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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. Exam 5, V1a Page 1 2014 by All 10, Inc. 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. Exam 5, V1a Page 2 2014 by All 10, Inc. Chapter 1 - Introduction BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Underwriting Expenses (U/W expenses) U/W expenses (a.k.a. operational and administrative expenses) are related to acquiring and servicing policies. Four categories for classifying these expenses are: 1. Commissions and brokerage are: amounts paid to insurance agents or brokers as compensation for generating business. paid as a percentage of premium written. vary between new and renewal business based on the quality of the business written or the volume of business written or both. 2. Other acquisition costs (other than commissions and brokerage expenses) include costs associated with media advertisements and mailings to prospective insureds. 3. General expenses include the remaining expenses associated with the insurance operations and other miscellaneous costs (e.g. costs associated with the general upkeep of the home office). 4. Taxes, licenses, and fees include all taxes and miscellaneous fees paid by the insurer excluding federal income taxes (e.g. premium taxes and licensing fees) Underwriting Profit (UW Profit) Since premiums may be insufficient to pay claims and expenses, capital must be maintained to support this risk, and the insurer is entitled to earn a reasonable expected return (profit) on that capital. Two main sources of profit for insurers are UW profit and investment income (II). 1. UW profit (i.e. operating income) is the total profit from all policies (a.k.a. income minus outgo). 2. II is generated from funds invested in securities held by the insurer. See chapter 7 to see how UW expense provisions are derived and how it’s incorporated in the ratemaking process. 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. Exam 5, V1a Page 3 2014 by All 10, Inc. 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. Exam 5, V1a Page 4 2014 by All 10, Inc. Chapter 1 - Introduction BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Severity (a measure of the average cost of claims): Severity Total Losses Number of Claims Assume total loss dollars are $300,000,000 and the number of claims is 100,000. Then severity is $3,000 (= $300,000,000 / 100,000). Values used in the numerator and denominator do vary: For example: Paid severity is calculated using paid losses on closed claims divided by closed claims. Reported severity is calculated using reported losses and reported claims. ALAE may be included or excluded from the numerator. Analyzing changes in severity: provides information about loss trends and highlights the impact of any changes in claims handling procedures. Pure Premium (or Loss Cost or Burning Cost): (a measure of the average loss per exposure) Pure Premium = Total Losses = Frequency x Severity Number of Exposures Pure premiums are the portion of the risk’s expected costs that is “purely” attributable to loss. Assume total loss dollars are $300,000,000 and the number of exposures is 2,000,000. Then pure premium is $150 (= $300,000,000 / 2,000,000) = 5.0% x $3,000. Pure premium is often calculated using reported losses (or ultimate losses) and earned exposures, and reported losses may or may not include ALAE and/or ULAE. Changes in pure premium show industry trends in overall loss costs due to changes in both frequency and severity. Average Premium While the pure premium focuses on the loss portion of the fundamental insurance equation, the average premium focuses on the premium side of the ratio. Average Premium = Total Premium No. of Exposures Let total premium equal $400,000,000 and total exposures equal 2,000,000 Then average premium is $200 (=$400,000,000 / 2,000,000). Note: premium and exposures must be on the same basis (e.g., written, earned, or in-force). Changes in average premium, adjusted for rate changes, show changes in the mix of business written (e.g., shifts toward higher or lower risk characteristics reflected in rates). Exam 5, V1a Page 5 2014 by All 10, Inc. 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. Exam 5, V1a Page 6 2014 by All 10, Inc. 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. Exam 5, V1a Page 7 2014 by All 10, Inc. 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 Exam 5, V1a Page 8 2014 by All 10, Inc. 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. Exam 5, V1a Page 9 2014 by All 10, Inc. 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." Exam 5, V1a Page 10 2014 by All 10, Inc. 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 Exam 5, V1a Page 11 2014 by All 10, Inc. 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% Exam 5, V1a Page 12 2014 by All 10, Inc. Chapter 1 - Introduction BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2011 Question 8 – Model Solution 3 OER = LAE/E Premium + UW Expense Ratio Underwriting expense ratio = 33.60/280 + 9.8/280 + 36.96/308 = 0.275 Combined ratio = Loss Ratio (1 + 0.082) + UW Expense/Written premium UW Expense/Written Premium = [33.60 + 9.8 + 36.96]/280 = 0.287 Combined ratio = LR(1.082) + 0.287 Solve for LR: LR = 0.65896 CR = 1.0 = 0.65896 + LAE/Earned premium + 0.287 Solve for LAE/EP: LAE/Earned Premium = 0.054 So operating expense ratio = 0.054 + 0.275 = 0.329 Questions from the 2012 exam 10a. (1 point) Werner and Modlin state that "It is important to consider the [fundamental insurance] equation at the individual or segment level" in addition to the aggregate level. Discuss two reasons it would be acceptable to maintain an imbalance in the fundamental insurance equation at the individual or segment level. Question 10 Model - Solution 1 – part a 1. Maintain competitive position. If changing rates would hurt your competitive position then it may be acceptable to take less of a change and have an unbalanced Fund. Ins Equation -> In other words hurting retention enough to offset increase. 2. If the relative cost of the change outweighs the benefit. If the operational cost of changing rating algorithms or data collection processes outweigh the change in premiums associated with the change then it could be appropriate to have an unbalanced Fund. Ins Equation Question 10 Model - Solution 2 – part a 1. It might due to a regulatory constraint. The regulator restrict the rate change (e.g. capped at +/- 25%) 2. Marketing Constraint. If the company’s marketing objective is to increase the market share on age group 50-55 drivers, it may reduce rate to attract this group of insureds. Company may have look at the long term profitability of the book using an asset share pricing technique. Examiners Comments This part of question was generally answered well. Common answers that received credit included marketing considerations (riding the market cycle, competitor pressure), regulatory considerations (e.g. cap on rate changes, restrictions on rating variables), and an asset share pricing approach that anticipates future profits at the expense of initial costs. Exam 5, V1a Page 13 2014 by All 10, Inc. Chapter 1 - Introduction BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2012 exam 10b. (1.5 points) Reconcile an imbalance in the fundamental insurance equation with the following quote from the Statement of Principles Regarding Property & Casualty Insurance Ratemaking: "A rate provides for the costs associated with an individual risk transfer." Question 10 - Model Solution – part b An actuarially sound indication many not always be implemented since an insurance company needs to balance other objectives, such as marketing, then actuarially balancing premium and loss. The actuary is allowed to deviate from this principle under influence of management, with the proper disclosure. Additionally asset sharing pricing techniques have demonstrated that under certain circumstances, it is ultimately profitable to write business that currently produce a net loss. Examiners Comments Part b was not answered well. By far the most common response was a mathematical balancing of the fundamental insurance equation, either by raising the premium or lowering expenses. However, the question was asking candidates to justify their reasoning for an imbalanced fundamental insurance equation from part A in light of the actuarial standards of practice. Successful candidates acknowledged that actuarial rate indications can balance the fundamental insurance equation but that management may decide to choose premiums that differ from actuarial indications, or that regulatory restrictions supersede all actuarial standards of practice. Exam 5, V1a Page 14 2014 by All 10, Inc. Chapter 2 – Rating Manuals BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 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. Exam 5, V1a Page 15 2014 by All 10, Inc. Chapter 2 – Rating Manuals BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 2 Rate Pages 14 - 15 Rate pages contain inputs (e.g. base rates, rating tables, and fees) to calculate premium. A base risk is a risk profile pre-defined by the insurer. The base risk can be a set of common risk characteristics or can be chosen based on marketing objectives. Example 1: The base risk for personal auto collision coverage may be an adult, married male, with a $500 deductible, who lives in a very populated area, etc. The insurer may have an objective to encourage new insureds to purchase a deductible of $500 or higher (even though it may have more policies with a $250 deductible). If the base is set at the $500 deductible, it will be used in the initial premium quote. But if the insured requests a comparison quote with a $250 deductible, a higher premium will result (relative to using a base set at a $250 deductible), which may deter the insured psychologically. Example 2: A multi-product discount for homeowners who have an auto policy with the same insurer. If the insurer sets the base equal to those who qualify for the discount, then there will be an increase in premium for those who do not qualify for the discount. Although the premium charged is the same whether buying a single or multi-product discount, a discount has more positive appeal than an increase in premium. The base rate is the rate that applies to the base risk (and is usually not the average rate). If the product contains multiple coverages priced separately (as in personal auto insurance), then there is a separate base risk, base rate, and rating tables for each coverage. Rates for all risk profiles, other than the base profile, will vary from the base rate. The rate variation for different risk characteristics occurs by modifying the base rate (e.g. applying multipliers, addends, etc. in the rating algorithm). Characteristics are rating variables (a.k.a. discounts/surcharges or credits/debits) and the rate variations are contained in rating tables. The variations from the base rate are referred to as relativities, factors, or multipliers (if applied to the rating algorithm multiplicatively) or addends (if applied to the base rate or some other figure in an additive or subtractive manner). Rating Variables for various lines of insurance are as follows: Type of Insurance Rating Variables Personal Automobile Driver Age and Gender, Model Year, Accident History Homeowners Amount of Insurance, Age of Home, Construction Type Workers Compensation Occupation Class Code Commercial General Liability Classification, Territory, Limit of Liability Medical Malpractice Specialty, Territory, Limit of Liability Commercial Automobile Driver Class, Territory, Limit of Liability Rate pages contain all the components needed to calculate rates. Exam 5, V1a Page 16 2014 by All 10, Inc. Chapter 2 – Rating Manuals BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Expenses: The premium charged must consider expenses incurred in acquiring and servicing policies. Some expenses vary by the amount of premium (e.g. commission is usually a % of the premium) Some expenses are fixed regardless of the premium (e.g. the cost of issuing a policy). An insurer may include an explicit expense fee in the rating algorithm to account for fixed expenses and incorporate a provision within the base rate to account for variable expenses. Otherwise, an insurer may incorporate all expenses via a provision within the base rates. In this case, the insurer may have a minimum premium so that the premium charged is adequate to cover expenses and an amount for minimal expected losses. 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. Exam 5, V1a Page 17 2014 by All 10, Inc. Chapter 2 – Rating Manuals BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Note: While UW criteria has been historically subjective in nature, there has been a trend over time (especially for personal lines products) to designate new explanatory variables as UW criteria, which can then be used for placement into rating tiers or separate companies. The trend to designate new explanatory variables as UW criteria has given some companies a competitive advantage by reducing the transparency of the rating algorithm. Examples of Underwriting Characteristics used in Various Lines of Insurance Type of Insurance Underwriting Characteristics Personal Automobile Insurance Credit Score, Homeownership, Prior Bodily Injury Limits Homeowners Insurance Credit Score, Prior Loss Information, Age of Home Workers Compensation Safety Programs, Number of Employees, Prior Loss Information Commercial General Liability Insurance Credit Score, Years in Business, Number of Employees Medical Malpractice Patient Complaint History, Years Since Residency, Number of Weekly Patients Commercial Automobile Driver Tenure, Average Driver Age, Earnings Stability 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. Exam 5, V1a Page 18 2014 by All 10, Inc. Chapter 2 – Rating Manuals BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Amount of Insurance (AOI) Rating Table AOI (in thousands) $ 80 $ 95 ::: $170 $185 $200 $215 ::: $410 $425 $440 $455 $470 $485 $500 Additional $15K Rate Relativity 0.56 0.63 ::: 0.91 0.96 1.00 1.04 ::: 1.51 1.54 1.57 1.60 1.63 1.66 1.69 0.03 If a policyholder purchases $425,000 of insurance for his home, a rate relativity of 1.54 is applied to the base rate. Straight-line interpolation is used for values not listed in the table. Territory The location of the home is a key rating variable. Homeowners insurers group similar geographic units (e.g. zip codes) to form rating territories. WGIC grouped zip codes into five distinct rating territories (with rate relativities shown below). Territory 3 is the base territory (and thus has a relativity of 1.00) and all other territories are expressed relative to Territory 3. Territory Rate Relativity 1 0.80 2 0.90 3 4 5 1.00 1.10 1.15 Protection Class and Construction Type WGIC’s homeowners rates vary by fire protection class and construction type. Class 1 indicates the highest quality protection while class 10 refers to the lowest quality protection. Within each class, there is a separate relativity based on construction type (frame and masonry). Frame construction is more susceptible to loss than masonry and therefore frame relativities are higher than the masonry relativities across every protection class. The base rate for this two-way variable is Protection Class 1-4 Frame (although Protection Class 5 Masonry coincidentally has a relativity of 1.00). Exam 5, V1a Page 19 2014 by All 10, Inc. Chapter 2 – Rating Manuals BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Protection Class / Construction Type Rating Table Protection Class Construction Type Frame Masonry 1-4 1.00 0.90 5 1.05 1.00 6 1.10 1.05 7 1.15 1.10 8 1.25 1.15 9 2.10 1.75 10 2.30 1.90 Underwriting Tier WGIC uses UW characteristics (used to place insurance policies into one of four distinct underwriting tiers based on the overall riskiness of the exposure to loss) that are not explicitly shown in the rating manual. Underwriting Tier Rating Table Tier Rate Relativity A 0.80 B 0.95 C 1.00 D 1.45 Tier D is considered the most risky and has the highest rate relativity. Deductible Policyholders choose their deductible. Rate relativities for each deductible are shown in the table below. Deductible Rate Relativity $250 1.00 $500 0.95 $1,000 0.85 $5,000 0.70 Miscellaneous Credits Wicked Good offers the following discounts: Miscellaneous Credit Credit Amount New Home Discount 20% 5-Year Claims-Free Discount 10% Multi-Policy Discount 7% Insurers offering a large number of discounts will have a maximum discount percentage that can be used, however Wicked Good does not limit the overall cumulative discount. Exam 5, V1a Page 20 2014 by All 10, Inc. Chapter 2 – Rating Manuals BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Additional Optional Coverages The basic homeowners policy includes: i. a $100,000 limit for liability coverage and a $500 limit for medical coverage (this split limit is often expressed as $100,000/$500). ii. a $2,500 inside limit to jewelry losses within the contents coverage. The following tables show the additional premium charged if the policyholder elects to purchase additional higher limits: Jewelry Coverage Rate Limit Additive $ 2,500 Included $ 5,000 $35 $10,000 $60 Liability/Medical Rate Limit Additive $100,000/$500 Included $300,000/$1,000 $25 $500,000/$2,500 $45 Expense Fee WGIC has an explicit expense fee to cover fixed expenses incurred in the acquiring and servicing policies. The expense fee is $50 per policy as shown in the table below. Policy Fee $50 Homeowners Rating Algorithm for WGIC The rating algorithm to calculate the final premium for a homeowners policy for WGIC is: Total Premium = All-Peril Base Rate x AOI Relativity x Territory Relativity x Protection Class / Construction Type Relativity x Underwriting Tier Relativity x Deductible Credit x [1.0 - New Home Discount – Claims-Free Discount] x [1.0 - Multi-Policy Discount] + Increased Jewelry Coverage Rate + Increased Liability/Medical Coverage Rate + Policy Fee. Rounding is common and WGIC rounds to the penny after each step and to the whole dollar at the final step. Exam 5, V1a Page 21 2014 by All 10, Inc. Chapter 2 – Rating Manuals BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Homeowners Rate Calculation Example for WGIC WGIC is preparing a renewal quote for a homeowner with the following risk characteristics: • Amount of insurance = $215,000 • The insured lives in Territory 4. • The home is frame construction located in Fire Protection Class 7. • Based on the insured’s credit score, tenure with the company, and loss history, the policy is in UW Tier C. • The insured opts for a $1,000 deductible. • The home falls under the definition of a new home as defined in Wicked Good’s rating rules. • The insured is eligible for the five-year claims-free discount. • There is no corresponding auto or excess liability policy written with WGIC. • The insured is eligible for the five-year claims-free discount. • There is no corresponding auto or excess liability policy written with WGIC. • The policyholder opts to increase coverage for jewelry to $5,000 and to increase liability/medical coverage limits to $300,000/$1,000. Entries from Rating Manual Base Rate $500 AOI Relativity 1.04 Territory Relativity 1.10 Protection Class / Construction Type Relativity 1.15 Underwriting Tier Relativity 1.00 Deductible Credit 0.85 New Home Discount 20% Claims-Free Discount 10% Multi-Policy Discount 0% Increased Jewelry Coverage Rate $35 Increased Liability/Medical Coverage Rate $25 Expense Fee $50 The rating algorithm from the rating manual can be applied to calculate the final premium for the policy: $501 $500 *1.04 *1.10 *1.15 *1.00 * 0.85 *[1.0 - 0.20 - 0.10]*[1.0 - 0] $35 $25 $50. Exam 5, V1a Page 22 2014 by All 10, Inc. Chapter 2 – Rating Manuals BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 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 Exam 5, V1a Page 23 2014 by All 10, Inc. Chapter 2 – Rating Manuals BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Claims-free Discount Individual insureds who have been with WGIC for at least three consecutive years preceding the effective date of the current policy may qualify for a claims-free discount. To qualify, the individual insured cannot have cumulative reported losses in X/S of $5,000 over the prior 3 years. The amount of the claims-free discount is 15%. Schedule Rating (SR) Commercial lines insurers incorporate SR into their rating algorithms to adjust manual premium based on objective criteria or underwriter judgment. WGIC’s schedule rating plan includes the following credits and debits. A. Continuing Education – A credit of up to 25% for attendance at approved continuing education courses and seminars. The total hours spent at courses and seminars must be at least 15 hours in the prior 12 months. B. Procedure – A debit of up to 25% for nurses who have professional licenses and/or scope of practice in high-risk exposure areas such as invasive surgery or pediatric care. C. Workplace Setting – A debit of up to 25% for nurses that work in high-risk workplace settings (e.g. surgical centers and nursing homes). A maximum aggregate schedule rating credit or debit of 25% is used by WGIG. Limit Factors WGIC offers different per claim and annual aggregate limits for its Nurse’s Professional Liability program. The following are relativities corresponding to each limit option: Limit Rating Table Limit Option Rate Relativity $100K/$300K 0.60 $500K/$1M 0.80 $1M/$3M 1.00 $2M/$4M 1.15 WGIC pays all ALAE in addition to the limit shown. Deductible Deductible options available to the insured reduce premium and the associated credit are shown below. Deductible Rating Table Deductible (Per Claim) Credit None 0% $1,000 5% $5,000 8% Exam 5, V1a Page 24 2014 by All 10, Inc. Chapter 2 – Rating Manuals BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Claims-made Factor WGIC writes claims-made MM policies as opposed to occurrence policies. For CM policies, the coverage trigger is the date the claim is reported rather than the date the event occurs. A policyholder who buys a CM policy for the first time is only offered coverage for claims occurring after the start of the policy and reported during the year. When the CM policy is renewed, coverage is provided for claims occurring after the original inception date and reported during the policy period. Also, an extended reporting endorsement covers claims that occur during the coverage period but are reported after the policy terminates (e.g. a doctor who retires may purchase an extended reporting endorsement to cover claims reported after the MM policy terminates). The extended reporting endorsement factors adjust the premium based Years of Prior Claims-made Coverage. See Chapter 16 for more details on CM coverage. Claims-Made Maturity Factors Maturity Factor 1st Year 0.200 2nd Year 0.400 3rd Year 0.800 4th Year 0.900 5th Year 0.950 6th Year 0.975 Mature 1.000 Extended Reporting Endorsement Factors Years of Prior Factor Claims-made Coverage 12 Month 0.940 24 Month 1.700 36 Month 2.000 48 Month 2.250 60 Month 2.400 Group Credit The size of the credit depends on the number of nurses that are insured under the policy. Group Credit Number of Credit Nurses 1 0% 2 – 14 5% 15+ 10% The final premium (including the group credit) should be calculated for each nurse and aggregated for all professionals to determine the premium for the group policy. Exam 5, V1a Page 25 2014 by All 10, Inc. Chapter 2 – Rating Manuals BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Minimum Premium The rating manual specifies that the minimum premium for each nurse, after all discounts, is $100. Medical Malpractice Rating Algorithm for WGIC Rating variables are applied multiplicatively, not additively, in consecutive order. Premium is rounded to the nearest penny after each step and to the nearest dollar amount at the end to determine the final premium per professional. Total Premium per Professional = [Max of Min Premium in the rating manual of $100 or (Base Rate per Nurse x Specialty Relativity x Part-time Status Relativity x Territory Relativity x (1.0 - Claims-free Discount) x (1.0 +/- Schedule Rating Debit/Credit) x Limit Relativity x (1.0 - Deductible Credit) x Claims-made Factor x (1.0 - Group Credit ))] The total policy premium for a policy with multiple professionals is the sum of the premium for the individual professionals on the policy. Medical Malpractice Rate Calculation Example for WGIC A practice of five nurses applied for MM coverage with WGIC. Quoted premium was $6,500 for a single policy covering the five professionals. The practice has recently added a psychiatric nurse, and has requested a new quote from WGIC to cover all six professionals on a single policy. Assume the following characteristics: The new nurse is an employed professional who works 15 hours per week. He was previously covered by an occurrence policy and is applying for a CM policy with WGIC. He practices in Wicked Good’s Territory 3. He attended five hours of approved continuing education courses in the prior 12 months. He holds a professional license in senior care, which is considered high risk. He also works in a senior care facility. The underwriter has chosen to apply debits of 25% for each of these criteria, but the maximum aggregate debit allowable is 25%. The policy has $1M/$3M of coverage with a $1,000 deductible per claim. Exam 5, V1a Page 26 2014 by All 10, Inc. Chapter 2 – Rating Manuals BASIC RATEMAKING – WERNER, G. AND MODLIN, C. The following rating tables from WGIC’s rating manual is used to calculate the premium Entries from Rating Manual Employed Annual Rate $2,500 Specialty Relativity 0.80 Part-time Status Relativity 0.50 Territory 3 Relativity 1.25 Schedule Rating (subject to 25% maximum) 0%+25%+25% (capped at 25%) Limit Relativity for $1M/$3M 1.00 Credit for $1000 Deductible 5% Claims-made Factor 0.20 Group Credit 5% Minimum Premium $100 Using the rating manual’s rating algorithm, the premium for the individual nurse is calculated as follows: $282 = $2,500 x 0.80 x 0.50 x 1.25 x [1.00 + 0.25] x 1.00 x [1.00 - 0.05] x 0.20 x [1.00 - 0.05]. Since this premium is greater than the minimum premium per nurse of $100, it applies The total premium for the six individuals combined is $6,782 = $6,500 + $282. 