2,000
2,200
2,500
2,650
3,000
3,150

2003
2004
2005
2006
2007
2008

940
1,200
1,250
1,400
1,500
2,250

1,620
1,690
1,725
1,550
1,900

1,700
1,710
1,800
1,900

1,750
1,800
1,950

1,750
1,800

1,750

Estimate the IBNR as of 12/31/08 using the following method: Expected Claims Technique.
Use an Expected Claim Ratio = 80% for all years.
To select claim development factors, use the volume-weighted averages for the latest three years.
See also Friedland Chapter 7 and 9 for other methods.

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Chapter 8 – Expected Claims Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2008 Exam Questions (modified):
Question 10.
Given the following for an accident year:
- Earned Premium:
- Reported Losses as of 12 months:
- Expected loss ratio:
- Expected reporting pattern:

$20,000,000
$10,000,000
70%

Age (months) % Reported
12
40%
24
60%
36
80%
48
90%
60
100%
a. (1.5 points) This portion of the questions associated with the Brosius article, now on Exam 7.
b. (1 point) Estimate the ultimate value of the claims currently aged at 12 months. Use the Expected Claims
Method, as described in Friedland.
c. (.75 points) See Mack/Benktander and Friedland Ch 9.
2009 Exam Questions
9. (2.5 points) Given the following information:
Cumulative Incurred Loss

Accident Year

12 Months

24 Months

36 Months

2006
2007
2008

$5,630,000
6,380,000
7,348,000

$7,106,000
8,051,000

$8,282,000

On-Level
Earned
Premium
$12,380,000
13,430,000
14,280,000

Selected Development Factors
12-Ultimate
24- Ultimate
36- Ultimate
1.570
1.250
1.070
•
•

The annual loss ratio trend is 7.0%.
Accident year 2008 paid losses as of December 31, 2008 total $6,100,000.

a. (1.5 points) Using the expected claims technique, calculate the IBNR for accident year 2008.
b. (1 point) Briefly describe two situations where the expected claims technique may be appropriate.

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Chapter 8 – Expected Claims Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2011 Exam Questions
25. (2.5 points) Given the following information as of December 31, 2010:
Accident
Year
2007
2008
2009
2010

12-24
2.400

Earned
Premium
$21,000,000
$22,050,000
$23,152,500
$23,525,000

24-36
1.800

Paid Claims
$11,700,000
$8,200,000
$4,900,000
$1,900,000

On-Level Earned
Premium Factors
1.093
1.061
1.030
1.000

Paid Loss Development Factors
36-48
48-60
1.500
1.200

60-UIt
1.020

• Loss trend is 4% per year
a. (2.25 points) Use the expected claim technique to estimate ultimate claims for accident year 2010.
b. (0.25 point) Briefly describe a disadvantage of the expected claim technique.
2012 Exam Questions
21. (3 points) Given the following as of December 31, 2011:
Accident
Year
2008
2009
2010
2011

Earned
Premium
$2,491
$2,853
$2,898
$2,800

Accident
Year

12
Months

2008
2009
2010
2011

1,100
$1,200
$1,100
$1,000

On-Level Earned
Premium
$2,616
$2,853
$2,753
$2,800
Cumulative Paid Claims ($000s)
24
36
Months
Months
$1,430
$1,560
$1,430

$1,573
$1,716

48
Months
$1,652

Cumulative Paid Claim Development Factors
12-Ult
24-Ult
36-Ult
48-Ult
1.502
1.155
1.050
1.000
•

Tort reform effective January 1, 2010 reduced expected losses by 5% for accident year 2010 and
subsequent years.

•

Loss trend is 0%.

• Case outstanding for accident year 2011 as of December 31, 2011 is $780.
a. (2.5 points) Use the expected claim technique to estimate IBNR for accident year 2011.

b. (0.5 point) Evaluate the reasonableness of negative IBNR.

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Chapter 8 – Expected Claims Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to Sample Questions:
1. For the “Reported Claim Development” method, ultimate claims for each accident year were estimated as the
product of the cumulative reported CDF for the valuation age & reported claims through the valuation date.
For the “Paid Claim Development” method, ultimate claims for each accident year were estimated as the
product of the cumulative paid CDF for the valuation age & paid claims through the valuation date.
In the “Expected Claims” technique, ultimate claims for an accident year are calculated by multiplying the
appropriate premium for the year by a selected “claim ratio.”
2. 2 ways an insurer may select an Expected Claim Ratio for use in Expected Claim methods:
Exposure-Based methods (may use adjusted historical data) & Statistical Modeling methods
3. List 2 challenges of the Expected Claims method, according to Friedland.
Determining an appropriate exposure base,
& Estimating the measurement of claims relative to that exposure base
4. What name does Brosius give to the method described in Friedland as the “Expected Claims” technique?
The “Budgeted Loss” method (for ultimate loss estimates with no credibility to actual experience)
Note: The term “Expected LOSS RATIO” method has also been used in some texts.
5. Summarize Friedland’s key points re: “When the Expected Claims Technique Works and When it Does Not.”
Friedland mentions 2 scenarios where the Expected Claims technique may be used:
1) Entering a new line of business or territory (using industry benchmarks for the claim ratios)
2) If the Cumulative CDFs are highly leveraged, an actuary may choose to use an Expected Claims
technique for the most recent years in the experience period.
3) If an insurer undergoes or is impacted by a major change, may use Expected Claims method for years
likely to be affected.
Friedland comments on potential advantages/disadvantages of the Expected Claims technique:
Since this method applies selected claim ratios to premiums, instead of developing the actual losses as in
Chapter 7’s Development method, we have:
1) Potential advantage in the stability of the projected ultimate losses.
2) Potential disadvantage in the lack of responsiveness to actual experience
However, by changing the selected claim ratios, the results of using the Expected Claims method can
become more responsive (and less stable). In some examples, such as being impacted by a major change,
an “actuary may be able to adjust the a priori expectation in advance of the changes being fully manifested in
the data (so) the expected claims method could prove to be more responsive that data-dependent methods.”
See Friedland Chapter 8.

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Chapter 8 – Expected Claims Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
6.

Accident
Year

1st
Report

2nd
Report

3rd
Report

4th
Report

5th
Report

6th
Report

2,000
2,200
2,500
2,650
3,000
3,150

2003
2004
2005
2006
2007
2008

940
1,200
1,250
1,400
1,500
2,250

1,620
1,690
1,725
1,550
1,900

1,700
1,710
1,800
1,900

1,750
1,800
1,950

1,750
1,800

1,750

Accident
Year

Earned
Premium

Expected
Claim
Ratio

Expected
Reported
Claims Claims at
(Ultimate) 12/31/2008

IBNR
(broadly
defined)

(1)
2,000
2,200
2,500
2,650
3,000
3,150

(2)
80.00%
80.00%
80.00%
80.00%
80.00%
80.00%

2003
2004
2005
2006
2007
2008
Total

(3)=(1)*(2)
1,600
1,760
2,000
2,120
2,400
2,520

(4) given
1,750
1,800
1,950
1,900
1,900
2,250

(5)=(4)-(3)
-150
-40
50
220
500
270
850

Solutions to 2008 Exam Questions (modified):
Question 10b.
Use Expected Claims = Earned Premium * "Expected Loss Ratio"
Then the Ultimate Estimate = $20M * 70% = 14,000,000
Note: Expected Claims method didn't use $ reported to date, or % reported, in order to estimate ultimate claims.

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Chapter 8 – Expected Claims Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2009 Exam Questions
Question 9 - Model Solution 1
a. Using the Expected Claims Technique calculate the IBNR for Accident Year 2008.
(1)
(2)
(3)=(1)×(2)
(4)
(5)=(3)×(4)
(6)
(7)=(5)/(6)
AY
Incurred
CDF Proj Ult Loss Trend Trended Ult On level EP Loss Ratio
Loss
2
2006
8,282,000 1.070
8,861,740
1.07
10,145,806
12,380,000
0.8195
2007
8,051,000 1.250
10,063,750
1.070
10,768,213
13,430,000
0.8018
2008
7,348,000 1.570
11,536,360
1.000
11,536,360
14,280,000
0.8079
Average:
0.8097
Selected Expected Loss Ratio = 80.97%
AY 2008 Projected Ultimate Claim = 2008 OL EP * Selected ELR = 80.97% × 14,280,000 = 11,562,516
IBNR for AY 2008 = AY 2008 Ultimate Claims – AY 2008 Reported Losses at 12 months
= 11,562,516 – 7,348,000 = 4,214,516
b. When credible data is not available to use other estimation techniques, for instance, entering a new
line of business and no historical claim experience.
When there is a significant change that makes historical claim experience irrelevant, example like
major change in regulation.
Question 9 - Model Solution 2 – Part b.
b. Expected claims technique would be useful when entering a new line of business – could use industry
benchmark ratio – where little is known about this line’s loss behavior.
It could also be used for longer tailed lines where LDF’s are highly leveraged, and an “a priori”
estimate of expected claims is thought to be more accurate than the development method.
Solutions to 2011 Exam Questions
25a. (2.25 points) Use the expected claim technique to estimate ultimate claims for accident year 2010.
25b. (0.25 point) Briefly describe a disadvantage of the expected claim technique.
Question 25 – Model Solution
a. AY 2010 Projected Ultimate Claim = 2010 OL EP * Selected ELR
Selected ELR = Average [from 2007 – 2009] Trended Ultimate losses/ Onlevel Earned premium
CY OL EP = EP * OLEP Factor; AY Trended Ult Loss = Paid Claims * LDF to ult * Loss Trend Factor
Trend historical loss data from AY ending 12/31/20XX to 12/31/2010 using the given loss trend of 4%
Eg: CY2008 OLEP = $22,050,000 * 1.061; AY 2008 Trend Ult Loss = $8,200,000 * 1.500 * 1.200 * 1.0201 *
.04
2007
2008
2009
2010

On-Level EP
22,953,000
23,395,050
23,847,075
23,525,000

Trend Period
3
2
1
0

Trended Ult. Loss
16,108,952
16,283,704
16,841,261
15,069,888

Avg. Loss ratio 2007-2009 = 70.13%
AY 2010 expected Ult. Loss = 23,525,000 x 70.13%=

Loss Ratio
.702
.696
.706
.641

$16,498,082.50

b. The expected claims technique is relatively unresponsive to recent changes in claims experience.

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Chapter 8 – Expected Claims Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2012 Exam Questions
21a. (2.5 points) Use the expected claim technique to estimate IBNR for accident year 2011.
21b. (0.5 point) Evaluate the reasonableness of negative IBNR.
Question 21 – Model Solution 1 (Exam 5B Question 6)
a.
Developed
On level
Paid losses at Paid CDF
Trend*
Tort reform
Projected
Loss
Earned premium
12/31/2011
To ultimate 7/1/2011 adjustments
losses
ratio
3
2008: 2616
1652
1.000
1.0
0.95
1569.40
0.60
2
2009: 2853
1716
1.050
1.0
0.95
1711.71
0.60
2010: 2753
1430
1.155
1.0
1.00
1651.65
0.60
2011: 2800
1000
1.502
1.0
1.00
1502.00
0.54
Total:11,022
6434.76
0.58
*: Trend from 7/1 of each accident year to 7/1/2011
Select = 60% based on average of accident years 2008-2010
Accident year 2011 IBNR = Expected Ultimate Claims – Reported Claims
= 2800 (0.6)-(1000+780) = -100
b. For certain lines of business, negative IBNR can be possible if case reserves are historically set too
strong in early maturities and develop downwards over time, or it is common in lines of business
expecting future salvage and subrogation recoveries such as auto physical damage. Without knowing
the specifics at the line of business in part A, it is difficult to tell if negative IBNR is reasonable.
However, since the line of business involves tort reform, I would expect it to be a liability line which
makes me believe negative IBNR for this line, especially since 2011 is only at 12 months of
development, is inappropriate.
Question 21 – Model Solution 2 (Exam 5B Question 6)
AY Ult. Claims (trend is 0)
Est. Ult Claim Ratio = Ult Claims/on-level EP
a.
08
1,652k
63.15%
09
1,716k (1.05) = 1,801.8k
63.15%
10
1,430k (1.155) = 1,651.65k
59.999% (which is 5% below ’08 & ’09)
Expected claim ratio for AY11= 59.99%
→ Ult. Claims for AY11=2800k(0.5999) = 1,679,720
→ IBNR for AY11= 1,679,720 - 1,000,000 - 780,000 = -100,280
b. Negative IBNR could be reasonable if this is reflecting anticipated recoveries such as salvage and
subrogation. In this case, it seems likely that case reserves are excessive given the tort reforms recently
taking hold.
Examiner’s comments
a. This part of the question was generally well-answered. However, there were certain steps at which
points were frequently lost. A number of candidates made no adjustment for tort reform. Among those
that did, some calculated incorrect adjustment factors and/or applied the factors to the wrong years.
Many candidates wrongly included AY 2011 in the calculation of the expected claim ratio. There were
also a fair number who explicitly excluded it, but for the wrong reasons (e.g., “immature,” “leveraged,”
“outlier”). One area of ambiguity in the question that was identified by some candidates was whether
or not the case o/s of $780 was expressed, as were paid losses, in thousands of dollars. Some
assumed that they were, while others assumed they were not. Though the expectation was that the
former would be assumed, no points were deducted for assuming the latter.
b. Most candidates received partial credit for this part of the question, as either general or AY 2011
specific comments were made, but not both.

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Chapter 9 – Bornhuetter-Ferguson Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Sec
1
2
3
4
5
6
7
8
9
1

Description
Introduction to the Bornhuetter-Ferguson (BF) Technique
Key Assumptions
Common Uses of the BF Technique
Mechanics of the BF Technique
Unpaid Claim Estimate Based on the BF Technique
When the BF Technique Works and When It Does Not
The BF Method and Cumulative CDFs Less than 1.00
XYZ Insurer
Influence of a Changing Environment on the BF Technique

Pages
152
152 - 153
153 - 154
154 - 155
155
156
157
157
157 - 160

Introduction to the Bornhuetter-Ferguson (BF) Technique

152

The BF technique:
 is a commonly used claims estimation technique.
 is a blend of the development and expected claims techniques, by splitting ultimate claims into two
components: actual reported (or paid) claims and expected unreported (or unpaid) claims.
 gives more weight to actual claims as experience matures, and less weight to expected claims .
 was developed to overcome the problems with the development and expected claims technique.
Problem with the development technique:
This technique can lead to erratic, unreliable projections when the CDF is large because a small swing in
reported claims or the reporting of an unusually large claim could result in a very large swing in projected
ultimate claims.
Problem with the expected claims technique:
It ignores actual reported results.
In "Loss Development Using Credibility," Brosius described the BF method as a credibility weighting between
the development method and the expected claims method.
 In the development method, full credibility (i.e. Z = 1) is given to actual claims experience; and in the
expected claims method, no credibility (i.e. Z = 0) is given to actual claims.
 In the BF method, credibility is equal to the % of claims developed at a particular stage of maturity,
which is determined as Z = 1.00/CDF.
 Therefore, more weight is given to the expected claims method in less mature years, and more weight
is given to the development method in more mature years of the experience period.

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Chapter 9 – Bornhuetter-Ferguson Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2

Key Assumptions

152 - 153

The BF method assumes that unreported (or unpaid) claims will develop based on expected claims.
 Reported claims do not contain informational value as to the amount of claims yet-to-be reported.
 This differs greatly from the development method where the primary assumption is that unreported (or
unpaid) claims will develop based on reported (or paid) claims to date.
The reporting and payment patterns used in the BF methods and the development methods are the same.
The expected claims used in the BF method using reported claims are the same as those used in the BF
method using paid claims.
How development factors are applied in the two methods differs.

Common Uses of the BF Technique

153 - 154

The BF technique:
 is most often applied to reported and paid claims
 can be used with number of claims and with ALAE.
 can be used with all lines of insurance (including short-tail lines and long-tail lines).
 is used with data organized in many different time intervals including:
* Accident year
* Policy year
* Underwriting year
* Report year
* Fiscal year
This technique can use data organized by month, quarter, or half-year.

Mechanics of the BF Technique

154 - 155

The BF technique is a blend of the development method and the expected claims method.
The following two formulae represent the reported and paid BF methods, respectively:
Ultimate Claims = Actual Reported Claims + Expected Unreported Claims
Ultimate Claims = Actual Reported Claims + (Expected Claims) x (% Unreported)
Ultimate Claims = Actual Paid Claims + Expected Unpaid Claims
Ultimate Claims = Actual Paid Claims + (Expected Claims) x (% Unpaid)
Implementing the BF Method:
The goal: To determine expected unreported and expected unpaid claims (since actual reported and paid
claims are both known quantities).
The key step: To select claim development patterns and develop an expected claims estimate.
A step by step approach (See Exhibit I, Sheet 1 for U.S. Industry Auto) follows:

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Chapter 9 – Bornhuetter-Ferguson Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
I. Compute Projected Ultimate Claims
Step 1: Use the reported and paid CDFs (from Chapter 7) to compute the %s unreported and %s unpaid (see
Columns (5) and (6)).
The % unreported = 1.00 – 1/reported CDF.
The % unpaid = 1.00 – 1/paid CDF.
The selected claim development factors for reported and paid claims as well as the associated
reporting and payment patterns in Table 1 below.

Age
(Month)
12
24
36
48
60
72
84
96
108
120

Table 1 - U.S. Industry Auto
Selected Reporting and Payment Patterns
Reported Claims
Paid Claims
CDF to
%
%
CDF to
%
%
Ultimate Reported Unreported Ultimate
Paid
Unpaid
1.292
77.4%
22.6%
2.390
41.8%
58.2%
1.11
90.1%
9.9%
1.404
71.2%
28.8%
1.051
95.1%
4.9%
1.184
84.5%
15.5%
1.023
97.8%
2.2%
1.085
92.2%
7.8%
1.011
98.9%
1.1%
1.040
96.2%
3.8%
1.006
99.4%
0.6%
1.020
98.0%
2.0%
1.003
99.7%
0.3%
1.011
98.9%
1.1%
1.001
99.9%
0.1%
1.006
99.4%
0.6%
1.000
100.0%
0.0%
1.004
99.6%
0.4%
1.000
100.0%
0.0%
1.002
99.8%
0.2%

Keep in mind that the primary assumption of the reported BF method is that unreported claims will
emerge in accordance with expected claims.
U.S. Auto Industry
Projection of Ultimate Claims Using Reported and Paid Claims ($000)

Accident
Expected
CDF to Ultimate
Percentage
Expected Claims
Year
Claims
Reported
Paid
Unreported Unpaid
Unreported
Unpaid
(7) = [(2) x(5)](8) =[(2) x (6)]
(1)
(2)
(3)
(4)
(5)
(6)
1998 51,430,657
1.000
1.002
0.00%
0.20%
0
102,656
:::
:::
:::
:::
:::
:::
:::
:::
2006 61,926,981
1.110
1.404
9.91%
28.77%
6,136,908 17,819,445
2007 61,864,556
1.292
2.390
22.60%
58.16% 13,981,773 35,979,805
Total 569,281,839
25,609,761 72,517,830

Exhibit I
Sheet 1
Projected Ultimate Claims
Claims at 12/31/07
using B-F Method with
Reported
Paid
Reported
Paid
(9)
(10)
(11) = [(7)+(9] (12)=[(8)+(10)]
47,742,304
47,644,187
47,742,304
47,746,843
:::
:::
:::
:::
54,641,339
43,606,497
60,778,247
61,425,942
48,853,563
27,229,969
62,835,336
63,209,774
543,481,587
498,050,368 569,091,348
570,568,198

Column Notes:
(2) Developed in Chapter 8, Exhibit III, Sheet 1.
(3)and (4) Developed in Chapter 7, Exhibit II, Sheets 1 and 2, capped at a minimum of 1.00.
(5) =[1.00 - (1.00 / (3))].
(6) =[1.00 - (1.00 / (4))].
(9) and (10) Based on Best's Aggregates & Averages U.S. Private Passenger Auto Experience

Step 2: Calculate the expected unreported claims by AY.
Column (7) expected unreported claims equal Column (2) expected claims multiplied by %
unreported in Column (5) for each AY.
Column (8) expected unpaid claims equal Column (2) expected claims multiplied by % unpaid in
Column (6) for each AY.

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Chapter 9 – Bornhuetter-Ferguson Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
A. Projected Ultimate Claims (continued):
Step 3: Calculate the projected ultimate claims.
Recall the BF formulae:
Ultimate Claims = Actual Reported Claims + Expected Unreported Claims
Note: For the reported BF projection, expected unreported claims equals estimated IBNR.

Ultimate Claims = Actual Paid Claims + Expected Unpaid Claims
Column (11) projected ultimate claims equal Column (9) actual reported claims plus Column (7)
expected unreported claims.
Column (12) projected ultimate claims equal Column (10) actual paid claims plus Column (8)
expected unpaid claims.

Unpaid Claim Estimate Based on the BF Technique

155

II. Compute Estimated IBNR and the Total Unpaid Claim Estimate (see Exhibit I, Sheet 2):
Step 4: Compute estimated IBNR and the total unpaid claim estimate
XYZ Insurer - Auto BI
Development of Unpaid Claim Estimate ($000)

Exhibit II
Sheet 2

Unpaid Claim Estimate at 12/31/07
Projected Ultimate Claims
Case
IBNR Based on
Total Based on
Claims at 12/31/07
Using B-F Method with
B- F Method with
B- F Method with
Outstanding
Reported
Paid
Reported
Paid
at 12/31/07
Reported
Paid
Reported
Paid
(2)
(3)
(4)
(5)
(6)= [(2) - (3)] (7)= [(4) - (2)] (8)= [(5) - (2)] (9)= [(6)+ (7)] (10)= [(6)+(8)]
47,742,304
47,644,187
47,742,304
47,746,843
0
4,539
98,117
102,656
98,117
:::
:::
:::
:::
:::
:::
:::
:::
:::
54,641,339
43,606,497
60,778,247
61,425,942
11,034,842
6,136,908
6,784,603
17,171,750
17,819,445
48,853,563
27,229,969
62,835,336
63,209,774
21,623,594
13,981,773
14,356,211
35,605,367
35,979,805
543,481,587 498,050,368 569,091,348 570,568,198
45,431,219
25,609,761
27,086,611
71,040,980
72,517,830

Accident
Year
(1)
1998
:::
2006
2007
Total

Column Notes:
(2) and (3) Based on Best's Aggregates & Averages U.S. Private Passenger Auto Experience
(4) and (5) Developed in Exhibit I, Sheet 1.

Columns (7 and 8) Estimated IBNR equals projected ultimate claims less reported claims
Projected ultimate claims come from Step 3 above
Columns (9 and 10) Total unpaid claim estimate equals Estimated IBNR + Case O/S reserves. Also,
Total unpaid claim estimate equals Estimated IBNR + (Reported – Paid Claims)
Total unpaid claim estimate equals projected ultimate claims minus paid claims.

When the BF Technique Works and When It Does Not

156

An advantage of the BF technique: Random fluctuations early in the life of an AY do not significantly
distort the projections.
Example: While several large/unusual reported claims for an AY would produce an overly conservative
ultimate claims estimate when using the reported claim development technique, such is not
the case when using the BF technique.

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Chapter 9 – Bornhuetter-Ferguson Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
The BF method can be used:
 for the most immature years associated with long-tail lines of insurance, due to the highly
leveraged nature of claim development factors for such lines.
 if the data is extremely thin or volatile or both.
For example, when an insurer enters a new line of business or a new territory and there is not
yet a credible volume of historical claim development experience.
The actuary would likely need to rely on benchmarks, either from similar lines at the same
insurer or insurance industry experience, for development patterns and expected claim ratios
(or pure premiums).
 for very short-tail lines (where the IBNR can be set equal to a multiple of the last few months EP).

The BF Method and Cumulative CDFs Less than 1.00

157

Downward development (i.e. CDFs < 1.00) does occur: Examples:
 For automobile physical damage and property, salvage and subrogation recoveries lag the
reporting and payment of claims, resulting in report-to-report factors that are less than 1.00.
 For insurers with a conservative case outstanding reserving, downward development of reported
claims as payments for claims may be less than the case outstanding set by claims adjusters. First,
revisit the original premise of the BF method for lines of business in which CDFs are less than 1.00.
Recall that the BF method can be considered a credibility-weighting of the results from the development
method and from the expected claims method.
The basic formula for calculating the credibility-weighted projection is:
[(Z) x (development method)] + [(1 - Z) x (expected claims method)] where, 0 ≤ Z ≤ 1
Z is the credibility assigned to the development method; Z = 1.00/CDF, and
(1 - Z) is the complement of credibility assigned to the expected claims method.
Adjustments than can be made when working with CDFs less that 1.00:
 Limit the CDFs to a minimum value of 1.00 when applying the BF technique (used in this text).
 Perform the BF calculations, but rely on another technique to select ultimate claims for the year(s)
in question (i.e. years with CDFs less than 1.00).

XYZ Insurer

157

Exhibit II, Sheets 1 and 2: Projected ultimate claims, IBNR and Total Unpaid claims based on the results of the
reported and paid BF methods using Chapter 8 expected claims are shown
Exhibit II, Sheet 3 (projected ultimate claims) compares the results of the BF method with the expected claims
method and the development method
Exhibit II, Sheet 4 (estimated IBNR) compares these results for the three projection methods.

Exam 5, V2

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Chapter 9 – Bornhuetter-Ferguson Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
XYZ Insurer - Auto BI
Projection of Ultimate Claims Using Reported and Paid Claims ($000)

Accident
Year
(1)
1998
:::
2007
2008
Total

CDF to Ultimate
Expected
Claims
Reported
Paid
(2)
(3)
(4)
15,660
1.000
1.010
:::
:::
:::
39,835
1.512
6.569
39,433
2.551
21.999
561,516

Exhibit II
Sheet 1

Percentage
Expected Claims
Unreported Unpaid
Unreported
Unpaid
(5)
(6)
(7) = [(2) x(5)] (8) =[(2) x (6)]
0.0%
1.0%
0.00
155
:::
:::
:::
:::
33.9%
84.8%
13,489
33,771
60.8%
95.5%
23,975
37,640
63,581
223,842

Projected Ultimate Claims
Claims at 12/31/08
using B-F Method with
Reported
Paid
Reported
Paid
(9)
(10)
(11) = [(7)+(9] (12)=[(8)+(10)]
15,822
15,822
15,822
15,977
:::
:::
:::
:::
31,732
11,865
45,221
45,636
18,632
3,409
42,607
41,049
449,626
330,627
513,207
554,469

Column Notes:
(2) Developed in Chapter 8, Exhibit III, Sheet 1.
(3) and (4) Developed in Chapter 7, Exhibit II, Sheets 1 and 2, capped at a minimum of 1.00.
(5) =[1.00 - (1.00 / (3))].
(6) =[1.00 - (1.00 / (4))].
(9) and (10) Based on data from XYZ Insurer.
(11) = [(7) + (9]
(12)= [(8)+ (10)].

XYZ Insurer - Auto BI
Development of Unpaid Claim Estimate ($000)

Accident
Year
(1)
2007
2008
Total

Claims at 12/31/08
Reported
Paid
(2)
(3)
31,732
11,865
18,632
3,409
449,626
330,629

Projected Ultimate Claims
Using B-F Method with
Reported
Paid
(4)
(5)
45,221
45,636
42,607
41,049
513,207
554,471

Exhibit II
Sheet 2

Case
Outstanding
at 12/31/08
(6)= [(2)-(3)]
19,867
15,223
118,997

Unpaid Claim Estimate at 12/31/08
IBNR Based on
Total Based on
B- F Method with
B- F Method with
Reported
Paid
Reported
Paid
(7)= [(4)-(2)] (8)= [(5)-(2)] (9)= [(6)+ (7)] (10)= [(6)+(8)]
13,489
13,904
33,356
33,771
23,975
22,417
39,198
37,640
63,581
104,845
182,578
223,842

Column Notes:
(2) and (3) Based on data from XYZ Insurer.
(4) and (5) Developed in Exhibit II, Sheet 1.
(6)= [(2) - (3)]
(7)= [(4) - (2)]
(8)= [(5) - (2)].
(9)= [(6)+ (7)].
(10)= [(6) + (8)],

XYZ Insurer - Auto BI
Summary of Ultimate Claims ($000)

Accident
Year
(1)
1998
1999
:::
2007
2008
Total

Claims at 12/31/08
Reported
Paid
(2)
(3)
15,822
15,822
25,107
24,817
:::
:::
31,732
11,865
3,409
18,632
449,626
330,629

Exhibit II
Sheet 3
Projected Ultimate Claims
Development Method
Expected
B-F Method
Reported
Paid
Claims
Reported
Paid
(4)
(5)
(6)
(7)
('8)
15,822
15,980
15,660
15,822
15,977
25,082
25,165
24,665
25,107
25,158
:::
:::
:::
:::
:::
47,990
77,938
39,835
45,227
45,636
47,536
74,994
39,433
42,609
41,049
514,929
605,030
561,516
513,207
554,471

Exam 5, V2

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Chapter 9 – Bornhuetter-Ferguson Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
XYZ Insurer - Auto BI
Summary of IBNR ($000)

Accident
Year
(1)
1998
:::
2007
2008
Total

Exhibit II
Sheet 4

Case
Outstanding
at 12/31/08
(2)
0
:::
19,867
15,223
118,997

Development Method
Reported
Paid
(3)
(4)
0
158.22
:::
:::
16,247
46,209
28,898
56,363
65,303
155,402

Estimated IBNR
Expected
Claims
(5)
-162
:::
8,103
20,801
111,890

B-F Method
Reported
Paid
(6)
(7)
0
155
:::
:::
13,489
13,904
23,975
22,417
63,581
104,843

Column Notes:
(2)Based on data from XYZ Insurer.
(3)and (4) Estimated in Chapter 7, Exhibit II, Sheet 4.
(5) Estimated in Chapter 8, Exhibit III, Sheet 3.
(6) and (7) Estimated in Exhibit II, Sheet 2.

Influence of a Changing Environment on the BF Technique

157 - 160

Similar to the analyses done in Chapters 7 and 8, we discuss the performance of the BF technique during
times of change.
Scenario 1 — U.S. PP Auto Steady-State
Exhibit III, Sheet 1, top section. Note: An expected claim ratio of 70% is used in Scenarios 1 through 4.
 Since the steady-state environment also has a 70% ultimate claim ratio, the BF technique
generates an accurate estimate of IBNR.
 The development and expected claims techniques also generated accurate IBNR values in a
steady-state environment.
Impact of Changing Conditions
U.S. PP Auto - Development of Unpaid Claim Estimate
Age of
Accident Accident Year Expected
Year
at 12/31/08
Claims
(1)
(2)
(3)
Steady-State
1999
120
700,000
2007
24
1,034,219
::
::
::
2008
12
1,085,930
Total
8,804,524

Claims at 12/31/08
Reported
Paid
(4)
(5)
700,000
700,000
930,797
734,295
::
::
836,166
456,090
8,365,888 7,573,547

Exhibit Ill
Sheet 1

CDF to Ultimate
Reported
Paid
(6)
(7)
1.000
1.111
::
1.299

1.000
1.408
::
2.381

Expected Percentage
Unreported Unpaid
(8 )
(9)
0.0%
10.0%
::
23.0%

0.0%
29.0%
::
58.0%

Projected Ultimate Claims
Using B-F Method with
Reported
Paid
(10)
(11)
700,000
1,034,219
::
1,085,930
8,804,526

700,000
1,034,218
::
1,085,929
8,804,523

Estimated IBNR
Using B-F Method with
Reported
Paid
(12)
(13)
0
103,422
::
249,764
438,638

0
103,421
::
249,763
438,635

Actual
IBNR
(14)
0
103,422
::
249,764
438,636

Diff from Actual IBNR
Using B-F Method with
Reported
Paid
(15)
(16)
0
-1
::
0
-2

Column Notes:
(2) Age of accident year at December 31, 2008.
(3) See Chapter 8, Exhibit IV, Sheet 1.
(4) and (5) From last diagonal of reported and paid claim triangles in Chapter 7, Exhibit III, Sheets 2 through 5
(6) and (7) CDF based on 5-year simple average age-to-age factors presented in Chapter 7, Exhibit III, Sheets 2 through 5.
(8) = [1.00 - (1.00 / (6))],
(9)= [1.00 -(1.00 / (7))].
(10) = [((3) x(8)) +(4)]
(11) = [((3) x (9)) + (5)]
(11) = [((3) x (9))+ (5)]
(12) = [(10)-(4)]
(13) = [(11)-(4)]
(12) = [(10) - (4)]
(13) = [(11) - (4)]
(14) Developed in Chapter 7, Exhibit Ill, Sheet 1.
(15) = [(14) - (12)]
(16) = [(14) - (13)]

Exam 5, V2

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0
0
::
1
1

Chapter 9 – Bornhuetter-Ferguson Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Scenario 2 — U.S. PP Auto Increasing Claim Ratios
The weakness of the expected claims method is also a weakness of the BF method.
Ultimate Claims = Actual Reported Claims + Expected Unreported Claims
Ultimate Claims = Actual Paid Claims + Expected Unpaid Claims
Projected ultimate claims increase between Scenarios 1 and 2, due to higher values of actual reported
and paid claims (and not higher estimates of the expected unreported and unpaid claims).
Impact of Changing Conditions
U.S. PP Auto - Development of Unpaid Claim Estimate
Age of
Accident Accident Year Expected
Year
at 12/31/08
Claims
(1)
(2)
(3)
Increasing Claim Ratios
1999
120
700,000
:::
:::
:::
2007
24
1,034,219
2008
12
1,085,930
Total
8,804,524

Claims at 12/31/08
Reported
Paid
(4)
(5)
700,000 700,000
:::
:::
1,263,224 996,544
1,194,523 651,558
9,647,367 8,575,113

Exhibit Ill
Sheet 1

CDF to Ultimate
Reported
Paid
(6)
(7)
1.000
:::
1.111
1.299

1.000
:::
1.408
2.381

Expected Percentage
Unreported Unpaid
(8 )
(9)
0.0%
:::
10.0%
23.0%

0.0%
:::
29.0%
58.0%

Projected Ultimate Claims
Using B-F Method with
Reported
Paid
(10)
(11)
700,000
:::
1,366,646
1,444,287
10,086,005

700,000
:::
1,296,467
1,281,397
9,806,090

Estimated IBNR
Using B-F Method with
Reported
Paid
(12)
(13)
0
:::
103,422
249,764
438,638

0
:::
33,243
86,874
158,723

Actual
IBNR
(14)
0
:::
140,358
356,805
601,982

Diff from Actual IBNR
Using B-F Method with
Reported
Paid
(15)
(16)
0
:::
36,936
107,042
163,345

0
:::
107,116
269,931
443,260

Since the expected claims estimate does not change, the expected unreported and unpaid claims remain
the same between Scenario 1 and Scenario 2.
Without a change in the expected claim ratio, this method does not respond to an increasing claim ratios
scenario.
 For the reported BF technique, the estimated IBNR is identical between the Scenario 1 and
Scenario 2.
 The paid BF performs even worse than the reported BF technique, since the expected unpaid
claims is understated to an even greater degree than the expected unreported claims. This is due
to the longer-term nature of the payment pattern than the reporting pattern.
Scenario 3 — U.S. PP Auto Increasing Case Outstanding Strength
Exhibit III, Sheet 2, top section
The reported BF technique produces an estimate of IBNR that is greater than the actual IBNR.
However, the overstatement is less for the reported BF method than for the reported claim development
method because we did not increase the expected claims.
The paid BF method is unaffected by changes only in case outstanding strength (similar to the paid claim
development technique)
Impact of Changing Conditions
U.S. PP Auto - Development of Unpaid Claim Estimate

Exhibit IIl
Sheet 2

Age of
Projected Ultimate Claims
Estimated IBNR
Diff from Actual IBNR
Claims at 12/31/08
CDF to Ultimate Expected Percentage Using B-F Method with Using B-F Method with Actual Using B-F Method with
Accident Accident Year Expected
Reported Paid Unreported Unpaid
Reported
Paid
Reported
Paid
IBNR Reported
Paid
Year
at 12/31/08
Claims
Reported
Paid
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8 )
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
Increasing Case Outstanding Strength
1999
120
700,000
700,000
700,000
1.000
1.000
0.00%
0.00%
700,000
700,000
0
0
0
0
0
:::
:::
:::
:::
:::
:::
:::
:::
:::
:::
:::
:::
:::
:::
:::
:::
2007
24
1,034,219
979,922
734,295
1.119
1.408
10.61%
29.00%
1,089,655
1,034,218 109,733
54,296
54,296 -55,437
0
2008
12
1,085,930
931,185
456,090
1.318
2.381
24.15%
58.00%
1,193,385
1,085,929 262,200
154,744 154,745 -107,455
1
Total
8,804,524
8,551,189
7,573,547
9,009,508
8,804,523 458,319
253,334 253,335 -204,984
1
Column Notes:
(2) Age of accident year at December 31, 2008.
(3) See Chapter 8, Exhibit IV, Sheet 2.
(4) and (5) From last diagonal of reported and paid claim triangles in Chapter 7, Exhibit Ill, Sheets 6 through 9
(6) and (7) CDF based on 5-year simple average age-to-age factors presented in Chapter 7, Exhibit III, Sheets 6 through 9
(8) = [1.00 - (1.00/(6))]
(9) = - [1.00 - (1.00/ (7))]
(10) = [((3) x (8)) + (4)]
(11) = [(3) x (9)) + (5)]
(12) = [(10) - (4)]
(13) = [(11) -( 4)]
(14) Development in Chapter 7, Exhibit III, Sheet 1
(15) = [(14) - (12)]
(16) = [(14) - (13)]

Exam 5, V2

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Chapter 9 – Bornhuetter-Ferguson Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Understanding the relative IBNR values being produced:
A. First review aspects of the development technique:
2 forces contribute to the excessive estimate of IBNR using the development technique.
1. Age-to-age factors increase due to increases in case reserves, and
2. The resulting higher CDFs are multiplied higher reported claims due to the increase in case
outstanding strength. This is known as the leveraging effect due to higher CDFs.
B. Higher CDFs will result in greater percentages of expected unreported claims.
Recall that the CDFs are an important input to the BF method, and that higher CDFs will result in
greater percentages of expected unreported claims.
However, the leveraging effect is not as great because the BF method uses expected claims, not actual
claims, as the basis for determining unreported claims, and expected claims have not changed.
Scenario 4 — U.S. PP Auto Increasing Claim Ratios and Case Outstanding Strength
Exhibit III, Sheet 2, bottom section. Keep in mind there is no change in the expected claims assumption.
Impact of Changing Conditions
U.S. PP Auto - Development of Unpaid Claim Estimate
Age of
Claims at 12/31/08
CDF to Ultimate
Accident Accident Year Expected
Year
at 12/31/08
Claims
Reported
Paid Reported Paid
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Increasing Claim Ratios and Case Outstanding Strength
1999
120
700,000
700,000 700,000 1.000
1.000
:::
:::
:::
:::
:::
:::
:::
2007
24
1,034,219 1,329,895 996,544 1.120
1.408
2008
12
1,085,930 1,330,264 651,558 1.320
2.381
Total
8,804,524 9,901,691 8,575,113

Exhibit IIl
Sheet 2

Expected Percentage
Unreported Unpaid
(8 )
(9)
0.0%
:::
10.7%
24.3%

0.0%
:::
29.0%
58.0%

Projected Ultimate Claims
Estimated IBNR
Diff from Actual IBNR
Using B-F Method with Using B-F Method with Actual Using B-F Method with
Reported
Paid
Reported
Paid
IBNR Reported
Paid
(10)
(11)
(12)
(13)
(14)
(15)
(16)
700,000
:::
1,440,327
1,593,780
10,362,125

700,000
:::
1,296,467
1,281,397
9,806,090

0
:::
110,432
263,516
460,434

0
:::
-33,428
-48,867
-95,601

0
0
0
:::
:::
:::
73,687 -36,745 107,115
221,064 -42,452 269,931
347,658 -112,776 443,260

Observations:
 Estimated IBNR based on the reported BF method is overstated.
 Estimated IBNR based on the paid BF projection is understated.
 The expected claims used in the example are too low for both projections.
Comments on the results from the paid BF method (given no change in expected claims):
 The paid BF method produces an IBNR estimate $443,260 lower than the actual IBNR, because
there is no change in expected claims.
This is the same difference between estimated and actual IBNR that we saw in Scenario 2, where
claim ratios increased and case outstanding strength remained stable.
 Since the payment pattern is unaffected by changes in case outstanding adequacy, there is no
effect on the paid BF method, and the understatement of expected claims the sole reason for the
inadequacy of the paid BF method.
Understanding the relative IBNR values being produced from reported BF technique:
 In Scenario 2 (increasing claim ratios and stable case outstanding strength), the reported BF
technique produces an estimated IBNR that is lower than the actual IBNR.
 In Scenario 3 (stable claim ratio and increasing case outstanding strength), the reported BF
technique produces an estimated IBNR that is higher than the actual IBNR.
These factors in Scenarios 2 and 3 work in opposite ways in Scenario 4.
While expected claims are too low, higher CDFs more than offsets this effect, leading to an
estimated IBNR is $112,773 higher than the actual IBNR.
Key: In general, the difference from the actual IBNR using the BF method could be positive or negative
depending on the extent of case outstanding strengthening and the deterioration in the claim ratio.

Exam 5, V2

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Chapter 9 – Bornhuetter-Ferguson Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
U.S. Auto Steady-State (No Change in Product Mix)
Exhibit IV, top section (See the exhibit at the end of this chapter)
The BF technique generates the correct IBNR requirement when there is no change in the product mix
(this is similar to the projections using the claim development and expected claims techniques. To
understand why, review the rationale given in these examples in prior chapters).
U.S. Auto Changing Product Mix (i.e. the volume of commercial auto insurance is increasing at a
greater rate than that of private passenger auto insurance).
Exhibit IV, bottom section (See the exhibit at the end of this chapter)
 Both the reported and paid BF methods produce estimated IBNR lower than the actual IBNR.
This is due to the expected claim ratio assumption being unchanged from the U.S. Auto Steady-State.
Adjustments needed:
 The expected claim ratio assumption needs to be modified (due to the commercial auto segment
growing at a greater rate than the private passenger auto segment).
 The reporting and payment patterns also require change. With an increasing proportion of
commercial auto, the reporting and payment patterns lengthen, and results in the requirement for
a higher IBNR value.
Benktander Technique (See Volume 1b for commentary on this technique)
An advantage of the BF technique versus the development technique is stability in the presence of sparse
data.
The Benktander method is a credibility-weighted average of the BF technique and the development technique.
The advantage of the method is that it will prove more responsive than the BF technique and more stable than
the development technique (see "Credible Claims Reserves: The Benktander Method").
The Benktander method is an iterative BF method. The only difference in the two methods is the
derivation of the expected claims.
 For the BF method, expected claim = expected claim ratio * earned premium.
 For the Benktander technique, expected claims are the projected ultimate claims from an initial
BF projection (thus, the reference to the Benktander method as an iterative BF method).
 The Benktander projection of ultimate claims will approach the projected ultimate claims produced by the
development technique after sufficient iterations (see Mack's 2000 ASTIN paper for the detailed proof.)
Exhibits V and VI: The Benktander technique is shown using the six examples of changing environments.
The same exhibit format used for the BF technique is used for the Benktander method.
The only difference in the two methods is the expected claims that are used (see prior page).
(See the exhibit at the end of this chapter)

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ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
The following table summarizes the differences from the true unpaid claims, in thousands of dollars, based on
the BF technique and the Benktander technique for the six examples related to changing environments.

Example Name
U.S. PP Auto Steady-State
U.S. PP Auto Increasing Claim Ratios
U.S. PP Auto Increasing Case
Outstanding Strength
U.S. PP Auto Increasing Claim Ratios
and Case Outstanding Strength
U.S. Auto Steady-State
U.S. Auto Changing Product Mix

Difference from True IBNR ($000) Using
Bornhuetter-Ferguson
Method
Benktander Method
Reported
Paid
Reported
Paid
0
0
0
0
163
443
29
196
-205
0
-239
0
-113

443

-300

196

0
223

0
400

0
233

0
498

Observations:
The Benktander technique is:
 significantly more responsive to changes in the underlying claim ratio
 less responsive to changes in the case outstanding adequacy.
 less responsive to changes in the product mix than the BF technique.
The Benktander method always gives greater credibility to the development technique.
Thus, given no changes in the underlying claim development patterns, we expect the Benktander
method to be more responsive than the BF method.
When claim development patterns are changing, the Benktander method may not produce the most appropriate
estimate (as seen in the examples with changing case outstanding adequacy and changes in product mix).
With the changing product mix, the Benktander method would have proven responsive to the changing
claim ratio but not to the changes in the underlying development patterns.

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Chapter 9 – Bornhuetter-Ferguson Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Chapter 9 - Bornhuetter-Ferguson Technique
Impact of Change in Product Mix Example
U.S. PP Auto - Development of Unpaid Claim Estimate

Exhibit IV

Age of
Claims at 12/31/08
CDF to Ultimate Expected Percentage
Accident Accident Year Expected
Reported Paid Unreported Unpaid
Year
at 12/31/08
Claims
Reported
Paid
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8 )
(9)
Steady-State (No Change in Product Mix)
1999
120
1,500,000 1,500,000 1,500,000 1.000 1.000
0.0%
0.0%
2000
108
1,575,000 1,575,000 1,566,600 1.000 1.005
0.0%
0.5%
2001
96
1,653,750 1,653,750 1,628,393 1.000 1.016
0.0%
1.5%
2002
84
1,736,438 1,736,438 1,700,551 1.000 1.021
0.0%
2.1%
2003
72
1,823,260 1,814,751 1,757,622 1.005 1.037
0.5%
3.6%
2004
60
1,914,422 1,885,068 1,786,794 1.016 1.071
1.5%
6.7%
2005
48
2,010,143 1,948,499 1,742,124 1.032 1.154
3.1%
13.3%
2006
36
2,110,651 1,937,577 1,581,581 1.089 1.335
8.2%
25.1%
2007
24
2,216,183 1,852,729 1,277,999 1.196 1.734
16.4%
42.3%
1.484 3.191
32.6%
68.7%
2008
12
2,326,992 1,568,393 729,124
Total
18,866,839 17,472,205 15,270,788
Changing Product Mix
1999
120
2000
108
2001
96
2002
84
2003
72
2004
60
2005
48
2006
36
2007
24
2008
12
Total

1,500,000
1,575,000
1,653,750
1,736,438
1,823,260
1,914,422
2,249,447
2,673,012
3,211,085
3,897,387
22,233,800

1,500,000
1,575,000
1,653,750
1,736,438
1,814,751
1,885,068
2,193,545
2,471,446
2,680,487
2,556,695
20,067,180

1,500,000
1,566,600
1,628,393
1,700,551
1,757,622
1,786,794
1,951,435
1,983,482
1,766,164
1,097,644
16,738,685

1.000
1.000
1.000
1.000
1.005
1.016
1.032
1.090
1.200
1.503

1.000
1.005
1.016
1.021
1.037
1.071
1.154
1.336
1.750
3.273

0.0%
0.0%
0.0%
0.0%
0.5%
1.5%
3.1%
8.3%
16.7%
33.5%

Projected Ultimate Claims
Estimated IBNR
Using B-F Method with Using B-F Method with
Reported
Paid
Reported
Paid
(10)
(11)
(12)
(13)

Actual
IBNR
(14)

1,500,000 1,500,000
0
0
1,575,000 1,575,000
0
0
1,653,750 1,653,751
0
0
1,736,438 1,736,437
0
-1
1,823,260 1,823,259
8,509
8,508
1,914,422 1,914,422
29,354
29,354
2,010,144 2,010,143
61,645
61,644
2,110,650 2,110,651 173,073 173,074
2,216,183 2,216,183 363,454 363,454
2,326,992 2,326,992 758,599 758,599
18,866,839 18,866,838 1,394,634 1,394,633

0
0
0
-1
8,508
29,354
61,644
173,074
363,454
758,599
1,394,634

0
0
0
-1
0
0
0
0
0
0
0

0
0
0
0
0
0
0
0
0
0
0

1,500,000 1,500,000
0
0
1,575,000 1,575,000
0
0
1,653,750 1,653,751
0
0
1,736,438 1,736,437
0
-1
1,823,260 1,823,259
8,509
8,508
1,914,422 1,914,422
29,354
29,354
2,262,528 2,251,361
68,983
57,816
2,692,025 2,656,353 220,579 184,907
3,216,658 3,142,864 536,171 462,377
3,860,965 3,804,379 1,304,270 1,247,684
22,235,046 22,057,827 2,167,866 1,990,647

0
0
0
-1
8,508
29,354
71,855
239,057
596,924
1,445,385
2,391,083

0
0
0
-1
0
0
2,871
18,478
60,754
141,115
223,217

0
0
0
0
0
0
14,039
54,150
134,547
197,701
400,438

0.0%
0.5%
1.5%
2.1%
3.6%
6.7%
13.3%
25.2%
42.9%
69.4%

Diff from Actual IBNR
Using B-F Method with
Reported
Paid
(15)
(16)

Column Notes:
(2) Age of accident year at December 31, 2008.
(3) See Chapter 8, Exhibit V.
(4) and (5) From last diagonal of reported and paid claim triangles in Chapter 7, Exhibit IV, Sheets 2 through 5.
(6) and (7) CDF based on 5-year simple average age-to-age factors presented in Chapter 7, Exhibit IV, Sheets 2 through 5.
(8) = [1.00 - (1.00 / (6))].
(9) = [1.00 - (1.00 / (7))].
(10) = [((3) x (8)) + (4)].
(11) = [((3) x (9)) + (5)].
(12) = [(10) - (4)].
(13) = [(11) - (4)].
(14) Developed in Chapter 7, Exhibit IV, Sheet 1.
(15) = [(14) - (12)]
(16) = [(14) - (13)].

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Chapter 9 – Bornhuetter-Ferguson Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND

Exam 5, V2

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Chapter 9 – Bornhuetter-Ferguson Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND

Exam 5, V2

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 2014 by All 10, Inc.

Exhibit VI

Expected Ultimate Claims
Projected Ultimate Claims
Estimated IBNR
Age of
Claims at 12/31/08
CDF to Ultimate Expected Percentage Using G-B Method with Using G-B Method with
Accident Accident Year Using B-F Method with
Reported
Paid
Reported
Paid
Year
at 12/31/08 Reported
Paid
Reported
Paid
Reported Paid Unreported Unpaid
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8 )
(9)
(10)
(11)
(12)
(13)
(14)
Steady-State (No Change in Product Mix)
1999
120
1,500,000 1,500,000 1,500,000 1,500,000 1.000
1.000
0.0%
0.0%
1,500,000
1,500,000
0
0
2000
108
1,575,000 1,575,000 1,575,000 1,566,600 1.000
1.005
0.0%
0.5%
1,575,000
1,575,000
0
0
2001
96
1,653,750 1,653,751 1,653,750 1,628,393 1.000
1.016
0.0%
1.5%
1,653,750
1,653,751
0
1
2002
84
1,736,438 1,736,437 1,736,438 1,700,551 1.000
1.021
0.0%
2.1%
1,736,438
1,736,437
0
-1
2003
72
1,823,260 1,823,259 1,814,751 1,757,622 1.005
1.037
0.5%
3.6%
1,823,260
1,823,259
8,509
8,508
2004
60
1,914,422 1,914,422 1,885,068 1,786,794 1.016
1.071
1.5%
6.7%
1,914,422
1,914,422
29,354
29,354
2005
48
2,010,144 2,010,143 1,948,499 1,742,124 1.032
1.154
3.1%
13.3%
2,010,144
2,010,143
61,645
61,644
2006
36
2,110,650 2,110,651 1,937,577 1,581,581 1.089
1.335
8.2%
25.1%
2,110,650
2,110,651 173,073
173,074
2007
24
2,216,183 2,216,183 1,852,729 1,277,999 1.196
1.734
16.4%
42.3%
2,216,183
2,216,183 363,454
363,454
1.484
3.191
32.6%
68.7%
2,326,992
2,326,992 758,599
758,599
2008
12
2,326,992 2,326,992 1,568,393 729,124
Total
18,866,839 18,866,838 17,472,205 15,270,788
18,866,839 18,866,838 1,394,634 1,394,633

0
0
0
-1
8,508
29,354
61,644
173,074
363,454
758,599
1,394,634

0
0
0
-1
0
0
0
0
0
0
0

0
0
-1
0
0
0
0
0
0
1
1

Changing Product Mix
1999
120
1,500,000 1,500,000 1,500,000
2000
108
1,575,000 1,575,000 1,575,000
2001
96
1,653,750 1,653,751 1,653,750
2002
84
1,736,438 1,736,437 1,736,438
2003
72
1,823,260 1,823,259 1,814,751
2004
60
1,914,422 1,914,422 1,885,068
2005
48
2,262,528 2,251,361 2,193,545
2006
36
2,692,025 2,656,353 2,471,446
2007
24
3,216,658 3,142,864 2,680,487
2008
12
3,860,965 3,804,379 2,556,695
Total
22,235,046 22,057,827 20,067,180

0
0
0
-1
8,508
29,354
71,855
239,057
596,924
1,445,385
2,391,083

0
0
0
-1
0
0
2,470
16,909
59,823
153,304
232,505

0
0
-1
0
0
0
13,783
58,343
163,795
262,295
498,217

1,500,000
1,566,600
1,628,393
1,700,551
1,757,622
1,786,794
1,951,435
1,983,482
1,766,164
1,097,644
16,738,685

1.000
1.000
1.000
1.000
1.005
1.016
1.032
1.090
1.200
1.503

1.000
1.005
1.016
1.021
1.037
1.071
1.154
1.336
1.750
3.273

0.0%
0.0%
0.0%
0.0%
0.5%
1.5%
3.1%
8.3%
16.7%
33.5%

0.0%
0.5%
1.5%
2.1%
3.6%
6.7%
13.3%
25.2%
42.9%
69.4%

1,500,000
1,575,000
1,653,750
1,736,438
1,823,260
1,914,422
2,262,929
2,693,594
3,217,588
3,848,776
22,225,758

1,500,000
0
1,575,000
0
1,653,751
0
1,736,437
0
1,823,259
8,509
1,914,422
29,354
2,251,616
69,384
2,652,160 222,148
3,113,616 537,101
3,739,785 1,292,081
21,960,046 2,158,578

0
0
1
-1
8,508
29,354
58,071
180,714
433,129
1,183,090
1,892,866

Actual
IBNR
(15)

Diff from Actual IBNR
Using G-B Method with
Reported
Paid
(16)
(17)

Column Notes:
(2) Age of accident year at December 31, 2008.
(3) and (4) Developed in Exhibit IV.
(5) and (6) Front last diagonal of reported and paid claim triangles in Chapter 7, Exhibit IV, Sheets 2 through 5.
(7) and (8) CDF based on 5-year simple average age-to-age factors presented in Chapter 7, Exhibit IV, Sheets 2 through 5.
(9)= (1.00 - (1.00 / (7))].
(10)= [1.00 - (1.00 /.(8))].
(1I) = [((3) x (9)) + (5)].
(12 = [((4) x (10)) + (6)]
(13) = [(11) - (5)]
(14) = [(12) - (5)]
(15) Developed in Chapter 7, Exhibit IV, Sheet I.
(16) = [(15) - (13)].
(17) = [(15) - (14)].

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Chapter 9 – Bornhuetter-Ferguson Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Sample Questions:
1. For the “Reported Claim Development” method, ultimate claims for each accident year were estimated as the
product of the cumulative reported CDF at the valuation age, and reported claims through the valuation date.
For the “Paid Claim Development” method, ultimate claims for each accident year were estimated as the
product of the cumulative paid CDF at the valuation age, and paid claims through the valuation date.
For the “Expected Claims” technique, ultimate claims for an accident year were calculated by multiplying the
appropriate premium for the year by a selected “claim ratio.”
How are ultimate claims estimated using the “Bornhuetter-Ferguson” technique, for one accident year?
2. The “Percentage Unpaid” as described for the Bornhuetter - Ferguson technique, is equal to which of the
following, where CDF = “Claim Development Factor”
 1.00/( cumulative paid CDF)
 1.00 - (1.00/(cumulative paid CDF))
 (cumulative paid CDF) - 1.00
 ((cumulative paid CDF) - 1.00)/( cumulative paid CDF)
1. 
2.  3. , 
4. , , 
5. Neither 1,2,3 or 4
3. Based on the following information, calculate the “Percentage Unreported” as described for the
Bornhuetter - Ferguson technique, where CDF = “Claim Development Factor”

Earned Premium

5,000

Cumulative Paid CDF (ultimate)

2.987

Cumulative Reported CDF (ultimate)

2.457

Reported Claims

3,500

Paid Losses

2,200

Reported Age-to-Age Factor (12-24 months)

1.5

True or False: Bornhuetter-Ferguson method produces Ultimate Claims that are a credibility weighting
between the Development method and the Expected Claims method, where the credibility assigned to the
Development method results is “the percent of claims developed at a particular stage of maturity.”

5. Summarize Friedland’s key points re: “When the B-F Technique Works and When it Does Not.”
Include an advantage over Development techniques, and potential uses of the B-F method.

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Chapter 9 – Bornhuetter-Ferguson Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
6. Based on the following data as of 12/31/08:
Earned
Premium

Accident
Year

2,000
2,200
2,500
2,650
3,000
3,150

2003
2004
2005
2006
2007
2008

Reported Claims including ALAE ($000's omitted)
1st
2nd
3rd
4th
5th
Report
Report
Report
Report
Report
940
1,200
1,250
1,400
1,500
2,250

1,620
1,690
1,725
1,550
1,900

1,700
1,710
1,800
1,900

1,750
1,800
1,950

1,750
1,800

6th
Report
1,750

6a. Estimate the IBNR as of 12/31/08 using the following method: Bornhuetter-Ferguson
Use Expected Claim Ratio = 80% for all years.
To select claim development factors, use the volume-weighted averages for the latest three years.
See also Friedland Chapter 7 and 8 for other methods.
6b. Show that the Bornhuetter-Ferguson method produces Ultimate Claims that are a credibility weighting
between the Development method and the Expected Claims method.
7. Based on the following information through 12/31/08:
Earned
Premium

Accident
Year

1,500
1,650
1,875
2,000
2,250
2,400

2003
2004
2005
2006
2007
2008

Reported Claims including ALAE ($000's omitted)
as of
as of
as of
as of
as of
12/31/03
12/31/04
12/31/05
12/31/06
12/31/07
500

800
700

1,000
1,200
850

1,200
1,250
1,275
550

as of
12/31/08

1,250
1,300
1450
1200
900

1,250
1275
1425
1550
1350
775

Estimate the IBNR using the Bornhuetter-Ferguson Technique if expected claim ratio = 60%
To select claim development factors, use the volume-weighted averages for the latest three years.
8. Based on the following information through 12/31/08:
Earned
Premium

Accident
Year

4,000
4,400
4,840
5,324
5,856

2004
2005
2006
2007
2008

Reported Claims including ALAE ($000's omitted)
12 mo,
24 mo,
36 mo,
48 mo,
60 mo,
Report 1
Report 2
Report 3
Report 4
Report 5
1,500
1,650
1,815
1,997
2,196

2,000
2,250
2,531
2,848

2,250
2,588
2,976

2,500
3,000

2,500

Estimate the IBNR as of 12/31/08 using B-F method if expected claim ratio = 80% for all years.
To select claim development factors, use the volume-weighted averages for the latest three years.

Exam 5, V2

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Chapter 9 – Bornhuetter-Ferguson Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
9. Based on the following information through 12/31/08:
Earned
Premium

Accident
Year

2,000
2,200
2,000
1,900
2,150
2,250

2003
2004
2005
2006
2007
2008

Reported Claims including ALAE ($000's omitted)
12 mo,
24 mo,
36 mo,
48 mo,
60 mo,
Report 1
Report 2
Report 3
Report 4
Report 5
200
290
209
205
228
143

800
403
2,225
1,519
1,126

1,200
1,269
1,224
1,486

1,650
1,639
960

72 mo,
Report 6

1,750
1,800

1,750

Expected Claim Ratio = 80% for all years.
a. List three aggregate methods (discussed in Friedland chapters 7-9) that you could use to
estimate IBNR losses as of 12/31/08.
b. Of the three methods listed in (a.), which do you feel is most appropriate? Why?
c. Estimate the IBNR as of 12/31/08 based on the selected method.
To select claim development factors, use the volume-weighted averages for the latest 3 years.
10. Based on the following information (amounts in 000's):
Accident
Year

Earned
Premium

Reported Claims
at 12-31-08

Expected
Claim Ratio

2005
2006
2007
2008

120,000
120,000
120,000
120,000

25,000
50,000
75000
90000

85%
85%
85%
85%

Selected ultimate reported claim development factors (CDFs):
4.00
12 months
24 months
2.25
36 months
1.25
48 months
1.10

What is the Bornhuetter-Ferguson IBNR estimate at 12/31/08?

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Chapter 9 – Bornhuetter-Ferguson Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
1995 Exam Questions (modified):
44.

You are given the following data:
Earned
Premium

Accident
Year

4,500
5,000
5,200
5,300
5,700

1990
1991
1992
1993
1994

Reported Claims including ALAE ($000's omitted) as of
12 mo,
24 mo,
36 mo,
48 mo,
60 mo,
Report 1
Report 2
Report 3
Report 4
Report 5
2,000
2,102
2,234
2,339
2,482

2,600
2,638
2,938
2,985

2,990
3,086
3,408

3,283
3,343

3,283

Assume that all claims reach ultimate settlement at 60 months, and the expected claim ratio is 75%.
a. (1.5 points) Using the Bornhuetter-Ferguson Technique described in Friedland, determine the
IBNR as of 12/31/94.
Select development factors using latest 3 years, volume-weighted. Show all work.
b. (0.5 points) See Friedland Chapter 7.
c. (1.5 points) See Friedland Chapter 15.
2001 Exam Questions (modified):
1. According to Friedland, it is not appropriate to derive the IBNR reserve as a function of expected
losses for a new line of business using Bornhuetter-Ferguson technique.
2002 Exam Questions (modified):
22. (4 points) You are given the following information:
Accident
Year

Earned
Premium

Reported Claims
at 12-31-01

Expected
Claim Ratio

1998
1999
2000
2001

200
1,000
1,500
1,500

100
1,000
900
600

80%
80%
80%
80%

Selected age-to-age reported claim development factors:
1.25
12 - 24 months
24 - 36 months
1.10
36 - 48 months
1.05
48 - 60 months
1.08

No further development after 60 months.
a. (1 point) See Friedland Chapter 7.
b. (1 point) Calculate the IBNR reserve as of December 31, 2001 using the Bornhuetter-Ferguson
technique. Show all work.
c. (0.5 pt) Identify one situation in which it would be preferable to use the Bornhuetter-Ferguson
method rather than the Development method to estimate the IBNR.
d. (1 point) See Friedland Chapter 15.

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ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2003 Exam Questions (modified):
23. (3 points) You are given the following information:
Earned
Premium

Accident
Year

1,000
1,000
1,500
1,800

1999
2000
2001
2002

Reported Claims including ALAE ($000's omitted)
at age
at age
at age
at age
12 mo
24 mo
36 mo
48 mo
250
500
750
825
200
350
490
300
450
400

•

The expected claim ratio is 75% (including adjustment expenses)

•

Claim development factors should be calculated using an all-years simple average.

•

The tail factor is 1.05 for development from 48 months to ultimate.

a. (1 point) See Friedland Chapter 7.
b. (1 point)

Using the Bornhuetter-Ferguson method, calculate the total IBNR reserve. Show all work.

c. (1 point) Briefly identify two situations when the use of the Bornhuetter-Ferguson method to develop an
IBNR reserve would be preferred over the Development method.
2004 Exam Questions (modified):
25. (2 points) You are given the following information:
•

An insurance company was formed to write workers compensation business in 2001.

•

Earned premium in 2001 was $1,000,000.

•

Earned premium growth through 2003 has been constant at 20% per year.

•

The expected claim ratio for accident year 2001 is 60%.

•

As of December 31, 2003, the company's reserving actuary believes the expected loss ratio
has increased two percentage points each accident year since the company's inception.

•

Selected incurred loss development factors are as follows:
12 to 24 months
1.500
24 to 36 months
1.336
36 to 48 months
1.126
48 to 60 months
1.057
60 to 72 months
1.050
72 to ultimate
1.000

Using the Bornhuetter-Ferguson method, calculate the total IBNR reserve as of December 31, 2003.
Show all work.

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ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2005 Exam Questions (modified):
10. (4 points) You are given the following information:
Earned
Premium

Accident
Year

19,000
20,000
21,000
22,000

2001
2002
2003
2004

at age
12 mo
4,850
5,150
5,400
7,200

Reported Claims by Development Age
at age
at age
at age
24 mo
36 mo
48 mo
9,700
14,100
16,200
10,300
14,900
10,800

Assume an expected Claim Ratio = 0.90 for all years.
Choose selected incremental development factors using a straight average of the age-to-age factors.
Assume no development past 48 months.
a. (1 point) See Friedland Chapter 7.
b. (0.5 point) Using the Bornhuetter-Ferguson method, calculate the indicated IBNR for accident year
2004 as of December 31, 2004.
c, d, and e: See Friedland Chapter 15.
2006 Exam Questions (modified):
1. Given the following information:
Written Premium
Earned Premium
Accident Year Paid Loss
Accident Year Case Reserve
Expected Loss Ratio
Incurred Loss Development Factor

$7,000,000
6,000,000
300,000
1,200,000
60%
1.800

Compute the ultimate claim ratio using the Bornhuetter-Ferguson method.
A.< 37.5%
B. > 37.5% but < 47.5%
C. > 47.5% but < 57.5%
D. > 57.5% but < 67.5%
E. > 67.5%

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ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2007 Exam Questions (modified):
46. (2 points) Given the following information for a large deductible policy effective January 1, 2004
through December 31, 2004:
 Premium:
$475,000
 Observed loss as of 36 months:
$350,000
 Age-to-ultimate development factor as of 36 months:
1.250
 Expected ultimate loss ratio for the insurer's large deductible book of business:
60%
 This policy written by the same insurance company for over 10 years.
a. (0.5 point) Use the Bornhuetter-Ferguson approach to estimate the ultimate loss for the 2004
policy as of December 31, 2006.
b. (0.5 point) Explain how much credibility the Bornhuetter-Ferguson formula assigns to the loss
development projection for this policy.
c. (0.5 point) Briefly describe a disadvantage of the Bornhuetter-Ferguson approach.
d. (0.5 point) Describe a possible improvement to the accuracy of this estimate while still using the
Bornhuetter-Ferguson approach.
2008 Exam Questions
Question 10. Given the following for an accident year:
- Earned Premium:
- Reported Losses as of 12 months:
- Expected loss ratio:
- Expected reporting pattern:

$20,000,000
$10,000,000
70%

Age (months) % Reported
12
40%
24
60%
36
80%
48
90%
60
100%
a. (1.5 points) This portion of the questions is associated with the Brosius article, now on exam 7.
b. (1 point) Estimate the ultimate value of the claims currently aged at 12 months using the BornhuetterFerguson Method on reported claims, as described in Friedland.
c. (.75 points) Explain how the Benktander formula can be described as a credibility weighted average.
(Note: See also Mack, now on exam 7)

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ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2009 Exam Questions
10. (3 points) Given the following information evaluated as of December 31, 2008:
Incurred Loss
Development
Accident Year
Exposure Units
Incurred Loss
Factor to Ultimate
2005
19,000
$3,500,000
1.37
2006
19,750
4,000,000
1.54
2007
21,000
3,800,000
2.22
2008
21,500
2,000,000
5.00
• The expected loss rate for all the accident years is $250 per exposure unit.
• The company has no exposure prior to 2005.
a. (1 point Use the Bornhuetter-Ferguson method to calculate the IBNR at 12/31/2008 for all accident years.
b. (1.5 points) Use the Cape Cod method to calculate the IBNR at December 31, 2008 for all accident years.
c. (0.5 point) For this book of business, briefly discuss whether the Bornhuetter-Ferguson method or the Cape
Cod method would be expected to produce a more accurate IBNR estimate.
2011 Exam Questions
27. (3 points) Given the following information as of December 31, 2010:
Accident
Earned
Claims
Selected CDF
Year
Premium
Reported
to Ultimate
2008
$950,000
$510,000
1.050
2009
$975,000
$520,000
1.120
2010
$1,000,000
$450,000
1.300
a. (1 point) Use the Bornhuetter-Ferguson technique and an expected claims ratio of 60% to estimate
the IBNR for accident year 2010.
b. (1.5 points) Use the Cape Cod technique to estimate the IBNR for accident year 2010.
c. (0.5 point) Describe the primary difference between the Bornhuetter-Ferguson technique and the
Cape Cod technique.
2012 Exam Questions
18. (3.25 points) Given the following information evaluated as of December 31, 2011:

Accident
Year
2009
2010
2011

Earned
Premium
$950,000
$975,000
$1,000,000

Earned
Premium
$978,500
$1,023,750
$1,000,000

On-Level
Claims
Reported
$510,000
$520,000
$465,000

Reported
CDF to
Ultimate
1.05
1.12
1.30

a. (0.75 point) Use the Bornhuetter-Ferguson technique and an expected claims ratio of 60.0% to
estimate the IBNR for accident year 2010.
b. (2 points) Use the Cape Cod technique to calculate the IBNR for accident year 2010.
c. (0.5 point) Describe the difference in the underlying assumption between the two techniques.

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Chapter 9 – Bornhuetter-Ferguson Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to Sample Questions:
1. For the “Reported Claim Development” method, ultimate claims for each accident year were estimated as the
product of the cumulative reported CDF at the valuation age, and reported claims through the valuation date.
For the “Paid Claim Development” method, ultimate claims for each accident year were estimated as the
product of the cumulative paid CDF at the valuation age, and paid claims through the valuation date.
For the “Expected Claims” technique, ultimate claims for an accident year were calculated by multiplying the
appropriate premium for the year by a selected “claim ratio.”
a. How are ultimate claims estimated using the “Bornhuetter-Ferguson” technique, for one accident year?
Friedland suggests it is easiest to split the ultimate losses in two components for the B-F method:
1) Actual claims reported to date +
2) Expected “IBNR” calculated using a-priori expected ultimate claims and the expected percent unreported.
(or Actual paid + Expected unpaid, if the B-F method is performed using paid instead of reported claims)
2. The “Percentage Unpaid” as described for the Bornhuetter - Ferguson technique, is equal to which of the
following, where CDF = “Claim Development Factor”
 1.00/( cumulative paid CDF)
No
 1.00 - (1.00/(cumulative paid CDF))
Yes
 (cumulative paid CDF) - 1.00
No
 ((cumulative paid CDF) - 1.00)/( cumulative paid CDF)
Yes
1. 
2.  3. , 
4. , , 
5. Neither 1,2,3 or 4
Note: “Cumulative” CDFs reflect ultimate levels, while Age-to-Age factors are incremental.
Also, there are two sets of CDFs we can calculate, for paid or reported claims. Here we use paid.
3. Based on the following information, calculate the “Percentage Unreported” as described for the Bornhuetter Ferguson technique, where CDF = “Claim Development Factor”
“Percentage Unreported” Factor =
1 - (1/2.457) = .593
Note: Here we use the ultimate CDF for reported claims.
Keep in mind: For B-F method, IBNR = (Expected Claims) * (% unreported)
4.

a. True or False: Bornhuetter-Ferguson method produces Ultimate Claims that are a credibility weighting
between the Development method and the Expected Claims method, where the credibility assigned to the
Development method results is “the percent of claims developed at a particular stage of maturity.”
TRUE: See sample question below. Similar statement can be made for IBNR. Friedland, Brosius, Mack,
Siewert and Patrik all discuss credibility weightings.

5. Summarize Friedland’s key points re: “When the B-F Technique Works and When it Does Not.”
Advantage: B-F method is less affected by random fluctuations in observed claim activity (that can
significantly distort Development method results).
Uses of B-F: Cases where data is thin and/or volatile, Long-tailed lines (particularly for the most recent
years)
Friedland also notes that the B-F method can be a useful method for short-tail lines as well.

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Chapter 9 – Bornhuetter-Ferguson Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
6. Based on the following data as of 12/31/08:

Earned
Premium

Accident
Year

2,000
2,200
2,500
2,650
3,000
3,150

2003
2004
2005
2006
2007
2008

Reported Claims including ALAE ($000's omitted)
1st
2nd
3rd
4th
5th
Report
Report
Report
Report
Report
940
1,200
1,250
1,400
1,500
2,250

1,620
1,690
1,725
1,550
1,900

1,700
1,710
1,800
1,900

1,750
1,800
1,950

1,750
1,800

6th
Report
1,750

a. Finding IBNR at 12/31/08
Selected CDF calculations
ATA: 3-yr Volume-weighted average
Note: 1st report at 12 months
Reported CDF to Ultimate

1st to 2nd
Report

2nd to 3rd
Report

3rd to 4th
Report

4th to 5th
Report

5th to 6th
Report

1.2470
at 12 mo
1.4344

1.0896*
at 24 mo
1.1503**

1.0557
at 36 mo
1.0557

1.0000
at 48 mo
1.0000

1.0000
at 60 mo
1.0000

* Example of Age-to-Age calculation for 2nd to 3rd report, using 3-year volume-weighted average:
(1900+1800+1750)/(1550+1725+1690) = 1.0896
** Example of Ultimate CDF calculation for claims at 24 months of development:
(1.0896 for 2nd-to-3rd) * (1.0557 for 3rd-to-4th) * (1.00 for 4th-to-5th) * (1.0 tail) = 1.1503

Accident
Year
2003
2004
2005
2006
2007
2008
Total

Exam 5, V2

Age of
Data at
12/31/08

Reported
CDF to
Ultimate

Percent
Reported
12/31/08

Percent
Unreport
12/31/08

(1)
72 months
60 months
48 months
36 months
24 months
12 months

(2) above
1.0000
1.0000
1.0000
1.0557
1.1503
1.4344

(3)=1.0/(2)
100.0%
100.0%
100.0%
94.7200%
86.9300%
69.7200%

(4)=1.-(3)
0.0%
0.0%
0.0%
5.2800%
13.0700%
30.2800%

Earned
Premium

A priori
Expected
Claim Ratio

A priori
Expected
Claims

(5) given
2,000
2,200
2,500
2,650
3,000
3,150

(6) given
80.0%
80.0%
80.0%
80.0%
80.0%
80.0%

(7)=(5)*(6)
1,600
1,760
2,000
2,120
2,400
2,520

Note: The Percent Unreported
= 1 minus inverse of Ult. CDF

"IBNR" Or Shortcut using
IBNR
Expected Expected Claims *
(broadly
Unreport Percent Unreported
defined)
(8)=(7)*(4)
(8)=(5)*(6)*[1.0-1.0/CDF]
0
0
0
0
0
0
111.9360
111.9360
313.6800
313.6800
763.0560
763.0560
1,188.6720
1,188.6720

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Chapter 9 – Bornhuetter-Ferguson Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND

b. Since Expected Ultimate Claims = Actual Reported + Expected Unreported:
Bornhuetter-Ferguson
Development Expected
Credibility
"B-F"
Method
Claims
Weighted
Reported
"IBNR"
Expected
Credibility
Expected Expected
Expected
Accident
Claims at
Expected
Ultimate
to "Actual"
Ultimate
Ultimate
Ultimate
Year
12/31/08
Unreport
Claims
by B-F
Claims
Claims
Claims
(9)
(10)=(8) (11)=(9)+(10)
(12)=(3) (13) Ch 7. (14) Ch. 8
(15)
2003
1,750
0
1,750
100.0%
1,750
1,600
1,750
2004
1,800
0
1,800
100.0%
1,800
1,760
1,800
2005
1,950
0
1,950
100.0%
1,950
2,000
1,950
2006
1,900
111.9360
2,011.9360
94.7200%
2,006
2,120
2,011.8582
2007
1,900
313.6800
2,213.6800
86.9300%
2,186
2,400
2,213.5960
2008
2,250
763.0560
3,013.0560
69.7200%
3,227
2,520
3,013.1993
Total
12,738.6720
(15) = (12)*(13) + [1.0-(12)]*(14)
12,738.6535
(13) & (14) See details in Ch. 7 & Ch. 8 Q & A.
Matches B-F Expected Ultimate Claims in (11)

Minor rounding issues do exist.

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Chapter 9 – Bornhuetter-Ferguson Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
7. Estimate the IBNR using the Bornhuetter-Ferguson Technique if expected claim ratio = 60%
To select claim development factors, use the volume-weighted averages for the latest three years.
Note: Friedland warns that data/triangles may be accumulated in different ways. First, we reorganize:
Reported Claims including ALAE ($000's omitted)
Earned
Accident
12 mo,
24 mo,
36 mo,
48 mo,
60 mo,
Premium
Year
Report 1
Report 2
Report 3
Report 4 Report 5
1,500
1,650
1,875
2,000
2,250
2,400

2003
2004
2005
2006
2007
2008

500
700
850
550
900
775

800
1,200
1,275
1,200
1,350

1,000
1,250
1,450
1,550

1,200
1,300
1,425

1,250
1,275

Selected CDF calculations
ATA: 3-yr Volume-weighted average

12: 24 mo
1.6630
at 12 mo
2.0606

24: 36 mo
1.1565
at 24 mo
1.2391

36:48 mo
1.0608
at 36 mo
1.0714

48:60 mo
1.0100
at 48 mo
1.0100

Reported CDF to Ultimate

Accident
Year
2004
2005
2006
2007
2008
Total

Exam 5, V2

Age of
Data at
12/31/08

Reported
CDF to
Ultimate

Percent
Reported
12/31/08

Percent
Unreport
12/31/08

(1)
60 months
48 months
36 months
24 months
12 months

(2) above
1.00
1.0100
1.0714
1.2391
2.0606

(3)=1.0/(2)
100.00%
99.01%
93.34%
80.71%
48.53%

(4)=1.-(3)
0.00%
0.99%
6.66%
19.29%
51.47%

Earned
Premium

A priori
Expected
Claim Ratio

A priori
Expected
Claims

(5) given
1,650
1,875
2,000
2,250
2,400

(6) given
60.0%
60.0%
60.0%
60.0%
60.0%

(7)=(5)*(6)
990.0
1,125.0
1,200.0
1,350.0
1,440.0

72 mo,
Report 6
1,250

tail
1.00

Note: The Percent Unreported
= 1 minus inverse of Ult. CDF

"IBNR" Or Shortcut using
IBNR
Expected Expected Claims *
(broadly
Unreport Percent Unreported
defined)
(8)=(7)*(4)
(8)=(5)*(6)*[1.0-1.0/CDF]
0.0000
0
11.1375
11.1375
79.9200
79.9200
260.4150
260.4150
741.1680
741.1680
1,092.6405
1,092.6405

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Chapter 9 – Bornhuetter-Ferguson Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
8. Estimate the IBNR using the Bornhuetter-Ferguson Technique if expected claim ratio = 80%
To select claim development factors, use the volume-weighted averages for the latest three years.
Selected CDF calculations
ATA: 3-yr Volume-weighted average
Reported CDF to Ultimate

Accident
Year
2004
2005
2006
2007
2008
Total

12: 24 mo
1.3967
at 12 mo
1.8295

24: 36 mo
1.1523
at 24 mo
1.3099

Age of
Data at
12/31/08

Reported
CDF to
Ultimate

Percent
Reported
12/31/08

Percent
Unreport
12/31/08

(1)
60 months
48 months
36 months
24 months
12 months

(2) above
1.0000
1.0000
1.1368
1.3099
1.8295

(3)=1.0/(2)
100.00%
100.00%
87.97%
76.34%
54.66%

(4)=1.-(3)
0.00%
0.00%
12.03%
23.66%
45.34%

Earned
Premium

A priori
Expected
Claim Ratio

A priori
Expected
Claims

(5) given
4,000
4,400
4,840
5,324
5,856

(6) given
80.0%
80.0%
80.0%
80.0%
80.0%

(7)=(5)*(6)
3,200.0
3,520.0
3,872.0
4,259.2
4,684.8

36:48 mo
1.1368
at 36 mo
1.1368

48:60 mo
1.0000
at 48 mo
1.0000

tail
1.0000

Note: The Percent Unreported
= 1 minus inverse of Ult. CDF

"IBNR" Or Shortcut using
IBNR
Expected Expected Claims *
(broadly
Unreport Percent Unreported
defined)
(8)=(7)*(4)
(8)=(5)*(6)*[1.0-1.0/CDF]
0.0000
0
0.0000
0.0000
465.8016
465.8016
1007.7267
1,007.7267
2124.0883
2,124.0883
3,597.6166
3,597.6166

9. a. List three aggregate methods (discussed in Friedland chapters 7-9) that you could use to estimate
IBNR losses as of 12/31/08.
1. Development Technique
2. Expected Claims Technique
3. Bornhuetter-Ferguson Technique
9b. Of the three methods listed in (a.), which do you feel is most appropriate? Why?
Bornhuetter-Ferguson Technique, because of the volatility of the data

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Chapter 9 – Bornhuetter-Ferguson Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
9c. Estimate the IBNR as of 12/31/08 based on the selected method.
Earned
Premium

Accident
Year

2,000
2,200
2,000
1,900
2,150
2,250

2003
2004
2005
2006
2007
2008

ATA factors by AY:

Reported Claims including ALAE ($000's omitted)
12 mo,
24 mo,
36 mo,
48 mo,
60 mo,
Report 1
Report 2
Report 3
Report 4
Report 5
200
290
209
205
228
143

800
403
2,225
1,519
1,126

1,200
1,269
1,224
1,486

1,650
1,639
960

1,750
1,800

72 mo,
Report 6
1,750

Accident
Year

1st to 2nd
Report
12:24 mo

2nd to 3rd
Report
24:36 mo

3rd to 4th
Report
36:48 mo

4th to 5th
Report
48:60 mo

5th to 6th
Report
50:72 mo

2003
2004
2005
2006
2007

4.0000
1.3897
10.6459
7.4098
4.9386

1.5000
3.1489
0.5501
0.9783

1.3750
1.2916
0.7843

1.0606
1.0982

1.0000

Selected CDF calculations
ATA: 3-yr Volume-weighted average

12: 24 mo
7.5857
at 12 mo
9.0399

24: 36 mo
0.9595
at 24 mo
1.1917

36:48 mo
1.1506
at 36 mo
1.2420

48:60 mo
1.0794
at 48 mo
1.0794

Reported CDF to Ultimate

Accident
Year
2004
2005
2006
2007
2008
Total

Exam 5, V2

Age of
Data at
12/31/08

Reported
CDF to
Ultimate

Percent
Reported
12/31/08

Percent
Unreport
12/31/08

(1)
60 months
48 months
36 months
24 months
12 months

(2) above
1.0000
1.0794
1.2420
1.1917
9.0399

(3)=1.0/(2)
100.00%
92.64%
80.52%
83.91%
11.06%

(4)=1.-(3)
0.00%
7.36%
19.48%
16.09%
88.94%

Earned
Premium

A priori
Expected
Claim Ratio

A priori
Expected
Claims

"IBNR"
Expected
Unreport

(5) given
2,200
2,000
1,900
2,150
2,250

(6) given
80.0%
80.0%
80.0%
80.0%
80.0%

(7)=(5)*(6)
1,760.0
1,600.0
1,520.0
1,720.0
1,800.0

(8)=(7)*(4)
0.000
117.760
296.096
276.748
1,600.920
2,291.524

Page 163

tail
1.00

Note: The Percent Unreported
= 1 minus inverse of Ult. CDF

Or shortcut using
IBNR
Est. Expected Claims
(broadly
defined)
x Percent Unreported
(8)=(5)*(6)*[1.0-1.0/CDF]
0.000
117.760
296.096
276.748
1,600.920
2,291.524

 2014 by All 10, Inc.

Chapter 9 – Bornhuetter-Ferguson Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND

Exam 5, V2

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Chapter 9 – Bornhuetter-Ferguson Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
10. What is the Bornhuetter-Ferguson IBNR estimate at 12/31/08?
To select claim development factors, use the volume-weighted averages for the 3 three years.

Accident
Year
2005
2006
2007
2008
Total

Exam 5, V2

Age of
Data at
12/31/08

Reported
CDF to
Ultimate

Percent
Reported
12/31/08

Percent
Unreport
12/31/08

(1)
48 months
36 months
24 months
12 months

(2) given
1.10
1.25
2.25
4.00

(3)=1.0/(2)
90.9091%
80.0000%
44.4444%
25.0000%

(4)=1.-(3)
9.0909%
20.0000%
55.5556%
75.0000%

Earned
Premium

A priori
Expected
Claim Ratio

A priori
Expected
Claims

"IBNR"
Expected
Unreport

(5) given
120,000
120,000
120,000
120,000

(6) given
85.0%
85.0%
85.0%
85.0%

(7)=(5)*(6)
102,000
102,000
102,000
102,000

(8)=(7)*(4)
9,272.718
20,400.000
56,666.712
76,500.000
162,839.430

Page 165

Note: The Percent Unreported
= 1 minus inverse of Ult. CDF

Or shortcut using
IBNR
Est. Expected Claims
(broadly
defined)
x Percent Unreported
(8)=(5)*(6)*[1.0-1.0/CDF]
9,272.718
20,400.000
56,666.712
76,500.000
162,839.430

 2014 by All 10, Inc.

Chapter 9 – Bornhuetter-Ferguson Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 1995 Exam questions (modified):
44.
You are given the following:
Earned
Premium

Accident
Year

4,500
5,000
5,200
5,300
5,700

1990
1991
1992
1993
1994

Reported Claims including ALAE ($000's omitted) as of
12 mo,
24 mo,
36 mo,
48 mo,
60 mo,
Report 1
Report 2
Report 3
Report 4
Report 5
2,000
2,102
2,234
2,339
2,482

2,600
2,638
2,938
2,985

2,990
3,086
3,408

3,283
3,343

3,283

12: 24 mo
1.2825
at 12 mo
1.6224

24: 36 mo
1.1600*
at 24 mo
1.2650**

36:48 mo
1.0905
at 36 mo
1.0905

48:60 mo
1.0000
at 48 mo
1.0000

tail
1.0000

Accident
Year
1990
1991
1992
1993
1994
Total

Exam 5, V2

Age of
Data at
12/31/94

Reported
CDF to
Ultimate

Percent
Reported
12/31/94

Percent
Unreport
12/31/94

(1)
60 months
48 months
36 months
24 months
12 months

(2) above
1.00
1.00
1.0905
1.2650
1.6224

(3)=1.0/(2)
100.0%
100.0%
91.7011%
79.0514%
61.6371%

(4)=1.-(3)
0.0000%
0.0000%
8.2989%
20.9486%
38.3629%

Earned
Premium

A priori
Expected
Claim Ratio

A priori
Expected
Claims

"IBNR"
Expected
Unreport

(5) given
4,500
5,000
5,200
5,300
5,700

(6) given
75.0%
75.0%
75.0%
75.0%
75.0%

(7)=(5)*(6)
3,375.0
3,750.0
3,900.0
3,975.0
4,275.0

(8)=(7)*(4)
0
0
323.6571
832.7069
1,640.0140
2,796.3779

Page 166

Note: The Percent Unreported
= 1 minus inverse of Ult. CDF

Or shortcut using
IBNR
Est. Expected Claims
(broadly
defined)
x Percent Unreported
(8)=(5)*(6)*[1.0-1.0/CDF]
0
0
323.6571
832.7069
1,640.0140
2,796.3779

 2014 by All 10, Inc.

Chapter 9 – Bornhuetter-Ferguson Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2001 Exam Questions (modified):
1 According to Friedland, it is not appropriate to derive the IBNR reserve as a function of expected losses for a
new line of business using Bornhuetter-Ferguson technique.
False. New lines of business are an example where Bornhuetter-Ferguson is likely to be appropriate.
Solutions to 2002 Exam Questions (modified):
Question 22. a. (1 point) See Friedland Chapter 7.
b. Calculate the IBNR reserve as of December 31, 2001 using the Bornhuetter-Ferguson technique:
Selected ATA factors (given)
(1)
Tail at 60 months
1.00
48 - 60 months
1.08
36 - 48 months
1.05
1.10
24 - 36 months
12 - 24 months
1.25

Reported Ultimate CDF
Tail Factor
at
at
at
at

Exp. % Unreported

Accident

(2) = product of (1)
1.0000

(3) = 1.0 - 1.0 / (2)
0.0%

1.0800
1.1340
1.2474
1.5593

7.4074%
11.8166%
19.8333%
35.8687%

Year
prior
1998
1999
2000
2001

IBNR
(broadly
defined)

48 mo.
36 mo.
24 mo.
12 mo.

(3) The Percent Unreported = 1 minus inverse of Ultimate Reported CDF

Accident
Year
1998
1999
2000
2001
Total

Earned
Premium

A priori
Expected
Claim Ratio

A priori
Expected
Claims

"IBNR"
Expected
Unreport

Or shortcut using
Est. Expected Claims
x Percent Unreported

(4) given
200
1,000
1,500
1,500

(5) given
80.0%
80.0%
80.0%
80.0%

(6)=(4)*(5)
160.0
800.0
1,200.0
1,200.0

(7)=(6)*(3)
11.8518
94.5328
237.9996
430.4244
774.8086

(7)=(Premium)*(Exp Claims %)*[1-1/CDF]

12
95
237.9996
430.4244
774.8086

Note: Compare back to Chapter 7 Q&A (Development Method) for part A.

c. (0.5 point) Identify one situation in which it would be preferable to use the Bornhuetter-Ferguson
method rather than the Development method to estimate the IBNR.
The B-F method is preferable to use when historical data is sparse or non-credible. Insurers face this
when writing new lines of business.
e. (1 point) See Friedland Chapter 15.
Thus, the amount of loss emergence during CY 2002 on AYs 1998 – 2001 is 335,726.

Exam 5, V2

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Chapter 9 – Bornhuetter-Ferguson Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2003 Exam Questions (modified):
23. (3 points)
a. (1 point) See Friedland Chapter 7.
b. (1 point)

Using the Bornhuetter-Ferguson method, calculate the total IBNR reserve. Show all work.

Earned
Premium

Accident
Year

1,000
1,000
1,500
1,800

1999
2000
2001
2002

at age
12 mo
250
200
300
400

Reported Claims including ALAE ($000's omitted)
at age
at age
at age
24 mo
36 mo
48 mo
500
750
825
350
490
450

12:24 mo

24:36 mo

36:48 mo

1999
2000
2001

2.0000
1.7500
1.5000

1.5000
1.4000

1.1000

ATA: Simple Average (all yr)

12: 24 mo
1.7500*

24: 36 mo
1.4500

36:48 mo
1.1000

ATA factors by AY:
Example:
12:24 for AY 2000
1.75 = 350/200

Selected ATA factors (given)
(1)
1.05
Tail at 48 months
36 - 48 months
1.10
24 - 36 months
1.45
1.75
12 - 24 months

Reported Ultimate CDF
(2) = product of (1)
at 48 mo.
1.05
at 36 mo.
1.1550
at 24 mo.
1.6748
at 12 mo.
2.9309
(3) The Percent Unreported = 1 minus inverse of Ultimate Reported CDF

Accident
Year
1999
2000
2001
2002
Total

Earned
Premium

A priori
Expected
Claim Ratio

A priori
Expected
Claims

"IBNR"
Expected
Unreport

(4) given
1,000
1,000
1,500
1,800

(5) given
75.0%
75.0%
75.0%
75.0%

(6)=(4)*(5)
750
750
1,125
1,350

(7)=(6)*(3)
35.7143
100.6493
453.2783
889.3908
1,479.0327

See tail factor

1.05 tail

Exp. % Unreported
(3) = 1.0 - 1.0 / (2)
4.7619%
13.4199%
40.2914%
65.8808%

Accident
Year
1999
2000
2001
2002

Or shortcut using
Est. Expected Claims
x Percent Unreported

IBNR
(broadly
defined)

(7)=(Premium)*(Exp Claims %)*[1-1/CDF]

35.7143
100.6493
453.2783
889.3908
1,479.0327

Note: Compare back to Chapter 7 Q&A (Development Method) for part A.

Exam 5, V2

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Chapter 9 – Bornhuetter-Ferguson Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
c. Two reasons to use Bornhuetter-Ferguson method over the Development method:
CAS:
1. Data lacks credibility (i.e. no data volume)
2. Loss development patterns are volatile (i.e. large standard error between selected factors and age to age
factors derived from data)
Friedland comments in Chapter 9:
“Actuaries frequently use the Bornhuetter-Ferguson method for long-tail lines of insurance, particularly for
the most immature years, due to the highly leveraged nature of claim development factors for such lines.
Actuaries may also use the Bornhuetter-Ferguson method if the data is extremely thin or volatile or both. For
example, when an insurer has recently entered a new line of business or a new territory . . .”
Solutions to 2004 Exam Questions (modified):
25. (2 points) Based on the given information, Use the Bornhuetter-Ferguson method to calculate the total IBNR
reserve as of December 31, 2003. Show all work.

Accident
Year
1999
2000
2001
2002
2003
Total

ATA
Factors
Given

Reported
CDF to
Ultimate

Percent
Reported
Given

Percent
Unreport
Given

(1)
1.050
1.057
1.126
1.336
1.500

(2) from (1)
1.0500
1.1099
1.2497
1.6696
2.5044

(3)=1.0/(2)
95.2381%
90.0982%
80.0192%
59.8946%
39.9297%

(4)=1.-(3)
4.7619%
9.9018%
19.9808%
40.1054%
60.0703%

Earned
Premium

A priori
Expected
Claim Ratio

A priori
Expected
Claims

"IBNR"
Expected
Unreport

(5) given
n/a
n/a
1,000,000
1,200,000
1,440,000

(6) given
n/a
n/a
60.0%
62.0%
64.0%

(7)=(5)*(6)
n/a
n/a
600,000
744,000
921,600

(8)=(7)*(4)
n/a
n/a
119,884.8000
298,384.1760
553,607.8848
971,876.8608

Note: The Percent Unreported
= 1 minus inverse of Ult. CDF

Or shortcut using
IBNR
Est. Expected Claims
(broadly
defined)
x Percent Unreported
(8)=(5)*(6)*[1.0-1.0/CDF]
n/a
n/a
119,884.8000
298,384.1760
553,607.8848
971,876.8608

Note: Many problems use the same Expected Claims Ratio for all years. This problem does not.

Exam 5, V2

Page 169

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Chapter 9 – Bornhuetter-Ferguson Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2005 Exam Questions (modified):
10. (4 points) You are given the following information:

Earned
Premium

Accident
Year

19,000
20,000
21,000
22,000

2001
2002
2003
2004

at age
12 mo
4,850
5,150
5,400
7,200

Reported Claims by Development Age
at age
at age
at age
24 mo
36 mo
48 mo
9,700
14,100
16,200
10,300
14,900
10,800

Assume an expected Claim Ratio = 0.90 for all years.
Choose selected factors using a straight average of the age to age factors.
Assume no development past 48 months.
a. (1 point) See Friedland Chapter 7
b. (0.5 point) Using the Bornhuetter-Ferguson method, calculate the indicated IBNR for accident year
2004 as of December 31, 2004.
AY

12:24 mo

24:36 mo

36:48 mo

2001
2002
2003

2.000
2.000
2.000
12: 24 mo

1.4536
1.4466

1.1489

24: 36 mo

36:48 mo

ATA: Simple Average (all yr)

2.0000

1.4501

1.1489

ATA factors by AY:

Selected ATA factors
(1)
1.00
Tail at 48 months
36 - 48 months
1.15
24 - 36 months
1.45
12 - 24 months
2.00

See tail factor

1.00 tail

Reported Ultimate CDF
(2) = product of (1)
at 48 mo.
1.00
at 36 mo.
1.15

Exp. % Unreported
(3) = 1.0 - 1.0 / (2)
0.0%
12.9602%

at 24 mo.
1.6660
at 12 mo.
3.3320
(3) The Percent Unreported = 1 minus inverse of Ultimate Reported CDF

39.9760%
69.9880%

Accident
Year
2001
2002
2003
2004
Total

Earned
Premium

A priori
Expected
Claim Ratio

A priori
Expected
Claims

"IBNR"
Expected
Unreport

(4) given
19,000
20,000
21,000
22,000

(5) given
90.0%
90.0%
90.0%
90.0%

(6)=(4)*(5)
17,100
18,000
18,900
19,800

(7)=(6)*(3)
0
2,332.8360
7,555.4640
13,857.6240
23,745.9240

Or shortcut using
Est. Expected Claims
x Percent Unreported

Accident
Year
2001
2002
2003
2004

IBNR
(broadly
defined)

(7)=(Premium)*(Exp Claims %)*[1-1/CDF]

0
2,332.8360
7,555.4640
13,857.6240
23,745.9240

Note: Only the calculations for Accident Year 2004 are required:
22,000 * 90% * (1 - 1/3.3320) = 19,800 * 70% =

13,857.6240

Note: Compare back to Chapter 7 Q&A (Development Method) for part A.

2005 #10 parts c, d, e: See Friedland Chapter 15

Exam 5, V2

Page 170

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Chapter 9 – Bornhuetter-Ferguson Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2006 Exam Questions (modified):
1. Given the following information:
Written Premium
Earned Premium
Accident Year Paid Loss
Accident Year Case Reserve
Expected Loss Ratio
Incurred Loss Development Factor

$7,000,000
6,000,000
300,000
1,200,000
60%
1.800

Compute the ultimate claim ratio using the Bornhuetter-Ferguson method.
A.< 37.5%
B. > 37.5% but < 47.5%
C. > 47.5% but < 57.5%

D. > 57.5% but < 67.5%

The formula for the ultimate loss ratio using the BF method is as follows:
 Paid Loss + Case Reserve + IBNR estimate 

Earned Premium



Ultimate Loss Ratio = 

The formula to compute IBNR using the BF method is as follows:



IBNR = Expected Loss * Expected Loss IBNR Factor = Expected Loss *  1.000-

1.000


LDF to ultimate 

Expected Loss = (Earned Premium)(Expected Loss Ratio) = (6,000,000)(.60) = 3,600,000
IBNR Factor =1.0 - 1/1.80=.4444
Thus, IBNR = (3,600,000)(.4444) = 1,600,000
Therefore, the ultimate loss ratio using the BF method is computed as follows:
 300,000 + 1,200,000 + 1,600,000 
=
= 51.67%
 .5167
6,000,000



Ultimate Loss Ratio = 

Answer C

Solutions to 2007 Exam Questions (modified):
Question 46
a. (0.5 point) Use the BF approach to estimate the ultimate loss for the 2004 policy as of 12/31/2006.
b. (0.5 point) Explain how much credibility the BF formula assigns to the loss dev projection for this
policy.
c. (0.5 point) Briefly describe a disadvantage of the Bornhuetter-Ferguson approach.
d. (0.5 point) Describe a possible improvement to the accuracy of this estimate while still using the BF
approach.
Question 46 - Model Solution 1
a. BF Ult = Observed + (1 – 1/LDF)*(Prem)*(Loss Ratio) = 350,000 + (1 – 1/1.25)*(475,000)*(0.6) = 407,000
b. Solve for Z: 350k*(1.25)*Z + 475k*(0.6)*(1 – Z) = 407,000; Z = 0.8
c. It reduces the use of actual losses to the extent of the complement of credibility
d. Use the Benktander iterative approach. (See Friedland Chapter 9)
Question 46 - Model Solution 2
a. BF Ult = 350,000 + (1 – 1/1.25)*(475,000)*(0.6) = 407,000
b. It gives credibility to the extent that losses were expected to emerge up to the given age ie Z = 1 /LDF.
c. This approach is tied to an arbitrary expected loss so it isn't as responsive to deteriorating emerging
losses as other methods
d. The Stanard-Buhlmann method uses the B-F method but gives a way of calculating the expected loss
method on historical losses so it is less arbitrary. (See Friedland Chapter 10)

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Chapter 9 – Bornhuetter-Ferguson Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2008 Exam Questions
Question 10

b) First, note:
1) Development Method Est. Ultimate: Already shown in Ch 7 as $25M
2) Expected Claims Method Est. Ultimate: Already shown in Ch 8 as $14M
We can show the Bornhuetter-Ferguson estimate of Ultimate Claims in two ways:
(i) B-F Ultimate = Reported Claims + IBNR (as Premium * LR * % unreported)
= 10M + 20M * .7 * (1-40%) = 18.4M
OR
(ii) B-F Ultimate = p * [Development Method Est] + [1-p]*[ Budgeted Loss Est. ]
where p = percent reported
= (40%) * 25M + (1-40%) * 14M = 18.4M
c) We can show the Benktander estimate of Ultimate Claims in two ways too:
(i) Est. Ultimate = Reported Claims + IBNR (as B-F Est. Ultimate * % unreported)
= 10M + 18.4M * (1-40%) = 21.04M
…Note: since the B-F output was an input for the Bentander method, it is
sometimes referred to as an "iterative Bornhuetter-Ferguson Method"
See Mack and Friedland Chapter 9 for more details.
(ii) The second way to find the Benktander estimated ultimate::
Benktander Ultimate = p * [Development Method Ult.] + [1-p]*[ B-F Est. Ultimate ]
where p = percent reported
= (40%) * 25M + (1-40%) * 18.4M = 21.04M
Alternatively, by APPLYING DIFFERENT WEIGHTS, we can also illustrate
a weighting of the Development Method and the Bedgeted Loss method.
Then, the weight applied to the Budgeted Loss Method is [% unreported] 2
and the rest to the Development Method (chain ladder)
Benktander Ult. = [1-q 2 ] * [Dev. Method Ult.] + [q 2 ]*[ Budgeted Loss Ult. ]
Note switch: Mack shows here it is easier to use q = percent un reported
= (1-60%2 ) * 25M + (60% 2 ) * 14M = 21.04M

Exam 5, V2

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Chapter 9 – Bornhuetter-Ferguson Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2009 Exam Questions
Question 10 - Model Solution
a. Use the Bornhuetter-Ferguson method to calculate the IBNR at December 31, 2008 for all accident years.
AY
(1) Exp Unit (2 )Expected Ult. Loss (3) % unreported
(4) IBNR
2005
19,000
4,750,000
0.27
1,282,500
2006
19,750
4,937,500
0.35
1,728,125
2007
21,000
5,250,000
0.55
2,887,500
2008
21,500
5,375,000
0.80
4,300,000
Total
10,198,125
(2) = (1) × 250, where 250 is the expected loss rate for all AYs. (3) = 1 – 1/LDFUlt
(4) IBNR = (2) × (3). Total IBNR = 10,198,125
b. Use the Cape Cod method to calculate the IBNR at December 31, 2008 for all accident years.
AY
(5) Used Up Exp Unit (6) Reported Loss
2005
13,870
3,500,000
2006
12,837.5
4,000,000
2007
9,450
3,800,000
2008
4,300
2,000,000
Total
40,457.5
13,300,000
(5) = (1) * (1.0 – (4))
Expected Loss per exposure = 13,300,000 / 40,457.5 = 328.74
Total SB IBNR = Expected loss per exposure * Unused Exposure Units
= 328.74 × (19,000 × 0.27 + 19,250 × 0.35 + 21,000 × 0.55 + 21,500 × 0.8) =13,410,126
= BFIBNR × 328.74/250 =13,410,126
c1. Since the calculated expected loss per exposure with the Cape Cod method is relatively different from
the a prior expectation, I would prefer to use the Cape Cod method since the a prior estimate might
be too low and is not responsive at all to in development pattern.
c2. Select Cape Cod method, because it seems there is deterioration in loss ratios as shown above.
Cape Cod method is more responsive in this case than BF method.

Solutions to 2011 Exam Questions
a. (1 point) Use the BF technique and an expected claims ratio of 60% to estimate the IBNR for AY 2010.
b. (1.5 points) Use the Cape Cod technique to estimate the IBNR for accident year 2010.
c. (0.5 point)Describe the primary difference between the BF technique and the Cape Cod technique.
Question 27 – Model Solution
-1

a. AY 2010 IBNR = EP * ELR *(1.0 – 1/LDF-ult) = 1,000,000 x 0.6 x (1 - 1.30 ) = 138,462 IBNR
b. Comments: Cape Cod ELR computation: "Used-up premium" = EP * % reported. = EP /LDF-Ult
Estimated claim ratios = Actual reported claims/ Used-up premium.
b. Cape Cod ELR = (510 + 520 + 450) / (950/1.05 + 915/1.12 + 1000/1.3) = 0.5816
-1

AY 2010 IBNR = EP * ELR *(1.0 – 1/LDF-ult) = 0.5816 x 1,000,000 x (1 – 1.30 ) = 134,215 IBNR
c. The BF method uses an a priori estimate of the claims ratio. The Cape Cod method uses a claims ratio
calculated from the actual experience.

Exam 5, V2

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Chapter 9 – Bornhuetter-Ferguson Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2012 Exam Questions
18a. (0.75 point) Use the Bornheutter-Ferguson technique and an expected claims ratio of 60.0% to
estimate the IBNR for accident year 2010.
18b. (2 points) Use the Cape Cod technique to calculate the IBNR for accident year 2010.
18c. (0.5 point) Describe the difference in the underlying assumption between the two techniques.
Question 18 – Model Solution 1 (Exam 5B Question 3)
a. IBNR 2010 = 975,000 * 60% (1.0 -1/1.12) = 62,679 as of 2010
b. Cape Cod. Compute the Estimated Claim Ratio (ECR)

=
ECR

000 )
( 510, 000 + 520, 000 + 465, =
∑ rpt
=
57.166%
∑ used-up premium ( 978, 500 *1 / 1.05 + 1, 023, 750 *1 / 1.12 + 1, 000, 000 *1 / 1.3)

Unadj ECR for AY 2010 = 57.166% * 1,023,750/975,000 = 60%
AY 2010 IBNR = 60% * 975,000 (1.0 - 1/1.12) = 62,704
c. the difference is the expected claim ratio. In B-F expected claim ratio is usually from independent
analysis or judgmentally selected. In cape cod ECR is derived from experience period.
Question 18 – Model Solution 2 (Exam 5B Question 3)
a.
BF IBNR = EP x LR x (1 -%RPT) = 975,000 x 60% x (1.0 – 1/1.12) = 62,678.57
b. Cape Cod method

=
ECR

∑ rpt
=
∑ AdjEP×%Rpt

ECR for AY 2010 =

( 510 + 520 + 465 ) × 1, 000

= 0.5717

( 978.5 × 1 / 1.05 + 1, 023.75 × 1 / 1.12 + 1, 000 × 1 / 1.3) × 1000

1, 023, 750

0.60
× 0.5717 =

975, 000
AY 2010 IBNR= 0.60 x 975,000 x (1.0 - 1/1.12) = 62,678.57
c.

For BF method the underlying assumption is future claim ratio will be the same as the prior selected
ratio which is independent from loss experience. Cape cod method estimate selected loss ratio from
historical loss experience and apply it to estimate reserves.

Examiner’s comments
a. Candidates generally understood the problem and calculation. Half the candidates used earned
premium, the other half used on level earned premium. Both answers were accepted.
b. Generally candidates did well on this part as well. Most candidates were very close to the concept of
calculating a different expected claims ratio to apply in a fashion similar to the BF method. Some
common mistakes included not calculating Used Up Premium in order to derive the Cape Cod ECR,
not completing the ECR calculations using all years of data, selecting a simple average instead of a
weighted average, or incorrectly using EP instead of OLEP.
c. Most candidates received full credit.

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Chapter 10 – Cape Cod Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Sec
1
2
3
4
5
6
7
1

Description
Key Assumptions
Common Uses of the Cape Cod Technique
Mechanics of the Cape Cod Technique
Unpaid Claim Estimate Based on Cape Cod Technique
When the Cape Cod Technique Works and When it Does Not
XYZ Insurer
Influence of a Changing Environment on the Cape Cod Method

Pages
174
174
174 - 175
176
176
176 - 177
177 - 179

Key Assumptions

174

The key assumption: Unreported claims will develop based on expected claims, which are computed
using reported (or paid) claims and earned premium.
This is the same assumption under the BF method.
This assumption differs from the primary assumption under the development method, which is unreported
claims will develop based on reported claims to date.

Common Uses of the Cape Cod Technique

174

Reinsurers often use the Cape Cod technique.
The technique can used:
 with reported claims and paid claims.
 for all lines of insurance including short-tail lines and long-tail lines.
Similar to the development and BF methods, the Cape Cod method can use data organized in the following
time intervals:
* Accident year
* Policy year
* Underwriting year
* Report year
* Fiscal year
This technique can be applied to monthly, quarterly, semiannual or annual data.

Mechanics of the Cape Cod Technique

174 - 175

Similarities to the BF technique:
It is a blend of the claim development method and the expected claims method.
The formula of the reported BF method is the same for the Cape Cod method:
Ultimate Claims = Actual Reported Claims + Expected Unreported Claims

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Chapter 10 – Cape Cod Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
The difference between the Cape Cod and BF technique is how expected claims are computed.
 The key innovation according to Patrik in "Reinsurance" is that the SB (Stanard-Buhlmann) Method is
the use of an ultimate expected loss ratio computed using an overall all years combined reported
claims experience. See Foundations of CAS, Chapter 7, “Reinsurance” for Patrik’s development of
the formulae underlying the Cape Cod technique (a.k.a. the SB Method).
 In the BF method, the ultimate expected loss ratio is selected judgmentally
A problem with both the SB Method and BF method is that the IBNR by year is highly dependent upon the
rate level adjusted premium by year (meaning each year's premium must be adjusted to reflect the rate level
cycle on a relative basis).
Exhibit I, Sheet 1: Development of Expected Claim Ratios:
U.S. Industry Auto
Development of Expected Claim Ratio

Accident
Year
(1)
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
Total

Earned
Premium
(2)
68,574,209
68,544,981
68,907,977
72,544,955
79,228,887
86,643,542
91,763,523
94,115,312
95,272,279
95,176,240
820,771,905

Age of
Accident
at 12/31/07
(3)
120
108
96
84
72
60
48
36
24
12

Exhibit I
Sheet 1
Reported
Year Claims at
12/31/2007
(4)
47,742,304
51,185,767
54,837,929
56,299,562
58,592,712
57,565,344
56,976,657
56,786,410
54,641,339
48,853,563
543,481,587

Reported
CDF to
Ultimate
(5)
1.000
1.000
1.001
1.003
1.006
1.011
1.023
1.051
1.110
1.292

% of
Ultimate
Used Up
Reported
Premium
(6) = [1.00 / (5)] (7) = [(2) x (6)]
100.0%
68,574,209
100.0%
68,544,981
99.9%
68,839,138
99.7%
72,327,971
99.4%
78,756,349
98.9%
85,700,833
97.8%
89,700,413
95.1%
89,548,346
90.1%
85,830,882
77.4%
73,665,820
781,488,943

Estimated
Claim
Ratios
(8) = [(4) / (7)]
69.6%
74.7%
79.7%
77.8%
74.4%
67.2%
63.5%
63.4%
63.7%
66.3%
69.5%

Column and Line Notes:
(2) Based on Best's Aggregates & Averages U.S. private passenger automobile experience.
(3) Age of accident year in (1) at December 31, 2007.
(4) Based on Best's Aggregates & Averages U.S. private passenger automobile experience.
(5) Developed in Chapter 7, Exhibit I, Sheet 1.
(6) = [1.00 / (5)].
(7) = [(2) x (6)].
(8) = [(4) / (7)]

Column (2) shows unadjusted earned premiums by year. Reinsurers often use ultimate premiums in instead.
Column (4) reported claims are the latest diagonal from the development triangle in Chapter 7, used to
derive the CDFs in Column (5)
Column (6) is the reporting pattern, where the % reported equals 1/CDF
Column (7), the "used-up premium" equals Column (2) EP * Column (6) % reported.
Column (8) estimated claim ratios, by AY, equal Column (4) actual reported claims/Column (7) used-up premium.
Notes: An alternative to the use of premium and claim ratios.
Use exposures and pure premiums instead of calculating used-up premium.
Calculate used-up exposures and calculate estimated pure premiums instead of estimated claim
ratios for each year in the experience period.
The used-up premium is the denominator in determining expected claim ratio.
This premium allocation represents the premium corresponding to claims expected to be reported
through the valuation date.

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Chapter 10 – Cape Cod Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
The U.S. Industry Auto example:
 There is a change in the claim ratios for the latest AYs compared with the earliest years (i.e. 1998
through 2002).
The average estimated claim ratio for AYs 1998 - 2002 is 75.2% and the claim ratios vary from a
low of 69.6% to a high of 79.7%.
The average estimated claim ratio for AYs 2003 – 2007 is 64.8%
 In the expected claims technique and the BF technique, we rely on different claim ratios for the earlier
years and the latest years in the experience period to best reflect our expectation of expected claims for
each year.
 In contrast, the Cape Cod method uses a weighted average claim ratio from all years.
Thus, a mechanical approach of developing expected claims is used in the Cape Cod method while
actuarial judgment is used the BF method to determine an a priori expected claim estimate.

Unpaid Claim Estimate Based on Cape Cod Technique

176

We follow a similar procedure for determining the unpaid claim estimate based on the Cape Cod technique as
presented in the prior chapters. Estimated IBNR is equal to projected ultimate claims less reported claims and
the total unpaid claim estimate is equal to the difference between projected ultimate claims and paid claims.
Exhibit 1, Sheet 2: Projection of Ultimate Claims using Reported Claims
U.S. Industry Auto
Projection of Ultimate Claims Using Reported Claims ($000)

Accident
Year
(1)
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
Total

Earned
Premium
(2)
68,574,209
68,544,981
68,907,977
72,544,955
79,228,887
86,643,542
91,763,523
94,115,312
95,272,279
95,176,240
820,771,905

Expected
Claim
Ratio
(3)
69.5%
69.5%
69.5%
69.5%
69.5%
69.5%
69.5%
69.5%
69.5%
69.5%

Estimated
Expected
Claims
(4)
47,689,504
47,669,177
47,921,621
50,450,934
55,099,233
60,255,708
63,816,367
65,451,904
66,256,509
66,189,720
570,800,677

Exhibit I
Sheet 2
Reported
CDF
Ultimate
(4) = [(2) x (3)]
1.000
1.000
1.001
1.003
1.006
1.011
1.023
1.051
1.110
1.292

Percentage
Unreported
(6)=1.0- (1.0/(5))
0.0%
0.0%
0.1%
0.3%
0.6%
1.1%
2.2%
4.9%
9.9%
22.6%

Expected
Unreported
Claims
(7) = [(4) x (6)]
0
0
47,874
150,900
328,624
655,601
1,434,777
3,176,068
6,565,960
14,959,286
27,319,090

Reported
Claims at
12/31/2007
(8)
47,742,304
51,185,767
54,837,929
56,299,562
58,592,712
57,565,344
56,976,657
56,786,410
54,641,339
48,853,563
543,481,587

Promected
Ultimate
Claims
(9) = [(7) + (8)]
47,742,304
51,185,767
54,885,803
56,450,462
58,921,336
58,220,945
58,411,434
59,962,478
61,207,299
63,812,849
570,800,677

Column Notes:
(2) Based on Best's Aggregates & Averages U.S. private passenger automobile experience.
(3) Based on total weighted estimated claim ratios developed in Exhibit I, Sheet 1.
(5) Developed in Chapter 7, Exhibit I, Sheet 1.
(8) Based on Best's Aggregates & Averages U.S. private passenger automobile experience.

Exhibit 1, Sheet 3: Calculations associated with the Development of Unpaid Claim Estimate
Columns (2) and (3) contain reported and paid claims data as of 12/31/2007.
Column (4) projected ultimate claims are from Exhibit I, Sheet 2 = expected unreported + reported claims.
Column (5) case outstanding = Columns (2) – Column (3)
Column (6) Estimated IBNR = Projected ultimate claims - Reported claims.
Column (7) total unpaid claim estimate = Case outstanding + Estimated IBNR.

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Chapter 10 – Cape Cod Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
5

When the Cape Cod Technique Works and When it Does Not

176

Note: Similar comments apply to the Cape Cod method as to the BF technique.
An advantage of the Cape Cod method (over the development technique) is that it may not be distorted by
random fluctuations early in the development of an AY.
A shortcoming of the Cape Cod method (compare with the BF technique): It is not necessarily as appropriate
as the BF method if the data is extremely thin or volatile or both.
Since expected claims are based on reported claims to date, there must be a sufficient volume of credible
reported claims to derive a reliable expected claims estimate.
Data adjustments applicable to the Cape Cod and BF methods:
EP adjustments (from a theoretical perspective): Include using historical rate level changes to adjust
historical premiums to an on-level basis.
Claims would also be adjusted for trend, benefit-level changes, and other similar factors.
From a practical perspective, such data is often unavailable, and one may continue to use both the BF and
Cape Cod methods to develop the unpaid claim estimate without the adjustment of premiums or claims.
When evaluating the results of various techniques and selecting final ultimate claims values, take into
account any simplifying assumptions (e.g. not adjusting premium for rate level changes) made.

XYZ Insurer

176 - 177

Weaknesses in Cape Cod method are due to the uncertainty in the selected development patterns for
reported claims.
 Due to the changes the insurer has faced, uncertainty lies in the applicability of historical claim
development patterns.
 Since the Cape Cod method uses these patterns to calculate used-up premium (a critical component
in computing the expected claim ratio), this method may not be appropriate
 Similar to the BF method, reported CDFs are limited to a minimum of 1.00 for the Cape Cod method.

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Chapter 10 – Cape Cod Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Exhibit II, Sheet 1:
XYZ Insurer - Auto BI
Development of Expected Claim Ratio

Accident
Year
(1)
1998
:::
2007
2008
Total

Exhibit II
Sheet 1

On-Level
Age of
Reported
Pure
Earned
On-Level
Earned Accident Year Claims Premium
Premium Adjustment Premium at 12/31/08 at 12/31/08 Trend
(2)
(3)
(4)
(5)
(6)
(7)
20,000
0.989
19,780
132
15,822
1.400
:::
:::
:::
:::
:::
:::
62,438
0.800
49,950
24
31,732
1.034
47,797
1.000
47,797
12
18,632
1.000
732,144
600,140
449,626

Tort
Reform
Factors
(8 )
0.670
:::
1.000
1.000

Adjusted
Claims
at 12/31/08
(9)
14,845
:::
32,819
18,632
374,740

Reported
% of
Used Up
Claim Ratios
CDF to
Ultimate
On-Level
Estimated
Selected
Ultimate
Reported
Premium
Adjusted
Adjusted
(11)=[1.00 /(10)] (12) = [(4) x (11)] (13) = [(9) / (12)]] (14)=[ToT in(13)]
(10)
1.000
100.0%
19,780
75.1%
70.8%
:::
:::
:::
:::
:::
1.512
66.1%
33,036
99.3%
70.8%
2.551
39.2%
18,737
99.4%
70.8%
529,484
70.8%

Estimated
Unadjusted
(15)
74.6%
:::
54.7%
70.8%

Column and Line Notes:
(2) Based on data from insurer.
(3) For 2002 and after, based on Chapter 8, Exhibit III , Sheet 2. For 1998-2001, assume a 2% rate change per annum.
(4) = [(2) x (3)].
(5) Age of accident year in (1) at December 31, 2008.
(6) Based on data from insurer.
(7) Assume an annual pure premium trend rate of 3.425%.
(8) Based on independent analysis of tort reform.
(9) = [(6) x (7) x (8)].
(10) Developed in Chapter 7, Exhibit II, Sheet 1, in which the CDF are limited to a minimum of 1.00.
(15) = [(14) x (3) / (7) / (8)].

1. Compute Column (4) On-Level EP using EP and Column (3) rate level adjustment factors from Exhibit III,
Sheet 2 from Chapter 8 for AYs 2002 and after. Assume a 2% rate change for prior years.
2. Compute Column (9) Adjusted Claims by multiplying reported claims by Column (7) pure premium trend
and Column (8) tort reform factors.
3. Compute Column (12) Used Up OLEP by multiplying OLEP by % of ultimate reported claims (based on the
Column (10) reported CDFs
4. Compute Column (13) "Estimated Adjusted Claim Ratios" by dividing adjusted claims by Used Up OLEP.
"Estimated Adjusted Claim Ratios" indicates that the reported claims are adjusted for inflation and tort
reform.
5. Compute Column (14) "Selected Adjusted Claim Ratios" by using the all years computed claim ratio from
Column (13)
6. Compute Column (15) "Estimated Unadjusted Claim Ratios", which are adjusted back to the rate level,
inflationary level, and tort environment for each AY. These become our starting point for projecting
expected claims in Exhibit II, Sheet 2. The computation is [Column (14)*Column(3)/ Column
(7)/Column(8)]
Exhibit II Sheet 2: Projection of Ultimate Claims Using Reported Claims ($000)
XYZ Insurer - Auto BI
Projection of Ultimate Claims Using Reported Claims ($000)

Accident
Year
(1)
1998
:::
2007
2008
Total

Earned
Premium
(2)
20,000
:::
62,438
47,797
732,144

Expected
Claim
Ratio
(3)
74.6%
:::
54.7%
70.8%

Estimated
Expected
Claims
(4)
14,920
:::
34,181
33,828
510,046

Exhibit II
Sheet 2

Reported
Expected
CDF to
Percentage Unreported
Ultimate Unreported Unreported
Claims
(5)
(6)
(7)
1.000
0.0%
0
:::
:::
:::
1.512
33.9%
11,575
2.551
60.8%
20,567
54,672

Reported
Claims
12/3/2008
(8 )
15,822
:::
31,732
18,632
449,626

Projected
Ultimate
Claims
(9)
15,822
:::
43,307
39,199
504,298

Column Notes:
(2) Based on data from XYZ Insurer.
(3) Selected based on estimated claim ratios developed in Exhibit II, Sheet 1.
(4) = [(2) x (3)].
(5) Developed in Chapter 7, Exhibit II, Sheet 1, limited to a minimum of 1.00.
(6) = [1.00 - (1.00 / (5))].
(7) = [(4) x (6)].
(8) Based on data from insurer.
(9) = [(7) + (8)]

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Chapter 10 – Cape Cod Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Exhibit II Sheet 3: Development of Unpaid Claim Estimate ($000)
XYZ Insurer - Auto BI
Development of Unpaid Claim Estimate ($000)

Accident
Year
(1)
1998
:::
2007
2008
Total

Claims at 12/31/08
Reported
Paid
(2)
(3)
15,822
15,822
:::
:::
31,732
11,865
18,632
3,409
449,626
330,627

Exhibit II
Sheet 3
Projected
Ultimate
Claims
(4)
15,822
:::
43,307
39,199
504,298

Case
Outstanding
at 12/31/08
(5) = [(2) - (3)]
0
:::
19,867
15,223
118,999

Unpaid Claim Estimate
Based on Cape Cod Method
IBNR
Total
(6) = [(4) - (2)] (7) = [(4) - (3)]
0
0
:::
:::
11,575
31,442
20,567
35,790
54,672
173,671

Column Notes:
(2) and (3) Based on data from XYZ Insurer.
(4) Developed in Exhibit II, Sheet 2.
(5) = [(2) - (3)]
(6) = [(4) - (2)].
(7) = [(4) - (3)].

Exhibit II, Sheet 4 (projected ultimate claims) compares the results of the Cape Cod method with the BF
method, the expected claims method, and the claim development method.
XYZ Insurer - Auto BI
Summary of Ultimate Claims ($000)

Accident
Year
(1)
1998
:::
2007
2008
Total

Exhibit II
Sheet 4

Claims at 12/31/08
Reported
Paid
(2)
(3)
15,822
15,822
:::
:::
31,732
11,865
18,632
3,409
449,626
330,627

Projected Ultimate Claims
Expected
Development Method
B-F Method
Reported
Paid
Claims
Reported
(4)
(5)
(6)
(7)
15,822
15,980
15,660
15,822
:::
:::
:::
:::
47,979
77,941
39,835
45,221
47,530
74,995
39,433
42,607
514,929
605,028
561,516
513,207

Paid
(8 )
15,977
:::
45,636
41,049
554,469

Cape
Cod
(9)
15,822
:::
43,307
39,199
504,298

Exhibit II, Sheet 5 (Estimated IBNR) compares the results of the Cape Cod method with the BF method, the
expected claims method, and the claim development method.
XYZ Insurer - Auto BI
Summary of IBNR ($000)

Accident
Year
(1)
1998
:::
2007
2008
Total

Case
Outstanding
at 12/31/08
(2)
0
:::
19,867
15,223
118,997

Exhibit II
Sheet 5

Development Method
Reported
Paid
(3)
(4)
0
158
:::
:::
16,247
46,209
28,898
56,363
65,303
155,402

Estimated IBNR
B-F Method
Expected
Claims
Reported
Paid
(5)
(6)
(7)
-162
0
155
:::
:::
:::
8,103
13,489
13,904
20,801
23,975
22,417
111,890
63,581
104,843

Cape
Cod
(8 )
0
:::
11,575
20,567
54,672

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Chapter 10 – Cape Cod Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
7

Influence of a Changing Environment on the Cape Cod Method

177 - 179

We continue from prior chapters in with these examples using the Cape Cod method.
Scenario 1 — U.S. PP Auto Steady-State
Exhibit III, Sheets 1 and 3 top sections
Exhibit III, Sheet 1: Scenarios 1 and 2 - Development of Expected Claim Ratio
Exhibit III, Sheet 3: U.S. PP Auto - Development of Unpaid Claim Estimate
Impact of Changing Conditions
Scenarios 1 and 2 - Development of Expected Claim Ratio

Exhibit III
Sheet 1

Age of
Reported
Reported
% of
Accident
Earned
Accident Year
Claims
CDF to
Ultimate
Used Up
Year
Premium
at 12/31/08
12/31/2008
Ultimate
Reported
Premium
(1)
(2)
(3)
(4)
(5)
(6) = [1.00 / (5)] (7) = [(2) x (6)]
Steady-State
1999
1,000,000
120
700,000
1.000
100.0%
1,000,000
:::
:::
:::
:::
:::
:::
:::
2007
1,477,455
24
930,797
1.111
90.0%
1,329,709
12
836,166
1.299
77.0%
1,194,523
2008
1,551,328
Total
12,577,893
8,365,888
11,951,266
Column Notes:
(2) Assume $1,000,000 for first year in experience period (1999) and 5% annual increase thereafter.
(3) Age of accident year at December 31, 2008.
(4) From last diagonal of reported claim triangles in Chapter 7, Exhibit III, Sheets 2 and 4.
(5) Developed in Chapter 7, Exhibit III, Sheets 2 and 4.

Estimated
Claim
Ratios
(8) = [(4) / (7)]

Impact of Changing Conditions
U.S. PP Auto - Development of Unpaid Claim Estimate

Accident
Earned
Year
Premium
(1)
(2)
Steady-State
1999
1,000,000
:::
:::
2007
1,477,455
2008
1,551,328
Total
12,577,893

Exhibit III
Sheet 3

Expected Estimated Reported
Expected
Reported
Projected
Claim
Expected CDF to Percentage Unreported Claim at
Ultimate
Estimated
Ratio
Claims
Ultimate Unreported
Claim
12/31/2008
Claims
IBNR
(4) = [(2) x (3)]
(7) = [(4) x (6)]
(9) = [(7) + (8)] (10) = [(9)-(8)]
(5)
(6)
(8)
(3)
70.0%
:::
70.0%
70.0%

700,000
:::
1,034,219
1,085,930
8,804,527

1.000
:::
1.111
1.299

70.0%
:::
70.0%
70.0%
70.0%

0.0%
:::
10.0%
23.0%

0
:::
103,422
249,764
438,639

700,000
:::
930,797
836,166
8,365,888

700,000
:::
1,034,219
1,085,930
8,804,527

0
:::
103,422
249,764
438,639

Difference
from
Actual IBNR

Actual
IBNR
(11)

(12)= [(11) - (10)]

0
:::
103,422
249,764
438,637

0
:::
-1
0
-2

Column Notes:
(2) Assume $1,000,000 for first year in experience period (1999) and 5% annual increase thereafter.
(3) Selected based on estimated overall claim ratio developed in Exhibit Ill, Sheet 1.
(5) Developed in Chapter 7, Exhibit Ill, Sheets 2 and 4.
(6) = [1.00 - (1.00 / (5))].
(8) From last diagonal of reported claim triangles in Chapter 7, Exhibit III, Sheets 2 and 4.
(11) Developed in Chapter 7, Exhibit III, Sheet I.
(12)= [(11) - (10)].

In Chapters 7 – 9, the development technique, expected claims technique, and BF techniques all generate an
accurate IBNR value in a steady-state environment. This is also the case for the Cape Cod Method.

Exam 5, V2

Page 180

 2014 by All 10, Inc.

Chapter 10 – Cape Cod Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Scenario 2 — U.S. PP Auto Increasing Claim Ratios
Exhibit III, Sheets 1 and 3 bottom sections
Advantage of the Cape Cod Method:
Column (8) estimated claim ratios responds to the changing environment in claims experience, since the
expected claim ratio based on reported claims through the valuation date. The total all years combined
estimated claim ratio is 80.7%
Impact of Changing Conditions
Scenarios 1 and 2 - Development of Expected Claim Ratio

Accident
Year
(1)
Increasing Claim
1999
:::
2005
2006
2007
2008
Total

Earned
Premium
(2)
Ratios
1,000,000
:::
1,340,096
1,407,100
1,477,455
1,551,328
12,577,893

Exhibit III
Sheet 1

Age of
Accident Year
at 12/31/08
(3)

Reported
Claims
12/31/2008
(4)

Reported
CDF to
Ultimate
(5)

120
:::
48
36
24
12

700,000
:::
1,116,300
1,203,071
1,263,224
1,194,523
9,647,367

1.000
:::
1.020
1.053
1.111
1.299

% of
Ultimate
Used Up
Reported
Premium
(6) = [1.00 / (5)] (7) = [(2) x (6)]
100.0%
:::
98.0%
95.0%
90.0%
77.0%

Estimated
Claim
Ratios
(8) = [(4) / (7)]

1,000,000
:::
1,313,294
1,336,745
1,329,710
1,194,523
11,951,266

70.0%
:::
85.0%
90.0%
95.0%
100.0%
80.7%

Column Notes:
(2) Assume $1,000,000 for first year in experience period (1999) and 5% annual increase thereafter.
(3) Age of accident year at December 31, 2008.
(4) From last diagonal of reported claim triangles in Chapter 7, Exhibit III, Sheets 2 and 4.
(5) Developed in Chapter 7, Exhibit III, Sheets 2 and 4.
Impact of Changing Conditions
U.S. PP Auto - Development of Unpaid Claim Estimate

Accident
Earned
Year
Premium
(1)
(2)
Increasing Claim Ratios
1999
1,000,000
:::
:::
2005
1,340,096
2006
1,407,100
2007
1,477,455
2008
1,551,328
Total
12,577,893

Expected
Claim
Ratio
(3)
80.7%
:::
80.7%
80.7%
80.7%
80.7%

Estimated
Expected
Claims
(4) = [(2) x (3)]

807,225
:::
1,081,759
1,135,847
1,192,640
1,252,272
10,153,196

Exhibit III
Sheet 3

Reported
Expected
Reported
CDF to Percentage Unreported Claim at
Ultimate Unreported
Claim
12/31/2008
(5)
(6)
(8)
(7) = [(4) x (6)]
1.000
:::
1.020
1.053
1.111
1.299

0.0%
:::
2.0%
5.0%
10.0%
23.0%

0
:::
21,635
56,792
119,264
288,022
505,829

700,000
:::
1,116,300
1,203,071
1,263,224
1,194,523
9,647,367

Projected
Ultimate
Claims
(9) = [(7) + (8)]

700,000
:::
1,137,935
1,259,863
1,382,488
1,482,545
10,153,196

Estimated
IBNR
(10) = [(9)-(8)]

0
:::
21,635
56,792
119,264
288,022
505,829

Actual
IBNR
(11)
0
:::
22,782
63,319
140,358
356,805
601,982

Difference
from
Actual IBNR
(12)= [(11) - (10)]

0
:::
1,146
6,527
21,094
68,783
96,154

Shortcoming of the Expected Claims Method and the BF Method:
The shortcoming is the lack of responsiveness to actual emerging claims.
Estimated IBNR (BF vs. Cape Cod Method)
 In the BF reported claim projection, there is no change in the estimated IBNR of $438,638 between
Scenario 1 and Scenario 2 since the expected claim ratio does not change.
 Using the Cape Cod method, the estimated IBNR is $505,828 for Scenario 2.
While this value is smaller than the actual IBNR required of $601,984, the Cape Cod technique is
more responsive than the BF method when the claim ratios are increasing.

Exam 5, V2

Page 181

 2014 by All 10, Inc.

Chapter 10 – Cape Cod Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Scenario 3 — U.S. PP Auto Increasing Case Outstanding Strength
Exhibit III, Sheets 2 and 4 top section
Exhibit III, Sheet 2 – Scenario 3 - Development of Expected Claim Ratio
Exhibit III, Sheet 4 – Scenario 3 - Development of Unpaid Claim Estimate
Impact of Changing Conditions
Scenarios 3 and 4 - Development of Expected Claim Ratio

Exhibit III
Sheet 2

Age of
Reported Reported
% of
Estimated
Accident
Earned
Accident Year Claims at
CDF to
Ultimate
Used Up
Claim
Year
Premium
at 12/31/08 12/31/2008 Ultimate
Reported
Premium
Ratios
(6) = [1.00 / (5)] (7) = [(2) x (6)] (8) = [(4) / (7)]
(1)
(2)
(3)
(4)
(5)
Increasing Case Outstanding Strength
1999
1,000,000
120
700,000
1.000
100.0%
1,000,000
70.0%
:::
:::
:::
:::
:::
:::
:::
:::
2005
1,340,096
48
933,377
1.020
98.1%
1,314,355
71.0%
2006
1,407,100
36
962,808
1.055
94.8%
1,334,351
72.2%
2007
1,477,455
24
979,922
1.119
89.4%
1,320,694
74.2%
2008
1,551,328
12
931,185
1.318
75.9%
1,176,757
79.1%
Total
12,577,893
8,551,189
11,923,151
71.7%
Column Notes:
(2) Assume $1,000,000 for first year in experience period (1999) and 5% annual increase thereafter.
(3) Age of accident year at December 31, 2008.
(4) From last diagonal of reported claim triangles in Chapter 7, Exhibit III, Sheets 6 and 8.
(5) Developed in Chapter 7, Exhibit III, Sheets 6 and 8.
Impact of Changing Conditions
U.S. PP Auto - Development of Unpaid Claim Estimate

Accident
Year
(1)

Earned
Premium
(2)

Expected
Claim
Ratio
(3)

Estimated
Expected
Claims
(4) = [(2) x (3)]

Reported
CDF to
Ultimate
(5)

Exhibit III
Sheet 4
Expected
Percentage Unreported
Unreported
Claims
(6)
(7) = [(4) x (6)]

Reported
Projected
Claims at
Ultimate
Estimated
12/31/2008
Claims
IBNR
(8)
(9) = [(7) + (8)] (10) = [(9)-(8)]

Difference
from
Actual IBNR

Actual
IBNR
(11)

(12)= [(11) - (10)]

0
:::
4,690
22,162
54,296
154,745
253,335

0
:::
-13,771
-30,014
-58,132
-113,895
-216,240

Increasing Case Outstanding Strength

1999
:::
2005
2006
2007
2008
Total

1,000,000
:::
1,340,096
1,407,100
1,477,455
1,551,328
12,577,893

71.7%
:::
71.7%
71.7%
71.7%
71.7%

717,192
:::
961,106
1,009,161
1,059,619
1,112,600
9,020,765

1.000
:::
1.020
1.055
1.119
1.318

0.0%
:::
1.9%
5.2%
10.6%
24.1%

0
:::
18,461
52,176
112,428
268,640
469,576

700,000
:::
933,377
962,808
979,922
931,185
8,551,189

700,000
:::
951,838
1,014,984
1,092,350
1,199,825
9,020,765

0
:::
18,461
52,176
112,428
268,640
469,576

Column Notes:
(2) Assume $1,000,000 for first year in experience period (1999) and 5% annual increase thereafter.
(3) Selected based on estimated overall claim ratio developed in Exhibit III, Sheet 2.
(4) = [(2) x (3)].
(5) Developed in Chapter 7, Exhibit III, Sheets 6 and 8.
(6) = [1.00 - (1.00 / (5))].
(8) From last diagonal of reported claim triangles in Chapter 7, Exhibit III, Sheets 6 and 8.
(11) Developed in Chapter 7, Exhibit III, Sheet 1.

Observations:
The Cape Cod method results in an estimated IBNR that overstates the actual IBNR by an even greater
amount than the reported BF technique.
 Expected claims for the BF method remain unchanged, the expected claims increase using the Cape
Cod method because the method reflects the higher level of reported claims.
 Projected ultimate claims are increasing for the Cape Cod method under Scenario 3 due to both
increasing expected claims and higher CDFs.

Exam 5, V2

Page 182

 2014 by All 10, Inc.

Chapter 10 – Cape Cod Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Scenario 4 — U.S. PP Auto Increasing Claim Ratios and Case Outstanding Strength
Exhibit III, Sheets 2 and 4 bottom section
Impact of Changing Conditions
Scenarios 3 and 4 - Development of Expected Claim Ratio

Exhibit III
Sheet 2

Age of
Reported Reported
% of
Accident
Earned
Accident Year Claims at
CDF to
Ultimate
Year
Premium
at 12/31/08 12/31/2008 Ultimate
Reported
(6) = [1.00 / (5)]
(1)
(2)
(3)
(4)
(5)
Increasing Claim Ratios and Case Outstanding Strength
1999
1,000,000
120
700,000
1.000
100.0%
:::
:::
:::
:::
:::
:::
2007
1,477,455
24
1,329,895
1.120
89.3%
12
1,330,264
1.320
75.7%
2008
1,551,328
Total
12,577,893
9,901,691

Used Up
Premium

Estimated
Claim
Ratios

(7) = [(2) x (6)] (8) = [(4) / (7)]

1,000,000
:::
1,319,695
1,174,877
11,920,130

70.0%
:::
100.8%
113.2%
83.1%

Impact of Changing Conditions
U.S. PP Auto - Development of Unpaid Claim Estimate

Accident
Year
(1)

Earned
Premium
(2)

Expected
Claim
Ratio
(3)

Estimated
Expected
Claims
(4) = [(2) x (3)]

Reported
CDF to
Ultimate
(5)

Exhibit III
Sheet 4

Percentage
Unreported
(6)

Increasing Claim Ratios and Case Outstanding Strength

1999
:::
2007
2008
Total

1,000,000
:::
1,477,455
1,551,328
12,577,893

83.1%
:::
83.1%
83.1%

830,670
:::
1,227,278
1,288,641
10,448,075

1.000
:::
1.120
1.320

0.0%
:::
10.7%
24.3%

Expected
Unreported
Claims
(7) = [(4) x (6)]

0
0
:::
131,047
312,707
546,384

Reported
Claims at
12/31/2008
(8)
700,000
:::
1,329,895
1,330,264
9,901,691

Projected
Ultimate
Claims
(9) = [(7) + (8)]

700,000
:::
1,460,942
1,642,971
10,448,075

Estimated
IBNR
(10) = [(9)-(8)]

0
:::
131,047
312,707
546,384

Actual
IBNR
(11)
0
:::
73,687
221,064
347,658

Difference
from
Actual IBNR
(12)= [(11) - (10)]

0
:::
-57,360
-91,643
-198,726

Cape Cod method Observations:
 the method can overstate the actual IBNR (e.g. the method responds effectively to the change in
claim ratios, but it overreacts to the change in case outstanding adequacy).
 the method significantly overstates the actual IBNR needed (i.e. indicating that the effect of increasing
case outstanding strength exceeds the influence of increasing claim ratios).
 The estimated claim ratios are higher than their true values by the combined effects of Scenario 4.

U.S. Auto Steady-State (No Change in Product Mix)
Exhibit IV, Sheets 1 and 2 top section:
Similar to our projections using the development and expected claims techniques, the Cape Cod technique
generates the correct IBNR requirement when there is no change in the product mix.

Exam 5, V2

Page 183

 2014 by All 10, Inc.

Chapter 10 – Cape Cod Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
U.S. Auto Changing Product Mix
Exhibit IV, Sheets 1 and 2 bottom section:
Impact of Change in Product Mix Example
Scenarios 5 and 6 - Development of Expected Claim Ratio
Age of
Accident
Earned
Accident Year
Year
Premium
at 12/31/08
(1)
(2)
(3)
Changing Product Mix
1999
2,000,000
120
:::
:::
:::
2005
2,999,262
48
2006
3,564,016
36
2007
4,281,446
24
12
2008
5,196,516
Total
29,645,066

Exhibit IV
Sheet 1

Reported
Claims at
12/31/2008
(4)

Reported
CDF to
Ultimate
(5)

%
Ultimate
Reported

Used Up
Premium

Extimated
Claim
Ratios

(6) = [1.00 / (5)]

(7) = [(2) x (6)]

(8) = [(4) / (7)]

1,500,000
:::
2,193,545
2,471,446
2,680,487
2,556,695
20,067,180

1.000
:::
1.032
1.090
1.200
1.503

100.0%
:::
96.9%
91.7%
83.3%
66.5%

2,000,000
:::
2,907,284
3,269,911
3,566,552
3,457,489
26,754,578

75.0%
:::
75.4%
75.6%
75.2%
73.9%
75.0%

Column and Line Notes:
(2) For no change scenario, assume $2,000,000 for first year in experience period (1999) and 5% annual
increase thereafter. For change scenario, assume annual increase of 30% for commercial auto beginning in 2005.
(3) Age of accident year at December 31, 2008.
(4) From last diagonal of reported claim triangles in Chapter 7, Exhibit IV, Sheets 2 and 4.
(5) Developed in Chapter 7, Exhibit IV, Sheets 2 and 4.
Impact of Change in Product Mix Example
Scenarios 5 and 6 - Development of Unpaid Claim Ratio

Accident
Earned
Year
Premium
(1)
(2)
Changing Product Mix
1999
2,000,000
:::
:::
2005
2,999,262
2006
3,564,016
2007
4,281,446
2008
5,196,516
Total
29,645,066

Expected
Claim
Ratio
(3)
75.0%
:::
75.0%
75.0%
75.0%
75.0%

Extimated
Expected
Claims
(4) = [(2) x (3)]

1,500,093
:::
2,249,586
2,673,178
3,211,284
3,897,629
22,235,181

Exhibit IV
Sheet 2

Reported
CDF to
Ultimate
(5)

Percentage
Unreported
(6)

1.000
:::
1.032
1.090
1.200
1.503

0.0%
:::
3.1%
8.3%
16.7%
33.5%

Expected
Unreported
Claims
(7) = [(4) x (6)]

Reported
Claims at
12/31/2008
(8)

0
:::
68,988
220,593
536,204
1,304,351
2,168,001

1,500,000
:::
2,193,545
2,471,446
2,680,487
2,556,695
20,067,180

Projected
Ultimate
Claims

Estimated
IBNR

(9) = [(7) + (8)]

(10) = [(9)-(8)]

1,500,000
:::
2,262,533
2,692,039
3,216,691
3,861,046
22,235,181

0
:::
68,988
220,593
536,204
1,304,351
2,168,001

Actual
IBNR
(11)
0
:::
71,855
239,057
596,924
1,445,385
2,391,083

Difference
from
Actual IBNR
(12)= [(11) - (10)]

0
:::
2,867
18,464
60,720
141,034
223,083

Column Notes:
(2) For no change scenario, assume $2,000,000 for first year in experience period (1999) and 5% annual increase thereafter.
For change scenario, assume annual increase of 30% for commercial auto beginning in 2005.
(3) Selected based on estimated overall claim ratios developed in Exhibit IV, Sheet I.
(5) Developed in Chapter 7, Exhibit IV, Sheets 2 and 4.
(6) = [1.00 - (1.00 / (5))].
(8) From last diagonal of reported claim triangles in Chapter 7, Exhibit IV, Sheets 2 and 4.
(11) Developed in Chapter 7, Exhibit IV, Sheet I.

The Cape Cod method produces estimated IBNR that is lower than the actual IBNR.
Although reported claims are increasing, there are also changes in the reporting pattern.
Thus, the Cape Cod method does not respond correctly to the changing product mix.

Exam 5, V2

Page 184

 2014 by All 10, Inc.

Chapter 10 – Cape Cod Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Sample Questions:
1. The Cape Cod method is very similar to the Bornhuetter-Ferguson (B-F) method. What is the difference
Friedland cites between the Cape Cod and the B-F methods?
2. What weakness is shared by both the Expected Claims Method and the Bornhuetter-Ferguson Method, but
not the Cape Cod technique?
3. State the primary assumption underlying the Development technique that Friedland notes is not true for the
Cape Cod method or the B-F method.
4. Summarize Friedland’s key points re: “When the Cape Cod Technique Works and When it Does Not.”
Include comments on data adjustments that might be made using this method and the B-F method.
5. What other name does Patrik use to describe the method Friedland calls the Cape Cod technique?
Note: Patrik is now on the exam 7 syllabus.
6. Based on the following data as of 12/31/08:

Earned
Premium

Accident
Year

2,000
2,200
2,500
2,650
3,000
3,150

2003
2004
2005
2006
2007
2008

Reported Claims including ALAE ($000's omitted)
1st
2nd
3rd
4th
5th
Report
Report
Report
Report
Report
940
1,200
1,250
1,400
1,500
2,250

1,620
1,690
1,725
1,550
1,900

1,700
1,710
1,800
1,900

1,750
1,800
1,950

1,750
1,800

6th
Report
1,750

Estimate the IBNR as of 12/31/08 using the following method: Cape Cod Technique
To select claim development factors, use the volume-weighted averages for the latest three
years.
See also Friedland Chapter 7, 8 and 9 for other methods.

1997 Exam Questions (modified):
11. Calculate the Cape Cod technique IBNR estimate at 12-31-96, given the following data:
Accident
Year
1993
1994
1995
1996
Total

Exam 5, V2

Premium
Actual
Adjusted
200
200
250
250
900

Reported Claims
at 12-31-96

Reported
Ult CDF

150
200
100
50
500

1.33
1.49
2.50
10.00

200
250
300
350
1,100

Page 185

 2014 by All 10, Inc.

Chapter 10 – Cape Cod Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2000 Exam Questions (modified):
68. (2 points) You are given the following information as of 12/31/99:
Accident Premium
Year Adjusted
1996
1997
1998
1999
Total

Case Outstanding
at 12-31-99

200,000
300,000
350,000
425,000
1,275,000

Paid Claims Reported
at 12-31-99
Percent

27,000
90,000
135,000
140,000
392,000

120,000
80,000
45,000
20,000
265,000

90%
75%
50%
35%

Use the Cape Cod method to calculate the IBNR as of 12/31/99.
2004 Exam Questions (modified):
49. (2 points) You are given the following information:
Accident
Year (AY)
1998
1999
2000
2001
2002
2003
Total

Premium
Actual
Adjusted
4,500
5,000
5,500
6,000
6,500
7,000
34,500

Reported Claims
at 12-31-03

Reported
Ult CDF

3,200
3,400
3,500
2,800
2,100
1,600
16,600

1.00
1.05
1.18
1.43
2.00
4.00

6,200
6,500
7,500
7,800
7,800
7,000
42,800

Calculate the Cape Cod estimates of IBNR and Ultimate Losses, for AY 2002 only.
2005 Exam Questions (modified):
38. (1.5 points) You are given the following information:
Accident
Year
2001
2002
2003
2004
Total

Premium
Actual
Adjusted
6,000
7,500
9,000
10,000
32,500

Reported Claims
at 12-31-04

Reported
CDF (ultimate)

5,000
4,000
4,000
2,000
15,000

1.143
1.333
2.000
3.333

8,000
8,000
10,000
10,000
36,000

Calculate the a priori “expected claim ratio” to be used in the Cape Cod technique.

Exam 5, V2

Page 186

 2014 by All 10, Inc.

Chapter 10 – Cape Cod Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2008 Exam Questions (modified):
36. (1.5 points)
Given the following as of December 31, 2001:
Calendar
Accident
Year
2003
2004
2005
2006
2007

Earned
Premium
10,000
11,000
13,000
15,000
17,000

Adjusted
Premium
9,000
9,000
11,000
13,000
15,000

Aggregate Aggregate
Reported
Loss
Loss Report Lag
8,000
0.95
8,000
0.88
7,000
0.75
6,000
0.55
4,000
0.30

Calculate the IBNR as of December 31, 2007 using the Stanard-Buhlmann method.
(Note Friedland terminology: using the Cape Cod Method. See also Patrik .)

2010 Exam Questions
18. (2 points) Given the following data for a reinsurer as of December 31, 2009:
Calendar/
Aggregate
Age-toAccident
Earned
Adjusted
Reported
Ultimate
Year
Premium
Premium
Loss
LDF
2005
$10,000
$12,000
$9,000
1.03
2006
11,000
12,000
9,000
1.11
2007
13,000
13,000
7,000
1.25
2008
15,000
14,000
10,000
1.47
2009
17,000
15,000
6,000
2.00
a. (1 point) Use the Stanard-Buhlmann method to calculate the IBNR for accident year 2008 as of
December 31, 2009.
b. (1 point) Discuss two problems that may affect the accuracy of a reinsurer's earned premium data.
2011 Exam Questions
27. (3 points) Given the following information as of December 31, 2010:
Accident
Earned
Claims
Selected CDF
Year
Premium
Reported
to Ultimate
2008
$950,000
$510,000
1.050
2009
$975,000
$520,000
1.120
2010
$1,000,000
$450,000
1.300
a. (1 point) Use the Bornhuetter-Ferguson technique and an expected claims ratio of 60% to estimate
the IBNR for accident year 2010.
b. (1.5 points) Use the Cape Cod technique to estimate the IBNR for accident year 2010.
c. (0.5 point)Describe the primary difference between the Bornhuetter-Ferguson technique and the
Cape Cod technique.

Exam 5, V2

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Chapter 10 – Cape Cod Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2012 Exam Questions
18. (3.25 points) Given the following information evaluated as of December 31, 2011:

Accident
Year
2009
2010
2011

Earned
Premium
$950,000
$975,000
$1,000,000

On-Level
Earned
Premium
$978,500
$1,023,750
$1,000,000

Claims
Reported
$510,000
$520,000
$465,000

Reported
CDF to
Ultimate
1.05
1.12
1.30

a. (0.75 point) Use the Bornheufter-Ferguson technique and an expected claims ratio of 60.0% to
estimate the IBNR for accident year 2010.
b. (2 points) Use the Cape Cod technique to calculate the IBNR for accident year 2010.
c. (0.5 point) Describe the difference in the underlying assumption between the two techniques.

Exam 5, V2

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Chapter 10 – Cape Cod Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to Sample Questions:
1. The Cape Cod method is very similar to the Bornhuetter-Ferguson method. What is the difference Friedland
cites between the Cape Cod and the B-F methods?
While the a-priori expected claim ratio for B-F can be selected judgmentally (allowing it to take on a wide
range of possible values), the a-priori expected claim ratio for Cape Cod is calculated in a specified way.
In particular, Cape Cod relies on the claims experience to date and requires the calculation of “used-up”
premium. See examples below.
2. What weakness is shared by both the Expected Claims Method and the Bornhuetter-Ferguson Method, but
not the Cape Cod technique?
The “lack of responsiveness to actual emerging claims” does not apply to Cape Cod, since the a-priori
expected claim ratio is derived using reported claims (with some adjustments where applicable)
3. State the primary assumption underlying the Development technique that Friedland notes is not true for the
Cape Cod method or the B-F method.
Development Method assumes that IBNR will develop based on reported (not expected) claims to date –
The B-F and Cape Cod methods both use the idea of an “a priori” expected claim estimate.
4. Summarize Friedland’s key points re: “When the Cape Cod Technique Works and When it Does Not.”
Compared to the Development technique, Cape Cod estimates may not suffer the same distortion in the
early development stages of an accident year.
Comments on data adjustments:
Friedland notes that, ideally, it would be best to make adjustments to actual data, for BOTH the BornhuetterFerguson AND Cape Cod techniques. These adjustments include bringing premium on-level (for rate
changes) as well as adjusting claims for trends and benefit-level changes.

5. What other name does Patrik use to describe the method Friedland calls the Cape Cod technique?
Stanard-Buhlmann
Note: Patrik is now on the exam 7 syllabus.

Exam 5, V2

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Chapter 10 – Cape Cod Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
6. Estimate the IBNR as of 12/31/08 using the following method: Cape Cod Technique
To select claim development factors, use the volume-weighted averages for the latest three years.
Selected CDF calculations

1st to 2nd
Report

2nd to 3rd 3rd to 4th
Report
Report

4th to 5th
Report

5th to 6th
Report
1.0000
at 60 mo
1.0000

ATA: 3-yr Volume-weighted average
Note: 1st report at 12 months
Reported CDF to Ultimate

1.2470
at 12 mo
1.4344

1.0896
at 24 mo
1.1503

1.0557
at 36 mo
1.0557

1.0000
at 48 mo
1.0000

Adjusted
Premium
if avail.**
(1) given
2,000
2,200
2,500
2,650
3,000
3,150
15,500

Reported
CDF to
Ultimate

Percent
Reported
to date

Used-Up Reported
Premium
Claims
to date as avail.**

(2) above
1.0000
1.0000
1.0000
1.0557
1.1503
1.4344

(3)=1.0/(2)
100.0000%
100.0000%
100.0000%
94.7239%
86.9338%
69.7156%

"CC"
Estimated
Claim Ratio
(6)=(5)/(4)
see total
see total
see total
see total
see total
see total

Accident
Year
2003
2004
2005
2006
2007
2008
Total

(4)=(1)*(3)
2,000
2,200
2,500
2,510.1834
2,608.0140
2,196.0414
14,014.2388

(5) given
1,750
1,800
1,950
1,900
1,900
2,250
11,550

82.4162%

** The Cape Cod technique allows/prefers use of "adjusted" data where available.
(4) Used-Up premium also equals (1)/(2): Adjusted Premium divided by Ult. CDF
(6) " … method requires the use of the weighted average claim ratio from all years.'

Accident
Year
2003
2004
2005
2006
2007
2008
Total

"CC"
Claim
Ratio

A priori
Expected
Claims

"IBNR"
Expected
Unreport

(6) total
82.4162%
82.4162%
82.4162%
82.4162%
82.4162%
82.4162%

(7)=(1)*(6)
1,648.3240
1,813.1564
2,060.4050
2,184.0293
2,472.4860
2,596.1103

(8)=(7)*[1-(3)]

0.0000
0.0000
0.0000
115.2316
323.0600
786.2164
1,224.5080

Or shortcut using
IBNR
C.C. Expected Claims
(broadly
defined)
x Percent Unreported
(8)=(Prem)*(CC %)*[1.0-1.0/CDF]
0
0
0
115.232
323.059
786.217
1,224.508

Note: See Ch. 9 Q&A. If B-F claim ratio = "CC" claim ratio, results would be identical.

Exam 5, V2

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Chapter 10 – Cape Cod Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 1997 Exam Questions (modified):
11. Calculate the Cape Cod technique IBNR estimate at 12-31-96, given the following data:
Accident
Year
1993
1994
1995
1996
Total

Accident
Year
1993
1994
1995
1996
Total

Premium
Actual
Adjusted
200
200
250
250
900

Adjusted
Premium
if avail.**
(1) given
200
250
300
350
1100

Reported Claims
at 12-31-96

Reported
Ult CDF

150
200
100
50
500

1.33
1.49
2.50
10.00

200
250
300
350
1,100

Reported
CDF to
Ultimate

Percent
Reported
to date

(2) given
1.3333
1.4925
2.5000
10.0000

(3)=1.0/(2)
75.0019%
67.0017%
40.0000%
10.0000%

Used-Up Reported
Premium
Claims
to date as avail.**
(4)=(1)*(3)
150.0038
167.5043
120.0000
35.0000
472.5081

(5) given
150
200
100
50
500

"CC"
Estimated
Claim Ratio
(6)=(5)/(4)
see total
see total
see total
see total
105.8183%

Accident
Year
1993
1994
1995
1996
Total

"CC"
Claim
Ratio

A priori
Expected
Claims

"IBNR"
Expected
Unreport

(6) total

(7)=(1)*(6)
211.6366
264.5458
317.4549
370.3641

(8)=(7)*[1-(3)]

105.8183%
105.8183%
105.8183%
105.8183%

52.9051
87.2956
190.4729
333.3277
664.0013

Or shortcut using
IBNR
C.C. Expected Claims
(broadly
defined)
x Percent Unreported
(8)=(Prem)*(CC %)*[1.0-1.0/CDF]
52.9052
87.2957
190.4729
333.3276
664.0014

Note: Patrik is now on the exam 7 syllabus.

Exam 5, V2

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Chapter 10 – Cape Cod Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2000 Exam Questions (modified):
68. (2 points) You are given the following information as of 12/31/99:
Accident Premium
Year Adjusted
1996
1997
1998
1999
Total

Case Outstanding
at 12-31-99

200,000
300,000
350,000
425,000
1,275,000

Paid Claims Reported
at 12-31-99
Percent

27,000
90,000
135,000
140,000
392,000

120,000
80,000
45,000
20,000
265,000

90%
75%
50%
35%

Use the Cape Cod method to calculate the IBNR as of 12/31/99:

Accident
Year
1996
1997
1998
1999
Total

Adjusted
Premium
if avail.**
(1) given
200,000
300,000
350,000
425,000
1,275,000

Reported
CDF to
Ultimate

Percent
Reported
to date

(2)=1.0/(3)
not used
not used
not used
not used

(3) given
90.0%
75.0%
50.0%
35.0%

Used-Up Reported
Premium
Claims
to date as avail.**
(4)=(1)*(3)
180,000
225,000
175,000
148,750
728,750

(5) See note

147,000
170,000
180,000
160,000
657,000

"CC"
Estimated
Claim Ratio
(6)=(5)/(4)
see total
see total
see total
see total
90.1544%

** The Cape Cod technique allows/prefers use of "adjusted" data where available.
(5) Be sure to add the Paid Claims + Case Outstanding = Reported Claims
(6) " … method requires the use of the weighted average claim ratio from all years.'

Accident
Year
1996
1997
1998
1999
Total

"CC"
Claim
Ratio

A priori
Expected
Claims

(6) total
90.1544%
90.1544%
90.1544%
90.1544%

(7)=(1)*(6)
180,308.80
270,463.20
315,540.40
383,156.20

"IBNR"
Expected
Unreport
(8)=(7)*[1-(3)]

18,030.88
67,615.80
157,770.20
249,051.53
492,468.41

Or shortcut using
IBNR
C.C. Expected Claims
(broadly
defined)
x Percent Unreported
(8)=(Prem)*(CC %)*[1.0-(3)]
18,030.88
67,615.80
157,770.20
249,051.53
492,468.41

Note: Patrik is now on the exam 7 syllabus.

Exam 5, V2

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Chapter 10 – Cape Cod Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2004 Exam Questions (modified):
49. (2 points) You are given the following information:
Accident
Year (AY)
1998
1999
2000
2001
2002
2003
Total

Premium
Actual
Adjusted
4,500
5,000
5,500
6,000
6,500
7,000
34,500

Reported Claims
at 12-31-03

Reported
Ult CDF

3,200
3,400
3,500
2,800
2,100
1,600
16,600

1.00
1.05
1.18
1.43
2.00
4.00

6,200
6,500
7,500
7,800
7,800
7,000
42,800

Calculate the Cape Cod estimates of IBNR and Ultimate Losses, for AY 2002 only.
Accident
Year
1998
1999
2000
2001
2002
2003
Total

Premium
if avail.**
(1) given
6,200
6,500
7,500
7,800
7,800
7,000
42,800

CDF to
Ultimate

Reported
to date

(2) given
1.00
1.05
1.18
1.43
2.00
4.00

(3)=1.0/(2)
100.0000%
94.9668%
85.0340%
69.9790%
50.0000%
25.0000%

Premium
Claims
to date as avail.**
(4)=(1)*(3)
6,200.0000
6,172.8420
6,377.5500
5,458.3620
3,900.0000
1,750.0000
29,858.7540

(5) given
3,200
3,400
3,500
2,800
2,100
1,600
16,600

Estimated
Claim Ratio
(6)=(5)/(4)
see total
see total
see total
see total
see total
see total
55.5951%

A priori
Expected
Claims

"IBNR"
Expected
Unreport

(6) total
55.5951%
55.5951%
55.5951%
55.5951%
55.5951%
55.5951%

(7)=(1)*(6)
3,446.8962
3,613.6815
4,169.6325
4,336.4178
4,336.4178
3,891.6570

(8)=(7)*[1-(3)]

Accident
Year
1998
1999
2000
2001
2002
2003
Total

0.0000
181.8838
624.0272
1,301.8360
2,168.2089
2,918.7428
7,194.6987

Or shortcut using
IBNR
C.C. Expected Claims
(broadly
defined)
x Percent Unreported
(8)=(Prem)*(CC %)*[1.0-1.0/CDF]
0
181.89
624.027
1,301.836
2,168.2089
2,918.7428
7,194.699

See also Patrik .
Note: To find AY 2002 IBNR, only the calculations for 2002 are required:
However, to find the 55.6% "CC" a-priori claim ratio, all years are used.
7800 * 55.5951% * (1 - 1/2.0) =

2,168.2089

Finally:
Cape Cod Estimated Ultimate Claims = Reported Claims + IBNR as above
AY 2002 Cape Cod Estimated Ultimate Claims = 2,100 + 2,168.2089

4,268.2089

Exam 5, V2

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Chapter 10 – Cape Cod Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2005 Exam Questions (modified):
38. (1.5 points) You are given the following information:
Accident
Year
2001
2002
2003
2004
Total

Premium
Actual
Adjusted
6,000
7,500
9,000
10,000
32,500

Reported Claims
at 12-31-04

Reported
CDF (ultimate)

5,000
4,000
4,000
2,000
15,000

1.1429
1.3333
2.0000
3.3333

8,000
8,000
10,000
10,000
36,000

Calculate the a priori “expected claim ratio” to be used in the Cape Cod technique:

Accident
Year
2001
2002
2003
2004
Total

Adjusted
Premium
if avail.**
(1) given
8,000
8,000
10,000
10,000
36,000

Reported
CDF to
Ultimate

Percent
Reported
to date

(2) given
1.1429
1.3333
2.0000
3.3333

(3)=1/(2)
87.4967%
75.0019%
50.0000%
30.0003%

Used-Up Reported
Premium
Claims
to date as avail.**
(4)=(1)*(3)
6,999.7360
6,000.1520
5,000.0000
3,000.0300
20,999.9180

(5) given
5,000
4,000
4,000
2,000
15,000

"CC"
Estimated
Claim Ratio
(6)=(5)/(4)
see total
see total
see total
see total
71.42890%

Note: Patrik is now on the exam 7 syllabus.

Exam 5, V2

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Chapter 10 – Cape Cod Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2008 Exam Questions (modified):
Question 36
Stanard-Buhlmann (Cape Cod) is just a B-F technique with a particular Expected Claims Ratio!
Adjusted
Percent
Used-Up Reported
"SB" or "CC"
Accident
Premium
Reported
Premium
Claims
Estimated
Year
"LAG"
to date 12/31/2007
if avail.**
Claim Ratio
(1) given
(3) given *
(4)=(1)*(3)
(5) given
(6)=(5)/(4)
2003
8,000
9,000
0.95
8,550
see total
2004
0.88
8,000
9,000
7,920
see total
2005
0.75
7,000
11,000
8,250
see total
2006
0.55
6,000
13,000
7,150
see total
2007
4,000
15,000
0.30
4,500
see total
Total
33,000
57,000
36,370
90.7341%
** The Cape Cod technique allows/prefers use of adjusted data where available.
*Recall, "lag" = the percent emerged = the inverse of CDF
(4) Used-Up premium also equals Adjusted Premium divided by Ult. CDF
(6) " … method requires the use of the weighted average claim ratio from all years."

Exp. Claim
Ratio

A priori
Expected
Claims

"IBNR"
Expected
Unreport

SB%=(6)total
90.7341%
90.7341%
90.7341%
90.7341%
90.7341%

(7)=(1)*SB%
8,166.0690
8,166.0690
9,980.7510
11,795.4330
13,610.1150

(8)=(7)*[1-(3)]

"SB" or "CC"

Accident
Year
2003
2004
2005
2006
2007
Total

408.3035
979.9283
2,495.1878
5,307.9449
9,527.0805
18,718.4450

Or shortcut using
IBNR
C.C. Expected Claims
(broadly
defined)
x Percent Unreported
(8)=(Adj. Prem)*(SB %)*[1.0-"lag"]
408.3035
979.9283
2,495.1878
5,307.9449
9,527.0805
OR

18,718.4450

Solutions to questions from the 2010 Exam:
18a. (1 point) Use the Stanard-Buhlmann method to calculate the IBNR for accident year 2008 as of
December 31, 2009.
18b. (1 point) Discuss two problems that may affect the accuracy of a reinsurer's earned premium data.
Question 18 - Solution 1
a. SB ELR = (9+9+7+10+6)/(12/1.03+12/1.11+13/1.25+14/1.47+15/2) = 0.82
0.82 × 14,000 × (1-1/1.47) = 3,678
b. i) Inaccurate rate change data
ii) Imprecise by line breakdown

Exam 5, V2

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Rounding; see next sol.

