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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

1

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.

2

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
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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.

3

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.

4

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
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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.

5

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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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.
Solutions to questions from the 1990 exam:
Question 4.
1. T
2. T
3. T

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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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
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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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Sec
1
2
3
4
5
6
7
8
1

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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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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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.

3

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.

4

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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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

5

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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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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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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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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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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6

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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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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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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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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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.

7

U. S. Workers Compensation Rating Manual Example

29 - 34

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

Premises

Classification
Peculiarities

+/-10%

+/-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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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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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.

8

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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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

1

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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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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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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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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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

Claim

Loss

Case

Paid

Salvage/

Number

Date

Date

Date

Status

Chars

Payment

Reserve

ALAE

Subro

A

1

01/10/10

01/15/10

01/15/10

Open

…

$

$10,000

$

$

A

1

01/10/10

01/15/10

03/01/10

Open

…

$1,000

$9,000

$

$

A

1

01/10/10

01/15/10

05/01/10

Closed

…

$9,000

$

$

$

C

2

10/01/10

10/15/10

10/15/10

Open

…

$

$18,000

$

$

C

2

10/01/10

10/15/10

12/15/10

Open

…

$2,000

$17,000

$

$

C

2

10/01/10

10/15/10

03/01/11

Open

…

$7,000

$15,000

$

$

C

2

10/01/10

10/15/10

03/01/12

Closed

…

$15,000

$

$

$1,000

C

3

02/01/11

02/15/11

02/15/11

Open

…

$

$15,000

$

$

C

3

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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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.

2

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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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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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).

3

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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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.

4

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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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
49. (4 points) Incurred losses can be related to earned premiums using several different time measurements
as follows: i. Calendar year ii. Calendar/accident year iii. Policy year iv. Report year
a. (2 points) Provide one advantage and one disadvantage of each for use in ratemaking.
b. (1 point) Name a line of insurance which uses each time measurement. Your answer should be
restricted to the material on the syllabus.
c. (1 point) For each line named in part b, state why the choice of time measurement is appropriate.

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.

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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

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

2

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).

3

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

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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

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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).

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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.

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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.

5

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.

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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.

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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.

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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.

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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.

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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.

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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.

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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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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).

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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).

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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.

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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.

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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

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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

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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

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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.

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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.

2

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.

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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.

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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.

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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

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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.

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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 

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1.1464

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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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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.).

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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.

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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).

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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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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).
One-step trending process is not appropriate to use when:
 changes in average premium vary significantly year-by-year and/or
 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
1/1/2011, the shift would have been completed by 12/31/2011, and the observed trend
would not continue into the future.
When situations like this occur, companies may use a two-step trending approach.

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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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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.

3

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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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%

• 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, chapter 2, "Ratemaking," Foundations of Casualty
Actuarial Science:
(a) (2 points) Calculate the on-level earned premium for the experience period 1992-1994.

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

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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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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

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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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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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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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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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.

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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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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.

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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.

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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

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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

.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

+.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.

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Answer D.

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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.

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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.

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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

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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

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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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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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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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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

1.00

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

0

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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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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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

1.00

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

0

3/1

2005

1/1

2006

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.
OLF 

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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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

1

0.92
1.00

0.92
0.92
0.92

0

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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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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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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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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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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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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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

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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

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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

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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

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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

1.0
1.00*1.07=1.07

0

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

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Current Rate Level 1.1182

 1.051
Average Rate Level 1.0637

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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

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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.

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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

A

B

1/2008

7/1 10/1

0%

b. OLF 

0.05

100%

C

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

A

1.00

C

1.155

B

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

C

A

0%
1/2008

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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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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.

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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

c.

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

0

4

8

12

16

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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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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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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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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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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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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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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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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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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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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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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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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2

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.

3

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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4

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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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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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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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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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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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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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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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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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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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

0%

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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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.

5

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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6

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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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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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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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 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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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)

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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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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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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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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

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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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.

X

= Exposures
__

P; P
V

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

QT

= 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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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%).

2

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

3

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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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.

5

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.

6

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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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.

7

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.

8

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.

9

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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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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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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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.
Solutions to questions from the 1996 exam
____

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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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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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

1

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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Recall the following notation:

X

= 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)

QT

= 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).

2

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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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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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).