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. Exam 5, V1a Page 27 2014 by All 10, Inc. Chapter 2 – Rating Manuals BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Class Rate The classification system groups employers with similar operations and similar loss exposures based on job duties performed by the employees. The table below shows class rates for specific operations (in this case, retirement centers) that WGIC writes, and are based on the NCCI class rates, adjusted for WGIC’s expenses and perceived differences in loss potential. Class Rates Rate per $100 of Class Payroll 8810-Clerical 0.49 8825-Food Service Employees 2.77 8824-Health Care Employees 3.99 8826-All Other Employees 3.79 To calculate manual premium: determine which classes best describe the activities of the company seeking insurance. estimate the amount of exposure ($100s of payroll) expected for each class during the policy period using the insured’s data. multiply the rate per $100 of payroll by the estimated payroll for each class, and aggregate across all classes for which the prospective insured has exposures to compute manual premium. Rating and Underwriting Characteristics Experience Rating (ER) Manual rates are averages reflecting the usual conditions found in each class. Manual rates are adjusted using ER to reflect that each risk within a class is different to some extent in terms of loss potential. ER applies for larger policies (which are believed to have more stable loss experience) and NCCI designates minimum aggregate manual premium for a company to be eligible for ER. Regulators mandate that ER be used if the employer meets the industry eligibility requirements. When using ER, manual premium is adjusted upward if the actual losses for the company are higher than expected and vice versa. See Chapter 15 for more information on ER. WGIC only insures small companies and thus ER is not applicable to its insureds. Schedule Rating (SR) WGIC has a set of credits and debits that require the underwriter to apply judgment in the UW process. The underwriter uses judgment (based on experience and internal guidelines) to select a value between the maximum and minimum for each attribute that may apply for an insured’s workplace operations. The range of schedule credits and debits that WG’s underwriters can apply is shown below: The overall maximum credit or debit that an underwriter can apply to a single policy is 25%. The policy must have an annual manual premium of at least $1,000 to qualify for schedule rating. Exam 5, V1a Page 28 2014 by All 10, Inc. Chapter 2 – Rating Manuals 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. Exam 5, V1a Page 29 2014 by All 10, Inc. Chapter 2 – Rating Manuals BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Workers Compensation Rating Algorithm for WGIC The rating algorithm to calculate the final premium for a given policy using the aforementioned rating manual variables is as follows: Total Premium = Higher of N [ (Classi rate x $ Payroll for classi / 100) where N number of classes i 1 x (1.0+ Schedule Rating Factor) x (1.0- Pre-Employment Drug Screening Credit) x (1.0- Employee Assistance Program Credit) x (1.0- Return-to-Work Program Credit) + Expense Constant] and, the Minimum Premium specified in the rating manual ($1,500 in WGs case). Premium is rounded to the nearest penny after each step and to the nearest dollar amount at the end to determine the total premium (as stated in the manual) ER factors and premium discounts do not appear in WGIC’s rating algorithm because these rating variables do not apply to its book of business. Workers Compensation Rate Calculation Example for WGIC A retirement living center with the following employee classes groups has requested a quote. Payroll by Class Class Payroll 8810 – Clerical $35,000 8825 - Food Service Employees $75,000 8824 - Health Care Employees $100,000 8826 - All Other Employees & Salespersons, Drivers $25,000 The center has trained its entire staff in first aid and first aid equipment is available in the building. The center has been inspected by Wicked Good and the premises are clean and well-maintained. The center requires all employees to be drug-tested prior to employment. Steps in computing manual premium. Step 1: Compute aggregate manual premium. Manual Premium by Class Class Payroll Payroll/$100 8810 Clerical 8825 - Food Service Employees 8824 - Health Care Employees 8826 - All Other Employees Total (1) $35,000 $75,000 $100,000 $25,000 $235,000 (2)=(1)/100 $350 $750 $1,000 $250 Rate per $100 of Class Manual Payroll Premium (3) (4)=(2)*(3) 0.49 $171.50 2.77 $2,077.50 3.99 $3,990.00 3.79 $947.50 $7,186.50 Total manual premium for the policy is $7,186.50 = $171.50 + $2,077.50 + $3,990.00 + $947.50. Exam 5, V1a Page 30 2014 by All 10, Inc. Chapter 2 – Rating Manuals BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Step 2: Underwriter determination of the following credits that should apply based on the retirement living center’s characteristics: Schedule Rating Modifications Modification Premises Classification Medical Safety Employees — Management Peculiarities Facilities Devices Selection, —Safety Training, Organization Supervision -10% 0% 0% -2.5% -5% 0% The total credit (reduction to manual premium) for SR is 10% + 2.5% + 5% = 17.5%. The credit takes into account the first aid equipment, staff training, and cleanliness of the premises. Since the credit is less than the maximum allowable credit of 25%, the entire 17.5% credit is applied to the manual premium. The schedule rating factor applied to manual premium is 0.825 =1.000 - 0.175. Step 3: Determine the following other factors that apply to the policy: Entries from Wicked Good’s Rating Manual Entries from Rating Manual Pre-employment Drug Screening Credit 5% Employee Assistance Program Credit 0% Return-to-Work Program Credit 0% Expense Constant $150 The Employee Assistance Program credit and Return-to-Work credit do not apply to the policy because the center does not have those programs. Thus, the total premium for the policy is $5,782 = $7,186.50 x 0.825 x (1.0 - 0.05) x (1.0 - 0) x (1.0 - 0) + $150. Since $5,782 is greater than the minimum premium per policy of $1,500, the total premium for the policy is $5,782. 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 Exam 5, V1a Page 31 2014 by All 10, Inc. Chapter 3 – Ratemaking Data BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Sec 1 2 3 4 Description Introduction and Internal Data Data Aggregation External Data Key Concepts Pages 36 - 42 42 - 44 44 - 47 47 - 47 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. Exam 5, V1a Page 32 2014 by All 10, Inc. Chapter 3 – Ratemaking Data BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Risk Data Insurer databases record policy exposure and premium separately from losses in a claims database, however the ratemaking analysis ultimately requires linking this information for ratemaking purposes. Policy Database A policy database captures records (i.e. individual policies or some subdivision of the policy) and fields (i.e. explanatory information about the record). A record is defined in a product’s policy database depending upon what exposure measure is used and how premium is calculated. Examples of policy database organization for different lines of business: In homeowners insurance, a record may be a home for an annual policy period. In U.S. WC insurance, rating is based on the payroll of industry classes so separate records are maintained at the class level. In personal auto insurance, separate records are created for: i. each coverage (though this could be handled via a coverage indicator field in the database). ii. each auto on a policy (if multiple autos are insured on one policy) or separate records may be maintained for individual operators on each auto. Example: An auto policy insuring two drivers on two cars for six coverages could involve 24 records (or four records if coverage is handled as a field). In addition, records are also subdivided according to any changes in the risk(s) during the policy period (i.e. if a policy is amended during the policy term, separate records are created for the partial policy periods before and after the change). See the examples provided later in this summary to better illustrate this. Fields often present for each record in the policy database are: • Policy identifier • Risk identifier(s): When there are multiple risks on a policy, unique risk identifiers are required (e.g. vehicle number and operator number may be necessary for personal auto databases). • Relevant dates: While each record contains the effective and expiration dates for the policy or coverage, separate records are maintained for individual risks and/or individual coverages on the policy, and the start date of each risk/coverage is recorded. (e.g. if collision coverage for a new car is added to an existing auto policy, a record is added with the relevant start date noted). • Premium: If the line of business has multiple coverages, premium is recorded by coverage as a separate record or via a coverage indicator field. (e.g. personal auto databases track premium separately for bodily injury, property damage, comprehensive, collision and earned and in-force premium can be calculated from the data on record). • Exposure: Is typically the written exposure but it can be recorded by coverage. • Characteristics: Include rating variables, UW variables, etc. Some characteristics describe the policy as a whole (e.g., the policy origination year), while others describe individual risks (e.g. make/model of automobile) and consequently vary between different records on the same policy. Exam 5, V1a Page 33 2014 by All 10, Inc. Chapter 3 – Ratemaking Data BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Example: Homeowners policies used to construct a policy database: Policy A is written on 1/1/2010 with an annual premium of $1,100. The home is located in Territory 1 and the insured has a $250 deductible. The policy remains unchanged for the full term of the policy. Policy B is written on 4/1/2010 with an annual premium of $600. The home is located in Territory 2 and the insured has a deductible of $250. The policy is canceled on 12/31/2010. Policy C is written on 7/1/2010 with an annual premium of $1,000. The home is located in Territory 3 and has a deductible of $500. On 1/1/2011, the insured decreases the deductible to $250. The full annual term premium after the deductible change is $1,200. Policy database construction: Policy A can be represented with one record since expired at its original expiration date and had no changes. Policy B is represented by two records because it was canceled before the policy expired. The first record for contains information known at policy inception (e.g. one exposure and $600 in WP). The second record represents an adjustment for the cancellation such that when aggregated, the two records show a result net of cancellation. As the policy was canceled 75% of the way through the policy period, the second record should show -0.25 exposure and -$150 (=25% x -$600) of written premium. Policy C is represented by three records since it has a mid-term adjustment The first record includes all the information at policy inception. The second record negates the portion of the original policy that is unearned at the time of the amendment (i.e. -0.50 exposure and -$500 premium and deductible equal to $500). The third record represents the information applicable to the portion of the policy written with the new deductible (i.e. +0.50 exposure and +$600 premium and deductible equal to $250). Policy Database Original Original Transaction Effective Termination Effective Policy Date Date Date A B B C C C 01/01/10 04/01/10 04/01/10 07/01/10 07/01/10 07/01/10 12/31/10 03/31/11 03/31/11 06/30/11 06/30/11 06/30/11 01/01/10 04/01/10 12/31/10 07/01/10 01/01/11 01/01/11 Ded Other Written Written Terr Chars Exposure Premium $250 $250 $250 $500 $500 $250 1 2 2 3 3 3 … … … … … … 1.00 1.00 -0.25 1.00 -0.50 0.50 $1,100 $600 -$150 $1,000 -$500 $600 This is ordered by policy rather than transaction effective date. In a more sophisticated data capture, information for: Policy B would be aggregated to one record that shows a “net” exposure of 0.75 and “net” written premium of $450. Policy C would be aggregated to two records representing before and after the deductible change. The first record would reflect the period of time with the $500 deductible and would have a “net” exposure of 0.50 and “net” written premium of $500. The second record reflecting the period of time with the $250 deductible would be identical to the third record in the original example. The exposure is 0.50 and written premium is $600. This type of transaction aggregation is required for statistical ratemaking analysis (e.g. GLMS see Chapter 10). Exam 5, V1a Page 34 2014 by All 10, Inc. Chapter 3 – Ratemaking Data BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Claims Database Each record represents a transaction tied to a specific claim (e.g. a payment or a change in reserve). Claims involving multiple coverages or causes of loss may be represented as separate records or via indicator fields. Fields often present for each record in a claims database are as follows: • Policy identifier • Risk identifier(s): If relevant, the claim database contains a way to identify the risk that had the claim. This will be necessary to match the claim to the corresponding record in the policy database. • Claim identifier: The claim database contains a unique identifier for each specific claim. This same identifier is used if the claim has multiple claim transaction records. • Claimant identifier: The claim database contains a unique identifier for each specific claimant on a particular claim. • Relevant loss dates: includes fields for the date of loss, the date the company was notified of the loss (i.e. the report date), and the date of the transaction for the specific record (e.g. date of a loss payment, reserve change, or claim status change). • Claim status: Tracks whether the claim is open (i.e. still an active claim) or closed (i.e. has been settled). For some policies, it may be common for claims to be re-opened, and it may be advantageous to add the re-opened and re-closed status descriptions. • Claim count: Identifies the number of claims by coverage associated with the loss occurrence. Alternatively, if each record or a collection of records defines a single claim by coverage, aggregating claim counts can be accomplished without this explicit field. • Paid loss: Captures the payments made for each claim record. If there are multiple coverages, perils or types of loss, the loss payments can be tracked in separate fields or separate records. If the product is susceptible to catastrophic losses (e.g. hurricanes for property coverage), then catastrophic payments are tracked separately either through a separate record or an indicator included on the record. • Event identifier: Identifies any extraordinary event (e.g. catastrophe) involving this particular claim. • Case reserve: Includes the case reserve or the change in the case reserve at the time the transaction is recorded (e.g. if a payment of $500 is made at a particular date, and this triggers a simultaneous change in the case reserve, a record is established for this transaction and the paid loss and case reserve fields are populated) The case reserve is recorded in separate fields or records by coverage, peril or type of loss and by catastrophe or non-catastrophe claim, if applicable (as with paid losses). • Allocated loss adjustment expense: If ALAE can be subdivided into finer categorization, additional fields may be used accordingly. Insurers may not set ALAE reserves and only payments are tracked on the database. If a case reserve for ALAE is set, it is maintained in the database, captured separately by coverage or peril and by catastrophe or non-catastrophe, if applicable. ULAE cannot be assigned to a specific claim and are handled elsewhere. • Salvage/subrogation: If an insurer replaces property, it assumes ownership of the damaged property, which may then be reconditioned and sold to offset part of the payments made for the loss; these recoveries are called salvage. When an insurer pays for an insured’s loss, the company receives the rights to subrogate (i.e. to recover any damages from a third party who was at fault to the loss event). Any salvage or subrogation that offsets the loss is tracked and linked to the original claim, if possible. • Claim characteristics: Insurers may collect characteristics associated with the claims (e.g. type of injury, physician information). While studying the impacts of these characteristics on average claim size may be interesting for certain purposes (e.g. loss reserve studies), only characteristics known for every policyholder at the time of policy quotation are usable in the rating algorithm. V Exam 5, V1a Page 35 2014 by All 10, Inc. Chapter 3 – Ratemaking Data BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Example: Homeowners policies used to construct a claims database: The following example policies can help clarify the data requirements. • Policy A: A covered loss occurs on 1/1/2010. The claim is reported to the insurer on 1/15/2010, and an initial case reserve of $10,000 is established. An initial payment of $1,000 is made on 3/1/2010, with a corresponding $1,000 reduction in the case reserve. A final payment of $9,000 is made on 5/1/2010, and the claim is closed. • Policy B: No claim activity. • Policy C: A covered loss occurs on 10/1/2010, is reported on 10/15/2010, and a case reserve of $18,000 is established. The insurer makes a payment of $2,000 on 12/15/2010, and reduces the case reserve to $17,000. An additional payment of $7,000 is made on 3/1/2011, and the case reserve is reduced to $15,000. The claim is closed on 3/1/2012, when the insurer makes a final payment of $15,000 and receives a $1,000 salvage recovery by selling damaged property. • Policy C: A second loss occurs on 2/1/2011. The claim is reported on 2/15/2011, and an initial reserve of $15,000 is set. On 12/1/2011, the company pays a law firm $1,000 for fees related to the handling of the claim. The claim is closed on that date with no loss payments made. Claims database construction: The claim from Policy A generates 3 separate records: one when the claim is reported and the initial reserve is set, one when the first payment is made, one when the last payment is made. There are no claim records for Policy B as no claims were reported. The two claims from Policy C generate six records: For claim 1, one record when the claim is reported and the initial reserve is set, and three for the three different dates that payments and reserve adjustments are made. For claim 2, one record on the date it is reported and the initial reserve is set and a subsequent record on the date the claim is closed. Claim Database Policy Claim Accident Report Transaction Claim 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. Exam 5, V1a Page 36 2014 by All 10, Inc. Chapter 3 – Ratemaking Data BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Accounting Information Some required data for ratemaking is not specific to any one policy. The salary of the CEO is an expense that cannot be allocated to line of business or individual policy. UW expenses and ULAE fall into this category and should be tracked at the aggregate level. UW expenses (incurred in acquiring and servicing policies) include general expenses, other acquisition expenses, commissions and brokerage, and taxes, licenses, and fees. Commissions can be assigned to specific policies. General expenses (e.g. costs associated with the company’s buildings, and other acquisition expenses like advertising costs) cannot be assigned to a specific claim and are tracked at the aggregate level. Loss adjustment expenses (LAE) are expenses incurred in the process of settling claims. Allocated loss adjustment expenses (ALAE) are directly attributable to a specific claim and are captured on the claim record. Unallocated loss adjustment expenses (ULAE) cannot be assigned to a specific claim, and include items like the cost of a claim center or salaries of employees responsible for maintaining claims records. Since ULAE cannot be assigned to a specific claim, these are tracked at the aggregate level. Insurers track UW and ULAE expenses paid by calendar year. Subdivision to line of business (LOB) and state may be approximated. Aggregate figures are used to determine expense provisions used in the ratemaking process. 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. Exam 5, V1a Page 37 2014 by All 10, Inc. Chapter 3 – Ratemaking Data BASIC RATEMAKING – WERNER, G. AND MODLIN, C. CY aggregation captures premium and loss transactions during a 12-month CY (without regard to policy effective date, accident date, or report date of the claim). CY earned premium (EP) and earned exposure are those earned during a 12 month period. At CY end, all premium and exposures are fixed. CY paid losses include all loss paid during the CY regardless of occurrence date or report date. CY Reported losses = paid losses + the change in case reserves during that twelve-month CY. At the end of the CY, all reported losses are fixed. Advantage of CY aggregation: data is quickly available at CY end. CY data is used for financial reporting so there is no additional expense to aggregate the data this way for ratemaking purposes. Disadvantage of CY aggregation: the mismatch in timing between premium and losses. CY EP come from policies in force during the year (written either in the previous or the current CY). Losses, however, may include payments and reserve changes on claims from policies issued years ago. CY year aggregation for ratemaking analysis may be most appropriate for lines of business or individual coverages in which losses are reported and settled relatively quickly (e.g. homeowners). AY aggregation of premium and exposures follows the same precept as CY premium and exposures, and thus the method is often referred to as CY-AY or FY-AY. AY aggregation of losses considers losses for accidents that have occurred during a twelve-month period, regardless of when the policy was issued or the claim was reported. AY paid losses include loss payments only for those claims that occurred during the year. AY reported losses = loss payments + plus case reserves only for those claims that occurred during the year. At AY end, reported losses change as additional claims are reported, claims are paid, or reserves are changed. Advantage: AY aggregation provides a better match of premium and losses than CY aggregation. Losses on accidents occurring during the year are compared to EP on policies during the same year. Since the AY is not closed (fixed) at year end, future development on known losses needs to be estimated. Selecting a valuation date several months after year end allows the emergence of some development in the data which may improve the estimation of ultimate losses. PY aggregation (a.k.a. UW year) considers all premium and loss transactions on policies that were written during a 12-month period, regardless of when the claim occurred or was reported, reserved, or paid. All premium and exposures earned on policies written during the year are part of that policy year’s earned premium and earned exposures. Premium and exposures are fixed after the expiration date of all policies written during the year. PY paid losses include payments made on those claims covered by policies written during the year. PY reported losses = payments + case reserves only for those claims covered by policies written during the year. At PY end, losses change as additional claims occur, claims are paid, or reserves are changed. Advantage: PY aggregation represents the best match between losses and premium (since losses on policies written during the year are compared with premium earned on those same policies). Disadvantage: Data takes longer to develop than both CY and AY, since PY exposures for a product with an annual policy term are not fully earned until 24 months after the start of the PY. Exam 5, V1a Page 38 2014 by All 10, Inc. Chapter 3 – Ratemaking Data BASIC RATEMAKING – WERNER, G. AND MODLIN, C. RY aggregation is: similar to CY-AY except losses are aggregated according to when the claim was reported (as opposed to when the claim occurred). used for commercial lines products using claims-made policies (e.g. medical malpractice). See Chapter 16. Overall versus Classification Analysis When reviewing the adequacy of the overall rate level, the premium, losses, and exposures can be highly summarized (aggregated by CY, AY, PY, or RY for the product and location (e.g. state) being analyzed). If a class analysis is being performed, then the data must be at a more refined level. For a univariate classification analysis, the data can be aggregated by year (AY or PY) for each level (e.g. territory) of the rating variable being studied. For a multivariate analysis, it is preferable to organize data at the individual policy or risk level. Limited Data Actuaries are sometimes required to perform ratemaking analysis and work with the data that is available and use actuarial judgment to overcome the data deficiencies (e.g. if EP by territory normally used for an analysis of auto territorial relativities is not available actuary may use in-force premium by territory to estimate the earned premium by territory). 