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Chapter 10 – Cape Cod Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to questions from the 2010 Exam:
Question 18 - Solution 2
a.
C/AY

"Used-Up" Premium

2005
2006
2007
2008
2009
TOTAL

11,650
10,811
10,400
9,524
7,500
49,885

SB IBNR

14,000 × (.8219)(1-1/1.47)=3,679

LR = 41,000/49,885 = 82.19%
B 1. The reinsurer relies on the insurer to report premium, there can be a lag in this reporting.
2. Earned premium is often reported in aggregate to reinsurer, so the reinsurer must make
assumptions to split premium.
Solutions to questions from the 2011 Exam:
27a. (1 point) Use the Bornhuetter-Ferguson technique and an expected claims ratio of 60% to estimate the
IBNR for accident year 2010.
27b. (1.5 points) Use the Cape Cod technique to estimate the IBNR for accident year 2010.
27c. (0.5 point) Describe the primary difference between the Bornhuetter-Ferguson technique and the Cape
Cod technique.
Question 27 – Model Solution
-1

a. AY 2010 IBNR = EP * ELR *(1.0 – 1/LDF-ult) = 1,000,000 x 0.6 x (1 - 1.30 ) = 138,462 IBNR
b. Comments: Cape Cod ELR computation:
Compute Estimated claim ratios = Actual reported claims/ Used-up premium.
Compute: "Used-up premium" = EP * % reported = EP /LDF-Ult
b. Cape Cod ELR = (510 + 520 + 450) / (950/1.05 + 915/1.12 + 1000/1.3) = 0.5816
Note: The model solution has a rounding problem. Using the values above, the CC ELR = .59414
-1

Exam 5, V2

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Chapter 10 – Cape Cod Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2012 Exam Questions
18a. (0.75 point) Use the Bornhuetter-Ferguson technique and an expected claims ratio of 60.0% to
estimate the IBNR for accident year 2010.
18b. (2 points) Use the Cape Cod technique to calculate the IBNR for accident year 2010.
18c. (0.5 point) Describe the difference in the underlying assumption between the two techniques.
Question 18 – Model Solution 1 (Exam 5B Question 3)
a. IBNR 2010 = 975,000 * 60% (1.0 -1/1.12) = 62,679 as of 2010
b. Cape Cod. Compute the Estimated Claim Ratio (ECR)

=
ECR

000 )
( 510, 000 + 520, 000 + 465, =
∑ rpt
=
57.166%
∑ used-up premium ( 978, 500 *1 / 1.05 + 1, 023, 750 *1 / 1.12 + 1, 000, 000 *1 / 1.3)

Unadj ECR for AY 2010 = 57.166% * 1,023,750/975,000 = 60%
AY 2010 IBNR = 60% * 975,000 (1.0 - 1/1.12) = 62,704
sol below)

 rounding; s/b 62,678.57 (see

c. the difference is the expected claim ratio. In B-F expected claim ratio is usually from independent
analysis or judgmentally selected. In cape cod, ECR is derived from experience period.
Question 18 – Model Solution 2 (Exam 5B Question 3)
a.
BF IBNR = EP x LR x (1 -%RPT) = 975,000 x 60% x (1 – 1/1.12) = 62,678.57
b. Cape cod method

=
ECR

∑ rpt
=
∑ AdjEP×%Rpt

ECR for AY 2010=

( 510 + 520 + 465 ) × 1000

= 0.5717

( 978.5 × 1 / 1.05 + 1023.75 × 1 / 1.12 + 1, 000 × 1 / 1.3) × 1000

1, 023, 750
975, 000

× 0.5717
= 0.60

AY 2010 IBNR= 0.6 x 975,000 x (1.0 -1/1.12) = 62,678.57
c.

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Chapter11 – Frequency-Severity Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Sec
1
2
3
4
5
6
7
8
9

Description
Introduction
Common Uses of Frequency-Severity Techniques
Types of Frequency-Severity Techniques
FS Approach #1 – Development Method with Claim Counts and
Severities
FS Approach #2 — Incorporating Exposures and Inflation in the
Method
FS Approach #3 – Disposal Rate Technique
When Frequency-Severity Techniques Work and When they Do Not
Enhancements for Frequency-Severity Techniques
Frequency-Severity Projection as Input to BF Technique

Pages
194
194
194 - 195
195 - 200
201 - 205
205 - 212
205 - 212
213
214

Note: To keep the number of pages in this manual to a minimum, we have uploaded all the exhibits
associated with chapter to our website. Login to your account and click on the addendum link to
download these exhibits.

Introduction

194

Frequency-Severity (FS) techniques provide additional unpaid claim estimates and help to understand the
drivers in claims activity.
In "Evaluating Bodily Injury Liabilities Using a Claims Closure Model," Adler and Kline discuss the rhythm
in the claims settlement process:
Claims emerge at an identifiable rate, they are settled at an identifiable rate, the payments grow at an
identifiable rate and the accuracy of individual case estimates improves at an identifiable rate.
Using the FS technique:
 projected ultimate claims = estimated ultimate number of claims (i.e. frequency) x estimated
ultimate average value (i.e. severity).
By analyzing frequency and severity, trends and patterns in the rates of claims emergence (i.e. reporting)
and settlement (i.e. closure) as well as in the average values of claims can be determined.
This can be valuable when an entity is undergoing change in operations, philosophy, or management.
 can help in validating or rejecting the findings from other actuarial projection techniques.

Common Uses of Frequency-Severity Techniques

194

FS techniques:
 can be used for projecting unpaid claim estimates for both primary layers of coverage and excess
layers of insurance.
 can be used with AY, PY, RY and CY data.
 are appropriate for all lines of insurance but are more often used for long-tail lines.
 are generally not used by reinsurers, since underwriting year data does not have the detailed
statistics regarding the number of claims.

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From a technical basis:
 frequency equals the number of claims per unit of exposure, and severity equals the average cost
per claim.
 historical data for claims, number of claims, and exposures is needed.
In practice, "FS methods" refers to projections of ultimate claim counts multiplied by ultimate severities (without
using an exposure measurement).

Types of Frequency-Severity Techniques

194 - 195

Three types of FS projection methods are examined in this chapter.
Note: The consolidated industry example is not analyzed because the number of claims is not available
from the industry source data (i.e. Best's Aggregates & Averages)
FS approach 1: The development technique applied separately to claim counts and average values.
This method is presented in Exhibit I for a Canadian portfolio of private passenger automobile collision
coverage (Auto Collision Insurer) and in Exhibit II for XYZ Insurer.
FS approach 2: Projecting ultimate claims for the most recent two accident years.
The expected claims and BF techniques are often used methods to supplement claim development method.
Recall that highly leveraged CDFs for the most recent AYs (from the development method) lead to greater
uncertainty in projections of ultimate claims; which leads to greater uncertainty in the unpaid claim estimate.
FS approach 3: The Disposal Rate Technique
 It builds upon the basic development triangle used with both claims and claim counts.
 The rate of claim count closure at each maturity age and the incremental paid severity by maturity
age are examined.
See Exhibit V as an example of this approach for a portfolio of general liability insurance (GL Insurer)
and Exhibit VI for XYZ Insurer.

FS Approach #1 – Development Method with Claim Counts and
Severities

195 - 200

Two Key Assumptions
1. Individual claim counts are defined in a consistent manner over the experience period.
Example: Do not group claimant counts and occurrence counts together (i.e. recording all claimants
under an occurrence as a single claim), unless the mix of the two ways of counting a claim
is consistent.
2. Claim counts are reasonably homogenous.
Example: Do not analyze first-dollar, low-limit claims with high-layer, multi-million dollar, excess claims.
Again, FS methods rely on the development technique, which assumes that claims reported (or paid) to
date will continue to develop in a similar manner in the future.
In a FS method, it is assumed that:
 claim counts reported (or closed) to date will continue to develop in a similar manner in the future, and that
 the relative change in a given year's severities from one evaluation point to the next is similar to the
relative change in prior years' severities at similar evaluation points.

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Mechanics of the Technique
Exhibit I, Sheets 1 through 8, present the first FS example for Auto Collision Insurer.
This first example has four basic steps:
1. Project and select ultimate claim counts
2. Project ultimate severity
3. Project ultimate claims
4. Develop unpaid claim estimate
In this example, we use semi-annual accident periods and valuations in intervals of six months.
1. Project and Select Ultimate Claim Counts
Exhibit I, Sheets 1 – 3 used the development technique to project both closed and reported claim
counts to an ultimate basis. (See exhibits located on our website).
 Closed claim counts include claim counts closed with payments or claim-related expense payments
or both, but do not include claim counts closed with no payment (CNP).
 Reported claim counts include the number of closed claims in addition to the number of open
claims with a case outstanding (for claim only or claim-related expense) greater than $0.
Since the reported claim counts exclude CNP counts, we observe negative (or downward) development
(i.e. age-to-age factors less than 1.00) in Exhibit I, Sheet 2.
a. private passenger collision is a very fast reporting and settling coverage of auto insurance.
b. due to the fast-reporting nature of this coverage, there are more claim counts closed without
payment in subsequent valuations than new claim counts reported.
Thus, we see age-to-age factors of less than 1.00 for every accident half-year at 6-to-12 months.
and similar behavior through 36 months for the reported claim count triangle of age-to-age factors.
The importance of understanding the type of data provided by the insurer
 If the closed counts exclude CNP counts but reported counts include the CNP counts, both cannot be
used to produce comparable estimates of the ultimate number of claims.
 If claims include all claim adjustment expense (with or without claim payments or case outstanding)
but counts do not include claims with claim adjustment expense only, an appropriate match cannot be
made of the number of claims and the dollars that are spent on the claims.
 Claimant count versus occurrence count: Does the insurer record one count or multiple counts for
accidents involving injuries to multiple parties involved in a single occurrence?
 How are claims recorded when the payment is below the deductible?
Exhibit I, Sheets 1 and 2 shown the development triangles for closed and reported claim counts
Selected age-to-age factors are based on the simple average for the latest three half-years for both counts.
Notice the variability from accident half-year to accident half-year at 6-to-12 months for the closed claim
counts, while the averages appear relatively close to one another (this is reviewed later)
In Exhibit I, Sheet 3, we project the ultimate number of claims by accident half-year.
Note: Accident half-year 2008-1, which represents the period from 1/1/2008 – 6/30/ 2008 is six months
old as of 6/30/2008; and accident half-year 2007-2, which represents the period from 7/1/2007 –
12/31/2007 is 12 months old at 6/30/2008 (begin counting with the beginning of the accident
half-year period).

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A method to determine if changes or patterns are taking place in the triangular data is to use the
development diagnostic: ratio of closed-to-reported claim counts
Exhibit I, Sheet 4: ratio of closed-to-reported claim counts (exhibits located on our website).
Look down the column at age six months, and evidence of seasonality in the relationship between closed
and reported counts can be seen.
 For accident half-years ending with a 2 (i.e. 7/1 – 12/31), the average ratio of closed-to-reported
counts is 0.71, (with minimal variability from period to period around this average).
 For accident half-years ending with a 1 (i.e. 1/1 – 6/30), the average ratio of closed-to-reported counts
is 0.81, (with minimal variability from period to period around this average).
* Reasons for a lower proportion of claim counts closed at six months for the accident half-years ending 12/31
than for those ending 6/30:
 A higher number of claims reported in Canada in November and December may be due to more
hazardous driving conditions at the beginning of winter.
 There is less time to settle these claims with a 12/31 closing date than those claims occurring in
January and February with a half-year closing date of 6/30.
 There may less time available to process and close November and December claims due to the
shorter work period for companies that close over the Christmas holidays.
Thus, discussions with the claims department management are needed to understand the reasons for
such patterns in the data.
Note: There are no material differences or patterns evident in any maturities beyond six months.
* Discern if any patterns exist in either the closed count triangle or the reported count triangle or both since a
distinctive pattern is observed in the ratio of closed-to-reported claim counts at six months
Part 2 of Exhibit I, Sheet 1
A closer review of age-to-age factors for closed claim counts shows differences in the age-to-age factors
for accident half-years ending June versus December.
There are no patterns in the reported claim count triangle at the 6-to-12 month interval (see the table below):

Accident Half-Year
2003-2
2004-1
2004-2
2005-1
2005-2
2006-1
2006-2
2007-1
2007-2

Age-to-Age Factors at 6-12 Months
Closed Claim
Reported Claim
Counts
Counts
1.281
0.932
1.153
0.934
1.275
0.910
1.154
0.956
1.327
0.942
1.181
0.966
1.353
0.956
1.212
0.983
1.312
0.995

Accident Half Years 1
Simple Average All Years
Simple Average Latest 3 Years

1.175
1.183

0.960
0.968

Accident Half Years 2
Simple Average All Years
Simple Average Latest 3 Years

1.310
1.331

0.947
0.964

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Based on the above commentary, our selected age-to-age factor for closed counts is revised from 1.292
(the simple average of the latest 3 accident half-years) to 1.183 (the simple average latest three accident
half-years ending at 6/30).
The new projected ultimate claim counts for accident half-year 2008-1 based on closed counts are:
[(closed claim counts at 6/302008) x (development factor to ultimate)] =
[(2,533) x (1.001 x 1.009 x 1.183)] = [(2,533) x (1.195)] = 3,027
The projected number of ultimate claims based on reported claim counts for accident half-year 2008-1 is
3,061 (very close to our new projected value of 3,027, based on closed claim counts).
2. Project Ultimate Severity
Exhibit I, Sheet 5 – Reported Claims and Severities
Exhibit I, Sheet 6 - The reported severity triangle is analyzed and development factors are selected.
 There appears to be greater development for accident half-years ending December rather than June,
and further explanation from claims management is needed to fully understand the factors influencing
the claim development patterns.
 A 6-to-12 month factor of 1.039 based on the medial average (i.e. average excluding high and low
values) is selected, assuming that the experience of the most recent few years is more representative
of future experience than the earlier periods.
 We also use the medial average to select the age-to-age factors for the remaining maturities.
3. Project Ultimate Claims
Exhibit I, Sheet 7 – Projected Ultimate Claims = [projected ultimate severities] * [projected ultimate claim counts]
4. Develop Unpaid Claim Estimate
Exhibit I, Sheet 8. Total unpaid claim estimates = Case O/S +estimated IBNR
Estimated IBNR = projected ultimate claims - reported claims.
For Auto Collision Insurer, the estimated IBNR is negative for all accident half-years except the latest
period, 2008-1.
Negative IBNR is a result of:
a. salvage and subrogation recoveries (S&S), which are included with the claim development data, or
b. a conservative philosophy towards setting case outstanding.
In this example, negative IBNR is a result of the downward (i.e., favorable) development of claim counts

Analysis for XYZ Insurer
Exhibit II - the FS approach for XYZ Insurer uses the same approach used in Exhibit 1 (see Exhibit II located
on our website)
Recall that based on interviews with management of XYZ Insurer and reviews of the diagnostic development
triangles, there have been significant changes in both their internal and external environments. (It may be
valuable to review the diagnostic triangles presented in Chapter 6 for XYZ Insurer.)
As a result, we select the volume-weighted average of the age-to-age factors for the latest two years to
reflect the most recent operating environment at XYZ Insurer.
Exhibit II, Sheet 3: Projected ultimate claim counts.
 The two projections of claim counts are close for AYs 1998 - 2005,
 There are significant differences in the projected number of ultimate claims for 2006 - 2008.
 For every year starting in 2000 - 2008, ultimate count projections based on closed counts are greater
than those based on reported counts.

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Exhibit II, Sheet 5 – Reported severities triangle.
 In the age-to-age factors triangle, the latest point in each column is usually the lowest point in the
column, which is consistent with management's assertion that there has been a significant increase in
case outstanding strength (in CY 2007).
 The latest two years are used for selected development factors (to best reflect the current
environment at this insurer).
Exhibit II, Sheet 6 - Projected Ultimate Claims = [projected ultimate severities] * [projected ultimate claim counts]
Exhibit II, Sheet 7 – Development of IBNR and the Unpaid Claim Estimate
Observations:
 The estimated IBNR and total unpaid claim estimate are:
a. higher than those generated from the reported claim development technique and
b. lower than those generated from the paid claim development technique.
 Recall Exhibit II, Sheet 3 – Projected ultimate claim counts:
a. based on closed counts are significantly greater than those based on reported counts.
b. is consistent with conclusions regarding an increased rate of claims settlement.
Thus, rely on the reported count projection which is not affected by changes in claims closure patterns.

FS Approach #2 — Incorporating Exposures and Inflation in the
Method

201 - 205

Key Assumptions
This second FS approach relies on the development technique, with critical assumptions that include:
1. Claim counts and reported claims to date will continue to develop in a similar manner in the future.
2. Claim counts are defined consistently over time.
3. The mix by claim type is consistent (since potential claims can vary significantly by type of claim).
New to this approach:
Three trend rates (exposure trend, frequency trend, and severity trend) are incorporated into the analysis of
both frequency and severity parameters.
Considerations when selecting trend rates:
1. Economic inflationary factors
2. Societal factors (that tend to increase both the number and claim size over time).
3. Rates varying by line of business (and by sub-coverage within a line of business).
4. Variation in trend rates for exposures, frequency, and severity by geographic region (e.g. country,
state/province and subdivisions within a state/province).
5. Variation based on the limits (i.e. retention) carried by the insurer or self-insurer.
Note: Beyond inflationary trend factors, WC often requires adjustments for statutory benefit changes.
Sources to use when selecting trend assumptions:
1. General insurance industry data
2. Government statistical organizations
3. Economic indices
4. Insurer-specific experience.
Note: When using regression on an insurer's own claims experience, the accuracy and appropriateness of
selected trend rates is critical for many FS methods, since the longer the projection period, the
greater the uncertainty (as trend factors become large and highly leveraged).

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Two examples shown: a self-insurer of U.S. workers compensation (WC Self-Insurer) and XYZ Insurer.

Mechanics of the Approach
This approach to FS has 5 basic steps:
1. Project and select ultimate claim counts
2. Compare ultimate claim counts to exposures and select frequency (new compared to approach 1)
3. Project ultimate severity
4. Project ultimate claims
5. Develop unpaid claim estimate
1. Project and Select Ultimate Claim Counts
Exhibit III, Sheets 1 – 3 - Project closed and reported claim counts and select ultimate claim counts by AY.
Select development factors based on the volume-weighted average for the latest five years.
a. For the closed claim count triangle (84-to-96 months), select a development factor of 1.003
(resulting in a smoother pattern than the one data point of 1.008)
b. Judgmentally select a tail factor for closed claim counts of 1.007 (based on a review of closed and
reported claim counts at ages of 72, 84, and 96 months).
Exhibit III, Sheet 3, selected ultimate claim counts are based on the average of the two projections.
2. Compare Ultimate Claim Counts to Exposures and Select Frequency
Exhibit III, Sheet 4.
New to this approach: The frequency analysis compares ultimate claim counts by AY to an exposure base.
For WC, the exposure base is payroll (in hundreds of dollars).
The goal: Determine the proper frequency (i.e., number of claims per exposure unit) for the latest two AYs.
 Since payroll is inflation-sensitive, adjust payroll for each AY to a common time period.
 To simplify, assume a 2.5% annual inflation rate for payroll for all years in the experience period and
trend all historical payroll to the cost level of AY 2008 (see Columns (5) - (7) )
Similarly, trend factors should be used to reflect changes in counts.
 It is ideal to analyze the self-insurer's own historical experience to determine the frequency trend rate.
 In this example, there insufficient historical data, and the actuary relies on knowledge of U.S. WC in
general and the specific industry of this self-insured organization; we assume a -1.0% annual trend in
the number of claims.(See Columns (2) - (4)).
Trended ultimate frequency equals ultimate trended claim counts divided by trended payroll in Column (7).
After a review of these rates by AY in Column (8), we see a change in frequency between 2001 - 2004 and
2006 - 2008.
Thus, the actuary should speak to management to understand what caused the change in frequency.
 Has there been a new cost containment program introduced?
 Has there been a change in the definition of a claim?
 Has there been a change in third-party administrators?
 Was there a change in the type of work performed by employees?
 There was a large increase in both claims and payroll between 2005 and 2006. Was this the result of
a corporate acquisition?

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In this example, the change in frequency is due to a major acquisition, resulting in the hiring of a new risk
manager and the introduction of new safety and risk control procedures.
 A 2008 frequency rate of 0.36% is selected (reflective of the new and improved environment with
respect to claims at this organization).
 The 2007 frequency rate of 0.37% equals 0.36% * 1.025 (the adjustment for payroll inflation) / .99
(the adjustment for claims trend).
3. Project Ultimate Severity
Exhibit III, Sheets 5 – 8: Projected paid severities and reported severities to an ultimate value
Exhibit III, Sheet 6 – 7: The analysis for paid and reported severities.
Development factors were selected based on the medial average (i.e. average excluding high and low
values) for the latest five years.
Tail factors at 96 months of 1.025 for reported severities and 1.15 for the paid severities were selected
(based on analysis of insurance industry benchmark development patterns for U.S. WC).
Exhibit III, Sheet 8: Comparison of the two projections and selection of ultimate severities for AYs 2001 – 2006.
Exhibit III, Sheet 9: Selection of 2008 and 2007 Severities
Adjust the severities for each historical AY year to the cost level of AY 2008.
 A 7.5% annual severity trend rate is selected.
 The authors chose to simplify the model by not incorporating an adjustment of claims by year to the
2008 statutory benefits level.
 A 2008 severity value of $7,100 was selected. The 2007 severity value $6,605 = $7,100/1.075.
4. Project Ultimate Claims
Exhibit III, Sheet 10 - Projection of Ultimate Claims and Development of Unpaid Claim Estimate for AY 2007
and 2008.
Payroll for both AYs was given
1. Multiply payroll by the selected frequency rates to compute the projected ultimate number of claims (Line (3)).
2. Multiply ultimate number of claims by the selected severities to derive the projected ultimate claims (Line (5)).
5. Develop Unpaid Claim Estimate
Total unpaid claim estimate = Case outstanding + estimated IBNR
Estimated IBNR = Projected ultimate claims - Reported claims.
Analysis for XYZ Insurer
Exhibit IV, Sheets 1 - 3 use FS#2 approach to review the experience of older, more mature accident years for
the purpose of determining estimates of both frequency and severity for 2007 and 2008..
In this approach, adjustments for rate level changes, inflation, and tort reform are incorporated.
Exhibit IV, Sheet 1 - Projection of Ultimate Frequency
 Selected ultimate claim counts for AYs 2002 – 2006 are obtained from the reported claim count
projection in Exhibit II, Sheet 3.
 An annual -1.5% claims frequency trend is used based on analysis of insurance industry trends.

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Exposure base:
Vehicle or policy count are ideal exposure bases when conducting an analysis of unpaid claims for auto
liability insurance, however this information is not available and so XYZ uses EP as an exposure base.
Columns (5) - (7) of Exhibit IV, Sheet 1: Adjust historical earned premiums to the 2008 rate level.
Column (8) trended ultimate frequency equals column (4) trended claim counts divided by OLEP.
 The 2008 selected frequency rate is 2.36%.
 The 2007 selected frequency rate is 2.36% divided by the annual claim count trend (-1.5%) and
multiplied by the rate level change that took place in 2008. 1.92% = 2.36% * 0.8/0.985
Exhibit IV, Sheet 2 - Selection of 2008 and 2007 Severities
Column (5) Trended ultimate severities equal Column (2) projected ultimate severities from Exhibit II, Sheet 6
multiplied by a 5% annual severity trend and by tort reform factors from Chapter 8, Exhibit III Sheet 2.
 The 2008 selected ultimate severity is 26,720 (after review column (5) averages).
 The 2007 selected ultimate severity is $25,448 (the 2008 value adjusted for one less year of trend).
Exhibit IV, Sheet 3 - Projection of Ultimate Claims and Development of Unpaid Claim Estimate
Projected ultimate claims for 2007 and 2008 are based on the multiplication of:
 Projected ultimate counts (EP * selected frequency %) and
 Projected ultimate severity values (the latter two from Exhibit IV, Sheets 1 and 2)
It’s advisable to compare the projection of ultimate claim counts, severities, and claims using FS #1
approach and FS #2 approach. The following table summarizes these values.

2007 Ultimate Claim Counts
Closed Counts Projection
Reported Counts Projection
Selected Value
2007 Severity
2008 Ultimate Claim Counts
Closed Counts Projection
Reported Counts Projection
Selected Value
2008 Severity
Projected Ultimate Claims ($000)
Accident Year 2007
Accident Year 2008

Approach # 1

Approach # 2

1,804
1,308
1,556
37,606

1,199
25,448

1,679
1,172
1,426
41,544

1,128
26,720

58,516
59,242

30,512
30,140

Notice that ultimate claims from the second approach are roughly half of the projections from the first
approach due to lower projections of both ultimate claim counts and average values per claim.
In Chapter 15, we compare and contrast the various projection methods for this example.

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6

FS Approach #3 – Disposal Rate Technique

205 - 212

Key Assumptions
It is assumed that historical patterns of claims emergence and settlement are predictive of future patterns of
reported and closed claim counts.
It is implicitly assumed that that there are no significant partial (i.e. interim) payments.
The assumed severity trend rate (to adjust for inflation) must be selected carefully.
A slight change in trend can result in a material change in the estimated of unpaid claims.
Mechanics of the Approach
The 7 steps in the FS method #3 are:
1. Project ultimate claim counts and select ultimate claim counts by accident year
2. Develop disposal rate triangle and select disposal rate by maturity age
3. Project claim counts by accident year and maturity (complete the square)
4. Analyze severities and select severities by maturity
5. Calculate severities by maturity age and accident year (complete the square)
6. Multiply claim counts by severities to determine projected claims
7. Determine unpaid claim estimate
1. Project Ultimate Claim Counts and Select Ultimate Claim Counts by Accident Year
For this example, a portfolio of occurrence basis, general liability insurance data (GL Insurer) is reviewed.
Exhibit V, Sheets 1 – 3: Development of closed, reported and projected ultimate claim counts
 Exhibit V, Sheet 2: Downward (i.e. negative) development in the age-to-age factors for reported
claim counts are shown, is most likely caused by the data excluding CNP counts.
 Selected development factors based on the volume-weighted averages for the latest 3 years.
 Select tail factors are based on experience for the oldest maturities, including the ratio of closed-toreported claim counts, and benchmark patterns for a similar portfolio of coverage.
Exhibit V, Sheet 3: Selected ultimate counts are based on the average of the paid and reported projections.
2. Develop Disposal Rate Triangle and Select Disposal Rate by Maturity Age
Exhibit V, Sheet 4: - Development of Disposal Rate
 Disposal rates are cumulative closed claim counts (for each AY-maturity age cell) / selected ultimate
claim count for a particular AY.
 Each ratio represents the % of ultimate claim counts that are closed at a given stage of maturity for a
given AY.
 The medial five-year average is used to select a disposal rate at each maturity age.
 There is considerable stability in the disposal rates at each maturity.
 Expect disposal rates to monotonically increase over time (see table below)
Maturity Age (Months)
12
24
36
48
60
72
84
96

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0.2
0.433
0.585
0.71
0.791
0.862
0.882
0.912

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3. Project Claim Counts by Accident Year and Maturity (Complete the Square)
Exhibit V, Sheet 5: Development of Closed Claim Counts
Top section: Closed Claim Counts
Bottom section: Incremental and Projected Incremental Closed Claim Counts
 Selected disposal rates by maturity and the selected ultimate claim counts by AY are used to
complete the square of the incremental closed claim count triangle.
 Incremental claim counts in the column labeled 12 represent counts that are closed in the first 12
months from the start of the AY. Those in the column labeled 24 represent the counts that are closed
in the 12-24 month period.
 Incremental Count Triangle:
The top left part of the "completed square" is computed based on the differences between successive
columns of the cumulative closed claim count triangle.
The bottom-right, highlighted (projected) part of the incremental closed claim count square, is
computed by first adjusting the cumulative closed claim counts at the latest valuation to an ultimate
basis and then applying the selected disposal rates for each age interval.
For example, for AY x at Age y, projected incremental closed claim counts are computed as follows:
[(ultimate claim counts for AY x – cumulative closed claim counts for AY x along latest diagonal) / (1.00 –
selected disposal rate at maturity of latest diagonal)] x [disposal rate at y – disposal rate at y-1]
Examples:
The estimated incremental closed claim counts for AY 2008 at 24 months are equal to:
[(609 – 127) / (1.000 - 0.200)] x [0.433 – 0.200] = 140
The estimated incremental closed claim counts for AY 2005 at 84 months are:
[(588 – 403) / (1.000 – 0.710)] x [0.882 – 0.862] = 13
Projected ultimate claims equal incremental closed claim counts * average incremental paid claims.
The use of incremental claim counts and incremental severities is unique to this FS method
4. Analyze Severities and Select Severities by Maturity
Exhibit V, Sheet 6 - Calculation of Severities
1. Compute the incremental paid claim triangle from the cumulative paid claim triangle.
2. Compute incremental paid severities: [incremental paid claims/ incremental closed claim counts]
Note: There are patterns in this incremental triangle of paid severities.
 In general, the paid severities increase as the claims mature, which is:
a. consistent with the belief that smaller claims settle quicker than more complicated/costly claims.
b. common for long-tail lines of insurance (e.g. U.S. general liability)
3. Adjust the severities to a common time period (i.e. cost level) before severities selections are made.
Exponential regression is often used to determine annual trend rates. Reasons include:
i. Its use implies a constant % increase in inflation.
ii. It is believed to be most indicative of the normal inflation process.
iii. A weighted exponential least squares fit gives greater weight to more recent experience.
iv. Linear projections are rarely used (due to the implied decreasing % trend).

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Exhibit V, Sheet 7: A summary of regression analyses for incremental paid severities.
i. To determine a severity trend, fit exponential curves to incremental paid severities at each maturity age.
ii. Run a variety of combinations of years and test for the goodness-of-fit of the regression.
Estimated annual rates of change (i.e. trend rate) and goodness-of-fit tests (i.e. R-squared) are shown.
Observations:
i. A good fit to the data is not found based on GL Insurer's experience alone
ii. Using industry-wide experience, supplemented with the insurer's limited data, a 5% annual severity
trend is selected.
iii. Notice that there is some evidence that trend rates differ and may be greater for the older maturities.
However, to simplify, a single trend rate for all maturities is used.
Exhibit V, Sheet 8, middle section: Restatement of all incremental paid severities at the 2008 cost level.
Examples:
i. The incremental paid severity for AY 2007 at 12 months is $10,086; after adjustment for trend to the
1
2008 cost level, the severity is $10,590 ($10,086 x 1.05 ).
ii. The incremental paid severity for AY 2003 at 72 months is $46,648; after adjustment for trend to the
5
AY 2008 cost level, the severity is $59,536 ($46,648 x 1.05 ).
Exhibit V, Sheet 8, bottom section - averages of trended severities
Observations:
i. An increasing pattern in paid severities exists by age from 12 months - 96 months.
ii. Selected incremental paid severities at the 2008 cost level for maturity ages 12- 60 months are made
but beyond this point, the data becomes sparse and a simple average of the latest 3 years is used.
Given variability in trended severities, consider combining the experience of several maturity ages.
Variability may be due to:
a. the result of 1 or more large claims closed at older ages.
b. a smaller number of claims in the data set at the oldest maturity ages.
By combining multiple years of experience, the influence of random large claims or other factors is reduced.
Exhibit V, Sheet 9 - Development of Trended Severity (at ages 60 and older and 72 and older)
1. A triangle of incremental closed claim counts for maturities 60 through 96 months is given (from E5S6).
2. A triangle of incremental paid claims for these same maturities is given (from E5S6).
3. Adjusted paid claims using the 5% annual severity trend to bring all payments to the 2008 cost level.
4. Estimated trended tail severity equals [sum trended claim payments]/[sum incremental closed claim counts].
Tail severities selection requires substantial judgment. Considerations as to the maturity age at which to
combine data for tail factor analysis depends on:
i. The age(s) at which the results become erratic
ii. The influence on the total projections of selecting a particular age
iii. The % of claims expected to be closed beyond the selected maturity age

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Tail Severity Selection Thought Process:
i. Greater variability in trended severities begins at age 60.
ii. The selected disposal rate at 60 months is 0.791 (i.e. we expect more than 20% of the claim counts to
remain open at this age). .791 is from E5S4.
iii. Given the large change in increment closed counts (227 at 60 months compared with 124 at age 72
months), it is clear that for ages 72 months and older, the experience should be combined for selecting
an incremental tail severity.
iv. What should be done at 60 months?
An incremental trended severity of $140,802 at 60 months based on the experience of 60-month data
only is selected, but this is not very different from the estimated severity of $144,160 for ages 60 and
older developed in Sheet 9.
The importance of selecting the appropriate point at which data should be combined for determining a
tail severity:
* A trended tail severity of $175,816 is selected based on the experience of ages 72 and greater.
* The affect of selecting a tail severity based on the experience of 60 months and greater would be
a reduction of the unpaid claim estimate of more than 10%.
The following table is a summary of the selected severities, at the 2008 cost level, by maturity.

Maturity Age (Months)
12
24
36
48
60
72 and older

Selected Severity at 2008 Cost Level
11,259
32,980
65,523
80,544
140,802
175,816

Final Notes:
 While the selected severities for GL Insurer are increasing for all maturities through 72 months, at
some point in time, the average value will likely not continue to increase.
 Consider the influence of large claims on the incremental average paid values.
Consider capping claims to a predetermined value or excluding large claims in their entirety.
In either case, a provision for large claims to the estimate of unpaid claims will need to be added.
5. Calculate Severities by Maturity Age and Accident Year (Complete the Square)
Exhibit V, Sheet 10 – Development of Severities
Given: 1. The top part of the square is the incremental paid severity triangle.
2. The bottom part is computed using selected severities at each age at the 2008 cost level and the
selected trend rate.
To complete the square for incremental paid severities:
Adjust the selected severities at the 2008 cost level to the cost level expected for each AY.
Examples:
For AY 2006 at age 48 months, $73,056 = $80,544 (selected 2008 cost level severity at 48
2
months)/1.05 .
6
For AY 2002 at 96 months, $131,197 = selected 2008 cost level severity of $175,816/1.05 .

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6. Multiply Claim Counts by Severities to Determine Projected Claims
Exhibit V, Sheet 11: Projection of Ultimate Claims
1. Projected incremental paid claims equals multiplication of the two completed squares
= [the incremental closed claim counts] * [incremental paid severities]
2. Cumulate the projected incremental paid claims to derive projected cumulative paid claims (i.e. projected
ultimate claims).
7. Determine Unpaid Claim Estimate
Exhibit V, Sheet 12: Estimated IBNR and the Total Unpaid Claim Estimates
Observations (reviewing the results for this technique):
An unusually low IBNR for AY 2004 (-$1,950) exists compared with the AY2003 value ($3,611) and the
AY 2005 value ($9,340).
Return to the data to see if there is anything unusual in either the claims or the severity for this year.
i. Closed claim counts in Exhibit V, Sheet 1 seem reasonable when compared with other years.
ii. Paid severity for AY 2004 at 60 months is low compared to prior AY at 60 months and compared to AY
2005 at 48 months.
iii. There is an unusually high case outstanding for AY 2004 in comparison with other years.
iv. However, the estimate of total unpaid claims for 2004 is reasonable when compared to other years.
v. AY 2003 seems to have similar issues. The incremental paid severity is unusually low when compared
to other AYs, and the IBNR is lower than usual when compared to AY 2002 and 2005.
Turn to claims department management to:
 understand the reasons for the high value of case outstanding and the low values for average
payments, and to
 determine if there are any factors that might preclude using this type of projection method.
Analysis for XYZ Insurer
Initial commentary:
Recall from Chapter 6 that:
 closed claim counts for XYZ Insurer exclude claims closed with no payment (CNP) and
 paid claims include partial payments as well as payments on closed claims.
Thus, the average paid claim triangle is a combination of payments on settled and on claims still open.
Due to the mismatch of collars and claim counts, management is contacted and since there is not a large
volume of partial payments, we proceed with the analysis.
Exhibit VI, Sheets 1 – 8: The disposal rate method for XYZ Insurer.
As in Approach #2, projected ultimate claim counts are derived from the reported claim count experience.
Exhibit VI, Sheet 1:
 Selected disposal rates are based on the simple average of the latest two years.
 Evidence of a change in disposal rates for the latest valuations, at 12, 24 and 36 months exists.
Exhibit VI, Sheet 2: Complete the square of projected incremental claim counts.
Exhibit VI, Sheet 3: Incremental paid severities are determined.

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Exhibit VI, Sheet 4: Select severity values at the 2008 cost level (after adjustment for trend and tort reform)
by maturity age.
 Assume a 5% severity trend
 Increasing severity values for each successive maturity age are observed.
 A look at the triangle of incremental paid severities shows that the severities along the latest diagonal
are the highest value in each column for 6 of the 8 AYs in the experience period.
 Has the speed-up in settlement resulted in a shift in the type of claim now being closed at each
maturity age? Consider the affect of this phenomenon on the projection method and the true unpaid
claims requirement for XYZ Insurer.
Exhibit VI, Sheet 5: Development of Trended Tail Severity for ages 84 and 96 months.
A tail severity of $70,432 for ages 84 and 96 is selected.
Exhibit VI, Sheets 6 and 7: The development of projected ultimate claims by AY-maturity age cell.
Exhibit VI, Sheet 8: The calculation of estimated IBNR and the total unpaid claim estimate.
Exhibit VI, Sheet 9 (projected ultimate claims) and Exhibit VI, Sheet 10 (estimated IBNR):
A comparison of the results of the 3 FS projections for XYZ Insurer with the results of the Cape Cod
method, the BF method, the expected claims method, and the development method are shown.

When Frequency-Severity Techniques Work and When they Do Not

205 - 212

Advantages to using a FS approach:
1. Its use in developing estimated unpaid claim estimates for the most recent AYs.
a. Both paid and reported claim development methods can prove unstable and inaccurate for the more
recent AYs.
b. This weaknesses can be addressed by separating estimates of ultimate claims into frequency and severity.
The number of reported claims reported is usually stable, and thus the projection of ultimate claim
counts produces reliable estimates.
Since severity estimates for the more mature AYs can be obtained with greater certainty, adjusting these
severities using tend factors can help in developing estimates of severities for the most recent AYs.
2. Its used to gain greater insight into the claims process (e.g. the rate of claims reporting and settlement and
the average dollar value of claims)
3. It can be used with paid claims data only. Thus, changes in case outstanding philosophy or procedures will
not affect the results.
4. Its ability to explicitly reflect inflation in the projection methodology instead of assuming that past
development patterns will properly account for inflationary forces.
A potential disadvantage in doing so is its highly sensitive to the inflation assumption.
Disadvantages to using a FS approach:
1. The unavailability of data.
2. Changes in the definition of claim counts, claims processing, or both may invalidate the assumption that
future claim count development will be similar to historical claim count development.
Thorne in his discussion of the Berquist and Sherman paper "Loss Reserve Adequacy Testing: A
Comprehensive, Systematic Approach" states: "A change in the meaning of a 'claim' can cause substantial
errors in the resulting reserve estimates when relying on the projection of ultimate severity for recent
accident years. These changes need not even be internal to the company. For example, changes in the
waiting periods, statutes of limitation, and no-fault coverage can have a significant effect on the meaning
of a 'claim' and thus on ultimate severity."
3. If the mix of claims is inconsistent, this will distort a FS analysis unless an adjustment is made for
the change in the mix of claim types or claim causes.

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8

Enhancements for Frequency-Severity Techniques

213

Considerations when using FS Techniques include:
1. The influence of seasonality on both the frequency and the severity of claims.
2. The influence of inflation on both the number of claims and the average value of claims.
3. Using more sophisticated trending analyses into the FS techniques.
4. Understanding the data (paid claims and claim counts) underlying the analysis of unpaid claims.
Questions include:
* Do paid claims data include significant partial payments?
* Are claim count statistics available for the number of paid claims or only closed claim counts?
* If only closed counts are available, is it reasonable to calculate an average paid value using paid
claims that contain substantial partial payments?
* How are reopened claims treated in the claims database? They may appear as a negative reported
claim count or as a new claim.
Reopened claims were ignored in the examples in this chapter.
i. Depending on how reopened claims are handled (e.g. is the claim assigned the original claim
identification number or a new claim identification number?) there could be distortions in the
claim count statistics due to reopened claims.
ii. This could affect both frequency and severity indications.
iii. Reopened claims are more prevalent in U.S. WC and in Canadian auto accident benefits, than
in other lines.
Thus, it may be wise to segregate reopened claims from other claims and analyze reopened
claims separately.

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214

Projected ultimate claims from a FS technique are often valuable as an alternative expected claims estimate
for the BF technique.
Further, actuary may feel more comfortable selecting frequency and severity values than an expected claim
ratio (or pure premium) value.
Thus, the unpaid claim estimate can be computed using one of the FS projections as used as the expected
claims with the BF technique.

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Sample Questions:
1. The methods discussed in Friedland’s chapters 7 through 10 could all be applied to data compiled on
an aggregate basis. How does the technique discussed in chapter 11 differ?
2. Describe the three types of Frequency-Severity methods that Friedland demonstrates.
3. Define the “Disposal Rate” that is used in the Frequency Severity Disposal Rate technique.
4. What does Friedland mean by “completing the square”?
5. Summarize Friedland’s key points re: “When the Frequency-Severity Techniques Work and When they Do
Not.” Include 4 advantages and 4 disadvantages/limitations.
6. Use the Frequency-Severity Development technique, along with select data from Friedland’s Chapter 11
Exhibits for the Auto Collision Insurer (as given below), to answer the following questions.
Note: No adjustments for exposures or severity trend are made.
6a. Given the following data, project ultimate claim counts for accident periods in 1/1/2006 - 6/30/2008.
Use a 3-period simple average to select age-to-age factors. Assume no development after 30 months.

Accident
Half Year

Period
Ending

2006
2006
2007
2007
2008

30-Jun
31-Dec
30-Jun
31-Dec
30-Jun

Reported Claim Counts: Data Triangle
1st report 2nd report 3rd report 4th report 5th report
6 mo.
12 mo.
18 mo.
24 mo.
30 mo.
2,808
2,799
2,578
2,791
3,139

2,712
2,675
2,533
2,778

2,704
2,670
2,529

2,701
2,668

2,700

6b. Given this additional data, project ultimate claim severities for accident periods in 1/1/2006 6/30/2008.

Accident
Half Year

Period
Ending

2006
2006
2007
2007
2008

30-Jun
31-Dec
30-Jun
31-Dec
30-Jun

Reported Claims ($1000): Data Triangle
1st report 2nd report 3rd report 4th report 5th report
6 mo.
12 mo.
18 mo.
24 mo.
30 mo.
11,947
12,503
11,662
12,647
14,071

11,856
12,762
11,523
12,854

11,820
12,706
11,492

11,772
12,697

11,760

Use a “5 period x 1 medial average” to select age-to-age factors (see additional data in solution).
Assume no development after 30 months.
6c. Project ultimate claim severities for accident periods in 1/1/2006 - 6/30/2008.
6d. Calculate the IBNR estimates for accident periods in 1/1/2006 - 6/30/2008.

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2001 Exam Questions (modified):
33. (2 points) You are given the following information:

Accident
Year
1998
1999
2000

Estimated
Ultimate
Claim Count
1,000
1,200
1,300

Cumulative Loss Payments
Age of Development (Months)
12
24
36
300,000
930,000
1,490,000
396,000
1,189,800
471,900

Age of Accident Year
12 Months
24 Months
36 Months

Disposal Rates
30%
72%
100%

Using a Frequency-Severity Disposal Rate technique, calculate the cumulative claim payments for
accident year 2000 at 24 and 36 months of development. Assume 5% future annual inflation and no
partial payments. Show all work and state any additional assumptions.
2004 Exam Questions (modified):
20. (3 points) You are given the following information:
Annual Average Severity Trend:

Accident
Year
2000
2001
2002
2003

Incremental Closed Claim Counts
(months of development)
0 to 12
12 to 24
24 to 36
36 to 48
10
17
15
13

15
12
12

10
17

Ultimate
Claim
Counts
50
60
70
70

Incremental Payments on Closed Claims
(months of development)
0 to 12
12 to 24
24 to 36
36 to 48
10,000
20,000
18,000
16,000

12,500
26,000
25,000

15,000
30,000

25,000

Using a Frequency-Severity Disposal Rate method, determine the projected ultimate payments for
accident year 2003.
Show all work and state any additional assumptions.

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2005 Exam Questions (modified):
14. (2 points) You are given the following information:

Accident
Year
2001
2002
2003
2004

Incremental Number of Closed Claims
(Age of Development in Months)
12
24
36
48
260
770
430
170
310
710
520
250
680
340

Projected
Ultimate
Claims
1,630
1,640
1,760
1,690

Average Paid Severity by Age at Closure
0-12 months
560
12-24
830
24-36
1,530
36-48
2,000

•

Assume no inflation.

•

Select the most recent diagonal of disposal rates for projections (see note in solution regarding
Friedland’s selections in chapter 11 exhibits).

Using a Frequency-Severity Disposal Rate approach, what is the estimate of Unpaid Claims as of
December 31, 2004 for accident year 2003? Show all work and state any additional assumptions.
2007 Exam Questions (modified):
35. (3 points) Given the following information:
Accident
Year
2003
2004
2005
2006

Accident
Year
2003
2004
2005
2006

Incremental Closed Claim Counts
(months of development)
0 to 12
12 to 24
24 to 36
36 to 48
40
48
36
44

80
96
72

60
72

Ultimate
Claim
Counts
200
240
180
220

Incremental Payments on Closed Claims
(months of development)
0 to 12
12 to 24
24 to 36
36 to 48
40,000
50,000
38,000
45,000

100,000
118,000
90,000

90,000
108,000

40,000

• The average annual severity trend is 10%.
Using a Frequency-Severity Disposal Rate approach, determine the projected ultimate payments for
accident year 2006. Show all work and state any additional assumptions.

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2008 Exam Questions (modified):
Question 3. Given the following information

Accident
Year
2004
2005
2006
2007

Incremental Closed Claim Counts
(months of development)
0 to 12
12 to 24
24 to 36
36 to 48
500
300
150
50
600
360
180
750
450
900

$000's
Accident
Year
2004
2005
2006
2007

Incremental Payments on Closed Claims
(months of development)
0 to 12
12 to 24
24 to 36
36 to 48
400.0
300.0
180.0
75.0
504.0
378.0
226.8
662.0
496.0
833.0

Ultimate
Claim
Counts
1000
1200
1500
1800

- The annual severity trend is 5%.
a. (2.5 points) Using a Frequency-Severity "Disposal Rate" method:
Estimate unpaid claims as of 12/31/07. Show all work.

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2009 Exam Questions
4. (4 points) Given the following information for an insurance carrier:
Cumulative Closed Claim Counts
Accident
Year

As of 12
Months

As of 24
Months

As of 36
Months

As of 48
Months

As of 60
Months

As of 72
Months

2003
2004
2005
2006
2007
2008

98
110
93
83
87
95

255
275
246
269
292

302
348
284
328

351
363
348

395
375

410

12
Months
0.250

Accident
Year
2003
2004
2005
2006
2007
2008

As of 12
Months
402
495
446
423
487
532

Selected Cumulative Disposal Rates
24
36
48
60
Months
Months
Months
Months
0.650
0.800
0.900
0.950

Cumulative Loss Paid ($000)
As of 24
As of 36
As of 48
Months
Months
Months
2,050
3,080
4,882
2,475
5,278
5,800
2,191
3,904
6,567
3,399
5,264
3,562

Selected
Ultimate Claim
Counts
418
400
395
417
447
413

72
Months
0.980

As of 60
Months
5,675
6,250

As of 72
Months
6,200

Use the disposal rate frequency-severity technique to answer the following:
a. (1 point) Calculate the expected incremental closed claim counts for periods 60-72 and 72-ultimate
for accident year 2004.
b. (1.5 points) Using a 6% annual trend factor, estimate the 60-ultimate tail severity at 2008 levels.
c. (1.5 points) Estimate the ultimate losses for accident year 2004.

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2010 Exam Questions
16. (4 points) Given the following information:
Reported Claim Counts Excluding Claims Closed With No Payment
Accident Year
12 Months
24 Months
36 Months
48 Months
2006
200
250
350
375
2007
250
350
370
2008
300
310
Reported Claims ($000)
Accident Year
12 Months
24 Months
36 Months
48 Months
2006
1,000
1,500
2,200
2,600
2007
1,100
1,900
2,300
2008
1,250
1,725
•
•
•
•
•
•

The 48-to-ultimate development factor for claim counts is 1.010.
The 48-to-ultimate development factor for reported severity is 1.025.
The selected annual frequency trend is +2.0% for 2006 to 2009.
The selected annual severity trend is -1.5% for 2006 to 2009.
Volume-weighted averages are used to calculate development factors.
Exposures have been constant and there is no exposure trend.

a. (3.25 points) Use the frequency-severity technique to calculate the expected ultimate claim cost
estimate for accident year 2009.
b. (0.75 point) State the three key assumptions underlying the frequency-severity technique.
2011 Exam Questions
26. (2.5 points) Given the following information as of December 31, 2010:
Accident
Year
2003
2004
2005

Incremental Closed Claim Counts
72 Months
84 Months
96 Months
2,000
2,000
1.000
3,000
2,000
3,000

Accident
Year
2003
2004
2005

Incremental Paid Claims (000s)
72 Months
84 Months
96 Months
$20,000
$28,000
$25,000
$33,000
$36,000
$36,000

• Selected annual severity trend = 10%
a. (2 points) Use the volume-weighted average to estimate the trended tail severity for maturity ages
of 72 months and older.
b. (0.5 point) Briefly describe two considerations in selecting the maturity age at which to combine
data for estimating a tail factor.

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2011 Exam Questions - continued:
28. (1.75 points) Given the following:

Accident
Year
2007
2008
2009
2010

Cumulative Total Closed Claim Counts
24
12 Months
36 Months
48 Months
Months
840
550
700
800
700
855
925
625
800
675

Cumulative Claim Counts Closed with No Payment
Accident
Year
2007
2008
2009
2010

Accident
Year
2007
2008
2009
2010

12 Months
30
55
35
40

24 Months
80
105
60

36 Months
105
130

48 Months
120

Projected Ultimate
Severity per Claim
Closed with Payment
$3350
$3400
$3275
$3450

• Assume no further closed claim development after 48 months.
• Use an all-year volume-weighted average for all factor selections.
Use the frequency-severity technique to estimate the ultimate claim amount for accident year 2010.

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2012 Exam Questions :
20. (3 points) Given the following:
Cumulative Reported Claim Counts
Accident
Year
2009
2010
2011

12 Months
210
221
212

24 Months
312
340

36 Months
320

Cumulative Reported Claims ($000s)
Accident
Year
2009
2010
2011

12 Months
$1,175
$1,210
$1,215

24 Months
$2,100
$2,305

36 Months
$2,375

• Assume no reported claim count development or claim development after 36 months.
Use a frequency-severity technique to estimate the ultimate claim amount for accident year 2011.

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Solutions to Sample Questions:
1. The methods discussed in Friedland’s chapters 7 through 10 could all be applied to data compiled on an
aggregate basis. How does the technique discussed in chapter 11 differ?
Chapter 11 discusses methods that examine claim frequency and severity components separately,
instead of looking only at total aggregate claims (as in the Development, Expected Claims, B-F and CC
methods). In addition, some f-s methods using incremental development techniques.
2. Friedland shows 3 Frequency-Severity approaches:
#1) Development Technique with Claim Counts and Severities,
Same procedure as applied to aggregate claims in Chapter 7, but applied separately to the frequency
and severity components. The ultimate counts and ultimate severities are multiplied to find the
estimated Ultimate Claims.
#2) Incorporation of Exposures and Inflation into Methodology #1:
For example, may involve trending both frequency and severity components.
#3) Disposal Rate Technique
The mechanics of this method are quite different from #1.
-In addition to calculating the ultimate claim count frequency, we also need “disposal rates.”
-Incremental paid severities are restated / incorporating trend.
See older (2007 and prior) exam questions for Adler-Kline and Fisher-Lange, as examples.
3. Define the “Disposal Rate” that is used in projecting the frequency of claims.
Friedland defines “the cumulative closed claim count for each accident year-maturity … divided by the
selected ultimate claim count for the particular accident year.” See past exam questions, but be careful
since a previous syllabus reading (Fisher-Lange) defined “disposal rates” as incremental ratios …
4. What does Friedland mean by “completing the square” ?
If the actual data (either frequencies or severities) through the most recent valuation is arranged as a
triangle: When we make estimates for the future values, we can use those projections to extend the
original triangle to form a square.
5. Summarize Friedland’s key points re: “When the Frequency-Severity Techniques Work and When they Do
Friedland discusses 4 advantages:
(1) Where development methods can be unstable, inaccurate, or unreliable for less mature years,
Frequency-Severity methods can provide an alternative.
(2) Freq-Sev methods offer insight into the claims process (claims reporting and settling)
(3) Since the ultimate claims are calculated without depending on case-outstanding reserves, any
changes in the reserving strategy or philosophy surrounding case reserves will not distort Freq-Sev
methods.
(4) Freq-Sev methods allow for inflation to be considered explicitly (which also leads to a disadvantage).
Friedland discusses 4 disadvantages:
(1) Freq-Sev methods can be highly sensitive to the inflation assumption.
(2) Freq-Sev methods require more data than aggregate methods, may be unavailable
(3) Also, the data available may not be relevant due to changes in the ways claims are defined or processed
(4) Freq-Sev methods can be distorted by a mix of claims (types/causes) that is not relatively consistent

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Chapter11 – Frequency-Severity Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
6. Use the Frequency-Severity Development technique, along with select data from Friedland’s Chapter 11
Exhibits for the Auto Collision Insurer (as given below), to answer the following questions.
Note: No adjustments for exposures or severity trend are made.
(A) Given
Frequency

(B) ATA
and CDF
Frequency

Accident
Half Year

Period
Ending

2006
2006
2007
2007
2008

30-Jun
31-Dec
30-Jun
31-Dec
30-Jun

Reported Claim Counts: Data Triangle
1st report
2nd report 3rd report
4th report
6 mo.
12 mo.
18 mo.
24 mo.
2,808
2,799
2,578
2,791
3,139

2,712
2,675
2,533
2,778

2,704
2,670
2,529

2,701
2,668

Reported Claim Counts: Age-to-Age Factors
1st to 2nd
2nd to 3rd 3rd to 4th
4th to 5th
6:12 mo
12:18 mo 18:24 mo
24:30 mo

5th report
30 mo.
2,700

Accident
Half Year

Period
Ending

2006
2006
2007
2007

30-Jun
31-Dec
30-Jun
31-Dec

0.9658
0.9557
0.9825
0.9953

0.9971
0.9981
0.9984

3-period simple avg ATA
Development Age

0.9778
6 mo.

0.998
12 mo.

0.999
18 mo.

0.9996
24 mo.

1.000
30 mo.

CDF to Ultimate

0.9745

0.9966

0.9987

0.9996

1.0000

Accident
Half Year

6 mo.
Period
Ending

Age of
Data at
6/30/08

Reported
Counts at
6/30/08

(1)
2006
2006
2007
2007
2008

30-Jun
31-Dec
30-Jun
31-Dec
30-Jun

30 months
24 months
28 months
12 months
6 months

(2) from (A)
2,700
2,668
2,529
2,778
3,139

0.9989
0.9993

See Tail
Below

0.9996

Given

CDF to
Ultimate
(3) from
(B)
1.000
0.9996
0.9987
0.9966
0.9745

Estimated
Ultimate
Counts
(4)=(2)*(3)
2,700.0000
2,666.9328
2,525.7123
2,768.5548
3,058.9555

Note: Friedland performs this work twice: on closed claims and on reported claims, and
considers both in selecting the Ultimate Claim Counts in Exhibit I, sheet 3.

Exam 5, V2

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Chapter11 – Frequency-Severity Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solution to Sample Question 6 (continued)
6b. Project Ultimate Severity using the Frequency-Severity Development technique
(D) Given
Claims

Accident
Half Year

Period
Ending

2006
2006
2007
2007
2008

30-Jun
31-Dec
30-Jun
31-Dec
30-Jun

(E) =(D)/(A) MUST DIVIDE FOR …
* 1000
Accident
Period
Half Year
Ending
Severities
2006
30-Jun
2006
31-Dec
2007
30-Jun
2007
31-Dec
2008
30-Jun
(F) ATA
and CDF
Severities

Reported Claims ($000s): Data Triangle
1st report
2nd report 3rd report
4th report
6 mo.
12 mo.
18 mo.
24 mo.
11,947
12,503
11,662
12,647
14,071

11,856
12,762
11,523
12,854

11,820
12,706
11,492

11,772
12,697

5th report
30 mo.
11,760

Reported Severity: Data Triangle (calculated)
1st report
2nd report 3rd report
4th report 5th report
6 mo.
12 mo.
18 mo.
24 mo.
30 mo.
4,254.6296
4,466.9525
4,523.6618
4,531.3508
4,482.6378

4,371.6814 4,371.3018
4,770.8411 4,758.8015
4,549.1512 4,544.0886
4,627.0698

4,358.3858 4,355.5556
4,758.9955

Severities: Age-to-Age Factors
2nd to 3rd 3rd to 4th
4th to 5th
12:18 mo 18:24 mo
24:30 mo
1.001
0.9985
0.9990

Accident
Half Year
* 2005 *

Period
Ending
31-Dec

1st to 2nd
6:12 mo
1.115

2006
2006
2007
2007

30-Jun
31-Dec
30-Jun
31-Dec

1.0275
1.0680
1.0056
1.0211

0.9999
0.9975
0.9989

Medial Avg 5x1 ATA *
Development Age

1.0389
6 mo.

0.999
12 mo.

0.997
1.000

See Tail
Below

0.9994

Given
0.9985
18 mo.

0.999
24 mo.

1.000
30 mo.

CDF to Ultimate
1.0361
0.9973
0.9979
0.9994
1.0000
* "Medial Avg 5x1" requires 5 periods, and excludes the highest and lowest ATA factor.
For the 5th period, ATA factors for the 12-31-05 period are added to the table above.
Example: 6 mo. Medial Avg 5x1 = [1.0275+1.0680+1.0211] / 3 = 1.0389
1.115 and 1.006 are tak en out, leaving only three medial factors.
(G) Est.
Ultimate
Severities

Accident
Half Year

6 mo.
Period
Ending

Age of
Data at
6/30/08

2006
2006
2007
2007
2008

30-Jun
31-Dec
30-Jun
31-Dec
30-Jun

30 months
24 months
28 months
12 months
6 months

(1)

Exam 5, V2

Page 224

Reported
Severities
6/30/08

CDF to
Ultimate

Estimated
Ultimate
Severities

(2) from (E) (3) from (F)
4,355.5556
1.0000
4,758.9955
0.9994
4,544.0886
0.9979
4,627.0698
0.9973
4,482.6378
1.0361

(4)=(2)*(3)
4,355.5556
4,756.1401
4,534.5460
4,614.5767
4,644.4610

 2014 by All 10, Inc.

Chapter11 – Frequency-Severity Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solution to Sample Question 6 (continued)
6c. Project Ultimate Claims using the Frequency-Severity Development technique
(H)=(C)*(G)
Ultimate
Claims

Accident
Half Year

6 mo.
Period
Ending

Estimated
Ultimate
Counts

(1) = (C4)
2006
30-Jun
2,700.0000
2006
31-Dec
2,666.9328
2007
30-Jun
2,525.7123
2007
31-Dec
2,768.5548
2008
30-Jun
3,058.9555
Estimated Ult. Claims for Accident Periods

Estimated Product of
Frequency
and
Severity
(/1000) =
Ultimate
Severities
Est. Ultimate Claims
(2) = (G4)
(3) = (1) * (2) / 1000
4,355.5556
11,760.0001
4,756.1401
12,684.3060
4,534.5460
11,452.9586
4,614.5767
12,775.7085
4,644.4610
14,207.1995
1/1/06 thru 6/30/08
62,880.1727

6d. Develop IBNR Estimates ($000) using the Frequency-Severity Development technique
(I)=(H)-(D)
IBNR

Accident
Half Year

6 mo.
Period
Ending

Estimated
Ultimate
Claims

Reported
Claims at
6/30/08

(1) = (H3)
(2) from (D)
2006
30-Jun
11,760
11,760.0001
2006
31-Dec
12,697
12,684.3060
2007
30-Jun
11,492
11,452.9586
2007
31-Dec
12,854
12,775.7085
2008
30-Jun
14,071
14,207.1995
Estimated IBNR for Accident Periods 1/1/02 thru 6/30/08

Estimated IBNR
(broadly defined
to include IBNER)
(3) = (1) - (2)
0.0001
-12.6940
-39.0414
-78.2915
136.1995
6.1727

Note: Compare to Exhibit 1, Sheet 8 in Friedland's Chapter 11, which also
includes a total Unpaid Claims estimate. Rounding differences exist.

Solutions to 2001 Exam Questions (modified):
33. Calculate the cumulative claim payments for accident year 2000 at 24 and 36 months of
development. Assume 5% future annual inflation and no partial payments. Show all work.
Note: Extra detail included to show the steps of the Disposal Rate method.
Disposal Rate Method STEP 1: Select Ultimate Claim Counts by year
(A) GIVEN (otherwise could same procedure as for development method)
Cumulative Claim Counts
Accident
Age of Development (Months)
12
24
36
Ultimate
Year
1998
1,000
1999
1,200
2000
1,300

Exam 5, V2

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Chapter11 – Frequency-Severity Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solution to 2001 #33 (continued):
Disposal Rate Method STEP 2A: Select Disposal Rates
(B) GIVEN (otherwise find as Cumulative closed counts / Ultimate counts)
Selected Disposal Rates
Accident
Age of Development (Months)
12
24
36
Ultimate
Year
1998
1999
2000
Selected
30%
72%
100%
100%
STEP 2B: Calculate CONDITIONAL factor from Disposal Rates (incremental)
Important: This step is included in the formula Friedland gives for calculating
projected incremental claim counts for accident year X at age Y using the
Freq Sev Disposal Rate Method: (ultimate claim counts for accident year X cumulative closed claim counts for accident year x along the latest diagonal)
/ (1 - selected disposal rate at maturity of latest diagonal)
x (disposal rate at y - disposal rate at [prior y])
(C) = [Difference in consecutive selections in (B)] / [1.0 minus earlier in (B) ]
Conditional Factor from Disposal Rates
Accident
Age of Development (Months)
0
to
12
12 to 24
24 to 36
Year
1998
30.0%
60.0%
100.0%
1999
30.0%
60.0%
100.0%
2000
30.0%
60.0%
100.0%
Selected*
30.0%
60.0%
100.0%
*Based on selected disposal rates. Example: 60% = (72% - 30%) / (1 - 30%)
STEP 3: Project Claim Counts (Incremental)
(D) = [Factor selected in (C)]* [(A) ultimate - all prior entries for (D)]
WARNING: This can be trick y to do in one step.
Accident
Year
1998
1999
2000

Incremental Claim Counts (incl projections)
Age of Development (Months)
0 to 12
12 to 24
24 to 36
300
420
280
360
504
336
390
546
364

Example: 390 = 30% * [1300 - 0] and 546 = 60% * [1300 - 390]
… and 364 = 100% * [1300 - 390 - 546]

Exam 5, V2

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Chapter11 – Frequency-Severity Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND

STEP 4A: To analyze severities, first need Incremental Claims to date
(E) Given cumulative claims, we can calculate the following:

Accident
Year
1998
1999
2000

Incremental Claims to date
Age of Development (Months)
12
24
36
300,000
630,000
560,000
396,000
793,800
471,900

Solution to 2001 #33 (continued):
STEP 4B: To analyze severities, next find Average Severities to date
(F) = (E)/(D)

Accident
Year
1998
1999
2000

Actual Average Severities to date
Age of Development (Months)
12
24
36
1,000
1,500
2,000
1,100
1,575
1,210

STEP 5: Project Severities, Incorporating trend
(G) Trend factors (given at 5% annually)

Accident
Year
1998
1999
2000

(H)=(F)*(G)

(I) from (H)
for AY
2000 only
including
selected
projections

Exam 5, V2

Trend Factors to 2000 (at given 5%)
Age of Development (Months)
12
24
36
1.103
1.103
1.103
1.050
1.050
1.000

Trended Average Severities to date
Accident
Age of Development (Months)
12
24
36
Year
1998
1,103
1,654
2,205
1999
1,155
1,654
2000
1,210
Selected at 2000 level
1,654
2,205

Accident
Year

2000

Trended Average Severities to date
Age of Development (Months)
12
24
36

1,210

Page 227

1,654

2,205

 2014 by All 10, Inc.

Chapter11 – Frequency-Severity Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND

STEP 6A: Multiply Severities by Counts for Incremental Paid Claims
(J) = (I) * (D)

Accident
Year
1998
1999
2000

Exam 5, V2

Estimated Total $ Claims (Projected)
Age of Development (Months)
0 to 12
12 to 24
24 to 36

471,900

902,948

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802,620

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Chapter11 – Frequency-Severity Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solution to 2001 #33 (continued):
STEP 6B: Add Across Incremental for Cumulative Paid Claims
(K) from (J)

Accident
Year
1998
1999
2000

Estimated Total $ Claims (Incl. Projected)
Age of Development (Months)
12
24
36

471,900

1,374,848

2,177,468

Cumulative loss payments for accident year 2000:
at 24 months of development
1,374,848
at 36 months of development
2,177,468

Solutions to 2004 Exam Questions (modified):
Question 20. Using a Frequency-Severity Disposal Rate method, determine the projected ultimate
payments for accident year 2003.
Show all work and state any additional assumptions.
Disposal Rate Method STEP 1: Select Ultimate Claim Counts by year
(A) Almost given (but need to cumulate the incremental counts)
Cumulative Claim Counts
Accident
Age of Development (Months)
12
24
36
48
Year
10
25
35
50
2000
2001
17
29
46
2002
15
27
2003
13

Disposal Rate Method STEP 2A: Select Disposal Rates
(B) Calculate as Cumulative closed counts / Ultimate counts
Selected Disposal Rates
Accident
Age of Development (Months)
12
24
36
48
Year
20.0%
50.0%
70.0%
100.0%
2000
2001
28.3%
48.3%
76.7%
2002
21.4%
38.6%
2003
18.6%
Selected*
18.6%
38.6%
76.7%
100.0%

Ultimate
50
60
70
70

Ultimate
100%
100%
100%
100%

*Selected from latest diagonal. Note: If not told how to select, state your
assumption. Friedland shows 3-yr and 5-yr simple average, and medial 5x1.

Exam 5, V2

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Chapter11 – Frequency-Severity Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solution to 2004 #20 (continued):
STEP 2B: Calculate CONDITIONAL factor from Disposal Rates (incremental)
DON'T FORGET THIS STEP.
(C) = [Difference in consecutive selections in (B)] / [1.0 minus earlier percent in (B)]
Accident
Year
2000
2001
2002
2003
Projection*

Conditional Factor from Disposal Rates (incremental)
Age of Development (Months)
0 to 12
12 to 24
24 to 36
36 to 48
20.0%
37.5%
40.0%
100.0%
use 100.0%
28.3%
27.9%
54.8%
use 62.0% use 100.0%
21.4%
21.8%
use
24.6%
use
62.0% use 100.0%
18.6%
18.6%
24.6%
62.0%
100.0%
to use

* Note: These projections are based on the selected disposal rates above.
Example: for 12 to 24 mo.: 24.6% = (38.6% - 18.6%) / (1 - 18.6%)

Disposal Rate Method STEP 3: Project Claim Counts (Incremental)
(D) = [Factor selected in (C)]* [(A) ultimate - all prior entries for (D)]
WARNING: This can be trick y to do in one step.
Incremental Claim Counts (incl projections)
Accident
Age of Development (Months)
0 to 12
12 to 24
24 to 36
36 to 48
Year
2000
10
15
10
15
2001
17
12
17
14.0
2002
15
12
26.7
16.3
2003
13
14.0
26.7
16.3
Example for 2003: 13 = 18.6%*[70 - 0] and 14 = 24.6%*[70 - 13] and
. . . and 26.7 = 62%*[70-13-14] and 16.3 = 100%*[70-13-14-26.7]
STEP 4A: To analyze severities, first need Incremental Claims to date
(E) Given

Accident
Year
2000
2001
2002
2003

0 to 12
10,000
20,000
18,000
16,000

Incremental Claims to date
Age of Development (Months)
12 to 24
24 to 36
36 to 48
12,500
15,000
25,000
26,000
30,000
25,000

STEP 4B: To analyze severities, next find Average Severities to date
(F) = (E)/(D)

Accident
Year
2000
2001
2002
2003

Exam 5, V2

0 to 12
1,000
1,176
1,200
1,231

Actual Average Severities to date
Age of Development (Months)
12 to 24
24 to 36
36 to 48
833
1,500
1,667
2,167
1,765
2,083

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Chapter11 – Frequency-Severity Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solution to 2004 #20 (continued):
STEP 5: Project Severities, Incorporating trend
(G) Trend factors (given at 5% annually)

Accident
Year
2000
2001
2002
2003
(H)=(F)*(G)

0 to 12
1.158
1.103
1.050
1.000

Trended Average Severities to date
Accident
Age of Development (Months)
0 to 12
12 to 24
24 to 36
36 to 48
Year
1,158
965
1,736
1,929
2000
2001
1,297
2,389
1,946
2002
1,260
2,188
2003
1,231
Selected Severity *
2,188
1,946
1,929
2003 level
*Selected from latest diagonal. Note: If not told how to select, state your
assumption. Friedland shows 3-yr and 5-yr simple average, and medial 5x1.

(I) from (H)
for AY
2003 only
including
selected
projections

Trend Factors to 2003 (at given 5%)
Age of Development (Months)
12 to 24
24 to 36
36 to 48
1.158
1.158
1.158
1.103
1.103
1.050

Accident
Year
2000
2001
2002
2003

0 to 12

1,231

Trended Average Severities to date
Age of Development (Months)
12 to 24
24 to 36
36 to 48

2,188

1,946

1,929

STEP 6A: Multiply Severities by Counts for Incremental Paid Claims
(J) = (I) * (D)

Accident
Year
2000
2001
2002
2003

0 to 12

16,000

Estimated Total $ Claims (Projected)
Age of Development (Months)
12 to 24
24 to 36
36 to 48

30,625

51,882

31,513

STEP 6B: Add Across Incremental for Cumulative Paid Claims
(K) from (J)

Accident
Year
2000
2001
2002
2003

Exam 5, V2

16,000

Estimated Total $ Claims (Incl. Projected)
Age of Development (Months)
24
36
48
SOLUTION

46,625

Page 231

98,507

130,020

 2014 by All 10, Inc.

Chapter11 – Frequency-Severity Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solution to 2005 Exam Questions (modified):
14. (2 points) Using a Frequency-Severity Disposal Rate approach, what is the unpaid claims estimate as of
December 31, 2004 for accident year 2003?
Disposal Rate Method STEP 1: Select Ultimate Claim Counts by year
(A) Almost given (but need to cumulate the incremental counts)
Cumulative Claim Counts
Accident
Age of Development (Months)
12
24
36
48
Year
260
1030
1460
1630
2001
2002
310
1,020
1,540
2003
250
930
2004
340

Ultimate
1,630
1,640
1,760
1,690

Disposal Rate Method STEP 2A: Select Disposal Rates
(B) Calculate as Cumulative closed counts / Ultimate counts
Selected Disposal Rates
Accident
Year
2001
2002
2003
2004
Selected*

12
16.0%
18.9%
14.2%
20.1%
20.1%

Age of Development (Months)
24
36
48
63.2%
89.6%
100.0%
62.2%
93.9%
52.8%
52.8%

93.9%

Ultimate
100%
100%
100%
100%

100.0%

*Selected from latest diagonal. Note: If not told how to select, state your
assumption. Friedland shows 3-yr and 5-yr simple average, and medial 5x1.

STEP 2B: Calculate CONDITIONAL factor from Disposal Rates (incremental)
DON'T FORGET THIS STEP . . . Only need 2003 here.
(C) = [Difference in consecutive selections in (B)] / [1.0 minus earlier percent in (B)]
Accident
Year

2003

Conditional Factor from Disposal Rates (incremental)
Age of Development (Months)
0 to 12
12 to 24
24 to 36
36 to 48

14.2%

45.0%

Projections*

use 87.1%

use 100.0%

87.1%

100.0%

to use

* Note: These projections are based on the selected disposal rates above.
Example for 24 to 36 mo.: 87.1% = (93.9% - 52.8%) / (1 - 52.8%)

Exam 5, V2

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Chapter11 – Frequency-Severity Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solution to 2005 #14 (continued):
Disposal Rate Method STEP 3: Project Claim Counts (Incremental)
(D) = [Factor selected in (C)]* [(A) ultimate - all prior entries for (D)]
WARNING: This can be trick y to do in one step . . . Only need 2003 here.
Accident
Year
2003

Incremental Claim Counts (incl projections)
Age of Development (Months)
0 to 12
12 to 24
24 to 36
36 to 48
250

680

722.7

107.3

Example: 250 = 14.2% * [1760 - 0] and 680 = 45% * [1760 - 250] . . .
722.7 = 87.1% * [1760 - 250 - 680] and 107.3 = 100%*[1760-250-680-722.7]

STEPS 4 & 5: Analyze and Select Severities (simplified here since given amounts to use)
(E) Given . . . Only need 2003 here.

Accident
Year
2001
2002
2003
2004

Average Severities (given 0% inflation here)
Age of Development (Months)
0 to 12
12 to 24
24 to 36
36 to 48
560
830
1,530
2,000
560
830
1,530
2,000
560
830
1,530
2,000
560
830
1,530
2,000

STEP 6: Multiply Severities by Counts for Incremental Paid Claims
(F) = (D) * (E) . . . For Accident Year 2003

Accident
Year
2003

Estimated Total $ Claims (Including Projected)
Age of Development (Months)
0 to 12
12 to 24
24 to 36
36 to 48
140,000

564,400

1,105,705

214,634

STEP 7: Determine Unpaid Claims Estimate
(G) = (F) Projections only . . . For Accident Year 2003
Estimated Total $ Claims (Future Projected Only)
Accident
Year

Age of Development (Months)
24
36

2003

Exam 5, V2

1,105,705

Page 233

SOLUTION
Est. Unpaid

+214,634=

1,320,339

 2014 by All 10, Inc.

Chapter11 – Frequency-Severity Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solution to 2007 Exam Questions (modified):
Question 35. Using a Frequency-Severity Disposal Rate approach, determine the projected ultimate
payments for accident year 2006.
Disposal Rate Method STEP 1: Select Ultimate Claim Counts by year
(A) Almost given (but need to cumulate the incremental counts)
Cumulative Claim Counts
Accident
Age of Development (Months)
12
24
36
48
Year
40
120
180
200
2003
2004
48
144
216
2005
36
108
2006
44

Disposal Rate Method STEP 2A: Select Disposal Rates
(B) Calculate as Cumulative closed counts / Ultimate counts
Selected Disposal Rates
Accident
Age of Development (Months)
12
24
36
48
Year
20.0%
60.0%
90.0%
100.0%
2003
2004
20.0%
60.0%
90.0%
2005
20.0%
60.0%
2006
20.0%
Selected
20.0%
60.0%
90.0%
100.0%

Ultimate
200
240
180
220

Ultimate
100%
100%
100%
100%

STEP 2B: Calculate CONDITIONAL factor from Disposal Rates (incremental)
DON'T FORGET THIS STEP.
(C) = [Difference in consecutive selections in (B)] / [1.0 minus earlier percent in (B)]
Accident
Year
2003
2004
2005
2006
Projection*

Conditional Factor from Disposal Rates (incremental)
Age of Development (Months)
0 to 12
12 to 24
24 to 36
36 to 48
20.0%
50.0%
75.0%
100.0%
use 100.0%
20.0%
50.0%
75.0%
use 75.0% use 100.0%
20.0%
50.0%
use 50.0%
use 75.0% use 100.0%
20.0%
20.0%
50.0%
75.0%
100.0%
to use

* Note: These projections are based on the selected disposal rates above.
Example: for 12 to 24 mo.: 50% = (60% - 20%) / (1 - 20%)

Exam 5, V2

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Chapter11 – Frequency-Severity Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solution to 2007 #35 (continued):
Disposal Rate Method STEP 3: Project Claim Counts (Incremental)
(D) = [Factor selected in (C)]* [(A) ultimate - all prior entries for (D)]
WARNING: This can be trick y to do in one step.
Incremental Claim Counts (incl projections)
Accident
Age of Development (Months)
0
to
12
12
to 24
24 to 36
36 to 48
Year
40
80
60
20
2003
2004
48
96
72
24.0
2005
36
72
54.0
18.0
2006
44
88.0
66.0
22.0
Example for 2006: 44 = 20%*[220 - 0] and 88 = 50%*[220 - 44] and
. . . and 66 = 75%*[220-44-88] and 22 = 100%*[220-44-88-66]

STEP 4A: To analyze severities, first need Incremental Claims to date
(E) Given

Accident
Year
2003
2004
2005
2006

0 to 12
40,000
50,000
38,000
45,000

Incremental Claims to date
Age of Development (Months)
12 to 24
24 to 36
36 to 48
100,000
90,000
40,000
118,000
108,000
90,000

STEP 4B: To analyze severities, next find Average Severities to date
(F) = (E)/(D)
Accident
Year
2003
2004
2005
2006

0 to 12
1,000
1,042
1,056
1,023

Actual Average Severities to date
Age of Development (Months)
12 to 24
24 to 36
36 to 48
1,250
1,500
2,000
1,229
1,500
1,250

STEP 5: Project Severities, Incorporating trend
(G) Trend factors (given at 10% annually)

Accident
Year
2003
2004
2005
2006

Exam 5, V2

0 to 12
1.331
1.210
1.100
1.000

Trend Factors to 2003 (at given 5%)
Age of Development (Months)
12 to 24
24 to 36
36 to 48
1.331
1.331
1.331
1.210
1.210
1.100

Page 235

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Chapter11 – Frequency-Severity Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2007 #35 (continued):
(H)=(F)*(G)

Trended Average Severities to date
Age of Development (Months)
12 to 24
24 to 36
36 to 48
1,664
1,997
2,662
1,487
1,815
1,375

Accident
0 to 12
Year
1,331
2003
2004
1,260
2005
1,161
2006
1,023
Selected Severity *
1,375
1,815
2,662
2006 level
*Selected from latest diagonal. Note: If not told how to select, state your
assumption. Friedland shows 3-yr and 5-yr simple average, and medial 5x1.
(I) from (H)
for AY
2006 only
including
selected
projections

Accident
Year
2003
2004
2005
2006

0 to 12

1,023

Trended Average Severities to date
Age of Development (Months)
12 to 24
24 to 36
36 to 48

1,375

1,815

2,662

STEP 6A: Multiply Severities by Counts for Incremental Paid Claims
(J) = (I) * (D)

Accident
Year
2003
2004
2005
2006

0 to 12

45,000

Estimated Total $ Claims (Projected)
Age of Development (Months)
12 to 24
24 to 36
36 to 48

121,000

119,790

58,564

STEP 6B: Add Across Incremental for Cumulative Paid Claims
(K) from (J)

Accident
Year
2003
2004
2005
2006

Exam 5, V2

45,000

Estimated Total $ Claims (Incl. Projected)
Age of Development (Months)
24
36
48
Solution

166,000

285,790

Page 236

344,354

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Chapter11 – Frequency-Severity Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solution to 2008 #3 (modified):
Using a Frequency-Severity "Disposal Rate" method:
Estimate unpaid claims as of 12/31/07. Show all work.
STEP 1: Create Cumulative Triangle & Select Ultimate Counts
(A) Almost given (but need to cumulate the incremental counts)
Cumulative Claim Counts
Accident
Age of Development (Months)
12
24
36
48
Year
500
800
950
1,000
2004
2005
600
960
1,140
2006
750
1,200
2007
900
STEP 2A: Select Disposal Rates
(B) Calculate as Cumulative closed counts / Ultimate counts
Selected Disposal Rates
Accident
Age of Development (Months)
12
24
36
48
Year
2004
2005
2006
2007
Selected

50.0%
50.0%
50.0%
50.0%

80.0%
80.0%
80.0%

95.0%
95.0%

100.0%

50.0%

80.0%

95.0%

100.0%

Ultimate
1,000
1,200
1,500
1,800

Ultimate
100%
100%
100%
100%

STEP 2B: Calculate CONDITIONA L factor from Disposal Rates (incremental)
DON'T FORGET THIS STEP.
(C)= [Difference in consecutive selections in (B)] / [1.0 minus earlier % in (B)]
For all
Conditional Factor from Disposal Rates (incremental)
Accident
Age of Development (Months)
0 to 12
12 to 24
24 to 36
36 to 48
Years
Projections*
to use
50.0%
60.0%
75.0%
100.0%
* Note: These projections are based on the selected disposal rates above.
Example: for 12 to 24 mo.: 60% = (80% - 50%) / (1 - 50%)

Disposal Rate Method STEP 3: Project Claim Counts (Incremental)
(D) = [Factor selected in (C)]* [(A) ultimate - all prior entries for (D)]
WARNING: Can be tricky to do in one step …
May want to practice combining steps 2 & 3.
Incremental Claim Counts (incl projections)
Accident
Age of Development (Months)
FYI
0 to 12
12 to 24
24 to 36
36 to 48
Ultimate
Year
500
300
150
50
2004
1000
2005
600
360
180
60
1200
2006
750
450
225
75
1500
2007*
900
540
270
1,800
90
* Example for 2007: 900 as given; 540 = 60%*[1800-900];
. . . 270 = 75%*[1800-900-540] and 90 = 100%*[1800-900-540-270]

Exam 5, V2

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Chapter11 – Frequency-Severity Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solution to 2008 #3 (continued):
STEP 4A: To analyze severities, first need Incremental Claims to date
(E) Given

Accident
Year
2004
2005
2006
2007

0 to 12
400
504
662
833

Incremental Claims to date
Age of Development (Months)
12 to 24
24 to 36
36 to 48
300
180
75
378
227
496

STEP 4B: To analyze severities, next find Average Severities to date
(F) = (E)/(D)

Accident
Year
2004
2005
2006
2007

0 to 12
0.800
0.840
0.883
0.926

Actual Average Severities to date
Age of Development (Months)
12 to 24
24 to 36
36 to 48
1.000
1.200
1.500
1.050
1.260
1.102

STEP 5: Project Severities, Incorporating trend
(G) Trend factors (given at 5% annually)
Trend Factors to 2007 (at given 5%)
Accident
Age of Development (Months)
0 to 12
12 to 24
24 to 36
36 to 48
Year
1.158
1.158
1.158
1.158
2004
2005
1.103
1.103
1.103
2006
1.050
1.050
2007
1.000

(H)=(F)*(G)
Accident
0 to 12
Year
0.926
2004
2005
0.926
2006
0.927
2007
0.926
Selected Severity *

Trended Average Severities to date
Age of Development (Months)
12 to 24
24 to 36
36 to 48
1.158
1.389
1.736
1.158
1.389
1.157
1.158

1.389

1.736

at '07 level

*Note: Nice here, but if not clear how to select, state your assumption.

Exam 5, V2

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Chapter11 – Frequency-Severity Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solution to 2008 #3 (continued):
(I) based on (H) for AY 2007 and de-trended at 5% for prior years
HERE, WE ONLY NEED TO LOOK AT PROJECTED AMOUNTS
Trended Average Severities to date
Notes
Accident
Age of Development (Months)
Selected
0 to 12
12 to 24
24 to 36
36 to 48
Year
07 Severity
Divided by
2004
2005
1.052
1.575
2006
1.323
1.051
1.654
2007
1.050 = 1
1.158
1.389
1.736
STEP 6A: Multiply Severities by Counts for Incremental Paid Claims
(J) = (I) * (D)

Accident
Year
2004
2005
2006
2007
2008

Estimated Total $ Claims (Projected)
Age of Development (Months)
0 to 12
12 to 24
24 to 36
36 to 48

625.065

297.675
375.071

94.500
124.031
156.279

Estimated
Unpaid
Claims
0.000
94.500
421.706
1,156.415
1,672.621
($000's)

Exam 5, V2

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Chapter11 – Frequency-Severity Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2009
Question 4 – Model Solution
a. Expected incremental closed claim counts for periods 60-72 and 72-ultimate for accident year 2004.
For AY x at Age y, projected incremental closed claim counts are computed as follows:
[(ultimate claim counts for AY x – cumulative closed claim counts for AY x along latest diagonal) / (1.00 –
selected disposal rate at maturity of latest diagonal)] x [disposal rate at y – disposal rate at y-1]
AY 2004 60 – 72:
(400 – 375) × (.98-.95) / (1-.95) = 15
AY 2004 72 – Ult:
25 × (1-.98)/(1-.95) = 10
b. (1.5 points) Using a 6% annual trend factor, estimate the 60-ultimate tail severity at 2008 levels.
2008-t
Solution 1: 60-ultimate tail severity = [Incremental paid losses (60-72)/incremental closed claims] * 1.06
Using AY 2003 data:
 incremental paid losses (60-72) = 6,200 – 5,675 = 525
 incremental closed claims (60-72) = (400 – 375) × (.98-.95) / (1-.95) = 15
5
2008 level 60-ultimate tail severity = [525/15] * 1.06 = 46,838
Solution 2: 60-ultimate tail severity = [Incremental paid losses (60-72)/incremental closed claims] * 1.06
Using AY 2003 and 2004 data:

2008-t

Incremental closed claim counts:
60
72
AY
2003
44
15
2004
12
44=395-351. The sum of these counts = 44+15+12=71
Trended incremental paid claims (000)
60
72
AY
2003
1061.21
702.57
2004
568.11
5

1061.21 = (5,675 – 4,882)×1.06 . The sum of these losses = 1,061.21+702.57+568.11 = 2,331.89
2008 level (in 000s) 60-ultimate tail severity = 2,331.89 / 71 = 32.84
c. (1.5 points) Estimate the ultimate losses for accident year 2004.
Solution 1: 2004 ultimate loss = Cumulative paid loss at 60 months + Incremental paid loss from 60-72
2004 Incremental paid loss from 60-72 = Tail severity at 2004 levels * Incremental cnts (60-72)
= ($6,200 – $5,675)/15 × 1.06 = $37,100; 400 – 375 = 25
2004 ultimate loss = $6,250,000 + $37,100 * 25 = $6,250,000 + $927,500 = 7,177,500
Solution 2: 2004 ultimate loss = Cumulative paid loss at 60 months + Incremental paid loss from 60-72
4
= 6,250 + [15 + 10] × 32.84/1.06 = 6,900.31 in (000s)

Exam 5, V2

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Chapter11 – Frequency-Severity Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2010 Exam Questions
16a. (3.25 points) Use the frequency-severity technique to calculate the expected ultimate claim cost
estimate for accident year 2009.
16b. (0.75 point) State the three key assumptions underlying the frequency-severity technique.
Question 16 - Solution 1
AY 2009 Expected Ultimate Claim Cost
= AY09 Trended and Developed Claim Count * AY09 Trended and Developed Severity
1. Compute Reported Claim Count Link Ratios
12-24
24-36
1.213
1.200
12-ult
24-ult
1.575
1.298

36-48
1.071
36-ult
1.082

48-ult
1.010
48-ult
1.010

2. Compute trended and developed claim counts.
AY
2006
2007
2008
Select

data
375
370
310

ATU
1.010
1.082
1.298

Trended
ult
402
416
410
409

trend
1.02^3
1.02^2
1.02

3. Compute Reported Severity = Reported Claims / Claim Count
AY
2006
2007
2008

12
5,000
4,400
$4,167

24
6,000
5,429
5,565

36
6,286
6,216

4. Compute Reported Severity Link Ratios
12-24
24-36
1.253
1.094
12-ult
24-ult
1.550
1.237

48
$6,933

36-48
1.103
36-ult
1.131

48-ult
1.025
48-ult
1.025

5. Compute trended and developed severities
AY
2006
2007
2008

Reported Sev
6,933
6,216
5,565

ATU
1.025
1.131
1.237

trend
0.985^3
0.985^2
0.985

Trended Ult
$6,791
$6,821
$6,781
$6,798

AY09 Trended and Developed Claim Count * AY09 Trended and Developed Severity
AY 09 Expected Ultimate Claim Cost = 409 × $6,798 = 2,780,382
b1. Definition of claim counts is consistent throughout experience period.
b2. Development of future claim is similar to development of prior claims.
b3. Mix of claim type is relatively consistent.

Exam 5, V2

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Chapter11 – Frequency-Severity Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2010 exam questions continued:
16a. (3.25 points) Use the frequency-severity technique to calculate the expected ultimate claim cost
estimate for accident year 2009.
16b. (0.75 point) State the three key assumptions underlying the frequency-severity technique.
Question 16 - Solution 2
a. Selected Rpt Claim Count Dev Factors
12-24 24-36 36-48
48– ult
1.213 1.20
1.071
1.01
To-ult 1.575 1.298 1.082
1.01
AY Ult. Claim Counts
06
(375)(1.01) = 379
07
(370)(1.082) = 400
08
(310)(1.298) = 402

Freq Trend to 2009 Level
379 × 1.02^3 = 402
400 × 1.02 ^2 = 416
402 × 1.02 = 410

Rept Severities
AY
12
24
36
48
06 5000 6000 6286 6933
07 4400 5429 6216
08 4167 5565
Selected Age-to-age
12-24
1.253
To-ult
1.550

24-36
1.094
1.237

36-48
1.103
1.131

48– ult
1.025
1.025

AY Ult. Severities
Sev Trend to 2009 Level
06
7106
7106× 0.985^3 = 6791
07
7030
7030 × 0.985 ^2 = 6821
08
6884
6884 × .985 = 6781
Selected ultimate claims = 409 (straight average of 06-08)
Selected ultimate Severity = 6798
2009 Estimated ult claim cost = 409 × 6798 = 2,780,382
b1. The definition of a claim is consistent over historical period used
b2. The mix of types of claims used is consistent
b3. Claims and claim count will continue to develop in a similar manner in the future as they have in
historical periods.

Exam 5, V2

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Chapter11 – Frequency-Severity Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to questions from the 2011 exam:
26a. (2 points) Use the volume-weighted average to estimate the trended tail severity for maturity ages of
72 months and older.
26b. (0.5 point) Briefly describe two considerations in selecting the maturity age at which to combine data
for estimating a tail factor.
Question 26 – Model Solution
Intial comments: See Exh V, S9 - Development of Trended Severity (at ages 60 and older and 72 and older)
1. A triangle of incremental closed claim counts for maturities 60 through 96 months is given (from E5S6).
2. A triangle of incremental paid claims for these same maturities is given (from E5S6).
3. Adjusted paid claims using the 5% annual severity trend to bring all payments to the 2008 cost level.
4. Estimated trended tail severity equals [sum trended claim payments]/[sum incremental closed claim counts].
a. Sum[incremental paid claims trended to 2010 @ 10% and divide by sum incremental closed claim counts]
7
7
7
6
6
5
[(20,000)(1.10) + (28,000)(1.10) + (25,000)(1.10) + (33,000)(1.10) + (36,000)(1.10) + (36,000)(1.10) ]/
[2,000 + 2,000 + 1,000 + 3,000 + 2,000 + 3,000] x 1,000 = 24,806
b. 1. Consider the age at which the data becomes erratic.
2. Consider the % of claims expected to closed beyond the selected age.

28. Use the frequency-severity technique to estimate the ultimate claim amount for accident year 2010.
Ult Claim Am for AY 2010 = $Proj Ult Severity per Clm Clsd with Payment * Ult Claim Count with Payment
Question 28 – Model Solution 1
1. Compute: Closed claim with payment triangle
AY
2007
2008
2009
2010

12
520
645
590
675-40 = 635

24
620
750
740

36
695
795

48
720 = 840-120

2. Compute volume weighted avg for LDF selections
12-24 = (620 + 750 + 740) / (520 + 645 + 590) = 1.2023
24-36 = (695 + 795) / (620 + 750) = 1.08759
36-48 = 720 / 695 = 1.0359
LDF to ULT (12-ULT) = 1.2023(1.08759)(1.0359) = 1.35465
3. Compute Ult claim count with payment for AY 2010 = 1.35465(635) = 860.2
4. Ult Claim Amount for AY 2010 = Proj Ult Severity per Clm Clsd with Payment * Ult claim count with
payment
AYT 2010 Ult Claim Amount = $3,450 * 860.2 = 2,967,690

Exam 5, V2

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Chapter11 – Frequency-Severity Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2011 exam questions:
Question 28 – Model Solution 2
Ult Claims = Ult Sev x Ult Count
Cum. Claim count closed with payment = total closed - total closed with no payment
AY
07
08
09
10

12
520
645
590
635

24
620
750
740

36
695
795

48
720

Compute age to age and age to ultimate LDFs
12-24
(620 + 750 + 740) / (520 + 645 + 590) = 1.202
CLDF 1.355
AY
07
08
09
10

Ult Severity
$3,350
3,400
3,275
$3,450

Count
720
795
740
635

Age-Ult LDF
1
1.036
1.127
1.355

24-36
1.088
1.127

36-48
1.036
1.036

Ult Count
720
824
834
860

Ult Claims = Ult Sev x Ult Count
$2,412,000
2,801,600
2,731,350
$2,967,000

AY10 Ult claim amount = $2,967,000

Exam 5, V2

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Chapter11 – Frequency-Severity Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2012 exam questions:
20. Use a frequency-severity technique to estimate the ultimate claim amount for accident year 2011.
Question 20 – Model Solution 1 (Exam 5B Question 5)
AY 2011 Ultimate claims = AY 2011 Ultimate Severity * AY 2011 Ultimate Claim Count

Reported
12-24
1.486
1.538

2009:
2010:
Selected:
Cumulative:

1.512
1.551

1.026
1.026

Claim Count
24-36
1.026

Age-to-age
36-ult.

1.000 => selected = straight average
1.000
Ultimate claim count 2011 = 212* 1.551 = 329

Severities= Cum. Reported Claims/Cum Reported Counts
12
24
36
2009:
5595
6731
7442
2010:
5475
6779
2011
5731
5,731 = 1,215,000/212
Severities Age-to-Age
2009:
2010:
Average=Selected:
Cumulative:

12-24
1.203
1.238

24-36
1.103

36-ult

1.221
1.346

1.103
1.103

1.000
1.000

*Ult Severities
2009

7,422 * 1.00 = 7,422

2010

6,779 * 1.10 = 7,477
5,731 * 1.346 =7,714

2011

*Ultimate claims 2011 = 7,714 * 329 = 2,537,906

Exam 5, V2

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Chapter11 – Frequency-Severity Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2012 exam questions:
20. Use a frequency-severity technique to estimate the ultimate claim amount for accident year 2011.
Question 20 – Model Solution 2 (Exam 5B Question 5)
Link Ratios (claim counts)
12-24
1.5128
1.5515

Selected (vol wtd)
CDF Ult

24-36
1.0256
1.0256

36-ult
1.000
1.000

Ult claims (AY11) = 1.5515 x 212 = 329
Disposal Rates = Cumulative Rptd/ Proj ult
AY
09
10
11
Selected

@24

@12
0.65625
0.6332
0.6444
0.6446

= 210/330

0.975
0.9742
0.975

@36

ult

1.0000
1.000

320
349=340x1.0256
329

Projected Rptd Counts: (AY11)
@12 =212
@24 =(.975-.6446/1-.6446)(329-212) = 109
@36 =329 – 109 – 212 = 8
Avg Severity = Incremental Rptd Claim/ Incremental closed counts
AY
@12
@24
@36
( 2305000 − 1210000 )
*=
09
5595
9069
34375
340 − 221
10
5475
*9202
11
5731
Selected (Simple Avg)
9136
34,375
5600
Ult Claims AY 2011 = 1,215,000 + 109 (9,136) + 8 (34,375) = 2,485,824
Examiner’s Comments
The question was straightforward with a majority of candidates receiving full credit.
Candidates lost points for:
using 12-24 age-to-age factor (as opposed to 12-ultimate factor) to derive ultimate counts or ultimate
severity, mixing incremental approach with cumulative approach,
using just the reported claim (loss) dollars triangle given in the question as severity triangle as opposed to
the approach of deriving the severity triangle, or derive 2011 ultimate counts by taking average of 2009
and 2010 (and sometimes also 2011) ultimate counts.

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Chapter12 – Case Outstanding Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Sec
1
2

Description
Case O/S Development Technique – Approach #1
Case O/S Development Technique – Approach #2

Pages
265 - 268
268 - 269

Case O/S Development Technique – Approach #1

265 - 268

Key Assumptions
 Claims activity related to IBNR is related in a consistent manner to claims already reported.
 Assumptions similar to those for the development techniques also apply to the case (O/S)
development technique.
Common Uses
 This method is appropriate when applied to lines of insurance for which most of the claims are
reported in the first accident period. Therefore, claims-made coverages and report year analysis
use the case O/S technique because the claims for a given AY are known at the end of the AY.
 The assumption that IBNR claim activity is related to claims already reported (i.e. development on
known claims versus pure IBNR) limits its use, and so it is not used extensively by actuaries.
Mechanics of the Method
Exhibit I, Sheet 1: The development triangles for case O/S and incremental paid claims.
These are derived from the reported and paid claim triangles in Chapter 7.
Chapter 12 - Case Outstanding Development Technique
U.S. Industry Auto
Case Outstanding and Incremental Paid Claims ($000)
Accident
Year
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007

12
18,478,233
18,544,291
19,034,933
19,401,810
20,662,461
21,078,651
21,047,539
21,260,172
20,973,908
21,623,594

24
9,937,970
9,955,034
10,395,464
10,487,914
11,176,330
11,098,119
11,150,459
11,087,832
11,034,842

36
5,506,911
5,623,522
5,969,194
5,936,461
6,198,509
6,398,219
6,316,995
6,141,416

Accident
Year
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007

12
18,539,254
20,410,193
22,120,843
22,992,259
24,092,782
24,084,451
24,369,770
25,100,697
25,608,776
27,229,969

24
14,691,785
15,680,491
16,855,171
17,103,939
17,702,531
17,315,161
17,120,093
17,601,532
17,997,721

36
6,830,969
7,168,718
7,413,268
7,671,637
8,108,490
7,670,720
7,746,815
7,942,765

Exhibit I
Sheet 1

Case Outstanding as of (months)

48
2,892,519
3,060,431
3,217,937
3,056,202
3,350,967
3,431,210
3,201,985

60
1,440,783
1,520,760
1,567,806
1,532,147
1,609,188
1,634,690

72
767,842
764,736
842,849
777,926
785,497

84
413,097
443,528
457,854
421,141

96
242,778
284,732
304,704

108
169,222
185,233

120
98,117

84
455,900
463,714
544,233
499,620

96
225,555
253,051
248,891

108
108,579
121,726

120
88,731

incremental paid Claims as of (months)

48
3,830,031
3,899,839
4,173,103
4,326,081
4,449,081
4,513,869
4,537,994

60
2,004,496
2,049,291
2,172,895
2,269,520
2,401,492
2,346,453

72
868,887
953,511
1,004,821
1,015,365
1,052,839

Exhibit I, Sheet 2: Ratio of Incremental Paid Claims to Previous Case Outstanding
Calculate the ratio of the incremental paid claims at age x to the case O/S at age x-12.
This ratio tells us the proportion of claims that were paid during the development interval (i.e. age x-12
to age x) on the claims O/S at the beginning of the age (i.e. age x-12).

Exam 5, V2

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Chapter12 – Case Outstanding Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Exhibit I
Sheet 2

Chapter 12 - Case Outstanding Development Technique
U.S. Industry Auto
Ratio of Incremental Paid Claims to Previous Case Outstanding
Accident
Year
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007

Ratio of Incremental Paid Claims to Previous Case Outstanding as of (months)

12
Simple Average
Latest 5
Latest 3
Medial Average
Latest 5x1

12
Selected

24
0.795
0.846
0.885
0.882
0.857
0.821
0.813
0.828
0.858

36
0.687
0.720
0.713
0.731
0.726
0.691
0.695
0.716

48
0.695
0.693
0.699
0.729
0.718
0.705
0.718

60
0.693
0.670
0.675
0.743
0.717
0.684

72
0.603
0.627
0.641
0.663
0.654

84
0.594
0.606
0.646
0.642

96
0.546
0.571
0.544

Averages of the Ratio of Incremental Paid Claims to Previous Case Outstanding
24
36
48
60
72
84
96

108
0.447
0.428

120
0.524

To Ult

108

120

To Ult

0.836
0.833

0.712
0.701

0.714
0.714

0.698
0.714

0.638
0.653

0.622
0.631

0.553
0.553

0.437
0.437

0.524
0.524

0.835

0.712

0.714

0.692

0.641

0.624

0.546

0.437

0.524

Selected Ratio of Incremental Paid Claims to Previous Case Outstanding
24
36
48
60
72
84
96
0.833

0.701

0.714

0.653

0.631

0.553

108

120

0.437

0.524

To Ult
1.100

AY 1998 ratio of incremental paid claims (12-24 months) to previous case O/S (at 12 months):
Incremental paid claims were $14,691,785 between the 12-24 month interval (labeled 24 months in
the development triangle).
Case O/S at 12 months was $18,478,233.
Thus, 79.5% (i.e. $14,691,785/$18,478,233) of the case O/S at 12 months results from the
incremental payment in the 12-to-24 month interval.
AY 2004 ratio of incremental paid claims (24-36 months) to previous case O/S (at 24 months):
Incremental paid claims were $7,746,815 between the 24-36 month interval.
Case O/S at 24 months was $11,150,459.
Thus, 69.5% (i.e. $7,746,815/$11,150,459) of the case O/S at 24 months results from the incremental
payment in the 24-36 month interval.
Selected ratios are based on the simple average of the latest 3 years.
A judgmentally selected ratio of 1.10 for the ratio to ultimate was made (this assumes that 10% more than
the case O/S at 120 months will ultimately be paid out).

Exam 5, V2

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Chapter12 – Case Outstanding Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Exhibit I, Sheet 3: Ratio of Case Outstanding to Previous Case Outstanding
The ratios = case O/S at age x / case O/S at age x-12.
Chapter 12 - Case Outstanding Development Technique
U.S. Industry Auto
Ratio of Case Outstanding to Previous Case Outstanding
Accident
Year
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007

12
Simple Average
Latest 5
Latest 3
Medial Average
Latest 5x1

12
Selected

Exhibit I
Sheet 3

Ratio of Case Outstanding to Previous Case Outstanding as of (months)
24
36
48
60
72
84
96
0.538
0.554
0.525
0.498
0.533
0.538
0.588
0.537
0.565
0.544
0.497
0.503
0.580
0.642
0.546
0.574
0.539
0.487
0.538
0.543
0.666
0.541
0.566
0.515
0.501
0.508
0.541
0.541
0.555
0.541
0.480
0.488
0.527
0.577
0.536
0.476
0.530
0.567
0.507
0.522
0.554
0.526

Averages of the Ratio of Case Outstanding to Previous Case Outstanding
24
36
48
60
72
84
96

108
0.697
0.651

120
0.580

To Ult

108

120

To Ult

0.529
0.526

0.564
0.566

0.528
0.528

0.488
0.486

0.514
0.511

0.551
0.555

0.632
0.632

0.674
0.674

0.580
0.580

0.527

0.562

0.530

0.488

0.515

0.542

0.642

0.674

0.580

108
0.674

120
0.580

24
0.526

Selected Ratio of Case Outstanding to Previous Case Outstanding
36
48
60
72
84
96
0.566
0.528
0.486
0.511
0.555
0.632

To Ult
0

AY 1998 at 24 months:
Case O/S for AY 1998 is $9,937,970 at 24 months and $18,478,233 at 12 months.
.538 = $9,937,970/$18,478,233 is the ratio of the case O/S at 24 months to case O/S at 12 months.
AY 2004 at 24 months:
Case O/S for is $6,316,995 at 36 months and $11,150,459 at 24 months.
.567 = $6,316,995/$11,150,459 is the ratio of the case O/S at 36 months to case O/S at 24 months.
Selected ratios are based on the simple average of the latest 3 years.
A judgmentally selected ratio of 0.10 for the ratio to ultimate was made (this assumes that there will be no
case O/S remaining for 132 months and later).
A challenge of this technique: Selection of the "to ultimate" ratios for both the ratio of incremental paid
claims to previous case O/S and the ratio of case O/S to previous case O/S.
The goal of the case O/S development method: To project ultimate claims based on completing the square
of incremental paid claims, which are related to the case O/S at the beginning of an interval.
Next step: Complete the square of the case O/S triangle (used to project the incremental paid claims).
Use selected ratios of case O/S to previous case O/S (from Exhibit I, Sheet 3) to project the case O/S for
each AY and age (in Exhibit I, Sheet 4.)

Exam 5, V2

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Chapter12 – Case Outstanding Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
U. S. Industry Auto
Projection of Paid Claims ($000)

Exhibit I
Sheet 4

Accident
Year
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007

12
18,478,233
18,544,291
19,034,933
19,401,810
20,662,461
21,078,651
21,047,539
21,260,172
20,973,908
21,623,594

24
9,937,970
9,955,034
10,395,464
10,487,914
11,176,330
11,098,119
11,150,459
11,087,832
11,034,842
11,374,010

36
5,506,911
5,623,522
5,969,194
5,936,461
6,198,509
6,398,219
6,316,995
6,141,416
6,245,721
6,437,690

Case Outstanding as of (months)
48
60
72
2,892,519 1,440,783
767,842
3,060,431 1,520,760
764,736
3,217,937 1,567,806
842,849
3,056,202 1,532,147
777,926
3,350,967 1,609,188
785,497
3,431,210 1,634,690
835,327
3,201,985 1,556,165
795,200
3,242,668 1,575,936
805,304
3,297,740 1,602,702
818,981
3,399,100 1,651,963
844,153

84
413,097
443,528
457,854
421,141
435,951
463,606
441,336
446,943
454,534
468,505

96
242,778
284,732
304,704
266,161
275,521
292,999
278,924
282,468
287,266
296,095

108
169,222
185,233
205,370
179,393
185,701
197,481
187,995
190,384
193,617
199,568

120
98,117
107,435
119,115
104,048
107,707
114,539
109,037
110,422
112,298
115,749

To Ult
0
0
0
0
0
0
0
0
0
0

Accident
Year
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007

12
18,539,254
20,410,193
22,120,843
22,992,259
24,092,782
24,084,451
24,369,770
25,100,697
25,608,776
27,229,969

24
14,691,785
15,680,491
16,855,171
17,103,939
17,702,531
17,315,161
17,120,093
17,601,532
17,997,721
18,012,454

36
6,830,969
7,168,718
7,413,268
7,671,637
8,108,490
7,670,720
7,746,815
7,942,765
7,735,424
7,973,181

Incremental Paid Claims as of (months)
48
60
72
3,830,031 2,004,496
868,887
3,899,839 2,049,291
953,511
4,173,103 2,172,895 1,004,821
4,326,081 2,269,520 1,015,365
4,449,081 2,401,492 1,052,839
4,513,869 2,346,453 1,067,453
4,537,994 2,286,217 1,016,176
4,384,971 2,315,265 1,029,087
4,459,444 2,354,587 1,046,564
4,596,511 2,426,958 1,078,732

84
455,900
463,714
544,233
499,620
495,649
527,091
501,771
508,147
516,777
532,661

96
225,555
253,051
248,891
232,891
241,081
256,374
244,059
247,160
251,357
259,083

108
108,579
121,726
133,156
116,312
120,403
128,041
121,890
123,439
125,535
129,394

120
88,731
97,062
107,614
94,002
97,307
103,480
98,509
99,761
101,455
104,574

To Ult
107,929
118,179
131,026
114,452
118,477
125,993
119,941
121,465
123,528
127,324

Accident
Year
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007

12
18,539,254
20,410,193
22,120,843
22,992,259
24,092,782
24,084,451
24,369,770
25,100,697
25,608,776
27,229,969

24
33,231,039
36,090,684
38,976,014
40,096,198
41,795,313
41,399,612
41,489,863
42,702,229
43,606,497
45,242,423

36
40,062,008
43,259,402
46,389,282
47,767,835
49,903,803
49,070,332
49,236,678
50,644,994
51,341,921
53,215,604

84
47,221,322
50,625,757
54,284,334
55,878,421
58,302,864
57,525,198
57,578,836
58,882,463
59,719,294
61,850,464

96
47,446,877
50,878,808
54,533,225
56,111,312
58,543,944
57,781,572
57,822,895
59,129,623
59,970,651
62,109,548

108
47,555,456
51,000,534
54,666,381
56,227,624
58,664,347
57,909,613
57,944,785
59,253,061
60,096,186
62,238,941

120
47,644,187
51,097,596
54,773,995
56,321,626
58,761,654
58,013,093
58,043,294
59,352,822
60,197,641
62,343,515

To Ult
47,752,116
51,215,775
54,905,021
56,436,079
58,880,132
58,139,086
58,163,235
59,474,287
60,321,169
62,470,839

Cumulative Paid Claims as of (months

48
43,892,039
47,159,241
50,562,385
52,093,916
54,352,884
53,584,201
53,774,672
55,029,965
55,801,366
57,812,115

60
45,896,535
49,208,532
52,735,280
54,363,436
56,754,376
55,930,654
56,060,889
57,345,230
58,155,952
60,239,072

72
46,765,422
50,162,043
53,740,101
55,378,801
57,807,215
56,998,107
57,077,065
58,374,316
59,202,517
61,317,804

AY 1999 projected case O/S at 120 months:
$107,435 equals 0.580 (selected ratio at 120 months) * $185,233 (case O/S at 108 months)
AY 2007 projected case O/S at 24 months:
$11,374,010 equals 0.526 (selected ratio at 24 months) * $21,623,594 (case O/S at 12 months)
Next Step:

Use the selected ratios of incremental paid claims to case O/S to project incremental paid
claims for all AYs and maturities. (See middle section of Exhibit I, Sheet 4.)
AY 2000 projected incremental payments for 120 months (i.e. the interval 108 to 120):
$107,614 = $205,370 x 0.524, where 0.524 is the selected ratio at 120 months and $205,370 is the
case O/S at 108 months
AY 2006 incremental paid claims at 48 months:
$4,459,444 equals 0.714 (the selected ratio at 48 months) * $6,245,721 (case O/S at 36 months)
Note: The highlighted cells are the projected values; the others values are from the original data triangles

Next Step: Calculation of cumulative paid claims (see bottom section of Exhibit I, Sheet 4)
Projected ultimate claims (the sum of the incremental paid claims) appear in the “To Ult” column.
Ultimate claims are carried forward to Column (4) of Exhibit I, Sheet 5.

Exam 5, V2

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Chapter12 – Case Outstanding Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Final Step: Exhibit I, Sheet 5: Calculate estimated IBNR and the total unpaid claim estimate
These are calculated in the same manner as those shown in the preceding chapters.
Chapter 12 - Case Outstanding Development Technique
U.S. Industry Auto
Development of Unpaid Claim Ratio ($000)

Accident
Year
(1)
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
Total

Claims at 12/31/08
Reported
Paid
(2)
(3)
47,742,304
47,644,187
51,185,767
51,000,534
54,837,929
54,533,225
56,299,562
55,878,421
58,592,712
57,807,215
57,565,344
55,930,654
56,976,657
53,774,672
56,786,410
50,644,994
54,641,339
43,606,497
27,229,969
48,853,563
543,481,587
498,050,368

Projected
Ultimate
Claims
(4)
47,752,116
51,215,775
54,905,021
56,436,079
58,880,132
58,139,086
58,163,235
59,474,287
60,321,169
62,470,839
567,757,738

Exhibit I
Sheet 5

Unpaid Claim Estimate
Based on Cape Outstanding
Case
Development Method
Outstanding
at 12/31/08
IBNR
Total
(5) = [(2) - (3)] (6) = [(4) - (2)] (7) = [(4) - (3)]
98,117
9,812
107,929
185,233
30,008
215,241
304,704
67,092
371,796
421,141
136,517
557,658
785,497
287,420
1,072,917
1,634,690
573,742
2,208,432
3,201,985
1,186,578
4,388,563
6,141,416
2,687,877
8,829,293
11,034,842
5,679,830
16,714,672
21,623,594
13,617,276
35,240,870
45,431,219
24,276,151
69,707,370

Column Notes:
(2) and (3) Based on Best's Aggregates & Averages U.S. private passenger automobile experience.
(4) Developed in Exhibit I , Sheet 4.

Estimated IBNR equals projected ultimate claims minus reported claims; Total unpaid claim estimate equals
projected ultimate claims less paid claims.
Compare the results of the case O/S development method with the reported and paid claim development
projections from Chapter 7.
XYZ Insurer (Example shown in Exhibit II, Sheets 1 through 5)
Exhibit II, Sheets 1 – 5, following the exact same format as Exhibit I.
1. Exhibit II, Sheet 1: Given case O/S and incremental paid claim triangles.
2. Exhibit II, Sheet 2: Calculate the ratios of incremental paid claims to previous case O/S
3. Exhibit II, Sheet 3: Calculate ratios of case O/S to previous case O/S
Given the operational and environmental changes noted in our discussions with management, selected ratios
are based on the latest two years of experience (to reflect the most current operating environment for XYZ)
4. Exhibit II, Sheet 4: Complete the square for both case O/S and incremental paid claims.
Projected ultimate claims using the case O/S development technique are based on the cumulative paid
claims through all maturities.
5. Exhibit II, Sheet 5: Calculate estimated IBNR and the total unpaid claim estimate in Columns (6) and (7).
Exhibit II, Sheet 6 (projected ultimate claims) and Exhibit II, Sheet 7 (estimated IBNR).
These exhibits compare the results of the case O/S development technique method with the FS method,
the Cape Cod method, the BF method, the expected claims method, and the development method.

Exam 5, V2

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 2014 by All 10, Inc.

Chapter12 – Case Outstanding Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Chapter 12 - Case Outstanding Development Technique
XYZ Insurer - Auto BI
Summary of Ultimate Claims ($000)

Accident
Year
(1)
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
Total

Development Method
Reported
Paid
(4)
(5)
15,822
15,980
25,082
25,164
36,948
37,922
38,488
40,599
48,314
49,592
44,950
49,858
74,786
80,537
76,661
80,332
58,370
72,108
47,979
77,941
47,530
74,995
514,929
605,028

Exhibit II
Sheet 6

Projected Ultimate Claims
Expected
B-F Method
Cape
Claims
Reported
Paid
Cod
(6)
(7)
(8 )
(9)
15,660
15,822
15,977
15,822
24,665
25,107
25,158
25,107
35,235
37,246
37,841
37,246
39,150
38,798
40,525
38,798
47,906
48,312
49,417
48,313
54,164
45,068
50,768
45,062
86,509
75,492
82,593
74,756
108,172
79,129
94,301
77,930
70,786
60,404
71,205
58,758
39,835
45,221
45,636
43,307
39,433
42,607
41,049
39,199
561,516
513,207 554,469
504,298

Frequency-Severity
Case O/S
Method 1 Method 2 Method 3
Dev.
(10)
(11)
(12)
(13)
15,822
15,822
25,084
25,054
37,071
36,913
38,772
39,192
38,804
48,666
46,869
48,796
46,105
44,510
45,093
76,606
71,947
74,874
80,740
71,700
77,725
64,510
50,077
58,666
58,527
30,487
31,831
46,198
59,214
30,172
29,847
46,005
551,117
513,949

Column Notes:
(2) and (3) Based on data from XYZ Insurer.
(4) and (5) Developed in Chapter 7, Exhibit II, Sheet 3.
(6) Developed in Chapter 8, Exhibit III, Sheet 1.
(7) and (8) Developed in Chapter 9, Exhibit II, Sheet 1.
(9) Developed in Exhibit II, Sheet 2.
(10) Developed in Chapter 11, Exhibit II, Sheet 6.
(11) Developed in Chapter 11, Exhibit IV, Sheet 3.
(12) Developed in Chapter 11, Exhibit VI, Sheet 7.
(13) Developed in Exhibit II, Sheet 4.

Chapter 12 - Case Outstanding Development Technique
XYZ Insurer - Auto BI
Summary of IBNR ($000)

Exhibit II
Sheet 7

Estimated IBNR
Case
B-F Method
Accident Outstanding Development Method Expected
Cape
Year
at 12/31/08 Reported
Paid
Claims
Reported
Paid
Cod
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8 )
1998
0
0
158
-162
0
155
0
1999
290
-25
57
-443
0
51
0
2000
465
-298
676
-2,011
0
595
0
2001
278
-310
1,801
352
0
1,727
0
2002
3,731
145
1,423
-263
143
1,248
144
2003
5,052
577
5,485
9,791
695
6,395
689
2004
17,477
4,498
10,249
16,221
5,204
12,305
4,468
2005
30,629
6,006
9,677
37,517
8,474
23,646
7,275
2006
25,985
9,566
23,304
21,982
11,600
22,401
9,954
2007
19,867
16,247
46,209
8,103
13,489
13,904
11,575
56,363
20,801
23,975
22,417
20,567
28,898
2008
15,223
Total
118,997
65,303 155,402 111,890
63,581
104,843
54,672

Frequency-Severity
Case O/S
Method 1 Method 2 Method 3
Dev.
(9)
(10)
(11)
(12)
0
0
-23
-53
-175
-333
-26
394
6
497
-1,300
627
1,732
106
720
6,318
1,618
4,586
10,085
1,029
7,070
15,706
1,109
9,862
26,795
-1,245
73
14,466
40,582
11,540
11,196
27,373
101,491
64,323

Column Notes:
(2) Based on data from XYZ Insurer.
(3) and (4) Estimated in Chapter 7, Exhibit II, Sheet 4.
(5) Estimated in Chapter 8, Exhibit III, Sheet 3.
(6) and (7) Estimated in Chapter 9, Exhibit II, Sheet 2.
(8) Estimated in Chapter 10, Exhibit II, Sheet 3.
(9) Estimated in Chapter 11, Exhibit II , Sheet 7.
(10) Estimated in Chapter 11, Exhibit IV, Sheet 3.
(11) Estimated in Chapter 11, Exhibit VI, Sheet 8.
(12) Estimated in Exhibit II, Sheet 5.

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Chapter12 – Case Outstanding Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
When the Case O/S Development Technique Works and When it Does Not
Limitations when using the case O/S development technique.
1. The assumption that future IBNR is related to claims already reported does not hold true for many lines
of insurance.
2. The infrequent use and the absence of benchmark data (for AY applications of this method).
3. A lack of intuitive sense and experiential knowledge as to what ratios are appropriate at each maturity for
both the incremental paid claims to previous case O/S and the case O/S to previous case O/S across
lines of insurance.

Case O/S Development Technique – Approach #2

268 - 269

Self-Insurer Case Only: Assume that the only data available for our self-insurer is case O/S.
 This situation is not common, but it can occur (especially for older years).
 The absence of historical cumulative paid claims can arise following times of transition (e.g. mergers and
acquisitions of corporations with self-insurance programs or consolidation of self-insured public entities.)
 Organizations that create self-insurance programs may only have current case O/S for claims in the
process of investigation and settlement available for years prior to the start of the self-insurance program.
Key Assumptions
 The assumptions from Chapter 7 regarding the development technique are applicable in this example.
 Industry-based reporting and payment development patterns are used to derive case O/S development
patterns.
 Claims recorded to date will develop in a similar manner in the future as our industry benchmark (i.e.,
the historical industry experience is indicative of the future experience for the self-insurer).
Common Uses
Used most often due to the absence of other reliable claims data for the purpose of developing an unpaid
claim estimate.
Mechanics of the Method
The standard development technique is used with case O/S to project an estimate of total unpaid claims
 In the Self-Insurer Case Only example, there are no historical paid claims.
 Insurance industry benchmark development patterns are used to project the GL case O/S values
that are available.
 Projected paid claims are estimates of unpaid claims and not ultimate claims.

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Chapter12 – Case Outstanding Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
The projection of the unpaid claim estimates for GL Self-Insurer Case Only is shown below:
Chapter 12 - Case Outstanding Development Technique
Self-Insurer Case Outstanding Only - General Liability
Development of Unpaid Claim Ratio ($000)

Accident
Year
(1)
1998
1999
2000
2001
2002
2003
Total

Case
Outstanding
at 12/31/08
(2)
500,000
650,000
800,000
850,000
975,000
1,000,000
4,775,000

CDF to Ultimate
Reported
(3)
1.015
1.020
1.030
1.051
1.077
1.131

Paid
(4)
1.046
1.067
1.109
1.187
1.306
1.489

Case
Outstanding
(5)
1.506
1.454
1.421
1.445
1.439
1.545

Exhibit III

Unpaid
Claim
Estimate
(6)
753,000
945,100
1,136,800
1,228,250
1,403,025
1,545,000
7,011,175

Column Notes:
(2) Based on data from Self-Insurer Case Outstanding Only.
(3) and (4) From Exhibit I, Sheet 2 in Chapter 8.
(5) = { [(3) - (1) * (4) ]/ ((4) -(3))} +1
(6) = [(2) * (5)].

The following formula is used to develop the case O/S development factor:

(Reported CDF to Ultimate - 1.00) x (Paid CDF to Ultimate)
+ 1.00
(Paid CDF to Ultimate - Reported CDF to Ultimate)
The case development factor includes provisions for case O/S and IBNR (the broad definition of IBNR,
which includes development on known claims). The estimated unpaid claims are shown in Column (6)
and equal the current estimate of case O/S * the derived case O/S CDF to ultimate.
Potential Limitations
1. Benchmarks may prove to be inaccurate in projecting future claims experience for the insurer.
2. It is inappropriate for the more recent, less mature years due to the increased variability of results related
to the highly leveraged development factors.
3. Large claims in the case O/S data can distort the results of projections based on this method.

Exam 5, V2

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Chapter12 – Case Outstanding Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Sample Questions:
1. Summarize Friedland’s key points re: ”When the Case Outstanding Development Technique Works and
When it Does Not.” Include 3 limitations of the method.
2. Friedland notes that the Case Outstanding Development technique is not extensively used. When is it
most appropriate to use this method?
3. Based on the following information and using the Case Outstanding Development Technique,
calculate an estimate of unpaid claims as of 12/31/08.
Assume all claims are closed by 60 months and that the final “Ratio of Incremental Paid Claims to
Previous Case Outstanding” is 1.0.
Report
Year
2005
2006
2007
2008
Report
Year
2005
2006
2007
2008

Reported Claims as of Months
12
24
36
23,000
24,840
26,082
27,386

24,000
25,920
27,216

24,800
26,784

Paid Claims (cumulative)
12
24
36
20,000
21,600
22,680
23,814

22,000
23,760
24,948

23,000
25,056

48
24,786

48
23,486

1995 Exam Questions (modified):
51. Given the following data, and assuming that all claims are reported within the first 12 months of an
accident year (i.e., during the accident year): Use the Case Outstanding Development Technique to
estimate ultimate claims by accident year.
Assume all claims are closed by 60 months and that the final “Ratio of Incremental Paid Claims to
Previous Case Outstanding” is 1.0.

Accident
Year
1990
1991
1992
1993
Accident
Year
1990
1991
1992
1993

Exam 5, V2

Reported Claims as of Months
12
24
36
131,800
136,900
135,000
126,500

189,145
197,635
195,030

204,764
214,780

Paid Claims (cumulative)
12
24
36
52,300
52,500
52,000
49,300

116,800
121,000
119,500

Page 255

159,900
166,500

48
212,850

48
196,400

 2014 by All 10, Inc.

Chapter12 – Case Outstanding Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
1997 Exam Questions (modified):
9. T/F. The selection of a tail factor is not an issue when projecting the “Ratio of Case Outstanding to
Previous Case Outstanding” described by Friedland.
2001 Exam Questions (modified):
32. (4 points) You are given the following information as of December 31, 2000:
Report
Year
1997
1998
1999
2000
Report
Year
1997
1998
1999
2000

Case Outstanding as of Months
12
24
36
200
300
350
400

150
250
275

75
150

Paid Claims (incremental)
12
24
36
100
125
175
225

75
80
110

70
100

48
50

Assume all claims are closed by 60 months and that the final “Ratio of Incremental Paid Claims to Previous
Case Outstanding” is 1.25. Based on the Case Outstanding Development Technique, calculate the Ultimate
Claim Estimates for report years 1997 through 2000. Show all work.
2003 Exam Questions (modified):

21. (1.5 points) You are given the following information:
•

The accident year 2002 reported claims as of December 31, 2002 = $16,500,000.

•

The accident year 2002 paid claims as of December 31, 2002 = $3,000,000.

Selected Ratios by Accident Year Development Interval:
“Ratio of Incremental Paid Claims to Previous Case Outstanding”
12-24
24-36
36-48
0.55
0.55
0.50

48-60
0.64

60 – Ult
1.03

“Ratio of Case Outstanding to Previous Case Outstanding”
12-24
24-36
36-48
0.90
0.60
0.55

48-60
0.40

60 – Ult
0.00

Using the Case Outstanding Development Technique, calculate the ultimate claims for Accident Year 2002.

Exam 5, V2

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Chapter12 – Case Outstanding Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2004 Exam Questions (modified):
8. You are given the following information:
•

2003 accident year reported claims as of December 31, 2003 = $20,000

•

2003 accident year case outstanding as of December 31, 2003 = $16,000

Selected Ratios by Accident Year Development Interval:
“Ratio of Incremental Paid Claims to Previous Case Outstanding”
12 to 24
24 to 36
36 to 48
0.50
0.60
0.48
“Ratio of Case Outstanding to Previous Case Outstanding”
12 to 24
24 to 36
36 to 48
0.80
0.70
0.55

48 to Ult
1.05
48 to Ult
0

Using the Case Outstanding Development Technique, find the ultimate claims for Accident Year 2003:
A. < $26,000
$28,000
D. > $28,000 but < $29,000

B. > $26,000 but < $27,000

C. > $27,000 but <
E. > $29,000

2005 Exam Questions (modified):
22. (2 points) You are given the following information:
Report
Year

2000
2001
2002
2003
2004

42,000
45,000
44,000
45,000
39,000

Report
Year

2000
2001
2002
2003
2004

22,000
24,000
25,000
27,000
24,000

Case Outstanding as of Months
24
36
48
29,000
33,000
30,000
32,000

16,000
19,000
18,000

4,000

9,000
10,000

4,000

Paid Claims (incremental)
24
36
16,000
17,000
19,000
18,000

13,000
14,000
15,000

8,000
9,000

Using the Case Outstanding Development Technique, calculate the estimated Unpaid Claims for Report
Year 2004 as of December 31, 2004.
Assume all claims are closed by 72 months and that the final “Ratio of Incremental Paid Claims to Previous Case
Outstanding” is 1.0.

Exam 5, V2

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Chapter12 – Case Outstanding Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2010 Exam Questions:
2. (4 points) Given the following information:
Case Outstanding ($000)
24
36
48
Months
Months
Months
4,630
4,500
3,565
4,680
4,390
5,230

Accident
Year
2006
2007
2008
2009

12
Months
3,860
4,020
4,150
4,300

Accident
Year
2006
2007
2008
2009

Cumulative Paid Claims ($000)
12
24
36
48
Months
Months
Months
Months
1,520
3,500
6,450
9,950
2,150
3,760
6,760
1,790
3,390
2,000

• Assume no further reported claim development after 48 months
• Use an all-year straight average for all factor selections
a. (3.5 points) Use Friedland's case outstanding development technique Approach #1 to estimate the
paid loss for accident years 2006 through 2009 as of 48 months.
b. (0.5 point) Explain whether the case outstanding development technique is generally more suitable
for accident year or report year analysis.

Exam 5, V2

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Chapter12 – Case Outstanding Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2011 Exam Questions:
29. (1.25 points) Given the following data as of December 31, 2010:
Case Outstanding (000s)
Accident
Year
2008
2009
2010

12 Months
$1,200
$1,500
$1,650

24 Months
$1,100
$1,350

36 Months
$450

Incremental Paid Claims (000s)
Accident
Year
2008
2009
2010
•

36 Months
$650

36 Months
0.40

48 Months
0.00

Selected ratio of incremental paid claims to previous case outstanding:
24 Months
0.80

•

24 Months
$960
$1,200

Selected ratio of case outstanding to previous case outstanding:
24 Months
0.90

•

12 Months
$1,500
$1,700
$1,650

36 Months
0.60

48 Months
1.05

Assume no further closed claim development after 48 months

a. (1 point) Use the case outstanding development technique to estimate the ultimate claims for
accident year 2009.
b. (0.25 point) Briefly describe when the case outstanding development technique is appropriate.

Exam 5, V2

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Chapter12 – Case Outstanding Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2012 Exam Questions:
19. (2 points) Given the following data as of December 31, 2011:
Case
Outstanding
Paid
Accident
Claims
Claims
Year
($000s)
($000s)
2009
$450
$3,200
2010
$1,350
$2,850
2011
$1,650
$1,900

•

Selected ratio of case outstanding to previous case outstanding:
24 Months
36 Months 48 Months
0.90

•

0.30

0.00

Selected ratio of incremental paid claims to previous case outstanding:
24 Months

36 Months

48 Months

0.80

0.60

1.05

Assume no further closed claim development after 48 months.

a. (1.5 points) Use the case outstanding development technique to estimate unpaid claims for
accident year 2011.
b. (0.5 point) Briefly describe two assumptions of this technique.

Exam 5, V2

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Chapter12 – Case Outstanding Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to Sample Questions:
1. Summarize Friedland’s key points re: ”When the Case Outstanding Development Technique Works and
When it Does Not.”
Limitation 1: The assumption that “future IBNR is related to claims already reported” is often not true.
Limitation 2: For accident year applications, the “infrequent use and absence of benchmark data” is also a
drawback.
Limitation 3: The “lack of intuitive sense and experiential knowledge” surrounding the ratios required by
these methods.
2.

Friedland notes that the Case Outstanding Development technique is not extensively used. It is most
appropriate for lines or classes for which most of the claims are reported in the first accident period:
Claims-Made coverages and Report Year analyses

3. Using the Case Outstanding Development Technique, estimate the unpaid claims:
(1)

Report
Year
2005
2006
2007
2008

(2)

Report
Year
2005
2006
2007
2008

(3) = (1) - (2)
Example : 2006 at 36m
= 26,784 -25,056
= 1,728

(4) = (2) - (2)prior
Example : 2006 at 36m
= 25,056-23,760
= 1,296

Exam 5, V2

Report
Year
2005
2006
2007
2008
Report
Year
2005
2006
2007
2008

Reported Claims as of Months
12
24
36
23,000
24,840
26,082
27,386

24,000
25,920
27,216

24,800
26,784

24,786

Paid Claims (cumulative)
12
24
36
20,000
21,600
22,680
23,814

22,000
23,760
24,948

23,000
25,056

23,486

Case Outstanding as of Months
12
24
36
3,000
3,240
3,402
3,572

2,000
2,160
2,268

1,800
1,728

Paid Claims (incremental)
12
24
36
20,000
21,600
22,680
23,814

Page 261

2,000
2,160
2,268

1,000
1,296

48
1,300

48
486

 2014 by All 10, Inc.

Chapter12 – Case Outstanding Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
(5) = (4) / (3)prior

Ratio: Paid Claims (incr) to PRIOR Case Outstanding
Year
12
24
36
48
2005
2006
2007
2008

Example : 2006 at 36m
= 1296/2160
= 60%

n/a
n/a
n/a
n/a

Selected
Three-Year Simple Averages
(6) = (5) & projections
where projections
= selected ratio
Example : 2007 at 48m
= 27%

66.7%
66.7%
66.7%

50.0%
60.0%

27.0%

66.7%

55.0%

27.0%

60 or Ult.

100%
given

Complete the square: Ratios with Incremental Paids
Final Ratio
Year
12
24
36
48 60 or Ult.
2005
2006
2007
2008

n/a
n/a
n/a
n/a

66.7%
66.7%
66.7%
66.7%

50.0%
60.0%
55.0%
55.0%

27.0%
27.0%
27.0%
27.0%

100.0%
100.0%
100.0%
100.0%

CAREFUL: These ratios apply to Case Outstanding, so we
also need to "complete the square" for Case Outstanding
before we can actually use these ratios.
To do so, we use another ratio - the Case Outstanding at a
given age, divided by the Case Outstanding at the prior age.
NOTE: It may be tempting to try to use 1 minus the ratio
above, (the incremental Paid / prior Case Outstanding), but
that logic only considers the effect that payments have on
Case Outstanding, and ignores other changes in estimates.

(7) = (3) / (3)prior

Example : 2006 at 36m
= 1728/2160
= 80%

Ratio: Case Outstanding to PRIOR Case Outstanding
Year
12
24
36
48
2005
2006
2007
2008

n/a
n/a
n/a
n/a

90.0%
80.0%

72.2%

66.7%

85.0%

72.2%

0%
definition

Complete the square: Ratios Case Outstanding / Prior
Year
12
24
36
48

0 at Ult.
60 or Ult.

Selected
Three-Year Simple Averages
(8) = (7) & projections
where projections
= selected ratio
Example : 2007 at 48m
= 72.2%

Exam 5, V2

60 or Ult.

66.7%
66.7%
66.7%
0.0%

2005
2006
2007
2008

n/a
n/a
n/a
n/a

Page 262

66.7%
66.7%
66.7%
66.7%

90.0%
80.0%
85.0%
85.0%

72.2%
72.2%
72.2%
72.2%

0.0%
0.0%
0.0%
0.0%

 2014 by All 10, Inc.

Chapter12 – Case Outstanding Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
(9) = (3) & projections
where projections
= (8) * (9) prior

Complete the square: Case Outstanding & projections
Year
12
24
36
48
2005
2006
2007
2008

Example : 2007 at 48m
.722 * 1,928 = 1,392
[.85 * 2,268 = 1,928]

3,000
3,240
3,402
3,572

2,000
2,160
2,268
2,381

1,800
1,728
1,928
2,024

0 at Ult.
60 or Ult.

1,300
1,248
1,392
1,462

0
0
0
0

Be sure to go out until the is no more case outstanding.
Otherwise, the cumulative paid will not equal ultimate claims.

NOW: We can go use those ratios we found in step (6), and
project out the future INCREMENTAL claim payments
(10) = (4) & projected
where projected
= (6) * (9) prior

Complete the square: Incremental Paid & projections
Year
12
24
36
48
2005
2006
2007
2008

Example : 2007 at 48m
.27 * 1,928 = 521

Showing Ultimates:
as in Friedland

20,000
21,600
22,680
23,814

2,000
2,160
2,268
2,381

1,000
1,296
1,247
1,310

486
467
521
547

Final Paid
60 or Ult.
1,300
1,248
1,392
1,462

To get to the Ultimate Claim amounts, we must use CUMULATIVE paids
since the Estimated Total Claims Payments are the Estimated Ultimate
Claims, by definition.

(11) = Sum across (10)
Year
2005
2006
2007
2008
Total Estimate of

Cumulative Paid & Projections at Ultimate
20,000
+ 2,000
+ 1,000
+ 486
+ 1,300
21,600
+ 2,160
+ 1,296
+ 467
+ 1,248
22,680
+ 2,268
+ 1,247
+ 521
+ 1,392
23,814
+ 2,381
+ 1,310
+ 547
+ 1,462
Ultimate Claims using Case Outstanding Dev. Method

Ultimate
= 24,786
= 26,771
= 28,108
= 29,513
109,178

FINALLY: To get the total "unpaid claim estimate," do the subtraction below:
Estimated
Ultimate
Year
Claims

Actual
Paid
to date

(12) in (11)
24,786
26,771
28,108
29,513

(13) in (2)
23,486
25,056
24,948
23,814

2005
2006
2007
2008
Total

Exam 5, V2

Page 263

Total Unpaid
Claims Estimate
(14) = (12) - (13)
1,300
1,715
3,160
5,699
11,874

 2014 by All 10, Inc.

Chapter12 – Case Outstanding Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Shortcut: If only ask ed for "Unpaid Claims Estimates," we can just add up the projected future
payment amounts . . . No need to show the Ultimate claims.
(10) Detail
Look ing only at the
Projected future
payments gives us an
Unpaid Claim Estimate

Sum of
60 or Ult. Projections
2005
1,300
1,300
2006
467
+1,248=
1,715
2007
1,247
+521
+1,392=
3,160
2008
2,381
+1,310
+547
+1,462=
5,699
Est. Unpaid Claims using Case Outstanding Dev. Method
11,874
Year

Incremental Paid: Projections only
24
36
48

Solutions to questions from the 1995 Exam (modified):
51. Use the Case Outstanding Development Technique to estimate ultimate claims by accident year.
Assume all claims are closed by 60 months and that the final “Ratio of Incremental Paid Claims to
Previous Case Outstanding” is 1.0.
(1) Given

Accident
Year
1990
1991
1992
1993

(2) Given

Exam 5, V2

52,300
52,500
52,000
49,300

204,764
214,780

116,800
121,000
119,500

79,500
84,400
83,000
77,200

72,345
76,635
75,530

159,900
166,500

Page 264

64,500
68,500
67,500

44,864
48,280

43,100
45,500

48
196,400

48
16,450

Paid Claims (incremental)
12
24
36
52,300
52,500
52,000
49,300

48
212,850

Case Outstanding as of Months
12
24
36

Accident
Year
1990
1991
1992
1993

189,145
197,635
195,030

Paid Claims (cumulative)
12
24
36

Accident
Year
1990
1991
1992
1993

(4) = (2) - (2)prior

131,800
136,900
135,000
126,500

Accident
Year
1990
1991
1992
1993

(3) = (1) - (2)

Reported Claims as of Months
12
24
36

48
36,500

 2014 by All 10, Inc.

Chapter12 – Case Outstanding Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to questions from the 1995 Exam #51 (continued):
(5) = (4) / (3)prior

Ratio: Paid Claims (incr) to PRIOR Case Outstanding
Year
12
24
36
48
1990
1991
1992
1993

n/a
n/a
n/a
n/a

Selected
Three-Year Simple Averages
(6) = (5) & projections
where projections
= selected ratio

81.1%
81.2%
81.3%

59.6%
59.4%

81.4%

81.2%

59.5%

81.4%

60 or Ult.

100%
given

Complete the square: Ratios with Incremental Paids
Final Ratio
Year
12
24
36
48 60 or Ult.
1990
1991
1992
1993

n/a
n/a
n/a
n/a

81.1%
81.2%
81.3%
81.2%

59.6%
59.4%
59.5%
59.5%

81.4%
81.4%
81.4%
81.4%

100.0%
100.0%
100.0%
100.0%

CAREFUL: These ratios apply to Case Outstanding, so we also need the
Case Outstanding projections before we can actually use these ratios.

(7) = (3) / (3)prior

Ratio: Case Outstanding to PRIOR Case Outstanding
Year
12
24
36
48
1990
1991
1992
1993

n/a
n/a
n/a
n/a

91.0%
90.8%
91.0%
0.0%

62.0%
63.0%

36.7%

90.9%

62.5%

36.7%

0%
definition

(8) = (7) & projections
where projections
= selected ratio

Complete the square: Ratios Case Outstanding / Prior
Year
12
24
36
48

0 at Ult.
60

36.7%
36.7%
36.7%
36.7%

0.0%
0.0%
0.0%
0.0%

(9) = (3) & projections
where projections
= (8) * (9) prior

Complete the square: Case Outstanding & projections
Year
12
24
36
48

0 at Ult.
60 or Ult.

Selected
Three-Year Simple Averages

1990
1991
1992
1993

n/a
n/a
n/a
n/a

79,500
84,400
83,000
77,200

91.0%
90.8%
91.0%
90.9%

72,345
76,635
75,530
70,200

62.0%
63.0%
62.5%
62.5%

44,864
48,280
47,211
43,880

16,450
17,703
17,311
16,089

0
0
0
0

Be sure to go out until the is no more case outstanding.
Otherwise, the cumulative paid will not equal ultimate claims.

Exam 5, V2

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Chapter12 – Case Outstanding Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to questions from the 1995 Exam #51 (continued):
NOW: We can go use those ratios we found in step (6), and
project out the future INCREMENTAL claim payments
(10) = (4) & projected
where projected
= (6) * (9) prior

Complete the square: Incremental Paid & projections
Year
12
24
36
48
1990
1991
1992
1993

52,300
52,500
52,000
49,300

64,500
68,500
67,500
62,691

43,100
45,500
44,921
41,751

All paid at
60 or Ult.

36,500
39,279
38,410
35,700

16,450
17,703
17,311
16,089

To get to the Ultimate Claim amounts, we must use CUMULATIVE paid
amounts, since Estimated Total Claims Payments are the Estimated
Ultimate Claims, by definition.
(11) = Sum across (10)
Year
1990
1991
1992
1993
Total Estimate of

Cumulative Paid & Projections at Ultimate
52,300
+ 64,500
+ 43,100
+ 36,500
+ 16,450
52,500
+ 68,500
+ 45,500
+ 39,279
+ 17,703
52,000
+ 67,500
+ 44,921
+ 38,410
+ 17,311
49,300
+ 62,691
+ 41,751
+ 35,700
+ 16,089
Ultimate Claims using Case Outstanding Dev. Method

=
=
=
=

Ultimate
212,850
223,482
220,141
205,531
862,004

Solutions to questions from the 1997 Exam (modified):
9. The selection of a tail factor is not an issue when projecting the “Ratio of Case Outstanding to Previous Case
Outstanding” described by Friedland.
True, the key ratio in this method is not a “tail factor” but the
“Ratio of Incremental Paid Claims to Previous Case Outstanding” factor for the last settlement interval.
Solutions to questions from the 2001 Exam (modified):
32. Based on the Case Outstanding Development Technique, calculate the Ultimate Claim Estimates for report
years 1997 through 2000. Show all work.
(1) GIVEN

Report
Year
1997
1998
1999
2000

(2) GIVEN

Report
Year
1997
1998
1999
2000

Exam 5, V2

Case Outstanding as of Months
12
24
36
200
300
350
400

150
250
275

75
150

Paid Claims (incremental)
12
24
36
100
125
175
225

Page 266

75
80
110

70
100

48
25

48
50

 2014 by All 10, Inc.

Chapter12 – Case Outstanding Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solution to questions from the 2001 Exam #32 (continued):
(3) = (2) / (1)prior

Ratio: Paid Claims (incr) to PRIOR Case Outstanding
Year
12
24
36
48
1997
1998
1999
2000

n/a
n/a
n/a
n/a

Selected
Three-Year Simple Averages
(4) = (3) & projections

37.5%
26.7%
31.4%

46.7%
40.0%

66.7%

31.9%

43.3%

66.7%

60 or Ult.

125%
given

Complete the square: Ratios with Incremental Paids
Final Ratio
Year
12
24
36
48 60 or Ult.
1997
1998
1999
2000

n/a
n/a
n/a
n/a

37.5%
26.7%
31.4%
31.9%

46.7%
40.0%
43.3%
43.3%

66.7%
66.7%
66.7%
66.7%

125.0%
125.0%
125.0%
125.0%

CAREFUL: These ratios apply to Case Outstanding, so we also need the
Case Outstanding projections before we can actually use these ratios.
(5) = (1) / (1)prior

Ratio: Case Outstanding to PRIOR Case Outstanding
Year
12
24
36
48
1997
1998
1999
2000

n/a
n/a
n/a
n/a

50.0%
60.0%

33.3%

79.0%

55.0%

33.3%

0%
definition

Complete the square: Ratios Case Outstanding / Prior
Year
12
24
36
48

0 at Ult.
60 or Ult.

Selected
Three-Year Simple Averages
(6) = (5) & projections

1997
1998
1999
2000
(7) = (1) & projections
where projected
= (6) * (7)prior

60 or Ult.

75.0%
83.3%
78.6%
0.0%

n/a
n/a
n/a
n/a

75.0%
83.3%
78.6%
79.0%

50.0%
60.0%
55.0%
55.0%

33.3%
33.3%
33.3%
33.3%

Complete the square: Case Outstanding & projections
Year
12
24
36
48
1997
1998
1999
2000

200
300
350
400

150
250
275
316

75
150
151
174

0.0%
0.0%
0.0%
0.0%

0 at Ult.
60 or Ult.

25
50
50
58

0
0
0
0

Complete the square: Incremental Paid & projections
Year
12
24
36
48

Final Paid
60 or Ult.

NOW: We can go use those ratios we found in step (4), and
project out the future INCREMENTAL claim payments
(8) = (2) & projected
where projected
= (4) * (7)prior

Exam 5, V2

1997
1998
1999
2000

100
125
175
225

Page 267

75
80
110
127

70
100
119
137

50
100
101
116

31
63
63
72

 2014 by All 10, Inc.

Chapter12 – Case Outstanding Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solution to questions from the 2001 Exam #32 (continued):
To get to the Ultimate Claim amounts, we must use CUMULATIVE paid
amounts, since Estimated Total Claims Payments are the Estimated
Ultimate Claims, by definition.
(9) = Sum across (8)
Year
Cumulative Paid & Projections at Ultimate
1997
100
+ 75
+ 70
+ 50
+
1998
125
+ 80
+ 100
+ 100
+
1999
175
+ 110
+ 119
+ 101
+
2000
225
+ 127
+ 137
+ 116
+
Total Estimate of Ultimate Claims using Case Outstanding Dev. Method

31
63
63
72

Ultimate
= 326
= 468
= 568
= 678
2,039

Solutions to questions from the 2003 Exam (modified):
21. Using the Case Outstanding Development Technique, calculate the ultimate claims for Accident Year 2002.
(1) = Reported - (2)
Year

Case Outstanding as of Months
12

2002 13,500,000 = 16,500,000 - 3,000,000
(2) Given
Year
2002

Paid Claims (incremental)
12
3,000,000

(3) Given
Ratio: Paid Claims (incr) to PRIOR Case Outstanding
Year
12
24
36
48
2002

Final Ratio
60 72 or Ult.

n/a

Selected

55.0%
given

50.0%
given

64%
given

103%
given

CAREFUL: These ratios apply to Case Outstanding, so we also need the
Case Outstanding projections before we can actually use these ratios.

(4) Given
Ratio: Case Outstanding to PRIOR Case Outstanding
Year
12
24
36
48
2002
Selected

0 at Ult.
72 or Ult.

55.0%
given

40%
given

0.0%
definition

0 at Ult.
72 or Ult.

7,290,000

4,009,500

1,603,800

n/a
90.0%
given

60.0%
given

(5) = (1) & projected, where projected = (4) * (5)prior
Compute the Case Outstanding Projections
Year
12
24
2002

13,500,000 12,150,000

NOW: We can use the ratios we found in step (3), and
project out the future INCREMENTAL claim payments

Exam 5, V2

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Chapter12 – Case Outstanding Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND

(6) = (2) & projected, where projected = (3)* (5)prior
Compute the Incremental Paid Projections
Year
12
24
2002

3,000,000

7,425,000

Final Paid
72 or Ult.

6,682,500

3,645,000

2,566,080

1,651,914

(7) = Sum across (6)
Year Calculate the Cumulative Paid Claim Projections at Ultimate
Ultimate
+ 7,425,000 + 6,682,500 + 3,645,000 + 2,566,080 + 1,651,914 24,970,494
2002 3,000,000
Total Estimate of Ultimate Claims using Case Outstanding Dev. Method
24,970,494

Solutions to questions from the 2004 Exam (modified):
8. Using the Case Outstanding Development Technique, find ultimate claims for Accident Year 2003:

(1) Given
Year
2003
(2) = Reported minus
Case Outstanding

Year
2003

Case Outstanding as of Months
12
16,000
Paid Claims (incremental)
12
4,000 = 20,000 - 16,000

(3) Given
Ratio: Paid Claims (incr) to PRIOR Case Outstanding
Final Ratio
Year
12
24
36
48 60 or Ult.
2003

n/a

Selected

50.0%
given

60.0%
given

48.0%
given

105%
given

CAREFUL: These ratios apply to Case Outstanding, so we also need the
Case Outstanding projections before we can actually use these ratios.
(4) Given
Ratio: Case Outstanding to PRIOR Case Outstanding
Year
12
24
36
48
2003

0 at Ult.
60 or Ult.

n/a

Selected

80.0%
given

70.0%
given

55.0%
given

0%
definition

0 at Ult.
60 or Ult.

8,960

4,928

(5) = (1) & projected, where projected = (4) * (5)prior
Compute the Case Outstanding Projections
Year
12
24
2003

16,000

12,800

NOW: We can use the ratios we found in step (3), and
project out the future INCREMENTAL claim payments

Exam 5, V2

Page 269

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Chapter12 – Case Outstanding Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
(6) = (2) & projected, where projected = (3)* (5)prior
Compute the Incremental Paid Projections
Year
12
24
2003

4,000

8,000

Final Paid
60 or Ult.

7,680

4,301

5,174

(7) = Sum across (6)
Total Estimate of Ultimate Claims using Case Outstanding Dev. Method
4,000
+ 8,000
+ 7,680
+ 4,301
For 2003:
+ 5,174
29,155
Answer: E

Solutions to questions from the 2005 Exam (modified):
22. Using the Case Outstanding Development Technique, calculate the estimated Total Unpaid Claims.
(1) GIVEN
Report
Year

2000
2001
2002
2003
2004

42,000
45,000
44,000
45,000
39,000

Report
Year

2000
2001
2002
2003
2004

22,000
24,000
25,000
27,000
24,000

Case Outstanding as of Months
24
36
48
29,000
33,000
30,000
32,000

16,000
19,000
18,000

8,000
9,000

4,000

9,000
10,000

4,000

(2) GIVEN
Paid Claims (incremental)
24
36
16,000
17,000
19,000
18,000

13,000
14,000
15,000

(3) = (2) / (1)prior
Ratio: Paid Claims (incr) to PRIOR Case Outstanding
Year
12
24
36
2000
2001
2002
2003
2004

n/a
n/a
n/a
n/a
n/a

Selected
Three-Year Simple Averages

Final Ratio
72

38.1%
37.8%
43.2%
40.0%

44.8%
42.4%
50.0%

56.3%
52.6%

50.0%

39.8%

45.8%

54.4%

50%

100%

Final Paid
72

54.4%

50.0%

100.0%

(4) = (3) & projections
Complete the square: Ratios with Incremental Paids
Year
12
24
36
2004

n/a

39.8%

45.8%

CAREFUL: These ratios apply to Case Outstanding, so we also need the
Case Outstanding projections before we can actually use these ratios.

Exam 5, V2

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Chapter12 – Case Outstanding Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
(5) = (1) / (1)prior
Ratio: Case Outstanding to PRIOR Case Outstanding
Year
12
24
36
2000
2001
2002
2003
2004

n/a
n/a
n/a
n/a
n/a

Selected
Three-Year Simple Averages

0 at Ult.
72 or Ult.

69.0%
73.3%
68.2%
71.1%

55.2%
57.6%
60.0%

50.0%
47.4%

50.0%

70.4%

57.6%

48.7%

50%

0 at Ult.
72 or Ult.

48.7%

50.0%

0.0%

0%
definition

(6) = (5) & projections (2004 only)
Complete the square: Ratios Case Outstanding / Prior
Year
12
24
36
2004

n/a

70.4%

(7) = (1) & projections (2004 only)

57.6%

where projections = (6) * (7)prior

Complete the square: Case Outstanding & projections
Year
12
24
36
2004

39,000

27,463

15,814

0 at Ult.
72 or Ult.

7,699

3,849

Be sure to go out until the is no more case outstanding.
Otherwise, the cumulative paid will not equal ultimate claims.
(8) = (2) & projected (2004 only)

where projections = (4) * (7)prior

Complete the square: Incremental Paid & projections
Year
12
24
36

Final Paid
72 or Ult.

8,609

3,849

Estimated Unpaid Claims = Sum of all Projected Future Payments
Year
24
36
48
60 72 or Ult.
15,508
12,565
8,609
3,849
3,849
2004

Unpaid
Total
= 44,381

2004

24,000

15,508

12,565

(9) = Sum across (8) for Unpaid Only (projections) for 2004

Exam 5, V2

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Chapter12 – Case Outstanding Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to questions from the 2010 Exam:
2a. (3.5 points) Use Friedland's case outstanding development technique Approach #1 to estimate the paid
loss for accident years 2006 through 2009 as of 48 months.
2b. (0.5 point) Explain whether the case outstanding development technique is generally more suitable for
accident year or report year analysis.
Question 2 – Model Solution
Step 1: Compute Case Outstanding (O/S) on Previous O/S
AY
12 - 24
24 - 36
36 - 48
2006
1.20
0.97
.79=3565/4500
2007
1.16
0.94
2008
1.26
Avg
1.21
0.96
0.79
Step 2: Compute Incremental Paid Claims
AY
12
24
36
2006
1,520
1,980
2,950
2007
2,150
1,610
3,000
2008
1,790
1,600 =3390-1790
2009
2,000

48
3,500

Step 3: Compute Incr. Paid Claims to Previous Case O/S
AY
12 - 24
24 - 36
36 - 48
2006
0.51
0.64
0.78
2007
0.4
0.64 =3000/4680
2008
0.39
Avg
0.43
0.64
0.78
Step 4: Compute Projected Case Outstanding
AY
12
24
36
48
2006
2007
2008
5,020 =5230 * 0.96
2009
5,203
4,994 =5203 * 0.96
Step 5: Projected Incremental Paid Claims
AY
12
24
36
48
2006
2007
3,424
2008
3,347
3,916
2009
1,849
3,330
3,896
=5203 * 0.64
Paid losses as of 48 Mos
AY
2006
9,950 (given)
2007
10,184 =6760 + 3424
2008
10,653 =3390 + 3347 + 3916
2009
11,075 =2000 + 1849 + 3330 +3896

Exam 5, V2

Page 272

=5020 * 0.78
=4994 * 0.78

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Chapter12 – Case Outstanding Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
b. Report year since there is no Pure IBNR.
b. It's more suitable for reported year since it is more suitable when most losses are reported during the 1st year.
Solutions to questions from the 2011 Exam:
29a. (1 point) Use the case outstanding development technique to estimate the ultimate claims for AY 2009.
29b. (0.25 point) Briefly describe when the case outstanding development technique is appropriate.
Question 29 – Model Solution
Intial comments:
AY 09 ultimate claims = AY 09 incremental paid claims at 12mos + AY 09 incremental paid claims at 24 mos
+ AY 09 incremental paid claims at 36mos + AY 09 incremental paid claims at 48 mos
AY 09 incremental paid claims at 36mos = AY09 Case O/S at 24 mos * Selected ratio of incremental paid
claims to previous case outstanding at 36mos
AY 09 Case O/S at 36mos = AY09 Case O/S at 24 mos * Selected ratio of case O/S to to previous case O/S at 36mos
AY 09 incremental paid claims at 48mos = AY09 Case O/S at 36 mos * Selected ratio of incremental paid claims to
previous case outstanding at 48mos
No further closed claim development after 48 months is assumed.
AY 09 incremental paid claims at 36mos = 1350 x .6 = 810
AY 09 Case O/S at 36mos = 1350 x .4 = 540
AY 09 incremental paid claims at 48mos = 540 x 1.05 = 567
a. ULT claims = 1700 + 1200 + 810 + 567 = 4277
b. Usually used with claims made analysis since there is no pure IBNR
Question 29 – Model Solution 1 – Part b
b. When all you have available is case o/s data, for example due to the acquisition of another company.
Question 29 – Model Solution 2 – Part b
b. When the IBNR is consistently related to reported claims
Question 29 – Model Solution 3 – Part b
b. Appropriate when most of the claims are reported in the first AY period.

Exam 5, V2

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Chapter12 – Case Outstanding Technique
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to questions from the 2012 Exam:
19a. (1.5 points) Use the case outstanding development technique to estimate unpaid claims for
accident year 2011.
19b.(0.5 point) Briefly describe two assumptions of this technique.
Question 19 – Model Solution 1 (Exam 5B Question 4)
a.
Projected case outstanding for 2011 (000s)
12
24
1485
AY 2011
1650
=1650 x 0.9
Paid Claims (000’s)
12
AY 2011
1800

24
1320
=1650 x 0.8

36
445.5 = 1485 x 0.3

36
891
=1485 x 0.6

48
467.78
= 445.5 x 1.05

Unpaid claims for AY 2011 = 1320 + 891 + 467.78 = 2678.78
b. –reported claims to date will continue to develop in a similar manner in future
-IBNR related to claims is consistently related to claims already reported.
Question 19 – Model Solution 2 (Exam 5B Question 4)
•

Selected ratio of case outstanding to previous case outstanding:
24 Months
36 Months 48 Months
0.90

•

0.30

0.00

Selected ratio of incremental paid claims to previous case outstanding:
24 Months

36 Months

48 Months

0.80

0.60

1.05

Case outstanding 12 x ratio to paid 24 + case 24 x ratio 36 + case36 x ratio 48
a. 1000 x 1650 (0.8+0.9(0.6+0.3(1.05))=2678.775 x 1000 = 2,678,775
b. (i) stable payment or claim settlement patterns
(ii) stable case reserving level
Examiner’s Comments
a. This part was generally well-answered. Some candidates incorrectly gave the projected ultimate (not
unpaid) claims as the answer. Some candidates incorrectly calculated the project unpaid claims for
all three years, not just AY 2011.
b. Candidates came up with a wide variety of answers to this question. The candidates did not score the
full credits if their answers were too vague or inaccurate. No credit if a candidate used the common
uses of the method (such as for lines of insurance for which most of the claims are reported in the first
accident period or for claims-made coverages) as the answer since the question was asking the
assumptions not the common uses of the method.

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Sec
1
2

Pages
283
283 - 284

3
4

Description
Introduction
Reacting to a Changing Environment through Data Selection and
Rearrangement
Treat Problem Areas through Data Adjustment
XYZ Insurer

Introduction

283

284 - 290
291 - 293

Berquist and Sherman developed a methodical actuarial approach for analyzing unpaid claims for insurers
who had undergone changes in operations and procedures.
They present two alternatives for the actuary in addressing such situations:
* Treat problem areas through data selection and rearrangement
* Treat problem areas through data adjustment

Reacting to a Changing Environment through Data Selection and
Rearrangement

283 - 284

Berquist and Sherman (B/S) recommend using data that is unaffected by changes in the insurer's claims and
underwriting procedures and operations.
Example:
If the insurer has changed its methods in establishing open case reserves, then the actuary may place
greater reliance on paid claims methods that will be unaffected by the changes in case O/S.
B/S suggests several ways for selecting alternative data to respond to potential problems related to a changing
environment:
*
Using earned exposures instead of the number of claims when claim count data is of questionable
accuracy or if there has been a major change in the definition of a claim count.
*
Substituting policy year data for accident year data when there has been a significant change in
policy limits or deductibles between successive policy years.
*
Substituting report year data for accident year data when there has been a dramatic shift in the social
or legal climate that causes claim severity to more closely correlate with the report year than with the
accident date.
*
Substituting accident quarter for accident year when the rate of growth of earned exposures changes
markedly, causing distortions in development factors due to significant shifts in the average accident
date within each exposure period.
Other adjustments that can be made to the data:
*
Divide the data into more homogeneous groups, which is valuable when there have been changes in
the composition of business by jurisdiction, coverage, class, territory, or size of risk.
When dividing the data into more homogeneous groups, retain sufficient volume of experience within
each group to ensure the data is credible.
*
Group claims data by size of the claim (see B/S paper).
A shift in emphasis by the claims department to settle large claims versus small claims is an operational
change that could affect many types of data used for estimating unpaid claims.

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Examples:
Greater attention to large claims could result in a slowdown in the rate of total claim settlements.
It may also speed-up the settlement of larger claims affecting both paid claims and case O/S triangles; if
large claims are settled earlier, then case O/S will no longer be present in the triangle at the later maturities
and the payments will appear in the triangles at earlier maturities than in the past.
Also, without appropriate monitoring, smaller claims may become larger claims more quickly than past
experience suggests.

Treat Problem Areas through Data Adjustment

284 - 290

B/S discuss 2 data adjustment techniques prior to applying traditional development methods.
The same examples discussed in the 1977 B/S paper and described below.
1. A portfolio of U.S. medical malpractice insurance for an experience period of 1969 to 1976 (Berq-Sher
Med Mal Insurer).
2. A portfolio of auto bodily injury liability also for an experience period of 1969 to 1976 (Berq-Sher Auto BI
Insurer).
1. Detecting Changes in the Adequacy Level of Case O/S and Reducing the Affect of Such Changes on
Reported Claims Projections
Exhibit I, Sheets 1 - 10: The analysis for Berq-Sher Med Mal Insurer
Exhibit I, Sheet 1: The unadjusted reported claim development triangle, in which B/S uses a simple
average for all years to project ultimate claims.
Chapter 13 - Berquist-Sherman Techniques
Berq-Sher Med Mal Insurer
Unadjusted Reported Claims
PART 1 - Data Triangle
Accident
Year
12
1969
2,897,000
1970
4,828,000
1971
5,455,000
1972
8,732,000
1973
11,228,000
1974
8,706,000
1975
12,928,000
1976
15,791,000
PART 2 - Age-to-Age Factors
Accident
Year
12 - 24
1969
1.781
1970
2.218
1971
2.189
1972
2.134
1973
1.778
1974
3.843
1975
3.783

Exhibit I
Sheet1

Projected Incremental Closed Claim Counts

24
5,160,000
10,707,000
11,941,000
18,633,000
19,967,000
33,459,000
48,904,000

24 - 36
2.076
1.579
1.736
1.725
2.511
1.897

PART 3 - Average Age-to-Age Factors
Accident
Year
12 - 24
24 - 36
Simple Average
All Years
2.532
1.921

36
10,714,000
16,907,000
20,733,000
32,143,000
50,143,000
63,477,000

48
15,228,000
22,840,000
30,928,000
57,196,000
73,733,000

Age-to-Age Factors
36 - 48
48 - 60
1.421
1.091
1.351
1.148
1.492
1.371
1.779
1.069
1.470

36 - 48
1.503

Averages
48 - 60

60
16,611,000
26,211,000
42,395,000
61,163,000

72
20,899,000
31,970,000
48,377,000

84
22,892,000
32,216,000

96
23,506,000

60 - 72
1.258
1.220
1.141

72 - 84
1.095
1.008

84 - 96
1.027

To Ult

60 - 72

72 - 84

84 - 96

To Ult

1.170

1.206

1.052

1.027

PART 4 - Selected Age-to-Age Factors

Selected
CDF to Ultimate
Percent Reported

Exam 5, V2

12 - 24
2.532
11.145
9.0%

24 - 36
1.921
4.402
22.7%

36 - 48
1.503
2.291
43.6%

Development Factor Selection
48 - 60
60 - 72
1.170
1.206
1.524
1.303
65.6%
76.7%

Page 276

72 - 84
1.052
1.080
92.6%

84 - 96
1.027
1.027
97.4%

To Ult
1.000
1.000
100.0%

 2014 by All 10, Inc.

Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Exhibit I, Sheet 2: The unadjusted paid claim triangle, in which B/S uses the volume-weighted average
for all years to project ultimate claims.
Chapter 13 - Berquist-Sherman Techniques
Berq-Sher Med Mal Insurer
Unadjusted Paid Claims
PART 1 - Data Triangle
Accident
Year
12
1,969
125,000
1,970
43,000
1,971
295,000
1,972
50,000
1,973
213,000
1,974
172,000
1,975
210,000
1,976
209,000
PART 2 - Age-to-Age Factors
Accident
Year
12 - 24
1969
3.248
1970
12.302
1971
3.888
1972
15.720
1973
3.911
1974
9.227
1975
7.452

Exhibit I
Sheet 2

Unadjusted Paid Claims as of (months)

24
406,000
529,000
1,147,000
786,000
833,000
1,587,000
1,565,000

36
1,443,000
2,016,000
2,479,000
3,810,000
3,599,000
6,267,000

48
2,986,000
3,641,000
5,071,000
9,771,000
11,292,000

24 - 36
3.554
3.811
2.161
4.847
4.321
3.949

36 - 48
2.069
1.806
2.046
2.565
3.138

48 - 60
1.496
2.066
2.248
1.895

24 - 36

36 - 48

48 - 60

3.709

2.455

1.952

60
4,467,000
7,523,000
11,399,000
18,518,000

72
8,179,000
14,295,000
17,707,000

84
12,638,000
18,983,000

96
15,815,000

72 - 84
1.545
1.328

84 - 96
1.251

To Ult

72 - 84

84 - 96

To Ult

1.407

1.251

72 - 84
1.407
2.616
38.2%

84 - 96
1.251
1.859
53.8%

Age-to-Age Factors

60 - 72
1.831
1.900
1.553

PART 3 - Average Age-to-Age Factors
12 - 24
Volume-weighted Average
All Years
6.185

Averages
60 - 72
1.718

PART 4 - Selected Age-to-Age Factors

Selected
CDF to Ultimate
Percent Paid

12 - 24
6.185
493.993
0.2%

24 - 36
3.709
79.870
1.3%

36 - 48
2.455
21.534
4.6%

Development Factor Selection
48 - 60
60 - 72
1.952
1.718
8.771
4.494
11.4%
22.3%

To Ult
1.486
1.486
67.3%

Exhibit I, Sheet 3: Project the unadjusted reported and unadjusted paid claims to an ultimate basis.
Chapter 13 - Berquist-Sherman Techniques
Berq-Sher Med Mal Insurer
Projection of Ultimate Claims Using Development Technique and Unadjusted Data

Accident
Year
(1)
1969
1970
1971
1972
1973
1974
1975
1976
Total

Age of
Acccident Year
Claims at 12/31/76
at 12/31/76 Reported
Paid
(2)
(3)
(4)
96
23,506,000
15,815,000
84
32,216,000
18,983,000
72
48,377,000
17,707,000
60
61,163,000
18,518,000
48
73,733,000
11,292,000
36
63,477,000
6,267,000
24
48,904,000
1,565,000
209,000
12
15,791,000
367,167,000
90,356,000

CDF to Ultimate
Reported
Paid
(5)
(6)
1.000
1.486
1.027
1.860
1.080
2.616
1.303
4.495
1.524
8.774
2.291
21.536
4.402
79.880
11.145
494.058

Exhibit I
Sheet 3

Projected ultimate Claims
Using Dev. Method with
Reported
Paid
(7) = [(3) x (5)] (8) = [(4) x (6)]]
23,506,000
23,501,090
33,085,832
35,308,380
52,247,160
46,321,512
79,695,389
83,238,410
112,369,092
99,076,008
145,425,807
134,966,112
215,275,408
125,012,200
103,258,122
175,990,695
837,595,383
650,681,834

Column Notes:
(2) Age of accident year in (1) at December 31, 1976.
(3) and (4) Based on data from Berq-Sher Med Mal Insurer.
(5) and (6) Based on CDF from Exhibit I, Sheets 1 and 2.
(7) = [(3) x (5)].
(8) = [(4) x (6)].

Significant differences in these projections exist by AY and in total. The paid claim development method
is shown for demonstration purposes, and is not a reliable projection method due to the highly leveraged
CDFs for most AYs in the experience period.