3

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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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):

PI

(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.

4

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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5

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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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

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:
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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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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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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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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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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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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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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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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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.
Solutions to questions from the 2002 exam:
Question 17.
c. (1 point) Calculate the indicated statewide rate level change

(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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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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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.
___

PI



.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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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 

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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;

X

= Exposures

Pc

= Premium at current rates

V

= Variable expense provision(EV divided by P)

QT

= 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

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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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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 

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):

PI

(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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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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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

B.

2 only

C.

3 only

D.

1, 3 only

E.

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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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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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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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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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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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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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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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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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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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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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.

2

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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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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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.

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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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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C. 3

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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.

3

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.

4

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.

5

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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Risk Classification Statement of Principles
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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Risk Classification Statement of Principles
AMERICAN ACADEMY OF ACTUARIES COMMITTEE ON RISK CLASSIFICATION
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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AMERICAN ACADEMY OF ACTUARIES COMMITTEE ON RISK CLASSIFICATION
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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AMERICAN ACADEMY OF ACTUARIES COMMITTEE ON RISK CLASSIFICATION
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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AMERICAN ACADEMY OF ACTUARIES COMMITTEE ON RISK CLASSIFICATION
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

2

Small

B

50

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

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Risk Classification Statement of Principles
AMERICAN ACADEMY OF ACTUARIES COMMITTEE ON RISK CLASSIFICATION
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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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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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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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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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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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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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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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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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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Risk Classification Statement of Principles
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.

1

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.

2

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.

3

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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Personal Vehicle Manual
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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Personal Vehicle Manual
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).

5

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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Personal Vehicle Manual
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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Personal Vehicle Manual
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

6

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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Personal Vehicle Manual
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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Personal Vehicle Manual
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

0.00

Less Than 15 Miles

1.05

1A

0.40

15 or More Miles

1.15

1B

0.50

Business Use

1.20

2

0.90

Farm Use

0.85

3

1.50

4

2.20

Collision Relativities

Comprehensive Relativities

Symbol

2012

2011

2012

2011

13

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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Personal Vehicle Manual
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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Personal Vehicle Manual
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.

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.

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

1

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

2

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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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.

4

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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Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
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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Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
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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Chapter 9 – Traditional Risk Classification
BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
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 

P

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%

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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

$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.

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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

3

0.90

Losses/LAE for each category during the experience period are:
Territory

Marital
Status

Loss &
LAE

1

Married

$7,760

1

Single

$5,789

2

Married

$8,307

2

Single

$16,038

3

Married

$8,233

3

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.

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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.

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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.

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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.

Exam 5, V1b

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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

Page 34

 2014 by All 10, Inc.

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.

Exam 5, V1b

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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

i

pure premium/  Class

i

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

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

Page 36

 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

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 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

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.

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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.

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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.

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Chapter 10 – Multivariate Classification
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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.

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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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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.

3

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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4

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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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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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.

6

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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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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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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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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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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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.

9

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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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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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).

11

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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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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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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BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
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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BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
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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BASIC RATEMAKING – WERNER, G. AND MODLIN, C.
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 – 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.
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 hold-out dataset have the same number of exposures.
Explain whether or not the model appears to be appropriate.

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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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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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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).

2

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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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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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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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



0

H

LAS ( H )   xf ( x)dx  H  f ( x)dx

The ILF for the limit H is represented as follows: ILF ( H ) 

H



0
B

H


0

B

 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.

3

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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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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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

Exam 5, V1b

=
=
=

(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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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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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) 



D

0



D

0



xf ( x)dx  D  f ( x)dx
D



xf ( x)dx  D  f ( x)dx





0

D

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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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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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
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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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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
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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
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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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6

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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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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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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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

Exam 5, V1b

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
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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

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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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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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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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%

(c) (1 point) This question is no longer applicable to the content covered in this chapter

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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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

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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?

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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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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.

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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.

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Chapter 11 – Special Classification
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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.

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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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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
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.

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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

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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

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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 

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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.

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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.

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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

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Pure prem
rate per $100
1.083

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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

F2

.

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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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.

2

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:

Exam 5, V1b

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







2

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 
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))

Exam 5, V1b

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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).

3

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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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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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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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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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

B

C

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:

C=

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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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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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 p