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. Exam 5, V1a Page 39 2014 by All 10, Inc. 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). Exam 5, V1a Page 40 2014 by All 10, Inc. Chapter 3 – Ratemaking Data BASIC RATEMAKING – WERNER, G. AND MODLIN, C. D. Other Third-Party Data (not specific to insurance) The most commonly used types are: 1. Economic data (e.g. Consumer Price Index (CPI)) Insurers may examine the CPI at the component level (e.g. medical cost and construction cost indices) to find trends relevant to the insurance product being priced. 2. Geo-demographic data (i.e. average characteristics of a particular area). i. Population density can be a predictor of accident frequency. ii. Weather indices, theft indices, and average annual miles driven. 3. Credit data is used by insurers to evaluate the insurance loss experience of risks with different credit scores. Insurers feel credit is an important predictor of risk and began to vary rates accordingly. 4. Other information related to different insurance products include: • Personal automobile insurance: vehicle characteristics, department of motor vehicle records • Homeowners insurance: distance to fire station • Earthquake insurance: type of soil • Medical malpractice: characteristics of hospital in which doctor practices • Commercial general liability: type of owner (proprietor, stock) • Workers compensation: OSHA inspection data. 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 Exam 5, V1a Page 41 2014 by All 10, Inc. 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. Exam 5, V1a Page 42 2014 by All 10, Inc. 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. Exam 5, V1a Page 43 2014 by All 10, Inc. 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. Exam 5, V1a Page 44 2014 by All 10, Inc. Chapter 3 – Ratemaking Data BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2007 exam: 53. (2.5 points) a. (1.5 points) Briefly define policy year, calendar year, and accident year loss experience. b. (0.5 point) Which of the three performs the best with respect to responsiveness? Explain. c. (0.5 point) Which of the three performs the best with respect to matching premiums and losses? Explain. CAS Model Solution a. PY: Losses are allocated to the year in which the policy was written. CY: Losses are allocated to the year in which payments were made and reserves were changed. AY: Losses are allocated to the year in which the accident occurred. b. Calendar year is the most recent and responsive because there is no delay due to developing losses. c. Policy year matches premiums and losses best because the losses are generated by the same policies for which premium was collected. Exam 5, V1a Page 45 2014 by All 10, Inc. 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. Exam 5, V1a Page 46 2014 by All 10, Inc. Chapter 4 – Exposures BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 2. Practical The exposure base should be practical, meaning it should be: 1. objective 2. relatively easy to use and 3. inexpensive to obtain and verify. The EB will be consistently measured by meeting these criteria. A well-defined and objective exposure should not be able to be manipulated (by policyholders and producers/underwriters). Moral Hazard Example: Asking a personal auto policyholder to state their estimated annual miles driven provides opportunity for dishonesty more so than the use of car-years as the exposure base. However, advances in technology may change the choice of EB for personal auto insurance. Example: Onboard diagnostic devices can accurately track driving patterns and transmit this data to insurers. Thus, some commercial long haul trucking carriers have implemented miles driven as an EB. For products liability, products currently in use is the exposure base that is most proportional to expected loss. However, it is difficult for most firms to accurately track how many of their products are actually being used during the period covered by the insurance policy. Therefore, gross sales is used as the EB as it is a reasonable and practical proxy for products in use. 3. Historical Precedence If there is a more accurate or practical EB than the one currently in use (e.g. miles driven versus car years), consider the following before implementing it. 1. Any change in the EB can lead to large premium swings for individual insureds. 2. A change in EB will require a change in the rating algorithm, which may require a significant effort to adjust the rating systems, manuals, etc. 3. Since ratemaking analysis is based on several years of data, a change in EB may necessitate significant data adjustments for future analyses. Example: WC has historically used payroll as an EB. In the 1980s, there was pressure to change the EB to hours worked for medical coverage to correct perceived inadequacies of the EB for union companies with higher pay scales. Although hours worked made intuitive sense, the EB was not changed at that time, given concerns regarding the transition. Instead, the rating variables and rating algorithm were adjusted to address the inequities (note that the debate over the choice of WC EB continues to reemerge). EBs currently used for different lines of business are shown below: Line of Business Typical Exposure Bases Personal Automobile Earned Car Year Homeowners Earned House Year Workers Compensation Payroll Commercial General Liability Sales Revenue, Payroll , Square Footage, Number of Units Commercial Business Property Amount of Insurance Coverage Physician's Professional Liability Number of Physician Years Professional Liability Number of Professionals (e.g., Lawyers or Accountants) Personal Articles Floater Value of Item Exam 5, V1a Page 47 2014 by All 10, Inc. 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. Exam 5, V1a Page 48 2014 by All 10, Inc. 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 Exam 5, V1a Page 49 2014 by Alll 10, Inc. 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 Exam 5, V1a Page 50 2014 by Alll 10, Inc. 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. Exam 5, V1a Page 51 2014 by All 10, Inc. 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. Page 52 2014 by Alll 10, Inc. 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 Exam 5, V1a Page 53 2014 by Alll 10, Inc. Chapter 4 – Exposures BASIC RATEMAKING – WERNER, G. AND MODLIN, C. In-force Exposure by Date In-Force Exposure a/o Policy A B C D E F Total Effective Date 10/01/10 01/01/11 04/01/11 07/01/11 10/01/11 01/01/12 Expiration Date 09/30/11 12/31/11 03/31/12 06/30/12 09/30/12 12/31/12 Exposure 01/01/11 06/15/11 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.00 1.00 1.00 0.00 0.00 1.00 0.00 0.00 1.00 0.00 0.00 6.00 2.00 3.00 01/01/12 0.00 0.00 1.00 1.00 1.00 1.00 4.00 Policy Terms Other Than Annual When policy terms are shorter or longer than a year, then aggregation for each type of exposure is calculated differently. If the policies are six-month policies, each policy would represent one-half of an exposure Six-Month Policies Effective Expiration Date Date Policy Exposure A 10/01/10 03/31/11 0.50 B 01/01/11 06/30/11 0.50 C 04/01/11 09/30/11 0.50 D 07/01/11 12/31/11 0.50 E 10/01/11 03/31/12 0.50 F 01/01/12 06/30/12 0.50 Example Policies Exam 5, V1a Page 54 2014 by All 10, Inc. Chapter 4 – Exposures BASIC RATEMAKING – WERNER, G. AND MODLIN, C. CY Written Exposures a/o 12/31/12 Written Exposures Policy A B C D E F Total Effective Date 10/01/10 01/01/11 04/01/11 07/01/11 10/01/11 01/01/12 Expiration Date Exposure CY 2010 CY 2011 CY 2012 03/31/11 0.00 0.00 0.50 0.50 06/30/11 0.50 0.00 0.50 0.00 09/30/11 0.50 0.00 0.50 0.00 12/31/11 0.50 0.00 0.50 0.00 03/31/12 0.50 0.00 0.50 0.00 06/30/12 0.00 0.00 0.50 0.50 3.00 0.50 2.00 0.50 CY Earned Exposures a/o 12/31/12 Earned Exposure Effective Date Policy A 10/01/10 B 01/01/11 C 04/01/11 D 07/01/11 E 10/01/11 F 01/01/12 Total Expiration Date 03/31/11 06/30/11 09/30/11 12/31/11 03/31/12 06/30/12 Exposure CY 2010 CY 2011 CY 2012 0.50 0.00 0.25 0.25 0.50 0.00 0.50 0.00 0.50 0.00 0.50 0.00 0.50 0.00 0.50 0.00 0.50 0.00 0.25 0.25 0.50 0.00 0.00 0.50 3.00 0.25 2.00 0.75 Policy Written Exposures a/o 12/31/12 Effective Expiration Policy Date Date Exposure 10/1/2010 3/31/2011 0.50 A 1/1/2011 6/30/2011 B 0.50 4/1/2011 9/30/2011 C 0.50 7/1/2011 12/31/2011 D 0.50 10/1/2011 3/31/2012 E 0.50 1/1/2012 6/30/2012 F 0.50 Total 3.00 Written Exposures PY 2010 PY 2011 PY 2012 0.50 0.00 0.00 0.00 0.50 0.00 0.00 0.50 0.00 0.00 0.50 0.00 0.00 0.50 0.00 0.00 0.00 0.50 0.50 2.00 0.50 Policy Year Earned Exposures a/o 12/31/12 Effective Expiration Policy Date Date Exposure 10/1/2010 3/31/2011 0.50 A 1/1/2011 6/30/2011 B 0.50 4/1/2011 9/30/2011 C 0.50 7/1/2011 12/31/2011 D 0.50 10/1/2011 3/31/2012 E 0.50 1/1/2012 6/30/2012 F 0.50 Total 3.00 Earned Exposures PY 2010 PY 2011 PY 2012 0.50 0.00 0.00 0.00 0.50 0.00 0.00 0.50 0.00 0.00 0.50 0.00 0.00 0.50 0.00 0.00 0.00 0.50 0.50 2.00 0.50 Exam 5, V1a Page 55 2014 by All 10, Inc. Chapter 4 – Exposures BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Assuming insured units are “number of homes” insured at a point in time, each semi-annual policy contributes one in-force exposure. In-force Exposure by Date Effective Expiration No. of Houses Policy Date Date Insured 10/1/2010 3/31/2011 A 1.00 1/1/2011 6/30/2011 B 1.00 4/1/2011 9/30/2011 C 1.00 7/1/2011 12/31/2011 D 1.00 10/1/2011 3/31/2012 E 1.00 1/1/2012 6/30/2012 F 1.00 Total 6.00 In-Force Exposures a/o CY 2010 CY 2011 CY 2012 1.00 0.00 0.00 1.00 1.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 1.00 2.00 2.00 2.00 Calculation of Blocks of Exposures Insurers may have policy information summarized on a monthly or quarterly basis and need to calculate exposures for a block of policies using this summarized data. In such a case: it is customary to treat all policies as if they were written on the mid-point of the period. when summarizing on a monthly basis, all policies are assumed to be written on the 15th of the month. (i.e. this is known as “15th of the month” rule or the “24ths” method.) this approximation applies as long as policies are written uniformly during each time period. if this approach is applied to longer periods (e.g. quarters or years), the assumption of uniform writings is less likely to be reasonable. To demonstrate how the rule applies, assume an insurer begins writing annual policies in 2010 and writes 240 exposures each month. It is reasonable to assume that some of the 240 exposures written in July were in-force as of the first day of the month. However, the “15th of the month” rule assumes that none of the exposures from the July policies contribute to in-force exposures as of 7/1/2010 because the rule assumes all the July policies are written on 7/15. (see the table below and look at in-force exposures as of 7/1/2010 and at 7/10/2010 written exposures). Exam 5, V1a Page 56 2014 by All 10, Inc. Chapter 4 – Exposures BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Aggregate In-force Calculation Written Assumed Month Effective Date Exposure Jan 10 240 01/15/10 Feb 10 240 02/15/10 Mar 10 240 03/15/10 Apr 10 240 04/15/10 May 10 240 05/15/10 June 10 240 06/15/10 240 July 10 07/15/10 Aug 10 240 08/15/10 Sep 10 240 09/15/10 Cot 10 240 10/15/10 Nov10 240 11/15/10 Dec 10 240 12/15/10 Total 2,880 07/01/10 240 240 240 240 240 240 0 0 0 0 0 0 1,440 01/01/11 240 240 240 240 240 240 240 240 240 240 240 240 2,880 07/01/11 0 0 0 0 0 0 240 240 240 240 240 240 1,440 Earned Exposure %’s calculation: Since policies for a given month are assumed to be written on the 15th of the month, the written exposures for annual policies will be earned over a 13-month calendar period: 1/24 of the exposure will be earned in the second half of the month in which it was written 1/12 (or 2/24) of the exposure will be earned in each of the next 11 months (i.e. months 2-12) and 1/24 of the exposure will be earned in the first half of month 13. Distribution of earned exposures to CYs 2010 and 2011: 1 2 3 4 5 (6) = (2) x (4) (7) = (2) x (5) Earned % Earned Exposures Written Exposures Assumed Month Written Effective date 2010 2011 2010 2011 Jan 10 Feb 10 Mar 10 Apr 10 May 10 Jun 10 Jul 10 Aug-10 Sep-10 Oct 10 Nov 10 Dec 10 Total 240 240 240 240 240 240 240 240 240 240 240 240 2,881 01/15/10 02/15/10 03/15/10 04/15/10 05/15/10 06/15/10 07/15/10 08/15/10 09/15/10 10/15/10 11/15/10 12/15/10 23/24 21/24 19/24 17/24 15/24 13/24 11/24 9/24 7/24 5/24 3/24 1/24 1/24 3/24 5/24 7/24 9/24 11/24 13/24 15/24 17/24 19/24 21/24 23/24 230 210 190 170 150 130 110 90 70 50 30 10 1,440 10 30 50 70 90 110 130 150 170 190 210 230 1,440 (4) = Portion of exposure earned in 2010. (5) = Portion of exposure earned in 2011. The same principles apply when using the “15th of the month” rule on PY aggregation. Exam 5, V1a Page 57 2014 by All 10, Inc. 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 Page 58 2014 by All 10, Inc. 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 Page 59 2014 by All 10, Inc. Chapter 4 – Exposures BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Section 2: Computing Exposures Questions from the 2000 exam 38. (4 points) Based on McClenahan, "Ratemaking," chapter 2 of Foundations of Casualty Actuarial Science, and the following data, answer the questions below. Personal Automobile Liability Data: Calendar Year 1997 Calendar Year 1998 Number of Autos Number of Autos Written on Written on Effective Date Effective Date Effective Date Effective Date January 1, 1997 100 January 1, 1998 900 April 1, 1997 300 April 1, 1998 1,100 July 1, 1997 500 July 1, 1998 1,300 October 1, 1997 700 October 1, 1998 1,500 Assume: • All policies are twelve-month policies. • Written premium per car during calendar year 1997 is $500. • A uniform rate increase of 15% was introduced effective July 1, 1998. a. (1/2 point) b. (1 point) c. (1/2 point) d. (1 point) e. (1 point) Calculate the number of in-force exposures on January 1, 1998. (chapter 4) Calculate the number of earned exposures for calendar year 1998. (chapter 4) List the two methods McClenahan describes that are used to adjust earned premiums to a current rate level basis. (chapter 5) Which of the two methods listed in part c. above would be more appropriate to use for this company's personal automobile liability business? Briefly explain why. (chapter 5) Using your selected method from part d. above, calculate the on-level earned premium for calendar year 1998. (chapter 5) Questions from the 2010 exam: 17. (2 points) Given the following activity on five annual personal automobile policies as of June 30, 2009: Policy 1 2 3 4 5 Effective Date July 1, 2007 October 1, 2007 January 1, 2008 March 1, 2008 July 1, 2008 Original Expiration Date June 30, 2008 September 30, 2008 December 31, 2008 February 28, 2009 June 30, 2009 Mid-term Cancellation Date N/A March 31, 2008 N/A June 30, 2008 N/A The exposure base is earned car years. a. (0.5 point) Calculate the 2008 calendar year written exposure. b. (0.5 point) Calculate the 2008 calendar year earned exposure. c. (0.5 point) Calculate the 2007 policy year written exposure. d. (0.5 point) Calculate the in-force exposure as of April 1, 2008. Exam 5, V1a Page 60 2014 by All 10, Inc. 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 Page 61 2014 by All 10, Inc. Chapter 4 – Exposures BASIC RATEMAKING – WERNER, G. AND MODLIN, C. The predecessor papers to the current syllabus reading “Basic Ratemaking” by Werner, G. and Modlin, C. were numerous. While past CAS questions were drawn from prior syllabus readings, the ones shown below remain relevant to the content covered in this chapter. Section 1: Criteria for Exposure Bases Solutions to questions from the 1992 exam 53. (a) 1. An accurate measure of the exposure to loss. 2. Easy to determine for the insurer. 3. Difficult to manipulate by the insured. Present Day Update: While the above 3 criteria were the right answers in 1992, the current reading by Werner and Modlin, list them differently: 1. Proportional to expected loss: The selected EB should be the factor most directly proportional to loss and be responsive to any change in exposure to risk. 2. Practical – Objective and Easy to Obtain/verify 3. Historical Precedence – changes in historical EB can cause large premium swings, changes in rating algorithms, and necessitate adjustments to historical data analyses. (b) It was caused by discontent among insureds over the inequities in the rating mechanism. If a unionized company pays more per employee, it will have higher payroll and pay more for its WC coverage. 1. To the extent that the unionized company's indemnity losses are higher, the premium difference is correct. 2. To the extent that losses are from medical payments, or are capped by max benefits, use of payroll is not justified. Solutions to questions from the 1995 exam Question 36. a1. Reasonableness: the exposure unit should be a reasonable measure of the exposure to loss. 2. Ease of Determination: the exposure unit must be subject to accurate determination. 3. Responsiveness to Change: It should react to change in the true exposure to loss. 4. Historical Practice: A change in an exposure unit could render the prior history unusable. Present Day Update: The list according to Werner and Modlin is a little different: 1. Proportional to expected loss: The selected EB should be the factor most directly proportional to loss and be responsive to any change in exposure to risk. 2. Practical – Objective and Easy to Obtain/verify 3. Historical Precedence – changes in historical EB can cause large premium swings, changes in rating algorithms, and necessitate adjustments to historical data analyses. b. Reasonableness: Car-years are a reasonable measure of the exposure to loss, but doesn’t differentiate by type of vehicle. It is easy to determine and somewhat responsive to change. Historically, it has been the industry measure for some time. Reasonableness: Miles driven are a reasonable measure of the exposure to loss, but doesn’t account for the location of the driving (urban or rural). It is not easy to determine since it subject to audit by the insurance company. It is responsive to change, since the relative exposure to loss increases as miles driven increases. It would be difficult to implement and would render the prior history unusable. Exam 5, V1a Page 62 2014 by All 10, Inc. Chapter 4 – Exposures BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 1997 exam Question 25. A Exposure units should: 1. Vary with the hazard. 2. Be practical. 3. Be verifiable. Present Day Update: The list according to Werner and Modlin is: 1. Proportional to expected loss: The selected EB should be the factor most directly proportional to loss and be responsive to any change in exposure to risk. 2. Practical – Objective and Easy to Obtain/verify 3. Historical Precedence – changes in historical EB can cause large premium swings, changes in rating algorithms, and necessitate adjustments to historical data analyses. B. Question no longer applicable to the content in this chapter. C. Large Risks are usually subject to either Composite Rating or Loss Rating. 1. Composite Rating is used to simplify the rating for insureds with multiple exposures (hundreds of vehicles in their auto fleets or many insured locations). First, a proxy exposure base (such as receipts or mileage for long haul trucking) is selected. Next, the rate per proxy unit is determined by dividing the risk’s premium, calculated normally, by proxy exposure base. The simplified equation for charged premium = (Number of expected proxy units) * (Rate per proxy unit). After policy expiration, the firm’s receipts are audited, so that the actual number of actual proxy units can be used to determine the firm’s final premium. 2. Under Loss rating, the exposure base is the risk itself, and the rate is its expected losses. The equation for charged premium = Expected Losses + Expense Load, for a very large risk. Exam 5, V1a Page 63 2014 by All 10, Inc. Chapter 4 – Exposures BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2009 exam Question 17 a1. varies with the hazard (WM would say be directly proportional to expected loss) 2. verifiable (WM would say this is a characteristic of being practical) 3. not subject to manipulation (WM would say this is a characteristic of being practical) 4. practical Present Day Update: The Werner and Modlin text uses the following list: 1. Proportional to expected loss: The selected EB should be the factor most directly proportional to loss and be responsive to any change in exposure to risk. 2. Practical – Objective and Easy to Obtain/verify 3. Historical Precedence – changes in historical EB can cause large premium swings, changes in rating algorithms, and necessitate adjustments to historical data analyses. b1. Miles driven certainly varies with the hazard; the more you drive the more likely you are to get in an accident. 2. Verifiable - may not be easy to verify. Someone would have to inspect each car at the end of the year to read the odometer. 3. Certainly subject to manipulation. If the insured was asked how many miles driven in a year without verification, he could easily lie. Even if the number was verified, there are still ways to turn the numbers on an odometer back. 4. Miles driven is practical and intuitive. Most insured would understand that miles driven would be directly correlated to probability of accidents. Overall, the change to miles driven should not be made since the downsides of costly verification and possibility of manipulation out weigh the benefits of varying with the hazard and practicality. Solutions to questions from the 2010 exam Question 16 a. (1 point) Identify and briefly describe two criteria for a good exposure base. b. (0.5 point) Evaluate "market value of the house" as an exposure base for homeowners insurance using the two criteria identified in part a. above. c. (0.5 point) Provide two reasons why a change in exposure base may be difficult. a1. 1. Directly proportional to loss. The exposure should have direct relationship to loss and vary proportionally to it (i.e. the expected loss of a policy with two exposures should be twice the expected loss of a similar policy with one exposure). a2. Practical. Exposure should be • Objective, not subjective, and definitively measurable • Verifiable. Can be checked b1. No. A house with $ 200K market value does not have 2 times expected loss than house with $100K market value. b2. No. Market value is somewhat subjective. No definite measure. c1. Rates are likely to change substantially when an exposure base changes. Insured may not be happy with changes. c2. System limitations: hard to build new system based on new exposure, and may not even have data for it. Exam 5, V1a Page 64 2014 by All 10, Inc. Chapter 4 – Exposures BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2011 exam: Question 2 – Model Solution 1 Car year to annual miles driven, 3 criteria: 1. Proportional to expected loss: Should select variable with the most direct relationship to loss. Should adjust based on modifications to exposure of the risk to a loss. Annual miles driven seems a better choice, since the more you drive, the more at risk you are to have a loss. 2. Practical: Should be objective, well-defined, and relatively easy to obtain and verify. Miles driven are objective and a well-defined exposure, but can be expensive to send inspectors to verify odometer. Also, if ask client, it is subject to moral hazard. 