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
1a. Testing the Assumptions of the Reported Claim Development Technique
Assume that the adequacy of the case O/S has not been changing over time. However if it has, then the
fundamental assumption of the development method does not hold and the method will not produce reliable
results of ultimate claims or unpaid claims.
Approaches to determine if an insurer has sustained changes in case O/S adequacy:
1. Meet with the claims department management to discuss the claims process
2. Calculate various claim development diagnostic tests, including: the ratio of paid-to-reported claims,
average case O/S, average reported claim, and average paid claims.
In their medical malpractice example, B/S compares the annual change in the average case O/S to
the annual change in the average paid claims to determine a shift in case O/S adequacy.
Begin testing the underlying assumptions in Exhibit I, Sheet 4 with a review of the average case O/S triangle.
 Average case O/S triangle is the unadjusted case O/S divided by the open claim counts
 Look down each column, the two latest points are significantly higher than the preceding values at
each maturity age (i.e., the latest two diagonals are higher than prior diagonals).
At 24 months, the average case O/S values for the last two AYs are $22,477 and $32,160 compared
to $13,785 and $11,433 for the preceding two AYs.
Chapter 13 - Berquist-Sherman Techniques
Berq-Sher Med Mal Insurer
Development Triangle - Unadjusted Data
Accident
Year
12
24
1969
3,701
5,660
1970
7,250
10,635
1971
5,877
8,122
1972
8,324
11,433
1973
10,124
13,785
1974
8,261
22,477
1975
11,176
32,160
1976
13,028

Exhibit I
Sheet4
Unadjusted Average Case Outstanding as of (months)

36
9,262
12,960
10,613
15,499
30,223
34,402

48
10,151
14,221
14,373
25,040
33,266

60
11,745
17,067
21,706
28,019

Annual Change based on Exponential Regression Analysis of Severities and Accident Year
15.62%
29.50%
31.11%
34.17%
32.96%
Goodness of Fit Test of Exponential Regression Analysis (R-Squared)
79.96%
89.46%
85.79%
94.05%
98.88%

72
16,627
23,411
29,044

84
19,238
24,551

32.16%

27.62%

98.31%

100.00%

96
21,423

Exponential regression is used to determine the annual trend rate in the average case O/S at each age.
 The average case O/S is fit at each maturity age with the AY.
 The fitted trend rate and the R-squared test (goodness of fit) for each age is shown.
 Annual trend rates of 30% for maturity ages 24 months through 72 months with R-squared values
of 85% or greater for all of these ages.
Testing the Assumptions of the Reported Claim Development Technique (continued):
Exhibit I, Sheet 5: Ratios of paid-to-reported claims and trend rates in the average paid claim triangle.
 If there has been an increase in the case O/S adequacy level, the ratios of paid-to-reported claims
should be decreasing along the latest two diagonals of the triangle.
 Some decreases are seen in this ratio triangle, but there is variability and it is hard to draw definitive
conclusions based on this diagnostic.
The test that B/S uses is to compare annual trend rates, using regression, of the average case O/S and
the average paid claims on closed counts.
The paid claim triangle can be used with the closed claim counts triangle to approximate the average
paid claims on closed counts (since partial payments are not common in Med Mal).

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Chapter 13 - Berquist-Sherman Techniques
Berq-Sher Med Mal Insurer
Development Triangles - Unadjusted Data
Accident
Year
12
24
1969
402
539
1970
110
919
1971
706
1,115
1972
161
862
1973
724
541
1974
518
1,394
1975
517
1,494
1976
525

Exhibit I
Sheet 5
Unadjusted Average Paid Claims as of (months)
36
48
60
2,971
8,620
9,199
5,487
9,129
12,403
5,644
4,928
12,994
5,782
9,477
14,085
4,003
11,709
7,635

Annual Change based on Exponential Regression Analysis of Severities and Accident Year
12.89%
11.98%
11.46%
6.72%
Goodness of Fit Test of Exponential Regression Analysis (R-Squared)
18.31%
35.27%
37.88%
10.14%

72
12,669
18,452
14,948

84
17,084
19,533

14.16%

8.62%

14.34%

84.57%

19.26%

100.00%

96
16,634

A comparison of annual rates of change between average case O/S and average paid claims:
* The annual trend rate appears to be 30% based on a review of the average case O/S triangle;
* Annual trend rate indications range from 7% to 14% using the average paid claim triangle.
B/S note that the trends for average paid claims are similar to industry benchmarks (at the time), and thus
they conclude that the higher trends for average case O/S are indicative of changes in case O/S adequacy.
Mechanics of the Berquist-Sherman Case O/S Adjustment
Two decisions requiring actuarial judgment must be made by the actuary:
1. Choose a diagonal from which all other values of the adjusted average case O/S triangle will be calculated.
The most common choice is the latest diagonal of the average case O/S triangle, since the latest diagonal
of the adjusted reported claim triangle will not change from the unadjusted data triangle.
2. Select an annual severity trend to adjust average case O/S values from the selected diagonal
B/S selected the latest diagonal as the starting point and a 15% annual severity trend.
Exhibit I, Sheet 6: Derivation of Adjusted Reported Claim Development Triangle
Top section: Adjusted average case O/S triangle.
Chapter 13 - Berquist-Sherman Techniques
Berq-Sher Med Mal Insurer
Derivation of Adjusted Reported Claim Development Triangle
Accident
Year
1969
1970
1971
1972
1973
1974
1975
1976




12
4,898
5,633
6,477
7,449
8,566
9,851
11,329
13,028

24
13,904
15,989
18,387
21,145
24,317
27,965
32,160

Exhibit I
Sheet 6

Adjusted Average Case Outstanding as of (months)
36
48
60
72
17,104
19,020
18,423
21,961
19,669
21,873
21,186
25,255
22,620
25,154
24,364
29,044
26,013
28,927
28,019
29,915
33,266
34,402

84
21,349
24,551

96
21,423

The last diagonal is the same as the one from the unadjusted average case O/S triangle (E1S4)
All other values are determined by de-trending from the latest diagonal. The calculations within each
column start with the latest point and the selected severity trend rate. Examples:
1
The 1975 adjusted average case O/S at 12 months is $11,329 ( = $13,028/1.15 ), representing 1 year
of trend.
The 1970 adjusted average case O/S at 48 months is $21,873 based on the 1973 average case O/S
3
of $33,266/1.15

The purpose of restating the average case O/S triangle is to have each diagonal in the triangle at the same
case O/S adequacy level as the latest diagonal (i.e. latest valuation).

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Exhibit I, Sheet 6: Derivation of Adjusted Reported Claim Development Triangle
Adjusted reported claims = adjusted average case O/S * number of open claims + unadjusted paid claims.
Accident
Year
1969
1970
1971
1972
1973
1974
1975
1976

12
3,793,504
3,760,482
5,982,185
7,819,355
9,533,246
10,348,458
13,102,479
15,791,000

24
12,084,942
15,830,500
25,583,831
33,794,110
34,585,431
41,241,243
48,904,000

Adjusted Reported Claims as of (months)
36
48
60
18,563,821
25,924,316
23,516,364
24,615,996
33,169,802
30,722,141
41,384,825
50,323,342
46,191,356
64559286
51,361,061
61,163,000
49,667,342
73,733,000
63,477,000

72
24,979,245
33,362,729
48,377,000

84
24,016,864
32,216,000

96
23,506,000

Exhibit I, Sheet 7: Adjusted reported claim triangle and development factor selections
Chapter 13 - Berquist-Sherman Techniques
Berq-Sher Med Mal Insurer
Adjusted Reported Claims

Exhibit I
Sheet 7

PART 1 - Data Triangle
Accident
Year
1969
1970
1971
1972
1973
1974
1975
1976

12
3,793,504
3,760,482
5,982,185
7,819,355
9,533,246
10,348,458
13,102,479
15,791,000

24
12,084,942
15,830,500
25,583,831
33,794,110
34,585,431
41,241,243
48,904,000

PART 2 - Age-to-Age Factors
Accident
Year
12 - 24
1969
3.186
1970
4.210
1971
4.277
1972
4.322
1973
3.628
1974
3.985
1975
3.732

Adjusted Reported Claims as of (months)
36
48
60
72
18,563,821
25,924,316
23,516,364
24,979,245
24,615,996
33,169,802
30,722,141
33,362,729
41,384,825
50,323,342
46,191,356
48,377,000
51,361,061
64,559,286
61,163,000
49,667,342
73,733,000
63,477,000

24 - 36
1.536
1.555
1.618
1.520
1.436
1.539

36 - 48
1.396
1.347
1.216
1.257
1.485

12 - 24

24 - 36

36 - 48

3.906

1.534

1.340

24 - 36
1.921
1.534
1.911
52.33%

36 - 48
1.503
1.340
1.246
80.27%

Age-to-Age Factors
48 - 60
60 - 72
0.907
1.062
0.926
1.086
0.918
1.047
0.947

84
24,016,864
32,216,000

96
23,506,000

72 - 84
0.961
0.966

84 - 96
0.979

To Ult

72 - 84

84 - 96

To Ult

0.964

0.979

72 - 84
1.052
0.964
0.944
105.96%

84 - 96
1.027
0.979
0.979
102.15%

PART 3 - Average Age-to-Age Factors

Simple Average
All Years

Averages
48 - 60
60 - 72
0.925

1.065

PART 4 - Selected Age-to-Age Factors

Unadj Selected
Adj Selected
CDF to Ultimate
Percent Reported

12 - 24
2.532
3.906
7.465
13.40%

Development Factor Selection
48 - 60
60 - 72
1.170
1.206
0.925
1.065
0.930
1.005
107.56%
99.49%

96- 108
1.000
1.000
1.000
100.00%

The selected development factors are lower based on adjusted data than on unadjusted data for all age-toage maturities except 12-to-24 months.
This is consistent with the belief that the case O/S adequacy had increased and an unadjusted reported
claim development projection would overstate future claim development.

Exam 5, V2

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Exhibit I, Sheet 8: Projection of Ultimate Claims Using Development Technique and Adjusted Data
Chapter 13 - Berquist-Sherman Techniques
Berq-Sher Med Mal Insurer
Projection of Ultimate Claims Using Development Technique and Adjusted Data

Claims at 12/31/76
Age of
Accident Accident Year
Adjusted
Year
at 12/31/76
Reported
Paid
Reported
(1)
(2)
(3)
(4)
(5)
1969
96
23,506,000
15,815,000
23,506,000
1970
84
32,216,000
18,983,000
32,216,000
1971
72
48,377,000
17,707,000
48,377,000
1972
60
61,163,000
18,518,000
61,163,000
1973
48
73,733,000
11,292,000
73,733,000
1974
36
63,477,000
6,267,000
63,477,000
1975
24
48,904,000
1,565,000
48,904,000
209,000
15,791,000
1976
12
15,791,000
Total
367,167,000
90,356,000
367,167,000

Exhibit I
Sheet 8

Projected ultimate Claims
Using Dev. Method with
CDF to Ultimate
Adjusted
Adjusted
Reported
Paid
Reported
Reported
Paid
Reported
(6)
(7)
(8)
(9) = [(3)x(6)] (10)=[(4)x(7)] (11)=[(5)x(8)]
1.000
1.486
1.000
23,506,000
23,501,090
23,506,000
1.027
1.859
0.979
33,085,832
35,289,131
31,539,464
1.080
2.616
0.944
52,266,704
46,321,512
45,656,084
1.303
4.494
1.005
79,693,384
83,222,528
61,474,940
1.524
8.773
0.930
112,403,868
99,064,716
68,550,870
2.291
21.538
1.246
145,443,637 134,976,860 79,081,019
4.402
79.887
1.911
215,253,430 125,022,648 93,459,963
11.145
494.099
7.465
175,986,370 103,266,710 117,875,378
837,639,227 650,665,195 521,143,718

Column Notes:
(2) Age of accident year in (1) at December 31, 1976.
(3) and (4) Based on data from Berq-Sher Med Mal Insurer.
(5) Developed in Exhibit I, Sheet 6.
(6) and (7) Based on CDF from Exhibit I, Sheets 1 and 2.
(8) Based on CDF from Exhibit I, Sheet 7.




Claim development projections based on unadjusted reported, paid claims and adjusted reported
claims are computed.
Projected ultimate claims based on the adjusted reported claim triangle are significantly less than the
ultimate claims produced by the unadjusted data.

Exhibit I, Sheet 9: Estimated IBNR and the total unpaid claim estimate using all three projection methods
Chapter 13 - Berquist-Sherman Techniques
Berq-Sher Med Mal Insurer
Development of Unpaid Claim Estimate

Accident
Year
(1)
1969
1970
1971
1972
1973
1974
1975
1976
Total

Claims at
Reported
(2)
23,506,000
32,216,000
48,377,000
61,163,000
73,733,000
63,477,000
48,904,000
15,791,000
367,167,000

12/31/76
Paid
(3)
15,815,000
18,983,000
17,707,000
18,518,000
11,292,000
6,267,000
1,565,000
209,000
90,356,000

Exhibit I
Sheet 9

Projected Ultimate Claims
Using Dev. Method with
Adjusted
Reported
Paid
Reported
(4)
(5)
(6)
23,506,000 23,501,090
23,506,000
33,085,832 35,289,131
31,539,464
52,266,704 46,321,512
45,656,084
79,693,384 83,222,528
61,474,940
112,403,868 99,064,716
68,550,870
145,443,637 134,976,860 79,081,019
215,253,430 125,022,648 93,459,963
175,986,370 103,266,710 117,875,378
837,639,227 650,665,195 521,143,718

Case
Outstanding
at 12/31/76
(7)
7,691,000
13,233,000
30,670,000
42,645,000
62,441,000
57,210,000
47,339,000
15,582,000
276,811,000

Unpaid Claim Estimate at 12/31/76
IBNR - Based on Dev. Method with
Total - based on Dev. Method with
Adjusted
Adjusted
Reported
Paid
Reported
Reported
Paid
Reported
(8)
(9)
(10)
(11)
(12)
(13)
0
-4,910
0
7,691,000
7,686,090
7,691,000
869,832
3,073,131
-676,536
14,102,832
16,306,131
12,556,464
3,889,704
-2,055,488
-2,720,916
34,559,704
28,614,512
27,949,084
18,530,384
22,059,528
311,940
61,175,384
64,704,528
42,956,940
38,670,868
25,331,716
-5,182,130 101,111,868
87,772,716
57,258,870
81,966,637
71,499,860
15,604,019 139,176,637
128,709,860
72,814,019
166,349,430
76,118,648
44,555,963 213,688,430
123,457,648
91,894,963
160,195,370
87,475,710 102,084,378 175,777,370
103,057,710
117,666,378
470,472,227 283,498,195 153,976,718 747,283,227
560,309,195
430,787,718

Column Notes:
(2) and (3) Based on data from Berq-Sher Med Mal Insurer.
(4) through (6) Developed in Exhibit I, Sheet 8.
(7) = [(2) - (3)].
(8) = [(4) - (2)].
(9) = [(5) - (2)].
(10) = [(6) - (2)].
(11) = [(7) + (8)].
(12) = [(7) + (9).
(13) = [(7) + (10)].

These amounts are summarized in the following table.
Claims Data Type
Unadjusted Reported
Unadjusted Paid
Adjusted Reported

Estimated IBNR
Total All Years ($ millions)
47
28
15

Total Unpaid Claim Estimate
Total All Years ($ millions)
74
56
43

Conclusion: The dramatically different results suggest that alternative estimation methods be used and
additional information be obtained to determine the most appropriate estimate of unpaid claims.
(See Chapter 15 — Evaluation of Techniques for further discussion.)

Exam 5, V2

Page 281

 2014 by All 10, Inc.

Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Potential Difficulty with the Adjustment
In Thorne’s review of the B/S paper, he states that the estimation of the underlying trend in severity requires a
great deal of care, due to the:
* the sensitivity of the reserve estimates to the selected rate, and
* the substantial judgment needed in selecting the rate
Estimating severity trends for Med Mal is complicated by the following factors:
* Slow claims payments reduces data available by AY (e.g. less than 3% of ultimate claims are paid
during the first 24 months and less than 30% during the first 60 months)
* Severity trends are distorted by irregular settlements and variation in the rate of claims closed without
payment.

2. Detecting Changes in the Rate of Settlement of Claims and Adjusting Paid Claims for Such
Changes
Exhibit II, Sheets 1 - 10: The analysis for Berq-Sher Auto BI Insurer
Exhibit II, Sheet 1: The unadjusted paid claim development triangle, in which B/S uses a volume-weighted
average for all years to project ultimate claims.
Test the data to determine if the rate of claims settlement is consistent over the experience period (i.e. the
underlying assumption of the paid claim development technique)
Chapter 13 - Berquist-Sherman Techniques
Berq-Sher Auto BI Insurer
Unadjusted Paid Claims ($000)

PART 1 - Data Triangle
Accident
Year
12
1969
1,904
1970
2,235
1971
2,441
1972
2,503
1973
2,838
1974
2,405
1975
2,759
1976
2,801

24
5,398
6,261
7,348
8,173
8,712
7,858
9,182

PART 2 - Age-to-Age Factors
Accident
Year
12 - 24
24 - 36
1969
2.835
1.389
1970
2.801
1.388
1971
3.010
1.451
1972
3.265
1.445
1973
3.070
1.461
1974
3.267
1.498
1975
3.328

Exhibit II
Sheet 1

Paid Claims
36
7,496
8,691
10,662
11,810
12,728
11,771

36 - 48
1.185
1.202
1.187
1.200
1.200

as of (months)
48
60
8,882
9,712
10,443
11,346
12,655
13,748
14,176
15,383
15,278

72
10,071
11,754
14,235

84
10,199
12,031

96
10,256

72 - 84
1.013
1.024

84 - 96
1.006

To Ult

60 - 72

72 - 84

84 - 96

To Ult

Age-to-Age Factors
48 - 60
60 - 72
1.093
1.037
1.086
1.036
1.086
1.035
1.085

PART 3 - Average Age-to-Age Factors
12 - 24
Simple Average
All Years
3.082
Latest 4
3.233
Volume-weighted Average
All Years
3.098
Latest 4
3.229

Averages
48 - 60

24 - 36

36 - 48

1.439
1.464

1.195
1.197

1.088
1.088

1.036
1.036

1.018
1.018

1.006
1.006

1.444
1.464

1.196
1.197

1.087
1.087

1.036
1.036

1.019
1.019

1.006
1.006

PART 4 - Selected Age-to-Age Factors

Selected
CDF to Ultimate
Percent Reported

Exam 5, V2

12 - 24
3.098
6.170
16.2%

24 - 36
1.444
1.991
50.2%

36 - 48
1.196
1.380
72.5%

Development Factor Selection
48 - 60
60 - 72
72 - 84
1.087
1.036
1.019
1.154
1.062
1.025
86.7%
94.2%
97.6%

Page 282

84 - 96
1.006
1.006
99.4%

To Ult
1
1.000
100.0%

 2014 by All 10, Inc.

Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Exhibit II, Sheet 2: Closed and reported claim counts and the ratio of closed-to-reported claim counts.

Chapter 13 - Berquist-Sherman Techniques
Berq-Sher Auto BI Insurer
Development Triangles - Unadjusted Data




Accident
Year
1969
1970
1971
1972
1973
1974
1975
1976

12
4,079
4,429
4,914
4,497
4,419
3,486
3,516
3,230

24
6,616
7,230
8,174
7,842
7,665
6,214
6,226

36
7,192
7,899
9,068
8,747
8,659
6,916

Accident
Year
1969
1970
1971
1972
1973
1974
1975
1976

12
6,553
7,277
8,259
7,858
7,808
6,278
6,446
6,115

24
7,696
8,537
9,765
9,474
9,376
7,614
7,884

36
7,770
8,615
9,884
9,615
9,513
7,741

Accident
Year
1969
1970
1971
1972
1973
1974
1975
1976

12
0.622
0.609
0.595
0.572
0.566
0.555
0.545
0.528

24
0.860
0.847
0.837
0.828
0.818
0.816
0.790

36
0.926
0.917
0.917
0.910
0.910
0.893

Exhibit II
Sheet 2

Closed Claim Counts as of (months)
48
60
72
84
7,494
7,670
7,749
7,792
8,291
8,494
8,606
8,647
9,518
9,761
9,855
9,254
9,469
9,093

Reported Claim Counts as of (months)
48
60
72
84
7,799
7,814
7,819
7,820
8,661
8,675
8,679
8,682
9,926
9,940
9,945
9,664
9,680
9,562

96
7,806

96
7,821

Ratio of Closed to Reported Claim Counts as of (months)
48
60
72
84
96
0.961
0.982
0.991
0.996
0.998
0.957
0.979
0.992
0.996
0.959
0.982
0.991
0.958
0.978
0.951

Looking down each column of the ratio triangle, a steady decrease in the rate of claim settlement over
the experience period is seen.
Thus, the primary assumption of the paid claim development method does not hold, and the method
would likely understate the true value required for unpaid claims.

Exam 5, V2

Page 283

 2014 by All 10, Inc.

Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Mechanics of the Berquist-Sherman Paid Claim Development Adjustment
1. Determine the disposal rates by AY and maturity, where the definition of disposal rates is the same as that
used in the FS approach of Chapter 11.
Exhibit II, Sheets 3 and 4: Projected number of ultimate claims based on reported claim counts
Chapter 13 - Berquist-Sherman Techniques
Berq-Sher Auto BI Insurer
Reported Claim Counts
PART 1 - Data Triangle
Accident
Year
12
1969
6,553
1970
7,277
1971
8,259
1972
7,858
1973
7,808
1974
6,278
1975
6,446
1976
6,115

24
7,696
8,537
9,765
9,474
9,376
7,614
7,884

PART 2 - Age-to-Age Factors
Accident
Year
12 - 24
24 - 36
1969
1.174
1.010
1970
1.173
1.009
1971
1.182
1.012
1972
1.206
1.015
1973
1.201
1.015
1974
1.213
1.017
1975
1.223

Exhibit II
Sheet 3

Reported Claim Counts as of (Months)
36
48
60
72
7,770
7,799
7,814
7,819
8,615
8,661
8,675
8,679
9,884
9,926
9,940
9,945
9,615
9,664
9,680
9,513
9,562
7,741

Age-to-Age Factors
36 - 48
48 - 60
1.004
1.002
1.005
1.002
1.004
1.001
1.005
1.002
1.005

60 - 72
1.001
1.000
1.001

72 - 84
1.000
1.000

84
7,820
8,682

84 - 96
1.000

96
7,821

To Ult

PART 3 - Average Age-to-Age Factors
12 - 24
Simple Average
All Years

24 - 36

1.196

1.013

Averages
36 - 48
48 - 60
1.005

60 - 72

1.002

1.001

72 - 84
1.000

84 - 96

To Ult

1.000

PART 4 - Selected Age-to-Age Factors
12 - 24
Selected
1.196
CDF to Ultimate
1.221
Percent Reported
81.9%

24 - 36
1.013
1.021
97.9%

Development Factor Selection
36 - 48
48 - 60
60 - 72
1.005
1.002
1.001
1.008
1.003
1.001
99.2%
99.7%
99.9%

Exhibit II
Sheet 4

Berq-Sher Auto BI Insurer
Reported Claim Counts

Accident
Year
(1)
1969
1970
1971
1972
1973
1974
1975
1976
Total

Age of
Accident Year
at 12/31/76
(2)
96
84
72
60
48
36
24
12

72 - 84
84 - 96
96- 108
1.000
1.000
1.000
1.000
1.000
1.000
100.0%
100.0%
100.0%

Reported
Claim Counts
at 12/31/76
(3)
7,821
8,682
9,945
9,680
9,562
7,741
7,884
6,115
67,430

CDF
to Ultimate
(4)
1.000
1.000
1.000
1.001
1.003
1.008
1.021
1.221

Projected
Ultimate
Claim Counts
(5) = [(3) x (4)]
7,821
8,682
9,945
9,690
9,591
7,803
8,050
7,466
69,047

Column Notes:
(2) Age of accident year in (1) at December 31, 1976.
(3) Based on data from Berq-Sher Auto BI Insurer.
(4) Based on CDF from Exhibit II, Sheet 3.

Exam 5, V2

Page 284

 2014 by All 10, Inc.

Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Exhibit II, Sheet 5: Disposal Rate and Development of Adjusted Closed Claim Counts
Chapter 13 - Berquist-Sherman Techniques
Berq-Sher Auto BI Insurer
Disposal Rate and Development of Adjusted Closed Claim Counts
Accident
Year
1969
1970
1971
1972
1973
1974
1975
1976

12
0.522
0.510
0.494
0.464
0.461
0.447
0.437
0.433

24
0.846
0.833
0.822
0.809
0.799
0.796
0.773

36
0.920
0.910
0.912
0.903
0.903
0.886

Selected Disposal Rate by Maturity Age
0.433
0.773
0.886
Accident
Year
1969
1970
1971
1972
1973
1974
1975
1976

12
3,383
3,756
4,302
4,192
4,149
3,376
3,482
3,230

Disposal Rate as of (months)
48
60
72
0.958
0.981
0.991
0.955
0.978
0.991
0.957
0.981
0.991
0.955
0.977
0.948

0.948

0.977

0.991

Adjusted Closed Claim Counts as of (months)
24
36
48
60
72
6,049
6,932
7,415
7,643
7,750
6,715
7,695
8,231
8,484
8,603
7,692
8,815
9,429
9,719
9,855
7,495
8,588
9,187
9,469
7,418
8,501
9,093
6,035
6,916
6,226

Exhibit II
Sheet 5

84
0.996
0.996

96
0.998

0.996

0.998

84
7,789
8,647

96
7,806

Projected
Ultimate
Claim Counts
7,821
8,682
9,945
9,690
9,591
7,803
8,050
7,466

Disposal rate equals cumulative closed claim counts for each AY-maturity age cell/ultimate claim counts
for a given accident year.
The disposal rates show a decrease in the rate of claims settlement.
B/S select the claims disposal rate along the latest diagonal as the basis for adjusting the closed
claim count triangle, since the latest diagonal of the adjusted paid claim triangle will not change from
the unadjusted paid claim triangle.
2. Adjusted triangle of closed claim counts equal disposal rate for each maturity * the ultimate number of claims
Examples:
*
For AY 1974 at 12 months, 3,376 (adjusted closed claim counts) equal 0.433 (disposal rate at 12
months) * 7,803 (projected ultimate claim counts for AY 1974)
*
For AY 1971at 60 months, 9,719 (adjusted closed claim counts) equal 0.977 (disposal rate at 60
months) * 9,945 (projected ultimate claim counts for AY 1971)
*

Note, slight differences which exist between values in the text and values in the exhibits are due to the fact that the exhibits carry a
greater number of decimals than shown.

3. B/S then use regression to identify a formula that approximates the relationship between the cumulative
number of closed claims (X) and cumulative paid claims (Y).
(bX)
fits extremely well.
B/S find that a curve of the form Y= ae

Exam 5, V2

Page 285

 2014 by All 10, Inc.

Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Exhibit II, Sheet 6: Summary of Regression Analysis
Chapter 13 - Berquist-Sherman Techniques
Berq-Sher Auto BI Insurer
Summary of Regression Analysis
Accident Year 1969
Cumulative
Closed
Paid
Predicted
Y Value
Months Claim Counts Claims
Development
X
Y
Y=ae^(bX)
(1)
(2)
(3)
(4)
12
4,079
1,904
1,851
24
6,616
5,398
5,888
36
7,192
7,496
7,658
48
7,494
8,882
8,789
60
7,670
9,712
9,523
72
7,749
10,071
9,873
84
7,792
10,199
10,068
96
7,806
10,256
10,133
R Squared
a
b

Exhibit II
Sheet 6

Accident Year 1970
Cumulative
Closed
Paid
Predicted
Claim Counts Claims
Y Value
X
Y
Y=ae^(bX)
(5)
(6)
(7)
4,429
2,235
2,185
7,230
6,261
6,718
7,899
8,691
8,785
8,291
10,443
10,280
8,494
11,346
11,152
8,606
11,754
11,664
8,647
12,031
11,858

Accident Year 1971
Cumulative
Closed
Paid
Predicted
Claim Counts
Claims
Y Value
X
Y
Y=ae^(bX)
(5)
(6)
(7)
4,914
2,441
2,404
8,174
7,348
7,724
9,068
10,662
10,637
9,518
12,655
12,496
9,761
13,748
13,632
9,855
14,235
14,098

0.99709
369.851
0.000401

0.99866
414.005
0.000358

0.99573
287.918
0.000456



Show the results of the regression for the three oldest AYs (1969, 1970, and 1971), including the Rsquared value and the estimated a and b values.
 Since exponential curves closely approximate the relationship between cumulative closed claim
counts and cumulative paid claims, B/S suggest that fitting exponential curves for every pair of two
successive points is appropriate as the basis for adjusting paid claims.
Exhibit II, Sheet 7 (left side): Triangles for unadjusted closed claim counts, unadjusted paid claims, and
adjusted closed claim counts.
Exhibit II, Sheet 7 (right side): The estimated parameters a and b for all two-point exponential regressions
Chapter 13 - Berquist-Sherman Techniques
Berq-Sher Auto BI Insurer
Derivation of Adjusted Paid Claims

Exhibit II
Sheet 7

Accident
Closed Claim Counts as of (months
Year
12
24
36
48
60
72
84
96
1969
4,079 6,616 7,192 7,494 7,670
7,749
7,792
7,806
1970
4,429 7,230 7,899 8,291 8,494
8,606
8,647
1971
4,914 8,174 9,068 9,518 9,761
9,855
1972
4,497 7,842 8,747 9,254 9,469
1973
4,419 7,665 8,659 9,093
1974
3,486 6,214 6,916
1975
3,516 6,226
1976
3,230

Accident
Year
1969
1970
1971
1972
1973
1974
1975
1976

Paid Claims ($000) as of (months)
12
24
36
48
60
1,904 5,398 7,496 8,882 9,712
2,235 6,261 8,691 10,443 11,346
2,441 7,348 10,662 12,655 13,748
2,503 8,173 11,810 14,176 15,383
2,838 8,712 12,728 15,278
2,405 7,858 11,771
2,759 9,182
2,801

Accident
72
84
10,071 10,199
11,754 12,031
14,235

Accident
Year
1969
1970
1971
1972
1973
1974
1975
1976

96
10,256

Adjusted Closed Claim Counts as of (months)
12
24
36
48
60
72
84
96
3,383 6,048 6,932 7,415 7,643
7,750
7,789
7,806
3,755 6,714 7,695 8,231 8,484
8,603
8,647
4,301 7,691 8,814 9,429 9,719
9,855
4,191 7,494 8,588 9,187 9,469
4,148 7,417 8,500 9,093
3,375 6,035 6,916
3,482 6,226
3,230

Year
1969
1970
1971
1972
1973
1974
1975
1976

Accident
Year
1969
1970
1971
1972
1973
1974
1975
1976

24
356
438
464
510
616
530
580

24
0.000411
0.000368
0.000338
0.000354
0.000346
0.000434
0.000444

12
1,430
1,744
1,984
2,246
2,584
2,292
2,718
2,801

24
4,276
5,180
6,241
7,225
7,997
7,269
9,182

36
124
181
244
337
468
220

Parameter a for Two-Point Exponential Fit
48
60
72
84
132
198
286
1,034
215
353
778
88
337
493
370
506
421
333

96
459

Parameter b for Two-Point Exponential Fit
36
48
60
72
84
96
0.000570 0.000562 0.000508 0.000459 0.000294 0.000398
0.000490 0.000468 0.000409 0.000315 0.000568
0.000416 0.000381 0.000341 0.000370
0.000407 0.000360 0.000380
0.000381 0.000421
0.000576

Adjusted Paid Claims ($000) as of (months)
36
48
60
72
6,463
8,497
9,579
10,077
7,864
10,156
11,301
11,744
9,594
12,233
13,550
14,235
11,071
13,837
15,383
11,981
15,278
11,771

84
10,191
12,031

96
10,256

Example: The exponential regression for AY 1969 between ages 12 and 24, such that X= (4,079; 6,616) and
Y= (1,904; 5,398), would result in a = 356 and b = 0.000411, which we place in the age 24 cell.

Exam 5, V2

Page 286

 2014 by All 10, Inc.

Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Adjusting the paid claims:
 Paid claims are adjusted based on the modifications that we have made to the closed claim count
triangle earlier.
 Three kinds of treatments: no adjustment, interpolation, and extrapolation.
* Since adjusted closed claim counts are the same as unadjusted closed claim counts along the
latest diagonal, the latest diagonal of the paid claim triangle does not require any adjustment.
* If the number of adjusted closed claims is within the range of any regression in its specific accident
year, we use interpolation. Example:
Since AY 1970 at age 48 has 8,231 adjusted closed claims, which is within the range of
unadjusted closed claims between ages 36 and 48 (7,899; 8,291), the paid claims for AY 1970 at
age 48 would be adjusted based on such regression with a = 215 and b = 0.000468.
(0.000468 x 8,231)
] } = 10,156.
Thus, the adjusted paid claims for AY 1970 at age 48 are equal to {215 x [e
* If the number of adjusted closed claims is not within the range of all regression in its specific AY,
then extrapolation is used to the regression that has the closest range. Example:
AY 1969 at age 12 has 3,383 adjusted closed claim counts, in which the regression between ages
12 and 24 has the closest unadjusted closed claim count range (4,079; 6,616) among all
regressions in year 1969.
(0.000411 x3,383)
]} =1,430.
Thus, adjusted paid claims for year 1969 at age 12 is calculated as {356 x [e
Exhibit II, Sheet 8: Adjusted Paid Claims ($000)
Chapter 13 - Berquist-Sherman Techniques
Berq-Sher Auto BI Insurer
Adjusted Paid Claims ($000)
PART 1 - Data Triangle
Accident
Year
12
1969
1,430
1970
1,744
1971
1,984
1972
2,246
1973
2,584
1974
2,292
1975
2,718
1976
2,801
PART 2 - Age-to-Age Factors
Accident
Year
12 - 24
1969
2.989
1970
2.969
1971
3.145
1972
3.217
1973
3.094
1974
3.172
1975
3.379

24
4,276
5,180
6,241
7,225
7,997
7,269
9,182

36
6,463
7,864
9,594
11,071
11,981
11,771

24 - 36
1.512
1.518
1.537
1.532
1.498
1.619

36 - 48
1.315
1.291
1.275
1.250
1.275

Exhibit II
Sheet 8

Adjusted Paid Claims as of (months)
48
60
72
84
8,497
9,579
10,077
10,191
10,156
11,301
11,744
12,031
12,233
13,550
14,235
13,837
15,383
15,278

Age-to-Age Factors
48 - 60
60 - 72
72 - 84
1.127
1.052
1.011
1.113
1.039
1.024
1.108
1.051
1.112

84 - 96
1.006

96
10,256

To Ult

PART 3 - Average Age-to-Age Factors
24-Dec 24 - 36
Simple Average
All Years
3.138
Latest 4
3.215
Volume-weighted Average
All Years
3.158
Latest 4
3.219

36 - 48

Averages
48 - 60
60 - 72

72 - 84

84 - 96

1.536
1.547

1.281
1.273

1.115
1.115

1.047
1.047

1.018
1.018

1.006
1.006

1.538
1.546

1.277
1.271

1.114
1.114

1.047
1.047

1.018
1.018

1.006
1.006

Development Factor Selection
36 - 48
48 - 60
60 - 72
1.196
1.087
1.036
1.277
1.114
1.047
1.527
1.195
1.073
65.5%
83.7%
93.2%

72 - 84
1.019
1.018
1.025
97.6%

84 - 96
1.006
1.006
1.006
99.4%

To Ult

PART 4 - Selected Age-to-Age Factors

Unadj Selected
Adj Selected
CDF to Ultimate
Percent Reported

Exam 5, V2

12 - 24
3.098
3.158
7.418
13.5%

24 - 36
1.444
1.538
2.349
42.6%

Page 287

96- 108
1.000
1.000
1.000
100.0%

 2014 by All 10, Inc.

Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Analyze the adjusted paid claim development triangle and select age-to-age development factors.
Compare the selected development factors from the unadjusted and adjusted paid claim triangle.
 For both adjusted and unadjusted paid claim triangles, selected factors are based on the volumeweighted average for all years.
 At all age-to-age maturities (except 72-to-84 and 84-to-96 months), selected development factors are
higher based on the adjusted data than on the unadjusted data. This is consistent with the claims
settlement rate decreasing in recent years
Thus, the unadjusted paid claim development projection understates future claim development and the
estimate of unpaid claims.
B/S provide alternatives for the derivation of CDFs. Two additional approaches for determining CDFs for the
adjusted paid claim triangle are as follows:
1. Exhibit II, Sheet 9: Linear Regression of Development Factors Using Adjusted Paid Claims
Using a linear regression of the CDFs at each maturity age and AY, the Y intercepts, slope, and Rsquared values for each maturity age are shown.
Chapter 13 - Berquist-Sherman Techniques
Berq-Sher Auto BI Insurer
Linear Regression of Development Factors Using Adjusted Paid Claims
Accident
Year
1969
1970
1971
1972
1973
1974
1975

12 - 24
2.989
2.969
3.145
3.217
3.094
3.172
3.378

24 - 36
1.511
1.518
1.537
1.532
1.498
1.619

36 - 48
1.315
1.291
1.275
1.250
1.275

Age-to-Age Factors
48 - 60
60 - 72
1.127
1.052
1.113
1.039
1.108
1.051
1.112

Exhibit II
Sheet 9

72 - 84
1.011
1.024

84 - 96
1.006

To Ult

Estimated Intercept from Linear Regression Analysis of Age-to-Age Factors and Accident Year
-104.01
-25.08
25.05
11.36
2.42
Estimated Slope from Linear Regression Analysis of Age-to-Age Factors and Accident Year
0.0543
0.0135
-0.0121
-0.0052
-0.0007
Goodness of Fit Test of Linear Regression Analysis (R-Squared)
70.3%
34.4%
63.7%
61.0%
1.0%

Accident
Year
1969
1970
1971
1972
1973
1974
1975
1976

Exam 5, V2

12 - 24

24 - 36
2.989
2.969
3.145
3.217
3.094
3.172
3.378
3.355

1.511
1.518
1.537
1.532
1.498
1.619
1.583
1.596

36 - 48
1.315
1.291
1.275
1.25
1.275
1.245
1.233
1.221

Age-to-Age Factors
48 - 60
60 - 72
1.127
1.113
1.108
1.112
1.102
1.097
1.091
1.086

Page 288

1.052
1.039
1.051
1.047
1.047
1.047
1.047
1.047

72 - 84
1.012
1.024
1.018
1.018
1.018
1.018
1.018
1.018

84 - 96
1.006
1.006
1.006
1.006
1.006
1.006
1.006
1.006

To Ult
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000

CDF
to Ultimate
1.000
1.006
1.024
1.073
1.182
1.471
2.329
7.815

 2014 by All 10, Inc.

Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2. Exhibit II, Sheet 10: Exponential Regression of Development Factors Using Adjusted Paid Claims
Using an exponential regression of the CDFs at each maturity age and AY, the Y intercepts, slope, and
R-squared values for each maturity age are shown.
* In both Sheets, the R-squared values are never greater than 75% for any maturity age.
* Extrapolated CDFs are used to complete the age-to-age triangles to derive the ultimate CDF for each AY.
Chapter 13 - Berquist-Sherman Techniques
Berq-Sher Auto BI Insurer
Exponential Regression of Development Factors Using Adjusted Paid Claims
Accident
Year
1969
1970
1971
1972
1973
1974
1975

12-24
2.989
2.969
3.145
3.217
3.094
3.172
3.378

24 - 36
1.511
1.518
1.537
1.532
1.498
1.619

36 - 48
1.315
1.291
1.275
1.250
1.275

Age-to-Age Factors
48 - 60
60 - 72
1.127
1.052
1.113
1.039
1.108
1.051
1.112

Exhibit II
Sheet 10

72 - 84
1.011
1.024

84 - 96 To
1.006

To Ult

Estimated Constant from Exponential Regression Analysis of Age-to-Age Factors and Accident Year
0
0
135,483,653
10,606
4
Estimated Growth from Exponential Regression Analysis of Age-to-Age Factors and Accident Year
1.017390463 1.0086231 0.99066952 0.99536201 0.99933607
Goodness of Fit Test of Exponential Regression Analysis (R-Squared)
70.6%
34.0%
63.3%
61.0%
1.0%

Accident
Year
1969
1970
1971
1972
1973
1974
1975
1976

12 - 24
2.989
2.969
3.145
3.217
3.094
3.172
3.378
3.359

24 - 36
1.511
1.518
1.537
1.532
1.498
1.619
1.582
1.596

36 - 48
1.315
1.291
1.275
1.250
1.275
1.245
1.234
1.222

48 - 60
1.127
1.113
1.108
1.112
1.102
1.097
1.092
1.087

60 - 72
1.052
1.039
1.051
1.047
1.047
1.047
1.047
1.047

72 - 84
1.012
1.024
1.018
1.018
1.018
1.018
1.018
1.018

84 - 96 To Ult
1.006
1.000
1.006
1.000
1.006
1.000
1.006
1.000
1.006
1.000
1.006
1.000
1.006
1.000
1.006
1.000

CDF
to Ultimate
1.000
1.006
1.024
1.073
1.182
1.472
2.329
7.823

Exhibit II, Sheet 11: Projection of Ultimate Claims Using Development Technique on Unadjusted and
Adjusted Data ($000)
Chapter 13 - Berquist-Sherman Techniques
Berq-Sher Auto BI Insurer
Projection of Ultimate Claims Using Development Technique and Adjusted Data ($000)

Accident
Year
(1)
1969
1970
1971
1972
1973
1974
1975
1976
Total

Age of
Accident Year Paid Claims Unadjusted
at 12/31/76
at 12/31/76
Pasid
(2)
(3)
(4)
96
10,256
1.000
84
12,031
1.006
72
14,235
1.025
60
15,383
1.062
48
15,278
1.154
36
11,771
1.380
24
9,182
1.991
6.170
12
2,801
90,937

Exhibit II
Sheet 11

CDF to Ultimate
Projected Ultimate Claims Using Dev Method with
Adjusted Paid
Adjusted Paid
Regression
Regression
Volume
Unadjusted
Volume
Exponential
Paid
Weighted
Linear
Exponential
Weighted
Linear
(8) = [(3) x (4)] (9) = [(3) x (5)] (10) = [(3)x(6)] (11) = [(3)x(7)]
(5)
(6)
(7)
1.000
1.000
1.000
10,256
10,256
10,256
10,256
1.006
1.006
1.006
12,103
12,107
12,103
12,103
1.025
1.024
1.024
14,586
14,589
14,578
14,578
1.073
1.073
1.073
16,330
16,510
16,500
16,500
1.195
1.182
1.182
17,629
18,263
18,056
18,058
1.527
1.471
1.472
16,238
17,972
17,320
17,328
2.348
2.329
2.329
18,286
21,560
21,387
21,385
7.416
7.815
7.823
17,281
20,771
21,890
21,913
122,710
132,028
132,090
132,120

Column Notes:
(2) Age of accident year in (1) at December 31, 1976.
(3) Developed in Exhibit II, Sheet 7.
(4) Based on CDF from Exhibit II, Sheet 1.
(5) through (7) Based on CDF from Exhibit II, Sheets 8 through 10, respectively.

Exam 5, V2

Page 289

 2014 by All 10, Inc.

Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
For the unadjusted paid claim triangle, the all-year volume-weighted average age-to-age factors are used.
For the adjusted paid claims, the all-year volume-weighted average as well as the development factors
derived from the linear and exponential regression analyses are used.
Exhibit II, Sheet 12: Development of Unpaid Claim Estimate ($000)
Chapter 13 - Berquist-Sherman Techniques
Berq-Sher Auto BI Insurer
Development of Unpaid Claim Estimate ($000)

Exhibit II
Sheet 12

Projected ultimate Claims Usinsg Dev Method with
Unpaid Claim Estimate at 12/31/76
Adjusted Paid
Adjusted Paid
Accident Paid Claims Unadjusted Volume
Regression
Unadjusted
Volume
Regression
Year
at 12/31/76
Paid
Weighted
Linear
Exponential
Paid
Weighted
Linear
Exponential
(7) = [(3) - (2)] (8) = [(4) - (2)] (9) = [(5) - (2)] (10) = [(6) - (2)]
(1)
(2)
(3)
(4)
(5)
(6)
1969
10,256
10,256
10,256
10,256
10,256
0
0
0
0
1970
12,031
12,103
12,107
12,103
12,103
72
76
72
72
1971
14,235
14,586
14,589
14,578
14,578
351
354
343
343
1972
15,383
16,330
16,510
16,500
16,500
947
1,127
1,117
1,117
1973
15,278
17,629
18,263
18,056
18,058
2,351
2,985
2,778
2,780
1974
11,771
16,238
17,972
17,320
17,328
4,467
6,201
5,549
5,557
1975
9,182
18,286
21,560
21,387
21,385
9,104
12,378
12,205
12,203
19,089
19,112
17,970
21,913
14,480
21,890
17,281
20,771
1976
2,801
Total
90,937
122,710
132,028
132,090
132,120
31,773
41,091
41,153
41,183
Column Notes:
(2) Based on data from Berq-Sher Auto BI Insurer.
(3) through (6) Developed in Exhibit II, Sheet 11.

Unpaid claim estimates based on the results of the unadjusted, adjusted volume weighted, adjusted linear
Regression and adjusted exponential regressions are shown:
* The estimated IBNR based on the adjusted paid claims projections are relatively close to one another.
* These estimates are about $10 million greater than those from the unadjusted development technique.
Potential Difficulty with the Adjustment
A key assumption of the B/S paid claims adjustment is that a higher % of closed claim counts relative to
ultimate claim counts is associated with a higher % of ultimate claims paid.
In Thorne’s review of the B/S paper, he notes: "Lack of recognition of the settlement patterns by size of loss
can be an important source of error ... it may be necessary to modify the technique to apply to size of loss
categories adjusted for inflation."
* Thorne’s detailed example shows the number of small claims (limited to $3,000) steadily decreasing while
the number of larger claims (limited to $20,000) is steadily increasing.
* He shows that the % of closed claim counts decreases and yet the % of paid claims increases due to the
shift to settling larger claims.
* Thus, he notes that the B/S technique actually adjusts paid claims to be less comparable among AYs and
increases the error in the estimate of unpaid claims.
* The example shows the recent trend toward an increasing proportion of severe, late closing claims.

Exam 5, V2

Page 290

 2014 by All 10, Inc.

Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
4

XYZ Insurer

291 - 293

Since the XYZ insurer has be subject to both operational and environmental changes, the B/S adjustments
are appropriate. Three sets of projections are made:
1 Adjustment due to changes in case O/S adequacy only
2 Adjustment for changes in settlement rate only
3 Adjustments for both the change in case O/S adequacy and settlement rates
1 Adjustment due to changes in case O/S adequacy only
Exhibit III, Sheet 1: Average Paid Claims - Unadjusted Data
Chapter 13 - Berquist-Sherman Techniques
XYZ Insurer - Auto BI
Average Paid Claims - Unadjusted Data
Accident
Year
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008

5,064
11,417
9,631
9,452
10,315
11,502
10,726
12,351

36
14,375
13,025
13,162
16,436
13,478
17,996
16,270
19,000

10,020
8,740
13,067
10,163
11,673
10,920
13,000
15,000

Exhibit III
Sheet 1

Average Paid Claims as of (months)
48
60
72
16,708
17,059
16,281
17,041
20,290
18,125
23,455
20,569

18,432
19,919
19,762
19,908
24,073
22,896
26,028

20,208
22,482
22,332
22,911
27,752
25,077

108

120

132

22,143
23,347
24,303
25,887
29,178

23,560
23,307
25,810
26,639

24,695
23,669
26,235

24,825
23,771

24,839

4.8%

3.1%

-4.2%

83.9%

34.2%

100.0%

Annual Change based on Exponential Regression Analysis of Severities and Accident Year
8.1%
5.4%
4.6%
4.3%
5.5%
5.1%
6.8%
Goodness of Fit Test of Exponential Regression Analysis (R-Squared)
46.4%
54.1%
57.2%
64.2%
85.2%
72.3%
95.1%

To determine a severity trend rate, unadjusted average paid claims are reviewed and an exponential
regression at each maturity age is performed.
Since there were not significant differences in the trend rate by maturity age for ages 24 through 72
months, a 5% severity trend rate for all maturities was selected.
Exhibit III, Sheet 2, top section: Adjusted average case O/S triangle.
The latest diagonal and the selected 5% severity trend rate are used to develop this triangle.
Chapter 13 - Berquist-Sherman Techniques
XYZ Insurer - Auto BI
Derivation of Case Adjusted Reported Claim Development Triangle
Accident
Year
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008

12
12,297
12,912
13,557
14,235
14,947
15,694
16,479
17,303
18,168
19,076
20,030

24
27,075
28,429
29,850
31,343
32,910
34,555
36,283
38,097
40,002
42,002

36
38,570
40,498
42,523
44,649
46,882
49,226
51,687
54,271
56,985

Selected Annual Severity Rate

Exam 5, V2

Adjusted Average Case Outstanding as of (months)
48
60
72
84
96
108
49,025
56,951
64,896
99,026
26,699
70,223
51,476
59,799
68,141
103,977
28,034
73,734
54,050
62,789
71,548
109,176
29,435
77,421
30,907
56,752
65,928
75,126
114,634
59,590
69,224
78,882
120,366
62,570
72,686
82,826
76,320
65,698
68,983

Exhibit III
Sheet 2

120
35,608
96,618

132

Page 291

 2014 by All 10, Inc.

Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Exhibit III, Sheet 2, bottom section:
Accident
Year
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008

15,789
19,342
20,450
30,186
33,703
24,715
19,992
18,632

27,527
29,145
37,781
40,864
57,792
56,945
41,339
31,732

36
23,631
31,913
35,225
46,968
46,599
66,886
65,226
48,804

Case Adjusted Reported Claims ($000) as of (months)
48
60
72
84
96
14,600
15,094
15,513
17,104
16,366
25,296
26,319
26,802
28,293
24,795
34,907
36,212
37,153
37,698
37,505
39,380
39,749
38,453
39,707
38,798
49,984
47,313
47,571
48,169
45,605
43,372
44,373
69,521
70,288
70,656

108
16,163
25,071
37,246

120
15,835
25,107

132
15,822

Adjusted reported claim development triangle, created as follows.
{[(adjusted average case O/S) x (open claim counts)] + (paid claims)}
* This is done for 12 months through 84 months.
* For 96 - 132 months, it is expected that the claims department has complete information and that the
case O/S is adequate and therefore the unadjusted reported claim triangle is appropriate without any
adjustment.

Exam 5, V2

Page 292

 2014 by All 10, Inc.

Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Exhibit III, Sheet 3: Adjusted reported claim development triangle, is analyzed as follows.
Chapter 13 - Berquist-Sherman Techniques
XYZ Insurer - Auto BI
Case Adjusted Reported Claims ($000)
PART 1 - Data Triangle
Accident
Year
12
1998
1999
2000
2001
15,789
2002
19,342
2003
20,451
2004
30,186
2005
33,704
2006
24,715
2007
19,992
2008
18,632
PART 2 - Age-to-Age Factors
Accident
Year
12 - 24
1998
1999
2000
2001
1.846
2002
1.953
2003
1.998
2004
1.915
2005
1.690
2006
1.673
2007
1.587
2008

Exhibit III
Sheet 3

27,527
29,146
37,781
40,865
57,792
56,945
41,339
31,732

24 - 36

1.159
1.209
1.243
1.140
1.157
1.145
1.181

23,630
31,913
35,224
46,968
46,599
66,886
65,226
48,804

36 - 48
1.071
1.094
1.118
1.064
0.979
1.039
1.083

Case Adjusted Reported Claims as of (months)
48
60
72
84
96
14,600
15,094
15,513
17,104
16,366
25,296
26,319
26,802
28,294
24,795
34,908
36,211
37,153
37,698
37,505
39,380
39,748
38,452
39,706
38,798
49,984
47,313
47,570
48,169
45,605
43,373
44,373
69,522
70,288
70,655

Age-to-Age Factors
48 - 60
60 - 72
72 - 84
1.034
1.028
1.103
1.040
1.018
1.056
1.037
1.026
1.015
1.009
0.967
1.033
0.947
1.005
1.013
0.951
1.023
1.011

108
16,163
25,071
37,246

120
15,835
25,107

132
15,822

84 - 96
0.957
0.876
0.995
0.977

96 - 108
0.988
1.011
0.993

108 - 120 109 - 120
0.980
0.999
1.001

To Ult

108 - 120 120 - 132

To Ult

PART 3 - Average Age-to-Age Factors
12 - 24
Simple Average
Latest 5
1.772
Latest 3
1.650
Medial Average
Latest 5x1
1.759
Volume-weighted Average
Latest 5
1.772
Latest 3
1.658

24 - 36

36 - 48

48 - 60

Averages
60 - 72

72 - 84

84 - 96

96 - 108

1.173
1.161

1.057
1.034

0.991
0.970

1.008
0.999

1.044
1.020

0.951
0.949

0.997
0.997

0.991
0.991

0.999
0.999

1.161

1.062

0.990

1.016

1.034

0.967

0.993

0.991

0.999

1.169
1.159

1.055
1.040

0.990
0.975

1.007
1.000

1.033
1.019

0.957
0.956

0.998
0.998

0.993
0.993

0.999
0.999

72 - 84
1.010
1.000
1.000
100.0%

84 - 96
1.011
1.000
1.000
100.0%

96 - 108
1.000
1.000
1.000
100.0%

PART 4 - Selected Age-to-Age Factors

Unadj Selected
Case Adj Selected
CDF to Ultimate
Percent Reported

12- 24
1.687
1.658
1.997
50.1%

24 - 36
1.265
1.159
1.205
83.0%

Development Factor Selection
36 - 48
48 - 60
60 - 72
1.102
1.020
1.050
1.040
1.000
1.000
1.039
1.000
1.000
96.2%
100.0%
100.0%

108 - 120 120 - 132
0.993
0.999
1.000
1.000
1.000
1.000
100.0%
100.0%

To Ult
1.000
1.000
1.000
100.0%

* At the 12-to-24 month interval (and somewhat for the 24-36 interval), a persistent downward trend in
the age-to-age factors is observed.
Is the trend rate appropriate?
Is there a potential shift in the type of claim that is now closed at 12 and 24 months?
* CDFs are selected based on the volume-weighted 3-year average (recognizing the decreasing age-toage factors in the most recent diagonals) for ages 12-to-24, 24-to-36, and 36-to-48.
Comparing these to the ones selected based on the unadjusted reported claim triangle (from Chapter 7):
i. the age-to-age factors are mostly less than those based on the unadjusted claims (as expected).
ii. smaller factors are expected since case O/S strengthening has occurred for XYZ Insurer
* A 1.000 factor is judgmentally selected for all remaining intervals to smooth out the remaining
variability.

Exam 5, V2

Page 293

 2014 by All 10, Inc.

Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2. Adjustment for changes in settlement rate only
Exhibit III, Sheet 4: Disposal Rate and Development of Adjusted Closed Claim Counts
Chapter 13 - Berquist-Sherman Techniques
XYZ Insurer - Auto Bl
Disposal Rate and Development of Adjusted Closed Claim Counts
Accident
Year
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008

0.209
0.131
0.111
0.104
0.123
0.183
0.251
0.236

0.462
0.468
0.391
0.377
0.375
0.466
0.54
0.605

36
0.655
0.662
0.643
0.541
0.577
0.637
0.693
0.716

Selected Disposal Rate by Maturity Age
0.236
0.605
0.716
Accident
Year
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008

12
150
247
332
343
367
385
534
567
396
309
276

24
385
633
852
880
940
987
1,369
1,453
1,016
791

36
456
750
1,008
1,042
1,113
1,168
1,620
1,720
1,201

48
0.801
0.782
0.778.
0.751
0.701
0.775
0.819
0.81

0.81

Exhibit III
Sheet 4

Disposal Rate as of (months)
60
72
84
0.859
0.903
0.939
0.869
0.936
0.962
0.864
0.918
0.971
0.842
0.933
0.984
0.854
0.942
0.980
0.924
0.962
0.897

0.897

0.962

0.980

Adjusted Closed Claim Counts
48
60
72
516
571
613
848
939
1,007
1,140
1,263
1,354
1,179
1,305
1,400
1,259
1,394
1,495
1,321
1,463
1,568
1,833
2,029
1,946

96
0.961
0.989
0.988
0.994

108
0.973
0.992
0.996

120
0.997
0.997

132
1

0.994

0.996

0.997

108
634
1,043
1,402

120
635
1,044

132
637

as of (months)
84
96
624
633
1,026
1,041
1,380
1,400
1,426
1,446
1,523

Proj. Ultimate
Reported
Claim Counts
637
1,047
1,408
1,455
1,554
1,631
2,263
2,402
1,679
1,308
1,172

* Disposal rates are selected based on the last diagonal of [closed counts/projected ultimate reported counts].
(projected ultimate reported claim counts are from Chapter 11)
Exhibits III, Sheets 5 and 6: Derivation of adjusted paid claims
(using the same format as in the previous example).
Chapter 13 - Berquist-Sherman Techniques
XYZ Insurer - Auto Bl
Summary of Regression Analysis

Exhibit III
Sheet 4

Accident Year 1998
Cumulative
Closed
Closed
Paid
Predicted
Months of
Claim Count
Claims
Y Value
Development
X
Y
Y=ae^(bX)
(1)
(2)
(3)
(4)
12
24
36
48
510
8,521
8,458
60
547
10,082
10,208
72
575
11,620
11,770
84
598
13,242
13,230
96
612
14,419
14,206
108
620
15,311
14,796
120
635
15,764
15,968
132
637
15,822
16,131
R Squared
a
b

Exam 5, V2

Accident Year 1999
Cumulative
Closed
Paid
Predicted
Claim Counts
Claims
Y Value
X
Y
Y=ae^(bX)
(5)
(6)
(7)

686
819
910
980
1,007
1,036
1,039
1,044

9,861
13,971
18,127
22,032
23,511
24,146
24,592
24,817

0.993716946
637.3038239
0.005084397

9,952
14,066
17,823
21,383
22,939
24,737
24,930
25,257

0.996197864
1659.37274
0.002601397

Page 294

Accident Year 2000
Cumulative
Closed
Paid
Predicted
Claim Counts
Claims
Y Value
X
Y
Y=-ae^(bX)
(5)
(6)
(7)
650
932
1,095
1,216
1,292
1,367
1,391
1,402

6,513
12,139
17,828
24,030
28,853
33,222
35,902
36,782

6,437
12,357
18,013
23,829
28,407
33,786
35,714
36,635

0.999543256
1417.357378
0.002312307

 2014 by All 10, Inc.

Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Chapter 13 - Berquist-Sherman Techniques
XYZ Insurer - Auto BI
Derivation of Adjusted Paid Claims
Accident
Year
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008

Accident
Year
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008

304
203
181
235
295
307
329
276

650
681
607
614
848
1,119
906
791

4,666
1,302 6,513
1,539 5,952
2,318 7,932
1,743 6,240
2,221 9,898
3,043 12,219
3,531 11,778
3,529 11,865
3,409

12
150
247
332
343
367
385
534
567
396
309
276

Exam 5, V2

24
385
633
852
880
940
987
1,369
1,453
1,016
791

36
686
932
936
841
941
1,442
1,664
1,201

36
6,309
9,861
12,139
12,319
13,822
12,683
25,950
27,073
22,819

36
456
750
1,008
1,042
1,113
1,168
1,620
1,720
1,201

Closed Claim Counts as of (months)
48
60
72
84
96
510
547
575
598
612
819
910
980
1,007 1,036
1,095 1,216 1,292 1,367 1,391
1,092 1,225 1,357 1,432 1,446
1,089 1,327 1,464 1,523
1,263 1,507 1,568
1,852 2,029
1,946

Exhibit III
Sheet 6

108
620
1,039
1,402

120
635
1,044

132
637

Paid Claims ($000) as of (months)
48
60
72
84
96
108
120
132
8,521 10,082 11,620 13,242 14,419 15,311 15,764 15,822
13,971 18,127 22,032 23,511 24,146 24,592 24,817
17,828 24,030 28,853 33,222 35,902 36,782
18,609 24,387 31,090 37,070 38,519
22,095 31,945 40,629 44,437
22,892 34,505 39,320
43,439 52,811
40,026

Adjusted Closed Claim Counts as of (months)
48
60
72
84
96
516
571
613
624
633
848
939 1,007 1,026 1,041
1,140 1,263 1,354 1,380 1,400
1,179 1,305 1,400 1,426 1,446
1,259 1,394 1,495 1,523
1,321 1,463 1,568
1,833 2,029
1,946

108
634
1,043
1,402

120
635
1,044

132
637

Accident
Year
1998
1999
2000
2,001
2,002
2,003
2,004
2005
2006
2007

Accident
Year
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008

Page 295

517
1,249
1,023
1,252
1,850
1,904
1,488

0.0036
0.003
0.0029
0.0024
0.0017
0.002
0.0026

12
1,661
3,123
3,230
1,773
3,816
3,178
4,604
4,814
4,225
3,346
3,409

24
4,835
8,593
10,170
12,166
21,877
18,707
35,259
21,474
14,688
11,865

1,551
853
1,878
1,647
2,500
2,386
1,545

0.0022
0.0029
0.0024
0.0022
0.0016
0.0015
0.0022

36
6,669
11,650
14,361
16,658
26,338
20,743
34,657
29,372
22,819

Parameter a for Two-Point Exponential Fit
48
60
72
84
96
108
838
629
443
349
146
1,635 1,341 1,435 2,084 9,319
43
1,348 1,196 1,288 2,543
400
1,680
1,037 2,021 2,561 1,289
734
2,817 4,090 3,111 4,399
2,258 2,737 1,369
4,239 5,625
2,695

120
4,588
3,706

132
4,912

Parameter b for Two-Point Exponential Fit
48
60
72
84
96
108
120
132
0.00455 0.0051 0.0057 0.006082 0.0075 0.00194 0.001836
0.0026 0.00286 0.0028 0.0024 0.000919 0.0061 0.00182
0.0024 0.00247 0.0024 0.0019 0.003233 0.0022
0.0026 0.00203 0.0018 0.0023 0.002739
0.0019 0.00155 0.0018 0.0015
0.0018 0.00168 0.0021
0.0013 0.0011
0.0014

Adjusted Paid Claims ($000) as of (months)
48
60
72
84
96
108
8,755 11,264 14,075 15,372 16,401 17,065
15,076 19,705 23,768 24,614 24,251 25,171
19,846 26,983 33,536 34,034 36,908 36,782
23,395 28,702 33,631 36,543 38,519
30,460 35,435 42,897 44,437
25,466 32,044 39,320
42,416 52,811
40,026

120
15,767
24,817

132
15,822

 2014 by All 10, Inc.

Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Exhibits III, Sheet 7: Adjusted paid claim development triangle
Chapter 13 - Berquist-Sherman Techniques
XYZ Insurer - Auto B1
Adjusted Paid Claims ($000)
PART 1 - Data Triangle
Accident
Year
12
1998
1,658
1999
3,120
2000
3,225
2001
1,769
2002
3,808
2003
3,171
2004
4,592
2005
4,805
2006
4,218
2007
3,341
2008
3,409

24
4,830
8,584
10,158
10,493
16,656
13,772
23,014
19,876
15,035
11,865

PART 2 - Age-to-Age Factors
Accident
Year
12 - 24
24 - 36
1998
1999
1.355
2000
3.150
1.428
2001
5.932
1.550
2002
4.374
1.374
2003
4.343
1.393
2004
5.012
1.408
2005
4.137
1.469
2006
3.564
1.518
2007
3.551
2008

36
6,659
11,634
14,502
16,264
22,893
19,187
32,407
29,202
22,819

36 - 48
1.316
1.306
1.376
1.365
1.256
1.316
1.309
1.371

Exhibit III
Sheet 7

Adjusted Paid
48
60
8,760
11,401
15,191
19,649
19,957
26,887
22,201
28,245
28,752
35,907
25,242
32,008
42,416
52,811
40,026

48 - 60
1.301
1.293
1.347
1.272
1.249
1.268
1.245

Claims as
72
14,476
23,499
32,415
34,318
42,543
39,320

of (months)
84
96
15,439
15,705
23,928
24,660
34,638
36,563
36,550
38,519
44,437

Age-to-Age Factors
60 - 72
72 - 84
84 - 96
1.270
1.067
1.017
1.196
1.018
1.031
1.206
1.069
1.056
1.215
1.065
1.054
1.185
1.045
1.228

96 - 108
1.002
1.004
1.006

108
15,742
24,751
36,782

120
15,769
24,817

132
15,822

108 - 120 109 - 120
1.002
1.003
1.003

To Ult

PART 3 - Average Age-to-Age Factors
12 - 24
Simple Average
Latest 5
4.121
Latest 3
3.751
Medial Average
Latest 5x1
4.015
Volume-weighted Average
Latest 5
4.152
Latest 3
3.783

24 - 36

36 - 48

48 - 60

60 - 72

Averages
72 - 84
84 - 96

96 - 108

108 - 120 120 - 132 To Ult

1.433
1.465

1.323
1.332

1.276
1.254

1.206
1.209

1.053
1.059

1.039
1.047

1.004
1.004

1.002
1.002

1.003
1.003

1.424

1.330

1.263

1.206

1.059

1.042

1.004

1.002

1.003

1.432
1.458

1.322
1.333

1.269
1.252

1.206
1.208

1.053
1.058

1.044
1.049

1.005
1.005

1.002
1.002

1.003
1.003

36 - 48
1.574
1.322
2.269
44.1%

48 - 60
1.316
1.269
1.716
58.3%

PART 4 - Selected Age-to-Age Factors

Unadj Selected
Adj Selected
CDF to Ultimate
Percent Reported

12 - 24
3.349
4.152
13.490
7.4%

24 - 36
2.079
1.432
3.249
30.8%

Development Factor Selection
60 - 72
72 - 84
84 - 96
1.203
1.136
1.059
1.206
1.053
1.044
1.352
1.121
1.065
73.9%
89.2%
93.9%

96 - 108
1.022
1.005
1.020
98.0%

108 - 120 120 - 132 To Ult
1.017
1.004
1.010
1.002
1.003
1.010
1.015
1.013
1.010
98.5%
98.7%
99.0%

* Selected CDFs are based on the 5-year volume-weighted average and are compared to the selected
factors in Chapter 7 based on the unadjusted paid claim triangle.
* At most ages, the selected factors are less than those based on the unadjusted claims, which is
consistent with knowing that the rate of settlement has increased.
3 Adjustments for both the change in case O/S adequacy and settlement rates
* Both the adjusted average paid claim triangle and the adjusted average case O/S triangle are used.
* One new adjusted triangle is needed for projection purposes: the adjusted number of open claims, which
equals reported claim counts - adjusted closed claim counts.
* The adjusted reported claim triangle is then equal to:
{[(adjusted average case O/S) x (adjusted open claim counts)] + (adjusted paid claims)}

Exam 5, V2

Page 296

 2014 by All 10, Inc.

Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Chapter 13 - Berquist-Sherman Techniques
XYZ Insurer - Auto BI
Derivation of Both Adjusted Open and Reported Claims
Accident
Year
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
Accident
Year
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008

15,458
18,385
18,678
27,628
30,761
23,780
20,191
18,632

276
389
407
435
462
614
647
456

502
541
574
629
799
840
629
473

962
975
988
1,398
1,500
1,077
883
760

25,147
27,441
35,541
35,516
52,000
51,870
40,204
31,732

36
22,826
31,038
34,446
43,302
41,939
64,127
64,325
48,804

Exhibit III
Sheet 8

Adjusted Open Claim Counts as of (months)
48
60
72
84
118
64
35
13
191
108
64
27
271
147
82
28
279
153
84
29
298
155
88
31
305
166
61
416
229
444

Both Adjusted Reported Claims ($000) as of (months)
48
60
72
84
96
14,546
15,024
16,744
16,701
15,807
25,019
26,098
27,841
26,729
24,836
34,579
36,118
38,258
37,712
36,812
38,060
38,323
40,658
39,886
38,797
46,525
46,641
49,495
48,168
44,319
44,073
44,372
69,744
70,288
70,654

108
4
6
8
9

120
3
4
6

108
15,921
25,060
37,247

132
2
3

120
15,837
25,107

132
15,822

Exhibit III, Sheet 9: Adjusted reported claim triangle
* The unadjusted selected age to age factors as well as the selected age to age factors from the case O/S
only adjustment are included in the review of the age to age factors from this triangle.
* The average age-to-age factors tend to be between these two sets of other two selected development
factors.
* Selected age to age factors are based on the 3-year volume-weighted average through 72 months; a 1.00
factor is selected for all remaining intervals, to smooth the indications for the older maturities.

Exam 5, V2

Page 297

 2014 by All 10, Inc.

Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Chapter 13 - Berquist-Sherman
XYZ Insurer - Auto
Both Adjusted Reported Claims ($000)
PART 1 - Data Triangle
Accident
Year
12
1998
1999
2000
2001
15,467
2002
18,395
2003
18,691
2004
27,647
2005
30,780
2006
23,796
2007
20,202
2008
18,632
PART 2 - Age-to-Age Factors
Accident
Year
12 - 24
1998
1999
2000
2001
1.775
2002
1.933
2003
1.902
2004
1.882
2005
1.686
2006
1.691
2007
1.571
2008

25,164
27,457
35,560
35,545
52,041
51,904
40,240
31,732

24 - 36

1.235
1.256
1.219
1.181
1.234
1.241
1.213

Exhibit Ill
Sheet 9

Both Adjusted Reported Claims as of (months)
36
48
60
72
14,541
15,031
15,934
22,847
25,010
26,111
26,443
31,068
34,566
36,137
36,285
34,478
38,046
38,343
38,516
43,338
46,509
46,664
47,093
41,992
44,319
44,123
44,373
64,203
69,745
70,288
64,391
70,655
48,804

36 - 48
1.095
1.113
1.103
1.073
1.055
1.086
1.097

48 - 60
1.034
1.044
1.045
1.008
1.003
0.996
1.008

Age-to-Age Factors
60 - 72
72 - 84
1.060
1.048
1.013
1.011
1.004
1.039
1.005
1.035
1.009
1.023
1.006

84
16,697
26,723
37,705
39,877
48,169

84 - 96
0.959
0.937
0.992
0.973

96
16,012
25,042
37,385
38,798

96 - 108
0.992
0.999
0.996

108
15,878
25,018
37,246

120
15,834
25,107

108 - 120 109 - 120
0.997
0.999
1.004

132
15,822

To Ult

PART 3 - Average Age-to-Age Factors
12 - 24
Simple Average
Latest 5
1.746
Latest 3
1.649
Medial Average
Latest 5x I
1.753
Volume-weighted Average
Latest 5
1.746
Latest 3
1.657

24 - 36

36 - 48

48 - 60

Averages
60 - 72
72 - 84

84 - 96

96 - 108

108 - 120 120 - 132 To Ult

1.217
1.229

1.083
1.080

1.012
1.002

1.007
1.006

1.031
1.032

0.965
0.967

0.996
0.996

1.000
1.000

0.999
0.999

1.222

1.086

1.006

1.032

0.966

0.996

1.000

0.999

1.220
1.230

1.084
1.083

1.010
1.003

1.007
1.007

1.030
1.032

0.969
0.970

0.996
0.996

1.001
1.001

0.999
0.999

PART 4 - Selected Age-to-Age Factors
12 - 24
Unadj Selected
1.687
Case Adj Selected
1.658
Both Adj Selected
1.657
CDF to Ultimate
2.229
Percent Reported
44.9%

24 - 36
1.265
1.159
1.230
1.345
74.3%

Development Factor Selection
36 - 48
48 - 60
60 - 72
1.102
1.020
1.050
1.040
1.000
1.000
1.083
1.003
1.007
1.093
1.010
1.007
91.5%
99.0%
99.3%

72 - 84
84 - 96
96 - 108 108 - 120 120 - 132 To Ult
1.010
1.011
1.000
0.993
0.999
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
100.0%
100.0%
100.0%
100.0%
100.0%
100.0%

Exhibit III, Sheets 10 and 11: Projected ultimate claims and computation of the unpaid claims estimates
* A comparison of the above amounts are made among the B/S Adjusted Reported Case, Adjusted Reported
Both, and the Adjusted Paid method.
* All three projections are relatively close to one another for all accident years.

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Chapter 13 - Berquist-Sherman
XYZ Insurer - Auto
Projection of Ultimate Claims Using Development Technique and Adjusted Data ($000)

Accident
Year
(1)
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
Total

Age of
Accident Year
at 12/31/08
(2)
132
120
108
96
84
72
60
48
36
25
12

Claims at 12/31/08
Reported
Paid
(3)
(4)
15,822
15,822
25,107
24,817
37,246
36,782
38,798
38,519
48,169
44,437
44,373
39,320
70,288
52,811
70,655
40,026
48,804
22,819
31,732
11,865
3,409
18,632
449,626
330,627

CDF to Ultimate
Adjusted
Adjusted Reported
Case
Both
Paid
(5)
(6)
(7)
1.000
1.000
1.010
1.000
1.000
1.013
1.000
1.000
1.015
1.000
1.000
1.020
1.000
1.000
1.065
1.000
1.000
1.121
1.000
1.007
1.352
1.000
1.010
1.716
1.039
1.093
2.269
1.205
1.345
3.249
1.997
2.229
13.490

Exhibit Ill
Sheet 10

Projected Ultimate Claims
Using Dev. Method with
Adjusted
Adjusted Reported
Case
Both
Paid
(8) = [(3)x(5)]

15,822
25,107
37,246
38,798
48,169
44,373
70,267
70,634
50,707
38,237
37,208
476,568

(9) = [(3)x(6)]

15,822
25,107
37,246
38,798
48,169
44,373
70,780
71,362
53,362
42,680
41,531
489,229

(10) = [(4)x(7)]

15,980
25,140
37,334
39,289
47,325
44,078
71,400
68,685
51,776
38,549
45,989
485,546

Column Notes:
(2) Age of accident year in (1) at December 31, 2008.
(3) and (4) Based on data from XYZ Insurer.
(5) through (7) Based on CDF from Exhibit III, Sheets 3, 9 and 7, respectively.

Chapter 13 - Berquist-Sherman
XYZ Insurer - Auto
Development of Unpaid Claim Estimate ($000)

Accident
Year
(1)
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
Total

Exhibit Ill
Sheet 11

Projected ultimate Claims
Unpaid Claim Estimate at 12/31/08
Using Dev. Method with
IBNR - Based on Dev. Method with
Total - Based on Dev. Method with
Case
Adjusted Reported
Adjusted Reported
Adjusted Reported
Adjusted Outstanding
Adjusted
Adjusted
Case
Both
Paid
at 12/31/08
Case
Both
Paid
Case
Both
Paid
(7)=[(2)-(3)] (8) = [(4) - (2)] (9)=[(5)-(2)] (10) = [(6) - (2)] (11) = [(7)+(8)] (12) = [(7) + (9)] (13) = [(7)+(10)]
(4)
(5)
(6)
15,822
15,822
15,980
0
0
0
158
0
0
158
25,107
25,107
25,140
290
0
0
33
290
290
323
37,246
37,246
37,334
464
0
0
88
464
464
552
38,798
38,798
39,289
279
0
0
491
279
279
770
48,169
48,169
47,325
3,732
0
0
-844
3,732
3,732
2,888
44,373
44,373
44,078
5,053
0
0
-295
5,053
5,053
4,758
70,267
70,780
71,400
17,477
-21
492
1,112
17,456
17,969
18,589
70,634
71,362
68,685
30,629
-21
707
-1,970
30,608
31,336
28,659
50,707
53,362
51,776
25,985
1,903
4,558
2,972
27,888
30,543
28,957
38,237
42,680
38,549
19,867
6,505
10,948
6,817
26,372
30,815
26,684
37,208
41,531
45,989
15,223
18,576
22,899
27,357
33,799
38,122
42,580
476,568
489,229
485,546
118,999
26,942
39,603
35,920
145,941
158,602
154,919

Column Notes:
(2) and (3) Based on data from XYZ Insurer.
(4) through (6) Developed in Exhibit III, Sheet 10.

Exhibit III, Sheets 12 and 13: Projected ultimate claims and Estimated IBNR.
* Compares the results of the B/S projections with all the other techniques presented for XYZ Insurer.
* There are significant differences when comparing the results from the unadjusted development technique to
those from the development technique applied to adjusted claims data.
These are summarized in the table below:
Estimated Total Unpaid
IBNR
Claim Estimate
65
184
155
274

($ Millions)
Unadjusted Reported Claims
Unadjusted Paid Claims
Adjusted (Case Only) Reported Claims
Adjusted (Case and Settlement) Reported Claims
Adjusted Paid Claims

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
In Chapter 15, evaluation and selection of ultimate claims for many of the examples in preceding
chapters, including XYZ Insurer, are discussed.
The actuary may wish to consider whether or not the results of the B/S analyses should be reflected in a
revised B/F projection for XYZ Insurer.
Specifically, the adjusted reporting and payment patterns could be used in place of the unadjusted reporting
and payment patterns, and any changes in the expected claim ratios due to B/S indications could be used
in determining initial expected claims.

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Sample questions:
1. Friedland explains how Berquist and Sherman approaches can be used for insurers who have undergone
changes (operational, procedural, etc.). List the two broad alternatives that may be available to an actuary
in these situations.
2. Of the two alternatives listed in 1), which is preferred?
3. Berquist/Sherman suggest two basic procedures for selecting data that is relatively unaffected by a particular
data problem. What are these two basic procedures (provide an example of each)?
4. For situations requiring data alterations, describe two types of adjustments that may be made, prior to
using a traditional method of estimating unpaid claims.
5. When using a Berquist-Sherman technique to adjust data for changes in Case Outstanding adequacy,
what are the two decisions that require an actuary’s judgment, according to Friedland?
6. A key part of the Berquist-Sherman process is analyzing the data triangles to see if a change (requiring
adjustment) has taken place. When testing for a change in the Rate of Claims Settlement, what is the
implication if there appears to be a steady decrease in the rate of claim settlement? If we ignored this
test, and applied the Paid Claim Development method, would the Unpaid Claim Estimate likely be high or
low?
Questions from the 1994 Exam (modified):
11. True/False: Based on the discussion in Friedland, Berquist and Sherman recommend the substitution of
policy year data for accident year data when the rate of growth of exposures changes markedly.
Questions from the 1997 Exam (modified):
37. According to Friedland’s discussion of Berquist-Sherman and Thorne’s review, which of the following are
true?
1. They recommend substituting Report Year for Accident Year data when there has been a significant
change in policy limits between successive years.
2. Thorne’s primary criticism of Berquist-Sherman’s method to adjust for changes in claims closure is
that it does not recognize the different settlement pattern by policy limit.
3. Thorne’s primary criticism of Berquist-Sherman’s method to adjust for changes in reserve strength it
that there is too much estimation involved in selecting the claim cost trend.
A. 1
B. 2
C. 3
D. 1, 2
E. 1, 3

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Questions from the 1998 Exam (modified):
53.
(3 points)
impact on

Given the following triangles, identify two trends in the data, and explain the potential
the ultimate claim estimate if the data remains unadjusted.

(Incremental) Paid Claims ($000's)
Accident
Development Period (months)
Year
12
24
36
48
1992
100
108
113
73
1993
110
113
115
69
1994
115
115
116
77
1995
105
112
120
1996
115
118
1997
125

(Incremental) Reported Claims ($000's)
Accident
Development Period (months)
Year
12
24
36
48
1992
275
166
85
45
1993
305
186
76
29
1994
315
203
69
21
1995
325
244
45
1996
340
256
1997
355
(Incremental) Reported Claim Counts
Accident
Development Period (months)
Year
12
24
36
48
1992
200
100
57
10
1993
195
98
60
8
1994
198
101
58
9
1995
205
102
55
1996
202
100
1997
200
(Cumulative) Ratio of Closed to Reported Claim Counts
Accident
Development Period
Year
12
24
36
1992
15.0%
36.0%
60.0%
1993
18.0%
38.0%
65.0%
1994
19.0%
40.0%
62.0%
1995
23.0%
42.0%
63.0%
1996
23.0%
45.0%
1997
26.0%

(months)
48
75.0%
78.0%
74.0%

60
36
34

72
28

60
27
16

72
15

60
5
6

72
1

60
90.0%
87.0%

72
93.0%

Hint: Test for changes in the Rate of Claims Settlement and changes in Adequacy of Case Outstanding.

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Questions from the 1998 Exam (modified):
49. (3 points) Given the data below, use the Berquist and Sherman method for adjusting for change in claim
settlement pattern, as described by Friedland, to calculate the revised cumulative paid claims for accident
year 1993 at each evaluation point, i.e., 12, 24, 36, 48, and 60. Assume that the relationship between the
incremental number of closed claim counts and the incremental paid claim $ is linear. Show all work.

Accident
Year
1993
1994
1995
1996
1997

12
4,000
4,800
5,000
5,500
6,400

Cumulative Claim Counts
Age of Development (Months)
24
36
48
60
7,300
8,500
9,200
10,000
8,000
10,000
11,400
9,500
11,900
10,650

Ultimate
10,000
12,000
14,000
15,000
16,000

(Data continues below)
Accident
Year
1993
1994
1995
1996
1997

Exam 5, V2

12
20,000
25,000
24,000
31,000
35,000

Cumulative Paid Claims ($000's)
Age of Development (Months)
24
36
48
60
35,000
45,000
52,000
56,000
39,000
48,000
55,000
42,000
50,000
50,000

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Questions from the 2000 Exam:
60. (3 points) You are given the following information as of December 31, 1999:

Accident Year
1996
1997
1998
1999

12 mos.
8,450
9,028
11,470
12,350

12 mos.
900
990
1,089
1,198

12 mos.
800
900
1,000
1,100

12 mos.
5,000
5,500
6,000
5,750

Cumulative Incurred Claims ($000)
Age of Development
24 mos.
36 mos.
12,755
30,230
30,203
46,625
36,300

48 mos.
43,390

Paid Claim $ per Closed Claim Count
Age of Development
24 mos.
36 mos.
1,500
2,000
1,650
2,200
1,815

48 mos.
5,050

Number of Open Claims (Count)
Age of Development
24 mos.
36 mos.
400
150
500
200
575

48 mos.
50

Cumulative Paid Claims ($000)
Age of Development
24 mos.
36 mos.
8,000
24,605
15,000
38,425
17,900

48 mos.
40,890

Friedland illustrates a method by Berquist and Sherman to reduce the impact on reported claim
projections due to changes in the adequacy level of Case Outstanding amounts. Using this
technique, calculate the adjusted 12-24 reported claim development factor for accident year 1997.
Show all work.

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Questions from the 2002 Exam:
26. (3 points) You are given the following information for a company that has recently undergone
changes affecting its claim settlement rates.

Accident
Year
1998
1999
2000
2001

12
16,250
18,375
20,625
23,000

Accident
Year
1998
1999
2000
2001

Cumulative Closed Claim Counts
Age (in Months)
Projected
24
36
48
Ultimate
35,000
50,000
50,000
50,000
39,375
52,500
52,500
44,000
55,000
57,500

Cumulative Paid Claims ($000)
Age (in Months)
12
24
36
121,875
262,500
375,000
137,813
295,313
393,750
154,688
330,000
172,500

48
375,000

Using the Berquist and Sherman method described by Friedland, calculate an estimate of the ultimate
Paid Claims for accident year 2000. Assume that the relationship between the incremental number of
closed claim counts (#) and the incremental paid claims ($) is linear. Use all-year simple averages to
select ATA factors. Show all work.
Questions from the 2004 Exam (modified):
22. (1 point) Based on Friedland’s discussion of Berquist and Sherman, state and briefly describe two
problems that can be mitigated by analyzing claim experience by separate size of loss categories.

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Questions from the 2004 Exam (modified):
23. (3 points) You are given the following information as of December 31, 2003:

Accident
Year
2001
2002
2003

Accident
Year
2001
2002
2003
Data continues below

Case Outstanding per Open Claim ($)
(Age of Development in Months)
12
24
36
$11,870
$18,840
$12,720
12,580
19,963
14,234

12
100
100
100

Number of Open Claims (Counts)
(Age of Development in Months)
24
90
90

36
50

Accident
Year
2001
2002
2003

Paid Claim ($) per Closed Claim Count
(Age of Development in Months)
12
24
36
$10,600
$21,200
$26,500
11,236
22,472
11,910

Accident
Year
2001
2002
2003

Cumulative Paid Claims ($)
(Age of Development in Months)
12
24
36
$933,000
$2,332,000
$4,198,000
989,000
2,473,000
1,049,000

Selected Reported CDF from 24 months to ultimate is 1.426.
a. (1.5 points) Based on Friedland’s explanation of Berquist and Sherman's method, demonstrate why
you might conclude that the relative level of Case Outstanding adequacy is different for accident
year 2003 as of 12 months than for earlier accident years. Show all work.
b. (1.5 points) Calculate an estimate for Ultimate Reported Claims for accident year 2003 using
Berquist and Sherman's technique for adjusting data to compensate for changing Case Outstanding
adequacy. Show all work.

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2005 Exam Questions (modified):
8. (3.5 points) You are given the following information as of December 31, 2004:
Cumulative Paid Claims
Accident
(Age of Development in Months)
Year
12
24
36
2002
$10,000
$32,800
$59,850
2003
13,125
36,120
2004
12,673

Accident
Year
2002
2003
2004

Closed Claim Counts
(Age of Development in Months)
12
24
36
20
41
57
25
43
23

Accident
Year
2002
2003
2004

Cumulative Reported Claims
(Age of Development in Months)
12
24
36
$18,000 $40,800
$62,250
23,205
42,420
23,761

Accident
Year
2002
2003
2004

Open Claim Counts
(Age of Development in Months)
12
24
36
10
8
2
12
6
11

•

Select ATA factors using all-years simple averages.

•

The selected tail factor for incurred development after 36 months is 1.100.

a. (1.5 points) Based on Berquist and Sherman's method, demonstrate that the relative level of the
Case Outstanding adequacy has changed for accident year 2004.
b. (2 points) Using Berquist and Sherman's technique for adjusting data to compensate for changing
Case Outstanding adequacy, calculate the ultimate reported claims for accident year 2004.

9. (2.5 points) Berquist and Sherman describe an approach to adjust the Paid Claim ($) triangle for
distortion.
a. (0.5 point) Identify the distortion for which this adjustment is intended.
b. (2 points) Describe the technique to make the necessary adjustment.

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2006 Exam Questions (modified):
20. (1.5 points)
a. (0.5 points) Why is it important for the actuary to engage in discussions with management in
business areas, such as claims and underwriting, when estimating Unpaid Claims?
b. (1 point) A review of the claims data for a company reveals that the time it takes to settle a claim
once it has been reported has increased over the last six months. Identify two questions that you
would ask of claims management of that company to better understand this trend.
21. (2 points) In the course of a reserve analysis, it is observed that the paid claim development triangles
are distorted by significant changes in the claims settlement rate. Briefly describe the procedure that
Berquist and Sherman propose to address this situation, as described in Friedland.
2007 Exam Questions (modified):
33. (2 points) Given the following data for a certain book of business:
Average Case Reserve ($) Per Open Claim
Accident
Year
2004
2005
2006

Age of Development in Months
12
24
36
6,354
12,493
25,192
8,196
17,400
10,000

Accident
Year
2004
2005
2006

Number of Open Claims
Age of Development in Months
12
24
36
400
250
75
550
325
450
Cumulative Paid Losses ($)

Accident
Year

Age of Development in Months
12

2004

1,600,000

2,740,000

2005

1,725,000

2,850,000

2006

1,775,000

36
4,000,000

 The annual severity trend for this book of business is 7%.
Construct the Berquist-Sherman adjusted incurred loss triangle, as explained in Friedland.
34. (1 point) Explain two reasons why using paid claim data to estimate a severity trend can be
inappropriate for medical malpractice losses.

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2007 Exam Questions (modified):
38. (2.25 points)
a. (0.75 point) Identify three operational changes that could affect the accuracy of Unpaid Claim
estimates for a book of business written by an insurance company.
b. (0.75 point) For each change identified in part a. above, briefly describe how it affects the
unadjusted chain ladder method (Development Method) for calculating Unpaid Claim estimates.
c. (0.75 point) For each response provided in part b. above, identify an adjustment that would result
in a more appropriate Unpaid Claim estimate.
39. (1.5 points)
a. (0.5 point) Identify and explain how a change external to a particular insurance company could
affect claim frequency for that company.
b. (0.5 point) Identify and explain how a change external to a particular insurance company could
affect claim severity for that company.
c. (0.5 point) Explain why frequency and severity changes cannot be considered independently.

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2008 Exam Questions (modified):
4. (Modified) 2.5 points
Given the following data as of December 31, 2007:
Cumulative Paid
AY
2004
2005
2006
2007

Loss ($000s)
12
24
30,729
103,361
24,573
85,337
22,567
88,009
27,761

36
125,237
105,979

Average Case O/S per Open Claim ($000s)
AY
12
24
36
2004
63.500
97.100
342.400
2005
62.100
115.000
394.200
2006
66.200
109.200
2007
79.800
Number of Open Claims (Counts)
AY
12
2004
810
2005
698
2006
654
2007
633

24
480
387
361

36
115
87

48
138,547

48
888.700

48
43

a. (2 points)
Using the method described by Berquist and Sherman to adjust for
changes in case reserve adequacy, calculate the ADJUSTED
cumulative reported loss triangle, assuming a 5% severity trend.
b. (.5 points)
Using all-year weighted average loss development factors, calculate
the Accident Year 2007 ultimate loss based on the ADJUSTED
cumulative reported loss triangle, assuming a 48-to-Ult CDF of 1.02.

5. modified (2 points)
When compiling data in preparation for a reserve analysis (estimates of unpaid claims), the
actuary must consider changes in the external environment as well as changes internal to the
insurance company. For each of a, b, c, and d, below, give an example of a situation in which it
would be preferable to use the suggested datasets and provide the rational of each example.
a.
b.
c.
d.

(.5
(.5
(.5
(.5

points)
points)
points)
points)

Exam 5, V2

Policy year data instead of accident year data
Accident quarter data instead of accident year data.
Report year data instead of accident year data.
Earned exposures instead of claim counts.

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2009 Exam Questions
11. (2.5 points) Given the following information:
Cumulative Paid Loss
Accident
Year
12 Months
24 Months
36 Months
2005
2006
2007
2008

$170,000
220,000
360,000
450,000

Accident
Year

12 Months

2005
2006
2007
2008

34
55
75
84

Accident
Year

12 Months

2005
2006
2007
2008
•
•

$320,000
420,000
650,000

$450,000
630,000

Number of Open Claims
24 Months
36 Months
20
35
50

15
24

Average Case Reserve
24 Months
36 Months

48 Months
$500,000

48 Months
8

48 Months

$2,500
$5,500
$8,000
$15,000
3,125
6,490
9,440
3,750
7,528
4,125
Selected case reserve severity trend at all maturities is 5%.
The 48 month to ultimate incurred loss development factor is 1.020.

a. (2 points) Use the Berquist-Sherman case reserve adjustment method to calculate ultimate losses
for accident year 2008.
b. (0.5 point) Briefly describe the purpose of the Berquist-Sherman case reserve adjustment.
2010 Exam Questions
1. (4 points) Given the following loss information as of 12 months maturity for AYs 2006 through 2009:
Accident
Paid Claims
Reported Claims
Closed Claim
Open Claim
Year
(S000)
($000)
Counts
Counts
2006
9,688
17,299
2,800
1,522
2007
17,778
38,345
5,000
3,639
2008
25,519
51,836
6,900
4,119
2009
34,093
74,115
8,875
5,544
a. (1.75 points) Test the above data for changes in case reserve adequacy and interpret the results.
b. (0.75 point) Describe the leveraging effect that a change in case reserve adequacy has on the IBNR
indicated by the reported loss development method.
c. (1.5 points) Use the Berquist-Sherman technique for case reserve adequacy to calculate the
adjusted reported claims for each accident year.

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ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2011 Exam Questions
31. (3.75 points) Given the following data as of December 31, 2010:

•
•
•

Accident
Year
2008
2009
2010

Cumulative Paid Claims (000s)
12 Months
24 Months
36 Months
$3,816
$9,771
$18,518
$3,600
$11,292
$6,268

Accident
Year
2008
2009
2010

Case Outstanding (000s)
12 Months
24 Months
36 Months
$21,936
$31,920
$27,424
$26,334
$28,648
$31,042

Accident
Year
2008
2009
2010

Cumulative Reported Claims (000s)
12 Months
24 Months
36 Months
$25,752
$41,691
$45,942
$29,934
$39,940
$37,310

Accident
Year
2008
2009
2010

12 Months
1,828
1,540
1,660

Open Claim Counts
24 Months
36 Months
1,900
1,522
1,600

Annual severity trend 10%
36-to-ultimate reported tail factor = 1.050
Use all-year volume-weighted average for development factor selection

a. (2.75 points) Use the Berquist-Sherman technique for case reserve adequacy to estimate the
ultimate claims for all accident years.
b. (0.25 point) Briefly describe the purpose of the Berquist-Sherman case reserve adjustment.
c. (0.75 point) Discuss whether changing the annual severity trend given above from 10% to 5%
would produce a higher or lower ultimate claims estimate under the Berquist-Sherman technique
for case reserve adequacy.

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2012 Exam Questions
22. (4.25 points) Given the following data as of December 31, 2011:
Cumulative Reported Claims ($000s)
Accident
Year
2009
2010
2011

12 Months
$9,931
$12,967
$12,924

24 Months
$11,583
$17,391

36 Months
$13,053

Cumulative Paid Claims ($000s)
Accident
Year
2009
2010
2011

12 Months
$3,711
$3,464
$3,128

24 Months
$8,747
$8,996

36 Months
$12,358

Open Claim Counts
Accident
Year
2009
2010
2011

12 Months
345
499
435

24 Months
167
350

36 Months
30

•

Assume no reported or paid development after 36 months.

•

The annual severity trend is 8%.

a. (0.75 point) Estimate the ultimate claims for accident year 2011 using the reported development technique.
b. (3 points) Estimate the ultimate claims for accident year 2011 using the Berquist-Sherman case
outstanding adjustment technique.
c. (0.5 point) Discuss the difference between the two estimates.

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2012 Exam Questions (continued)
24. (3.5 points) Given the following information:
Cumulative Closed Claim Counts
Accident
Year
12 Months
24 Months
36 Months
2009
500
1,200
1,695
2010
750
1,325
2011
825
Cumulative Paid Claims ($000s)
Accident
Year
2009
2010
2011

12 Months
$2,893
$4,339
$4,773

24 Months
$8,727
$9,636

36 Months
$12,919

Cumulative Reported Claim Counts
Accident
Year
2009
2010
2011

12 Months
1,500
1,650
1,600

24 Months
1,900
2,100

36 Months
2,050

Case Outstanding Claims ($000s)
Accident
Year
2009
2010
2011

12 Months
$8,715'
$7,844
$6,755

24 Months
$9,211
$10,197

36 Months
$3,944

•

There are no partial payments.

•

Assume no reported development after 36 months.

a. (2 points) Evaluate whether a Berquist-Sherman Case Outstanding Adjustment would be appropriate.
b. (1.5 points) Use disposal rates to evaluate whether a Berquist-Sherman Paid Claim Development
Adjustment would be appropriate.

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ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to Sample Questions:
1. Friedland explains how Berquist and Sherman approaches can be used for insurers who have undergone
changes (operational, procedural, etc.). Explain the two broad alternatives that may be available to an
actuary in these situations.
1) Avoid the problem: through data selection and/or rearrangement
2) Adjust the data to account for the changes
2. Whenever possible, it is better to avoid the problem by using data that is relatively unaffected by the changes
that the insurer faces.
3. Berquist/Sherman suggest two basic procedures for selecting data that is relatively unaffected by a particular
data problem. What are these two basic procedures (provide an example of each)?
Two procedures are described to obtain data that is relatively unaffected by a given problem.
These procedures are as follows:
1) Substitute data (example: using quarterly data in place of annual accident year data when there has
been substantial growth in premium volume), or
2) Subdivide the data (example: separating large claims from small claims).
4. For situations requiring data alterations, describe two types of adjustments that may be made, prior to
using a traditional method of estimating unpaid claims.
1) If there have been changes in the Adequacy of Case Outstanding amounts:
Below we will see how to test and adjust the Case Outstanding data triangle.
2) If there have been changes in the Rate of Claims Settlement:
Below we will see how to test and adjust the Paid Claim data triangle.
5. When using a Berquist-Sherman technique to adjust data for changes in Case Outstanding adequacy,
Friedland notes two decisions that require an actuary’s judgment:
1) Must “choose a diagonal from which he or she will calculate all other values of the adjusted
average case outstanding triangle.”
2) Must “select an annual severity trend to adjust the (values) from the selected diagonal …”
6. A key part of the Berquist-Sherman process is analyzing the data triangles to see if a change (requiring
adjustment) has taken place. When testing for a change in the Rate of Claims Settlement, what is the
implication if there appears to be a steady decrease in the rate of claim settlement? If we ignored this
test, and applied the Paid Claim Development method, would the Unpaid Claim estimate likely be high or
low?
The primary underlying assumption of the Paid Claim Development Technique would be contradicted.
Using this method would likely understate the actual amount needed to reserve for unpaid claims.
Solutions to 1994 Exam questions (modified):
11. True/False: Based on the discussion in Friedland, Berquist and Sherman recommend the substitution of
policy year data for accident year data when the rate of growth of exposures changes markedly.
False, the substitution is accident quarter for accident year.

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ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 1997 Exam questions (modified):
37. According to Friedland’s discussion of Berquist-Sherman and Thorne’s review, which of the following are
true?
1. They recommend substituting Report Year for Accident Year data when there has been a significant
change in policy limits between successive years.
• False, the recommendation would be policy year for accident year.
2. Thorne’s primary criticism of Berquist-Sherman’s method to adjust for changes in claims closure is that it
does not recognize the different settlement pattern by policy limit.
• False, the criticism is the failure to reflect changes in settlement pattern by size of claim.
3. Thorne’s primary criticism of Berquist-Sherman’s method to adjust for changes in reserve strength it that
there is too much estimation involved in selecting the claim cost trend.
• True, too much judgment is involved.
A. 1
B. 2
C. 3
D. 1, 2
E. 1, 3

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 1998 Exam questions (modified):
53.

(3 points) Given the following data:
(A) Given (Incremental) Paid Claims ($000's)
Accident
Development Period (months)
Year
12
24
36
48
1992
100
108
113
73
1993
110
113
115
69
1994
115
115
116
77
1995
105
112
120
1996
115
118
1997
125

(B) Given

(D) Given

Exam 5, V2

(Incremental) Reported Claims ($000's)
Accident
Development Period (months)
Year
12
24
36
48
1992
275
166
85
45
1993
305
186
76
29
1994
315
203
69
21
1995
325
244
45
1996
340
256
1997
355
(Incremental) Reported Claim Counts
Accident
Development Period (months)
Year
12
24
36
48
1992
200
100
57
10
1993
195
98
60
8
1994
198
101
58
9
1995
205
102
55
1996
202
100
1997
200
(Cumulative) Ratio of Closed to Reported Claim Counts
Accident
Development Period (months)
Year
12
24
36
48
1992
15.0%
36.0%
60.0%
75.0%
1993
18.0%
38.0%
65.0%
78.0%
1994
19.0%
40.0%
62.0%
74.0%
1995
23.0%
42.0%
63.0%
1996
23.0%
45.0%
1997
26.0%

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60
36
34

72
28

60
27
16

72
15

60
5
6

72
1

60
90.0%
87.0%

72
93.0%

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
1998 # 53 TEST 1: Test for change in the Rate of Claim Settlement .
TEST 1, first step: Estimate ultimate claim counts using reported claim counts.
(E)
(Cumulative) Reported Claim Counts
From (C)
Accident
Development Period (months)
Year
12
24
36
48
1992
200
300
357
367
1993
195
293
353
361
1994
198
299
357
366
1995
205
307
362
1996
202
302
1997
200
(F)
From (E)

Age-to-Age Factors for (Cumulative) Reported Claim Counts
Accident
Development Period (months)
Year
12:24
24:36
36:48
48:60
1992
1.500
1.190
1.028
1.014
1993
1.503
1.205
1.023
1.017
1994
1.510
1.194
1.025
1995
1.498
1.179
1996
1.495

3-yr Simple Avg ATA
CDF to Ultimate

1.501
1.868

1.193
1.245

1.025
1.044

1.015
1.018

60
372
367

72
373

60:72
1.003

Let 72:Ult

1.003
1.003

1.000
1.000

(G)
Ultimate Reported Claim Counts = [Latest Diagonal from (D)] * [CDF to Ultimate]
Accident
Development Period (months)
Year 12
24
36
48
60
72
Ultimate
373 x1.00=
373.0
1992
1993
367
x1.003=
368.0
1994
366
x1.018=
372.5
1995
362
x1.044=
377.8
1996
302
x1.245=
375.9
1997 200
x1.868=
373.6
TEST 1 (cont): Create claim disposed ratios (cumulative closed claims /ultimate claims)
(H) =
(D) * (E)

(Cumulative) CLOSED Claim Counts
Accident
Development Period
Year
12
24
36
1992
30.0
108.0
214.2
1993
35.1
111.3
229.5
1994
37.6
119.6
221.3
1995
47.2
128.9
228.1
1996
46.5
135.9
1997
52.0

(months)
48
275.3
281.6
270.8

60
334.8
319.3

(I) =
Disposal Rates (Cumulative Closed Counts, divided by Ultimate Counts)
(H) / (G)ult
Accident
Development Period (months)
Year
12
24
36
48
60
1992
0.080
0.290
0.574
0.738
0.898
1993
0.095
0.303
0.624
0.765
0.868
1994
0.101
0.321
0.594
0.727
1995
0.125
0.341
0.604
1996
0.124
0.362
1997
0.139

72
346.9

72
0.930

Test 1 Conclusion: Appears to be a speed up in the Rate of Claim Settlement (down columns).
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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
1998 #53 TEST 2: Test for change in Adequacy of Case Outstanding
TEST 2, first step: Calculate case outstanding amounts.
(J)
From (A)

(K)
From (B)

(L) =
(K) - (J)

(Cumulative) Paid Claims ($000's)
Accident
Development Period (months)
Year
12
24
36
48
1992
100
208
321
394
1993
110
223
338
407
1994
115
230
346
423
1995
105
217
337
1996
115
233
1997
125
(Cumulative) Reported Claims ($000's)
Accident
Development Period (months)
Year
12
24
36
48
1992
275
441
526
571
1993
305
491
567
596
1994
315
518
587
608
1995
325
569
614
1996
340
596
1997
355
Case Outstanding = Reported Claims - Paid Claims ($000's)
Accident
Development Period (months)
Year
12
24
36
48
1992
175
233
205
177
1993
195
268
229
189
1994
200
288
241
185
1995
220
352
277
1996
225
363
1997
230

60
430
441

72
458

60
598
612

72
613

60
168
171

72
155

TEST 2 (cont): Calculate Open Claim Counts
(M) =
(E) - (H)

Open Claim Counts = Cumulative Reported Counts - Cumulative Paid Counts
Accident
Development Period (months)
Year
12
24
36
48
60
1992
170.0
192.0
142.8
91.8
37.2
1993
159.9
181.7
123.6
79.4
47.7
1994
160.4
179.4
135.7
95.2
1995
157.9
178.1
133.9
1996
155.5
166.1
1997
148.0

72
26.1

TEST 2 (cont): Calculate Average Case Outstanding per Open Claim
(N) = 1000*
(L) / (M)

Accident
Year
1992
1993
1994
1995
1996
1997

Average Case Outstanding $ per Open Claim Count (Severity)
Development Period (months)
12
24
36
48
60
1,029
1,214
1,436
1,929
4,516
1,220
1,475
1,854
2,380
3,584
1,247
1,605
1,777
1,944
1,394
1,977
2,068
1,447
2,185
1,554

72
5,936

Test 2 Conclusion: Appears there may be evidence of Case Outstanding Adequacy strengthening in
the earlier development intervals. (Really want to compare growth % indications here against the
implied trends from paid data. See more recent exam questions below as better examples.)
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ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solution to 1998 Exam Question 53 (continued): Final Observations
Test 1 Conclusion: Appears to be a speed up in the Rate of Claim Settlement (down columns).
If paid claims are used as the basis for the estimate of ultimate claims, this could create an
overstatement in the ultimate amounts (assuming that more closed claims mean a greater
percentage of ultimate claim have been paid).
Test 2 Conclusion: Appears there may be evidence of Case Outstanding Adequacy strengthening in
the earlier development intervals. (Really want to compare growth % indications here against the
implied trends from paid data. See more recent exam questions below as better examples.)
If reported claims are used as the basis for the estimate of ultimate claims this could create an
overstatement in the ultimate amounts for accident years 1995 - 1997.

Note: This question only asks us to test if data adjustments are needed.
Below we see how to make the data adjustments, given that they are needed.
Most often the question will require one of the two adjustments (either an adjustment for a change in the rate
of claim settlement or an adjustment for a change in the adequacy of case outstanding amounts.)
The 1998 questions are shown to reverse order to present the tests (53) before the actual data adjustments.

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ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solution to 1998 Exam Question 49 (modified):
Calculate the revised cumulative paid claims for accident year 1993 at each evaluation point:
(A) Given
Accident
Year
1993
1994
1995
1996
1997

12
4,000
4,800
5,000
5,500
6,400

Cumulative Claim Counts
Age of Development (Months)
24
36
48
7,300
8,500
9,200
8,000
10,000
11,400
9,500
11,900
10,650

60
10,000

(A)
Projected
Ultimate
10,000
12,000
14,000
15,000
16,000

(B) Given
Accident
Year
1993
1994
1995
1996
1997

12
20,000
25,000
24,000
31,000
35,000

Cumulative Paid Claims ($000's)
Age of Development (Months)
24
36
48
60
35,000
45,000
52,000
56,000
39,000
48,000
55,000
42,000
50,000
50,000

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ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2) And use these selected Disposal Rates to restate (adjust) the historical count data:
(D) = Ultimate Counts in (A), multiplied by the selected Disposal Rates in (C)
ADJUSTED Cumulative Claim Counts
Accident
Age of Development (Months)
12
24
36
48
60
Year
4,000
7,100
8,500
9,500
10,000
1993
1994
1995
1996
1997
3) To move from adjusted claim counts to adjusted claim dollars , Friedland notes:
The authors "identify a mathematical formula that approximates the relationship … "
See the text for details on regression analysis, but for this question we are told to
to assume the relationship is linear , based on the unadjusted data points (by year).
Original AY 2003 Data
ADJUSTED AY 2003 Data
Counts # Claim $000
Counts #
Claims $000's (Cumulative)
Age
From (A)
From (B)
From (D)
Linearly Interpolated from left
12
4,000
20,000
4,000
20,000 directly = original data
24
7,300
35,000
7,100
34,091 See below
36
8,500
45,000
8,500
45,000 directly = original data
48
9,200
52,000
9,500
55,000
60
10,000
56,000
10,000
56,000 directly = original data
For example, adjusted paid losses at 24 months are calculated as:
20,000 + (7,100 – 4,000)/(7,300 – 4,000) x ( 35,000 – 20,000) = 34,091
A similar process could be followed for each accident year, creating entire triangles of
adjusted data. This data could then be used with a method for estimating unpaid claims.

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ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2000 Exam questions (modified):
Question 60: Propose a technique to reduce the impact on Reported Claim projections due to changes in
the Adequacy Level of Case Outstanding amounts. Using this technique, calculate the adjusted 12-24
reported age-to-age claim development factor for accident year 1997.
Step 1: Compute severity trends in paid losses per closed claim:
AY

12 mos

24 mos

96-97

990
= 1.10
900

1650
= 1.10
1500

97-98

1089
= 1.10
990

1815
= 1.10
1650

98-99

1198
= 1.10
1089

36 mos
2200
= 1.10
2000

Based on the above, the severity trend in paid losses per closed claims equals 10%
Step 2: a. Compute “Case Outstanding per Open Claim Count” for the latest diagonal ($000s):
AY

12 mos

24 mos

1997

calculation not
necessary
36,300 −17,900
= 32, 000
.575

1998

1999

36 mos

12,350 − 5,750
= 6, 000
1.1

b. Compute the remaining values in the triangle above by de-trending the above values by the
severity trend, 10%, determined in Step 1. (numbers in $000s):
AY

12 mos

24 mos

1997

5,455
= 4, 959
1.1

32,000
= 29, 091
1.1

1998

6,000
= 5, 455
1.1

32, 000

1999

6, 000

Step 3: Compute adjusted Reported Claims $ at 12 and 24 months respectively for AY 1997
Adjusted Reported Losses = [Adjusted Case Outstanding as above]*[Open Count#]/1000 + Paid Losses
AY 1997 at 12 mos.:

4,959*900
29,091*500
+ 5, 500 =
9, 963 . AY 1997 at 24 mos.:
+ 15, 000 =
29, 545 .
1,000
1,000

Step 4: Compute the adjusted 12-24 Reported loss development factor for AY 1997.
The adjusted 12-24 incurred ATA CDF for accident year 1997 =

Exam 5, V2

Page 323

29,545
= 2.965
9,963

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2002 Exam questions (modified):
26. (3 points) Calculate an estimate of the ultimate Paid Claims for accident year 2000.
(A) Given
Accident
Year
1998
1999
2000
2001

12
16,250
18,375
20,625
23,000

Cumulative Claim Counts (#)

(A)

Age of Development (Months)
24
36
48
35,000
50,000
50,000
39,375
52,500
44,000

Projected
Ultimate
50,000
52,500
55,000
57,500

(B) Given
Cumulative Paid Claims ($000's)
Accident
Year
1998
1999
2000
2001

12
121,875
137,813
154,688
172,500

24
262,500
295,313
330,000

Age of Development (Months)
36
48
375,000
375,000
393,750

2) And use these Selected Disposal Rates to restate (adjust) the historical count data:
(D) = Ultimate Counts in (A), multiplied by the selected Disposal Rates in (C)
ADJUSTED Cumulative Claim Counts (#)
Accident
Age of Development (Months)
12
24
36
48
Year
20,000
40,000
50,000
50,000
1998
1999
21,000
42,000
52,500
2000
22,000
44,000
2001
23,000

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solution to 2002 #26 (continued):
3) To move from adjusted claim counts to adjusted claim dollars , Friedland notes:
The authors "identify a mathematical formula that approximates the relationship … "
We a ssume the relationship is linear , based on the unadjusted data points (by year).
Note: we will only calculate the factors we need for this exam question:
Since we only need an estimate for 2000 AY unpaid, which is at 24 months, we need
enough data to develop a CDF from 24 to Ultimate. (1998 and 1999, 24 mo and after)
FOR 1998
Age
24
36
48

Original AY 1998 Data
Counts # Claim $000
From (A)
From (B)
35,000
262,500
50,000
375,000
50,000
375,000

ADJUSTED AY 1998 Data
Counts #
Claims $000's (Cumulative)
From (D)
Linearly Interpolated from left
40,000
300,000 see below
50,000
375,000 as for unadjusted
50,000
375,000 as for unadjusted

For example, adjusted paid losses at 24 months are calculated as:
262,500 + (40-35)/(50-35) x (375,000-262,500) = 300,000
ADJUSTED AY 1999 Data
Original AY 1999 Data
Counts # Claim $000
Counts #
Claims $000's (Cumulative)
Age
From (A)
From (B)
From (D)
Linearly Interpolated from left
24
39,375
295,313
42,000
315,000 see below
36
52,500
393,750
52,500
393,750 as for unadjusted
For example, adjusted paid losses at 24 months are calculated as:
295,313 + (42,000-39,375)/(52,500-39,375) x (393,750-295,313) = 315,000
FOR 1999

(E) from calculations immediately above, we have ADJUSTED PAID ($) DATA
ADJUSTED Cumulative Paid Claims ($000's)
Accident
Age of Development (Months)
12
24
36
48
Year
300,000
375,000
375,000
1998
1999
315,000
393,750
2000

4) Use ADJUSTED PAID LOSS DATA to develop a CDF to apply to AY 2000
(F) Based on the adjusted data in table (E)
ATA calculations
24 to 36
1998
1.25
1999
1.25

36 to 48
1.00
n/a

48 to Ult.

1.25

1.00

Selected (Simple Average)

at 24 mo

CDF calculations
CDF to Ultimate

1.25

5) Apply the CDF to AY 2000
Accident
Year
2000

Exam 5, V2

12
Latest

Cumulative Paid Claim $000's
Age of Development (Months)
24
36
48
330,000

CDF

multiplied by the selected 1.25 =

Page 325

ANSWER
Projected
Ultimate
412,500

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2004 Exam questions (modified):
22. (1 point) Based on Berquist and Sherman, state and briefly describe the two problems that can be
mitigated by analyzing loss experience by separate size of loss categories.
1. The claims department may have shifted their focus from settling small claims to settling large
claims. This could potentially disrupt the assumption that as the rate of closure goes up, the losses paid
on those claims goes up.
2. Claim adjusters may change the way they handle small, trivial claims. This change in operations
could distort the combined large and small database of claims.
23. a. (1.5 points) Based on Friedland’s explanation of Berquist and Sherman's method, demonstrate
why you might conclude that the relative level of Case Outstanding adequacy is different for accident
year 2003 as of 12 months than for earlier accident years. Show all work.
2004 # 23a : Test for change in Adequacy of Case Outstanding
Given Average Case Outstanding per Open Claim
(A) given

Average Case Outstanding $ per Open Claim Count (Severity)
Accident
Development Period (months)
Year
12
24
36
$18,840
$12,720
2001 $11,870
2002
12,580
19,963
2003
14,234

Now, compare the growth % indications here to the implied trend %s from paid data.
Friedland says, "Berquist and Sherman note that the observed trends for the average paid claims are
similar to industry benchmarks (at the time), and thus they conclude that the (different) trend rates for
average case oustanding are indicative of changes in case outsanding adequacy." We make a
similar assumption here.
Calculate growth rate % in Average Case Outstanding per Open Claim

(B)
from (A)

Average Case Outstanding $ per Open Claim Count (Severity)
Accident
Development Period (months)
Year
12
24
36
2002
6.0%
6.0%
2003 13.1% **
Example: 14,234 / 12,580 - 1 = 13.1%

Given Average Paid Claim ($) per Closed Claim Count

(C)

Exam 5, V2

Accident
Year
2001
2002
2003

Average Paid $ per Closed Claim Count (Severity)
Development Period (months)
12
24
36
10,600
21,200
26,500
11,236
22,472
11,910

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2004 # 23a : Test for change in Adequacy of Case Outstanding - continued
Calculate trend rate % in Average Paid Claim ($) per Closed Claim Count

(D)
from (C)

Calculation of the Trend Factor we tak e to be "true"
Accident
Development Period (months)
Year
12
24
36
2002
6.00%
6.00%
2003
6.0% **
Example: 22,474 / 21,200 - 1 = 6%

Compare growth rate %s in average outstanding data (B) to the trend rate %s in the paid data (D).
** The average open claim amount has risen from 6% to 13% compared to a 6% increase in average
paid severities over time. This demonstrates why we may conclude that the relative level of case
outstanding adequacy is changing. **

Solution to 2004 # 23 b. (1.5 points) Calculate an estimate for Ultimate Reported Claims for accident
year 2003 using Berquist and Sherman's technique for adjusting data to compensate for changing
Case Outstanding adequacy. Show all work.

Step 1: Begin by restating the average open severity using a 6% trend (De-trending)
Average Case Oustanding $ per Open Claim: ADJUSTED
Start with most recent diagonal given in (A) and DE-TREND at 6%
(E)
Accident
Development Period (months)
Notes on Calculations
Year
12
24
36
2001
12,668
18,833
12,720
13,428 = 14,234 / 1.06
2002
13,428
19,963
2003
14,234
12,668 = 13,428 / 1.06
Step 2: Multiply re-stated averages above by the open counts for re-stated Case Outstanding $
Re-stated Total $ Case Outstanding = Adjusted Average Case Outstanding * Open Counts (#)
(F) =
Accident
Development Period (months)
Notes on Calculations
(E) * open
Year
12
24
36
counts
2001 1,266,821 1,694,972
636,000
(given)
2002 1,342,830 1,796,670
1,342,800 = 13,428 * 100
2003 1,423,400
Step 3: Add re-stated case outstanding $ to cumulative paid claim $ for Reported Claims
Adjusted Reported Claims = Re-stated Case Outstanding + Cumulative Paid Claims
Accident
Development Period (months)
Notes on Calculations
(G) =
Year
12
24
36
(F) + Paid
2001 2,199,821 4,026,972 4,834,000
(given)
2002 2,331,830 4,269,670
4,269,670 = 1,796,670 + 2,473,000
2003 2,472,400

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solution to 2004 #23 (continued):
2004 # 23b : Example with change Adequacy of Case Outstanding - continued
Step 4: Compute ATA factor for 12-24 months using the adjusted reported claims, and
Use the given 24 to ultimate CDF to compute the 12 month age to ultimate CDF
ATA
2001
2002
Selected
Ult CDF

Development Period (months)
12:24
1.831
1.831
1.831

Reported CDF from 24 months to ultimate is 1.426 (given)

12-to-ultimate = 1.831 * 1.426 =

2.611

Finally, we estimate AY 2003 ultimate claims = 2,472,400 * 2.611 = 6,455,436

ANSWER

Solutions to 2005 Exam questions (modified):
8. (3.5 points)
a. (1.5 points) Based on Berquist and Sherman's method, demonstrate that the relative level of the
Case Outstanding adequacy has changed for accident year 2004.
2005 # 8a : Test for change in Adequacy of Case Outstanding
Given Average Case Outstanding per Open Claim
(A) = [Reported $ - Paid $ (cumulative)] / [Open Counts]
Accident
Development Period (months)
Year
12
24
36
800
1,000
1,200
2002
2003
840
1,050
2004
1,008
Now, compare the growth % indications here to the implied trend %s from paid data.
Friedland says, "Berquist and Sherman note that the observed trends for the average paid claims are
similar to industry benchmarks (at the time), and thus they conclude that the (different) trend rates for
average case oustanding are indicative of changes in case outsanding adequacy." We make a
similar assumption here.
Calculate growth rate % in Average Case Outstanding per Open Claim

(B)
from (A)

Average Case Outstanding $ per Open Claim Count (Severity)
Accident
Development Period (months)
Year
12
24
36
2003
5.0%
5.0%
2004 20.0% **
Example: 1008 / 840 - 1 = 20%

Continues below.

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2005 # 8a : Test for change in Adequacy of Case Outstanding - continued
Given Average Paid Claim ($) per Closed Claim Count

(C)

Accident
Year
2002
2003
2004

Average Paid $ per Closed Claim Count (Severity)
Development Period (months)
12
24
36
500
800
1,050
525
840
551

Calculate trend rate % in Average Paid Claim ($) per Closed Claim Count

(D)
from (C)

Calculation of the Trend Factor we tak e to be "true"
Accident
Development Period (months)
Year
12
24
36
2003
5.00%
5.00%
2004
5.0% **

Compare growth rate %s in average outstanding data (B) to the trend rate %s in the paid data (D).
** The average open claim amount has risen from 5% to 20% compared to a 5% increase in average
paid severities over time. This demonstrates why we may conclude that the relative level of case
outstanding adequacy is changing. **

Solution to 2005 8 b. (2 points) Using Berquist and Sherman's technique for adjusting data to
compensate for changing Case Outstanding adequacy, calculate the ultimate reported claims for
accident year 2004.
Step 1: Begin by restating the average open severity using a 5% trend (De-trending)
Average Case Oustanding $ per Open Claim: ADJUSTED
Start with most recent diagonal given in (A) and DE-TREND at 5%
(E)
Accident
Development Period (months)
Example Calculation
Year
12
24
36
2002
914.3
1,000
1,200
1,008 / 1.05 2 = 914.3
2003
960.0
1,050
2004
1,008
Step 2: Multiply re-stated averages above by the open counts for re-stated Case Outstanding $
Re-stated Total $ Case Outstanding = Adjusted Average Case Outstanding * Open Counts (#)
(F) =
Accident
Development Period (months)
Example Calculation
(E) * open
Year
12
24
36
counts
2002
9,143
8,000
2,400
914.3 * 10 (given) = 9,143
(given)
2003
11,520
6,300
2004
11,088
Step 3: Add re-stated case outstanding $ to cumulative paid claim $ for Reported Claims
Adjusted Reported Claims = Re-stated Case Outstanding + Cumulative Paid Claims
Accident
Development Period (months)
Example Calculation
(G) =
Year
12
24
36
(F) + Paid
2002
19,143
40,800
62,250
9,143 + 10,000 (given) = 19,143
(given)
2003
24,645
42,420
2004
23,761

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2005 # 8b : Example with change Adequacy of Case Outstanding - continued
Step 4: Compute ATA factors for 12-36 months using the adjusted reported claims, and
Use the given 36 to ultimate CDF to compute the 12 month age to ultimate CDF
ATA
2002
2003
Selected (Simple Avg)
CDF to Ultimate

12:24
2.131
1.721
1.926

Development Period (months)
Example Calculation
12:36
1.526
40,800 / 19,143 = 2.131
1.526

Reported CDF from 36 to ultimate is 1.1(given)

12-to-ult = 1.926 * 1.526 * 1.1 =

3.233

Finally, we estimate AY 2004 ultimate claims = 23,761 * 3.233 = 76,819

ANSWER

Solutions to 2005 Exam questions (modified) - continued:
9. (2.5 points) Berquist and Sherman describe an approach to adjust the paid loss triangle for distortion.
a. (0.5 point) Identify the distortion for which this adjustment is intended.
b. (2 points) Describe the technique to make the necessary adjustment.
Question 9 – Based on Model Answer 2
a. Change in Claim Settlement Rate
b. 1. Calculate disposal ratios (cumulative claims closed per ultimate count).
2. Use the latest diagonal of ratios as a base and restate claims closed triangle based on the disposal rates.
3. Once all claims are restated to latest diagonal closed claim percentages for each accident year,
use linear interpolation to determine amount of cumulative paid claim $ that corresponds to the # of
claims closed in each interval.
4. The result is the adjusted paid triangle.
Solutions to 2006 Exam questions (modified):
Solution to 2006 Question 20 – Based on Model Answer 1
a. It is important to know about any changes made to claims department processes. This will affect claim
closure patterns and reserve adequacy among other things.
It is important to talk with underwriting so that you know about any changes in the mix of business or
new exposures. These will also affect loss emergence and loss magnitudes.
b. 1) Has there been a change in priorities as far as settling large claims versus small claims?
2) Has there been a change in philosophy regarding trivial or very small claims?
Solution to 2006 Question 20 – Based on Model Answer 2
a. Claims handling practices may have changed over the experience period.
For example: change in priority on small vs. large claims, change in procedures for handling small
claims, increase or decrease in number of adjusters, change in amount of assistance from outside claims
adjusters. These changes can have an impact on the timing of loss and LAE payments and the ultimate
amount of the loss and LAE.
b. Underwriting policies and procedures may have changed over experience period.
1) Has there been a change in priority in handling small vs. large claims?
2) Has the caseload per claim department adjuster changed?

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ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2006 Exam questions (modified) - continued:
21. (2 points) In the course of a reserve analysis, it is observed that the paid claim development triangles
are distorted by significant changes in the claims settlement rate. Briefly describe the procedure that
Berquist and Sherman propose to address this situation, as described in Friedland.
1. If you are given an accident year history of cumulative paid claims ($) and cumulative closed and
ultimate claim counts (#), examine ultimate Claims Disposal Rates (cumulative closed claims
divided by projected ultimate claims). If shifts in claims disposed ratios are present, select
appropriate Claims Disposal Rates by age of development.
2. The selected rates from (1) are applied to projected ultimate claims to obtain the number of
cumulative closed claim counts which would be equivalent to the indicated claims disposed ratio for
that age of development by accident year. These are (adjusted or) restated closed claim counts(#).
3. To approximate the adjusted cumulative paid claims ($) which correspond to the restated counts, a curve
is fit between 2 points (claim countX, cumulative paid claimsX) & (claim countX+1, cumulative paid
claimsX+1). Although Friedland shows how Berquist & Sherman determined that an exponential curve fit
the data well, the instructions in many problems state that one should assume that the relationship is
linear.
Solutions to 2007 Exam questions (modified):
Question #33 – Model Solution 1. Note this can be determined in a 2 step process as follows:
Step 1: Construct the Adjusted Average Case Outstanding per open claim triangle by de-trending the
latest calendar year diagonal, using the annual paid severity trend.
Adjusted Average Case Outstanding
Accident Year
12
24
36
Examples
2004

8,734.40

16,261.10

2005

9,345.80

17,400.00

2006

10,000.00

25,192.00

9,345.8 = 10,000/1.07
2
8,734.4 = 10,000/1.07

Step 2: Construct the Adjusted Reported Claims ($) Triangle by multiplying the above triangle by open
claim counts and adding the cumulative Paid Claims:
Adjusted Reported Claims
Examples
Accident Year
12
24
36
5,889,400 = 25,192 x 75 + 4,000,000
2004
5,093,760
6,805,275
5,889,400 6,805,275 =16,261 x 250 + 2,740,000

Exam 5, V2

2005

6,865,190

2006

6,275,000

8,505,000

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2007 Question #33 – Model Solution 2. Note this can be determined in a 3 step process as follows:
Step 1: De-trend the current diagonal of average Case Outstanding per open claim count by 7% per yr:
AY
04
05
06

12
8,734.39
9,345.79
10,000.00

24
16,261.68
17,400.00

Note: 16,216.68 = 17,400/1.07

36
25,192.00

8,734.39 = 10,000/1.07

9,345.79=10,000/1.07

Step 2: Multiply each value by the number of open claims (counts) to get adjusted Case Outstanding
data:
AY
04
05
06

12
3,493,756
5,140,185
4,500,000

24
4,065,421
5,655,000

36
1,889,400

Step 3: Add the above triangle to current paid data to get Adjusted Reported Claims:
AY
04
05
06

12
5,093,756
6,865,185
6,275,000

24
6,805,421
8,505,000

36
5,889,400

Solution to 2007 Question 34 – Based on Model Solution 1
1. A slow payment / emergence pattern reduces the utility of available claim experience data, and
2. Irregular payment patterns and the variation of the portion of claims closed without payment distorts trends.
Solution to 2007 Question 34 – Based on Model Solution 2
Estimating severity trends for malpractice claims using paid claims is inappropriate because:
1. The slow payment of claims substantially reduces the experience available by accident year for trending
2. Trends in severity are distorted by irregular settlements and variation in the rate of claims closed
without payment.
Solution to 2007 Question #38 – Based on Model Solution 1:
a. 1. An increase in Claims Settlement rate.
2. An improvement in Case Outstanding adequacy
3. Writing business with better loss ratios (e.g. this affects the BF, Expected Claims method, etc.)
b. 1. A sudden increase in LDF→this overstates the reserve estimate.
2. A sudden increase in LDF→this overstates the reserve estimate.
3. This has no effect since the CL method uses current loss experience to estimate loss reserves.
c. 1. Apply the B&S adjustment – to restate closed claim #s using the current settlement rate percentages.
2. Apply B&S adjustment – adjust past case outstanding to reflect current level of case outstanding
adequacy.
3. No adjustment is needed for CL, but for the BF method, the expected loss ratio should be adjusted
accordingly

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ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solution to 2007 Question #38 – Based on Model Solution 2:
a.
1. Case Outstanding Adequacy change
2. Claim Settlement Rate change
3. Mix of business change (higher policy limit)
b.

1. case reserve strengthening → chain ladder overstates
2. rate of claim settlement increase →chain ladder overstates
3. people buying higher policy limit → chain ladder overstates
1. Use Berquist/Sherman approach, detrend average open with appropriate trend and multiply with
open to get adj case outstanding, add pd loss to get adj reported claim $ triangle
2. Use Berquist/Sherman approach, calculate disposal rate (= closed claim/ultimate claim) and
restate the closed claim # triangle and interpolate to estimate restated paid claims $ triangle
3. Use PY instead of AY, trend loss data and policy limit

Solution to 2007 Question #39 – Based on Model Solution 1:
a. Claims Consciousness increases in a given area. This can increase claim frequency as more people
will be more aware of how to sue or file a claim.
b. More liberal awards given out by juries in a given county. If awards for certain juries are more liberal in
amounts awarded, larger claim amounts will be submitted and claim severities will increase.
c. They are related to each other. They have a combined impact on pure premiums. For example, an
increase in the frequency of certain claims can rapidly impact the severity of those claims.
Solution to 2007 Question #39 – Based on Model Solution 2:
a. A new law that allows claims that were not previously considered such as allowing work stress as a
workers comp claim when previously not considered an injury. Frequency would increase.
b. A new judicial ruling that sets a precedent for high environmental damages for pollution that was
previously an unheard amount (i.e., the need to clean up to a new degree of "clean" or to compensate
people at a further distance around the site than previously considered).
c. The total loss to the company is the average frequency times average severity. By only considering
one aspect you are ignoring the compound effect on the aggregate distribution from the correlation
between the two. Ex: Two additional small claims have much less of an impact than 2 additional large
claims have on total claims.
Solution to 2008 #4 part a (Berquist Sherman, as in Friedland's Chapter 13)
3 STEPS TO ADJUSTING THE DATA (for changes in Case Outstanding adequacy)
1: Begin by restating the average Case O/S per Open using appropriate trend
2: Multiply re-stated averages above by the open counts for re-stated Case O/S $
3: Add re-stated Case Outstanding $ to cumulative Paid Claim $ for Reported Claims
This adjusted Reported Claims triangle will then be used in Part b.

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ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
WORKING IN $000's
Step 1: Begin by restating the average Case O/S per Open using appropriate trend
(A) Start with most recent diagonal given in (E) and DE-TREND at 5% as given
Accident
Average Case Outstanding $ per Open Claim: ADJUSTED (de-trended)
Year
12
24
36
48 Notes on De-trending
2004
68.934
99.048
375.429
888.700 *** Divide latest by 1.05 3
2005
72.381
104.000
394.200
** Divide latest by 1.05 2
2006
76.000
109.200
* Divide latest by 1.05
2007
79.800
Ex: 375.43 = 394.2/1.05
Starting values are from the latest diagonal given for the Avg Case O/S per open claim.
Step 2: Multiply re-stated averages above by the open counts for re-stated Case O/S $
(B) Recall, given Open Counts
AY
12
2004
810
2005
698
2006
654
2007
633

24
480
387
361

36
115
87

48
43

Then, Re-stated Total $ Case O/S = Adjusted Average Case O/S * Open Counts (#)
(C) = (A) * (B)
(Adjusted) Re-stated Case Outstanding Amounts
Notes
AY
12
24
36
48 Ex: 24-mo calculation
2004
55,837
47,543
43,174
38,214 for AY 2005:
2005
50,522
40,248
34,295
Multiply 387 (# open) by
2006
49,704
39,421
adj. Average Case O/S
of 104.00, for 40,248
2007
50,513
Step 3: Add re-stated Case O/S $ to cumulative Paid Claim $ for Reported Claims
(D) Recall, given
AY
2004
2005
2006
2007

Cumulative Paid Losses
12
24
36
30,729
103,361
125,237
24,573
85,337
105,979
22,567
88,009
27,761

48
138,547

Finally, (E) = (C) + (D)
Adjusted Reported Claims = Re-stated Case Outstanding + Cumulative Paid Claims
AY
12
24
36
48 Ex: 24-mo calculation
2004
86,566
150,904
168,411
176,761 for AY 2005:
2005
75,095
125,585
140,274
Add the adjusted O/S of
2006
72,271
127,430
40,280 to the paid of
85,337 for 125,585
2007
78,274

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
* Note, latest diagonal here does not match the cumulative reported amounts that were
given in the actual exam, due to the fact that the cumulative reported amounts
that were shown for original exam did NOT = Paid + (Avg Case O/S) * (Number Open).
The CAS solution forces the diagonal to match by not showing certain calculations.
Either way, this triangle can now be used to calculate CDF's in part (b) …

Solution to 2008 Exam question #4 (continued):
(b) To estimate the AY 2007 Ultimate Claims, follow 2 steps:
(i) Use the adjusted data triangle to develop ATA and CDFs to ultimate
(ii) Apply the factors to reported claims to estimate Ultimate Claims
(i) We're told to use all-year volume-weighted ATA selections,
and 1.02 as the CDF-to-ult at 48 months, so:
weighted
12: 24 mo 24: 36 mo 36: 48 mo
1.727
1.116
1.050
ATA
at
12
mo
at
24
mo
at
36
mo
Ultimate
at 48 mo
2.064
1.195
1.071
1.020 (1.02 given)
CDF
Example: weighted ATA factor for 12-to-24:
1.727 = (150,904+125,585+127,430)/(86,566+75,095+72,271) from (a)
(ii) Apply the factors to estimate Ultimate Claims
Age of Reported
Reported
Accident
Data at Claims at
CDF to
Year
12/31/07
12/31/07
Ultimate
(1) (2) See note
2007 12 months
78,274

(3) above
2.064

Expected
Ultimate
Claims
(4)=(2)*(3)
161,540

Comments:
* See note re: reported loss triangle provided on the original exam.
Due to reconcilliation issues, this might not be the best illustration.
$78,294 is the value given as reported for AY 2007.
$78,274 is from the calculated sum of the paid + case O/S
If the amounts reconciled, we'd use $78,294 value given.
Other years differ more and also impact the calculations of CDFs.
As mentioned, the CAS answer forces the latest diagonal to match the
values shown in the reported triangle. The sample solution is 161,920.

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solution to 2008 #5 using quotes from first page of Friedland's Chapter 13
Berquist and Sherman recommend that, wherever possible, the actuary should use data that is relatively
unaffected by changes in the insurer’s claims and underwriting procedures and operations.
a) "Substituting policy year data for accident year data when there has been a significant change in
policy limits or deductibles between successive policy years."
b) "Substituting accident quarter for accident year when the rate of growth of earned exposures changes
markedly, causing distortions in development factors due to significant shifts in the average accident
date within each exposure period. "
c) "Substituting report year data for accident year data when there has been a dramatic shift in the social
or legal climate that causes claim severity to more closely correlate with the report year than with the
accident date. "
d) "Using earned exposures instead of the number of claims when claim count data is of questionable
accuracy or if there has been a major change in the definition of a claim count."

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solution to 2009 Questions
Question 11 - Model Solution 1
a. Step 1: Compute Adjusted Avg Case Reserves. Use the latest diagonal from the given avg case reserve
triangle and compute each value along preceding diagonals as the current diagonal value divided by
the given severity trend factor of 1.05 (e.g. 7,170 = 7,528/1.05; 6,828 = 7,170/1.05)
AY
2005
2006
2007
2008

Adjusted Avg Case Reserves
12
24
36
3,563.3301
6,828.1179
8,990.4762
3,741.4966
7,169.5238
9,440.0000
3,928.5714
7,528.0000
4,125.0000

48
15,000.0000

Step 2: Compute Adjusted Case Reserves=Adjusted Avg Case Reserves × Number of Open Claims (given)
AY
2005
2006
2007
2008

12
24
36
48
121,153.2234 136,562.3580 134,857.1430 120,000.0000
205,782.3130 250,933.3330 226,560.0000
294,642.8550 376,400.0000
346,500.0000

Step 3: Compute Adjusted Reported Loss = Adjusted Case Reserves + Cumulative Paid Loss (given)
AY
2005
2006
2007
2007

12
24
36
48
291,153.2234 456,562.3580 584,857.1430 620,000.0000
425,782.3130 670,933.3330 856,560.0000
654,642.8550 1,026,400.0000
796,500.0000

Step 4: Selected Adjusted Reported Loss Link Ratios (using a-t-a ratios along latest diagonal)
12-24
1.5679
LDFs
2.1645

24-36
1.2767

36-48
1.0601

48-ult
1.02

1.3805

1.0813

1.02

AY2008 Ult Loss = 796,500 × 2.165 =

1,724,024

b. To adjust for case reserve inadequacy by setting all case reserves to the current calendar year level.
Question 11 - Model Solution 2. A more efficient way to compute Adjusted Incurred triangle
= Adjusted Avg Case Reserves × Open Claims Count + Paid Loss
AY
5
6
7
8

12
24
36
48
291,153.2234 456,562.3580 584,857.1430 620,000.0000
425,782.3130 670,933.3330 856,560.0000
654,642.8550 1,026,400.0000
796,500.0000

b. The purpose of the Berq Sher case reserve adjustment is to restate the case incurred triangles as if the
same level of case outstanding adequacy had been maintained for all years. Hence we consider the case
reserves along the latest diagonal and trend them back for older AY and restate the case incurred losses.

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solution to 2010 Questions
1a. (1.75 points) Test the above data for changes in case reserve adequacy and interpret the results.
1b. (0.75 point) Describe the leveraging effect that a change in case reserve adequacy has on the IBNR
indicated by the reported loss development method.
1c. (1.5 points) Use the Berquist-Sherman technique for case reserve adequacy to calculate the adjusted
reported claims for each accident year.
Question 1 – Model Solution 1
a)

AY
2006
2007
2008
2009

Average Paid
Severity
=Paid/Closed
Counts
3.4600
3.5556
3.6984
3.8415

Change
from
prior AY
--1.0276
1.0402
1.0387

Case O/S
=Rep. - Paid

Average Case O/S
=Case O/S/Open
Counts

Change
from
prior AY

7,611
20,567
26,317
40,022

5.0007
5.6518
6.3892
7.2190

--1.1302
1.1305
1.1299

The average case outstanding increases much more rapidly that the average paid severity. This
indicates an increase in case outstanding adequacy.
b)

AY
2006
2007
2008
2009

Higher case adequacy leads to higher LDFs. Moreover, these higher LDFs are applied to higher
reported amounts, thus overestimating both ultimate losses and IBNR.
Detrend at 4% per year, consistent with inflation on paid severity
(1)
Restated
Avg Case
O/S
5.7052
6.4176
6.9413
7.2190

(2)
Open
Counts

(3)
Paid
Claims

(1) × (2) + (3)
Adjusted
Reported

1,522
3,639
4,119
5,544

9,688
17,778
25,519
34,093

18,371.3144
41,131.6464
54,110.2147
74,115.1360

6.4176 = 7.219 / 1.04^2
6.9413 = 7.219 / 1.04

Question 1 – Model Solution 2
a) Test the above data for changes in case reserve adequacy and interpret the results.
Average Paid
Trend in
AY
Severity
Average Paid
2006
3.4600
2006-2007
2.7630%
= 3.5556 / 3.4600 - 1
2007
3.5556
2007-2008
4.0162%
2008
3.6984
2008-2009
3.8692%
2009
3.8415
Selected Average
3.5495%
Average Case
Trend in Average
Outstanding
Case Outstanding
2006
5.0007
=(17,299-9,688) / 1,522
2006-2007
13.0202%
2007
5.6518
2007-2008
13.0472%
2008
6.3892
2008-2009
12.9875%
2009
7.2190
Since average case is increasing 13% per year, but average paid is only increasing 3.5% per year, there is a
strengthening of case adequacy.
AY

b) If there is an increasing case reserve adequacy, it will overstate the IBNR using reported development
method. This is due to historical LDFs (too high) being applied to reported claims.
c) Use the Berquist-Sherman technique for case reserve adequacy to calculate the adjusted reported claims for each AY
Restated
Restated
AY
Avg Case
AY
Reported
2006
6,511.1243
2006
19,597,931.18
2007
6,739.0137 =6974.8792 / 1.035
2007
42,301,270.85
2008
6,974.8792 = 7,219 / 1.035
2008
54,248,527.42
=(6974.9 x 4,119) + 25,519 x 1,000
2009
7,219.0000
2009
74,115,136.00

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solution to 2011 Questions
31a. (2.75 points) Use the Berquist-Sherman technique for case reserve adequacy to estimate the ultimate
claims for all accident years.
31b. (0.25 point) Briefly describe the purpose of the Berquist-Sherman case reserve adjustment.
31c. (0.75 point) Discuss whether changing the annual severity trend given above from 10% to 5% would
produce a higher or lower ultimate claims estimate under the Berquist-Sherman technique for case
reserve adequacy.
Question 31 – Model Solution 1 – part a
Step 1: Compute Avg Case o/s (Case o/s / open claim count)
AY
12
24
36
8
12,000.0000 16,800.0000
18,018.4000
9
17,100.0000 17,905.0000
10
18,700.0000
Step 2: Compute Adj. Avg Case o/s (10% Trend)
AY
12
24
36
8
15454.55
16277.27
18018.00
9
17000.00
17905.00
10
18700.00
Detrend current diagonal by 10%
Step 3: Compute Adj Reported [(adj Avg Case) x (open count) + (paid)]
AY
12
24
36
8
32,066.92
40,697.81
45,941.40
9
29,780.00
39,940.00
10
37,310.00
Selected ATAF (USING weighted avg): Σ(x+12) / Σx
AY
12-24
24-36
36-Ult
Selected
1.3052
1.1288
1.05 (given)
Cum.
1.5469
1.1852
1.05

AY
8
9
10

(1)
Reported @
$45,942
$39,940
$37,310

(2)
LDF
1.0500
1.1852
1.5469

(3) = (1) x (2)
Ultimate Losses
48,239.10
47,336.89
57,714.84
153,259,771.00

b. If there has been an increase in reserves, the unadjusted LDF’s will overestimate
development. The B-S technique adjusts the Avg O/S to avoid this.

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solution to 2011 Questions
31c. (0.75 point) Discuss whether changing the annual severity trend given above from 10% to 5% would
produce a higher or lower ultimate claims estimate under the Berquist-Sherman technique for case
reserve adequacy.
Question 31 – Model Solution 1 – part c
Compute Adj. Avg Case o/s (using a 5% severity trend factor)
24
36
12
AY
8
16961.45
17052.38
18018.00
9
17809.52
17905.00
17,809.52 = 18,700 / 1.05
10
18700.00
Compute Ajd Reported [Adj Avg Case x Open Count + Paid]
24
36
12
AY
8
34821.53
42170.52
45941.40
9
31026.66
39940.00
31,026.66 = 17,809.52 * 1,540/1000 + 3,600
10
37310.00
Selected Factors (using weighted avg)
12-24
24-36
36-48
A-t-A (5%)
1.2492
1.0894
1.05
LDF ult (5%)
1.4290
1.1439
1.05
LDF ult (10%)
1.5469
1.1852
1.05
For ages 12 & 24, the ultimates would be lower using a 5% trend opposed to a 10% trend.

Question 31 – Model Solution 2 – part b
b 1. Trend back from the latest diagonal so like that our case adequacy will all be at the same level.
2. Using the new adjusted avg case o/s calculate the adj reported claims
Avg case o/s x open claim counts + unadj. Paid claims
Question 31 – Model Solution 2 – part c
If we trend at a lower rate, then the avg case o/s will be higher and thus our adjusted rptd claims will be
higher for the years we bring to the same adequacy level. Thus our CDF ult will be lower and our ultimate
claims will be lower.

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to questions from the 2012 exam
22a. (0.75 point) Estimate the ultimate claims for accident year 2011 using the reported development technique.
22b. (3 points) Estimate the ultimate claims for accident year 2011 using the Berquist-Sherman case outstanding
adjustment technique.
Question 22 – Model Solution (Exam 5B Question 7)
(a) => Age-to-age factors for reported claims:
AY
12-24
24-36
Selected
9 1.1663
1.1269
10 1.3412
12-24 =
24-36 =
36-ult =
Straight avg. = 1.2538, 1.1269
Volume weighted avg. = 1.2654, 1.1269
AY2011 Ultimate claims = 12,924 (1.4129) =

1.1269
1.2538
1.0000
12-to-ultimate =
18,260

1.4129

(b) => Case O/S triangle ($000) = Cumulative Reported- Cumulative Paid AY 12 mths
AY
12 mths
24 mths
36 mths
09
6,220.00 2,836.00
695.00
10
9,503.00 8,395.00
11
9,796.00
=> Average Case O/S triangle = Case O/S / open claim counts
AY
12
24
36
09
18,028.99 16,982.04 23,166.67
10
19,044.09 23,985.71
11
22,519.54
=> Adjusted average case O/S triangle (using 8% trend and trending back from latest diagonal):
AY
12
24
36
09
19,306.88 22,208.99 23,166.67
10
20,851.43 23,985.71
11
22,519.54
=>Adjusted Reported Triangle ($000):=(Adjusted average case O/S * open claim counts) + Cumulative Paid Claims
AY
12
24
36
09
10,371.87 12,455.90 13,053.00
10
13,868.86 17,391.00
11
12,924.00
AY
12-24
24-36
9
1.2009
1.0479
10
1.254
Selected
1.2275
1.0479
→ Ultimate mate claims ($000) = 12,924 (1.228 x 1.048) = 16,624.11

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to questions from the 2012 exam
22c. (0.5 point) Discuss the difference between the two estimates.
Question 22 – Model Solution (Exam 5B Question 7)
(c) The development method from part (a) overestimates ultimate claims because it does not recognize
the increase in case adequacy that can be seen when the annual change in average case O/S is
analyzed (at 12 months). That is 18.25% is much greater than 5.63%
AY
2009
2010
2011

12 months
18,028.99
19,044.09
22,519.54

Change
5.63%
18.25%

The method from part (b) restates historical data at the curr case adequacy level, whereas the
development factors in part (a) are too high.
Examiner’s Comments
Overall, the candidates did well on this question. For many candidates, only a minor omission in the
discussion or a computation error in the methods kept them from achieving full marks.
a. Most candidates appropriately demonstrated the reported development method. The most common
errors found were computation errors. A few candidates opted to use the latest 12-24 age-to-age
factor rather than some sort of average. Although this exacerbated the problem for the method, this
selection was accepted where clearly indicated.
b. The candidates are generally able to demonstrate the Berquist-Sherman method, with computation
errors being the most common type of error. Where candidates struggled with the methodology, they
generally recognized the method makes adjustments at the average case outstanding level. The
struggle is usually with the application of the trend to the average outstanding and with the process to
go from the adjusted average case outstanding back to the adjusted reported claims.
c. A common mistake found in the discussion is the claim that the reported development method does
not account for trend. This is imprecise. The reported development method is a reasonable method in
a stable environment, including stable trends. It is the change in the pattern that causes problems
with the reported development method, and some candidates failed to make this
distinction. The candidates were expected to highlight the changing patterns and make the
connection this causes issues for the reported development method, which the Berquist-Sherman
method attempts to address.

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Chapter13 – Berquist-Sherman Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to questions from the 2012 exam
24a. (2 points) Evaluate whether a Berquist-Sherman Case Outstanding Adjustment would be
appropriate.
24b. (1.5 points) Use disposal rates to evaluate whether a Berquist-Sherman Paid Claim Development
Adjustment would be appropriate.
Question 24 – Model Solution (Exam 5B Question 8)
a.
Open claim count = Reported - Closed
12
24
36
09
1,000
700
355
10
900
775
11
775

Avg O/S = O/S Claims / Open Count
12
24
36
09
8.7150
13.1586
11.1099
10
8.7156
13.1574
11
8.7161
Note: Observed trend of close to 0%

Avg Paid over Closed claim count
12
24
36
09
5.7860
7.2725
7.6218
10
5.7853
7.2725
11
5.7855
Observed trend of close to 0%.

Since both avg paid and avg o/s are stable with similar trend, it does not seem necessary to adjust
historical case o/s w/ Berquist Sherman method.
b.
DF on reported claim count
12-24
24-36
09
1.2667
1.0789
10
1.2727
Average
1.2697
1.0789

Ultimate Claim Count
09
2,050.00
10
2,265.69 = 2100 * 1.0789
11
2,191.81 = 1600 * 1.2697 * 1.0789

Disposal rate = Closed Count over Ult Claim Count)
12
24
36
09
0.2439
0.5854
0.8268
10
0.3310
0.5848
11
0.3764

From the disposal rate, there appears to be a speeding up in settlement rate in the first 12 months, so it
is appropriate to adjust w/ Berquist Sherman paid claim adj. method.
Examiner’s Comments

a. Many candidates correctly calculated case outstanding and observed no trend, but failed to calculate trend

in paid severity for comparison. Some candidates calculated both case outstanding and paid severity and
to correctly state than no adjustment was needed, but then failed to explain why no adjustment was
needed.
Candidates who failed to receive full credit commonly did 1 of 3 things:
•

Calculated ultimate claims by developing paid claim counts to 36 months

•

Calculated disposal rates as (reported claim counts / ultimate claim counts)

•

Instead of disposal rates, calculated the ratio of closed claims to reported claims.

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Chapter14 – Recoveries: Salvage & Subro and Reins
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Sec
1
2
3
1

Description
Salvage, Subrogation, and Collateral Sources
Estimating S&S Recoveries-Auto Physical Damage Insurer
Reinsurance and Aggregate Limits

Pages
329
329 - 330
330 - 332

Salvage, Subrogation, and Collateral Sources

329

Ways in which data is maintained:
1. Detailed data may be maintained for case outstanding estimates and payments for the different
types of recoveries (e.g. salvage, subrogation, deductibles, and collateral sources).
2. Claims data may be combined for all types of recoveries.
3. Payments are recorded but estimates of case O/S recoveries may not be recorded.
4. Recoveries may be treated as a negative claim payment (separate data for recoveries is not
maintained).
To quantify the potential affect of S&S, the actuary must understand how the insurer processes such
recoveries and what data is available for analysis.
The development technique is used to quantify the affect of S&S recoveries on estimates of total unpaid
claims (when S&S data is available).
 Salvage is commonly associated with property coverages and tends to be fast reporting and settling.
 Subrogation, associated with liability coverage, can take years to realize, well after claims are paid,
resulting in age-to-age factors less than 1.00 for older maturities for some lines of business.

Estimating S&S Recoveries-Auto Physical Damage Insurer

329 - 330

Data from Auto Physical Damage Insurer is used to demonstrate two methods to quantify S&S recoveries.
This insurer maintains payment activity and case outstanding estimates for S&S.
1. The Development Method
Exhibit I, Sheets 1 and 2: Reported and Received Salvage and Subrogation ($000)
Comments on the term “received”:
i. The term “paid” S&S is often used instead of “received” S&S.
ii. Paid S&S represents a payment made by a third-party to the insurer.
Both auto physical damage and S&S associated with this coverage have quick reporting patterns.

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Chapter14 – Recoveries: Salvage & Subro and Reins
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
The reported S&S development factors are stable and indicate an age-to-age factor of 1.068 at 12-to24 months and less than 1.00 at 24-to-36 months.
The development factors for received S&S are also stable.
Selected factors are based on the latest 5-year volume-weighted average factors.
Chapter 14 - Recoveries: Salvage and Subrogation and Reinsurance
Auto Physical Damage Insurer
Reported Salvage and Subrogation ($000)

Exhibit I
Sheet 1

PART 3 Only - Average Age-to-Age Factors
12-24
Simple Average
Latest 5
1.067
Latest 3
1.074
Medial Average
Latest 5x1
1.072
Volume-weighted Average
Latest 5
1.068
Latest 3
1.074

24 - 36

36 - 48

48 - 60

0.998
0.997

1.000
1.002

1.000
1.001

0.999

1.000

0.998
0.997

1.000
1.002

Average
60 - 72

72 - 84

84 - 96

96 - 108

108 - 120 120 - 132

1.000
1.001

1.002
1.000

1.000
1.000

0.999

1.000

1.000
1.001

1.001
1.001

1.000
1.000

72 - 84
1.001
1.001
99.9%

84 - 96
1.000
1.000
100.0%

96 - 108
1.000
1.000
100.0%

To Ult

PART 4 - Selected Age-to-Age Factors

Selected
CDF to Ultimate
Percent Reported

12-24
1.068
1.067
93.7%

24 - 36
0.998
0.999
100.1%

Development
36 - 48
1.000
1.001
99.9%

Factor Selection
48 - 60
60 - 72
1.000
1.000
1.001
1.001
99.9%
99.9%

108 - 120 120 - 132
1.000
1.000
1.000
1.000
100.0%
100.0%

To Ult
1.000
1.000
100.0%

1. The Development Method (continued):
Exhibit I, Sheet 3: Projection of Ultimate Salvage and Subrogation ($000)
Chapter 14 - Recoveries: Salvage and Subrogation and Reinsurance

Exhibit I

Auto Physical Damage Insurer

Sheet 3

Projection of Ultimate Salvage and Subrogation ($000)
Age of

Projected Ultimate S & S
Using Dev Method with

Accident

Accident Year

Year

at 12/31/08

Reported

Received

Reported

Received

Reported

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

1998

132

793

1.000

793

2006

5,715

5,655

1.001

1.006

5,721

5,689

2007

6,031

5,957

0.999

1.022

6,025

6,088

2008

5,414

2,710

1.067

1.938

5,777

5,252

44,719

41,879

45,097

44,639

Total

S & S 12/31/2008

CDF to Ultimate

Received

Column Notes:
(2) Age of accident year in (1) at December 31, 2008.
(3) and (4) Based on data from Auto Physical Damage Insurer.
(5) and (6) Based on CDF from Exhibit I, Sheets 1 and 2.
(7) [(3) x (5)]
(8) = [(4) x (6)].

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Chapter14 – Recoveries: Salvage & Subro and Reins
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2. The Ratio Method
The first step: Estimate the ultimate claims gross of S&S.
Exhibit I, Sheets 4 – 5: Reported and Paid Claims Gross of S&S based on reported and paid claims.
CDFs are computed based on the 5-year volume-weighted averages
Exhibit I, Sheet 6. Projected ultimate claims Gross of S&S based on reported and paid claims.
Given this fast reporting and settling line of insurance, the projections are very similar, as expected.
Chapter 14 - Recoveries: Salvage and Subrogation and Reinsurance
Auto Physical Damage Insurer
Projection of Ultimate Claims Gross of S&S Using Reported and Paid Claims ($000)

Accident
Year
(1)
1998
:::
2007
2008
Total

Age of
Accident Year
at 12/31/08
(2)
132
:::
24
12

S & S 12/31/2008
Reported
Received
(3)
(4)
2,864
2,864
:::
:::
16,862
16,822
14,727
12,889
129,369
127,456

CDF to Ultimate
Reported
Received
(5)
(6)
1.000
1.000
:::
:::
1.001
1.005
1.115
1.279

Exhibit I
Sheet 6

Projected Ultimate S & S
Selected
Using Dev Method with
Ult. Claims
Reported
Received
Gross of S&S
(7)
(8)
(9)
2,864
2,864
2,864
:::
:::
:::
16,879
16,906
16,897
16,422
16,485
16,466
131,081
131,153
131,149

Column Notes:
(2) Age of accident year in (1) at December 31, 2008.
(3) and (4) Based on data from Auto Physical Damage Insurer.
(5) and (6) Based on CDF from Exhibit I, Sheets 4 and 5.
(7) = [(3) x (5)]
(8) = [(4) x (6)].
(9) = [Average of (7) and (8)].

The second step: Project Ultimate S&S
Exhibit I, Sheet 7: Using the development technique to analyze the ratio of received S&S to paid claims.
Chapter 14 - Recoveries: Salvage and Subrogation and Reinsurance
Auto Physical Damage Insurer
Ratio of Received Salvage and Subrogation to Paid Claims

Exhibit I
Sheet 7

PART 3 - Average Age-to-Age Factors

Simple Average
Latest 5
Latest 3
Medial Average
Latest 5x1

12-24

24 - 36

36 - 48

48 - 60

60 - 72

Average
72 - 84

84 - 96

96 - 108

1.496
1.474

1.012
1.013

1.000
0.999

1.002
1.003

1.001
1.001

1.005
0.999

1.000
1.000

1.485

1.010

1.000

36 - 48
1.000
1.000

48 - 60
1.000
1.000

108 - 120 120 - 132

To Ult

PART 4 - Selected Age-to-Age Factors

Selected
CDF to Ultimate

12-24
1.486
1.499

24 - 36
1.009
1.009

Development Factor Selection
60 - 72
72 - 84
84 - 96
1.000
1.000
1.000
1.000
1.000
1.000

96 - 108
1.000
1.000

108 - 120 120 - 132
1.000
1.000
1.000
1.000

To Ult
1.000
1.000

Advantages to using the ratio approach:
1. Development factors are not as highly leveraged as those based on received S&S dollars.
2. Relates to selecting ultimate S&S ratio(s) for the most recent year(s) in the experience period.

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Chapter14 – Recoveries: Salvage & Subro and Reins
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Exhibit I, Sheet 8: Projection of Ultimate Salvage and Subrogation ($000)
AY 2008 projected ultimate S&S ratio (.315) based on:
.210 (Exhibit I, Sheet 7 ratio of received S&S) * 1.499 (Exhibit I, Sheet 7 CDF)
However, compared to the immediate preceding years, 0.315 seems low.
This may be due to a change in recording S&S or a large claim.
Average ultimate S&S ratios for the last 5 years (ex 2008) is 0.347 and for the last 3 years (ex 2008) is 0.344.
Thus, an ultimate S&S ratio for 2008 of 0.345 is selected.
Ultimate S&S equals selected ultimate claims (Exhibit I, Sheet 6) * selected ultimate S&S ratio (Column (6)).
Chapter 14 - Recoveries: Salvage and Subrogation and Reinsurance
Auto Physical Damage Insurer
Projection of Ultimate Salvage and Subrogation ($000)

Accident
Year
(1)
1998
2007
2008
Total

Age of
Accident Year
at 12/31/08
(2)
132
:::
24
12

Ratio of
Received S&S to

Paid Claims
at 12/31/08
(3)
0.277
:::
0.354
0.210

CDF
to Ultimate
(4)
1.000
:::
1.010
1.499

Projected
Ultimate
Ratio
(5)
0.277
:::
0.357
0.315

Exhibit I
Sheet 8

Selected Ultimate
S&S
Ratio
(6)
0.277
:::
0.357
0.345

Claims
Gross of S&S
(7)
2,864
:::
16,897
16,466

Projected
Ultimate
S&S
(8)
793
:::
6,039
5,681

131,149

44,924

Column Notes:
(2) Age of accident year in (1) at December 31, 2008.
(3) From latest diagonal of triangle in Exhibit I, Sheet 7.
(4) Based on CDF from Exhibit I, Sheet 7.
(5) = [(3) x (4)].
(6) = (5) for all years except accident year 2008. Judgmentally selected 0.345 for 2008 based on prior years.
(7) Developed in Exhibit I, Sheet 6.
(8) = [(6) x (7)].

Exhibit I, Sheet 9: Development of Unpaid Claim Estimate ($000)
* Shows the results of all 3 projections (i.e. using dev method with reported and received, and ratio method)
* Shows estimated S&S recoverable equal to projected ultimate S&S minus received S&S.
The estimated S&S recoverable is the reduction to the total estimate of unpaid claims for the insurer.

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Chapter14 – Recoveries: Salvage & Subro and Reins
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
3

Reinsurance and Aggregate Limits

330 - 332

Estimating unpaid claims can be applied to gross, ceded, or net of reinsurance claims experience using
the techniques shown in Chapters 7 – 13.
Approaches to estimating unpaid claims on a net of reinsurance basis:
1. Analyze gross (i.e. direct and assumed) and ceded experience separately;
2. Analyze gross and net experience separately.
Choosing a gross versus net versus ceded analysis may depend upon:
* Data availability, gross versus ceded program characteristics, and the actuary’s personal
preferences.
i. If ceded claims are coded in the same database as gross data, net data is available.
In this case, the actuary is more likely to conduct both gross and net analyses.
ii. If ceded claims data are coded to a different system, matching gross and ceded data to derive net
claim triangles may be more difficult.
In this case, the actuary will likely prepare separate gross and ceded analyses.
* The volume and quality of the data.
Key: When conducting a net (of reinsurance) or ceded analysis, the actuary needs to be aware of the
implied relationships between gross, ceded, and net claims, at all stages of the analysis:
* At the beginning (when the actuary is reviewing and reconciling the data).
* During the analysis (when the actuary uses judgment in developing an unpaid claim estimate).
* At the end of the analysis (when the actuary evaluates projection methods and selects ultimate
claims and unpaid claim estimates).
When conducting a net (of reinsurance) or ceded analysis (continued):
1. Checks to conduct at the beginning of the analysis:
Check if net claim and net premium data are equal to or less than the gross data.
i. Reinsurance arrangements are often quota share or excess of loss.
For QS treaties create a development triangle ratio-ing net-to-gross claims to test the Q.S %’s by year.
Confirm that the ratios are consistent with relationships between net and gross premium.

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Chapter14 – Recoveries: Salvage & Subro and Reins
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Exhibit II, Sheet 1: Shows 3 triangles for an insurer having a QS for the past four years.
For 2005, the insurer had a 70% SQ, and increased the % to 85% in 2007 and to 90% in 2008.
The gross reported claims, the net reported claims, and the ratio of net to gross reported claims
are shown below:
Chapter 14 - Recoveries: S&S and Reinsurance
Exhibit II
Impact of Quota Share Reinsurance
Sheet 1
Accident
Gross Reported Claims ($000) as of (months)
Year
12
24
36
48
2005
2006
2007
2008
Accident
Year
2005
2006
2007
2008
Accident
Year
2005
2006
2007
2008

35,839
42,290
47,365
49,733
37,452
44,568
49,024
39,324
46,009
41,212
Net Reported Claims ($000) as of (months)
12
24
36
48
25,087
29,603
33,155
34,813
26,216
31,197
34,317
33,426
39,108
37,091
Ratio of Net to Gross Reported Claims as of (months)
12
24
36
48
0.700
0.700
0.850
0.900

0.700
0.700
0.850

0.700
0.700

0.700

ii. For XOL treaties, examine large claims to confirm that retentions and limits for ceded claims by year
are consistent with the corresponding XOL reinsurance contracts.
Verifying treatment of large claims helps to ensure that the ceded and/or net claim triangles are correct.
Exhibit II, Sheet 2:
The insurer maintains $1 million excess of loss reinsurance.
In AY 2005, the insurer sustained two large claims in excess of $1 million
In AY 2007, one large claim in excess of $1 million.
Chapter 14 - Recoveries: S&S and Reinsurance
Exhibit II
Impact of Excess of Loss Reinsurance
Sheet 2
Gross Reported Claims ($000) as of (months)
Accident
Year
12
24
36
48
2005
2006
2007
2008
Accident
Year

12,199
15,615
18,425
20,268
12,992
16,890
20,267
13,901
17,655
14,735
Net Reported Claims ($000) as of (months)
12
24
36
48

2005
2006
2007
2008
Accident
Year

11,752
14,076
16,502
18,056
12,992
16,890
20,267
13,644
17,303
14,735
Ceded Reported Claims ($000) as of (months)
12
24
36
48

2005
2006
2007
2008

Exam 5, V2

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0
257
0

1,539
0
352

1,923
0

2,212

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Chapter14 – Recoveries: Salvage & Subro and Reins
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2. Checks to conduct during of the analysis:
Ensure that key assumptions, and actuarial judgment, are consistent between the gross and net or gross and
ceded analyses. Examples:
 the tail factor for net claims should be smaller than for gross claims, since net claims are often
capped due to excess or aggregate coverage,
 Net claim development patterns should be less than or equal to gross claim development patterns.
Determine the order in which to conduct the gross or net claim development analyses
 Some actuaries will conduct a gross analysis first since these triangles contain a greater volume of
claims experience, and may have greater credibility.
The gross CDFs may be used as input for the selection of ceded or net CDFs.
 However, gross claims are subject to more random variation due to large claims, and thus a net
analysis may be conducted first, and the net selected CDFs can be used as input for the selection of
gross CDFs.
Thus, there should be a reasonable relationship between the selected development factors for net and gross
claims.
Areas of reasonableness in the net and gross or ceded and gross analyses:
1. Among the trend assumptions as well as expected claim ratios, frequency, and severity assumptions.
2. When selecting ultimate claims, ensuring that the implied relationship between the net and gross
claims and resulting estimates of unpaid claims to ceded claims are reasonable,
3. Ensuring net IBNR in each AY is generally not greater than gross IBNR.
Times when the net IBNR will be greater than the gross IBNR include:
i. When an estimate of uncollectible reinsurance is included in the net IBNR but not in the gross IBNR and there are significant
billed reinsurance amounts for which significant collectibility issues exist.
ii. For a runoff book with reinsurance disputes for items such as asbestos.

Aggregate or stop-loss coverage
* Used by many insurers to protect their financial results across multiple lines of coverage.
* Can apply on an accident year, policy year, or calendar year basis.
* Critical to understand how the coverage operates, and how the insurer treats prior recoveries from
aggregate coverage in the source data used in the analysis of unpaid claims.
i. Determine whether or not to take stop-loss or aggregate programs into account within the claim
development triangles or at a later stage of the analysis.
ii. Often, the actuary would want data prior to the application of stop-loss or aggregate coverage since
the actuary can adjust for such coverage as a final step in the developing the unpaid claim estimate.
Exhibit II, Sheet 3: Self-Insurance Pool with Excess of Loss and Stop Loss Reinsurance
The following is a simple approach to adjust for the affect of excess of loss and stop-loss reinsurance.
 Self-Insurance Pool is a group of self-insured municipalities that has maintained a $500,000 per
occurrence excess of loss coverage since the pool inception.
 Stop-loss coverage has varied over time depending on the availability and price of such
coverage.
There was a $4 million combined stop-loss (i.e., the stop-loss limit of $4 million applied to the sum
of ultimate claims for policy years 2002-03 through 2004-05).
The stop-loss limit was $1.5 million for policy years 2005-06 and 2006-07.
There was no stop-loss coverage purchased for 2007-08.

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Chapter14 – Recoveries: Salvage & Subro and Reins
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Exhibit II, Sheet 3: Self-Insurance Pool with Excess of Loss and Stop Loss Reinsurance
Chapter 14 - Recoveries: Salvage and Subrogation and Reinsurance
Impact of Reinsurance Programs

Exhibit II
Sheet 3

Self-Insurance Pool with Excess of Loss and Stop Loss Reinsurance
Ultimate Claims

Net of Excess of Loss, Net of Stop Loss
Claims at 12/31/08
Estimated
Reported
Paid
IBNR

Year

Net of Excess of Loss
Gross of Stop Loss

Stop Loss
Limit

Ultimate
Claims

(1)
2002 - 03
2003 - 04
2004 - 05
2005 - 06
2006 - 07
2007 - 08
Total

(2)
1,184,999
1,770,725
1,306,107
2,168,077
1,137,216
1,364,048
8,931,172

(3)

(4)

(5)

(6)

(7)

(8)

[4,000,000]

[3,753,248]

[3,253,624]

[246,752]

[746,376]

1,500,000
1,500,000
N/A

1,500,000
1,137,216
1,364,048
8,001,264

1,500,000
914,262
432,679
6,600,189

1,016,783
629,296
257,877
5,157,580

0
222,954
931,369

483,217
507,920
1,106,171

1,401,075

2,843,684

Policy

Unpaid Claim
Estimate

Column Notes:
(2) Selected based no review of various projection techniques.
(3) Based on Self-Insurance Pool stop-loss reinsurance program.
(4) = [minimum of (2) and (3)].
(5) and (6) Based on Self-Insurance Pool experience.
(7) = [(4) - (5)]
(8) = [(4) - (6)]

As shown above, the actuary estimates ultimate claims using reported and paid claims limited to the per
occurrence retention (i.e. $500,000 per occurrence) in Column (2).
The stop-loss limits are shown in Column (3).
Column (4) ultimate claims take into account both the ultimate excess of loss claims and stop-loss
coverages.
Estimated IBNR and the total unpaid claim estimate net of both excess of loss and stop-loss
coverage are computed in Columns (7) and (8)

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Chapter14 – Recoveries: Salvage & Subro and Reins
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Questions from the 2009 Exam
1. (2 points) Given the following information as of December 31, 2008:
Accident
Year

Paid Claims
Gross of S&S

2006
2007
2008

$15,513
15,568
9,441

Selected
Ratio of
Development Factor to
Ultimate Claims Received S&S
Ultimate for
Gross of S&S to Paid Claims
S&S Ratio
$17,000
17,250
16,500

0.361
0.379
0.286

1.000
1.007
1.300

a. (1.5 points) Use the ratio method to estimate the recoverables for salvage and subrogation (S&S)
for accident years 2006 - 2008.
b. (0.5 point) Briefly discuss one advantage in using the ratio method to determine salvage and
subrogation recoverables.
Questions from the 2011 Exam
32. (3 points) Given the following data as of December 31, 2010:
Accident
Year
2007
2008
2009
2010

Cumulative Paid Claims Gross of Salvage and Subrogation
12 months
24 months
36 months
48 months
$12,200
$13,260
$13,280
$13,280
$12,180
$13,300
$13,320
$12,880
$14,040
$11,980

Accident
Year
2007
2008
2009
2010

Cumulative Received Salvage and Subrogation
12 months
24 months
36 months
48 months
$3,074
$4,670
$4,720
$4,746
$3,098
$4,558
$4,602
$3,180
$4,732
$2,858

• Assume no further development after 48 months.
• Use all-year simple averages for tail factor selections.
Use a ratio approach to estimate ultimate salvage and subrogation recoveries.

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Chapter14 – Recoveries: Salvage & Subro and Reins
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Questions from the 2011 Exam
33. (1.5 points) Given the following information:
Accident
Year
2006
2007

12 months
$55,963
$57,584

Gross Reported Claims (000s)
24 months
36 months 48 months
$62,679
$66,439
$66,439
$62,191
$65,922
$65,922

Accident
Year
2006
2007

12 months
$50,367
$37,430

Net Reported Claims (000s)
24 months
36 months 48 months
$50,870
$51,125
$51,125
$40,424
$42,849
$42,849

• Insurer has either a quota share reinsurance contract or an excess of loss reinsurance contract in
place each accident year.
a. (1 point) Analyze the gross and net reported claims data to determine which type of reinsurance
was purchased for each accident year. Explain your reasoning.
b. (0.5 point) Briefly explain how the selection of tail factors for both net and gross reported claims
should be impacted by the presence of an excess of loss reinsurance contract.
Questions from the 2012 Exam
25. (1.75 points) Given the following data as of December 31, 2011:
Cumulative Paid Claims Gross of
Salvage and Subrogation ($000s)
Accident
Year
12 Months
24 Months
36 Months
2009
$15,117
$16,953
$16,953
2010
$15,092
$16,862
2011
$14,727
Cumulative Received
Salvage and Subrogation ($000s)
Accident
Year
12 Months
24 Months
36 Months
2009
$2,104
$4,493
$4,605
2010
$1,995
$4,657
2011
$2,025
Selected cumulative development factors for ratio of received salvage and subrogation to paid claims:
Age
CDF to
(months) Ultimate
36
1.000
24
1.025
12
2.047
Assume no development after 36 months.
Use a ratio approach to estimate the ultimate salvage and subrogation for accident year 2011.

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Chapter14 – Recoveries: Salvage & Subro and Reins
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Questions from the 2012 Exam)
26. (1.25 points) Given the following information for a self-insured pool as of December 31, 2011:

Policy
Year
2008
2009
2010
2011

Reported Claims Reported Claims
Gross
Gross of Excess
Net of Excess
Cumulative
of Loss
of Loss
Development
($000s)
($0005)
Factors
$1,635
$634
1.440
$3,109
$625
1.760
$2,358
$728
2.140
$1,897
$674
2.710

Net
Cumulative
Development
Factors
1.380
1.620
1.940
2.450

Stop Loss
Limit
($000s)
$1,000
$1,250
$1,250
$1,500

•

The pool has maintained $1 million per occurrence excess of loss reinsurance since inception.

•

The pool has also maintained stop loss coverage over limits that vary over time shown above.

Estimate the pool's ultimate claims net of both excess of loss and stop loss for policy years 2008 through
2011.

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Chapter14 – Recoveries: Salvage & Subro and Reins
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to questions from the 2009 Exam
Question 1 – Model Solution 1
a.
[1]
[2] Given
Accident Received S&S
Year
Paid Claims
2006
0.361
2007
0.379
2008
0.286
[7] Given
Paid Claims
15,513
15,568
9,441

Notes
[4] = [2]
[6] = [4]
[8] = [2]
[9] = [6]

[3] Given
Dev. Factor
to Ult
1.000
1.007
1.300
[8]
Paid S&S
5,600.1930
5,900.2720
2,700.1260

[4]
Ult Rec
S&S Ratios
0.3610
0.3817
0.3718

[5] Given
Ult
Claims
17,000
17,250
16,500

[6]
Ult
S&S
6,137.0000
6,584.3250
6,134.7000
18,856.0250

[9]
S&S Recoverables
536.8070
684.0530
3,434.5740
4,655.4340
= S&S Recoverables

× [3]
× [5]
× [7]
- [8]

b. The development factors for the ratio method are less leveraged at early maturities than development
factors would be in the reported recoveries or received recoveries development methods.
Question 1 – Model Solution 2
a.

AY
2006
2007
2008

(1)=Given
Paid Claims
Gross S/S
15,513
15,568
9,441

(2)=Given
Selected Ult
Gross S/S
17,000
17,250
16,500

(3)=Given
Ratio Received S/S
Paid Claims
0.361
0.379
0.286

(4)= Given
Dev to Ult for
Ratio
1.000
1.007
1.300

(5) = (3) x (4)
Ultimate S/S
0.3610
0.3817
0.3718
Total

(6) = (1) x (3)
Received S/S
5,600.1930
5,900.2720
2,700.1260
14,201

(7) = (2) x (5)
Ultimate S/S
6,137.0000
6,584.3250
6,134.7000
18,856

(8) = (7) – (6)
S/S
536.8070
684.0530
3,434.5740
4,655.4340

b.. The ratio method provides Ult S&S ratios to paid claims, which can be used as a diagnostic, so that the
actuary may use judgment in selecting a more reasonable S/S ratio for AY’s that show odd behavior.

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Chapter14 – Recoveries: Salvage & Subro and Reins
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to questions from the 2011 Exam
32. Use a ratio approach to estimate ultimate salvage and subrogation recoveries.
Question 32
Step 1: Compute ratios of received S&S to paid claims gross of S&S
AY
12 months 24 months
36 months
48 months
07
0.2520
0.3522
0.3554
0.3574
08
0.2544
0.3427
0.3455
09
0.2469
0.3370
10
0.2386
Step 2: Compute age to age and age to ult factors of ratios of
received S&S to paid claims gross of S&S
AY
12-24
24-36
36-48
48 - Ult
2007
1.3976
1.0091
1.0056
2008
1.3471
1.0082
2009
1.3649
Selected
1.3699
1.0087
1.0056
1.0000
Age to Ult
1.3895
1.0143
1.0056
1.0000
Step 3: Compute age to age and age to ult paid claims factors
AY
12-24
24-36
36-48
48 - Ult
2007
1.0869
1.0015
1.0000
2008
1.0920
1.0015
2009
1.0901
Selected
1.0901
1.0015
1.0000
1.0000
Age to Ult
1.0917
1.0015
1.0000
1.0000
Step 4: Compute Ultimate S&S Recoveries
Paid Losses Received S&S Received S&S
Paid Losses LDF to Ult
to Paid
LDF to Ult
AY
(1)
(2)
(3)
(4)
2007
13,280
1.0000
0.3574
1.0000
2008
13,320
1.0000
0.3455
1.0056
2009
14,040
1.0015
0.3370
1.0143
2010
11,980
1.0917
0.2386
1.3895

Exam 5, V2

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Ult S&S
Recoveries
(5)=(1)*(2)*(3)*(4)

4746.2720
4627.8315
4806.3389
4335.9985

 2014 by All 10, Inc.

Chapter14 – Recoveries: Salvage & Subro and Reins
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to questions from the 2011 Exam
• The insurer has either a quota share reinsurance contract or an excess of loss reinsurance contract in
place each accident year.
33a. (1 point) Analyze the gross and net reported claims data to determine which type of reinsurance was
purchased for each accident year. Explain your reasoning.
33b. (0.5 point) Briefly explain how the selection of tail factors for both net and gross reported claims
should be impacted by the presence of an excess of loss reinsurance contract.
Question 33 – Model Solution
a. Compute the ratio of net claims-to-gross claims
2006
2007

12
0.9000
0.6500

24
0.8116
0.6500

36
0.7695
0.6500

48
0.7695
0.6500

For AY 2006, it’s a excess of loss reinsurance since the ratio from one maturity to another are not
consistent. The ratio depends of the amount of excess loss that has been ceded.
For AY 2007, it’s a quota share. They cede 35% of their business to reinsurance. It’s a quota share since
the ratios are consistent.
b. The tail factor for the gross reported claims would be chosen as normal. However, for the net reported
claims, it would vary from years to years. The net tail factor is less than the gross tail factor.

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Chapter14 – Recoveries: Salvage & Subro and Reins
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to questions from the 2012 Exam
25. Use a ratio approach to estimate the ultimate salvage and subrogation for AY 2011.
Question 25 – Model Solution 1 (Exam 5B Question 10)
Development of Cumulative Paid Claim Gross
AY
12-24
24-36
36 - ult
2009
1.1215
1.0000
1.0000
2010
1.1173
Selected
1.1194
1.0000
1.0000

1.1215 = $16,953 / $15,117

(2011 cumulative paid gross) x (S+S factor) = 14.727 x 1.1194 = $16,485.4038
Ratio of S+S Received to Paid Gross
AY
12-24
24-36
36 - ult
2009
0.1392
0.265
0.2716
2010
0.1322
0.2762
2011
0.1375

0.1392 = $2,104 / $15,117

(2011 Ratio of Received to Paid Gross) x (Selected CDF to Ult) x Ultimate Gross
= 0.1375 x 2.047 x Ult gross = $4,640.0230

Question 25 – Model Solution 2 (Exam 5B Question 10)
Development of Cumulative Paid Claim Gross
AY
12-24
24-36
36-ult
2009
1.1215
1.0000
2010
1.1173
Simple all-year average.
ATA
1.1194
1.0000
1.0000
LDF
1.1194
1.0000
1.0000
Ratio of S+S Received to Paid Gross
AY
12
24
36
2009
0.1392
0.2650
0.2716
2010
0.1322
0.2762
2011
0.1375

AY
2011

(1)
Ratio
0.1375

(2)
Ratio CDF
2.047

(3)
Cumul.
14,727

(4)
Paid CDF
1.1194

(5)=(1)(2)(3)(4)
Ult S/S
4,640.0230

Examiner’s Comment
This was a fairly simple and straightforward question. A majority of candidates achieved full credit on this
problem. Some candidates failed to project claims to ultimate value and as a result salvage and
subrogation (S&S) to ultimate. Other candidates lost credit by taking the average ratio of S&S to paid
claims at 12 months, then applying a development factor to that average. This ignored older accident year
data and was felt to be inappropriate.

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Chapter14 – Recoveries: Salvage & Subro and Reins
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to questions from the 2012 Exam
Estimate the pool's ultimate claims net of both excess of loss and stop loss for policy years
2008 through 2011.
Question 26 –Model Solution 1 (Exam 5B Question 11)
The pool's ultimate claims net of both excess of loss and stop loss = the minimum of the net ultimate
losses and the stop loss limit for each of the policy years under consideration.
(2)
Net LDF

(3) = 1 x 2
Net Ult

(4)
Stop loss

(5) Min((3),(4))
Net Ult final

PY
2008

(1)
Rept
Claims net
634

1.38

874.92

1,000

874.92

2009

625

1.62

1,012.50

1,250

1,012.50

2010

728

1.94

1,412.32

1,250

1,250.00

2011

674

2.45

1,651.30

1,500

1,500.00

Total

4,637.42

4637.42 * 1000 = 4,637,420

Question 26 –Model Solution 2 (Exam 5B Question 11)
Compute net ultimate loss and then select the minimum of the net ultimate loss and the stop loss
634 * 1.38 = 874.92
’08 Net Ultimate
1012.5 <1250
’09 Stop Loss Limit
1412 > 1250
PY 2010
1651 > 1500
PY 2011
Total = 4,637.42
Question 26 –Model Solution 3 (Exam 5B Question 11)
Assume stop loss applies to each PY independently. Assume stop loss applies to loss net of XOL.

PY
2008
2009
2010
2011
Total

(1)
Ult Loss
Of XOL
874.92
1,012.50
1,412.32
1,651.30
4951.04

(2)
Stop Loss
1,000
1,250
1,250
1,500

(3)
Stop Loss
Cessions
0.00
0.00
162.32
151.30
313.62

(4)
Loss Net XOL
& Stop Loss
874.92
1,012.50
1,250.00
1,500.00
4,637.42

(1) = Rpt Loss Net XOL
(3) = (1) – min[(1),(2)]
(4) = (1) – (3)
Examiner’s Comments
A little over half of the candidates received full credit and about a third received no credit, most of which
completely skipped the question. Those who received partial credit received some credit for
demonstrating some understanding of reserving and reinsurance, but did not apply stop gap correctly, did
not use reported claims net of XOL and net development factors, and/or made math errors.

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Sec
1
2
3
4

Pages
345 - 346
346
347 - 348
349

5
6

Description
Introduction
U. S. Industry Auto
XYZ Insurer
Changing Conditions - Changes in Claim Ratios and Case
Outstanding Adequacy and Changes in Product Mix
Berq–Sher Insurers
Monitoring and Interim Techniques for Unpaid Claim Estimates

Introduction

345 - 346

350
350 - 353

The methods used for estimating unpaid claims presented in Chapters 7 through 14 are reviewed.
Actuaries should use more than one method when analyzing unpaid claims, since no single method can
produce the best estimate in all situations.
Berquist and Sherman:
 recommend that where possible, an analysis of unpaid claims should use methods that
incorporate the following:
* Projections of reported claims
* Projections of paid claims
* Projections of ultimate reported claim counts and severities
* Estimates of the number and average amount of outstanding claims
* Claim ratio estimates
 further recommend that wherever possible, the concepts of credibility, regression analysis, and
data smoothing be incorporated into the actuarial methods used.
At times credibility maybe used in the selection process, while at other times actuarial judgment
will prevail.
When incorporating regression analysis into a method, used some measure of the goodness-of-fit
to evaluate the appropriateness of that method's projections.
 state: "The methods applied should range from those which are highly stable (i.e. representative
of the average of experience over several years) to those which are highly responsive to trends
and to more recent experience."
Selection of the most appropriate estimate of unpaid claims is the actuary’s responsibility.
Patrik (in Reinsurance):
 There is no single right method.
 Use as many legitimate methods and compare and contrast the estimates from these methods.
 Review the spread of estimates to better understand the range and distribution of possibilities,
and the sensitivity of our answers to varying assumptions and estimation methods.
 Testing the method retroactively is one method to evaluate a particular technique by determining
the historical accuracy of the method and whether or not the particular method is free from bias in
projecting future results.

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Ronald Wiser (in "Loss Reserving"):
 explain significant differences between the projections of various methods, typically due to
changes in company procedures or to changes in the external environment.
 while attempting to reconcile a number of different estimates is difficult, it often yields new
insights for the actuary.
Calculate claim ratios, severities, pure premiums, and claim frequencies as a final check in the selection
of ultimate claims (especially for the most recent years)
 ultimate amounts can be evaluated in contexts outside their original analysis.
 if exposures are not available, compare ultimate claim counts with premiums as a proxy for
frequency.
Calculate also the implied average case outstanding on open and unreported claims.
Review these statistics:
 for reasonableness from the perspective of year-to-year changes,
 with knowledge gained from meetings with management, and
 with knowledge of the industry in general.
Such a review should give the actuary greater confidence in the unpaid claim estimate or lead to seeking
additional information before reaching a conclusion.

U. S. Industry Auto

346

The results of the projection techniques are all consistent given the volume of business.

$ Billions
Development — Reported
Development — Paid
Expected Claims
Bornhuetter-Ferguson — Reported
Bornhuetter-Ferguson — Paid
Cape Cod
Case Outstanding Development

Estimated Unpaid Claims
as of 12/31/07
IBNR
Total
26
71
29
74
26
71
26
71
27
73
27
73
24
70

In total and by AY, the methods produce unpaid claims that are similar to one another.

XYZ Insurer

347 - 348

We expect to see significant differences in the various estimates of unpaid claims in results for XYZ Insurer,
since we know that the underlying assumptions of some of the methods do not hold true, due to recent
changes in both its internal operations as well as the external environment.

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Exhibit I, Sheet 1: Projected ultimate claims from the following methods (including the influence of the B/S
adjustments on the projected ultimate claims):
* Reported and paid claim development techniques based on unadjusted reported and paid claims
* BF technique based on unadjusted reported and paid claim development patterns
* Cape Cod method based on unadjusted reported claim development pattern
Adjusted projections based on the following:
* Reported and paid claim development techniques incorporating B/S adjustments to case outstanding
only, paid claims only, as well as to both case outstanding and paid claims
* BF based on adjusted reported and paid claim development patterns as well as revised expected
claim ratios
Chapter 15 - Evaluation of Techniques
XYZ Insurer - Auto BI
Summary of Ultimate Claims ($000)

Accident
Year
(1)
1998
1999
:::
2007
2008
Total

Unajusted projections for Ultimate Claims
Development Method
B-F Method
Reported
Paid
Reported
Paid
(2)
(3)
(4)
(5)
15,822
15,980
15,822
15,977
25,082
25,164
25,107
25,158
:::
:::
:::
:::
47,979
77,941
45,221
45,636
47,530
74,995
42,607
41,049
514,929
605,028
513,207
554,469

Exhibit I
Sheet 1

Cape Cod
Method
(6)
15,822
25,107
:::
43,307
39,199
504,298

Adjusted projections for Ultimate Claims
Development Method
B-F Method
Case Rptd
Both Rptd
Paid
Reported
Paid
(7)
(8)
(9)
(10)
(11)
15,822
15,822
15,980
15,822
15,975
25,107
25,107
25,140
25,107
25,128
:::
:::
:::
:::
:::
38,237
42,680
38,549
40,300
34,988
37,227
41,531
45,989
36,842
33,988
476,636
489,258
485,546
482,909
484,647

Column Notes:
(2) and (3) Developed in Chapter 7, Exhibit II, Sheet 3.
(4) and (5) Developed in Chapter 9, Exhibit II, Sheet 1.
(6) Developed in Chapter 10, Exhibit II, Sheet 2.
(7) through (9) Developed in Chapter 13, Exhibit III, Sheet 10.
(10) and (11) Developed using projected ultimate claims in (8) as the new intial expected claims estimates.

Note: Calculations for the revised BF incorporating the B/S adjustments on development patterns
and the expected claim ratio are not included, but the user is encouraged to reproduce these
calculations to ensure a greater understanding of the mechanics of each method.
Next: Removed from consideration are:
 the first 3 techniques, because using unadjusted data does not satisfy the underlying
assumptions for these techniques.
 the B/S adjustment for case outstanding only since this projection does not reflect the changes
observed in settlement rates.
Exhibit I, Sheets 2 through 6: Exhibits used to assist in selecting ultimate claims by accident year.
* Exhibit I, Sheet 2 — Summary of Ultimate Claims
* Exhibit I, Sheet 3 — Comparison of Estimated Ultimate Claim Ratios
* Exhibit I, Sheet 4 — Comparison of Estimated Ultimate Severities
* Exhibit I, Sheet 5 — Comparison of Estimated Average Case Outstanding and Unreported Claims
* Exhibit I, Sheet 6 — Comparison of Estimated IBNR
Each exhibit contains details by AY.
 For the frequency-severity approaches (#2 and #3), only ultimate claims for the recent AYs are
estimated.
 For other techniques, projected ultimate claims for all AYS in the experience period (i.e. 1998 through
2008) are shown.

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Exhibit I, Sheets 2 – 6: Summarized results for the following methods:
* Reported and paid claim development techniques incorporating B/S adjustments to paid claims only
as well as to both case outstanding and paid claims
* BF based on adjusted reported and paid claim development patterns as well as revised expected
claim ratios
* All three frequency-severity projections (from Chapter 11)
Chapter 15 - Evaluation of Techniques
XYZ Insurer - Auto BI
Summary of Ultimate Claims ($000)

Accident
Year
(1)
1998
1999
::
2007
2008
Total

Claims as of 12/31/08
Reported
Paid
(2)
(3)
15,822
15,822
25,107
24,817
::
::
31,732
11,865
18,632
3,409
449,626
330,627

Exhibit I
Sheet 2

Adjusted Projections for Ultimate Claims
Development Method
B-F Method
Both Rptd
Paid
Reported
Paid
(4)
(5)
(6)
(7)
15,822
15,980
15,822
15,975
25,107
25,140
25,107
25,128
::
::
::
::
42,680
38,549
40,300
34,988
41,531
45,989
36,842
33,988
489,258
485,546
482,909
484,647

Projections for Ultimate Claims
Frequency-Severity
(8)
15,822
25,084
::
58,527
59,214
551,117

(9)
0
0
::
30,487
30,172

(10)
0
0
::
11,865
3,409

Selected
Ultimate
Claims
(11)
15,822
25,107
::
40,300
33,507
483,796

Column Notes:
(2) and (3) Based on data from XYZ Insurer.
(4) and (5) Developed in Chapter 13, Exhibit III, Sheet 10.
(6) and (7) Developed using projected ultimate claims in (4) as the new intial expected claims estimates.
(8) Developed in Chapter 11, Exhibit II, Sheet 6.
(9) Developed in Chapter 11, Exhibit IV, Sheet 3.
(10) Developed in Chapter 11, Exhibit VI, Sheet 8.
(11) = (4) for accident years 2004 and prior; (11) = [Average of (6) and (7) for 2005 and 2006]; (11) = (6) for 2007; (11) = [Average of (6) and (9)] for 2008.

FS Method # 1 (from Chapter 11):
 Incorporating closed claim counts into the selection of ultimate claim counts may overstate the
true value of projected ultimate claims.
 Column (8) estimate of total ultimate claims for all AYs combined is $551,155;
Total ultimate claims for all other methods are less than $490,000.
Thus, FS Method # 1 is excluded from further consideration.
For the oldest seven years (1998 - 2004), the results are consistent results from the various projection
methods. However, beginning in 2005, the differences become more substantial.
The selection of ultimate claims can be assisted by a review of the estimated ultimate claim ratios and
ultimate severities as well as the estimated IBNR.
Chapter 15 - Evaluation of Techniques
XYZ Insurer - Auto BI
Comparison of Estimated Ultimate Claim Ratios

Accident
Year
(1)
1998
1999
:::
2007
2008
Total

Earned
Premium
(2)
20,000
31,500
:::
62,438
47,797
732,144

Development
Both Rptd
(3)
79.1%
79.7%
:::
68.4%
86.9%
66.8%

Method
Paid
(4)
79.9%
79.8%
:::
61.7%
96.2%
66.3%

Exhibit I
Sheet 3

Estimated Ultimate Claim Ratios Based on
B-F Method
Frequency - Severity
Reported
Paid
Method 1
Method 2
Method 3
(5)
(6)
(7)
(8)
(9)
79.1%
79.9%
79.1%
79.7%
79.8%
79.6%
:::
:::
:::
:::
:::
64.5%
56.0%
93.7%
48.8%
19.0%
77.1%
71.1%
123.9%
63.1%
7.1%
66.0%
66.2%
75.3%

Selected
Ult Claims
Ratios
(10)
79.1%
79.7%
:::
64.5%
70.1%
66.1%

Column Notes:
(2) Based on data from XYZ Insurer.
(3) through (10) = [(projected ultimate claims in Exhibit I, Sheet 2) / (2)].

Exam 5, V2

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Chapter 15 - Evaluation of Techniques
XYZ Insurer - Auto BI
Comparison of Estimated Ultimate Claim Ratios

Accident
Year
(1)
1998
1999
:::
2007
2008
Total

Ultimate
Claim Counts
(2)
637
1,047
:::
1,556
1,425
17,378

Development
Paid
(3)
24,838
23,978
:::
27,424
29,138
28,154

Method
Reported
(4)
25,087
24,010
:::
24,770
32,266
27,940

Exhibit I
Sheet 4

Estimated Ultimate Severites Based On
B-F Method
Frequency-Severity
Reported
Paid
Method 1
Method 2
Method 3
(5)
(6)
(7)
(8)
(9)
24,838
25,078
24,839
23,978
23,998
23,956
:::
:::
:::
:::
:::
25,895
22,482
37,607
19,590
7,624
25,848
23,846
41,545
21,169
2,392
27,788
27,888
31,713

Selected
Ultimate
Severities
(10)
24,838
23,978
:::
25,895
23,509
27,840

Column Notes:
(2) Developed in Chapter 11, Exhibit II, Sheet 3.
(3) through (10) = [(projected ultimate claims in Exhibit I, Sheet 2) x 1000 / (2)].

Chapter 15 - Evaluation of Techniques
XYZ Insurer - Auto BI
Comparison of Estimated IBNR

Accident
Year
(1)
1998
1999
:::
2007
2008
Total

Case
Outstanding
at 12/31/08
(2)
0
290
:::
19,867
15,223
118,997

Development Method
Both Rptd
Paid
(3)
(4)
0
158
0
42
:::
:::
10,948
6,817
22,899
27,357
39,632
35,920

Exhibit I
Sheet 6

Estimated IBNR Based On
B-F Method
Frequency-Severity
Reported
Paid
Method 1
Method 2
Method 3
(5)
(6)
(7)
(8)
(9)
0
157
0
0
42
-23
:::
:::
:::
:::
:::
8,568
3,256
26,795
-1,245
-19,867
18,210
15,356
40,582
11,540
-15,223
33,283
35,021
101,491

Selected
IBNR
(10)
0
0
:::
8,568
14,875
34,170

Column Notes:
(2) Based on data from XYZ Insurer.
(3) through (10) = [(projected ultimate claims in Exhibit I, Sheet 2) - ((2) in Exhibit I, Sheet 2)].

Estimated average case outstanding and unreported claim on open and IBNR claims is another valuable statistic.
Chapter 15 - Evaluation of Techniques
XYZ Insurer - Auto BI
Comparison of Estimated Average Case Outstanding and Unreported Claims

Accident
Year
(1)
1998
1999
:::
2007
2008
Total

Open
IBNR Counts
at 12/31/08
(2)
0
3
:::
765
1,150
3,515

Estimated Average Case Outstanding and Unreported Claims Based on
Development Method
B-F Method
Frequency-Severity
Both Rptd
Paid
Reported
Paid
Method 1
Method 2
(3)
(4)
(5)
(6)
(7)
(8)
‑
96,667
107,540
96,667
103,667
88,845
0
:::
:::
:::
:::
:::
:::
40,280
34,882
37,170
30,226
60,996
24,343
33,149
37,026
29,072
26,590
48,526
23,272
530,573
542,227
524,635
554,689
631,163

Exhibit I
Sheet 5

Method3
(9)
0
:::
0
0

Selected
ultimate
Average
(10)
96,667
:::
37,170
26,172
529,352

Column Notes:
(2) Based on data from XYZ Insurer.
(3) through (10) = { [(estimated IBNR in Exhibit I, Sheet 6) + ((2) in Exhibit I, Sheet 6)] x 1000 / (2)}.

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Acceptable ways to select ultimate claims:
 Select one method and use it for all years.
The B/S adjusted reported claim (both case and paid adjustments) method may be a reasonable
selection for all years for XYZ Insurer.
 Select different methods for different AYs. For example, select the B/S adjusted reported claim
method for AY 1998 - 2006 and the BF method for 2007 and 2008.
 Use a weighted average based on assigned weights to the various methods; these weights may be
consistent for all years or may vary by AY.
Recall that there is no single "right" way for the actuary to select ultimate claims, and thus unpaid claims.
 Review the results of the various techniques, diagnostic tests (e.g. implied claim ratios and severities),
and information gained during the unpaid claims estimation process.
 Retroactive tests are also valuable when selecting which methods to rely on for selecting ultimate claims.
In the example above, selected ultimate claims were based on:
 the B/S adjusted reported claim for AYs 1998 - 2004;
 the average of the adjusted reported and paid BF techniques for AYs 2005 and 2006;
 the adjusted reported BF technique for AY 2007; and
 the average of the adjusted reported BF technique and FS approach #2 for AY 2008.
The key statistics in selections by AY are the estimated IBNR, the estimated ultimate severities, and the
estimated claim ratios.

Changing Conditions - Changes in Claim Ratios and Case
Outstanding Adequacy and Changes in Product Mix

349

Chapters 7 – 10: Four scenarios regarding the U.S. private passenger automobile example are given.
#1. U.S. PP Auto Steady-State: All techniques produced an accurate estimate of unpaid claims.
# 2 U.S. PP Auto Increasing Claim Ratios, # 3 U.S. PP Auto Increasing Case Outstanding Strength, and
# 4 U.S. PP Auto Increasing Claim Ratios and Case Outstanding Strength; the techniques varied in their
ability to accurately respond to the changing conditions.
#5 U.S. Auto Steady-State (a combined PP and commercial auto portfolio) produce the actual IBNR value.
However, when the product mix changes, the methods respond differently to the changing conditions.
The following table summarizes the estimated IBNR for each of the projection techniques

Estimation Technique
True IBNR
Development – Reported
Development – Paid
Expected Claims
Bornhuetter-Ferguson – Reported
Bornhuetter-Ferguson – Paid
Benktander – Reported
Benktander – Paid
Cape Cod

Exam 5, V2

Estimated IBNR ($000)
Increasing
Increasing
Case
Claim
Outstanding
Ratios
Strength
602
253
602
501
602
253
-843
253
439
458
159
253
573
492
406
253
506
470

Page 365

Increasing
Ratios and
Changing
Case outstanding Product
Strength
Mix
348
2,391
694
2,153
348
1,723
-1,097
2,167
460
2,168
-96
1,991
648
2,159
151
1,893
546
2,168

 2014 by All 10, Inc.

Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
For each of these scenarios:
 there is considerable variability between the methods in total and by AY.
 it is important to understand what the drivers are for the differences between methods (may need
more information from management as well as further quantitative analysis to determine which
method is most appropriate).
 the availability of claim counts and the ability to test the estimated ultimate severities could prove
valuable to the actuary.

Berq–Sher Insurers

350

Exhibit II: Summarized results of the various projection methods for Berq-Sher Med Mal Insurer
XYZ Insurer - Auto BI
Comparison of Estimated IBNR

Accident
Year
(1)
1969
:::
1975
1976
Total

Claims as of 12/31/76
Reported
Paid
(2)
(3)
23,506,000
15,815,000
:::
:::
48,904,000
1,565,000
15,791,000
209,000
367,167,000
90,356,000

Exhibit II

Projected Ultimate Claims
Berq-Sher
Development Method
Reported
Paid
Adj Rptd
(4)
(5)
(6)
23,506,000
23,501,090
23,506,000
:::
:::
:::
215,253,430
125,022,648
93,459,963
175,986,370
103,266,710
117,875,378
837,639,227
650,665,195
521,143,718

Estimated IBNR Based on
Berq-Sher
Development Method
Reported
Paid
Adj Rptd
(7)
(8)
(9)
0
-4,910
0
:::
:::
:::
166,349,430
76,118,648
44,555,963
160,195,370
87,475,710
102,084,378
470,472,227
283,498,195
153,976,718

Column Notes:
(2) and (3) Based on medical malpractice insurance experience.
(4) through (6) Developed in Chapter 13, Exhibit I, Sheet 8.
(7) = [(4) - (2)].
(8) = [(5) - (2)].
(9) = [(6) - (2)].

Ultimate claims are estimated using:
 the development technique applied to unadjusted reported and paid claims, and
 adjusted reported claims (claims adjusted to reflect changes in case outstanding adequacy).
Note: the diagnostics that can be performed is limited for both the B/S examples since complete claim
count data is not available.
Observations:
 An increase in case outstanding strength has occurred during the experience period.
Thus, the development method based on unadjusted reported claims is not appropriate since its
underlying assumption is not valid (i.e. case O/S adequacy has not remained constant)
 Since the unadjusted paid claim development and adjusted reported claim development methods
produce significant differences, the actuary should seek additional information, including the use of other
methods, before making a final determination as to ultimate claims and thus the unpaid claim estimate.

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Exhibit III: Summarized the results for Berg-Sher Auto BI Insurer.
XYZ Insurer - Auto BI
Summary of Ultimate Claims and Estimated Ultimate Severities

Accident
Year
(1)
1969

Paid
Claims
as 12/31/76
(2)
10,256

Paid Claims
Dev Method
(3)
10,256

:::
1975
1976
Total

:::
9,182
2,801
90,937

:::
18,286
17,281
122,710

Exhibit III

Projected Ultimate Claims
Berquist-Sherman Adjusted Paid
Dev Method
Lin Reg
Exp Reg
(4)
(5)
(6)
10,256
10,256
10,256
:::
21,560
20,771
132,028

Column Notes:
(2) Based on automobile bodily injurty experience.
(3) through (6) Developed in Chapter 13, Exhibit II, Sheet 11.
(7) Developed in Chapter 13, Exhibit II, Sheet 4.
(8) = [(3) x 1000 / (7)].
(9) = [(4) x 1000 / (7)].
(10) = [(5) x 1000 / (7)].
(11) = [(6) x 1000 / (7)].

:::
21,387
21,890
132,090

:::
21,385
21,913
132,120

Ultimate
Claim Counts
(7)
7,821
:::
8,050
7,466
69,047

Estimated Ultimate Severities Based on
Berquist-Sherman Adjusted Paid
Paid Claims
Dev Method
Dev Method
Lin Reg
Exp Reg
(8)
(9)
(10)
(11)
1,311
1,311
1,311
1,311
:::
2,272
2,315
1,777

:::
2,678
2,782
1,912

:::
2,657
2,932
1,913

:::
2,657
2,935
1,913

#DIV/0!

Four estimates of ultimate claims using the development technique are developed.
1. Project ultimate claims based on unadjusted paid claims data.
2. Adjust the paid claims data for changes in the rate of claims settlement and develop three alternative
sets of claim development factors.
3. Summarize ultimate claims and estimated ultimate severities for each of the four projections.
Observations:
All three projections based on the adjusted paid claims are similar to one another, in total and by AY.
Note: These projections are not necessarily independent since they are based on the same source data.
Incorporate other techniques to verify the results of the B/S adjusted paid claims methodology.
The results of the B/S adjustment are consistent with our expectations regarding a decrease in the rate of
claims settlement.

Monitoring and Interim Techniques for Unpaid Claim Estimates

350 - 353

The final part of Wiser's four-phase approach to estimating unpaid claims is to monitor projections of the
development of unpaid claims over subsequent calendar periods.
 Deviations of actual development from projected development of claims or claim counts are
useful to evaluate the accuracy of the unpaid claim estimate.
 Comparing actual-to-expected claims helps the actuary to evaluate the appropriateness of prior
selections and make revisions as necessary if actual claims do not emerge as expected.
Monitoring unpaid claims is useful:
 from a financial reporting perspective,
 for budgeting and planning purposes,
 for pricing and other strategic decision-making, and
 for planning for the next complete analysis of unpaid claims.

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Comparing actual and expected claims by AY and between successive annual valuations is shown in
Exhibit IV.
For DC Insurer, ultimate claims at 12/31/2007 are derived based on the reported claim development technique.
DC Insurer
Reported Claims ($000)

Exhibit IV
Sheet 1

PART 3 - Average Age-to-Age Factors
3-6
Simple Average
All Years
1.830
Latest 7
1.888
Latest 5
1.898
Medial Average
Latest 5x1
1.896
Volume-weighted Average
All Years
1.838
Latest 7
1.889
Latest 5
1.895

6-9

9-12

12-15

Averages
15-18

18-21

21-24

24-27

27-30

30-33

33-36

1.487
1.503
1.470

1.339
1.347
1.311

1.094
1.119
1.119

1.002
1.006
1.008

1.003
1.004
1.005

1.001
1.002
1.002

1.000
1.001
1.001

1.000
1.000
1.001

1.000
1.000
1.000

1.474

1.322

1.126

1.007

1.005

1.002

1.001

1.000

1.480
1.485
1.464

1.326
1.335
1.309

1.091
1.119
1.118

1.003
1.006
1.008

1.003
1.004
1.005

1.001
1.002
1.002

1.000
1.001
1.001

1.001
1.000
1.001

1.000
1.000
1.000

Development Factor Selection
9-12
12-15
15-18
1.309
1.118
1.008
1.487
1.136
1.016
67.2%
88.0%
98.4%

18-21
1.005
1.008
99.2%

21-24
1.002
1.003
99.7%

24-27
1.001
1.001
99.9%

27-30
1.000
1.000
100.0%

30-33
1.000
1.000
100.0%

33-36
1.000
1.000
100.0%

To Ult

PART 4 - Selected Age-to-Age Factors

Selected
CDF to Ultimate
Percent Reported

3-6
1.895
4.125
24.2%

6-9
1.464
2.177
45.9%

DC Insurer
Projection Ultimate Claims Using Reported Claims ($000)

Accident
Year
(1)
1997
:::
2005
2006
2007
Total

Age of
Accident Year
at 12/31/07
(2)
132
:::
36
24
12

Reported
Claims
at 12/31/07
(3)
3,376
:::
2,814
2,949
2,463
28,575

CDF
to Ultimate
(4)
1.000
:::
1.000
1.001
1.136

To Ult
1.000
1.000
100.0%

Exhibit IV
Sheet 2

Projected
Ultimate
Claims
(5)
3,376
:::
2,814
2,952
2,798
28,913

Column Notes:
(2) Age of accident year in (1) at December 31, 2007.
(3) Based on data from DC Insurer.
(4) Based on selected CDF in Exhibit IV, Sheet 1.
(5) = [(3) x (4)].

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Exhibit IV, Sheet 3: Use selected ultimate claims and the selected reporting pattern to compare actual
reported claims one year later (i.e. 12/31/2008) with our expected claims for the year.
DC Insurer
Annual Monitoring Test ($000)

Accident
Year
(1)
1997
1998
:::
2006
2007
Total

Selected
Ultimate
Claims
(2)
3,376
2,788
:::
2,952
2,798
28,913

Exhibit IV
Sheet 3

Expected % Reported at
12/31/2007
12/31/2008
(3)
(4)
100.0%
100.0%
100.0%
100.0%
:::
:::
99.9%
100.0%
88.0%
99.9%

Reported Claims at
12/31/2007
(5)
3,376
2,788
:::
2,949
2,463
28,575

12/31/2008
(6)
3,376
2,788
:::
3,030
2,733
28,983

Claims Reported Between
12/31/07 and 12/313/08
Actual
Expected
Difference
(7)
(8)
(9)
0
0
0
0
0
0
:::
:::
:::
81
3
78
270
332
-62
408
335
73

Column Notes:
(2) Developed in Exhibit IV, Sheet 2.
(3) and (4) Based on selected CDF in Exhibit IV, Sheet 1.
(5) and (6) Based on data from DC Insurer.
(8) = [(6) - (5)].
(7) = {[(2) - (5)] / [1.0 - (3)] x [(4) - (3)]}.
(9) = [(7) - (8)].

For each AY, expected reported claims in the calendar year are equal to:
[(ultimate claims selected at 12/31/2007 - actual reported claims at 12/31/2007) / (% unreported at 12/31/2007)]
x (% reported at 12/31/2008 - % reported at 12/31/2007)]
The % unreported is computed as [1.00 - (1.00 / cumulative claim development factor)].
Examples:
The expected reported claims for accident year 2007 during calendar year 2008 are equal to:
AY07 Expected ClaimCY08 = {[($2,798 - $2,463) / (1 - 0.880)] x (0.999 - 0.880)} = $332
The expected reported claims for accident year 2006 during calendar year 2008 are equal to:
AY06 Expected ClaimCY08 = {[($2,952 - $2,949) / (1- 0.999)] x (1.000 - 0.999)} = $3
Actuaries often rely on techniques other than the development technique to select ultimate claims.

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
A method often used to derive payment patterns is to compare the historical paid claim development
triangle to the final value of selected ultimate claims, as shown in Exhibit V, Sheet 1.
Exhibit V
Sheet 1

XYZ Insurer - Auto BI
Ratio of Paid Claims to Selected Ultimate Claims ($000)
PART 1 - Data Triangle
Accident
Year
12
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008

1,302
1,539
2,318
1,743
2,221
3,043
3,531
3,529
3,409

PART 2 - Ratios
Accident
Year
12
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008

0.035
0.040
0.048
0.039
0.031
0.041
0.064
0.088
0.102

24
4,666
6,513
5,952
7,932
6,240
9,898
12,219
11,778
11,865

24
0.186
0.175
0.153
0.165
0.141
0.140
0.164
0.214
0.294

Paid Claims as of (months)
36
48
60
6,309
9,861
12,139
12,319
13,822
12,683
25,950
27,073
22,819

8,521
13,971
17,828
18,609
22,095
22,892
43,439
40,026

10,082
18,127
24,030
24,387
31,945
34,505
52,811

108

120

132

11,620
22,032
28,853
31,090
40,629
39,320

13,242
23,511
33,222
37,070
44,437

14,419
24,146
35,902
38,519

15,311
24,592
36,782

15,764
24,817

15,822

Ratio of Paid Cliams to Selected Ultimate Claims as of (months)
36
48
60
72
84
96
0.399
0.393
0.326
0.318
0.287
0.286
0.367
0.362
0.415

0.539
0.556
0.479
0.480
0.459
0.516
0.614
0.536

0.637
0.722
0.645
0.629
0.663
0.778
0.746

108

120

132

0.734
0.878
0.775
0.801
0.843
0.886

0.837
0.936
0.892
0.955
0.923

0.911
0.962
0.964
0.993

0.968
0.979
0.988

0.996
0.988

1.000

120

132

Selected
Ultimate
15,822
25,107
37,246
38,798
48,169
44,373
70,780
74,726
54,968
40,300
33,491

PART 3 - Average Age-to-Age Factors

Average
12

108

0.065
0.085
0.095

0.191
0.224
0.254

0.343
0.381
0.389

0.521
0.555
0.575

0.692
0.729
0.762

0.837
0.844
0.865

0.909
0.923
0.939

0.957
0.973
0.978

0.978
0.978
0.984

0.992
0.992
0.992

1.000
1.000
1.000

0.064

0.173

0.339

0.510

0.685

0.841

0.917

0.963

0.979

0.992

1.000

PART 4 - Selected Age-to-Age Factors
Development Factor Selection
12-24
24 - 36
36 - 48
48 - 60
60 - 72
Selected
0.085
0.224
0.381
0.555
0.729

72 - 84
0.844

84 - 96
0.923

96 - 108
0.973

Simple Average
Latest 5
Latest 3
Latest 2

Medial Average
Latest 5x1

108 - 120 120 - 132
0.978
0.992

To Ult
1.000

Various averages of the % paid at each maturity are calculated and a payment pattern is selected.
Similar calculations for the reporting pattern are shown in Exhibit V, Sheet 2.

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Shown below are the implied payment and reporting patterns based on the unadjusted development
patterns, the development patterns after B/S adjustments, and the final selections from Exhibit V, Sheets
1 and 2.

Maturity
Age
12
24
36
48
60
72

Comparison of Reporting and Payment Patterns
Reporting
Payment
Unadjusted Adjusted
Unadjusted Adjusted
CDF
CDF
Selected
CDF
CDF
39.2%
44.9%
51.1%
4.5%
7.4%
66.1%
74.3%
75.8%
15.2%
30.8%
83.6%
91.4%
88.7%
31.6%
44.1%
92.2%
99.0%
95.8%
49.8%
58.3%
94.0%
99.3%
97.1%
65.6%
74.0%
98.7%
100.0%
98.9%
78.9%
89.2%

Selected
8.5%
22.4%
38.1%
55.5%
72.9%
84.4%

It can be challenging to develop a system for quarterly or monthly monitoring given an estimation
process that focuses only on annual CDF.
 Some insurers maintain claim development data on a quarterly basis. For these organizations,
development factors are readily available for quarterly analyses, and linear interpolation between
quarters is likely sufficient for monthly monitoring purposes.
 For insurers who only have annual claim development data, linear interpolation of annual
development patterns is not appropriate, particularly for the most immature AYs.
According to B/F In the paper "The Actuary and IBNR":
In the absence of data, it might be reasonable to assume that the:
i. cumulative distribution of development by quarter for the most recent AY is skewed say 40% at 3
months, 70% at 6 months, 85% at 9 months, 100% at 12 months, and that the
ii. distribution for prior AYs is uniform: 25%, 50%, 75%, 100%.
Upon further study, the authors found that their data revealed prior year's development were also skewed;
approximate distribution: 33%, 60%, 80%, 100%. The data reviewed were excess of loss and it is
recognized that distributions observed may not be typical of ordinary business.
DC Insurer has the systems capability to capture claim development data on a quarterly basis, and built a
model for monthly claims monitoring based on linear interpolation of the quarterly CDFs.
Exhibit IV, Sheet 4: A template for January and February 2008.
DC Insurer
Monthly Monitoring Test ($000)

Accident
Year
(1)
1997
:::
2006
2007
Total

Selected
Ultimate
Claims
(2)
3,376
:::
2,952
2,798
28,913

Expected % Reported at
12/31/2007
1/31/2008
2/29/2008
(3)
(4)
(5)
100.0%
100.0%
100.0%
:::
:::
:::
99.9%
99.9%
100.0%
88.0%
91.5%
95.0%

Exhibit IV
Sheet 4

Actual Reported Claims at
12/31/2007
1/31/2008
2/29/2008
(6)
(7)
(8)
3,376
3,376
3,376
:::
:::
:::
2,949
2,951
2,986
2,463
2,473
2,538
28,575
28,616
28,727

Claims Reported Between
12/31/07 and 12/31/08
Actual
Expected
Difference
(9)
(10)
(11)
0
0
0
:::
:::
:::
2
0
2
10
97
-87
41
98
-57

Claims Reported Between
01/31/08 and 02/29/08
Actual
Expected
Difference
(12)
(13)
(14)
0
0
0
:::
:::
:::
35
3
32
65
98
-33
111
100
11

Column Notes:
(2) Developed in Exhibit IV, Sheet 2.
(3) Based on selected CDF in Exhibit IV, Sheet 1.
(4) and (5) Based on linear interpolation of selected CDF in Exhibit IV, Sheet 1.
(6) through (8) Based on data from DC Insurer.
(9) = [(7) - (6)]
(10) = {{(2) - (6)] / [1.0 - (3)] x [(4) - (3)11.
(11) = [(9) - (10)].
(12) = [(8) - (7)].
(13) = {[(2) - (6)] / [1.0 - (3)] x [(5) - (4)1}.
(14) = [(12) - (13)].

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
In his review of the B/F paper "The Actuary and IBNR", Hugh White offered a problem that is still relevant
for actuaries monitoring unpaid claims today. Mr. White stated:
You are trying to establish the reserve for commercial auto BI and the reported proportion of expected
losses as of statement date for the current AY is 8% higher than it should be. Do you:
1. Reduce the bulk (i.e. IBNR) reserve a corresponding amount (because you sense an acceleration
in the rate of reporting)?;
2. Leave the bulk reserve at the same % level of expected losses (because you sense a random
fluctuation such as a large loss)?; or
3. Increase the bulk reserve in proportion to the increase of actual reported over expected reported
(because you don't have 100% confidence in your "expected losses")?
None of these suggested "answers" is satisfactory without further extensive investigation, and yet, all
are reasonable.
The actuary must obtain a comprehensive understanding of the situation, achieved through meetings with
management and other parties who understand the situation and through detailed analyses of the claims
and claims experience.

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Sample Questions:
1. Chapter 15 expands on the last two steps in a four-phase approach to the process of estimating unpaid
claims described by Wiser. List the four steps, as shown in Friedland’s Introduction to Part 2.
2. In the discussion of Monitoring and Interim Techniques, Friedland emphasizes the importance of
comparing the difference between actual and expected claims reported (or paid) in period of time
(such as a month or quarter). She states that these comparisons are important “so that the actuary
can understand the appropriateness of prior selections and make revisions as necessary if actual
claims do not emerge as expected.” Define the term emergence.
3. Given the following data as of 12/31/07 (taken from Friedland’s Exhibit IV, Sheets 2 & 3):
CDF to Ultimate at 12 months
CDF to Ultimate at 24 months

1.136
1.001

Reported Claims to date for AY 07
Est. Ultimate Claims for AY 07

2,463
2,798

Calculate the expected reported claims (emergence) for AY 2007 during Calendar Year 2008.

1994 Exam Questions (modified):
53.

You are given the following data:
Reported CDF to Ult. at 12 months
Reported CDF to Ult. at 24 months
Reported CDF to Ult. at 36 months
Accident Year 1994 data:
Reported Claims as of 12/31/94
Earned Premium
Expected Claims Ratio

4.00
2.00
1.50

130,000
1,000,000
65%

Assuming the Bornhuetter-Ferguson method is used to estimate ultimate claims, calculate the
following amounts for Accident Year 1994:
a. (.5 point) The estimate of IBNR as of 12/31/94.
b. (1 point) The amount of IBNR expected to emerge in 1995.

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
1995 Exam Questions (modified):
44. (Continued from Chapter 9 question)
You are given the following information (amounts are in $000s):

Earned
Premium

Accident
Year

4,500
5,000
5,200
5,300
5,700

1990
1991
1992
1993
1994

Reported Claims including ALAE ($000's omitted) as of
12 mo,
24 mo,
36 mo,
48 mo,
60 mo,
Report 1
Report 2
Report 3
Report 4
Report 5
2,000
2,102
2,234
2,339
2,482

2,600
2,638
2,938
2,985

2,990
3,086
3,408

3,283
3,343

3,283

Also given Cumulative Distribution of development by quarter, WITHIN a calendar year:
1st Qtr
2nd Qtr
3rd Qtr
4th Qtr
Most recent prior Accident Year
40%
65%
85%
100%
Earlier Accident Years
35%
60%
80%
100%

Assume that all claims reach ultimate settlement at 60 months, and the expected claim ratio is 75%.
For purposes of allocating expected development, use distribution of development by quarter.
a. See Chapter 9 for illustration of B-F method.
b. See Chapter 7 for illustration of Development method.
c. (1.5 points) Using Estimated IBNR developed using the B-F method at 12/31/94, determine the
expected IBNR balance as of 6/30/95 for accident years 1994 and prior. Show all work.
1996 Exam Questions (modified):
48. (1 point) What are the two important considerations discussed by Friedland with respect to the review of
IBNR estimates at the close of interim accounting periods?
List three reasons an Insurance Company may be interested in monitoring its Unpaid Claims Estimates at
various times throughout a year.

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ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
1997 Exam Questions (modified):
52.

You are given the following information:
"AY"
Accident
Year

Reported
Claims
& ALAE

"EP"
Earned
Premium

1993
1994
1995
1996

5,000,000
7,000,000
6,000,000
8,000,000

9,000,000
9,450,000
9,922,500
10,418,625

Ultimate Claim
Development Factors (CDFs)
by development age
48
36
24
12

to
to
to
to

Ultimate
Ultimate
Ultimate
Ultimate

1.05
1.15
1.30
1.75

Expense Assumptions (as a % of Earned Premium):
General Expense
5%
Acquisition Expense
10%
Taxes, Licenses, Fees
3%
Underwriting Profit Load
0%
Other Assumptions:
Earned premium is growing at 5% per year.
There is no expected claim development beyond 60 months.
Calculate the following, using the Bornhuetter and Ferguson methodology outlined by Friedland:
a. (1 point) The expected claims to be reported for accident years 1993 to 1996 (including detail by
AY) during calendar year 1997.
b. (1 point) An estimate of expected IBNR as of 12/31/97.
Show all work.

2000 Exam Questions (modified):
54. (1 point) According to Friedland, what are two reasons for reviewing estimates of Unpaid Claims
(reserves) between annual calculations?

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2002 Exam Questions (modified): (Continued from Chapter 7 & 9)
22. You are given the following information:
Accident
Year

Earned
Premium ($000's)

Reported Claims
at 12-31-01 ($000's)

Expected
Claim Ratio

1998
1999
2000
2001

200
1,000
1,500
1,500

100
1,000
900
600

80%
80%
80%
80%

Selected Age-to-Age reported claim development factors:
12 - 24 months
1.250
24 - 36 months
1.100
36 - 48 months
1.050
48 - 60 months
1.080
No further development after 60 months
a,b,c. See Chapter 7 & 9
d. (1 point) Using the Bornhuetter-Ferguson method, calculate the amount of claim development to be
expected during Calendar Year 2002, on Accident Years 1998 through 2001. Show all work.
2003 Exam Questions (modified):
2. You are given the following information:
Accident
Year
2000
2001
2002

Selected Ultimate
Claims
$310,000
290,000
300,000

Age in
Months
12
24
36
48
60

Reported
CDF to
Ultimate
2.10
1.55
1.25
1.10
1.05

What is the total dollars of claims expected to be reported (emerge) during calendar year 2003, on accident
years 2000 through 2002?
A. < $120,000
D. > $160,000 but < $180,000

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C. > $140,000 but < $160,000
E. > $180,000

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2004 Exam Questions (modified):
2. You are given the following accident year Age-to-Age claim development factors:
ATA Factors by Months of Development
12 to 24
24 to 36
36 to 48
48. to 60
60 to Ultimate
1.60
1.30
1.10
1.05
1.00
What percentage of the IBNR at 12 months will be expected to emerge (be reported) during the
following calendar year?
A. < 25%
B. > 25% but < 35%
C. > 35% but < 45%
D. > 45% but < 55%
E. > 55%

2005 Exam Questions (modified):
10. You are given the following information:

•
•

Accident

Earned

Year
2001
2002
2003
2004

Premium
$19,000
20,000
21,000
22,000

Reported Claims
(Age of Development in Months)
12
$4,850
5,150
5,400
7,200

24
$ 9,700
10,300
10,800

36
$14,100
14,900

48
$16,200

Expected Claim Ratio = 0.90.
Assume no development past 48 months.

a. & b. See chapters 7 & 9.
c. (1 point) Using the Bornhuetter-Ferguson method, calculate the expected IBNR for Accident Year 2004
expected to be reported (emerge) during calendar year 2005.
d. (0.5 point) State two possible causes for reported claims at 12 months for accident year 2004 being
approximately 25% higher than would have been expected, based solely on premium growth.
e. (1 point) For each possible cause you identified in part d. above, how would you adjust your estimate of
the expected IBNR emergence for accident year 2004 during calendar year 2005?

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2008 Exam Questions (modified):
Question 2 (1.50 points).

Given the following for policy year 2006 for a line of business:
Premium
Expected loss emerged at 24 months
Expected loss emerged at 36 months
Reported loss as of December 31, 2007
Bornhuetter-Ferguson estimate of ultimate loss

a. (0.5 point)
See chapter 9.

1,600,000
68%
82%
800,000
1,133,000

b. (0.5 point)
See chapter 7.

c. (0.5 point)
Calculate the expected calendar year 2008 development for policy year 2006
based on the Bornhuetter-Ferguson Method.
Question 11 (1.50 points).
The loss ratio for a book of business is improving. There have been no changes in either claim
emergence patterns or the company’s claim reserving practices. IBNR has been estimated based
on two different methods, and is summarized as follows:
Loss Development Method
24,000,000
Bornhuetter-Ferguson Method
31,000,000
a. (.75 points)
Discuss the issues surrounding the expected accuracy of each of the methods, given the situation.
b. (.75 point)
After the period of improvement, the loss ratio stabilizes. Briefly describe the adjustments, if any,
that should be made to the methods used to estimate IBNR, to arrive at an accurate estimate.

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2008 Exam Questions (modified):
Question 27 (2.0 points).

Given the following to be used in a Review of Unpaid Claim Estimates:

Accident
Year
2004
2005
2006

Reported
Claims as
of 12-31-06
4,500,000
4,300,000
3,700,000

Selected
IBNR as
of 12-31-06
1,100,000
2,300,000
4,800,000

Reported
Claims as
of 12-31-07
4,750,000
5,200,000
5,000,000

Age (mo.) Reported CDF to Ultimate
12
2.222
24
1.538
36
1.250
48
1.176
a. (1.5 points)
Calculate the expected claim emergence during calendar year 2007 for
accident years 2004 - 2006, based on the selected IBNR at 12-31-06.
b. (.5 point)
Using numerical support, describe the conclusion that should be
drawn regarding the accuracy of the IBNR reserving process used at
12-31-06, based on a comparison of actual versus expected claim
emergence during calendar year 2007.

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2009 Exam Questions
15. (3.25 points) Given the following information as of December 31, 2008:
Estimated
Claims
Earned
Ultimate
Accident
Reported
Claims Paid
Premium
Claim
Year
($000)
($000)
($000)
Counts
2006
2007
2008

51,450
33,600
19,950

24,150
12,600
4,200

113,400
65,100
50,400

1,890
1,680
1,470

Open and
IBNR
Counts
630
840
1,260

Ultimate claims estimates ($000) resulting from four different development methods:
Both Case and
Payment-Rate
Payment-Rate
Unadjusted
Unadjusted
Adjusted
Adjusted
Accident
Reported
Paid
Reported
Paid
Year
Development
Development
Development
Development
2006
55,100
72,000
55,650
54,600
2007
45,600
85,800
45,150
40,950
2008
51,700
82,500
45,580
46,200
The claims department implemented a new program in 2007, which resulted in the adjusters paying
claims faster.
a. (0.5 point) Taking into account the new claims program, identify which one of the above development
methods should be rejected and explain why.
b. (2.25 points) For each of the remaining three methods, calculate the ultimate claims ratio, ultimate
severity and unpaid severity tests for accident year 2008.
c. (0.5 point) Describe a course of action that the reserving actuary might take in light of the results of
the diagnostic tests in part b. above.

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2009 Exam Questions (cont’d)
16. (2.5 points) Given the following information as of December 31, 2007:

Accident
Year

Reported
Claims
($000)

Selected
Ultimate
($000)

2005
2006
2007

1,200
1,113
1,166

1,200
1,325
1,446

Selected cumulative development factors:
12-Ultimate
24-Ultimate
36-Ultimate
1.212
1.154
1.010
Accident year 2007 reported claims ($000) as of December 31, 2008 total 1,250.
a. (1 point) Based on the data and selections as of December 31, 2007, calculate the difference
between the actual reported claims versus the expected claims emergence in calendar year 2008
for accident year 2007.
b. (1 point) Using linear interpolation of the given development pattern, project the expected emerged
claims for accident year 2007 from January 1, 2008 through May 31, 2008.
c. (0.5 point) Identify whether using linear interpolation in part b. above will overestimate or
underestimate the projection and explain why.
2010 Exam Questions:
12. (2 points) Given the following information as of December 31, 2009:
Expected
Percentage
Unreported
0%
10%
X%
40%

Accident
On-Level
Reported
Year
Earned Premium
Claims
2006
$100,000
$62,000
2007
120,000
60,000
2008
140,000
50,000
2009
160,000
40,000
Total
520,000
212,000
• The expected claim ratio is 65%.
• The projected ultimate claims using the Bornhuetter-Ferguson technique is $279,600 for all years
combined.
Calculate X, the expected percentage unreported for accident year 2008.

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2011 Exam Questions
30. (2.5 points) Given the following data for a line of business as of December 31, 2010:
Accident
Year
2006
2007
2008
2009
2010

12 Months 24
$10,000
$10,300
$10,609
$10,927
$14,205

Accident
Year
2006
2007
2008
2009
2010

12 Months 24
$5,000
$5,150
$5,305
$5,464
$5,628

Average Case Reserves on Open Claims
24 Months
36 Months
48 Months 48
$12,000
$15,000
$22,000
$12,360
$15,450
$28,600
$12,731
$20,085
$16,550

60 Months
$35,000

Average Cumulative Paid on Closed Claims
24 Months
36 Months
48 Months 48
$6,000
$7,500
$11,000
$6,180
$7,725
$11,330
$6,365
$7,957
$6,556

60 Months
$16,000

a. (1 point) Fully discuss whether the reported claim development technique is appropriate to
estimate unpaid claims for this line of business.
b. (1 point) Fully discuss whether the paid claim development technique is appropriate to estimate
unpaid claims for this line of business.
c. (0.5 point) Discuss whether the expected claim technique is appropriate to estimate unpaid claims
for this line of business.

37. (1.75 points) The following table summarizes the estimated IBNR from various estimation techniques
for four different books of business.
Estimation Technique
Development – Reported
Development – Paid
Expected Claims
Bornhuetter-Ferguson – Reported
Bornhuetter-Ferguson – Paid

Book A
$610
$610
$14
$445
$161

Book B
$495
$250
$250
$453
$250

Book C
$450
$923
$450
$450
$781

Book D
$806
$645
$811
$809
$745

a. (0.5 point) The true IBNR for Book A is $610. Based on the estimated IBNR for each method shown, discuss
a change in Book A that would explain the discrepancies between the estimates and the true IBNR.
b. (0.5 point) The true IBNR for Book B is $250. Based on the estimated IBNR for each method shown, discuss
a change in Book B that would explain the discrepancies between the estimates and the true IBNR.
c. (0.5 point) The true IBNR for Book C is $450. Based on the estimated IBNR for each method shown, discuss
a change in Book C that would explain the discrepancies between the estimates and the true IBNR.
d. (0.25 point) For Book D, briefly discuss a next step the actuary should take to understand the
difference in results.

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2012 Exam Questions
29. (1 point) Given the following accident year 2011 information as of December 31, 2011:
Initial Expected Losses
$50,000,000
Reported Losses
$10,000,000
Paid Losses
$2,000,000
Selected 12-Ult Reported Development Factor
10.000
Selected 12-Ult Paid Development Factor
22.500
Technique
Development – Reported
Development – Paid
Expected Claims
Bornheutter-Ferguson – Reported
Bornheutter-Ferguson – Paid
•

Unpaid Claim Estimate
$98,000,000
$43,000,000
$48,000,000
$53,000,000
$47,777,778

According to the claims department, an extraordinarily large claim has been reported but not yet paid.

Determine and fully justify a reasonable unpaid claim estimate for accident year 2011 claims as of
December 31, 2011.
30. (1.5 points) Given the following information as of December 31, 2010:
Cumulative Reported Claims ($000s)
Accident
Year
2006
2007
2008
2009
2010

Pattern I
Pattern II

12
Months

24
Months

36
Months

48
Months

60
Months
$147,194

$148,459
$135,337
$140,800
$115,050
Reported Claim Development Factors
12-Ult
24-Ult
36-Ult
48-Ult
1.502
1.155
1.050
1.000
1.452
1.134
1.060
1.000

The following claims are reported during calendar year 2011 ($000s):
Claims Reported in Calendar Year 2011 ($000s)
Accident Year
CY 2011
2010
2009
2008
2007
CY 2011
$114,800
$34,200
$10,100
$8,104
$1,000

Total
$168,204

Determine which of the two reported claim development patterns shown above best reflect the actual
emergence of claims. Justify your selection.

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to Sample Questions:
1. Friedland’s Chapter 15 expands on the last two steps in a four-phase approach to the process of
estimating unpaid claims, as described by Wiser. The four steps, as in the Introduction to Part 2:
- Exploring the data to identify its key characteristics and possible anomalies. Balancing data to other
verified sources should be undertaken at this time.
- Applying appropriate techniques for estimating unpaid claims.
- Evaluating the conflicting results of the various methods used, with an attempt to reconcile or explain
the different outcomes. At this point, the projected ultimate amounts are evaluated in contexts outside
their original frame of analysis.
- Monitoring projections of claim development over subsequent calendar periods. Deviations of actual
development from projected development of counts or amounts are one of the most useful diagnostic
tools in evaluating the accuracy of unpaid claim estimates.
2. Define the term emergence:
Emergence refers to the reporting or development of claims and/or claim counts over time.
Note: In Friedland’s Monitoring Tests, she compares Actual Claims Reported (Emergence) against
Expected Claims Reported (Emergence) based on the selected ultimate claim estimates.
3. Calculate the expected reported claims (emergence) for AY 2007 during Calendar Year 2008.
Step 1: Calculate the expected percent to be reported in the interim period:
Since the Percent ( of total Ultimate Claims ) Reported = Inverse of Ultimate CDF **
The expected Percent Reported at 12 months = 1/1.136=
0.8803
& expected Percent Reported at 24 months = 1/1.001=
0.9990
Then the expected percent reported between ages 12 and 24 months =
11.87%
** Don't forget to work with inverses (%), instead of the CDFs directly
Proceeding the safe way:
We need a factor for the Unreported part of Ultimate Claims, not total Ultimate Claims.
That is, we must mak e this % conditional upon the development remaining, so we
divide by (1 - Percent Reported at beginning of Interval) = (1 - .8803) =
0.1197.
Thus, the factor we apply to IBNR (broadly defined) = .1187/.1197 =
99.2%
to give the percent of IBNR expected to emerge in period between 12 and 24 months.
Step 2: Apply the appropriate percent to the estimated IBNR
IBNR (broadly defined) = Estimated Ultimate Claims - Reported Claims =
Estimated IBNR = 2,798 - 2,463 =
x Expected emergence of IBNR

335
x 99.2%
332

Note: Friedland shows this calculation, using notation:
AY 07 Expected Claims CY08 = { [ ($2,798 - $2,463) / (1 - .880) ] * (.999 - .880) } = $332

Continues below:

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solution to sample question 3 (continued):
OR, proceeding for SPECIAL CASES only:
When the CDFs are established AND applied such that the Estimated Ultimate Claims
exactly mimic the actual Reported Claims, we can use a shortcut:
Step 2: Apply the (unconditional) percent to the estimated Ultimate Claims
Estimated Ultimate Claims =
2,798
x Expected emergence of Ultimate
x 11.87%
332
Another special case is the B-F method, but we need to replace "Estimated Ultimate
Claims" with the "A-Priori Expected Claims" since under the B-F method, IBNR
exactly follows this a-priori expected amount. And then apply the unconditional percent.

Solutions to 1994 Exam Questions (modified):
53. Calculate the following amounts for accident year 1994:
a. (.5 point) The estimate of IBNR as of 12/31/94
$1,000,000 * .65 * (1-1/4.00) = $487,500
b. (1 point) The amount of IBNR expected to emerge in 1995
$1,000,000 * .65 * [(1-1/4.00) - (1-1/2.00] = $162,500
Extra detail shown below, for illustrative purposes:
a) Recall, for the B-F method:
IBNR = (A-priori Expected Claims) * (Percent Unreported)
Age of
Data at
12/31/94

Reported
CDF to
Ultimate

Percent
Reported
12/31/94

Percent
Unreport
12/31/94

(1)
1994 12 months
Total

(2) given
4.00

(3)=1.0/(2)
25.0%

(4)=1.-(3)
75.0%

A priori
Earned
Expected
Premium Claim Ratio

A priori
Expected
Claims

"IBNR"
Expected
Unreport

(5) given
1,000,000

(7)=(5)*(6)
650,000

(8)=(7)*(4)
487,500

Accident
Year

Accident
Year
1994

(6) given
65.0%

Note: The Percent Unreported
= 1 minus inverse of Ult. CDF

Or Shortcut using
IBNR
Expected Claims *
(broadly
defined)
Percent Unreported
(8)=(1M)*(65%)*[1.0-1.0/4.00]
487,500

Continues below:

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 1994 Exam Question #53 (continued):
b)
Step 1: Calculate the expected percent to be reported in the interim period:
Since the Percent ( of total Ultimate Claims ) Reported = Inverse of Ultimate CDF **
The expected Percent Reported at 12 months = 1 / 4.00 =
0.2500
& expected Percent Reported at 24 months = 1 / 2.00 =
0.5000
Then the expected percent reported between ages 12 and 24 months =
25.00%
** Don't forget to work with inverses (%), instead of the CDFs directly
Proceeding the safe way:
We need a factor for the Unreported part of Ultimate Claims, not total Ultimate Claims.
That is, we must mak e this % conditional upon the development remaining, so we
divide by (1 - Percent Reported at beginning of Interval) = (1 - .25) =
0.7500.
Thus, the factor we apply to IBNR (broadly defined) = .25 / .75 =
33.3%
to give the percent of IBNR expected to emerge in period between 12 and 24 months.
Step 2: Apply the appropriate percent to the estimated IBNR
IBNR (broadly defined) = Estimated Ultimate Claims - Reported Claims =
Estimated IBNR from a =
x Expected emergence of IBNR

OR, proceeding for SPECIAL CASES only, a shortcut:
A special case is the B-F method, where we use "A-Priori Expected Claims" since
under the B-F method, IBNR exactly follows this a-priori expected amount.
And then apply the unconditional percent:
Step 2: Apply the (unconditional) percent to the estimated Ultimate Claims
A-priori Expected Claims =
x Expected emergence of Ultimate

487,500
x 33.3%
162,500

650,000
x 25.00%
162,500

Solutions to 1995 Exam Questions (modified):
44.

Continued (from chapter 9)
c. Using Estimated IBNR developed using the B-F method at 12/31/94, determine the expected IBNR
balance as of 6/30/95 for accident years 1994 and prior. Show all work.
Initial Comments:
Part c is asking us to take 6 months of expected emergence out of the IBNR estimates we found in part a.
Those IBNR estimates were calculated in Chapter 9, but are repeated below for convenience.
Step 1: Start by finding the percent of emergence expected for an annual period, as in the questions above.
Step 2: Multiply by the IBNR estimates for each year.
Step 3: Pro-rate those amounts to 6 mo., using the development patterns we are given for within each year.
Step 4: For the balance, we subtract the 6 mo. emerged IBNR estimates from the 12/31/94 IBNR amounts.

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solution to 1995 Exam Question # 44 (continued):
Preliminary Info for 1995 # 44 Recall B-F IBNR estimates from part a (see chapter 9):
Accident
Year

Age of
Data at
12/31/94

Reported
CDF to
Ultimate

Percent
Reported
12/31/94

Percent
Unreport
12/31/94

1990
1991
1992
1993
1994

(1)
60 months
48 months
36 months
24 months
12 months

(2) above
1.0000
1.0000
1.0905
1.2650
1.6224

(3)=1.0/(2)
100.0000%
100.0000%
91.7011%
79.0514%
61.6371%

(4)=1.-(3)
0.0000%
0.0000%
8.2989%
20.9486%
38.3629%

A priori
Earned
Expected
Premium Claim Ratio

A priori
Expected
Claims

"IBNR"
Expected
Unreport

(7)=(5)*(6)
3,375
3,750
3,900
3,975
4,275

(8)=(7)*(4)
0.0000
0.0000
323.6571
832.7069
1,640.0140
2,796.3780

Accident
Year
1990
1991
1992
1993
1994
Total

(5) given
4,500
5,000
5,200
5,300
5,700

(6) given
75.0%
75.0%
75.0%
75.0%
75.0%

Note: The Percent Unreported
= 1 minus inverse of Ult. CDF

Or shortcut using
IBNR
Est. Expected Claims
(broadly
defined)
x Percent Unreported
(8)=(7)*[1.0-1.0/CDF]
0.0000
0.0000
323.6571
832.7069
1,640.0140
2,796.3780

Steps 1 and 2 produce the expected emergence in the next annual period, as in the questions above.
Percent
Reported

Step 1
Percent
Reported

at 1/1/95

at 12/31/95

(9) FYI (10) See (3)
36 to 48 91.7011%
24 to 36 79.0514%
12 to 24 61.6371%

(11) see(3)
100.0000%
91.7011%
79.0514%

Estimated
IBNR

Ages in
Acc
NEXT yr
Year at 12/31/94 (CY 1995)
(8) above
1992
323.6571
1993
832.7069
1994 1,640.0140
Total 2,796.3780

Percent
Unreported
at 1/1/95
(12)= 1-(10)

8.2989%
20.9486%
38.3629%

Percent of current IBNR
expected to emerge
between 1/1 - 12/31/95
(13) = [(11)-(10)] / (12)
100.0000%
60.3845%
45.3936%

Step 2
Est. IBNR
to emerge
during '95
(14)=(8)*(13)

323.6571
502.8259
744.4614
1,570.9444

* Here the period we evaluate for emerged losses is the year immediately following our original estimate

Note, as an ALTERNATIVE, we could use short-cut using a prior Expected Claims since we're using B-F
A-Priori
Percent
Percent
Percent of Ultimate
Acc
Expected
Reported
Reported
expected to emerge
Year
Claims
at 1/1/95 at 12/31/95
between 1/1 - 12/31/95
1992
1993
1994
Total

(7) above
3,900
3,975
4,275
12,150

(10) See (3)
91.7011%
79.0514%
61.6371%

(11) see(3)
100.0000%
91.7011%
79.0514%

(13') = [(11)-(10)]
8.2989%
12.6497%
17.4143%

Est. IBNR
to emerge
during '95
(14)=(7)*(13')

323.6571
502.8256
744.4613
1,570.9440

Continues on next page.

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solution to 1995 Exam Question # 44 (continued):
Step 1: Start by finding the percent of emergence expected for an annual period, as in the questions above.
Step 2: Multiply by the IBNR estimates for each year.
Step 3: Pro-rate those amounts to 6 mo., using the development patterns we are given for within each year.
Step 4: For the balance, we subtract the 6 mo. emerged IBNR estimates from the 12/31/94 IBNR amounts.
Relevant
Accident
Years
1992
1993
1994
Total

Step 3
Amount of IBNR
Pro-ration
expected to emerge
for 6 mo.
between 1/1 - 12/31/95
within yr.
(14)
323.6571
502.8256
744.4613
1,570.9440

Amount of IBNR
expected to emerge
between 1/1 - 6/30/95

Step 4
Estimated Estimated
IBNR
IBNR
at 12/31/94 at 6/30/95

(16) = (14) * (15)
194.1943
301.6954
483.8998
979.7895

(17) above (18)=(17)-(16)
323.6571
129.4628
832.7069
531.0115
1,640.0140 1,156.1142
2,796.3780 1,816.5885

(15)
60%
60%
65%

Note: these types of questions intentionally exclude new loss occurrences.
So, claims occurring in the first half accident year 1995 are excluded, to compare "apples-to-apples."

Solutions to 1996 Exam Questions (modified):
48. (1 point) What are the two important considerations discussed by Friedland with respect to the review of
IBNR estimates at the close of interim accounting periods?
CAS answer (pre-Friedland)
1. The need to compare actual with expected claim emergence. If there are material differences,
adjustments to the Unpaid Claims Estimates (reserve) may have to be made.
2. The need to consider material changes in exposures or premium rate adequacy.
Related Friedland quotes:
“In addition to measuring changes in claims for historical periods, the actuary must incorporate the effect of
changes in the exposure for the current period …”
“Monitoring unpaid claims can be important for insurers from a financial reporting perspective, for budgeting
and planning purposes, for pricing and other strategic decision-making, and for planning for the next
complete analysis of unpaid claims.”

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 1997 Exam Questions (modified):
52
Calculate the following, using the Bornhuetter and Ferguson methodology outlined by Friedland:
a. (1 point) The expected claims to be reported for accident years 1993 to 1996 (including detail by AY)
during calendar year 1997.
b.
(1 point) An estimate of expected IBNR as of 12/31/97.
a.
Preliminary Comments:
To find the Expected Claims Ratio to use in our A-Priori B-F values, we use 1 - (Expenses & Profit Load)
A-priori Expected Claims Ratio = 100% - 5% - 10% - 3% - 0% = 82% to apply to Earned Premium (EP)

Accident
Year

A-Priori
Expected
Claims

Step 1 (using shortcut described above for B-F special case)
Ages in
Percent
Percent
Percent of Ultimate
NEXT yr
Reported
Reported
expected to emerge
(CY 1997)
at 1/1/97 at 12/31/97
in the '97 Calendar Year

(1)=.82*EP
(2) FYI
(3)=1/CDF
(4)
1993 7,380,000.0
48 to 60
95.2381% 100.0000%
1994 7,749,000.0
36 to 48
86.9565%
95.2381%
1995 8,136,450.0
24 to 36
76.9231%
86.9565%
1996 8,543,272.5
12 to 24
57.1429%
76.9231%
Total
Note: these types of questions intentionally exclude new loss occurrences.

Step 2
Est. IBNR
to emerge
during '97

(5) = [ (4) - (3) ]
(6)=(1)*(5)
4.7619%
351,428.2200
8.2816%
641,741.1840
10.0334%
816,362.5743
19.7802% 1,689,876.3870
3,499,408.3653
answer a.

So, claims occurring in accident year 1997 are excluded in part a., to compare "apples-to-apples."
However, part b. of this question ask s us to look at AY 1997 claims as well . . .

AY
1993
1994
1995
1996
1997
Total

Start with activity for AY's '96 and prior
+ New for AY 1997
Estimated IBNR at
IBNR from 12/31/96
Estimated IBNR at
end of THIS period
expected to emerge in '97
end of NEXT period (97)
(so, for AY's '96 and prior)
for AY '97 claims only
(that is, at 12/31/96)
(8) from (6) above
(9) See calculation below
(7) = (1) * [ 1.0 - (3)]
351,428.2200
n/a
351,428.2200
641,741.1840
n/a
1,010,740.8150
816,362.5743
n/a
1,877,640.4301
1,689,876.3870
n/a
3,661,398.8386
not applicable 8,543,272.5 * [1-57.1429%] * 1.05 growth = 3,844,468.7805

= Combined for All AY's
Estimated
IBNR
at 12/31/97
(10) = (7) minus (8) + (9)
0.0000
368,999.6310
1,061,277.8558
1,971,522.4516
3,844,468.7805
answer to b.
7,246,268.7189

Solutions to 2000 Exam Questions (modified):
54. (1 point) According to Friedland, what are two reasons for reviewing estimates for Unpaid Claims
(reserves) between annual calculations?
CAS answer (pre-Friedland)
1. To see if claims are emerging as had been expected.
2. To see if a change in Estimated Unpaid Claims is necessitated by a change in exposures.
See also Friedland comments included above with 1996 # 48.

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2002 Exam Questions (modified):
#22. Given the following information:

Accident
Year

Earned
Premium

Reported Claims
at 12-31-01

Expected
Claim Ratio

1998
1999
2000
2001

200
1,000
1,500
1,500

100
1,000
900
600

80%
80%
80%
80%

Selected Age-to-Age reported claim development factors:
12 - 24 months
24 - 36 months
36 - 48 months
48 - 60 months

1.250
1.100
1.050
1.080

d. Using the Bornhuetter-Ferguson method, calculate the amount of claim development to be expected during
calendar year 2002 on accident years 1998 through 2001. Show all work.
Note: See Chapter 9 for part b calculations. Results shown in column (1) below.
$000's
Estimated
Acc
IBNR
Year at 12/31/01
(1) from b
11.8518
94.5328
237.9996
430.4244
774.8086

1998
1999
2000
2001
Total

Step 1 (proceeding the safe way, using our answer to part b in Ch 9)
Ages in
Percent
Percent
Percent
Percent of current IBNR
expected to emerge
NEXT yr
Reported
Reported Unreported
(CY 2002)
at 1/1/02
at 12/31/02
at 1/1/02
between 1/1 - 12/31/02
(2) FYI
48 to 60
36 to 48
24 to 36
12 to 24

(3)=1/CDF
92.5926%
88.1834%
80.1667%
64.1313%

(4)=1/CDF
100.0000%
92.5926%
88.1834%
80.1667%

(5)=1.0-(3)
7.4074%
11.8166%
19.8333%
35.8687%

(6) = [(4)-(3)] / (5)
100.0000%
37.3136%
40.4204%
44.7058%

Note: these types of questions intentionally exclude new loss occurrences.
So, claims occurring in accident year 2002 are excluded, to compare "apples-to-apples."

Step 2
Est. IBNR
to emerge
during '02
(7)=(1)*(6)

11.8518
35.2736
96.2004
192.4247
335.7505
x 1000 =
335,750.5

OR, an alternative solution:
$000's
Accident
Year

A-Priori
Expected
Claims
(1)=.80*EP

1998
1999
2000
2001
Total

160
800
1,200
1,200

Step 1 (using shortcut described above for B-F special case)
Percent
Percent
Percent of Ultimate
Reported
Reported
expected to emerge
at 1/1/02 at 12/31/02
in the '02 Calendar Year
(3)=1/CDF
92.5926%
88.1834%
80.1667%
64.1313%

(4)=1/CDF
100.0000%
92.5926%
88.1834%
80.1667%

(5') = [ (4) - (3) ]
7.4074%
4.4092%
8.0167%
16.0354%

Step 2
Est. IBNR
to emerge
during '02
(6)=(1)*(5')
11.8518
35.2736
96.2004
192.4248
335.7506
x 1000 =
335,750.6

Exam 5, V2

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2003 Exam Questions (modified):
2. You are given the following information:
Accident
Year
2000
2001
2002

Selected Ultimate
Claims
$310,000
290,000
300,000

Age in
Months
12
24
36
48
60

Reported
CDF to
Ultimate
2.10
1.55
1.25
1.10
1.05

What is the total amount of incurred claims that is expected to emerge during calendar year 2003 on
accident years 2000 through 2002?
A. < $120,000
D. > $160,000, but < $180,000

B. > $120,000 but < $140,000

C . > $140,000 but < $160,000
E. > $180,000

Preliminary Comments:
2003 # 2 does not specify the methodology used to select ultimate claim estimates, nor does it specify the reported
claims as of 12/31/02. The expected emergence depends on the methodology used. For example, if the B-F
technique were used, the percents below would be applied to the A-Prior Expected Claims, which are not equal to
the Selected Ultimate Claims (generally). However, the most reasonable answer using the data given is below:
$000's
Accident
Year

Selected
Ultimate
Claims
(1) given

2000
2001
2002
Total

310,000
290,000
300,000

Step 1 (using method applied to selected Ultimate Loss estimates)
Ages in
Percent
Percent
Percent of Ultimate
NEXT
Reported
Reported
expected to emerge
period ('03)
at 1/1/03 at 12/31/03
in the '03 Calendar Year
(2) FYI
36 to 48
24 to 36
12 to 24

(3)=1/CDF
80.0000%
64.5161%
47.6190%

(4)=1/CDF
90.91% **
80.0000%
64.5161%

(5) = [ (4) - (3) ]
10.9091%
15.4839%
16.8971%

Step 2
Estimated
Emergence
during '03
(6)=(1)*(5)
33,818.21
44,903.31
50,691.30
129,412.82

** Example: 2000 AY will be 48 months old at 12/31/03. 48-to-ult CDF = 1.1 and 1/1.1 = 90.91%

Answer B.

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2004 Exam Questions (modified):
2. You are given the following accident year age-to-age claim development factors:
Months of Development
12 to 24
24 to 36
36 to 48
48. to 60
60 to Ultimate
1.60
1.30
1.10
1.05
1.00
What percentage of the IBNR estimate at 12 months will be expected to emerge during the following
calendar year? C. > 35% but < 45% Answer: C
Months of Development
ATA
CDF to Ult

12 to 24

24 to 36

36 to 48

48. to 60

60 to Ultimate

1.6

1.3

1.1

1.05

2.4024

1.5015

1.155

1.05

Calculate the expected percent to be reported in the interim period:
Since the Percent ( of total Ultimate Claims ) Reported = Inverse of Ultimate CDF **
The expected Percent Reported at 12 months = 1 / 2.4024 =
& expected Percent Reported at 24 months = 1 / 1.5015 =
Then the expected percent reported between ages 12 and 24 months =
And normalize, since the question asks for an IBNR factor:
We need a factor for the Unreported part of Ultimate Claims, not total Ultimate Claims.
That is, we must mak e this % conditional upon the development remaining, so we
divide by (1 - Percent Reported at beginning of Interval) = (1 - .4163) =
Thus, the factor we apply to IBNR (broadly defined) = .2497/.5837=
to give the percent of IBNR expected to emerge in period between 12 and 24 months.

0.4163
0.6660
24.97%

0.5837.
42.78%

** Don't forget to work with inverses (%), instead of the CDFs directly

Solutions to 2005 Exam Questions (modified):
2005 #10 continued from Chapters 7 and 9, where we are given the following data:

Exam 5, V2

Accident
Year
2000
2001
2002

Selected Ultimate
Claims
$310,000
290,000
300,000

Age in
Months
12
24
36
48
60

Reported
CDF to
Ultimate
2.10
1.55
1.25
1.10
1.05

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solution to 2005 Exam Question #10 (continued):
c. (1 point) Using the Bornhuetter-Ferguson method, calculate the expected IBNR emergence for
accident year 2004 case incurred claims during calendar year 2005. … continued below
Preliminary Info for 2005 # 10 Recall B-F IBNR estimates from part b (see chapter 9):
Selected ATA factors
Reported Ultimate CDF
(2) = product of (1)
(1)
Tail at 48 months
1.0000 at 48 mo.
1.0000
36 - 48 months
1.1489 at 36 mo.
1.1489
24 - 36 months
1.4501 at 24 mo.
1.6660
12 - 24 months
2.0000 at 12 mo.
3.3320

Exp. % Unreported
(3) = 1.0 - 1.0 / (2)
0.0000%
12.9602%
39.9760%
69.9880%

Accident
Year
2001
2002
2003
2004

(3) The Percent Unreported = 1 minus inverse of Ultimate Reported CDF

Accident
Year
2001
2002
2003
2004
Total

A priori
Earned
Expected
Premium Claim Ratio
(4) given
19,000
20,000
21,000
22,000

A priori
Expected
Claims

(5) given
90.0000%
90.0000%
90.0000%
90.0000%

"IBNR"
Expected
Unreport

(6)=(4)*(5)
(7)=(6)*(3)
17,100
0.0000
18,000 2,332.8360
18,900 7,555.4640
19,800 13,857.6240
23,745.9240

Or shortcut using
Est. Expected Claims

IBNR
(broadly

x Percent Unreported
defined)
(7)=(3)*(4)*[1.0-1.0/CDF]
0.0000
2,332.8360
7,555.4640
13,857.6240
23,745.9240

Note: Only the calculations for Accident Year 2004 are required:
22,000 * 90% * (1 - 1/3.33) = 19,800 * 70% =

c) Details shown for completeness
Ages in
Acc Estimated
NEXT yr
Year
IBNR
(CY 2005)
2001
2002
2003
2004
Total

(7)
0.0000
2,332.8360
7,555.4640
13,857.6240

Percent
Reported
at 1/1/05

(8) FYI (9) See (3)
48 to 60 100.0000%
36 to 48 87.0398%
24 to 36 60.0240%
12 to 24 30.0120%

Percent
Reported
at 12/31/05
(10) see(3)
100.0000%
100.0000%
87.0398%
60.0240%

Percent
Unreported

at 1/1/05
(11)= 1-(9)

0.0000%
12.9602%
39.9760%
69.9880%

13,857.6240

Percent of current IBNR
expected to emerge
between 1/1 - 12/31/05

Est. IBNR
to emerge
during '05

(12) = [(10)-(9)] / (11)
n/a
100.0000%
67.5800%
42.8816%

(13)=(7)*(12)

0.0000
2,332.8360
5,105.9862
5,942.3760
13,381.1982

* Here the period we evaluate for emerged losses is the year immediately following our original estimate
But, for exam purposes:
Solution to c:

Exam 5, V2

Only the calculations for Accident Year 2004 are required:
13,858 * (60.02% - 30.01%) / (1 - 30.01%) =

5,942.3760

OR since B-F, can also use shortcut with A-prior Expected Claims for AY 2004:
19,800 * (60.02% - 30.01%) =

5,942.3760

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solution to 2005 Exam Question #10 (continued):
2005 #10 d. (0.5 point) State two possible causes for case incurred claims at 12 months for accident year
2004 being approximately 25% higher than would have been expected based solely on premium growth.
Question 10 – Based on Model Solution 1:
1. An increase in the rate of claim reporting
2. An unexpected large claim being reported
Question 10 – Based on Model Solution 2:
1. Reserve (Allowance for Unpaid Claims) Strengthening
2. Deterioration of Expected Claim Ratio
2005 #10 e. (1 point) For each possible cause you identified in part d. above, how would you adjust your
estimate of the expected IBNR emergence for accident year 2004 during calendar year 2005?
Question 10 – Based on Model Solution 1:
1. Reduce IBNR since there is an expectation of fewer claims to occur in the future.
2. No change needs to be made to IBNR since there is no expectation of large claims occurring again
in the future.
Question 10 – Based on Model Solution 2:
1. If reserves have been strengthened, lower than expected IBNR emergence would be anticipated.
2. If the Expected Claims Ratio has deteriorated, greater than expected IBNR emergence would be anticipated.
Solution to 2008 Exam Question
Question 2, Solution
a) See chapter 9.
b) See chapter 7.
c) Finding Expected emergence in CY 2008, for policy year 2006 only, using B-F method:
Using "the safe way" … FOR POLICY YEAR 2006 ONLY: Step 1
Step 2
Estimated
Percent
Percent
Percent
Percent of current IBNR
Est. IBNR
IBNR
Reported
Reported
Unreported
expected to emerge
to emerge
between
1/1
12/31/08
during '08
at 12/31/07
at 1/1/08
at 12/31/08
at 1/1/08
(2) given
(3) given
(4)= 1-(2)
(5) = [(3)-(2)] / (4)
(6)=(1)*(5)
(1) note
332,800
68.00%
82.00%
32.00%
145,600
0.4375
Note: B-F expected IBNR = (1,600,000)*(65%)*[1.0-.68] = 332,800
OR, we could use short-cut using a prior Expected Claims since we're using the B-F method
FOR POLICY YEAR 2006 ONLY: Step 1
Step 2
A-Priori
Percent
Percent
Percent of Ultimate
Est. IBNR
Expected
Reported
Reported
expected to emerge
to emerge
between 1/1 - 12/31/08
during '08
Claims
at 1/1/08
at 12/31/08
(2)
given
(3)
given
(5')
=
[(3)-(2)]
(6)=(1')*(5')
(1') note
1,040,000
68.00%
82.00%
145,600
0.14
Note: B-F a-priori Expected Claims = Premium * ECR = (1,600,000)*(65%) = 1,040,000

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solution to 2008 Exam Question
Question 11, Solution
a. Given an improving Loss Ratio with no change in reporting/settlement/payment patterns
(i) Loss Development Method
OK, since the development patterns will accurately reflect the emerging experience
(ii) Bornhuetter-Ferguson Method
Would overstate IBNR if B-F method is applied without adjustment
(since emerging experience would be better than reflected in B-F a priori claim ratio)
b. After the experience stabilizes at the new (lower loss ratio) levels:
(i) Loss Development Method
OK, since the development patterns have accurately reflected the emerging experience
(ii) Bornhuetter-Ferguson Method
The a-priori expected claim ratio would need to be selected to reflect the new outlook
(that is, lower than the original ratio selected based on older years with worse experience)

Solution to 2008 Exam Question
Question 27, Solution
Part a) Projecting Expected Emergence during CY 2007, as if it's 12-31-06
(in $000s)
Step 1: Find the one-year claim emergence EXPECTED based on IBNR estimates at 12/31/06
Percent
Percent
Expected %
$ of
Reported
Reported
of IBNR
EXPECTED
Estimated
at 1/1/07
at 12/31/07
Percent
expected to
IBNR to
Accident
IBNR
(inverse of
(inverse of Unreported
emerge in
emerge in
Year
at 12/31/06 CDFs given)
CDFs given)
at 1/1/07
CY 2007
CY 2007
(5) =
(1) given
(2) by age
(3) by age+1
(4)= 1-(2)
[(3)-(2)] / (4)
(6)=(1)*(5)
2004
1,100
80.0000%
85.0340%
20.0000%
25.1700%
276.8700
2005
2,300
65.0195%
80.0000%
34.9805%
42.8253%
984.9819
2006
4,800
45.0045%
65.0195%
54.9955%
36.3939%
1,746.9072
Answer to (a)
Sum of EXPECTED
3,008.7591
Part b) A retrospective look at CY 2007, given it's 12-31-07
Step 2: Calculate IBNR that was ACTUALLY reported in CY 2007
Amounts
Amounts ACTUAL IBNR
Accident
Reported
Reported
that emerged
Year
at 12/31/06 at 12/31/07
in CY 2007
(7) given
(8) given
(9)= (8)-(7)
2004
4,500
4,750
250
2005
4,300
5,200
900
2006
3,700
5,000
1,300
Sum of ACTUAL
2,450

(in $000s)
Step 3: Compare
ACTUAL development
minus
EXPECTED development
(10) = (9)-(6)
-26.8700
-84.9819
-446.9072
Difference =
-558.7591

When the "actual" emergence is significantly less than the "expected" emergence,
the methodology used to estimate Ultimate Claims may be overstating the IBNR (esp AY '06).
The actuary may conclude that other methods should be considered and tested.

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2009 Exam Questions
Question 15 - Model Solution 1
a. Unadjusted paid development because the old LDFs assume more is paid later so older LDFs will be
too since the new is in place, overestimating ultimate losses.
b. Ultimate claim ratio (relative to EP), Ultimate severity (relative to ultimate claim counts), Unpaid
severity (relative to open and IBNR counts)
1. Unadjusted reported development:
i. Ultimate claim ratio = 51,700 / 50,400 = 102.58%
ii. Ultimate severity = 51,700,000 / 1,470 = 35,170
iii. Unpaid severity = (51,700 – 4,200) × 1,000 / 1,260 = 37,698
2. Case & payment rate adjusted reported development
i. Ultimate claim ratio = 45,580 / 50,400 = 90.44%
ii. Ultimate severity = 45,580,000 / 1,470 = 31,007
iii. Unpaid severity = (45,580 – 4,200) × 1,000 / 1,260 = 32,841
3. Payment rate adjusted paid development
i. Ultimate claim ratio = 46,200 / 50,400 = 91.67%
ii. Ultimate severity = 46,200,000 / 1,470 = 31,429
iii. Unpaid severity = (46,200 – 4,200) × 1,000 / 1,260 = 33,333
c. Since unadjusted reported development estimates are high compared to the other methods, might talk
to claims department about if there were any changes in case outstanding adequacy as well.
Question 15 - Model Solution 2
a. The unadjusted paid development since the claims settlement practices of the company have changed from
2007 and the historical settlement patterns will no longer be able to project future claims activity accurately.
b.
1. Ultimate claims ratio for AY 2008
i. unadjusted reported development: 51,700 / 50,400 = 102.58%
ii. both case and payment rate adjusted reported development: 45,580 / 50,400 = 90.4%
iii. payment rate adjusted paid development: 46,200 / 50,400 = 91.67%
2. Ultimate severity = [ultimate loss for each method] / [ultimate claim count]
i. 51,700 ×1,000 /1,470 = 35,170
ii. 45,580 ×1,000 /1,470 = 31,007
iii. 46,200 ×1,000 /1,470 = 31,429
3. Unpaid severity = [unpaid under each method] / [open claim count]
i. [ultimate loss estimate – paid loss] / [open claims] = (51,700 – 4,200) / 1,260 × 1,000 = 37,698
ii. (45,580– 4,200) /1,260 ×1,000 = 32,841
iii. (46,250– 4,200) /1,260 ×1,000 = 33,333
c. The unadjusted reported development appears to be overstated given the limited information.
Actuary may prefer to use average of the case and payment rate adjusted reported and payment rate
adjusted paid estimates.

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2009 Exam Questions
Question 16 - Model Solution 1
a. AY 2007 in CY 2008:
Actual = AY 2007 reported claims ($000) as of 12/31/2008 – AY 2007 reported claims ($000) as of 12/31/2007
= 1,250 - 1,166 = 84
Expected = (Ult – Reptd) / % Unreptd × (% Reptd at 2008 - % Reptd at 2007)
= (1,446-1,166) × (1/1.154 - 1/1.212)/(1.0 - 1/1.212) = 66.38
Actual – Expected = 84 – 66.38 = 17.62
b. 17-Ult = 1.212 - [(17-12)/12 × (1.212 – 1.154)] =1.188 (linear interpolation on the given 12-ult and 24-ult LDFs)
Expected Reported Losses = (1,446-1,166) × (1/1.188 - 1/1.212)/(1 - 1/1.212)= 26.68
c. Likely to underestimate.
Because the “to-ult” factor calculated based on lin. interp is likely overstated because development
tends to slow down overtime, more losses should be reported during first half of year, which means a
smaller dev-to-ult factor than linear interp suggests.
Question 16 - Model Solution 2
a. Expected emergence = (Ult–Rptd)/% Unrptd*(% Rptd at ‘08 - % Rptd at ‘07) = 280/.1749 ×.041 = 66.39
actual 1,250 – 1,166 = 84
Diff = 84 - 66.39 = 17.61
b. Expect 66.39 over the year. (66.39) × (5/12) = 27.66
c. Linear interpolation assumes that the claims will emerge evenly throughout the year.
However, claims are usually reported earlier. Since there will be less dev in the future, we could expect
more emergence earlier in the year. Use of linear interpolation would underestimate the projection.
Solutions to 2010 Exam Questions
12. The projected ultimate claims using the Bornhuetter-Ferguson technique is $279,600 for all years combined.
Calculate X, the expected percentage unreported for accident year 2008.
Question 12 – Solution 1
(1)
(2)
(3)
(4)
(5)
65% × (1)
(2) + (3) × (4)
AY
On-Level
Rpt Claims
Exp %
Expected
Projected Ult.
EP ($000)
($000)
Unrprted
Claims
Claims
2006
100
62
0%
65
62
2007
120
60
10%
78
62.8
2008
140
50
X%
91
50 + 91X%
2009
160
40
40%
104
81.6
Total
520
212
338
261.4 + 91X%
261,400 + 91,000X% = 279,600
X% = 20%

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2011 Exam Questions
30a. (1 point) Fully discuss whether the reported claim development technique is appropriate to estimate
unpaid claims for this line of business.
30b. (1 point) Fully discuss whether the paid claim development technique is appropriate to estimate
unpaid claims for this line of business.
30c. (0.5 point) Discuss whether the expected claim technique is appropriate to estimate unpaid claims
for this line of business.
Question 30 – Model Solution 1
Avg Case Trend

Avg paid/closed

AtA for Pd

12 mon
06: 3%
07: 3%
08: 3%

12mon
3%
3%
3%
3%

12-24
1.2
1.2
1.2
1.2

24-26
1.25
1.25
1.25

09: 14,205/10,927 = 1.3 30%
10

a. Reported claims are not appropriate without a case reserve adjustment. Avg cases reserves trending higher
as they did from 09 to 10 is a clue that case reserves have been strengthened. If not accounted for, this
would overstate losses.
b. Paid claim development is appropriate. The reserve changes will not show up in the paid estimates.
Also, the development factors are very stable.
c. Since there appears to be a good volume of paid data and the development factors are consistent, I
would not use the expected claims technique which is best when actual losses are not reliable.
Question 30 – Model Solution 2
a. Check the trend in O/S average reserves.
10,000
 +3%
10,300
>+ 3%
10,609
> +3%
10,927
> +30% ←increase in trend, unadjusted development technique is not appropriate.
14,205
b. Check trend in avg. paid claims
5000 %
6000 +3%
5150 >+3%
6180 +3%
5305 >+3%
6365 +3%
5464 >+3%
6556 +3%
5625 > +2.9%
Trend is consistent, no change in settlement patterns in evident. It is appropriate to use the paid claim
development technique.
c. Expected claims technique is appropriate to use since the technique is not affected by changes in
reserve adequacy.

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2011 Exam
Question 37
a. (0.5 point) The true IBNR for Book A is $610. Based on the estimated IBNR for each method shown, discuss
a change in Book A that would explain the discrepancies between the estimates and the true IBNR.
b. (0.5 point) The true IBNR for Book B is $250. Based on the estimated IBNR for each method shown, discuss
a change in Book B that would explain the discrepancies between the estimates and the true IBNR.
c. (0.5 point) The true IBNR for Book C is $450. Based on the estimated IBNR for each method shown, discuss
a change in Book C that would explain the discrepancies between the estimates and the true IBNR.
d. (0.25 point) For Book D, briefly discuss a next step the actuary should take to understand the
difference in results.
Question 37 – Model Solution 1
a. A deteriorating LR will cause the Rpt Dev, Paid Dev, methods to estimate correct IBNR and Exp claims,
BF methods to understate b/c ELR will be too low for most recent years.
b. Case Reserve Strengthening. Paid methods are not affected by changing case reserve adequacy and
Exp. Claims does not depend on experience. Reported methods will overstate IBNR.
c. Change in Claim Closure Rate. If rate is increasing, paid methods will overstate IBNR.
d. Meet with Claims Dept, UW, and management to discuss any internal changes that may be causing
the difference b/w methods.
Question 37 – Model Solution 2
a. A change in claim ratio ie. Increased claim ratios will cause expected claims method to underestimate
IBNR. Development (reported and paid) are accurate. The Bornhuetter-Ferguson methods
underestimate but not as much as the expected claims technique as it weights expected claims
method with development technique.
b. A change in case reserving methods ie. Case strengthening. The reported Development and B-F
Methods over-state as the CDFs calculated are too high and these high CDFs are applied to high
reported claims. So only the paid development, BF paid and expected claims are accurate. BF
reported is less over-stated than Rpt Development as it weighs in the expected claims method which is
accurate.
c. A change in claim settlement rates, ie. Increased claim closure rates. The paid development and B-F
paid techniques over-state as the CDFs selected are high and these CDFs are applied to high paid
claims. BF-paid is less over-stated because it weighs in the expected claims technique.
d. Should calculate loss ratios, pure premiums, average paid and average case outstanding, etc. bearing
in mind the year-to-year changes and pick the most appropriate reserve, knowledge from the
management and knowledge of the industry.

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2012 Exam Questions
29. Determine and fully justify a reasonable unpaid claim estimate for AY 2011 claims as of 12/31/ 2011.
Question 29 – Model Solution 1 (Exam 5B Question 14)
Given our expected loses and reported LDF, our expected reported claims at 12m would be $50m /10,000
= $5m. that implies the impact of the large claim is also $5m (10-5 = 5). Therefore a reasonable approach
would be to separate the large claim impact and apply it to a paid claim based development method
Since the paid IDF is so highly leveraged, the BF paid method would probably be more suitable.
Therefore BF paid estimate + impact of large claim = unpaid estimate $47,777,778 + $5m = 52,777,778
Question 29 – Model Solution 2 (Exam 5B Question 14)
This extraordinary large claim will skew any method relying on reported claims without proper removal
and separate handling. So we will not use reported development or B/F reported.
The paid development factor is 22.5, which is very leveraged and it is sensitive to initial paid losses. Se
we will not use paid development.
The remaining methods are expected claims and B/F-paid. Since B/F-paid is not sensitive to volatility in early
maturities, I think it is appropriate to use it since it is essentially a credibility weighting of paid development and
expected claims. I believe this to be slightly preferable to the expected claims method, which is not sensitive to
changing conditions. Also, our volume of data seems high enough to obtain some weight.
Select B/F-paid $47,777,778
Question 29 – Model Solution 3 (Exam 5B Question 14)
Use the reported BF technique, estimate is $53M.
Paid development is not appropriate because the large claim is not in the data. Also has highly leveraged
factor for 12-ult.
Reported development is not appropriate because the extraordinarily large claim is in reported data, but
should not be expected to have similar claims throughout the year, so reported experience @ 12 mos. is
not predictive of unreported claims.
Expected claims tech. doesn’t account for the large claim, nor does paid B-F. REPORTED B-F accounts
for the large claim, but uses expected claims to determine the unreported portion, which is a reasonable
estimate of unreported claims.
Examiner’s Comments
Candidates who understood the theory of each of the reserving methods tended to score well on this question as
they were able to structure their answers to provide pros and/or cons of each method to support their final
selection. A small number of candidates proposed an alternative method to the five methods presented in the
question and were awarded credit based on the support provided for their proposal.
Candidates sometimes failed to: address the impact of the large loss, address all of the methods presented in
the question, fully support an alternately proposed method.
Candidates sometimes incorrectly stated that: the large loss would distort the loss development factors used in
the paid and/or reported development methods, highly leveraged loss development factors were a reason not to
use the Bornhuetter-Ferguson methods, or the large loss would distort the a priori loss ratio used in the
Bornhuetter-Ferguson methods.

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2012 Exam Questions
30. Determine which of the two reported claim development patterns shown above best reflect the actual
emergence of claims. Justify your selection.
Question 30 – Model Solution 1 (Exam 5B Question 15)
Expected emergence:
Difference from actual
Patt 1
Patt 2
CY
Pattern 2
Pattern 1
2007
2008
2009
2010

0
8,120
9,829
32,263

0
6,767
14,080
34,365

Total

-1000
-1337
3980
365

-1000
16
-271
-1937

2,008

-3,192

I would select pattern 1, since it is closer to actual emergence overall.
Question 30 – Model Solution 2 (Exam 5B Question 15)
Expected emergence = IBNR (%rept – prior % rept/starting % unrept)
IBNR
Patt 1
Patt 2
IBNR
06
07
08
09
10

06
07
08
09
10
Total

% rept
100%
100%
95.2%
86.6%
66.6%

% rept
100%
100%
94.3%
88.2%
68.9%
↑
1/1.452

patt 1
0
0
6767
21824
57755

patt 2
0
0
8120
18867
52003
↑
0.452* 115,050

Patt 1

Patt 2

emergence

exp’d emergence

0
0
6767
14006
34584
55357

0
0
8120
9753 = 18867*(0.943-0.882)/(1-0.882)
50145

Actual emergence was 53,404. Pattern 1 was much better at predicting most recent but worse for
other periods. To be conservative and since it was a bit closer to actual estimate, choose pattern 1.

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Chapter15 – Evaluation of Techniques
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2012 Exam Questions
Question 30 – Model Solution 3 (Exam 5B Question 15)

06
07
08
09
10

Pattern 1
IBNR
0
0
6769
21,824
57,775

2011
emergence
0
0
6769
14,080
34,565

Pattern 2
IBNR
0
0
8120
18,867
52,003

2011
emergence
0
0
8120
9,829
32,263

Pattern 2 is more predictably closer to actual claim emergence. 1 is too erratic.
Pattern
1
2
Actual

2010
34,565
32,263
34,200

2009
14,080
9,829
10,100

2008
6769
8,120
8,104

Diff
1
2

+1.07%
-5.7%

+39.4%
-2.7%

-16.5%
-2%

2007
0
0
1000

2006
0
0
0

Examiner’s Comments
The majority of candidates were able to put the expected and/or actual values in a comparable form, and
make a valid selection based on those values with some justification. Candidates lost points for:
•

Using the wrong formula for the loss emergence % or applying it incorrectly to ultimate IBNR, ultimate
loss or cumulative reported loss

•

Describing the selected method as “closer” without supplying numeric justification for the
response

•

Including 2011 in the total expected loss emergence when comparing to actual

•

Only compared the emergence for 1 or 2 accident years to draw the conclusion

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Chapter16 – Estimating Unpaid Claim Adj Expenses
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Sec
1
2
3
1

Description
Introduction
Example – Auto Property Damage Insurer
Choosing a Technique for Estimating Unpaid ALAE

Pages
371
371 – 374
374

Introduction

371

ALAE vs. ULAE
 ALAE are costs the insurer can assign to a particular claim (e.g. legal and expert witness expenses)
 ULAE cannot be allocated to a specific claim (e.g. salaries, rent, and computer expenses for the
claims department of an insurer).
Actuaries in Canada still separate LAE into ALAE and ULAE (a.k.a. internal loss adjusting expense, or ILAE).
In 1998, the NAIC promulgated two new categories of claim adjustment expenses for U.S. insurers reporting
on Schedule P of the P&C statutory Annual Statement:
 Defense and cost containment (DCC) which includes all defense litigation and medical cost
containment expenses regardless of whether internal or external to the insurer.
 Adjusting and other (A&O), includes all claims adjusting expenses, whether internal or external to the
insurer.
The authors choose to use the term ALAE in this chapter and state that the development methods presented in
this chapter can also be used for DCC. Key factors include whether expenses:
* can be organized by accident year (policy, underwriting, or report year)
* tend to track AY, PY, U/W Y, or RY or are more dependent on CY.
While ALAE often demonstrate a close relationship with claims experience, ULAE or A&O are often related to
the size of the insurer's claims department.

Example – Auto Property Damage Insurer

371 – 374

Auto Property Damage Insurer is used to demonstrate four projection techniques for ALAE.
Techniques to develop ALAE:
1. The development technique using paid ALAE.
2. The development technique using reported ALAE (when case O/S for ALAE exists), which for Auto
Property Damage Insurer maintains.
3. The development of the ratio of paid ALAE-to-paid claims only.
Exhibit I, Sheets 1 – 3: The ALAE development method for reported and paid ALAE.
Exhibit I, Sheets 4 – 8: The development method applied to the ratio of paid ALAE-to-paid claims.
Exhibit I, Sheets 9 – 10: The projection of ultimate ALAE, using ratio of paid ALAE-to-paid claims, but
using additive development factors instead of multiplicative factors to project ultimate ALAE.
In "Loss Reserving," Wiser notes: "If the ratios are very small at early maturities, the additive approach
seems to be more stable."
All assumptions underlying the development technique (see Chapter 7) apply to the following example for
ALAE.

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Chapter16 – Estimating Unpaid Claim Adj Expenses
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Begin with the projection of reported and paid ALAE in Exhibit I, Sheets 1 and 2.
 Notice the increasing reported and paid ALAE for AYs 2006 - 2008.
 Looking down the age-to-age factors columns for reported ALAE, we see a changing pattern of
development at 12-to-24 months and 24-to-36 months. The age-to-age factors are smaller for the
more recent AY compared to the earlier AYs.
Auto Property Damage Insurer
Reported ALAE ($000)
PART 1 - Data Triangle
Accident
Year
12
1998
684
1999
625
:::
:::
PART 2 - Age-to-Age Factors
Accident
Year
12 - 24
1998
1.393
1999
1.486
2000
1.350
2001
1.421
2002
1.411
2003
1.252
2004
1.231
2005
1.202
2006
1.154
2007
1.162
2008

Exhibit I
Sheet I

24
953
929
:::

36
1,031
1,006
:::

24 - 36
1.082
1.083
1.065
1.055
1.092
1.092
1.067
1.056
1.051

36 - 48
1.030
1.027
1.028
1.041
1.056
1.055
1.026
1.031

48
1,062
1,033
:::

Reported ALAE as of (months)
60
72
84
1,080
1,084
1,089
1,041
1,046
1,049
:::
:::
:::

Age-to-Age Factors
48 - 60
60 - 72
1.017
1.004
1.008
1.005
1.017
1.003
1.015
1.005
1.028
1.058
1.003
1.044
1.031

96
1,092
1,051
:::

108
1,092
1,051
:::

120
1,092
1,051

132
1,092

72 - 84
1.005
1.003
1.001
1.001
1.016

84 - 96
1.003
1.002
1.000
1.004

96 - 108
1.000
1.000
1.000

108 - 120 120 - 132
1.000
1.000
1.000

To Ult

108 - 120 120 - 132

To Ult

PART 3 - Average Age-to-Age Factors
12 - 24
Simple Average
Latest 5
1.200
Latest 3
1.173
Medial Average
Latest 5x1
1.198
Volume-weighted Average
Latest 5
1.193
Latest 3
1.170

24 - 36

36 - 48

48 - 60

Averages
60 - 72

72 - 84

84 - 96

96 - 108

1.071
1.058

1.042
1.037

1.019
1.021

1.023
1.036

1.005
1.006

1.002
1.002

1.000
1.000

1.071

1.042

1.020

1.018

1.003

1.002

1.000

1.070
1.057

1.042
1.038

1.018
1.020

1.024
1.036

1.005
1.006

1.002
1.002

1.000
1.000

Development Factor Selection
48 - 60
60 - 72
72 - 84
1.020
1.036
1.006
1.066
1.045
1.008
93.8%
95.7%
99.2%

84 - 96
1.002
1.002
99.8%

96 - 108
1.000
1.000
100.0%

PART 4 - Selected Age-to-Age Factors

Selected
CDF to Ultimate
Percent Reported

12 - 24
1.170
1.367
73.1%

24 - 36
1.057
1.169
85.5%

36 - 48
1.038
1.106
90.4%

108 - 120 120 - 132
1.000
1.000
1.000
1.000
100.0%
100.0%

To Ult
1.000
1.000
100.0%

These observations lead us ask the following:
 Is ALAE increasing because the portfolio of insureds is increasing?
 Were there operational/policy changes over the experience period impacting ALAE case O/S (since
the same magnitude of change is not evident when looking down columns of the age-to-age factors for
paid ALAE)?
Age-to-Age and tail factor selections:
 Age-to-age factors are selected based on the 3-year volume-weighted average (for both reported
ALAE and paid ALAE) to reflect the most recent experience.
 A tail factor of 1.00 for reported ALAE is selected since there is no further development beyond
96 months.
 A tail factor of 1.005 for paid ALAE is selected based on a review of the ratios of reported ALAEto-paid ALAE from 96 months to 132 months and paid development during this period.

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Chapter16 – Estimating Unpaid Claim Adj Expenses
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Exhibit I, Sheet 3: Projection of Ultimate ALAE ($000)
Auto Property Damage Insurer
Projection of Ultimate ALAE ($000)

Accident
Year
(1)
1998
1999
:::
2006
2007
2008
Total

Age of
Accident Year
at 12/31/08
(2)
132
120
:::
36
24
12

Exhibit I
Sheet 3

ALAE at 12/31/08
Reported
Paid
(3)
(4)
1,092
1,084
1,051
1,045
:::
:::
1,198
1,132
1,596
1,454
952
1,556
12,679
11,685

Projected Ultimate ALAE
Using Dev Method with
Reported
Paid
(7)
(8)
1,092
1,089
1,051
1,050
:::
:::
1,325
1,307
1,866
1,804
2,035
2,128
13,774
13,516

CDF to Ultimate
Reported
Paid
(5)
(6)
1.000
1.005
1.000
1.005
:::
:::
1.106
1.155
1.169
1.241
1.367
2.138

Column Notes:
(2) Age of accident year in (1) at December 31, 2008.
(3) and (4) Based on data from Auto Property Damage Insurer.
(5) and (6) Based on CDF from Exhibit I, Sheets 1 and 2.
(7)= [(3) x (5)]
(8)= [(4) x (6)].

Key Observations:
 The reported and paid ALAE projections are similar.
 There is a significant increase in the ultimate ALAE for AYs 2006 - 2008.
The second approach for ultimate ALAE projection is shown in Exhibit I, Sheets 4 - 8.
 The approach uses the development technique applied to the ratio of paid ALAE-to-paid claims only.
 The first step is to estimate ultimate claims, shown in Exhibit I, Sheets 4 and 5, based on reported
claims only and paid claims only, respectively.
Auto Property Damage Insurer
Reported Claims Only ($000)
PART 1 - Data Triangle
Accident
Year
12
1998
109,286
1999
120,639
2000
115,422
2001
129,430
2002
134,190
2003
152,678
2004
144,595
2005
137,791
2006
159,818
2007
162,205
2008
176,030

Exhibit I
Sheet 4

24
111,832
119,607
119,143
139,925
143,852
166,131
154,830
154,230
178,399
178,425

PART 2 - Age-to-Age Factors
Accident
Year
12 - 24
24 - 36
1998
1.023
0.989
1999
0.991
0.978
2000
1.032
0.996
2001
1.081
0.987
2002
1.072
0.995
2003
1.088
0.999
2004
1.071
0.997
2005
1.119
1.000
2006
1.116
1.006
2007
1.100
2008

36
110,648
116,924
118,641
138,161
143,093
166,015
154,295
154,307
179,384

36 - 48
0.987
0.996
0.986
0.994
0.995
0.997
1.000
0.998

48
109,174
116,482
117,008
137,395
142,360
165,579
154,228
153,981

48 - 60
0.997
0.999
0.998
0.999
0.997
0.998
0.997

Reported Claims Only as of (months)
60
72
84
96
108
120
132
108,849 108,779 108,786 108,646 108,736 108,735 108,732
116,332 116,230 116,236 116,161 116,160 116,125
116,782 116,919 116,860 116,825 116,472
137,269 137,033 136,998 137,056
142,004 141,715 141,627
165,229 163,508
153,750

Age-to-Age Factors
60 - 72
72 - 84
0.999
1.000
0.999
1.000
1.001
0.999
0.998
1.000
0.998
0.999
0.990

84 - 96
0.999
0.999
1.000
1.000

96 - 108
1.001
1.000
0.997

108 - 120
1.000
1.000

120 - 132 To Ult
1.000

There is some evidence of an increasing volume of claims, but it is not as significant as the increase in ALAE.
 Notice the age-to-age factors that are less than 1.00 (a.k.a. downward or negative development) for
reported claims only.
 Auto Property Damage Insurer does not consider S&S when setting case O/S, given that large
recoveries due to S&S are common for this line of business.

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Chapter16 – Estimating Unpaid Claim Adj Expenses
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Next: Compare the development patterns for ALAE and claims only.
Reported ALAE
Age
(Months)
12
24
36
48
60
72
84
96
108

CDF
1.367
1.169
1.106
1.066
1.045
1.008
1.002
1.000

Reported Claims Only

Implied%
Reported
73.2%
85.5%
90.4%
93.8%
95.7%
99.2%
99.8%
100.0%

Paid ALAE
Age
(Months)
12
24
36
48
60
72
84
96
108
120
132

CDF
2.138
1.241
1.155
1.096
1.058
1.028
1.013
1.009
1.007
1.005
1.005

Implied%
Reported
46.8%
80.6%
86.6%
91.2%
94.5%
97.3%
98.7%
99.1%
99.3%
99.5%
99.5%

Implied%
CDF
Reported
1.101
90.8%
0.990
101.0%
0.989
101.1%
0.991
100.9%
0.993
100.7%
0.998
100.2%
0.999
100.1%
0.999
100.1%
1.000
100.0%

Paid claims Only
Implied%
CDF
Reported
1.584
63.1%
1.029
97.2%
1.007
99.3%
1.004
99.6%
1.002
99.8%
1.001
99.9%
1.001
99.9%
1.001
99.9%
1.000
100.0%

Note that ALAE reported and paid patterns lag the claims only patterns, which could be related to the
S&S and the expenses incurred in obtaining these recoveries.
Again, the 3-year volume-weighted averages are used to reflect the most recent experience.
Exhibit I, Sheet 6: Projection of Ultimate Claims Using Reported and Paid Claims Only ($000)
Auto Property Damage Insurer
Projection of Ultimate Claims Using Reported and Paid Claims Only ($000)

Accident
Year
(1)
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
Total

Age of
Accident Year
at 12/31/08
(2)
132
120
108
96
84
72
60
48
36
24
12

Claims only at 12/31/08
Reported
Paid
(3)
(4)
108,732
108,730
116,125
116,033
116,472
116,807
137,056
136,995
141,627
141,461
163,508
163,257
153,750
152,613
153,981
153,154
179,384
175,602
178,425
171,505
124,470
176,030
1,625,090
1,560,627

CDF to Ultimate
Reported
Paid
(5)
(6)
1.000
1.000
1.000
1.000
1.000
1.000
0.999
1.001
0.999
1.001
0.998
1.001
0.993
1.002
0.991
1.004
0.989
1.007
0.990
1.029
1.101
1.584

Exhibit I
Sheet 6
Projected Ultimate Claims
Using Dev Method with
Reported
Paid
(7)
(8)
108,732
108,730
116,125
116,033
116,472
116,807
136,919
137,132
141,485
141,602
163,181
163,420
152,675
152,918
152,599
153,767
177,418
176,834
176,646
176,508
193,794
197,147
1,636,048
1,640,900

Selected
Ultimate
Claims Only
(9)
108,731
116,079
116,640
137,025
141,544
163,301
152,797
153,183
177,126
176,577
195,471
1,638,474

Column Notes:
(2) Age of accident year in (1) at December 31, 2008.
(3) and (4) Based on data from Auto Property Damage Insurer
(5) and (6) Based on CDF from Exhibit I, Sheets 4 and 5.
(7) = [(3) x (5)].
(8) = [(4) x (6)].
(9) = [Average of (7) and (8)].

Note: The reported and paid claims only projections are similar for this stable, short-tail line of insurance.

Exam 5, V2

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Chapter16 – Estimating Unpaid Claim Adj Expenses
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
In Exhibit I, Sheet 7, the development technique is used to analyze the ratio of paid ALAE-to-paid
claims only.
 An important assumption is that the relationship between ALAE and claims only is stable over the
experience period.
 Confirm this assumption during the data gathering process and discussions with management.
 A change in defense strategy or a new policy with respect to the use of external versus internal defense
counsel limit the use of historical relationships to project future ALAE experience.
Auto Property Damage Insurer
Ratio of Paid ALAE to Paid Claims Only
PART 1 - Ratio Triangle
Accident
Year
12
1998
0.0066
1999
0.0065
:::
:::

24
0.0081
0.0077
:::

PART 2 - Age-to-Age Factors
Accident
Year
12 - 24
24 - 36
1998
1.227
1.086
1999
1.185
1.078
2000
1.125
1.079
2001
1.154
1.083
2002
1.163
1.105
2003
1.078
1.091
2004
1.057
1.089
2005
1.035
1.051
2006
1.051
1.032
2007
1.250
2008

Exhibit I
Sheet 7

Ratio of Paid ALAE to Paid Claims Only as of (months)
36
48
60
72
84
0.0088
0.0093
0.0097
0.0098
0.0099
0.0083
0.0085
0.0088
0.0088
0.0089
:::
:::
:::
:::
:::

36 - 48
1.057
1.024
1.029
1.062
1.079
1.067
1.033
1.032

96
0.0099
0.0090
:::

108
0.0100
0.0090
:::

84 - 96
1.000
1.011
1.014
1.000

96 - 108
1.010
1.000
1.000

108 - 120
1.000
1.000

120 - 132 To Ult
1.000

72 - 84

84 - 96

96 - 108

108 - 120

120 - 132

Age-to-Age Factors
48 - 60
60 - 72
72 - 84
1.043
1.010
1.010
1.035
1.000
1.011
1.043
1.000
1.000
1.029
1.014
1.014
1.044
1.056
1.040
1.031
1.000
1.032

120
0.0100
0.0090

132
0.0100

PART 3 - Average Age-to-Age Factors

Simple Average
Latest 5
Latest 3
Medial Average
Latest 5x1

Averages
60 - 72

12 - 24

24 - 36

36 - 48

48 - 60

1.094
1.112

1.074
1.057

1.055
1.044

1.036
1.036

1.014
1.023

1.015
1.018

1.006
1.008

1.003
1.003

1.000
1.000

1.062

1.077

1.054

1.035

1.005

1.012

1.006

1.000

To Ult

PART 4 - Selected Age-to-Age Factors
Development Factor Selection
12 - 24
24 - 36
36 - 48
48 - 60
60 - 72
72 - 84
84 - 96
96 - 108
108 - 120 120 - 132 To Ult
Selected
1.109
1.054
1.049
1.035
1.028
1.014
1.004
1.001
1.002
1.000
1.000
CDF to Ultimate
1.332
1.201
1.140
1.086
1.050
1.021
1.007
1.003
1.002
1.000
1.000
Note: Selected factors differ than those appearing in Exhibit I, Sheet 7, in order to replicate CDF to Ultimate's factors

Advantages and Disadvantages to using the ratio method:
Advantages:
1. It recognizes the relationship between ALAE and claims only.
2. The ratio development factors are not as highly leveraged as those based on paid ALAE dollars.
Age-to-age factors based on the simple average of the latest 3 years are selected.
A tail factor of 1.00 for the ratio of paid ALAE-to-paid claims is selected based on the absence of
development at 108-to-120 months.
This method produces projected ultimate ALAE less than the reported and paid ALAE projections (a key
reason for this is the absence of a tail factor).
Note that paid ALAE lagged paid claims only, and if these implied patterns are correct, then there should
be a tail factor for the ratio of paid ALAE-to-paid claims only.
3. The ability to interject actuarial judgment in the projection analysis, especially for the selection of the
ultimate ALAE ratio for the most recent year(s) in the experience period.

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Chapter16 – Estimating Unpaid Claim Adj Expenses
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Disadvantages:
1. Any error in the estimate of ultimate claims only could affect the estimate of ultimate ALAE.
2. When large amounts of ALAE are spent on claims that ultimately settle with no claim payment, the
projection process is distorted.
This suggests reviewing large claims and projecting estimates of unpaid large claims separately.
This also applies to the analysis of unpaid ALAE with respect to large expenses as for large claims.
Exhibit I, Sheet 8: Ratio of Paid ALAE to Paid Claims Only
Auto Property Damage Insurer
Ratio of Paid ALAE to Paid Claims Only

Accident
Year
(1)
1998
1999
:::
2006
2007
2008
Total

Age of
Accident Year
at 12/31/08
(2)
132
120
:::
36
24
12

Ratio of
Paid ALAE to
Paid Claims Only
at 12/31/08
(3)
0.0100
0.0090
:::
0.0064
0.0085
0.0076

Exhibit I
Sheet 8

CDF to
Ultimate
(4)
1.000
1.000
:::
1.140
1.201
1.332

Projected
Ultimate
Ratio
(5)
0.0100
0.0090
:::
0.0073
0.0102
0.0102

Selected Ultimate
Paid-to-Paid
Ratio
Claims Only
(6)
(7)
0.0100
108,731
0.0090
116,079
:::
:::
0.0073
177,126
0.0077
176,577
0.0077
195,471
1,638,474

Projected
Ultimate
Paid ALAE
(8)
1,087
1,045
:::
1,292
1,360
1,505
12,477

Column Notes:
(2) Age of accident year in (1) at December 31, 2008.
(3) From latest diagonal of triangle in Exhibit I, Sheet 7.
(4) Based on CDF from Exhibit I, Sheet 7.
(5) = [(3) x (4)].
(6) = (5), except for 2007 and 2008 which are judgmentally
selected based on review of prior years.
(7) Developed in Exhibit I, Sheet 6.
(8) = [(6) x (7)].

The development technique is used to project an initial estimate of the ALAE ratio to claim amount of
0.0102 for AYs 2007 and 2008.
 However, 0.0102 seems high (compared to the immediate preceding years), and may be due to a
change in procedures for recording ALAE or unusually large expenses.
The average of the ultimate ALAE ratios for all the years up to 2006 is .0077, and the average for
the latest three years excluding 2007 and 2008 is 0.0071.
 An ultimate ALAE ratio for 2007 and 2008 of 0.0077 is selected (based on the average for all years).
 Ultimate ALAE is based on selected ultimate claims (from Exhibit I, Sheet 6) times ultimate ALAE ratio
(from Column (6)).
A third approach is to use additive rather than multiplicative development factors to ultimate.
In Exhibit I, Sheets 9: Ratio of Paid ALAE to Paid Claims Only - Additive Method
In Exhibit I, Sheets 10: Projection of Ultimate ALAE ($000) - Additive Method

Exam 5, V2

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Chapter16 – Estimating Unpaid Claim Adj Expenses
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Exhibit 1, Sheet 9, Top section: The ratio of paid ALAE-to-paid claims only is shown.
Exhibit 1, Sheet 9, Middle section: Age-to-age factors based on the difference between ratios of paid
ALAE-to-paid claims only at successive ages are shown.
Auto Property Damage Insurer
Ratio of Paid ALAE to Paid Claims Only - Additive Method
PART 1 - Ratio Triangle
Accident
Year
12
1998
0.0066
1999
0.0065
:::
:::
2007
0.0068
2008
0.0076
PART 2 - Age-to-Age Factors
Accident
Year
12 - 24
1998
0.0015
1999
0.0012
:::
:::
2007
0.0017
2008

Exhibit I
Sheet 9

24
0.0081
0.0077
:::
0.0085

36
0.0088
0.0083
:::

Ratio of Paid ALAE to Paid Claims Only as of (months)
48
60
72
84
96
0.0093
0.0097
0.0098
0.0099
0.0099
0.0085
0.0088
0.0088
0.0089
0.0090
:::
:::
:::
:::
:::

24 - 36
0.0007
0.0006
:::

36 - 48
0.0005
0.0002
:::

48 - 60
0.0004
0.0003
:::

Age-to-Age Factors - Additive
60 - 72
72 - 84
84 - 96
0.0001
0.0001
0.0000
0.0000
0.0001
0.0001
:::
:::
:::

96 - 108
0.0001
0.0000
:::

108
0.0100
0.0090
:::

108 - 120
0.0000
0.0000
:::

120
0.0100
0.0090

132
0.0100

120 - 132 To Ult
0.0000

Examples:
the 12-to-24 month factor for AY 1998 is equal to .0015 = the paid ratio of 0.0081 at 24 months minus
the paid ratio of 0.0066 at 12 months.
the 36-to-48 month factor for AY 2002 is equal to 0.0005 = the paid ratio at 48 months of 0.0068 less
the paid ratio at 36 months of 0.0063, or 0.0005.
Exhibit 1, Sheet 9, Bottom section: average age-to-age factors are calculated.
 Additive age-to-age factors based on the simple average for the latest 3 years are selected.
 The age-to-ultimate factor is based on cumulative addition (not multiplication) beginning with
the selected factor for the oldest age.
PART 3 - Average Age-to-Age Factors

Simple Average
Latest 5
Latest 3
Medial Average
Latest 5x1

Averages - Additive
60 - 72
72 - 84

12 - 24

24 - 36

36 - 48

48 - 60

0.0006
0.0007

0.0004
0.0003

0.0003
0.0003

0.0002
0.0002

0.0001
0.0002

0.0003

0.0004

0.0003

0.0002

0.0000

36 - 48
0.0003
0.0009

48 - 60
0.0002
0.0006

84 - 96

96 - 108

108 - 120

120 - 132

0.0001
0.0001

0.0000
0.0001

0.0000
0.0000

0.0001

0.0000

108 - 120
0.0000
0.0000

120 - 132
0.0000
0.0000

To Ult

PART 4 - Selected Age-to-Age Factors

Selected
CDF to Ultimate

Exam 5, V2

12 - 24
0.0007
0.0019

24 - 36
0.0003
0.0012

Development Factor Selection - Additive
60 - 72
72 - 84
84 - 96
96 - 108
0.0002
0.0001
0.0001
0.0000
0.0004
0.0002
0.0001
0.0000

Page 409

To Ult
0.0000
0.0000

 2014 by All 10, Inc.

Chapter16 – Estimating Unpaid Claim Adj Expenses
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Exhibit 1, Sheet 10: Projection of Ultimate ALAE ($000) - Additive Method
Auto Property Damage Insurer
Projection of Ultimate ALAE ($000) - Additive Method

Accident
Year
(1)
1998
1999
:::
2007
2008
Total

Ratio of
Age of
Paid ALAE to
Accident Year Paid Claims Only
at 12/31/08
at 12/31/08
(2)
(3)
132
0.0100
120
0.0090
:::
:::
24
0.0085
12
0.0076

Exhibit I
Sheet 10

Additive
CDF
to Ultimate
(4)
0.0000
0.0000
:::
0.0012
0.0019

Projected
Ultimate
Ratio
(5)
0.0100
0.0090
:::
0.0097
0.0095

Selected Ultimate
Paid-to-Paid
Ratio
Claims Only
(6)
(7)
0.0100
108,731
0.0090
116,079
:::
:::
0.0097
176,577
0.0095
195,471
1,638,474

Projected
Ultimate
Paid ALAE
(8)
1,087
1,045
:::
1,719
1,863
13,241

Column Notes:
(2) Age of accident year in (1) at December 31, 2008.
(3) From latest diagonal of triangle in Exhibit I, Sheet 9.
(4) Based on additive CDF from Exhibit I, Sheet 9.
(5) = [(3) + (4)]
(6) = (5)
(7) Developed in Exhibit I, Sheet 6.
(8) = [(6) x (7)]

The only difference between this projection and the projection in Exhibit I, Sheet 8, is that we add the
paid ALAE ratio from the latest diagonal of the triangle to the CDF instead of multiplying by the CDF.
Note: In Exhibit I, Sheet 9, we do not modify the ALAE ratio for the latest years, but allow the initial
projected ratio values for 2007 and 2008 to be used to project ultimate ALAE.
Exhibit I, Sheet 11: The results of the 4 projections are shown.
Auto Property Damage Insurer
Development of Estimated Unpaid ALAE ($000)

Accident
Year
(1)
1998
1999
:::
2007
2008
Total

Age of
Accident Year
at 12/31/08
(2)
132
120
:::
24
12

Paid
ALAE
at 12/31/08
(3)
1,084
1,045
:::
1,454
952
11,685

Exhibit I
Sheet 11

Projected Ultimate ALAE
Using Dev Method with Using Ratio Method with
Reported
Paid
Mult.
Additive
(4)
(5)
(6)
(7)
1,092
1,089
1,087
1,087
1,051
1,050
1,045
1,045
:::
:::
:::
:::
1,866
1,804
1,360
1,719
2,128
2,035
1,505
1,863
13,774
13,516
12,477
13,241

Column Notes:
(2) Age of accident year in (1) at December 31, 2008.
(3) Based on data from Auto Property Damage Insurer.
(4) and (5) Developed in Exhibit I, Sheet 3.
(6) Developed in Exhibit I, Sheet 8.
(7) Developed in Exhibit I, Sheet 10.
(8) = [(4) - (3)].
(9) = [(5) - (3)].

(10) = [(6) - (3)].

Estimated Unpaid ALAE
Using Dev Method with
Using Ratio Method with
Reported
Paid
Mult.
Additive
(8)
(9)
(10)
(11)
8
5
3
3
6
5
0
0
:::
:::
:::
:::
412
350
-94
265
1,176
1,083
553
911
2,089
1,831
792
1,556

(11) = [(7) - (3)].

Estimated unpaid ALAE = total unpaid ALAE (including both case O/S for ALAE and ALAE IBNR).
Notes: Without a tail factor, projected ALAE, based on the standard development technique, applied to the ratio
of paid ALAE-to-paid claims only appears low.
This method is not sufficient, even if the tail factor is changed to 1.005.
The challenge is in selecting the ultimate ALAE ratio for the most recent two AYs, and with a selected
ratio of 0.0077, the estimate of unpaid ALAE is negative for AY 2007.
This does not seem correct based on knowledge of the property damage line of insurance and the
operations of XYZ Insurer.
Conduct similar evaluation analyses (Chapter 15) in selecting which method is appropriate for each AY.

Exam 5, V2

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Chapter16 – Estimating Unpaid Claim Adj Expenses
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
3

Choosing a Technique for Estimating Unpaid ALAE

374

The choice of a technique to estimate unpaid ALAE depends upon:
 types of data available,
 the credibility of the data, and an
 understanding as to how the insurer's environment affects the various projection techniques.
These comments that apply for ALAE are similar to those that apply for claims with respect to when the
various estimation techniques work and when they do not.

Exam 5, V2

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Chapter16 – Estimating Unpaid Claim Adj Expenses
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Sample Questions:
1. In Chapter 1, Friedland briefly describes the NAIC’s requirements for U.S. insurers to report their Claim
Adjustment Expenses in Schedule P of the P&C statutory Annual Statement, using two separate expense
groupings. Name and explain the 2 categories.
2. In the Introduction to Part 4, Friedland explains that, despite the NAIC groupings used to split Claims
Adjustment Expenses for the purpose of statutory filings, it is common for U.S. insurers to use an “ALAE and
ULAE categorization” for the purpose of determining Unpaid Claims Adjustment Expense estimates.
Define ALAE and ULAE, and provide examples of each.
3. Friedland points out that, in many cases, ALAE data is simply combined with claims data when estimating
unpaid amounts, such that the term “claims” often refers to “claims and ALAE.” Why is ULAE not typically
combined in the same way?
4. In Chapter 16, Friedland shows methods of estimating Unpaid ALAE for instances when ALAE is not simply
combined with claims. (In these cases, estimates of Unpaid Claims will have been developed separately.)
List the 4 techniques that Friedland demonstrates in Chapter 16.
2000 Exam Questions (modified):
39. (2 points) You have developed an estimate of Unpaid Claims, and are now estimating Unpaid ALAE.
You are given the following triangle of paid ALAE per $100 of paid claims, and additional data below.

Development
Month

Cumulative Paid ALAE per $100 of Cumulative Paid Claims
Accident Year

12
24
36
48
60

1995
2.45
2.80
2.96
3.00
3.00

1996
2.50
2.90
3.10
3.16

1997
2.40
2.77
2.95

1998
2.20
2.60

1999
2.10

Paid Claims at
December 31,
1999

$100,000,000

$110,000,000

$115,000,000

$120,000,000

$100,000,000

Ultimate Claims
Estimate

$100,00,000

$110,000,000

$120,000,000

$150,000,000

$200,000,000

There is no development beyond 48 months.
Using the Additive Approach described by Friedland, what is the estimate of Unpaid ALAE as of
December 31, 1999? Show all work.

Exam 5, V2

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Chapter16 – Estimating Unpaid Claim Adj Expenses
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2003 Exam Questions (modified):
20. (3 points) You are given the following information:

Accident
Year
1998
1999
2000
2001
2002

Cumulative Paid Claims (excluding ALAE)
Age of Development in Months
12
24
36
48
$9,200
$9,300
$9,300
$8,000
8,900
9,060
9,060
9,200
9,900
9,980
8,300
9,400
9,500

Accident
Year
1998
1999
2000
2001
2002

Cumulative Paid ALAE
Age of Development in Months
12
24
36
48
$690
$753
$764
$500
760
853
861
550
650
710
555
770
630

Selected
Ultimate
Claims
$9,300
9,060
9,980
9,520
10,680

• Use a simple all-years average to select ATA factors.
• Assume there is no further development of claims or ALAE after 48 months.
Using the Development Method applied to Ratio of Paid-ALAE-to-Paid-Claims described by Friedland,
estimate the Unpaid ALAE for accident year 2002 as of December 31, 2002. Show all work.
2006 Exam Questions (modified):
17. (1.5 points) Historically, an insurance company's only method of estimating unpaid ALAE was utilizing
development factors based on combined claim and ALAE data. It has been discovered that the claims
department of that company implemented a revised strategy two years ago to use outside counsel earlier
in the claim settlement process in hopes of lowering claim payments.
a. (1 point) Briefly describe two techniques for estimating unpaid ALAE that would address the new
claims-handling strategy.
b. (0.5 points) Explain why these techniques may or may not be appropriate.

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Chapter16 – Estimating Unpaid Claim Adj Expenses
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2007 Exam Questions (modified):
47. (1.5 points) Given the following information for an insurance company as of December 31, 2006:
Ultimate
Claims
$65,000
62,500
66,000
64,500

Accident
Year
2003
2004
2005
2006

Paid
DCC
$3,000
2,100
1,200
500

Ratio of Cumulative Paid DCC to
Cumulative Paid Claims
Age of Development in Months
Accident
Year
2003
2004
2005
2006

12
2.4%
2.0%
2.0%
2.2%

24
3.0%
2.5%
2.6%

36
4.0%
3.7%

Ultimate
4.8%

Estimate the Unpaid Defense and Cost Containment (DCC) for accident years 2003 through 2006.
State any assumptions.

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Chapter16 – Estimating Unpaid Claim Adj Expenses
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2009 Exam Questions
13. (2 points) Given the following information:
Accident
Ratio of Paid ALAE to Paid Claims Only
Year
12 Months
24 Months
36 Months
Ultimate
2005
0.0060
0.0077
0.0080
0.0080
2006
0.0070
0.0081
0.0090
2007
0.0065
0.0079
2008
0.0062
Estimated Ultimate Claims Only ($000)
Accident
Ultimate
Year
Claims Only
2005
210,000
2006
218,400
2007
227,140
2008
236,220
a. (1 point) Using the multiplicative paid ALAE-to-paid claims only method, and using all-year, simple
average age-to-age development factors, estimate ultimate ALAE for accident year 2008.
b. (1 point) Briefly describe one advantage and one disadvantage of the multiplicative paid ALAE-topaid claims only method.
2010 Exam Questions
7. (2 points) Given the following information as of December 31, 2009:
Accident
Reported
Reported
Year
Claims Only
ALAE
2007
$163,900
$1,253
2008
179,200
1,490
2009
176,300
1,567
Cumulative Development Factors to Ultimate
Ratio of
Reported
Reported
Reported ALAE
Age
Claims Only
ALAE
to Reported Claims
36
1.000
1.003
1.003
24
0.998
1.103
1.106
12
1.103
1.469
1.332
a. (1.5 points) Use the development method applied to the reported ALAE-to-reported claims ratio to
calculate the expected unreported ALAE for each accident year as of December 31, 2009
b. (0.5 point) Briefly describe one advantage and one disadvantage of the method used in part a. above.

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Chapter16 – Estimating Unpaid Claim Adj Expenses
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
2011 Exam Questions
34. (1.75 points) Given the following data as of December 31, 2010:
Paid Claims Only (excludes expense)
Accident
Year
2007
2008
2009
2010

2007
2008
2009
2010
•
•
•

12 months
$55,683
$62,489
$69,791
$75,187

12 months
$2,985
$3,581
$3,979
$4,315

24 months
$68,489
$75,495
$80,489

36 months
$76,486
$82,168

Paid ALAE
24 months
36 months
$4,288
$5,217
$4,968
$5,908
$5,289

48 months
$77,685

48 months
$5,609

Accident year 2010 ultimate paid claims estimate = $101,535
Assume no further development after 48 months.
Use all-year simple averages for all factor selections.

a. (1.5 points) Use the paid ALAE-to-paid claims only additive method to estimate ultimate ALAE for AY 2010.
b. (0.25 point) Briefly describe an advantage of using a ratio approach to estimate ultimate ALAE.
2012 Exam Questions
27. (2 points) Given the following information for a line of business as of December 31, 2011:
Ratio of Paid ALAE to Paid Claims Only
Accident
Year
12 Months 24 Months
36 Months
48 Months
2008
0.0052
0.0057
0.0061
0.0064
2009
0.0054
0.0058
0.0061
2010
0.0068
0.0074
2011
0.0074

Accident
Year
2008
2009
2010
2011

Ultimate
Claims Only
(000s)
$152
$160
$170
$185

• Assume no development after 48 months.
Estimate the ultimate ALAE for accident year 2011.

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Chapter16 – Estimating Unpaid Claim Adj Expenses
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to Sample Questions:
1. In Chapter 1, Friedland briefly describes the NAIC’s requirements for U.S. insurers to report their Claim
Adjustment Expenses in Schedule P of the P&C statutory Annual Statement, using two separate expense
groupings. Name and explain the 2 categories.
(1) DCC: Defense and Cost Containment expenses, including all defense litigation and medical cost
containment expenses
(2) A&O: Adjusting and Other expenses, including all claims-adjusting expenses
2. In the Introduction to Part 4, Friedland explains that, despite the NAIC groupings used to split Claims
Adjustment Expenses for the purpose of statutory filings, it is common for U.S. insurers to use an “ALAE and
ULAE categorization” for the purpose of determining unpaid claims adjustment expenses.
Define ALAE and ULAE, and provide examples of each.
(1) ALAE: Allocated Loss Adjustment Expenses include the costs that an insurer is able to allocate to a
particular claim, such as lawyers/legal fees and expert witness expenses (aligns more closely to DCC)
(2) ULAE: Unallocated expenses include amounts not easily allocated to a specific claim, such as payroll of
claims adjusters and rent & computer expenses of claims department (aligns more closely to A&O)
3. Friedland points out that, in many cases, ALAE is simply combined with the claims data when estimating
unpaid amounts, so the term “claims” often refers to “claims and ALAE.” Why is ULAE not typically
combined in the same way?
ALAE amounts (which can be allocated to specific claims) are often closely related to the claims amounts. On
the other hand, ULAE amounts are usually less closely related to the claims amounts, and more closely
related to the size the insurance company’s claims department (since salaries and office rent for claims
adjusters are unallocated).
Note: The methods for estimating Unpaid ULAE can therefore be quite different than the methods for
estimating Unpaid Claims and/or ALAE. (ULAE methods will be covered in Chapter 17). See also Conger on
ULAE.
4. In Chapter 16, Friedland shows methods of estimating Unpaid ALAE for instances when ALAE is not simply
combined with claims. List the 4 techniques that Friedland demonstrates in Chapter 16.
(1) Development method applied to reported ALAE $ (as in analogous version for claims in chapter 7)
(2) Development method applied to paid ALAE $ (as in analogous version for claims in chapter 7)
(3) Development method applied to RATIO of paid ALAE, to paid claims (“the ratio approach”)
Example: See prior exam question 2003 #20
(4) Additive development approach applied to ratio of paid ALAE to paid claims (“the additive approach”)
Example: See prior exam question 2000 #39

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Chapter16 – Estimating Unpaid Claim Adj Expenses
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2000 Exam questions (modified):
Question 39. Using the Additive Approach described by Friedland, what is the estimate of Unpaid ALAE as of
December 31, 1999?
Step 1: Write an equation to estimate the Unpaid ALAE:
Estimated Unpaid ALAE = Ultimate ALAE – Paid ALAE to date
Step 2: Compute “additive increments”.
Since there is no development beyond 48 months, only years 97 – 99 are needed.
To determine the additive increment, compute a 3 year average of the difference in cumulative
paid ALAE per $100. Add the additive increment to cumulative paid ALAE per $100 Claims to
date to compute ultimate ALAE ratios.
Month developed to

3 year (if possible) average of the difference in
cumulative paid ALAE

[(2.6 − 2.2) + (2.77 − 2.40) + (2.9 − 2.5)]
=.39
3

[(2.95− 2.77) + (3.10 − 2.90) + (2.96 − 2.80)]
= .18
3

[(3.16 − 3.10) + (3.00 − 2.96)]
= .05
2

Ultimate ALAE Ratios

1997

1998

1999

2.95 + (.05)

2.60 + (.18 + .05)

2.10 + (.39 + .18 + .05)

= 3.0

= 2.83

= 2.72

Step 3: Compute Ultimate ALAE (by AY) and Paid ALAE to date (by AY)
Ultimate ALAE = Ultimate ALAE ratio * Ultimate Claims
Paid ALAE = Paid ALAE ratio * Paid Claims
1997
1. Ultimate ALAE
2. Paid ALAE

3. Unpaid ALAE

3.0 *

1998

[120 M ]
= 3.6 M
100

2.95 *

[115 M ]
= 3.3925 M
100

207,500

1999

2.83 *

[150 M ]
= 4.245 M
100

2.72 *

[200 M ]
= 5.44 M
100

2.60 *

[120 M ]
= 3.12 M
100

2.10 *

[100 M ]
= 2.1M
100

1,125,000

3,340,000

= (1. – 2.)

So, Total Estimated Unpaid ALAE as of 12/31/99 = 207,500 + 1,125,000 + 3,340,000 = 4,672,500

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Chapter16 – Estimating Unpaid Claim Adj Expenses
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2003 Exam questions (modified):
20. (3 points) Using the Development Method applied to Ratio of Paid-ALAE-to-Paid-Claims described by
Friedland, estimate the unpaid ALAE for accident year 2002 as of December 31, 2002. Show all work.

Step 1: Compute RATIOS of Paid ALAE to Paid Claim amounts
Accident
Year
1998
1999
2000
2001
2002

RATIO by development age
12
24
36
0.0750
0.0810
0.0625
0.0854
0.0942
0.0598
0.0657
0.0711
0.0669
0.0819
0.0663 = 630 / 9500, for example

48
0.0822
0.0950

Note: Rounded to 4 digits

Step 2: Compute and select ATA factors, using the ratios above
Accident
Year
1998
1999
2000
2001

ATA Factors by development age
12:24
24:36
36:48
48:ult
1.0800
1.0148
1.0000
1.3664
1.1030
1.0085
(given)
1.0987
1.0822
1.2242 = .0819 /.0669, for example

All-yr Avg
CDF to Ult

1.2298
1.3541

1.0884
1.1011

1.0117
1.0117

1.0000
1.0000

Step 3: Compute projected ULTIMATE RATIO (of paid ALAE to Paid Claims) for AY 2002
… Do not forget this step!
Accident Year 2002 Projection of Ultimate Ratio:
Ultimate RATIO = (2002 RATIO at 12 mo.) * (Ultimate CDF)
Ultimate RATIO = (.0663) * (1.3541) = .0898

Step 4: Apply this ultimate ratio to ultimate claims to estimate the ultimate ALAE and unpaid ALAE:

Accident
Year
2002

Exam 5, V2

Ultimate
Paid-to-Paid
Ratio

Estimated
Ultimate
Claims

Estimated
Ultimate
ALAE

Paid
ALAE

Estimated
Unpaid
ALAE

(1) above

(2) given

(3)=(1)*(2)

(4) given

(5)=(3)-(4)

0.0898

10,680

959

630

329

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Chapter16 – Estimating Unpaid Claim Adj Expenses
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2006 Exam questions (modified):
17. (1.5 points) Historically, an insurance company's only method of estimating unpaid ALAE was utilizing
development factors based on combined claim and ALAE data. It has been discovered that the claims
department of that company implemented a revised strategy two years ago to use outside counsel earlier
in the claim settlement process in hopes of lowering claim payments.
a. (1 point) Briefly describe two techniques for estimating unpaid ALAE that would address the new
claims handling strategy.
b. (0.5 points) Explain why these techniques may or may not be appropriate.
Solution Based on CAS Model Solution 1, with additional comments from the Friedland text:
a1. Use paid-ALAE-to-Paid-Claim development ratios to develop the ultimate paid ALAE to paid loss
ratios, then apply these ratios to the ultimate claims to project unpaid ALAE estimates.
Note: This if the technique Friedland refers to as the Ratio method (%).
a2. Develop ALAE separately by triangle method.
Note: This if the technique Friedland calls the Development method, applied to ALAE dollars ($).
b1. Developing Paid-ALAE-to-Paid-Claim ratios could be appropriate since it recognizes the
relationship between the two, and also uses the paid ALAE data. The issue that should be
emphasized is if ultimate claims projection has an error, then the ALAE projection will be in error
too.
Notes: Other advantages of this Ratio Method include “the ability to easily interject actuarial
judgment in the projection analysis, particularly for the selection of the ultimate ALAE ratio for the
most recent year(s)” and the fact that “ratio development factors tend not to be as highly
leveraged as the development factors based on paid ALAE dollars,” according to Friedland (pg
373).
Another potential challenge of the Ratio Method “exists for some lines of business where large
amounts of ALAE may be spent on claims that ultimately settle with no claims payment.”
b2. Developing ALAE separately may not be appropriate, since the ALAE payments are generally closely
related to paid claims, and using this type of development method would not reflect this relationship.

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Chapter16 – Estimating Unpaid Claim Adj Expenses
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2007 Exam questions (modified):
47. Estimate the Unpaid Defense and Cost Containment (DCC) for accident years 2003 through 2006.
Accident
Year
2003
2004
2005
2006

Ultimate
Loss
$65,000
62,500
66,000
64,500

Paid
DCC
$3,000
2,100
1,200
500

Ratio of Cumulative Paid DCC to
Cumulative Paid Loss
Accident Age of Development in Months
Year
12
24
36
Ultimate
2003
2.4% 3.0%
4.0%
4.8%
2004
2.0% 2.5%
3.7%
2005
2.0% 2.6%
2006
2.2%
Preliminary note: Friedland notes “while we choose to use the term ALAE in this chapter, we point out that
the development methods presented in Chapter 16 can also be used for DCC.”
Based on Model Solution 1
Initial comments: The “Ratio Approach” in this model solution uses Paid-DCC-to-Paid-Claim development
ratios to compute ultimate DCC-to-Paid-Claim ratios. These ratios are then applied to ultimate claims to
determine ultimate DCC, from which Paid DCC is subtracted to estimate the unpaid DCC.
Age to Age Factors
AY
12-24 24-36 36-ult
03
1.25
1.333 1.2
04
1.25
1.480
05
1.3
Simple average
1.267 1.407 1.2
CDF to ultimate
2.139 1.688 1.2
Paid Est. Unpaid
Paid DCC to
CDF to
Ult Paid to Ultimate Claims *
Paid Claims
Paid ratio Ult Pd to Pd ratio
DCC
DCC
Ultimate
Ratio
AY
(1) given*
(2) above
(3)=(1)*(2)
(4) see note
(5) given (6) = (4)-(5)
03
4.8
1.000
4.80
3,120
3,000
120
04
3.7
1.200
4.44
2,775
2,100
675
05
2.6
1.688
4.39
2,897.4
1,200
1,697.4
06
2.2
2.139
4.71
3,037.95
500
2,537.95
5,030.35
* See latest diagonal of ratios in the triangle of ratios provided.
(4) = (3) * ultimate losses given in the problem.

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Chapter16 – Estimating Unpaid Claim Adj Expenses
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2007 Exam questions (modified) – continued with an alternative solution.
Preliminary note: This question does not specify the method to use, so different answers were accepted.
Question #47 – Model Solution 2 (continued)
Initial comments: The approach used in this model solution is to use the “Additive Approach” to estimate
the Unpaid DCC. To determine the additive increment, compute a 3 year average of the difference in
cumulative paid ALAE per $100. Add the additive increment to cumulative paid ALAE per $100 to date to
compute ultimate ALAE ratios. These ratios are then applied to ultimate claims to determine ultimate
DCC from which Paid DCC is subtracted to estimate the Unpaid DCC.
Additive Development of DCC-to-Claims Ratio Method:
AY
03
04
05

12-24
0.6%=(3-2.4)
0.5%
0.6%

24-36
1%
1.2%

36-ult
0.8%

Selected
Ultimate

0.57%
2.47%

1.1%
1.9%

0.8%
0.8%

Note: Selected DCC-to-Claims ratios are based on a straight average

AY
03
04
05
06

Ult
Loss
(1) given
65,000
62,500
66,000
64,500

Ult. DCC
Ult
Pd
DCC
To Claims Ratio
DCC
DCC
Reserve
(2) see note
(3) = (1)*(2) (4) given (5) = (3)-(4)
4.8%
3,120
3,000
120
4.5%=3.7+.8
2,812.50
2,100
712.59
4.5%=2.6+1.9
2,970
1,200
1,770
4.67% = 2.2+2.47 3,012.15
500
2,512.15
5,114.74

(2) Based on data given in the problem and DCC to claims ratios computed above.

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Chapter16 – Estimating Unpaid Claim Adj Expenses
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2009 Exam questions
Question 13 - Model Solution 1
a. (1 point) Using the multiplicative paid ALAE-to-paid claims only method, and using all-year, simple
average age-to-age development factors, estimate ultimate ALAE for accident year 2008.
Step 1: Estimated ultimate ALAE for AY 2008 = Estimated ult pd ALAE to pd Loss ratio * ultimate losses
Step 2: Calculate pd ALAE to pd Loss link ratios
AY
12-24
24-36
36-Ult
05
1.2833
1.0389
1.000
06
1.157
1.1111
07
1.215
08
-Step 3: Calculate the all year simplage average pd ALAE to pd Loss link ratios
* All year simple avg age to age link ratio: 12-24: 1.21850
24-36: 1.07495
* LDF to ultimate:
12–ult = 1.21850 * 1.07495 = 1.3099
24 – ult = 1.07495
(24-ult was computed but not needed to solve the problem)
Step 4: Using the equation in Step 1, the results from Step 3 and the given data in the problem, estimate
ultimate ALAE for AY 2008
A (given)
B
C=A×B
D (given)
E = CxD
AY
Pd to Pd Ratio
Ult LDF
Ult Ratio
Ult Loss
Ult ALAE
2008
.0062
1.3099
.00812
236,220,000
1,918,106
b. (1 point) Briefly describe one advantage and one disadvantage of the multiplicative paid ALAE-to-paid claims
only method.
Adv: Development factors are usually not as highly leveraged for the development as dollar development
Disadv: If ultimate loss estimates are not accurate, the ultimate ALAE estimates will be inaccurate as well.
Question 13 - Model Solution 2
a. Note: This is a more efficient solution (shown to demonstrate the minimum needed to obtain full credit)
Development Factors
12
24
36
Ult
Age-Age
1.219
1.075
1.000
1.000
Cumulative
1.310
1.075
1.000
1.000
AY 2008 Ultimate ALAE =.0062 x 1.310 x 236,220,000 = 1,918,579
b. Note: No advantage given in this model solution (perhaps because it was the same as that stated in
model solution 1)?
Disadv: The analysis is distorted when large amounts of ALAE are spent on claims that ultimately
settle with no pymt to the claimant.

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Chapter16 – Estimating Unpaid Claim Adj Expenses
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2010 Exam questions
7a. (1.5 points) Use the development method applied to the reported ALAE-to-reported claims ratio to
calculate the expected unreported ALAE for each accident year as of December 31, 2009
7b. (0.5 point) Briefly describe one advantage and one disadvantage of the method used in part a. above.
Question 7 – Model Solution 1
a.
(1)

1253/163,900=
1490/179,200=
1567/176,300=

Ratio
0.0076
0.0083
0.0089

(2)
(3) = (1) × (2)
(4)
CDF
Ultimate
Reported
to Ultimate
Ratio
to Ultimate
1.003
0.0077
1.000
1.106
0.0092
0.998
1.332
0.0118
1.103

(5)
Ultimate
Claims
163,900
178,842
194,459

(6) = (3) x (5)
(7)
(8) = (6) - (7)
Ultimate
Reported Unreported
ALAE
ALAE
ALAE
1,257
1,253
4
1,645
1,490
155
2,302
1,567
735
Total
894

(1) = Reported ALAE / Reported Claims
(5) = Reported Claims in $ x (4)

b. Advantage: Development factors of ALAE to claims are less leveraged for reported ALAE only
Disadvantage: There may be claims with zero reported indemnity but substantial ALAE payments - ie
for defense expenses
Question 7 – Model Solution 2
a.
(1)
AY

(2)
Reported

2007
2008
2009

163,900
179,200
176,300

(3)
Reported
Claim ATU
1.000
0.998
1.103

(4) = (2) x (3)
(5)
(6) = (5) / (2)
(7)
(8) = (6) × (7) (9) = (4) × (8) (10) = (9) - (5)
Ultimated Reported Reported ALAE Reported ALAE Ultimate ALAE Ultimate
Unreported
Claims
ALAE
-to-claim ratio to Reported Claims to Claim Ratio
ALAE
ALAE
163,900
$1,253
0.7645%
1.003
0.7668%
1,257
$4
178,842
$1,490
0.8315%
1.106
0.9196%
1,645
$155
194,459
$1,567
0.8888%
1.332
1.1839%
2,302
$735
Total
$894

b. Advantage - allows for interjection of actuarial judgment in selection of ultimate ALAE to reported claim
ratio to reflect operational or judicial/external changes
Disadvantage - an error in the estimation of ultimate claim will lead to an error in the estimation of
ultimate ALAE to reported claim ratio to reflect operational or judicial/external changes

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Chapter16 – Estimating Unpaid Claim Adj Expenses
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2011 Exam questions
34a. (1.5 points) Use the paid ALAE-to-paid claims only additive method to estimate ultimate ALAE for
accident year 2010.
34b. (0.25 point) Briefly describe an advantage of using a ratio approach to estimate ultimate ALAE.
Question 34 – Model Solution 1
Compute the ratio or Paid ALAE to Paid Claims
Accident
RATIO by development age
Year
12
24
36
48
2007
0.0536
0.0626
0.0682
0.0722
2008
0.0573
0.0658
0.0719
2009
0.0570
0.0657
2010
0.0574 = 4,315/ 75,187 for example

Compute Additive age to age factors
Accident
Additive ATA Factors by development age
Year
12:24
24:36
36:48
48:ult
2007
0.0090
0.0056
0.0040
0.0000
2008
0.0085
0.0061
(given)
2009
0.0087 = .0657 - .0570, for example
All-yr Avg
0.0087
0.0058
0.0040
0.0000
0.0186
0.0098
0.0040
0.0000
CDF to Ult
Note:
.0186=.0087+.0098
.0098=0.0058+0.004
Accident Year 2010 Projection of Ultimate Ratio:
Ultimate RATIO = (2010 RATIO at 12 mo.)+ (Ult CDF)
Ultimate RATIO = (.0574) + (.0186) = 0.076

Accident
Year
2010
Answer is

Ultimate Estimated Estimated
Paid-to-Paid Ultimate Ultimate
Ratio
Claims
ALAE
(1) above (2) given (3)=(1)*(2)
0.0760 101,535 $7,717

Estimated
Paid
Unpaid
ALAE
ALAE
(4) given (5)=(3)-(4)
$4,315
$3,402

7,713. (4) and (5) shown for information purposes only

b. Using a ratio instead of a straight dollar development method reduces the chance of highly leverages
CDFs at early maturities.

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Chapter16 – Estimating Unpaid Claim Adj Expenses
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to 2011 Exam questions
34b. (0.25 point) Briefly describe an advantage of using a ratio approach to estimate ultimate ALAE.
Question 34 – Model Solution 2 – Part b
Less leveraged LDFs
Question 34 – Model Solution 3 – Part b
An advantage is that the development factors are more stable when compared to the factors that result
from the ALAE only data.
Question 34 – Model Solution 4 – Part b
Advantage : recognizes the inherent relationship between claims only and ALAE.
Question 34 – Model Solution 4 – Part b
Allows actuary to interject opinion and selections directly into the reserving process

Solutions to 2012 Exam questions
Question 27 - Model Solution 1 (Exam 5B Question 12)
Ratios
12-24
24-36
36-48
08
1.096
1.070
1.049
09
1.074
1.052
0
1.088
Avg.
1.086
1.061
1.049
CDF to Ult
1.209
1.113
1.049
Ult 2011 ratio = (.0074)(1.209) = .0089
Ult 2011 ALAE = Ult ratio*Ult claims = (.0089)(185,000) = 1,646.5
Question 27 - Model Solution 2 (Exam 5B Question 12)
Using the additive approach: Additive ALAE a-t-a factors
AY
12-24
24-36
36-48
2008

0.0005

0.0004

2009
2010

0.0004
0.0006

0.0003

= .0064 - .0061

Simple Avg
Selection
Cumulative factors

0.0005
0.00035
0.0003
0.00115
0.00065
0.0003
=0.0005 + 0.00065
AY 2011 ultimate ALAE = (0.0074 + 0.00115) * 185,000 = 1,581.75
Examiner’s Comments
Most candidates used the development of the paid ALAE to paid claims only approach to arrive at a
reasonable answer. Of those candidates who attempted the problem, the most common error involved
simple calculation mistakes.

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Chapter17 – Estimating Unpaid ULAE
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Sec
1
2
3
4
5
1

Description
Introduction
Dollar-Based Techniques
Count Based Techniques
Triangle-Based Techniques
Comparison Example

Pages
386
387-402
402-406
406
407

Introduction

386

ULAE refer to general overhead expenses associated with claims-handling (e.g. the costs of investigating,
handling, paying, and resolving claims)
 ALAE: costs that can be assigned to a specific claim (e.g. legal fees, the cost of expert witnesses,
police reports, engineering reports, and independent adjusters if assigned to a particular claim)
 ULAE: costs that cannot be assigned to a unique claim (e.g. costs associated with operating the claims
department, including rent, technology, salaries, as well as management and administrative expenses).
Two broad techniques for estimating unpaid ULAE: dollar-based and count based methods.
While these techniques rely on different assumptions, and vary significantly in the amount of data and
calculations required, they may produce similar results.
They are used for an entire population of claims, and need to be correct only for the “average” claim being
reported, handled, paid, or closed during a time period (not for each individual claim).
ULAE liabilities have a “market value” in the fees that a third-party claims administrator (TPA) would require to
manage the book of claims.
Self-insurers use such market values to determine the unpaid ULAE for financial reporting purposes.

Dollar-Based Techniques

387-402

Dollar-based techniques assume that ULAE track with claim dollars with regards to both timing and relative
amount. This assumption:
 means that the timing of ULAE expenditures follows the timing of the reporting or payment of claim dollars.
 implies that a $1,000 claim requires ten times as much ULAE as a $100 claim.
4 commonly used dollar-based techniques:
1. Classical (a.k.a. traditional)
2. Kittel refinement
3. Conger and Nolibos method – generalized Kittel approach
4. Mango-Allen refinement

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Dollar Based Techniques:
1. Classical (or Traditional) Technique
Unpaid ULAE is estimated using a CY paid ULAE-to-CY paid claims ratio.
Key Assumptions of Classical Technique
• The insurer’s ULAE-to-claim relationship has reached a steady-state so that the ratio of paid ULAE-topaid claims approximates ultimate ULAE-to-ultimate claims.
• The volume and cost of future claims management on not-yet-reported claims and reported-but-not-yetclosed claims will be proportional to IBNR and case O/S, respectively.
Assume that ½ of ULAE are sustained when opening a claim and ½ is sustained when closing the claim.
Thus,
i. 50% of the ULAE ratio is applied to case O/S (since for known claims, ½ of the unallocated work was
already completed at the time of opening);
ii. 100% of the ULAE ratio is applied to IBNR, since all unallocated work remains to be completed (i.e. the
work associated with opening and closing the claims).
Mechanics of Classical Technique
4 steps in the classical technique for estimating unpaid ULAE:
1. Calculate ratios of historical CY paid ULAE-to-CY paid claims
2. Review historical paid ULAE-to-paid claims ratios for trends or patterns
3. Select a ratio of ULAE-to-claims applicable to future claims payments
4. Apply 50% of the selected ULAE ratio to case O/S and 100% of the selected ULAE ratio to IBNR

Calendar
Year
(1)
2004
2005
2006
2007
2008
Total

Paid
ULAE
(2)
14,352,000
15,321,000
16,870,000
17,112,000
17,331,000
80,986,000

Paid
Claims
(3)
333,000,000
358,000,000
334,000,000
347,000,000
391,000,000
1,763,000,000

(5) Selected ULAE Ratio
(6) Case Outstanding at 12/31/08
(7) Total IBNR at 12/31/08
(8) Pure IBNR at 12/31/08
(9) Estimated Unpaid ULAE at 12/31/08
Using Total IBNR
(10) Estimated Unpaid ULAE at 12/31/08
Using Pure IBNR

Ratio of
Paid ULAE to
Paid Claims
(4)
0.043
0.043
0.051
0.049
0.044
0.046
0.045
603,000,000
316,000,000
19,000,000
27,787,500
21,105,000

Column and Line Notes:
(2) and (3) Based on data from XYZ Insurer.
(4) = [(2) / (3)].
(5) Selected based on ULAE ratios in (4).
(6) Based on data from XYZ Insurer.
(7) Based on actuarial analysis at 12/31/08 for all lines combined.
(8) Estimated assuming pure IBNR is equal to 5% of accident year 2008 ultimate claims
Ultimate claims for all lines combined for accident year 2008 are $380 million for XYZ Insurer.
(9) = {[(5) x 50% x (6)] + [(5) x 100% x (7)]}.
(10) = {[(5) x 50% x ((6) + (7) - (8))] + [(5) x 100% x (8)]}.