3. Historical precedent: Car years have historically been used. Changing to miles driven could cause: significant variation in premium -need to modify systems -need to collect new data (cost of survey or inspections) Based on the 3 criteria, the costs of implementing this new structure and practical issues overweight the benefits of the 1st one. Should keep earned car years as exposure base. Question 2 – Model Solution 2 Exposure base should be: 1. proportional to loss 2. practical (verifiable, objective, easy to admin) 3. Have historical precedence Annual miles driven satisfies 1 in that it is proportional to loss. More miles driven = more exposure. Annual miles driven does not satisfy 2 in that it is difficult to verify and can be easily manipulated. Annual miles driven does not satisfy 3 since it hasn’t been used in the past. Changing the exposure base may cause prem. swings. Also, the data needed may not be readily available to create a database. Exam 5, V1a Page 65 2014 by All 10, Inc. Chapter 4 – Exposures BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2012 exam: 2. (1.5 points) An insurance company is considering changing its exposure base for workers compensation from payroll to number of employees. Evaluate the merits of this change based on each of three different criteria of a good exposure base. Question 2 (Exam 5A Question 2) 1. Directly proportional to expected loss: Number of employees does reflect exposure to loss, but payroll is more reflective of exposure loss. For example, having twice as many employees does not mean that the expected losses will double, but only that frequency of loss would double (severity would depend on the payroll distribution). Payroll is responsive to changes in both frequency and severity. 2. Practical: Numbers of employees is a well-defined and objective measure. However, it may not be as easy to obtain as payroll information because payroll is tracked for numerous financial reports whereas number of employees is not. It may be harder to administer because insured could manipulate information regarding number of employees more easily than that regarding payroll. 3. Considers historical precedence: Number of employees does not meet this criteria because payroll has been used historically as the exposure base for WC. Changing to numbers of employees may lead to the following issues: 1. Lead to large premium swings. 2. Require significant systems changes. 3. Require a change in rating algorithm. 4. Necessitate significant data adjustments for future ratemaking analysis. CONCLUSION: Given these constraints, I would NOT recommend changing the exposure base to number of employees. Examiner Comments Candidates scored well on this question. Some candidates lost points for either not supporting the reason or restating the criteria as the reason. Exam 5, V1a Page 66 2014 by All 10, Inc. Chapter 4 – Exposures BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Section 2: Computing Exposures Solutions to questions from the 2000 exam Question 38. Parts a and b. the number of in-force exposures on January 1, 1998, and earned exposures for CY 1998. Number of Autos Number of Written on Inforce Exposures 1998 Earned 1998 Earned Effective Date Effective Date Factor Exposures on 1/1/98 (1) (2) (3) (4)=(1)*(3) January 1, 1997 100 0 0.0 0 April 1, 1997 300 300 .25 75 July 1, 1997 500 500 .50 250 October 1, 1997 700 700 .75 525 January 1, 1998 900 900 1.0 900 April 1, 1998 1,100 0 .75 825 July 1, 1998 1,300 0 .50 650 October 1, 1998 1,500 0 .25 375 Total 2,400 3,600 * In-force exposures are the number of insured units exposed to having a claim at a given point in time. Inforce exposure counts a full car year for each 12 month policy in force as of 1/1/98, regardless of the length of the remaining term. * Earned exposures are the portion of written exposures for which coverage has already been provided as of a certain point in time. For example: 3 of the 12 months of coverage for the 300 exposures written on 4/1/97 occur during CY 1998. Assuming there are no policy cancellations, this portion (3/12) of the total exposures written will be earned during CY 1998, and thus the 1998 Earned Factor is .25. Parts c., d. and e. See Chapter 5. Exam 5, V1a Page 67 2014 by All 10, Inc. 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 Page 68 2014 by All 10, Inc. Chapter 4 – Exposures BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2011 exam: Question 3 Policy Effective Date Original Expiration Date Cancellation Date A February 1, 2009 July 31, 2009 B May 1, 2009 October 31, 2009 C August 1, 2009 January 31, 2010 D November 1, 2009 April 30, 2010 January 31, 2010 E January 1, 2010 June 30, 2010 F July 1, 2010 December 31, 2010 a. (0.25 point) Calculate the written car years in calendar year 2010. b. (0.25 point) Calculate the written car years in policy year 2010. c. (0.25 point) Calculate the earned car years in calendar year 2010. d. (0.25 point) Calculate the earned car years in policy year 2010. e. (0.25 point) Calculate the number of in-force policies as of January 1, 2010. Initial comments: Since we are asked to compute CY and PY written car years, CY and PY earned car years and in-force policies for six different policies, it is best to set up a table similar to the one below to answer the question in the most efficient way possible. Since the given policies are six-month policies, each would represent one-half of a written exposure. Since insured units are defined as number of autos insured at a point in time, each semi-annual policy can contribute to one in-force exposure. Since the exposures needing to be calculated are associated with 2010, it is clear that policy A and policy B contribute 0 exposures to questions a., b. c. d. and e. Definitions of the type of exposures being asked to compute are as follows: Written exposures arise from policies issued (i.e. underwritten or written) during a specified period of time (e.g. a calendar quarter or a CY). CY 2011 written exposures are the sum of the exposures for all policies that had an effective date in 2011. If a policy cancels midterm, the policy will contribute a written exposure to two different CYs if the policy cancellation date is in a different CY year than the original policy effective date. Policy D is cancelled on 1/31/2010, one half way through its policy period. Policy D will contribute 1/2 written exposure to CY 2009 and -(1/2)*(1/2) = -0.25 written exposure to CY 2010. Earned exposures are the portion of written exposures for which coverage has already been provided as of a certain point in time. The % of Policy C earned in CY 2010 is 1/6 (January only). Thus, Policy C contributes 1/2*1/6 = 1/12 earned exposures to CY 2010. The % of Policy D earned in CY 2010 is 1/6 (January only). Thus, Policy D contributes 1/2*1/6 = 1/12 earned exposures to CY 2010. Note: Unlike CY earned exposure, exposure for one policy cannot be earned in two different PYs. In-force exposures are the number of insured units exposed to having a claim at a given point in time. Policies A and B are not exposed to loss as of 1/1/2010 (due to policy expiration). Policy F is not exposed to loss as of 1/1/2010 (since it is not effective until 7/1/2010). Exam 5, V1a Page 69 2014 by All 10, Inc. Chapter 4 – Exposures BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2011 exam: Question 3 – CAS Model Solution (a) (b) (c) (d) Policy A 0 0 0 0 B 0 0 0 0 C 0 0 1/12 0 D -1/4 0 1/12 0 E 1/2 1/2 1/2 1/2 F 1/2 1/2 1/2 1/2 Total .75 1 14/12 1 (e) 0 0 1 1 1 0 3 Assume that a full policy = ½ car year (semi annual) (a) .75 = -1/4 + 1/2 + 1/2 (b) 1 = 1/2 + 1/2 (c) 14/12 = 1/12 + 1/12 + 1/2 + 1/2 (d) 1 = 1/2 + 1/2 (e) 3 = 1 + 1 + 1 (recall that each semi-annual policy can contribute to one in-force exposure). Exam 5, V1a Page 70 2014 by All 10, Inc. Chapter 4 – Exposures BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2012 exam: 3a. (1 point) Calculate the 2011 earned exposures assuming policies are written uniformly during each quarter. 3b. (0.5 point) Discuss the appropriateness of the assumption in part a. above given the exposure data. Question 3 – Model Solution 1 (Exam 5A Question 3) a. Pol Eff dates (1) 1/1 thru 3/31 4/1 thru 6/30 7/1 thru 9/30 10/1 thru 12/31 Avg eff date (2) 2/15 5/15 8/15 11/15 % yr rem (3) 0.875 0.625 0.375 0.125 exp (4) 100 200 300 400 EE (5)=(3)*(4) 87.5 125.0 112.5 50.0 375.0 2011 Earned Exposures: 375.0 3/12=.25/2=.125. [6/12+3/12]/2 = [.5+.25]/2=.375. [9/12+6/12]/2 = [.75+.5]/2=.625. [12/12+9/12]/2 = [1.0+.75]/2=.875. b The assumption of uniform writings throughout the quarter seems inappropriate, given that there is such a dramatic increase in writings from one quarter to the next. It’s more likely that writings increase throughout the quarter as well. Question 3 – Model Solution 2 (Exam 5A Question 3) Proportion Earned Jan– 23/24 F - 21/24 100 M - 19/24 A - 17/24 M - 15/24 200 J - 13/24 J- 11/24 A - 9/24 300 S - 7/24 O - 5/24 N - 3/24 400 D - 1/24 2011 Earned Exposure = Avg No. of Policies Written per month * monthly Proportion Earned by year end = 100/3 [(23 +21+19) /24] + 200/3[(17+15+13)/24] + 300/3 [(11+9+7)/24] + 400/3 [(5+3+1)/24] = 87.5 + 125 + 112.5 + 50 = 375 b. Exposure is increasing each quarter. It is likely that this is the case within quarter ie March has more exposure than January. We assume uniform exposure which does not appear correct with this increasing observed exposure trend. Exam 5, V1a Page 71 2014 by All 10, Inc. Chapter 4 – Exposures BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2012 exam Question 3 – Model Solution 3 (Exam 5A Question 3) a Policy eff dates exposures Average written 1/1 – 3/31 100 2/15 4/1 – 6/30 200 5/15 7/1 – 9/30 300 8/15 10/1 – 12/31 400 11/15 Earned year 10.5/12 7.5/12 4.5/12 1.5/12 earned 87.5 125 112.5 50. 375 (Answer for a)) b. Appropriate to assume that policies are written uniformly during each quarter? → As written exposures are steadily increasing. It won’t be appropriate to assume policies are uniformly written during the year. → Quarterly periods are fairly granular enough to assume that polices are written uniformly in the period. Examiners Comments Candidates scored well on this question. Some candidates used the same assumptions but applied/calculated on a monthly basis. This was given full credit as well. Common mistakes include making the exposures uniform throughout the year and effective at the beginning of the month instead of uniform throughout the quarter. Exam 5, V1a Page 72 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Sec 1 2 3 1 Description Premium Aggregation Adjustments To Premium Key Concepts Pages 63 - 70 70 - 87 88 - 88 Premium Aggregation 63 - 70 The goal of ratemaking is to balance the fundamental insurance equation: Premium = Losses + LAE + UW Expenses + UW Profit. The ratemaking process begins with applying a series of adjustments to historical premium. 1. Bring historical premium to the rate level currently in effect. Without this adjustment, any rate changes during or after the historical period with not be fully reflected in the premium and will distort the projections 2. Develop premium to ultimate levels if the premium is still changing. 3. Project the historical premium to the premium level expected in the future. This accounts for changes in the mix of business that have occurred or are expected to occur after the historical experience period. Appendices A, C, and D provide examples from various lines of business of the premium adjustments made in ratemaking analysis. Two approaches to evaluate the adequacy of rates underlying an insurer’s premium are the: Pure premium approach and Loss ratio approach. The loss ratio approach requires that premium to be collected during a future time period be estimated (this is not the case when using the pure premium approach). When using the pure premium approach, the adjustments in this chapter are not needed. This chapter covers: ways to define and aggregate premium techniques used to adjust historical premium to current rate level techniques used to develop historical premium to ultimate level techniques used to measure and apply premium trend Exam 5, V1a Page 73 2014 by All 10, Inc. Chapte er 5 – Prem ium BASIC RATTEMAKING – WERNER, G G. AND MOD DLIN, C. Methods of Aggregattion for Annu ual Terms Two meth hods to aggregate premium ms are CY (the same as Ca alendar-AY) a and PY. Recall the 4 common n methods of data aggrega ation are CY, AY, PY, and RY. Homeow wners policies s are used to demonstrate e these conce epts Effective Expiration Policy y Date Date Premium A 10/01/10 09/30/11 $200 B 01/01/11 12/31/11 $250 C 04/01/11 03/31/12 $300 D 07/01/11 06/30/12 $400 E 10/01/11 09/30/12 $350 F 01/01/12 12/31/12 $225 These policies p are illustrated belo ow. The x-axis s represents time t and the y-axis represents the perccentage of the e policy term tthat has expirred (this representation is not applicable to products p like warranty w that don’t earn evvenly). CY and AY A Aggregatiion consider all a premium trransactions d during the 12--month CY wiithout regard to the date of po olicy issuance e (since CY an nd AY premiu ums are equivvalent, the texxt uses the term CY premiu um). At A the end of th he CY, CY prremiums are fixed. f Since CY captures transacttions occurring on or after tthe first day o of the year, an nd on or before the last da ay of the yearr, CY is repre esented graph hically as a sq quare (as sho own below). CY Ag ggregation Exam 5, V1a Page 74 2014 by Alll 10, Inc. Chapte er 5 – Prem ium BASIC RATTEMAKING – WERNER, G G. AND MOD DLIN, C. PY (a.k.a. UW year) ag ggregation considers c all premiums p on policies with effective date es during the year. PY is reprresented grap phically using a parallelog gram starting with a policy written on the e first day of tthe PY and ending with a policy wrritten on the la ast day of the e PY. gregation PY Agg Since a PY takes 24 months to co omplete, and CY premium is fixed at 12 2 months, mosst ratemaking g analysis focuses s on CY premiums (and AY Y losses). Four typ pes of premium 1. Written premium arise a from policies issued (i.e. ( underwrittten) during a specified perriod of time (e e.g. a calendar quarter or a CY). CY 2011 written pre emium is the sum s of premiums for policiies having an n effective datte in 2011. Since polic cies B, C, D and E all have e effective dattes (shown ass large circless on the horizo ontal axis) in 2011, theirr entire premiu um contribute es to CY 2011 1 written prem mium. However, policies p A and d F have effec ctive dates in years 2010 a and 2012, and d thus do not contribute to CY 2011 written w premium. CY Wrritten Premiu um The dis stribution of written w premiu um to each ca alendar year iss shown belo ow: Calend dar Year Written Premium m a/o 12/31/1 12 Written Premium m Efffective Expiration E Premium Date Date Policy P CY 2010 CY 2011 CY 2012 A 10 0/01/10 09/30/11 0 $200.00 $200.00 $ 250.00 $250.00 B 01 1/01/11 12/31/11 1 $ 300.00 C 04 4/01/11 03/31/12 0 $300.00 $ 400.00 $400.00 D 07 7/01/11 06/30/12 0 $ 350.00 $350.00 E 10 0/01/11 09/30/12 0 $225.00 $225.00 F 01 1/01/12 12/31/12 1 $1,300.00 $1725.00 $200.00 Total T $225.00 Exam 5, V1a Page 75 2014 by Alll 10, Inc. 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. Exam 5, V1a Page 76 2014 by All 10, Inc. Chapte er 5 – Prem ium BASIC RATTEMAKING – WERNER, G G. AND MOD DLIN, C. 2. Earned d premium are a the portion n of written prremium for wh hich coverage e has been prrovided and the insurer is entittled to retain as a of a certain n point in time e. To bettter understan nd the differen nce between CY C and PY ea arned exposu ure, look at th he CY diagram m: CY Ea arned premiu um For Po olicy C, 75% of o the policy period p is earne ed in 2011 an nd 25% of the e policy period d is earned in n 2012. Policy C contributes s $225 (75% * $300) of earrned premium m to CY 2011 and $75 earn ned premium to CY 2012. Distriibution of CY Y Earned Pre emium a/o 12 2/31/12 Effective E Ex xpiration Date Date Policy Premium A 10/01/10 09 9/30/11 $200.00 $ B 01/01/11 0 12 2/31/11 $250.00 $ 04/01/11 0 03 3/31/12 $300.00 $ C D 07/01/11 0 06 6/30/12 $400.00 $ E 10/01/11 09 9/30/12 $350.00 $ F 01/01/12 0 12 2/31/12 $225.00 $ Total Earned Premium CY 2011 CY 2012 $150.00 $250.00 $75.00 $225.00 $200.00 $200.00 $87.50 $262.50 $225.00 $50.00 $912.50 $762.50 CY Y 2010 $50.00 arned Premiu um: PY Ea Exam 5, V1a Earned pre emium is assigned to the year y the policyy was written and increase es over time. At the end of a PY (i.e. 24 months affter the start o of a PY having annual policcies), PY earned and written pre emium are equivalent. Unlike CY Y earned prem mium, premiium for one p policy canno ot be earned in two differrent PYs. Premiums for lines subjject to premiu um audits con ntinue to deve elop after the end of the po olicy period. Page 77 2014 by Alll 10, Inc. 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 Exam 5, V1a Page 78 2014 by Alll 10, Inc. 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. Exam 5, V1a Page 79 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Simple Example Assume policies have annual terms and premium is calculated according to the following rating algorithm: Premium = Exposure x Rate per Exposure x Class Factor + Policy Fee. The class factor has three values, or levels (X, Y, and Z), each with a distinct rate differential. The following three rate changes occurred during or after the historical experience period. • 7/1/2010: the base rate was increased and resulted in an overall average rate level increase of 5%.* • 1/1/2011: the base rate and policy fee were adjusted resulting in an overall average rate level increase of 10%. • 4/1/2012: the policy fee and class Y and Z rate relativities were changed resulting in an overall average rate level decrease of -1%. * The reader may be confused by the overall average rate changes provided in this example [e.g., how a 5.6% (=950/900-1.00) change in rate per exposure results in an overall average rate change of 5.0%]. The overall average rate change considers the average change in the total premium per policy, which is a function of the rate per exposure, the number of exposures per policy, the applicable class factors, and the policy fee. These detailed inputs have not been provided; the overall average rate change should be taken as a given for the purpose of illustrating premium at current rate level techniques. Rate Change History Rate Level Effective Group Date 1 Initial 2 07/01/10 3 01/01/11 4 04/01/12 Overall Average Rate change -5.0% 10.0% -1.0% Rate Per Exposure $900 $950 $1,045 $1,045 X 1.00 1.00 1.00 1.00 Class Factor Y 0.60 0.60 0.60 0.70 Z 1.10 1.10 1.10 1.10 Policy Fee $1,000 $1,000 $1,100 $1,090 Method 1: Extension of Exposures This method rerates every policy to restate historical premium to the amount that would be charged under the current rates. Advantage: It is the most accurate current rate leveling method, given the level of current computing power to perform the number of calculations required to rerate each policy. Disadvantage: The rating variables, risk characteristics and rating algorithm needed to rerate each policy during the historical period are often not readily available. Assume the following: We wish to adjust the historical premium for PY 2011 to the current rate level. One such policy was effective on 3/1/2011 and had 10 class Y exposures. The actual premium charged for the policy was based on the rates effective on 1/1/2011, and was $7,370 (= 10 x $1,045 x 0.60 + $1,100). To put the premium on-level: Substitute the current base rate, class factor, and policy fee in the calculations; this results in an onlevel premium of $8,405 (= 10 x $1,045 x 0.70 + $1,090). Perform the same calculation for every policy written in 2011 and then aggregate across all policies. Notes: Policies with the exact same rating characteristics can be grouped for the purposes of the extension of exposures technique, but is only relevant in lines with simple rating algorithms and few rating variables. In commercial lines products, where subjective debits and credits can be applied to manual premium, complicates the use of the extension of exposures technique since it may be difficult to determine what debits and credits would be applied under today’s schedule rating guidelines. Exam 5, V1a Page 80 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Method 2: Parallelogram Method (a.k.a. the geometric method) The parallelogram method: is performed on a group of policies is less accurate than extension of exposures. assumes that premium is written evenly throughout the time period involves adjusting aggregated historical premium by an average factor to put the premium on-level. application varies by policy term, method of aggregation (CY vs. PY), and whether the rate change affects policies midterm or only policies with effective dates occurring after the change. Standard Calculations The objective: Replace the average rate level for a given historical year with the current rate level. The major steps are as follows: 1. Determine the timing and amount of the rate changes during and after the experience period and group the policies into rate level groups according to the timing of each rate change. 2. Calculate the portion of the year’s earned premium corresponding to each rate level group. 3. Calculate the cumulative rate level index for each rate level group. 4. Calculate the weighted average cumulative rate level index for each year. 5. Calculate the on-level factor as the ratio of the current cumulative rate level index and the average cumulative rate level index for the appropriate year. 6. Apply the on-level factor to the earned premium for the appropriate year. For the parallelogram method, exact rates are not required. Step 1: Obtain the effective date and overall rate changes for the policies under consideration. Recall that annual policies have been issued and rate changes apply to policies effective on or after the date (i.e. do not apply to policies in mid-term). Rate Overall Level Average Effective Group Rate Date 1 Initial 2 07/01/10 5.0% 3 01/01/11 10.0% 4 04/01/12 -1.0% Step 2: View these rate changes in graphical format. Assume the actuary is trying to adjust each CY’s EP premium to current rate level. CYs are represented by squares. Each rate change is represented by a diagonal line, the slope of which depends on the term of the policy (which is annual in this case) The numbers 1, 2, 3, and 4 represent the rate level group in effect. Exam 5, V1a Page 81 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Rate Changes assuming CY EP with Annual Policies Next calculate the portion of each CY’s EP (the area within the square) that corresponds to each rate level. For CY 2011, there are three areas representing EP on policies written: after 1/1/2010 and prior to the 7/1/2010 rate change (area of rate level group 1 in CY 2011). on or after 7/1/2010 and before 1/1/2011 (area of rate level group 2 in CY 2011). on or after 1/1/2011 and before 1/1/2012 (area of rate level group 3 in CY 2011). Geometry and the assumption that the policies written are uniformly distributed are used to calculate the portion of the square represented by each rate level area. Note: The following geometric formulae may be used in the parallelogram method: Area of a triangle: ½ x base x height Area of a parallelogram: base x height Area of a trapezoid: ½ x (base1 + base 2) x height Area 1 in CY 2011 is a triangle with area equal to ½ x base x height. The base and height are both 6 months (1/1/2011 to 6/30/2011) so the area (in months) is 18 (= ½ x 6 x 6). This area’s portion of the entire CY square is 0.125 (=18 /(12 x 12)). Simplify by restating the base and height as portions of a year (0.125 = ½ x ½ x ½). In some areas (e.g. area 2 in CY 2011), it is easier to calculate as 1.0 - the sum of the remaining areas. CY 2011 rate levels area are shown below: Area 1 in CY 2011: 0.125 =0.50 x 0.50 x 0.50 Area 2 in CY 2011: 0.375 =1.00 - (0.125 + 0.500) Area 3 in CY 2011: 0.500 =0.50 x 1.00 x 1.0 Exam 5, V1a Page 82 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Step 3: Calculate the cumulative rate level index for each rate level group. The first rate level group is assigned a rate level of 1.00. The cumulative rate level index of each subsequent group is the prior group’s cumulative rate level index multiplied by the rate level for that group. i. the cumulative rate level index for the second rate level group is 1.05 (= 1.00 x 1.05). ii. the cumulative rate level index for the third rate level group is 1.155 (= 1.05 x 1.10). 