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Exhibit 1 comments:
 In estimating unpaid ULAE, the experience for the insurer as a whole (i.e. all lines of coverage
combined) is used.
 It is good to have five years of complete and accurate data.
 It is surprising to see relatively stable ULAE ratios given all the changes we know transpired at XYZ
Insurer during the experience period.
 A ULAE ratio of 0.045 is selected based on a review of the historical experience as well as discussions
with company management regarding future expectations.
These discussions include expectations regarding claims department caseload, the relationship between
claim and salary inflation, as well as management’s expectations of the future use of independent
adjusters and TPAs.
For XYZ Insurer, case O/S at 12/31/ 2006 is $603 million and selected IBNR is $316 million.
Using the classical technique, we estimate unpaid ULAE at 12/31/2008 to be
$27.8 million = [(0.045 x 50% x $603 million) + (0.045 x 100% x $316 million)]
Challenges of the Classical Technique
One challenge: “closing” a claim and “paying” a claim do not necessarily mean the same thing. Examples:
• For glass coverage, a single payment is the norm, and payment represents settlement (i.e. closure) of the
claim, and therefore the end of the claims handling activity.
• For WC, a claim payment and closing of the claim often differ, since regular payments can replace lost
wages for an extended period of time.
Address this challenge by adjusting the %’s applied to the case O/S and the IBNR. Example:
For an insurer with a portfolio of long-tail professional liability coverage, with substantial claims-handling
work during the life of the claim, unpaid ULAE ratios of 25% are applied to case O/S and 75% to IBNR
(assumes a greater % of expenses are related to closing the claims rather than opening claims).
Another challenge is the definition of IBNR.
The broad definition of IBNR includes liability for both claims that are not yet reported as well as future case
development on known claims.
The narrow definition of IBNR is incurred but not yet reported (IBNYR, a.k.a. pure IBNR), while future case
development on known claims is referred to as incurred but not enough reported (IBNER).
Using the classical technique, apply 100% of the ULAE ratio to IBNYR (pure IBNR) and 50% of the ULAE
ratio to the sum of case reserves and IBNER.
Pure IBNR maybe estimated as a % of total IBNR or a % of the selected ultimate claims for the latest AY(s).
Assume:
 pure IBNR for XYZ Insurer is equal to 5% of the latest AYs (2008) ultimate claims.
 ultimate claims for AY 2008 of $380 million
Calculate the unpaid ULAE for XYZ Insurer as follows:
Unpd ULAE = [(ULAE ratio x 50% x unpd known claims) + (ULAE ratio x 100% x Pure IBNR)]
= [(0.045 x 50% x (case outstanding + IBNER)) + (0.045 x 100% x IBNYR)]
Calculate: IBNYR claims of $19 million (0.05 x $380 million) and derive the IBNER claims as total IBNR less
IBNYR or $297 million ($316 million - $19 million).
Calculate: Estimated unpaid ULAE for XYZ Insurer as follows:
Unpd ULAE = [(0.045 x 50% x ($603 million + $297 million)) + (0.045 x 100% x $19 million)] = $21.1 million
This estimate of unpaid ULAE is significantly less than the initial estimate of $27.8 million for XYZ insurer.

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Comments:
Most actuaries assume 5% of the most recent AY ultimate claims approximates pure IBNR.
Test this assumption by calculating the pure IBNR claims and determine the ratio to total unpaid claims.
First estimate the number of IBNR claim counts (projected ultimate claim counts - reported claim counts).
Multiply IBNR counts for each AY by an ultimate severity value for each AY to estimate ultimate claims
associated with pure IBNR.
Perform the analysis for each line of business, and the total ultimate claims associated with pure IBNR can be
compared to total ultimate claims for both IBNR and reported claim counts for the latest AY.
When the Classical Technique Works and When it Does Not
In “Determination of Outstanding Liabilities for ULAE,” Wendy Johnson states that the classical technique “will
only give good results for very short-tailed, stable lines of business.”
Kay Kellogg Rahardjo in “A Methodology for Pricing and Reserving for Claim Expenses in Workers
Compensation” states:
 The paid to paid method assumes that claims incur expenses only when initially opened and when
closed, which is not true for liability claims.
 The paid to paid ratio itself is subject to distortion when a company is growing or shrinking or when a line
of business is in “transition” (e.g. consider WC in the early 1990s as many large customers moved to
deductible policies or towards self-insurance).
Additional challenges include
* Choosing between the use of:
 paid claims or closed claims and
 total IBNR or pure IBNR.
* Assuming that 50% of ULAE payments are sustained when a claim is opened and 50% when a claim is
closed may not accurately describe an insurer’s application of resources to the life cycle of its claims.
* Use of the classical technique leading to inaccurate results when the volume of claims is growing.
Mango and Allen in “Two Alternative Methods for Calculating the Unallocated Loss Adjustment Expense
Reserve”, note that:
i. the numerator in the ratio (i.e. CY paid ULAE) tends to react relatively quickly to an increase in exposure
or an increase in the number of claims being reported.
ii. the denominator (i.e. paid claims) reflects claim payments made on claims reported at the former, lower,
exposure base and will not be as responsive to the growth in volume.
Thus, the resulting paid ULAE-to-paid claims ratio may misrepresent the true situation.
A similar mismatch between paid ULAE and paid claims can occur if the volume is decreasing.
* Inflation can also create distortions in the classical technique.
In his 1973 paper “Unallocated Loss Adjustment Expense Reserves in an Inflationary Economic
Environment,” Kittel notes that the classical technique does include an inflation adjustment to the degree that
total unpaid claims take inflation into account.
i. if the costs underlying ULAE inflate at the same rate as claim costs, then inflation is accounted for.
ii. however, if different rates of inflation underlie the claims experience and ULAE, the estimated unpaid
ULAE may not be predictive of future experience.

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Mango and Allen expand on this point:
“... the paid-to-paid ratio is distorted upwardly under inflationary conditions because the impact of inflation on
the denominator of the ratio lags its impact on the numerator. This lag is due to the fact that most of the losses
paid in a CY were incurred in a prior year, and thus are largely unaffected by the most recent inflation.”
In summary, the classical technique may not be appropriate for:
1. Long-tail lines of business
2. Times of changing inflationary forces, either in the past or expected in the future
3. When an insurer is experiencing a rapid change in volume (either expansion or decrease in the size of its
portfolio)
4. Where the 50/50 assumption is not an appropriate representation of the claims handling workflow
Kittel Refinement
Kittel describes a weakness in the classical technique:
The Loss Department doesn’t just close claims but it also opens them.
Paid losses don’t accurately represent the work done by the Loss Department since they do not take into
account claims opened during the year which remain open at year end.
This can be significant when loss reserves vary from year to year (e.g. a growing line with rapidly inflating loss
costs could have loss reserves increase at 30% - 40% per year).
Key Assumptions of Kittel Refinement to the Classical Technique:
• ULAE is sustained as claims are reported even if no claim payments are made.
• ULAE payments for a specific calendar year are related to both the reporting and payment of claims.
Thus, Kittel’s refinement is the use of the ratio of paid ULAE-to-the average of paid claims and incurred (as
a reasonable approximation of the relationship of ultimate ULAE- to-ultimate claims).
CY incurred claims = CY paid claims + change in total claim liabilities (including both case O/S and IBNR).
Derivation of Kittel’s formula:
Use the 50/50 assumption, ignore partial payments, the loss dollars processed with the CY paid ULAE are:
½ unit of work
x
payments on prior outstanding reserves
1 complete unit
x
losses opened and paid during the year
½ unit of work
x
losses opened remaining open
If reserves are accurate, CY incurred = AY incurred = losses opened and paid + opened remaining open.
So,
Calendar paid = opened and paid + paid on prior O/S reserves
Calendar incurred = opened and paid + opened remaining opened
½ (calendar paid + incurred) = Losses opened and paid
+
½ payments on prior O/S
+
½ losses opened remaining open
Kittel accepts the second key assumption of the classical technique as valid for the Kittel refinement
(i.e. ½ of expenses are sustained when opening a claim and ½ of expenses when closing a claim).

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Mechanics of the Kittel Refinement
Kittel’s refinement to the classical technique is shown in Exhibit II. The four steps in this technique are:
1. Develop ratio of historical CY paid ULAE-to-average of CY paid and CY incurred claims
2. Review historical ratios for trends or patterns
3. Select a ratio of ULAE-to-claims applicable to future claims payments
4. Apply 50% of the selected ULAE ratio to case outstanding and 100% of the selected ULAE ratio to IBNR
(identical to the classical technique)
Kittel's Refinement

Exhibit 2

Calendar
Year
(1)
2004
2005
2006
2007
2008

Paid
ULAE
(2)
14,352,000
15,321,000
16,870,000
17,112,000
17,331,000

Paid
Claims
(3)
333,000,000
358,000,000
334,000,000
347,000,000
391,000,000

Incurred
Claims
(4)
535,213,000
492,265,000
435,985,000
432,966,000
475,300,000

Average
Paid and Inc.
Claims
(5)
434,106,500
425,132,500
384,992,500
389,983,000
433,150,000

Total

80,986,000

1,763,000,000

2,371,729,000

2,067,364,500

ULAE Ratio
Paid
Claims
(6)
0.043
0.043
0.051
0.049
0.044

Paid ULAE to
Avg Paid and
Inc. Claims
('7)
0.033
0.036
0.044
0.044
0.040

0.046

0.039

0.04
603,000,000
316,000,000
19,000,000
24,700,000
18,760,000

(8) Selected ULAE Ratio
(9) Case Outstanding at 12/31/08
(10) Total IBNR at 12/31/08
(11) Pure IBNR at 12/31/08
(12) Estimated Unpaid ULAE at 12/31/08 Using Total IBNR
(13) Estimated Unpaid ULAE at 12/31/08 Using Pure IBNR

(2) through (4) Based on data from XYZ Insurer.
(5) = [Average of (3) and (4)].
(6) = [(2) / (3)].
(7) = [(2) / (5)].
(8) Selected based on ULAE ratios in (7).
(9) Based on data from XYZ Insurer.
(10) Based on actuarial analysis at 12/31/08 for all lines combined.
11) Estimated assuming pure IBNR is equal to 5% of accident year 2008 ultimate claims.
Ultimate claims for all lines combined for accident year 2008 are $380 million for XYZ Insurer.
(12) = {[(8) x 50% x (9)] + [(8) x 100% x (10)]}.
(13) = {[(8) x 50% x ((9) + (10) - (11))] + [(8) x 100% x (11)]}.

Using Kittel’s refinement, we observe lower ULAE ratios than with the classical technique (traditional paid-topaid approach).
 This is expected when incurred claims are greater than paid claims on a CY basis.
 Based on Kittel’s refinement, a ULAE ratio of 0.040 is selected.
Using Kittel’s refined technique, estimate unpaid ULAE for XYZ Insurer to be $24.7 million using the formula
with total IBNR and $18.8 million using the formula with an adjustment to determine pure IBNR.
$24.7 million = [(0.04 x 50% x $603 million) + (0.04 x 100% x $316 million)]
$18.8 million = [(0.04 x 50% x ($603 million + $297 million)) + (0.04 x 100% x $19 million)]
The Kittel refinement address the challenge in the classical technique related to sustaining ULAE for activities
beyond simply paying a claim.
However, the refinement does not explicitly address the issue associated with the definition of IBNR (i.e.
modifying the formula to differentiate between IBNYR and IBNER).

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Problems associated with the Kittel Refinement
 The use of traditional 50/50 assumption regarding ULAE expenditures does not allow for allocation of
ULAE costs between opening, maintaining, and closing claims which may vary from insurer to insurer.
 There is no potential for using different rates of inflation between ULAE and claims.
Conger and Nolibos Method – Generalized Kittel Approach
Conger and Nolibos sought to define a procedure to estimate unpaid ULAE that would:
a. Recognize an insurer’s rapid growth
b. Be consistent with patterns of the insurer’s ULAE expenditures over the life of a claim
c. Reproduce key concepts underlying the Johnson technique
d. Use commonly available and reliable aggregate payment and unpaid claims data
e. Include an extension to the Kittel refinement which would allow for alternatives to the 50/50 rule
The generalized approach uses weighted claims, which recognizes that claims use up different amounts of
ULAE at different stages of their life cycle, from opening to closing.
 Newly opened, open, and newly closed claims are each given different weights when determining the
claims basis to which ULAE payments during a past or future calendar period are related.
 Since handling costlier claims warrants more resources than handling smaller claims, they use claim
dollars instead of claim counts in their generalized approach.
The claim basis for a particular time period is defined to be the weighted average of the:
• Ultimate cost of claims reported during the period (ultimate includes reported amounts and
future development on known claims)
• Ultimate cost of claims closed during the period (includes any future payment made after the claim closing)
• Claims paid during the period
Compare:
Kittel’s weights are fixed at 50% for incurred claims and 50% for paid claims.
rd
The generalized method introduces a 3 claim measure that allows distinguishing the cost of maintenance from
the cost of closing.
Key Assumptions of Generalized Approach
• Expenditure of ULAE resources is proportional to the dollars of claims being handled (in contrast to
Johnson’s assumption that ULAE costs are independent of claim size and nature).
• ULAE amounts spent opening claims are proportional to the ultimate cost of claims being reported.
• ULAE amounts spent maintaining claims are proportional to payments made.
• ULAE amounts spent closing claims are proportional to the ultimate cost of claims being closed.
Mechanics of Generalized Approach
Conger and Nolibos define U1 + U2 + U3 = 100%, where:
• U1 is the % of ultimate ULAE spent opening claims
• U2 is the % of ultimate ULAE spent maintaining claims
• U3 is the % of ultimate ULAE spent closing claims
Determine reasonable ranges for U1, U2, and U3 and test the sensitivity of the final estimate of unpaid ULAE to
variations within those ranges.

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The values of U1, U2, and U3 could vary significantly from insurer to insurer and between lines of business.
 For a litigation-intense liability book of business, a strong concentration of activity close to the time of claim
settlement and payment exists.
 For WC, greater front-end costs exist.
For time period T, Conger and Nolibos define M, the total amount spent on ULAE during a time period T, to be
M = (U1 x R x W) + (U2 x P x W) + (U3 x C x W), where
• R is the ultimate cost of claims reported during T
• P is the claims paid during T
• C is the ultimate cost of claims closed during T
• W is the ratio of ultimate ULAE to ultimate claims (L)
* T could be activity between t1 and t2 related to an AY or for all AYs, where t1 and t2 are points in time.
* Algebraically derive the ratio W = M/B by defining B, the claims basis for the time period T to be:
B = (U1 x R) + (U2 x P) + (U3 x C)
Thus, M = B x W, and W = M/B.
Each component of the claims basis is a value of the claims underlying the ULAE payments. Thus,
• U1 x R: the claims basis for ULAE spent setting up new claims
• U2 x P: the claims basis for ULAE spent maintaining open claims
• U3 x C: the claims basis for ULAE spent closing existing claims
Insurers measure and report M, the ULAE payments during a period, on a CY basis.
 Once U1, U2, and U3 are estimated or selected, the claims basis B can be calculated from claim amounts
R, P, and C, that can be determined from data underlying an analysis for estimating unpaid claims.
 M (total ULAE payments) and B (claim basis) can be calculated for historical calendar periods. By
computing the ratio W (= M/B, where both M and B are expressed on a CY basis), we obtain ratios of
ULAE to claims by CY.
 Select an overall ratio of ULAE-to-claims, W*, which is used to estimate future ULAE payments.
Ultimate ULAE (U) for a group of AYs can be estimated as:
U = W* x L, where
• W* is the selected ultimate ULAE-to-claims ratio
• L is the independently estimated ultimate claims for the same group of AY

3 ways to estimate unpaid ULAE for a group of AYs.
1. Compute Unpaid ULAE by subtracting ULAE already paid (M) from the estimate of ultimate ULAE (U).
Unpaid ULAE = (W* x L) - M
Practical and Conceptual Problems with this Method:
 Practically, it may be difficult to quantify the historical paid ULAE that corresponds only to the AYs claims
represented by L.
 Conceptually, this shares the potential distortions of an expected claims ratio approach to estimating
unpaid claims (unpaid claims equal a predetermined expected claims ratio time earned premium less
claims paid to date). The unpaid claim estimate is distorted if actual paid claims do not approach
expected ultimate claims.

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2. Conger and Nolibos Preferred Method:
The method is similar to a BF technique in that an a priori provision of unpaid ULAE is calculated.
Unpaid ULAE = W* x (L - B)
Deriving the estimate (for a group of AYs). Assume that
R(t) – ultimate cost of claims known at time t
P(t) – total amount paid at time t
C(t) – ultimate cost of claims closed at time t
Compute Unpaid ULAE = W* x {U1 x [L – R(t)] + U2 x [L – P(t)] + U3 x [L – C(t)]},
Each component of the unpaid ULAE formula represents a provision for the ULAE associated with:
• Opening claims not yet reported
• Making payments on currently active claims and on those claims that will be reported in the future
• Closing “unclosed” claims (i.e. those claims that are open at time t and those claims that will be reported
and opened in the future)
Rearranging the equation, one obtains:
Unpaid ULAE = W* x (L - B)
This method assumes that the amount of ULAE paid to date and the unpaid ULAE are not directly related,
except to the extent that these payments influence the selection of the ratio W*.
This is similar to the assumption underlying the BF technique.
3. Compute Unpaid ULAE in a similar way to the claims development method.
Unpaid ULAE could be estimated by the following formula:
Unpaid ULAE = M x (L/B – 1.00)
This implies that unpaid ULAE are proportional to paid amounts reported to date.
Practical problems and concerns:
 The practical difficulty of establishing the ULAE amounts paid that correspond to accidents occurring
during a particular period
 This method may be overly responsive to random fluctuations in ULAE emergence.
Application of Generalized Approach to Claim Counts
The formula for a claim count basis used in the determination of unpaid ULAE is:
b = (v1 x r) + (v2 x o) + (v3 x c), where
• r represents reported claim counts
• o represents open claim counts
• c represents closed claim counts
• v1 is the estimate of the relative cost of handling the reporting of a claim (for one year)
• v2 is the estimate of the relative cost of managing an open claim (for one year)
• v3 is the estimate of the relative cost of closing a claim (for one year)
It is not necessary to determine the actual costs of the various claim activities but instead their relative
magnitudes.

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Example: Johnson assumes that v1 = 2, v2 = 1, and v3 = 0.
Using estimated v1, v2, and v3, select w*, the ratio of ULAE to the claim count basis, based on the historical
data w = M/b, where M still represents ULAE payments.
After selecting w* (or a series of w*i which reflect future inflation adjustments),
Compute: Unpaid ULAE = Σ w*i x [(v1 x ri) + (v2 x oi) + (v3 x ci)], where
• ri represents the number of claims to be reported in each CY i
• oi represents the number of open claims at the end of CY i
• ci represents the claims to be closed during CY i
• i represents the series of future CY-ends until all claims are closed
Comments:
 Only claims occurring on or before the valuation date should be considered.
 A claim that stays open for a number of years is counted multiple times in the summation, and is
consistent with the assumption that there are ULAE payments each year as long as a claim stays open.
 The formula could be adapted to reflect the Rahardjo and Mango-Allen concepts of cost varying over time
by stratifying the claims activities more finely than just reporting, opening, and closing.
Simplification of Generalized Approach
The estimation of R (ultimate cost of reported claims) and C (ultimate cost of closed claims) may not be easy.
R can be computed as the ultimate for the accident period ending on that date - pure IBNR amounts, which
represent the ultimate cost of not yet reported claims.
C represents the final cost of claims closed as of the valuation date including any subsequent payments (i.e.
paid on closed if the line of business does not have subsequent payments.)
Consider a simplification where estimates of R and C are not required.
1. Estimate ultimate claims for the AY as a proxy for the ultimate costs of claims reported in the CY.
The CY amount equals the sum of the corresponding AY ultimate claims + pure IBNR at the beginning of the
year - pure IBNR at the end of the year.
The error in this approximation is based on a review of changes in exposures between AYs and the
characteristics of the coverage being analyzed to make adjustments based on judgment.
2. Assume U3 = 0, if no additional effort is required to close an existing claim
This assumption is not appropriate for professional liability or employment practices liability are lines of
business where a significant portion of the claims-related expenses will be incurred with its settlement.
If it is ok to assume that U3 = 0, then U1 + U2 = 100%, and compute B, the claims basis for each CY as
Est. B = (U1 x A) + (U2 x P), where A represents the ultimate claims for the AY.
Then calculate observed W values for each year as W = M/Est. B
After a review of the observed ULAE ratios, select an appropriate ratio W* for estimating unpaid ULAE.
Next, estimate pure IBNR (perhaps by analyzing claim reporting patterns and ultimate severities) and deduct
this estimate from L to obtain an estimate of the ultimate costs of claims reported to date (R).
Finally, compute ULAE is either of two ways:
Unpaid ULAE = W* x {L – [(U1 x R) + (U2 x P)]}, which can be expressed as
Unpaid ULAE = W* x [U1 x (L – R) + U2 x (L – P)}

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Practical Difficulties with the Generalized Approach
 The estimation of R and C, the ultimate cost of reported and closed claims, is not simple.
 It is not known about the relative accuracy of the generalized method (as compared to other dollar-based
methods) in an inflationary environment.
 The effect of reopened claims on the accuracy of the estimates of unpaid ULAE is not known.
 How to modify the approach to properly reflect the change over time in the quantity or cost of resources
dedicated to the handling of a claim, as that claim ages is not known.
Mango-Allen Variation of the Kittel Refinement to the Classical Technique
 Their variation applies when working with a line of business where the actual historical calendar period
claims are volatile, due to random reporting or settlement of large claims (i.e. for lines of business with a
relatively small number of claims of widely varying sizes).
 They suggest replacing actual calendar period claims with expected (by applying selected reporting and
payment patterns to a set of AY estimated ultimate claims) claims for those historical calendar periods.
 They explain that the actuary can estimate the expected paid claims by applying selected reporting and
payment patterns to a set of accident year estimated ultimate claims.
Key Assumptions of Mango-Allen Refinement to the Classical Technique
 An insurer’s ULAE-to-claim relationship is derived based on a review of the ratio of paid ULAE-toexpected paid claims (vs. the classical technique where paid ULAE is compared to actual paid claims).
 Uses the second key assumption of the classical technique (i.e. one-half of expenses are sustained when
opening a claim and one-half of expenses when closing a claim).
Mechanics of Mango-Allen Refinement to the Classical Technique
Shown in Exhibit III for New Small Insurer, a new insurer specializing in lawyers’ professional liability
coverage. 5 steps in this technique:
1. Estimate calendar year expected paid claims
2. Develop ratio of historical calendar year paid ULAE-to-expected calendar year paid claims
3. Review historical ratios for trends or patterns
4. Select a ratio of ULAE-to-claims applicable to future claims payments
5. Apply 50% of the selected ULAE ratio to case outstanding and 100% of the selected ULAE ratio to IBNR
1. Begin the analysis by estimating expected paid claims for each of the four CYs in the experience period.
* Expected CY payments are based on EP * an expected claims ratio * the percentage expected to be paid
in each year.
* Since New Small Insurer is a new company without credible historical claims experience, rely on the
claims ratio underlying the pricing analyses as well as insurance industry benchmark payment patterns.

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Exhibit III, Sheet 1
Accident

Direct
Earned

Expected
Claims

Expected

Year
(1)
2005
2006
2007
2008

Premium
(2)
4,300,000
4,250,000
4,420,000
3,985,000

Ratio
(3)
55%
55%
55%
55%

Claims
(4)
2,365,000
2,337,500
2,431,000
2,191,750

Total

16,995,000

Expected Payment Percentage in Calendar Year
2005
(5)
12%

2006
(6)
15%
12%

2007
(7)
15%
15%
12%

2008
(8)
15%
15%
15%
12%

9,325,250

Expected Claims Paid in Calendar Year
2005
(9)
283,800

2006
(10)
354,750
280,500

2007
(11)
354,750
350,625
291,720

2008
(12)
354,750
350,625
364,650
263,010

283,800

635,250

997,095

1,333,035

Column Notes:
(2) Based on information provided by New Small Insurer.
(3) Based on actuarial analysis conducted for pricing purposes.
(4) = [(2) x (3)].

(5) through (8) Based on actuarial analysis of insurance industry benchmark paid claims development experience.

(9) = [(4) x (5)].
(10) = [(4) x (6)].
(11) = [(4) x (7)].
(12) = [(4) x (8)].

2. Proceeds in a similar fashion as the classical technique (See Exhibit III, Sheet 2.)
Calendar
Year
(1)
2005
2006
2007
2008
Total

Paid
ULAE
(2)
55,000
62,500
70,000
80,000
267,500

Paid Claims
Actual
Expected
(3)
(4)
1,253,450
283,800
86,000
635,250
410,650
997,095
309,600
1,333,035
2,059,700 3,249,180

ULAE Ratio
Paid ULAE-to-Paid Claims
Actual
Expected
(5)
(6)
0.044
0.194
0.727
0.098
0.170
0.070
0.258
0.060
0.130
0.082

(7) Selected ULAE Ratio
(8) Case Outstanding at 12/31/08
(9) Total IBNR at 12/31/08
(10) Pure IBNR at 12/31/08
(11) Estimated Unpaid ULAE at 12/31/08 Using Total IBNR
(12) Estimated Unpaid ULAE at 12/31/al Using Pure IBNR

0.07
225,000
6,430,000
109,588
457,975
236,761

(2) and (3) Based on data from New Small Insurer.
(4) Developed in Exhibit III, Sheet 1.
(5) = [(2) / (3)].
(6) = [(2) / (4)].
(7) Selected based on ULAE ratios in (6) and input of New Small Insurer Mgt.
(8) Based on claims data from New Small Insurer.
(9) Based on actuarial analysis at 12/31/08.
(10) Estimated assuming pure IBNR is equal to 5% of AY expected claims.
(11) = {[(7) x 50% x (8)] + [(7) x 100% x (9)]}.
(12) = {[(7) x 50% x ((8) + (9) - (10))] + [(7) x 100% x (10)]}.

Observe that:
 the ratios of paid ULAE-to-actual paid claims are much more volatile than the ratios of paid ULAE-toexpected paid claims.
 a pronounced downward trend in the paid ULAE-to-expected paid claims ratios.
Understanding the reasons behind this trend
i. reviewing the assumptions underlying the development of expected paid claims
ii. discuss with management about actual paid ULAE.

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Possible explanations:
* The industry-based payments pattern for developing expected paid claims may be too fast for the insurer.
* The variability and downward trends could be related to large claims (from a review of claims data that there
are several open claims for the most recent AYs in litigation with large case O/S and small payments to date).
After discussion with management about its expectations for the upcoming years, and a review of current claims
data, we select a ratio of 0.07 for estimating unpaid ULAE.
Estimated unpaid ULAE at 12/31/08 of $457,975 using total IBNR and $236,761 using pure IBNR is as follows:
$457,975 = [(0.070 x 50% x $225,000) + (0.070 x 100% x $6,430,000)]
$236,761 = {[0.070 x 50% x ($225,000 + (6,430,000 – 109,588))] + [0.070 x 100% x $109,588]}
When the Mango-Allen Refinement Works and When it Does Not
 The Mango-Allen refinement is a good alternative for insurers with limited or highly volatile claims
experience.
 However, for insurers with a large volume of paid claims experience, the additional calculations to
estimate expected paid claims to improve the accuracy of projected unpaid ULAE may not justify the time
and costs involved.

Count Based Techniques

402-406

2 drawbacks to the use of claims (vs. claim counts) as a base for estimating unpaid ULAE.
1. ULAE is not solely dependent on the magnitude of its accompanying claim dollars. ULAE is also
dependent on the average claim size. For example,
i. the ULAE required to settle a one million-dollar claim is less than the ULAE required to settle ten
$100,000 claims.
ii. However, the classical technique with its use of a paid-to-paid ratio does not recognize this difference.
2. The estimate of unpaid ULAE becomes a “rider” on the estimate of unpaid claims, responding to
whatever volatility is present in the estimate of ultimate claims.
Unpaid ULAE is not expected to respond fully to fluctuations in claim amounts. If there is a sudden
drop in claim counts or in the value of claims, we would not expect an immediate drop in the overhead
expenses or the number of claims management personnel.
Key assumptions in count-based techniques:
 is that the same kind of transaction costs the same amount of ULAE regardless of the claim size.
 a claim that stays open longer will cost proportionately more than a quick-closing claim, with
respect to some component of ULAE.
Early Count Techniques
R.E. Brian suggested breaking the ULAE process into five kinds of transactions:
1. Setting up new claims
2. Maintaining outstanding claims
3. Making a single payment
4. Closing a claim
5. Reopening a claim

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In the Brian technique:
 the actuary projects the future number of each type of transaction.
 each of these transactions would carry a similar cost, and suggested estimating the cost per
transaction using ratios of historical ULAE expenditures to the number of claim transactions
occurring during the same calendar periods.
Assumptions and Weaknesses of the technique:
* The primary assumption (which Conger and Nolibos identify as a weakness) is that each of the five
kinds of claims transactions requires similar ULAE resources and expenditures.
However, the weakness could easily be remedied by refining the formula to allow for different costs for
the different types of transactions.
* A more significant weakness of this technique is the difficulty in estimating both the number of future
transactions and the average cost of each transaction.
Reliable and consistent claim count and claim transaction data supporting these projections is often not
readily available.
Wendy Johnson Technique (similar to Brian’s approach)
Wendy Johnson’s approach focuses on two key transactions: reporting and maintenance.
 Johnson then projects the future number of newly reported claims, as well as the number of claims that
will be in a pending status each year (i.e. will require maintenance work during the year).
 Johnson then estimates the cost of each transaction by comparing historical aggregate ULAE
expenditures to the number of transactions occurring in the same time period.
 Johnson’s technique allows for an explicit differential in the amount of ULAE cost required for different
types of claim transactions (e.g. opening a claim costs $x and maintaining existing claims costs an
additional $x).
The benefit of Johnson’s approach is that it only requires the actuary to estimate the relative amount of
resources for each transaction type (detailed time-and-motion studies to calculate the actual cash cost of each
transaction type are not needed).
Mango-Allen Claim Staffing Technique (in response to shortcomings in Johnson’s method)
The technique is a “transaction-based method”, using future claim staff workload levels and a new projection
base, equal to the sum of calendar year opened, closed, and pending claims (OCP claims).
The following four components are computed
1. Future CY OCP claims
2. Future CY claim staff workloads, which are expressed as OCP claims per staff member
3. Future CY claim staff count
4. Future CY ULAE per claim staff member
Future CY ULAE payments = (future claim staff count)*(future ULAE per claim staff member), and consider
inflation.
Estimated unpaid ULAE is the sum of future CY ULAE payments.

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3 characteristics of OCP claims that make their use as a base for the claim staffing method appealing:
1. It is a reasonable proxy for claims department activity (i.e. directly proportional to levels of claim activity,
especially number of staff and workload levels of the staff).
2. It is claim count based. Claims counts (if case complexity issues are addressed) bear a more direct
relationship to claim staff activity.
3. It is derivable from typical reserve study information. Projected opened, closed and pending claims are
derivable from ultimate claim counts, a claim reporting pattern and a claim closing pattern.
Conger and Nolibos note:
 that the estimate of unpaid ULAE is likely to be sensitive to the magnitude of the selected parameters.
 the estimates will be influenced by parameters not explicitly considered in the article (e.g. the implicit
assumption that equal amounts of ULAE resources are required to open, close, and handle one average
claim for a year).
Rahardjo
Kay Kellogg Rahardjo:
 discusses the different levels of work effort required for handling claims in the first 30 days than for
claims that have been open for five years.
 focuses on the length of time for which WC claims remain open, which she defines to be the “duration.”
She states: “As duration increases, so does the expense of handling the claim for the remainder of the
claim’s life.”
Spalla
Joanne Spalla asserts:
 that manual time-and-motion studies are not needed to determine the costs of claim-related activities
and transactions.
 the use of modern claim department information systems to track time spent on individual claims by level
of employee since many claims-related activities are computer-supported.
 these average claim costs, loaded for overhead and other costs that are not captured by the
computerized tracking systems, can be applied within frameworks as described by Rahardjo and MangoAllen (claim staffing technique).
A benefit of working with the underlying cost data is that it allows for more detailed analysis of the claim activity
costs, to determine which types of claim transactions and which stages of the claim life cycle have relatively
similar (or different) costs.
Conger and Nolibos suggest when Spalla’s method, consider evaluating a ‘reality check’: if the selected costs
per transaction were applied to the numbers of transactions that were undertaken last year, would the result
match that period’s actual total ULAE expenditures?”
While Spalla describes determining the actual cost, the approach could also be used to quantify the relative
amount of cost per transaction compared to the cost of other kinds of claim transactions.
 This relativity is less subject to annual change than the dollar cost per transaction or per activity.
 With relativities, the actuary could use the general approaches described in Rahardjo and Mango-Allen,
but now with some quantitative basis for the magnitude of the parameters.

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4

Triangle-Based Techniques

406

1. Actuaries can estimate ULAE using triangle-based development techniques.
To analyze ULAE in triangular format, a method used to assign ULAE to individual cells (AY by evaluation
year) of the triangle is needed.
Since “actual” ULAE by AY is not observable, at least not for all categories of ULAE, the actuary will need to
form assumptions for the creation of the paid ULAE triangle. ULAE payments are usually allocated using the
pattern of claim payments.
Note: AY triangles of ULAE may be distorted if either the method of allocating calendar ULAE to accident
years changes over time or if the claims payment patterns change.
2. R.S. Slifka suggests using a time-and-motion study to estimate the claim department’s allocation of
resources/costs between current AY claims and prior AY claims.
For example, assume that a time and motion study suggests that:
• 60% of the current accident year’s ULAE remains unpaid
• 15% of the prior accident year’s ULAE remains unpaid
• 5% of the second prior accident year’s ULAE remains unpaid
Total unpaid ULAE is estimated as 80% (60% + 15% + 5%) of a typical CYs ULAE payment.
This technique presumes a steady state, and can be refined to reflect volume growth as well as the effects
of inflation.
3. Construct paid ULAE triangles based on time and motion studies.
For example, assume that time and motion studies suggest that 50% of ULAE is paid at the time a claim is
reported and the remaining 50% is paid in proportion to claim payments.
An actuary can then assign historical calendar ULAE to accident year-calendar year cohorts:
i. 50% according to the distribution of reported claims across current AY, prior AY, second prior AY, and so
on; and
ii. 50% according to the distribution of paid claims, as indicated by an appropriate AY claims payment
pattern.
Once the ULAE triangle is constructed, apply the traditional development technique to estimate ultimate
ULAE and indicated unpaid ULAE.
In practice, ULAE triangle projections are rarely used by actuaries.

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5

Comparison Example

407

Conger and Nolibos provide an example of a U.S. WC insurer who has been in operations for 6 years.
In Exhibit IV, Sheet 1, CY and AY experience data from their example is shown for PQR Insurer.
Chapter 17 - Unallocated Loss Adjustment Expenses
PQR Insurer
Summary of Input Parameters ($000)
Calendar Year
Ult on
Paid
Paid
Reported Reported in
Year
ULAE
Claims
Claims
Calendar
(1)
(2)
(3)
(4)
(5)
2003
1,978
4,590
19,534
27,200
2004
4,820
14,600
57,125
76,700
2005
8,558
38,390
85,521
106,900
2006
12,039
58,297
128,672
154,300
2007
13,143
86,074
145,070
163,100
2008
15,286
105,466
163,626
176,400
Total

55,824

307,417

599,548

704,600

Exhibit IV
Sheet 1

Ultimate
Claims
(6)
28,600
79,200
108,400
156,700
163,400
177,100

Accident Year
IBNR at
12/31/2008
(7)
257
1,742
5,095
16,140
34,477
56,141

713,400

113,852

Reported
Claims
(8)
28,343
77,458
103,305
140,560
128,923
120,959
599,548

Note: Claims include allocated claim adjustment expenses.
Column Notes:
(2) through (4) Based on data from PQR Insurer. Reported claims represent paid claims,
case outstanding, and estimated IBNR. (5) through (7) Based on actuarial analysis at year-end 2008.
(8) Based on data from PQR Insurer. Includes paid claims, case outstanding, and estimated IBNR.

Exhibit IV, Sheet 2:
 Over the six years of operations, paid ULAE averaged about 18% of claims, and given the downward
trend in the paid-to-paid ratios in Column (6), a ULAE ratio of 16% may be selected.
 Based on the above, an actuary using the:
i. traditional technique would derive estimated unpaid ULAE of $41.6 million.
 ii. Kittel refinement, and a 11.5% ULAE ratio would derive estimated unpaid ULAE of $29.9 million (see (7).
Chapter 17 - Unallocated Loss Adjustment Expenses
PQR Insurer
Classical and Kittel Techniques ($000)

Calendar
Year
(1)
2003
2004
2005
2006
2007
2008
Total

Paid
ULAE
(2)
1,978
4,820
8,558
12,039
13,143
15,286
55,824

Paid
Claims
(3)
4,590
14,600
38,390
58,297
86,074
105,466
307,417

Reported
Claims
(4)
19,534
57,125
85,521
128,672
145,070
163,626
599,548

Average of
Paid and
Claims
(5)
12,062
35,863
61,956
93,485
115,572
134,546
453,484

(8) Selected ULAE Ratio
(9) Case Outstanding at 12/31/08
(10) IBNR at 12/31/08
(11) Estimated Unpaid ULAE at 12/31/08

Exhibit IV
Sheet 2

ULAE RatioPaid ULAE to
Paid
Avg Paid &
Claims
Rptd Claims
Traditional
Kittel
(6)
(7)
0.431
0.164
0.33
0.134
0.223
0.138
0.207
0.129
0.153
0.114
0.145
0.114
0.182
0.123
0.16
292,130
113,853
41,587

0.115
292,130
113,853
29,891

Column and Line Notes:
(2) through (4) From Exhibit IV, Sheet 1.
(5) = [Average of (3) and (4)].
(6) = [(2) / (3)].
(7) = [(2) / (5)].
(8) Selected based on ULAE ratios in (6) and (7).
(9) Based on data from PQR Insurer.
(10) Based on actuarial analysis at 12/31/08 for all lines combined.
(11) = {[(8) x 50% x (9)] + [(8) x 100% x (10)]}.

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For PQR Insurer, Conger and Nolibos found that:
 ULAE expenditures are concentrated more heavily towards the front end of the claim than are the claim
payments.
 the growth of PQR Insurer will result in an overstatement of the estimated unpaid ULAE using the
traditional technique.
 discussions with PQR management and examination of the flows of work and allocation of resources in
the claims department suggest 60% to 70% of the work for a claim is concentrated at the time the claim
is reported, and 30% to 40% of the work is spread over the remaining life of the claim.
 no particular extra degree of effort is required to close the claim.
Since ULAE expenses are heavier at the beginning of the claim’s life cycle, the estimated unpaid ULAE
using the Kittel refinement results in a lower estimate of unpaid ULAE ($29.9 million) than the traditional
technique ($41.6 million).
Exhibit IV, Sheet 3, shows the Conger and Nolibos generalized method with U1= 60%, U2= 40%, and U3 = 0%
Chapter 17 - Unallocated Loss Adjustment Expenses
Sheet 3
PQR Insurer
Exhibit IV
Conger and Nolibos Generalized Approach - 60/40 Assumption ($000)
Ult on Claims
Calendar
Paid
Reported in
Paid
Claims
ULAE
Year
ULAE
Calendar Year
Claims
Basis
Ratio
(1)
(2)
(3)
(4)
(5)
(6)
2003
1,978
27,200
4,590
18,156
0.109
2004
4,820
76,700
14,600
51,860
0.093
2005
8,558
106,900
38,390
79,496
0.108
2006
12,039
154,300
58,297
115,899
0.104
2007
13,143
163,100
86,074
132,290
0.099
2008
15,286
176,400
105,466
148,026
0.103
Total
55,824
704,600
307,417
545,727
0.102
(7) Selected ULAE Ratio
(8) Ultimate Claims
(9) Indicated Unpaid ULAE Using:
(a) Expected Claim Method
(b) Bornhuetter-Ferguson Method
(c) Development Method
Column and Line Notes:
(2) through (4) From Exhibit IV, Sheet 1.
(5) = {[(3) x 60%] + [(4) x 40%]}.
(6) = [(2) / (5)].
(7) Selected based on ULAE ratios in (6).
(8) From Exhibit IV, Sheet 1.
(9a) = {[(7) x (8)] - (Total in (2))}.
(9b) = {(7) x [(8) - (Total in (5))]}.
(9c) = {{[(8) / (Total in (5))] - 1.00} x (Total in (2))}.

0.100
713,400
15,516
16,767
17,152

The claims basis in Column (5) is equal to 60% of the ultimate on claims reported in the year (R) and 40% of
paid claims (C). A ULAE ratio of 10% is selected based on a review of the historical experience by year.
The estimated unpaid ULAE in Line (9) is computed using the 3 approaches described in the previous section:
• Expected claim method = [(selected ULAE ratio x ultimate claims) – total paid ULAE to date]
• Bornhuetter-Ferguson method = [selected ULAE ratio x (ultimate claims – total claims basis)]
• Development method = {[(ultimate claims / total claims basis) – 1.00] x total paid ULAE to date}

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Exhibit IV, Sheet 4, shows the Conger and Nolibos generalized method with U1= 70%, U2= 30%, and U3 = 0%
Exhibit IV, Sheet 5 presents the Conger and Nolibos simplified generalized approach.
Shown is a range of estimated unpaid ULAE assuming pure IBNR = 4% of the latest AY ultimate claims or
6% of the latest AY ultimate claims.
Chapter 17 - Unallocated Loss Adjustment Expenses
Exhibit IV
PQR Insurer
Sheet 5
Conger and Nolibos Simplified Generalized Approach - 60/40 Assumption ($000)

Year
(1)
2003
2004
2005
2006
2007
2008
Total

Cal Year
Paid
ULAE
(2)
1,978
4,820
8,558
12,039
13,143
15,286
55,824

Acc Year
Ultimate
Claims
(3)
28,600
79,200
108,400
156,700
163,400
177,100
713,400

Cal Year
Paid
Claims
(4)
4,590
14,600
38,390
58,297
86,074
105,466
307,417

(7) Selected ULAE Ratio
(8) Ultimate Claims
(9) Estimated Pure IBNR Based on
(a) 4% of Latest Accident Year Ultimate Claims
(b) 6% of Latest Accident Year Ultimate Claims
(10) Indicated Unpaid ULAE Using
(a) 4% of Latest Accident Year Ultimate Claims
(b) 6% of Latest Accident Year Ultimate Claims

Claims
Basis
(5)
18,996
53,360
80,396
117,339
132,470
148,446
551,007

ULAE
Ratio
(6)
0.104
0.090
0.106
0.103
0.099
0.103
0.101
0.10
713,400
7,084
10,626
16,664
16,877

Column and Line Notes:
(2) through (4) From Exhibit IV, Sheet 1.
(5) = {[(3) x 60%] + [(4) x 40%]}.
(6) = [(2) / (5)].
(7) Selected based on ULAE ratios in (6).
(8) From Exhibit IV, Sheet 1.
(9a) = [4% x (accident year 2008 ultimate claims in (3))].
(9b) = [6% x (accident year 2008 ultimate claims in (3))].
(10a) = {(7) x [60% x (9a)] + {40% x [(8) - (Total in (4))]}}.
(10b) = {(7) x [60% x (9b)] + {40% x [(8) - (Total in (4))]}}.

Many actuaries only use one method to estimate unpaid ULAE.
When determining which method to use, a selection criterion for assessing alternative methods should be used.
 One approach is to evaluate the results in terms of the number of years of payments indicated by the
unpaid estimate.
 The expected number of future year payments will vary depending on the types of insurance in insurer’s
portfolio. For example:
i. for short-tail lines of insurance, the actuary may expect the estimate of unpaid ULAE to represent 1-2
years of additional CY payments.
ii. for long-tail lines of coverage, the estimated unpaid ULAE may be expected to represent 3-4 years of
payments.

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ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
`Note: With the release of the 2011 CAS Exam 6 syllabus, "Unallocated Loss Adjustment Expense Reserves in an
Inflationary Environment " by Kittel and "Determination of Outstanding Liabilities for Unallocated Loss
Adjustment Expenses" by Johnson are no longer part of the required syllabus readings. These articles
have been replaced by "Estimating Unpaid ULAE Liabilities " by Friedland.
However, in the article by Friedland, numerous references (and numerical examples) are made to the
articles authored by Kittel and Johnson. Therefore, the following past CAS questions drawn from the
content within the Kittel and Johnson paper have been provided (with cautions noted *).

1994 Exam Questions:
58.You are given the following information:

Item
Calendar year 1993 paid losses
Total loss reserves @12/31/92
Total loss reserves @12/31/93
IBNR for losses @12/31/92
IBNR for losses @12/31/93
Calendar year 1993 paid ULAE
Calendar year 1993 incurred ULAE

Amount
900,000
9,000,000
10,000,000
3,600,000
4,000,000
90,000
210,000

a. (1 point)
Compute the 12/31/93 ULAE reserve using the traditional paid-to-paid ratio. Show all
work.
b. (1 point)
Compute the 12/31/93 ULAE reserve using the Kittel method. Show all work.
* Conger shows Kittel's Refined Method to use "B" = 50% (Paid Loss) + 50% (Reported Loss).
Instead of using reported loss, this old exam solution will use incurred.
1996 Exam Questions:
30. According to Johnson in 'Determination of Outstanding Liabilities for ULAE,' which of the following
statements are FALSE?
1. Johnson’s method is based on the assumption that ULAE have little or nothing to do with the
nature of particular claims.
2. Johnson's method assumes that the ULAE payments are proportional to the loss payments.
1999 Exam Questions:
13. T/F Johnson’s ULAE Model assumes that unallocated loss adjustment expenses have a direct
correlation to the nature of particular claims.
Questions from the 2004 Exam:
5. You are given the following information:
2003 calendar year paid loss = $15,000
Total loss reserves as of December 31, 2003 = $20,000
Total loss reserves as of December 31, 2002 = $18,000
IBNR reserve as of December 31, 2003 = $3,000
2003 calendar year paid ULAE = $1,000
50% of the ULAE occurs when the claim is reported and 50% when it is closed
Using Kittel's paid to paid/ incurred method, what is the ULAE reserve at December 31, 2003?
A. < $450

B. > $450 but < $575 C. > $575 but < $700 D. > $700 but < $825 E.

> $825

Note: This question is not consistent with the way Conger presents Kittel's Refined Method.

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Questions from the 2005 Exam:
2. You are given the following information:
• Calendar year 2004 paid ULAE = $25,000
• Loss reserve at December 31, 2003 = $1,000,000
• Calendar year 2004 loss payments = $250,000
• Calendar year 2004 incurred losses = $500,000
• IBNR percentage of loss reserve at December 31, 2004 = 20%
• 60% of a clairn's ULAE expense is paid when opened with the remaining expense paid at closing.
Using the paid ULAE to paid loss method, what is the ULAE reserve as of December 31, 2004?
A. < $55,000 B. > $55,000, but < $62,500
C. > $62,500 but < $70,000
D. > $70,000, but < $77,500
E. > $77,500
* Wiser's discussion of the paid-to-paid method shows an adaption away from the 50-50 assumption.
Accordingly, when applying the ratio, we'd use the 60% for opening % (and 40% at closing), as given.
But, Conger shows Traditional Method and Kittel's Refined Methods to use the "50-50" assumption.
So, if the question asked for the "Traditional" Method or "Kittel's Refined Method" as in Conger,
Then, to be consistent with Conger's Exhibits B or C, the solution would not use the 60% given.
Questions from the 2006 Exam:
18. (2.5 points) Given the following information:
2005 Calendar Year Paid Loss
Outstanding Case Reserves as of December 31, 2005
IBNR Reserve as of December 31, 2005
2005 Calendar Year Paid ULAE
Outstanding Case Reserves as of December 31, 2004
IBNR Reserve as of December 31, 2004

$ 2,000,000
12,000,000
4,000,000
90,000
10,000,000
3,600,000

 Estimated percentage of work at closing
70%
 Estimated percentage of work at opening
30%
a. (0.5 points) Compute the ULAE reserve at December 31, 2005 using the traditional paid-to-paid ratio.
*Conger shows Traditional Method and Kittel's Refined Methods to use the "50-50" assumption.
To be consistent with Conger's Exhibit B, this question would be solved ignoring the 30% & 70% given.
But, Wiser's discussion of the paid-to-paid method shows an adaption away from the 50-50 assumption.
So, under Wiser, we could use the 30% for opening % (and 70% at closing), as given.
b. (1 point) Compute the ULAE reserve at December 31, 2005 using the method described by Kittel.
* Conger shows Kittel's Refined Method to use "B" = 50% (Paid Loss) + 50% (Reported Loss).
Instead of Reported Loss, the old exam solution will use Incurred. The difference is the IBNYR.
Additionally, as in part a), Conger's example of Kittel's Refined method uses the "50-50" assumption.
To be consistent with Conger's Exhibit C, this question would be solved ignoring the 70% & 30% given.
c. (1 point) Identify two problems with the ULAE reserving methods that use calendar year paid-to-paid
loss ratios as a starting point.

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Questions from the 2008 Exam:
16. (1 point)
a. (0.25 point) Identify the fundamental assumption underlying a dollar-based approach to estimating
ULAE liabilities.
b. (0.25 point Identify the fundamental assumption underlying a count-based approach to estimating
ULAE liabilities.
c. (0.5 point) Identify two considerations that could influence an actuary's decision to choose a dollarbased versus a count-based approach when estimating ULAE liabilities.

Questions from the 2009 Exam:
14. (4 points) Given the following information:

Calendar
Year

Paid
ULAE

Paid Loss &
ALAE

Reported
Loss &
ALAE

2006
2007
2008

$11,000
14,000
16,000

$60,000
90,000
110,000

$134,000
152,000
170,000

Estimated Ultimate
Loss & ALAE on
Claims Reported in
Calendar Year
$159,000
170,000
183,000

Estimates as of December 31, 2008 for all accident years combined:
•
Case reserves
$293,000
•
IBNR
$114,000
•
Ultimate Loss & ALAE
$667,000
The claims department indicates that 60% of its work is expended when the claim is reported, and 40% of
its work is spread over the life of the claim. No additional work is expended in closing the claim.
a. (1 point) Use the traditional paid-to-paid method to calculate the ULAE reserve as of Dec. 31, 2008.
b. (1 point) Use Kittel's refinement to the traditional paid-to-paid method to calculate the ULAE reserve
as of December 31, 2008.
c. (1 point) Use the Bornhuetter-Ferguson method applied to Conger's generalized approach to
calculate the ULAE reserve as of December 31, 2008.
d. (1 point) Describe two issues with other methods that led Conger to develop the generalized approach.

Questions from the 2011 Exam:
35. (2 points) Given the following information as of December 31, 2010:
Ultimate
Calendar
Paid
Reported
Paid
Year
ULAE
Claims
Claims
2007
$2,000
$21,000
$2,100
2008
$2,750
$21,500
$12,650
2009
$4,500
$35,000
$22,650
2010
$5,500
$40,000
$30,100
• 40% of ultimate ULAE is spent on maintaining claims.
• 60% of ultimate ULAE is spent on opening claims.
• Ultimate value of claims for 2007 through 2010 = $117,500
Use the Conger-Nolibos expected claim method to estimate the unpaid ULAE.

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Questions from the 2012 Exam:
28. (1.5 points) Given the following data:
Paid
Paid
Calendar
ULAE
Claims
Year
($000s)
($000s)
2009
$11,000
$56,000
2010
$12,000
$85,500
2011
$14,000
$102,000

Reported
Claims
($000s)
$125,600
$145,000
$162,500

•

The case outstanding as of December 31, 2011 is $150,000,000

•

The IBNR estimate as of December 31, 2011 is $50,000,000

Use the Kittel technique to estimate unpaid unallocated loss adjustment expenses (ULAE).

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Solutions to 1994 Exam Questions:
58. a.(1 point) Compute the 12/31/93 ULAE reserve using the traditional paid-to-paid ratio. Show all work.

(1a)
(1b)
(1c)=(1b)/(1a)
(1d)
(1e)

Paid-to-paid method
Calendar year 1993 paid losses
Calendar year 1993 paid ULAE
Paid-to-paid ratio
Case loss reserves @12/31/93
IBNR loss reserves

(1f)=(1c)x(1e)+(1c)x(1d)/2 Estimated ULAE reserve

b. (1 point)

900,000
90,000
10.0%
6,000,000
4,000,000
700,000

Compute the 12/31/93 ULAE reserve using the Kittel method. Show all work.

(2a)
(2b)
(2c)=avg(2a,2b)
(2d)
(2e)=(2d)/(2c)
(2f)=(1d)
(2g)=(1e)

Kittel method
Calendar year 1993 paid losses
Calendar year 1993 incurred losses
Average of paid and incurred losses
Calendar year 1993 paid ULAE
Kittel ratio
Case loss reserves @12/31/93
IBNR loss reserves

(2h)=(2e)x(2g)+(2e)x(2f)/2 Estimated ULAE reserve

900,000
1,900,000
1,400,000
90,000
6.43%
6,000,000
4,000,000
450,000

Solutions to 1996 Exam Questions:
30. According to Johnson, which of the following statements are FALSE?
1. Johnson’s method is based on the assumption that ULAE have little or nothing to do with the nature of
particular claims.
True
2. Johnson's method assumes that the ULAE payments are proportional to the loss payments. False
Solutions to 1999 Exam Questions:
13. T/F Johnson’s ULAE Model assumes that unallocated loss adjustment expenses have a direct
correlation to the nature of particular claims.
False, Johnson assumes no correlation.

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ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to questions from the 2004 Exam:
5. You are given the following information:
2003 calendar year paid loss = $15,000
Total loss reserves as of December 31, 2003 = $20,000
Total loss reserves as of December 31, 2002 = $18,000
IBNR reserve as of December 31, 2003 = $3,000
2003 calendar year paid ULAE = $1,000
50% of the ULAE occurs when the claim is reported and 50% when it is closed
Using Kittel's paid to paid/incurred method, what is the ULAE reserve at December 31, 2003?
A. < $450

B. > $450 but < $575 C. > $575 but < $700 D. > $700 but < $825 E.

> $825

Step 1: Compute the “Paid to Paid” ratio under the “alternative method” (e.g. paid/incurred method) and
write equations to determine the CY 2003 incurred losses and the ULAE reserve.
Alternative (paid to paid/incurred) method
Paid to Paid ratio

CY Paid ULAE
.50*[CY Paid +CY Incurred Loss ]

CY 2003 Incurred losses equals CY 2003 Paid Loss + [2003 Total Reserves - 2002 Total Reserves]
= 15,000 + 20,000 – 18,000 = 17,000
ULAE Reserves = .50 * Paid to Paid Ratio * (Case Reserves) + Paid to Paid Ratio*(IBNR Reserves)
= .50 * Paid to Paid Ratio * [(Total Reserves) + (IBNR Reserves)]
Step 2: Using the formulas in Step 1 and the values provided in the problem, solve for the absolute value
of the difference between these two estimates:
Alternative (paid to paid/incurred) method
Paid to Paid ratio

1,000
= .0625
.50*[15,000+17,000]

2003 ULAE Reserve

.50 * .0625 * [20,000 + 3,000] = 718.75

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Chapter17 – Estimating Unpaid ULAE
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to questions from the 2005 Exam:
2. You are given the following information:
• Calendar year 2004 paid ULAE = $25,000
• Loss reserve at December 31, 2003 = $1,000,000
• Calendar year 2004 loss payments = $250,000
• Calendar year 2004 incurred losses = $500,000
• IBNR percentage of loss reserve at December 31, 2004 = 20%
• 60% of a claim's ULAE expense is paid when opened with the remaining expense paid at closing.
Using the paid ULAE to paid loss method, what is the ULAE reserve as of December 31, 2004?
A. < $55,000 B. > $55,000, but < $62,500
D. > $70,000, but < $77,500
E. > $77,500

C. > $62,500 but < $70,000

Step 1: Write an equation to determine the ULAE reserve as of December 31, 2004, using the paid ULAE
to paid loss method.
ULAE Reserves = Paid to Paid Ratio * AF * (Case Reserves) + Paid to Paid Ratio*(IBNR Reserves),
where, AF is the adjustment factor, which is equal to 1.0 - % of a claim’s ULAE expense is paid when
opened with the remaining expense paid at closing. Since 60% of a claim's ULAE expense is paid
when opened with the remaining expense paid at closing, AF = 1.0 – 0.60 = 0.40.

Step 2: Write equations to determine case reserves at 12/13/2004, total reserves at 12/31/2004, IBNR
reserves at 12/31/2004, and solve for each.
Case reserves at 12/31/2004 = Total reserves at 12/31/2004 – IBNR at 12/31/2004
CY 2004 incurred losses = [CY 2004 loss payments + Total reserves at 12/31/2004
- Total reserves at 12/31/2003]
500,000 = 250,000 + [Total reserves at 12/31/2004 – 1,000,000]
Thus, total reserves at 12/31/2004 = 500,000 – 250,000 + 1,000,000 = 1,250,000
IBNR reserves at 12/31/2004 = .20 * 1,250,000 = 250,000
Case reserves at 12/31/2004 = 1,250,000 - .20 * 1,250,000 = 1,000,000

Step 3: Using the equation in Step 1, the data given in the problem, and the results from Step 2, compute
the ULAE reserve as of December 31, 2004, using the paid ULAE to paid loss method.
ULAE Reserves = Paid to Paid Ratio * AF* (Case Reserves) + Paid to Paid Ratio*(IBNR Reserves),
= 25,000/250,000 * 0.40 * 1,000,000 + 25,000/250,000 * 250,000
= 40,000 + 25,000 = 65,000
Answer C > $62,500 but < $70,000

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ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to questions from the 2006 Exam:
18. (2.5 points) Given the following information:
2005 Calendar Year Paid Loss
Outstanding Case Reserves as of December 31, 2005
IBNR Reserve as of December 31, 2005
2005 Calendar Year Paid ULAE
Outstanding Case Reserves as of December 31, 2004
IBNR Reserve as of December 31, 2004
Estimated percentage of work at closing
Estimated percentage of work at opening

$2,000,000
12,000,000
4,000,000
90,000
10,000,000
3,600,000

70%
30%

a. (0.5 points) Compute the ULAE reserve at December 31, 2005 using the traditional paid-to-paid ratio.
b. (1 point) Compute the ULAE reserve at December 31, 2005 using the method described by Kittel.
c. (1 point) Identify two problems with the ULAE reserving methods that use calendar year paid-to-paid loss
ratios as a starting point.
a. Write an equation to determine the ULAE reserve as of December 31, 2005, using the paid ULAE to
paid loss method.
ULAE Reserve = Paid to Paid Ratio * % at closing * (Case Reserves) + Paid to Paid Ratio*(IBNR Reserves)
Paid to Paid ratio = paid ULAE/paid loss = 90,000 / 2,000,000 = .045
ULAE Reserve =.045 * .70 * ($12,000,000) + .045 * ($4,000,000) = $ 558,000
b. Write an equation to determine the ULAE reserve as of December 31, 2005, using the Kittel method.
ULAE Reserve = Paid to Paid Ratio * % at closing * (Case Reserves) + Paid to Paid Ratio*(IBNR Reserves)
Paid to Paid ratio = paid ULAE/(paid loss + % at opening (change in total reserves))
Change in reserves = (12M + 4M) – (10M + 3.6M) = 2,400,000
Paid to Paid ratio = 90,000/(2,000,000 + .3* 2,400,000) = .033
ULAE Reserve =.033 * .70 * ($12,000,000) + .033 * ($4,000,000) = $409,200
Question 18 - Model answer 1
c1. It doesn’t take into consideration open claims at the end of the year
c2. It will overstate reserves if the company is growing.
Question 18 - Model answer 2
c1. When the company is growing, using the paid to paid ratio to estimate ULAE will over estimate the
ULAE ratio
c2. If the claim department changes its claim settlement pattern, the paid to paid ratio will change. Thus,
the estimate of ULAE based on historical paid to paid will not be accurate.

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ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to questions from the 2008 Exam:
16a. (0.25 point) Identify the fundamental assumption underlying a dollar-based approach to estimating
ULAE liabilities.
16b. (0.25 point Identify the fundamental assumption underlying a count-based approach to estimating
ULAE liabilities.
16c. (0.5 point) Identify two considerations that could influence an actuary's decision to choose a dollarbased versus a count-based approach when estimating ULAE liabilities.
Question 16 - Model answer 1
a. Dollar-based – ULAE payments follow the paid losses & ALAE.
b. Count-based – ULAE payment for some type of transaction will be same irrespective of claim size
and nature.
c. (1)If there is enough data (claims paid, transactions done) to identify the cost associate with each
transaction type to make the count-based method reliable. That is, if the ULAE paid is large
enough to warrant a detailed analysis of transactions/cost-based operation.
(2) If the ULAE to paid loss ratio is stable then better to use a dollar-based system.
Question 16 - Model answer 2
a. ULAE dollars track with claim dollars i.e., a $1000 claim has 10 times as much ULAE as a $100 claim.
b. ULAE dollars track with the type of transaction regardless of dollars, so similar types of
transactions have same ULAE expenditures.
c. (1) Whether the volume of losses is growing considerably.
(2) The application of the company’s resources to various stages of the life of a claim.

Solutions to questions from the 2009 Exam:
Question 14 - Model Solution 1
a. ULAE Ratio estimate = CY paid ULAE/[CY paid loss + ALAE]
= (11,000+14,000+16,000)/(60,000+90,000+110,000) = (41,000)/260,000 = 0.158
ULAE Reserve = 0.158 × (IBNR + ½ Case Reserve)
= 0.158 × (114,000 + ½ × 293,000) = 41,159
b. Use Kittel ULAE Ratio = (CY paid ULAE) / [½ ×(CY paid + CY reported)]
= 41,000 / [½ × (260,000 +456,000)] = 0.115
ULAE Reserve = 0.1115 × (114,000 + ½ × 293,000) = 29,834
c. B = u1 × R + u2 × P, where R = Sum of estimated ultimate loss & ALAE on claims reported in CY
u1 = 60 %, u2 = 40%, where u1 = % of work expended when the claim is reported. u2 = 1.0 – u1
B = 0.6 × 512,000 + 0.4 × 260,000; B = 411,200
W = M/B = 41,000/411,200 = 0.0997
By B-F approach, ULAE reserve = W × (L-B), where L = ultimate loss & ALAE for all years as of 12/31/08
ULAE Reserve = 0.0997 × (667,000 – 411,200) = 25,503
d1. estimates may be distorted if book is growing
d2. 50-50 assumption may not be true

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ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to questions from the 2009 Exam (continued):
Question 14 - Model Solution 2
The difference in Solution 1 vs. Solution 2 lies in the assumption made by the candidate answering this
question, which is stated in part a. below.
a. As a historical matter, the traditional and Kittle refinement used a 50/50 split between opened and
remainder, we will calculate our answers based on the 60/40 split specified.
(1)
(2)
(3) = (1) / (2)
Paid ULAE
Paid Loss & ALAE
Paid-Paid
06
11,000
60,000
0.183
07
14,000
90,000
0.156
08
16,000
110,000
0.145
41,000
260,000
0.158
selected ratio
RULAE = (ratio)* (IBNR + (.4) Case) = (.158) (114,000+ (.4) (293,000)) = 36,530
b.

06
07
08

(4)
Reported Loss & ALAE
134,000
152,000
170,000

(5) = 1/2 [(2) + (4)]
Kittle basis
97,000
121,000
140,000
358,000

(6) = (1) / (5)
ratio
0.113
0.116
0.114
0.115

selected ratio

RULAE = (ratio) (IBNR + (.4) Case) = (.115) (114,000+ (.4) (293,000)) = 26,588
c. Note: This solution is the same as shown in Solution 1 (only rounding differences exist: 0.0997 vs 0.100)
(7)
(8) = (.6) × (7) +
Ult on CY reported
(.4) × (2)
(9) = (1) / (8)
Conger Basis
ratio
06
159,000
119,400
0.092
07
170,000
138,000
0.101
08
183,000
153,800
0.104
411,200
0.100
selected ratio
"B-F" method: RULAE = (ratio)(Ult L+ALAE for all years as of 12/31/08 – Conger Basis)
= (.100)(667,000 - 411,200) = 25,580
d1. When business is growing, the paid-paid method is inaccurate due to the mismatch between ULAE
payments and their associated claims.
d2. The traditional 50-50 split between cost to open and cost to maintain-and-close may not hold.
The generalized method allows separation of cost to maintain and cost to close.

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Chapter17 – Estimating Unpaid ULAE
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to questions from the 2011 Exam:
35. Use the Conger-Nolibos expected claim method to estimate the unpaid ULAE.
Intial comments
Conger and Nolibos define U1 + U2 + U3 = 100%, where:
• U1 is the % of ultimate ULAE spent opening claims
• U2 is the % of ultimate ULAE spent maintaining claims
• U3 is the % of ultimate ULAE spent closing claims
For time period T, Conger and Nolibos define M, the total amount spent on ULAE during a time period T, to be
M = (U1 x R x W) + (U2 x P x W) + (U3 x C x W), where
• R is the ultimate cost of claims reported during T
• P is the claims paid during T
• C is the ultimate cost of claims closed during T
• W is the ratio of ultimate ULAE (U) to ultimate claims (L)
* T could be activity between t1 and t2 related to an AY or for all AYs, where t1 and t2 are points in time.
* Algebraically derive the ratio W = M/B by defining B, the claims basis for the time period T to be:
B = (U1 x R) + (U2 x P) + (U3 x C)
Thus, M = B x W, and W = M/B.
Ultimate ULAE (U) for a group of AYs can be estimated as:
U = W* x L, where
• W* is the selected ultimate ULAE-to-claims ratio
• L is the independently estimated ultimate claims for the same group of AY
Compute Unpaid ULAE by subtracting ULAE already paid (M) from the estimate of ultimate ULAE (U).
Unpaid ULAE = (W* x L) - M
Note: Calculations involving U3 or C are not needed in solving this problem since U3 or C is not given.
Question 35 - Model Solution 1
Compute W*
CY
B
07
13440
08 17,960=0.60*$21,500+0.40*$12,650
09
30060
10
36040
97,500

M (given)
2000
2750
4500
5500
14,750

Selected W* = 0.15
ULAE Ratio = W = M / B = M / [(u1 x r) + (u2 x p)], where u1 = 0.60,
Ult ULAE = W xL = 0.15(117,500) = 17,625
Unpaid ULAE = Ult ULAE – Paid ULAE = 17,625 – 14,750 = 2875
Question 35 - Model Solution 2
B = [117.5 x 60% + 67.5 x 40%] = 97,500; M = 14,750 (given)
W = M / B = 15.13%
L = 117,500 (given)
Unpaid ULAE = W x L – M
= 15.13% x 117,500 – 14,750 = 3027.75

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W = M/B
0.1488
0.1531
0.1497
0.1526
0.1513
u2 = 0.40

 2014 by All 10, Inc.

Chapter17 – Estimating Unpaid ULAE
ESTIMATING UNPAID CLAIMS USING BASIC TECHNIQUES - FRIEDLAND
Solutions to questions from the 2012 Exam:
28..Use the Kittel technique to estimate unpaid unallocated loss adjustment expenses (ULAE).
Question 28 – Model Solution (Exam 5B Question 13)
Ratio of paid ULAE to avg. (rept. + paid)

09
10
11
Total

Pd ULAE
11,000
12,000
14,000
37,000

Avg pd + rept
90,800
115,250
132,250
338,300

Ratio
.1211
.1041
.1059
.1094 ← Selected – not enough data to say
that there is a clear downward trend

ULAE = Ratio * [.50 * Case o/s + IBNR)] = (0.1094)[(.5)(150,000,000) + (50,000,000)] = 13,675,000
Examiner’s Comments
Most candidates that attempted the question received full credit.
The most common deductions were for candidates that incorrectly calculated the ULAE ratio (did not
recognize that the ratio should be Paid ULAE/avg [paid claims, reported claims]),
or that calculated the unpaid ULAE but treated it in the wrong fashion, for example as total ULAE and
then subtracted paid ULAE.

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Unpaid Claim Estimates
ASOP 43
Sec
1
2
3
4
1

Description
Purpose and Scope
Definitions
Analysis of Issues and Recommended Practices
Communications and Disclosures

Pages
1
2-3
3-8
9-10

Purpose and Scope

1.1 Purpose—This actuarial standard of practice (ASOP) provides guidance to actuaries estimating loss
and loss adjustment expense for unpaid claims for P&C coverages.
Any reference to “unpaid claims” includes unpaid claim adjustment expense
1.2 Scope—This standard applies to:
 developing unpaid claim estimates only for events that have already occurred or will have occurred,
as of an accounting date (exclusive of estimates developed solely for ratemaking purposes).
 estimating unpaid claims for all classes of entities, including self-insureds, insurance companies,
reinsurers, and governmental entities.
 estimates of gross amounts before recoverables (e.g. deductibles, ceded reinsurance, and salvage
and subrogation)
 estimates of amounts after such recoverables, and
 estimates of amounts of such recoverables.
The actuary should comply with this standard except to the extent it may conflict with applicable law
(statutes, regulations, and other legally binding authority).

Definitions

2-3

Definitions of certain terms below used in this actuarial standard of practice.
Actuarial Central Estimate—An estimate of the expected value over the range of reasonably possible outcomes.
Claim Adjustment Expense—The costs of administering, determining coverage for, settling, or defending
claims even if it is ultimately determined that the claim is invalid.
Event—The incident or activity that triggers potential for claim or claim adjustment expense payment.
Model Risk—The risk that the methods are not appropriate to the circumstances or the models, are not
representative of the specified phenomenon.
Parameter Risk— The risk that the parameters used in the methods or models are not representative of
future outcomes.
Process Risk— The risk associated with the projection of future contingencies that are inherently variable,
even when the parameters are known with certainty.

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ASOP 43
Principal— The actuary’s client or employer. Where the actuary has both a client and an employer, as is
common for consulting actuaries, the facts and circumstances will determine whether the client or the
employer (or both) is the principal with respect to any portion of this standard.
Unpaid Claim Estimate— The actuary’s estimate of the obligation for future payment resulting from claims
due to past events.
Unpaid Claim Estimate Analysis— The process of developing an unpaid claim estimate.

Analysis of Issues and Recommended Practices

3-8

3.1 Purpose or Use of the Unpaid Claim Estimate
Potential purposes or uses of unpaid claim estimates include establishing liability estimates for
 external financial reporting
 internal management reporting
 various special purpose uses (e.g. appraisal work and scenario analyses).
3.2 Constraints on the Unpaid Claim Estimate Analysis
At times constraints exist in performing an actuarial analysis, such as those due to limited data, staff,
time or other resources. When constraints create a significant risk that a more in-depth analysis
would produce a materially different result, the actuary should notify the principal of that risk and
communicate the constraints on the analysis to the principal.
3.3 Scope of the Unpaid Claim Estimate - the actuary should identify the following:
a. the intended measure of the unpaid claim estimate
1. Examples of types of measures for the unpaid claim estimate include high estimate, low
estimate, median, mean, mode, actuarial central estimate, mean plus risk margin, actuarial
central estimate plus risk margin, or specified percentile.
An actuarial central estimate may or may not be the result of the use of a probability distribution or
a statistical analysis (which is meant to clarify the concept rather than assign a precise statistical
measure, as commonly used actuarial methods typically do not result in a statistical mean).
The terms “best estimate” and “actuarial estimate” are not sufficient descriptions of the intended
measure, since they describe the source or the quality of the estimate but not the objective of the
estimate.
2. The actuary should consider whether the intended measure is appropriate to the intended
purpose or use of the unpaid claim estimate.
3. The description of the intended measure should include whether any amounts are discounted.
b. whether the unpaid claim estimate is to be gross or net of recoverables;
c. whether collectibility risk is to be considered when the unpaid claim estimate is affected by
recoverables;
d. the specific types of unpaid claim adjustment expenses covered in the unpaid claim estimate (e.g.
coverage dispute costs, defense costs, and adjusting costs);
e. the claims to be covered by the unpaid claim estimate (e.g. type of loss, line of business, year, and
state); and
f. any other items that, in the actuary’s professional judgment, are needed to describe the scope sufficiently.
3.4 Materiality — Should be evaluated based on professional judgment, taking into account the
requirements of applicable law and the intended purpose of the unpaid claim estimate.

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Unpaid Claim Estimates
ASOP 43
3.5 Nature of Unpaid Claims — Aspects of unpaid claims (including any material trends) that may require
an understanding include:
a. coverage;
b. conditions or circumstances that make a claim more or less likely or the cost more or less
severe;
c. the underlying claim adjustment process; and
d. potential recoverables.
3.6 Unpaid Claim Estimate Analysis—
The actuary should consider the following items when performing the unpaid claim estimate analysis:
3.6.1 Methods and Models
The actuary should select specific methods or models, modify such methods or models, or
develop new methods or models based on relevant factors including the following:
a. the nature of the claims and underlying exposures;
b. the development characteristics associated with these claims;
c. the characteristics of the data;
d. the applicability of methods or models to the available data; and
e. the reasonableness of the assumptions underlying each method or model.
The actuary should consider:
a. whether a method or model is appropriate in light of the purpose, constraints, and scope of
the assignment.
For example, while an unpaid claim estimate produced by a simple method may be appropriate
for an immediate internal use, it may not be for external financial reporting purposes.
b. whether different methods or models should be used for different components of the unpaid
claim estimate. For example, different coverages within a line of business may require
different methods.
c. the use of multiple methods or models appropriate to the purpose, nature and scope of the
assignment and the characteristics of the claims unless, in the actuary’s professional
judgment, reliance upon a single method or model is reasonable given the circumstances.
3.6.2 Assumptions — The actuary should:
a. consider the reasonableness of the assumptions underlying each method or model used.
Assumptions involve professional judgment as to the appropriateness of the methods and
models used and the parameters underlying the application of such methods and models.
Assumptions may be implicit or explicit and may involve interpreting past data or projecting
future trends.
b. use assumptions that have no known significant bias to underestimation or overestimation
of the identified intended measure.
c. The actuary should consider the sensitivity of the unpaid claim estimates to reasonable
alternative assumptions.
3.6.3 Data — The actuary should refer to ASOP No. 23, Data Quality, with respect to the selection of
data to be used, relying on data supplied by others, reviewing data, and using data.
3.6.4

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Recoverables—Consider interaction among the different types of recoverables and adjust the
analysis of unpaid claims to reflect that interaction in a manner the actuary deems appropriate.

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Unpaid Claim Estimates
ASOP 43
3.6.5 Gross vs. Net— Consider the facts and circumstances of the assignment when choosing which
components (the gross estimate, the estimated recoverables, and the net estimate) to estimate.
3.6.6

External Conditions— Consider external conditions (e.g. potential economic changes,
regulatory actions, judicial decisions, or political or social forces) that are known by qualified
actuaries in the same practice area and that are likely to have a material effect on the actuary’s
unpaid claim estimate analysis.

3.6.7

Changing Conditions— Consider changes in conditions with regard to claims, losses, or
exposures, that are likely to be insufficiently reflected in the experience data or in the
assumptions used to estimate the unpaid claims.
Examples include reinsurance program changes and changes in the practices by the entity’s
claims personnel to the extent such changes are likely to have a material effect on the results
of the actuary’s unpaid claim estimate analysis.
Consider obtaining supporting information from the principal or the principal’s duly authorized
representative and may rely upon their representations unless, in the actuary’s professional
judgment, they appear to be unreasonable.

3.6.8 Uncertainty— The actuary should consider:
a. the uncertainty associated with the unpaid claim estimate analysis.
Note: The standard does not require or prohibit the actuary from measuring this uncertainty.
b. the purpose and use of the unpaid claim estimate in deciding whether or not to measure this
uncertainty.
when measuring uncertainty, consider the types and sources of uncertainty being measured
and choose the methods, models, and assumptions that are appropriate for the measurement
of such uncertainty.
for example, when measuring the variability of an unpaid claim estimate covering multiple
components, consider whether the components are independent of each other or whether they
are correlated.
types and sources of uncertainty surrounding unpaid claim estimates include uncertainty due
to model risk, parameter risk, and process risk.
3.7 Unpaid Claim Estimate—Take into account the following with respect to the unpaid claim estimate:
3.7.1 Reasonableness— the reasonableness of the unpaid claim estimate:
i. includes using appropriate indicators or tests that, in the actuary’s professional judgment,
provide a validation that the unpaid claim estimate is reasonable.
ii. should be determined based on facts known to, and circumstances known to or
reasonably foreseeable by, the actuary at the time of estimation.
3.7.2 Multiple Components — When the actuary’s unpaid claim estimate comprises multiple components,
consider whether the estimates of the multiple components are reasonably consistent.
3.7.3

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Presentation—The unpaid claim estimate may be presented in a variety of ways (e.g. as a
point estimate, a range of estimates, a point estimate with a margin for adverse deviation, or
a probability distribution of the unpaid claim amount.)

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Unpaid Claim Estimates
ASOP 43
4

Communications and Disclosures

9-10

4.1 Actuarial Communication—the actuary should disclose the following in an appropriate actuarial
communication:
a.
the intended purpose(s) or use(s) of the unpaid claim estimate, including adjustments that the
actuary considered appropriate in order to produce a single work product for multiple purposes
or uses (as described in section 3.1)
b.
significant limitations, if any, which constrained the actuary’s unpaid claim estimate analysis
such that, in the actuary’s professional judgment, there is a significant risk that a more in-depth
analysis would produce a materially different result (as described in section 3.2)
c.
the scope of the unpaid claim estimate (as described in section 3.3)
d.
the following dates:
(1) the accounting date of the unpaid claim estimate, which is the date used to separate paid
versus unpaid claim amounts;
(2) the valuation date of the unpaid claim estimate, which is the date through which
transactions are included in the data used in the unpaid claim estimate analysis; and
(3) the review date of the unpaid claim estimate, which is the cutoff date for including
information known to the actuary in the unpaid claim estimate analysis, if appropriate.
An example is as follows: “This unpaid claim estimate as of 12/31/2005 was based on data
evaluated as of 11/30/2005 and additional information provided to me through 1/17/2006.”
e.
specific significant risks and uncertainties, if any, with respect to whether actual results may
vary from the unpaid claim estimate; and
f.
significant events, assumptions, or reliances, if any, underlying the unpaid claim estimate that,
in the actuary’s professional judgment, have a material effect on the unpaid claim estimate,
including assumptions provided by the actuary’s principal or an outside party or assumptions
regarding the accounting basis or application of an accounting rule.
If the actuary depends upon a material assumption, method, or model that the actuary does not
believe is reasonable or cannot determine to be reasonable, the actuary should disclose the
dependency of the estimate on that assumption/method/model and the source of that
assumption/method/model.
4.2 Additional Disclosures—In certain cases, the actuary may need to make the following disclosures in
addition to those in section 4.1:
a.
In the case when the actuary specifies a range of estimates, the actuary should disclose the:
i. basis of the range provided, for example, a range of estimates of the intended measure
(each of such estimates considered to be a reasonable estimate on a stand-alone basis);
ii. a range representing a confidence interval within the range of outcomes produced by a
particular model or models; or
iii. a range representing a confidence interval reflecting certain risks, such as process risk and
parameter risk.
b.
In the case when the unpaid claim estimate is an update of a previous estimate, the actuary
should disclose changes in assumptions, procedures, methods or models that the actuary
believes to have a material impact on the unpaid claim estimate and the reasons for such
changes to the extent known by the actuary.
This standard does not require the actuary to measure or quantify the impact of such changes.

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ASOP 43
4.3 Prescribed Statement of Actuarial Opinion—This ASOP does not require a prescribed statement of
actuarial opinion as described in the Qualification Standards for Prescribed Statements of Actuarial
Opinion promulgated by the American Academy of Actuaries.
4.4 Deviation from Standard—If, in the actuary’s professional judgment, the actuary has deviated
materially from the guidance set forth elsewhere in this standard, the actuary can still comply with this
standard by applying the following sections as appropriate:
4.4.1 Material Deviations to Comply with Applicable Law—If compliance with applicable law requires
the actuary to deviate materially from the guidance set forth in this standard, the actuary should
disclose that the assignment was prepared in compliance with applicable law, and the actuary
should disclose the specific purpose of the assignment and indicate that the work product may
not be appropriate for other purposes.
4.4.2 Other Material Deviations—The actuary’s communication should disclose any other material
deviation from the guidance set forth in this standard.
The actuary should:
i. consider whether, in the actuary’s professional judgment, it would be appropriate and
practical to provide the reasons for, or to quantify the expected impact of, such deviation.
ii. be prepared to explain the deviation to a principal, another actuary, or other intended users
of the actuary’s communication.
iii. be prepared to justify the deviation to the actuarial profession’s disciplinary bodies.

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Statement of Principles: Loss and LAE Reserves
CAS
The purpose of the CAS Statement of Principles is to present essential guidelines for any
comprehensive and systematic approach to testing the adequacy of loss reserves.
1. Definitions:
Five elements of a total loss reserve:
1. Case reserves assigned to specific claims.
2. A provision for future development on known claims.
3. A provision for claims that re-open after they have been closed.
4. A provision for claims that have occurred but have not yet been reported to the insurer.
5. A provision for claims that have been reported to the insurer but have not yet been recorded.
Differing loss reserve categories and definitions
Category 1: The reserve for known claims:
a. is the amount that will be required for future payments of claims that have already been
reported to the insurer.
b. is equal to (1) + (2) + (3)
Category 2: The reserve for unknown claims (a.k.a. IBNR reserve)
a. (4) is also known as “pure” IBNR claims while (5) is known as claims in transit.
b. in practice, (2) + (3) + (4) + (5) are often called IBNR.
Dates are important in the loss reserve estimation process. The 5 key dates are:
1. Accident Date: the date the loss occurred.
2. Report Date: the date the loss is first reported to an insurer.
3. Recorded Date: the date the loss is first recorded on the insurer's books.
4. Accounting Date: the “as of” date for the loss reserve estimate.
It is generally a date when a financial statement is prepared (e.g. month end, quarter end or year end).
5. Valuation Date: the date the evaluation of the loss liability is made.
The valuation date can be before, after or the same as the accounting date.
Loss reserve terminology:
1. The required loss reserve as of a given accounting date:
a. is the amount that must be paid to settle all claim liabilities.
b. can only be known when all claims have been settled.
c. is a fixed number that does not change at different valuation dates.
2. The indicated loss reserve:
a. results from an actuarial analysis of a reserve inventory as of a given accounting date conducted
as of a certain valuation date.
b. is the analyst's opinion of the amount of the required loss reserve.
c. changes at different valuation dates and will converge to the required loss reserve
3. The carried loss reserve: the amount of unpaid claim liability shown in financial statements.
4. The loss reserve margin: = the carried reserve - the required reserve.
Since the required reserve is unknown, we only have an indicated margin.
Thus, the indicated loss reserve margin = the carried loss reserve - the indicated loss reserve.

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Statement of Principles: Loss and LAE Reserves
CAS
2. Principles. While there are four ‘stated’ principles, there are 3 unique principles in that principle 1 and
principle 2 are identical, with the exception that principle 1 applies to loss reserves while principle 2
applies to LAE reserves.
1. “An actuarially sound loss reserve for a defined group of claims as of a given valuation date is a provision,
based on estimates derived from reasonable assumptions and appropriate actuarial methods for the
unpaid amount required to settle all claims, whether reported or not, for which liability exists on a particular
accounting date.”
2. “An actuarially sound loss adjustment expense reserve for a defined group of claims as of a given valuation
date is a provision, … for the unpaid amount required to investigate, defend, and effect the settlement of
all claims… for which loss adjustment expense liability exists on a particular accounting date.”
3. The uncertainty inherent in the estimation of required provisions… implies that a range of reserves can be
actuarially sound. The true value of the liability for losses or loss adjustment expenses at any accounting
date can be known only when all attendant claims have been settled.
4. The most appropriate reserve within a range of actuarially sound estimates depends on both the relative
likelihood of estimates within the range and the financial reporting context in which the reserve will be
presented.
3. Considerations.
The CAS Statement of Principles provides the following summarized list of these "considerations"
(1) Data Organization
Five key dates relative to data organization:
1.
2.
3.
4.
5.

Accident Date
Report Date
Recorded Date
Accounting Date (AD)
Valuation Date (VD)

(2) Homogeneity
Subdivide experience into groups exhibiting similar characteristics, such as:
• Loss development patterns, and
• Size of loss distributions.
(3) Credibility
Credibility is a measure of the predictive value that an actuary attaches to a body of data.
Credibility is generally increased by:
• Making groups more homogenous, and
• Increasing the amount of the experience within a homogenous group. [Larger insurers can
therefore refine/partition their data in finer detail than smaller insurers.]
(4) Data Availability
• Data should meet requirements for proper reserve evaluation.
• Data should reconcile to financial statements (now a required “part” of the Actuarial Opinion).
(5) Emergence Patterns
Delay between occurrence and reporting of a claim.
(6) Settlement Pattern
Length of time between report and settlement of a claim.
(7) Loss Development Pattern
Should be carefully reviewed. Insurer's claim practices affect the manner in which claims develop.

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Statement of Principles: Loss and LAE Reserves
CAS
(8) Frequency/Severity
High frequency/low severity versus low frequency/high severity exposures.
(9) Reopened Claim Potential
Tendency for claims to reopen varies by line of business. Workers' compensation claims are
generally the most likely to reopen.
(10)Claims-Made Coverages
Eliminates "pure" IBNR claim potential. Only “pipeline” IBNR.
(11)Aggregate Limits
Use data modeling techniques to estimate their impact.
(12)Salvage/Subrogation
Need to be considered for proper evaluation of reserves on a GAAP basis. Optional on a
statutory basis.
(13)GAAP
Different from statutory accounting. For example, reserves are reduced by anticipated
salvage/subrogation under GAAP.
(14)Reinsurance
• Evaluate impact of changes in net retention.
• Analyze direct and ceded experience separately.
• Recoverability of ceded amounts is generally evaluated separately (now “part” of the
Actuarial Opinion).
(15)
Portfolio Transactions, Commutations, Structured Settlements
• These transactions generally recognize the time value of money.
• Their impact on loss reserves and development patterns should be evaluated.
(16)Pools and Associations
Consider the appropriateness of reserves reported by pools and associations.
(17)Operational Changes
• Examples include:
1. New computer system
2. Accounting change
3. Reorganization of claims department
• Reserve computation should reflect impact of these changes.
(18)Changes in Contracts
• Examples include:
1. Policy limits
2. Deductibles
3. Coverage attachment points
• These changes may alter the frequency/severity of claims.
(19)External Influences Examples include:
1. Judicial environment
2. Regulation
3. Legislative changes
4. Residual market
5. Economic variables

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Statement of Principles: Loss and LAE Reserves
CAS

(20)Discounting
Always perform reserve analysis on an undiscounted basis, then apply the effect of
discounting.
(21)Provision for Uncertainty
• When a reserve is carried at full-value it may include an implicit provision for uncertainty.
• A reserve carried at present-value may require an explicit provision for uncertainty.
• A reserve with a high degree of variability, even if carried at full-value, may require an
explicit provision for uncertainty.
(22)Reasonableness
• Incurred losses implied by the reserves should be measured for reasonableness against
relevant indicators such as premiums, exposures, frequency/severities, or number of
policies.
• Material departures from expected results should be explained.
(23)Loss-Related Balance Sheet Items
Examples include:
• Contingent commissions
• Retrospective premium adjustments
• Policyholder dividends
• Premium deficiency reserves
• Statutory reserves
• Provision for reinsurance
(24)Loss Reserving Methods
• The actuary has the responsibility for the selection of the most appropriate reserving
methods.
• Generally the actuary should examine the reserve indications of more than one method.
(25)Standards of Practice
This is technically the 25th consideration.

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Statement of Principles: Loss and LAE Reserves
CAS
Sample Questions:
1. Which of the following is a consideration in the estimation of loss and LAE reserves according to the CAS
Statement of Principles?
 Reasonableness
 Salvage and Subrogation
 Claim settlement patterns
 Risk margins for loss and LAE reserves
1. 
2. 
3. , 
4. , , 
5. Neither 1,2,3 or 4
2. Which of the following is/are true?
 Subdividing loss experience into more homogeneous categories tends to increase the credibility of
the loss experience.
 The date a claim is recorded in the insurance company's financial systems is one of the five key
dates mentioned in the Statement of Principles regarding Loss and LAE reserves.
 Evaluating reserves for high frequency/low severity lines of insurance generally requires more
detailed and extensive analysis than the evaluation of reserves for low frequency/high severity
lines of insurance.
1. 
2. 
3. , 
4. , , 
5. Neither 1,2,3 or 4
3. Which of the following is not a consideration cited in the "Statement of Principles?"
 The impact of reinsurance plans and retentions
 The impact of commutations and structured settlements
 The impact of higher rates of economic inflation
1. 
2. 
3. , 
4. , , 
5. Neither 1,2,or 3
4. Which of the following are principles of reserving identified by the CAS?
 A range of reserves
 An actuarially sound loss reserve
 Provision for uncertainty
1. 
2. 
3. , 
4. , , 

5. Neither 1,2,3 or 4

5. What are the five key dates identified in the Statement of Principles for the organization of a reserving
database?
6. Which of the following is a key criteria to recognize when accumulating data for actuarial analysis
according to Wiser?
 Policy limit
 Claim frequency
 Claim severity
 Loss development pattern
1. 
2. 
3. , 
4. , , , 5. Neither 1,2,3 or 4

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Statement of Principles: Loss and LAE Reserves
CAS
Questions from the 1994 Exam:
4. True/False: According to the CAS Statement of Principles, data used in the analysis of reserves must
reconcile to the insurer’s financial records.

Questions from the 1998 Exam:
61. (1 point) Based on the CAS “Statement of Principles Regarding Property and Casualty Loss and Loss
Adjustment Expense Reserves,” the most appropriate reserve within a range of actuarially sound
estimates depends on two considerations. List these two considerations.

Questions from the 1999 Exam:
9. True/False. According to the CAS “Statement of Principles Regarding Property and Casualty Loss and
Loss Adjustment Expense Reserves,” if reserves in a data triangle have been established at present
values, the development history should be restated to remove the effect of discounting.

Questions from the 2000 Exam:
7. True/False. According to the CAS “Statement of Principles Regarding Property and Casualty Loss and
Loss Adjustment Expense Reserves,” the provision for claims in transit is an element of the IBNR reserve.
8. True/False. According to the CAS “Statement of Principles Regarding Property and Casualty Loss and
Loss Adjustment Expense Reserves,” a valuation date must be coincident with or subsequent to the
accounting date.

Questions from the 2001 Exam:
4. According to CAS “Statement of Principles Regarding Property and Casualty Loss and Loss Adjustment
Expense Reserves,” the five elements of a total loss reserve should be individually quantified.

Questions from the 2002 and 2003 Exams:
There were no questions drawn from this article appearing on the above referenced exams

Questions from the 2004 Exam:
28. (2 points) What are the four principles of loss and LAE reserving contained in "Statement of Principles
Regarding Property and Casualty Loss & Loss Adjustment Expense Reserves"?

Questions from the 2005 Exam:
There were no questions drawn from this article appearing on the above referenced exam.

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Statement of Principles: Loss and LAE Reserves
CAS
Questions from the 2006 Exam:
22. (1.5 points) Although paid losses are an objective measure of past losses, the projection of future
payment patterns from past ones may be subject to distortions from a number of sources.
Identify and briefly describe three such sources of distortion.

Questions from the 2007 Exam:
There were no questions drawn from the content within this article appearing on the above referenced exam.

Questions from the 2008 Exam:
1. (1 point) A company's reserving actuary observes that one segment for a particular line of business has
much higher severity and a longer-tailed settlement pattern than the remaining segments. Exposures in the
high-severity, longer-tailed segment are growing faster than in the other segments.
a. (0.5 point) Explain how the guidance provided by the Statement of Principles Regarding Property and
Casualty Loss and Loss Adjustment Expense Reserves applies to this situation.
b. (0.5 point) Describe a potential bias that could result if the actuary analyzes these segments on a
combined basis.

Questions from the 2009 Exam:
There were no questions drawn from the content within this article appearing on the above referenced exam.

Questions from the 2010 Exam:
6. (2 points) According to the Casualty Actuarial Society's "Statement of Principles Regarding Property and
Casualty Loss and Loss Adjustment Expense Reserves":
a. (1 point) Identify four broad categories of operational changes within an insurance company that could
affect an unpaid claim estimate.
b. (1 point) Provide a specific example for each broad category of operational changes identified in part a.
above.

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Statement of Principles: Loss and LAE Reserves
CAS
Solutions to sample questions:
1. All are considerations
1. 
2. 

3. , 

4. , , 

5. Neither 1,2,3 or 4

2. Which of the following is/are true?
 Subdividing loss experience into more homogeneous categories tends to increase the credibility of
the loss experience.
True.
 The date a claim is recorded in the insurance company's financial systems is one of the five key
dates mentioned in the Statement of Principles regarding Loss and LAE reserves.
True.
 Evaluating reserves for high frequency/low severity lines of insurance generally requires more
detailed and extensive analysis than the evaluation of reserves for low frequency/high severity
lines of insurance.
False, just the opposite.
1. 
2. 
3. , 
4. , , 
5. Neither 1,2,3 or 4
3. Which of the following is not a consideration cited in the "Statement of Principles"
 The impact of reinsurance plans and retentions
 The impact of commutations and structured settlements
 The impact of higher rates of economic inflation
All are considerations
1. 
2. 
3. , 
4. , , 
5. Neither 1,2,or 3
4. Which of the following are principles of reserving identified by the CAS?
 A range of reserves
This is a principle of reserving
 An actuarially sound loss reserve
This is a principle of reserving
 Provision for uncertainty
This is not a principle of reserving, it is a consideration.
1. 
2. 
3. , 
4. , , 

5. Neither 1,2,3 or 4

5. What are the five key dates identified in the Statement of Principles for the organization of a reserving
database?
1.
2.
3.
4.
5.

Exam 5, V2

Accident Date
Report Date
Recorded Date
Accounting Date
Valuation Date

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Statement of Principles: Loss and LAE Reserves
CAS
Solutions to sample questions: (continued)
6. Which of the following is a key criteria to recognize when accumulating data for actuarial analysis
according to Wiser?
To answer this question, without reading Wiser, remember that Wiser basically rehashes the entire CAS
statement of principles. In particular, he describes the "considerations" relevant to data used for actuarial
analysis.
 Policy limit
This is a CAS consideration
 Claim frequency
This is a CAS consideration
 Claim severity
This is a CAS consideration
 Loss development pattern
This is also a CAS consideration
1. 
2. 
3. , 
4. , , , 5. Neither 1,2,3 or 4
Similar to question on 1993, Part 7, exam

Solutions to questions from the 1994 Exam:
4. True/False: According to the CAS Statement of Principles, data used in the analysis of reserves must
reconcile to the insurer’s financial records.
True, recall that this is also required in the Statement of Actuarial Opinion.

Solutions to questions from the 1998 Exam:
61. (1 point) Based on the CAS “Statement of Principles Regarding Property and Casualty Loss and Loss
Adjustment Expense Reserves,” the most appropriate reserve within a range of actuarially sound
estimates depends on two considerations. List these two considerations.
The most appropriate reserve within a range of actuarially sound estimates depends on both the:
• Relative likelihood of estimates within the range, and
• Financial reporting context in which the reserve will be presented.

Solutions to questions from the 1999 Exam:
9. True, since the “unwinding” of discount as claims move closer to their ultimate settlement date produces
the appearance of loss development in a development triangle.

Solutions to questions from the 2000 Exam:
7. True/False. According to the CAS “Statement of Principles Regarding Property and Casualty Loss and
Loss Adjustment Expense Reserves,” the provision for claims in transit is an element of the IBNR reserve.
True. See page 57.
8. True/False. According to the CAS “Statement of Principles Regarding Property and Casualty Loss and
Loss Adjustment Expense Reserves,” a valuation date must be coincident with or subsequent to the
accounting date. False. A valuation date may be prior to, coincident with or subsequent to the
accounting date. See page 57.

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Statement of Principles: Loss and LAE Reserves
CAS
Solutions to questions from the 2001 Exam:
4. According to CAS “Statement of Principles Regarding Property and Casualty Loss and Loss Adjustment
Expense Reserves,” the five elements of a total loss reserve should be individually quantified. False.
Although a total loss reserve is composed on five elements, the five elements may not necessarily be
individually quantified. See page 57.

Solutions to questions from the 2002 and 2003 Exams:
There were no questions drawn from this article appearing on the above referenced Exams

Solutions to questions from the 2004 Exam:
28. (2 points) What are the four principles of loss and LAE reserving contained in "Statement of Principles
Regarding Property and Casualty Loss & Loss Adjustment Expense Reserves"?
Question 28 – Model Solution 1:
1. An actuarially sound loss reserves, for a given group of claims as of a valuation date, is a provision, based
on reasonable assumptions and appropriate actuarial techniques, for the unpaid amount to settle all
claims, whether reported or not, for which liability exists as of an accounting date.
2. An actuarially sound loss adjustment expenses reserves, for a given group of claims as of a valuation
date, is a provision, based on reasonable assumptions and appropriate actuarial techniques, for the
unpaid amount to investigate, influence or adjust the loss amount of all claims, whether reported or not, for
which liability exists as of an accounting date.
3. The inherent uncertainty of the reserving process implies that a range or reserves may be actuarially
sound.
4. The most appropriate actuarially sound reserve amount within a range depends on:
a. The relative likelihood of the reserves in the range
b. The financial context the reserves is to be presented.
Question 28 – Model Solution 2:
1. An actuarially sound loss reserve for a defined group of claims as of a given valuation date, is a provision
base on reasonable assumptions and appropriate actuarial methods for the unpaid amount required to
settle all claims, whether reported or not for which a liability exists on a particular accounting date.
2. An actuarially sound LAE reserve, for a defined group of claims on a given valuation date, is a provision
based on reasonable assumption and appropriate actuarial methods, for the unpaid amount needed to
investigate, defend, and effect the settlement of all claims, whether reported or not for which liability exists
on a particular accounting date.
3. The uncertainty inherent in the loss reserving process implies that a range of estimates can be actuarially
sound. The ultimate value the amounts unpaid cannot be known until all attendant claims are settled.
4. The most appropriate loss reserve from a range of reasonable estimates will depend on the relative
likelihood of the estimates within the range and the financial reporting context in which the reserves will be
presented.

Solutions to questions from the 2005 Exam:
There were no questions drawn from this article appearing on the above referenced exam.

Exam 5, V2

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Statement of Principles: Loss and LAE Reserves
CAS
Solutions to questions from the 2006 Exam:
22. (1.5 points) Although paid losses are an objective measure of past losses, the projection of future
payment patterns from past ones may be subject to distortions from a number of sources. Identify and
briefly describe three such sources of distortion.
Question 22 – Model Answer 1
Model Answer #1
1. The claims department may have had a backup of claims after a catastrophe or major event. One would
not want to project this pattern into the future
2. A new type of claim may emerge that was not present in the past (for example, asbestos). You would not
want to use older year settlement patterns in the future
3. A large loss could impact one year’s losses. This one loss should not be used to project into the future.
Question 22 – Model Answer 2
1. Legal environment changes. Juries may start awarding larger payments to the claimant.
2. Claims department philosophy changes. Larger claims may be settled first, slowed down or sped up.
3. Changes in exposures. If the type of exposures underlying the losses are different than historical
Additional comments:
Review the concepts of Data Homogeneity, Frequency and Severity, Operational changes and External
influences as they apply to claims department practices, shifts in types of business/exposures being
underwritten, frequency and severity in types of claims that arise and the impact of the judicial environment.
Also review the Bouska paper on PEBLEs, and the section on “When Triangles Fail”.

Solutions to questions from the 2007 Exam:
There were no questions drawn from the content within this article appearing on the above referenced exam.

Solutions to questions from the 2008 Exam:
1. A company's reserving actuary observes that one segment for a particular line of business has much
higher severity and a longer-tailed settlement pattern than the remaining segments. Exposures in the highseverity, longer-tailed segment are growing faster than in the other segments.
a. (0.5 point) Explain how the guidance provided by the Statement of Principles Regarding Property and
Casualty Loss and Loss Adjustment Expense Reserves applies to this situation.
b. (0.5 point) Describe a potential bias that could result if the actuary analyzes these segments on a
combined basis.
Question 1 – Model Answer 1
a. The statement of principles says that both severity and development patterns can be used to identify
homogeneous lines of business. Thus, this high-severity, long-tailed line can be analyzed separately
from other lines.
b. If the actuary analyzes these segments together, the analysis implicitly assumes growth is the same
across all lines. This will lead to inadequate IBNR for the longer-tailed line. There is a bias to
understate IBNR.

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Statement of Principles: Loss and LAE Reserves
CAS
Question 1 – Model Answer 2
a. The statement offers guidance on when to analyze business segments separately vs. when to combine
them. In this case the higher severity, longer tailed line should be analyzed separately as long as there
is enough data in the two groups (long & short tailed) to have credible analyses.
b. Because the longer-tailed line is growing faster, it should have more weight in future loss payment
patterns than it had in past ones. A bias could result in the analysis that assumes the past development
patterns are predictive of future patterns. This would be a downward bias.

Solutions to questions from the 2010 Exam:
Question 6 - Solution 1
a) Change in underwriting guidelines;
Change in reinsurance structure;
Change in claims handling philosophy;
Change in information technology.
b)

Write more large risks;
Change reinsurance limit and attachment point;
Change in settlement rate;
Change in accounting process because of new technology implementation.

Solution 2
a) Underwriting;
Marketing;
Reserving;
Information technology.
b)

May loose underwriting standards causing a deteriorating loss ratio;
May emphasize a specific product causing a change in the mix of business;
May decide to strengthen case reserves;
May implement a new database causing temporary lags while employees get used to the new system.

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ALL 10, Inc.
Comprehensive Study Materials and Internet-Based Training

Actuarial Notes for the
Spring 2014 CAS
Exam 5A and Exam 5B
5A - Basic Techniques for Ratemaking
5B – Estimating Claim Liabilities

Volume 3
Independently Authored and Modified Past CAS Tests
Multiple Choice Questions
and
Independently Authored Preparatory Tests
Computational and Essay Based Questions

Exam 5 – Independently Authored and Modified Past CAS Questions
T/F and Multiple Choice Questions - Preparatory Test 1
General information about this exam






This practice test contains 20 questions consisting of true/false and multiple choice questions.
This practice test contains past CAS questions that have been modified (or completely re-written),
because the content of past CAS questions asked are no longer applicable to the content covered by the
Werner/Modlin text.
This practice test should be taken after working all past CAS questions associated with the articles shown
below, to demonstrate your understanding of the content covered in the chapters/articles listed below.
After answering the multiple choice questions in this test, candidates should consider how such questions can
be re-written in essay based format, since essay based questions will constitute approximately 25-30% of all
questions appearing on the exam.

Articles covered on this exam:
Article .................................................... Author .................................. Syllabus Section
A. Chapter 4: Exposures .....................................................Modlin, Werner .......... A. Basic Techniques for Ratemaking
A. Chapter 5: Premium ........................................................Modlin, Werner ........... A. Basic Techniques for Ratemaking

A. Chapter 8: Overall Indication ....................................Modlin, Werner ........... A. Basic Techniques for Ratemaking
A. Statement of Principles Re PC Ins Ratemaking.... CAS ................................... A. Basic Techniques for Ratemaking

A. Actuarial Standard No. 13 – Trending Proc. ........ CAS .................................... A. Basic Techniques for Ratemaking
A. Chapter 9: Traditional Risk Classification.............. Modlin, Werner ........... A. Basic Techniques for Ratemaking

A. Chapter 12: Credibility ...................................................Modlin, Werner ........... A. Basic Techniques for Ratemaking
A. Chapter 13: Other Considerations ............................ Modlin, Werner ........... A. Basic Techniques for Ratemaking
A. Chapter 14: Implementation .......................................Modlin, Werner ........... A. Basic Techniques for Ratemaking

A. Chapter 15: Commercial Lines Rating Mech ......... Modlin, Werner ........... A. Basic Techniques for Ratemaking

Exam 5A – Modified Past CAS Questions - Test 1
Question 1
Based on the "Actuarial Standard of Practice No. 13, Trending Procedures in Property/Casualty Insurance
Ratemaking," which of the following are examples of biases or distortions which should be considered when
examining historical insurance data for trend?
1. The impact of school vacations on automobile miles driven.
2. An automatic insurance to value program at policy renewal.
3. The introduction of higher policy limits.
1

1, 2

1, 2, 3

Question 2
(1 point) According to Werner and Modlin in “Basic Ratemaking”, which of the following statements regarding
the pure premium ratemaking method are true?
1. The pure premium method would be preferable to the loss ratio method for developing rates for a new
Homeowners endorsement covering sewer backup.
2. The pure premium method requires the calculation of on-level factors.
3. The pure premium method produces indicated rate changes.
1

1, 2

1, 3

1, 2, 3

Question 3
(1 point) According to Werner and Modlin in “Basic Ratemaking”, which of the following are true?
1. Most states have statutes that require that rates shall not be inadequate, excessive or unfairly
discriminatory between risks of like kind and quality.
2. Some states’ statutes may require certain rates to be “actuarially sound.”
3. The description of the goal of the ratemaking process does not consider generating a reasonablereturn on funds provided by investors.
1

1, 3

1, 2

1, 2, 3

Question 4
(1 point) According to Werner and Modlin in “Basic Ratemaking”, actuarial criteria are used to achieve which
of the following goals when establishing a classification system?
1. Causality
2. Homogeneity
3. Affordability
1

1, 2

1, 3

Exam 5A – Modified Past CAS Questions - Test 1
Question 5
According to Werner and Modlin in “Basic Ratemaking”, which of the following are true?
1. Based on the criterion that the exposure base should be the factor most directly proportional to the
expected loss, number of house years is the preferred exposure base, and amount of insurance
should be used as a rating variable.
2. For products liability, the exposure base that is intuitively the most proportional to expected loss is the
number of products currently in use, and is the exposure base currently used.
3. Workers compensation has historically used hours worked as an exposure base.
1 only

3 only

1, 2 only

2, 3 only

1, 2, 3

Question 6
According to Werner and Modlin in “Basic Ratemaking”, which of the following are true?
1. If there is a more accurate or practical exposure base than the one currently in use, the actuary
should take steps to implement it.
2. Amount of Insurance Coverage is the typical exposure base for homeowners insurance.
3. In composite rating, the premium is initially calculated using estimates for each exposure measure
along with relevant rating algorithms for each coverage.
3

1, 2

1, 3

2, 3

1, 2, 3

Question 7
According to Werner and Modlin in “Basic Ratemaking”, which of the following should exist for an exposure
base to be practical.
1. It should be objective
2. It should be relatively easy to use and
3. It should be inexpensive to obtain and verify.
3 only

1 only

1, 3 only

2, 3 only

None of the given answer choices

Question 8
(1 point) According to the Statement of Principles Regarding Property and Casualty Insurance Ratemaking,
which of the following are true?
1. Informed actuarial judgment should not be used in ratemaking, unless there is a lack of credible data.
2. Consideration should be given in ratemaking to the effects of subrogation and salvage.
3. A rate is an estimate of the expected value of present costs.
1

1, 3

2, 3

1, 2, 3

Question 9
(1 point) According to the CAS Committee on Ratemaking Principles, "Statement of Principles Regarding
Property and Casualty Insurance Ratemaking," which of the following are NOT stated principles?
1. A rate provides for all costs associated with the transfer of risk.
2. A rate is an estimate of the expected value of future costs.
3. A rate provides for the costs associated with an individual risk transfer.
1
1, 2
1, 3
2, 3
None of the given answer choices
Copyright  2014 by All 10, Inc.
Page 3

Exam 5A – Modified Past CAS Questions - Test 1
Question 10
(1 point) According to the "Statement of Principles Regarding Property and Casualty Insurance Ratemaking",
which of the following are true?
1. The cost of reinsurance should be considered in the ratemaking process
2. Changes in the underwriting process should be considered in the ratemaking process.
3. Affordability is specifically stated as an important factor that should be considered in the ratemaking
process.
1 only

1, 2 only

3 only

2, 3 only

1, 2, 3

Question 11
According to the Statement of Principles Regarding Property and Casualty Insurance Ratemaking, which of
the following are false?
1. Credibility is increased either by making groups more homogeneous or by decreasing the size of the
group analyzed.
2. When considering trends, consideration should only be given to past changes in claims costs, claim
frequencies, exposures expenses and premiums.
3. When an individual risk's experience is sufficiently credible, the premium for that risk should be
modified to reflect the individual experience.
1
3
1, 2
1, 3
1, 2, 3
Question 12
The CAS Statement of Principles on Ratemaking describes a number of considerations that commonly apply
to any ratemaking methodology. In its discussion of "Risk", the Statement distinguishes between (i) the
charge for the risk of random variation from expected costs and (ii) the charge for any systematic variation of
the estimated costs from expected costs.
Which of the following statements apply to the charge for any systematic variation of the estimated costs from
expected costs?
1. It should influence the underwriting profit provision.
2. It should be reflected in the determination of the total return.
3. It should be reflected in the contingency provision.
1 only

3 only

1, 3

2, 3

None of the given answer choices

Question 13
According to Werner and Modlin in “Basic Ratemaking”, which are examples that can cause changes in the
average premium level?
1. A rating characteristic can cause average premium to change.
2. Moving all existing insureds to a higher deductible
3. Acquiring the entire portfolio of another insurer writing higher policy limits.
2

1, 3

2, 3

1, 2, 3

Exam 5A – Modified Past CAS Questions - Test 1
Question 14
According to the "Actuarial Standard of Practice No. 13, Trending Procedures in Property/Casualty Insurance
Ratemaking," which of the following are true?
1. It is inappropriate to analyze only factors which have an impact on trend in one direction.
2. When selecting a trending procedure, the actuary should first look to the Proceedings or the Syllabus
of Examinations of the CAS before considering alternate procedures described in other publications.
3. Any trending procedure requires the actuary to exercise informed judgment.
1

1, 2

1, 3

1, 2, 3

Question 15
According to Werner and Modlin in “Basic Ratemaking”, which of the following are true?
1. The description of the goal of the ratemaking process includes consideration of generating a reasonablereturn on funds provided by investors.
2. Regulatory review generally requires that rates shall not be inadequate, excessive or unfairly
discriminatory between risks of like kind and quality.
3. The two basic approaches used in manual ratemaking are the pure premium method and the loss ratio
method.
A. 1.

B. 2

C. 1, 3

D. 2, 3

E. 1, 2, 3

Question 16
According to Werner and Modlin in “Basic Ratemaking”, which of the following are true?
1. In-force exposures are the total exposures arising from policies issued during a specified time period.
2. Written exposures are the number of units exposed to loss at a given point in time.
3. Annual payroll in hundreds of dollars is the typical exposure unit for U.S. workers compensation insurance
A. 1.

B. 3

C. 1, 3

D. 2, 3

E. 1, 2, 3

Question 17
According to Werner and Modlin in “Basic Ratemaking”, the uncollectability of deductible payments is not an
additional risk associated with deductible policies.
True
False

Question 18
According to Werner and Modlin in “Basic Ratemaking”, which of the following statements are true?
1. The off-balance exists because the indicated classification relativities produce an average
classification relativity different from the average classification relativity underlying the current rates.
2. If projected and indicated premiums are not in balance, balance may be achieved through expense
reductions.
3. If projected and indicated premiums are not in balance, a company can also achieve balance by
reducing the average expected loss.
1 only

2 only

1 and 3

1, 2, 3

None of the given answer choices

Exam 5A – Modified Past CAS Questions - Test 1
Question 19
According to Werner and Modlin in “Basic Ratemaking”, which of the following are true?
1. Regulators may prohibit the use of a characteristic for rating even if it can be demonstrated to be
statistically strong predictors of risk.
2. Regulators may limit the amount of an insurer’s rate change to either the overall average rate change
for the jurisdiction or to the change in premium for any individual or group of customers, or both, even
if the actuary can justify all methods used in his or her ratemaking procedures.
3. In the case of banned or restricted usage of a variable (e.g. insurance credit scores), an insurer can
use a different allowable rating variable (e.g. payment history with the company) it believes can
explain some or all of the effect associated with the restricted variable.
1 only

2 only

3 only

2, 3 only

1, 2, 3

Question 20
According to Werner and Modlin in “Basic Ratemaking”, which of the following statements are true of the
characteristics of the different methods for determining the complement of credibility?
1. Because Harwayne's method uses data from the same class in other states and attempts to adjust for
state-to-state differences, the complement is unbiased.
2. In the Rate Change From the Larger Group Applied to Present Rates method, although the
complement is a significant improvement over the Bayesian complement, the complement remains
largely biased.
3. In the Trended Present Rates method, the complement is more accurate for loss costs with high
process variance because the process variance is reflected in last year's rates.
1

1,3

1,2

Exam 5A – Solutions to Modified Past CAS Questions - Test 1
Question 1 discussion:
Answer: 1, 2, 3
See section 3.2
The actuary should consider the effect of known biases or distortions on the data relied upon (e.g. the
impact of catastrophic influences, seasonality, coverage changes, nonrecurring events, claim practices, and
distributional changes in deductibles, types of risks, and policy limits).
Question 2 discussion:
1. True. See chapter 8
2. False. The loss ratio method requires the calculation of on-level factors. See chapter 8
3. False. The loss ratio method produces indicated rate changes. See chapter 8
Answer: 1
Question 3 discussion: 1. True. See chapter 9
2. True. See chapter 9
3. False. See chapter 1
Answer: 3
Question 4 discussion: Blooms:
1. False. This is one of the social criteria. See chapter 9
2. True. See chapter 9
3. False. This is one of the social criteria. See chapter 9
Answer: 2
Question 5 discussion: Blooms:
1. True. See chapter 4.
2. False. Gross Sales. See chapter 4.
3. False. Payroll. See chapter 4.
Answer: 1 only
Question 6 discussion: Blooms: 1. False. The actuary should consider historical preference before
implementing it. See chapter 4.
2. False. Earned House Years
3. True.
Answer: 3
Question 7 discussion: Blooms: 1. True. See chapter 4.
2. True. See chapter 4.
3. True. See chapter 4.
Answer: None of the given answer choices

Exam 5A – Solutions to Modified Past CAS Questions - Test 1
Question 8 discussion: Blooms:
1. False.
2. True.
3. False. A rate is an estimate of the expected value of future costs.
Answer: 2
Question 9 discussion: Blooms:
1. True. This is a stated principle.
2. True. This is a stated principle.
3. True. This is a stated principle.
Answer: None of the given answer choices
Question 10 discussion:
1. True. See reinsurance as a consideration.
2. True. See operation changes as a consideration.
3. False.
Answer: 1, 2 only
Question 11 discussion:
1. False. Credibility is increased either by making groups more homogeneous or by increasing the size of
the group analyzed.
2. False. When considering trends, consideration should be given to past and prospective changes in
claims costs, claim frequencies, exposures expenses and premiums.
3. True.
Answer: 1, 2
Question 12 discussion:
"The rate should also include a charge for any systematic variation of the estimated costs from the
expected costs. This charge should be reflected in the determination of the contingency provision."
Answer: 3 only
Note: With respect to the (i) charge for the risk of random variation from the expected costs, this risk charge
should be reflected in the determination of the appropriate total return consistent with the cost of capital
and, therefore, influences the underwriting profit provision.
Question 13 discussion:
1. True. See chapter 5.
2. True. See chapter 5.
3. True. See chapter 5.
Answer: 1, 2, 3

Exam 5A – Solutions to Modified Past CAS Questions - Test 1
Question 14 discussion:
1. True. See section 5.6
2. False. There is no requirement to look to the Proceedings or to the Syllabus of Examinations of the CAS
first. See page 2
3. True. See section 5.8
Answer: 1, 3
Question 15 discussion:
Answer: 1, 2, 3. Statement 3 = See chapter 8.
Question 16 discussion:
1. F. Written exposures are the total exposures arising from policies issued during a specified time period.
2. F. In-force exposures are the number of units exposed to loss at a given point in time.
3. T. Annual payroll in hundreds of dollars is the typical exposure unit for U.S. workers compensation
insurance
Answer: 3 See Chapter 4.

Question 17 discussion:
False. See chapter 15
Deductible processing:
i. When the insurer is responsible for paying the entire claim and seeks reimbursement for amounts below
the deductible from the insured, the premium should reflect the cost of invoicing and monitoring deductible
activity as well as a provision for the risk that the insured may become bankrupt and be unable to pay for any
future deductible invoices (i.e. credit risk).
ii. Even if collateral is received to cover potentially uncollectible deductible amounts, it is rare that this credit
risk is fully collateralized.
Question 18 discussion:
1. True. See chapter 14
2. True. See chapter 14
3. True. See chapter 14
Answer: 1, 2, 3 only

Question 19 discussion:
1. True. See chapter 13.
2. True. See chapter 13.
3. True. See chapter 13.
Answer: 1, 2, 3
Question 20 discussion:
1. True. See chapter 12
2. False. ...the complement is largely unbiased.
3. False. The complement is less accurate for loss costs with high process variance.
Answer: 1
Copyright  2014 by All 10, Inc.
Page 9

Exam 5 – Modified Past CAS Questions – T/F and Multiple Choice
Preparatory Test 2
General information about this exam






This practice test contains 21 questions consisting of true/false and multiple choice questions.
This practice test contains past CAS questions that have been modified (or completely re-written),
because the content of past CAS questions asked are no longer applicable to the content covered by the
Werner/Modlin text.
This practice test should be taken after working all past CAS questions associated with the articles shown
below, to demonstrate your understanding of the content covered in the chapters/articles listed below.
After answering the multiple choice questions in this test, candidates should consider how such questions can
be re-written in essay based format, since essay based questions will constitute approximately 25-30% of all
questions appearing on the exam.

Articles covered on this exam:
Article .................................................... Author .................................. Syllabus Section
A. Chapter 15: Commercial Lines Rating Mech ......... Modlin, Werner ........... A. Basic Techniques for Ratemaking

A. Chapter 16: Claims Made Ratemaking ..................... Modlin, Werner ........... A. Basic Techniques for Ratemaking
Personal Auto Premiums: Asset Share Pricing .......... Feldblum ........................ A. Basic Techniques for Ratemaking

Exam 5A– Modified Past CAS Questions - Test 2
Question 1:
(1 point) According to Feldblum, "Personal Automobile Premium: An Asset Share Pricing Approach For PropertyCasualty Insurance," which of the following are evidence that property-casualty insurance is taking on attributes
that motivate asset share pricing?
1. Insurers rarely cancel or non-renew policies.
2. Expected loss costs are greater for renewal business than for new business.
3. A greater emphasis is being placed on the investment income component of rates.
1
2
1, 3
2, 3
1, 2, 3

Question 2
According to Feldblum, "Personal Automobile Premiums: An Asset Share Pricing Approach for Property-Casualty
Insurance," which of the following are true?
1. It is preferable to review persistency rates by original driver classification rather than by the current driver
classification.
2. Analysis of persistency rates is important, but not a key part of asset share pricing models.
3. Agency ownership of policy renewals affects persistency rates.
1
2
1, 2
1, 3
1, 2, 3

Question 3
According to Feldblum in "Personal Automobile Premiums: An Asset Share Pricing Approach for PropertyCasualty Insurance," the fundamental issue in asset share pricing methods is the predictability of long term
profitability.
True
False

Question 4
According to Werner and Modlin in “Basic Ratemaking”, which of the following is not one of the principles of
claims-made (C-M) ratemaking?
A. C-M policies have less risk of case reserve inadequacies than do occurrence policies.
B. Substantially less investment income is earned on C-M policies than under occurrence policies.
C. Sudden unexpected shifts in the reporting pattern will have less of an impact on the cost of mature C-M
coverage than on the cost of occurrence coverage.
D. A C-M policy should always cost less than an occurrence policy as long as pure premiums are increasing.
E. Whenever there is a sudden, unpredictable increase or decrease in the underlying trend, C-M policies priced
on the basis of the prior trend will be closer to the correct price than occurrence policies priced the same way.

Exam 5A– Modified Past CAS Questions - Test 2
Question 5
According to Werner and Modlin in “Basic Ratemaking”, which of the following statements are true about claimsmade coverage
1. The long period between the occurrence of a claim and the settlement of a claim can be driven by a reporting
lag, a settlement lag, or both.
2. From a loss development perspective, reporting lag relates to IBNER (claims that are incurred but not enough
reported), and settlement lag relates to pure IBNR (claims that are incurred but not reported).
3. The major difference between claims-made and occurrence coverage is that the coverage trigger is the date
the claim is reported rather than the date the event occurs.
1
3
1, 2
1, 3
1, 2, 3
Question 6
According to Werner and Modlin in “Basic Ratemaking”, which of the following are true of experience rating
plans?
1. The experience rating adjustment for the future policy period manual premium is equal to a credibility
weighting of the adjusted past experience and some expected results.
2. The experience period usually ranges from two to five policy years, ending with the last complete year.
3. Many experience rating plans apply per occurrence caps on the losses in order to exclude unusual or
catastrophic losses.
1
3
1, 2
1, 3
1, 2, 3

Question 7
According to Werner and Modlin in “Basic Ratemaking”, which of the following are true?
1. In the NCCI ER plan, primary losses are capped at $10,000 and the excess losses are calculated as the
portion of each individual loss above $10,000.
2. The D-ratio is the loss elimination ratio at the primary loss limit
3. In the general liability experience rating plan, the maximum single loss (MSL) is applied to loss and
allocated loss adjustment expense combined.
2
3
1, 3
2, 3
1, 2, 3
Question 8
According to Werner and Modlin in “Basic Ratemaking”, the basic premium in the NCCI retrospective rating plan
provides for which of the following costs?
1. An allowance for profit and contingencies
2. Premium taxes
3. The cost of limiting the retrospective premium to be between the minimum and maximum premium
negotiated under the policy.
1
2
1, 2
1, 3
1, 2, 3

Exam 5A– Modified Past CAS Questions - Test 2
Question 9
According to Werner and Modlin in “Basic Ratemaking”, which of the following are true?
1. A retrospective rating plan uses the insured’s actual experience during the policy period as the basis for
determining the premium for that same period.
2. Conceptually, retrospectively rated insurance is similar to self-insurance with the exception that
retrospectively rated insurance policies contain provisions that cause the insurer to retain some risk and
that affect the timing of payments for costs incurred under the policy.
3. The total premium charged may be subject to a minimum and maximum amount to help stabilize the yearto-year cost
2
3
1, 3
2, 3
1, 2, 3
Question 10
According to Werner and Modlin in “Basic Ratemaking”, which of the following statements are true regarding
claims-made ratemaking?
1. The investment income earned under claims-made policies is substantially less than the investment
income earned under occurrence policies.
2. An occurrence policy will generally cost less than a claims-made policy.
3. Claims-made policies incur some liability for IBNR claims.
1
2
3
1, 2
1, 3
Question 11
A claim occurred in May 2001 and was reported in September 2003. Which of the following would cover this
claim?
1. A one-year occurrence policy effective January 1, 2003
2. A second-year claims-made policy effective January 1, 2002
3. Tail coverage effective January 1, 2003 for a physician retiring after 10 years of practice covered by
claims-made coverage
2
3
1
2, 3
1, 2
Question 12
According to Feldblum, "Personal Automobile Premiums: An Asset Share Pricing Approach for Property Casualty
Insurance,' which of the following are true?
1. The principal benefit to asset share pricing is the determination of profitability over the entire time a
policyholder stays with the company.
2. The asset share pricing model is inappropriate to use for high risk drivers, such as young males, because
they do not tend to remain with one company long enough to permit completion of a long term analysis.
3. Level premiums and losses in property and casualty insurance is a reason why property and casualty
actuaries have relied on asset share pricing.
1
3
1, 3
2, 3
1, 2, 3

Exam 5A– Modified Past CAS Questions - Test 2
Question 13
According to Feldblum, “Personal Automobile Premiums: An Asset Share Pricing Approach for Property-Casualty
Insurance,” which of the following is false?
A. Life insurance policy claim rates are more certain than property-casualty policy claim rates.
B. It is inappropriate to assume the same pattern of persistency ratios for both direct writers and independent
agency companies.
C. A level commission structure is inappropriate for the persisting and profitable risks.
D. The dominant market share of the direct writers makes asset share pricing a more appropriate model for
Personal auto insurance.
E. Asset share pricing determines rate revisions, not rates.
Question 14
According to Werner and Modlin in “Basic Ratemaking”, which of the following are true?
1. The longer the settlement lag, the greater will be the difference in investment income between claims-made
and occurrence policies.
2. A claims-made policy should always cost less than or equal to an occurrence policy.
3. The confidence interval about the projected losses for an occurrence policy is generally wider than for a
claims-made policy priced at the same time.
1
2
3
1, 2
2, 3
Question 15
According to Werner and Modlin in “Basic Ratemaking”, which of the following are true regarding how primary and
excess credibility factors are expressed in NCCI’s formula?
1. The primary credibility factor is a function of the ballast value (B).
2. The excess credibility factor is a function of both (B) and (w).
3. The ballast value and weighting value are obtained from a table based upon the policy’s expected losses and
both increase as expected losses increase.
1
2
3
1, 2
2, 3
Question 16:
According to Feldblum, "Personal Automobile Premiums: An Asset Share Pricing Approach for Property-Casualty
Insurance, " which of the following are true?
1. In practice, persistency rates depend upon the premium discount that is offered.
2. The exposure to road hazards is higher for older drivers than it is for younger drivers.
3. Older drivers, and in particular retired drivers, have more time on their hands to compare prices and thus,
have more of an impetus to price shop at renewal time.
1 only
1 and 2 only
2 and 3 only
1, 2, and 3 None of the given answer choices
Question 17
According to Feldblum, "Personal Automobile Premiums: An Asset Share Pricing Approach for Property-Casualty
Insurance, " which of the following are true?
1. Although there is an intuitive relationship between duration and persistency for life insurance, this is not
the case for casualty insurance.
2. If persistency differences are ignored in traditional ratemaking methods, then rate relativities are too high
for the poorly persisting classes and too low for the long-persisting classes.
3. Asset share pricing helps the actuary determine the true profitability of the insurance writings.
1 only
3 only
1 and 3 only
1, 2, and 3
None of the given answer choices
Copyright  2014 by All 10, Inc.
Page 14

Exam 5A– Modified Past CAS Questions - Test 2
Question 18
According to Werner and Modlin in “Basic Ratemaking”, which of the following are true?
1. A tail policy covers all losses with accident dates and report dates occurring after the insured's last
claims-made policy expired.
2. The term "lag", as used by the authors, is the difference between the date the accident occurred and the
date the accident was reported.
3. The coverage trigger for a CM policy is the accident date.
1
2
3
1, 2
None of the given answer choices
Question 19
According to Werner and Modlin in “Basic Ratemaking”, which of the following are true?
1. Claims-made rates are both more accurate and more responsive to changing conditions.
2. The major difference between the claims-made and the occurrence policy lies not in the coverage
provided, but in the timing of pricing decisions affecting that coverage.
3. A mature claims made policy written at the beginning of a year contributes one exposure to all matrix
elements within a report year column of a report year by lag matrix.
1, 2
2, 3
1, 2, 3
1, 3
None of the given answer choices
Question 20
According to Werner and Modlin in “Basic Ratemaking”, which of the following are true?
1. Rating techniques for large commercial risks: large deductible plans, loss-rated composite rating, and
retrospective rating plans
2. If the premium collected under experience rating plans does not equal the expected premium in total,
then the plan has an "off-balance."
3. Experience rating is used when an individual insured’s past experience, with adjustments, can be
predictive of the future experience.
1 only
2 only
1, 3
1, 2
1, 2, and 3

Question 21
According to Werner and Modlin in “Basic Ratemaking”, which of the following are true?
1. In ISO's CGL Experience and Schedule Rating Plan, the maximum single limit per occurrence applies to
total limits losses and unlimited ALAE.
2. Schedule rating is the only individual risk rating system that does not directly reflect an entity’s claim
experience.
3. A unique characteristic of the NCCI experience rating plan is that it divides losses for each claim into a
primary portion and an excess portion.
1 only
3 only
1, 3
1, 2
2 and 3

Exam 5A– Solutions to Modified Past CAS Questions - Test 2
Question 1 discussion:
Answers to this question are found on page 195, Proceedings, November 1996.
1. True. Cancellations. Insurers rarely cancel or non-renew policies, since profitability depends on the stability of
the book of business.
2. False. Expected loss costs are greater for new business than for renewal business. See page 195.
3. False. This is not mentioned.
Answer: 1
Feldblum lists 3 attributes about P&C insurance that motivate the use of asset share pricing:
1. Commissions. Commission rates tend to be higher in the 1st year than in renewal years.
2. Cancellations. Insurers rarely cancel or non-renew policies, since profitability depends on the stability of the
book of business.
3. Loss costs. Feldblum states that this phenomenon is valid for personal auto insurance as well as for other lines
of business.
Question 2 discussion
1. True. See page 239. Although persistency rates by duration are easily determined for current classifications
(the % of young male drivers in their 5th policy year who persist into their 6th year), it is persistency rates by
original classification, not current classification, that it is needed.
Notice the difference: the persistency of young male drivers in their 5th policy year does not tell us the
expected 5th year persistency of young male drivers.
2. False. Feldblum states "Persistency rates (retention rates) are the crux of asset share pricing models. They are
most important when the net insurance income varies by duration since policy inception." See page 207.
3. True. On page 208, Feldblum compares persistency rates among direct writers and independent agency
companies:
Answer: 1, 3
Question 3 discussion
False Persistency rates (the term "retention rates" are used interchangeably in this paper) are the crux (and hence
the fundamental issue) of asset share pricing models. See page 207.
Question 4 discussion
Answer: A. See chapter 6.
Question 5 discussion
1. True. See chapter 16
2. False. From a loss development perspective, reporting lag relates to pure IBNR (claims that are incurred but not
reported), and settlement lag relates to IBNER (claims that are incurred but not enough reported).
2. True. See chapter 16
Question 6 discussion
1. True. See chapter 16
2. True. See chapter 16
3. True. See chapter 16
Answer: 1, 2, 3

Exam 5A– Solutions to Modified Past CAS Questions - Test 2
Question 7discussion
1. False. In the NCCI ER plan, primary losses are capped at $5,000 and the excess losses are calculated as the
portion of each individual loss above $5,000.
2. True.
3. True. See chapter 15.
Answer: 2, 3
Question 8 discussion
According to Modlin, the following elements are included in the basic premium.
1. Profit and contingency allowance.
2. Expenses, excluding expenses provided for by the LCF.
3. Net charge for limiting the retro premium between the minimum and the maximum.
Thus, 1 is True, 2 is False, and 3 is True
Answer: 1, 3
Question 9 discussion
1. True. See chapter 15.
2. True. See chapter 15.
3. True. See chapter 15.
Answer: 1, 2, 3
Question 10 discussion
1. The investment income earned under claims-made policies is substantially less than the investment income
earned under occurrence policies. True. This is principle number 5.
2. An occurrence policy will generally cost less than a claims-made policy. False. This is a misstatement of
principle number 1. A claims-made policy should always cost less than an occurrence policy, as lone as claim
costs are increasing.
3. False. Claims-made policies incur NO liability for IBNR claims. This is principle number 4. Claims-made
policies incur no liability for IBNR claims so the risk of reserve inadequacy is greatly reduced.
Question 11 discussion
1. A one-year occurrence policy effective January 1, 2003. False. Occurrence policies cover claims occurring
during the policy period. An accident occurring on 5/1/2001 would not be covered by a policy covering the
period 1/1/2003 – 12/31/2003
2. A second-year claims-made policy effective January 1, 2002. False. Since the claim was reported in 2003, it
would not be covered by a claims-made policy effective in 2002.
3. Tail coverage effective January 1, 2003 for a physician retiring after 10 years of practice covered by claimsmade coverage. True. A claims made policy covers claims reported (made) (in this example, 9/1/2003) during
the policy period (i.e. 1/1/2003 – 12/31/2003), regardless of when the accident date occurred. See chapter 16.
Answer: 3 only

Exam 5A– Solutions to Modified Past CAS Questions - Test 2
Question 12 discussion
1. T. It examines the profitability from inception to termination (including renewals) of the policy. See page 192
2. F. Feldblum demonstrates how asset share pricing is used to determine class relativities for young drivers. See
page 217.
3. F. P&C premiums are variable. Level premiums associated with whole life policies have lead life actuaries to
place greater reliance on asset-share pricing models than P&C actuaries (which work with premiums that
fluctuate widely). See page 197.
Question 13 discussion
A. Life insurance policy claim rates are more certain than property-casualty policy claim rates. True. Claim rates in
casualty insurance are more variable and less well understood. See page198.
B. It is inappropriate to assume the same pattern of persistency ratios for both direct writers and independent
agency companies. True. Direct writers, like Sate Farm, have high retention rates because they offer low
premium rates and provide renewal discounts. Many independent agency companies have low retention rates
because they can move the insured to whichever company offers the lowest rate. See page 208.
C. A level commission structure is inappropriate for the persisting and profitable risks. True. A level commission
structure works wells for risks that terminate quickly. It works poorly for risks that endure with the carrier. See
page 206.
D. The dominant market share of the direct writers makes asset share pricing a more appropriate model for
personal automobile insurance. True. In the personal lines of business, direct writers are steadily gaining
market share, See page 206.
E. False. Asset share pricing determines rates, not rate revisions. See page 215.
Question 14 discussion
The answer to each of these questions can be found by reviewing the 5 principles of claims made ratemaking.
1. F. The longer the reporting lag or the shorter the settlement lag, the greater the difference will be. See
chapter 16.
2. F. A CM policy should always cost less than an occurrence policy, as long as claim costs are rising. See
chapter 16.
3. T. See chapter 16.
Question 15 discussion
1. True. See chapter 15.
2. True. See chapter 15.
3. True. See chapter 15.
Answer: 1, 2, 3
Question 16 discussion:
1. True. See pages 250 -- 251.
2. False. The exposure to road hazard declines as drivers age. See page 243.
3. False. Older drivers, with lower premiums and often with less information about competing carriers, have less
incentive and less opportunity to price shop. See page 244.
Answer: 1 only

Exam 5A– Solutions to Modified Past CAS Questions - Test 2
Question 17 discussion:
1. False. There is an intuitive relationship between duration and persistency for both life and casualty insurance.
See page 209.
2. False. If persistency differences are ignored in traditional ratemaking methods, then rate relativities are too low
for the poorly persisting classes and too high for the long-persisting classes. See page 217.
3. True. See page 217.
Answer: 3 only
Question 18 discussion
1. False. A tail policy covers all losses whose accident date lies in the period during which the claims-made
coverage was in force, and whose reported date is after the insured's last claims-made policy expired.
See chapter 16.
2. False. The term "lag", as used by the authors, is the difference between the year accident occurred and the
year the accident was reported. See chapter 16
3. False. The coverage trigger for a CM policy is the report date.
Answer: None of the given answer choices
Question 19 discussion
1 True. See chapter 16.
2. True. See chapter 16.
3. True. See chapter 16.
Answer: 1, 2, 3

Question 20 discussion
1. True. See chapter 15.
2. True. See chapter 15.
3. True. See chapter 15.
Answer: 1, 2, 3

Question 21 discussion
1. False. The maximum single limit per occurrence applies to basic limits losses and unlimited ALAE. See
chapter 15.
2. True. See chapter 15.
3. True. See chapter 15.
Answer: 2 and 3 only

Exam 5 – Modified Past CAS Questions – T/F and Multiple Choice
Preparatory Test 3
General information about this exam






Articles covered on this exam:
Article .................................................... Author .................................. Syllabus Section
A. Chapter 5: Premium ........................................................Modlin, Werner ........... A. Basic Techniques for Ratemaking

A. Chapter 6: Losses and LAE..........................................Modlin, Werner ........... A. Basic Techniques for Ratemaking

A. Actuarial Standard No. 13 – Trending Proc. ........ CAS .................................... A. Basic Techniques for Ratemaking
A. Chapter 11: Special Classification .............................. Modlin, Werner .......... A. Basic Techniques for Ratemaking
A. Chapter 15: Commercial Lines Rating Mech ......... Modlin, Werner ........... A. Basic Techniques for Ratemaking
Personal Auto Premiums: Asset Share Pricing .......... Feldblum ........................ A. Basic Techniques for Ratemaking

ISO Personal Auto Manual ..................................................... ISO ................................... A. Basic Techniques for Ratemaking

Exam 5A – Modified Past CAS Questions - Test 3
Question 1:
According to "Actuarial Standard of Practice No. 13, Trending Procedures in Property/Casualty Insurance
Ratemaking," in the absence of strong contrary indications, the actuary should rely on extrapolations of the
historical insurance data from mathematical models.
True

False

Question 2
According to Werner and Modlin in “Basic Ratemaking”, which of the following are true?
1. The loss ratio method produces indicated rate changes whereas the pure premium method produces
indicated rates.
2. The pure premium method and the loss ratio method will produce identical results when consistently
applied to the same data.
3. The extension of exposures technique is a part of the pure premium method.
1

1, 2

2, 3

1, 2, 3

Question 3
According to Werner and Modlin in “Basic Ratemaking”, which of the following are true?
1. The goal of the ratemaking process includes consideration of generating a reasonable-return on funds
provided by investors.
2. Rate regulation generally requires that rates shall not be inadequate, excessive or unfairly discriminatory
between risks of like kind and quality.
3. The two basic approaches used in manual ratemaking are experience rating and schedule rating.
A. 1.

B. 2

C. 1, 2

D. 2, 3

E. 1, 2, 3

Question 4
1. An insurer writes the following policies during 1992:
Effective
Date
May 1
August 1
November 1

Policy
Term
6 months
12 months
6 months

Premium
$6,000
$12,000
$2,400

What is the insurer's unearned premium reserve on December 31, 1992?
A. <$6,000

B. >$6,000 but <$7,000

C. >$7,000 but <$8,000

D.> $8,000, but < $9,000 E. > $9,000.

Question 5
According to the "Statement of Principles Regarding Property and Casualty Insurance Ratemaking," which
of the following are true?
1. Historical premium, exposure, loss and expense experience is usually the starting point of ratemaking.
2. Policy year is the best acceptable method of organizing data to be used in ratemaking.
3. Marketing, underwriting, legal and other business considerations should NOT be a factor when applying
the principles set forth in the above statement.
1

1, 2
Copyright  2014 by All 10, Inc.
Page 21

None of the given answer choices

Exam 5A – Modified Past CAS Questions - Test 3
Question 6. You are given:
Effective Date
4/1/94
7/1/95
4/1/96

Rate Change
+5.0%
+13.0%
-3.0%

• All policies are 12 month policies.
• Policies are written uniformly throughout the year.
Using the parallelogram method described Werner and Modlin in “Basic Ratemaking”, in what range does the
on-level premium factor fall, to bring calendar year 1995 earned premium to current rate level?
A. < 1.07

B. > 1.07 but < 1.09

C. > 1.09 but < 1.11

D. > 1.11 but < 1.13

E. > 1.13

Question 7
According to Feldblum, "Personal Automobile Premiums: An Asset Share Pricing Approach for Property
Casualty Insurance,' which of the following are true?
1. The principal benefit to asset share pricing is the determination of profitability over the entire time a
policyholder stays with the company.
2. The asset share pricing model is appropriate to use for high risk drivers, such as young males, because
they do not tend to remain with one company long enough to permit completion of a long term analysis.
3. Level premiums and losses in property and casualty insurance is a reason why property and casualty
actuaries are now relying on asset share pricing.
3

1, 2

1, 3

2, 3

1, 2, 3

Question 8
According to Feldblum in "Personal Automobile Premiums: An Asset Share Pricing Approach for PropertyCasualty Insurance," asset share modeling is considered particularly valuable when differences in
termination rates influence expected profits.
True

False

Question 9
According to Werner and Modlin in “Basic Ratemaking”, which of the following are true?
1. The D-ratio adjusts for the impact of the MSL by reducing the expected basic limits losses and ALAE for
expected losses and ALAE higher than the MSL.
2. In the NCCI retrospective rating plan, the basic premium provides for a net charge for limiting the
retrospective premium between the minimum and the maximum retrospective premiums.
3. In the ISO experience rating plan, both the maximum single loss (MSL) and basic limits are applied to
losses and allocated loss adjustment expenses (ALAE).
3

1, 2

1, 3

2, 3

1, 2, 3

Exam 5A – Modified Past CAS Questions - Test 3
Question 10
According to Werner and Modlin in “Basic Ratemaking”, which of the following are reasons why deductibles
are popular among both insureds and insurers?
1. Premium reduction
2. Provides incentive for loss control
3. Controls catastrophic exposure, for insurers writing a large number of policies in cat prone areas.
3

1, 2

1, 3

2, 3

1, 2, 3

Question 11
10. A 12-month policy is written on March 1, 2002 for a premium of $900. As of December 31, 2002,
which of the following is true?

A.
B.
C.
D.
E.

Calendar Year
2002 Written
Premium

Calendar Year
2002 Earned
Premium

Inforce
Premium

$900
$750
$900
$750
$900

$900
$750
$750
$750
$750

$900
$900
$750
$750
$900

Question 12. You are given:
• Full estimated policy premium is booked at inception.
• Premium develops upward by 7% at final audit, six months after the policy expires.
• All policies are written for an annual period.
• Premium is written uniformly throughout the year.
Based on Werner and Modlin in “Basic Ratemaking”, in what range does the policy year premium
development factor fall for 24 to 36 months?
A. < 1.01

B. > 1.01 but < 1.02

C. > 1.02 but < 1.03

D. > 1.03 but < 1.04

E. > 1.04

Question 13. Given the information below, determine the written premium trend period.
• Experience period is April 1, 2001 to March 31, 2002
• Planned effective date is April 1, 2003
• Policies have a 6-month term
• Rates are reviewed every 18 months
• Historical premium is earned premium
A. < 1.8 years
D. ≥ 2.4 years, but < 2.7 years

B. ≥ 1.8 years, but < 2.1 years
E. ≥ 2.7 years

C. ≥ 2.1 years, but < 2.4 years

Exam 5A – Modified Past CAS Questions - Test 3
Question 14. Given the following data and using the loss development method as described by Werner and
Modlin in “Basic Ratemaking”, calculate the projected ultimate accident year 2001 losses.
Accident Year

As of December 31, 2002
Paid Losses

Case Reserves

1999
2000
2001
2002

$11,000
$6,000
$3,500
$1,000

$1,000
$2,000
$4,000
$4,000

•

Projected ultimate accident year 2000 losses = $9,240

•

12-24 case-incurred link ratio = 1.71

•

24-36 case-incurred link ratio = 1.20

A. < $8,700
B. ≥ $8,700, but < $9,200
D. ≥ $9,700, but < $10,200

C. ≥ $9,200, but < $9,700
E. ≥ $10,200

Question 15
According to Werner and Modlin in “Basic Ratemaking”, which of the following are true regarding
coinsurance?
1. A coinsurance penalty corrects for inequity caused by similar homes insured to different insurance to
value levels by adjusting the indemnity payment in the event of a loss.
2. Another way to achieve equity is to calculate and use rates based on the level of insurance to value.
3. A rate can be calculated given the expected frequency, the size of loss distribution, and the full value of
the property.
1

2, 3

1, 3

1, 2, 3

Question 16
According to Werner and Modlin in “Basic Ratemaking”,which of the following statements regarding
insurance to value is false?
A. Coinsurance can adjust the premium rate to the amount of insurance.
B. The pure premium rate, which equates pure premiums and expected indemnity, falls as the policy faces
increases, regardless of whether small or large losses predominate.
C. The possibility of losses less than the co-insurance requirement creates the pricing problem known as
"insurance to value."
D. If large losses outnumber small ones, pure premium rates should decrease at an increasing rate.
E. If losses less than the policy face are possible, the pure premium rate decreases as the policy face
increases.

Exam 5A – Modified Past CAS Questions - Test 3
Question 17
According to Werner and Modlin in “Basic Ratemaking”, which of the following statements are true?
1. A coinsurance penalty is the amount by which a coinsurance requirement exceeds the amount of the
carried insurance.
2. A coinsurance deficiency is the amount by which the indemnity payment resulting from a loss is reduced
due to the coinsurance clause.
3. Given an insured with a coinsurance deficiency, a loss need not occur to be a coinsurer.
1 and 2

1 and 3

1, 2, and 3

Question 18. Given the following data, calculate the trended loss ratio.
Number of
Insureds

Earned
Premium

Developed
Incurred
Losses

$50,000

$35,000

•
•
•
•
•

Years of Trend = 2.5
Annual Exposure Trend = 2.0%
Annual Premium Trend = 2.9%
Annual Frequency Trend = -1 .0%
Annual Severity Trend = 6.0%

A. < 68%

B. > 68% but < 71%

C. > 71 % but < 74%

D. > 74%, but < 77%

E. > 77%

Question 19. Based on Insurance Services Office, Inc., Personal Automobile Manual (Effective 6-98), which
of the following is false?
A. The Manual describes the types of vehicles eligible for coverage.
B. The Manual specifies which drivers must be categorized as "Youthful Operators".
C. The Manual sets forth rating factor adjustments for companies electing not to use the Safe Driver
Insurance Plan.
D. The Manual describes the primary and secondary classifications applicable.
E. The Manual specifies that all Liability and Physical Damage policies must have a policy period of no
longer than 12 months.
Question 20. According to Insurance Services Office, Inc., Personal Vehicle Manual (Edition 6-98), which of
the following are true?
1. Expense Fees are added separately to the premium for the Single Limit Liability or BI and PD Liability,
Comprehensive, Collision and No-Fault Coverages applying to each auto.
2. Expense Fees are not subject to modification by the provisions of any rating plans or other rating rules
(e.g. Classifications, Safe Driver Insurance Plan).
3. Expense Fees are subject to the Cancellation and Suspension provisions of this manual.
1

1, 2

1, 3

1, 2, 3

Exam 5A – Modified Past CAS Questions - Test 3
Question 21. According to Insurance Services Office, Inc., Personal Vehicle Manual (Edition 6-98), which
of the following are true with respect to classification changes?
1. A policy may not be changed mid-term because of the attained age of an operator of the auto.
2. A policy may not be changed mid-term to effect a change in the Driving Record Sub Classification.
3. A policy may not be changed mid-term due to a change in symbol assignment based on a review of loss
experience.
1

1, 2

1, 3

1, 2, 3

Exam 5A – Solutions to Modified Past CAS Questions - Test 3
Question 1 discussion:
True. See page 2, section 4.2 (Models)
Question 2 discussion:
1. True. See chapter 8
2. True. See chapter 8
3. False. See chapter 8.
Answer: 1, 2
Question 3 discussion:
1. True. See chapter 1.
2. True. See chapter 9.
3. False. The two basic approaches used in manual ratemaking are the pure premium method and the loss
ratio method. See chapter 8
Answer: 1, 2
Question 4 discussion:
The premium for the policy effective 5/1 is fully earned by 11/1/92. There is no unearned premium at 12/31/92.
5/12 ths of the premium for the policy effective 8/1 is earned by 12/31/92.
The unearned premium is = (7/12) * $12,000 = $7,000.
2/6 ths of the premium for the policy effective 11/1 is earned by 12/31/92.
The unearned premium is = (4/6) * $2,400 = $1,600.
Thus, the total unearned premium = $7,000 + 1,600 = 8,600.
Answer D. See Chapter 5
Question 5 discussion:
1. True.
2. False. There are several acceptable methods of organizing data including calendar year, accident year,
report year and policy year. Each presents certain advantages and disadvantages; but, if handled properly,
each may be used to produce rates.
3. False.
Answer: 1

Exam 5A – Solutions to Modified Past CAS Questions - Test 3
Question 6 discussion:
To facilitate the calculation of CY on-level factors, setup a diagram similar to the one below:

Calculate the numerator of the on-level factor. This is equal to (1.05)*(1.13)*(1-.03) = 1.150905.
Calculate the average rate level factor for the calendar year. This is a weighted average of the rate level
factors in the calendar year. The weights will be relative proportions of the square.
First calculate the area of all triangles (area = .50 * base * height) within a unit square and then determine
the remaining proportion of the square by subtracting the sum of the areas of the triangles from 1.0.
For CY 1995, the average rate level factor = (1/2)(3/12)(3/12)*1.0 + (1/2)(1/2)(1/2)*1.1865+ (1.0 - .15625)*1.05
= .03125 + .1483125 + .8859375 = 1.0655
The on-level factor = 1.150905 / 1.0655 = 1.0801549.

Answer B. See Chapter 5

Question 7 discussion:
1. True. It examines the profitability from inception to termination (including renewals) of the policy. See page
192
2. True. Feldblum demonstrates how asset share pricing is used to determine class relativities for young
drivers. See page 217.
3. False. Level premiums associated with whole life policies have lead life actuaries to place greater reliance
on asset-share pricing models than P&C actuaries (which work with premiums that fluctuate widely). See
page 197.
Answer: 1, 2
Question 8 discussion:
True. Termination rates more clearly distinguish persistency patterns by classification. Probabilities of
termination, in certain analyses, provide a better portrayal of the insurer's profitability. See pages 210 - 211.
Question 9 discussion:
1. True. See page 1.
2. True.
3. False. Limit paid losses to basic per occurrence limits (25,000) and then limit the latter, including unlimited
ALAE, by the MSL
Answer: 1, 2 See chapter 15
Question 10 discussion:
1. True. See Chapter 11
2. False. See Chapter 11
3. True. See Chapter 11
Answer: 1, 3

Exam 5A – Solutions to Modified Past CAS Questions - Test 3
Question 11 discussion:
A 12-month policy is written on March 1, 2002 for a premium of $900. As of December 31, 2002, which of the
following is true?
Step 1: Answering this question is best understood in terms of exposures
Written exposures are those units of exposures on policies written during the period in question,
Earned exposures are the exposure units actually exposed to loss during the period, and
Inforce exposures are those exposure units exposed to loss at a given point in time.….
Step 2: Based on the definitions in Step 1, only earned premium differs from written premium and inforce
premium and therefore needs to be computed.
Thus, earned premium at 12/31/02 equals $900 * 10/12 = $750.
Answer E. See Chapter 5
Question 12 discussion:
Question 12. Assume that policy year 199X premium is being booked at $P per month.
Developed premium, due to final audits, is not known until 6 months after the policy expires.
At 12/31/9X+1, developed premium for only those policies issued during the 1st 6 months of PY 199X is known.
At 12/31/9X+2, developed premium for all policies issued during PY 199X is known.
Reported Premium for polices issued during the
Evaluation
Date

1st 6 months of PY
199X

Last 6 months of PY
199X

Total PY
199X

12/31/9X
12/31/9X+1
12/31/9X+2

6 months * ($P/month)
6 * P * 1.07
6 * P * 1.07

6 months * ($P/month)
6*P
6 * P * 1.07

12P
12.42P
12.84P

Therefore, the PY premium development factor for 24 to 36 months is 12.84P/12.42P = 1.034
See Chapter 5

Answer D.

Question 13 discussion:
Step 1: Determine the average written date during the experience period. For the experience period 4/1/01 –
3/31/02, and given that 6 month policies are being written, the average earned date is 10/1/01 and the average
written date is 7/1/01, or ½ the policy term earlier from the average earned date.
Step 2: Determine the average written date during the exposure period. The average written date during the
future policy period is a function of the length of time that the rates are expected to remain in effect. In this
example, since rates are reviewed every 18 months, this would make the average written date 9 months after
the proposed effective date of 4/1/03, which is 1/1/04. Thus, the written premium trend period is 2.50 years.
Answer: D. ≥ 2.4 years, but < 2.7 years See Chapter 6

Question 14 discussion:
Step 1: Determine AY 2001 case incurred losses at 12/31/2002 projected to 36 months.
Case incurred losses at 12/31/2002 = $3500 + $4,000 = $7,500. Note that at 12/31/02, AY 2001 case
incurred losses are at 24 months of development. The loss development factor from 24-36 months is
given as 1.20. Thus, AY 2001 case incurred losses projected to 36 months equals $9,000.
Step 2: Determine AY 2001 case incurred losses at 12/31/2002 projected to ultimate.
AY 2000 36-48 months case incurred loss development factor is $9,240/$8,000 = 1.155. Thus, at
12/31/02, AY 2001 cased incurred losses are at ultimate equals $9,000 * 1.155 = $10,395.
Answer E. ≥ $10,200 See Chapter 6
Copyright  2014 by All 10, Inc.
Page 29

Exam 5A – Solutions to Modified Past CAS Questions - Test 3
Question 15 discussion:
1. True. See chapter 11.
2. True. See chapter 11.
3. True. See chapter 11.
Answer: 1, 2, 3

Question 16 discussion:
A. True. See chapter 11.
B. True. See chapter 11.
C. False. The possibility of losses less than the policy face creates the pricing problem known as "insurance to
value". See chapter 11.
D. True. See chapter 11.
E. True. See chapter 11.
Question 17 discussion:
1. False. A coinsurance deficiency is the amount by which a coinsurance requirement exceeds the amount of
the carried insurance. See chapter 11.
2. False. A coinsurance penalty is the amount by which the indemnity payment resulting from a loss is reduced
due to the coinsurance clause. See chapter 11.
3. True. See chapter 11.
Answer: 3
Question 18 discussion:
When working with inflation-sensitive exposure bases, incorporate the exposure trend into the estimate of the
expected future loss ratio. To maintain a valid loss ratio projection, the adjustments made to the numerator of the
loss ratio should be on a consistent basis with those made to the denominator. …The numerator of the loss ratio is
adjusted for frequency trend and severity trend, while the denominator is adjusted for average premium trend.”
Step 1: Based on the givens of the problem, write an equation to determine the trended loss ratio.

 Developed Incurred Losses   Freq Trend*Sev Trend 
Trended Loss Ratio = 
*

Earned Premium
Premium Trend

 


Years of Trend

Step 2: Using the equation in Step 1, and the data in the problem, solve for the trended loss ratio.
2.5
 $35,000   .99 * 1.06 
Trended Loss Ratio = 
 = .7352 Answer C: > 71 % but < 74%
*
 $50,000   1.029 

Exam 5A – Solutions to Modified Past CAS Questions - Test 3
Question 19 discussion
A. The Manual describes the types of vehicles eligible for coverage. True. See page G-1.
B. The Manual specifies which drivers must be categorized as "Youthful Operators". True. See section 4:
Classifications, page G-5.
C. The Manual sets forth rating factor adjustments for companies electing not to use the Safe Driver Insurance
Plan. True. See section 5: Safe Driver Insurance Plan, section 2 page G-8.
D. The Manual describes the primary and secondary classifications applicable. True. See section 4:
Classifications, page G-2.
E. The Manual specifies that all Liability and Physical Damage policies must have a policy period of no longer than
12 months. False. "No policy may be written for a period longer than 12 mos. for Liab. Coverage or 36 mos. for
Physical Damage."
Question 20 discussion
1. True. See page G-2.
2. True. See page G-2.
3. True. See page G-2.
Answer: 1, 2, 3
Question 21 discussion
1. True. See page G-3.
2. True. See page G-3.
3. True. See page G-3.
Answer: 1, 2, 3

Exam 5A – Independently Authored Questions - Preparatory Test 1
General information about this exam
This practice test contains 30 questions consisting of computational and essay based questions.

Total Number of Qs:
Total Number of Points:

Essay
Questions
18
24.5

Computational
Questions
12
35.5

Total
30
60

1. The recommend time for this exam is 3:30:00. Make sure you have sufficient time to take this practice test.
2. Consider taking this exam after working all past CAS questions first.
3. Make sure you have a sufficient number of blank sheets of paper to record your answers for
computational questions.

Chapter 2: Rating Manuals .................................................Modlin, Werner .......... A. Basic Techniques for Ratemaking

Chapter 3: Ratemaking Data..............................................Modlin, Werner .......... A. Basic Techniques for Ratemaking
Chapter 4: Exposures ...........................................................Modlin, Werner .......... A. Basic Techniques for Ratemaking
Chapter 5: Premium ..............................................................Modlin, Werner ........... A. Basic Techniques for Ratemaking
Statement of Principles Re PC Ins Ratemaking .......... CAS ................................... A. Basic Techniques for Ratemaking

Exam 5A – Independently Authored Questions - Test 1
Question 1 (1.25 points)
According to Werner and Modlin in “Basic Ratemaking”, answer the following questions.
a. (.25 points). State the basic economic relationship for the price of any product.
b. (1.0 point). Transform the equation in part a. into the fundamental insurance equation, and briefly describe
each component of the fundamental insurance equation using basic insurance terminology.

Question 2 (1.25 points)
a. (.25 points). Define the term ‘exposure’.
b. (1.0 point). Briefly describe four ways insurers measure exposures.

Question 3 (1.50 points)
a. (1.0 point). Briefly describe two reasons why reported losses may differ from ultimate losses.
b. (.50 points). Based on your response in part a, write an equation that relates reported losses to ultimate
losses.

Question 4 (1.50 points)
a. (.50 points). Briefly describe the goal of ratemaking as it relates to the fundamental insurance equation.
b. (1.0 point). List and briefly describe two key points to consider in achieving balance in the fundamental
insurance equation.

Question 5 (5.25 points)
You are given the following information about the ABC insurance company
Number of Reported Claims
Number of Earned Exposures
Total Reported Losses
Total Reported LAE
Total Earned Premium
Total Written Premium
Commissions and brokerage
Other acquisition costs
General expenses
Taxes, licenses, and fees
Number of Potential Renewal Policies
Number of Policies Renewed
Number of Quotes
Number of Accepted Quotes

2,000
40,000
6,000,000
1,200,000
8,000,000
8,400,000
840,000
420,000
640,000
336,000
2,000
1,700
6,000
1,200

a. (0.75 points). Compute ABC’s frequency, severity and pure premium
b. (1.0 point). Briefly describe what can be identified when analyzing changes in claim frequencies and claim
severities.
c. (1.0 point). Compute ABC’s average premium, loss ratio, LAE ratio and underwriting expense ratio.
d. (1.0 point). Compute ABC’s operating ratio and combined ratio
e. (1.5 points). Compute ABC’s Retention ratio and Close ratio and briefly describe why computing such ratios
are important.
Copyright  2014 by All 10, Inc.
Page 2

Exam 5A – Independently Authored Questions - Test 1
Question 6 (1.0 point)
According to Werner and Modlin in “Basic Ratemaking”, list four elements that are necessary to calculate the
premium for a given risk: for most lines of business.

Question 7 (1.0 point)
According to Werner and Modlin in “Basic Ratemaking”, Rating algorithms describes how to combine the
components in the rules and rate pages to calculate the premium charged for any risk not pre-printed in a rate
table. List four possible types of instructions included within a rating algorithm

Question 8 (1.0 point)
According to Werner and Modlin in “Basic Ratemaking”, while underwriting criteria has been historically
subjective in nature, there has been a trend over time (especially for personal lines products) to designate new
explanatory variables as underwriting criteria, which can then be used for placement into rating tiers or separate
companies.
Briefly describe three underwriting characteristics that are currently used for three different types of insurance.

Exam 5A – Independently Authored Questions - Test 1
Question 9 (5.25 points)
You are given the following information from ABC insurance company’ rating manual:
Protection Class and Construction Type
Construction Type
Protection
Class
Frame
Masonry
1-4
1.00
0.90
5
1.05
1.00
6
1.10
1.05
7
1.15
1.10
8
1.25
1.15
9
2.10
1.75
10
2.30
1.90
Underwriting Tier
Tier
Rate Relativity
A
0.80
B
0.95
C
1.00
1.45
D

Territory
Territory
1
2
3
4
5

Rate Relativity
0.80
0.90
1.00
1.10
1.15

Deductible
Deductible
$250
$500
$1,000
$5,000

Rate Relativity
1.00
0.95
0.85
0.70

Amount of Ins (AOI) Rating Table

Miscellaneous Credits
Miscellaneous Credit
Credit Amount
New Home Discount
20%
5-Year Claims-Free Discount
10%
Multi-Policy Discount
7%
Add'l Optional Coverages
Jewelry Coverage Rate
Limit
Additive
$2,500
Included
$5,000
$35
$10,000
$60

Expense Fee
Policy Fee
$50

AOI (in 000s)

Rate Relativity

$80
$95
:::
$170
$185
$200
$215

0.56
0.63
:::
0.91
0.96
1.00
1.04

Liability/Medical
Limit
$100,000/$500
$300,000/$1,000
$500,000/$2,500

Additive
Included
$25
$45

ABC is preparing a renewal quote for a homeowner with the following risk characteristics:
• Amount of insurance = $185,000. Base rate = $750.
• The insured lives in Territory 2.
• The home is frame construction located in Fire Protection Class 7.
• Based on the insured’s credit score, tenure with the company, and loss history, the policy is in UW Tier C.
• The insured opts for a $1,000 deductible.
• The home falls under the definition of a new home as defined in ABC’s rating rules.
• The insured is eligible for the five-year claims-free discount.
• There is no corresponding auto or excess liability policy written with ABC.
• The policyholder opts to increase coverage for jewelry to $5,000 and to increase liability/medical
coverage limits to $300,000/$1,000.
The rating algorithm calls for rating variables to be applied in a multiplicative manner, except for the following
which are to be applied in an additive manner: Increased Jewelry Coverage; Increased Liability/Medical
Coverage; Policy Fee
Calculate the final premium for the policy.

Exam 5A – Independently Authored Questions - Test 1
Question 10 (3.0 points) You are given the following information from a retirement center and ABC insurance
company’ rating manual. A retirement living center with the following employee classes groups has requested a
quote.
Class

Rate per $100 of
Payroll
(from rating manual)
(from insured)
0.59
$40,000

Payroll

8810 – Clerical
8825 - Food Service Employees
8824 - Health Care Employees
8826 - All Other Employees & Salespersons, Drivers

$85,000
$100,000
$30,000

2.88
4.00
3.75

The following underwriter determined schedule credits apply:
 The center has trained its entire staff in first aid and first aid equipment is available in the building: -2.5%
 The center has been inspected by ABC and the premises are clean and well-maintained: -10%
 The center follows careful procedures in selecting, training, and supervising its workers: - 5%
Other factors that apply to the policy from ABC’s Rating Manual are as follows:
Entries from Rating Manual
Pre-Employment drug screening test

Expense Constant

$250

The minimum premium for the policy of $1,500.
The rating algorithm calls for rating variables to be applied in a multiplicative manner, except for the
following expense constant.
Compute total premium for the policy.
Question 11 (1.0 point) According to Werner and Modlin in “Basic Ratemaking”, list and provide examples
of two types of internal data involved in a ratemaking analysis
Question 12 (3.0 points)
You are given the following information about three homeowner’s policies written by the ABC insurance
 Policy A is written on 1/1/2012 with an annual premium of $1,300. The home is located in Territory 1 and
the insured has a $250 deductible. The policy remains unchanged for the full term of the policy.
 Policy B is written on 4/1/2012 with an annual premium of $800. The home is located in Territory 2 and
the insured has a deductible of $250. The policy is canceled on 12/31/2012.
 Policy C is written on 7/1/2012 with an annual premium of $1,500. The home is located in Territory 3
and has a deductible of $500. On 1/1/2013, the insured decreases the deductible to $250. The full
annual term premium after the deductible change is $1,800.
Using the policy data above, complete ABC’s policy database entries. Determine whether one or multiple
records for each policy are needed when constructing the database.
Original
Original
Transaction
Effective Termination Effective
Policy
Date
Date
Date

Ded

Written
Written
Terr Exposure Premium

Exam 5A – Independently Authored Questions - Test 1
Question 13 (2.0 points)
According to Werner and Modlin in “Basic Ratemaking”, list the four type of data aggregation methods and
briefly describe the advantages and disadvantages of their use.

Question 14 (1.5 points)
According to Werner and Modlin in “Basic Ratemaking”, list three types of third party data, not specific to
insurance, and briefly describe how they are used for insurance/ratemaking purposes.

Question 15 (3.0 points)
According to Werner and Modlin in “Basic Ratemaking”, answer the following questions.
a. (.75 points) Define the term exposure. List three criteria that a good exposure base should meet.
b. (2.25 points) With respect to homeowners insurance, should number of house years or amount of insurance
be the exposure base. Briefly explain the rationale behind your choice for exposure base.

Question 16 (1.5 points) According to Werner and Modlin in “Basic Ratemaking” a well-defined and
objective exposure should not be able to be manipulated by policyholders and producers/underwriters.
While the use of estimated annual miles driven as an exposure base for auto insurance has been cited as
an opportunity for insureds to be dishonest, presenting an a moral hazard for insurers , briefly explain why it
may not and given an example supporting your position.