1 2 3 4 Overall Average Rate Level Cumulative Rate Effective Rate Level Date Index Rate Level Change Group Index 1 Initial -1.00 1.0000 2 7/1/10 5.0% 1.05 1.0500 3 1/1/11 10.0% 1.10 1.1550 4 4/1/12 -1.0% 0.99 1.1435 (4)= (Previous Row 4) x (3) Step 4: Calculate the average rate level index for each year (i.e. the weighted average of the cumulative rate level indices in Step 3, using the areas calculated in Step 2 as weights). The average rate level index for CY 2011 is 1.0963 =1.000 x 0.125 + 1.0500 x 0.375 + 1.1550 x 0.500. Step 5: Calculate the on-level factor as follows: On - Level Factor for Historical Period Current Cumulative Rate Level Index Average Rate Level Index for Historical Period The numerator is the most recent cumulative rate level index The denominator is the result of Step 4. The on-level factor for CY 2011 EP (assuming annual policies) is 1.0431 1.1435 1.0963 Step 6: The on-level factor is applied to the CY 2011 EP to bring it to current rate level. CY 2011 EP at current rate level= CY 2011 EP x 1.0431. Exam 5, V1a Page 83 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Standard CY Calculations for Six-Month Policies If the policy term is six months (common in personal automobile coverage), then the rate level groups can be depicted as follows: Step 2: The areas for CY 2011 are: Area 1 in CY 2011: N/A Area 2 in CY 2011: 0.250 Area 3 in CY 2011: 0.750 = 0.50 x 0.50 x 1.00 = 1.00 - 0.250 Step 3: The cumulative rate level indices are the same as those used for the annual policies. Step 4: The average rate level index for CY 2001 assuming semi-annual policies: 1.1288 = 1.0500 x 0.250 + 1.1550 x 0.750 Step 5: The on-level factor to adjust CY 2011 EP to current rate level is: 1.0130 = 1.1435 (and is 1.1288 smaller than for annual policies because the semi-annual rate changes earn more quickly). Standard PY Calculations for Annual Policies Since PY 2011 only had one rate level applied to the whole year, PY 2012 will be reviewed. The area of each parallelogram is base x height. Area 3 in Policy Year 2012 has a base of 3 months (or 0.25 of a year) and the height is 12 months (or 1.00 year). Step 2: The relevant areas for PY 2012 are as follows: • Area 3 in PY 2012: 0.25 = 0.25 x 1.00 • Area 4 in PY 2012: 0.75 = 0.75 x 1.00 Step 3: The cumulative rate level indices are the same as those used in the CY example. Step 4: The average rate level index for PY 2012 is: 1.1464 = 1.1550 x 0.25 + 1.1435 x 0.75. Step 5: The on-level factor to adjust PY 2012 EP to current rate level is 0.9975 Exam 5, V1a Page 84 1.1435 1.1464 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Rate Changes Mandated by Law Rate changes mandated by law changes apply the rate change to all policies on or after a specific date (including those in-force). The rate level change is represented as a vertical line. Assume a law change mandates a rate decrease of 5% on 7/1/2011 applicable to all policies. The vertical line splits rate level groups 2 and 3 into two pieces each. The -5% law change impacts rate level indices associated with the portion of areas 2b, 3b, and 4. The areas for CY 2011 are as follows: • Area 1 in CY 2011: 0.125 = • Area 2a in CY 2011: 0.250 = • Area 2b in CY 2011: 0.125 = • Area 3a in CY 2011: 0.125 = • Area 3b in CY 2011: 0.375 = 0.50 x 0.50 x 0.50 0.50 - 0.125 - 0.125 0.50 x 0.50 x 0.50 0.50 x 0.50 x 0.50 0.50 - 0.125 The cumulative rate level indices associated with each group are as follows: Step 3 (with Benefit Change) Rate Level Cumulative Rate Group Level Index 1 2a 2b 3a 3b 4 1.0000 1.0500 0.9975 1.1550 1.0973 1.0863 CY 2011 on-level factor: 1.0171 Exam 5, V1a 1.0863 1.0000 x 0.125 1.0500 x 0.250 0.9975 x 0.125 1.1550 x 0.125 1.0973 x 0.375 Page 85 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Comments on the Parallelogram Method Two problems with the parallelogram method: 1. The method is not useful if the assumption that policies are evenly written throughout the year is not true. Example: Boat owners policies are usually purchased prior to the start of boat season and thus are not uniformly written throughout the year. Ways to partially circumvent the need for uniform writings: a. Use a more refined period of time than a year (e.g. quarters or months). b. Calculate the actual distribution of writings and use these to determine more accurate weightings to compute the historical average rate level. Aggregate policies based on which rate level was applicable rather than based on a time period, and the premium for each rate level group is adjusted together based on subsequent rate changes. 2. Premium for certain classes will not be on-level if the implemented rate changes vary by class. Even if the overall premium may be adjusted to a current rate level, adjusted premium will not be appropriate for class ratemaking. This major shortcoming has caused insurers to favor of the extension of exposures approach. Premium Development When working with an incomplete year of data or when premiums for a line of business are subject to premium audits, premium development methods are used for ratemaking purposes. To incorporate responsiveness into the ratemaking analysis, the actuary may choose to use data for a year that is not yet complete (more common for PY analysis due to the time it takes for the PY to close). Assume a ratemaking analysis is performed on PY 2011 data as of 12/31/2011. While WP is known, it is not known which policies may have changes or will be cancelled during the policy term. To estimate how premium will develop to ultimate, historical patterns of premium development are analyzed to understand the effect of cancellations and mid-term adjustment on PY premium. For Lines that utilize premium audits: The insured will pay premium based on an estimate of the total exposure. Once the policy period is complete and the actual exposure is known, the final premium is calculated. For example, WC premium depends on payroll and the final WC premium is determined by payroll audits that occur 3 - 6 months after the policy expires. Premium development depends on several factors including: The type of plan (permitted by the jurisdiction or offered by the carrier). The stability between the original premium estimate and the final audited premium. Internal company operations (e.g. auditing procedures, marketing strategy, accounting policy, etc.). Exam 5, V1a Page 86 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. PY Premium Development Example: A WC carrier writes one policy per month in 2011. Estimated premium for each policy is booked at policy inception for $500,000. Premium develops upward by 8% at the first audit (6 months after the policy expires). At 12/31/2012, the six policies written in the first half of 2011 have completed their audits, but the six policies written in the second half of the year have not. PY 2011 premium as of 12/31/2012 is: $6,240,000 = 6 x $500,000 x 1.08 + 6 x $500,000 At 12/31/2013, all twelve policies have completed their final audits and premium is final. PY 2011 premium as of 12/31/2013 is: $6,480,000 = 12 x $500,000 x 1.08 From 12/31/2012 (24 months after the start of the PY) to 12/31/2013 (36 months after the start of the PY), the premium development factor is 1.0385 (= $6.48 million / $6.24 million). Premium development does not typically apply to CY premium since CY premium is fixed. However, some actuaries may adjust CY premium if audit patterns are changing and a CY analysis is being performed. Note: Rates changes, Inflationary changes and Policy Characteristic Distributional changes impact the average premium level Exposure Trend The average premium level can change over time due to inflation in lines of business with exposure bases that are inflation-sensitive, like payroll (for WC and GL) or receipts (GL). Trends are used to project inflation-sensitive exposures (and thus premium) and are determined using internal company data (e.g. WC payroll data) or industry or government indices (e.g. average wage index). Premium Trend The average premium level can change over time due to changes in the characteristics of the policies written (a.k.a. distributional changes) and the resulting change in average premium level is known as premium trend. Examples that can cause changes in the average premium level: • A rating characteristic can cause average premium to change (e.g. HO premium varies based on the amount of insurance purchased, which is indexed and increases automatically with inflation; therefore, average premium increases as well). • Moving all existing insureds to a higher deductible (e.g. if an insurer moves each insured to a higher deductible upon renewal, and renewals are spread throughout the year, there will be a decrease in average premium over the entire transition period). Trend is not necessary once the transition is complete. • Acquiring the entire portfolio of another insurer writing higher policy limits (e.g. a HO insurer acquires a book of business that includes predominantly high-valued homes, the acquisition will cause a very abrupt increase in the average premium due to the increase in average home values). After the books are consolidated, no additional shifts in the business are expected. Exam 5, V1a Page 87 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. To adjust for premium trend, the actuary needs to: determine how to measure any changes that have occurred decide whether observed distributional shifts were caused by a one-time event or a shift that is expected to continue in the future judgmentally incorporate any additional shifts that are reasonably expected to happen in the future. Actuaries examine changes in historical average premium per exposure to determine premium trend. Average premium should be calculated on an exposure basis rather than a policy basis, using the exposure base underlying the rate. A decision to use earned or written premium must be made. Written premium is a leading indicator of trends that will emerge in earned premium and the trends observed in written premium are appropriate to apply to historical earned premium. Assuming adequate data is available, the actuary will use quarterly average written premium (as opposed to annual average written premium) to make the statistic as responsive as possible. Data used to estimate premium trend due to distributional changes: Change in Average WP (1) (2) (3) (4) Average Written Premium at Rate Level (5) Quarter Written Premium at Current Rate Written Exposures 1Q09 $323,189.17 453 $713.44 -- 2Q09 $328,324.81 458 $716.87 -- 3Q09 $333,502.30 463 $720.31 -- 4Q09 $338,721.94 468 $723.76 -- 1Q 10 $343,666.70 472 $728.11 2.1% 2Q10 $348,696.47 477 $731.02 2.0% 3Q10 $353,027.03 481 $733.94 1.9% 4Q10 $358,098.58 485 $738.35 2.0% 1Q11 $361,754.88 488 $741.30 1.8% 2Q11 $367,654.15 493 $745.75 2.0% 3Q11 $372,305.01 497 $749.10 2.1% 4Q11 $377,253.00 501 $753.00 2.0% Annual Change (4) = (2) / (3) (5) = (4) / (Prior Year4) - 1.0 Changes in the quarterly average WP are used to determine the amount historical premium needs to be adjusted for premium trend. Note the premium used has been adjusted to the current rate level (if this is not done, the data will show an abrupt change in the average written premium corresponding to the effective date of the rate change). Exam 5, V1a Page 88 2014 by All 10, Inc. 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 Exam 5, V1a Page 89 2014 by Alll 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Items affecting the length of the trend period: 1. If the historical period consists of policies with terms other than 12 months, the “trend from” date will be different than discussed above. Example: If the policies in the prior example were six-month policies, then the “trend from” date is 4/1/2011. The “trend to” date is unchanged. Trend Period for 1-Step Trending with 6-Month Policies 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. Exam 5, V1a Page 90 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Two-Step Trending Two-step trending is used when the insurer expects premium trend to change over time. Adjust the historical premium to the level present at the end of the historical period, and then apply a separate adjustment to project premium into the future. Two step trending may be used by a homeowners’ insurer that observes large increases in amount of insurance during the experience period that are not expected to continue into the future. Step 1: Adjust the historical premium to the current trend level using the following adjustment factor: Current Premium Trend Factor = Latest Average WP at Current Rate Level Historical Average EP at Current Rate Level If average EP for CY 2011 is $740.00 and the average WP for the latest available quarter (Calendar Quarter 4Q 2011) is $753.00, then the current premium trend factor is 1.0176 (= 753.00/740.00). The latest average WP is for the fourth quarter of 2011; thus, the average written date is 11/15/2011 (this will be “trend from” date for the second step in the process). If the average been based on the average WP for CY 2011 (as opposed to the fourth quarter), then the average written date would have been 6/30/2011. When average premium is volatile, select a current trend versus using the actual change in average premium. The current trend factor is calculated by trending (1.0 + selected current trend) from the average written date of premium earned in the experience period (i.e. 1/1/2011) to the average written date of the latest period in the trend data (i.e. 11/15/2001). Step 2: Compute the projected premium trend factor. Select the amount the average premium is expected to change annually from the “trend from” date to the projected period. The “trend from” date is 11/15/2011. The “trend to” date is the average written date during the period the proposed rates are to be in effect, which is still 6/30/2013. Thus, the projected trend period is 1.625 years long (11/15/2011 to 6/30/2013). Given a projected annual premium trend of 2%, the projected trend factor is 1.0327 (= (1.0 + 0.02)1.625). Trend Period for 2-Step Trending Exam 5, V1a Page 91 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. The total premium trend factor for two-step trending is the product of the current trend factor and the projected trend factor (i.e. 1.0509 (= 1.0176 x 1.0327)). That number is applied to the average historical EP at current rate level to adjust it to the projected level: CY11 EP at projected rate level = CY11 EP at current rate level x Current Trend Factor x Projected Trend Factor. Two-Step Trending (1) CY 2011 Earned Premium at Current Rate Level (2) CY 2011 Earned Exposures (3) CY 2011 Average Earned Premium at Current Rate Level (4) 4th Quarter of 2011 Average Written Premium at Current Rate Level (5)Step 1 Factor (6) Selected Projected Premium Trend (7) Projected Trend Period (8) Step 2 Factor (9) Total Premium Trend Factor (10) Projected Premium at Current Rate Level $1,440,788 1,947 $740.00 $753.00 1.0176 2.0% 1.6250 1.0327 1.0509 $1,514,124 (3) = (1) / (2) (5) = (4) / (3) (7) (8) = (1.0 + (6)) (9) = (5) x (8) (10)= (1) x (9) Appendices A-D provide realistic examples of ratemaking analysis, including the premium adjustments, intended to reinforce the concepts covered in this chapter. 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 Exam 5, V1a Page 92 2014 by All 10, Inc. 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 Exam 5, V1a Page 93 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 1994 exam 1. An insurer writes the following policies during 1992: Effective Date May 1 August 1 November 1 Policy Term 6 months 12 months 6 months Premium $6,000 $12,000 $2,400 What is the insurer's unearned premium reserve on December 31, 1992? A. <$6,000 B. >$6,000 but <$7,000 C. >$7,000 but <$8,000 D.> $8,000, but < $9,000 E. > $9,000. Questions from the 1996 exam Question 30. (4 points) You are given: Wisconsin Personal Automobile Bodily Injury 20/40 Basic Limits Calendar/ Accident Year 1992 1993 1994 Combined Ultimate Loss & ALAE 325,000 575,000 800,000 1,700,000 Written Premium 750,000 1,000,000 1,250,000 3,000,000 Earned Premium 375,000 875,000 1,125,000 2,375,000 Rate Level History Effective % Rate Date Change 1/1/91 +7.0% 10/1/93 +5.0% 7/1/94 +3.0% 1/1/95 +5.0% • 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 Page 94 D. > 1.11 but < 1.13 E. > 1.13 2014 by All 10, Inc. 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) Exam 5, V1a Page 95 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2000 exam 40. (4 points) Using the techniques described by McClenahan in "Ratemaking," chapter 2 of Foundations of Casualty Actuarial Science, and the following data, answer the questions below. You are given the following information for your company's homeowners business in a single state: Calendar/ Ultimate Loss Accident Year and ALAE Written Premium Earned Premium 1997 635,000 1,000,000 975,000 1998 595,000 1,050,000 1,000,000 Effective Date July 1, 1996 January 1, 1998 July 1, 1999 Rate Change +4.0% +1.8% +3.0% Target Loss and ALAE Ratio Proposed effective date Effective period for rates Credibility Alternative indication Policy period Severity trend Frequency trend 0.670 July 1, 2000 One year 0.60 0.0% Twelve months +3.0% +1.0% a. (1 1/2 points) Calculate the on-level factors for each of the two calendar years 1997 and 1998. (chapter 5) b. (1 1/2 points) Calculate the trended projected ultimate on-level loss and ALAE ratio for the combined experience period 1997-1998. (chapter 6) c. (1 point) Calculate the credibility-weighted indicated rate level change. (chapter 8) Questions from the 2001 exam Question 38. (2 points) Using the parallelogram method described by McClenahan in “Ratemaking,” chapter 2, Foundations of Casualty Actuarial Science, determine the calendar year 1999 on-level earned premium. Show all work. Calendar Year Earned Premium Effective Date Rate Change 1997 $10,000 July 1, 1997 +5.2% 1998 $11,500 No Change 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. Page 96 2014 by All 10, Inc. 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 Page 97 2014 by All 10, Inc. 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. Exam 5, V1a Page 98 2014 by All 10, Inc. 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. Exam 5, V1a Page 99 2014 by All 10, Inc. 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. Exam 5, V1a Page 100 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Section 2: Premium Aggregation – For Workers’ Compensation Questions from the 1994 exam 48. (3 points) Answer this question using the Feldblum Study Note Reading, "Workers Compensation Ratemaking," and the information below. The adjustments to rates that affect the experience period are shown below. • Experience rate change of 10% on 7/1/92. • Law amendment change of 2% on 1/1/93. • Experience rate change of 15% on 7/1/93. • Law amendment change of 3% on 1/1/94. Premium writings are evenly distributed throughout the year. (a) (1.5 points) What adjustment factor is needed to bring calendar year 1993 premiums to current level? (Show a diagram representing the appropriate time periods.) (b) (1.5 points) What adjustment factor is needed to bring policy year 1993 premiums to current level? (Show a diagram representing the appropriate time periods.) Questions from the 1996 exam Question 36. (3 points) Rate Implementation Change Date Type of Change +8% 5/1/94 Experience +15% 7/1/95 Law Amendment -10% 7/1/95 Experience +5% 4/1/96 Experience • Policies are written uniformly throughout the year. According to Feldblum, "Workers' Compensation Ratemaking:" (a) (2 points) Calculate the premium adjustment factor to bring policy year 1995 premium to current rate level. (b) (1 point) How are experience rate changes and law amendment rate changes different in their purpose and their effect? Questions from the 1997 exam 12. You are given: • Full estimated policy premium is booked at inception. • Premium develops upward by 7% at final audit, six months after the policy expires. • All policies are written for an annual period. • Premium is written uniformly throughout the year. Based on Feldblum, "Workers' Compensation Ratemaking," in what range does the policy year premium development factor fall for 24 to 36 months? A. < 1.01 Exam 5, V1a B. > 1.01 but < 1.02 C. > 1.02 but < 1.03 D. > 1.03 but < 1.04 Page 101 E. > 1.04 2014 by All 10, Inc. 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. Exam 5, V1a Page 102 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2002 exam 27. (6 points) Based on Feldblum, "Workers' Compensation Ratemaking," and the information shown below, answer the following questions. Show all work. • Through the use of deviations and schedule rating, your company has been charging 25% below its manual rates for workers compensation. • Policy year 2000 earned premium as of December 31, 2001 = $90 million. • Policy year 2000 reported loss as of December 31, 2001 = $40 million. • Written premium is distributed uniformly by month. • Policy term is 12 months. • Policy audits occur 6 months after expiration and produce a 10% increase in premium. • The following rate changes have been implemented: Date Amount July 1, 1999 - 6.0% July 1, 2000 +10.0% July 1, 2001 + 7.0% • There was a 5% increase in the benefit levels effective January 1, 2001. There was no rate change to account for this. • Loss development factor = 1.80. • Annual loss trend = 8%. • Annual wage trend = 4%. • The effective date for this analysis is July 1, 2002. • Rates will be effective for a period of one year. • Loss adjustment expense = 20% of loss. • The target loss and loss adjustment expense ratio is 72%. a. (2 points) What is the policy year 2000 earned premium after all appropriate adjustments for premium development, current rate level, premium trend, and benefit changes? (chapter 5) b. (2 points) What are the policy year 2000 losses after the appropriate adjustments for loss development, loss trend, and benefit changes? (see chapter 6, but will be computed in this chapter) c. (½ point) What is the projected loss and loss adjustment expense ratio for policy year 2000? (See chapter 6), but this will be computed in this chapter) d. (½ point) What is the indicated rate change based on experience from policy year 2000? (See chapter 8 for the computations needed to answer this question) e. (1 point) What should the ratio of charged to manual premium be in order to produce the target loss and loss adjustment expense ratio? (See chapter 8) Exam 5, V1a Page 103 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2003 exam 33. (2 points) Using the information shown below, calculate the factor needed to adjust policy year 2002 written premium to current level. Show all work. • Policies are written uniformly throughout the year and have a term of 12 months. • The law amendment change affects all policies in force. Assume the following rate changes: • Law amendment change on July 1, 2002 = +10% • Experience rate change on October 1, 2002 = +5% • Experience rate change on January 1, 2003 = +7% Questions from the 2004 exam 11. Given the following data, calculate the policy year 2001 premium development factor from 24 to 36 months. • Full estimated policy year premium is booked at inception, $10 million a month in 2001. • Premium develops upward by 5% at the final audit, three months after the policy expires. • All policies are annual. A. < 1.010 B. > 1.010 but < 1.015 C. > 1.015 but < 1.020 D. > 1.020 but < 1.025 E. > 1.025 31. (4 points) Given the following information, answer the questions below. Show all work. • Policies are written uniformly throughout the year. • Polices have a term of 12 months. • The law amendment change affects all policies in force. Assume the following rate changes: • Experience rate change on October 1, 2001 =+7% • Experience rate change on July 1, 2002 =+10% • Law amendment change on July 1, 2003 = -5% a. (2 points) Calculate the factor needed to adjust calendar year 2002 earned premium to current level. b. (2 points) Calculate the factor needed to adjust policy year 2002 earned premium to current level. Exam 5, V1a Page 104 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2007 exam 37. (2.0 points) Assume the following information about a worker's compensation insurer: All policies are annual. April 1, 2004: The company implemented a 10% experience rate change. October 1, 2004: The company implemented a 5% rate change due to a law change that impacted all in-force policies. a. (1.0 point) Draw the diagram underlying the calculation of the current rate level factor used to adjust policy year 2004 premium to current rate level. Label the starting and ending dates of the historical period. Label the rate change and law change. Calculate the relative rate level of each area and label the diagram. Do not calculate the percentage each area represents of the year. b. (1.0 point) Draw the diagram underlying the calculation of the current rate level factor used to adjust calendar year 2004 earned premium to current rate level. Label the starting and ending, dates of the historical period. Label the rate change and law change. Calculate the relative rate level of each area and label the diagram. Do not calculate the percentage each area represents of the year. Show all work. Questions from the 2009 exam 19. (2.5 points) Given the following information: • All policies are semi-annual. • A +5% rate change was implemented effective October 1, 2007. • A benefit change of +10% was enacted affecting premium on all outstanding policies on July 1, 2008. a. (0.75 point) Draw and label a diagram of the parallelogram method for calendar year 2008 earned premium. b. (1.25 points) Calculate the on-level factor for calendar year 2008 earned premium. c. (0.5 point) Explain why the parallelogram method may not be appropriate for calculating on-level factors for snowmobile insurance. Questions from the 2010 exam 19. (3 points) Given the following information for Company XYZ book of business in State X: • All policies are semi-annual. • A law change is effective on July 1, 2008 and applies to all in-force and future policies. The estimated overall premium impact of the law change is +10%. • A 5% overall rate increase is implemented on October 1, 2008. • 2008 calendar year earned premium is $1,000,000. a. (1 point) Draw and fully label a diagram for calendar year 2008 earned premium reflecting the parallelogram method. b. (1 point) Calculate the on-level factor for calendar year 2008 earned premium. c. (1 point) Draw and fully label a diagram for policy year 2008 earned premium reflecting the parallelogram method. Exam 5, V1a Page 105 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Section 3: Premium Aggregation – Using the One and Two Step Procedures Questions from the 2003 exam 11. Given the information below, determine the written premium trend period. • Experience period is April 1, 2001 to March 31, 2002 • Planned effective date is April 1, 2003 • Policies have a 6-month term • Rates are reviewed every 18 months • Historical premium is earned premium A. < 1.8 years B. 1.8 years, but < 2.1 years D. 2.4 years, but < 2.7 years E. 2.7 years C. 2.1 years, but < 2.4 years Questions from the 2004 exam: 35. (3 points) You are given the following information. Using a two-step trending procedure as described in Jones, "An Introduction to Premium Trend," answer the questions below. Show all work. • The experience period is January 1, 2001 through December 31, 2003. • Planned effective date is July 1, 2005. • Rates are reviewed annually. • Policies have a 6-month term. • The trend will apply to calendar-accident year 2002 earned premium at current rate level. a. (1 point) Calculate the beginning and ending dates for each of the Step 1 and Step 2 trend periods, assuming the selected trend is based on average written premium. b. (1 point) Calculate the beginning and ending dates for each of the Step 1 and Step 2 trend periods, assuming the selected trend is based on average earned premium. c. (1 point) Describe a situation when it may be more appropriate to use a two-step trending procedure, rather than a one-step trending procedure. Exam 5, V1a Page 106 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2005 exam: 37. (4 points) Given the information below, answer the following questions. Show all work. Calendar/Accident Year Average Written Premium 2002 $1,000.00 2003 $933.33 2004 $882.00 • • • • • The planned effective date for a rate change is January 1, 2006. Rates are reviewed every 18 months. All policies are annual, and are written uniformly throughout the year. A 20% rate decrease was implemented effective July 1, 2003. A separate analysis has determined that a shift in the limit distribution from 2002-2004 has resulted in a +3% annual premium trend. This shift is not expected to continue past 2004. a. (3.5 points) Using two-step trending, determine the total premium trend factors for each year above. b. (0.5 point) Why is two-step trending a more suitable procedure for trending premium than for trending loss frequency or severity? Questions from the 2006 exam: 26. (3.5 points) As the actuary for Company XYZ, you are performing a physical damage rate review for State X. Use the following information to answer the questions below. Experience period consists of calendar year premium for 2002 through 2004. Current level earned premium for calendar year 2002 is $42,500,000. Planned effective date of rate revision is June 1, 2006. Anticipate annual rate revisions every 12 months. Each year, insureds purchase newer, more expensive vehicles, resulting in upward premium drift. Historically, the premium drift has averaged 5% through 2004. However, given current trends and expectations regarding future car sales, the insurer expects a 3% premium drift in the future. The insurer uses exponential premium trend. a. (1.5 points) Assume all policies have a six-month term. Use 2-step trending with average written premium to calculate the trended premium for calendar year 2002. Show all work. b. (1.5 points) Assume all policies have an annual term. Use 2-step trending with average written premium to calculate the trended premium for calendar year 2002. Show all work. c. (0.5 point) Explain one advantage of using 2-step trending in this example over 1-step trending. 27. (1 point) a. (0.5 point) Explain why using average premiums is better than total premiums when analyzing premium trend. b. (0.5 point) Give one argument for using average earned premiums in the premium trend analysis and one argument for using average written premiums. Exam 5, V1a Page 107 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2007 exam: 36. (3.0 points) You are given the following information: All policies are annual. The future policy period begins January 1, 2007. The future annual premium trend is 3% per year. The proposed rates will be in effect for one year. Calendar Earned Average Written Year Exposures Premium At Current Rate Level 2003 1.000 $3,777 2004 1,050 3,688 2005 1,100 3,998 Average Earned Premium At Current Rate Level $3,605 3,749 3,899 Calculate the trended premium for each year, using the two-step trending method. Show all work. Questions from the 2008 exam: 15. (2.0 points) a. (0.75 point) Question no longer applicable to the content covered in this chapter. b. (1.25 points) You are given the following information. Accident Year 2004 2005 2006 2007 Average Earned Premium at Current Rate Level $ 98 $102 $106 $110 Average Written Premium at Current Rate Level $100 $104 $108 $112 The projected premium trend is 4%. The proposed effective date of new rates is January 1, 2009. The proposed rates will remain in effect for one year. All policies are semi-annual. Calculate the premium trend factor needed to project 2006 calendar/accident year earned premium to prospective rate levels, using the two-step trending procedure. Exam 5, V1a Page 108 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2010 exam: 18. (2 points) Given the following information: Calendar Year 2008 2009 Earned Written Exposures Exposures 1,000 1,100 1,200 1,300 On-Level Earned On-Level Written Premium $ 487,500 $ 615,000 Premium $ 550,000 $ 682,500 • All policies are annual. • Proposed effective date is January 1, 2011. • Rates are expected to be in effect for one year. • Projected premium trend is 5%. Calculate the calendar year 2008 earned premium at prospective levels using two-step trending. Questions from the 2011 exam: 5. (2.25 points) Given the following information: • Policy term: six months • Proposed rates in effect: January 1, 2012, to June 30, 2013 • Selected projected premium trend: 5% Calendar Average Earned Premium Average Written Premium Year at Current Rate Level at Current Rate Level 2009 $375 $380 2010 $390 $395 a. (2 points) Calculate the total premium trend factor for each of calendar years 2009 and 2010 using two-step trending. b. (0.25 point) Briefly discuss when it is appropriate to use two-step trending. Questions from the 2012 exam: 6. (2 points) Given the following information for a Homeowners company: The 4th Calendar Quarter of 2011 (4Q11) Average Written Premium is $560. The proposed effective date of the next rate change is July 1, 2012. Assume a +5% prospective annual premium trend. Rate review is performed every 2 years. Calendar Year Ending Earned Exposures (House-Years) Earned Premium at Current Rates December 31, 2009 10,000 $5,000,000 December 31, 2010 10,000 $5,250,000 December 31, 2011 10,000 $5,512,500 a. (1 point) Use the two-step trending method to calculate the projected earned premium for the calendar year ending December 31, 2009. b. (1 point) After completing the analysis, the actuary determines that the assumed annual increase in the amount of insurance to account for inflation was materially reduced post-January 1, 2012. Discuss any necessary adjustments to the completed analysis in part a. above Exam 5, V1a Page 109 2014 by All 10, Inc. 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 Page 110 2014 by All 10, Inc. 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. Exam 5, V1a Page 111 Answer D. 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 1996 exam Question 30 (a) To calculate the on-level earned premium for the experience period 1992-1994, CY on-level factors must be computed first. 1.05 1.00 1.00 1.00 1.082 1.05 1/91 (i) (ii) (iii) (iv) (v) (vi) (vii) 1/92 1/93 1/94 1.136 1.082 1/95 The rate change in 1991 is not relevant to the calculation. Calculate the numerator of the on-level factor. This is equal to (1.05)(1.03)(1.05) = 1.136 Calculate the average rate level factor for the calendar year. This is a weighted average of the rate level factors in the calendar year. The weights will be relative proportions of the square. First calculate the area of all triangles (area = .5*base*height) within a unit square and then determine the remaining proportion of the square by subtracting the sum of the areas of the triangles from 1.0. For CY 1992, the average rate level factor = 1.00. The on-level factor = 1.136 / 1.00 = 1.136. For CY 1993, the average rate level factor = (1/2)(.25)(.25)*1.05 + (1.0 - .0325)*1.00 = 1.002. The on-level factor = 1.136 / 1.002 = 1.134 For CY 1994, the average rate level factor = (1/2)(.75)(.75)*1.00 + (1/2)(.5)(.5)*1.082+ (1.0 .40625)*1.05 = 1.04 The on-level factor = 1.136 / 1.04 = 1.092 Thus, the on-level premium is computed On level On level as CY EP factor EP 1992 375,000 1.1355 425,812 1993 875,000 1.1337 991,987 1994 1,125,000 1.0920 1,228,500 Total 2,646,299 Solutions to questions from the 1997 exam Question 19. (a) To facilitate the calculation of CY on-level factors, setup a diagram similar to the one below: Calculate the numerator of the on-level factor. This is equal to (1.05)*(1.13)*(1-.03) = 1.150905. Calculate the average rate level factor for the calendar year. This is a weighted average of the rate level factors in the calendar year. The weights will be relative proportions of the square. First calculate the area of all triangles (area = .50 * base * height) within a unit square and then determine the remaining proportion of the square by subtracting the sum of the areas of the triangles from 1.0. For CY 1995, the average rate level factor = (1/2)(3/12)(3/12)*1.0 + (1/2)(1/2)(1/2)*1.1865+ (1.0 - .15625)*1.05 = .03125 + .1483125 + .8859375 = 1.0655 The on-level factor = 1.150905 / 1.0655 = 1.0801549. Exam 5, V1a Page 112 Answer B. 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 1998 exam Question 41. Note: View the earning of CY EP using a unit square. View the earning of PY EP using a parallelogram. Compute on-level factors as follows: [Current rate level factor / average rate level factor (during the period in question). Rate Changes + 0.05 + .10 0.5 0.028 -0.05 0.15 0.25 0.222 0.75 0 9/1/94 1/1/95 1/1/96 1/1/97 Step 1: Current rate level factor =1.05 * 1.10 * .95 * 1.15 = 1.262. This is the numerator for each on-level factor. Step 2: Calculate the denominators for each on-level factor. The denominators are the average rate level factor for each calendar/ policy year. This is a weighted average of the rate level factors in each calendar / policy year. The weights will be relative proportions of each square / parallelogram. Note: It may be convenient to think of CY 95 with a base of 12 units and a height of 6 units. To compute the relative proportion of the unit square, calculate the areas of as many triangles as possible, and then compute the remaining area by subtracting the sum of the areas of the two triangles from 1.0. Shape Dotted Triangle Bold Triangle Difference Remainder Area (1/2) * (2/12) * (2/6) = .028 (1/2) * (6/12) * (6/6) = .25 .25 - .028 = .222 1 - .028 - .222 = .75 Rate Level 1.0 1.05 1.155 Step 3: Compute EP at present rates by multiplying EP by the CY on-level factor. a. The weighted rate level for 1995 is 1.0 * (.028) + 1.05 * (.222) +1.155 * (.75) = 1.127 b. The 1995 CY on-level factor is 1.262 / 1.127 = 1.120 c. CY 1995 On-Level EP = $1,200 * 1.120 = $1,344 Quicker Solution: 1.00 1.05 1.155 9/1 1/1 1995 The dotted line refers to the 6 month term. Focus on only the 1995 square. As above, numerator is 1.00 * 1.05 * 1.155 = 1.262 Note that small area is ½ * 2/12 * 4/12 = 1/36 Denominator is 1.155(.75) + 1.00(1/36) * 1.05 (1-0.75-1/36) = 1.127 1.262/1.127 = 1.12 (on-level factor for 1995) 1.12 * 1200 = 1,344. Exam 5, V1a Page 113 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 1999 exam: Question 58. Given: The Company issues annual policies, calculated an RLAF of 1.080 for CY 1998 earned premium It was assumed that the only rate change that took place in the last few years was effective 1/1/98, but it was later determined that it was actually effective 3/1/98. It is assumed that all months have an equal number of days and that premium writings are evenly distributed through the year. Step 1: Based on the given information, construct a diagram similar to the one below: To recalculate the RLAF using the 3/1/98 effective date, first calculate the rate change at 1/1/98. X% 1.00 1+X 1/97 1/98 3/1 1/99 (during the period in question) RLAF Current Rate Level Factor Avg Rate Level Factor Since we are assuming only one rate change effective 1/1/98, the current rate level factor is 1+X. The average rate level factor for the calendar year is the weighted average of the rate level factors in the calendar year. The weights will be relative proportions of the square. Solve for X. 1 X Thus, 1.08 , .54 + .54(1+X)= (1+X). .08 = .46X; X = .174 [(.50*1.00) (.50*1 X )] Step 2: To recalculate the RLAF using the 3/1/98 effective date, re-compute the average rate level factor. 1.174 1.174 RLAF 1.1071 [.50(.10/12)(.10/12)*1.174 (1.0 .50(10/12)(10/12))*1.00] 1.0604 Solutions to questions from the 2000 exam: Question 38. c. List two methods used to adjust earned premiums to a current rate level basis. 1. Extension of The best method. Re-rate each policy using current rates. Exposure: 2. Parallelogram: a. Assumes exposures are uniformly written over the Calendar Year (CY) b. Each CY of EP is viewed as a unit square, 1 year wide, 100% of exposure high. d. The more appropriate method to use for this company's personal automobile liability business would be the extension of exposures method. The company's writings show an increasing trend in written exposures which violates the parallelogram method's assumption that exposures are uniformly written over the calendar year. e. Using your selected method from part d. above, calculate the on-level earned premium for calendar year 1998. When using the extension of exposure technique, on-level earned premium equals current rate per unit of exposure * number of earned exposures. In this example: the current rate per unit of exposures is $500 * 1.15 = $575 the number of earned exposures in 1998 = 3,600 Thus, on-level earned premium for calendar year 1998 equals $575 * 3,600 = $2,070,000 Exam 5, V1a Page 114 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2000 exam: Question 40. a. Calculate the on-level factors for each of the two calendar years 1997 and 1998. Step 1: Draw a unit square for each calendar year and diagonal lines at points in time representing historical rate changes. Step 2: Calculate the numerator of the on-level factor. This is the product of all rate changes. Step 3: Calculate the average rate level factor for each calendar year. This is a weighted average of the rate level factors in each calendar year. The weights will be relative proportions of each square. First calculate the area of all triangles (area = .50 * base * height) within a unit square and then determine the remaining proportion of the square by subtracting the sum of the areas of the triangles from 1.0. Step 4: Divide the result of step 1 by the result of step 3: Rate Changes +0.04 +0.018 +0.03 1.0 1.00*1.04=1.04 1.04*1.018=1.0587 1.0587*1.03=1.0905 7/1/96 1/1/97 1/1/98 7/1/99 On-level factor for CY 1997: 1.04*1.018*1.03 1.0905 1.0536 (1/ 2)*(6/12)*(6/12)*(1) (1.0 36/ 288)*(1.04) 1.035 On-level factor for CY 1997 equals 1.0536 * 975,000 = 1,027,260 On-level factor for CY 1998: 1.04*1.018*1.03 1.0905 1.0392 (1/ 2)*(12/12)*(12/12)*(1.04) (1/ 2)*(1)*(1)*(1.0587) 1.0494 On-level factor for CY 1998 equals 1.0392 * 1,000,000 = 1,039,200 Quicker Solution: Numerator is 1.04 * 1.018 * 1.03 = 1.0905 1997 Denominator : (1/8) 1.00 + (7/8) 1.04 = 1.035 On-level factor = 1.0905/1.035 = 1.054 1998 Denominator: (1/2) 1.04 + (1/2) 1.0587 = 1.049 On-level factor = 1.0905/1.049 = 1.039 Exam 5, V1a Page 115 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2001 exam: Question 38. (2 points) Using the parallelogram method described by McClenahan in “Ratemaking,” determine the calendar year 1999 on-level earned premium. Show all work. Step 1: Draw a rectangle (normally a unit square if 1-year policies were issued) for each calendar year and diagonal lines at points in time representing historical rate changes. Step 2: Calculate the numerator of the on-level factor. This is the product of all rate changes. Step 3: Calculate the average rate level factor for calendar year 1999. This is a weighted average of the rate level factors in calendar year 1999. The weights will be relative proportions of each rectangle. First calculate the area of all triangles (area = .5 * base * height) within a unit rectangle and then determine the remaining proportion of the rectangle by subtracting the sum of the areas of the triangles from 1.0. Note: Since 2-year policies are issued, the ratio of the height to the base is 2:1. Step 4: Divide the result of step 1 by the result of step 3: Rate Changes +0.052 +0.074 2 1.00 1.052 1.052 * 1.074 0 7/1/97 1/1/1998 4/1/99 1/1/2000 Area of triangle: 1/2 * base * height Rate level Area 1.00 1/2 * 6/12 * 6/24 = 0.0625 1.129848 1/2 * 9/12 * 9/24 = 0.140625 1.052 1.0 - 0625 - .140625 = 0.7968750 On-level factor for CY 1997: 1.052*1.074 1.129848 1.0661987 (1/ 2)*(6/12)*(6/ 24)*(1.0) (1/ 2)*(9/12)*(9/ 24)*(1.129848) (.796875)*(1.052) 1.0596974 On-level earned premium for CY 1999 equals 1.0661987 * $14,000 = $14,927 Exam 5, V1a Page 116 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2002 exam: Question 17. b. (2 points) Calculate the experience period on-level earned premium. Step 1: Draw a rectangle (normally a unit square for a calendar year if 1-year policies were issued) for each period and diagonal lines at points in time representing historical rate changes. +10% -15% Rate Level: 1.00 .50* b*h = .50*12*12 = 72 .50* b*h = .50*6*12 = 36 Area = (12*18) – (72 +36) =216 Rate Level: 1.10 1/1/2000 .935 = 1.10*.85 1/1/2001 6/30/200 1/1/2002 No of Earned Exposures:200 100 Step 2: Calculate the rate level at various levels during the experience period. This is the product of all rate changes at a given point in time (i.e. 1.00; 1.00 * 1.10 = 1.10; 1.10 * .85 = .935). Step 3: Calculate the on-level factor for the experience period. This is the current rate level divided by the weighted average of the rate level factors in the experience period. The weights will be relative proportions of each rectangle or triangle. First calculate the area of all triangles (area = .5 * base * height) within a unit rectangle and then determine the remaining proportion of the rectangle by subtracting the sum of the areas of the triangles from 1.0. AvgRateLevel Factor .50*12*12*1.0.50*6*6*.935(2167236)*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. Exam 5, V1a Page 117 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2003 exam: 10. A 12-month policy is written on March 1, 2002 for a premium of $900. As of December 31, 2002, which of the following is true? Step 1: Answering this question is best understood in terms of exposures Written exposures are those units of exposures on policies written during the period in question, Earned exposures are the exposure units actually exposed to loss during the period, and Inforce exposures are those exposure units exposed to loss at a given point in time.…. Step 2: Based on the definitions in Step 1, only earned premium differs from written premium and inforce premium and therefore needs to be computed. Thus, earned premium at 12/31/02 equals $900 * 10/12 = $750. Answer E. Solutions to questions from the 2005 exam: 38. (1.5 points) The parallelogram method is used to adjust calendar year 2003 earned premium to current rate level. Given the following information, will the parallelogram method understate, overstate, or accurately state the on-level factor applied to calendar year 2003 earned premium? Explain your answer. • There was a 10% rate increase effective on January 1, 2003. • The written exposures grew 5% each month in 2003. The parallelogram method assumes a uniform distribution of policies is written over an entire calendar year. Using the parallelogram method, the on-level factor for CY 2003 is computed as Current Rate Level 1.10 1.048 Average Rate Level .50*(1.0) .50*(1.1) However, if exposures are growing 5% each month, more weight should be given to the current rate level factor, 1.10. For example, the on-level factor could be computed as 1.10 , where z is less than 50%. z *(1.0) (1 z )*(1.1) This would produce a lower on-level factor compared to that produced by the traditional method. Hence, the parallelogram method would overstate the on-level factor applied to CY 2003 premiums. Exam 5, V1a Page 118 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2006 exam: 28. (3 points) Company XYZ reduced rates 8% effective May 1, 2004, which was their first rate change since January 1, 2000. Assume all policies have annual terms. a. (1 point) Using the parallelogram method, calculate the 2005 on-level factor. Show all work. b. (0.5 point) Assume that this change was for a boatowners line and that 50% of the policies are written uniformly throughout May and June, with the other 50% written uniformly throughout the rest of the year. Is the calculation above reasonable for this line? Explain. c. (1.5 points) Based on the assumptions given in part b. above, calculate the 2005 on-level factor. Show all work. a. The parallelogram method assumes a uniform distribution of policies is written over an entire calendar year. Step 1: Draw a unit square to represent a calendar year, since 1-year policies were issued, for each period under consideration and draw diagonal lines at points in time representing historical rate changes. Rate Change -.08 % of Exposure 1 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. Exam 5, V1a Page 119 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2006 exam: Question 28 (part c. continued): Now, consider a policy year divided into twenty four equal parts, with the first month and the last month of the policy year earning only 1/24 of the premium (earned premium is spread over thirteen months). Thus, we assume that the average policy for each month was written in the middle of the month, such that only 1/24th of the January 2004 policies were still unearned as of 1/1/2005, 3/24th of the February 2004 policies were still unearned as of 1/1/2005, 5/24th of the March 2004 policies were still unearned as of 1/1/2005 and 7/24th of the April 2004 policies were still unearned as of 1/1/2005. Therefore, the proportion of CY 2005 earned exposures from policies written in 2004 at a 1.00 rate level can be computed as follows: January 2004 policies: .05 * (1/24) = 0.0021 February 2004 policies: .05 * (3/24) = 0.0063 March 2004 policies: .05 * (5/24) = 0.0104 April 2004 policies: .05 * (7/24) = 0.0146 Total = 0.0021 + 0.0063 + 0.0104 + 0.0146 = 0.0334 Average Rate Level for 2005 = 0.0334(1.00) + .9666(0.92) = 0.9227 Current Rate Level = 0.92 On-level Factor for 2005 = 0.92/0.9227 = 0.9971 **Finally compare .9227 to .9244, which was computed in part a, and commented on in part b.