Question 17 (1.5 points) Based on Werner and Modlin in “Basic Ratemaking”, use the homeowners policy data
below to answer the following questions

Policy
A
B
C
D
E
F

Effective Expiration
Date
Date
Exposure
10/1/2012 9/30/2013
10.00
1/1/2013 12/31/2013
10.00
4/1/2013 3/31/2014
10.00
7/1/2013 6/30/2014
10.00
10/1/2013 9/30/2014
10.00
1/1/2014 12/31/2014
10.00

a. (.75 points). Compute the number of CY 2012, CY 2013 and CY 2014 written exposures.
b. (.75 points). Compute the number of written exposures policy D will contribute to CY 2013 and CY 2014 if
policy D is cancelled 3/31/2014.
c. (.75 points). Compute the number of written exposures policy D will contribute to PY 2013 and PY 2014 if
policy D is cancelled 3/31/2014.
d. (.75 points). Compute the number of CY 2012, CY 2013 and CY 2014 earned exposures.
e. (.75 points). Assuming the above policies were written for 6 month terms (as opposed to annual terms),
compute the number of CY 2012, CY 2013 and CY 2014 earned exposures.

Exam 5A – Independently Authored Questions - Test 1
Question 18 (1.5 points) Based on Werner and Modlin in “Basic Ratemaking”, use the information below to
answer the following questions.
An insurer begins writing annual policies in 2012 and writes 480 exposures each month during 2012 only.
The insurer is using the “15th of the month” rule to compute In-force exposures
a. (.75 points). Compute the aggregate In-force exposures as of 7/01/2012, 1/01/2013 and 7/01/2013
b. (.75 points). Compute the aggregate earned exposures for CY 2012 and CY 2013
Question 19 (1.5 points) According to Werner and Modlin in “Basic Ratemaking” list and briefly describe three
adjustments to historical premium to produce projected future premium.
Question 20 (4.0 points) According to Werner and Modlin in “Basic Ratemaking” answer the following question
based on the information given below. Assume ABC issues annual policies and premium is calculated
according to the rating algorithm: Premium = Exposure x Rate per Exposure x Class Factor + Policy Fee.
Rate Change History
Rate
Level
Group
1
2
3
4

Effective
Date
Initial
07/01/12
01/01/13
04/01/14

Overall
Average
Rate change
-5.0%
10.0%
-1.0%

Rate
Per
Exposure
$1,800
$1,900
$2,090
$2,090

X
1.00
1.00
1.00
1.00

Class Factor
Y
0.75
0.75
0.75
0.80

Z
1.15
1.15
1.15
1.15

Policy
Fee
$1,000
$1,000
$1,100
$1,090

Assume ABC issued one policy effective on 3/1/2013 that had 10 class Y exposures.
Compute the actual premium that was charged using the extension of exposures method and the PY 2014
premium at current rate level using the extension of exposures method.
Question 21 (3.0 points) According to Werner and Modlin in “Basic Ratemaking” answer the following questions
based on the given data below.
Annual policies have been issued and rate changes apply to policies effective on or after the date
Rate Level
Group
1
2
3
4

Effective
Date
Initial
07/01/10
01/01/11
04/01/12

Overall
Average Rate
5.0%
10.0%
-1.0%

a. (1.0 point). Compute the on-level factor applied to the CY 2011 EP to bring it to current rate level.
b. (1.0 point). Compute the on-level factor applied to the PY 2012 EP to bring it to current rate level.
c. (1.0 point). Assuming six month policies were issued, compute the on-level factor applied to the CY
2011 EP to bring it to current rate level.
d. (1.0 point). Assume a law change mandates a rate decrease of 5% on 7/1/2011 applicable to all
policies, compute the on-level factor applied to the CY 2011 EP to bring it to current rate level.
Question 22 (1.0 point) According to Werner and Modlin in “Basic Ratemaking”, list and briefly describe two
problems with the parallelogram method.

Exam 5A – Independently Authored Questions - Test 1
Question 23 (1.5 points) According to Werner and Modlin in “Basic Ratemaking”, answer the following question.
 A WC carrier writes one policy per month in 2013.
 Estimated premium for each policy is booked at policy inception for $750,000.
 Premium develops upward by 8% at the first audit (6 months after the policy expires).
Compute the premium development factor from 12/31/2014 (24 months after the start of the PY) to 12/31/2015
(36 months after the start of the PY)

Question 24 (1.0 point) According to Werner and Modlin in “Basic Ratemaking”, list and briefly describe three
examples that can cause changes in an insurer’s average premium level.
Question 25 (1.5 points) According to Werner and Modlin in “Basic Ratemaking”, answer the following question.
Assume the following:
 CY 2011 EP is being used to estimate the rate need for annual policies that are to be in effect
from 1/1/2013 – 12/31/2013.
 WP is used as the basis of the trend selection and EP for the overall rate level indications
 The actuary selects a trend factor of 2%, the amount average premium is expected to change annually.
Compute the one step trend factor.

Question 26 (1.5 points) According to Werner and Modlin in “Basic Ratemaking”, list and briefly describe three
examples that can affect the length of the trend period. Provide graphical representations of the adjustments if
needed.

Question 27 (1.0 point) According to Werner and Modlin in “Basic Ratemaking”, list and briefly describe two
examples when a one-step trending process is not appropriate to use.

Question 28 (3.0 points) According to Werner and Modlin in “Basic Ratemaking”, and given the information
below, determine the written premium trend period using the one-step trending procedure.
• Experience period is April 1, 2001 to March 31, 2002
• Planned effective date is April 1, 2003
• Policies have a 6-month term
• Rates are reviewed every 18 months
• Historical premium is earned premium

Exam 5A – Independently Authored Questions - Test 1
Question 29 (3.0 points) Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and
given the information below, determine the projected premium at current rate level using the two-step trending
procedure.
Use the latest average written premium at current rate level, the historical average EP at current rate level, and
average written date during the period the proposed rates are to be in effect.
• Experience period is 1/1/2013 to 12/31/2013
• Planned effective date is 1/1/2015
• Policies have a 12-month term
• Rates are reviewed every 12 months
th
• Latest average written premium at current rate level for the 4 quarter 2013 is 953.00
• Average earned premium for CY 2013 is 940.00
• CY 2013 Earned Premium at Current Rate Level is $1,880,788
• CY 2013 Earned Exposures is 2,150
• Selected Projected Premium Trend is 2.0%

Question 30 (1.0 point) The CAS Statement of Principles on ratemaking describes a number of
considerations that commonly apply to any ratemaking methodology. Under the heading of Other
Influences, the Statement lists five external influences which might have an impact on future experience.
List four of these five external influences.

Exam 5A – Solutions to Independently Authored Questions - Test 1
Question 1 discussion: Blooms: Knowledge; Difficulty 1, LOKS: Describe the information
requirements for ratemaking related to premiums and demonstrate the use of premiums in
ratemaking
a. The basic economic relationship for the price of any product is Price = Cost + Profit.
b. Premium is the “price” of the insurance product.
“Cost” is the sum of the losses, LAE, and UW expenses.
UW profit is income minus the outgo from issuing policies (and Profit is also derived from II)
The fundamental insurance equation is Premium = Losses + LAE + UW Expenses + UW Profit. See chapter 1
Question 2 discussion: Blooms: Comprehension; Difficulty 1, LOKS: Describe the information
requirements for ratemaking related to exposures and demonstrate the use of exposures in
ratemaking.
a. An exposure is a unit of risk that underlies the premium.
b. Four ways insurers measure exposures are as follows:
1. Written exposures are the total exposures arising from policies issued during a specified time period
(e.g. a calendar year or quarter).
2. Earned exposures are the portion of written exposures for which coverage has already been provided
(as of a certain point in time).
3. Unearned exposures are the portion of written exposures for which coverage has not yet been provided
(as of that point in time).
4. In-force exposures are the number of units exposed to loss at a given point in time.
See chapter 1
Question 3 discussion: Blooms: Comprehension; Difficulty 1, LOKS: Describe the information
requirements for ratemaking related to loss and loss adjustment expenses and demonstrate the use
of loss and loss adjustment expenses in ratemaking.
a1. When there are unreported claims, the estimated amount to settle these claims is known as incurred
but not reported (IBNR) reserve.
a2. The incurred but not enough reported (IBNER) reserve (a.k.a. development on known claims) is the
difference between the aggregate reported losses at the time the losses are evaluated and the
aggregate amount estimated to ultimately settle these reported claims.
b. Ultimate Losses = Reported Losses + IBNR Reserve + IBNER Reserve.
See chapter 1
Question 4 discussion: Blooms: Comprehension; Difficulty 1, LOKS Describe the information
requirements for ratemaking related to premiums and demonstrate the use of premiums in
ratemaking
a. The goal of ratemaking is to assure that the fundamental insurance equation is balanced (e.g. rates
should be set so premium is expected to cover all costs and achieve the target UW profit).
Two key points in achieving balance in the fundamental equation are:
b1. Ratemaking is prospective, and this involves estimating the components of the fundamental
insurance equation to determine whether or not the estimated premium is likely to achieve the target
profit during the period the rates will be in effect.
b2. Balance should be attained at the aggregate level (otherwise rates will either be redundant or
inadequate and individual levels) and at the individual level (otherwise failure to recognize differences
in risk will lead to rates that are not equitable, which violates principle 3 of the CAS Statement of
Ratemaking Principles).
See chapter 1

Exam 5A – Solutions to Independently Authored Questions - Test 1
Question 5 discussion: Blooms: Application. Difficulty 3. LO: Calculate the underwriting expense
provisions underlying the overall rate level indication.
a1.

Number of Claims
is 5% (= 2,000 / 40,000).
Number of Exposures

Frequency =

a2. Severity =

Total Losses
is $3,000 (= $6,000,000 / 2,000).
Number of Claims

a3. Pure Premium =

Total Losses
= Freq x Sev is $150 (= $6,000,000 / 40,0000) = 5.0% x $3,000.
No. of Exposures

b. Analyzing changes in claims frequency can help identify, industry trends associated with the incidence of
claims, utilization of insurance coverage, and the effectiveness of specific underwriting actions.
Analyzing changes in severity:
 provides information about loss trends and
 highlights the impact of any changes in claims handling procedures.

Average Premium =
Loss Ratio =

Total Premium
is $200 (=$8,000,000 / 40,000).
No. of Exposures

Total Losses
Pure Premium
is 75% (= $6,000,000 / $8,000,000).
=
Total Premium Average Premium

LAE Ratio =

Total Loss Adjustment Expenses
is 20% (= $1,200,000 / $8,000,000).
Total Losses

UW Expense Ratio =

Total UW Expenses
is 27%
Total Premium

Underwriting Expense Ratio = Total Underwriting Expense / Total Premium
(1)
Commissions and brokerage
840,000
Total Written Premium
Other acquisition costs
420,000
Total Written Premium
Total Earned Premium
General expenses
640,000
Taxes, licenses, and fees
336,000
Total Written Premium
TOTAL

(2)
$8,400,000
$8,400,000
$8,000,000
$8,400,000

U/W Exp Ratio
(3)=(1)/(2)
10.0%
5.0%
8.0%
4.0%
27.0%

LAE
is 27% + ($1,200,000 / $8,000,000) = 42%
Total Earned Premium
LAE
Underwriting Expenses
= 75%+15%+27% = 117%
Combined Ratio = Loss Ratio +
+
Earned Premium
Written Premium
Number of Policies Renewed
e. Retention Ratio =
85% (= 1,700 / 2,000).
Number of Potential Renewal Policies
d.

OER = UW Expense Ratio +

Close Ratio =

Number of Accepted Quotes
is 20% (= 1,200 /6,000).
Number of Quotes

Retention ratios are used to gauge the competitiveness of rates and are closely examined following rate
changes or major changes in service, and as a key parameter in projecting future premium volume.
Closed ratios are used to determine the competitiveness of rates for new business.
See chapter 1
Copyright  2014 by All 10, Inc.
Page 11

Exam 5A – Solutions to Independently Authored Questions - Test 1
Question 6 discussion: Blooms: Knowledge; Difficulty 1, LOKS: Calculate a policy premium for a
specified risk using the rate pages provided.
Rules
Found in the insurer’s rating manual
Rate pages (i.e. base rates, rating tables, and fees)
Found in the insurer’s rating manual
Rating algorithm
Found in the insurer’s rating manual
Underwriting guidelines
Found in the insurer’s UW manual
See chapter 2
Question 7 discussion: Blooms: Knowledge; Difficulty 1, LOKS: Calculate a policy premium for a
specified risk using the rate pages provided
The algorithm includes instructions such as:
 the order in which rating variables should be applied
 how rating variables are applied in calculating premium (e.g. multiplicative, additive, or some unique
mathematical expression)
 maximum and minimum premiums (or in some cases the maximum discount or surcharge to be applied)
 specifics with how rounding takes place.
See chapter 2
Question 8 discussion: Blooms: Comprehension; Difficulty 1, LOKS: Calculate a policy premium
for a specified risk using the rate pages provided
Personal Automobile
Insurance Credit Score, Homeownership, Prior Bodily Injury Limits
Homeowners
Insurance Credit Score, Prior Loss Information, Age of Home
Workers Compensation
Safety Programs, Number of Employees, Prior Loss Information
Commercial General Liability
Insurance Credit Score, Years in Business, Number of Employees
Medical Malpractice
Patient Complaint History, Years Since Residency,
Commercial Automobile
Driver Tenure, Average Driver Age, Earnings Stability
See chapter 2
Question 9 discussion: Blooms: Application; Difficulty 3, LOKS: Calculate a policy premium for a
specified risk using the rate pages provided
Total Premium = All-Peril Base Rate x AOI Relativity * Territory Relativity * Protection Class / Construction Type
Relativity
* Underwriting Tier Relativity * Deductible Credit
* [1.0 - New Home Discount – Claims-Free Discount] * [1.0 - Multi-Policy Discount]
+ Increased Jewelry Coverage Rate + Increased Liability/Medical Coverage Rate + Policy Fee.
Entries from Rating Manual
Base Rate
AOI Relativity
Territory Relativity
Protection Class / Construction Type Relativity
Underwriting Tier Relativity
Deductible Credit
New Home Discount
Claims-Free Discount
Multi-Policy Discount
Increased Jewelry Coverage Rate
Increased Liability/Medical Coverage Rate
Expense Fee

$750
0.96
0.90
1.15
1.00
0.85
20%
10%
0%
$35
$25
$50

The rating algorithm from the rating manual can be applied to calculate the final premium for the policy:

$522.36 $750*.96*.90*1.15*1.00*0.85*[1.0 - 0.20 - 0.10]*[1.0 -.07] + $35 + $25 + $50.
Copyright  2014 by All 10, Inc.
Page 12

Exam 5A – Solutions to Independently Authored Questions - Test 1
Question 10 discussion: Blooms: Application; Difficulty 2, LOKS: Calculate a policy premium for a
specified risk using the rate pages provided
The rating algorithm to calculate the final premium for a given policy using the aforementioned rating manual
variables is as follows:
Total Premium = Higher of
N

[∑ (Classi rate x $ Payroll for classi / 100)

where N = number of classes

i =1

8810 Clerical
8825 - Food Service Employees
8824 - Health Care Employees
8826 - All Other Employees
Total

Payroll

Payroll/$100

(1)
$40,000
$85,000
$100,000
$30,000
$255,000

(2)=(1)/100
$400
$850
$1,000
$300

Rate per $100 of Class Manual
Payroll
Premium
(3)
(4)=(2)*(3)
0.59
$236.00
2.88
$2,448.00
4.00
$4,000.00
3.75
$1,125.00
$7,809

Step 2: Determine the total reduction to manual premium based on the given schedule credits
Schedule Rating Modification
Premises
Classification
Medical
Safety
Employees —
Management
Peculiarities
Facilities
Devices
Selection,
—Safety
Training,
Organization
Supervision
-10%
0%
0%
-2.5%
-5%
0%
The total credit (reduction to manual premium) for SR is 10% + 2.5% + 5% = 17.5%.
Step 3: Using the formula in Step 1, the results from Step 2 and the data given in the problem, compute the
total premium for the policy.
Thus, the total premium for the policy is $6,370.30 = $7,809.00 x 0.825 x (1.0 - 0.05) + $250.
Since $6,370.30 is greater than the min premium per policy of $1,500, the total premium for the policy is
$6,370.30.
Question 11 discussion: Blooms: Comprehension; Difficulty 1, LOKS: Calculate the underwriting
expense provisions underlying the overall rate level indication.
Two types of internal data involved in a ratemaking analysis are:
1. risk information (e.g. exposures, premium, claim counts, losses, and claim or policy characteristics).
2. accounting information (e.g. UW expenses and ULAE, which is often available only at an aggregate level).
See chapter 3
Copyright  2014 by All 10, Inc.
Page 13

Exam 5A – Solutions to Independently Authored Questions - Test 1
Question 12 discussion: Blooms: Application; Difficulty 2, LOKS: Calculate the underwriting
expense provisions underlying the overall rate level indication.
Policy A can be represented with 1 record since expired at its original expiration date and had no changes.
Policy B is represented by two records because it was canceled before the policy expired.
The first record for contains information known at policy inception (e.g. 1 exposure and $800 in WP).
The second record represents an adjustment for the cancellation such that when aggregated, the two records
show a result net of cancellation. As the policy was canceled 75% of the way through the policy period, the
second record should show -0.25 exposure and -$200 (=25% x -$600) of written premium.
Policy C is represented by three records since it has a mid-term adjustment

Policy
A
B
B
C
C
C

Original
Original Transaction
Effective Termination Effective
Date
Date
Date
1/1/2012 12/31/2012 1/1/2012
4/1/2012 3/31/2013
4/1/2012
4/1/2012 3/31/2013 12/31/2012
7/1/2012 6/30/2013
7/1/2012
7/1/2012 6/30/2013
1/1/2013
7/1/2012 6/30/2013
1/1/2013

Other
Ded
$250
$250
$250
$500
$500
$250

Terr
1
2
2
3
3
3

…
…
…
…
…
…

Written
Written
Exposure Premium
1
$1,300
1
$800
-0.25
($200)
1
$1,500
-0.5
($750)
0.5
$900

See chapter 3
Question 13 discussion: Blooms: Comprehension Difficulty 2, LOKS: Organization of data:
calendar year, policy year, accident year
Four types of data aggregation methods are calendar year (CY), AY (AY), policy year (PY), and report year (RY).
CY aggregation captures premium and loss transactions during a 12-month CY (without regard to policy effective
date, accident date, or report date of the claim).
Advantage of CY aggregation: data is quickly available at CY end. CY data is used for financial reporting so there
is no additional expense to aggregate the data this way for ratemaking purposes.
Disadvantage of CY aggregation: the mismatch in timing between premium and losses.
CY EP comes from policies in force during the year (written either in the previous or the current CY).
Losses, however, may include payments and reserve changes on claims from policies issued years ago.
CY year aggregation for ratemaking analysis may be most appropriate for lines of business or individual coverages
in which losses are reported and settled relatively quickly (e.g. homeowners).
AY aggregation of premium and exposures follows the same precept as CY premium and exposures, and thus the
method is often referred to as CY-AY or FY-AY.
Advantage: AY aggregation provides a better match of premium and losses than CY aggregation.
Losses on accidents occurring during the year are compared to EP on policies during the same year.
Since the AY is not closed (fixed) at year end, future development on known losses needs to be estimated.
PY aggregation (a.k.a. UW year) considers all premium and loss transactions on policies that were written during a
12-month period, regardless of when the claim occurred or was reported, reserved, or paid.
Advantage: PY aggregation represents the best match between losses and premium (since losses on policies
written during the year are compared with premium earned on those same policies).
Disadvantage: Data takes longer to develop than both CY and AY, since PY exposures for a product with an
annual policy term are not fully earned until 24 months after the start of the PY.
RY aggregation is similar to CY-AY except losses are aggregated according to when the claim was reported (as
opposed to when the claim occurred).
RY data is used for commercial lines products using claims-made policies (e.g. medical malpractice).
See chapter 3
Copyright  2014 by All 10, Inc.
Page 14

Exam 5A – Solutions to Independently Authored Questions - Test 1
Question 14 discussion: Blooms: Comprehension; Difficulty 1, LO5, KS: Sources of data and
selection criteria
The most commonly used types are:
1. Economic data (e.g. Consumer Price Index (CPI))
Insurers may examine the CPI at the component level (e.g. medical cost and construction cost indices) to find
trends relevant to the insurance product being priced.
2. Geo-demographic data (i.e. average characteristics of a particular area).
i. Population density can be a predictor of accident frequency.
ii. Weather indices, theft indices, and average annual miles driven.
3. Credit data is used by insurers to evaluate the insurance loss experience of risks with different credit scores.
Insurers feel credit is an important predictor of risk and began to vary rates accordingly.
See chapter 3
Question 15 discussion: Blooms: Knowledge & Comprehension; Difficulty 2, LO2, KS: Definition of
exposure base and b. Characteristics of exposure bases
a. An exposure is the basic unit that measures a policy’s exposure to loss.
a1. be directly proportional to expected loss
a2. be practical
a3. consider preexisting exposure bases used within the industry.
b. It should be clear that the expected loss for one home insured for 2 years is two times the expected loss of
the same home insured for 1 year.
Also, the expected loss for homes also varies by amount of insurance purchased.
However, while the expected loss for a $200,000 home is higher than that for a $100,000 home, it may not
necessarily be two times higher.
Since the EB should be the factor most directly proportional to the expected loss, number of house years is
the preferred EB, and amount of insurance should be used as a rating variable.
See chapter 4
Question 16 discussion Blooms: Comprehension; Difficulty 1, LO2, KS: b. Characteristics of
exposure bases
Asking a personal auto policyholder to state their estimated annual miles driven provides opportunity for
dishonesty more so than the use of car-years as the exposure base.
However, advances in technology may change the choice of EB for personal auto insurance.
Example: Onboard diagnostic devices can accurately track driving patterns and transmit this data to insurers.
Thus, some commercial long haul trucking carriers have implemented miles driven as an EB.
See chapter 4

Exam 5A – Solutions to Independently Authored Questions - Test 1
Question 17 discussion: Blooms: Application; Difficulty 1, LO2, KS: Written exposure versus
earned exposure versus
a.

Calendar Year Written Exposures a/o 12/31/14
Effective Expiration
Policy
Date
Date
Exposure
10/1/2012 9/30/2013
A
10.00
1/1/2013 12/31/2013
B
10.00
4/1/2013 3/31/2014
C
10.00
7/1/2013 6/30/2014
D
10.00
10/1/2013 9/30/2014
E
10.00
1/1/2014 12/31/2014
F
10.00
Total
60.00

Written Exposures
CY 2012
CY 2013
CY 2014
10.00
0.00
0.00
0.00
10.00
0.00
0.00
10.00
0.00
0.00
10.00
0.00
0.00
10.00
0.00
0.00
0.00
10.00
10.00
40.00
10.00

b. If Policy D is cancelled on 3/31/2014 (i.e. after 75% of the policy has expired), then Policy D will contribute
10 written exposures to CY 2013 and -2.5 written exposures to CY 2014.
c. If Policy D is cancelled on 3/31/2014 (i.e. after 75% of the policy has expired), then Policy D will contribute
10 written exposures to PY 2013 and -2.5 written exposures to PY 2013. In case of cancellation, the
original written exposure and the written exposure due to the cancellation are all booked in the same PY
(since PY written exposures are aggregated by policy effective dates).
d.

Calendar Year Earned Exposures a/o 12/31/14
Effective Expiration
Policy
Date
Date
Exposure
10/1/2012 9/30/2013
A
10.00
1/1/2013 12/31/2013
B
10.00
4/1/2013 3/31/2014
C
10.00
7/1/2013 6/30/2014
D
10.00
10/1/2013 9/30/2014
E
10.00
1/1/2014 12/31/2014
F
10.00
Total
60.00

Earned Exposures
CY 2010
CY 2011
CY 2012
2.50
7.50
0.00
0.00
10.00
0.00
0.00
7.50
2.50
0.00
5.00
5.00
0.00
2.50
7.50
0.00
0.00
10.00
2.50
32.50
25.00

Calendar Year Earned Exposures a/o 12/31/14
Effective Expiration
Earned Exposures
Policy
Date
Date
Exposure CY 2012 CY 2013 CY 2014
10/1/2012
3/31/2013
5.00
5.00
0.00
A
10.00
1/1/2013
6/30/2013
0.00
10.00
0.00
B
10.00
4/1/2013
9/30/2013
0.00
10.00
0.00
C
10.00
7/1/2013 12/31/2013
0.00
10.00
0.00
D
10.00
10/1/2013
3/31/2014
0.00
5.00
5.00
E
10.00
1/1/2014
6/30/2014
0.00
0.00
10.00
F
10.00
Total
60.00
5.00
40.00
15.00

Exam 5A – Solutions to Independently Authored Questions - Test 1
Question 18 discussion: Blooms: Application; Difficulty 1, LO 2KS: Written exposure versus
earned exposure versus in-force exposure
a. (.75 points). Compute the aggregate In-force exposures as of 7/01/2012, 1/01/2013 and 7/01/2013

Aggregate In-force Calculation
Written
Assumed
Month
Exposure Effective Date
480
Jan-12
01/15/12
480
Feb-12
02/15/12
480
Mar-12
03/15/12
480
Apr-12
04/15/12
480
May-12
05/15/12
480
Jun-12
06/15/12
480
Jul-12
07/15/12
480
Aug-12
08/15/12
480
Sep-12
09/15/12
480
Oct-12
10/15/12
480
Nov-12
11/15/12
480
Dec-12
12/15/12
Total
5,760

In-Force Exposures a/o
07/01/12
01/01/13
07/01/13
480
480
0
480
480
0
480
480
0
480
480
0
480
480
0
480
480
0
0
480
480
0
480
480
0
480
480
0
480
480
0
480
480
480
480
0
2,880
5,760
2,880

b. (.75 points). Compute the aggregate earned exposures for CY 2012 and CY 2013

Aggregate Earned Exposure Calculation
(1)
(2)
(3)
Assumed
Written
Exposures
Effective
Month
Written
date
480
Jan-10
01/15/10
480
Feb-10
02/15/10
480
Mar-10
03/15/10
480
Apr-10
04/15/10
480
May-10
05/15/10
480
Jun-10
06/15/10
480
Jul-10
07/15/10
480
Aug-10
08/15/10
480
Sep-10
09/15/10
480
Oct-10
10/15/10
480
Nov-10
11/15/10
480
Dec-10
12/15/10
Total
5760

(4)

(5)
Earning
Percentage Percentage
2012
2013
23/24
1/24
7/8
1/8
19/24
5/24
17/24
7/24
5/8
3/8
13/24
11/24
11/24
13/24
3/8
5/8
7/24
17/24
5/24
19/24
1/8
7/8
1/24
23/24

(6)=(2)*(4) (7)=(2)*(5)
Earned
Exposure
Exposure
2012
2013
460
20
420
60
380
100
340
140
300
180
260
220
220
260
180
300
140
340
100
380
60
420
20
460
2,880
2,880

Question 19 discussion: Blooms: Comprehension; Difficulty 1, LO 3 KS: Written premium versus
earned premium versus in-force premium
Historical premium must be:
1. Brought to current rate level. This involves adjusting premium for rate increases (decreases) that occurred
during or after the historical experience period. This is known as adjusting the premium “to current rate
level” or putting the premium “on-level”. Two current rate level methods are extension of exposures and the
parallelogram method.
2. Developed to ultimate. This is relevant when an analyzing incomplete policy years or premium that has yet
to undergo audit.
3. Adjusted for actual or expected distributional changes. This is done through premium trending, and both the
one-step and two-step trending are discussed in this section.
See chapter 5
Copyright  2014 by All 10, Inc.
Page 17

Exam 5A – Solutions to Independently Authored Questions - Test 1
Question 20 discussion: Blooms: Application; Difficulty 3, LO 3, KS: Determinations of and
application of premium trend
The actual premium charged for the policy was based on the rates effective on 1/1/2013, and was
$16,775 (= 10 x $2,090 x 0.75 + $1,100).
To put the premium on-level, substitute the current base rate, class factor, and policy fee in the calculations; this
results in an on-level premium of $17,810 (= 10 x $2,090 x 0.80 + $1,090).
Note: Perform the same calculation for every policy written in 2011 and then aggregate across all policies.
See chapter 5
Question 21 discussion: Blooms: Application; Difficulty 2, LO 3, KS: Determinations of and
application of premium trend
a Step 1: Obtain the effective date and overall rate changes for the policies under consideration.
a Step 2: View these rate changes in graphical format.
CY 2011 rate levels area are shown below:
Area 1 in CY 2011:
0.125
=0.50 x 0.50 x 0.50
Area 2 in CY 2011:
0.375
=1.00 - (0.125 + 0.500)
Area 3 in CY 2011:
0.500
=0.50 x 1.00 x 1.0

a Step 3: Calculate the cumulative rate level index for each rate level group.
 The first rate level group is assigned a rate level of 1.00.
 The cumulative rate level index of each subsequent group is the prior group’s cumulative rate
level index multiplied by the rate level for that group.
i. the cumulative rate level index for the second rate level group is 1.05 (= 1.00 x 1.05).
ii. the cumulative rate level index for the third rate level group is 1.155 (= 1.05 x 1.10).
1
2
3
4
Rate
Level
Group

(4)=

Effective
Overall
Rate Level Cumulative Rate
Date
Average Rate
Index
Level Index
Change

Initial

1.00

1.0000

2
3

7/1/10

5.0%

1.05

1.0500

1/1/11

10.0%

1.10

1.1550

4/1/12

-1.0%

0.99

1.1435

(Previous Row 4) x (3)

a Step 4: Calculate the average rate level index for each year (i.e. the weighted average of the
cumulative rate level indices in Step 3, using the areas calculated in Step 2 as weights).
The average rate level index for CY 2011 is 1.0963 =1.000 x 0.125 + 1.0500 x 0.375 + 1.1550 x 0.500.

Exam 5A – Solutions to Independently Authored Questions - Test 1
a Step 5: Calculate the on-level factor as follows:

On - Level Factor for Historical Period =



Current Cumulative Rate Level Index
Average Rate Level Index for Historical Period

The numerator is the most recent cumulative rate level index
The denominator is the result of Step 4.

The on-level factor for CY 2011 EP (assuming annual policies) is 1.0431 =

1.1435
1.0963

a Step 6: The on-level factor is applied to the CY 2011 EP to bring it to current rate level.
CY 2011 EP at current rate level= CY 2011 EP x 1.0431.
b. Standard PY Calculations for Annual Policies

Since PY 2011 only had one rate level applied to the whole year, PY 2012 will be reviewed.
The area of each parallelogram is base x height.
Area 3 in Policy Year 2012 has a base of 3 months (or 0.25 of a year) and the height is 12 months (or 1.00 year).
b Step 2: The relevant areas for PY 2012 are as follows:
• Area 3 in PY 2012: 0.25 = 0.25 x 1.00
• Area 4 in PY 2012: 0.75 = 0.75 x 1.00
b Step 3: The cumulative rate level indices are the same as those used in the CY example.
b Step 4: The average rate level index for PY 2012 is: 1.1464 = 1.1550 x 0.25 + 1.1435 x 0.75.
b Step 5: The on-level factor to adjust PY 2012 EP to current rate level is 0.9975 =

1.1435
1.1464

c. CY Calculations for Semi-Annual Policies
c Step 2: The areas for CY 2011 are:
Area 1 in CY 2011: N/A
Area 2 in CY 2011: 0.250 = 0.50 x 0.50 x 1.00
Area 3 in CY 2011: 0.750 = 1.00 - 0.250
c Step 3: The cumulative rate level indices are the same as those used for the annual policies.
c Step 4: The average rate level index for CY 2001 assuming semi-annual policies:
1.1288 = 1.0500 x 0.250 + 1.1550 x 0.750
c Step 5: The on-level factor to adjust CY 2011 EP to current rate level is: 1.0130 =

1.1435
(and is
1.1288

smaller than for annual policies because the semi-annual rate changes earn more quickly).
See chapter 5

Exam 5A – Solutions to Independently Authored Questions - Test 1
Question 21 discussion (continued)
d. The rate level change is represented as a vertical line.
Assume a law change mandates a rate decrease of 5% on 7/1/2011 applicable to all policies.

0.50 x 0.50 x 0.50
0.50 - 0.125 - 0.125
0.50 x 0.50 x 0.50
0.50 x 0.50 x 0.50
0.50 - 0.125

The cumulative rate level indices associated with each group are as follows:
Step 3 (with Benefit Change)
Rate Level
Group

Cumulative Rate
Level Index

1.0000

1.0500

0.9975

1.1550

1.0973

1.0863

CY 2011 on-level factor:

1.0171 =

1.0863
1.0000 * 0.125 + 1.0500 * 0.250 + 0.9975 * 0.125 + 1.1550 * 0.125 + 1.0973 * 0.375

See chapter 5

Exam 5A – Solutions to Independently Authored Questions - Test 1
Question 22 discussion: Blooms: Comprehension; Difficulty 1, LO 3 KS: Determinations of and
application of premium trend
1. The method is not useful if the assumption that policies are evenly written throughout the year is not true.
Example: Boat owners policies are usually purchased prior to the start of boat season and thus are not
uniformly written throughout the year.
Ways to partially circumvent the need for uniform writings:
a. Use a more refined period of time than a year (e.g. quarters or months).
b. Calculate the actual distribution of writings and use these to determine more accurate weightings to
compute the historical average rate level.
Aggregate policies based on which rate level was applicable rather than based on a time period, and
the premium for each rate level group is adjusted together based on subsequent rate changes.
2. Premium for certain classes will not be on-level if the implemented rate changes vary by class.
Even if the overall premium may be adjusted to a current rate level, adjusted premium will not be
appropriate for class ratemaking.
This major shortcoming has caused insurers to favor of the extension of exposures approach.
See chapter 5
Question 23 discussion: Blooms: Application; Difficulty 1, LO 3, KS: Determinations of and
application of premium trend
At 12/31/2014, the six policies written in the first half of 2013 have completed their audits, but the six policies
written in the second half of the year have not.
PY 2013 premium as of 12/31/2014 is: $9,360,000 = 6 x $750,000 x 1.08 + (6 x $750,000)
At 12/31/2015, all twelve policies have completed their final audits and premium is final.
PY 2013 premium as of 12/31/2015 is: $9,720,000 = 12 x $750,000 x 1.08
From 12/31/2014 (24 months after the start of the PY) to 12/31/2015 (36 months after the start of the PY), the
premium development factor is 1.0385 (= $9.72 million / $9.36 million).
See chapter 5
Question 24 discussion: Blooms: Comprehension; Difficulty 1, LO 3, KS: Determinations of and
application of premium trend
Examples that can cause changes in the average premium level:
• A rating characteristic can cause average premium to change (e.g. HO premium varies based on
the amount of insurance purchased, which is indexed and increases automatically with inflation;
therefore, average premium increases as well).
• Moving all existing insureds to a higher deductible (e.g. if an insurer moves each insured to a
higher deductible upon renewal, and renewals are spread throughout the year, there will be a decrease
in average premium over the entire transition period).
Trend is not necessary once the transition is complete.
• Acquiring the entire portfolio of another insurer writing higher policy limits (e.g. a HO insurer
acquires a book of business that includes predominantly high-valued homes, the acquisition will cause a
very abrupt increase in the average premium due to the increase in average home values).
After the books are consolidated, no additional shifts in the business are expected.
See chapter 5

Exam 5A – Solutions to Independently Authored Questions - Test 1
Question 25 discussion: Blooms: Application; Difficulty 1, LO 3 KS: Determinations of and
application of premium trend
The trend period as the length of time from the average written date of policies with premium earned
during the historical period to the average written date for policies that will be in effect during the time the
*
rates will be in effect.
* Some insurers determine the trend period as the average date of premium earned in the experience
period to the average date of premium earned in the projected period. This simply shifts both dates by the
same amount, so the trend period is the same length.
The historical and projected periods can be represented as follows:

Historical period: CY 2011 EP contains premium from policies written 1/1/2010 to 12/31/2011.
Thus, the average written date for premium earned is 1/1/2011.
Projected period: Policies will be written from 1/1/2013 – 12/31/2013.
Thus, the average written date during the projected period is 6/30/2013.
Therefore, the trend period is 2.5 years (i.e. 1/1/2011 - 6/30/2013).
2.5
The adjustment to account for premium trend is: 1.0508 (= (1.0 + 0.02) ).
Trend Period for 1-Step Trending

See chapter 5

Exam 5A – Solutions to Independently Authored Questions - Test 1
Question 26 discussion: Blooms: Comprehension; Difficulty 2, LOKS: Determinations of and
application of premium trend
Items affecting the length of the trend period:
1. If the historical period consists of policies with terms other than 12 months, the “trend from” date will be
different than discussed above.
Example: If the policies in the prior example were six-month policies, then the “trend from” date is 4/1/2011.
The “trend to” date is unchanged.
Trend Period for 1-Step Trending with 6-Month Policies

2. If the historical premium is PY 2011 (rather than CY 2011) then the “trend from” date is later and
corresponds to the average written date for PY 2011 (i.e. 7/1/2011).
3. If the proposed rates are expected to be in effect for more or less than one year, then the “trend to” date will
be different (e.g. if the proposed rates are expected to be in effect for two years, then the “trend to” date will
be 12/31/2013).
See chapter 5
Question 27 discussion: Blooms: Comprehension; Difficulty 1, LO 3, KS: Determinations of and
application of premium trend
One-step trending process is not appropriate to use when:
1. Changes in average premium vary significantly year-by-year and/or
2. Historical changes in average premium are very different than the changes expected in the future.
Example: If the insurer forced all insureds to a higher deductible at their first renewal on or after 1/1/11, the
shift would have been completed by 12/31/11, and the observed trend would not continue into the future.
When situations like this occur, companies may use a two-step trending approach.
See chapter 5
Question 28 discussion: Blooms: Application; Difficulty 2, LO 3, KS: Determinations of and
application of premium trend
Step 1: Determine the average written date during the experience period. For the experience period
4/1/01 – 3/31/02, and given that 6 month policies are being written, the average earned date is 10/1/01
and the average written date is 7/1/01, or ½ the policy term earlier from the average earned date.
Step 2: Determine the average written date during the exposure period. The average written date during the
future policy period is a function of the length of time that the rates are expected to remain in effect. In
this example, since rates are reviewed every 18 months, this would make the average written date 9
months after the proposed effective date of 4/1/03, which is 1/1/04.
Thus, the written premium trend period is 2.50 years.
See chapter 5

Exam 5A – Solutions to Independently Authored Questions - Test 1
Question 29 discussion: Blooms: Application; Difficulty 2, LO 3 KS: Determinations of and
application of premium trend
Step 1: Adjust the historical premium to the current trend level using the following adjustment factor:

Current Premium Trend Factor =

Latest Average WP at Current Rate Level
Historical Average EP at Current Rate Level

If average EP for CY 2013 is $940.00 and the average WP for the latest available quarter (Calendar
Quarter 4Q 2013) is $953.00, then the current premium trend factor is 1.0138 (= 953.00/940.00).
The latest average WP is for the fourth quarter of 2013; thus, the average written date is
11/15/2013 (this will be “trend from” date for the second step in the process).
If the average been based on the average WP for CY 2013 (as opposed to the fourth quarter), then the
average written date would have been 6/30/2013.
When average premium is volatile, select a current trend versus using the actual change in average premium.
The current trend factor is calculated by trending (1.0 + selected current trend) from the average written
date of premium earned in the experience period (i.e. 1/1/2013) to the average written date of the latest
period in the trend data (i.e. 11/15/2003).
Step 2: Compute the projected premium trend factor.
Select the amount the average premium is expected to change annually from the “trend from” date to the
projected period.
The “trend from” date is 11/15/2013.
The “trend to” date is the average written date during the period the proposed rates are to be in effect,
which is still 6/30/2015.
Thus, the projected trend period is 1.625 years long (11/15/2013 to 6/30/2015).
1.625
).
Given a projected annual premium trend of 2%, the projected trend factor is 1.0327 (= (1.0 + 0.02)
Trend Period for 2-Step Trending
The total premium trend factor for two-step trending is the product of the current trend factor and the
projected trend factor (i.e. 1.0467 (= 1.0138 x 1.0327)).
That number is applied to the average historical EP at current rate level to adjust it to the projected level:
CY13 EP at projected rate level = CY13 EP at current rate level x Current Trend Factor x Projected
Trend Factor.

Exam 5A – Solutions to Independently Authored Questions - Test 1
Two-Step Trending
(1) Calendar Year 2013 Earned Premium at Current Rate Level
(2) Calendar Year 2013 Earned Exposures
(3) Calendar Year 2013 Average Earned Premium at Current Rate Level
(4) 4th Quarter of 2013 Average Written Premium at Current Rate Level
(5) Step 1 Factor
(6) Selected Projected Premium Trend
(7) Projected Trend Period
(8) Step 2 Factor
(9) Total Premium Trend Factor
(10) Projected Premium at Current Rate Level

$1,880,788

2,150
$940.00
$953.00
1.01383
2.0%
1.6250
1.0327
1.0470
$1,969,156

The latest average WP is for the fourth quarter of 2013; thus, the average written date is 11/15/2013 (this will be
“trend from” date for the second step in the process).
The “trend to” date is the average written date during the period the proposed rates are to be in effect, which is still
6/30/2015.
Thus, the projected trend period is 1.625 years long (11/15/2013 to 6/30/2015).

(5) = (4) / (3)
(7)
(8) = (1.0 + (6))
(9) = (5) x (8)
(10)= (1) x (9)
See chapter 5
Question 30 discussion: Blooms: Knowledge; Difficulty 1, LO 6, KS: Mechanics associated with
each method (including organization of the data)

1. Judicial environment
2. Regulatory and legislative changes
3. Guaranty funds
4. Economic variables
5. Residual market mechanisms

Exam 5A – Independently Authored Questions - Preparatory Test 2
General information about this exam
This practice test contains 29 questions consisting of computational and essay based questions.

Total Number of Qs:
Total Number of Points:

Essay
Questions
13
20.25

Computational
Questions
16
40.5

Total
29
60.75

1. The recommend time for this exam is 2:30:00. Make sure you have sufficient time to take this practice test.
2. Consider taking this exam after working all past CAS questions first.
3. Make sure you have a sufficient number of blank sheets of paper to record your answers for
computational questions.

Chapter 7: Other Expenses and Profit ........................... Modlin, Werner ........... A. Basic Techniques for Ratemaking

Chapter 8: Overall Indication ............................................Modlin, Werner ........... A. Basic Techniques for Ratemaking

Chapter 9: Traditional Risk Classification .................... Modlin, Werner ........... A. Basic Techniques for Ratemaking
Actuarial Standard No. 13 – Trending Proc. .............. CAS .................................... A. Basic Techniques for Ratemaking

Exam 5A – Independently Authored Questions - Test 2
Question 1 (2.0 points)
You are given the following payment and reserve information about two different claims on two different
policies:
Policy
Effective
Date
07/01/11

Date of
Loss
11/01/11

Report
Date
11/19/11

09/10/11

02/14/12

Transaction
Date
11/19/11
02/01/12
09/01/12
01/15/13
02/14/12
11/01/12
03/01/13

Incremental
Payment
$0
$2,000
$14,000
$6,000
$10,000
$16,000
$2,000

Case
Reserve
$20,000
$18,000
$5,000
$0
$20,000
$8,000
$0

a. (0.5 point) Calculate the calendar-year reported losses for 2012 and 2013.
b. (0.5 point) Calculate the accident-year reported for 2011 and 2012 evaluated as of 12/31/2013.
c. (0.5 point) Calculate the policy-year reported losses for 2011 and 2012 evaluated as of 21/31/2013.
d. (0.5 point) Briefly describe how losses are aggregated under a report year basis, what types of reserves can
be analyzed, and for what type of business is this method of loss aggregation used.

Question 2 (0.75 points)
According to Werner and Modlin in “Basic Ratemaking”, briefly describe three types preliminary adjustments to
losses prior to projecting losses to the cost level expected when the rates will be in effect.

Question 3 (2.0 points)
You are given the following reported losses, number of claims with reported losses excess of $1,000,000, and
ground-up excess losses:

Accident

Year
1996
1997
1998
1999
2000
Total

(1)

(2)

(3)

Reported

Number of
Excess

Ground –Up
Excess

Losses
$86,369,707
$85,938,146
$87,887,865
$86,488,983
$90,329,298
$437,013,999

Claims
5
1
3
0
7
16

Losses
$6,212,939
$1,280,000
$3,903,023
$0
$12,918,382
$24,314,344

Using the procedure described by Werner and Modlin in “Basic Ratemaking”, compute the excess loss factor.

Exam 5A – Independently Authored Questions - Test 2
Question 4 (3.0 points)
Ratio to
Average
Weekly
Wage
<50%
50-75%
75-100%
100-125%
125-150%
>150%
Total

#
Workers
7
24
27
19
12
11
100

Total
Weekly
Wage
$3,000
$16,252
$23,950
$23,048
$16,500
$17,250
$100,000

The state average weekly wage (SAWW) is $1,000
Current Workers' Compensation Law
• Compensation rate is 66.7% of worker's pre-injury wage.
• Maximum benefit limit = 100% of state average weekly wage.
• Minimum benefit limit = 50% of state average weekly wage.
Revised Workers' Compensation Law
• Compensation rate is 66.7% of worker's pre-injury wage.
• Maximum benefit limit = 83.3% of state average weekly wage.
• Minimum benefit limit = 50% of state average weekly wage.
Using the procedure described Werner and Modlin in “Basic Ratemaking”, calculate the direct effect of the benefit
level change.

Question 5 (2.0 points)
Assume a law change implemented on August 15, 2010 only affects losses on policies written on or after August
15, 2010. The direct effect of the change for annual policies on an accident year basis is estimated at +5%.
rd

a. (0.50 points) Calculate the law change adjustment factor to be applied to 3 quarter 2010 calendar
accident quarter reported losses.
rd
b. (0.50 points) Calculate the law change adjustment factor to be applied to 3 quarter 2010 policy quarter
reported losses.
Now assume a benefit change affects losses on claims that occur on or after August 15, 2010, regardless of the
effective date of the policy. The direct effect of the change for annual policies on an accident year basis is
estimated at +5%.
rd

c. (0.50 points) Calculate the benefit change adjustment factor to be applied to 3 quarter 2010 calendar
accident quarter reported losses.
rd
d. (0.50 points) Calculate the benefit change adjustment factor to be applied to 3 quarter 2010 policy
quarter reported losses.

Exam 5A – Independently Authored Questions - Test 2
Question 6 (2.0 points) You are given the following:

Claim
Number
1
2
3
4
5
Total

(1)
Total
Limits
Loss
$9,000
$13,000
$24,000
$29,000
$48,000
$123,000

Assume
 basic limits losses are capped at 25,000.
 total limits losses are subject to a 10% severity trend.
Compute:
a. (1.0 point). Basic limits loss trend.
b. (1.0 point). Excess limits loss trend.

Question 7 (2.0 points) According to Werner and Modlin in “Basic Ratemaking”, when loss experience being
analyzed is subject to the application of limits, it is important that the leveraged effect of those limits on the
severity trend be considered.
For each category of initial loss size shown below, complete the table below by stating or demonstrating
algebraically the magnitude that ‘Trend’ has on Basic Limits Losses, Total Limits Losses and Excess Losses.

Initial Loss Size

Basic Limits

Total Losses Excess Losses

Loss< [Limit/ (1+Trend)]
[Limit/(1+Trend)] < Loss < Limit
Limit < Loss

Question 8 (2.0 points) According to Werner and Modlin in “Basic Ratemaking”, while it is true that loss
development incorporates inflationary pressures that cause payments for reported claims to increase in the time
after reporting, this does not prove an overlap either.
Given the following:
 The historical experience period is CAY 2010.
 Assume it is typical for claims to settle within 18 months.
 The projection period is policy year beginning 1/1/ 2012
 Rates are expected to be in effect for annual policies written from 1/1/2012 – 12/31/2012.
Using the above information, create a graphical timeline illustration of how losses are trended and developed
which demonstrates there is no overlap between loss development and loss trend.

Exam 5A – Independently Authored Questions - Test 2
Question 9 (1.0 point) According to Werner and Modlin in “Basic Ratemaking”, and assuming that ULAE
expenditures track with loss dollars consistently over time, both in terms of rate of payment and in proportion to the
amount of losses paid, calculate the ratio of CY paid ULAE to CY paid loss plus ALAE.
Calendar
Year
2010
2011
2012

Paid Loss
And ALAE
$963,467
$1,118,918
$1,284,240

Paid ULAE
$149,026
$159,170
$190,968

Question 10 (1.5 points) ABC writes HO insurance and determines the following on a per policy basis:
 The average expected loss and LAE for each policy is $360.
 ABC incurs $40 in fixed expenses each time it writes a policy.
 15% of each dollar of premium covers expenses that vary with the amount of premium
 Company management has determined that the target profit provision should be 5% of premium.
a. (1 point). Re-write the equation Premium = Losses + LAE + UW Expenses + UW Profit, using the notation in
“Basic Ratemaking”, to determine the average premium per policy.
b. (0.50 points). Using the values given in the problem, and the equation in part a., compute the premium ABC
should charge.

Question 11 (1.5 points) According to Werner and Modlin in “Basic Ratemaking”, answer the following questions.
a. (1 point). List and briefly describe four categories of underwriting expenses.
b. (0.50 points). List and briefly describe two groups the underwriting expense provision is divided into.

Question 12 (1.5 points) According to Werner and Modlin in “Basic Ratemaking”, the data used in the all variable
expense method can be either countrywide or state based and premiums used can be either earned or written
premiums.
For each of the four expense categories below, fill in the table below and briefly describe the type of data that is
used and why it is used.
Expense

Data Used

General Expense
Other Acquisition
Commissions and Brokerage
Taxes, Licenses, and Fees

Divided By

Exam 5A – Independently Authored Questions - Test 2
Question 13 (3.0 points). According to Werner and Modlin in “Basic Ratemaking”, answer the following
questions about the all variable expense method.
a. (0.50 points). List two possible distortions in computing the correct premium when the all variable expense
method is used.
b. (1.5 points). Assume ABC insurer determines the following on a per policy basis:
 The average expected loss and LAE for each policy is $360.
 ABC incurs $40 in fixed expenses each time it writes a policy.
 15% of each dollar of premium covers expenses that vary with the amount of premium
 Company management has determined that the target profit provision should be 5% of premium.
Using your response in part a., show mathematically the difference in premiums computed assuming that the
correct premium always results from using a fixed expense of $40 and a variable expense and profit provision of
20% compared to assuming that all expenses are variable.
c. (1.0 points). Briefly describe two approaches used by insurers that use the all variable expense method to
circumvent the incorrect premiums produced when using this method.

Question 14 (3.0 points). According to Werner and Modlin in “Basic Ratemaking”, answer the following
questions.
a. (1 point). Briefly describe the shortcoming when using the all variable expense method and the advantage
to using the premium-based projection method.
b. (1 point). Assuming that the selected ratio of fixed vs. variable expenses are 75% to 25% respectively, and
using the data below and the procedure described in the text, compute the fixed and variable expense
percentage provisions.

a Countrywide Expenses
b1 Countrywide Earned Premium
b2 Countrywide Written Premium

2013
$24,331,974
$445,000,000
$455,000,000

2014
$26,502,771
$485,950,000
$490,000,000

2015
$30,975,169
$525,000,000
$545,000,000

c. (1 point). Briefly describe the shortcoming of using this approach and list three situations that can cause
such a shortcoming to exist.

Question 15 (3.0 points). According to Werner and Modlin in “Basic Ratemaking”, answer the following questions.
a. (1 point). Briefly describe the difference in how the exposure/policy-based projection method is performed
compared to the premium-based projection method.
b. (1 point). Assuming that the selected ratio of fixed vs. variable expenses are 75% to 25% respectively, and
using the data below and the procedure described in the text, compute the fixed and variable expense
percentage provisions using the exposure/policy-based projection method.

Countrywide Expenses
% Assumed Fixed
Countrywide Earned Exposures
Countrywide Earned Premium

2013
$24,331,974

2014
$26,502,771

2015
$30,975,169

4,323,500
$445,000,000

4,610,500
$485,950,000

4,817,000
$525,000,000

Selected
75.0%

c. (1 point). List three shortcomings when using the exposure/policy-based projection method.

Exam 5A – Independently Authored Questions - Test 2
Question 16 (2.0 points). According to Werner and Modlin in “Basic Ratemaking”, answer the following questions.
a. (1 point) Using the fundamental insurance equation, Premium = Losses + LAE + UW Expenses + UW Profit,
and the notation used in the text, derive the pure premium indicated rate formula.
b. (1 point) Given the following data, and using the pure premium indicated rate formula from part a., compute the
indicated average rate per exposure.
• Projected pure premium including LAE
= $400
• Projected fixed UW expense per exposure
= $35
• Variable expense ratio
= 25%
• Target profit percentage
= 10%

Question 17 (3.0 points). According to Werner and Modlin in “Basic Ratemaking”, answer the following questions.
a. (2 points) Using the fundamental insurance equation, Premium = Losses + LAE + UW Expenses + UW Profit,
and the notation used in the text, derive the loss ratio indicated rate change formula.
b. (1 point) Given the following data, and using the pure premium indicated rate formula from part a., compute the
loss ratio indicated rate change
• Projected ultimate loss and LAE ratio
= 70%
• Projected fixed expense ratio
= 5.5%
• Variable expense ratio
= 20%
• Target profit percentage =
=10%

Question 18 (3.0 points). According to Werner and Modlin in “Basic Ratemaking”, answer the following questions.
a. (2 points) List and briefly describe two major differences between the loss ratio and pure premium approaches.
b. (1 point) List and briefly describe when it is preferable to use the loss ratio and pure premium approaches
respectively.

Question 19 (2.0 points). Using the procedure shown by Werner and Modlin in “Basic Ratemaking”, demonstrate
the equivalency of the loss ratio and pure premium methods.
Both formulae can be derived from the fundamental insurance equation (thus two approaches are
mathematically equivalent).

Question 20 (1.0 point). According to According to “Actuarial Standard of Practice No. 13: Trending Procedures
in Property/Casualty Insurance Ratemaking,” list four ways in which an actuary may present the trend estimate
resulting from the trending procedure

Question 21 (1.0 point). According to According to “Actuarial Standard of Practice No. 13: Trending
Procedures in Property/Casualty Insurance Ratemaking,” the actuary should select data appropriate for the
trends being analyzed.
List four factors the actuary should consider when selecting historical insurance and non-insurance data.

Exam 5A – Independently Authored Questions - Test 2
Question 22 (1.0 point). According to According to “Actuarial Standard of Practice No. 13: Trending in
Property/Casualty Insurance Ratemaking,” list two criteria that an actuary should consider when determining the
trending period.

Question 23 (1.0 point). According to According to “Actuarial Standard of Practice No. 13: Trending in
Property/Casualty Insurance Ratemaking,” list two disclosures an actuary should make in an actuarial
communication.

Question 24 (2.0 points). According to Werner and Modlin in “Basic Ratemaking”, one criterion to evaluate the
appropriateness of a rating variable is statistical.
a. (1 point) List three statistical criterion to help ensure the accuracy and reliability of a potential rating variable.
b. (1 point) Briefly describe what it means for a rating variable should be a statistically significant risk
differentiator:

Question 25 (2.0 points). According to Werner and Modlin in “Basic Ratemaking”, one criterion to evaluate the
appropriateness of a rating variable is operational.
a. (1 point) List three operational criterion for a rating variable to be considered practical.
b. (1 point) Briefly explain whether the skill level of a surgeon for medical malpractice insurance is an objective
rating variable and if not, list two other objective rating variables for a surgeon.

Question 26 (1.5 points). According to Werner and Modlin in “Basic Ratemaking”, it is desirable for insurance to
be affordable for all risks. List three situations which help to ensure that insurance will be affordable.

Exam 5A – Independently Authored Questions - Test 2
Question 27 (8.0 points). You are given the following data from ABC insurer’s homeowners book of business:
 All UW expenses are variable. The variable expense provision is 30% of premium, the target profit
percentage is 5% of premium
 There are only 2 rating variables: amount of insurance (AOI) and territory.
Exposure Distribution
Territory
AOI
Low
Medium
High
Total



1
8
106
180
294

3
139
130
40
309

Total
272
365
351
988

AOI
Low
Medium
High
Total

1
1%
11%
18%
30%

Territory
2
3
13%
14%
13%
13%
13%
4%
39%
31%

Total
28%
37%
35%
100%



2
125
129
131
385

True
Relativity
0.7300
1.0000
1.4300

Charged
Relativity
0.8000
1.0000
1.3500

Terr
1
2
3

True
Relativity
0.6312
1.0000
1.2365

Charged
Relativity
0.6000
1.0000
1.3000

The base levels are Medium AOI and Territory 2:
The exposure, premium, and loss information needed for the analysis is summarized as follows:
Simple Example Data

AOI
Low
Medium
High
Low
Medium
High
Low
Medium
High
TOTAL

Terr
1
1
1
2
2
2
3
3
3

Exposure
8
106
180
125
129
131
139
130
40
988

Loss & LAE
$220.93
$4,448.05
$10,565.98
$6,156.12
$8,289.95
$12,063.68
$8,391.25
$10,238.70
$4,625.34
$65,000.00

Premium @
Current Rate
Level
$335.99
$6,479.87
$14,498.71
$10,399.79
$12,599.75
$17,414.65
$14,871.70
$16,379.68
$7,019.86
$100,000.00

a. (2.0 points). Using the pure premium method, compute the indicated territory pure premium relativities to the
base level.
b. (1.0 point). Briefly describe why the indicated relativities in part a. do not match the true relativities.
c. (2.0 points). Using the loss ratio method, compute the indicated territory pure premium relativities to the
base level.
d. (1.0 point). Briefly describe why the indicated relativities using the loss ratio method are closer to the true
relativities.
e. (2.0 points). Using the adjusted pure premium method, compute the indicated territory pure premium
relativities to the base level.

Exam 5A – Independently Authored Questions - Test 2
Question 28 (2 points).
Using the one-step trending procedure described Werner and Modlin in “Basic Ratemaking”, and the data
below, compute the indicated rate change for rates with an effective date of 1/1/2005.
Assumptions:
 All policies issued throughout the experience period were 12-month policies.
 The premium figures shown below are based on a book of business that has remained constant.
 The only rate increase implemented during the experience period was for 10% and occurred on
1/1/2002.
 The annual loss trend is 5%.
 The expense and profit ratio, including an allowance for investment income, is 0.254.

Year
(1)
1
2
3
Total

Developed
Earned
Incurred
Premium Losses
(2)
(3)
100,000
71,200
105,000
79,800
83,930
110,000
315,000 234,930

Question 29 (1 Point) According to "Actuarial Standard of Practice No. 13 - Trending Procedures in
Property/Casualty Insurance," list the three criteria that should be considered when determining the
trending period.

Exam 5A – Solutions to Independently Authored Questions - Test 2
Question 1 discussion: Blooms: Application; Difficulty 1, LO 4, KS: Organization of data: CY, PY, AY
Calendar Year 2012 reported losses: $35,000 =2,000+14,000+10,000+16,000+5,000+8,000-20,000
Calendar Year 2013 reported losses: -$5,000 = 6,000+2,000-5,000-8,000
AY 2011 reported losses as of December 31, 2013: $22,000 =2,000+14,000+6,000+0
AY 2012 reported losses as of December 31, 2013: $28,000 =10,000+16,000+2,000+0
PY 2011 reported losses as of December 31, 2013: $50,000 =2,000+14,000+6,000+10,000+16,000+2,000+0
PY 2012 reported losses as of December 31, 2013: $0 Neither of the two policies is issued was 2012
RY Loss aggregation method:
Losses are aggregated according to when the claim is reported (as opposed to when the claim occurs for AY).
Accident dates are maintained so the lag in reporting can be determined, since report year losses can be
subdivided based on the report lag.
This type of aggregation results in no IBNR claims, but a shortfall in case reserves (i.e. IBNER) can exist.
RY aggregation is limited to the pricing of claims-made (CM) policies. See chapter 6
Question 2 discussion: Blooms: Knowledge; Difficulty 1, LO 4, KS: Organization of data: calendar
year, policy year, accident year
1. Removing individual shock losses and catastrophe losses from historical losses and replacing them with a
long-term expectations provision.
2. Developing immature losses to ultimate.
3. Restating losses to the benefit and cost levels expected during the future policy period. See chapter 6
Question 3 discussion: Blooms: Application; Difficulty 1, LO 4, KS: Loss Development
(1)
Accident

Year
1996
1997
1998
1999
2000
Total

(2)

(3)

Number of
Excess

Ground –Up
Excess

(4)
Losses

(5)=(1) - (4)

Excess of
Non-Excess
Losses
Claims
Losses
$1,000,000
Losses
$86,369,707
5
$6,212,939
$1,212,939
$85,156,768
$85,938,146
1
$1,280,000
$280,000
$85,658,146
$87,887,865
3
$3,903,023
$903,023
$86,984,842
$86,488,983
0
$0
$0
$86,488,983
$90,329,298
7
$12,918,382
$5,918,382
$84,410,916
$437,013,999
16
$24,314,344
$8,314,344
$428,699,655
(4)= (3) - [$1,000,000 x (2)]
(7) Excess Loss Factor
(7)= 1.0 + (Tot6)
Reported

(6)=(4) / (5)
Excess

Ratio
1.42%
0.33%
1.04%
0.00%
7.01%
1.94%
1.0194

Question 4 discussion: Blooms: Application; Difficulty 2, LO 4 KS: Adjustment for coverage and
benefit level changes
The key is to calculate the benefits provided before and after the change.
The minimum benefit is 50% of the SAWW ($1,000) which equals $500 (= $1,000 x 50%).
The minimum benefit of $500 applies to workers who earn less than 75% of the SAWW
(i.e. $500 = 66.7% x 75% x $1,000), given the current compensation rate of 66.7%.
The aggregate benefits for 31 (= 7 + 24) employees in this category are $15,500 (= 31 x $500).
The maximum benefit is 100% of the SAWW ($1,000) and thus equals $1,000 (= $1,000 x 100%).
The maximum benefit of $1,000 applies to workers who earn more than 150% of the SAWW
(i.e. $1,000 = 66.7% x 150% x $1,000), given the current compensation rate of 66.7%.
The aggregate benefits for the 11 employees in this category are $11,000 (= 11 x $1,000).
The remaining 58 (= 27 + 19 + 12) employees fall between the minimum and maximum benefits.
This means their total benefits are 66.7% of their actual wages or $42,354 ( = ( 66.7% x 23,950 )
+ ( 66.7% x 23,048 ) + ( 66.7% x 16,500 ) ).

Exam 5A – Solutions to Independently Authored Questions - Test 2
The sum total of benefits is $68,854 (= $15,500 + $11,000 + $42,354) under the current benefit structure.
Question 4 discussion (continued):
Once the maximum benefit is reduced from 100% to 83.3% of the SAWW, more workers will be subjected
to the new maximum benefit.
Workers earning approximately >125% of the SAWW are subject to the maximum (i.e. $833.75 = (66.7%
x 125% x $1,000) > $833). These 23 (= 11 + 12) workers will receive $19,159 (= 23 x $833) in benefits.
Workers subject to the min benefit, 31, are not impacted by the change, and their benefits remain $15,500.
There are now only 46 (= 27 + 19) employees that receive a benefit equal to 66.7% of their pre-injury wages or:
$31,348 (= (66.7% x 23,950) + (66.7% x 23,048)) because more workers are now impacted by the maximum.
The new sum total of benefits is $66,007 (= 19,159 + 15,500 + 31,348).
The direct effect from revising the maximum benefit is -4.1% (= 66,007 / 68,854 – 1.0).
See chapter 6
Question 5 discussion: Blooms: Application; Difficulty 1, LO 4 KS: Effect of law changes
a. Focusing on the third quarter of 2010, the portion of losses assumed to be pre- and post-change are as
follows:
• 3Q 2010 Post-change: 0.0078 = 0.50 x 0.125 x 0.125
• 3Q 2010 Pre-change: 0.2422 = 0.25 - 0.0078

=
The adjustment factor for 3rd quarter 2010 reported
losses is Adjustment

1.05
= 1.0484
0.2422


 0.0078 
1.00*
 +1.05*

 0.2500 
 0.2500 

The adjustment factors for the reported losses from all other quarters are calculated similarly.
b. Affect on Losses on New Annual Policies (PY Basis)

The adjustment factor applicable to the third quarter 2010 policy quarter reported losses is:

Adjustment

1.05
= 1.0244
 0.50 * 0.25 
 0.50 * 0.25 
1.00 * 
 + 1.05* 

 0.25 
 0.25 




Exam 5A – Solutions to Independently Authored Questions - Test 2
Question 5 discussion (continued)
c. Example: A benefit change affecting all losses occurring on or after 8/15/2010 (regardless of the
policy effective date).
Affects all New Losses (AY Basis)

The adjustment factor applicable to the third accident quarter 2010 losses is as follows:

Adjustment

1.05
= 1.0244
 0.50 * 0.25 
 0.50 * 0.25 
1.00 * 
 + 1.05* 

 0.25 
 0.25 

d. Affects all New Losses (PY Basis)

ii. The adjustment factor applied to third policy quarter 2010 losses is
1.05
=
Adjustment = 1.0015
 0.0078 
 0.2422 
1.00*
 +1.05*

 0.2500 
 0.2500 
See chapter 6

Question 6 discussion: Blooms: Application; Difficulty 1, LO 4 KS: Relationship between trend and
loss development
Effect of Limits on Severity Trend
(1)
(2)
Total
Losses
Claim
Limits
Capped @
Number
Loss
$25,000
1
$9,000
$9,000
2
$13,000
$13,000
3
$24,000
$24,000
4
$29,000
$25,000
5
$48,000
$25,000
Total
$123,000
$96,000
(2)=min [(1), $25,000]

(3)= (1) - (2)

(3)

(4)

(5)
Total Limits

Excess
Losses
$0
$0
$0
$4,000
$23,000
$27,000

Loss
$9,900
$14,300
$26,400
$31,900
$52,800
$135,300

Trend
10.0%
10.0%
10.0%
10.0%
10.0%
10.0%

(6)
(7)
Trended Losses
Capped @ $25,000
Loss
Trend
$9,900
10.0%
$14,300
10.0%
$25,000
4.2%
$25,000
0.0%
$25,000
0.0%
$99,200
3.3%

(4)= (1) x 1.10 (5)=(4)/(1)-1.0 (6)=min [ (4) , $25,000]

See chapter 6

(8)
(9)
Excess Losses
Loss
$0
$0
$1,400
$6,900
$27,800
$36,100

(7)= (6)/(2)-1.0 (8)= (4) - (6)

Trend
N/A
N/A
N/A
72.5%
20.9%
33.7%
(9)=(8)/(3)-1.0

Exam 5A – Solutions to Independently Authored Questions - Test 2
Question 7 discussion: Blooms: Application; Difficulty 1, LO 4 KS: Relationship between trend and
loss development
Initial Loss Size
Loss< Limit/ (1+Trend)
Limit/ (1+Trend) < Loss < Limit
Limit < Loss

Basic Limits

Total Losses Excess Losses

Trend

Undefined

Limit/Loss-1.0

Trend

Undefined

Trend

{ [Loss*(1.0+Trend)]-Limit} / (Loss-Limit)

See chapter 6
Question 8 discussion: Blooms: Synthesis Difficulty 1, LO 4 KS: Relationship between trend and
loss development

Based on the given information, we know that:
 The average date of claim occurrence is 7/1/2010.
 Since it is typical for claims to settle within 18 months, the “average claim” will settle on 12/31/2011.
 Since rates are expected to be in effect for annual policies written from 1/1/2012 – 12/31/2012, the
average hypothetical claim in the projected period will occur on 1/1/2013, and will settle 18 months later
on 6/30/2014 (i.e. consistent with the settlement lag of 18 months).
Therefore:
 Trend adjusts the average historical claim from the loss cost level that exists on 7/1/2010 to the
average loss cost level expected on 1/1/2013 (30 months)
 Development adjusts the trended, undeveloped claim on 1/1/2013 (at 30 months from 7/1/2010) to the
ultimate level, expected to occur by 6/30/2014 (which constitutes an additional 18 months of
development).
This 48 month period represents 30 months of trend to adjust the cost level to that anticipated during
the forecast period and the 18 months of development to project this trended value to its ultimate
settlement value.
Thus, there is no overlap between the use of loss trend and loss development.
See chapter 6

Exam 5A – Solutions to Independently Authored Questions - Test 2
Question 9 discussion: Blooms: Application; Difficulty 1, LO 4 KS: Organization of data: calendar
year, policy year, accident year
Note: Calculate the ratio of CY paid ULAE to CY paid loss plus ALAE over several years (e.g. three years or
longer, depending on the line of business).
 This ratio is applied to each year’s reported loss plus ALAE to incorporate ULAE.
 The ratio is calculated on losses that have not been adjusted for trend or development as this data is
readily available for other financial reporting.
 The resulting ratio of ULAE to loss plus ALAE is then applied to loss plus ALAE that has been adjusted
for extraordinary events, development, and trend.
ULAE Ratio
(1)
Paid Loss
And ALAE
$963,467
$1,118,918
$1,284,240
$3,366,625

Calendar
Year
2010
2011
2012
Total

(2)
Paid ULAE
$149,026
$159,170
$190,968
$499,164
(4) ULAE Factor

(3)
ULAE
Ratio
15.5%
14.2%
14.9%
14.8%
1.148

(3) = (2) / (1)
(4) = 1.0 + (Tot3)

See chapter 6
Question 10 discussion: Blooms: Application; Difficulty 1, LO 5 KS: Differences in procedures for
loss adjustment expenses versus underwriting expenses
a. Premium = Losses + LAE + UW Expenses + UW Profit

P=
L + EL + ( EF + V * P) + QT * P
P - (V + QT ) * P = L + EL + EF
P=

[ L + EL + EF ]
[1.0 - V - QT ]
−

−

[ L + EL + EF ] / X [ L + EL + EF ]
P =
=
[1.0 - V - QT ]
[1.0 - V - QT ]
−

−

L + EL + EF
[$360 + $40]
=
=
= $500
b. P
[1.0 − V − QT ] [1.0 − 0.15 − 0.05]
−

The company should charge $500, composed of $360 of expected losses and LAE, $40 of fixed expenses,
$75.00 (= 15% x $500) of variable expenses, and $25.00 (= 5% x $500) for the target UW profit.
See chapter 7

Exam 5A – Solutions to Independently Authored Questions - Test 2
Question 11 discussion: Blooms: Comprehension; Difficulty 1, LO 5 KS: Expenses categories (e.g.,
commission, general, other acquisition, taxes, licenses and fees
a1. Commissions and brokerage:
 are paid as a percentage of premium written.
 may vary between new and renewal business.
a2. Other acquisition costs (e.g. media advertisements, mailings to prospective insureds, and salaries of
sales employees who do not work on a commission) are other expenses to acquire business.
a3. Taxes, licenses, and fees (e.g. premium taxes and licensing fees) include all taxes and miscellaneous
fees due from the insurer excluding federal income taxes.
a4. General expenses (e.g. overhead associated with the insurer’s home office and salaries of certain
employees (e.g. actuaries)) include the expenses associated with insurance operations.
The u/w expense provision is further divided into two groups: fixed and variable.
Fixed expenses (e.g. overhead costs associated with the home office) are assumed to be the same for
each risk, regardless of premium size (i.e. the expense is a constant dollar amount for each risk or policy).
Variable expenses (e.g. premium taxes and commissions) vary directly with premium and thus are
constant percentage of the premium.
See chapter 7

Question 12 discussion: Blooms: Comprehension; Difficulty 1, LO 5 KS: Expenses categories (e.g.,
commission, general, other acquisition, taxes, licenses and fees
Expense

Data Used

Divided By

General Expense

Countrywide

Earned Premium

Other Acquisition

Countrywide

Written Premium

Countrywide/State

Written Premium

State

Written Premium

Commissions and Brokerage
Taxes, Licenses, and Fees

WP is used when expenses are incurred at policy inception (it reflects the premium at the onset of the policy).
EP is used when expenses are assumed to be incurred throughout the policy (it reflects the gradual payment of
expenses that can be proportional to the earning of premium over the policy term).
Other acquisition costs and general expenses are assumed to be uniform across all locations, so C/W data from
the IEE are used to calculate these ratios.
The data used to derive commissions and brokerage expense ratios varies from carrier to carrier (e.g. some
insurers use state-specific data and some use C/W data, depending on whether the insurer’s commission plans
vary by location).
TL&F vary by state and the expense ratios are based on state data from the Annual Statement.
See chapter 7

Exam 5A – Solutions to Independently Authored Questions - Test 2
Question 13 discussion: Blooms: Application Difficulty 2, LO 5 KS: Fixed expenses and variable
expenses
a. By treating all expenses as variable, this understates the premium need for risks with a relatively small policy
premium and overstates the premium need for risks with relatively large policy premium.
b. Results of All Variable Expense Method
Loss Cost
$270
$360
$450

Fixed
Expense
$40
$40
$40

Correct Premium
Variable Expense
And Profit
20%
20%
20%

All Variable Expense Method
Fixed
Var Expense
Expense
And Profit
Premium
$0
28%
$375.00
$0
28%
$500.00
$0
28%
$625.00

Premium
$387.50
$500.00
$612.50

%Diff
-3.2%
0.0%
2.0%

Note: The $40 as a ratio to premium is 8% (= $40 / $500). The variable expense method produces
the correct premium only when variable expenses are 28% and when loss costs are $270.00
The All Variable Expense Method undercharges risks with premium less than the average and
overcharges the risks with premium more than the average.
c1. WC insurers that use this approach may implement a premium discount structure that reduces the
expense loadings based on the amount of policy premium charged.
c2. Some insurers using the All Variable Expense Method may also implement expense constants to
cover policy issuance, auditing, and handling expenses that apply uniformly to all policies.
See chapter 7
Question 14 discussion: Blooms: Comprehension & Application; Difficulty 2, LO 5 KS: Fixed
expenses and variable expenses
a. For insurers with a significant amount of both fixed and variable u/w expenses, the premium based projection
method is used since it recognizes the two types of expenses separately.
The enhancement is that this approach calculates fixed and variable expense ratios separately (as opposed
to a single variable expense ratio) so that each can be handled more appropriately within the indication
formulae.
b. Step 1: Determine the % of premium attributable to each expense type by dividing historical underwriting
expenses by EP or WP for each year during the historical experience period. Here, general expenses are
assumed to be incurred throughout the policy period, and thus are divided by EP.
b. Step 2: Choose a selected ratio (e.g. if the ratios are stable over time, a 3-year average may be chosen; if
the ratios demonstrated a trend over time, the most recent year’s ratio or some other value may be
selected). In this problem, the fixed % is given as 75%.
b. Step 3: Divide the selected expense ratio into fixed and variable ratios (using detailed expense data so that
this division can be made directly, or using activity-based cost studies that help split each expense category
appropriately). Since the problem states that 75% of the general expenses are fixed, that percentage is
used to split the selected general expense ratio of 5.9% into a fixed expense provision of 4.4% and a
variable expense provision of 1.5%.
General Expense Provisions Premium-Based Projection Method

a Countrywide Expenses
b1 Countrywide Earned Premium
b2 Countrywide Written Premium
c Ratio % [(a)/(b1)]
d % Assumed Fixed
e Fixed Expense % [(c ) x (d)]
f Variable Expense % [(c ) x (1.0-(d))]

2013
$24,331,974
$445,000,000
$455,000,000
5.5%

2014
$26,502,771
$485,950,000
$490,000,000
5.5%

2015
$30,975,169
$525,000,000
$545,000,000
5.9%

3-Year
Average

5.6%

Selected

5.6%
75.0%
4.2%
1.4%

Exam 5A – Solutions to Independently Authored Questions - Test 2
b. Step 4 (not needed to solve the problem, but is useful additional information): Sum the fixed and variable
expense ratios across the different expense categories to determine total fixed and variable expense
provisions.
If the average fixed expense per exposure (required for the pure premium approach discussed in Chapter 8)
is needed, the fixed expense provision can be multiplied by the projected average premium.
Fixed Expense Per Exposure = Fixed Expense Ratio x Projected Average Premium
c. The fixed expense ratio will be distorted if the historical and projected premium levels are different.
Situations that can cause such a difference to exist:
c1. Recent rate increases (or decreases) implemented during or after the historical period will tend to overstate
(or understate) the expected fixed expenses.
c2. Distributional shifts that have increased the average premium (e.g. shifts to higher amounts of insurance) or
decreased the average premium (e.g. shifts to higher deductibles) will tend to overstate or understate the
estimated fixed expense ratios, respectively.
c3. Countrywide expense ratios that applied to state projected premium to determine the expected fixed
expenses can create inequitable rates for regional or nationwide carriers.
See chapter 7
Question 15 discussion: Blooms: Comprehension & Application; Difficulty 2, LO 5 KS: Fixed
expenses and variable expenses
a. Variable expenses are treated the same way as the Premium-based Projection Method, but historical
fixed expenses are divided by historical exposures or policy count rather than premium.
b. General Expense Provisions Using Exposure-Based Projection Method

a Countrywide Expenses
b % Assumed Fixed
c Fixed Expense $ [(a) x (b)]
d Countrywide Earned Exposures
e Fixed Expense Per Exposure [(c) / (d)]
f Variable Expense $ [(a) x (1.0-(b))]
g Countrywide Earned Premium
h Variable Expense % [(f) / (g)]






2013
$24,331,974

2014
$26,502,771

2015
$30,975,169

$18,248,981
4,323,500
$4.22
$6,082,994
$445,000,000
1.4%

$19,877,078
4,610,500
$4.31
$6,625,693
$485,950,000
1.4%

$23,231,377
4,817,000
$4.82
$7,743,792
$525,000,000
1.5%

3-Year
Average

Selected
75.0%

$4.45

1.4%

c1. First, the method requires the actuary to judgmentally split the expenses into fixed and variable portions
c2. The method allocates countrywide fixed expenses to each state based on the exposure or policy
distribution by state (as it assumes fixed expenses do not vary by exposure or policy).
However, average fixed expense levels may vary by location (e.g. advertising costs may be higher in some
locations than others).
c3. Some expenses considered fixed actually vary by certain characteristics (e.g. fixed expenses may vary
between new and renewal business).
See chapter 7

Exam 5A – Solutions to Independently Authored Questions - Test 2
Question 16 discussion: Blooms: Application; Difficulty 1, LO 6 KS: Mechanics associated with
each method (including organization of the data)
a. Derivation of Pure Premium Indicated Rate Formula
Premium = Losses + LAE + UW Expenses + UW Profit.

PI =
L + EL + ( EF + V * PI ) + (QT * PI ).
PI − V * PI − QT * PI =
( L + EL ) + EF .

PI × [1.0 − V − QT ] = ( L + EL ) + EF ; PI =

( L + EL + EF )
[1.0 − V − QT ]

Dividing by the number of exposures converts each of the component terms into averages per exposure, and
the formula becomes the pure premium indication formula:
_________ ____
 ( L + EL ) + EF   L + E + E 
L
F
___
X
X  

PI
 P
=
=
=
X
[1.0 − V − QT ]
[1.0 − V − QT ] I

b. Given the following information:
• Projected pure premium including LAE
• Projected fixed UW expense per exposure
• Variable expense ratio
• Target profit percentage

= $400
= $35
= 25%
= 10%

 _________ ____ 
 L + EL + EF 
[$400 + $35]
The indicated average rate per exposure equals
=
=$669.23
[1.0 − V − QT ] [1.0 - 0.25 - 0.10]
See chapter 8

Exam 5A – Solutions to Independently Authored Questions - Test 2
Question 17 discussion: Blooms: Application; Difficulty 2, LO 6 KS: Mechanics associated with
each method (including organization of the data)
a. Start with the fundamental insurance equation: Premium = Losses + LAE + UW Expenses + UW Profit.

PC = Premium at current rates; QC = Profit percentage at current rates , the fundamental insurance
equation can be rewritten as follows:
Rearranging the terms leads to
Dividing both sides by

PC =L + EL + ( EF + V * PC ) + QC * PC

QC * PC =
PC - ( L + EL ) - ( EF + V * PC )

PC yields QC = 1.0 -

(L + EL )+(EF +V * PC )
L
= 1.0 PC
PC

 E + EF

- L
+V 
 PC


Substitute (QT) for (QC) and the indicated premium (PI) for the projected premium at current rates (PC)

QT = 1.0 -

(L + EL )+ EF
-V
PC * Indicated Change Factor

Rearranging terms leads to: 1.0 -V - QT =

(L + EL )+ EF
PC * Indicated Change Factor

E
+ F
PC
PC
,
(1.0 -V - QT )
(L + EL ) + F 
PC


which is equivalent to the loss ratio indication formula: Indicated Change Factor =
[1.0 -V - QT ]

L + EL + EF
Dividing through by PC yields Indicated Change Factor =
=
PC * (1.0 -V - QT )

(L + EL )

 ( L + EL ) + F 
PC


[70% + 5.5%]
b. Indicated Change = 
=
− 1.0
=
− 1.0 7.9%
[1.0 - V − QT ]
[1.00 − 0.20 − 0.10]
Thus, the overall average rate level is inadequate and should be increased by 7.9%.
See chapter 8
Question 18 discussion: Blooms: Comprehension; Difficulty 2, LO 6 KS: Assumption of each method
a. Two major differences between the two approaches.
1. The loss measure used in each approach:
 The loss ratio indication formula requires premium at current rate level and the pure premium indication
formula does not.
 The pure premium formula requires exposures whereas the loss ratio indication formula does not.
2. The output of the two formulae.
 The loss ratio formula produces an indicated change to rates currently charged.
 The pure premium formula produces an indicated rate (thus, the pure premium method must be used
with a new line of business for which there are no current rates to adjust).
b. Preference:
 The pure premium approach is preferable if premium is not available or if it is difficult to calculate
premium at current rate level
 The loss ratio method is preferable if exposure data is not available or if the product being priced does
not have clearly defined exposures
See chapter 8
Copyright  2014 by All 10, Inc.
Page 45

Exam 5A – Solutions to Independently Authored Questions - Test 2
Question 19 discussion: Blooms: Application; Difficulty 1, LO 6 KS: Mechanics associated with
each method (including organization of the data)

(L + EL ) + F 


PC
1. Start with the loss ratio indication formula: Indicated Change Factor =
[1.0 -V - QT ]
(L + EL ) + EF 

PC
PC 
Restate the formula as: Indicated Change Factor = 
[1.0 -V - QT ]
2. The indicated adjustment factor, the ratio of the indicated premium (PI ) to the projected premium at current

P
rates (PC), yields the following: I

(L + EL ) + EF 

PC
PC 
=
PC
[1.0 -V - QT ]

3. Multiplying both sides by the projected average premium at current rates ( PC

/ X ) results in the pure

premium indication formula (proving the two methods are equivalent):

(L + EL ) + EF 
_________
____
X
X  [ L + EL + EF ]

PI

=
=
X
[1.0 -V - QT ]
[1.0 -V - QT ]
See chapter 8
Question 20 discussion: Blooms: Knowledge; Difficulty 1, LO 3 KS: Determinations of and
application of premium trend
The actuary may present the trend estimate resulting from the trending procedure in a variety of ways (e.g. a
point estimate, a range of estimates, a point estimate with a margin for adverse deviation, or a probability
distribution of the trend estimate).
Question 21 discussion: Blooms: Knowledge; Difficulty 1, LO 3 KS: Organization of data: calendar
year, policy year, accident year
When selecting data, the actuary should consider the following:
1. the credibility assigned to the data by the actuary;
2. the time period for which the data is available;
3. the relationship to the items being trended; and
4. the effect of known biases or distortions on the data relied upon (e.g. the impact of catastrophic
influences, seasonality, coverage changes, nonrecurring events, claim practices, and distributional changes
in deductibles, types of risks, and policy limits).
Question 22 discussion: Blooms: Knowledge; Difficulty 1, LO 4 KS: Organization Approaches to
determining trend (e.g., exponential and linear analyses)
The actuary should consider the following when determining the trending period:
1. the lengths of the experience and forecast periods
2. changes in the mix of data between the experience and forecast periods

Exam 5A – Solutions to Independently Authored Questions - Test 2
Question 23 discussion: Blooms: Knowledge; Difficulty 1, LO 3 KS: Organization of data: calendar
year, policy year, accident year
In addition, the actuary should disclose the following, as applicable, in an actuarial communication:
1. the intended purpose(s) or use(s) of the trending procedure, including adjustments that the actuary
considered appropriate in order to produce a single work product for multiple purposes or uses
2. significant adjustments to the data or assumptions in the trend procedure, that may have a material impact
on the result or conclusions of the actuary’s overall analysis.
Question 24 discussion: Blooms: Comprehension; Difficulty 1, LO 8 KS: Risk Classification of
Principles, AAA
a. The following statistical criterion helps to ensure the accuracy and reliability of a potential rating variable:
 Statistical significance
 Homogeneity
 Credibility
b. The rating variable should be a statistically significant risk differentiator:
 Expected cost estimates should vary for the different levels of the rating variable
 Estimated differences should be within an acceptable level of statistical confidence
 Estimated differences should be relatively stable from one year to the next.
See chapter 9

Question 25 discussion: Blooms: Comprehension; Difficulty 1, LO KS: Risk Classification of
Principles, AAA
a. For a rating variable to be practical, it should be
* Objective
* Inexpensive to administer
* Verifiable
b. Estimated costs for medical malpractice insurance vary by the skill level of a surgeon. However, the skill
level of a surgeon is difficult to determine and subjective (thus, it is not a practical choice for a rating variable).
More objective rating variables like board certification, years of experience, and prior medical malpractice
claims can serve as proxies for skill level.
See chapter 9
Question 26 discussion: Blooms: Knowledge; Difficulty 1, LO 8 KS: Risk Classification of
Principles, AAA
Affordability: It is desirable for insurance to be affordable for all risks. This is true when:
* it is required by law (e.g. states require “proof of financial responsibility” from owners of vehicles)
* it is required by a third party (e.g. lenders require homeowners insurance)
* it facilitates ongoing operation (e.g. stores purchase commercial general liability insurance).
See chapter 9

Exam 5A – Solutions to Independently Authored Questions - Test 2
Question 27 discussion: Blooms: Application; Difficulty 3, LO 9 KS: Formulae and process for
each rating differential or relativity
a. Pure Premium Method:
Pure Premium Method
(1)
(2)

(3)

(4)
(5)
(6)
Indicated
Indicated
Pure
Indicated
Relativity to
Terr
Exposure
Loss & ALE
Premium
Relativity
Base
1
294
$15,234.96
$51.82
0.7877
0.7526
2
385
$26,509.75
$68.86
1.0466
1.0000
3
309
$23,255.29
$75.26
1.1439
1.0930
Total
$65.79
988
$65,000.00
1.0000
0.9555
(4)= (3)/(2);
(5)= (4)/(Tot4);
(6)= (5)/(Base5)
b. The pure premium for each level is based on the experience of each level and assumes a uniform
distribution of exposures across all other rating variables.
 If one territory has a disproportionate number of exposures of high or low AOI homes, this assumption
is invalid.
 By ignoring this exposure correlation between territory and AOI, the loss experience of high or low AOI
homes can distort the indicated territorial relativities resulting in a “double counting” effect.
i. Territory 1 indicated PP relativity is higher than the true relativity due to a disproportionate share of
high-value homes in Territory 1.
ii. Territory 3 indicated PP relativity is lower than the true relativity due to a disproportionate share of
low-value homes in Territory 3.
c. Loss Ratio Method:
(1)

(2)

Terr
1
2
3
Total

(3)

Premium @
Current Rate
Level
Loss & LAE
$21,314.57
$15,234.96
$40,414.19
$26,509.75
$38,271.24
$23,255.29
$100,000.00 $65,000.00

(4)= (3)/(2);

(4)

Loss & LAE
Ratio
71.5%
65.6%
60.8%
65.0%

(5)
Indicated
Relativity
Change
Factor
1.0996
1.0092
0.9348
1.0000

(5)= (4)/(Tot4) ;

(6)

(7)

(8)

Current
Relativity
0.6000
1.0000
1.3000

Indicated
Relativity
0.6598
1.0092
1.2153

Indicated
Relativity
@Base
0.6538
1.0000
1.2043

(7)= (5)x(6);

(8)= (7)/(Base7)

d.
* Since the PP approach relies on exposures (i.e. one exposure for each house year), the risks in each territory
are treated the same regardless of the AOI.
* In contrast, LR approach relies on premium (in the denominator of the loss ratio) which reflects the fact that
the insurer collects more premium for homes with higher AOI.
Using the current premium helps adjust for the distributional bias.
* Regardless, the LR method did not produce the correct relativities (the distortion coming from the variation in
AOI relativities being charged rather than the true variation).
If the current AOI relativities equaled the true AOI relativities, then the LR method will produce the true
territorial relativities.
See chapter 9

Exam 5A – Solutions to Independently Authored Questions - Test 2
Question 27 discussion (continued):
e. The calculation of the current exposure-weighted average AOI relativities by territory is shown below:
Weighted AOI Relativity
Charged
AOI

Exposures by Territory

AOI

Factor

Low

0.80

125

139

Medium

1.00

106

129

130

High

1.35

180

131

294

385

309

1.2088

1.0542

0.9553

Total
Wtd Avg AOI Rel by Terr

Adjusted Pure Premium Method
(1)

(2)

Terr
1
2
3
Total

Earned
Exposures
294
385
309
988

(4)= (2)*(3)
See chapter 9

(3)
Wtd Avg
AOI
Relativity
1.2088
1.0542
0.9553

(6)= (5)/(4);

(4)

(5)

Adjusted
Exposures
355.40
405.85
295.2
1,053.79

Loss & LAE
$15,234.96
$26,509.75
$23,255.29
$65,000.00

(7)= (6)/(Tot6);

(6)
Indicated
Pure
Premium
$42.87
$65.32
$78.78
$61.68

(7)
Indicated
Relativity
0.6950
1.0590
1.2772
1.0000

(8)
Indicated
Relativity
@Base
0.6563
1.0000
1.2061
0.9443

(8)= (7)/(Base7)

Question 28 discussion: Blooms: Application; Difficulty 1, LO 6 KS: Mechanics associated with
each method (including organization of the data)

See chapter 8

Exam 5A – Solutions to Independently Authored Questions - Test 2
Question 29 discussion: Blooms: Knowledge; Difficulty 1, LO 3 KS: Organization of data: calendar
year, policy year, accident year
1. The length of the experience period.
2. The expected length of the forecast period.
3. The changes in the mix of data between the experience and forecast periods.
(Section 3.5)

Exam 5A – Independently Authored Questions - Preparatory Test 3
General information about this exam
This practice test contains 30 questions consisting of computational and essay based questions.

Total Number of Qs:
Total Number of Points:

Essay
Questions
13
21.75

Computational
Questions
17
34.5

Total
30
56.25

Chapter 11: Special Classification ....................................Modlin, Werner ........... A. Basic Techniques for Ratemaking

Chapter 12: Credibility..........................................................Modlin, Werner ........... A. Basic Techniques for Ratemaking

Exam 5A – Independently Authored Questions - Test 3
Question 1 (1.50 points)
According to Werner and Modlin in “Basic Ratemaking”, briefly describe the major shortcomings of the following
three univariate approaches to classification ratemaking: the pure premium method, the loss ratio method and
the adjusted pure premium method.

Question 2 (4.0 points) An insurer is revising its current relativities for two rating variables used in pricing its
auto coverage because there is an uneven distribution of business along other classification dimensions not
being analyzed in the current review.
 There are only two rating variables: gender and territory.
 Gender has values male (with a rate relativity g1) and female (g2).
 Territory has values urban (t1) and rural (t2).
 The base levels relative to multiplicative indications are female and rural (hence g2 = 1.00 and t2 = 1.00).
The company actuary has determined that it is appropriate to use the balance principle applied to a
multiplicative minimum bias model and has complied the following data:
The actual loss costs (pure premiums) are as follows:
Urban
Rural
Total
Male
$650
$300
$528
Female
$250
$240
$244
Total
$497
$267
$400
The exposure distribution is as follows:
Urban
Rural
Male
170
90
Female
105
110
Total
275
200

Total
260
215
475

Using initial territorial relativities, t1= 1.86, and t2 = 1.0, calculate the revised relativities for Territory and
Gender after one full iteration.

Question 3 (2.25 points) According to Werner and Modlin in “Basic Ratemaking”, answer the following
questions:
a. (0.75 points) Briefly describe four benefits associated with the use of multivariate methods.
b. (0.75 points) Briefly describe how univariate methods stack up to the list of benefits in part a.
c. (0.75 points) Briefly describe how minimum bias methods stack up to the list of benefits in part a.

Question 4 (2.0 points) According to Werner and Modlin in “Basic Ratemaking”, GLM analysis is typically
performed on loss cost data, or preferably frequency and severity separately. This is unlike univariate
analysis of claims experience that is typically performed on either loss ratios or loss costs.
Briefly describe four, either statistical or practical reasons, supporting this practice:

Exam 5A – Independently Authored Questions - Test 3
Question 5 (1.50 points) According to Werner and Modlin in “Basic Ratemaking”, list three elements a modeler
must have access to solve a generalized linear model.
Question 6 (3.0 points) According to Werner and Modlin in “Basic Ratemaking”, answer the following questions.
a. (1.0 point) Briefly describe three ways in which statistical diagnostics generated from a generalized linear
model assist a modeler in evaluating potential rating variables.
b. (1.0 point) What is common statistical diagnostic for deciding whether a variable has a systematic effect on
losses and briefly describe how is it used?
c. (1.0 point) Briefly describe what deviance measures and how deviance measures are used?

Question 7 (1.0 point) According to Werner and Modlin in “Basic Ratemaking”, insurers using GLMs seek to
augment data that has already been collected and analyzed about their own policies with external data.
List four types of external data that can augment a multivariate analysis.
Question 8 (2.75 points) According to Werner and Modlin in “Basic Ratemaking”, answer the following questions.
a. (0.50 points) List the two general phases of territorial ratemaking.
b. (0.75 points) List three types of geographic units and briefly describe the advantage of using each type
c. (0.50 points) Historically, actuaries use univariate techniques (e.g. pure premium approach) to develop an
estimator for each geographic unit. Briefly describe two major issues with this approach.
d. (1.0 point) Briefly describe a better approach to using univariate techniques to develop an estimator for each
geographic unit.
Question 9 (1.75 points) According to Werner and Modlin in “Basic Ratemaking”, answer the following questions.
a. (0.75 points) Insurance providing protection against third-party liability claims are offered at the lowest limit,
i.e. basic limits (BL), and at higher limits, i.e. increased limits (IL). List three reasons to establish rate
relativities (i.e. to use increased limits ratemaking) for various limits.
b. (1.0 points) Lines of business in which IL ratemaking is used include private passenger and commercial auto
liability, umbrella, any commercial product offering liability coverage (e.g. contractor’s liability, professional
liability, etc). List and describe two types of policy limits offered, and how the limits are applied.
Question 10 (2.50 points) Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and the
data below, answer the following questions:
You are given the following 5,000 reported uncensored claims categorized by the size of the loss
Size of Loss Distribution
Size of Loss
X <= $ 100,000
$ 100,000 < X <= $ 250,000
$ 250,000 < X <= $ 500,000
$ 500,000 < X <= $ 1,000,000
Total

Reported
Claims
2,299
1,948
680
73
5,000

Reported
Losses
$107,629,223
$317,599,929
$222,743,514
$43,097,470
$691,070,136

Exam 5A – Independently Authored Questions - Test 3
Question 11 (2.50 points) Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and the
data below, answer the following questions:
An insurer writes policies at three policy limits ($100,000, $250,000, and $500,000) and the historical database
contains only censored losses. 5,000 claims censored at the 3 policy limits are shown below:
Censored Loss Distribution of Policies with Policy Limit
$100,000 Limit
Size of Loss
Claims
Losses
X <= $ 100,000
2019
$156,657,898
$ 100,000 < X <= $ 250,000
$ 250,000 < X <= $ 500,000
$ 500,000 < X <= $ 1,000,000
Total
2,019
$156,657,898

$250,000 Limit
Claims
Losses
690
$34,903,214
773
$142,767,479

1,463

$177,670,693

$500,000 Limit
Claims
Losses
712
$35,768,111
574
$90,009,422
232
$81,092,725
1,518

$206,870,258

a. (1.0 point) Compute the limited average severity at 100,000, i.e. LAS (100K)
b. (1.0 point) Compute the limited average severity at 250,000, i.e. LAS (250K)
c. (0.50 points) Compute the limited average severity at 500,000, i.e. LAS (500K)
Question 12 (1.0 point) According to Werner and Modlin in “Basic Ratemaking”, list four reasons why
deductibles are popular among both insureds and insurers:
Question 13 (1.50 points) Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and the
data below, compute the LER at $250 for the size of loss distribution of ground-up homeowners losses.
Size of Loss Distribution
(1)

Size of Loss
X <= $ 100
$ 100 < X <= $ 250
$ 250< X <= $ 500
$ 500 < X <= $ 1,000
$ 1,000 < X
Total

(2)
Reported
Claims
3,200
1,225
1,137
1,895
2,543
10,000

(3)
Ground-Up
Reported
Losses
$225,365
$199,588
$453,954
$1,531,938
$10,640,545
$13,051,390

Question 14 (1.50 points) Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and the
data below, compute the LER (i.e. the credit) to change from a $250 to a $500 deductible.

Deductible
Full Cov
$100
$250
$500
$1,000
Total

Reported
Claims
525
655
1,344
2,244
254
5,022

Net Reported
Losses
$700,220
$1,248,403
$2,910,672
$5,299,242
$909,755
$11,068,292

Net Reported
Losses
Assuming
$500 Ded
$547,924
$1,029,848
$2,594,621
$5,299,242
Unknown

Net Reported
Losses
Assuming
$250 Ded
$608,134
$1,156,269
$2,910,672
Unknown
Unknown

Exam 5A – Independently Authored Questions - Test 3
Question 15 (1.50 points) Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and the
data below, calculate of the premium discount for a policy with standard premium of $425,000.
Premium Range
$0
$5,000
$5,000
$100,000
$100,000 $500,000
$500,000
above

Prod
16.0%
11.0%
8.0%
6.0%

General
12.0%
9.0%
5.0%
4.0%

Taxes
3.0%
3.0%
3.0%
3.0%

Profit
5.0%
5.0%
5.0%
5.0%

Question 16 (2.50 points)
a. (1.0 points). Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and the data
below, compute the loss constant. Assume that loss constants are to be applied to risks with annual
premium of $2,500 or less in order to achieve a 70% loss ratio for both small and large risks.

Premium Range
$1
$2,500
$2,501
above

Policies
2,000
2,000

Premium
$2,000,000
$10,000,000

Reported
Loss
$1,500,000
$7,000,000

b. (0.50 points). Small WC risks tend to have less favorable loss experience, as a % of premium, than large
risks. List three reasons why this is the case.
Question 17 (2.5 points) Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and the
data below, answer the following questions.
Assume a home valued at $500,000 is insured only for $300,000 despite a coinsurance requirement of 80%.
a. (0.50 points). The indemnity payments and coinsurance penalties for a $200,000 loss.
b. (0.50 points). The indemnity payments and coinsurance penalties for a $300,000 loss.
c. (0.50 points). The indemnity payments and coinsurance penalties for a $350,000 loss.
d. (0.50 points). The indemnity payments and coinsurance penalties for a $450,000 loss.
e. (0.50 points). Briefly describe the magnitude of the co-insurance penalty for losses in the following ranges:
$0 - $F,
F - $cV,
$cV
Question 18 (1.5 points) According to Werner and Modlin in “Basic Ratemaking”, answer the following questions.
a. (0.50 points). Define the term credibility.
b. (1 point). List the three criteria upon which credibility (Z) is given to observed experience, assuming
homogenous risks.
Question 19 (1.50 points) According to Werner and Modlin in “Basic Ratemaking”, answer the following questions:
An actuary is using the classical credibility approach to determine the expected number of claims for the
observed experience to be fully credible. Assume the following about the observed experience:
• Full credibility is set so that the observed value is to be within +/-10% of the true value 90% of the time.
• Exposures are homogeneous, claim occurrence follows a Poisson distribution, and no variation in claim
costs exists.
• The observed pure premium of $250 is based on 30 claims.
• The pure premium of the related experience is $350.
a. (0.50 points). Compute the expected number of claims needed for full credibility.
b. (0.50 points). Calculate the credibility associated with the observed pure premium
c. (0.50 points). Calculate the credibility-weighted pure premium estimate.

Exam 5A – Independently Authored Questions - Test 3
Question 20 (1.0 point) According to Werner and Modlin in “Basic Ratemaking”, list three advantages to using
classical credibility approach to computing credibility and one disadvantage to using it.
Question 21 (2.0 points) Using the procedures describe by Werner and Modlin in “Basic Ratemaking”, and the data
below, answer the following questions.
•
The observed value is $250 based on 21 observations.
•
EVPV = 3.00, VHM = 0.75 and the prior mean is $275.
•
The observed pure premium of $250 is based on 100 claims.
a. (0.50 points). Briefly describe the goal of the Bühlmann credibility approach to estimating credibility.
b. (0.50 points). State the formula for how Z is calculated, and describe what each term represents.
c. (0.50 points). State four assumptions under which Bühlmann credibility applies
d. (0.50 points). Calculate the Bühlmann credibility-weighted estimate.
Question 22 (1.0 point) According to Werner and Modlin in “Basic Ratemaking”, Bühlmann Credibility and
Bayesian Credibility are uniquely related. List two specific unique relationships between Bühlmann Credibility
and Bayesian Credibility.
Question 23 (1.50 points) According to Werner and Modlin in “Basic Ratemaking”, list and briefly describe six
desirable qualities for a complement of credibility to possess.
Question 24 (1.0 point) Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and the
data below, calculate the complement of credibility for class 1 based on the rate change-for the larger group
applied to the present rate.
Indicated Present
Pure
Pure
Class
Exposure Losses
Premium Premium
1
100
$ 70,000
$700
$720
2
200
$180,000
$900
$920
3
300
$200,000
$667
$700
Total
600
$450,000
$750
$772
Notes: Both indicated and present pure premiums are at current cost levels.
Underlying losses are extension of exposures by present premiums.
Total present premium is ratio of total underlying to total exposures.

Underlying
Losses
$ 72,000
$184,000
$210,000
$463,200

Exam 5A – Independently Authored Questions - Test 3
Question 25 (3.0 points) Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and the
data below, calculate the complement of credibility for class 1 in state S using Harwayne’s method.

Question 26 (1.5 points) Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and the
data below, calculate the complement of credibility using the trended present rates method.
Consider the following data for 2006 policy rates:
Present pure premium rate
Annual inflation (trend)
Amount requested in last rate change
Effective date requested for last rate change
Amount approved by state regulators
Effective date actually implemented

$120
10%
+20%
1/1/04
+15%
3/1/04

Question 27 (1.0 point) Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and the
data below, calculate the complement of credibility using the competitor’s rates method.
Consider a competitor’s rate of $200.
 A Schedule P analysis suggests the competitor will run a 70% loss ratio.
 One’s own company has less underwriting expertise.
 One’s own company expects twelve percent more losses per exposure than the competitor.
Question 28 (2.0 points) Suppose one wishes to estimate the layer between $500,000 and $1,000,000 given losses of
$1,750,000 capped at $500,000 each. Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and the
data below, calculate the complement of credibility using the Increased Limits Factor Method.

Limit of Liability
$ 50,000
$ 100,000
$ 250,000
$ 500,000
$1,000,000

Increased Limits Factor
1.00
1.10
1.25
1.40
2.40

Exam 5A – Independently Authored Questions - Test 3
Question 29 (2.0 points) Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and the
data below, calculate the complement of credibility for the layer between $500,000 and $750,000 using a lower
limit analysis.
Assume losses capped at $500 are too sparse and thus you have chosen to used losses capped at a limit lower
than the attachment point (i.e. losses at $250,000 limit)
Assume losses capped at $250,000 are $1,750,000, and the ILFs below apply.

Limit of
Liability
$100,000
$250,000
$500,000
$750,000
$ 1,000,000

Increased
Limits
Factor
1.00
1.75
2.75
3.25
3.40

Question 30 (2.0 points) Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and the
data below, calculate the expected loss for the layer between $500,000 and $750,000 assuming a total limits
loss ratio of 60% and using a Limits Analysis.

Limit of
Liability (d )
$100,000
$250,000
$500,000
$750,000
$1,000,000
Total

Premium
$1,000,000
$500,000
$200,000
$200,000
$75,000
$1,975,000

ILF @
d
1.00
1.75
2.50
3.00
3.40

Exam 5A – Solutions to Independently Authored Questions - Test 3
Question 1 discussion: Blooms: Knowledge; Difficulty 1, LO 9 KS: Formulae and process for each
rating differential or relativity
In general, the major shortcoming of univariate approaches is their failure to accurately account for the effect of
other rating variables.
1. The PP approach does not consider exposure correlations with other rating variables.
2. The LR approach uses current premium to adjust for an uneven mix of business to the extent the premium
varies with risk, but premium is only an approximation since it deviates from true loss cost differentials.
3. The adjusted pure premium approach multiples exposures by the exposure-weighted average of all other
rating variables’ relativities to standardize data for the uneven mix of business before calculating the oneway relativities. But, this is an approximation to reflect all exposure correlations. See chapter 10
Question 2 discussion: Blooms: Application; Difficulty 2, LO 9 KS: : Fundamentals of univariate
and multivariate relativity analyses
Step 1: Write four equations with observed weighted loss costs on the left and indicated weighted loss
costs (the base rate, the exposure, and the indicated relativities) on the right.
Males
170 x $650 +90 x $300 = ($100 x 170 xg1 x t1 ) +( $100 x 90 x g1 x t2 )
Females
105 x $250 + 110 x $240 = $100 x 105 x g2 x t1 + $100 x 110 x g2 x t2
Urban
170 x $650+ 105 x $250 = $100 x 170 x g1 x t1 + $100 x 105 x g2 x t1
Rural
90 x $300+ 110 x $240 = $100 x 90 x g1 x xt2 + $100 x 110 x g2 x t2
Step 2: Choose initial (or seed) relativities for the levels of one of the rating variables.
A sensible seed is the univariate PP relativities.
The urban relativity is the total urban loss costs divided by the total rural loss costs:
t1 = 1.86 = ($497.27/$267.00)
t2 = 1.00.
Step 3: Substituting these seed values into the first two equations, solve for the first values of g1 and g2:
170 x $650 + 90 x $300 = ($100 x 170 x g1 x 1.86) + ($100 x 90 x g1 x 1.00)
$137,500 = ($31,620 x g1) + ($9,000 x g1)
$137,500 = $40,620 x g1
g1 = 3.39.
105 x $250 + 110 x $240 = ($100 x 105 x g2 x 1.86) + ($100 x 110 x g2 x 1.00)
$52,650 = ($19,530 x g2) + ($11,000 x g2)
$52,650 = $30,530 x g2
g2 = 1.72.
Step 4: Using these seed values for gender, g1 and g2, set up equations to solve for the new intermediate
values of t1 and t2:
170 x $650 + 105 x $250 = ($100 x 170 x 3.39 x t1) + ($100 x 105 x 1.72 x t1)
$136,750 = ($57,630 x t1) + (18,060 x t1)
$136,750 = $75,690 x t1
t1 = 1.81.
90 x $300 + 110 x $240 = ($100 x 90 x 3.39 x t2) + ($100 x 110 x 1.72 x t2)
$53,400 = ($30,510 x t2) + ($18,920 x t2)
$53,400 =$49,430 x t2
t2 = 1.08.
See chapter 10
Copyright  2014 by All 10, Inc.
Page 59

Exam 5A – Solutions to Independently Authored Questions - Test 3
Question 3 discussion: Blooms: Knowledge; Difficulty 2, LO 9 KS: Fundamentals of univariate and
multivariate relativity analyses
a1. The main benefit is the consideration of all rating variables simultaneously and automatically adjust
for exposure correlations between rating variables
a2. Allows for randomness in determining what is driving the cost of claims and to what degree. Raw data
contains systematic effects (a.k.a. signal) and unsystematic effects (a.k.a. noise), and the multivariate
method seeks to remove the noise and capture the signal. It allows assumptions to be modified depending
on what is being modeled (e.g. claim frequency, loss severity, or the probability a policy will be renewed).
a3. The methods produce model diagnostics (i.e. additional information about the certainty of results and the
appropriateness of the model fitted).
a4. They allow interaction between two or more rating variables.
b. Univariate methods:
 are distorted by distributional biases.
 require no assumptions about the nature of the underlying experience.
 produce a set of answers with no additional information about the certainty of the results.
 can incorporate interactions but only by expanding the analysis into two-way or three-way tables.
 scores high in terms of transparency (but is plagued by the inaccuracies of the method).
c. Minimum bias methods:
 account for an uneven mix of business but iterative calculations are computationally inefficient.
 require no assumptions about the structure of the model and the bias function.
 do not produce diagnostics
 scores high on transparency and outperforms univariate analysis in terms of accuracy (but does not
provide all of the benefits of full multivariate methods).
See chapter 10
Question 4 discussion: Blooms: Comprehension; Difficulty 1, LO 9 KS: Fundamentals of univariate
and multivariate relativity analyses
1. Modeling loss ratios requires premiums to be adjusted to current rate level at the granular level and that
can be practically difficult.
2. Experienced actuaries already have an a priori expectation of frequency and severity patterns (e.g., youthful
drivers have higher frequencies). In contrast, the loss ratio patterns are dependent on the current rates.
Thus, the actuary can better distinguish the signal from the noise when building models.
3. Loss ratio models become obsolete when rates and rating structures are changed.
4. There is no commonly accepted distribution for modeling loss ratios.
See chapter 10
Question 5 discussion: Blooms: Knowledge; Difficulty 1, LO 9 KS: Fundamentals of univariate and
multivariate relativity analyses
1. A modeling dataset with a suitable number of observations of the response variable and associated
predictor variables to be considered for modeling.
2. A link function must be chosen to define the relationship between the systematic and random components
(i.e. to define the relationship between the expected response variable (e.g., claim severity) and the linear
combination of the predictor variables (e.g., age of home, amount of insurance, etc.)).
3. A distribution of the underlying random process must be chosen, typically a member of the exponential
family of distributions (e.g., normal, Poisson, gamma, binomial, inverse Gaussian)
See chapter 10

Exam 5A – Solutions to Independently Authored Questions - Test 3
Question 6 discussion: Blooms: Comprehension; Difficulty 2 LO 9 KS: Fundamentals of univariate
and multivariate relativity analyses
a. Statistical significance is an important criterion for evaluating rating variables, and statistical diagnostics
are a major byproduct of GLMs. Statistical diagnostics:
* aid the modeler in understanding the certainty of the results and the appropriateness of the model.
* can determine if a predictive variable has a systematic effect on losses (and be retained in the model).
* assess the modeler’s assumptions around the link function and error term.
b. A common statistical diagnostic for deciding whether a variable has a systematic effect on losses is the
standard errors calculation.
* “standard errors are an indicator of the speed with which the log-likelihood falls from the maximum given a
change in parameter.”
* 2 standard errors from the parameter estimates are akin to a 95% confidence interval.
i. the GLM parameter estimate is a point estimate
ii. standard errors show the range in which the modeler can be 95% confident the true answer lies within.
c. Deviance measures (an additional diagnostic) assess the statistical significance of a predictor variable.
* Deviance measures of how much fitted values differ from the observations.
* Deviance tests are used when comparing nested models (one is a subset of the other) to assess whether the
additional variable(s) in the broader model are worth including.
See chapter 10
Question 7 discussion: Blooms: Knowledge; Difficulty 1, LO 9 KS: Fundamentals of univariate and
multivariate relativity analyses
1. geo-demographics (e.g. population density of an area, average length of home ownership of an area);
2. weather (e.g. average rainfall or number of days below freezing of a given area);
3. property characteristics (e.g. square footage of a home or business, quality of the responding fire
department);
4. information about insured individuals or business (e.g. credit information, occupation).
See chapter 10
Question 8 discussion: Blooms: Comprehension; Difficulty 2, LO 11 KS: Common policy provisions
a. Territorial ratemaking generally involves two phases:
I. Establishing territorial boundaries
II. Determining rate relativities for the territories
b. Three common types of geographical units are postal/zip codes, counties and census blocks.
i. zip codes have the advantage of being readily available but the disadvantage of changing over time.
ii. counties have the advantage of being static and readily available, but due to their large size, tend to contain
very heterogeneous risks.
iii. census blocks are static over time, but require a process to map insurance policies to the census blocks.
c. Two major issues with using the a univariate technique to develop an estimator for each geographic unit are:
1. The geographic estimator reflects both the signal and the noise.
Since geographic units tend to be small, the data is sparse and the resulting loss ratios or pure premiums or
both will be too volatile to distinguish the noise from the signal.
2. Since location is highly correlated with other non-geographic factors, the resulting estimator is biased.
Copyright  2014 by All 10, Inc.
Page 61

Exam 5A – Solutions to Independently Authored Questions - Test 3
Question 8 discussion (continued):
d. A better approach involves using a multivariate model (e.g. a GLM) on loss cost data using a variety of nongeographic and geographic explanatory variables.
1. Non-geographic variables include rating variables (e.g. age of insured, claim history) as well as other
explanatory variables not used in rating.
2. Geographic variables include geo-demographic variables (e.g. population density) and geo-physical
variables (e.g. average rainfall).
See chapter 11
Question 9 discussion: Blooms: Knowledge; Difficulty 1, LO 11 KS: Common policy provisions
a. Reasons to establish rate relativities (i.e. to use increased limits ratemaking) for various limits:
1. As personal wealth grows, individuals have more assets to protect and need more insurance coverage.
2. Inflation drives up costs and trends in costs have a greater impact on IL losses than on BL losses.
3. The propensity for lawsuits and the amount of jury awards have increased significantly (i.e. social inflation)
and this has a disproportionate impact on IL losses.
b. Two types of policy limits offered:
1. Single limits: Refers to the total amount the insurer will pay for a single claim (e.g. if an umbrella policy has a
limit of $1,000,000, then the policy will only pay up to $1,000,000 for any one claim).
2. Compound limits: Applies two or more limits to the covered losses. Examples:
i. A split limit: includes a per claimant and a per occurrence limit (e.g. in personal auto insurance, a split limit
for bodily injury liability of $15,000/$30,000 means that if the insured causes an accident, the policy will pay
each injured party up to $15,000 with total payment to all injured parties not to exceed $30,000).
ii. An occurrence/aggregate limit: limits the amount payable for any one occurrence and for all occurrences
incurred during the policy period (e.g. if an annual professional liability policy has a limit of
$1,000,000/$3,000,000, the policy will not pay more than $1,000,000 for any single occurrence and will not
pay more than $3,000,000 for all occurrences incurred during the policy period).
See chapter 11
Question 10 discussion: Blooms: Application; Difficulty 2, LO 11 KS: Formula
a. LAS ($100,000) is calculated by capping every claim at $100,000 and dividing by the total number of claims.
 All 2,299 claims in the first interval have individual sizes of loss less than $100,000, so they are uncapped.
 The other 2,701 claims in the other three intervals have individual sizes of loss that exceed $100,000 and
are capped at $100,000 [$270,100,000 (= 2,701 x $100,000)].
 LAS ($100,000) = the Sum ($377,729,223 = $107,629,223 + $270,100,000)/ total claim count.

LAS ($100 K )

$107, 629, 223 + (1,948 + 680 + 73) x $100, 000
= $75,546
5, 000

b. Using this technique, the ILF for $250,000 is calculated as follows: Indicated ILF($250K)=

LAS ($250 K )

$107, 629, 223 + $317,599,929 + (680 + 73) x $250, 000
= $122, 696
5, 000

c. Indicated ILF($250K)=

LAS($250K) $122,696
=
= 1.62
LAS($100K) $75,546

See chapter 11

LAS($250K)
LAS($100K)

Exam 5A – Solutions to Independently Authored Questions - Test 3
Question 11 discussion: Blooms: Application; Difficulty 2, LO 11 KS: Formula
a. To calculate LAS by limit, calculate a LAS for each layer of loss and combine the estimates for each layer
taking into consideration the probability of a claim occurring in the layer. The LAS of each layer is based
solely on loss data from policies with limits as high as or higher than the upper limit of the layer.
When calculating the LAS ($100K), use the experience from all policies limits censored at $100,000:

LAS ($100 K ) =
=

$156, 657,898 + $34,903, 214 + $35, 768,111 + $100, 000 * (773 + 574 + 232)
5, 000
$385, 229, 223
= $77, 046
5, 000

b. Calculating LAS ($250,000)
Step 1: Determine the losses in the $100K - $250 K layer.
i. Policies with a limit of $100,000 cannot contribute any losses to that layer and the data is not used.
ii. Of the 1,463 claims with policies having a $250K limit, 773 claims have losses in the $100K to $250K layer.
Total censored losses for those 773 claims are $142,767,479.
Eliminating the first $100K of each of those losses results in losses in the $100K to $250K layer.
$142,767,479 - 773 x $100,000 = $65,467,479
iii. Policies with a limit of $500K also contribute loss dollars to the $100K to $250K layer.
Of the 1,518 claims associated with a limit of $500K limit, 574 have losses in the $100K to $250K layer.
These claims contribute $32,609,422 (=$90,009,422 – 574 x $100,000) of losses to the layer.
Another 232 claims exceed $250,000, and each contributes $150,000 to the $100K to $250K layer.
$34,800,000 = 232x ($250,000- $100,000)
The sum of the above values are the losses in the $100K to $250K layer:
$65,467,479+ $32,609,422+ $34,800,000 = $132,876,901.
These losses were from 1,579 (=773+574+232) claims. Thus, LAS(100K-250K) =

$84,153 =

$132,876,901
1,579

Step 2: Before combining this with the LAS ($100K), adjust for the fact that these losses are based on a subset
of the claims used to calculate the LAS ($100K).
The adjustment involves calculating the probability that the loss will exceed $100K, given that a claim occurs.
Since the actuary cannot know whether or not the claims from the policies with a $100K limit would have
exceeded $100K, that data is not used for this calculation. To adjust this, the LAS for the $100K to $250K
layer can be multiplied by the following probability:

Pr($100K ≤ X ≤ $250K) =

773+ 574 + 232 1,579
.
=
1,463+1,518
2,981

The values above are the numbers of claims from the 250K policy limit and 500K policy limit for losses > 100K.
This is equivalent to dividing the losses in the layer by the total claim count for those policies:

1,579 $132,876,901
=
2,981
2,981
Thus, LAS($250K) = $77,046 + $44,575 = $121,621
$44,575 = $84,153 *

c. Calculating LAS ($500,000) using the same techniques: For losses in the $250K to $500K layer, only policies with a

$81,092,725 - 232 * $250,000
1,518
LAS($500K) = $77,046 + $44,575 + $15,213 = $136,834 See chapter 11

$500K limit or greater can be used:
Thus,

$15,213 =

Exam 5A – Solutions to Independently Authored Questions - Test 3
Question 12 discussion: Blooms: Knowledge; Difficulty 1, LO 11 KS: Layers of loss
1. Premium reduction: A deductible reduces the rate as the insured pays a portion of the losses.
2. Eliminates small nuisance claims: Deductibles minimize the filing of small claims (and the expense
associated with investigating and handling small claims, which is often greater than the claim amount).
3. Provides incentive for loss control: Since the insured is responsible for the first layer of loss, the insured
has a financial incentive to avoid losses.
4. Controls catastrophic exposure: For insurers writing a large number of policies in cat prone areas, the
use of large cat deductibles can reduce its exposure to loss.
See chapter 11
Question 13 discussion: Blooms: Application Difficulty 1, LO 11 KS: Layers of loss
To calculate LER ($250), compute the amount of losses in each layer that will be eliminated by the deductible.
 The first two rows contain losses less than $250 and are completely eliminated by the deductible.
 The remaining rows contain individual losses that are at least $250; thus $250 will be eliminated for each
of the 5,575 claims (=1,137+1,895+2,543).
The LER = losses eliminated/ total losses:

LER ($250)

($225,365 + $199,588) + $250 × (1,137 + 1,895 + 2,543)
= 0.139
$13, 051,390

See chapter 11
(1)

Size of Loss
X <= $ 100
$ 100 < X <= $ 250
$ 250< X <= $ 500
$ 500 < X <= $ 1,000
$ 1,000 < X
Total

(4)Losses<250=
(4)Losses>=250=
(5)

(2)

Reported
Claims
3,200
1,225
1,137
1,895
2,543
10,000

(3)
Ground-Up
Reported
Losses
$225,365
$199,588
$453,954
$1,531,938
$10,640,545
$13,051,390
(5) LER =

(4)
Losses
Eliminated By
$250
Deductible
$225,365
$199,588
$284,250
$473,750
$635,750
$1,818,703
0.139

(3)
(2) x $250
(Tot4) / (Tot3)

Exam 5A – Solutions to Independently Authored Questions - Test 3
Question 14 discussion: Blooms: Application; Difficulty 1, LO 11 KS: Layers of loss
Data from policies with deductibles greater than the deductible being priced cannot be used to calculate the
LER. For example:
 data from policies with a $500 deductible cannot be used to determine LERs for a $250 or $100
deductible, however
 data from policies with deductibles less than the deductible being priced can be used to determine
LERs (e.g. data from policies with a $500 deductible can be used to determine the LER associated with
moving from a $750 deductible to a $1,000 deductible).
Calculating the credit to change from a $250 to a $500 deductible.
(1)

(2)

Deductible
Full Cov
$100
$250
$500
$1,000
Total

Reported
Claims
525
655
1,344
2,244
254
5,022

(3)

(4)
Net Reported
Losses
Assuming
$500 Ded
$547,924
$1,029,848
$2,594,621
$5,299,242
Unknown

(5)
Net Reported
Losses
Assuming
$250 Ded
$608,134
$1,156,269
$2,910,672
Unknown
Unknown

Net Reported
Losses
$700,220
$1,248,403
$2,910,672
$5,299,242
$909,755
$11,068,292
(7) Net Reported Losses for Ded <=$250
(8) Losses Eliminated <=$250 Ded
(9)LER

(6)
Losses
Eliminated
Moving from
$250 to $500
$60,210
$126,421
$316,051
Unknown
Unknown
$4,675,075
$502,682
0.108

(3)= Net of the deductible
(4) =(3) Adjusted to a $500 deductible (5)=(3) Adjusted to a $250 deductible
(6)= (5) - (4) (7)= Sum of (5) for $0, $100, $250 Deductibles (8)=Sum of (6) for $0, $100, $250 Deductibles
 Each row contains data for policies with different deductible amounts.
 The analysis can only use policies with deductibles of $250 or less (since the goal is to determine the
losses eliminated when changing from a $250 to a $500 deductible)
 Columns 4 and 5 contain the net reported losses in Column 3 restated to $500 and $250 deductible
levels, respectively.
Columns 4 and 5 are not Column 3 minus the product of Column 2 and the assumed deductible.
This is because not every reported loss exceeds the assumed deductible.
The losses in Columns 4 and 5 are based on an assumed distribution of losses by deductible and size
of loss, and cannot be recreated given the data shown.
See chapter 11
Question 15 discussion: Blooms: Application; Difficulty 1, LO 11 KS: :Formula
Workers Compensation Premium Discount Example
(1)
(2)
(3)
(4)
(5)
Premium
Premium Range
in Range
Prod
General
16.0%
12.0%
$0
$5,000
$5,000
$5,000
$100,000 $95,000
11.0%
9.0%
$100,000 $500,000 $325,000
8.0%
5.0%
$500,000
above
6.0%
4.0%
Standard Premium
$425,000

(6)

(7)

(8)

Taxes
3.0%
3.0%
3.0%
3.0%

Profit
5.0%
5.0%
5.0%
5.0%

Total
36.0%
28.0%
21.0%
18.0%

(3)= Min of [(2) - (1), Standard Premium - Sum Prior(3)]
(10)= (9)/[1.0 -(6) - (7)]
(9)= (8Row 1)-(8)
See chapter 11

(9)
Expense
Reduction
0.0%
8.0%
15.0%
18.0%

(10)
Discount
%
0.0%
8.7%
16.3%
19.6%

(11)=

(11)
Premium
Discount
$0
$8,261
$52,989
$0
$61,250

(3) x (10)

Exam 5A – Solutions to Independently Authored Questions - Test 3
Question 16 discussion: Blooms: Application; Difficulty 2, LO 11 KS: Formula
a.
Workers Compensation Loss Constant Example
(1)
(2)
(3)
(4)

Premium Range
$1
$2,500
$2,501
above

Policies
2,000
2,000

Premium
$2,000,000
$10,000,000

(5)
Reported
Loss
$1,500,000
$7,000,000

(6)
Initial
Loss
Ratio
75.0%
70.0%

(7)
Target
Loss
Ratio
70.0%
70.0%

(8)

(9)

Premium
Shortfall
$142,857

Loss
Constant
$71.43
$0.00

b. Small companies:
1. have less sophisticated safety programs because of the large amount of capital to implement and
maintain.
2. may lack programs to help injured workers return to work.
3. premiums are unaffected or slightly impacted by experience rating; small insureds may not be eligible
for ER and may have less incentive to prevent or control injuries than large insureds.
See chapter 11
Question 17 discussion: Blooms: Application Difficulty 2, LO 11 KS: Formula
a. The coinsurance requirement of 80% is $400,000. Since F is $300,000 a coinsurance deficiency exists and
a (the apportionment ratio) = 0.75 (=$300,000 / $400,000).
The indemnity payments and coinsurance penalties for a $200,000 loss are:

F
$300, 000
=$200, 000 ×
=$150, 000
cV
$400, 000
=
e L=
- I $200, 000 - $150, 000
= $50, 000
I =L ×

b. The indemnity payments and coinsurance penalties for a $300,000 loss:

F
$300, 000
=
$300, 000 ×
=
$225, 000
cV
$400, 000
=
e L=
- I $300, 000 - $225, 000
= $75, 000
I =×
L

c. The following are the indemnity payments and coinsurance penalties for a $350,000 loss:

F
$300, 000
=$350, 000 ×
= $262,500
cV
$400, 000
=
e F=
- I $300, 000 - $262,500
= $37,500
I =L ×

d. The following are the indemnity payments and coinsurance penalties for a $450,000 loss:

F
$300, 000
=
$450, 000 ×
=
$337,500, but $337,500 > F , so I =
F=
$300, 000
cV
$400, 000
e = F − I = $300, 000 - $300, 000 = $0.
I=
L×

e. The magnitude of the co-insurance penalty:
 the dollar coinsurance penalty increases linearly between $0 and F (where the penalty is the largest).
 the penalty decreases for loss sizes between F and cV.
 there is no penalty for losses larger than the cV, but the insured suffers a penalty in that the payment
does not cover the total loss.
See chapter 11
Copyright  2014 by All 10, Inc.
Page 66

Exam 5A – Solutions to Independently Authored Questions - Test 3
Question 18 discussion: Blooms: Knowledge; Difficulty 1, LO 9 KS: Credibility and complements of
credibility
a. Credibility is “a measure of the predictive value in a given application that the actuary attaches to a particular
body of data.”
b. The credibility (Z) given to observed experience, assuming homogenous risks, is based on three criteria:
1. 0 < Z < 1 (i.e. no negative credibility and capped at fully credible).
2. Z should increase as the number of risks increases (all else being equal).
3. Z should increase at a non-increasing rate.
See chapter 12
Question 19 discussion: Blooms: Application; Difficulty 1, LO 9 KS: Credibility and complements
of credibility
a. Since the actuary regards the loss experience fully credible if there is a 90% probability that the
observed experience is within 10% of its expected value.
 This is equivalent to a 95% probability that observed losses are no more than 10% above the mean.
In the SN table, the 95th percentile is 1.645 standard deviations above the mean; therefore, the expected
2

 1.645 
number of claims needed for full credibility =
is: E (Y ) =
 270
 0.10 


If the number of observed claims > the standard for full credibility (270 in the example), the measure of
credibility (Z)
is 1.00: Z 1.00 where Y ≥ E (Y )
=

b. If the number of observed claims is < the standard for full credibility, the square root rule is applied to

=
calculate Z: Z

Y
, where Y < E (Y ).
E (Y )

In the example, if the observed number of claims is 30,
=
Z

30
= 0.334.
270

c. The credibility-weighted estimate is $316.6 (=0.334 x $250 + (1-0.334) x $350). See chapter 12
Question 20 discussion: Blooms: Knowledge; Difficulty 1, LO 9 KS: Credibility and complements of
credibility
3 Advantages:
1. It is the most commonly used and thus generally accepted.
2. The data required is readily available.
3. The computations are straightforward.
Disadvantage: Simplifying assumptions may not be true in practice (e.g. no variation in the size of losses).
See chapter 12
Question 21 discussion: Blooms: Knowledge & Application; Difficulty 1, LO 9 KS: Credibility and
complements of credibility
a. The goal of Bühlmann credibility (a.k.a. least squares credibility): minimize the square of the error
between the estimate and the true expected value of the quantity being estimated.
b. Z is defined as follows: Z =

N
, where N represents the number of observations and K is the ratio of
N+K

the expected value of the process variance (EVPV) to the variance of the hypothetical means (VHM) (i.e.
the ratio of the average risk variance to the variance between risks).

Exam 5A – Solutions to Independently Authored Questions - Test 3
Question 21 discussion (continued):
c. The assumptions under the Bühlmann credibility formula are as follows:
* (1.0 - Z) is applied to the prior mean.
* Risk parameters and risk process do not shift over time.
* The EVPV of the sum of N observations increases with N.
* The VHM of the sum of N observations increases with N.
d. K
=

EVPV 3.00
21
== 4.00,
=
Z
= 0.84; and
VHM
.75
21 + 4.00

Bühlmann Credibility-weighted Estimate = Estimate = Z x Observed Experience + (1.0 - Z) x Prior Mean.
Bühlmann Credibility-weighted Estimate = 0.84 x $250 + (1- 0.84) x $275 = $254.
See chapter 12
Question 22 discussion: Blooms: Knowledge; Difficulty 1, LO 9 KS: Credibility and complements of
credibility
1. Bühlmann credibility is the weighted least squares line associated with the Bayesian estimate.
2. The Bayesian estimate is equivalent to the LSC estimate in certain mathematical situations.
See chapter 12
Question 23 discussion: Blooms: Comprehension Difficulty 1, LO 9 KS: Credibility and
complements of credibility
1. Accurate: A COC that causes rates to have a low error variance around the future expected losses
being estimated is considered accurate.
2. Unbiased: Differences between the complement and the observed experience should average to 0 over time.
Accurate vs. Unbiased:
 An accurate statistic may be consistently higher or lower than the following year’s losses, but it is
always close.
 An unbiased statistic varies randomly around the following year’s losses over many successive years,
but it may not be close.
3. Independent: The complement should also be statistically independent from the base statistic (otherwise,
any error in the base statistic can be compounded).
4 and 5. Available and Easy to Compute: If not, the COC is not practical and justification to a third party (e.g.
regulator) for approval is needed.
6. Logical relationship (to the observed experience): is easier to support to any third party reviewing the
actuarial justification.
Question 24 discussion: Blooms: Application; Difficulty 1, LO 9 KS: Credibility and complements
of credibility



Larger Group Indicated Loss Cost
C = Current Loss Cost of Subject Experience × 

 Larger Group Current Average Loss Cost 
Using this formula and the given data, the complement for Class 1 would be: $720 * ( $750/ $772] =
$699.4 8
See chapter 12

Exam 5A – Solutions to Independently Authored Questions - Test 3
Question 25 discussion: Blooms: Application; Difficulty 3, LO 9 KS: Credibility and complements
of credibility
For Harwayne’s full method, one first computes
PT = [100 * 3.67 + 180 * 4.00]/[100 + 180] = 3.88
PU = [100 * 2.22 + 180 * 4.09]/[100 + 180] = 3.42
Then, one computes the state adjustment factors:
FT = 2.86/3.88 = .737 and FU = 2.86*3.42 = .836.
The next step is to compute the other states’ adjusted Class 1 rates:
P'1,T = .737 * 3.67 = 2.70 and P'1,U = .836/2.22 = 1.86.
The last step is to weight the two states’ adjusted rates with their Class 1 exposures to produce
C= [2.70 * 150 + 1.86 * 90]/[150 + 90] =2.39. This is Harwayne’s complement of the credibility.
See chapter 12
Question 26 discussion: Blooms: Application Difficulty 1, LO 9 KS: Credibility and complements of
credibility
2 1.20 
* (1.1) 
$152
=
=
The complement of the credibility
would be C $120
1.15 

See chapter 12
Question 27 discussion: Blooms: Application Difficulty 1, LO 9 KS: Credibility and complements of
credibility
The complement would be $200 * .70 * 1.12 = $156.80
See chapter 12
Question 28 discussion: Blooms: Application; Difficulty 1, LO 9 KS: Credibility and complements
of credibility
Suppose one wishes to estimate the layer between $500,000 and $1,000,000 given losses of $2,000,000
capped at $500,000 each. The complement using increased limits would be

 ILFA + L 
 2.4 
C PA 
=
=
− 1 $1,750,000  =
− 1 $1,250,000
 ILFA

 1.4 
See chapter 12
Question 29 discussion: Blooms: Application; Difficulty 1, LO 9 KS: Credibility and complements
of credibility
___ 
ILFA+L − ILFA 
C= Ld × 
 ,where
ILFd



•

Ld is the loss cost capped at the lower limit, d; • ILFA is the ILF for the attachment point A;

•

ILFd is the ILF for the lower limit, d;

•

ILFA+L is the ILF for the sum of the attachment point A and the excess insurer’s limit of liability L (i.e.
this sum is the top of the excess layer being priced).

 3.25 - 2.75 
C = $1, 750, 000 × 
 = $500, 000
 1.75

See chapter 12

Exam 5A – Solutions to Independently Authored Questions - Test 3
Question 30 discussion: Blooms: Application; Difficulty 1, LO 9 KS: Credibility and complements
of credibility
When insurers sell policies with a wide variety of policy limits.
 Some policy limits fall below the attachment point and some extend beyond the top of the excess layer.
 Thus, each policy’s limit and ILF needs to be considered in the calculation of the complement.
i. Policies at each limit of coverage are analyzed separately.
ii. Estimated losses in a layer are computed using the premium and expected loss ratio in that layer.
iii. An ILF analysis on each first dollar limit’s loss costs is performed.

C=
LR × ∑ Pd ×

( ILFmin( d , A+ L ) - ILFA+ L )
ILFd

d≤A

, where

LR = Total loss ratio, and Pd= Total premium for policies with limit d.
Thus, expected loss for the layer $500,000 to $750,000 are computed as follows:
(1)

(2)

(3)

(4) = (2)*(3)

(5)

(6)

(7)

(8)

(9) = (4)*(8)
Expected

Limit of
Liability (d)

Premium

Expected
Loss Ratio

Expected
Capped
Losses

ILF @
d

ILF @
A

ILF @
A+L

% Loss
In Layer

Loss in
Layer

100,000

$1,000,000

60.0%

$ 600,000

1.00

2.50

3.00

0.0%

$
$
$

250,000
500,000
750,000

$ 500,000
$ 200,000
$ 200,000

60.0%
60.0%
60.0%

$ 300,000
$ 120,000
$ 120,000

1.75
2.50
3.00

2.50
2.50
2.50

3.00
3.00
3.00

0.0%
0.0%
16.7%

$ 1,000,000
Total

$ 75,000
$1,975,000

60.0%

$ 45,000

3.40

2.50

3.00

14.7%

(8): if d< =A then 0.0%; if A < d < A +L then [(5)- (6)]/(5); if d >A+L then [(7)- (6)]/(5)

See chapter 12

$20,040
$6,615
$26,655

Exam 5 – Independently Authored Questions - Preparatory Test 4
General information about this exam
This practice test contains 20 questions consisting of computational and essay based questions.

Total Number of Qs:
Total Number of Points:

Essay
Questions
6
9

Computational
Questions
14
40

Total
20
49

Articles covered on this exam:
Article .................................................... Author ..............................................................
Chapter 13: Other Considerations ...................................Modlin, Werner ........... A. Basic Techniques for Ratemaking
Chapter 14: Implementation ..............................................Modlin, Werner ........... A. Basic Techniques for Ratemaking
Personal Auto Premiums: Asset Share Pricing .......... Feldblum ........................ A. Basic Techniques for Ratemaking

Appendix D: Workers Compensation Indication....... Modlin, Werner ........... A. Basic Techniques for Ratemaking

Exam 5 – Independently Authored Questions - Test 4
Question 1 (2.0 points) According to Werner and Modlin in “Basic Ratemaking”, answer the following questions:
a. (1.0 points). List four regulatory constraints that cause insurers to implement rates different from those
indicated by their ratemaking analyses.
b. (1.0 points). List four insurer actions that can be taken with respect to regulatory restrictions.

Question 2 (1.5 points) According to Werner and Modlin in “Basic Ratemaking”, answer the following questions:
a. (1.0 points). Operational constraints can make it difficult for an insurer to implement the actuarially indicated
rate change. List two types of operational constraints that insurers can face.
b. (.50 points). Briefly describe the best course of action an insurer can undertake when operational
constraints arise.

Question 3 (1.25 points) According to Werner and Modlin in “Basic Ratemaking”, list five factors that affect an
insured’s propensity to renew an existing policy or purchase a new policy.

Question 4 (4.0 points) Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and the data
below, answer the following questions:
 Assume an insurer issues 30,000 quotes in the month of April and generates 7,500 new policies.
 Assume 40,000 policies are up for renewal in the month of May and 35,000 renew
 Assume there were 240,000 policies at the beginning of the June, 7,600 new policies were added and
5,200 policies were lost during June.
a. (1.0 point). Compute the insurer’s close ratio, and briefly describe why it is important to understand the
denominator of the ratio.
b. (1.0 point). Compute the insurer’s retention ratio, and briefly describe two desirable aspects of renewal
policyholders vs. new policyholders from the insurer’s perspective.
c. (1.0 point). Briefly describe why insurers rely on closely monitoring close ratios and retention ratios and what
impact rate changes can have upon close and retention ratios
d. (1.0 point). Compute the insurer’s growth ratio, and describe one way that growth can be impacted other
than by price.
Question 5 (1.25 points) According to Werner and Modlin in “Basic Ratemaking”, briefly describe two types of
non-pricing solutions an insurer can implement when the indicated average premium per exposure does not equal
the projected average premium per exposure. For each type, list several ways in which balance can be achieved.
Question 6 (1.0 points) Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and the data
below, compute the proposed fixed expense fee.
Expense Type
Total Expense
% Fixed
Commission
0.19
0%
Other Acquisition
0.04
85%
General
0.03
95%
Taxes, Licenses and Fees
0.02
85%
Profit and Contingencies
0.08
0%
Projected Average Premium per Exposure = $250.00

Exam 5 – Independently Authored Questions - Test 4
Question 7 (4.5 points) Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and the data
below, answer the following questions.
Assume the insurer relies on the following data to select proposed rate differentials for each rating variable:
R1

Current
Differential

Indicated
Differentia

Competitor
Differential

Proposed
Differential

Current
Indicated Competitor Proposed
Differential Differential Differential Differential

1
2
3

0.8000
1.0000
1.2000

0.9000
1.0000
1.2500

0.9200
1.0000
1.2500

0.9000
1.0000
1.2500

A
B
C

1.0000
1.0500
1.2000

1.0000
0.9000
1.3000

1.0000
0.9500
1.6500

1.0000
0.9500
1.3000

Current
Discount

Indicated
Discount

Competitor
Discount

Proposed
Discount

Current
Discount

Indicated
Discount

Competitor
Discount

Proposed
Discount

Y
N

5.0%
0.0%

4.0%
0.0%

5.0%
0.0%

Y
N

10.0%
0.0%

2.5%
0.0%

7.5%
0.0%

5.0%
0.0%

Exposures and proposed rate differentials and discounts are given below
Exposures

10,000
7,500
3,000
9,000
20,000
5,000

1
2
3
1
2
3

A
A
A
B
B
B

Y
Y
Y
Y
Y
Y

Assume the following:

The proposed rating algorithm for a given risk is defined as follows:

PP , ijkm = [ BP × R1P , i × R 2 P , j × (1.0 − D1P , k − D 2 P ,m ) + AP ] × X ijkm

AP = 25 , and the proposed average premium is $250.



The ‘seed’ base rate is $215.00,



The current average premium using extension of exposures on current rates is $242.13.

a. (3 points). Compute the proposed base rate using the extension of exposure technique.
b. (1.5 points). Compute the proposed base rate using the loss ratio method, assuming the indicated % change
in average premium is 3.25%.

Exam 5 – Independently Authored Questions - Test 4
Question 8 (2.0 points) Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and the data
below, answer the following question.
Assume an insurer relies on the following data to compute the average proposed rate differential across all rating
variables:
Proposed Differentials Wtd by Exposures
(1)
(2)
R1
1
2
3
Total

Exposures
152,500
570,000
147,000
869,500

(1)

(2)

R2
A
B
C
Total

Exposures
235,000
480,000
154,500
869,500

(1)

(2)

D1
Y
N
Total

Exposures
156,625
712,875
869,500

(1)

(2)

D2
Y
N
Total

Exposures
153,625
715,875
869,500

(3)
Proposed
Differential
0.9000
1.0000
1.2500
1.0247
(3)
Proposed
Differential
1.0000
0.9500
1.3000
1.0257
(3)
Proposed
Discount
0.0500
0.0000
0.0090
(3)
Proposed
Discount
0.0500
0.0000
0.0088

___



Let
___

S P can be approximated as the product of the average differential of each of the rating variables:

SP ≈


∑X

× R1P ,i

∑X

× R 2 Pj


 ∑ X k × D1P ,k ∑ X m × D 2 P ,m  


× 1.0 −  k
+ m


X
X


 

AP = 25 and the proposed average premium is $250.

Compute the proposed base rate using the Approximated Average Rate Differential Method.

Exam 5 – Independently Authored Questions - Test 4
Question 9 (2.0 points) Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and the data
below, answer the following question.
 The Current Base Rate = $210.00
 The Current Average Premium = $242.13
 The Target Change in Average Premium = 3.25%
 The Proposed Additive Premium per Policy = $25.00
Proposed Average Change in Differentials (Using Exposures)
(1)
(2)
(3)
(4)
Current
Proposed
D1
Exposures
Discount
Discount
Y
156,625
0.0500
0.0500
N
712,875
0.0000
0.0000
Total
869,500
0.0090
0.0090
(1)

(2)

D2
Exposures
Y
153,625
N
715,875
Total
869,500
(Tot3) = (3) Weighted by (2)
(5)

(6)

R1
1
2
3
Total

Exposures
152,500
570,000
147,000
869,500

(5)

(6)

R2
A
B
C
Total

Exposures
235,000
480,000
154,500
869,500

(10)

(11)

1-D1-D2
Total

Exposures
235,000

(3)
(4)
Current
Proposed
Discount
Discount
0.1000
0.0500
0.0000
0.0000
0.0177
0.0088
(Tot4) = (4) Weighted by (2)
(7)
Current
Differential
0.8000
1.0000
1.2000
0.9987

(8)
Proposed
Differential
0.9000
1.0000
1.2500
1.0247

(7)
Current
Differential
1.0000
1.0500
1.2000
1.0631

(8)
Proposed
Differential
1.0000
0.9500
1.3000
1.0257

(12)
Current
Differential
0.9733

(13)
Proposed
Differential
0.9822

Compute the proposed base rate using the Approximated Change in Average Rate Differential Method.

Exam 5 – Independently Authored Questions - Test 4
Question 10 (4.5 points) Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and the
data below, answer the following question.
An actuary has decided to limit the premium impact caused by the change in rate differentials for a rating
variable. In particular, the proposed rate relativity for any level that produces a premium impact that exceeds
the desired maximum premium increase of 20% must be adjusted.
 Assume there is no additive premium.
 You are given the following information:
Level
1
2
3
Total



Premium
$158,000
$644,000
$198,000
$1,000,000

Current
0.8200
1.0000
1.1500

Selected
0.9300
1.0000
1.2000

The actuary selects an overall change for all levels of 15%.

a. (3 points). Compute the base rate adjustment.
b. (1.5 points). Compute the proposed Level 1 relativity adjusted for base rate offset

Question 11 (1.5 points) According to Werner and Modlin in “Basic Ratemaking”, when writing a new insurance
product, insurers often do not have the data to generate rates, and often rely on similar products sold by
competitors. List and briefly describe four types of adjustments an insurer may make if it uses a competitor’s
manual as a starting point.

Question 12 (1.50 points).
According to Feldblum, "Personal Automobile Premiums: An Asset Share Pricing Approach for PropertyCasualty Insurance," asset share pricing is not yet common in property/casualty insurance for several reasons.
List three reasons cited by the author.

Exam 5 – Independently Authored Questions - Test 4
Question 13 (5.50 points)
The Non-Standard Auto Insurance Company is trying to compute the proper premium rate relativity for its
young male driver class using an asset share pricing approach with a 3-year time horizon. The following
information is known for their two classes of drivers:
Adult Drivers
First year average premium 1,000
First year average Loss & LAE 500
Adult Drivers
Variable expense ratio
Fixed expense ratio
Persistency Rates
Year
Adult Drivers
1
100%
2
90%
3
92%





New
10%
15%

Renewal
5%
5%

Young Male Drivers
100%
80%
82%

Cost of capital:
Loss and LAE trend:
Fixed expense trend:
Expected rate increases:

10%
5%
3%
6%

Using the method described in Feldblum "Personal Automobile Premium: An Asset Share Pricing Approach
For Property-Casualty Insurance," Answer the following questions.
a. (3.0 points). Compute the present value of premium for adult drivers.
b. (.50 points). Compute the rate of return on premium.
c. (2 points). Given a present value of profit of $1500 on young male drivers, compute the premium rate
relativity for young male drivers, such that the Non-Standard Auto Company will earn the same return
on premium for both classes.

Exam 5 – Independently Authored Questions - Test 4
Question 14 (3 points)
You are given the following for an average policy:





PV Loss is the present value at the beginning of each policy year.
Assume all policies are annual and have January 1 effective dates.
The policy count at year 0 is 1,000.

Using the asset share pricing model described by Feldblum, "Personal Automobile Premiums: An Asset
Share Pricing Approach for Property-Casualty Insurance:"
a. (2 points) If you increase rates 5% on January 1 of year 1 and then keep rates constant throughout the
five-year period, you project a 25% policy count decrease in year 1 and all other patterns will
remain the same.
Calculate the revised present value 5-year aggregate profit.
b. (1 point) If you increase rates 5% on January 1 of year 1 and then keep rates constant throughout the fiveyear period, what decrease (in decimals) in year 1 policy counts would result in the original estimated
present value 5-year aggregate profit of $152,390, assuming all other patterns will remain the same?

Question 15 (1 point)
Based on Feldblum, "Personal Automobile Premiums: An Asset Share Pricing Approach for PropertyCasualty Insurance," and the following information, calculate the termination rate for the third year.
 Number of policies originally issued = 1,000
 Number of first-year lapses = 300
 Number of second-year lapses = 150
 Number of third-year lapses = 100

Exam 5 – Independently Authored Questions - Test 4
Question 16 (2.5 points) You have been asked to compute the indicated rate change for a workers'
compensation book of business. Using the procedure described by Werner and Modlin in “Basic Ratemaking”,
and the data given below, answer the following question.

Accident
Year
2012
2013
2014
2015
2016
Total

Expected
Indemnity
Loss Ratio
27.9%
33.2%
34.4%
28.0%
35.6%
31.9%

Expected
Medical
Loss Ratio
59.1%
52.0%
53.4%
55.6%
49.8%
53.9%

Expected
ALAE
Ratio
10.2%
10.2%
10.2%
10.2%
10.2%
10.2%

Expected
ULAE
Ratio
7.7%
7.7%
7.7%
7.7%
7.7%
7.7%

Compute the indicated overall rate change.

Question 17 (2.5 points) You have been asked to calculate the adjustment an individual company should make
to the advisory loss costs to account for underwriting expenses, profit targets, and operational differences that
would affect loss cost levels for a workers' compensation book of business.
Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and given the data below, answer
the following question.
 General Expenses
9.3%
 Other Acquisition Costs
6.8%
 Taxes, License and Fees
2.7%
 Commissions and Brokerage Fees
8.8%
 Target Profit Provision
1.7%
 Expected Loss Cost Difference
-6.0%
 Operational Adjustment
0.940
 Current Deviation
1.350
 Industry Deviation
8.2%
Compute the company change to industry advisory loss costs.

Exam 5 – Independently Authored Questions - Test 4
Question 18 (2.5 points) You have been asked to calculate the medical benefit cost level factors for a workers'
compensation book of business. Using the procedure described by Werner and Modlin in “Basic Ratemaking”,
and given the data below, answer the following question.

Accident
Year
2012
2013
2014
2015
2016

Medical
Fee
Schedule
Change
0.0%
0.0%
-15.0%
0.0%
5.0%

Annual
"Other Medical"
Level
Change
2.2%
1.7%
3.7%
3.8%
3.6%

Protion of
Medical Losses
Subject to Fee
Schedules
70.0%
70.0%
65.0%
65.0%
65.0%

Projected

0.0%

6.1%

65.0%

Compute the factors needed to adjust historical accident year reported medical losses to the projected loss cost
levels.

Question 19 (2.5 points) You have been asked to calculate the expected medical loss ratios for each accident
year in the experience period for a workers' compensation book of business. Using the procedure described by
Werner and Modlin in “Basic Ratemaking”, and given the data below, answer the following question.

Year
2012
2013
2014
2015
2016
Total

Projected
Loss
Cost
Premium
$3,888,921,656
$4,039,795,024
$4,086,617,127
$4,212,122,582
$4,297,583,764
$20,525,040,153

Reported
Medical
Losses
$1,862,884,241
$1,624,586,453
$1,341,387,071
$1,233,856,180
$812,155,751
$6,874,869,695

Medical
Loss
Development
Factor
1.343
1.426
1.564
1.843
2.795

Factor to Adjust
Medical
Benefits to
Projected Cost
0.995
0.990
1.082
1.067
1.021

Compute the expected medical loss ratios for each accident year in the experience period.

Exam 5 – Independently Authored Questions - Test 4
Question 20 (2.5 points) You have been asked to calculate the projected loss cost premium for each accident
year in the experience period for a workers' compensation book of business. Using the procedure described by
Werner and Modlin in “Basic Ratemaking”, and given the data below, answer the following question.

Accident
Year
2012
2013
2014
2015
2016

Industry
Loss
Cost
Premium
$3,250,810,701
$3,457,177,017
$3,611,917,078
$3,883,157,640
$3,996,217,983

Annual
Payroll
Level
Change
2.2%
2.7%
3.4%
3.9%
3.2%

Exposure Trend
Expected
Future Wage
Level Change
6.1%
6.1%
6.1%
6.1%
6.1%

Historical
Average
Experience
Modification
0.987
0.981
0.977
0.978
0.953

Expected
Average
Experience
Modification
0.950
0.950
0.950
0.950
0.950

Compute the projected loss cost premium for each accident year in the experience period.

Exam 5 – Solutions to Independently Authored Questions - Test 4
Question 1 discussion: Blooms: Knowledge; Difficulty 1, LO 7 KS: Regulatory constraints
a1. Regulations may limit the amount of an insurer’s rate change (to either the overall average rate change for
the jurisdiction or to the change in premium for any individual or group of customers, or both)
a2. Regulation requiring insurers to provide written notice to its insureds regarding the magnitude of the
requested change.
a3. Regulations prohibiting the use of a characteristic for rating (even if it can be demonstrated to be
statistically strong predictors of risk).
a4. Regulations prescribing the use of certain ratemaking techniques (e.g. the use of multivariate classification
analysis).
a5. Regulators disagreeing with the insurer’s actuarial ratemaking assumptions (e.g. a regulator may disagree
with the method the actuary used to calculate loss trend, or may disagree with the trend selected).
Insurer actions that can be taken with respect to regulatory restrictions:
b1. An insurer can take legal action to challenge the regulation.
b2. An insurer may revise its U/W guidelines to limit business written at what it considers to be inadequate rate
levels (although some locations require insurers to “take all comers” for personal lines).
b3. An insurer may change marketing directives to minimize new applicants whose rates are thought to be
inadequate (e.g. concentrate its advertising on areas in which it believes the rate levels to be adequate).
b4. In the case of banned or restricted usage of a variable (e.g. insurance credit scores), an insurer can use a
different allowable rating variable (e.g. payment history with the company) it believes can explain some or
all of the effect associated with the restricted variable.
See chapter 13
Question 2 discussion: Blooms: Comprehension; Difficulty 1, LO 7 KS: Operational constraints
a1. Modifying rating algorithms can require significant systems changes, and the complexity of the change
depends on the extent of the changes and the number of systems.
a2. Implementing a new rating variable may require data that has not been previously captured, and may
require getting the data directly, either through a questionnaire sent to insureds or by visually inspecting the
insured item. These approaches can call for additional staff with unique skills.
b. When an operational constraint arises, a cost-benefit analysis can determine the appropriate course of
action. The cost of implementing the change is the cost associated with modifying the system. The benefit
is the incremental profit that can be generated by charging more accurate rates, and attracting more
appropriately priced customers.
See chapter 13
Question 3 discussion: Blooms: Knowledge; Difficulty 1, LO 7 KS: Marketing constraints:
competitive comparisons, close ratios, retention ratios, growth, distributional analysis, policyholder
dislocation analysis
1. Price of competing products: If the same product is offered at a lower price, they are likely to purchase the
competing product.
2. Overall cost of the product: If the product is costly, insureds are likely to compare prices to determine any
potential savings (and vice versa).
3. Rate changes: Significant increases (or decreases) in premium for an existing policy can cause existing
insureds to look for better options.
4. Characteristics of the insured (e.g. a young policyholder may shop (and change insurers) more frequently
than an older policyholder).
5. Customer satisfaction and brand loyalty: Poor claims handling or a bad customer service experience may
cause existing insureds to explore other options.
See chapter 13
Copyright  2014 by All 10, Inc.
Page 82

Exam 5 – Solutions to Independently Authored Questions - Test 4
Question 4 discussion: Blooms: Application; Difficulty 3, LO 7 KS: Marketing constraints:
competitive comparisons, close ratios, retention ratios, growth, distributional analysis, policyholder
dislocation analysis
a. Close =
Ratio

Number of Accepted Quotes 7,500
= = 25%
Total Number of Quotes
30, 000

Assume Insurer A includes all quotes issued, while insurer B may only include one quote per applicant.
Insurer A will have a lower close ratio if applicants request more than one quote before making a decision
(e.g. if an applicant gets several quotes with different limits).
b. Retention Ratio =

Number of Policies Renewed
35, 000
= = 875%
Total Number of Potential Renewal Policies 40, 000

Renewal customers are less expensive to service and generate fewer losses than new customers.
c. Insurers rely on close ratios and retention ratios as primary signals of the competitiveness of rates
for new business and renewal customers, respectively. Rate changes affect renewal business
directly (since any change can motivate existing customers to shop elsewhere) and influence the
insurer’s competitive position (e.g. If an insurer takes a rate decrease, the expectation is that the
close and retention ratios will improve, and vice versa)
d. %PolicyGrowth =

(New Policies Written - Lost Policies) Policies at End of Period
=
- 1.0 ,
Policies at Onset of Period
Policies at Onset of Period

Monthly policy growth is 1.0% (= [7,600 - 5,200] / 240,000).
If an insurer tightens or loosens the underwriting standards, growth can be affected.

See chapter 13

Question 5 discussion: Blooms: Comprehension; Difficulty 1, LO 10 KS: Non-pricing solutions
1. Balance can be achieved through expense reductions (i.e. reduction in UW or LAE expenses, by reducing
the marketing budget or staffing levels).
2. Balance can be achieved by reducing the average expected loss by changing the make-up of the portfolio of
insureds, by reducing the coverage provided by the policy (a.k.a. a coverage level change) or by instituting
better loss control procedures. See chapter 14
Question 6 discussion: Blooms: Application; Difficulty 1, LO 10 KS: Fixed expenses
Using the given information (cols (1) and (2)) below, compute cols (3) and (4).
Expense Type
Commission
Other Acquisition
General
Licenses and Fees
Profit and Contingencies

Total
Expense
(1)
0.19
0.04
0.03
0.02
0.08

V+Q
H
% Fixed
Fixed
Variable
(2)
(3)=(1)*(2) (4)=(1)*(1.0 - (2))
0.00%
0.000
0.190
85.00%
0.034
0.006
95.00%
0.029
0.002
85.00%
0.017
0.003
0.00%
0.000
0.080

H=Fixed Expense Fee Ratio =

0.0795

0.2805

F
H *R
, where F = fixed expense per exposure, H = fixed expense ratio
Expense
=
Fee =
1−V − Q 1−V − Q
(fixed expenses as a % of the rate). Calculation of $Fee (Using the Fixed Expense Ratio)
Calculation of $Fee (Using the Fixed Expense Ratio)
(1) Fixed Expense Ratio
7.950%
(2) Projected Average Premium per Exposure
$250.00
(3)=(1)*(2)
(3) Average Fixed Expense
$19.88
From col (4)
(4) Variable Expense %
20.05%
From col (4)
(5) Target Profit %
8.0%
(6)=1.0-(4)-(5)
(6) Variable Permissible Loss Ratio
71.95%
(7)=(3)/(6)
(7) Proposed Fee
$27.62 See chapter 14
Copyright  2014 by All 10, Inc.
Page 83

Exam 5 – Solutions to Independently Authored Questions - Test 4
Question 7 discussion: Blooms: Application; Difficulty 3, LO 10 KS: Calculation of final base rate
___

a. The proposed base rate is given by the following: B=
BS ×
p

( PP − AP )
___

( PS − AP )

Extension of Exposures (Assuming Seed Base Rate = $215)

(1)

(2)

(3)

(4)

(5)

Exposures

A
A
A
B
B
B

Y
Y
Y
Y
Y
Y

10,000
1
7,500
2
3,000
3
9,000
1
20,000
2
5,000
3
54,500
(7) Avg Prop Prem (Base Seed = $215)

Proposed Base Rate (Extension of Exposures)
(1) Seed Base Rate
(2) Average Premium assuming Seed Base Rate
(3) Proposed Fixed Fee per Policy
(4) Proposed Average Premium
(5)Proposed Base Rate

(6)
Proposed Premium
(assuming Seed
Base Rate = $215)
1,991,500.00
1,638,750.00
800,625.00
1,713,982.50
4,176,500.00
1,273,906.25
11,595,264
212.76

$215.00
$212.76
$25.00
$250.00
$257.65

(2)= from Row (7)
(5)= (1) x [(4) - (3)] /[(2) - (3)]
___

b. The proposed base rate is given by the following: BP =
BS ×

PP − AP

___

PS − AP

Proposed Base Rate (Extension of Exposures, Loss Ratio Method)
(1) Target % Change in Average Premium
(2) Current Average Premium
(3) Proposed Average Premium
(4) Seed Base Rate
(5) Average Premium assuming Seed Base Rate
(6) Proposed Fixed Fee per Policy
(7) Proposed Base Rate
(3)= (1.0 + (1)) x (2)
(7)= (4) x [(3) - (6)] /[(5) - (6)]

See chapter 14

___

=
BS ×

(1 + ∆%) × PC − AP

3.25%
$242.13
$250.00
$215.00
$212.76
$25.00
$257.65

___

PS − AP

Exam 5 – Solutions to Independently Authored Questions - Test 4
Question 8 discussion: Blooms: Application; Difficulty 1, LO 10 KS: Calculation of final base rates
Step 1: Compute the proposed differentials weighted by exposures.
Proposed Differentials Wtd by Exposures
(1)
(2)
R1
1
2
3
Total

Exposures
152,500
570,000
147,000
869,500

(1)

(2)

R2
A
B
C
Total

Exposures
235,000
480,000
154,500
869,500

(1)

(2)

D1
Y
N
Total

Exposures
156,625
712,875
869,500

(1)

(2)

D2
Y
N
Total

Exposures
153,625
715,875
869,500

(3)
Proposed
Differential
0.9000
1.0000
1.2500
1.0247
(3)
Proposed
Differential
1.0000
0.9500
1.3000
1.0257
(3)
Proposed
Discount
0.0500
0.0000
0.0090
(3)
Proposed
Discount
0.0500
0.0000
0.0088

___

(Tot3) = (3) weighted by (2).

(4)

S p = 1.0323 (4) = (Tot3R1) x (Tot3R2) x (1.0 - Tot3D1 - Tot3D2)
___

Step 2: Solve for the proposed base rate: BP =

PP − AP
___

SP
The proposed base rate, assuming the exposure-weighted average proposed rate differential across all rating variables from
___

the table above,=
is: BP

PP − AP $250 − $25
=
= $217.96
___
1.0323
SP

See chapter 14

Exam 5 – Solutions to Independently Authored Questions - Test 4
Question 9 discussion: Blooms: Application; Difficulty 1, LO10 KS: Calculation of final base rates
Step 1: Compute the proposed average change in differentials using exposures
Proposed Average Change in Differentials (Using Exposures)
(1)
(2)
(3)
(4)
Current
Proposed
D1
Exposures
Discount
Discount
Y
156,625
0.0500
0.0500
N
712,875
0.0000
0.0000
Total
869,500
0.0090
0.0090
(1)

(2)

D2
Exposures
Y
153,625
N
715,875
Total
869,500
(Tot3) = (3) Weighted by (2)
(5)

(6)

R1
1
2
3
Total

Exposures
152,500
570,000
147,000
869,500

(5)

(6)

R2
A
B
C
Total

Exposures
235,000
480,000
154,500
869,500

(10)

(11)

1-D1-D2
Total

Exposures
235,000

(3)
(4)
Current
Proposed
Discount
Discount
0.1000
0.0500
0.0000
0.0000
0.0177
0.0088
(Tot4) = (4) Weighted by (2)
(7)
Current
Differential
0.8000
1.0000
1.2000
0.9987

(8)
Proposed
Differential
0.9000
1.0000
1.2500
1.0247

(9)
Proposed /
Current
1.1250
1.0000
1.0417
1.0260

(7)
Current
Differential
1.0000
1.0500
1.2000
1.0631

(8)
Proposed
Differential
1.0000
0.9500
1.3000
1.0257

(9)
Proposed /
Current
1.0000
0.9048
1.0833
0.9648

(12)
Current
Differential
0.9733

(13)
Proposed
Differential
0.9822

(14)
Proposed /
Current
1.0091

(15) Average Change in Differential
(9)= (8) / (7)
(Tot9)= (9) Weighted by (6)
(12)= 1 - (Tot3D1) - (Tot3D2)
(13)= 1 - (Tot4D1) - (Tot4D2)
(14)= (13) / (12)
(15) = (Tot9R1) x (Tot9R2) x (Tot14)

0.9989

Exam 5 – Solutions to Independently Authored Questions - Test 4
___

Step 2: Using the results from the prior table
and (1.0 + ∆ B %)
=

(1.0 + ∆%) × PC − AP
___

PC − AC

proposed base rate can be calculated as shown in the following table.
Proposed Base Rate (Approximated Method)
(1) Current Base Rate (given)
(2) Current Average Premium (given)
(3) Target Change in Average Premium (given)
(4) Proposed Average Premium
(5) Proposed Additive Premium per Policy (given)
(6) Average Rating Differential Adjustment
(7) Proposed Base Rate Adjustment
(8) Proposed Base Rate
(4)= (1.0 + (3)) x (2)

$210.00
$242.13
3.25%
$250.00
$ 25.00
0.9989
1.0374
$217.85

(7)= [ (4) - (5) ] / [ (2) - (5) ] x [ 1.0 /(6) ]

See chapter 14

(8)=(1) x (7)

1.0
, the
(1.0 + ∆ S %)

Exam 5 – Solutions to Independently Authored Questions - Test 4
Question 10 discussion: Blooms: Application; Difficulty 3, LO 10 KS: Calculation of final base rates
a. Step 1: Compute whether the proposed rate relativity for any level produces a premium impact that exceeds
the desired maximum premium increase of 20%.
Rate Change after Capping Non-Base Level at 20%
(1)
(2)
(3)
(4)
(5)

Level
1
2
3
Total

Premium
$158,000
$644,000
$198,000
$1,000,000

Current
0.8200
1.0000
1.1500

(10)
(11)
(12)
(13)

(6)

OffDifferential Balance
Selected Change
Factor
0.9300
13.41%
0.9711
1.0000
0.00%
0.9711
1.2000
4.35%
0.9711
2.98%
0.9711

(7)

(8)

(9)

Selected
Overall
Change
15.0%
15.0%
15.0%
15.0%

Total
Change
26.65%
11.67%
16.53%
15.00%

Premium
Above 20%
Cap
10,510
0
0
10,510

Proposed Premium from Non-capped Levels (2, 3)
Proposed Level 1 Relativity to Comply with Cap
Base Rate Adjustment to cover Shortfall
Proposed Lev 1 relativity adjusted for base rate offset

$949,890
0.8812
1.0111
0.8715

(5)=
(4) / (3) - 1.0
(Tot5)=
(5) weighted by (2)
(6)=
[1.0] / [1.0 + (Tot5)]
(8)=
[1.0 + (5)] x (6) x [1.0 + (7)] - 1.0
(9)= max of [(2) x ((1.0 + (8))] - [ (2) x (1.0 + 20%)] and 0
(10)= (2) x (1+(8)) summed over Levels 2 and 3
(11)= [(1.0 + 20%) / ((6Row 1) x (1.0 + (7Row 1))] x (3Row 1)
(12)= 1.0 + (Tot9) / (10)
(13)= (11) / (12)

a Step 2: Compute the base rate adjustment.
The base rate needs to be adjusted upward to cover the premium shortfall caused by the cap:

Base Rate Adj = 1.0 +

Premium Above Cap
10,510
= 1+
= 1.011
Proposed Premium from all Non - Capped Levels
949,890

b. The relativity for the capped level (Level 1) needs to be reduced to account for:
i. the amount the change exceeds the cap and for
ii. the amount the base rate will be increased by the base rate adjustment:

Differential Adjustment = [

1.0 +%Cap
1.0
]× Curr Rel×
1.0 + Overall Rate Change× OBF
Base Rate Adj

Adjustment to Level 1 Differential due to Capping
[1.0 + 20%]/[1.15 * .9711] *.82 = .8812
Thus, the Proposed Level 1 relativity adjusted for base rate offset = .8812/1.0111 = .8715
See chapter 14

Exam 5 – Solutions to Independently Authored Questions - Test 4
Question 11 discussion: Blooms: Comprehension; Difficulty 1, LO 10 KS: Rating variables and
differentials
1. Estimate whether its fixed expenses will be higher or lower than those of the target competitor and increase
or decrease the competitor’s expense fee by the appropriate percentage.
2. Estimate whether its variable expenses will be higher or lower than those of the target competitor, and adjust
the base rate and the expense fee by the ratio of [the target competitor’s variable permissible loss ratio/ the
expected variable permissible loss ratio].
3. Estimate whether its expected loss costs will be different than the target competitor’s due to operational
differences or a lack of experience with the product, and change the base rate.
4. Target a certain segment of the market that the competitor does not seem to be targeting.
If the insurer chooses to reduce the rate differential in that territory, it can adjust the base rate to offset the
change in the average territorial differential.
See chapter 14
Question 12 discussion: Blooms: Knowledge; Difficulty 1, LO 13 KS: Model characteristics and
formulae
1. The data needed are not always available.
2. Casualty pricing techniques are still somewhat undeveloped.
3. The casualty insurance policy allows great flexibility in premiums and benefit levels.
4. Liability claim costs are uncertain, both in magnitude and in timing.
See page 196.