** Solutions to questions from the 2007 exam: 34. Calculate: a. (0.5 point) 2004 written premium. b. (0.5 point) 2004 earned premium. c. (0.5 point) 2004 policy year premium. d. (0.5 point) Premium in-force as of March 31, 2005. Model Solution a. WP includes all premium written during a calendar period. Thus, 2004 WP = 1,200+ 2,400 + 3,600 = 7,200 b. EP includes that portion of calendar year written premium which has been earned as of 12/31 of the calendar year. 2004 EP = 1,200 + 2,400(1/2) + 3,600(1/6) = 3,000 c. PY premium includes all premium associated with policies issued during a given time period. Policy year data is based upon the year in which the policy giving rise to exposures, premiums, claims and losses is effective. Thus, 2004 PY Premium = 1,200 + 2,400 + 3,600 = 7,200 d. In-force premium includes the full-term premium for each policy that has not expired at a point in time. All individual policy premiums are aggregated to arrive at a total in-force premium for the insurer. Inforce Premium as of 3/31/05 = 2,400 + 3,600 = 6,000 Exam 5, V1a Page 120 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2008: Model Solution - Question 14 14. (2.5 points) Assume a -8% rate change was implemented effective March 1, 2005 and that all policies have annual terms. a. (1.0 point) Calculate the on-level factors for calendar years 2005 and 2006 earned premiums using the parallelogram method. Initial comments. Note that the question fails to state whether policies are uniformly written throughout the policy period. When computing on-level factors using the parallelogram method, such an assumption must be made. Therefore if the question does not state that polices are uniformly written throughout the policy period, it is wise to state that on your answer sheet prior to solving the problem. a. Calculate the on-level factors for CYs 2005 and 2006 earned premiums using the parallelogram method. Step 1: Draw a unit square to represent a calendar year, since 1-year policies were issued, for each period under consideration and draw diagonal lines at points in time representing historical rate changes. Rate Change -.08 % of Exposure 1 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] Exam 5, V1a Page 121 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2008 (continued): Model Solution - Question 14 (continued): b. (1.0 point) Calculate the on-level factors for policy years 2005 and 2006 earned premiums using the parallelogram method. Step 1: Draw a parallelogram to represent a policy year, since 1-year policies were issued. For PYs 2005 and 2006, draw diagonal lines at points in time representing historical rate changes. Rate Change -.08 % of Exposure 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. Exam 5, V1a Page 122 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2009 exam: Question#: 18 a. WP includes all premium written during a calendar period. Thus, CY 2007 WP = 750 + 1,200 + 900 + 800 = $3,650 b. EP includes that portion of calendar year written premium which has been earned as of 12/31 of the calendar year. CY 2008 EP = 1,200 (3/12) + 900 (6/12) + 800 (9/12) + 850 = 300 + 450 + 6 00 + 850 = $2,200 c. PY EP premium includes all premium associated with policies, issued during a given time period, as of a given evaluation date. Thus, PY 2007 earned premium as of 3/31/08 = 750 + 1,200+ 900 (9/12)+ 800(6/12) = 750 + 1,200+ 675 + 400 = $3,025 d. In-force premium includes the full-term premium for each policy that has not expired at a point in time. All individual policy premiums are aggregated to arrive at a total in-force premium for the insurer. In - force premium as of 7/1/08 = 800 + 850 = $1,650 Solutions to questions from the 2011 exam: 4a. (1 point) Calculate the calendar year 2010 earned premium. 4b. (0.5 point) Calculate the on-level factor that applies to the calendar year 2010 earned premium to bring premiums to current rate level. Question 4 – Model Solution 1 Givens: Policy Premium = Base Rate x Class Factor + Policy Fee; Base Rate = $1,000; Policy Fee = $50 Class Teens: Class factor = 2.00; Class Adults: Class factor = 1.00 ABC writes 10 policies per quarter, each with an effective date of the beginning of the quarter. On 7/1, the company increased the base rate to $1,100 and revised the class factor for adults to 0.90. The company writes an even distribution of teen and adult classes each quarter. a. 10 pols issued per quarter equally = 5 adult and 5 teen policies issued each quarter Quarter 1: Adult = 1000 * (1) + 50 = 1050; * 5 policies = 5,250 Teens = 1000 * (2) + 50 = 2050; * 5 policies = 10,250 Quarter 2: same as quarter 1 Quarter 3: Adult = 1100 * (.90) + 50 = 1040; * 5 policies = 5,200 Teens = 1100 * (2) + 50 = 2250; * 5 policies = 11,250 Quarter 4: same as quarter 3 2010 EP = (5,250 + 10,250) + (5,250 + 10,250) * .75 + (5200 + 11250) * .5 + (5200+11250) * .25 = 15,500 + 11,625 + 8,225 + 4,112.50 = 39,462.50 b. EP for 2010 if all @ CRL = [Latest EP for Adult and Teens] * % earned per quarter = (5200 + 11250)(1 + .75 + .5 + .25) = (16450)*(2.5) = 41,125 OLF = EP @CRL/CY 2010 EP = 41,125/39,462.5 = 1.0421286 Question 4 – Model Solution 2 a. Q1 EP: (1000 * 2 + 50) * 5 + (1,000 * 1 + 50) * 5= 15,500 Q2 EP: 15,500 * 3/4 = 11625 Q3 EP: [(1,100 * 2 + 50) * 5 + (1,100 * .9 + 50) * 5] * 1/2 = 16,450 * 1/2 = 8,225 Q4 EP: 16450 * 1/4 = 4112.5 2010 EP = 15,500 + 11,625 + 8,225 + 4,112.5 = 39,462.5 b. 16450 * (1 + ¾ + ½ + ¼ ) = 41,125 On level factor = 41,125/ 39,462.5 = 1.042 Exam 5, V1a Page 123 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2012 exam: 4a. 4b. 4c. 4d. (0.5 point) Calendar year 2011 written premium will be fixed (i.e. not change) at December 31, 2011. (0.5 point) Calendar year 2011 earned premium will be fully earned (i.e. not change) at 12/31/ 2011. (0.5 point) Policy year 2011 written premium will be fixed (i.e. not change) at December 31, 2011. (0.5 point) Policy year 2011 earned premium will be fully earned (i.e. not change) at December 31, 2011. Question 4 – Model Solution 1 (Exam 5A Question 4) a. True, because calendar year written premium is based off of transactions that occur in that year. For example, if a policy that was effective in 2011 is cancelled sometime in 2012 before expiration, this would not impact calendar year 2011 written premium, but would be reflected in calendar year 2012 written premium. b. True, because calendar year earned premium comes from policy transactions that are effective before 1/1/2012. Similar to part (a), if a policy that was effective in 2011 is cancelled in 2012 (prior to expiration), this would not impact CY 2011 Earned Premium, but would be reflected in CY 2012 Earned Premium. c. False, because Policy Year 2011 written premium is based off all transactions for policies that were effective in 2011. So, if a policy written in 2011 is cancelled in 2012 prior to expiration, this would be reflected in PY 2011 written premium (it would not impact PY 2012 written premium). d. False, because Policy Year 2011 earned premium accounts for all transactions for policies that were effective in 2011 (regardless of transaction date). Same would hold true for Earned Prem as holds true for written premium in the example from part (c). Question 4 – Model Solution 2 (Exam 5A Question 4) a. True – CY WP is fixed at year end. CY WP includes all transactions in the calendar period. b. True – CY EP is fixed at year end. CY EP = CY WP + Starting UEPR – Ending UEPR. All these are fixed at year end. c. False – PY11 WP is not fixed @ 12/31/2011. Endorsements and audit premiums in CY2012 and (possibly) beyond will change WP. d. False – PY11 EP cannot be fully earned at 12/31/2011. A policy written 12/1/2011 is only 1/12 earned a/o 12/31/11. Question 4 – Model Solution 3 (Exam 5A Question 4) a. Yes. Includes new prem written + midterm adjustments during calendar year 2011. b. True, calendar year earned premium is premium associated with coverage provided during calendar year 2011. c. Policy year 2011 written premium will not be fixed as of 12/31/2011, because any midterm changes associated with policies effective during 2011, even if change happens in 2012 or later, should be included. E.g. policy effective 7/1/2011, add a new vehicle on 4/1/2012, this contributes to PY 2011 written. d. PY 2011 earned prem will not be fixed as of 12/31/11. This is the earned premium associate with all policies with effective dates in 2011. If they are annual policies, all coverage has not been provided Exam 5, V1a Page 124 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2012 exam: Examiners Comments - Exam 5 Question 4 (Exam 5A Question 4) a. Many candidates answered this correctly. However, some just repeated the question explaining that calendar year 2011 written premium will be fixed at 12/31/11, which isn’t enough for the explanation. There were also candidates who mentioned this includes premium written in 2011 and any cancellations, which isn’t enough of an explanation as need to give some indication as to when cancellation occurred to differentiate from policy year premium. Many candidates mentioned that any transactions occurring for in 2012 will count towards calendar year 2012 written premium, which is enough of an explanation. b. Many candidates answered this correctly. However, some just repeated the question explaining that calendar year 2011 earned premium will be fixed at 12/31/11, which isn’t enough for the explanation. Some candidates mentioned what is earned afterwards in 2012 will go towards calendar year 2012 earned premium, which is enough of an explanation. Similar to a), occasionally a candidate would explain that calendar year data is fixed, which is not enough of an explanation, because need to indicate when it is fixed (i. e. at end of year). c. Of all the parts, part c. was the one most frequently answered incorrectly. Many candidates answered this correctly. However, there were also a significant amount of candidates who did not indicate when the cancellation or midterm adjustment occurred, which is not enough of an explanation as it does not differentiate from calendar year premium. Many times a candidate would say this part is correct because it only includes premium written during the year, which receives 0 points. Occasionally a candidate would say this is fixed at 12/31/12, which isn’t enough of an explanation to receive full credit as it is not necessarily true (i.e. audits). d. Many candidates answered this correctly. Some candidates said this was incorrect because any cancellation or mid-term adjustments would change policy year 2011 earned premium, which is not enough of an explanation to receive full credit as it does not differentiate from calendar year premium (need to mention when cancellation or mid-term adjustment occurs). Questions from the 2012 exam: 5a. (0.5 point) Discuss whether or not it is appropriate to perform a classification ratemaking analysis using premiums adjusted with aggregate on-level factors. 5b. (0.5 point) State one advantage and one disadvantage of the parallelogram method relative to the extension of exposures method. Exam 5 Model Solution 1 – Part a (Exam 5A Question 5a) No. If a rate change disproportionately effects a certain class more than others, the on-level factors will vary by class. Therefore aggregate OLF should not be used. Exam 5 Model Solution 2 – Part a (Exam 5A Question 5a) It would be appropriate only if all classes have had the same rate change history. If not, then we need rate change info for each class, so that the true rate adjustment for each class can be determined. Examiner’s Comments: The answers to part (a) often lacked sufficient detail to demonstrate the candidates understanding of why the aggregate on level factors may/may not be appropriate for class ratemaking. Exam 5, V1a Page 125 2014 by All 10, Inc. 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. Exam 5, V1a Page 126 2014 by All 10, Inc. 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. Exam 5, V1a Page 127 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to the questions from the 1996 exam Question 36. (a). The premium adjustment factor is also known as an on-level factor. The numerator of the on-level factor considers rate changes which impact both PY 1995, represented by the parallelogram below, and rate changes up and through the current level. The denominator of the on-level factor considers only those rate changes which impact PY 1995. Calculate the numerator of the on-level factor. This is equal to (1.0)(1.15)(.90)(1.05) = . 1.08675 Calculate the average rate level factor for the policy year. This is a weighted average of the rate level factors in the policy year. The weights will be relative proportions of the parallelogram. First calculate the area of all triangles (area = .50 * base * height) within the parallelogram and then determine the remaining proportion of the parallelogram by subtracting the sum of the areas of the triangles from 1.0. Notice the area of the parallelogram at the 1.035 level. Its area is calculated as base * height = .50*1.0 = .50. The average rate level factor for the policy year = (1/2)(.5)(.5)*1.0 + (1/2)(.5)(.5)*1.15 +.50*1.0*1.035 + (1.0 - .125 - .125 - .50)*1.15 = 1.07375. +15% -10% 1.15 1.00 1.15 1.00 1/94 5/1 1/95 1.035 1.035 7/1 1/96 The on-level factor = 1.08675 / 1.07375 = 1.012. (b) Experience rate changes are represented graphically as diagonal lines, and are computed to adjust current rates for changes anticipated in projected experience level. These affect new and renewal policies only. Law amendment changes are represented graphically as straight lines, and since they affect all policies inforce at a given point in time. These changes adjust premiums for statutory modifications to benefits. Exam 5, V1a Page 128 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 1997 exam Question 12. Assume that policy year 199X premium is being booked at $P per month. Developed premium, due to final audits, is not known until 6 months after the policy expires. At 12/31/9X+1, developed premium for only those policies issued during the 1st 6 months of PY 199X is known. At 12/31/9X+2, developed premium for all policies issued during PY 199X is known. Reported Premium for polices issued during the Evaluation Date 1st 6 months of PY 199X Last 6 months of PY 199X Total PY 199X 12/31/9X 12/31/9X+1 12/31/9X+2 6 months * ($P/month) 6 * P * 1.07 6 * P * 1.07 6 months * ($P/month) 6*P 6 * P * 1.07 12P 12.42P 12.84P Therefore, the PY premium development factor for 24 to 36 months is 12.84P/12.42P = 1.034 Answer D. Solutions to questions from the 1999 exam Question 37 Note: At 12/31/9X+1, premium for PY 199X is at 24 months of development. At 12/31/9X+2, premium for PY 199X is at 36 months of development. a. Reported Premium for polices issued during the Evaluation Date 1st 6 months of PY 199X Last 6 months of PY 199X Total PY 199X 12/31/9X 12/31/9X+1 12/31/9X+2 6 months * ($1M/month) 6 * ($1M/month)*.20 6 months * ($1.1M/month) 12.6M 12.6M + 1.2M = 13.8M 13.8M + 1.32M =15.12M 6 * ($1.1M/month)*.20 Therefore, the PY premium development factor for 24 to 36 months is 15.12M/13.8M = 1.096 b. CY premiums include audit premium from past policies. As long as premium volume remains steady, next year's audit premiums associated with current exposures should approximate this year's audit premiums due to from prior year's exposures, so the PDF is approximately = 1.00 Exam 5, V1a Page 129 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2001 exam Question 15. Compute the policy year reported premium development factor from 12 to 24 months. Assume that policy year 199X premium is being booked at $P per month. Final audit occurs 3 months after policy expiration. On average, audits result in 15% additional premium. Developed premium, due to final audits, is not known until 3 months after the policy expires. At 12/31/9X+1, developed premium for policies issued during the 1st 9 months of PY 199X is known. At 12/31/9X+2, developed premium for all policies issued during PY 199X is known. Reported Premium for polices issued during the Evaluation Date 1st 9 months of PY 199X Last 3 months of PY 199X Total PY 199X 12/31/9X 12/31/9X+1 12/31/9X+2 9 months * ($P/month) 9 * P * 1.15 9 * P * 1.15 3 months * ($P/month) 3*P 3 * P * 1.15 12P 13.35P 13.80P Therefore, the PY premium development factor for 12 to 24 months is 13.35P÷12.00P = 1.1125 Answer D. Solutions to the questions from the 2001 exam Question 47. a. Describe how premiums and losses are compiled under each of the three experience periods: 1. Policy year experience compiles premiums and losses arising from policies issued in a given period (typically a one year period). Thus, premiums and losses arising from a given block of policies can be directly matched. 2. Calendar year experience reflects financial statement transactions for a given year. Earned premium is defined as written premium for the year plus the unearned premium reserve at beginning of this year minus UEP reserve at end of the year. Calendar year incurred losses are paid losses during the year plus loss reserves at the end of the year minus loss reserves at the beginning of the year. 3. Calendar/Accident year – Premiums are computed as calendar year earned premiums or can be adjusted for audits or earned but not reported (EBNR) premium changes. Losses include payments and reserves for accidents occurring in a given period. b. (1½ points) State one advantage and one disadvantage associated with each type of experience period. Experience period Policy year Advantage It matches premiums and losses from a given block of policies Calendar year It is more “mature” than similarly aged policy year or cal/acc year experience. Calendar/Acc year Accident year losses can be matched to the corresponding exposure year earned premium. Exam 5, V1a Page 130 Disadvantage Policy year experience is less “mature” than similarly aged calendar year or cal/acc year experience. It is not available for individual classifications and premium and loss experience are not related to a given block of policies. Premium must be adjusted for exposure audits or retrospective adjustments 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2002 exam Question 27. a. (2 points) What is the policy year 2000 earned premium after all appropriate adjustments for premium development, current rate level, premium trend, and benefit changes? Step 1: Draw a diagram similar to the one below which identifies periods in time in which rate changes take place. Benefit level change +5% Rate change -6% 10% 7% 1.0 7/1/99 1/1/2000 7/1/00 1/1/2001 1.1 7/1/01 Policy year 2000 is represented by the dashed line parallelogram. Further, rate level changes are shown separately from benefit level changes, since the problem states that although a 5% increase in benefit levels were effective 1/1/01, no rate change to account for the benefit level change took place. Step 2: To determine premium development, a development factor to account for premium audits needs to be determined. At 12/31/01, policies issued between 1/1/00 – 6/30/00 have completed their audits whereas policies issued between 7/1/00 – 12/31/00 have not. At 12/31/01, the factor 1.10 1.1 to account for future premium development is 1.047619 .5(1.10) .5(1.0) 1.05 Step 3: To determine the current rate level, we can ignore the -6% rate level change that was effective 7/1/99, establish a base rate level of 1.0, and determine that the current rate level is (1.0 * 1.10 * 1.07) 1.177. The average rate level for policy year 2000 is 1.05 (.50*1.0+.50*1.10) and therefore: The on-level factor for policy year 2000 is Current Rate Level 1.177 1.121 Average Rate Level 1.050 Step 4: To determine the premium trend period, one must determine the time between the average date of writing during policy year 2000 (7/1/00) and the corresponding projected date in the forecast period. Since we are told that the effective date of the analysis is 7/1/02, and that rates will be effective for a period of one year, average written date during the forecast period is 1/1/03. Thus, the premium trend period is 2.5 years (7/1/00 – 1/1/03), and the premium trend factor is 1.04 2.5 = 1.103. Step 5: Using the policy year 200 earned premium given in the problem, and the results of Steps 2 – 4, compute on-leveled, developed and trended earned premium. On-leveled, developed and trended policy year 2000 earned premium is 90M * 1.0476 * 1.121 * 1.103 = 116.58M Exam 5, V1a Page 131 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2002 exam (continued) b. (2 points) What are the policy year 2000 losses after the appropriate adjustments for loss development, loss trend, and benefit changes? Step 1: A development factor to account for benefit level changes needs to be determined. Since a 5% increase in benefit levels affects all policies inforce as of its effective date (shown as the solid vertical line at 1/1/01 in the graph above), the factor to account for this benefit level change is 1.05 1.024 .5(1.0) .5(1.05) Step 2: To determine the loss trend period, one must determine the time between the average accident during the experience period (which for policy year 2000 is 1/1/01) and the average accident date during the effective period of the rates (which for a one year effective period beginning 7/1/02 is 7/1/03). Thus, the loss trend factor is 1.082.5 = 1.212 Therefore, losses adjusted for development, benefit changes, trend and loss adjustment expenses are 40M * 1.80 * 1.024 * 1.212 * 1.20 = 107.28M c. (½ point) What is the projected loss and loss adjustment expense ratio for policy year 2000? The projected loss and LAE ratio for policy year is the ratio of the result from questions (b) to (a) 107.28 above: .92 116.58 d. (½ point) What is the indicated rate change based on experience from policy year 2000? The indicated rate change based on experience from policy year 2000 is the ratio of the projected .92 loss and LAE ratio to the garget loss and LAE ratio minus one: 1 .278 .72 e. (1 point) What should the ratio of charged to manual premium be in order to produce the target loss and loss adjustment expense ratio? Since the company has been charging 25% below its manual rates for workers compensation, and since the target loss and loss adjustment expense ratio is based on the anticipated expense costs during the future policy period, the ratio of charged to manual premium to produce the target loss and loss adjustment expense ratio should be 1.278 * (1.0 - .25) = .96 Exam 5, V1a Page 132 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2003 exam 33. (2 points) Calculate the factor needed to adjust policy year 2002 written premium to current level. Show all work. Step 1: Draw a diagram similar to the one below which identifies periods in time in which rate changes take place. Law amendment change +10% Rate change + 5% +7% 1.10 1.155 1.0 7/1/99 1/1/2002 7/1/02 10/1/02 1/1/03 Policy year 2002 is represented by the dashed line parallelogram. Further, rate level changes are shown separately from law amendment changes. Step 2: To determine the current rate level, establish a base rate level of 1.0, and determine that the current rate level is (1.10 * 1.05 * 1.07) 1.236. Since PY 2002 had 3 rate levels in effect, we need to determine the respective area weights to apply to the rate levels. For the 1/1/02 level, the weight is ½ *½ *½ = 1/8. For the 10/1/02 level, the weight is ¼ * 1.0 = ¼. Thus, the weight for the 7/1/02 level is 1.00 – 1/8 – ¼ = 5/8. The average rate level for policy year 2002 is (1/8 * 1.0 + 5/8 * 1.10 + ¼ * 1.155) 1.101. Current Rate Level 1.236 Therefore, the on-level factor for policy year 2002 is 1.122 Average Rate Level 1.101 Solutions to questions from the 2004 exam 11. Given the following data, calculate the policy year 2001 premium development factor from 24 to 36 months. • Full estimated policy year premium is booked at inception, $10 million a month in 2001. • Premium develops upward by 5% at the final audit, three months after the policy expires. • All policies are annual. We are told that developed premium, due to final audits, is not known until 3 months after the policy expires. At 12/31/02, developed premium for policies issued during the 1st 9 months of PY 2001 is known. At 12/31/03, developed premium for all policies issued during PY 2001 is known. This can be demonstrated mathematically as follows: Reported Premium for polices issued during the Evaluation Date 1st 9 months of PY 2001 Last 3 months of PY 2001 Total PY 2001 12/31/01 12/31/02 12/31/03 9 months * $10M/month 9 * $10M * 1.05 9 * $10M * 1.05 3 months * $10M/month 3 * 10M 3 * $10M * 1.05 120M 124.5M 126M Therefore, the PY premium development factor for 24 to 36 months is $126M/$124.5M = 1.012 Answer B: > 1.010 but < 1.015 Exam 5, V1a Page 133 2014 by All 10, Inc. 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 Exam 5, V1a Page 134 Current Rate Level 1.1182 1.051 Average Rate Level 1.0637 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2004 exam (continued): Question 31 (continued): b. (2 points) Calculate the factor needed to adjust policy year 2002 earned premium to current level. Step 1: Draw a diagram similar to the one below which identifies periods in time in which rate changes take place. Law amendment change Experience rate changes -5% + 7% +10% 1.045 1.0 1.10 10/1/01 7/1/02 7/1/03 Policy year 2002 is represented by the dashed line parallelogram. Further, rate level changes are shown separately from law amendment changes. Step 2: To determine the current rate level, establish a base rate level of 1.0, and determine that the current rate level is (1.00 *01.10 * .95) = 1.045. Since PY 2002 had 3 rate levels in effect, we need to determine the respective area weights to apply to the rate levels. Prior to the 7/1/02 experience rate change, the weight associated with the PY 2002, 1.0 rate level, is .50 (half the area of the parallelogram). The relative weight associated with the 7/1/03 law amendment change, with a rate level of 1.10 * .95 = 1.045, is ½ *½ *½ = 1/8. Thus, the weight for the 7/1/02,1.10 rate level, is 1.00 – 1/8 – 1/2 = 3/8. The average rate level for policy year 2002 is (.50 * 1.00 + .375 * 1.10 + .125 * 1.045) = 1.0431. Therefore, the on-level factor to adjust policy year 2002 earned premium to current level is Current Rate Level 1.045 1.002 Average Rate Level 1.0431 Exam 5, V1a Page 135 2014 by All 10, Inc. 