Exam 5 – Solutions to Independently Authored Questions - Test 4
Question 13 discussion Blooms: Application; Difficulty 3, LO 13 KS: Premium
Step 1. Compute the PV of Premium for adult male drivers:
Policy
Year

(1)
1
2
3
Total
Policy
Year

(1)
1
2
3
Total

Premium

(2)
1,000.00
1,060.00
1,123.60
3,183.60

Profit

(11)
250.00
387.45
383.47

Annual Losses
subsequent
Year 1

(3)
500.00

Discount
Factor

(12)
1.000
1.100
1.210

(4)
0.00
525.00
551.25

Fixed Expense
New
Renewal

(5)
150

Variable Expense
Year 1
Renewal

(6)
0
51.50
53.05

(7)
100

(8)
0
53.00
56.18

Persistency
Rate

Cumulative
Persistency

(9)
1.00
0.90
0.92

(10)
1.00
0.90
0.83

Present Value of
Profit
Premium

(13)
250.00
352.23
316.92
919.14

(14)
1,000.00
867.27
768.88
2,636.15

Column 2 is an average premium per car of 1000 with a 6% annual growth due to annual rate increases
Column (3) is column (2) * 0.5 new business loss ratio. Column (4) is column (3) * 1.05 net trend.
Column (5). First year fixed expenses are .15*1,000. Fixed renewal expenses in renewal year 1.
equal fixed renewal expenses in policy year 1 times fixed expense trend: 52 = 1,000*.05*1.03
Variable expenses are 10% of 'premium' in the 1st 'year', and 5% in the following years
Column (9) is 1.0 - termination rates
Column (10) = the downward product of column (9).
Column (11) = Column (10) *{Column (2) - Sum of Columns (3, 4, 5, 6, 7 and 8)}.
Column (12) uses a rate of 10% per year compounded annually.
Column (13) = column (11) / column (12).
Column (14) = column (2) * column (10) / column (12) .

Step 2. Compute the PV of Premium for Young Male Drivers
The return on premium is 919.14/2,636.15 = 34.87%
The PV of Profit for young male drivers is $1,500, therefore to earn a 34.87% return, the PV of premium
must be equal to 1,500/.3487 = 4,302.08
Step 3. Compute the premium rate relativity.
First, we need the initial premium for young male drivers. Let Initial premium for Young Male drivers = P
Set up an equation for the present value of premium for young male drivers by multiplying initial premium by
annual increases and cumulative persistency and dividing by discount factors.
P + P*1.06*.8/1.1 + P*(1.062)*.656/1.21 = 4,302.08
P + .77P + .609P = 4302.08
P = 1808.36
Premium relativity for young male drivers = 1,816.78/1,000 = 1.817

Exam 5 – Solutions to Independently Authored Questions - Test 4
Question 14 discussion: Blooms: Application; Difficulty 2 LO 13 KS: Model characteristics and
formulae
The following is given:
PV of
Policy
Year

Premium

Loss

Variable Expense
Year 1
Renewal

Fixed Expense
Persistency
Year 1 Renewal
Rate

Cumulative
Persistency

Profit

Discount
Factor

Present Value of
Profit
Premium

(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
1,000 800.00
250
0
150
0
1.000
1.000 -200.00
1.000 -200.00
1
50
40
0.850
0.850 113.90
1.100 103.55
2
1,000 776.00
3
1,000 752.72
50
40
0.850
0.723 113.63
1.210 93.91
4
1,000 730.14
50
40
0.850
0.614 110.46
1.331 82.99
5
1,000 708.23
50
40
0.850
0.522 105.33
1.464 71.94
Total
152.38
Only the values in bold need to be adjusted in accordance with the 5% rate increase. This produces the following impact:
Policy
PV of
Variable Expense Fixed Expense Persistency Cumulative
Discount PV of
Rate
Persistency Profit
Factor
Profit
Year Premium Loss
Year 1 Renewal Year 1 Renewal
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
1
1,050 800.00
263
0
150
0
1.000
1.000 -162.50
1.000 -162.50
2
1,050 776.00
53
40
0.850
0.850 154.28
1.100 140.25
3
1,050 752.72
53
40
0.850
0.723 147.95
1.210 122.28
4
1,050 730.14
53
40
0.850
0.614 139.63
1.331 104.90
5
1,050 708.23
53
40
0.850
0.522 130.12
1.464 88.87
Total
293.80

(13)
1,000.00
772.73
597.11
461.40
356.54
3,187.77

Therefore, the revised present value 5-year aggregate profit = 1,000 * .75 * $293.80 = $220,350
b.

5-year aggregate profit = (Policy count at time 0)*(policy count impact)*(aggregate present value profit).
$152,390 = 1,000 * (1 - x) * $293.8. x = .48.

Question 15 discussion Blooms: Application; Difficulty 1, LO 13 KS: Termination rates
General information:
"Persistency may be analyzed either by termination rates or by probabilities of termination.
The termination rate is the number of terminations during a given renewal period divided by the sum of
terminations during that period plus policies persisting through that period.
Termination rates more clearly distinguish persistency patterns by classification."
Solution:
The termination rates by year are:
30.00% [= 300 / 1000] the 1st year
21.43% [=150 / (1,000 -- 300) = 700] the 2nd year, and
18.18% [=100 / (1,000 -- 300 -- 150) = 550] the 3rd year.
See page 210.

Exam 5 – Solutions to Independently Authored Questions - Test 4
Question 16 discussion: Blooms: Application; Difficulty 2, LO 3 KS: Effect of rate changes

Accident
Year
2012
2013
2014
2015
2016
Total

(1)
Expected
Indemnity
Loss Ratio
27.9%
33.2%
34.4%
28.0%
35.6%
31.9%

(2)
Expected
Medical
Loss Ratio
59.1%
52.0%
53.4%
55.6%
49.8%
53.9%

(3)
Expected
ALAE
Ratio
10.2%
10.2%
10.2%
10.2%
10.2%
10.2%

(5) = [ (1) + (2)] * [ 1.0 + (3) + (4) ]
(6) Selected
(7) = (6) - 1.0

(4)
Expected
ULAE
Ratio
7.7%
7.7%
7.7%
7.7%
7.7%
7.7%
(6) Selected
(7) Indication

(5)
Expected
Loss & LAE
Ratio
102.6%
100.5%
103.6%
98.5%
100.7%
101.2%
101.2%
1.2%

Interpreting the results. The objective of the overall analysis is to determine advisory loss costs, the premium
(derived in another exhibit) does not include any underwriting expenses or profit; therefore, the target loss
ratio is 100%. Subtracting one from the selected loss ratio produces the overall indicated change to the
current advisory loss cost premium.
See Appendix D.

Question 17 discussion: Blooms: Application; Difficulty 2, LO 3 KS: Organization of data: calendar
year, policy year, accident year
(1) General Expenses
(2) Other Acquisition Costs
(3) Taxes, License and Fees
(4) Commissions and Brokerage Fees
(5) Target Profit Provision
(6) Total Expense and Profit
(7) Expense and Profit Adjustment

9.3%
6.8%
2.7%
8.8%
1.7%
29.3%
1.413

(8) Expected Loss Cost Difference
(9) Operational Adjustment
(10) Proposed Deviation

-6.0%
0.940
1.329

(11) Current Deviation
(12) Industry Deviation
(13) Company Change

1.350
8.2%
6.5%

(6) = (1) + (2) + (3) + (4) + (5)
(10) = (7) * (9)
(7) = 1.0 / [ 1.0 - (6) ]
(9) = 1.0 + (8)
(13) = (10) / (11) * [ 1.0 + (12) ] - 1.0

Interpreting the results. Row 8 is the expected difference in loss costs due to any known operational
differences between the individual company and the industry. An overall average adjustment of -6% was
selected to reflect an expectation of lower losses attributable to the company’s more stringent
underwriting and claims handling practices.
Row 10: Combines the adjustment for expenses and profit with the adjustment for operational differences, and
represents the deviation factor that the company should apply to the industry advisory loss costs.
Row 11 (the current company deviation factor); Row 12 (the industry loss cost change).
See Appendix D.
Copyright  2014 by All 10, Inc.
Page 92

Exam 5 – Solutions to Independently Authored Questions - Test 4
Question 18 discussion: Blooms: Application; Difficulty 2, LO 3 KS: Organization of data: calendar
year, policy year, accident year

Accident
Year
2012
2013
2014
2015
2016

(1)
Medical
Fee
Schedule
Change
0.0%
0.0%
-15.0%
0.0%
5.0%

(2)
Annual
"Other Medical"
Level
Change
2.2%
1.7%
3.7%
3.8%
3.6%

(3)
Protion of
Medical Losses
Subject to Fee
Schedules
70.0%
70.0%
65.0%
65.0%
65.0%

(4)

Combined
Effect
0.7%
0.5%
-8.5%
1.3%
4.5%

(5)
Factor to Adjust
Medical Benefits
to Projected
Cost Level
0.995
0.990
1.082
1.067
1.021

Projected

0.0%

6.1%

65.0%

2.1%

1.000

(1) Based on evaluations of the cost impact of changes to the Fee Schedule
(1 Proj) Selected
(2) Based on medical component of the Consumer Price Index
(2 Proj) Selected (3% annual trend)
(3) Selected Based on separate study
(4) = (1) * (3) + [ (2) * ( 1 - (3) ]
(5) = [ 1.0 + (4NextRow) ] * (5NextRow)

Note: These factors are used in computing projected ultimate medical losses. See Appendix D.

Question 19 discussion: Blooms: Application; Difficulty 2, LO 3 KS: Organization of data: calendar
year, policy year, accident year

Year
2012
2013
2014
2015
2016
Total

(1)
Projected
Loss
Cost
Premium
$3,888,921,656
$4,039,795,024
$4,086,617,127
$4,212,122,582
$4,297,583,764
$20,525,040,153

(2)
Reported
Medical
Losses
$1,862,884,241
$1,624,586,453
$1,341,387,071
$1,233,856,180
$812,155,751
$6,874,869,695

(3)
Medical
Loss
Development
Factor
1.343
1.426
1.564
1.843
2.795

(4)
Factor to Adjust
Medical
Benefits to
Projected Cost
0.995
0.990
1.082
1.067
1.021

(1) From Premium Exhibit
(2) Input
(3) From Medical Sheet 1 (Development)
(4) From Medical Sheet 2 (Cost Change)
(5) = (2) * (3) * (4)
(6) = (5) / (1)

See Appendix D.

(5)
Projected
Ultimate
Medical
Losses
$2,489,604,052
$2,294,295,718
$2,268,860,623
$2,427,333,777
$2,318,553,868
$11,798,648,038

(6)
Expected
Medical
Loss
Ratio
64.0%
56.8%
55.5%
57.6%
54.0%
57.5%

Exam 5 – Solutions to Independently Authored Questions - Test 4
Question 20 discussion: Blooms: Application; Difficulty 2, LO 3 KS: Organization of data: calendar
year, policy year, accident year

Accident
Year
2012
2013
2014
2015
2016
Total

(1)
Industry
Loss
Cost
Premium
$3,250,810,701
$3,457,177,017
$3,611,917,078
$3,883,157,640
$3,996,217,983
$18,199,280,418

(2)
Annual
Payroll
Level
Change
2.2%
2.7%
3.4%
3.9%
3.2%

(3)
(4)
Exposure Trend
Factor to
Expected
Current Wage Future Wage
Level
Level Change
1.139
6.1%
1.109
6.1%
1.072
6.1%
1.032
6.1%
1.000
6.1%

(5)
(6)
(7)
Factor to
Historical
Expected
Adjust to
Average
Average
Future Wage Experience Experience
Level
Modification Modification
1.208
0.987
0.950
1.176
0.981
0.950
1.138
0.977
0.950
1.095
0.978
0.950
1.061
0.953
0.950

See Appendix D.

(8)
Projected
Loss
Cost
Premium
$3,779,712,874
$3,937,916,001
$3,995,181,436
$4,129,751,491
$4,226,241,632
$20,068,803,435

Exam 5 – Independently Authored Questions - Preparatory Test 5
General information about this exam
This practice test contains 20 questions consisting of computational and essay based questions.

Total Number of Qs:
Total Number of Points:

Essay
Questions
3
5

Computational
Questions
17
47.5

Total
20
52.5

Chapter 16: Claims Made Ratemaking ........................... Modlin, Werner ........... A. Basic Techniques for Ratemaking

Appendix B: Homeowners indication ............................ Modlin, Werner ........... A. Basic Techniques for Ratemaking

Appendix C: Medical Malpractice Indication............... Modlin, Werner ........... A. Basic Techniques for Ratemaking

Exam 5 – Independently Authored Questions - Test 5
Question 1 (4.5 points) Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and the
data below, answer the following questions. Assume the following:
• The policy being experience rated is an occurrence policy with an annual term, and the effective date
is 7/1/2010.
• The experience period consists of the last three completed policies effective 7/1 to 6/30 (i.e. annual
policies originating in July 2006, 2007, and 2008), evaluated at 3/31/2010.
• Expected percentage of unreported losses at 3/31/2010 for the three years are 42.0%, 32% and 21.2%
• Losses are capped at basic limits, and ALAE are unlimited.
• A MSL is applied to the basic limits losses and unlimited ALAE combined.
• The Z of the company is 0.44.
• The expected experience ratio (EER) is 0.85
• Reported Losses and ALAE at 3/31/10 Limited by Basic Limits and MSL = $130,000
• Current company B/L Losses and ALAE = $74,000
• Loss trend equals 4.3%
a. (3 points). Compute the Expected Unreported Losses and ALAE at 3/31/10 Limited by Basic Limits and MSL
b. (1.5 points). Compute the experience modification.
Question 2 (4.5 points) Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and the
data below, answer the following questions regarding a WC experience rating plan..
 The effective date of the policy being rated is 9/1/2010
 The policy is comprised of only one class code.
st
 The 1 table below lists the actual losses from the last three complete policy years.
nd
 The 2 table shows payroll and expected loss costs rates for the prospective period.
Policy Year
9/1/06-07

9/1/07-08

9/1/08-09

Claim #
1
2
3
1
2
3
1
2

Total
Policy
Year
9/1/06-07
9/1/07-08
9/1/08-09
Total




Payroll
$1,778,182
$1,934,545
$2,106,364
$5,819,091

Reported
Losses
$20,000
$105,000
$30,000
$45,000
$50,000
$7,500
$12,000
$55,000
$324,500
Expected
Loss Cost
4.35
3.48
2.67

The D-ratio is 0.26
B = $35,000 and w = 0.30

a. (3 points). Compute actual and expected primary losses, and actual and expected excess losses.
b. (1.5 points). Compute the experience modification.
Copyright  2014 by All 10, Inc.
Page 96

Exam 5 – Independently Authored Questions - Test 5
Question 3 (1.5 points) According to Werner and Modlin in “Basic Ratemaking”, briefly describe two purposes
why schedule rating is used and provide an example which demonstrates these two purposes.

Question 4 (3.5 points) Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and the data
below, answer the following questions regarding composite rating.
 Bob’s Rentals sells new and used equipment, operates a repair and service shop, and offers leases
and rentals on equipment it owns.
 Bob’s Rentals is large enough to meet ISO’s Composite Rating Plan eligibility requirements for loss
rating and desires coverage up to $1,000,000 per occurrence with $2,000,000 general aggregate.
 The last three years of reported losses and ALAE over all 3operations, separated into BI and PD is
shown below. Amounts are capped at $250,000 per occurrence.
 The selected composite exposure base is total receipts.

* Loss and ALAE annual trend (for bodily injury and property damage) is 5%.
* Exposure annual trend rate is 3%.
* Expected loss & ALAE ratio is 68%.
 Total receipts for the proposed policy period are estimated to be $152,000
Reported Loss and ALAE as of 12/31/2008
Policy
Year
7/1/05-06
7/1/06-07
7/1/07-08
Total

Incurred Loss and ALAE
BI
PD
1,356,511
517,616
1,355,545
623,184
1,193,012
568,669
3,905,068
1,709,469

Total
Receipts
$122,387,756
$126,490,456
$131,443,738
$380,321,950
Development Factors

Age to
Ultimate
42-Ult
30-Ult
18-Ult

Bodily
Injury
1.50
1.75
1.95

Property
Damage
1.23
1.38
1.53

a. (3.0 points). Compute the loss-rated composite rate for Bob’s Rentals for its upcoming annual policy
effective 7/1/2009.
b. (0.50 points). Compute the Deposit Premium for the proposed policy

Exam 5 – Independently Authored Questions - Test 5
Question 5 (3.5 points) Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and the data
below, compute the large deductible CGL policy premium.
Assume the following
* The deductible is $500,000 per occurrence.
* The insurer will handle all claims (including those that fall below the deductible)
* The deductible is not expected to reduce ALAE costs, which are estimated to be 12% of total losses.
* The deductible applies to losses only.
* Total ground-up losses without recognition of a deductible are estimated to be $1,000,000.
* Fixed expenses are assumed to be $50,000.
* Variable expenses are assumed to be 13% of premium.
* The insurer makes payments on all claims and seeks reimbursement for amounts below the deductible
from the insured. The cost to process deductibles is 4% of the losses below the deductible.
* Deductible recoveries are not fully collateralized, and the credit risk is estimated to be 1% of the expected
deductible payments.
* The desired UW profit for full-coverage (i.e. no deductible) premium is 2%.
* An additional risk margin of 10% of excess losses for policies with a deductible of $500,000 is charged.
* The % of total losses below the deductible (i.e. the LER) are summarized below.
Loss Elimination Ratios

Loss Limit

LER

$100,000
$250,000
$500,000

60%
75%
90%

Question 6 (3.5 points) Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and the data
below, compute the retrospective premium at first adjustment.
The 1st computation of the retrospective premium occurs 6 months after the end of the policy period
The policy is an annual policy and limited reported losses valued as of 18 months are $162,000.
(1) Minimum retrospective premium ratio (negotiated)

62.0%

(2) Maximum retrospective premium ratio (negotiated)

135.0%

(3) Loss Conversion Factor (negotiated)

1.10

(4) Per Accident Loss Limitation (negotiated)

$100,000

(5) Expense Allowance (excludes tax multiplier)

20%

(6) Expected Loss Ratio

65%

(7) Tax Multiplier

1.03

(8) Standard Premium

$769,231

(9) Insurance Charge for Maximum Premium

0.42

(10) Insurance Savings for Minimum Premium

0.03

Question 7 (1.5 points) According to Werner and Modlin in “Basic Ratemaking”, briefly describe three elements
provided by the basic premium in retrospective rating

Exam 5 – Independently Authored Questions - Test 5
Question 8 (2.0 points) According to Werner and Modlin in “Basic Ratemaking”, briefly describe the five
principles of claims-made ratemaking.
Question 9 (1.0 point) According to Werner and Modlin in “Basic Ratemaking”, answer the following questions:
a. (.50 points). Using report year by lag notation, write an equation representing an annual occurrence policy
written on January 1, 2012. Assume all claims are reported within 5 years of occurrence.
b. (.50 points). Using report year by lag notation, write an equation representing a mature claims made policy
written on January 1, 2012. Assume all claims are reported within 5 years of occurrence.

Question 10 (3.5 points) Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and the
data below, answer the following questions:
Assume the following:
 Exposure levels are constant.
 The average loss cost for RY 2015 is $2,000.
 Loss costs increase by 6% each report year.
 Loss costs do not vary by report year lag. Any trends affecting settlement lag have been ignored.
a. (1.50 points). If an equal number of incurred claims are reported each year and all claims are reported within
5 years of occurrence, compute the cost of an annual occurrence policy written on January 1, 2016.
b. (1.50 points). If an equal number of incurred claims are reported each year and all claims are reported
within 5 years of occurrence, compute the cost of a mature claims made policy written on January 1, 2016.
c. (1.50 points). If 5% of the claims are reported one year later than expected, but all claims are reported within
five years, compute the cost of an annual occurrence policy written on January 1, 2016

Question 11 (2.5 points) According to Werner and Modlin in “Basic Ratemaking”,
a. (0.50 points). Define the term retroactive date.
b. (2.0 points). Draw a report year by report year lag matrix for an insured switching from a mature CM policy
to an occurrence policy in 2011. Assume all claims are reported within 5 years of occurrence. Use report
year by report year lag notation to indicate what portion of the matrix:
i. would be covered by the 2010 claims made policy,
ii. would be covered by the continued purchase of occurrence policies beginning in 2011,
iii. represents coverage for an extending reporting endorsement purchased by the insured after switching
from a mature CM policy to an occurrence policy in 2011.
Question 12 (1.0 points) You have been asked to compute the annual fixed expense trend for a homeowners
book of business.
Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and given the following data,
compute the average annual fixed expense trend percentage for CY 2015.
(1) Employment Cost Index - Finance, Insurance & Real Estate, excluding Sales Opportunity (annual change over latest 2 years)
U.S. Department of Labor
(2) % of Other Acquisition and General Expense used for Salaries and Employee Relations & Welfare Insurance Expense Exhibit, 2015
(3) Consumer Price Index, All Items (annual change over latest 2 years)

3.8%

50.0%

2.2%

Exam 5 – Independently Authored Questions - Test 5
Question 13 (2.5 points) You have been asked to compute the projected fixed expense provision and the
variable expense provision for a homeowners book of business, to be used directly in the pure premium
indication formula for new rates in State XX effective 1/1/2017.
Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and given the data below, answer
the following questions:
2015
(1) General
Countrywide Expenses
% Assumed Fixed
Countrywide Earned Exposures
Countrywide Earned Premium
(2) Other Acquisition
Countrywide Expenses
% Assumed Fixed
Countrywide Written Exposures
Countrywide Written Premium
(3) Taxes, Licenses and Fees
Fixed Expense Per Exposure
Variable Expense % [(f)/(g)]
(4) Commission and Brokerage
Fixed Expense Per Exposure
Variable Expense %

Selected

$2,211,221
75.0%
52,752
$49,059,360
$2,647,322
75.0%
53,015
$50,213,747
$3.00
1.40%
$0.00
10.20%

Compute:
a. The dollar amount of projected fixed expenses.
b. The variable expense provision %.

Question 14 (2.5 points) Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and
given the data below, answer the following question.
A reinsurance contract was purchased with an effective date of January 1, 2017 and a twelve-month term
covering a Homeowners book of business.
 Expected Reinsurance Recoveries under the contract = $408,672
 Cost of Reinsurance (Expected Ceded Premium) = $613,248
 AY 2015 Earned Exposures = 12,911
 Expected Increase in Annual Exposures = 1.5%
Compute the Projected Net Reinsurance Cost Per Exposures

Exam 5 – Independently Authored Questions - Test 5
Question 15 (2.5 points) Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and
given the data below, answer the following question:
You are an in-house actuary and are charged with computing the non-modeled catastrophe pure premium for
your company's Homeowners book of business using the data below.
Amount of
Insurance
Calendar
Years
Year
($000s)
1996
$1,592,745
1997
$1,737,727
1998
$1,918,827
1999
$2,121,436
2000
$2,267,800
2001
$2,314,018
2002
$2,392,245
2003
$2,489,736
2004
$2,598,391
2005
$2,661,682
2006
$2,669,491
2007
$2,657,573
2008
$2,645,909
2009
$2,676,445
2010
$2,651,309
2011
$2,423,000
2012
$2,519,920
2013
$2,620,716
2014
$2,725,545
2015
$2,916,501
(4) All-Year Arithmetic Average





Reported
Cat Losses
and
Paid ALAE
$4,011
$23,851
$141,702
$35,172
$132,264
$206,471
$202,240
$757,560
$157,863
$2,426,190
$88,165
$233,412
$49,394
$432,295
$1,118
$63,908
$440,935
$26,386
$63,516
$162,000

Cat-to-AIY
Ratio
0.003
0.014
0.074
0.017
0.058
0.089
0.085
0.304
0.061
0.912
0.033
0.088
0.019
0.162
0.000
0.026
0.175
0.010
0.023
0.056
0.110

ULAE Factor
Non-Modeled Cat Provision Per AIY
Selected Average AIY Per Exposure

1.023
0.113
$247.20

Compute the Non-Modeled Cat Pure Premium

Exam 5 – Independently Authored Questions - Test 5
Question 16 (2.5 points) You have been asked to compute the underwriting expense and ULAE ratio for a
medical malpractice book of business, to be used directly in the pure premium indication formula for new rates,
with a proposed effective date in State XX of 5/1/2016
Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and given the data below, answer
the following question.
2013

2014

(1) General Expenses
a Countrywide Expenses
b Countrywide Earned Premium
(2) Other Acquisition
a Countrywide Expenses
b Countrywide Written Premium
(3) Taxes, Licenses, and Fees
a Countrywide Expenses
b Countrywide Written Premium
(4) Commission and Brokerage
a Countrywide Expenses
b Countrywide Written Premium

2015
$32,039
$498,269
$13,730
$523,866
$11,114
$523,866
$111,101
$523,866

The company has derived a ULAE ratio of 2.8%
Compute the Underwriting Expense and ULAE ratio

Question 17 (2.5 points) You have been asked to compute the severities and adjusted frequencies for a medical
malpractice book of business, so that exponential trends can be fit to the severity and adjustment frequency
data. Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and given the data below,
answer the following questions.

Accident
Year
2011
2012
2013
2014

Selected
Ultimate
Loss &
ALAE
$10,181,756
$5,716,706
$16,597,848
$21,238,428

Reported
Claim
Count
59
63
52
26

Reported
Age-toUltimate
Factor
1.0488
1.1953
1.4992
2.6041

Earned
Premium
$13,176,857
$13,129,499
$13,486,005
$16,604,630

Current
Rate
Level
Factor
1.2058
1.2724
1.3018
1.2390

a. Compute the ultimate severities for accident years 2011 – 2014
b. Compute the adjusted frequencies for accident years 2011 – 2014

Exam 5 – Independently Authored Questions - Test 5
Question 18 (2.5 points) You have been asked to compute the two year average ultimate Loss and ALAE ratio
for a medical malpractice book of business, so that reported Bornhuetter-Ferguson method can be used to
develop losses and ALAE to ultimate for the three most recent accident years (2012 – 2014)
Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and given the data below, answer
the following question.

Accident
Year
2010
2011

Adjustment
Ultimate
to Avg
Earned
Loss and
Rate Level
Premium
ALAE
in 2011
11,923,731 $9,727,917
0.9876
11,595,634 $9,333,276
1.0000

Selected
BF Net
Trend
11.3%
11.3%

Compute: The two year average ultimate Loss and ALAE ratio.
Question 19 (2.5 points) You have been asked to compute the B-F ultimate Loss and ALAE ratio for a medical
malpractice book of business, for the three most recent accident years (2012 – 2014). Using the procedure
described by Werner and Modlin in “Basic Ratemaking”, and given the data below, answer the following
question.
2- Year Avg
Ultimate Loss
and ALAE
Average
Accident
Ratio
Earned
Rate
Level
Year (2010-2011) Premium
2012
86.2% 11,553,959 0.9329
2013
86.2% 11,867,684 0.9115
2014
86.2% 14,612,074 0.9583

Rate
Level
2011
0.9876
0.9876
0.9876

Selected
Reported
BF
Reported Losses
Net
Age-to-Ult and ALAE
Trend
Factor a/o 9/30/15
11.3%
1.8690 $1,628,500
11.3%
3.9128 $3,228,250
11.3%
21.3756 $1,082,250

Compute: the B-F ultimate Loss and ALAE ratio for accident years 2012 - 2014
Question 20 (2.5 points) You have been asked to compute the indicated rate change for a medical malpractice
book of business. Using the procedure described by Werner and Modlin in “Basic Ratemaking”, and given the
data below, answer the following questions:
Calendar

Current

Ultimate

Net

Accident

Earned

Rate Level

Loss

Trend

Year

Premium

Factor

and ALAE

Factor

2010

$12,420,553

1.1969

$9,338,800

1.7842

2011

$12,078,786

1.1998

$8,959,945

1.6379

2012

$12,035,374

1.2664

$5,030,701

1.5035

2013

$12,362,171

1.2958

$14,606,107

1.3802

2014

$15,220,911

1.2330

$18,689,817

1.2669

 Expense and ULAE Ratio = 35.1%
 Profit and Contingency Provision = -5%
 Number of Reported Claims = 313
 Claims Required for Full Credibility Standard = 683
 Countrywide Indicated Rate Change = 20.3%
Compute the Credibility - Weighted Indicated Rate Change

Exam 5 – Solutions to Independently Authored Questions - Test 5
Question 1 discussion: Blooms: Application; Difficulty 3 LO 12 KS: Experience modification
a.
Calculation of Expected Unreported Losses and ALAE and Company Subject Loss Costs
(1)
(2)
(3)
(4)
(5)
(6)

Policy Period
7/1/06-07
7/1/07-08
7/1/08-09
Total

Current
Company B/L
Loss & ALAE
Coverage
Costs
Prem/Ops+Prod 74,000.00
Prem/Ops+Prod 74,000.00
Prem/Ops+Prod 74,000.00
222,000.00

(4)= 1.0/ [1.043]^No of years of trend;

Detrend
Factors
0.845
0.881
0.919

(7)
(8)
Expected
Percentage B/L Expected B/L
Company
Subject B/L Expected Losses & ALAELosses & ALAE
Loss & ALAE Experience Unreported at Unreported at
Costs
Ratio
3/31/2010
3/31/2010
62,530.87
0.850
0.212
11,268.06
65,219.70
0.850
0.320
17,739.76
68,024.15
0.850
0.420
24,284.62
195,774.72
53,292.44

(5)= (3) x (4)

(6),(7) = given

(8)= (5) x (6) x (7)

b.
Experience Credit/Debit Calculation
(1) Experience Components
(a)
Reported Losses and ALAE at 3/31/10 Limited by Basic Limits and MSL
(b)
Expected Unreported Losses+ ALAE at 3/31/10 Limited by BL and MSL
(c)
Projected Ultimate Losses and ALAE Limited by Basic Limits and MSL
(d)
Company Subject Basic Limit Loss and ALAE Costs
(e)
Actual Experience Ratio
(2)
Expected Experience Ratio
(3)
Credibility
(4)
Experience (Credit)/Debit
(1a)= Given (1b)=Table 2
(2),(3)= Given

(1c)=(1a) + (1b) (1d)=Table 2
(4)=[((1e) - (2)) / (2)] x (3)

(1e)= (1c)/(1d)

See chapter 15
Question 2 discussion: Blooms: Application; Difficulty 3, LO 12 KS Layers of loss
a.

Policy Year
9/1/06-07

9/1/07-08

9/1/08-09

Claim #
1
2
3
1
2
3
1
2

Total
(2) = Minimum [ (1), $5,000 ]

(1)
Reported
Losses
$20,000
$105,000
$30,000
$45,000
$50,000
$7,500
$12,000
$55,000
$324,500
(3) = (1) - (2)

(2)
Primary
Losses
$5,000
$5,000
$5,000
$5,000
$5,000
$5,000
$5,000
$5,000
$40,000

See chapter 15

(3)
Excess
Losses
$15,000
$100,000
$25,000
$40,000
$45,000
$2,500
$7,000
$50,000
$284,500

130,000.00
53,292.44
183,292.44
195,774.72
0.936
0.850
0.440
4.5%

Exam 5 – Solutions to Independently Authored Questions - Test 5
Question 2 discussion (continued):
(1)
Policy
Year
9/1/06-07
9/1/07-08
9/1/08-09
Total

Payroll
$1,778,182
$1,934,545
$2,106,364
$5,819,091

(3) = [ (1) / $100 ] x (2)

M=
M

(2)

(3)

(4)

Expected
Loss Cost
4.35
3.48
2.67

Expected
Losses
$77,350.91
$67,322.18
$56,239.91
$200,913.00

D-Ratio
0.26
0.26
0.26

(5) = (3) x (4)

(5)
(6)
Expected
Expected
Primary
Excess
Losses
Losses
$20,111.24 $57,239.67
$17,503.77 $49,818.41
$14,622.38 $41,617.53
$52,237.38 $148,675.62

(6) = (3) - (5)

AP + w × Ae + (1.0 − w) × Ee + B
,
E+B
40, 000 + [0.30 × $284,500] + [(1.0 - 0.30) × $148, 675.62] + $35, 000
= 1.121
$52, 237.38 + $148, 675.62 + $35, 000

The e-mod factor of 1.121 is applied multiplicatively to policy standard premium.

See chapter 15

Question 3 discussion: Blooms: Knowledge; Difficulty 1, LO 12 KS: Purpose of individual risk
rating
Schedule Rating is used to modify the manual rate, in commercial lines pricing, to reflect characteristics that
are:
1. expected to have a material effect on the insured’s future loss experience but that are not actually reflected
in the manual rate, or
2. not adequately reflected in the prior experience (if ER applies).
Example: If an insured implements a new loss control program, it is expected that losses will be lower than that
indicated by the actual historical experience (hence an underwriter can use SR to reflect this).
See chapter 15

Exam 5 – Solutions to Independently Authored Questions - Test 5
Question 4 discussion: Blooms: Application Difficulty 3, LO 12 KS: Composite loss-rated risks
a. Step 1: Develop trend factors to be applied to the loss and ALAE and the exposure base.
 The AAD of the proposed policy period is 12/31/2009, and the AAD of each policy year from the
experience period is 12/31.
 Based on the assumed trend rates, the trend factors are calculated as follows:
Trend Factors
(1)

Policy
Year
7/1/05-06
7/1/06-07
7/1/07-08

(2)
Annual
Loss &
ALAE
Trend
5.00%
5.00%
5.00%

Trend
Period
4
3
2

(3)
Loss &
ALAE
Trend
Factor
1.2155
1.1576
1.1025

(4)

(5)

Annual
Exposure
Trend
3.00%
3.00%
3.00%

Exposure
Trend
Factor
1.1255
1.0927
1.0609

(3) =[1.0 + (2)]^(1)
(5) =[1.0+ (4)]^(1)
a. Step 2: Estimate the trended ultimate loss and ALAE.
Trended Ultimate Loss & ALAE
(1)
Policy
Year
7/1/05-06
7/1/06-07
7/1/07-08
Total

(2)

(3)

Incurred Loss and ALAE
BI
PD
1,356,511
517,616
1,355,545
623,184
1,193,012
568,669
3,905,068
1,709,469

(4)

Development Factors
BI
PD
1.50
1.23
1.75
1.38
1.95
1.53

(5)
Loss &
ALAE
Trend Factor
1.2155
1.1576
1.1025

(6)
Trended
Ultimate Loss &
ALAE
3,247,145
3,741,673
3,524,072
10,512,890

(6) = [ (1) x (3) + (2) x (4) ] x (5)
a. Step 3: Compute trended composite exposures.
(1)
Policy
Year
7/1/05-06
7/1/06-07
7/1/07-08
Total

Total Receipts
($000's)
122,388
126,490
131,444
380,322

(2)
Exposure
Trend
Factor
1.1255
1.0927
1.0609

(3) = (1) x (2)
Trended
Exposure
137,748
138,220
139,449
415,417

a. Step 4: Compute the composite rate:
(1) Trended Ultimate Loss & ALAE
(2) Expected Loss & ALAE Ratio
(3) Adjusted Premium
(4) Trended Composite Exposure
(5) Composite Rate

(3) = (1) / (2)

$10,512,890
68.0%
$15,460,132
$415,417
$37.22

(5) = (3) / (4)

b. Compute the Deposit premium:
Assuming total receipts for the proposed policy period are estimated to be $152,000, then the deposit
premium is $$5,656,826.11 (= $152,000 x 37.22).
See chapter 15
Copyright  2014 by All 10, Inc.
Page 106

Exam 5 – Solutions to Independently Authored Questions - Test 5
Question 5 discussion: Blooms: Application Difficulty 2, LO 12 KS: Formulae
The formula for the large deductible policy premium is

Premium =

Losses above Deductible + ALAE + Fixed Expense + Credit Risk + Risk Margin
(1.0 -Variable Expense Provision - Profit Provision)

Step 1: Estimate losses above the $500,000 deductible.
(1) Expected total ground-up losses
(2) Excess ratio = [1.0 - .90 (500K LER)]
(3) Estimated losses above deductible (1) x (2)

$1,000,000
10%
$ 100,000

$100,000
$120,000
$50,000
$36,000
$9,000
$10,000
15%
$382,353

(1) = prior table; (2) = 12% x $1,000,000 (3a) = Provided (3b) = 4% x $1,000,000 x .90 (LER at 500K)
(4) = 1% x $1,000,000 x .90 (LER at 500K)
(5) = 10% x (1)
(7) = [(1) + (2) + (3a) + (3b) + (4) + (5)] / [1.0 - (6)]
See chapter 15

Exam 5 – Solutions to Independently Authored Questions - Test 5
Question 6 discussion: Blooms: Application; Difficulty 3, LO 12 KS: Retrospective rating
The basic formula for retrospective premium is as follows:
Retro Premium = [Basic Premium + Converted Losses] x Tax Multiplier, where the retro premium is subject to a
maximum and minimum.
Basic Premium = [Expense Allowance - Expense Provided Through LCF + Net Ins Charge] x Standard Premium
LCF = Loss Conversion Factor
Expense Provided Through LCF = Expected Loss Ratio x (LCF -1.0)
Net Insurance Charge = [Insurance Charge - Insurance Savings] x Expected Loss Ratio x LCF.
Converted Losses: Converted Losses = Reported Losses x LCF.
Minimum/Maximum Retrospective Premium
Minimum Retro Premium = Standard Premium x Minimum Retro Premium Ratio.
Maximum Retro Premium= Standard Premium x Maximum Retro Premium Ratio.
(11) Basic Premium
(12) Converted Losses
(13) Preliminary Retrospective Premium
(14) Minimum Retrospective Premium
(15) Maximum Retrospective Premium
(16) Retrospective Premium

$318,346
$178,200
$511,443
$476,923
$1,038,462
$511,443

(11) = [(5)-(6) x [ (3)-1.0 ]+[ (9)-(10)] x (6) x (3)] x (8)
(12) = $162,000 x (3)
(13) = [(11)+(12) ] x (7)
(14) = (1) x (8)
(15) = (2) x (8)
(16) = Min [Max[(13),(14)] , (15) ]
See chapter 15
Question 7 discussion: Blooms: Knowledge; Difficulty 1, LO 4 KS: Organization of data: calendar
year, policy year, accident year
The Basic Premium provides for:
1. The insurer’s target UW profit and expenses (excluding expenses provided for by the LCF and the tax
multiplier), and
2. The cost of limiting the retrospective premium (to be between the minimum and maximum premium
negotiated under the policy), and
3. The cost of limiting each occurrence to a negotiated loss limitation (if applicable).
See chapter 15
Question 8 discussion: Blooms: Knowledge; Difficulty 1, LO 4 KS: Claims made coverage: report
lag, coverage triggers, principles of claims-made policies, retroactive date, tail coverage
1. A claims-made policy should always cost less than an occurrence policy as long as claim costs are
increasing.
2. If there is a sudden, unexpected change in the underlying trends, a claims-made policy priced based on the
prior trend will be closer to the correct price than an occurrence policy based on the prior trend.
3. If there is a sudden, unexpected shift in the reporting pattern, the cost of a mature claims-made policy (i.e. a
policy that covers claims reported during the policy period regardless of accident date) will be affected
relatively little, if at all, relative to the occurrence policy.
4. Claims-made policies incur no liability for IBNR, so the risk of reserve inadequacy is greatly reduced.
5. Investment income earned from claims-made policies is substantially less than under occurrence policies.
See chapter 16
Copyright  2014 by All 10, Inc.
Page 108

Exam 5 – Solutions to Independently Authored Questions - Test 5
Question 9 discussion: Blooms: Comprehension; Difficulty 1, LO 4 KS: Claims made coverage:
report lag, coverage triggers, principles of claims-made policies, retroactive date, tail coverage
a. Occurrence policies cover claims that occur during the policy period regardless of when the claim is
reported, and are aggregated by accident year (i.e. each diagonal in the table). Example:
 An annual occurrence policy written on 1/1/2012 covers claims incurred during the policy period and
reported either during or after the policy period.
 This policy covers claims reported in 2012 with no report lag, claims reported in 2013 with a one-year
report lag, claims reported in 2014 with a two-year report lag, etc.
Thus, Occurrence Policy (2012) = L(2012,0)+ L(2013,1)+ L(2014,2)+ L(2015,3)+ L(2016,4).
b. The coverage trigger for a CM policy is the report date. A CM policy is represented by the entries in a row.
A CM policy written on 1/1/2012 covers all claims reported in 2012 (regardless of the report lag):
CM Policy (2012) = L(2012,0)+ L(2012,1)+ L(2012,2)+ L(2010,3)+ L(2012,4).
See chapter 16
Question 10 discussion: Blooms: Application; Difficulty 3, LO 4 KS: Claims made coverage: report
lag, coverage triggers, principles of claims-made policies, retroactive date, tail coverage
a. and b.
Report
Year
2015
2016
2017
2018
2019
2020
2021
2022
2023

0
$400.00
$420.00
$441.00
$463.05
$486.20
$510.51
$536.04
$562.84
$590.98

Accident
Year
2015
2016
2017
2018
2019

Occurrence
Loss Costs
$2,210.25
$2,320.77
$2,436.80
$2,558.64
$2,686.58

Loss Costs by Report Year Lag
1
2
3
$400.00
$400.00
$400.00
$420.00
$420.00
$420.00
$441.00
$441.00
$441.00
$463.05
$463.05
$463.05
$486.20
$486.20
$486.20
$510.51
$510.51
$510.51
$536.04
$536.04
$536.04
$562.84
$562.84
$562.84
$590.98
$590.98
$590.98

4
$400.00
$420.00
$441.00
$463.05
$486.20
$510.51
$536.04
$562.84
$590.98

Using Loss Costs by Report Year Lag from above
=400 + 420 + 441 + 463.05 + 486.20
=420 + 441 + 463.05 + 486.20 + 510.51

Claims Made
Loss Costs
$2,000.00
$2,100.00
$2,205.00
$2,315.25
$2,431.01
$2,552.56
$2,680.19
$2,814.20
$2,954.91

Exam 5 – Solutions to Independently Authored Questions - Test 5
Question 10 discussion (continued):
c.
Report
Year
2015
2016
2017
2018
2019
2020
2021
2022
2023
Accident
Year
2015
2016
2017
2018
2019

0
$300.00
$315.00
$330.75
$347.29
$364.65
$382.88
$402.03
$422.13
$443.24
Occurrence
Loss Costs
$2,231.80
$2,343.39
$2,460.56
$2,583.59
$2,712.77

4
$500.00
$525.00
$551.25
$578.81
$607.75
$638.14
$670.05
$703.55
$738.73

Claims Made
Loss Costs
$2,000.00
$2,100.00
$2,205.00
$2,315.25
$2,431.01
$2,552.56
$2,680.19
$2,814.20
$2,954.91

Using Loss Costs by Report Year Lag from above
=300 + 420 + 441 + 463.05 + 607.75
=315.00 + 441 + 463.05 + 486.20 + 638.14

“If there is a sudden, unexpected shift in the reporting pattern, the cost of a mature CM policy will be affected
relatively little, if at all, relative to the occurrence policy.”
Example: Assume that 5% of the claims are reported one year later than expected, but all claims are reported
within five years (e.g. in 2010, $100 of the loss cost shifts from lag 0 to lag 1, $100 of the loss costs
from lag 1 shift to lag 2, and so on).
Since an equal amount of loss costs are shifting in and out of lag periods 1, 2, and 3, the only impact
is on the first and last lag periods.
Conclusions:

There is no impact on the loss cost estimates for the CM policies

Estimates for the occurrence policies have changed (e.g. for AY 2016 loss cost estimate for the occurrence
policies has changed by .0097 (= ($2,343.39 / $2,320.67) – 1.0).

See chapter 16

Exam 5 – Solutions to Independently Authored Questions - Test 5
Question 11 discussion: Blooms: Application; Difficulty 2, LO 4 KS: Claims made coverage: report
lag, coverage triggers, principles of claims-made policies, retroactive date, tail coverage
a. A retroactive date is the date associated with a claims-made policy for which coverage is provided for
claims occurring on or after the retroactive date.
To obtain complete coverage without overlap, the retroactive date should coordinate with the expiration of the
last occurrence policy.
b. Insurers offer an extended reporting endorsement (or tail coverage) that covers claims that occurred
but were not reported before the expiration of the last CM policy.
Switching from Claims-Made to Occurrence Policy with Tail Coverage
Report Year Lag
1
2
3

Report Year

2010

L(2010,0) L(2010,1 ) L(2010,2) L(2010,3) L(2010,3)

2011
2012
2013
2014
2015

L(2011,0)
L(2012,0)
L(2013,0)
L(2014,0)
L(2015,0)

L(2011,1)
L(2012,1)
L(2013,1)
L(2014,1)
L(2015,1)

L(2011,2)
L(2012,2)
L(2013,2)
L(2014,2)
L(2015,2)

L(2011,3)
L(2012,3)
L(2013,3)
L(2014,3)
L(2015,3)

L(2011,4)
L(2012,4)
L(2013,4)
L(2014,4)
L(2015,4)

CM = within dotted rectangle Tail Coverage = within the dotted triangle
Occurrence Policy Coverage = shaded
See chapter 16
Question 12 discussion: Blooms: Application; Difficulty 3, LO 3 KS: Organization of data: calendar
year, policy year, accident year
(4) Annual Expense Trend = [ (1) * (2) ] + [ (3) * { 100% - (2) } ]= [ .038 * .50 ] + [ .022 * { 100% - .50 } = 3.0%
See Appendix B:

Exam 5 – Solutions to Independently Authored Questions - Test 5
Question 13 discussion: Blooms: Application; Difficulty 2, LO 3 KS: Organization of data: calendar
year, policy year, accident year
2015
(1) General
a Countrywide Expenses
b % Assumed Fixed
c Fixed Expense $ [(a)*(b)]
d Countrywide Earned Exposures
e Fixed Expense Per Exposure [(c)/(d)]
f Variable Expense % [(a)*(1.0-(b))]
g Countrywide Earned Premium
h Variable Expense % [(f)/(g)]
(2) Other Acquisition
a Countrywide Expenses
b % Assumed Fixed
c Fixed Expense $ [(a)*(b)]
d Countrywide Written Exposures
e Fixed Expense Per Exposure [(c)/(d)]
f Variable Expense % [(a)*(1.0-(b))]
g Countrywide Written Premium
h Variable Expense % [(f)/(g)]
(3) Taxes, Licenses and Fees
a. Fixed Expense Per Exposure
b.Variable Expense % [(f)/(g)]
(4) Commission and Brokerage
a. Fixed Expense Per Exposure
b. Variable Expense %
(5) Total Fixed Expenses
(6) Fixed Expense Trend
(7) Trend Period
(8) Fixed Expense Trend Factor
(9) Projected Fixed Expense
(10) Variable Expense Provision

Selected

$2,211,221
75.0%
$1,658,416
$52,752
$31.44
$552,805
$49,059,360
1.1%

$31.44

1.1%

$2,647,322
75.0%
$1,985,491
$53,015
$37.45
$661,830
$50,213,747
1.3%

$37.45

1.3%

1.3%
$3.00
1.40%
$0
10.20%

(1e) + (2e) + (3a) + (4a)
From 07/01/2015 to 07/01/2017
[1.0 + (6)]^ (7)
(5) * (8)
(1h) + (2h) + (3h) + (4h)

See Appendix B:

$71.89
3.75%
2
1.076
77.38
14.0%

Exam 5 – Solutions to Independently Authored Questions - Test 5
Question 14 discussion: Blooms: Application; Difficulty 2, LO 3 KS: Extension of exposures
(1) Expected Reinsurance Recoveries
(2) Cost of Reinsurance (Expected Ceded Premium)
(3) Net Cost of Reinsurance
(4) Latest Year Exposures
(5) Expected Annual Exposure Increase
(6) Projection Period
(7) Projected Exposures
(8) Projected Net Reinsurance Cost Per Exposure

$408,672
$613,248
$204,576
12,911
1.5%
2.0
13,301
$15.38

(3) = (2) - (1)
(6) From Midpoint of Latest Year to Midpoint of Reinsurance
Contract [ (07/01/2015) to (07/01/2017) ]
(7) = (4) * [ 1.00 + (5) ] ^ (6)
(8) = (3) / (7)

See Appendix B:

Question 15 discussion: Blooms: Application; Difficulty 2, LO 3 KS: Organization of data: calendar
year, policy year, accident year
(5) From ULAE Ratio Exhibit
(6) = (4) * (5)
(7) From AIY Projection Exhibit
(8) = (6) * (7)

(4) All-Year Arithmetic Average
(5) ULAE Factor
(6) Non-Modeled Cat Provision Per AIY
(7) Selected Average AIY Per Exposure
(8) Non-Modeled Cat Pure Premium

0.110
1.023
0.113
$247.20
$27.91

See Appendix B:
Question 16 discussion: Blooms: Application; Difficulty 2, LO 3 KS: Organization of data: calendar
year, policy year, accident year
2013

2014

(1c) + (2c) + (3c) + (4c)
(5) + (6)

2015

Selected

$32,039
$498,269
6.4%

6.4%

$13,730
$523,866
2.6%

2.6%

$11,114
$523,866
2.1%

2.1%

$111,101
$523,866
21.2%

21.2%
32.4%
2.8%
35.1%

Exam 5 – Solutions to Independently Authored Questions - Test 5
Question 17 discussion: Blooms: Application; Difficulty 2, LO 10 KS: Rating algorithms

Accident
Year
2011
2012
2013
2014
(4) =
(5) =
(8) =
(9) =

(1)
Selected
Ultimate
Loss &
ALAE
$10,181,756
$5,716,706
$16,597,848
$21,238,428

(2)
Reported
Claim
Count
59
63
52
26

(3)
Reported
Age-toUltimate
Factor
1.0488
1.1953
1.4992
2.6041

(4)

(5)

(6)

Developed
Claim
Count
62
75
78
68

Severity
$164,546
$75,917
$212,912
$313,683

Earned
Premium
$13,176,857
$13,129,499
$13,486,005
$16,604,630

(7)
Current
Rate
Level
Factor
1.2058
1.2724
1.3018
1.2390

(8)
Earned
Premium
at Current
Rate Level
$15,888,176
$16,706,042
$17,556,618
$20,573,637

(9)

Adjusted
Frequency
3.89
4.51
4.44
3.29

(2) * (3)
(1) / (4)
(6) * (7)
[ (4) / (8) ] * 1,000,000

See Appendix C.

Question 18 discussion: Blooms: Application; Difficulty 3, LO 10 KS: Rating algorithms

Accident
Year
2010
2011

(1)

(2)

(3)

Earned
Premium
11,923,731
11,595,634

Ultimate
Loss and
ALAE
9,727,917
9,333,276

Ultimate
Loss and
ALAE Ratio
81.6%
80.5%

(4)
Adjustment
to Avg
Rate Level
in 2011
0.9876
1.0000

(5)

(6)

Selected
BF Net
Trend
11.3%
11.3%

Trend
Length
1.00
0.00

(3) = (2) / (1)
(6) From 07/01/20XX to 07/01/2011
(7) = [ 1 + (5) ] ^ (6)
(8) = (3) / (4) * (7)
(9) Straight Average of (8)

(7)
Net
Trend
Adjustment
to 2011
1.1130
1.0000

2- Year Avg
Ultimate Loss and
ALAE Ratio
(9) (2010-2011)

See Appendix C.

(8)
Ultimate
Loss and
ALAE Ratio
as of 2011
91.9%
80.5%

86.2%

Exam 5 – Solutions to Independently Authored Questions - Test 5
Question 19 discussion: Blooms: Application; Difficulty 2, LO 10 KS: Rating algorithms
(1)
(2)
2- Year Avg
Ultimate Loss
and ALAE
Accident
Ratio
Earned
(2010-2011) Premium
Year
2012
86.2% 11,553,959
2013
86.2% 11,867,684
2014
86.2% 14,612,074

(3)

(4)

Average
Rate
Level
0.9329
0.9115
0.9583

Rate
Level
2011
0.9876
0.9876
0.9876

(5)

(6)

Average
Selected
Rate
BF
Level
Net
Adjustment
Trend
0.9446
11.3%
0.9229
11.3%
0.9703
11.3%

(7)

(8)

Trend
Length
from
2011
1.00
2.00
3.00

Net
Trend
Adjustment
1.1130
1.2388
1.3787

(5) = (3) / (4)
(8) = [ 1.0 + (6) ] ^ (7)
(9)
Expected
Losses
and ALAE
Ratio
101.6%
115.7%
122.5%

(10)

(11)

(12)

(13)

(14)
(15)
Expected Losses
B-F
Expected
Reported
and ALAE
Ultimate
Losses
Reported
Losses
Not Yet
Losses
and
Age-to-Ult Percent and ALAE
Reported
and
ALAE
Factor Unreported a/o 9/30/15
a/o 9/30/15
ALAE
11,735,508 1.8690
46.5% $1,628,500
$5,456,348
$7,084,848
13,730,899 3.9128
74.4% $3,228,250 $10,221,716 $13,449,966
17,896,871 21.3756
95.3% $1,082,250 $17,059,615 $18,141,865

(9) = (1) / (5) * (8)
(10) = (2) * (9)
(12) = 1 - 1 / (11)
(14) = (10) * (12)
(15) = (13) + (14)

See Appendix C.

Exam 5 – Solutions to Independently Authored Questions - Test 5
Question 20 discussion: Blooms: Application; Difficulty 2, LO 3 KS: Effect of rate changes
(1)

(2)

(3)

(4)

(5)

(6)

(7)

Projected

Current

Earned

Ultimate

Net

Ultimate

Accident

Earned

Rate Level

Premium

Loss

Trend

Loss

Loss and

Year

Premium

Factor

@ CRL

and ALAE

Factor

and ALAE

ALAE Ratio

Calendar

2010

$12,420,553

1.1969

$14,866,160

$9,338,800

1.7842

$16,662,243

112.1%

2011

$12,078,786

1.1998

$14,491,689

$8,959,945

1.6379

$14,675,397

101.3%

$5,030,701

2012

$12,035,374

1.2664

$15,241,660

1.5035

$7,563,861

49.6%

2013

$12,362,171

1.2958

$16,019,393 $14,606,107

1.3802

$20,158,923

125.8%

2014

$15,220,911

1.2330

$18,767,842 $18,689,817

1.2669

$23,677,834

126.2%

Total

$64,117,795

$82,738,259

104.2%

$79,386,744 $56,625,370

(3) = (1)*(2)

(8) Selected Loss and ALAE Ratio

(6) = (4)*(5)

(9) Expense and ULAE Ratio

(7) = (6)/(3)

(10) Profit and Contingency Provision

(11) = 100% - (9) - (10)

(11) Permissible Loss Ratio

(12)= [(8)/ (11)] - 1.0

(12) Statewide Indicated Rate Change
(13) Number of Reported Claims
(14) Claims Required for Full Credibility Standard

(15) = Min { [ (13) / (14) ] ^ 0.5, 1.0 }

(15) Credibility

(17) = (12) * (15) + (16) * [ 1.0 - (15) ]

(17) Credibility - Weighted Indicated Rate Change

(16) Countrywide Indicated Rate Change
(18) Selected Rate Change

See Appendix C.

104.2%
35.1%
-5.0%
69.9%
49.1%
313
683
67.7%
20.3%
39.8%
39.8%

ALL 10, Inc.
Comprehensive Study Materials and Internet-Based Training

Actuarial Notes for the
Spring 2014 CAS Exam 5B
Estimating Claim Liabilities
Volume 3
Independently Authored and Modified Past CAS
Multiple Choice Questions Tests

and

Independently Authored Preparatory Tests
Computational and Essay Based Questions

Exam 5B – Independently Authored and Modified Past CAS MC Questions
Preparatory Test 1
General information about this exam
1. This test contains 21 multiple choice questions.
2. The recommend time for this exam is 40 min.
3. Consider taking this exam after working all past CAS questions first.

Articles covered on exam:
Article .................................................... Author .................................. Syllabus Section

Chapter 1 – Overview ....................................................... Friedland ..............B: Estimating Claim Liabilities
Chapter 2 – The Claims Process ...................................... Friedland ..............B: Estimating Claim Liabilities
Chapter 3 – Understanding the Types of Data Used ........ Friedland ..............B: Estimating Claim Liabilities
Chapter 7 – Development Technique .............................. Friedland ..............B: Estimating Claim Liabilities
Chapter 8 – Expected Claims Technique ........................ Friedland ..............B: Estimating Claim Liabilities
Chapter 9 – Bornhuetter-Ferguson Technique ................ Friedland ..............B: Estimating Claim Liabilities
Statement of Principles: Loss and LAE Reserves ............ CAS ......................B: Estimating Claim Liabilities
ASOP No. 9 – Documentation and Disclosure ................. AAA ......................B: Estimating Claim Liabilities

Exam 5B – Independently Authored and Modified
Past CAS MC Questions - Test 1
Question 1
According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, which of the following are
true?
1. A loss liability's valuation date can be before, after or the same as its accounting date.
2. The accounting date is the date that defines the group of claims for which liability may exist
3. For claims-made policies, the accident date may not be defined as the date the claim was reported.
A. 1

B. 1, 2

C. 1, 2, 3

D. 2, 3

E. None of the given answer choices

Question 2
According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, which of the following are
true?
1. An insured loss that occurred on 1/5/2009, for a policy written on 12/31/2008, would be included in the
unpaid claim estimate for the accounting date 12/31/2008.
2. The valuation date is used to define the group of claims to be included in the liability estimate.
3. The valuation date does not depend on when the actuary does his/her analysis.
A. 2

B. 2, 3

C. 1, 2, 3

D. 1, 3

E. None of the given answer choices

Question 3
According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, which of the following are
true?
1. The carried reserve is the result of the application of a particular estimation technique.
2. The unpaid claim for unpaid claims is the amount reported in a published statement or in an internal
statement of financial condition.
3. The unpaid claims estimate includes four components: case outstanding on known claims, provision for
future development on known claims, estimate for reopened claims, and provision for claims incurred but
not reported.
A. 1

B. 2

C. 1, 2

D. 2, 3

E. None of the given answer choices

Question 4
According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, which of the following are
true regarding the types of work an independent adjuster (IA) is hired to handle?
1. To handle an individual claim or a group of claims.
2. The book of claims from small to mid-sized commercial insurers and self-insurers.
3. A specific type of claim or a claim in a particular region where the insurer does not have the necessary
expertise.
A. 1

B. 2

C. 1, 2

D. 2, 3

E. 1, 2, 3

Exam 5B – Independently Authored and Modified
Past CAS MC Questions - Test 1
Question 5
According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, which of the following is/are
NOT approaches used by insurers to set case outstanding?
1. Establish case outstanding for ALAE (or DCC) only and other insurers for ULAE (or A&O) only.
2. Set the case outstanding using the advice of legal counsel.
3. Set the case outstanding based on the maximum value, which would be the policy limit.
4. Establish the case outstanding based on the best estimate of the ultimate settlement value of the claim
including consideration of future inflationary forces.
A. 1

B. 2

C. 3

D. 4

E. All are approaches used by insurers.

Question 6
According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, which of the following are
true?
1. Some of the key assumptions of the expected claims method include consistent claim processing and a
stable mix of types of claims.
2. Using the development technique, unpaid claim estimate equal the difference between projected
ultimate claims and actual reported claims.
3. Reporting patterns are derived from the cumulative paid claim development factors.
A. 2

B. 2, 3

C. 1, 2, 3`

D. 1, 3

E. None of the given answer choices

Question 7
According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, which of the following are
true?
1. Reporting patterns can be used in monitoring the development of claims during the year as well as for
present value (i.e., discounting) calculations.
2. The development technique is used for high-frequency, low-severity lines with stable and timely
reporting of claims throughout the accident year.
3. When using reported claims and the loss development technique, it is assumed that there have been no
significant changes during the experience period in the speed of claims closure and payment.
A. 2
B. 2, 3
C. 1, 2, 3
D. 1, 3
E. None of the given answer choices
Question 8
According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, which of the following are
true with respect to impact of changing conditions on the U.S. PP Auto Insurance example (steady state)?
1. In the case of increasing claim ratios (and no case reserve strengthening), the paid development method
produces an IBNR estimate greater than the IBNR estimate produced under the reported development
method.
2. In the case of increasing case outstanding (and stable claim ratios), the paid development method
produces an IBNR estimate smaller that the IBNR estimate produced under the reported development
method, but greater than the actual IBNR estimate.
3. Without adjustment, the reported claim development method understates the projected ultimate claims
and thus the IBNR in times of increasing case outstanding strength.
A. 1
B. 1, 2
C. 2
D. 2, 3
E. None of the given answer choices

Exam 5B – Independently Authored and Modified
Past CAS MC Questions - Test 1
Question 9
According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, which of the following are
true with respect to impact of changing conditions on the U.S. PP Auto Insurance example (steady state)?
1. In the case of increasing claim ratios and case reserve strengthening, the paid development method
produces an IBNR estimate greater than the IBNR estimate produced under the reported development
method.
2. In the case of increasing claim ratios and case reserve strengthening, the reported development method
produces an IBNR estimate greater than the actual IBNR estimate.
3. In the case of increasing claim ratios and case reserve strengthening, the reported development method
produces an IBNR estimate greater than the IBNR estimate produced under the increasing case
outstanding (and stable claim ratios) scenario
A. 2

B. 2, 3

C. 1, 2, 3

D. 1, 3

E. None of the given answer choices

Question 10
According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, which of the following are
true regarding the use of the expected claims method?
1. It is often used when there is stable reinsurance retention limits throughout the experience period.
2. It is often used when there is a stable mix of types of claims and policy limits.
3. It is often used when an insurer enters a new line of business or a new territory.
A. 3

B. 2, 3

C. 1, 2, 3

D. 1, 3

E. None of the given answer choices

Question 11
According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, which of the following are
true under the Bornhuetter-Ferguson technique?
1. Ultimate Claims = Actual Reported Claims + (Expected Claims) x (% Unreported)
2. Ultimate Claims = Actual Paid Claims + (Expected Claims) x (% Unpaid)
3. Ultimate Claims = Actual Paid Claims + Expected Unreported Claims
A. 2

B. 1, 2

C. 3

D. 1, 3

E. None of the given answer choices

Question 12
According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, which of the following are
true?
1. The expected loss method ignores actual results.
2. For the reported BF technique, the estimated IBNR is identical under both the steady state situation and
in the increasing claim ratios scenario.
3. The loss development method yields the correct answer in an increasing claim ratio situation but is
vulnerable to distortion from case outstanding strengthening.
A. 2

B. 1, 2

C. 3

D. 1, 3

E. None of the given answer choices

Exam 5B – Independently Authored and Modified
Past CAS MC Questions - Test 1
Question 13
According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, which of the following are
true?
1. When claim ratios are increasing and there is no reserve strengthening, the expected loss method
produces a lower IBNR estimate than that produced in the steady-state situation.
2. When claim ratios are stable, but there is increasing case outstanding strengthening, the expected loss
method produces the same IBNR estimate than that produced in the steady-state situation.
3. When claim ratios are increasing and there is reserve strengthening, the loss development method
produces an IBNR estimate which is overstated.
A. 3

B. 3

C. 1, 2, 3

D. 1, 3 E. None of the given answer choices

Question 14
According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, which of the following are
true when analyzing traditional loss development triangles?
1. The volume (or scale) of the accident year cohort changes horizontally from one accident year to the next.
2. The value of cumulative paid claims for an accident year changes vertically from age to age.
3. Loss development can be negative
A. 3

B. 3

C. 1, 2, 3

D. 1, 3 E. None of the given answer choices

Question 15
According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, which of the following are
true?
1. Actuaries often rely on report year development triangles for the analysis of claims-made coverages
such as U.S. medical malpractice and errors and omissions liability.
2. Reinsurers often organize claims data by accident year.
3. For self-insurers, the policy year, fiscal year, and accident year are often the same.
A. 3

B. 3

C. 1, 2, 3

D. 1, 3 E. None of the given answer choices

Question 16
According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, which of the following are
true?
1. The Benktander technique is significantly more responsive to changes in the underlying claim ratio
but is less responsive to changes in the case outstanding adequacy.
2. The Benktander technique is also less responsive to changes in the product mix than the
Bornhuetter-Ferguson technique.
3. Thus, where there are no changes in the underlying claim development patterns, we expect the
Benktander method to be more responsive than the Bornhuetter-Ferguson method.
A. 1

B. 2

C. 1, 2

D. 1, 2, 3

E. None of the given answer choices

Exam 5B – Independently Authored and Modified
Past CAS MC Questions - Test 1
Question 17
According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, which of the following are
true?
1. The Benktander method is a credibility mixture of Bornhuetter-Ferguson and Expected Claims
techniques
2. The Benktander method is often considered an iterative Bornhuetter-Ferguson method.
3. In the Benktander technique, the expected claims are the projected ultimate claims from an initial
Bornhuetter-Ferguson projection.
A. 1

B. 2

C. 1, 2

D. 1, 2, 3

E. None of the given answer choices

Question 18
According to “ASOP No. 9, Documentation and Disclosure”, which of the following are true?
1. A required actuarial document is an actuarial communication which the formal content is prescribed by
law or regulation.
2. The term “actuarial work product” applies only to written actuarial communications.
3. A Statement of Actuarial Opinion is a formal statement of the actuary's professional opinion on a defined
subject. It outlines the scope of the work but normally does not include descriptive details.
A. 1

B. 2

C. 1, 2

D. 2, 3

E. None of the given answer choices

Question 19
According to “ASOP No. 9, Documentation and Disclosure”, which of the following are true?
1. If someone other than an actuary conveys information prepared by the actuary to indirect users of the
work product, the actuary should take steps to rectify misquotation, misinterpretation, or other misuse of
the work product by indirect users.
2. If aware of any significant conflict between the interests of indirect users and the interests of the client or
employer, the actuary should advise the client or employer of the conflict and should include appropriate
qualifications or disclosures in any related actuarial communication.
3. Ownership of documentation is normally established by client or employer.
A. 1

B. 2

C. 1, 2

D. 2, 3

E. 1, 2, 3

Question 20
According to “Statement of Principles Regarding Property and Casualty Loss and LAE Reserves,” which of
the following are true?
1. Line and coverage definitions suitable for the establishment of reserves for large insurers can be in
much finer detail than in the case of small insurers.
2. It may be necessary to augment claims-made statistics with appropriate report period statistics
generated under occurrence programs.
3. If reserves are established in less detail than necessary for reporting requirements, procedures for
properly assigning the reserves to required categories must be developed.
A. 1, 2

B. 2, 3

C. 1, 2, 3

D. 1, 3,

E. None of the given answer choices

Exam 5B – Independently Authored and Modified
Past CAS MC Questions - Test 1
Question 21
According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, which of the following are
true?
1. An inadequate estimate of unpaid claims could drive an insurer to raise its rates unnecessarily, resulting
in a loss of market share and a loss of premium revenue to the insurer.
2. Unpaid claims estimates impact financial decision-making such as capital management.
3. An inaccurate estimate can have a negative impact on the insurer's decisions regarding its reinsurance
needs and claims management procedures and policies.
A. 1

B. 1, 2

C. 1, 2, 3

D. 2, 3

E. None of the given answer choices

Solutions to Exam 5B – Independently Authored and
Modified Past CAS MC Questions - Test 1
Question 1 discussion:
1. True. See chapter 3
2. True. See chapter 3
3. False. See chapter 3
Answer B: 1, 2

Question 2 discussion:
1. False. See chapter 3
2. False. The accounting date is used to define the group of claims to be included in the liability estimate.
See chapter 3
3. True. See Chapter 3
Answer E: None of the given answer choices.

Question 3 discussion:
1. False. The unpaid claim estimate is the result of the application of a particular estimation technique.
See chapter 1
2. False. The carried reserve for unpaid claims is the amount reported in a published statement or in an
internal statement of financial condition. See chapter 1
3. False. The unpaid claims estimate includes five components: case outstanding on known claims,
provision for future development on known claims, estimate for reopened claims, provision for claims
incurred but not reported, and provision for claims in transit (i.e., claims reported but not recorded).
See chapter 1
Answer E: None of the given answer choices
Question 4 discussion:
1. True. See chapter 2.
2. False. This is typically handled by a third party administrator. See chapter 2.
3. True. See chapter 2.
Answer: B
Question 5 discussion:
1. False. This is a common approach used by insurers. See chapter 2
2. False. This is a common approach used by insurers. See chapter 2
3. False. This is a common approach used by insurers. See chapter 2
4. False. This is a common approach used by insurers. See chapter 2
Answer: E
Question 6 discussion:
1. False. This statement is true of the development method. See chapter 7
2. False. Using the development technique, unpaid claim estimate equal the difference between
projected ultimate claims and actual paid claims. See chapter 7
3. False. Reporting patterns are derived from the cumulative reported claim development factors. See
chapter 7
Answer: None of the given answer choices
Copyright  2014 by All 10, Inc.
Page 8

Solutions to Exam 5B – Independently Authored and
Modified Past CAS MC Questions - Test 1
Question 7 discussion:
1. False. Payment patterns can be used in monitoring the development of claims during the year as well
as for present value (i.e., discounting) calculations. See chapter 7
2. True. See chapter 7
3. False. When using paid claims and the loss development technique, it is assumed that there have
been no significant changes during the experience period in the speed of claims closure and payment.
See chapter 7
Answer: A

Question 8 discussion:
1. False. See chapter 7
2. False. See chapter 7.
3. False. See chapter 7.
Answer: None of the given answer choices

Question 9 discussion:
1. False. See chapter 7
2. True. See chapter 7.
3. True. See chapter 7.
Answer: B

Question 10 discussion:
1. False. See chapter 8.
2. False. See chapter 8
3. True. See chapter 8.
Answer: A
Question 11 discussion:
1. True. See chapter 9.
2. True. See chapter 9.
3. False. Ultimate Claims = Actual Paid Claims + Expected UnPaid Claims See chapter 9.
Answer: B
Question 12 discussion:
1. True. See chapter 9
2. True. See chapter 9
3. True. See chapter 7.
Answer: E

Solutions to Exam 5B – Independently Authored and
Modified Past CAS MC Questions - Test 1
Question 13 discussion:
1. True. The estimated IBNR will be lower than actual IBNR as in the steady state situation. See chapter 8
2. True. It produces an IBNR estimate that is correct. See page 163. See chapter 8
3. True. See section 10, chapter 7.
Answer: C

Question 14 discussion:
1. False. The volume (or scale) of the accident year cohort changes vertically from one accident year to
the next. See chapter 5.
2. False. The value of cumulative paid claims for an accident year changes horizontally from age to age.
See chapter 5.
3. True. See chapter 5.
Answer: B
Question 15 discussion:
1. True. See chapter 5.
2. False. Reinsurers often use underwriting year data. See chapter 5.
3. True. See chapter 5.
Answer: D

Question 16 discussion:
1. True. See chapter 9.
2. True. See chapter 9.
3. True. See chapter 9.
Answer: D

Question 17 discussion:
1. False. The Benktander method is a credibility mixture of Bornhuetter-Ferguson and Development
techniques See chapter 9
2. True. See page chapter 9.
3. True. See page chapter 9.
Answer E: None of the given answer choices

Question 18 discussion:
1. True. See page 1.
2. False. The term “actuarial work product” applies to written and oral actuarial communications. See
page 1.
3. True. See page 2.
Answer: None of the given answer choices

Solutions to Exam 5B – Independently Authored and
Modified Past CAS MC Questions - Test 1
Question 19 discussion:
1. False. The actuary should take reasonable steps to ensure that an actuarial work product is presented
fairly, that the presentation as a whole is clear in its actuarial aspects, and that the actuary is identified
as the source of the actuarial aspects and as the individual who is available to answer questions. See
section 5.3
2. True. See section 5.6.
3. False. Ownership of documentation is normally established by the actuary and the client or employer,
in accordance with law.
Answer: B

Question 20 discussion:
1. True. See commentary under the "Credibility" consideration.
2. True. See commentary under the "Claims Made" consideration.
3. True. See commentary under the "Data Availability" consideration.
Answer: C
Question 21 discussion:
1. False. An inadequate estimate of unpaid claims could drive an insurer to reduce its rates not realizing
that the estimated unpaid claims were insufficient to cover historical claims. An excessive estimate of
unpaid claims could cause the insurer to increase rates unnecessarily, resulting in a loss of market
share and a loss of premium revenue to the insurer. See chapter 1
2. True. See chapter 1
3. True. See chapter 1
Answer D.

Exam 5B – Independently Authored and Modified Past CAS MC Questions
Preparatory Test 2
General information about this exam
1. This test contains 20 multiple choice questions.
2. The recommend time for this exam is 40 min.
3. Consider taking this exam after working all past CAS questions first.

Articles covered on exam:
Article .................................................... Author .................................. Syllabus Section
Chapter 10 – Cape Cod Technique ................................. Friedland ............. B: Estimating Claim Liabilities
Chapter 11 – Frequency-Severity Techniques ................. Friedland ..............B: Estimating Claim Liabilities
Chapter 12 – Case Outstanding Development TechniqueFriedland ..............B: Estimating Claim Liabilities
Chapter 13 – Berquist-Sherman Techniques.................... Friedland ..............B: Estimating Claim Liabilities

Exam 5B – Independently Authored and Modified
Past CAS MC Questions - Test 2
Question 1
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the following
are true regarding the Cape Cod technique?
1. The Cape Cod method splits ultimate claims into two components: actual reported (or paid) and
expected unreported (or unpaid).
2. In the Bornhuetter-Ferguson technique, the expected claim ratio is obtained from the reported claims
experience instead of an independent and often judgmental selection as in the Cape Cod technique.
3. The key assumption of the Cape Cod method is that unreported claims will develop based on expected
claims, which are derived using reported (or paid) claims and earned premium.
A. 1

B. 2

C. 1, 3

D. 2, 3

E. 1, 2, 3

Question 2
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the following
are true regarding the Cape Cod technique?
1. Reinsurers are among the most frequent users of the Cape Cod technique.
2. Actuaries generally use the Cape Cod method in a reported claims application, but they can also use it
with paid claims.
3. The technique is appropriate for mainly for short-tailed lines and not long-tail lines.
A. 1
B. 2
C. 1, 2
D. 2, 3
E. 1, 3

Question 3
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the following
are true regarding the Bornhuetter-Ferguson technique and the Cape Cod technique?
1. The Cape Cod method is a blend of two other methods: the Bornhuetter-Ferguson method and the
expected claims method.
2. Under the Cape Cod method: Ultimate Claims = Actual Reported Claims + Expected Unreported Claims
3. The major difference between the Cape Cod technique and the Bornhuetter-Ferguson
technique is the source of the expected claims.
A. 1
B. 2
C. 1, 2
D. 2, 3
E. 1, 3

Question 4
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the following
are true?
1. A problem with the SB (Stanard-Buhlmann) Method is that the IBNR by year is highly dependent upon
the rate level adjusted premium by year.
2. The key innovation of the SB Method is that the ultimate expected loss ratio for all years combined is
estimated from the overall reported claims experience, instead of being selected judgmentally, as in the
BF Method.
3. A problem which affects the SB method, unlike the BF method, is that the user must adjust each year's
premium to reflect the rate level cycle on a relative basis.
A. 2

B. 1, 2

C. 1, 2, 3

D. 3

E. None of the given answer choices

Exam 5B – Independently Authored and Modified
Past CAS MC Questions - Test 2
Question 5
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the following
are true regarding the Cape Cod technique?
1. Used-up premium is equal to the earned premium multiplied by the percentage of claims unreported.
2. Instead of calculating used-up premium, the actuary could calculate used-up exposures and calculate
estimated pure premiums instead of estimated claim ratios for each year in the experience period.
3. Reinsurers often use ultimate premiums in computing used up premium instead of earned premium.
A. 2
B. 2, 3
C. 1, 2, 3
D. 1, 3
E. None of the given answer choices

Question 6
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the following
are true regarding frequency and severity techniques?
1. Projections based on frequency-severity techniques can be extremely valuable, not only in providing
additional estimates of unpaid claims, but also in understanding the drivers in claims activity.
2. When actuaries use frequency-severity techniques in their simplest form, they project ultimate claims by
multiplying the estimated ultimate number of claims by the estimated ultimate average value divided by
estimated ultimate exposures.
3. One of the problems with frequency-severity methods is that they cannot be used to validate or reject the
findings from other actuarial projection techniques.
A. 3

B. 2, 3

C.1, 2, 3

D. 1, 3 E. None of the given answer choices

Question 7
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the following
are true regarding frequency and severity techniques?
1. Frequency-severity techniques can use accident year, policy year, report year, and calendar year data.
2. Reinsurers often use frequency-severity methods with underwriting year data.
3. Frequency-severity techniques are appropriate for all lines of insurance but are more often used for
medium tail lines.
A. 1
B. 1, 2
C. 1, 3
D. 2, 3
E. None of the given answer choices

Question 8
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the following
are true regarding frequency and severity techniques?
1. The simplest frequency-severity approach is based on a disposal rate analysis.
2. In the second frequency-severity approach discussed, the authors' focus on projecting ultimate claims
for the most recent two accident years, since the development method can often result in substantial
development factors to ultimate for the most recent accident years.
3. In the third frequency-severity approach discussed, the authors' examine the rate of claim count closure
at each maturity age and the incremental paid severity by maturity age.
A. 1
B. 2
C. 1, 2
D. 2, 3
E. None of the given answer choices

Exam 5B – Independently Authored and Modified
Past CAS MC Questions - Test 2
Question 9
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the following
are true regarding frequency and severity techniques?
1. Two of the major requirements of frequency-severity techniques are that the individual claim counts being
grouped are defined in a consistent manner over the experience period and that the claim counts are
reasonably homogenous.
2. Since many frequency-severity methods rely on the development technique applied separately to claim
counts and average values, a key assumption of the development technique is also applicable to this type
of frequency-severity analysis.
3. The actuary using the development technique on severities assumes that the relative change in a given
year's severities from one evaluation point to the next is similar to the relative change in prior years'
severities at similar evaluation points.
A. 2
B. 2, 3
C. 1, 2, 3
D. 1, 3
E. None of the given answer choices

Question 10
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the following
are true with respect to the disposal rate technique to estimating unpaid claims?
1. Similar to the previous two frequency-severity approaches, it is assumed that historical patterns of claims
emergence and settlement are predictive of future patterns of reported and closed claim counts.
2. While there is an implicit assumption of this method is that there are no significant partial (i.e. interim)
payments, the method ultimately adjusts itself for such payments.
3. The selected trend rate to account for inflation adjustment in severity is important, but a slight change in
trend will not result in a material change in the estimated of unpaid claims.
A. 1

B. 1, 2

C. 1, 2, 3

D. 1, 3 E. None of the given answer choices.

Question 11
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the following
are advantages to using a frequency-severity approach?
1. Changes in the definition of claim counts, claims processing, or both are offset by the relative impact it
has upon paid and reported claims, resulting in relatively stable frequencies and severities.
2. The data to perform such analysis is often available.
3. The ability to explicitly reflect inflation in the projection methodology instead of assuming that past
development patterns will properly account for inflationary forces.
A. 1

B. 3

C. 2, 3

D. 1, 2, 3

E. None of the given answer choices

Exam 5B – Independently Authored and Modified
Past CAS MC Questions - Test 2
Question 12
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the following
are true with respect to the case outstanding development technique?
1. The case reserve development method attempts to analyze the adequacy of both case and IBNR
reserves based on the history of payments against those case reserves.
2. It is assumed that claims activity related to IBNR is related consistently to claims already reported.
3. Assumptions for the case outstanding development technique are similar to those for other loss
development techniques.
A. 1

B. 2

C. 2, 3

D. 1, 2

E. None of the given answer choices

Question 13
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the following
are true with respect to the case outstanding development technique?
1. The case outstanding development technique is used extensively by actuaries.
2. The assumption that IBNR claim activity is related to development on known claims versus pure IBNR
limits its use.
3. The case outstanding development method is appropriate for claims-made coverages and report year
analysis because the claims for a given accident year are known at the end of the accident year.
A. 1
B. 2, 3
C 1, 2, 3
D. 1, 3 E. None of the given answer choices

Question 14
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the following
are true with respect to the case outstanding development technique?
1. Ratios of the incremental paid claims at age x to the case outstanding at age x+12 are computed.
2. Ratios of the case outstanding to the previous case outstanding are computed.
3. A challenge of this technique is the selection of the "to ultimate" ratios for both the ratio of incremental
paid claims to subsequent case outstanding and the ratio of case outstanding to previous case
outstanding.
A. 1
B. 2
C. 1, 2
D. 1, 3
E. 1, 2, 3

Question 15
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the following
are true with respect to limitations of the case outstanding development technique?
1. The assumption that future IBNR is related to claims already reported does not hold true for many P&C
lines of insurance.
2. The infrequent use and the absence of benchmark data (for accident year applications of this method).
3. A lack of intuitive sense and experiential knowledge as to what ratios are appropriate at each maturity for
both the incremental paid claims to case outstanding and the case outstanding to previous case
outstanding across P&C lines of insurance.
A. 2
B. 2, 3
C. 1, 2, 3
D. 1, 3 E. None of the given answer choices

Exam 5B – Independently Authored and Modified
Past CAS MC Questions - Test 2
Question 16
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, Berquist and Sherman
cite which of the following examples for selecting alternative data to respond to potential problems related to
a changing environment?
1. Using written exposures instead of the number of claims when claim count data is of questionable
accuracy or if there has been a major change in the definition of a claim count.
2. Substituting policy year data for calendar year data when there has been a significant change in policy
limits or deductibles between successive policy years.
3. Substituting policy year data for accident year data when there has been a dramatic shift in the social or
legal climate that causes claim severity to more closely correlate with the policy year than with the accident
date.
A. 1

B. 3

C. 1, 3

D. 2, 3

E. None of the given answer choices

Question 17
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the following
are true for selecting data to respond to potential problems related to a changing environment?
1. One way to adjust the data for changes in operations is to divide the data into less homogeneous groups,
and is valuable when there have been changes in the composition of business by jurisdiction, coverage,
class, territory, or size of risk.
2. While dividing the data into less homogeneous groups, the actuary must seek to retain sufficient volume of
experience within each grouping to ensure the credibility of the data.
3. If greater attention is directed at the handling of large claims, there may be a speed-up in the settlement of
these particular claims that could affect both the paid claims and case outstanding triangles; if the large
claims are settled earlier then the case outstanding will no longer be present in the triangle at the later
maturities and the payments will appear in the triangles at earlier maturities than in the past.
A. 1, 2

B. 3

C. 1, 2, 3

D. 1, 3

E. None of the given answer choices

Question 18
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the following
are true regarding the approaches an actuary can use to determine if an insurer has sustained changes in
case outstanding adequacy?
1. A meeting with claims department management to discuss the claims process should be a prerequisite
to any analysis of unpaid claims.
2. The actuary can also calculate various claim development diagnostic tests, including: the ratio of paid-toreported claims, average case outstanding, average reported claim, and average paid claims.
3. In their medical malpractice example, Berquist and Sherman compare the annual change in the average
case outstanding to the annual change in the average reported claims to confirm a shift in case
outstanding adequacy.
A. 2

B. 1, 2

C. 1, 2, 3

D. 1, 3

E. None of the given answer choices

Exam 5B – Independently Authored and Modified
Past CAS MC Questions - Test 2
Question 19
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the following
are true with respect to the Mechanics of the Berquist-Sherman Paid Claim Development Adjustment?
1. The first step of the Berquist-Sherman paid claims adjustment is to determine the disposal
rates by policy year and maturity.
2. To determine the disposal rates, we first project the number of ultimate claims based on
reported claim counts.
3. The disposal rate is equal to the cumulative closed claim counts for each policy-maturity age
cell divided by the ultimate claim counts for the particular policy year.
A. 2

B. 2, 3

C. 1, 2, 3

D. 1, 3

E. None of the given answer choices

Question 20
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the following
are true regarding the authors' last projection for XYZ Insurer, which adjusts the data for changes in both
case outstanding adequacy and the rate of claims settlement?
1. The authors' use both an adjusted average paid claim triangle and the adjusted average case
outstanding triangle.
2. There is one new adjusted triangle we need to create for this projection: the adjusted number of open claims.
3. The authors' derive the adjusted open claim count triangle by subtracting the adjusted closed claim
count triangle from paid claim counts.
A. 1

B. 2

C. 1, 2

D. 2, 3

E. 1, 2, 3

Solutions to Exam 5B – Independently Authored and
Modified Past CAS MC Questions - Test 2
Question 1 discussion:
1. True. See chapter 10
2. False. In the Cape Cod technique, the expected claim ratio is obtained from the reported claims
experience instead of an independent and often judgmental selection as in the BornhuetterFerguson technique. See chapter 10
3. True. See chapter 10
Answer: C

Question 2 discussion:
1. True. See chapter 10
2. True. See chapter 10
3. False. The technique is appropriate for all lines of insurance including short-tail lines and long-tail lines.
See chapter 10
Answer: C

Question 3 discussion:
1. False. The Cape Cod method is a blend of two other methods: the claim development method and the
expected claims method. See chapter 10
2. True. See chapter 10
3. True. See chapter 10
Answer: D

Question 4 discussion:
1. True. See chapter 10
2. True. See chapter 10
3. False. This is a problem for the BF method as well. See chapter 10
Answer: B

Question 5 discussion:
1. False. Used-up premium is equal to the earned premium multiplied by the percentage of claims
unreported. See chapter 10
2. True. See chapter 10
3. True. See chapter 10
Answer: B

Solutions to Exam 5B – Independently Authored and
Modified Past CAS MC Questions - Test 2
Question 6 discussion:
1. True. See chapter 11
2. False. When actuaries use frequency-severity techniques in their simplest form, they project ultimate
claims by multiplying the estimated ultimate number of claims by the estimated ultimate average value.
3. False. Frequency-severity methods can also be important to validate or reject the findings from other
actuarial projection techniques. See chapter 11
Answer: E. None of the given answer choices
Question 7 discussion:
1. True. See chapter 11
2. False. Generally reinsurers do not use frequency-severity methods with underwriting year data simply
because they do not have access to detailed statistics regarding the number of claims. See
chapter 11
3. False. Frequency-severity techniques are appropriate for all lines of insurance but are more often used
for short tail lines. See chapter 11
Answer: A

Question 8 discussion:
1. False. The first and simplest frequency-severity approach is the development technique applied
separately to claim counts and average values. See chapter 11
2. True. See chapter 11
3. True. See chapter 11
Answer: D

Question 9 discussion:
1. True. See chapter 11
2. True. See chapter 11
3. True. See chapter 11
Answer: C

Question 10 discussion:
1. True. See chapter 11
2. False. There is no mention of this. See chapter 11
3. False. "… a slight change in trend can result in a material change in the estimated of unpaid claims, and
therefore the trend rate must be selected carefully. See chapter 11
Answer: A

Question 11 discussion:
1. False. This is not mentioned. See chapter 11
2. False. See chapter 11
3. True. See chapter 11
Answer: B
Copyright  2014 by All 10, Inc.
Page 20

Solutions to Exam 5B – Independently Authored and
Modified Past CAS MC Questions - Test 2
Question 12 discussion:
1. False. The case reserve development method attempts to analyze the adequacy of case reserves
based on the history of payments against those case reserves. See chapter 12.
2. True. See chapter 12.
3. True. See chapter 12.
Answer: C

Question 13 discussion:
1. False. See chapter 12.
2. True. See chapter 12.
3. True. See chapter 12.
Answer: B

Question 14 discussion:
1. False. Ratios of the incremental paid claims at age x to the case outstanding at age x-12 are computed.
2. True. See chapter 12.
3. False. A challenge of this technique is the selection of the "to ultimate" ratios for both the ratio of
incremental paid claims to previous case outstanding and the ratio of case outstanding to previous case
outstanding. See chapter 12.
Answer: B

Question 15 discussion:
1. True. See chapter 12.
2. True. See chapter 12.
3. True. See chapter 12.
Answer: C

Question 16 discussion:
1. False. Using earned exposures… See chapter 13.
2. False. Substituting policy year data for accident year data. See chapter 13.
3. False. Substituting report year data for accident year data when there has been a dramatic shift in the
social or legal climate that causes claim severity to more closely correlate with the report year than with
the accident date. See chapter 13.
Answer: E. None of the given answer choices

Question 17 discussion:
1. False. "… the data into more homogeneous groups,…". See chapter 13.
2. False. "… the data into more homogeneous groups,…". See chapter 13.
3. True. See chapter 13.
Answer: B
Copyright  2014 by All 10, Inc.
Page 21

Solutions to Exam 5B – Independently Authored and
Modified Past CAS MC Questions - Test 2
Question 18 discussion:
1. True. See chapter 13.
2. True. See chapter 13.
3. False. In their medical malpractice example, Berquist and Sherman compare the annual change in the
average case outstanding to the annual change in the average paid claims to confirm a shift in case
outstanding adequacy. See chapter 13.
Answer: B

Question 19 discussion:
1. False. he first step of the Berquist-Sherman paid claims adjustment is to determine the disposal rates by
accident year and maturity. See chapter 13.
2. True. See chapter 13.
3. False. The disposal rate is equal to the cumulative closed claim counts for each accident-maturity age
cell divided by the ultimate claim counts for the particular accident year. See chapter 13.
Answer: A

Question 20 discussion:
1. True. See chapter 13.
2. True. See chapter 13.
3. False. We derive the adjusted open claim count triangle by subtracting the adjusted closed claim count
triangle from reported claim counts. See chapter 13.
Answer: C

Question 21 discussion:
1. True. See chapter 13
2. True. See chapter 13
3. True. See chapter 13
Answer: 1, 2, and 3

Exam 5B – Independently Authored and Modified Past CAS MC Questions
Preparatory Test 3
General information about this exam
1. This test contains 20 multiple choice questions.
2. The recommend time for this exam is 40 min.
3. Consider taking this exam after working all past CAS questions first.

Articles covered on exam:
Article .................................................... Author .................................. Syllabus Section
Chapter 14 – Recoveries: Salvage & Subro and Reins .... Friedland ..............B:
Chapter 15 – Evaluation of Techniques ............................ Friedland ..............B:
Chapter 16 — Estimating Unpaid Claim Adj Expenses .... Friedland ..............B:
Chapter 17 - Estimating Unpaid ULAE ............................. Friedland ..............B:

Estimating Claim Liabilities
Estimating Claim Liabilities
Estimating Claim Liabilities
Estimating Claim Liabilities

Exam 5B – Independently Authored and Modified
Past CAS MC Questions - Test 3
Question 1
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the
following are true with respect to salvage and subrogation?
1. Salvage refers to an insurer's right to recover the amount of claim payment to a covered insured from
a third-party responsible for the injury or damage.
2. Subrogation represents any amount that the insurer is able to collect from the sale of such damaged
property.
3. Recoveries due to salvage, can take years to realize, well after the underlying claims are paid,
resulting in age-to-age factors less than one for older maturities for some lines of business.
A. 1

B. 2, 3

C 1, 2, 3

D. 1, 3

E. None of the given answer choices

Question 2
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the
following are true with respect to salvage and subrogation?
1. Actuaries frequently use the both the development technique and the Bornhuetter-Ferguson technique
to quantify the affect of S&S recoveries on estimates of total unpaid claims.
2. Paid S&S represents a payment made by the insured to the insurer.
3. Many actuaries also use a ratio approach when analyzing S&S.
A. 1

B. 3

C 1, 2, 3

D. 1, 3

E. None of the given answer choices

Question 3
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the
following are true with respect to salvage and subrogation?
1. One advantage of the ratio approach is that the development factors tend not to be as highly
leveraged as the development factors based on received S&S dollars.
2. One advantage or the ratio approach is related to the selection of the ultimate S&S ratio(s) for the
most recent year(s) in the experience period, especially when the development approach produces an
ultimate S&S ratio which is not consistent with the more recent reported ratios.
3. Ultimate S&S using the ratio approach is determined by multiplying selected ultimate claims and the
selected ultimate S&S ratio.
A. 1 only

B. 2 only

C. 3 only

D. 1, 2 only

E. 1, 2, 3

Exam 5B – Independently Authored and Modified
Past CAS MC Questions - Test 3
Question 4
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the
following are true with respect to gross and net data?
1. If ceded claims are coded in the same database as gross data, net data is available. In this
case, the actuary is more likely to conduct both gross and net analyses.
2. Some insurers code the ceded claims data to a different system; thus matching the gross and ceded
data to derive net claim triangles may be more difficult. In this case, the actuary will likely prepare
separate gross and ceded analyses.
3. The choice of gross versus net versus ceded analysis may be a function of data volume and quality.
A. 1

B. 2

C. 3

D. 1, 2

E. 1, 2, 3

Question 5
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the
following are true with respect to gross and net data?
1. If the reinsurance program consists of quota share arrangements, the actuary can create a
development triangle with the ratio of net-to-gross claims and thus test the quota share
percentage(s) by year.
2. If the reinsurance program consists of excess of loss arrangements, the actuary may want to examine
large claims to confirm that retentions and limits for ceded claims by year are consistent with the
corresponding excess of loss reinsurance contracts or with information provided.
3. Since net claims are often capped due to excess or aggregate coverage, we frequently observe net
claim development patterns that are less than or equal to gross claim development patterns
A, 2

B. 2, 3

C. 1, 2, 3

D. 1, 3 E. None of the given answer choices

Question 6
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the
following are true with respect to gross and net IBNR
1. Net IBNR in each AY is generally not greater than gross IBNR.
2. When an estimate of uncollectible reinsurance is included in the net IBNR but not in the gross IBNR
and there are significant billed reinsurance amounts for which significant collectibility issues exist, net
IBNR will be greater than the gross IBNR
3. For a runoff book with reinsurance disputes for items such as asbestos, net IBNR will be
greater than the gross IBNR
A, 2

B. 2, 3

C. 1, 2, 3

D. 1, 3 E. None of the given answer choices

Exam 5B – Independently Authored and Modified
Past CAS MC Questions - Test 3
Question 7
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, Berquist and Sherman
recommend that where possible, the actuary conducting an analysis of unpaid claims should use methods
that incorporate which the following?
1. Projections of reported claims
2. Projections of paid claims
3. Projections of ultimate reported claim counts and severities
4. Estimates of the number and average amount of outstanding claims
5. Claim ratio estimates
A. 1, 2

B. 1, 2, 3

C. 1, 2, 4

D. 1, 2, 3, 4

E. 1, 2, 3, 4, 5

Question 8
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the following
are true?
1. When selecting the most appropriate estimate of unpaid claims, actuaries may incorporate the concept
of credibility into the selection process, while at other times actuarial judgment will prevail.
2. If there is sufficient claim history available, testing a reserving method retroactively can help the
actuary to determine the historical accuracy of the method and whether or not the particular method
is free from bias in projecting future results
3. An important final check of the selected ultimate claims, particularly for the oldest years,
should include calculation of claim ratios, severities, pure premiums, and claim frequencies.
A. 1

B. 1, 2

C. 3

D. 2, 3

E. None of the given answers

Question 9
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the following
are true with respect to selecting ultimate claims when the results from a variety of reserving methods are
fairly consistent for older accidents years, but are more variable for more recent accident years?
1. Some actuaries may select one method and use it for all years.
2. Some actuaries may select different methods for different accident years.
3. Some actuaries may use a weighted average of the results from various methods based on
assigned weights to those methods; these weights may be consistent for all years or may vary
by accident year.
A. 1

B. 1, 2

C. 3

D. 2, 3

E. 1, 2, 3

Exam 5B – Independently Authored and Modified
Past CAS MC Questions - Test 3
Question 10
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the following
are true when monitoring unpaid claim estimates.
1. It is typically a simple exercise to develop a model that allows comparisons of actual and expected
claims by accident year between successive annual valuations.
2. Expected paid claims in the calendar year 2008 for AY 2007 are equal to [(ultimate claims selected at
December 31, 2007 — actual reported claims at December 31, 2007) / (% unreported at December 31,
2007)] x (% reported at December 31, 2008 - % reported at December 31, 2007)
3. When actuaries rely on techniques other than the development technique to select ultimate claims, it is
often valuable to look at an alternative method for deriving reporting and payment patterns (other than
the inverse of the age to age development factor).
A. 2

B. 2, 3

C. 1, 2, 3

D. 1, 3

E. None of the given answer choices

Question 11
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the following
are true regarding an appropriate system for quarterly or monthly monitoring of loss development?
1. It is a relatively easy task to develop a system for quarterly or monthly monitoring given an estimation
process that focuses only on annual claim development patterns.
2. Insurers that maintain claim development data on a quarterly basis have development factors that are
readily available for quarterly analyses, and linear interpolation between quarters is likely sufficient for
monthly monitoring purposes.
3. For insurers who only have annual claim development data, linear interpolation of annual development
patterns is usually appropriate.
A. 1

B. 2

C. 1, 3

D. 2, 3

E. 1, 2, 3

Question 12
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the following
are common techniques to develop ALAE?
1. The development technique using paid ALAE.
2. The development technique using reported ALAE (when case O/S for ALAE exists)
3. The development of the ratio of reported ALAE-to-reported claims only.
A. 2

B. 1

C. 1, 2

D. 2, 3

E. None of the given answer choices

Question 13
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the following
are advantages to using the ratio method to develop ALAE?
1. It recognizes the relationship between ALAE and claims only.
2. The ratio development factors are not as highly leveraged as those based on paid ALAE dollars.
3. The ability to interject actuarial judgment in the projection analysis, especially for the selection of the
ultimate ALAE ratio for the most recent year(s) in the experience period.
A. 1
B. 1, 2
C. 1, 2, 3
D. 1, 3
E. None of the given answer choices
Copyright  2014 by All 10, Inc.
Page 27

Exam 5B – Independently Authored and Modified
Past CAS MC Questions - Test 3
Question 14
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the following
are true?
1. An important assumption underlying the ratio analysis is that the relationship between ALAE and claims
only is relatively stable over the experience period.
2. A disadvantage of the ratio method is that any error in the estimate of ultimate claims only could affect
the estimate of ultimate ALAE.
3. A potential challenge with a ratio method exists for some lines of business where large amounts of ALAE
may be spent on claims that ultimately settle with no claim payment.
A. 1

B. 1, 2

C. 1, 2, 3

D. 1, 3

E. None of the given answer choices

Question 15
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the following
are true regarding dollar-based and count based techniques for estimating unpaid ULAE?
1. These techniques may produce similar results.
2. Many self-insurers use market values, fees paid to third-party claims administrator to manage a book of
claims, to determine the unpaid ULAE for financial reporting purposes.
3. These techniques, which rely on fundamentally different assumptions, vary significantly in the amount of
data and calculations required.
A. 1
B. 3
C. 1, 3
D. 2, 3
E. 1, 2, 3

Question 16
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the following
are true regarding the key assumptions of the Classical (traditional) technique to computing unpaid ULAE?
1. ULAE is sustained as claims are reported even if no claim payments are made.
2. ULAE payments for a specific calendar year are related to both the reporting and payment of claims.
3. The volume and cost of future claims management on not-yet-reported claims and reported-but-not-yetclosed claims will be proportional to IBNR and case O/S, respectively.
A. 1

B. 1, 2

C. 3

D. 2, 3

E. None of the given answers

Question 17
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the following
are true regarding the Classical and Kittel refinement techniques to estimating unpaid claims?
1. One challenge with the Kittel refinement technique is that “closing” a claim and “paying” a claim do not
necessarily mean the same thing.
2. The definition of IBNR poses a challenge for actuaries using the classical technique.
3. According to Johnson, the classical technique “will only give good results for very short-tailed, stable
lines of business.”
A. 1
B. 1, 2
C. 3
D. 2, 3
E. None of the given answers

Exam 5B – Independently Authored and Modified
Past CAS MC Questions - Test 3
Question 18
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the following
are true regarding the Classical and Kittel refinement techniques to estimating unpaid claims?
1. The paid to paid methodology assumes that claims incur expenses only when initially opened and when
closed, which is an unreasonable assumption for claims from short-tailed lines.
2. In the Kittel refinement, calendar year incurred claims are defined to be calendar year reported claims
plus the change in total claim liabilities, including both case outstanding and IBNR.
3. The classical technique makes the implicit simplifying assumption that paid claims are approximately
equal to reported claims, and thus the two quantities can be used interchangeably.
A. 1

B. 1, 2

C. 3

D. 2, 3

E. None of the given answers

Question 19
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the following
are true regarding the Kittel's refinement technique to estimating unpaid claims?
1. The relative volume and cost of future claims management activity on not-yet-reported claims and
reported-but-not-yet-closed claims is expected to be proportional to the dollars of IBNR and case
outstanding, respectively.
2. One-half of expenses are sustained when opening a claim and one-half of expenses when closing a
claim.
3. The Kittel refinement fails to addresses the distortion created when using the classical technique for a
growing insurer.
A. 1

B. 2

C. 3

D. 2, 3

E. 1, 2, 3

Question 20
According to Friedland et al. in “Estimating Unpaid Claims Using Basic Techniques”, which of the following
are true regarding the Conger and Nolibos Method – Generalized Kittel Approach?
1. The claim basis for a particular time period is defined to be the weighted average of the ultimate cost of
claims reported during the period, the ultimate cost of claims closed during the period, and the amount of
claims outstanding at the end of the period.
2. Since Conger and Nolibos believe that handling costlier claims warrants and requires relatively more
resources than handling smaller claims, they use claim counts instead of claim dollars in their
generalized approach.
3. The values of U1, U2, and U3 could vary significantly from insurer to insurer and between lines of business.
A. 1

B. 3

C. 1, 3

D. 1, 2

E. 1, 2, 3

Solutions to Exam 5B – Independently Authored and
Modified Past CAS MC Questions - Test 3
Question 1 discussion:
1. False. Statement 1 refers to Salvage. See chapter 14
2. False. Statement 2 refers to Subrogation. See chapter 14
3. False. Statement 2 refers to Subrogation. See chapter 14
Answer: None of the given answer choices
Question 2 discussion:
1. False. The development technique… See chapter 14
2. False. Paid S&S represents a payment made by a third-party to the insurer. See chapter 14
3. True. See chapter 14
Answer: B
Question 3 discussion:
1. True. See chapter 14
2. True. See chapter 14
3. True. See chapter 14
Answer: E

Question 4 discussion:
1. True. See chapter 14
2. True. See chapter 14
3. True. See chapter 14
Answer: E

Question 5 discussion:
1. True. See chapter 14
2. True. See chapter 14
3. True. See chapter 14
Answer: C

Question 6 discussion:
1. True. See chapter 14
2. True. See chapter 14
3. True. See chapter 14
Answer: C

Solutions to Exam 5B – Independently Authored and
Modified Past CAS MC Questions - Test 3
Question 7 discussion:
Answer: E – See chapter 15

Question 8 discussion:
1. True. See chapter 15
2. True. See chapter 15
3. False. An important final check of the selected ultimate claims, particularly for the most recent years,
should include calculation of claim ratios, severities, pure premiums, and claim frequencies. See chapter 15
Answer: B

Question 9 discussion:
1. True. See chapter 15
2. True. See chapter 15
3. True. See chapter 15
Answer: E

Question 10 discussion:
1. True. See chapter 15
2. False. " Expected reported claims… See chapter 15
3. False. " the inverse of the cumulative development factor).See chapter 15
Answer: E

Question 11 discussion:
1. False. It can be a challenging task to develop a system for quarterly or monthly monitoring given an
estimation process that focuses only on annual claim development patterns. See chapter 15
2. True. See chapter 15
3. False. For insurers who only have annual claim development data, linear interpolation of annual
development patterns is usually not appropriate, particularly for the most immature accident years. See
chapter 15
Answer: B

Question 12 discussion:
1. True. See chapter 16
2. True. See chapter 16
3. False. The development of the ratio of paid ALAE-to-paid claims only. See chapter 16
Answer: C

Solutions to Exam 5B – Independently Authored and
Modified Past CAS MC Questions - Test 3
Question 13 discussion:
1. True. See chapter 16
2. True. See chapter 16
3. True. See chapter 16
Answer: C

Question 14 discussion:
1. True. See chapter 16
2. True. See chapter 16
3. True. See chapter 16
Answer: C

Question 15 discussion:
1. True. See chapter 17.
2. True. See chapter 17.
3. True. See chapter 17.
Answer: E
Question 16 discussion:
1. False. This is an assumption of the Kittel refinement technique. See chapter 17.
2. False. This is an assumption of the Kittel refinement technique. See chapter 17.
3. True. See chapter 17.
Answer: C
Question 17 discussion:
1. False. This is an assumption of the Classical refinement technique. See chapter 17.
2. True. See chapter 17.
3. True. See chapter 17.
Answer: D
Question 18 discussion:
1. False. See chapter 17.
2. False. In the Kittel refinement, calendar year incurred claims are defined to be calendar year paid claims
plus the change in total claim liabilities, including both case outstanding and IBNR. See chapter 17.
3. True. See chapter 17.
Answer: C
Question 19 discussion:
1. True. See chapter 17.
2. True. See chapter 17.
3. False. See chapter 17.
Answer: C

Solutions to Exam 5B – Independently Authored and
Modified Past CAS MC Questions - Test 3
Question 20 discussion:
1. False. "… and the claims paid during the period". See chapter 17.
2. False. Since Conger and Nolibos believe that handling costlier claims warrants and requires relatively
more resources than handling smaller claims, they use claim dollars instead of claim counts in their
generalized approach. See chapter 17.
3. True. See chapter 17.
Answer: B

Exam 5B – Independently Authored Preparatory Test 1
General information about this exam
1. This test contains 22 computational and essay questions.
2. The recommend time for this exam is 2:30:00. Make sure you have sufficient time to take this practice test.
3. Consider taking this exam after working all past CAS questions, associated with the articles below, first.
4. Many of the essay questions may require lengthy responses.
5. Make sure you have a sufficient number of blank sheets of paper to record your answers.

Articles covered on exam:
Article .................................................... Author .................................. Syllabus Section
Chapter 1 – Overview ....................................................... Friedland ..............B: Estimating Claim Liabilities
Chapter 2 – The Claims Process ...................................... Friedland ..............B: Estimating Claim Liabilities
Chapter 3 – Understanding the Types of Data Used ........ Friedland ..............B: Estimating Claim Liabilities
Chapter 4 - Meeting with Management ............................. Friedland ..............B: Estimating Claim Liabilities
Chapter 5 - The Development Triangle ............................. Friedland ..............B: Estimating Claim Liabilities
Chapter 6 - Development Triangle as a Diagnostic Tool .. Friedland ..............B: Estimating Claim Liabilities

Chapter 7 – Development Technique .............................. Friedland ..............B: Estimating Claim Liabilities
Chapter 9 – Bornhuetter-Ferguson Technique ................ Friedland ..............B: Estimating Claim Liabilities
Statement of Principles: Loss and LAE Reserves ............ CAS ......................B: Estimating Claim Liabilities
ASOP No. 9 – Documentation and Disclosure ................. AAA ......................B: Estimating Claim Liabilities

Exam 5B - Independently Authored Preparatory Test 1
Question 1. (4 points) You are given the following information:

Accident
Year
2002
2003
2004
2005

Earned
Premium
($000)
400
2000
3000
3000

Case Incurred
Losses ($000),
Valued as of
12/31/2005
200
2000
1800
1200

Expected
Loss
Ratio
80%
80%
80%
80%

Selected age-to-age incurred loss development factors:
12 - 24 months
1.250
24 - 36 months
1.100
36 - 48 months
1.050
48 - 60 months
1.080
No further development after 60 months
a. (1 point)
b.
c.
d.
e.

Calculate the IBNR reserve as of December 31, 2005 using case incurred loss development.
Show all work.
(1 point) Calculate the IBNR reserve as of December 31, 2005 using the Bornhuetter-Ferguson
method. Show all work.
(0.5 point) Identify one situation in which it would be preferable to use the case incurred method rather
than the Bornhuetter-Ferguson method to develop the IBNR.
(0.5 point) Identify one situation in which it would be preferable to use the Bornhuetter-Ferguson method
rather than the case incurred loss development method to estimate the IBNR.
(1 point) Using the Bornhuetter-Ferguson method, calculate the amount of loss development to be
expected during calendar year 2006 on accident years 2002 through 2005. Show all work.

Question 2.
(1.5 points) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, accurately
estimating unpaid claims is essential for proper decision-making.
With respect to either rates, market share, or underwriting, strategic and financial decision making, briefly
describe how improper estimates could ruin an insurer’s financial condition in terms of an:
a. (.50 points) inadequate estimate of unpaid claims.
b. (.50 points) excessive estimate of unpaid claims.
c. (.50 points) inaccurate estimate of unpaid claims.

Question 3.
(1.5 points) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, there are several
approaches to establishing case outstanding reserves. List and briefly describe three approaches.

Question 4.
(1.5 points) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, list and briefly
describe three approaches, for many insurers, determining the case outstanding for reinsurance recoveries is
a fairly straightforward exercise. Briefly describe how ceded case outstanding is set for both proportional and
non-proportional reinsurance.
Copyright  2014 by All 10, Inc.
Page 2

Exam 5B - Independently Authored Preparatory Test 1
Question 5.
(2.25 points) Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, describes the computation of
cumulative reported claims, cumulative paid claims and case outstanding reserves. Assume the following:
* An automobile insurer issues a 1 year policy effective 1/1/2007 – 11/30/2008.
* An accident occurred on 11/15/2008, but the insurer does not receive notice of the claim until 2/20/2009.
* Over the life of the claims, a claims professional records a number of transactions which include:
Date
Transaction
February 20, 2009
Case 0/S of $15,000 established for claim only
April 1, 2009
Claim payment of $1,500 - case 0/S reduced to
$13,500 (case 0/S change of -$1,500)
May 1, 2009
Expense payment to IA of $500; no change in case O/S
September 1, 2009
March 1, 2010

January 25, 2011
April 15, 2011

September 1, 2011

March 1, 2012

Case 0/S for claim increased to $30,000
(case 0/S change of +$16,500)
Claim thought to be settled with additional
payment of $24,000 – case 0/S reduced to $0
and claim closed (case 0/S change of -$30,000)
Claim reopened with case 0/S of $10,000 for
claim and $10,000 for defense costs
Partial payment of $5,000 for defense litigation
and case 0/S for defense costs reduced to
$5,000 – no change in case 0/S for claim
Final claim payment for an additional $12,000
case 0/S for claim reduced to $0 (case 0/S
change of -$10,000)
Final defense cost payment for an additional
$6,000 – case 0/S for defense costs reduced to
$0 and claim closed (case 0/S change of -$5,000)

a. (.75 points) Compute cumulative reported claims as of 3/1/2010.
b. (.75 points) Compute cumulative paid claims as of 4/15/2011.
c. (.75 points) Compute Case outstanding as of 9/1/2011.

Question 6
(1 point). Based on the CAS “Statement of Principles Regarding Property and Casualty Loss and Loss
Adjustment Expense Reserves,” define what constitutes an actuarially sound loss reserve, for a defined
group of claims as of a given valuation date.

Question 7
(1 point). Based on the CAS “Statement of Principles Regarding Property and Casualty Loss and Loss
Adjustment Expense Reserves,” one of the considerations when establishing a loss reserve is to
examine the impact of external influences. List 5 types of external influences.

Exam 5B - Independently Authored Preparatory Test 1
Question 8
(2.25 points) Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, describes the
computation of cumulative reported claims, cumulative paid claims and case outstanding reserves.
Assume the following:

Claim
Number
‘(1)

At December 31, 2007
Cumulative
Paid
Case
Reported
Claims
O/S
Claims
‘(2)
‘(3)
‘(4)

Transactions During 2008
At December 31, 2008
Change
Cumulative
Paid
in
Reported
Paid
Case
Reported
Claims Case O/S Claims
Claims
O/S
Claims
‘(5)
‘(6)
‘(7)
‘(8)
‘(9)
‘(10)

500

5,000

5,500

200

5,000

15,000

20,000

4,500

2,000

10,000

12,000

1,000

(1,000)

5,000

a. (.75 points) Compute cumulative reported claims as of 12/31/2008 for claim number 1.
b. (.75 points) Compute case outstanding as of 12/31/2008 for claim number 2.
c. (.75 points) Compute cumulative reported claims as of 12/31/2008 for claim number 3.

Question 9
(3 points). According to ASB "Actuarial Standard of Practice No. 9, Documentation and Disclosure in
Property and Casualty Insurance Ratemaking, Loss Reserving, and Valuations," documentation of an
actuarial work product is required whether or not there is a legal or regulatory requirement for the
documentation. In addition, appropriate records, worksheets, and other documentation of the actuary's work
should be maintained by the actuary and retained for a reasonable period of time.
List and briefly describe three other requirements regarding the extent of documentation required in an
actuarial work product.

Question 10
(1 point) A total loss reserve is composed of five elements, although the five elements may not
necessarily be individually quantified. List the five elements.

Question 11
(1.5 points). According to ASB "Actuarial Standard of Practice No. 9, Documentation and Disclosure in
Property and Casualty Insurance Ratemaking, Loss Reserving, and Valuations," briefly discuss the extent of
an actuary’s responsibility when relying on another actuary’s work product.

Question 12
(1 point). According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, list and briefly
describe one reason why a small insurer and a larger insurer would have a need for external data.

Exam 5B - Independently Authored Preparatory Test 1
Question 13
(1.5 points) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, list and briefly
describe three key claims characteristics that actuaries focus on when separating data into groups prior to
the analysis of unpaid claims.

Question 14
(1.5 points) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, list and briefly
describe three claims characteristics actuaries consider when establishing a large claim threshold.

Question 15
(1.5 points) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, the presence of
unusually large claims can distort some of the methods used for estimating unpaid claims.
Briefly describe three steps an actuary can perform to handle the projection of large claims.
The actuary may choose to exclude the large claims from the initial projection and then, at the end of the unpaid
claims analysis, add a case specific projection for the reported portion of large claims and a smoothed provision
for the IBNR portion of large claims.

Question 16
(3 points). Based on Frieldland in “Estimating Unpaid Claims Using Basic Techniques” and using the data
below, answer the following questions:

You are given the following information:
Incurred Loss and ALAE
Age of Development in Months
Earned
Premiums
$2,000
2,000
3,000
3,600

Accident
Year
2002
2003
2004
2005

12
$500
400
600
800

24
$1,000
700
900

36
$1,500
980

48
$1,650

The expected loss and ALAE ratio is 70%.
Loss development factors should be calculated using a simple average.
The tail factor is 1.05 for development from 48 months to ultimate.
a. (1.5 point) Using the incurred age-to-age development factor method, calculate the total IBNR reserve.
Show all work.
b. (1.5 point) Using the Bornhuetter-Ferguson method, calculate the total IBNR reserve. Show all work.

Question 17
(1.5 points) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, list and briefly
describe two advantages and two disadvantages of using calendar year data.

Exam 5B - Independently Authored Preparatory Test 1
Question 18
(1.5 points) You are the reserving actuary for the XYZ insurance company. You are about to conduct a year
end review of unpaid claims.
According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, what types of questions
would you ask actuaries in your ratemaking unit prior to conducting your reserve review?
Question 19. (3.5 points) Friedland in ““Estimating Unpaid Claims Using Basic Techniques” provide a detailed
example of how to create paid claims and reported claims triangles using claims transaction data.
Using the following data:
a. (2 points) Create a cumulative paid claim triangle for accident years 2005 – 2008.
b. (1.5 points) Create a case outstanding claim triangle for accident years 2005 – 2008.

Claim
ID

Accident
Date

Report
Date

1
2
3
4

Jan-5-05
May-4-05
Aug-20-05
Oct-28-05

Feb-1-05
May-15-05
Dec-15-05
May-15-06

5
6
7

Mar-3-06
Sep-18-06
Dec-1-06

Jul-1-06
Oct-2-06
Feb-15-07

8
9
10
11

Mar-1-07
Jun-15-07
Sep-30-07
Dec-I2-07

Apr-1-07
Sep-9-07
Oct-20-07
Mar-10-08

12
13
14
15

Apr-12-08
May-28-08
Nov-12-08
Oct-15-08

Jun-18-08
Jul-23-08
Dec-5-08
Feb-2-09

Claims Transaction Data
2005 Transactions
2006 Transactions
Ending
Ending
Total
Case
Total
Case
Payments
0/S
Payments
0/S
400
200
0

200
300
400

2007 Transactions
Ending
Total
Case
Payments
0/S

2008 Transactions
Ending
Total
Case
Payments
0/S

220
200
200
0

0
0
200
1,000

0
0
300
0

0
0
0
1,200

0
0
0
300

0
0
0
1,200

260
200

190
500

190
0
270

0
500
420

0
230
0

0
270
650

200
460
0

200
390
400

200
0
400
60

0
390
400
530

400
300
0

200
300
540

Question 20
(1.5 points) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, answer the
following:
a. (.50 points) What does a review of the ratio of paid to reported claims help the actuary determine?
b. (.50 points) How can changes in the ratio of paid-to-reported claims be taking place, but such changes
cannot be observed?
c. (.50 points) What does a review of the ratio of paid claims to on-level earned premiums help the actuary
determine?
Question 21
(1.5 points) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, list and briefly
describe factors affecting the reporting and closing of claims.
Question 22
(1.5 points) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, actuaries
compute average reported claims, average paid claims, and average outstanding claims.
List and briefly describe two important issues related to computing average values.
Copyright  2014 by All 10, Inc.
Page 6

Solutions to Exam 5B - Independently Authored Preparatory Test 1
Question 1 discussion:
a. (1 point) Calculate the IBNR reserve as of December 31, 2005 using case incurred loss development.
Accident
Year
2002
2003
2004
2005

Case Incurred
at 12/31/05
(1)
200
2,000
1,800
1,200

Age to Age
LDFs
(2)
1.080
1.050
1.100
1.250

LDFs to
Ultimate
(3)
1.080
1.134
1.247
1.559

IBNR
Factor
(4)=(3)-1.0
0.0800
0.1340
0.2474
0.5593

IBNR at
12/31/2005
(5) = (1)*(4)
16.00
268.00
445.32
671.10
1400.42

Notes: (1) and (2) are given
(3) = downward multiplicative product of (2)

Thus, the IBNR reserve at 12/31/05 is 1,400,420.
b. (1 point) Calculate the IBNR reserve as of December 31, 2005 using the Bornhuetter-Ferguson method.
Show all work.
Accident
Year
2002
2003
2004
2005

Earned
Premium
(1)
400
2,000
3,000
3,000

ELR
(2)
0.80
0.80
0.80
0.80

Age to Age
LDFs
(3)
1.080
1.050
1.100
1.250

LDFs to
Ultimate
(4)
1.080
1.134
1.247
1.559

% Unreported
at 12/31/05
(5)=1.0 - 1.0/(4)

0.0741
0.1182
0.1983
0.3587

IBNR at
12/31/2005
(6) = (1)*(2)*(5)
23.70
189.07
476.00
860.80
1549.57

Notes: (1) and (2) and (3) are given
(4) = downward multiplicative product of (3)

Thus, the IBNR reserve using the BF method at 12/31/05 is 1,549,570.
c. (0.5 point) Identify one situation in which it would be preferable to use the case incurred method
rather than the Bornhuetter-Ferguson method to develop the IBNR.
When there is a deteriorating loss ratio, but a consistent loss emergence pattern, the case incurred
method is preferable to the BF method to develop IBNR. Key: Since we assume no change in the
adequacy of case outstanding, there are no changes in the age-to-age factors, and thus no changes
in the cumulative claim development factors between the increasing claim ratio scenario and the
steady-state environment.
The higher value of projected ultimate claims is solely due to higher values of claims reported and
paid
The estimated IBNR is the same for both the reported and paid claim development methods, and is
equal to the actual IBNR.
Thus, conclude that the development technique is responsive to changes in the underlying claim
ratios assuming no changes in the underlying claims reporting or payment pattern.
d. (0.5 point) Identify one situation in which it would be preferable to use the Bornhuetter-Ferguson
method rather than the case incurred loss development method to estimate the IBNR.
The B-F method is preferable to use when historical data is extremely thin or volatile or both. Insurers
face this when writing new lines of business.

Solutions to Exam 5B - Independently Authored Preparatory Test 1
Question 1 discussion (continued):
e. (1 point) Using the Bornhuetter-Ferguson method, calculate the amount of loss development to be
expected during calendar year 2006 on accident years 2002 through 2005. Show all work.
Loss emergence during CY 2006 from AY's 2001 - 2005
Accident
Year
2002
2003
2004
2005

Accident
Year
2002
2003
2004
2005

Earned
Premium
(1)
400
2,000
3,000
3,000
Estimated
IBNR at
12/31/2005
(7) = (3)*(6)
23.70
189.07
476.00
860.80

ELR
(2)
0.80
0.80
0.80
0.80

Expected
Losses
(3) = (1)*(2)
320
1,600
2,400
2,400

Age to Age
LDFs
(4)
1.080
1.050
1.100
1.250

LDFs to
Ultimate
(5)
1.080
1.134
1.247
1.559

% Reported
at 12/31/05

% Reported
at 12/31/06

(8)=1.0/(5)

(9) using (8)

92.59%
88.18%
80.17%
64.13%

100.00%
92.59%
88.18%
80.17%

% Reported
during CY 2006
(10) = (9) - (8)
7.41%
4.41%
8.02%
16.03%

Expected IBNR
emergence
during CY 2006
(11) = (3) * (10)
23.704
70.547
192.400
384.800
671.451

% Unreported
at 12/31/05
(6)=1.0 - 1.0/(5)

0.0741
0.1182
0.1983
0.3587

Thus, the amount of loss emergence during CY 2006 on AYs 2002 – 2005 is 671,451.

Question 2 discussion: Blooms: Comprehension; Difficulty 1, LO 1, KS: Importance of
accurate estimates of unpaid claims
a. An inadequate estimate of unpaid claims could cause an insurer to reduce its rates not realizing that
the estimated unpaid claims were insufficient to cover historical claims.
The new lower rates would be insufficient to pay the claims arising from the new policies. If the
insurer gains market share as a result of the lower rates, the premiums collected would prove to be
inadequate to cover future claims, and could lead to a situation where the future solvency of the
insurer is at risk.
b. An excessive estimate of unpaid claims could cause the insurer to increase rates unnecessarily,
resulting in a loss of market share and a loss of premium revenue to the insurer, negatively impacting
the insurer’s financial strength.
c. An inaccurate estimate of unpaid claims could lead to poor underwriting, strategic, and financial
decisions, because financial results influence an insurer decisions (e.g. where to increase business
and whether to exit an underperforming market).
An inaccurate estimate can have a negative impact on the insurer's decisions regarding its
reinsurance needs and claims management procedures and policies.
See chapter 1

Solutions to Exam 5B - Independently Authored Preparatory Test 1
Question 3 discussion: Blooms: Comprehension; Difficulty 1, LO 3, KS: Key terms: case
outstanding, paid claims, reported claims, incurred but not reported, ultimate claims, claims
related expenses, reported and closed claim counts, claim counts closed with no payment,
insurance recoverables, exposures, experience period, maturity or age, and components of
unpaid claim estimates
Approach 1: Establish case O/S based on the best estimate of the ultimate settlement value of such a
claim including inflation.
Approach 2: Set case O/S equal to the maximum value (i.e. the $1 million policy limit)
Approach 3: Seek the advice of legal counsel.
Assume that legal counsel estimates that there is an 80% chance that the claim will settle without
payment and a 20% chance of a full policy limit claim.
1. Set the case O/S based on the mode ($0 in this case).
2. Set the case O/S based on the expected value calculation or $200,000 = [(80% x $0) + (20% x $1 million)].
Approach 4: Establish case O/S for the estimated claim amount only.
Approach 5: Establish case O/S for the estimated claim amount and all claim-related expenses.
Approach 6: Establish case O/S for ALAE (or DCC) only, Establish case O/S for ULAE (or A&O) only.
See chapter 2
Question 4 discussion: Blooms: Comprehension; Difficulty 1, LO 3, KS: Key terms: case
outstanding, paid claims, reported claims, incurred but not reported, ultimate claims, claims
related expenses, reported and closed claim counts, claim counts closed with no payment,
insurance recoverables, exposures, experience period, maturity or age, and components of
unpaid claim estimates
When the reinsurance is proportional (i.e., quota share), insurers determine the ceded case outstanding
based on the reinsurers share of the total case outstanding.
If the reinsurance is excess of loss, the reinsurance ceded case outstanding for a claim that exceeds
the insurer's retention is simply the total case outstanding estimate (provided that the claims adjuster
estimates the case outstanding on a total limits basis) less the insurer's retention.
See chapter 2

Solutions to Exam 5B - Independently Authored Preparatory Test 1
Question 5 discussion: Blooms: Comprehension; Difficulty 2, LO 3, KS: Mechanics associated
with each technique (including organization of the data)
Date
February 20, 2009
April 1, 2009

May 1, 2009

September 1, 2009
March 1, 2010

January 25, 2011

April 15, 2011

September 1, 2011

March 1, 2012

Transaction
Case 0/S of $15,000 established for claim
only
Claim payment of $1,500 - case 0/S
reduced to
$13,500 (case 0/S change of -$1,500)
Expense payment to IA of $500; no
change in case O/S
Case 0/S for claim increased to $30,000
(case 0/S change of +$16,500)
Claim thought to be settled with additional
payment of $24,000 – case 0/S reduced
to $0
and claim closed (case 0/S change of $30,000)
Claim reopened with case 0/S of $10,000
for
claim and $10,000 for defense costs
Partial payment of $5,000 for defense
litigation
and case 0/S for defense costs reduced
to
$5,000 – no change in case 0/S for claim
Final claim payment for an additional
$12,000
case 0/S for claim reduced to $0 (case
0/S
change of -$10,000)
Final defense cost payment for an
additional
$6,000 – case 0/S for defense costs
reduced to
$0 and claim closed (case 0/S change of $5,000)

Reported Value
of Claim to Date
$15,000

Cumulative
Paid to Date
$0

$15,000

$1,500

Case O/S
$15,000
$13,500

$15,500

$2,000
$13,500

$32,000

$2,000

$30,000

$26,000

$46,000

$26,000
$20,000

$46,000

$31,000
$15,000

$48,000

$43,000
$5,000

$49,000

a. 26,000
b. 31,000
c. 5,000
See chapter 2

Question 6 discussion: Blooms: Comprehension; Difficulty 3, LO 1, KS: Statement of Principles,
CAS

An actuarially sound loss reserve, for a defined group of claims as of a given valuation date, is a
provision, based on estimates derived from reasonable assumptions and appropriate actuarial
methods, for the unpaid amount required to settle all claims, reported or not, for which liability exists
on a particular accounting date.

Solutions to Exam 5B - Independently Authored Preparatory Test 1
Question 7 discussion: Blooms: Comprehension; Difficulty 1, LO 3, KS: Statement of Principles
Examples include:
1. Judicial environment
2. Regulation
3. Legislative changes
4. Residual market
5. Economic variables

Question 8 discussion: Blooms: Comprehension; Difficulty 2, LO 3 KS: Mechanics associated with
each technique (including organization of the data)

Claim
Number
‘(1)

At December 31, 2007
Cumulative
Paid
Case
Reported
Claims
O/S
Claims
‘(2)
‘(3)
‘(4)

Transactions During 2008
At December 31, 2008
Change
Cumulative
Paid
in
Reported
Paid
Case
Reported
Claims Case O/S Claims
Claims
O/S
Claims
‘(5)
‘(6)
‘(7)
‘(8)
‘(9)
‘(10)

500

5,000

5,500

200

5,000

15,000

20,000

4,500

2,000

10,000

12,000

1,000

(1,000)

5,000

(800)

700

4,000

4,700

4,500

9,500

15,000

24,500

6,000

3,000

15,000

18,000

See chapter 2

Question 9 discussion:
1. Documentation should be sufficient for another actuary practicing in the same field to evaluate the work.
2. The documentation should describe clearly the sources of data, material assumptions, and methods.
3. Any material changes in sources of data, assumptions, or methods from the last analysis should be
documented. The actuary should explain the reason(s) for and describe the impact of the changes.
See Section 5.2
Question 10 discussion:
1. case reserve
2. provision for future development on known claims
3. reopened claims reserve
4. provision for claims incurred but not reported
5. provision for claims in transit (incurred and reported but not recorded)

Solutions to Exam 5B - Independently Authored Preparatory Test 1
Question 11 discussion:
“Reliance on Another—An actuary who makes an actuarial communication assumes responsibility for it,
except to the extent the actuary disclaims responsibility by stating reliance on another person. Reliance
on another person means using that person's work without assuming responsibility therefore. A
communication should define the extent of any such reliance.” See section 5.8
Question 12 discussion: Blooms: Comprehension; Difficulty 1, LO 1, KS: Types of data and their
sources
* Smaller insurers may have less internal data because of a limited volume of business written or
because the organizations’ system does not provide such data. Thus, actuaries must turn to external
sources of data.
* Large insurers who have entered a new line of insurance or have focused on a new geographical
region may also need external sources of information when developing estimates of unpaid claims.
See chapter 3

Question 13 discussion: Blooms: Comprehension; Difficulty 1, LO 1, KS: Types of data and their
sources
* Volume of claim counts
* Length of time to report the claim once an insured event has occurred (i.e. reporting patterns)
* Ability to develop a case outstanding estimate from earliest report through the life of the claim
* Length of time to settle the claim (i.e. settlement, or payment, patterns)
* Likelihood of claim to reopen once it is settled
* Average settlement value (i.e. severity)
* Consistency of coverage triggered by the claims in the group (i.e. group claims subject to the same or
similar laws, policy terms, claims handling, etc.)
See chapter 3

Question 14 discussion: Blooms: Comprehension; Difficulty 1, LO 1, KS: Types of data and their
sources
* Size of claim relative to policy limits
* Size of claim relative to reinsurance limits
* Number of claims over the threshold each year
* Credibility of internal data regarding large claims
* Availability of relevant external data
See chapter 3

Question 15 discussion: Blooms: Comprehension; Difficulty 1, LO 1, KS: Importance of accurate estimates of
unpaid claims

The actuary may choose to exclude the large claims from the initial projection and then, at the end of
the unpaid claims analysis, add a case specific projection for the reported portion of large claims and a
smoothed provision for the IBNR portion of large claims.
See chapter 3.

Solutions to Exam 5B - Independently Authored Preparatory Test 1
Question 16 discussion: Blooms: Comprehension; Difficulty 3
A.

Step 1: Compute loss development factors (age to age and factors to ultimate):
24-36
36-48
48-ULT
AY
12-24
2002
2.0000
1.5000
1.1000
1.0500
2003
1.7500
1.4000
1.0500
2004
1.5000
1.0500
2005
1.0500
3 yr avg
1.7500
1.4500
1.1000
1.0500
2.9309
1.6748
1.1550
1.0500
Factor to Ult
Step 2: Isolate CY 2005 incurred losses from the given data
2004
2003
2002
2005
800
900
980
$1,650
Step 3: Compute ultimate losses and IBNR for AY's 1999 - 2002
2004
2003
2002
AY
2005
2,345
1,507
1,132
1,733
Ultimate losses
IBNR
1544.72
607.32
151.9
82.5
2386.44
Total IBNR for AYs 1999 - 2002

Step 1: Set up a table similar to the one below:
ELR
EL
Factor to Ult
AY
EP
(1)
(2)
(3)=(1)*(2)
(4)
2002
$2,000
0.70
1,400
1.0500
2003
$2,000
0.70
1,400
1.1550
2004
$3,000
0.70
2,100
1.6748
2005
$3,600
0.70
2,520
2.9309
Total
(1) and (2) are given
(4) from Part A. Above

IBNR
(5)=(3)*(1.0-1/(4))

66.67
187.88
846.12
1,660.20
2,760.86

Question 17 discussion: Blooms: Comprehension; Difficulty 1, LO 1, KS: Organization of
data: calendar year, accident year, policy year, underwriting year, report year
Advantages of CY data:
* no future development as the value remains fixed as time goes unlike claims and exposures
aggregated based on accident year, policy year, and even report year bases.
* readily available because most insurers conduct financial reporting on a CY basis.
Disadvantage of calendar year data:
* cannot be used for loss development purposes.
* very few techniques for estimating unpaid claims are based on CY claims.
See chapter 3

Solutions to Exam 5B - Independently Authored Preparatory Test 1
Question 18 discussion: Blooms: Comprehension; Difficulty 1, LO 1, KS: Types of data and their
sources
1. Have there been any changes in company operations or procedures which have caused you to depart
from standard ratemaking procedures? If so, please describe those changes and how they were
treated.
2. What data used for ratemaking purposes could also be used in testing unpaid claims?
3. Has there been any significant shifts in the business by type of risk or type of claim within the past
several years?
4. Do you have any of the following sources of information which may be of value in reserve testing:
a. External economic indices,
b. Combined claims data for several companies (e.g., data obtainable from bureau rate filings),
c. Special rating bureau studies,
d. Changes in state laws or regulations, and
e. Size of loss or cause of loss studies?
5. Could we obtain copies of recent rate filings?
6. Were there any changes in statues, court decisions, extent of coverage that necessitated some
reflection in the rate analysis?
7. How are new programs priced? If you are relying on another insurer's filing, how similar are the
underlying books of business?
See chapter 4

Question 19 discussion: Blooms: Comprehension; Difficulty 3, LO 2 , KS: Development triangle
as a diagnostic tool
a Step 1: Consolidate claims transaction data into incremental paid claims by CY

Claim
ID
1
2
3
4

Claims Transaction Paid Claims Data
Incremental Payments in Calendar Year
Accident
Report
Date
Date
2005
2006
2007
2008
Jan-5-05
Feb-1-05
400
220
0
0
May-4-05
May-15-05
200
200
0
0
Aug-20-05 Dec-15-05
0
200
300
0
Oct-28-05 May-15-06
0
0
300

5
6
7

Mar-3-06
Sep-18-06
Dec-1-06

Jul-1-06
Oct-2-06
Feb-15-07

8
9
10
11

Mar-1-07
Jun-15-07
Sep-30-07
Dec-12-07

Apr-1-07
Sep-9-07
Oct-20-07
Mar-10-08

12
13
14
15

Apr-12-08
May-28-08
Nov-12-08
Oct-15-08

Jun-18-08
Jul-23-08
Dec-5-08
Feb-2-09

260
200

190
0
270

0
230
0

200
460
0

200
0
400
60
400
300
0

Solutions to Exam 5B - Independently Authored Preparatory Test 1
a Step 2: Consolidate (sum down the column) the incremental paid claims in Step 1 into an AY
incremental paid claim triangle

Incremental Paid Claim Triangle
Incremental Paid Claims as of (months)
12
24
36
48
600
620
300
300
460
460
230
660
660
700

Accident
Year
2005
2006
2007
2008

a Step 3: Using the incremental paid claim triangle from Step 2, create the cumulative paid claim
triangle below.

Cumulative Paid Claim Triangle
Cumulative Paid Claims as of (months)
36
48
12
24
600
1,220
1,520
1,820
460
920
1,150
660
1,320
700

Accident
Year
2005
2006
2007
2008

b Step 1: Consolidate claims o/s transaction data into ending case o/s by CY

Claim
ID
1
2
3
4

Claims Transaction Ending Case Outstanding Data
Ending Case Outstanding
Accident
Report
Date
Date
2005
2006
2007
2008
Jan-5-05
Feb-1-05
200
0
0
0
May-4-05
May-15-05
300
0
0
0
Aug-20-05
Dec-15-05
400
200
0
0
Oct-28-05
May-15-06
1000
1200
1200

5
6
7

Mar-3-06
Sep-18-06
Dec-1-06

Jul-1-06
Oct-2-06
Feb-15-07

8
9
10
11

Mar-1-07
Jun-15-07
Sep-30-07
Dec-12-07

Apr-1-07
Sep-9-07
Oct-20-07
Mar-10-08

12
13
14
15

Apr-12-08
May-28-08
Nov-12-08
Oct-15-08

Jun-18-08
Jul-23-08
Dec-5-08
Feb-2-09

190
500
0

0
500
420

0
270
650

200
390
400
0

0
390
400
530
200
300
540

b Step 2: Consolidate (sum down the column) the ending case o/s in Step 1 into an AY case o/s triangle
Accident
Year
2005
2006
2007
2008

Case Outstanding Triangle
Case Outstanding as of (months)
12
24
36
48
900
1,200
1,200
1,200
690
920
920
990
1,320
1,040

Solutions to Exam 5B - Independently Authored Preparatory Test 1
Question 20 discussion: Blooms: Comprehension; Difficulty 1, LO 2 KS: Purposes of the
development triangle
a. The ratio of paid to reported helps determine whether there might have been changes in case
outstanding adequacy or in settlement patterns. Determine whether there are changes in paid claims
(i.e., the numerator) occurring or whether changes in case outstanding, which impact reported claims
(i.e., the denominator), taking place
b. Changes in the ratio of paid-to-reported claims may be taking place, but cannot be observed, because
offsetting changes in both claim settlement practices and the adequacy of case outstanding can result
in no change to the ratio of paid-to-reported claims.
c. This diagnostic triangle can help to determine whether there was a speedup in claims payment or
possibly deterioration in underwriting results.
See Chapter 6
Question 21 discussion: Blooms: Comprehension; Difficulty 1, LO 1, KS: Examples and uses of
diagnostic development triangles: * Claim and claim count * Ratio of premium to claims *
Average values * Ratios of claims and counts
Factors affecting the reporting and closing of claims include:
* A change in guidelines on how claims are established
* A decrease in the statute of limitations duration (from a major tort reform action)
* Delegating higher claim settlement limits to a TPA
* Restructuring of the claim field offices (e.g. merging or adding of new offices).
* Introducing a new claim call center
A change resulting in a temporary increase in closing patterns occurs when a claim department makes
an extra effort to get the backlog as low as possible before making a transition to a new system.
A speed-up due to faster processing occurs when the new system leads to a slowdown in closing, due to a
learning curve necessary before the new system is fully operational.
See Chapter 6
Question 22 discussion: Blooms: Comprehension; Difficulty 1, LO , KS: Examples and uses of
diagnostic development triangles: * Claim and claim count * Ratio of premium to claims *
Average values * Ratios of claims and counts
Two important issues related to average values:
1. Have a clear understanding of the definition of closed and reported claim counts.
Some insurers include claims with no payment (CNP) in the definition of closed claim counts and some
include claims with no case outstanding and no payments in the definition of reported claim counts.
Including CNPs in closed claim count statistics or claims with no case outstanding or payments in
reported claim counts produces a much lower average value.
A change in the definition of claim counts can impact the results of diagnostic analyses using claim
counts and on estimation techniques that rely on the number of claims.
2. Large claims. Both the presence and absence of such claims can distort average claims. Methods to
deal with large claims include:
a. Removing large claims from the database before conducting both ratio and average value calculations
and handling the unpaid large claim estimate separately.
b. Use development triangles using limited claims (e.g. claims can be limited to $500,000 or $1 million
per occurrence in the reported and paid claim development triangles).
See Chapter 6

Exam 5B – Independently Authored Preparatory Test 2
General information about this exam
1. This test contains 25 computational and essay questions.
2. The recommend time for this exam is 2:30:00. Make sure you have sufficient time to take this practice test.
3. Consider taking this exam after working all past CAS questions, associated with the articles below, first.
4. Many of the essay questions may require lengthy responses.
5. Make sure you have a sufficient number of blank sheets of paper to record your answers.

Articles covered on exam:
Article .................................................... Author .................................. Syllabus Section
Chapter 5 - The Development Triangle ............................. Friedland ..............B: Estimating Claim Liabilities
Chapter 6 - Development Triangle as a Diagnostic Tool .. Friedland ..............B: Estimating Claim Liabilities

Chapter 7 – Development Technique .............................. Friedland ..............B: Estimating Claim Liabilities
Chapter 8 – Expected Claims Technique ........................ Friedland ..............B: Estimating Claim Liabilities
Chapter 9 – Bornhuetter-Ferguson Technique ................ Friedland ..............B: Estimating Claim Liabilities
Chapter 10 – Cape Cod Technique ................................. Friedland ............. B: Estimating Claim Liabilities

Exam 5B - Independently Authored Preparatory Test 2
Question 1
(5.5 points) Based on Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, you are
given the following data as of 12/31/08:

Earned
Premium
4,000
4,400
5,000
5,300
6,000
6,300

Accident
Year
2003
2004
2005
2006
2007
2008

Reported Claims including ALAE ($000's omitted)
1st
2nd
3rd
4th
5th
Report
Report
Report
Report
Report
1,880
3,240
3,400
3,500
3,500
2,400
3,380
3,420
3,600
3,600
2,500
3,450
3,600
3,900
2,800
3,100
3,800
3,000
3,800
4,500

6th
Report
3,500

a. Estimate the IBNR as of 12/31/08 using the Development Technique
To select claim development factors, use the volume-weighted averages for the latest three years.
b. Using the data above and based on the discussion by Friedland, what is the 12-24 month ageto-age factor using:
(i) Simple (arithmetic) average of the last three years
(ii) Geometric average of the last four years
(iii) Medial average for the latest five years excluding one high and low value, “Medial latest 5x1”
c. Estimate the IBNR as of 12/31/08 using the Expected Claims Technique.
Use an Expected Claim Ratio = 80% for all years.
To select claim development factors, use the volume-weighted averages for the latest three years.
d. Estimate the IBNR as of 12/31/08 using the Bornhuetter-Ferguson Technique
Use Expected Claim Ratio = 80% for all years.
To select claim development factors, use the volume-weighted averages for the latest three years.
e. Show that the Bornhuetter-Ferguson method produces Ultimate Claims that are a credibility
weighting between the Development method and the Expected Claims method.
f. Estimate the IBNR as of 12/31/08 using the Cape Cod Technique
To select claim development factors, use the volume-weighted averages for the latest three years.

Question 2
(1.25 points) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, list and
briefly describe the underlying assumption made when using the development technique and four
other key assumptions associated with this technique.

Exam 5B - Independently Authored Preparatory Test 2
Question 3
(2 points). Based on the following information:

Case Incurred Calendar Year
Accident
Losses
Earned
at 12/31/02
Premium
Year
2002
30,000
150,000
Expected Loss Ratio =
0.75

Development
Factor to
Ultimate
at 12/31/02
4.00

a. (.75 points). What is the Bornhuetter-Ferguson IBNR estimate at 12/02?
b. (.75 points). What is the Chain Ladder IBNR estimate at 12/02?
c. (.75 points). What is the Benktander IBNR estimate at 12/02?

Question 4
(1 point). Based on the following information for Accident Year 2003 at 12 months of development:
Method
Chain Ladder
Bornhuetter Ferguson
Expected Loss Ratio

Reserve Estimate
8.50
10.72
12.20

Estimated % of claims paid at 12 months

.35%

Estimate the Benktander reserve RGB.

Question 5
(2 points) Friedland in “Estimating Unpaid Claims Using Basic Techniques” describes two approaches to
employing case outstanding development methods:
1. Accident year Case Outstanding Development Technique - Approach #1
2. Accident Year Case Outstanding Development Technique using Industry Benchmark Factors Approach #2
a. (.50 point) What are the key assumptions under the Approach #1 technique?
b. (.50 points) Key limitation when using the case O/S development technique
c. (1 point) What are the key assumptions under the Approach #2 technique?

Question 6
(1.0 point) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, list and briefly
describe two reasons why selection of a tail factor is of utmost importance.
Question 7
(1.50 points) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, answer the
following questions:
a. (0.75 points) List and briefly describe the type of insurance environment and types of claims experience
in which the development technique works well.
b. (0.75 points) Describe two examples when the development

Exam 5B - Independently Authored Preparatory Test 2
Question 8
(1.5 points) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, list and briefly
describe three situations in which the expected claims method is often used.

Question 9
(1.0 point) According to Friedland in “Estimating Unpaid Claims Using Basic Techniques”, list and briefly
describe two methods for selecting expected claim ratios using the expected claims method.

Question 10
(1.0 point) According to Friedland in “Estimating Unpaid Claims Using Basic Techniques”, briefly describe
one advantage and one disadvantage to using the expected claims method.

Question 11
(1.50 points) According to Friedland in “Estimating Unpaid Claims Using Basic Techniques”, answer the
following questions:
a. (0.50 points) What techniques are the Bornhuetter-Ferguson Technique a blend of?
b. (0.50 points) How are weights distributed to these two techniques?
c. (0.50 points) Why was the Bornhuetter-Ferguson Technique developed?
Question 12
(1.50 points) According to Friedland in “Estimating Unpaid Claims Using Basic Techniques”, list three
scenarios in which the BF technique is most often used.

Question 13
(1.0 point) According to Friedland in “Estimating Unpaid Claims Using Basic Techniques”, the Benktander
method is an iterative BF method. The only difference in the two methods is the derivation of the expected
claims. Describe the difference in how expected claims are derived in the BF and the Benktander methods.

Question 14
(1.0 point) According to Friedland in “Estimating Unpaid Claims Using Basic Techniques”, answer the
following questions:
a. (.50 points) Briefly describe the key assumption underlying the Cape Cod Technique.
b. (.50 points) How does this assumption differ from the primary assumption underlying the development
method?

Question 15
(1.0 point) According to Friedland in “Estimating Unpaid Claims Using Basic Techniques”, briefly describe one
advantage over the development technique and one disadvantage relative to the BF technique to using the
Cape Cod method

Exam 5B - Independently Authored Preparatory Test 2
Question 16
(1.0 point) According to Friedland in “Estimating Unpaid Claims Using Basic Techniques”, briefly describe
data adjustments that are recommended when using the Cape Cod technique.
Question 17
(1.5 points) According to Friedland in “Estimating Unpaid Claims Using Basic Techniques”, there are three
important dimensions in a development triangle: Row; Diagonals; and Columns
Briefly describe what each represents for a triangle comprised of four accident years beginning with AY 20XX
Question 18
(2.5 points) According to Friedland in “Estimating Unpaid Claims Using Basic Techniques”, you are given the
following information:

Claim
ID
1
2
3
4

Accident
Date
Jan-5-05
May-4-05
Aug-20-05
Oct-28-05

Table 5 – Detailed Example – Claims Transaction Data
2005 Transactions
2006 Transactions
2007 Transactions
Ending
Ending
Ending
Report
Total
Case
Total
Case
Total
Case
Date
Payments
0/S
Payments
0/S
Payments
0/S
Feb-1-05
400
200
220
0
0
0
May-15-05
200
300
200
0
0
0
Dec-15-05
0
400
200
200
300
0
May-15-06
0
1,000
0
1,200

5
6
7

Mar-3-06
Sep-18-06
Dec-1-06

Jul-1-06
Oct-2-06
Feb-15-07

8
9
10
11

Mar-1-07
Jun-15-07
Sep-30-07
Dec-I2-07

Apr-1-07
Sep-9-07
Oct-20-07
Mar-10-08

12
13
14
15

Apr-12-08
May-28-08
Nov-12-08
Oct-15-08

Jun-18-08
Jul-23-08
Dec-5-08
Feb-2-09

260
200

190
500

190
0
270
200
460
0

2008 Transactions
Ending
Total
Case
Payments
0/S
0
0
0
0
0
0
300
1,200

0
500
420

0
230
0

0
270
650

200
390
400

200
0
400
60

0
390
400
530

400
300
0

200
300
540

Compute the following:
a. the amount the insurer paid during CY 2005 during the first 12 months for AY 2005
b. the amount the insurer paid during the second 12 months (CY 2006) for AY 2005
rd
th
c. the amount the insurer paid during the 3 and 4 12 months periods (CY 2007 and CY 2008) for AY 2005
d. the amount the insurer paid during the CY 2006 for accidents occurring during 2006
e. the amount the insurer paid during the CY 2007 for accidents occurring during 2006

Question 19
(1.5 points) According to Friedland in “Estimating Unpaid Claims Using Basic Techniques”, claim count and
loss development can be positive or negative. Briefly describe how claims counts and loss development can
develop in a negative manner.

Exam 5B - Independently Authored Preparatory Test 2
Question 20
(3.0 points) Use the procedure described by Friedland in “Estimating Unpaid Claims Using Basic Techniques”
to answer the following questions:

Table 9 – Detailed Example – Claims Transaction Ending Case Outstanding Data
Ending Case Outstanding
Claim
Accident
Report
ID
Date
Date
2005
2006
2007
2008
1
Jan-5-05
Feb-1-05
200
0
0
0
2
May-4-05
May-15-05
300
0
0
0
3
Aug-20-05
Dec-15-05
400
200
0
0
4
Oct-28-05
May-15-06
1,000
1,200
1,200
5
6
7

Mar-3-06
Sep-18-06
Dec-1-06

Jul-1-06
Oct-2-06
Feb-15-07

8
9
10
11

Mar-1-07
Jun-15-07
Sep-30-07
Dec-12-07

Apr-1-07
Sep-9-07
Oct-20-07
Mar-10-08

12
13
14
15

Apr-12-08
May-28-08
Nov-12-08
Oct-15-08

Jun-18-08
Jul-23-08
Dec-5-08
Feb-2-09

a. Compute Case O/S for AY 2005 at 12 months
b. Compute Case O/S for AY 2005 at 24 months
c. Compute Case O/S for AY 2005 at 36 months
d. Compute Case O/S for AY 2006 at 12 months
e. Compute Case O/S for AY 2006 at 24 months
f. Compute Case O/S for AY 2006 at 36 months

190
500

0
500
420

0
270
650

200
390
400

0
390
400
530
200
300
540

Exam 5B - Independently Authored Preparatory Test 2
Question 21
(2.5 points) Using the procedure described by Friedland in “Estimating Unpaid Claims Using Basic
Techniques”, and the data below, create a reported claim count triangle for AYs 2005 - 2008

Claim
ID
1
2
3
4

Accident
Date
Jan-5-05
May-4-05
Aug-20-05
Oct-28-05

5
6
7

Mar-3-06
Sep-18-06
Dec-1-06

Jul-1-06
Oct-2-06
Feb-15-07

8
9
10
11

Mar-1-07
Jun-15-07
Sep-30-07
Dec-I2-07

Apr-1-07
Sep-9-07
Oct-20-07
Mar-10-08

12
13
14
15

Apr-12-08
May-28-08
Nov-12-08
Oct-15-08

Jun-18-08
Jul-23-08
Dec-5-08
Feb-2-09

260
200

190
500

190
0
270
200
460
0

2008 Transactions
Ending
Total
Case
Payments
0/S
0
0
0
0
0
0
300
1,200

0
500
420

0
230
0

0
270
650

200
390
400

200
0
400
60

0
390
400
530

400
300
0

200
300
540

Question 22
(2.5 points) According to Friedland in “Estimating Unpaid Claims Using Basic, types of claims data commonly
appearing in development triangles include:
* Reported claims
* Case O/S
* Cumulative total paid claims
* Cumulative paid claims on closed claim counts
* Incremental paid claims Reported claim counts
* Claim counts on closed with payment
* Claim counts on closed with no payment
* Total closed claim counts
* O/S claim counts
Using the data types above, describe 5 types of triangles of ratios and average claim values that
actuaries often analyze.

Exam 5B - Independently Authored Preparatory Test 2
Question 23
(2.0 points) According to Friedland in “Estimating Unpaid Claims Using Basic, actuaries often review the ratio
of paid claims to on-level earned premiums. You are given the following triangle

Table 7 - Ratio of Cumulative Paid Claims to On-Level Earned Premium
Accident

Year
2002
2003
2004
2005
2006
2007
2008
a.
b.
c.
d.

Ratio of Cumulative Paid Claims to On-Level Earned Premium as of (months)

12
0.041
0.029
0.028
0.031
0.051
0.071
0.071

24
0.142
0.104
0.123
0.126
0.171
0.238

36
0.247
0.211
0.323
0.278
0.331

48
0.395
0.38
0.54
0.412

60
0.571
0.573
0.657

72
0.727
0.653

84
0.795

What does this diagnostic triangle help the actuary determine?
Identify one notable observation occurring in the data.
State one question that an actuary may wish to explore based on your response to b.
What type of additional data may an actuary need to answer the question posed in c.?

Question 24
(1.5 points) According to Friedland in “Estimating Unpaid Claims Using Basic, actuaries often review
triangles of reported and closed claim counts.
However, before commencing the analysis of the claim count development triangles, it is important that the
actuary understand the types of data contained within such triangles.
State three questions an actuary should ask before analyzing reported and closed claim counts.

Question 25
(1.0 point) According to Friedland in “Estimating Unpaid Claims Using Basic, actuaries often review triangles
of average case outstanding.
State two questions an actuary should ask before analyzing case outstanding adequacy.

Solutions to Exam 5B - Independently Authored Preparatory Test 2
Question 1 discussion: Blooms: Comprehension; Difficulty 3 LO 5,
1a. Estimate the IBNR as of 12/31/08 using the Development Technique
To select claim development factors, use the volume-weighted averages for the latest three years.

Earned
Premium
4,000
4,400
5,000
5,300
6,000
6,300

Accident
Year
2003
2004
2005
2006
2007
2008

6th
Report
3,500

Selected CDF calculations

Accident
Year
2003
2004
2005
2006
2007
2008
Total

Age of
Data at
12/31/08
(1)
72 months
60 months
48 months
36 months
24 months
12 months

Reported
Claims at
12/31/08
(2)
3,500
3,600
3,900
3,800
3,800
4,500

Reported Expected
IBNR
CDF to
Ultimate
(broadly
Ultimate
Claims
defined)
(3) above (4)=(2)*(3) (5)=(4)-(2)
1.00
3,500
0
1.00
3,600
0
1.00
3,900
0
1.06
4,012
212
1.15
4,371
571
1.43
6,455
1,955
2,737

OR:
Shortcut

IBNR
(broadly
defined)
(5)=(2)*[(3) - 1.0]
0
0
0
212
571
1,955
2,737

Solutions to Exam 5B - Independently Authored Preparatory Test 2
1b. What is the 12-24 month age-to-age factor using:
(i) 12-24 month age-to-age factor using Simple (arithmetic) average of the last three years =1.25
(ii) 12-24 month age-to-age factor using Geometric average of the last four years =1.28
(iii) 12-24 month age-to-age factor using Medial latest 5x1 =1.35

Age-to-Age Development Factors by Accident Year
Note: Did not need these "Link Ratios" to calculate the volume-weighted ATA selections:
ATA factors by AY:

Accident
Year

Example:
12:24 month ATA
for AY 2005 =
(3450)/(2500) = 1.38
between 1st and 2nd
annual valuation dates

2003
2004
2005
2006
2007

Alternative ATA Selections
3-year Simple (Arithmetic) Average
4-year Geometric Average
"Medial latest 5x1"

1st to 2nd 2nd to 3rd 3rd to 4th 4th to 5th 5th to 6th
Report
Report
Report
Report
Report
12:24 mo 24:36 mo 36:48 mo 48:60 mo 50:72 mo
1.723
1.049
1.029
1.000
1.000
1.408
1.012
1.053
1.000
1.380
1.043
1.083
1.107
1.226
1.267

12:24 mo
1.25
1.28
1.35

Calculation Details
= (1.380+1.107+1.267)/3
= (1.408*1.38*1.107*1.267)^(1/4)
= (1.408+1.38+1.267)/3

Note: "Medial latest 5x1" excludes the highest and lowest values (1.723 and 1.107) in 5-yr period
1c. Estimate the IBNR as of 12/31/08 using the following method: Expected Claims Technique.

Accident
Year
2003
2004
2005
2006
2007
2008
Total

Earned
Premium
(1)
4,000
4,400
5,000
5,300
6,000
6,300

Expected
Claim
Ratio
(2)
80.00%
80.00%
80.00%
80.00%
80.00%
80.00%

Expected
Reported
Claims
Claims at
(Ultimate) 12/31/2008
(3)=(1)*(2)
(4) given
3,200
3,500
3,520
3,600
4,000
3,900
4,240
3,800
4,800
3,800
5,040
4,500

IBNR
(broadly
defined)
(5)=(4)-(3)
-300
-80
100
440
1000
540
1,700

Solutions to Exam 5B - Independently Authored Preparatory Test 2
1d. Estimate the IBNR as of 12/31/08 using the following method: Bornhuetter-Ferguson

Selected CDF calculations

Accident
Year
2003
2004
2005
2006
2007
2008
Total

Age of
Data at
12/31/08
(1)
72 months
60 months
48 months
36 months
24 months
12 months

Reported
CDF to
Ultimate
(2) above
1.00
1.00
1.00
1.06
1.15
1.43

Percent
Reported
12/31/08
(3)=1.0/(2)
100.0%
100.0%
100.0%
94.7%
86.9%
69.7%

Earned
Premium
(5) given
4,000
4,400
5,000
5,300
6,000
6,300

A priori
Expected
Claim Ratio
(6) given
80.0%
80.0%
80.0%
80.0%
80.0%
80.0%

A priori
Expected
Claims
(7)=(5)*(6)
3,200
3,520
4,000
4,240
4,800
5,040

Percent
Unreport
12/31/08
(4)=1.-(3)
0.0%
0.0%
0.0%
5.3%
13.1%
30.3%

"IBNR" Or Shortcut using
IBNR
Expected Expected Claims *
(broadly
Unreport Percent Unreported
defined)
(8)=(7)*(4)
(8)=(5)*(6)*[1.0-1.0/CDF]
0
0
0
0
0
0
224
224
627
627
1,526
1,526
2,377
2,377

Note: The Percent Unreported
= 1 minus inverse of Ult. CDF

Solutions to Exam 5B - Independently Authored Preparatory Test 2
1e. Show that the Bornhuetter-Ferguson method produces Ultimate Claims that are a credibility weighting
between the Development method and the Expected Claims method.

b. Since Expected Ultimate Claims = Actual Reported + Expected Unreported:
Bornhuetter-Ferguson
Development
Expected Credibility
"B-F"
Method
Claims Weighted
Reported
"IBNR"
Expected
Credibility
Expected
Expected Expected
Accident
Claims at
Expected
Ultimate
to "Actual"
Ultimate
Ultimate
Ultimate
Year
12/31/08
Unreport
Claims
by B-F
Claims
Claims
Claims
(9)
(10)=(8) (11)=(9)+(10)
(12)=(3) (13) Ch 7. (14) Ch. 8
(15)
2003
3,500
0
3,500
100.0%
3,500
3,200
3,500
2004
3,600
0
3,600
100.0%
3,600
3,520
3,600
2005
3,900
0
3,900
100.0%
3,900
4,000
3,900
2006
3,800
224
4,024
94.7%
4,012
4,240
4,024
2007
3,800
627
4,427
86.9%
4,371
4,800
4,427
2008
4,500
1,526
6,026
69.7%
6,455
5,040
6,026
Total
25,477
(15) = (12)*(13) + [1.0-(12)]*(14)
25,477
(13) & (14) See details in Ch. 7 & Ch. 8 Q & A.
Matches B-F Expected Ultimate Claims in (11)
Note: Credibility assigned to Development Method (relies on "actual" reported losses) = % reported.
Compliment of credibility assigned to Expected Claim technique.

Solutions to Exam 5B - Independently Authored Preparatory Test 2
1f. Estimate the IBNR as of 12/31/08 using the Cape Cod Technique

Selected CDF calculations

1st to 2nd 2nd to 3rd 3rd to 4th 4th to 5th 5th to 6th
Report
Report
Report
Report
Report
ATA: 3-yr Volume-weighted average
1.25
1.06
1.00
1.00
1.09*
Note: 1st report at 12 months
at 12 mo
at 24 mo
at 36 mo
at 48 mo
at 60 mo
Reported CDF to Ultimate
1.43
1.06
1.00
1.00
1.15**
Adjusted
Reported
Percent Used-Up Reported
"CC"
Premium
CDF to Reported Premium
Claims Estimated
Ultimate
to date
to date as avail.** Claim Ratio
if avail.**
(1) given
(2) above (3)=1.0/(2) (4)=(1)*(3)
(5) given (6)=(5)/(4)
2003
4,000
1.00
100.0%
4,000
3,500
see total
2004
4,400
1.00
100.0%
4,400
3,600
see total
2005
5,000
1.00
100.0%
5,000
3,900
see total
2006
5,300
1.06
94.7%
5,021
3,800
see total
2007
6,000
1.15
86.9%
5,216
3,800
see total
2008
6,300
1.43
69.7%
4,392
4,500
see total
Total
31,000
28,029
23,100
82.42%
** The Cape Cod technique allows/prefers use of "adjusted" data where available.
(4) Used-Up premium also equals (1)/(2): Adjusted Premium divided by Ult. CDF
(6) " … method requires the use of the weighted average claim ratio from all years.'
Accident
Year

Accident
Year
2003
2004
2005
2006
2007
2008
Total

"CC"
Claim
Ratio
(6) total
82.4%
82.4%
82.4%
82.4%
82.4%
82.4%

A priori
Expected
Claims
(7)=(1)*(6)
3,297
3,626
4,121
4,368
4,945
5,192

"IBNR"
Expected
Unreport
(8)=(7)*[1-(3)]

0
0
0
230
646
1,572
2,449

Or shortcut using
IBNR
C.C. Expected Claims
(broadly
defined)
x Percent Unreported
(8)=(Prem)*(CC %)*[1.0-1.0/CDF]
0
0
0
230
646
1,572
2,449

Note: See Ch. 9 Q&A. If B-F claim ratio = "CC" claim ratio, results would be identical.

Solutions to Exam 5B - Independently Authored Preparatory Test 2
Question 2 discussion: Blooms: Comprehension; Difficulty 1 LO 3, KS: Assumptions of each
estimation technique, KS:
The underlying assumption in the development technique is that claims recorded to date will continue
to develop in a similar manner in the future (i.e. the past is indicative of the future).
Other key assumptions of the development method include:
* consistent claim processing,
* a stable mix of types of claims,
* stable policy limits, and
* stable reinsurance (or excess insurance) retention limits throughout the experience period.

Question 3 discussion: Blooms: Comprehension; Difficulty 1, LO 3, KS: Mechanics
associated with each technique (including organization of the data)

Method
1. Bornhuetter-Ferguson: EP*ELR*(1-1/LDF)
2. Chain Ladder: Losses * LDF
3. Pk (credibility to CL, % reported)
4. Pk (credibility to BF, % unreported)
5. Benktander Reserve (Hovinien calc): 2.*3. +1.*4.

Estimated IBNR
84,375
90,000
0.25
0.75
85,781

Alternative calculation for Benktander # 1
6. Bornhuetter-Ferguson ultimate: 1. + case incurred
7. Benktander Reserve: 6. * 4.
Alternative calculation for Benktander # 2
8. A prior ultimate: EP * ELR
9. Chain Ladder ultimate

10.
11.
12.
13.

114,375
85,781

112,500
120,000
0.25
0.75
0.4375
0.5625
115,781
85,781

Credibility to CL: 1 - 4.^2
Credibility to BF: 4.^2
Benktander Ultimate
Benktander Reserve

Question 4 discussion: Blooms: Comprehension; Difficulty 1, LO 1, KS:
Using the Hovinen method:

qkRBF +(1 –qk)RCL
(.35 x 8.5) + (.65 x 10.72) = 9.943

Solutions to Exam 5B - Independently Authored Preparatory Test 2
Question 5 discussion: Blooms: Comprehension; Difficulty 1, LO 3, KS: Assumptions of each
estimation technique
a. Key assumptions
Claims activity related to IBNR is related in a consistent manner to claims already reported.
Assumptions similar to those for the development techniques also apply to the case outstanding
(O/S) development technique.
b. Key limitation when using the case O/S development technique is the assumption that future IBNR
is related to claims already reported does not hold true for many lines of insurance.
c1. The assumptions regarding the development technique are applicable in this example.
c2. Claims recorded to date will develop in a similar manner in the future as our industry benchmark (i.e.,
the historical industry experience is indicative of the future experience for the self-insurer). Thus,
industry-based reporting and payment development patterns are used to derive case O/S
development patterns.
Question 6 discussion: Blooms: Comprehension; Difficulty 1, LO 3, KS: The claim process
1. It influences the unpaid claim estimate for all accident years (in the experience period).
2. It can have a disproportionate amount of leverage on the total estimated unpaid claims.
Question 7 discussion: Blooms: Comprehension; Difficulty 1, LO 3, KS: When each
techniques works and when it does not
a1. The development method is appropriate for insurers in a stable environment.
a2. The development technique requires a large volume of historical claims experience.
a3. It works best when the presence or absence of large claims does not greatly distort the data.
b. The development technique may not be suitable when there is not a sufficient volume of credible
data, as in the following situations:
b1. When entering a new line of business or new territory
b2. For smaller insurers with limited portfolios.

Question 8 discussion: Blooms: Comprehension; Difficulty 1, LO 3, KS: The claim process
1. when an insurer enters a new line of business or a new territory.
2. when operational or environmental changes make recent historical data irrelevant for projecting
future claims activity for that cohort of claims.
3. for the most recent years in the experience period, since cumulative CDFs are highly leveraged.
4. when data is unavailable for other methods
5. for the latest year in the experience period after major changes in the legal environment take
place.

Question 9 discussion: Blooms: Comprehension; Difficulty 1, LO 3, KS: The claim process
1. Use insurance industry experience for benchmark claim ratios.
2. Use an average of the projected ultimate unadjusted reported and paid claims to earned premium
ratios using the development method.

Solutions to Exam 5B - Independently Authored Preparatory Test 2
Question 10 discussion: Blooms: Comprehension; Difficulty 1, LO 3, KS: The claim process
An advantage to using the expected claims technique is that the technique maintains stability over
time since actual claims do not enter into the calculations. This is because the claim ratios can be
judgmentally adjusted based on historical experience due to a belief that either the pricing or
underwriting or both are changing.
A disadvantage is that it is not responsive when actual claims experience differs from the initial
expectations.

Question 11 discussion: Blooms: Comprehension; Difficulty 1, LO 3, KS: Assumptions of each
estimation technique
a. It is a blend of the development and expected claims techniques, by splitting ultimate claims into
two components: actual reported (or paid) claims and expected unreported (or unpaid) claims.
b. It gives more weight to actual claims as experience matures, and less weight to expected claims .
c. It was developed to overcome the problems with the development and expected claims technique.
Question 12 discussion: Blooms: Comprehension; Difficulty 1, LO 3, KS: Assumptions of each
estimation technique
1. For the most immature years associated with long-tail lines of insurance, due to the highly leveraged
nature of claim development factors for such lines.
2. If the data is extremely thin or volatile or both. For example, when an insurer enters a new line of
business or a new territory and there is not yet a credible volume of historical claim development
experience. The actuary would likely need to rely on benchmarks, either from similar lines at the same
insurer or insurance industry experience, for development patterns and expected claim ratios (or pure
premiums).
3. For very short-tail lines, where the IBNR can be set equal to a multiple of the last few months EP.

Question 13 discussion: Blooms: Comprehension; Difficulty 1, LO 3, KS: Assumptions of each
estimation technique
For the BF method, expected claims equal an expected claim ratio time earned premium.
For the Benktander technique, expected claims are the projected ultimate claims from an initial BF
projection (thus, the reference to the Benktander method as an iterative BF method). Also, the
Benktander projection of ultimate claims will approach the projected ultimate claims produced by the
development technique after sufficient iterations.

Question 14 discussion: Blooms: Comprehension; Difficulty 1, LO 3, KS: Assumptions of each
estimation technique
a. The key assumption: Unreported claims will develop based on expected claims, which are computed
using reported (or paid) claims and earned premium.
This is also the same assumption underlying the BF method.
b. This assumption differs from the primary assumption under the development method, which is
unreported claims will develop based on reported claims to date.

Solutions to Exam 5B - Independently Authored Preparatory Test 2
Question 15 discussion: Blooms: Comprehension; Difficulty 1, LO 3, KS: When each
techniques works and when it does not
An advantage of the Cape Cod method (over the development technique) is that it may not be
distorted by random fluctuations early in the development of an AY.
A shortcoming of the Cape Cod method compared to the BF technique is that it is not necessarily as
appropriate as the BF method if the data is extremely thin or volatile or both.
Since expected claims are based on reported claims to date, there must be a sufficient volume of
credible reported claims to derive a reliable expected claims estimate.
Question 16 discussion: Blooms: Comprehension; Difficulty 1, LO 3, KS: Assumptions of each
estimation technique
1. EP adjustments. Include using historical rate level changes to adjust historical premiums to an on-level
basis.
2. Claims would also be adjusted for trend, benefit-level changes, and other similar factors.

Question 17 discussion: Blooms: Comprehension; Difficulty 1, LO 2, KS: Purposes of the
development triangle
Each row in the triangle represents one AY. By grouping the data into AYs, each row consists of a fixed group of
claims.
Each subsequent diagonal in the reported claim triangle represents a successive valuation date.
The first diagonal, which starts in the upper left corner of the triangle, is at a December 31, 20XX valuation date
and represents accident year 20XX at 12 months of maturity.
The second diagonal in the triangle is at the December 31, 20XX+1.
The last diagonal of the triangle, at a valuation date of December 31, 20XX+3
Each column in the claim development triangle represents an age (or maturity) and is directly related to the
combination of accident year (row) and valuation date (diagonal) used to create the triangle. In our example, we
present accident year data using annual valuations, and thus the ages in the columns are 12 months, 24 months,
36 months, and 48 months. Different valuations can be used by the actuary (e.g., 6 months, 12 months, 18
months, etc.). See Chapter 5
Question 18 discussion: Blooms: Comprehension; Difficulty 2 LO 2, KS: Examples and uses
of diagnostic development triangles: * Claim and claim count * Ratio of premium to claims *
Average values * Ratios of claims and counts
Compute the following:
a. the amount the insurer paid during CY 2005 during the first 12 months for AY 2005
b. the amount the insurer paid during the second 12 months (CY 2006) for AY 2005
rd
th
c. the amount the insurer paid during the 3 and 4 12 months periods (CY 2007 and CY 2008) for AY 2005
d. the amount the insurer paid during the CY 2006 for accidents occurring during 2006
e. the amount the insurer paid during the CY 2007 for accidents occurring during 2006
a, b, c. For claims that occurred during 2005, the insurer paid a total of:

$600 (400 +200) during the first 12-month period (2005),

$620 (220+200+200) during the second 12-month period (2006), and

$300 in each of the following two 12-month periods (2007 and 2008).
d. and e, For claims that occurred during CY 2006, the insurer paid

$460 (260+200) during CY 2006,

$460 (190+270) during CY 2007 and

Solutions to Exam 5B - Independently Authored Preparatory Test 2
Question 19 discussion: Blooms: Comprehension; Difficulty 1, LO 2, KS: Development
triangle as a diagnostic tool
(1.5 points) According to Friedland in “Estimating Unpaid Claims Using Basic Techniques”, claim count and loss
development can be positive or negative. Briefly describe how claims counts and loss development can develop in
a negative manner.


the number of claims can decrease from one valuation point to another (see chapter 11, private
passenger auto collision coverage)
 reported claim development can show downward patterns if the insurer:
i. settles claims for a lower value than the case O/S estimate or
ii. includes recoveries with the claims data.
See Chapter 5

Question 20 discussion: Blooms: Comprehension; Difficulty 3, LO 2, KS: Examples and uses
of diagnostic development triangles: * Claim and claim count * Ratio of premium to claims *
Average values * Ratios of claims and counts
Table 10 – Case Outstanding Triangle
Accident
Case Outstanding as of (months)
Year
12
24
36
48
2005
900
1,200
1,200
1,200
2006
690
920
920
2007
990
1,320
2008
1,040
Case O/S for AY 2005 at 12 months is computed by adding the ending case O/S values for Claim IDs 1, 2,
and 3 (200+300+400) to derive the case O/S value of $900.
Claim ID 4 case O/S is not included since it is not reported until 5/15/2006.
Case O/S for AY 2005 at 24 months equal case O/S values for Claim IDs 3 and 4 or $1,200 ($200 +
$1,000).
Note that case O/S for Claim IDs 1 and 2 are both $0 at December 31, 2006.
Case O/S for AY 2005 at 36 months and 48 months equal the ending case O/S for Claim ID 4 of $1,200.
Case O/S for AY2006 at 12 months (i.e., valuation date December 31, 2006) equals $690 which is equal to
the sum of the ending case O/S for Claim IDs 5 and 6 ($190 + $500).
Case O/S at 24 months equals the sum of case O/S on all three AY 2006 claims ($0 + $500 + 420).
Case O/S for AY 2006 at 36 months equals $920 which is equal to the case O/S for Claim IDs 6 and 7
See Chapter 5

Solutions to Exam 5B - Independently Authored Preparatory Test 2
Question 21 discussion: Blooms: Comprehension; Difficulty 2, LO 2, KS: Examples and uses
of diagnostic development triangles: * Claim and claim count * Ratio of premium to claims *
Average values * Ratios of claims and counts
The data in Table 5 can be used to build a reported claim count triangle. A description of how to build the
claim count development triangle by using AYs 2005 and 2008 follows.

Table 12 — Reported Claim Count Development Triangle
Accident
Reported Claim Counts as of (months)
Year
12
24
36
48
2005
3
4
4
4
2006
2
3
3
2007
3
4
2008
3
There are 4 claims for 2005, but only 3 of them were reported as of 12/31/2005.
Thus, the first cell in the reported claim count triangle which represents AY2005 as of 12/31/2005
shows 3 claims reported.
By 12/31/2006, all 4 claims were reported.
No further claims were reported for AY 2005, and thus the number of reported claims remains
unchanged at 4 for ages 36 months and 48 months.
There are 4 claims for AY 2008, but as of 12 months, only 3 claims were reported for AY 2008 (claim ID
15 was not reported until 2009 and thus is not included in the triangle).
See Chapter 5

Question 22 discussion: Blooms: Comprehension; Difficulty 1, LO 2 , KS: Purposes of the
development triangle
Actuaries use the data types previously listed to create triangles of ratios and average claim values, which
include:
* Ratio of paid-to-reported claims
* Ratio of total closed claim counts-to-reported claim counts
* Ratio of claim counts on closed with payment-to-total closed claim counts
* Ratio of claim counts on closed without payment-to-total closed claim counts
* Average case O/S (case O/S divided by O/S claim counts)
* Average paid on closed claims (cumulative paid claims on closed claims divided by claim counts closed
with payment)
Cumulative paid claims on closed claim counts may be difficult to obtain; Actuaries may determine that
interim or pre-closing payments are immaterial enough to justify the inexact match from including all
payments, even those from open claims/closed claim counts.
* Average paid (cumulative total paid claims divided by total closed claim counts)
* Average reported (reported claims divided by reported claim counts)
See Chapter 5

Solutions to Exam 5B - Independently Authored Preparatory Test 2
Question 23 discussion: Blooms: Comprehension; Difficulty 1, LO 2, KS: Examples and uses
of diagnostic development triangles: * Claim and claim count * Ratio of premium to claims *
Average values * Ratios of claims and counts
a. What does this diagnostic triangle help the actuary determine?
This diagnostic triangle can help to determine whether there was a speedup in claims payment or
possibly deterioration in underwriting results.
b. Identify one notable observation occurring in the data.
There seems to be evidence of a possible speed-up in payments, particularly at 12 and 24 months.
c. State one question that an actuary may wish to explore based on your response to b.
Has there been a shift in the type of claim settled at each age?
d. What type of additional data may an actuary need to answer the question posed in c.?
Reported and closed claim counts development diagnostic triangles are needed for further review.
See Chapter 6

Question 24 discussion: Blooms: Comprehension; Difficulty 1, LO 2, KS: Purposes of the
development triangle
1. How does the insurer treat reopened claims? Are they coded as a new claim or is a previously
closed claim re-opened? If the insurer treats reopened claims in the latter, there could potentially be
a decrease across a row in the closed claim count development triangle.
2. Does the insurer include claims closed with no payment (CNP) in the reported and closed claim
count triangles?
3. How are claims classified that have only expense payments and no claim payment?
See Chapter 6

Question 25 discussion: Blooms: Comprehension; Difficulty 1, LO 2 KS: Purposes of the
development triangle
1. Has there been a change in case outstanding practices, policies, philosophy, staff, or senior management
of the claims department?
2. Has there been changes in the mix of business in the portfolio that have nothing to do with changes in
case outstanding strength?
See Chapter 6

Exam 5B – Independently Authored Preparatory Test 3
General information about this exam
1. This test contains 23 computational and essay questions.
2. The recommend time for this exam is 2:30:00. Make sure you have sufficient time to take this practice test.
3. Consider taking this exam after working all past CAS questions, associated with the articles below, first.
4. Many of the essay questions may require lengthy responses.
5. Make sure you have a sufficient number of blank sheets of paper to record your answers.

Chapter 11 – Frequency-Severity Techniques ................. Friedland ..............B: Estimating Claim Liabilities
Chapter 12 – Case Outstanding Development TechniqueFriedland ..............B: Estimating Claim Liabilities
Chapter 13 – Berquist-Sherman Techniques.................... Friedland ..............B: Estimating Claim Liabilities

Exam 5B - Independently Authored Preparatory Test 3
Question 1
(1.0 point) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, list and
briefly describe two key assumptions underlying the use of FS Approach #1 – Development Method
with Claim Counts and Severities. Provide an example supporting each assumption.

Question 2
a. (1.5 points) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, list
and briefly describe three advantages to using a Frequency-Severity Technique.
b. (1.5 points) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, list
and briefly describe three disadvantages to using a Frequency-Severity Technique.

Question 3
(5.0 points) Use the Frequency-Severity Development technique, along with the data below, to answer
the following questions.
Note: No adjustments for exposures or severity trend are needed to be made.
3a. Given the following data, project ultimate claim counts for accident years 2004 - 2008
Use a 3-period simple average to select age-to-age factors.
Assume a 1.002 tail factor after 60 months.
Accident
Year
2004
2005
2006
2007
2008

Period
Ending
31-Dec
31-Dec
31-Dec
31-Dec
31-Dec

1st report
12 mo.
1,932
2,067
1,473
1,192
1,036

Reported Claim Counts: Data Triangle
2nd report 3rd report 4th report 5th report
24 mo.
36 mo.
48 mo.
60 mo.
2,168
2,234
2,249
2,258
2,293
2,367
2,390
1,645
1,657
1,264

3b. Given this additional data, project ultimate claim severities for accident years 2004 - 2008
Accident
Year
2004
2005
2006
2007
2008

Period
Ending
31-Dec
31-Dec
31-Dec
31-Dec
31-Dec

1st report
12 mo.
16,995
28,674
27,066
19,477
18,632

Reported Claims ($000s): Data Triangle
2nd report 3rd report 4th report 5th report
24 mo.
36 mo.
48 mo.
60 mo.
40,180
58,866
71,707
74,002
47,432
70,340
70,655
46,783
48,804
31,732

Use a 2-period simple average to select age-to-age factors.
Assume a 10% tail factor after 60 months.
3c. Project ultimate claim severities for accident years 2004 – 2008.
3d. Calculate IBNR estimates for accident years 2004 – 2008.

Exam 5B - Independently Authored Preparatory Test 3
Question 4
(4 points) You are given the following information:
 Annual Average Severity Trend:
5%
 Tort reform factors: AY 2005 = 0.67; AY 2006 = 0.75; AYs 2007-2008 = 1.00
 Reported claims through 12/31/2008 for AY 2008 =
6,669
 Annual Average Severity Trend:
5%

Accident
Year
2005
2006
2007
2008

Incremental Closed Claim Counts
(months of development)
0 to 12
12 to 24
24 to 36
36 to 48
295
824
545
282
307
599
295
329
462
276

Accident
Year
2005
2006
2007
2008

Incremental Payments on Closed Claims
(months of development)
0 to 12
12 to 24
24 to 36
36 to 48
3,043
9,176
14,854
12,953
3,531
8,247
11,041
3,529
8,336
3,409

Ultimate
Claim
Counts
2,402
1,680
1,309
1,172

a. (3 points) Using a Frequency-Severity Disposal Rate method, determine the projected ultimate
payments for accident year 2008.
b. (1.0 point) Compute the estimated IBNR for accident year 2008.
Show all work and state any additional assumptions.

Question 5
(1.0 point) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, briefly
describe when the Case Outstanding Development Technique is used, and what assumption it is
based on that limits its use.

Question 6
(1.0 point) According to Friedland in “Estimating Unpaid Claims Using Basic Techniques”, briefly describe
three limitations in using Case O/S Development Technique – Approach #2, which assumes that the only
available data is case outstanding.

Exam 5B - Independently Authored Preparatory Test 3
Question 7
(4 points) You are given the following information:

Report
Year
2005
2006
2007
2008
Report
Year
2005
2006
2007
2008

Reported Claims as of Months
12
24
36
28,674
47,432
70,340
27,066
46,783
48,804
19,477
31,732
18,632

12
3,043
3,531
3,529
3,409

Paid Claims (cumulative)
24
36
12,219
27,073
11,778
22,819
11,865

48
70,655

48
40,026

Using the case outstanding development technique, compute the estimated reserves for unpaid
losses as of 12/31/2008.

Exam 5B - Independently Authored Preparatory Test 3
Question 8
(3.5 point) Based on Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, you are
given the following information as of December 31, 2008:




Accident
Year
2006
2007
2008

Cumulative Reported Claims
(Age of Development in Months)
12
24
36
$36,000
$81,600
$124,500
46,410
84,840
47,522

Accident
Year
2006
2007
2008

Open Claim Counts
(Age of Development in Months)
12
24
36
20
16
4
24
12
22

Accident
Year
2006
2007
2008

Closed Claim Counts
(Age of Development in Months)
12
24
36
40
82
114
50
86
46

Accident
Year
2006
2007
2008

Cumulative Paid Claims
(Age of Development in Months)
12
24
36
$20,000
$65,600
$119,700
26,250
72,240
25,346

Select ATA factors using all-years simple averages.
The selected tail factor for incurred development after 36 months is 1.150.

a. (1.5 points) Based on Berquist and Sherman's method, demonstrate that the relative level of
the Case Outstanding adequacy has changed for accident year 2008.
b. (2 points) Using Berquist and Sherman's technique for adjusting data to compensate for
changing Case Outstanding adequacy, calculate the ultimate reported claims for accident
year 2008.

Exam 5B - Independently Authored Preparatory Test 3
Question 9
(3 points) You are given the following information for a company that has recently undergone
changes affecting its claim settlement rates.

Accident
Year
2005
2006
2007
2008

Cumulative Closed Claim Counts
Age (in Months)
48
36
24
12
32,500
36,750
41,250
46,000

70,000
78,750
88,000

100,000
105,000

100,000

Projected
Ultimate
100,000
105,000
110,000
115,000

Cumulative Paid Claims ($000)
Accident
Age (in Months)
48
36
24
12
Year
2005
243,750
525,000
750,000
750,000
2006
275,626
590,626
787,500
2007
309,376
660,000
2008
345,000
 Assume that the relationship between the incremental number of closed claim counts (#)
and the incremental paid claims ($) is linear.
 Use all-year simple averages to select ATA factors.
Using the Berquist and Sherman method described by Friedland, calculate an estimate of ultimate
paid claims for accident year 2007. Show all work.

Question 10
(2 points) According to Friedland, Berquist and Sherman B/S suggest several ways for selecting
alternative data to respond to potential problems related to a changing environment.
For each of the following data types, identify what alternative data type they suggest using and the
situation that would give rise to using the alternative data type.
* Number of claims
* Accident year (describe three situations that could arise and what alternative data types could be used)

Question 11.
(1.0 point) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, proper
estimating of unpaid claims is more than just a necessity for managing, investing in, and regulating
insurers.
For U.S. based insurers, list and briefly describe two further requirements for maintaining accurate
reserves.

Exam 5B - Independently Authored Preparatory Test 3
Question 12.
(1.0 point) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, insurers
used to categorize claim adjustment expenses as allocated loss adjustment expenses (ALAE) and
unallocated loss adjustment expenses (ULAE). In 1998, the NAIC promulgated two new
categorizations of adjustment expenses (effective January 1, 1998) for U.S. insurers reporting on
Schedule P of the P&C statutory Annual Statement.
List and briefly describe these two new categories and examples of expense types included in
these categories.

Question 13.
(1.0 point) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, a range
of estimates of the unpaid and a statement of confidence that the actual unpaid claims will be within
the stated range are valuable to management, regulators, policyholders, investors, and even the
general public.
As such, briefly explain why a point estimate of the unpaid claims is necessary, and what guidance
is given to the actuary in developing such a point estimate.

Question 14.
(1.0 point) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, briefly
describe the difference between the terms 'unpaid claim estimate' and 'carried reserve'

Question 15.
(1.0 point) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”,
actuaries refer to the sum of the following four components (i.e., provision for future development
on known claims, estimate for reopened claims, provision for claims incurred but not reported, and
provision for claims in transit) as the broad definition of incurred but not reported (IBNR).
Briefly describe one of the most important reasons for separating IBNR into its components.

Question 16.
(1.5 points) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, large
commercial insurers generally maintain internal claims departments with many claims adjusters
managing the claims. Small to mid-sized commercial insurers and self-insurers often hire third-party
claims administrators (TPAs) or independent adjuster (IA) to manage claims.
Briefly describe/differentiate the types of work performed by TPAs and IAs and how compensation
for their services is arrived at.

Question 17.
(1.5 points) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, a
single claim may extend over a period of several years. List four types of claims transactions that
could occur over the life of the claim.

Exam 5B - Independently Authored Preparatory Test 3
Question 18
(1.5 points) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, one
area that requires the actuary's close attention is the treatment of ALAE in excess of loss
reinsurance contracts.
Briefly describe three possible treatments of ALAE in excess of loss reinsurance contracts.

Question 19
(1.0 point) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, list one
advantage and one disadvantage to using report year aggregation.

Question 20
(1.0 point) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, list one
advantage and one disadvantage to using policy year aggregation.

Question 21
(1.5 points) You are a consulting actuary for the XYZ insurance company. You are about to
conduct a year end review of unpaid claims.
According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, what are three
questions would you ask the 'in-house' prior to conducting your reserve review?

Question 22
(1.5 points) You are a consulting actuary for the XYZ insurance company. You are about to
conduct a year end review of unpaid claims.
According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, what are three
questions would you ask those managing the reinsurance for the company prior to conducting
your reserve review?

Question 23
(1.5 points) You are a consulting actuary for the XYZ insurance company. You are about to
conduct a year end review of unpaid claims.
According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, what are three
questions would you ask an underwriting executive for the company prior to conducting your
reserve review?

Solutions to Exam 5B - Independently Authored Preparatory Test 3
Question 1 discussion: Blooms: Comprehension; Difficulty 1, LO 3, KS: Assumptions of
each estimation technique
1. Individual claim counts are defined in a consistent manner over the experience period.
Example: Do not group claimant counts and occurrence counts together (i.e. recording all
claimants under an occurrence as a single claim), unless the mix of the two ways of counting
a claim is consistent.
2. Claim counts are reasonably homogenous.
Example: Do not analyze first-dollar, low-limit claims with high-layer, multi-million dollar,
excess claims.

Question 2 discussion: Blooms: Comprehension; Difficulty 1, LO 3, KS: When each techniques
works and when it does not
1. Its use in developing estimated unpaid claim estimates for the most recent AYs.
a. Both paid and reported claim development methods can prove unstable and
inaccurate for the more recent AYs.
b. This weaknesses can be addressed by separating estimates of ultimate claims into
frequency and severity.
The number of reported claims reported is usually stable, and thus the projection of
ultimate claim counts produces reliable estimates.
Since severity estimates for the more mature AYs can be obtained with greater certainty,
adjusting these severities using tend factors can help in developing estimates of
severities for the most recent AYs.
2. Its used to gain greater insight into the claims process (e.g. the rate of claims reporting
and settlement and the average dollar value of claims)
3. It can be used with paid claims data only. Thus, changes in case outstanding philosophy
or procedures will not affect the results.
4. Its ability to explicitly reflect inflation in the projection methodology instead of assuming
that past development patterns will properly account for inflationary forces.
A potential disadvantage in doing so is its highly sensitive to the inflation assumption.
1. The unavailability of data.
2. Changes in the definition of claim counts, claims processing, or both may invalidate the
assumption that future claim count development will be similar to historical claim count
development.
3. If the mix of claims is inconsistent, this will distort a FS analysis unless an adjustment is
made for the change in the mix of claim types or claim causes.

Solutions to Exam 5B - Independently Authored Preparatory Test 3
Question 3 discussion: Blooms: Comprehension; Difficulty 3, LO 3, KS: Mechanics
associated with each technique (including organization of the data)
Assume a 1.002 tail factor after 60 months.
(A) Given
Frequency

(B) ATA
and CDF
Frequency

Accident
Year
2004
2005
2006
2007
2008

Period
Ending
31-Dec
31-Dec
31-Dec
31-Dec
31-Dec

1st report
12 mo.
1,932
2,067
1,473
1,192
1,036

Reported Claim Counts: Data Triangle
2nd report 3rd report 4th report 5th report
24 mo.
36 mo.
48 mo.
60 mo.
2,168
2,234
2,249
2,258
2,293
2,367
2,390
1,645
1,657
1,264

Closed Claim Counts: Age-to-Age Factors
Accident
Period 1st to 2nd
2nd to 3rd 3rd to 4th 4th to 5th
Year
Ending
12:24 mo
24:36 mo
36:48 mo
48:60 mo
2004
31-Dec
1.122
1.030
1.007
1.004
2005
31-Dec
1.109
1.032
1.010
2006
31-Dec
1.117
1.007
2007
31-Dec
1.060
3-period simple avg ATA
1.096
1.023
1.008
1.004
Development Age
12 mo.
24 mo.
36 mo.
48 mo.
CDF to Ultimate
1.137
1.038
1.014
1.006

See Tail
Below

Given
1.002
60 mo.
1.002

Age of
Reported
Estimated
Data at
Counts at
CDF to
Ultimate
12/31/08
12/31/08
Ultimate
Counts
(1)
(2) from (A) (3) from (B) (4)=(2)*(3)
2004
31-Dec 60 months
2,258
1.002
2,263
2005
31-Dec 48 months
2,390
1.006
2,404
2006
31-Dec 36 months
1,657
1.014
1,681
2007
31-Dec 24 months
1,264
1.038
1,312
2008
31-Dec 12 months
1,036
1.137
1,178
Note: Friedland performs this work twice: on closed claims and on reported claims, and
considers both in selecting the Ultimate Claim Counts in Exhibit II, Sheet 3.
Accident
Year

Period
Ending

Solutions to Exam 5B - Independently Authored Preparatory Test 3
Question 3 discussion:
3b. Given this additional data, project ultimate claim severities for accident years 2004 – 2008
Use a 3-period simple average to select age-to-age factors.
Assume a 10% tail factor after 60 months.
(D) Given
Claims

(E) =(D)/(A)
* 1000
Severities

(F) ATA
and CDF
Severities

(G) Est.
Ultimate
Severities

Accident
Year
2004
2005
2006
2007
2008

Period
Ending
31-Dec
31-Dec
31-Dec
31-Dec
31-Dec

MUST DIVIDE FOR …
Accident
Period
Year
Ending
2004
31-Dec
2005
31-Dec
2006
31-Dec
2007
31-Dec
2008
31-Dec

1st report
12 mo.
16,995
28,674
27,066
19,477
18,632

Reported Severity: Data Triangle (calculated)
1st report
2nd report 3rd report 4th report 5th report
12 mo.
24 mo.
36 mo.
48 mo.
60 mo.
8,797
18,533
26,350
31,884
32,773
13,872
20,686
29,717
29,563
18,375
28,440
29,453
16,340
25,104
17,985

Accident
Period 1st to 2nd
Year
Ending
6:12 mo
2004
31-Dec
2.107
2005
31-Dec
1.491
2006
31-Dec
1.548
2007
31-Dec
1.536
2-period simple avg ATA
1.542
Development Age
12 mo.
CDF to Ultimate
2.376

Accident
Year

Period
Ending

2004
2005
2006
2007
2008

31-Dec
31-Dec
31-Dec
31-Dec
31-Dec

Reported Claims ($000s): Data Triangle
2nd report 3rd report 4th report 5th report
24 mo.
36 mo.
48 mo.
60 mo.
40,180
58,866
71,707
74,002
47,432
70,340
70,655
46,783
48,804
31,732

Age of
Data at
12/31/08
(1)
60 months
48 months
36 months
24 months
12 months

Severities: Age-to-Age Factors
2nd to 3rd 3rd to 4th 4th to 5th
12:18 mo
18:24 mo
24:30 mo
1.422
1.210
1.028
1.437
0.995
1.036
1.236
24 mo.
1.541

1.102
36 mo.
1.246

1.028
48 mo.
1.131

Reported
Severities at
CDF to
12/31/08
Ultimate
(2) from (E) (3) from (F)
32,773
1.100
29,563
1.131
29,453
1.246
25,104
1.541
17,985
2.376

Estimated
Ultimate
Severities
(4)=(2)*(3)
36,050
33,426
36,713
38,681
42,732

See Tail
Below

Given
1.100
60 mo.
1.100

Solutions to Exam 5B - Independently Authored Preparatory Test 3
3c. Project ultimate claim severities for accident years 2004 - 2008

(H)=(C)*(G)
Ultimate
Claims

Estimated
Estimated Product of Frequency
and Severity (/1000) =
Ultimate
Ultimate
Counts
Severities
Est. Ultimate Claims
(1) = (C4)
(2) = (G4)
(3) = (1) * (2) / 1000
2004
31-Dec
36,050
81,565
2,263
2005
31-Dec
33,426
80,368
2,404
2006
31-Dec
36,713
61,701
1,681
2007
31-Dec
38,681
50,748
1,312
2008
31-Dec
42,732
50,338
1,178
Estimated Ult. Claims for Accident Years 2004 - 2008
324,720
Accident
Year

Period
Ending

3d. Calculate the IBNR estimates for accident periods in accident years 2004 – 2008

(I)=(H)-(D)
IBNR

Estimated
Reported
Estimated IBNR
(broadly defined
Ultimate
Claims at
Claims
12/31/08
to include IBNER)
(1) = (H3)
(2) from (D)
(3) = (1) - (2)
2004
31-Dec
74,002
7,563
81,565
2005
31-Dec
70,655
9,713
80,368
2006
31-Dec
48,804
12,897
61,701
2007
31-Dec
31,732
19,016
50,748
2008
31-Dec
18,632
31,706
50,338
Estimated IBNR for Accident Years 2004 - 2008
80,895
Note: Compare to Exhibit II, Sheet 7 in Friedland's Chapter 11, which also
includes a total Unpaid Claims estimate. Rounding differences exist.
Accident
Year

Period
Ending

Solutions to Exam 5B - Independently Authored Preparatory Test 3
Question 4 discussion: Blooms: Comprehension; Difficulty 3, LO 3
(4 points) You are given the following information:
a. (3 points) Using a Frequency-Severity Disposal Rate method, determine the projected
ultimate payments for accident year 2008.
Disposal Rate Method STEP 1: Select Ultimate Claim Counts by year
(A) Almost given (but need to cumulate the incremental counts for use later)
Cumulative Claim Counts
Accident
Age of Development (Months)
12
24
36
48
Year
295
1119
1664
1946
2005
307
906
1,201
2006
329
791
2007
276
2008
Disposal Rate Method STEP 2A: Select Disposal Rates
(B) Calculate as Cumulative closed counts / Ultimate counts
Selected Disposal Rates
Accident
Age of Development (Months)
12
24
36
48
Year
12.3%
46.6%
69.3%
81.0%
2005
2006
18.3%
53.9%
71.5%
2007
25.1%
60.4%
2008
23.5%
Selected*
24.3%
57.2%
70.4%
81.0%

Ultimate
2402
1,680
1,309
1,172

Ultimate
100%
100%
100%
100%

*Selected from avg latest two. Note: If not told how to select, state your
assumption. Friedland shows 3-yr and 5-yr simple average, and medial 5x1.
STEP 2B: Calculate CONDITIONAL factor from Disposal Rates (incremental)
DON'T FORGET THIS STEP.
(C) = [Difference in consecutive selections in (B)] / [1.0 minus earlier percent in (B)]
Accident
Year
2005
2006
2007
2008
Projection*

Conditional Factor from Disposal Rates (incremental)
Age of Development (Months)
0 to 12
12 to 24
24 to 36
36 to 48
12.3%
39.1%
42.5%
38.2%
use 35.9%
18.3%
43.6%
38.1%
use 30.8%
use 35.9%
25.1%
47.1%
use 43.4%
use 30.8%
use 35.9%
23.5%
24.3%
43.4%
30.8%
35.9%
to use

* Note: These projections are based on the selected disposal rates above.
Example: for 12 to 24 mo.: 43.4% = (57.2% - 24.3%) / (1 - 24.3%)
Disposal Rate Method STEP 3: Project Claim Counts (Incremental)

(D) = [Factor selected in (C)]* [(A) ultimate - all prior entries for (D)]
WARNING: This can be tricky to do in one step.
Incremental Claim Counts (incl projections)
Accident
Age of Development (Months)
0 to 12
12 to 24
24 to 36
36 to 48
Year
295
824
545
282
2005
2006
307
599
295
172.0
2007
329
462
159.7
128.6
2008
276
388.9
156.4
125.9
Example for 2008: 276 = 24.3%*[1,172 - 0] and 389 = 43.4%*[1,172 - 276] and
. . . and 156.4 = 30.8%*[1,172-276-389] and 125.9 = 35.9%*[1,172-sum prev]
Copyright  2014 by All 10, Inc.
Page 49

Solutions to Exam 5B - Independently Authored Preparatory Test 3
4. (4 points) (continued):
STEP 4A: To analyze severities, first need Incremental Claims to date
(E) Given

Accident
Year
2005
2006
2007
2008

0 to 12
3,043
3,531
3,529
3,409

Incremental Claims to date
Age of Development (Months)
12 to 24
24 to 36
36 to 48
9,176
14,854
12,953
8,247
11,041
8,336

STEP 4B: To analyze severities, next find Average Severities to date
(F) = (E)/(D)

Accident
Year
2005
2006
2007
2008

0 to 12
10.315
11.502
10.726
12.351

Actual Average Severities to date
Age of Development (Months)
12 to 24
24 to 36
36 to 48
11.136
27.255
45.933
13.768
37.427
18.043

STEP 5: Project Severities, Incorporating trend
(G) Trend factors (given at 5% annually)

Accident
Year
2005
2006
2007
2008
(H)=(F)*(G)
TortFact
0.67
0.75
1.0
1.0

0 to 12
1.158
1.103
1.050
1.000

Trend Factors to 2008 (at given 5%)
Age of Development (Months)
12 to 24
24 to 36
36 to 48
1.158
1.158
1.158
1.103
1.103
1.050

Trended and Tort Reform Average Severities to 2008 level
Accident
Age of Development (Months)
0 to 12
12 to 24
24 to 36
36 to 48
Year
8.001
8.637
21.139
35.626
2005
2006
9.510
11.384
30.948
2007
11.263
18.945
2008
12.351
2008 level
Selected Severity *
15.165
26.043
35.626
Tort reform factors: 2005=0.67; 2006=.75; 2007-2008=1.00
Trend* Tort reform factors * Severities
*Selected from latest two average. Note: If not told how to select, state your
assumption. Friedland shows 3-yr and medial 5x1.

Solutions to Exam 5B - Independently Authored Preparatory Test 3
4. (4 points) (continued):

(I) from (H)
for AY
2008 only
including
selected
projections

Accident
Year
2005
2006
2007
2008

0 to 12

Trended Average Severities to date
Age of Development (Months)
12 to 24
24 to 36
36 to 48

12.351

15.165

26.043

35.626

STEP 6A: Multiply Severities by Counts for Incremental Paid Claims
(J) = (I) * (D)
Accident
Year
2005
2006
2007
2008

0 to 12

Estimated Total $ Claims (Projected)
Age of Development (Months)
12 to 24
24 to 36
36 to 48

$3,409

$5,897

$4,072

$4,487

24-36: $4,072 = (Count of 156 closing) * ($26.043 paid per closing)
STEP 6B: Add Across Incremental for Cumulative Paid Claims
(K) from (J)
Estimated Total $ Claims (Incl. Projected)
Age of Development (Months)
24
36
48
SOLUTION

Accident
12
Year
2005
2006
2007
2008
$
3,409 $
9,306
$17,865=$3,409 + $9,306 + $13,379

13,379

17,865

b. (1.0 point) Compute the estimated IBNR for accident year 2008.

Step 7: Compute 2008 IBNR
2008 IBNR = 2008 Estimated Ultimate Claims - 2008 Reported claims as of 12/31/2008
2008 IBNR = 17,865 - 6,669 =
$11,196.00
Question 5 discussion: Blooms: Comprehension; Difficulty 1 LO 3, KS: Assumptions
of each estimation technique,
This method is appropriate when applied to lines of insurance for which most of the claims are
reported in the first accident period. Therefore, claims-made coverages and report year analysis
use the case O/S technique because the claims for a given AY are known at the end of the AY.
The assumption that IBNR claim activity is related to claims already reported (i.e., development on
known claims versus pure IBNR) limits its use, and so it is not used extensively by actuaries.

Solutions to Exam 5B - Independently Authored Preparatory Test 3
Question 6 discussion: Blooms: Comprehension; Difficulty 1, LO 3, KS: Assumptions of each
estimation technique,
Potential Limitations
1. Benchmarks may prove to be inaccurate in projecting future claims experience for the insurer.
2. It is inappropriate for the more recent, less mature years due to the increased variability of
results related to the highly leveraged development factors.
3. Large claims in the case O/S data can distort the results of projections based on this method.
Question 7 discussion: Blooms: Comprehension; Difficulty 3, LO 3, KS: Mechanics associated
with each technique (including organization of the data)
Using the case O/S development technique, what are the estimated unpaid claims as of 12/31/2008?
(3) = (1) - (2)
Report
Case Outstanding as of Months
Year
12
24
36
48
2005
25,631
35,213
43,267
30,629
Example : 2006 at 36m
2006
23,535
35,005
25,985
= 48,804 - 22,819
2007
15,948
19,867
= 25,985
2008
15,223
(4) = (2) - (2)prior
Example : 2006 at 36m
= 22,819-11,778
= 11,041

(5) = (4) / (3)prior
Example : 2006 at 36m
= 11,041/35,005
= 31.54%

(6) = (5) & projections
where projections
= selected ratio
Example : 2007 at 48m
= 29.94%

Report
Year
2005
2006
2007
2008

Paid Claims (incremental)
12
24
36
3,043
9,176
14,854
3,531
8,247
11,041
3,529
8,336
3,409

48
12,953

Ratio: Paid Claims (incr) to PRIOR Case Outstanding
Year
12
24
36
48
2005
n/a
35.80%
42.18%
29.94%
2006
n/a
35.04%
31.54%
2007
n/a
52.27%
2008
n/a
Selected
43.7%
36.9%
29.9%
Three-Year Simple Averages

60 or Ult.

100%
given

Complete the square: Ratios with Incremental Paids
Final Ratio
Year
12
24
36
48 60 or Ult.
2005
n/a
35.80%
42.18%
29.94%
100.0%
2006
n/a
35.04%
31.54%
29.94%
100.0%
2007
n/a
52.27%
36.86%
29.94%
100.0%
2008
n/a
43.66%
36.86%
29.94%
100.0%
CAREFUL: These ratios apply to Case Outstanding, so we
also need to "complete the square" for Case Outstanding
before we can actually use these ratios.
To do so, we use another ratio - the Case Outstanding at a
given age, divided by the Case Outstanding at the prior age.
NOTE: It may be tempting to try to use 1 minus the ratio
above, (the incremental Paid / prior Case Outstanding), but
that logic only considers the effect that payments have on
Case Outstanding, and ignores other changes in estimates.

Solutions to Exam 5B - Independently Authored Preparatory Test 3
Question 7 discussion (continued):
(7) = (3) / (3)prior

Example : 2006 at 36m
= 25,985/35,005
= 74.23%

(8) = (7) & projections
where projections
= selected ratio
Example : 2007 at 48m
= 70.79%
(9) = (3) & projections
where projections
= (8) * (9) prior
Example : 2007 at 48m
.7079 * 19,579 = 13,860
[.9855 * 19,867 = 19,579]

Ratio: Case Outstanding to PRIOR Case Outstanding
Year
12
24
36
48
2005
n/a
137.38%
122.87%
70.79%
2006
n/a
148.74%
74.23%
2007
n/a
124.57%
2008
n/a
0.00%
Selected
136.65%
98.55%
70.79%
Three-Year Simple Averages
Complete the square: Ratios Case Outstanding / Prior
Year
12
24
36
48
2005
n/a
137.38%
122.87%
70.79%
2006
n/a
148.74%
74.23%
70.79%
2007
n/a
124.57%
98.55%
70.79%
2008
n/a
136.65%
98.55%
70.79%

60 or Ult.

0%
definition
0 at Ult.
60 or Ult.
0.0%
0.0%
0.0%
0.0%

Complete the square: Case Outstanding & projections
0 at Ult.
Year
12
24
36
48 60 or Ult.
2005
25,631
35,213
43,267
30,629
0
2006
23,535
35,005
25,985
18,395
0
2007
15,948
19,867
19,579
13,860
0
0
2008
15,223
20,803
20,502
14,513
Be sure to go out until the is no more case outstanding.
Otherwise, the cumulative paid will not equal ultimate claims.
NOW: We can go use those ratios we found in step (6), and
project out the future INCREMENTAL claim payments

(10) = (4) & projected
where projected
= (6) * (9) prior
Example : 2007 at 48m
.2994 * 19,579 = 5,862

Complete the square: Incremental Paid & projections
Year
12
24
36
48
2005
3,043
9,176
14,854
12,953
2006
3,531
8,247
11,041
7,779
2007
3,529
8,336
7,323
5,862
2008
3,409
6,646
7,668
6,138

Final Paid
60 or Ult.
30,629
18,395
13,860
14,513

Solutions to Exam 5B - Independently Authored Preparatory Test 3
Question 7 discussion (continued):
Showing Ultimates:
as in Friedland

To get to the Ultimate Claim amounts, we must use CUMULATIVE paids
since the Estimated Total Claims Payments are the Estimated Ultimate
Claims, by definition.

(11) = Sum across (10)
Year
Cumulative Paid & Projections at Ultimate
2005
3,043
+ 9,176
+ 14,854
+ 12,953
+ 30,629
2006
3,531
+ 8,247
+ 11,041
+ 7,779
+ 18,395
2007
3,529
+ 8,336
+ 7,323
+ 5,862
+ 13,860
+ 14,513
2008
3,409
+ 6,646
+ 7,668
+ 6,138
Total Estimate of Ultimate Claims using Case Outstanding Dev. Method

Ultimate
= 70,655
= 48,993
= 38,910
= 38,374
196,933

FINALLY: To get the total "unpaid claim estimate," do the subtraction below:

Year
2005
2006
2007
2008
Total

Estimated
Ultimate
Claims
(12) in (11)
70,655
48,993
38,910
38,374

Actual
Paid
to date
(13) in (2)
40,026
22,819
11,865
3,409

Total Unpaid
Claims Estimate
(14) = (12) - (13)
30,629
26,174
27,045
34,965
118,814

Shortcut: If only asked for "Unpaid Claims Estimates," we can just add up the projected future
payment amounts . . . No need to show the Ultimate claims.
(10) Detail
Looking only at the
Projected future
payments gives us an
Unpaid Claim Estimate

Incremental Paid: Projections only
Sum of
Year
24
36
48 60 or Ult. Projections
2005
30,629
30,629
2006
7,779 +18,395=
26,174
2007
7,323
+5,862 +13,860=
27,045
34,965
2008
6,646
+7,668
+6,138 +14,513=
Est. Unpaid Claims using Case Outstanding Dev. Method
118,814

Solutions to Exam 5B - Independently Authored Preparatory Test 3
Question 8 discussion: Blooms: Comprehension; Difficulty 3, LO 4,
a. (1.5 points) Based on Berquist and Sherman's method, demonstrate that the relative level
of the Case Outstanding adequacy has changed for accident year 2008.
a. Test for change in Adequacy of Case Outstanding
Calculate Average Case Outstanding per Open Claim
(A) = [Reported $ - Paid $ (cumulative)] / [Open Counts]
Accident
Development Period (months)
Year
12
24
36
800
1,000
1,200
2006
840
1,050
2007
1,008
2008
Now, compare the growth % indications here to the implied trend %s from paid data.
Friedland says, "Berquist and Sherman note that the observed trends for the average paid claims are
similar to industry benchmarks (at the time), and thus they conclude that the (different) trend rates for
average case oustanding are indicative of changes
Calculate growth rate % in Average Case Outstanding per Open Claim
Average Case Outstanding $ per Open Claim Count (Severity)
Accident
Development Period (months)
Year
12
24
36
2007
5.00%
5.00%
2008
20.00%
Example: 1008 / 840 - 1 = 20%
8a : Test for change in Adequacy of Case Outstanding - continued
Given Average Paid Claim ($) per Closed Claim Count
(B)
from (A)

(C)

Accident
Year
2006
2007
2008

Average Paid $ per Closed Claim Count (Severity)
Development Period (months)
12
24
36
500
800
1,050
525
840
551

Calculate trend rate % in Average Paid Claim ($) per Closed Claim Count

(D)
from (C)

Calculation of the Trend Factor we take to be "true"
Accident
Development Period (months)
Year
12
24
36
2007
5.000%
5.000%
2006
4.952%

Solutions to Exam 5B - Independently Authored Preparatory Test 3
Question 8 discussion (continued):
b. (2 points) Using Berquist and Sherman's technique for adjusting data to compensate for changing
Case Outstanding adequacy, calculate the ultimate reported claims for accident year 2008.

Step 1: Begin by restating the average open severity using a 5% trend (De-trending)
Average Case Oustanding $ per Open Claim: ADJUSTED
Start with most recent diagonal given in (A) and DE-TREND at 5%
(E)
Accident
Development Period (months)
Year
12
24
36 Example Calculation
2
2006
914.3
1,000
1,200 1,008 / 1.05 = 914.3
2007
960.0
1,050
2008
1,008
Step 2: Multiply re-stated averages above by the open counts for re-stated Case Outstanding $

Re-stated Total $ Case Outstanding = Adjusted Average Case Outstanding * Open Counts (#)
(F) =
Accident
Development Period (months)
(E) * open
Year
12
24
36 Example Calculation
counts
2006
18,286
16,000
4,800 914.3 * 20 (given) = 18,286
(given)
2007
23,040
12,600
2008
22,176
Step 3: Add re-stated case outstanding $ to cumulative paid claim $ for Reported Claims
Adjusted Reported Claims = Re-stated Case Outstanding + Cumulative Paid Claims
Accident
Development Period (months)
(G) =
Year
12
24
36 Example Calculation
(F) + Paid
2006
38,286
81,600
124,500 18,286 + 20,000 (given) = 38,286
(given)
2007
49,290
84,840
2008
47,522
b : Example with change Adequacy of Case Outstanding - continued
Step 4: Compute ATA factors for 12-36 months using the adjusted reported claims, and
Use the given 36 to ultimate CDF to compute the 12 month age to ultimate CDF
ATA
2007
2008
Selected (Simple Avg)
CDF to Ultimate

12:24
2.131
1.721
1.926

Development Period (months)
12:36
Example Calculation
1.526
81,600 / 38,286 = 2.131
1.526 Reported CDF from 36 to ultimate is 1.15(given)

12-to-ult = 1.926 * 1.526 * 1.15 =

3.380

Finally, we estimate AY 2008 ultimate claims = 47,522 * 3.380 = 160,618

ANSWER

Solutions to Exam 5B - Independently Authored Preparatory Test 3
Question 9 discussion: Blooms: Comprehension; Difficulty 3, LO 4
Using the Berquist and Sherman method described by Friedland, calculate an estimate of the
ultimate Paid Claims for accident year 2007. Show all work.

Data
(A) Given
Accident
Year
2005
2006
2007
2008

12
32,500
36,750
41,250
46,000

Cumulative Claim Counts (#)
Age of Development (Months)
24
36
48
70,000
100,000
100,000
78,750
105,000
0
88,000
0
0

(A)
Projected
Ultimate
100,000
105,000
110,000
115,000

(B) Given
Accident
Year
2005
2006
2007
2008

12
243,750
275,626
309,376
345,000

Cumulative Paid Claims ($000's)
Age of Development (Months)
24
36
48
525,000
750,000
750,000
590,626
787,500
0
660,000
0
0
0
0
0

1) Adjusting for changes in settlement rates ... Friedland states:
"Berquist and Sherman select the disposal rate along the latest diagonal as the basis
for adjusting the closed claim count triangle." Accordingly, we find this diagonal:
(C) = Values of (A) along diagonal, divided by the corresponding Ultimate values in (A)
Disposal Rates = Cumulative Claim Counts / Ultimate Claims Counts
Accident
Age of Development (Months)
12
24
36
48
Calculations
Year
= 100,000 / 100,000
100.0%
2005
= 105,000 / 105,000
100.0%
2006
= 88,000 / 110,000
80.0%
2007
2008
= 46,000 / 115,500
40.0%
Selected
40.0%
80.0%
100.0%
100.0%
2) And use these Selected Disposal Rates to restate (adjust) the historical count data:
(D) = Ultimate Counts in (A), multiplied by the selected Disposal Rates in (C)
ADJUSTED Cumulative Claim Counts (#)
Accident
Age of Development (Months)
12
24
36
48
Year
40,000
80,000
100,000
100,000
2005
2006
42,000
84,000
105,000
44,000
2007
88,000
2008
46000

Solutions to Exam 5B - Independently Authored Preparatory Test 3
Question 9 discussion (continued):
3) To move from adjusted claim counts to adjusted claim dollars , Friedland notes:
The authors "identify a mathematical formula that approximates the relationship … "
We a ssume the relationship is linear , based on the unadjusted data points (by year).
Note: we will only calculate the factors we need for this exam question:
Since we only need an estimate for 2007 AY unpaid, which is at 24 months, we need
enough data to develop a CDF from 24 to Ultimate. (2006 and 2007, 24 mo and after)
ADJUSTED AY 2006 Data
Original AY 2006 Data
Counts # Claim $000
Counts #
Claims $000's (Cumulative)
Linearly Interpolated from left
Age
From (A)
From (B)
From (D)
24
70,000
525,000
80,000
600,000 see below
36
100,000
750,000
100,000
750,000 as for unadjusted
48
100,000
750,000
100,000
750,000 as for unadjusted
For example, adjusted paid losses at 24 months are calculated as:
525,000 + (80-70)/(100-70) x (750,000-525,000) = 600,000
FOR 2006

ADJUSTED AY 2007 Data
Original AY 2007 Data
Counts # Claim $000
Counts #
Claims $000's (Cumulative)
Age
From (A)
From (B)
From (D)
Linearly Interpolated from left
24
78,750
590,626
84,000
630,001 see below
36
105,000
787,500
105,000
787,500 as for unadjusted
For example, adjusted paid losses at 24 months are calculated as:
590,626 + (84,000-78,750)/(105,000-78,750) x (787,500-590,626) = 630,000
FOR 2007

(E) from calculations immediately above, we have ADJUSTED PAID ($) DATA
ADJUSTED Cumulative Paid Claims ($000's)
Accident
Age of Development (Months)
12
24
36
48
Year
600,000
750,000
750,000
2005
2006
630,001
787,500
2007

4) Use ADJUSTED PAID LOSS DATA to develop a CDF to apply to AY 2007
(F) Based on the adjusted data in table (E)
ATA calculations
24 to 36
2006
1.25
2007
1.25
Selected (Simple Average)
1.25

36 to 48
1.00
n/a
1.00

48 to Ult.

1.00

at 24 mo

CDF calculations
CDF to Ultimate

1.25

5) Apply the CDF to AY 2007
Accident
Year
2007

12
Latest

Cumulative Paid Claim $000's
Age of Development (Months)
24
36
48
660,000

CDF

multiplied by the selected 1.25 =

ANSWER
Projected
Ultimate
825,000

Solutions to Exam 5B - Independently Authored Preparatory Test 3
Question 10 discussion: Blooms: Comprehension; Difficulty 2 LO 4, KS: How internal
operating changes affect estimates of unpaid claims: * Claims processing *
Underwriting and policy provisions * Marketing * Coding of claim counts and/or claim
related expenses * Treatment of recoveries such as policyholder deductibles and
salvage and subrogation * Reinsurance
* Using earned exposures instead of the number of claims when claim count data is of questionable
accuracy or if there has been a major change in the definition of a claim count.
* Substituting policy year data for accident year data when there has been a significant change in policy
limits or deductibles between successive policy years.
* Substituting report year data for accident year data when there has been a dramatic shift in the social or
legal climate that causes claim severity to more closely correlate with the report year than with the accident
date.
* Substituting accident quarter for accident year when the rate of growth of earned exposures changes
markedly, causing distortions in development factors due to significant shifts in the average accident date
within each exposure period.
Question 11 discussion: Blooms: Comprehension; Difficulty 1, LO 3:
1. It is required by law (e.g. NY Law states that every insurer shall maintain reserves in an amount
estimated in the aggregate to provide for the payment of all losses or claims incurred on or prior to the date
of settlement.
2. The NAIC requires that most P&C insurers in the U.S. obtain a Statement of Actuarial Opinion signed by
a qualified actuary. See chapter 1
Question 12 discussion: Blooms: Comprehension; Difficulty 1, LO 3:
Defense and cost containment (DCC) and adjusting and other (A&O). Generally, DCC expenses include all
defense litigation and medical cost containment expenses regardless of whether internal or external to the
insurer; A&O expenses include all claims adjusting expenses, whether internal or external to the insurer.
See chapter 1
Question 13 discussion: Blooms: Comprehension; Difficulty 1, LO 3:
The insurer's balance sheet requires the insurer to record a point estimate of the unpaid claims, as required
by the NAIC Further, Actuarial Standard of Practice No. 43 defines the actuarial central estimate as an
estimate that represents an expected value over the range of reasonably possible outcomes.
See chapter 1
Question 14 discussion: Blooms: Comprehension; Difficulty 1, LO 3:
The unpaid claim estimate is the result of the application of a particular estimation technique, and different
estimation techniques will often generate different unpaid claim estimates. In addition, the unpaid claims
estimate will likely change from one valuation date to another for the same portfolio.
The carried reserve for unpaid claims is the amount reported in a published statement or in an internal
statement of financial condition.
See chapter 1
Question 15 discussion: Blooms: Comprehension; Difficulty 1, LO 3:
One of the most important reasons for separating IBNR into its components is to test the adequacy of case
outstanding over time. This can be an important management tool and a useful tool for the actuary when
determining which methods are most appropriate for estimating unpaid claims.
See chapter 1
Copyright  2014 by All 10, Inc.
Page 59

Solutions to Exam 5B - Independently Authored Preparatory Test 3
Question 16 discussion: Blooms: Comprehension; Difficulty 2, LO 3:
TPAs frequently handle a specific book of claims from beginning to end (i.e., from the initial report to the
final payment). Insurers usually require the TPA to report details of the claims on a predetermined basis
(e.g., monthly or quarterly). In certain circumstances, a TPA manages all the claims of an insurer, and the
insurer only has a minimal number of claims personnel reviewing the activities of the TPA. The
compensation for services of a TPA is generally based on a contract for the entire book of business and not
by individual claim, though compensation varies among TPAs
An insurer may hire an independent adjuster (IA) to handle an individual claim or a group of claims. The
insurer, who may have an active claims department, may need an IA to handle a specific type of claim or a
claim in a particular region where the insurer does not have the necessary expertise. Also when a disaster
occurs, such as a hurricane or earthquake, the insurer may hire a number of IAs (or a firm of IAs) to handle
the large volume of claims. The compensation for the services of IAs is generally based on a fee per claim.
See chapter 2
Question 17 discussion: Blooms: Comprehension; Difficulty 2, LO 3:
The different types of claim transactions over the life of the claim could include:
* Establishment of the initial case outstanding estimate
* Notification to the reinsurer if the claim is expected to exceed the insurer's retention
* A partial claim payment to injured party
* Expense payment for independent adjuster
* Change in case outstanding estimate
* Claim payment (assumed to be final payment)
* Takedown of case outstanding and closure of claim
* Re-opening of the claim and establishment of a new case outstanding estimate
* Partial payment for defense litigation
* Final claim payment
* Final payment for defense litigation
* Closure of claim
See chapter 2

Question 18 discussion: Blooms: Comprehension; Difficulty 1, LO 3:
1. Included with the claim amount in determining excess of loss coverage (which is the most
common treatment)
2. Not included in the coverage
3. Included on a pro rata basis; the ratio of the excess portion of the claim to the total claim amount
determines coverage for ALAE
See chapter 3

Solutions to Exam 5B - Independently Authored Preparatory Test 3
Question 19 discussion: Blooms: Comprehension; Difficulty 1, LO 3:
Advantages of Report Year Aggregation
The number of claims is fixed at the close of the year (other than for claims reported but not recorded).
The RY approach substitutes a known quantity (i.e. the number of reported claim counts) for an estimate.
Thus, a RY approach will often result in more stable data and more readily determinable development
patterns than an AY approach, since the number of AY claims is subject to change at each successive
valuation.
Disadvantage of Report Year Aggregation
RY estimation techniques only measure development on known claims (and not pure IBNR)
See chapter 3
Question 20 discussion: Blooms: Comprehension; Difficulty 2, LO 3:
Advantages of PY Aggregation:
The key advantage is the true matching of claims and premiums.
* PY experience is very important when underwriting or pricing changes occur (e.g. a shift from full
coverage to large deductible policies, a change in emphasis on certain classes of business, or an
increase/decrease in the price charged leading to a change in expected claim ratios and possibly a change
in the type of insured).
* PY aggregation is useful for self-insureds, who often issue a single policy.
Disadvantages of PY Aggregation
* The primary disadvantage is the extended time to gather complete data (i.e. it can take up to 24 months
to gather all reported claims) and to reliably estimate ultimate claims.
* PY data can make it difficult to understand and isolate the affect of a single large event (e.g. a major
catastrophe or court ruling), which changes how insurance contracts are interpreted.
See chapter 3
Question 21 discussion: Blooms: Comprehension; Difficulty 1, LO 3:
What are three questions would you ask the 'in-house' prior to conducting your reserve review?
1. Could we obtain copies of any and all actuarial studies done by consultants, auditors or internal
actuaries?
2. What areas of disagreement are there between these different studies?
3. What specific background information did you take into account in making your selections?
See chapter 4

Solutions to Exam 5B - Independently Authored Preparatory Test 3
Question 22 discussion: Blooms: Comprehension; Difficulty 2, LO 3:
What are three questions would you ask those managing the reinsurance for the company prior to
conducting your reserve review?
* Please provide details of reinsurance treaties for both assumed and ceded business.
* Please provide details of all reinsurance ceded treaties including:
i. Retention level or Q.S. %
ii. Reinsurers involved (including participation)
iii. Details of any sliding scale premium, commission, or profit commission (including currently booked
amounts)
iv. Any problems or delays encountered in collecting reinsurance
* Please provide details of any internal or sister company reinsurance agreements (cover notes, relevant
amounts, and by-line breakdowns).
* Have the reinsurance programs for next year been secured? If so, under what terms?
See chapter 4
Question 23 discussion: Blooms: Comprehension; Difficulty 2, LO 3:
What are three questions would you ask an underwriting executive for the company prior to conducting your
reserve review?
1. What changes have occurred in your company's book of business and mix of business in the past 5-7
years? How are the risks insured today different from those of the past?
2. Do you underwrite any large risks which are not characteristic of your general book of business?
3. Have any significant changes occurred in your underwriting guidelines in recent years?
4. Has the proportion of business attributable to excess coverages for self-insurers changed in recent
years? Can a distribution of such business be obtained by line, retention limit, class, etc.?
Is a record of self-insured losses and claims available?
5. How many different programs or types of risk are premium and claims experience tracked and
compiled into claim ratio runs?
6. Are any details of excess policies (e.g. attachment points, exclusions, per occurrence, sunset clauses,
aggregate caps, etc.) available?
7. How frequent are experience summaries run? How far back are these available?
8. How are new programs priced? If you are relying on another insurer's filings, how similar are the
underlying books of business?
See chapter 4

Exam 5B – Independently Authored Preparatory Test 4
General information about this exam
1. This test contains 23 computational and essay questions.
2. The recommend time for this exam is 2:30:00. Make sure you have sufficient time to take this practice test.
3. Consider taking this exam after working all past CAS questions, associated with the articles below, first.
4. Many of the essay questions may require lengthy responses.
5. Make sure you have a sufficient number of blank sheets of paper to record your answers.

Articles covered on exam:
Article .................................................... Author .................................. Syllabus Section
Chapter 7 – Development Technique .............................. Friedland ..............B: Estimating Claim Liabilities
Chapter 8 – Expected Claims Technique ........................ Friedland ..............B: Estimating Claim Liabilities
Chapter 9 – Bornhuetter-Ferguson Technique ................ Friedland ..............B: Estimating Claim Liabilities
Chapter 10 – Cape Cod Technique ................................. Friedland ............. B: Estimating Claim Liabilities
Chapter 17 – Estimating Unpaid ULAE ............................. Friedland ..............B: Estimating Claim Liabilities

Exam 5B - Independently Authored Preparatory Test 4
Question 1
(1.5 points) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, answer the
following questions:
a. List and briefly describe the key assumptions under the classical technique for setting ULAE reserves.
b. Assuming that ½ of ULAE are sustained when opening a claim and ½ is sustained when closing the claim,
describe how the %'s of the ULAE ratio are applied and to what reserves to compute the ULAE reserve.

Question 2
(1.0 point) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, list the four steps
involved in applying the classical technique to estimating unpaid ULAE

Question 3
(1.0 point) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, answer the
following questions.
a. One challenge of the classical technique is that “closing” a claim and “paying” a claim do not necessarily
mean the same thing. Briefly describe an example of this and a method to correct this shortcoming.
b. Another challenge of the classical technique is the use of broad definition of IBNR. Briefly describe why this
is a challenge and a way to correct this shortcoming.

Question 4
(1.5 points) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, state Wendy
Johnson's and Kay Kellogg Rahardjo's rationale for when the classical technique works and when it does not.

Question 5
(1.5 points) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, answer the
following questions:
a. Briefly describe Kittel's refinement to the classical technique (i.e. the weakness in the classical technique)
b. State the two key assumptions of Kittel refinement to the classical technique:

Question 6
(1.5 points) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, briefly describe the
two key problems associated with the Kittel Refinement.

Question 7
(2.5 points) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, answer the
following questions:
a. Briefly describe the four key assumptions underlying the Generalized Approach to computing ULAE reserves
b. Briefly describe what U1, U2, and U3

Exam 5B - Independently Authored Preparatory Test 4
Question 8
(2.5 points) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, the following are
two methods to estimate unpaid ULAE for a group of AYs.
Method 1. Compute Unpaid ULAE as follows: Unpaid ULAE = (W* x L) - M
Method 2. Compute Unpaid ULAE as follows: Unpaid ULAE = M x (L/B – 1.00)
a. Briefly describe what the variables W*, L, B and M represent
b. Briefly describe the practical and conceptual problems and concerns with both methods

Question 9
(2.5 points) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, the following
formula is used in Conger and Nolibos' preferred method to estimate unpaid ULAE for a group of AYs: Unpaid
ULAE = W* x (L - B). Using this formula, the variables R(t), P(t), and C(t), and the U1, U2, and U3 percentages,
answer the following questions:
a. Rewrite the formulate to compute unpaid ULAE given in the problem.
b. Briefly describe what this method assumes.

Question 10
(2.5 points) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, briefly describe four
practical difficulties with the generalized approach

Question 11
(1.5 points) Using the procedure described by Friedland in ““Estimating Unpaid Claims Using Basic
Techniques”, and the data given below, compute the expected claim payments in calendar years 2005 – 2008.
Accident
Year
2005
2006
2007
2008



Direct
Earned
Premium
2,866,667
2,833,333
2,946,667
2,656,667

Expected Payment Percentage in Calendar
2005
2006
2007
2008
12%
15%
15%
15%
12%
15%
15%
12%
15%
12%

Expected claims ratio is 60% each year

Exam 5B - Independently Authored Preparatory Test 4
Question 12
(1.5 points) Using the Mango-Allen Refinement to the Classical Technique as described by Friedland in
““Estimating Unpaid Claims Using Basic, and the data given below, answer the following:
Calendar
Year
(1)
2005
2006
2007
2008

Paid
ULAE
(2)
36,667
41,667
46,667
53,333

Paid Claims
Actual
Expected
(3)
(4)
835,633
206,400
57,333
462,000
273,767
725,160
206,400
969,480

 The selected ULAE Ratio should be based on an all years' average ratio
 Case Outstanding at 12/31/08 = 213,750
 Total IBNR at 12/31/08 = 6,108,500
 Pure IBNR at 12/31/08 = 5% of AY 2008 Expected Claims. AY 2008 Expected Claims = 1,594,000
Compute:
a. Estimated Unpaid ULAE at 12/31/08 Using Total IBNR
b. Estimated Unpaid ULAE at 12/31/08 Using Pure IBNR

Question 13
(1.5 points) Using the procedure described by Friedland in ““Estimating Unpaid Claims Using Basic, and the
data given below, answer the following:
Calendar
Year
2005
2006
2007
2008

Paid
ULAE
7,274
10,233
11,172
12,993

Paid
Claims
32,632
49,552
73,163
89,646

Reported
Claims
72,693
109,371
123,310
139,082

 The selected ULAE Ratio should be based on an all years' average ratio
 Case Outstanding at 12/31/08 = 248,311
 IBNR at 12/31/08 = 96,775
Compute:
a. Estimated Unpaid ULAE at 12/31/08 using the traditional paid to paid method.
b. Estimated Unpaid ULAE at 12/31/08 using Kittel's refined method.

Exam 5B - Independently Authored Preparatory Test 4
Question 14
(4.5 points) Using the Conger and Nolibos generalized method with U1= 60%, U2= 40%, and U3 = 0% as
described by Friedland in ““Estimating Unpaid Claims Using Basic, and the data given below, answer the
following:
Calendar
Year
(1)
2005
2006
2007
2008

Paid
ULAE
(2)
7,274
10,233
11,172
12,993

Ult on Claims
Reported in
Calendar Year
(3)
90,865
131,155
138,635
149,940

Paid
Claims
(4)
32,632
49,552
73,163
89,646

Claims
Basis
(5)
67,572
98,514
112,446
125,822



The selected ULAE Ratio should be based on an all years' average ratio, rounded to the nearest
ths
10 place
 Total ultimate claims for all AYs = 514,760
Compute:
a. Estimated Unpaid ULAE at 12/31/08 using the expected claims method
b. Estimated Unpaid ULAE at 12/31/08 using the Bornhuetter-Ferguson method.
c. Estimated Unpaid ULAE at 12/31/08 using the development method.

Question 15
(4.0 points) Using the Conger and Nolibos simplified generalized method with
U1= 60%, U2= 40%, and U3 = 0% as described by Friedland in ““Estimating Unpaid Claims Using Basic,
and the data given below, answer the following questions.
Calendar
Year
(1)
2005
2006
2007
2008

Paid
ULAE
(2)
7,274
10,233
11,172
12,993

Acc Year
Ultimate
Claims
(3)
90,865
131,155
138,635
149,940

Paid
Claims
(4)
32,632
49,552
73,163
89,646

Claims
Basis
(5)
67,572
98,514
112,446
125,822
ths

 The selected ULAE Ratio should be based on an all years' average ratio, rounded to the nearest 10 place
 Total ultimate claims for all AYs = 514,760
Compute:
a. Estimated Unpaid ULAE at 12/31/08 assuming pure IBNR = 4% of Latest Accident Year Ultimate Claims
b. Estimated Unpaid ULAE at 12/31/08 assuming pure IBNR = 6% of Latest Accident Year Ultimate Claims

Exam 5B - Independently Authored Preparatory Test 4
Question 16
(4.0 points) Using the procedure described by Friedland in ““Estimating Unpaid Claims Using Basic, and the data
given below, answer the following questions.
PART 1 - Data Triangle
Accident
Year
12
2003
45,163,102
2004
45,417,309
2005
46,360,869
2006
46,582,684
2007
48,853,563
PART 2 - Age-to-Age
Accident
Year
2003
2004
2005
2006
2007

24
52,497,731
52,640,322
53,790,061
54,641,339

Reported Claims as of (months)
36
48
55,468,551
57,015,411
55,553,673
56,976,657
56,786,410

60
57,565,344

Factors
12-24
1.162
1.159
1.160
1.173

24 - 36
1.057
1.055
1.056

36 - 48
1.028
1.026

Age-to-Age Factors
48 - 60
To Ult
1.010
1.000

Compute:
a. (1.0 point). The geometric average of the age to age factors for the latest four years at 12-24 months
b. (1.0 point). Percent reported at 12 months, assuming selected cumulative loss development factors to
ultimate are based on simple averages of the latest three years.
c. (1.0 point). Projected ultimate claims using the cumulative loss development factors computed in b. for
accident years 2003 – 2007.
d. (1.0 point). IBNR for accident years 2003 – 2007.

Question 17
(4 points) You are given the following information:
Earned
Premium

Accident
Year

38,000
40,000
42,000
44,000

2005
2006
2007
2008

Reported Claims by Development Age
at age
at age
at age
at age
12 mo
24 mo
36 mo
48 mo
9,700
19,400
28,200
32,400
10,300
20,600
29,800
10,800
21,600
14,400

Assume an expected Claim Ratio = 0.90 for all years.
Choose selected factors using a straight average of the age to age factors.
Assume no development past 48 months.
a. (1 point) Using the Development method, calculate the indicated IBNR for accident year 2008 as of
December 31, 2008.
b. (0.5 point) Using the Bornhuetter-Ferguson method, calculate the indicated IBNR for accident year
2008 as of December 31, 2008.
c. (1 point) Using the Bornhuetter-Ferguson method, calculate the expected IBNR for Accident Year
2008 expected to be reported (emerge) during calendar year 2009.

Exam 5B - Independently Authored Preparatory Test 4
Question 18
(3.0 points) Using the procedure described by Friedland in ““Estimating Unpaid Claims Using Basic, and
the data given below, answer the following questions.
You are asked to develop an estimate of unpaid claims for an insurer writing private passenger automobile
bodily injury in one jurisdiction.

Accident
Year

Claims at 12/31/098
Reported
Paid

2004
2005
2006
2007
2008

16,500,000 11,200,000
18,500,000 10,200,000
16,500,000 6,000,000
14,000,000 3,000,000
8,700,000
750,000

CDF to Ultimate
Reported Paid
1.200
1.400
1.800
2.900
4.000

1.750
2.500
5.000
15.000
90.000

On-Level
Earned
Premium

Trend at
14.50%
to 7/1/08

Factor to
Adjust
for Tort
Reform

32,000,000
47,000,000
50,000,000
57,000,000
62,000,000

1.719
1.501
1.311
1.145
1.000

0.750
1.000
1.000
1.000
1.000

Compute:

a. (1.0 point). Compute initial selected ultimate claims as the average of the reported and paid claim
development projections.
b. (1.0 point). Compute the selected claim ratio for AY 2008 as the average of trended adjusted claim ratios for
2004 – 2008, excluding high and low ratios.
c. (1.0 point). Compute estimated IBNR for AY 2008

Question 19
(3.0 points) Using the procedure described by Friedland in ““Estimating Unpaid Claims Using Basic, and the
data given below, answer the following questions.
You are asked to develop an estimate of unpaid claims for an insurer writing private passenger automobile
bodily injury in one jurisdiction.
Accident
Year
(1)
2004
2005
2006
2007
2008
Total

Claims at
Reported
(2)
70,288
70,655
48,804
31,732
18,632
240,111

12/31/08
Paid
(3)
52,811
40,026
22,819
11,865
3,409
130,930

Earned Claim Ratio
Premium Selected
(4)
(5)
99,322
87.1%
138,151
78.3%
107,578
65.8%
62,438
63.8%
47,797
82.5%

a. (2.0 points). Compute estimated IBNR for AYs 2004 - 2008
b. (1.0 points). Compute estimated total unpaid claims for AYs 2004 - 2008

Exam 5B - Independently Authored Preparatory Test 4
Question 20
(3.0 points) Using the procedure described by Friedland in ““Estimating Unpaid Claims Using Basic, and the
data given below, answer the following questions.
You are asked to develop projected ultimate claims using the B-F method using paid and reported claims
Accident Expected
CDF to Ultimate
Year
Claims
Reported
Paid
(1)
(2)
(3)
(4)
2003
56,318,302
1.011
1.040
2004
59,646,290
1.023
1.085
2005
61,174,953
1.051
1.184
2006
61,926,981
1.110
1.404
2007
61,864,556
1.292
2.390
Total
300,931,082

Claims at 12/31/07
Reported
Paid
(5)
(6)
57,565,344
55,930,654
56,976,657
53,774,672
56,786,410
50,644,994
54,641,339
43,606,497
48,853,563
27,229,969
274,823,313
231,186,786

a. (2.0 points). Compute projected ultimate claims using the B-F method using reported claims for AYs
2003 - 2007
b. (1.0 points). Compute projected ultimate claims using the B-F method using paid claims for AY 2007

Question 21
(3.0 points) Using the procedure described by Friedland in ““Estimating Unpaid Claims Using Basic, and the
data given below, answer the following questions.
Age of
Expected ultimate Claims
Claims at 12/31/08
CDF to Ultimate
Accident Accident Year Using B-F Method with
Year
at 12/31/08
Reported
Paid
Reported
Paid
Reported
Paid
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8 )
Steady-State
2004
60
893,397
893,397
884,463
857,661
1.000
1.000
2005
48
938,068
938,067
919,306
863,022
1.010
1.043
2006
36
984,970
984,970
935,722
827,375
1.042
1.143
2007
24
1,034,219
1,034,218 930,797
734,295
1.100
1.352
1,085,929 836,166
456,090
1.286
2.286
2008
12
1,085,930
Total
4,936,584
4,936,581 4,506,454 3,738,443

Exam 5B - Independently Authored Preparatory Test 4
Question 22
(3.0 points) Using the procedure described by Friedland in ““Estimating Unpaid Claims Using Basic, and the data
given below, answer the following questions.

Accident
Year
(1)
2003
2004
2005
2006
2007

Earned
Premium
(2)
86,643,542
91,763,523
94,115,312
95,272,279
95,176,240

Age of
Accident
at 12/31/07
(3)
60
48
36
24
12

Reported
Year Claims at
12/31/2007
(4)
57,565,344
56,976,657
56,786,410
54,641,339
48,853,563

Reported
CDF to
Ultimate
(5)
1.000
1.010
1.037
1.095
1.274

a. (1.50 points). Compute estimated claim ratios using the Cape Cod Method for AYs 2003 – 2007.
b. (1.50 points). Compute projected ultimate claims using the Cape Code Method for AYs 2003 – 2007.
Question 23
(4.0 points) Using the procedure described by Friedland in ““Estimating Unpaid Claims Using Basic, and the
data given below, answer the following questions.
You are given detailed rate change information for the ABC insurance company as well as information
regarding the affect of legal reform on the insurance product. You are asked to incorporate this
information into the Cape Cod projection method. You are also asked to adjust the current reported
claims for the influences of inflation (through claims trend factors) and tort reform.
Accident
Year
2004
2005
2006
2007
2008
Total

Reported
Pure
Earned
On-Level
Claims Premium
Premium Adjustment at 12/31/08 Trend
99,322
138,151
107,578
62,438
47,797
455,286

0.810
0.704
0.640
0.800
1.000

70,288
70,655
48,804
31,732
18,632
240,111

1.144
1.106
1.070
1.034
1.000

Tort
Reform
Factors

Reported
CDF to
Ultimate

0.670
0.670
0.750
1.000
1.000

1.064
1.085
1.196
1.512
2.551

a. (2.50 points). Compute used-up on-level premium using the Cape Cod Method for AYs 2004 – 2008.
b. (1.50 points). Compute estimated unadjusted claim ratios using the Cape Code Method for AYs 2004 – 2008.

Solutions to Exam 5B - Independently Authored Preparatory Test 4
Question 1 discussion: Blooms: Comprehension; Difficulty 1, LO 7, KS Organization of the data
a. Key Assumptions of Classical Technique
• The insurer’s ULAE-to-claim relationship has reached a steady-state so that the ratio of paid ULAE-topaid claims approximates ultimate ULAE-to-ultimate claims.
• The volume and cost of future claims management on not-yet-reported claims and reported-but-not-yetclosed claims will be proportional to IBNR and case O/S, respectively.
b. Assume that ½ of ULAE are sustained when opening a claim and ½ is sustained when closing the claim.
Thus,
i. 50% of the ULAE ratio is applied to case O/S (since for known claims, ½ of the unallocated work was
already completed at the time of opening);
ii. 100% of the ULAE ratio is applied to IBNR, since all unallocated work remains to be completed (i.e. the
work associated with opening and closing the claims).
See chapter 17
Question 2 discussion: Blooms: Comprehension; Difficulty 1, LO 7, KS Organization of the data
4 steps in the classical technique for estimating unpaid ULAE:
1. Calculate ratios of historical CY paid ULAE-to-CY paid claims
2. Review historical paid ULAE-to-paid claims ratios for trends or patterns
3. Select a ratio of ULAE-to-claims applicable to future claims payments
4. Apply 50% of the selected ULAE ratio to case O/S and 100% of the selected ULAE ratio to IBNR
See chapter 17
Question 3 discussion: Blooms: Comprehension; Difficulty 1, LO 7, KS Key assumptions of
estimation techniques
a. closing” a claim and “paying” a claim
• For glass coverage, a single payment is the norm, and payment represents settlement (i.e. closure) of the
claim, and therefore the end of the claims handling activity.
• For WC, a claim payment and closing of the claim often differ, since regular payments can replace lost
wages for an extended period of time.
Address this challenge by adjusting the %’s applied to the case O/S and the IBNR. Example:
For an insurer with a portfolio of long-tail professional liability coverage, with substantial claims-handling
work during the life of the claim, unpaid ULAE ratios of 25% are applied to case O/S and 75% to IBNR
(assumes a greater % of expenses are related to closing the claims rather than opening claims).
b. Another challenge is the definition of IBNR.
The broad definition of IBNR includes liability for both claims that are not yet reported as well as future case
development on known claims.
The narrow definition of IBNR is incurred but not yet reported (IBNYR, a.k.a. pure IBNR), while future case
development on known claims is referred to as incurred but not enough reported (IBNER).
Using the classical technique, apply 100% of the ULAE ratio to IBNYR (pure IBNR) and 50% of the ULAE
ratio to the sum of case reserves and IBNER.
See chapter 17

Solutions to Exam 5B - Independently Authored Preparatory Test 4
Question 4 discussion: Blooms: Comprehension; Difficulty 1, LO 7, KS Strengths and
weaknesses of the estimation techniques for claim related expenses
When the Classical Technique Works and When it Does Not
Johnson states that the classical technique “will only give good results for very short-tailed, stable lines of
business.”
Rahardjo states:
 The paid to paid method assumes that claims incur expenses only when initially opened and when
closed, which is not true for liability claims.
 The paid to paid ratio itself is subject to distortion when a company is growing or shrinking or when a line
of business is in “transition” (e.g. consider WC in the early 1990s as many large customers moved to
deductible policies or towards self-insurance).
See chapter 17
Question 5 discussion: Blooms: Comprehension; Difficulty 1, LO 7, KS Strengths and
weaknesses of the estimation techniques for claim related expenses
a. Kittel Refinement. Kittel describes a weakness in the classical technique:
The Loss Department doesn’t just close claims but it also opens them.
Paid losses don’t accurately represent the work done by the Loss Department since they do not take into
account claims opened during the year which remain open at year end.
This can be significant when loss reserves vary from year to year (e.g. a growing line with rapidly inflating loss
costs could have loss reserves increase at 30% - 40% per year).
b. Key Assumptions of Kittel Refinement to the Classical Technique:
• ULAE is sustained as claims are reported even if no claim payments are made.
• ULAE payments for a specific calendar year are related to both the reporting and payment of claims.
See chapter 17
Question 6 discussion: Blooms: Comprehension; Difficulty 1, LO 7, KS Strengths and
weaknesses of the estimation techniques for claim related expenses
Problems associated with the Kittel Refinement
 The use of traditional 50/50 assumption regarding ULAE expenditures does not allow for allocation of
ULAE costs between opening, maintaining, and closing claims which may vary from insurer to insurer.
 There is no potential for using different rates of inflation between ULAE and claims.
See chapter 17
Question 7 discussion: Blooms: Comprehension; Difficulty 2, LO 7, KS Estimation of unpaid ULAE
a. Key Assumptions of Generalized Approach
• Expenditure of ULAE resources is proportional to the dollars of claims being handled
• ULAE amounts spent opening claims are proportional to the ultimate cost of claims being reported.
• ULAE amounts spent maintaining claims are proportional to payments made.
• ULAE amounts spent closing claims are proportional to the ultimate cost of claims being closed.
b. Conger and Nolibos define U1 + U2 + U3 = 100%, where:
• U1 is the % of ultimate ULAE spent opening claims
• U2 is the % of ultimate ULAE spent maintaining claims
• U3 is the % of ultimate ULAE spent closing claims
See chapter 17
Copyright  2014 by All 10, Inc.
Page 73

Solutions to Exam 5B - Independently Authored Preparatory Test 4
Question 8 discussion: Blooms: Comprehension; Difficulty 2, LO 7, KS Estimation of unpaid ULAE
a. W* is the selected ultimate ULAE-to-claims ratio; L is the independently estimated ultimate claims for the same
group of AYs; M is the total amount spent on ULAE during a time period T, defined as M = (U1 x R x W) + (U2 x P x
W) + (U3 x C x W), where:
• R is the ultimate cost of claims reported during T
• P is the claims paid during T
• C is the ultimate cost of claims closed during T
• W is the ratio of ultimate ULAE to ultimate claims (L)
• U1, U2, and U1 are the %s of ultimate ULAE spent opening, maintaining and closing claims respectively
and B is the claims basis for the time period T is computed as B = (U1 x R) + (U2 x P) + (U3 x C)
b1. Method 1 Practical and Conceptual Problems
 Practically, it may be difficult to quantify the historical paid ULAE that corresponds only to the AYs claims
represented by L.
 Conceptually, this shares the potential distortions of an expected claims ratio approach to estimating
unpaid claims (unpaid claims equal a predetermined expected claims ratio time earned premium less
claims paid to date). The unpaid claim estimate is distorted if actual paid claims do not approach
expected ultimate claims.
b2. Method 2 Practical Problems and Concerns:
 The practical difficulty of establishing the ULAE amounts paid that correspond to accidents occurring
during a particular period
 This method may be overly responsive to random fluctuations in ULAE emergence.
See chapter 17
Question 9 discussion: Blooms: Comprehension; Difficulty 2, LO 7, KS Estimation of unpaid ULAE
a. Assume that:
R(t) – ultimate cost of claims known at time t
P(t) – total amount paid at time t
C(t) – ultimate cost of claims closed at time t
Compute unpaid ULAE = W* x {U1 x [L – R(t)] + U2 x [L – P(t)] + U3 x [L – C(t)]},
Each component of the unpaid ULAE formula represents a provision for the ULAE associated with:
• Opening claims not yet reported
• Making payments on currently active claims and on those claims that will be reported in the future
• Closing “unclosed” claims (i.e. those claims that are open at time t and those claims that will be reported
and opened in the future)
Rearranging the equation, one obtains: Unpaid ULAE = W* x (L - B)
b. This method assumes that the amount of ULAE paid to date and the unpaid ULAE are not directly related,
except to the extent that these payments influence the selection of the ratio W*.
This is similar to the assumption underlying the BF technique in that an a priori provision of unpaid ULAE is
calculated.
See chapter 17

Solutions to Exam 5B - Independently Authored Preparatory Test 4
Question 10 discussion: Blooms: Comprehension; Difficulty 2, LO 7, KS Strengths and
weaknesses of the estimation techniques for claim related expenses
Practical Difficulties with the Generalized Approach
 The estimation of R and C, the ultimate cost of reported and closed claims, is not simple.
 It is not known about the relative accuracy of the generalized method (as compared to other dollar-based
methods) in an inflationary environment.
 The effect of reopened claims on the accuracy of the estimates of unpaid ULAE is not known.
 How to modify the approach to properly reflect the change over time in the quantity or cost of resources
dedicated to the handling of a claim, as that claim ages is not known.
See chapter 17
Question 11 discussion: Blooms: Comprehension; Difficulty 1, LO 7, KS Organization of the data

Accident
Year
(1)
2005
2006
2007
2008

Direct
Earned
Premium
(2)
2,866,667
2,833,333
2,946,667
2,656,667

Total

11,303,333

Expected
Claims
Ratio
(3)
60%
60%
60%
60%

Expected
Claims
(4)
1,720,000
1,700,000
1,768,000
1,594,000

Expected Payment Percentage in Calendar
2005
2006
2007
2008
(5)
(6)
(7)
(8)
12%
15%
15%
15%
12%
15%
15%
12%
15%
12%

6,782,000

Column Notes:
(4) = [(2) x (3)].
(9) = [(4) x (5)].
(10) = [(4) x (6)].
(11) = [(4) x (7)].
(12) = [(4) x (8)].

See chapter 17

Expected Claims Paid in Calendar Year
2005
(9)
206,400

2006
(10)
258,000
204,000

2007
(11)
258,000
255,000
212,160

2008
(12)
258,000
255,000
265,200
191,280

206,400

462,000

725,160

969,480

Solutions to Exam 5B - Independently Authored Preparatory Test 4
Question 12 discussion: Blooms: Comprehension; Difficulty 1, LO 7, KS Estimation of unpaid ULAE
Calendar

Paid

Paid Claims

Year
(1)
2005
2006
2007
2008
Total

ULAE
(2)
36,667
41,667
46,667
53,333
178,333

Actual
Expected
(3)
(4)
835,633
206,400
57,333
462,000
273,767
725,160
206,400
969,480
1,373,133 2,363,040

ULAE Ratio
Paid ULAE-to-Paid Claims
Actual
Expected
(5)
(6)
0.044
0.178
0.727
0.090
0.170
0.064
0.258
0.055
0.130
0.075

0.075
213,750
6,108,500
79,700
469,060
241,570

(5) = [(2) / (3)].
(6) = [(2) / (4)].
(7) = is based on (6) total
(10) Estimated assuming pure IBNR = 5% * 1,594,000 (5% of AY 2008 expected claims.)
(11) = {[(7) x 50% x (8)] + [(7) x 100% x (9)]}.
(12) = {[(7) x 50% x ((8) + (9) - (10))] + [(7) x 100% x (10)]}.
See chapter 17
Question 13 discussion: Blooms: Comprehension; Difficulty 1, LO 7, KS Estimation of unpaid ULAE

Calendar
Year
2005
2006
2007
2008

Paid
ULAE
7,274
10,233
11,172
12,993

Paid
Claims
32,632
49,552
73,163
89,646

Reported
Claims
72,693
109,371
123,310
139,082

Average of
Paid and Rptd
Claims
52,662
79,462
98,236
114,364

41,672

244,993

444,456

344,724

ULAE RatioPaid ULAE to
Paid
Avg Paid &
Claims
Rptd Claims
Traditional
Kittel
0.223
0.138
0.207
0.129
0.153
0.114
0.145
0.114
0.170
0.121
0.170
248,311
96,775
37,579

(8) Selected ULAE Ratio
(9) Case Outstanding at 12/31/08
(10) IBNR at 12/31/08
(11) Estimated Unpaid ULAE at 12/31/08
(6) = [(2) / (3)].
(7) = [(2) / (5)].
(11) = {[(8) x 50% x (9)] + [(8) x 100% x (10)]}.

See chapter 17

0.121
248,311
96,775
26,707

Solutions to Exam 5B - Independently Authored Preparatory Test 4
Question 14 discussion: Blooms: Comprehension; Difficulty 3, LO 7, KS Estimation of unpaid ULAE
Calendar
Year
(1)
2005
2006
2007
2008
Total

Paid
ULAE
(2)
7,274
10,233
11,172
12,993
41,672

Ult on Claims
Reported in
Calendar Year
(3)
90,865
131,155
138,635
149,940
510,595

Paid
Claims
(4)
32,632
49,552
73,163
89,646
244,993

Claims
Basis
(5)
67,572
98,514
112,446
125,822
404,354

ULAE
Ratio
(6)
0.108
0.104
0.099
0.103
0.103
0.100
514,760

(7) Selected ULAE Ratio
(8) Ultimate Claims
(9) Indicated Unpaid ULAE Using:
(a) Expected Claim Method
(b) Bornhuetter-Ferguson Method
(c) Development Method
Column and Line Notes:
(5) = {[(3) x 60%] + [(4) x 40%]}.
(6) = [(2) / (5)].
(7) Selected based on ULAE ratios in (6).
(9a) = {[(7) x (8)] - (Total in (2))}.
(9b) = {(7) x [(8) - (Total in (5))]}.
(9c) = {{[(8) / (Total in (5))] - 1.00} x (Total in (2))}.

9,804
11,041
11,378

See chapter 17

Solutions to Exam 5B - Independently Authored Preparatory Test 4
Question 15 discussion: Blooms: Comprehension; Difficulty 3, LO 7, KS Estimation of unpaid ULAE
Calendar
Year
(1)
2005
2006
2007
2008
Total

Paid
ULAE
(2)
7,274
10,233
11,172
12,993
41,672

Acc Year
Ultimate
Claims
(3)
90,865
131,155
138,635
149,940
510,595

Paid
Claims
(4)
32,632
49,552
73,163
89,646
244,993

Claims
Basis
(5)
67,572
98,514
112,446
125,822
404,354

ULAE
Ratio
(6)=(2)/(5)
0.1077
0.1039
0.0994
0.1033
0.1031
0.100
514,760
5,998
8,996
11,151
11,330

Column and Line Notes:
(5) = {[(3) x 60%] + [(4) x 40%]}.
(6) = [(2) / (5)].
(7) Selected based on ULAE ratios in (6).
(9a) = [4% x (accident year 2008 ultimate claims in (3))].
(9b) = [6% x (accident year 2008 ultimate claims in (3))].
(10a) = (7) x {[60% x (9a)] + {40% x [(8) - (Total in (4))]}}.
(10b) = (7) x {[60% x (9b)] + {40% x [(8) - (Total in (4))]}}.

See chapter 17

Solutions to Exam 5B - Independently Authored Preparatory Test 4
Question 16 discussion Blooms: Comprehension; Difficulty 3, LO 3, KS Mechanics associated with each
technique (including organization of the data)
Compute:
a. (1.0 point). The geometric average for the latest four years at 12-24 months = (1.173 x 1.160 x 1.159 x
.25
1.162) = 1.163.
b. (1.0 point). Percent reported at 12 months, assuming selected cumulative loss development factors to
ultimate are based on simple averages of the latest three years.

Selected
CDF to Ultimate
Percent Reported

12-24
1.164
1.274
78.5%

24 - 36
1.056
1.095
91.4%

Development Factor Selection
36 - 48
48 - 60
1.027
1.010
1.037
1.010
96.5%
99.0%

To Ult
1.000
1.000
100.0%

c. (1.0 point). Projected ultimate claims using the cumulative loss development factors computed in b. for
accident years 2003 – 2007.
d. (1.0 point). IBNR for accident years 2003 – 2007.
Accident
Year
(1)
2003
2004
2005
2006
2007
Total

Age of
Year
(2)
60
48
36
24
12

Claims at 12/31/07 CDF to Ultimate
Reported
Reported
(3)
(4)
57,565,344
1.000
56,976,657
1.010
56,786,410
1.037
54,641,339
1.095
1.274
48,853,563
274,823,313

Projected Ultimate Claims
Using Dev. Method with
Reported
(5)=(3)*(4)
57,565,344
57,526,216
58,867,822
59,809,264
62,251,442
296,020,088

See chapter 7

IBNR
(6)=(5)-(3)
0
549,559
2,081,412
5,167,925
13,397,879
21,196,775

Solutions to Exam 5B - Independently Authored Preparatory Test 4
Question 17– Chapter 7, 9 and 15 discussion Blooms: Comprehension; Difficulty 3,
a. (1 point) Using the Development method, calculate the indicated IBNR for accident year 2008 as of
December 31, 2008.

Earned
Premium

Accident
Year

38,000
40,000
42,000
44,000

2005
2006
2007
2008

ATA factors by AY:

Reported Claims by Development Age
at age
at age
at age
at age
12 mo
24 mo
36 mo
48 mo
9,700
19,400
28,200
32,400
10,300
20,600
29,800
10,800
21,600
14,400
AY

12:24 mo

24:36 mo

36:48 mo

2005
2006
2007

2.000
2.000
2.000

1.454
1.447

1.149

12: 24 mo
2.00
at 12 mo
3.33

24: 36 mo
1.45
at 24 mo
1.67

36:48 mo
1.15
at 36 mo
1.15

ATA: Simple Average (all yr)
Reported CDF to Ultimate

Accident
Year
2005
2006
2007
2008
Total

See tail factor

at 48 mo given
1.00 tail

Age of

Reported

Expected

IBNR

OR:

IBNR

Data at
12/31/04

Claims at
12/31/04

CDF to
Ultimate

Ultimate
Claims

(broadly
defined)

Shortcut

(broadly
defined)

(1)
48 months
36 months
24 months
12 months

(2)
32,400
29,800
21,600
14,400

(3) above
1.00
1.15
1.67
3.33

(4)=(2)*(3)
32,400
34,238
35,987
47,983

(5)=(4)-(2)
0
4,438
14,387
33,583
52,409

(5)=(2)*[(3) - 1.0]
0
4,438
14,387
33,583
52,409

Note: Only the calculations for Accident Year 2008 are required:
14,400 * (3.33 - 1) =

33,583

b. (0.5 point) Using the Bornhuetter-Ferguson method, calculate the indicated IBNR for accident year 2008
as of December 31, 2008.
Selected ATA factors
(1)
1.00
Tail at 48 months
36 - 48 months
1.15
24 - 36 months
1.45
12 - 24 months
2.00

Reported Ultimate CDF
(2) = product of (1)
at 48 mo.
1.00
at 36 mo.
1.15

Exp. % Unreported
(3) = 1.0 - 1.0 / (2)
0.0%
13.0%

at 24 mo.
1.67
at 12 mo.
3.33
(3) The Percent Unreported = 1 minus inverse of Ultimate Reported CDF

Accident
Year
2005
2006
2007
2008
Total

Earned
Premium

A priori
Expected
Claim Ratio

A priori
Expected
Claims

(4) given
38,000
40,000
42,000
44,000

(5) given
90.0%
90.0%
90.0%
90.0%

(6)=(4)*(5)
34,200
36,000
37,800
39,600

"IBNR"
Expected
Unreport

40.0%
70.0%

Or shortcut using
Est. Expected Claims

Accident
Year
2005
2006
2007
2008

IBNR
(broadly
defined)

x Percent Unreported
(7)=(6)*(3) =(Premium)*(Exp Claims %)*[1-1/CDF]
0
0
4,667
4,667
15,112
15,112
27,716
27,716
47,494
47,494

Note: Only the calculations for Accident Year 2008 are required:
22,000 * 90% * (1 - 1/3.33) = 19,800 * 70% =

27,716

Solutions to Exam 5B - Independently Authored Preparatory Test 4
Question 17 – Chapter 7, 9 and 15 discussion Blooms: Comprehension; Difficulty 1, LO 3, KS
c. (1 point) Using the Bornhuetter-Ferguson method, calculate the expected IBNR for Accident Year 2008
expected to be reported (emerge) during calendar year 2009.
Selected ATA factors
(1)
1.00
Tail at 48 months
36 - 48 months
1.15
24 - 36 months
1.45
2.00
12 - 24 months

Reported Ultimate CDF
(2) = product of (1)
1.00
at 48 mo.
at 36 mo.
1.15
1.67
at 24 mo.
3.33
at 12 mo.

Exp. % Unreported
(3) = 1.0 - 1.0 / (2)
0.0%
13.0%

Accident
Year
2005
2006
2007
2008

40.0%
70.0%

(3) The Percent Unreported = 1 minus inverse of Ultimate Reported CDF

Accident
Year
2005
2006
2007
2008
Total

A priori
Earned
Expected
Premium Claim Ratio
(4) given
38,000
40,000
42,000
44,000

(5) given
90.0%
90.0%
90.0%
90.0%

A priori
Expected
Claims

"IBNR"
Expected
Unreport

(6)=(4)*(5)
34,200
36,000
37,800
39,600

(7)=(6)*(3)
0
4,667
15,112
27,716
47,494

Or shortcut using
Est. Expected Claims

IBNR
(broadly
defined)

x Percent Unreported
(7)=(3)*(4)*[1.0-1.0/CDF]
0
4,667
15,112
27,716
47,494

Note: Only the calculations for Accident Year 2008 are required:
44,000 * 90% * (1 - 1/3.33) = 19,800 * 70% =
c) Details shown for completeness
Ages in
Percent
Acc Estimated NEXT yr
Reported
Year
IBNR (CY 2009)
at 1/1/09
2005
2006
2007
2008
Total

(7)
0
4,667
15,112
27,716

(8) FYI
48 to 60
36 to 48
24 to 36
12 to 24

(9) See (3)
100.00%
87.04%
60.02%
30.01%

Percent
Reported
at 12/31/09
(10) see(3)
100.00%
100.00%
87.04%
60.02%

27,716

Percent Percent of current IBNR
expected to emerge
at 1/1/09 between 1/1 - 12/31/09

Est. IBNR
to emerge
during '09

Unreported
(11)= 1-(9)

0.00%
12.96%
39.98%
69.99%

(12) = [(10)-(9)] / (11)
n/a
100.00%
67.58%
42.88%

(13)=(7)*(12)

0
4,667
10,212
11,884
26,763

* Here the period we evaluate for emerged losses is the year immediately following our original estimate
But, for exam purposes:
Solution to c:

Only the calculations for Accident Year 2008 are required:
27,716 * (60.02% - 30.01%) / (1 - 30.01%) =

11,884

OR since B-F, can also use shortcut with A-prior Expected Claims for AY 2008:
39,600 * (60.02% - 30.01%) =

11,884

Solutions to Exam 5B - Independently Authored Preparatory Test 4
Question 18 discussion Blooms: Comprehension; Difficulty 3, LO 3, KS Mechanics associated
with each technique (including organization of the data)
a. (1.0 point). Compute initial selected ultimate claims as the average of the reported and paid claim
development projections.
Accident
Year
(1)
2004
2005
2006
2007
2008

Claims at
Reported
(2)
16,500,000
18,500,000
16,500,000
14,000,000
8,700,000

Projected Ultiamate
Initial Selected
12/31/098
CDF to Ultimate
Claims Based On
Ultimate
Paid
Reported Paid
Reported
Paid
Claims
(3)
(4)
(5)
(6) = [(2) x (4)] (7) = [(3) x(5)] (8) =[(6)+(7)]/2
11,200,000 1.200
1.750
19,800,000
19,600,000
19,700,000
10,200,000 1.400
2.500
25,900,000
25,500,000
25,700,000
6,000,000
1.800
5.000
29,700,000
30,000,000
29,850,000
3,000,000
2.900
15.000
40,600,000
45,000,000
42,800,000
750,000
4.000
90.000
34,800,000
67,500,000
51,150,000

b. (1.0 point). Compute the selected claim ratio for AY 2008 as the average of trended adjusted claim ratios for
2004 – 2008, excluding high and low ratios.
c. (1.0 point). Compute estimated IBNR for AY 2008

Initial Selected
On-Level
Trend at Adjusted
Trended
Ultimate
Earned
14.50%
for Tort
Adj. Ultimate
Claims
Premium
to 7/1/08 Reform
Claims
(12) = [(8) x(10) x(11)]
(8) =[(6)+(7)]/2
(9)
(10)
(11)
19,700,000
32,000,000
1.719
0.750
25,398,225
25,700,000
47,000,000
1.501
1.000
38,575,700
29,850,000
50,000,000
1.311
1.000
39,133,350
42,800,000
57,000,000
1.145
1.000
49,006,000
51,150,000
62,000,000
1.000
1.000
51,150,000
(14) Average Claim Ratio at 7/1/2008 Cost Level
Average 2004 to 2008 Excluding High and Low
80.9%
(15) Selected Claim Ratio at 7/1/2008 Cost Level
80.9%
(16) Expected Claims for 2008 Accident Year
50,158,000
(17) Unpaid Claim Estimate for 2008 Accident Year
Total
49,408,000
IBNR
41,458,000

See Chapter 8

Trended
Adjusted
Claim Ratio
(13) = [(12) / (9))]

79.4%
82.1%
78.3%
86.0%
82.5%

(16) = [ (15) x (9) for 2008].
(17) tot = [ (16) - (3) for 2008].
(17) IBNR = [ (16) - (2) for 2008].

Solutions to Exam 5B - Independently Authored Preparatory Test 4
Question 19 discussion Blooms: Comprehension; Difficulty 3, LO 3, KS Mechanics associated
with each technique (including organization of the data)
a. (2.0 points). Compute estimated IBNR for AYs 2004 - 2008
b. (1.0 points). Compute estimated total unpaid claims for AYs 2004 - 2008
Accident
Year
(1)
2004
2005
2006
2007
2008
Total

Claims at
Reported
(2)
70,288
70,655
48,804
31,732
18,632
240,111

12/31/08
Paid
(3)
52,811
40,026
22,819
11,865
3,409
130,930

Case
Earned Claim Ratio Expected Outstanding
Premium Selected
Claims at 12/31/08
(4)
(5)
(6)=(4)*(5) (7)=[(2)-(3)]
99,322
87.1%
86,509
17,477
138,151
78.3%
108,172
30,629
107,578
65.8%
70,786
25,985
62,438
63.8%
39,835
19,867
47,797
82.5%
39,433
15,223
344,736
109,181

Unpaid Claim Estimate Based
on Expected Claims Method
IBNR
Total
(8) = [(6)-(2))
(9) = [(6) - (3)]
16,221
33,698
37,517
68,146
21,982
47,967
8,103
27,970
20,801
36,024
104,625
213,806

See Chapter 8

Question 20 discussion Blooms: Comprehension; Difficulty 3, LO 3, KS Mechanics associated with each
technique (including organization of the data)
a. (2.0 points). Compute projected ultimate claims using the B-F method using reported claims for AYs 2003 2007
b. (1.0 points). Compute projected ultimate claims using the B-F method using paid claims for AY 2007
CDF to Ultimate
Percentage
Expected Claims
Accident Expected
Year
Claims
Reported
Paid
Unreported Unpaid Unreported
Unpaid
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8 )
2003
56,318,302
1.011
1.040
1.09%
3.85%
613,869
2,168,255
2004
59,646,290
1.023
1.085
2.25%
7.83%
1,342,042
4,670,305
2005
61,174,953
1.051
1.184
4.85%
15.54%
2,966,985
9,506,588
2006
61,926,981
1.110
1.404
9.91%
28.77%
6,136,964
17,816,393
2007
61,864,556
1.292
2.390
22.60%
58.16% 13,981,390
35,980,426
Total
300,931,082
25,041,250
70,141,965
Column Notes:
(5) =[1.00 - (1.00 / (3))].
(6) =[1.00 - (1.00 / (4))].
(7) = [(2) x(5)]
(8) =[(2) x (6)]
(11) = [(7) + (9]
(12)= [(8)+ (10)].

See Chapter 9

Claims at 12/31/07
Reported
Paid
(9)
(10)
57,565,344
55,930,654
56,976,657
53,774,672
56,786,410
50,644,994
54,641,339
43,606,497
48,853,563
27,229,969
274,823,313
231,186,786

Projected Ultimate Claims
using B-F Method with
Reported
Paid
(11)
(12)
58,179,213
58,098,909
58,318,699
58,444,977
59,753,395
60,151,582
60,778,303
61,422,890
62,834,953
63,210,395

Solutions to Exam 5B - Independently Authored Preparatory Test 4
Question 21 discussion Blooms: Comprehension; Difficulty 3, LO 3, KS Mechanics associated with each
technique (including organization of the data)
Question 21 below is restated for convenience purposes only
(3.0 points) Using the procedure described by Friedland in ““Estimating Unpaid Claims Using Basic, and the data
given below, answer the following questions.
Age of
Expected ultimate Claims
Accident Accident Year Using B-F Method with
Claims at 12/31/08
CDF to Ultimate
Year
at 12/31/08
Reported
Paid
Reported
Paid
Reported
Paid
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8 )
Steady-State
2004
60
893,397
893,397
884,463
857,661
1.000
1.000
2005
48
938,068
938,067
919,306
863,022
1.010
1.043
2006
36
984,970
984,970
935,722
827,375
1.042
1.143
2007
24
1,034,219
1,034,218 930,797
734,295
1.100
1.352
456,090
1.286
2.286
1,085,929 836,166
2008
12
1,085,930
Total
4,936,584
4,936,581 4,506,454 3,738,443

a. (2.0 points). Compute projected ultimate claims using the Gunnar Benktander Method using reported claims
for AYs 2004 - 2008
b. (1.0 points). Compute estimated IBNR using the Gunnar Benktander Method using reported claims for AYs
2004 - 2008
Solution to question 21
Expected ultimate Claims
Age of
Claims at 12/31/08
CDF to Ultimate
Accident Accident Year Using B-F Method with
Year
at 12/31/08
Reported
Paid
Reported
Paid
Reported
Paid
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8 )
Steady-State
2004
60
893,397
893,397
884,463
857,661
1.000
1.000
2005
48
938,068
938,067
919,306
863,022
1.010
1.043
2006
36
984,970
984,970
935,722
827,375
1.042
1.143
2007
24
1,034,219
1,034,218 930,797
734,295
1.100
1.352
1,085,929 836,166
456,090
1.286
2.286
2008
12
1,085,930
Total
4,936,584
4,936,581 4,506,454 3,738,443

Expected Percentage
Unreported Unpaid
(9)
(10)
0.0%
1.0%
4.0%
9.1%
22.2%

(9) = [1.00 - (1.00 / (7))].
(10) = [1.00 - (1.00 / (8))]
(11) = [((3) x (9)) + (5)]
(12) = [((4) x (10)) + (6)]
(13) = [ (11) - (5)].
(14) = [(12) - (5)].
See Chapter 9

0.0%
4.2%
12.5%
26.0%
56.2%

Projected Ultimate Claims
Estimated IBNR
Using G-B Method with Using G-B Method with
Reported
Paid
Reported
Paid
(11)
(12)
(13)
(14)
884,463
928,782
975,519
1,024,817
1,077,484
4,891,064

857,661
902,108
950,496
1,003,622
1,066,925
4,780,812

0
9,476
39,797
94,020
241,318
384,610

-26,802
-17,198
14,774
72,825
230,759
274,358

Solutions to Exam 5B - Independently Authored Preparatory Test 4
Question 22 discussion Blooms: Comprehension; Difficulty 3, LO 3, KS Mechanics associated with each
technique (including organization of the data)
a. (1.5 points). Compute estimated claim ratios using the Cape Cod Method for AYs 2003 – 2007.
Accident
Year
(1)
2003
2004
2005
2006
2007
Total

Earned
Premium
(2)
86,643,542
91,763,523
94,115,312
95,272,279
95,176,240
462,970,896

Age of
Accident
at 12/31/07
(3)
60
48
36
24
12

Reported
Year Claims at
12/31/2007
(4)
57,565,344
56,976,657
56,786,410
54,641,339
48,853,563
274,823,313

Reported
CDF to
Ultimate
(5)
1.000
1.010
1.037
1.095
1.274

% of
Ultimate
Used Up
Reported
Premium
(6) = [1.00 / (5)] (7) = [(2) x (6)]
100.0%
86,643,542
99.0%
90,886,888
96.5%
90,787,641
91.4%
87,040,109
78.5%
74,692,221
430,050,401

Estimated
Claim
Ratios
(8) = [(4) / (7)]
66.4%
62.7%
62.5%
62.8%
65.4%
63.9%

b. (1.5 points). Compute projected ultimate claims using the Cape Code Method for AYs 2003 – 2007.

Accident
Year
(1)
2003
2004
2005
2006
2007
Total

Earned
Premium
(2)
86,643,542
91,763,523
94,115,312
95,272,279
95,176,240
462,970,896

Expected
Claim
Ratio
(3)
63.9%
63.9%
63.9%
63.9%
63.9%

Estimated
Reported
Expected
Expected
CDF
Percentage
Unreported
Claims
Ultimate
Unreported
Claims
(4)
(4) = [(2) x (3)] (6)=1.0- (1.0/(5)) (7) = [(4) x (6)]
55,369,476
1.000
0.0%
0
58,641,395
1.010
1.0%
560,213
60,144,303
1.037
3.5%
2,126,545
60,883,662
1.095
8.6%
5,260,761
60,822,289
1.274
21.5%
13,090,293
295,861,125
21,037,812

See Chapter 10

Reported
Promected
Claims at
Ultimate
12/31/2007
Claims
(8)
(9) = [(7) + (8)]
57,565,344
57,565,344
56,976,657
57,536,870
56,786,410
58,912,955
54,641,339
59,902,100
48,853,563
61,943,856
274,823,313
295,861,125

Solutions to Exam 5B - Independently Authored Preparatory Test 4
Question 23 discussion Blooms: Comprehension; Difficulty 3, LO 3, KS Mechanics associated with each
technique (including organization of the data)
a. (2.50 points). Compute used-up on-level premium using the Cape Cod Method for AYs 2004 – 2008.
b. (1.50 points). Compute estimated unadjusted claim ratios using the Cape Code Method for AYs 2004 – 2008.
Accident
Year
(1)
2004
2005
2006
2007
2008
Total

Reported
CDF to
Ultimate
(10)
1.064
1.085
1.196
1.512
2.551

On-Level
Age of
Reported
Pure
Earned
On-Level
Earned Accident Year Claims Premium
Premium Adjustment Premium at 12/31/08 at 12/31/08 Trend
(2)
(3)
(4)
(5)
(6)
(7)
99,322
0.810
80,451
60
70,288
1.144
138,151
0.704
97,258
48
70,655
1.106
107,578
0.640
68,850
36
48,804
1.070
62,438
0.800
49,950
24
31,732
1.034
1.000
47,797
12
18,632
1.000
47,797
455,286
344,306
240,111

% of
Ultimate
Reported
(11)
94.0%
92.2%
83.6%
66.1%
39.2%

Used Up
On-Level
Premium
(12)
75,640
89,650
57,590
33,029
18,734
274,643

Estimated
Adjusted
(13)
71.2%
58.4%
68.0%
99.4%
99.5%
71.7%

Claim Ratios
Selected
Adjusted
(14)
71.7%
71.7%
71.7%
71.7%
71.7%

Tort
Reform
Factors
(8 )
0.670
0.670
0.750
1.000
1.000

Adjusted
Claims
at 12/31/08
(9)
53,884
52,371
39,153
32,819
18,632
196,859

Estimated
Unadjusted
(15)
75.7%
68.1%
57.2%
55.4%
71.7%

(4) = [(2) x (3)].
(9) = [(6) x (7) x (8)].
(11)=[1.00 /(10)].
(12) = [(4) x (11)].
(13) = [(9) / (12)].
(14) = [Total in (13)].
(15) = [(14) x (3) / (7) / (8)].
We use the label "Estimated Adjusted Claim Ratios" to indicate that the reported claims are adjusted for
inflation and tort reform. We rely on the claim ratio for all years combined, 71.7%, from Column (13) (also
shown in Column (14) for each year) as our starting point for developing estimated unadjusted claim
ratios in Column (15). These claim ratios, which are adjusted back to the rate level, inflationary level, and
tort environment for each accident year, become our starting point for projecting expected claims.
See Chapter 10

Exam 5B – Independently Authored Preparatory Test 5
General information about this exam
1. This test contains 20 computational and essay questions.
2. The recommend time for this exam is 2:30:00. Make sure you have sufficient time to take this practice test.
3. Consider taking this exam after working all past CAS questions, associated with the articles below, first.
4. Many of the essay questions may require lengthy responses.
5. Make sure you have a sufficient number of blank sheets of paper to record your answers.

Articles covered on exam:
Article .................................................... Author .................................. Syllabus Section
Chapter 9 – Bornhuetter-Ferguson Technique ................ Friedland ..............B: Estimating Claim Liabilities
Chapter 11 – Frequency-Severity Techniques ................. Friedland ..............B: Estimating Claim Liabilities
Chapter 12 – Case Outstanding Development TechniqueFriedland ..............B: Estimating Claim Liabilities
Chapter 13 – Berquist-Sherman Techniques.................... Friedland ..............B: Estimating Claim Liabilities
Chapter 14 – Recoveries: Salvage & Subro and Reins .... Friedland ..............B: Estimating Claim Liabilities
Chapter 15 – Evaluation of Techniques ............................ Friedland ..............B: Estimating Claim Liabilities
Chapter 16 – Estimating Unpaid Claim Adj Expenses ...... Friedland ..............B: Estimating Claim Liabilities
ASOP No. 43 – Unpaid Claim Estimates .......................... AAA ......................B: Estimating Claim Liabilities

Exam 5B - Independently Authored Preparatory Test 5
Question 1
(1.0 point) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, list two advantages
to using the ratio method compared to the development method when developing projected ultimate claims for
Salvage and Subrogation.

Question 2
(1.0 point) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, one of the areas of
reasonableness in the net and gross or ceded and gross analyses is to ensuring net IBNR in each AY is
generally not greater than gross IBNR.
Describe two times when the net IBNR will be greater than the gross IBNR.
Question 3
(2 points) You are given the following information:

Accident
Year
2004
2005
2006

Selected Ultimate
Loss
$620,000
580,000
600,000

Age in
Months
12
24
36
48
60

Incurred
Development to
Ultimate
2.20
1.65
1.35
1.10
1.05

What is the total amount of incurred loss that is expected to emerge during calendar year 2007 on accident
years 2004 through 2006?

Question 4
(1.5 points) Based on the following information:
 Accident Year 2003 earned premium = $2,000
 Accident Year 2003 expected loss ratio= 75%

Loss Development Factors
12-24
2.00
24-36
1.25
36-48
1.10
48-60
1.00
Based on the Bornhuetter-Ferguson expected loss method, what is the expected loss emergence in
calendar year 2006 for accident year 2003 losses?

Exam 5B - Independently Authored Preparatory Test 5
Question 5
(1.5 point) According to Friedland in “Estimating Unpaid Claims Using Basic Techniques”, briefly describe
three acceptable ways to select ultimate claims.

Question 6
(1.0 point) According to Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, Wiser's final
phase to his four-phase approach to estimating unpaid claims is to monitor projections of the
development of unpaid claims over subsequent calendar periods. Briefly describe what actions the
actuary takes in this phase.

Question 7
(2.0 points) Based on Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, you are given the
following information:

Accident
Year
(1)
2005
2006
2007
Total

Selected
Ultimate
Claims
(2)
2814
2,952
2,798
28,913

Expected % Reported at
12/31/2007
12/31/2008
(3)
(4)
100.0%
100.0%
99.9%
100.0%
88.0%
99.9%

Reported Claims at
12/31/2007
(5)
2814
2,949
2,463
28,575

12/31/2008
(6)
2885
3,030
2,733
28,983

Using selected ultimate claims and the selected reporting pattern above to compare actual reported
claims one year later (i.e. 12/31/2008) with our expected claims for the year, compute the following:
a. (1 point). Expected reported claims for accident year 2007 during calendar year 2008
b. (1 point). Expected reported claims for accident year 2006 during calendar year 2008

Exam 5B - Independently Authored Preparatory Test 5
Question 8
(4.0 points). Based on Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, you are given
the following information based on U.S. PP Auto Increasing Claims Ratio example:

Accident
Year
2003
2004
2005
2006
2007
2008

1st
Report
982,737
1,179,284
1,315,640
1,462,682
1,621,139
1,791,785

Accident
Year
2003
2004
2005
2006
2007
2008

Paid Claims
through
12/31/2008
1,250,756
1,470,276
1,571,933
1,595,652
1,494,816
977,337




Reported Claims including ALAE ($000's omitted)
2nd
3rd
4th
5th
Report
Report
Report
Report
1,148,654
1,212,468
1,250,756
1,263,519
1,378,385
1,454,961
1,500,908
1,516,223
1,537,760
1,623,191
1,674,450
1,709,627
1,804,607
1,894,836

6th
Report
1,263,519

Earned
Premium
1,823,259
1,914,423
2,010,144
2,110,650
2,216,183
2,326,992

Select 3-yr volume weighted ATA factors.
Assume reported CDF-to-Ultimate of 1.01 at 72 Months, and 1.0 at 84 months.

This insurer operates in an environment of:
Increasing claim ratios: 70% in AY 2003, 80% in 2004, 85% in 2005, 90% in 2006, 95% in 2007, 100% in AY 2008
No change in Case Outstanding adequacy levels.
a. Using the Reported Development method, estimate Ultimate Claims.
b. Using the Reported Development method, estimate IBNR (broadly defined).
c. Using the Reported Development method, estimate total unpaid claims, including Case O/S.
d. Based on your answers above, what is the expected amount of IBNR to be reported in Calendar
Year 2009, on claims for Accidents Years 2008 and prior?
e. Given the operating environment of this insurer, does this technique seem appropriate?

Exam 5B - Independently Authored Preparatory Test 5
Question 9
(4.0 point) Based on Friedland in ““Estimating Unpaid Claims Using Basic Techniques”, you are given the
following data extracted from the Auto Property Damage example in Chapter 16: Allocated Claim Adjustment
Expenses (ALAE):
Accident
Year
2003
2004
2005
2006
2007
2008

1st
Report
1,144
1,114
1,126
1,272
1,548
1,904

Accident
Year
2003
2004
2005
2006
2007
2008

1st
Report
1,514
1,486
1,578
1,976
2,746
3,112

Accident
Year
2003
2004
2005
2006
2007
2008

1st
Report
223,310
212,064
196,540
214,274
228,674
248,940

Paid ALAE ($000's omitted)
2nd
3rd
Report
Report
1,311
1,461
1,260
1,382
1,323
1,412
1,596
1,698
2,181

2nd
Report
1,422
1,373
1,422
1,710
2,394

4th
Report
1,562
1,440
1,481

Reported ALAE ($000's omitted)
3rd
4th
Report
Report
1,553
1,638
1,464
1,502
1,502
1,548
1,797

5th
Report
1,604
1,484

6th
Report
1,628

5th
Report
1,643
1,548

6th
Report
1,715

Paid Claims only ($000's omitted) i.e. excluding
2nd
3rd
4th
5th
Report
Report
Report
Report
237,138
242,735
243,834
244,203
223,736
227,594
228,344
228,920
224,256
229,343
229,731
256,998
263,403
257,258

6th
Report
244,886

Selected Ult.
Claims Only
by AY
326,602
305,592
306,362
354,242
353,120
390,970

For parts a - c, use 3-year volume-weighted averages in selecting age-to-age factors.
Assume a reported CDF-to-Ultimate tail of 1.01 at 72 Months.
Assume a paid CDF-to-Ultimate tail of 1.035 at 72 Months.
9a.
9b.
9d.
9c.
9e.

Using the Reported Development method, estimate Ultimate ALAE.
Using the Reported Development method, estimate unpaid ALAE.
Using the Paid Development method, estimate unpaid ALAE.
Using the Paid Development method, estimate unpaid ALAE.
Using the Ratio method, estimate Ultimate ALAE
For selections, use a 3-year simple average of ratios and assume the CDF to ultimate is 1.05 at 72
months.
9f. Using the Additive method, estimate Ultimate A
For selections, use a 3-year simple average of ratios and assume the additive CDF to ultimate for this
ratio is .06% at 72 months.

Question 10
(1.5 point) Based on Friedland in “Estimating Unpaid Claims Using Basic Techniques”, list and briefly describe
three advantages and two disadvantages to using the Ratio Method to develop ALAE.

Exam 5B - Independently Authored Preparatory Test 5
Question 11
There is no question 11
Question 12
(1.5 points). According to ASOP 43, “P&C Unpaid Claim Estimates”, when considering the scope of the unpaid
claim estimate, the actuary should identify the following the intended measure of the unpaid claim estimate.
a. (.75 points) Identify several examples of measures for the unpaid claim estimate.
b. (.75 points) What does the standard say about using the terms “best estimate” and “actuarial estimate”
when describing the intended measure?

Question 13
(1.5 points). According to ASOP 43, “P&C Unpaid Claim Estimates”, the actuary should consider changes in
conditions with regard to claims, losses, or exposures, that are likely to be insufficiently reflected in the
experience data or in the assumptions used to estimate the unpaid claims.
a. (.75 points) Identify two types of changes that are likely to have a material effect on the results of the
actuary’s unpaid claim estimate analysis.
b. (.75 points) How should the actuary obtain supporting information to validate the presence of these
changes?
Question 14
(4.0 points) Using the procedure described by Friedland in ““Estimating Unpaid Claims Using Basic, and the
data given below, answer the following questions.
As the company actuary in the workers compensation unit, your goal is to determine the appropriate frequency
(i.e., number of claims per exposure unit) for the latest two accident years. Since payroll is an inflationsensitive exposure base, we must adjust the payroll for each accident year to a common time period. Assume
a 2.5% annual inflation rate for payroll for all years in the experience period and trend all historical payroll to
the cost level of accident year 2008 (1.0 level).
Similarly, the claim counts should be adjusted using trend factors to reflect changes in counts. Assume a -1.0%
annual trend in the number of claims and trend all counts to cost level of accident year 2008 (1.0 level).
Selected
Accident Ultimate
Payroll
Year Claim Cnts ($00)
2004
2005
2006
2007
2008
Total

1,734
1,637
2,966
2,888
2,651
11,875

280,000
350,000
790,000
780,000
740,000
2,940,000

Compute
a. The selected frequency at 2008 level
b. The selected frequency at 2007 level

Exam 5B - Independently Authored Preparatory Test 5
Question 15
(3.0 points) Using the procedure described by Friedland in ““Estimating Unpaid Claims Using Basic, and the
data given below, answer the following questions.
You have been asked to use a frequency-severity approach to project ultimate claims and the unpaid claim
estimate for the latest two accident years. The selected frequency below was obtained by a frequency analysis
comparing the ultimate claim counts by accident year to the traditional exposure base used for WC.
The following data is from a self-insurer of U.S. workers compensation.
Accident Year
2007
2008
780,000
740,000
0.37%
0.36%
5,674
6,100
14,400,000 10,300,000
5,357,000 6,130,000

Payroll ($00)
Selected Frequency
Selected Severity
Reported Claims at 12/31/08
Case Outstanding at 12/31/08

Compute the unpaid claim estimate at 12/31/08 for AY's 2007 and 2008.

Question 16
(1.5 points) Using the procedure described by Friedland in ““Estimating Unpaid Claims Using Basic, and the
data given below, answer the following questions.
Assume the only data available for a self-insurer of general liability coverage is case outstanding.
You have been asked to use the standard development technique with case outstanding to project an
estimate of total unpaid claims for a self-insured entity of general liability coverage.
You will use an industry-based reporting and payment development patterns to derive case outstanding
development patterns. You implicitly assume that claims recorded to date for the self-insurer will develop
in a similar manner in the future as our industry benchmark (i.e., the historical industry experience is
indicative of the future experience for the self-insurer).
Accident
Year

Case
Outstanding
at 12/31/08

1999
2000
2001
2002
2003
Total

650,000
800,000
850,000
975,000
1,000,000
4,275,000

CDF to Ultimate
Reported

Paid

1.020
1.030
1.051
1.077
1.131

1.067
1.109
1.187
1.306
1.489

Compute the unpaid claim estimate for AYs 1999 – 2003.

Exam 5B - Independently Authored Preparatory Test 5
Question 17
(1.5 points) Using the procedure described by Friedland in ““Estimating Unpaid Claims Using Basic, and the
data given below, answer the following questions.
Case Outstanding and Incremental Paid Claims ($000)
Accident
Year
2003
2004
2005
2006
2007

12
21,078,651
21,047,539
21,260,172
20,973,908
21,623,594

Accident
Year
2003
2004
2005
2006
2007

12
24,084,451
24,369,770
25,100,697
25,608,776
27,229,969

Case Outstanding as of (months)

24
11,098,119
11,150,459
11,087,832
11,034,842

36
6,398,219
6,316,995
6,141,416

48
3,431,210
3,201,985

60
1,634,690

Incremental paid Claims as of (months)

24
17,315,161
17,120,093
17,601,532
17,997,721

36
7,670,720
7,746,815
7,942,765

48
4,513,869
4,537,994

60
2,346,453

Assume the Selected Ratio of Incremental Paid Claims to Previous Case Outstanding factor
from 60-Ult = 1.0
Assume that the Selected Ratio of Case Outstanding to Previous Case Outstanding factor
from 60-Ult = 1.0
Compute: Cumulative Paid Claims at ultimate (i.e. 72 months) for AYs 2003 - 2007

Exam 5B - Independently Authored Preparatory Test 5
Question 18 – Chapter 13
(1.5 points) Using the procedure described by Friedland in ““Estimating Unpaid Claims Using Basic, and the
data given below, answer the following questions.
Accident
Year
1972
1973
1974
1975
1976
Accident
Year
1,972
1,973
1,974
1,975
1,976

Unadjusted Reported Claims

12
8,732,000
11,228,000
8,706,000
12,928,000
15,791,000

24
36
18,633,000 32,143,000
19,967,000 50,143,000
33,459,000 63,477,000
48,904,000

48
57,196,000
73,733,000

60
61,163,000

Unadjusted Paid Claims as of (months)

12
50,000
213,000
172,000
210,000
209,000

24
786,000
833,000
1,587,000
1,565,000

36
3,810,000
3,599,000
6,267,000

48
9,771,000
11,292,000

Open Claim Counts as of (months)
Accident
Year
12
24
36
1972
1,043
1,561
1,828
1973
1,088
1,388
1,540
1974
1,033
1,418
1,663
1975
1,138
1,472
1976
1,196
• Selected Annual Severity Trend Rate = 1.15

60
18,518,000

48
1,894
1,877

Using the Berquist-Sherman Technique, compute Adjusted Reported Claims triangle

60
1,522

Exam 5B - Independently Authored Preparatory Test 5
Question 19 – Chapter 13
(2.5 points) Using the procedure described by Friedland in ““Estimating Unpaid Claims Using Basic, and the
data given below, answer the following questions.
Accident
Year
1972
1973
1974
1975
1976

12
2,503
2,838
2,405
2,759
2,801

24
8,173
8,712
7,858
9,182

Paid Claims as of (months)
36
48
60
11,810
14,176
15,383
12,728
15,278
11,771

Accident Closed Claim Counts as of (months)
Year
12
24
36
48
1972
4,497
7,842
8,747
9,254
1973
4,419
7,665
8,659
9,093
1974
3,486
6,214
6,916
1975
3,516
6,226
1976
3,230
Accident Reported Claim Counts
Year
12
24
1972
7,858
9,474
1973
7,808
9,376
1974
6,278
7,614
1975
6,446
7,884
1976
6,115

as of (months)
36
48
9,615
9,664
9,513
9,562
7,741

60
9,469

60
9,680

Assume the selected reported claim counts factor from 60-Ult = 1.001.
Using the Berquist and Sherman disposal rate method, compute the adjusted closed claim counts triangle

Exam 5B - Independently Authored Preparatory Test 5
Question 20 – Chapter 13
(2.5 points) Using the procedure Berquist Sherman procedure described by Friedland in ““Estimating Unpaid
Claims Using Basic, and the data given below, answer the following questions.
Accident
Closed Claim Counts as of (months
Year
12
24
36
48
60
72
84
96
1969
4,079 6,616 7,192 7,494 7,670 7,749 7,792 7,806
1970
4,429 7,230 7,899 8,291 8,494 8,606 8,647
1971
4,914 8,174 9,068 9,518 9,761 9,855
1972
4,497 7,842 8,747 9,254 9,469
1973
4,419 7,665 8,659 9,093
1974
3,486 6,214 6,916
1975
3,516 6,226
1976
3,230

Accident
Year
1969
1970
1971
1972
1973
1974
1975
1976

Paid Claims
12
24
1,904 5,398
2,235 6,261
2,441 7,348
2,503 8,173
2,838 8,712
2,405 7,858
2,759 9,182
2,801

($000) as of (months)
36
48
60
7,496 8,882 9,712
8,691 10,443 11,346
10,662 12,655 13,748
11,810 14,176 15,383
12,728 15,278
11,771

Accident
72
84
96
10,071 10,199 10,256
11,754 12,031
14,235

Accident
Year
1969
1970
1971
1972
1973
1974
1975
1976

Adjusted Closed Claim Counts as of (months)
12
24
36
48
60
72
84
96
0
0
0
3,332 6,038 6,932 7,415 7,643
3,699 6,703 7,695 8,231 8,484
0
0
4,237 7,678 8,815 9,429 9,718
0
4,128 7,480 8,588 9,187 9,469
4,086 7,404 8,500 9,093
3,324 6,024 6,916
3,430 6,215
3,181

Accident Parameter a for Two-Point Exponential Fit
Year
12
24
36
48
60
1969
356
124
132
198
1970
438
181
215
353
1971
464
244
337
493
1972
510
337
506
421
1973
616
468
333
1974
530
220
1975
580
1976

Year
1969
1970
1971
1972
1973
1974
1975
1976

72
286
778
370

84
1,034
88

96
459

Parameter b for Two-Point Exponential Fit
12
24
36
48
60
72
84
96
0.000411 0.000570 0.000562 0.000508 0.000459 0.000294 0.000398
0.000368 0.000490 0.000468 0.000409 0.000315 0.000568
0.000338 0.000416 0.000381 0.000341 0.000370
0.000354 0.000407 0.000360 0.000380
0.000346 0.000381 0.000421
0.000434 0.000576
0.000444

Example: The exponential regression for AY 1969 between ages 12 and 24, such that X= (4,079; 6,616) and Y= (1,904;
5,398), would result in a = 356 and b = 0.000411, which we place in the age 24 cell.

Compute: adjusted paid claims for AY 1970 at age 48
Compute: adjusted paid claims for year 1969 at age 12

Solutions to Exam 5B - Independently Authored Preparatory Test 5
Question 1 discussion: Blooms: Comprehension; Difficulty 1, LO 4, KS How internal operating
changes affect estimates of unpaid claims: * Claims processing * Underwriting and policy
provisions * Marketing * Coding of claim counts and/or claim related expenses * Treatment of
recoveries such as policyholder deductibles and salvage and subrogation * Reinsurance
Advantages to using the ratio approach:
1. Development factors are not as highly leveraged as those based on received S&S dollars.
2. Relates to selecting ultimate S&S ratio(s) for the most recent year(s) in the experience period.
See chapter 14
Question 2 discussion: Blooms: Comprehension; Difficulty 1, LO 4, KS How internal operating
changes affect estimates of unpaid claims: * Claims processing * Underwriting and policy
provisions * Marketing * Coding of claim counts and/or claim related expenses * Treatment of
recoveries such as policyholder deductibles and salvage and subrogation * Reinsurance
1. When an estimate of uncollectible reinsurance is included in the net IBNR but not in the gross IBNR and
there are significant billed reinsurance amounts for which significant collectibility issues exist.
2. For a runoff book with reinsurance disputes for items such as asbestos.
See chapter 14

Question 3 discussion: Blooms: Comprehension; Difficulty 1, LO 3, KS Mechanics associated with each
technique (including organization of the data)

Loss emergence during CY 2007 from AY's 2004 - 2006
Accident
Year

Selected Ultimate
Loss
(1)

2003
2004
2005
2006

Accident
Year
2004
2005
2006
Total

620,000
580,000
600,000

LDFs to
Ultimate
(2)
1.100
1.350
1.650
2.200

Selected Ultimate % Reported
at 12/31/06
Loss
(4) = (1)
(5)=1.0-(3)
620,000
74.07%
580,000
60.61%
600,000
45.45%

% Unreported
at 12/31/06
(3)=1.0 - 1.0/(2)

0.0909
0.2593
0.3939
0.5455

% Reported
at 12/31/07
(6) based on (3), (5)

90.91%
74.07%
60.61%

% Reported
during CY 2007
(7) = (6) - (5)
16.84%
13.47%
15.15%

See chapter 15

Expected Inc.Loss
emergence
during CY 2007
(8) = (4) * (7)
104,377
78,114
90,909
273,401

Solutions to Exam 5B - Independently Authored Preparatory Test 5
Question 4 discussion: Blooms: Comprehension; Difficulty 1, LO 1, KS Organization of data: calendar year,
accident year, policy year, underwriting year, report year

Interval
12-24
24-36
36-48
48-60

Age-to-Age
Factors
(1)
2.00
1.25
1.10
1.00

Age-to-Ultimate
Percent
Percent Estimated IBNR
Factors
Reported Unreported
Losses
(2) based on (1) (3)=1.00/(2)
(4)=1-(3) (5)=(4) x 1500
2.75
36.36%
63.64%
955
1.38
72.73%
27.27%
409
1.10
90.91%
9.09%
136
1.00
100.00%
0.00%
0

Accident Year 2003 earned premium
Accident Year 2003 expected loss ratio
Accident Year 2003 expected losses

2,000
75%
1500

Calendar Year 2006 emergence

$136

Time Index
(6)
12/31/03
12/31/04
12/31/05
12/31/06

Question 5 discussion: Blooms: Comprehension; Difficulty 1, LO 3, KS Key terms: case
outstanding, paid claims, reported claims, incurred but not reported, ultimate claims, claims
related expenses, reported and closed claim counts, claim counts closed with no payment,
insurance recoverables, exposures, experience period, maturity or age, and components of
unpaid claim estimates
1. Select one method and use it for all years. The B/S adjusted reported claim (both case and paid adjustments)
method may be a reasonable selection for all years (for an insurer like XYZ).
2. Select different methods for different AYs. For example, select the B/S adjusted reported claim method for
AY 1998 - 2006 and the BF method for 2007 and 2008.
3. Use a weighted average based on assigned weights to the various methods; these weights may be
consistent for all years or may vary by AY.
See chapter 15
Question 6 discussion Blooms: Comprehension; Difficulty 1, LO 3, KS The claim process
Computing deviations of actual development from projected development of claims or claim counts are
useful to evaluate the accuracy of the unpaid claim estimate. Therefore, comparing actual-to-expected
claims helps the actuary to evaluate the appropriateness of prior selections and make revisions as
necessary if actual claims do not emerge as expected.
See chapter 15

Question 7 discussion: Blooms: Comprehension; Difficulty 1, LO 3, KS The claim process
For each AY, expected reported claims in the calendar year are equal to:
[(ultimate claims selected at 12/31/2007 - actual reported claims at 12/31/2007) / (% unreported at 12/31/2007)]
x (% reported at 12/31/2008 - % reported at 12/31/2007)]
The % unreported is computed as [1.00 - (1.00 / cumulative claim development factor)].
The expected reported claims for accident year 2007 during calendar year 2008 are equal to:
AY07 Expected ClaimCY08 = {[($2,798 - $2,463) / (1 - 0.880)] x (0.999 - 0.880)} = $332
The expected reported claims for accident year 2006 during calendar year 2008 are equal to:
AY06 Expected ClaimCY08 = {[($2,952 - $2,949) / (1- 0.999)] x (1.000 - 0.999)} = $3
See chapter 15

Solutions to Exam 5B - Independently Authored Preparatory Test 5
Question 8 discussion: Blooms: Comprehension; Difficulty 3, LO 3, KS Mechanics associated with each
technique (including organization of the data)
Step 2: Calculate and Select Development Factors
1st to 2nd
2nd to 3rd
12:24 mo,
24:36 mo
Age-to-Age on reported
1.056*
1.169
3-yr volume-wtd avg.
at 12 mo
at 24 mo
1.111**
1.299
Reported CDF to Ult.

3rd to 4th
36:48 mo
1.032
at 36 mo
1.053

4th to 5th
48:60 mo
1.010
at 48 mo
1.020

5th to 6th
60:72 mo
1.000
at 60 mo
1.010

Tail at
72 months

1.01

*Example of Age-to-Age calculation for 2nd to 3rd report, using 3-year volume-weighted average:
(1,804,607+1,632,191+1,454,961)/(1,709,627+1,537,760+1,378,385)=1.056 as shown
**Example of Ultimate CDF calculation for claims at 24 months of development:
(1.056 for 2nd-to-3rd) * (1.032 for 3rd-to-4th) * (1.01 for 4th-to-5th) * (1.01 tail) = 1.111
Steps 3 & 4: Develop the claims & compute estimates
Age of
Reported
Reported
Estimated
Acciden
Data at
Claims
CDF to
Ultimate
Year 12/31/2008
12/31/08
Ultimate
Claims
(1) FYI
(2)
(3) above
(4)=(2)*(3)
2003
72 months
1,263,519
1.01
1,276,154
2004
60 months
1,516,223
1.010
1,531,385
2005
48 months
1,674,450
1.020
1,708,452
2006
36 months
1,804,607
1.053
1,899,396
2007
24 months
1,894,836
1.111**
2,105,163
2008
12 months
1,791,785
1.299
2,327,528
Answer A
10,848,078
Total

For unpaid, subtract: Ultimate - Paid:
Estimated
Accident
Ultimate
Paid
Year
Claims
Claims
above
given
2003
1,276,154 -1,250,756=
2004
1,531,385 -1,470,276=
2005
1,708,452 -1,571,933=
2006
1,899,396 -1,595,652=
2007
2,105,163 -1,494,816=
2008
2,327,528
-977,337=
Total Answer C

IBNR
OR:
IBNR
(broadly
(broadly
Shortcut
defined)
defined)
(5)=(4)-(2)
(5)=(2)*[(3) - 1.0]
12,635
12,635
15,162
15,162
34,002
34,002
94,790
94,790
210,327
210,327
535,744
535,744
902,660 Answer B
902,660

OR, add Case Outstanding to est. IBNR:
IBNR
Case O/S
Total Est.
(broadly
=Reported
Unpaid
defined)
minus Paid
Claims
(i)
(ii)
(ii)=(i) + (ii)
12,635
+12,764=
25,399
15,162
+45,947=
61,109
34,002
+102,518=
136,519
94,790
+208,955=
303,744
210,327
+400,020=
610,347
535,744
+814,448=
1,350,191
Answer C
2,487,309

Total Est.
Unpaid
Claims
difference
25,399
61,109
136,519
303,744
610,347
1,350,191
2,487,309

Example (ii) Case O/S 2008 = 1,791,785 - 977,337 = 814,448

Solutions to Exam 5B - Independently Authored Preparatory Test 5
Solution to d) Emergence during CY 2009 for AY's 2008 and prior, using Development Method on Reported Claims
Step 1 (proceeding the safe way, using our answer to part b)
% of IBNR
Estimated
Ages in
Percent
Percent Percent
to emerge b/t
Acc
IBNR
NEXT yr
Reported
Reported Unreported
1/1 - 12/31/09
Year at 12/31/08
(CY 2009)
at 1/1/09
at 12/31/09 at 1/1/09

2003
2004
2005
2006
2007
2008
Total

(1) from b
12,635
15,162
34,002
94,790
210,327
535,744
902,660

(2) FYI
72-84
60-72**
48-60
36-48
24-36
12-24

(4)=1/CDF (5)=1.0-(3)
100.00%
0.990%
99.01%
0.990%
99.01%
1.990%
98.01%
4.991%
95.01%
9.991%
90.01%
23.018%

(3)=1/CDF
99.01%
99.01%
98.01%
95.01%
90.01%
76.98%

(5)
100.000%
0.000%
50.252%
60.120%
50.050%
56.594%

Step 2

Est. IBNR
to emerge
during '09
(7)=(1)*(6)
12,635
0
17,086
56,988
105,269
303,200
495,178

** Note: since the selected ultimate CDFs are equal for 60 months and 72 months, there is no emergence here!
Step 2

Step 1 (proceeding via short-cut, for development method only...)

Acc
Year
2003
2004
2005
2006
2007
2008
Total

Reported
Claims
at 12/31/08
(1) given
1,263,519
1,516,223
1,674,450
1,804,607
1,894,836
1,791,785

Ages in
NEXT yr
(CY 2009)
(2) FYI
72-84
60-72**
48-60
36-48
24-36
12-24

Selected
ATA factor
for ages
(3) above
1.01
1.00
1.01
1.03
1.06
1.17

Subtract
Subtract
ATA-1
(4)=(3)-(1)
0.01
0.00
0.01
0.03
0.06
0.17

Est. IBNR
to emerge
during '09
(5)=(1)*(4)
12,635
0
17,086
56,988
105,269
303,200
495,178

** Note: since the ATA for 60 to 72 months = 1.0, there is no emergence here!
Solution to e) Given the operating environment of this insurer, is this technique appropriate to use?
Start by assuming there is no reason to dis-credit the method other than via the 2 comments given:
1) We're told "No change in Case Outstanding adequacy levels," so we shouldn't need to worry there.
2) We're given a set of claims ratios that accurately reflects the company's position:
70% in AY 2003, 80% in 2004, 85% in 2005, 90% in 2006, 95% in 2007, 100% in AY 2008
... If the Ultimate Claim projections are not consistent, then we may need to consider other methods.
Let's test: Recall Friedland mentions analysis of Claim Ratio Estimates in "Evaluation of Techniques."
Estimated
Implied
Accident
Ultimate
Earned
Claim
Year
Claims (a)
Premium
Ratio
(i)
(ii) given
(iii)=(i)/(ii)
2003
1,276,154
1,823,259
69.99%
2004
1,531,385
1,914,423
79.99%
2005
1,708,452
2,010,144
84.99%
2006
1,899,396
2,110,650
89.99%
2007
2,105,163
2,216,183
94.99%
2008
2,327,528
2,326,992
100.02%

All implied Claim Ratios are consistent with those given in the question. So, we conclude that the
Development Method was a good choice here.

Solutions to Exam 5B - Independently Authored Preparatory Test 5
Question 9 discussion:
Solution to a) Using the Reported Development method, estimate Ultimate ALAE
Solution to b) Using the Reported Development method, estimate Upaid ALAE
1st to 2nd
2nd to 3rd
3rd to 4th
4th to 5th
5th to 6th
Age-to-Age on reported
3-yr volume-wtd avg.
Reported CDF to Ult.

12:24 mo
0.877
at 12 mo
1.031

Accident
Year
2003
2004
2005
2006
2007
2008
Total
Solution
Solution

Age-to-Age on paid
3-yr volume-wtd avg.
Paid CDF to Ult.

1st to 2nd
12:24 mo
1.292
at 12 mo
1.581

Accident
Year
2003
2004
2005
2006
2007
2008
Total

24:36 mo
1.057
at 24 mo
1.175

36:48 mo
1.038
at 36 mo 1
1.112

48:60 mo
1.016
at 48 mo
1.071

Age of
Data at
12/31/2008
(1) FYI
72 months
60 months
48 months
36 months
24 months
12 months

Reported
ALAE at
12/31/2008
(2)
1,715
1,548
1,548
1,797
2,394
3,112
Answer A

Reported
CDF to
Ultimate
(3) above
1.01
1.054
1.071
1.112
1.175
1.031

60:72 mo
1.044
at 60 mo
1.054

Tail at
72 months

1.01

Estimated
Paid
Ultimate ALAE
ALAE at
ALAE
12/31/2008
(4)=(2)*(3)
(5) given
1,732
1,628
1,632
1483.5
1,659
1480.5
1,998
1,698
2,814
2,181
3,208
1,904
13,041 Answer B

Estimated
Unpaid
ALAE
(6)=(4)-(5)
104
149
178
300
633
1,304
2,667

to c) Using the Paid Development method, estimate Ultimate ALAE
to d) Using the Paid Development mentod, estimate Unpaid ALAE
2nd to 3rd
24:36 mo
1.075
at 24 mo
1.223

3rd to 4th
36:48 mo
1.054
at 36 mo
1.138

4th to 5th
48:60 mo
1.028
at 48 mo
1.080

Age of
Data at
12/31/2008
(1) FYI
72 months
60 months
48 months
36 months
24 months
12 months

Paid
ALAE at
12/31/2008
(2)
1,628
1483.5
1480.5
1,698
2,181
1904
Answer C

Paid
CDF to
Ultimate
(3)
1.035
1.050
1.080
1.138
1.223
1.581

5th to 6th
60:72 mo
1.015
at 60 mo
1.050

Tail at
72 months

1.035

Estimated
Ultimate
ALAE
(4)=(2)*(3)
1,684
1,558
1,600
1,933
2,668
3,010
12,454 Answer D

Estimated
Unpaid
ALAE
(5)=(4)-(2)
57
75
119
235
487
1,106
2,079

Solutions to Exam 5B - Independently Authored Preparatory Test 5
Question 9 discussion: Blooms: Comprehension; Difficulty 3, LO 7, KS Estimation of unpaid ALAE
Solution to e) using the Ratio method, estimate Ultimate ALAE
Step 1: Create a triangle of the ratio of paid ALAE, over Paid Claims only (excl ALAE)

AY
2003
2004
2005
2006
2007
2008

Ratio: [Paid ALAE] / [Paid Claims excluding ALAE]
at 12 mos.
at 24 mos.
at 36 mos.
at 48mos.
at 60 mos.
0.51%
0.55%
0.60%
0.64%
0.66%
0.53%
0.56%
0.61%
0.63%
0.65%
0.57%
0.59%
0.62%
0.64%
0.59%
0.62%
0.64%
0.68%
0.85%
0.76%

at 72 mos.
0.66%

Step 2: Calculate and Select Development Factors for the Ratio examined in step 1.
AY
2003
2004
2005
2006
2007

ATA
3-yr avg
CDF
to Ult.

12:24 mo 24:36 mo
1.079
1.089
1.072
1.078
1.030
1.043
1.046
1.038
1.252

36:48 mo
1.064
1.039
1.047

12:24 mo
1.109
at 12 mo
1.338

36:48 mo
1.050
36 mo
1.145

24:36 mo
1.053
at 24 mo at
1.206

48:60 mo
1.025
1.028

60:72 mo
1.012

Tail at 72 mo

48:60 mo
1.026
at 48 mo
1.091

60:72 mo
1.012
at 60 mo
1.063

at 72mo
1.050

Steps 3 & 4: DEVELOP the ratio & apply it to Ultimate Claims for Projected Ultimate ALAE

Accident
Year
2003
2004
2005
2006
2007
2008
Total

Age of
Data at
12/31/2008
(1) FYI
72 months
60 months
48 months
36 months
24 months
12 months

Ratio
as of

Selected
CDF to

12/31/2008
Ultimate
(2) above
(3) above
0.66%
1.050
0.65%
1.063
0.64%
1.091
0.64%
1.145
0.85%
1.206
0.76%
1.338
DON'T FORGET THIS STEP

Ultimate Selected Ult.
Ratio Claims only

Projected
Ultimate

to use (excl ALAE)
ALAE
(4) =(2)*(3) (5) given
(6)=(4)*(5)
0.007
326,602
2,279
0.007
305,592
2,105
0.007
306,362
2,154
0.007
354,242
2,616
0.010
353,120
3,611
0.010
390,970
4,002
16,766

Note: Structurally, this process is analogous to the Ratio method for S&S in Chapter 14.
Next, we will see that the Additive Method is very similar, but the development factors based on
addition, instead of the multiplication we usually perform. See 3f immediately below for an illustration.

Solutions to Exam 5B - Independently Authored Preparatory Test 5
Question 9 discussion:
Solution to f) Using the Additive method, estimate Ultimate ALAE.
Step 1: Create a triangle of the ratios -- SAME AS FOR RATIO METHOD

AY
2003
2004
2005
2006
2007
2008

at 12 mo
0.51%
0.53%
0.57%
0.59%
0.68%
0.76%

Ratio: [Paid ALAE] / [Paid Claims excluding ALAE]
at 24 mo
at 36 mo
at 48 mo
at 60 mo
0.55%
0.60%
0.64%
0.66%
0.56%
0.61%
0.63%
0.65%
0.59%
0.62%
0.64%
0.62%
0.64%
0.85%

Step 2: Calculate and Select Development Factors using DIFFERENCES
(While we'd normally divide the value for one-age to the prior, here we Subtract)
AY
12:24 mo
24:36 mo
36:48 mo
48:60 mo
2003
0.04%
0.05%
0.04%
0.02%
2004
0.04%
0.04%
0.02%
0.02%
2005
0.02%
0.03%
0.03%
2006
0.03%
0.02%
0.17%
2007
ATA
3-yr avg
CDF to Ult.
ADDITIVE*

12:24 mo
0.07%
at 12 mo
0.22%

24:36 mo
0.03%
at 24 mo
0.15%

36:48 mo
0.03%
at 36 mo
0.12%

48:60 mo
0.02%
at 48 mo
0.08%

at 72 mo
0.66%

60:72 mo
0.01%

Tail at 72 mo

60:72 mo
0.01%
at 60 mo
0.07%

at 72mo
0.06%

*Careful: For the Additive CDF, we literally add up the ATA factors (based on differences) ...
For example at 36 months: .12% = .03% + .01% + .02% + .06% ... not multiplying as usual
Steps 3 & 4: DEVELOP the ratio & apply it to Ultimate Claims for Projected Ultimate ALAE
SelectedUlt.
Age of
Ratio
Selected
Ultimate
Claims only
Accident
Data at
as of
CDF to
Ratio
Year
12/31/2008
12/31/2008
Ultimate
to use (excel ALAE)
(5) given
(1) FYI
(2) above
(3) above
(4)=(2) + (3)
326,602
2003
72 months
0.66%
0.06%
0.72%
305,592
2004
60 months
0.65%
0.07%
0.72%
306,362
2005
48 months
0.64%
0.08%
0.73%
354,242
2006
36 months
0.64%
0.12%
0.76%
24 months
353,120
2007
0.85%
0.15%
0.99%
0.76%
0.22%
390,970
2008
12 months
0.98%
Total
Don't Forget to ADD Here

Projected
Ultimate
ALAE
(6)=(4)*(5)
2,367
2,188
2,234
2,692
3,510
3,842
16,832

Solutions to Exam 5B - Independently Authored Preparatory Test 5
Question 10 discussion: Blooms: Comprehension; Difficulty 1, LO 7, KS Strengths and
weaknesses of the estimation techniques for claim related expenses
1. The development technique using paid ALAE.
2. The development technique using reported ALAE (when case outstanding for ALAE exists), which for Auto
Property Damage Insurer maintains.
3. The development of the ratio of paid ALAE-to-paid claims only.
See chapter 16

Question 12 discussion: Blooms: Comprehension; Difficulty 1, LO 7, KS Strengths and
weaknesses of the estimation techniques for claim related expenses
Advantages:
1. It recognizes the relationship between ALAE and claims only.
2. The ratio development factors are not as highly leveraged as those based on paid ALAE dollars. Age-toage factors based on the simple average of the latest 3 years is selected.
This method produces projected ultimate ALAE less than the reported and paid ALAE projections (a key
reason for this is the absence of a tail factor).
3. The ability to interject actuarial judgment in the projection analysis, especially for the selection of the
ultimate ALAE ratio for the most recent year(s) in the experience period.
Disadvantages:
1. Any error in the estimate of ultimate claims only could affect the estimate of ultimate ALAE.
2. When large amounts of ALAE are spent on claims that ultimately settle with no claim payment, the
projection process is distorted.
See chapter 16

Question 13 discussion: Blooms: Comprehension; Difficulty 1, LO 3, KS Standards of Practice,
ASOP Nos. 9 and 43
a. Examples of types of measures for the unpaid claim estimate include high estimate, low estimate, median,
mean, mode, actuarial central estimate, mean plus risk margin, actuarial central estimate plus risk margin,
or specified percentile.
b. The terms “best estimate” and “actuarial estimate” are not sufficient descriptions of the intended measure,
since they describe the source or the quality of the estimate but not the objective of the estimate.

Question 13 discussion: Blooms: Comprehension; Difficulty 1, LO 3, KS Standards of Practice,
ASOP Nos. 9 and 43
a. Examples include reinsurance program changes and changes in the practices by the entity’s claims
personnel to the extent such changes are likely to have a material effect on the results of the actuary’s
unpaid claim estimate analysis.
b. Obtain supporting information from the principal or the principal’s duly authorized representative and may
rely upon their representations unless, in the actuary’s professional judgment, they appear to be
unreasonable.

Solutions to Exam 5B - Independently Authored Preparatory Test 5
Question 14 discussion Blooms: Comprehension; Difficulty 3, LO 3, KS Mechanics associated
with each technique (including organization of the data)

Accident
Year
(1)
2004
2005
2006
2007
2008
Total

Claim Counts
Trend to
Selected
2008 at
Trend
Ultimate
-1.00%
Ultimate
(2)
(3)
(4)
1,734
0.961
1,665
1,637
0.970
1,589
2,966
0.980
2,907
2,888
0.990
2,859
1.000
2,651
2,651
11,875
11,670

Payroll
($00)
(5)
280,000
350,000
790,000
780,000
740,000
2,940,000

Trend
2008 at
2.50%
(6)
1.104
1.077
1.051
1.025
1.000

(9) Selected frequency at 2008 level
(10) Selected frequency at 2007 level

Trended
Payroll
($00)
(7)
309,068
376,912
829,994
799,500
740,000
3,055,473

Trended
Ultimate
Frequency
(8)
0.54%
0.42%
0.35%
0.36%
0.36%
0.38%
0.36%
0.37%

(3) Assume -1% annual claim count trend.
(4) =[ (2) * (3)]
(6) Assume 2.50% annual payroll trend.
(7) = [(5) x (6)].
(8) = [(4) / (7)].
(9) Judgmentally selected.
(10) = { (9) * [1 + (annual payroll trend of 2.50%)] / [1 + (annual claim count trend of -1.00%)] } .
Divide the ultimate trended claim counts in Column (4), by the trended payroll in Column (7). After
examining the frequency rates by accident year in Column (8), we recognize a change in frequency
between the earliest years in the experience period and the most recent years.
See chapter 11

Solutions to Exam 5B - Independently Authored Preparatory Test 5
Question 15 discussion Blooms: Comprehension; Difficulty 2, LO 3, KS The claim process
Compute the unpaid Claim Estimate at 12/31/08 for AY's 2007 and 2008
Accident Year
2007
2008
780,000
740,000
0.37%
0.36%
2,886
2,664
5,674
6,100
16,376,372 16,250,400
14,400,000 10,300,000
5,357,000 6,130,000
1,976,372 5,950,400
7,333,372 12,080,400

(1) Payroll ($00)
(2) Selected Frequency
(3) Projected Ultimate Claim Counts
(4) Selected Severity
(5) Projected Ultimate Claims
(6) Reported Claims at 12/31/08
(7) Case Outstanding at 12/31/08
(8) Estimated IBNR at 12/31/08
(9) Unpaid Claim Estimate at 12/31/08
Line Notes:
(3) = [(1) * (2)]
(5) = [(3) * (4)]
(8) = [(5) - (6)]
(9) = [(7) + (8)]

Calculate the projected ultimate claims for accident years 2007 and 2008. The self-insured organization
provided us with the payroll for both accident years. We multiply the payroll by the selected frequency
rates to determine the projected ultimate number of claims (Line (3)). We then multiply the ultimate
number of claims by the selected severities to derive the projected ultimate claims (Line (5)).
(3) = [(1) * (2)]
(5) = [(3) * (4)]
(8) = [(5) - (6)]
(9) = [(7) + (8)]
See chapter 11
Question 16 discussion Blooms: Comprehension; Difficulty 1, LO 3 KS KS The claim process
Compute the unpaid claim estimate for AYs 1999 – 2003.

Accident
Year
(1)
1999
2000
2001
2002
2003
Total

Case
Outstanding
at 12/31/08
(2)
650,000
800,000
850,000
975,000
1,000,000
4,275,000

CDF to Ultimate
Reported
(3)
1.020
1.030
1.051
1.077
1.131

Paid
(4)
1.067
1.109
1.187
1.306
1.489

Case
Outstanding
(5)
1.454
1.421
1.445
1.439
1.545

Column Notes:
(2) Based on data from Self-Insurer Case Outstanding Only.
(5) = { [((3) - 1.0) * (4) ]/ ((4) -(3))} +1
(6) = [(2) * (5)].

See chapter 12

Unpaid
Claim
Estimate
(6)
945,128
1,136,911
1,228,356
1,403,157
1,544,858
6,258,410

Solutions to Exam 5B - Independently Authored Preparatory Test 5
Question 17 discussion Step 1: Compute the following ratios
Accident
Year
2003
2004
2005
2006
2007

Ratio of Incremental Paid Claims to Previous Case Outstanding as of (months)

24
0.821
0.813
0.828
0.858

36
0.691
0.695
0.716

48
0.705
0.718

60
0.684

To Ult

Averages of the Ratio of Incremental Paid Claims to Previous Case Outstanding
12
24
36
48
60
To Ult
Latest 3
0.833
0.701
0.712
0.684
Selected Ratio of Incremental Paid Claims to Previous Case Outstanding
12
24
36
48
60
To Ult
0.833
0.701
0.712
0.684
1.000
Selected

Step 2: Compute the following ratios
Accident Ratio of Case Outstanding to Previous Case Outstanding as of (months)
Year
12
24
36
48
60
To Ult
2003
0.527
0.577
0.536
0.476
2004
0.530
0.567
0.507
2005
0.522
0.554
2006
0.526
2007
Averages of the Ratio of Case Outstanding to Previous Case Outstanding
12
24
36
48
60
To Ult
0.526
0.566
0.522
0.476
Latest 3
Selected Ratio of Case Outstanding to Previous Case Outstanding
12
24
36
48
60
To Ult
Selected
0.526
0.566
0.522
0.476
1.000

Step 3: Complete the square for the following triangles
Accident
Year
2003
2004
2005
2006
2007

12
21,078,651
21,047,539
21,260,172
20,973,908
21,623,594

24
11,098,119
11,150,459
11,087,832
11,034,842
11,374,010

36
6,398,219
6,316,995
6,141,416
6,245,721
6,437,690

Accident
Year
2003
2004
2005
2006
2007

12
24,084,451
24,369,770
25,100,697
25,608,776
27,229,969

24
17,315,161
17,120,093
17,601,532
17,997,721
18,012,454

36
7,670,720
7,746,815
7,942,765
7,735,424
7,973,181

Accident
Year
2003
2004
2005
2006
2007

12
24,084,451
24,369,770
25,100,697
25,608,776
27,229,969

24
41,399,612
41,489,863
42,702,229
43,606,497
45,242,423

36
49,070,332
49,236,678
50,644,994
51,341,921
53,215,604

Case Outstanding as of (months)
48
60
To Ult
3,431,210 1,634,690
0
3,201,985 1,524,145
0
3,205,819 1,525,970
0
3,260,266 1,551,887
0
3,360,474 1,599,586
0
Incremental Paid Claims as of (months)
48
60
To Ult
4,513,869 2,346,453 1,634,690
4,537,994 2,190,158 1,524,145
4,372,688 2,192,780 1,525,970
4,446,953 2,230,022 1,551,887
4,583,635 2,298,564 1,599,586
Cumulative Paid Claims as of (months)

48
53,584,201
53,774,672
55,017,682
55,788,874
57,799,239

60
55,930,654
55,964,830
57,210,462
58,018,896
60,097,804

72
57,565,344
57,488,975
58,736,432
59,570,783
61,697,389

AY 2007 projected case O/S at 24 months: $11,374,010 equals 0.526 (selected ratio at 24 months) * $21,623,594 (case
O/S at 12 months)
AY 2006 incremental paid claims at 36 months: $7,735,424 = 0.701 (selected ratio at 36 months) * $11,034,842 (case
O/S at 24 months). Cumulative paid at 72 for AY 2003 = 57,565,344 = 55,930,654 + 1,634,690
Copyright  2014 by All 10, Inc.
Page 108

Solutions to Exam 5B - Independently Authored Preparatory Test 5
Question 18 discussion Blooms: Comprehension; Difficulty 1, LO 4, KS How internal operating
changes affect estimates of unpaid claims: * Claims processing…Treatment of recoveries such as
policyholder deductibles and salvage and subrogation * Reinsurance
Compute: Adjusted Reported Claims triangle
Step 1: Compute unadjusted case outstanding and unadjusted average case outstanding
8,682,000 = 8,732,000 – 50,000. 8,324 = 8,682,000 / 1,043
Accident
Year
1972
1973
1974
1975
1976

Unadjusted Case Outstanding as of (months)

Accident
Year
1972
1973
1974
1975
1976

Open Claim Counts as of (months)

Accident
Year
1972
1973
1974
1975
1976

Unadjusted Average Case Outstanding as of (months)

12
8,682,000
11,015,000
8,534,000
12,718,000
15,582,000

24
17,847,000
19,134,000
31,872,000
47,339,000

12
1,043
1,088
1,033
1,138
1,196

12
8,324
10,124
8,261
11,176
13,028

24
1,561
1,388
1,418
1,472

24
11,433
13,785
22,477
32,160

36
28,333,000
46,544,000
57,210,000

48
47,425,000
62,441,000

60
42,645,000

36
1,828
1,540
1,663

48
1,894
1,877

60
1,522

48
25,040
33,266

60
28,019

36
15,499
30,223
34,402

Step 2 Compute the following: Adj Avg Case O/S at 12 mos for AY 1975 = 11,328 = 13,028/1.15.
13,102,402 = Open counts * Adj Av Case O/S + Unad Paid Claims = [1,138 * 11,329 + 210,000]
Accident
Year
1972
1973
1974
1975
1976

Adjusted Average Case Outstanding as of (months)
12
24
36
48
7,449
21,145
26,013
28,927
8,566
24,317
29,915
33,266
9,851
27,965
34,402
11,329
32,160
13,028

Selected Annual Severity Trend Rate
Accident
Year
1972
1973
1974
1975
1976

60
28,019

15%

Adjusted Reported Claims as of (months)
12
24
36
7,819,307
33,793,345
51,361,764
9,532,808
34,584,996
49,668,100
10,348,083
41,241,370
63,477,000
13,102,402
48,904,000
15,791,000

48
64,559,286
73,733,000

60
61,163,000

Solutions to Exam 5B - Independently Authored Preparatory Test 5
Question 19 discussion Blooms: Comprehension; Difficulty 2, LO 7, KS How internal operating
changes affect estimates of unpaid claims: * Claims processing… * Treatment of recoveries such
as policyholder deductibles and salvage and subrogation * Reinsurance
Step 1: Compute CDFs to ultimate for reported claim counts and compute projected ultimate claims
PART 2 - Age-to-Age Factors - Reported Claim Counts
Accident
Age-to-Age Factors
Year
12 - 24
24 - 36
36 - 48
48 - 60
To Ult
1972
1.206
1.015
1.005
1.002
1973
1.201
1.015
1.005
1974
1.213
1.017
1975
1.223
PART 4 - Selected Age-to-Age Factors
Development Factor Selection
12 - 24
24 - 36
36 - 48
48 - 60
To Ult
Selected - Simple Avg
1.211
1.015
1.005
1.002
1.001
CDF to Ultimate
1.239
1.023
1.008
1.003
1.001
Percent Reported
80.7%
97.7%
99.2%
99.7%
99.9%

Accident
Year
(1)
1972
1973
1974
1975
1976
Total

Age of
Accident Year
at 12/31/76
(2)
60
48
36
24
12

Reported
Claim Counts
at 12/31/76
(3)
9,680
9,562
7,741
7,884
6,115
40,982

CDF
to Ultimate
(4)
1.001
1.003
1.008
1.023
1.239

Projected
Ultimate
Claim Counts
(5) = [(3) x (4)]
9,690
9,591
7,803
8,066
7,577
42,726

Step 2: Compute disposal rates, select disposal rates by age, and use them to compute adj closed claim counts.
Accident Disposal Rate as of (months)
Year
12
24
36
48
60
1972
0.464
0.809
0.903
0.955
0.977
1973
0.461
0.799
0.903
0.948
1974
0.447
0.796
0.886
1975
0.436
0.772
1976
0.426 =3,230/7,575 = closed claim cnts/projected ult cnts
Selected Disposal Rate by Maturity Age
0.426
0.772
0.886

0.948

Projected
Ultimate
Claim Counts
9,690
9,591
7,803
8,066
7,577

0.977

Accident Adjusted Closed Claim Counts as of (months)
Year
12
24
36
48
60
1972
4,128
7,481
8,588
9,187
9,469
1973
4,086
7,404
8,501
9,093
1974
3,324
6,024
6,916
1975
3,436
6,227 =.772 * 8,068 = sel disposal rate * projected ult cnts
1976
3,228

Solutions to Exam 5B - Independently Authored Preparatory Test 5
Question 20 discussion Blooms: Comprehension; Difficulty 2, LO 4 KS How internal operating
changes affect estimates of unpaid claims: * Claims processing * Underwriting and policy
provisions * Marketing * Coding of claim counts and/or claim related expenses * Treatment of
recoveries such as policyholder deductibles and salvage and subrogation * Reinsurance
Compute: adjusted paid claims for AY 1970 at age 48
Compute: adjusted paid claims for year 1969 at age 12
Adjusting the paid claims:
* If the number of adjusted closed claims is within the range of any regression in its specific accident year, we
use interpolation. Example:
Since AY 1970 at age 48 has 8,231 adjusted closed claims, which is within the range of unadjusted closed
claims between ages 36 and 48 (7,899; 8,291), the paid claims for AY 1970 at age 48 would be adjusted
based on such regression with a = 215 and b = 0.000468.
(0.000468 x 8,231)
] } = 10,156.
Thus, the adjusted paid claims for AY 1970 at age 48 are equal to {215 x [e
* If the number of adjusted closed claims is not within the range of all regression in its specific AY, then
extrapolation is used to the regression that has the closest range. Example:
AY 1969 at age 12 has 3,334 adjusted closed claim counts, in which the regression between ages 12 and
24 has the closest unadjusted closed claim count range (4,079; 6,616) among all regressions in year 1969.
(0.000411 x3,334)
]} =1,402
Thus, adjusted paid claims for year 1969 at age 12 is calculated as {356 x [e
See chapter 13
Accident
Closed Claim Counts as of (months
Year
12
24
36
48
60
72
84
96
1969
4,079 6,616 7,192 7,494 7,670 7,749 7,792 7,806
1970
4,429 7,230 7,899 8,291 8,494 8,606 8,647
1971
4,914 8,174 9,068 9,518 9,761 9,855
1972
4,497 7,842 8,747 9,254 9,469
1973
4,419 7,665 8,659 9,093
1974
3,486 6,214 6,916
1975
3,516 6,226
1976
3,230

Accident
Year
1969
1970
1971
1972
1973
1974
1975
1976

Paid Claims
12
24
1,904 5,398
2,235 6,261
2,441 7,348
2,503 8,173
2,838 8,712
2,405 7,858
2,759 9,182
2,801

($000) as of (months)
36
48
60
7,496 8,882 9,712
8,691 10,443 11,346
10,662 12,655 13,748
11,810 14,176 15,383
12,728 15,278
11,771

Accident
72
84
96
10,071 10,199 10,256
11,754 12,031
14,235

Accident Parameter a for Two-Point Exponential Fit
Year
12
24
36
48
60
1969
356
124
132
198
1970
438
181
215
353
1971
464
244
337
493
1972
510
337
506
421
1973
616
468
333
1974
530
220
1975
580
1976

Year
1969
1970
1971
1972
1973
1974
1975
1976

84
1,034
88

96
459

Accident Adjusted Paid Claims ($000) as of (months)
Year
12
24
36
48
60
1969
1,401
4,257
6,463
8,496
9,578
1970
1,708
5,157
7,864
10,125
11,300
1971
1,941
6,213
9,594
12,232
13,549
1972
2,197
7,192
11,071
13,836
15,383
1973
2,529
7,961
11,981
15,278
1974
2,242
7,236
11,771
1975
2,655
9,182
1976
2,801

72
286
778
370

72
286
778
14,235

84
1,034
12,031

96
10,256

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