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 Page 136 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2009 exam: Question 19: a. Since a rate change was effective on 10/1/07 and applies to all future policies sold, a diagonal line is drawn at 10/1 to graphically depict the impact of the change when computing the on-level factor. Since a law change was effective on 7/1/08 and applies to all in-force and future policies, a solid vertical line is drawn at 7/1 to graphically depict the impact of the change when computing the on-level factor. 0.05 0.10 1.0 1.05 10/07 01/08 04/08 b. OLF 1.155=1.05*1.10 07/08 Current Rate Level Factor Avg Rate Level Factor The current rate level factor equals the product of all rate changes occurring during CY 2008 CRLF = 1.0 * 1.05 * 1.10 = 1.155 The average rate level factor is a weighted average of the varying rate levels that occurred in CY 2008. The weights will be relative proportions of the CY square. First calculate the area of all triangles (area = .5 * base * height) or rectangles within a unit square and then determine the remaining proportion of the square by subtracting the sum of the areas of the triangles and rectangles from 1.0. Since all policies are semi-annual, the diagonal line is representative of a policy written 10/1/2007 and expiring 3/31/2008. CY 2008 Average rate level = (.50)(3/12)(6/12) * 1.0 + [(1/2) - (.50)(3/12)(6/12)] * 1.05 + (.50)*1.155 = .0625 +.459375 +.5775 = 1.099375 On-level factor for 2008 CY EP = 1.155/1.099375 = 1.05059693 c. Snowmobile insurance is not uniformly earned throughout the year. The parallelogram method assumes uniform earnings. Exam 5, V1a Page 137 2014 by All 10, Inc. 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 Exam 5, V1a 7/1 10/1 Page 138 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Section 3: Premium Aggregation – Using the One and Two Step Procedures Solutions to questions from the 2003 exam 11. Determine the written premium trend period. Step 1: Determine the average written date during the experience period. For the experience period 4/1/01 – 3/31/02, and given that 6 month policies are being written, the average earned date is 10/1/01 and the average written date is 7/1/01, or ½ the policy term earlier from the average earned date. Step 2: Determine the average written date during the exposure period. The average written date during the future policy period is a function of the length of time that the rates are expected to remain in effect. In this example, since rates are reviewed every 18 months, this would make the average written date 9 months after the proposed effective date of 4/1/03, which is 1/1/04. Thus, the written premium trend period is 2.50 years. Answer: D. 2.4 years, but < 2.7 years Exam 5, V1a Page 139 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2004 exam Question 35. a. (1 point) Calculate the beginning and ending dates for each of the Step 1 and Step 2 trend periods, assuming the selected trend is based on average written premium. Preliminary information. The solution below includes a graphic depicting the beginning and ending dates for each of the Step 1 and Step 2 trend periods, assuming the selected trend is based on average written or average earned premium. The graphic is included in our solution for instructional purposes only. What are the trending periods to apply to CY/AY 2002 earned premium at current rate level using a twostep trending procedure? 2001 2002 Average Written Date 4/1/02 Average Earned Date 7/1/02 Step 1 2003 2006 Average Date for Latest Trend Point 7/01/003 Future Effective Date 7/1/05 Average Written Date 1/1/06 Average Earned Date 4/1/06 Step 2 a. AWP Step 1: Determine the trend period from the average written date of the experience period to the average date for the last data point in the average written date series: To determine the average written date, recognize that the first policies that contribute to calendar year 2002 earned premium would be ones written on 7/2/01, since these policies would be effective until the end of the day on 1/1/02. The last policies that would contribute to CY 2002 earned premium would be ones written on 12/31/02. The total amount of time between the two written dates is 18 months, so the average written date is 4/1/02. In establishing the ending point for the first part of the trending period (step 1), it is important to recognize that the average written premium measures in the series are 12-month averages. This means that each figure provides a measure of the average premium at the midpoint of its 12-month period. In other words, since the latest trend point in the series is for the year ending 12/31/03, then the measure of the average premium for that point corresponds to 7/1/03, not 12/31/03. Thus, the average written date of the experience period is 4/1/02 and the average date for the last data point in the average written date series is 7/1/03. Step 2: Determine the trend period from the average written date for the last data point in the average written date series to the average written date under the effective period of the rates. As stated before, the average written date for the last data point in the average written date series under the experience period is 7/1/03. The average written date for polices effective during the planned effective period is January 1, 2006. This is because the average written date in the future policy period does not depend on the length of the policies. Instead, it is the length of time the rates are assumed to be in effect before the next revision. Therefore, the beginning and ending dates for Step 2 trend is 7/1/03 – 1/1/06. Exam 5, V1a Page 140 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2004 exam (continued): b. (1 point) Calculate the beginning and ending dates for each of the Step 1 and Step 2 trend periods, assuming the selected trend is based on average earned premium. Preliminary information. It is important to realize that whether the selected trend is based on average written premium or average earned premium, the two alternatives have the same length trending periods. However, these periods are not identical. The trending period for the average earned premium approach is shifted in time so that it is a half a policy period later than the trending period for the average written premium approach. 2001 2002 Average Written Date 4/1/02 2003 2006 Average Date for Latest Trend Point 7/01/003 Average Earned Date 7/1/02 Step 1 Future Effective Date 7/1/05 Average Written Date 1/1/06 Average Earned Date 4/1/06 Step 2 a. AWP Step 1 Step 2 b. AEP 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. Exam 5, V1a Page 141 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2005 exam: 37a. (3.5 points) Using two-step trending, determine the total premium trend factors for each year above. Initial comments: the two-step trending method simply divides the latest average written premium at current by the average earned premium at current for each year in the experience period. This produces conversion factors for adjusting the total earned premium at current rate level for each year to the latest period’s average written premium level. In establishing the ending point for the first part of the trending period (step 1), it is important to recognize that the average written premium measures in the series are 12-month averages. This means that each figure provides a measure of the average premium at the midpoint of its 12-month period. In other words, if the latest trend point in the series is for the year ending 12/31/01, then the measure of the average premium for that point corresponds to 7/1/01, not 12/31/01. Therefore, the first step of the two-step trending procedure trends the premium to the midpoint of the latest trend data point in the series. The second step of the two-step trending procedure trends the premium from the midpoint of the latest trend data point to the average written date for the future policy period. If the target effective date were 1/1/03, then the average written date for the future policy year would be half way through, or 7/1/03, with the standard assumption that the proposed rates will be in effect for one year. The trending period in this example would need to extend from the midpoint of the latest average written premium measure (7/1/01) to the average written date for the future policy period (7/1/03). Therefore, the trending period for the second step would be two years. Problem Specific: First, one needs to adjust the historical premiums for the 20% rate decrease on 7/1/03. For CAY 2004 – The average written premium does not need to be adjusted For CAY 2003 – One half of the written premium needs to be adjusted down by 20%. Thus, the adjusted CAY 2003 average written premium is ½(933.33) + ½(933.33)(0.8) = 840 For CAY 2002 – The entire premium needs to be adjusted downward by 20%: 1,000 × 0.80 = 800 The first step in the two-step trending is to divide the latest year’s average written premium by each year’s average written premium. The ratios are the trend factors for step 1. They are used to trend the premiums to 7/1/04 and are computed as follows: CAY Trend Factor 2002 882/800 = 1.1025 2003 882/840 = 1.05 2004 882/882 = 1.0 This factor already includes the 3% trend due to shifts in limit distributions from 2002-2004. Exam 5, V1a Page 142 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2005 exam (continued): Question 37 (continued): In step 2, project the average premiums for each year to the anticipated future level. A prospective trend is not given, so I will use the historical trend of 1.05 reduced for the 3% trend not continuing past 2004. Thus, the prospective trend = 1.05/1.03 = 1.019 = 1.9% The step 2 trending period extends from 7/1/04 to the average written date of effective period. As rates are reviewed every 18 months, and given that the planned effective date for a rate change is January 1, 2006, the average written date will be 9 months past the effective date, or 10/1/06. Trend factor for step 2 = (1.019)2.25 = 1.043 Thus, the total premium trend factor is calculated as follows: CAY Step 1 Step 2 Total (1) (2) (3)=(1)*(2) 2002 1.1025 1.043 1.15 2003 1.05 1.043 1.095 2004 1.0 1.043 1.043 See page 28. b. (0.5 point) Why is two-step trending a more suitable procedure for trending premium than for trending loss frequency or severity? This procedure relies on the assumption that the latest year’s average written premium is a time value. For premiums, this assumption holds because premiums are relatively stable. Loss severity and frequency values vary greatly over time and the assumption does not hold. Alternatively, “Consider the theoretical implications of two-step trending. This trending method rests on the assumption that the last data point of the trend series is a “true” number. For loss frequency or severity, this can be a dubious assumption because of random fluctuations around the true expected value. For average premium, on the other hand, the individual data points are more believable because there is not as large a random element.” Exam 5, V1a Page 143 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2006 exam: Question 26 a. (1.5 points) Assume all policies have a six-month term. Use 2-step trending with average written premium to calculate the trended premium for calendar year 2002. Show all work. Step 1: Determine the trend period from the average written date of the experience period to the average date for the last data point in the average written date series: To determine the average written date, recognize that the first policies that contribute to calendar year 2002 earned premium would be ones written on 7/2/01, since these policies would be effective until the end of the day on 1/1/02. The last policies that would contribute to CY 2002 earned premium would be ones written on 12/31/02. The total amount of time between the two written dates is 18 months, so the average written date is 4/1/02. In establishing the ending point for the first part of the trending period (step 1), it is important to recognize that the average written premium measures in the series are 12-month averages. This means that each figure provides a measure of the average premium at the midpoint of its 12-month period. In other words, since the latest trend point in the series is for the year ending 12/31/04, then the measure of the average premium for that point corresponds to 7/1/04, not 12/31/04. Thus, the average written date of the experience period is 4/1/02 and the average date for the last data point in the average written date series is 7/1/04. This is the period where premium will be trended by the historic premium drift of 5%. Step 2: Determine the trend period from the average written date for the last data point in the average written date series to the average written date under the effective period of the rates. As stated before, the average written date for the last data point in the average written date series under the experience period is 7/1/04. The average written date for polices effective during the planned effective period is December 1, 2006. This is because the average written date in the future policy period does not depend on the length of the policies. Instead, it depends on the length of time the rates are assumed to be in effect before the next revision. Therefore, the beginning and ending dates for Step 2 trend is 7/1/04 – 12/1/06. This is the period where premium will be trended by the expected future premium drift of 3%. Thus, the trended premium for calendar year 2002 is computed as follows: Exam 5, V1a Page 144 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2006 exam: Question 26, part b: b. (1.5 points) Assume all policies have an annual term. Use 2-step trending with average written premium to calculate the trended premium for calendar year 2002. Show all work. Note: The only difference in solving this problem, compared with the problem in part a, is the starting date for the trend period. The rationale given for all other points in time in as stated in part a, for both steps, holds. To determine the average written date, given annual policies, recognize that the first policies that contribute to calendar year 2002 earned premium would be ones written on 1/2/01, since these policies would be effective until the end of the day on 1/1/02. The last policies that would contribute to CY 2002 earned premium would be ones written on 12/31/02. The total amount of time between the two written dates is 24 months, so the average written date is 1/1/02. Thus, the trended premium for calendar year 2002 is computed as follows: c. (0.5 point) Explain one advantage of using 2-step trending in this example over 1-step trending. 1-step-trending assumes uniform trend from the experience period to the future policy period. This assumption does not apply to certain situations where there are differences in trend between the past and the future. The 2-step trending procedure solves this problem. Exam 5, V1a Page 145 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2006 exam (continued): 27. (1 point) a. (0.5 point) Explain why using average premiums is better than total premiums when analyzing premium trend. b. (0.5 point) Give one argument for using average earned premiums in the premium trend analysis and one argument for using average written premiums. CAS Model Solution Part a. Total premiums are affected by exposure changes, while average premiums have averaged out the exposure effects. Thus changes in average premium are more related to the actual trend in premium. Part b. 1 – The premiums being trended are earned premiums, thus it is better to use average earned premiums in the premium trend analysis. 2 – Average written premiums are more responsive to recent changes. As Jones states “Since these trends will apply to historical earned premium at current rate level, we should evaluate trends based on shifts in average earned premium.” “Even though the historical premium is earned premium, we can determine the average written date for that block of premium and then observe changes in average written premium to establish the trend. Therefore, basing the trend analysis on average written premium is a valid approach. Furthermore, average written premium has an important advantage in that it allows us to capture more recent data than average earned premium. This is because of the simple fact that the premium for a given policy is not earned until well after it is written. In fact, at any given point in time, the latest quarter’s average earned premium is based on a group of policies that is a half a policy period older than the group of policies comprising the latest quarter’s average written premium. Using average earned premium would unnecessarily postpone the recognition of the effects of the most recent changes in the mix of business.” Solutions to questions from the 2007 exam: Question 36 - Calculate the trended premium for each year, using the two-step trending method. Model Solution - Initial comments. The two-step trending method requires the use of average earned premium at current rate level for each year in the experience period. The components are total earned premium at current rate level and earned exposures. In this problem, we are given the average earned premium at current rate level. How the two-step trending method is used. The two-step trending method simply divides the latest average written premium at current level by the average earned premium at current for each year in the experience period. This produces conversion factors for adjusting the total earned premium at current rate level for each year to the latest period’s average written premium level. Exam 5, V1a Page 146 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2007 exam: Question 36 - Calculate the trended premium for each year, using the two-step trending method. Model Solution Step 1: Bring the average earned premium at current rate level to the latest level available in the series of average written premiums at current rate level. This accounts for shifts in the mix of business and any other factors not already accounted for with a direct adjustment to the historical experience. For Step 1, we don’t need to consider exposures because average written premiums at current level are used. Step 2: Project the average written premiums at current level for each year to the anticipated future rate level. A three percent annual trend (stated in the problem (see (3)) is applied over a two-year period. The Step 2 trend period is 2 years (from 7/1/05 to 7/1/07) at 3%. Latest Total Value of Premium Step 1 Step 2 Avg EP Avg WP Trend @CRL @CRL (7/05) Trend Factor Trend Factor Factor CY (1) (2) (3) = (2)/(1) (4) (5) = (3)*(4) 2003 3,605 3,998 1.1090 1.177 1.032 2 2004 3,749 3,998 1.0664 1.131 1.03 2 1.088 2005 3,899 3,998 1.0254 1.03 CY 2003 2004 2005 Trended Average Premium (6) = (1)*(5) 4,242 4,242 4,242 Earned Exposures (7) 1,000 1,050 1,100 Trended Total Premium (8) = (6)*(7) 4,242,000 4,454,100 4,666,200 (4) = The selected annual trend for Step 2 (given in the problem as 3%) is applied from the midpoint of (2) to the average written date in the future policy period (which is 7/1/2007 in this problem). Note that the total premium trend factors in column (5) are used to compute trended average premium in (6), and are used in place of those developed by the one-step procedure. Exam 5, V1a Page 147 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2008 exam: Model Solution – part a. – question 15 a. Question no longer applicable to the content covered in this chapter. Model Solution – part b – question 15. - Initial comments. The two-step trending method requires the use of average earned premium at current rate level for each year in the experience period. This problem is based upon the example in Appendix 2 - the Two-Step Trending Method. Keep in mind that all policies are semi-annual and thus, Jones’ comments on “What about six month policies on pages 17 – 18 apply. In particular “For a six-month policy term, the first step of the procedure will involve a shorter trending period than the one used for 12-month policies. This is because the average written and average earned dates are closer together for shorter policies. The break point between the first and second step is still the same since we use 12month moving averages of written premium in both analyses. The second step of the procedure results in the same length trending period as was used for 12-month policies. This is because the average written date in the future policy period does not depend on the length of the policies. Instead, it is the length of time the rates are assumed to be in effect before the next revision.” In step 1, bring the average earned premium at current rate level to the latest level available in the series of average written premiums at current rate level. In step 2, project the average premiums for each year to the anticipated future level. In this example, a 4 percent annual trend is applied over a two-year period. NOTE: The following is not needed to solve the problem but is provided to give you a broader understanding of what is happening in this example. The first policies that contribute to calendar year 2006 earned premium would be ones written on 7/2/05, since these policies would be effective until the end of the day on 1/1/06. The last policies that would contribute to 2006 earned premium would be ones written on 12/31/06. The total amount of time between the two written dates is 18 months, so the average written date is 4/1/06. In establishing the ending point for the first part of the trending period (step 1), it is important to recognize that the average written premium measures in the series are 12-month averages. This means that each figure provides a measure of the average premium at the midpoint of its 12-month period. In other words, since the latest trend point in the series is for the year ending 12/31/07, then the measure of the average premium for that point corresponds to 7/1/07, not 12/31/07. Therefore, the first step of the two-step trending procedure trends the premium to the midpoint of the latest trend data point in the series. The second step of the two-step trending procedure trends the premium from the midpoint of the latest trend data point to the average written date for the future policy period. Since the target effective date is 1/1/09, then the average written date for the future policy year would be half way through, or 7/1/09, with the standard assumption that the proposed rates will be in effect for one year. The trending period in this example would need to extend from the midpoint of the latest average written premium measure (7/1/07) to the average written date for the future policy period (7/1/07). Therefore, the trending period for the second step would be two years. Exam 5, V1a Page 148 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2008 exam: Model Solution – part b. – question 15 Thus, the Step 1 trend factor is 112/106 = 1.056 and Step 2 trend factor = 1.042 = 1.0816, and The trend factor to 2006 calendar/accident year = 1.0566 x 1.0816 = 1.1428 This can also be demonstrated as shown below. (1) Year Avg EP @CRL (2) Latest Value of Avg WP @CRL 2004 2005 2006 2007 $98 $102 $106 $110 $112 $112 $112 $112 (3) (4) Step 1 Trend Factor (3)=(2)/(1) 1.1429 1.0980 1.0566 1.0182 Step 2 Trend Factor 1.0816 1.0816 1.0816 1.0816 (5) Total Premium Trend Factor (5)=(3)*(4) 1.2361 1.1876 1.1428 1.1013 Solutions to questions from the 2010 exam: Question 18 Calculate CY 2008 earned premium at prospective levels using two-step trending. Step 1: Adjust the historical premium to the current trend level using the following adjustment factor: Current Premium Trend Factor = Latest Average WP at Current Rate Level Historical Average EP at Current Rate Level Latest Avg WP at Current Rate Level is 682,500/1,300 = 525 Historical Avg EP at Current Rate Level is 487,500/1,000 = 487.50 Thus, the current premium trend factor is 1.0769 (= 525/487.50). The latest average WP is for CY 2009; thus, the average written date is 7/1/2009 (this will be “trend from” date for the second step in the process). Step 2: Compute the projected premium trend factor. Select the amount the average premium is expected to change annually from the “trend from” date to the projected period. The “trend from” date is 7/01/2009. The “trend to” date is the average written date during the period the proposed rates are to be in effect, which is 7/01/2011. Thus, the projected trend period is 2 years long (7/1/2009 to 7/1/2011). Given a projected annual premium trend of 5%, the projected trend factor is 1.1025 (= (1.0 + 0.05)2). The total premium trend factor for two-step trending is the product of the current trend factor and the projected trend factor (i.e. 1.18728 (= 1.0769 x 1.1025)). That number is applied to the average historical EP at current rate level to adjust it to the projected level: CY08 EP at projected rate level = CY08 EP at current rate level x Current Trend Factor x Projected Trend Factor. CY 2008 earned premium at prospective levels = (487,500) (1.0769) (1.052) = 578,800.10 Exam 5, V1a Page 149 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2011 exam: 5. (2.25 points) Given the following information: • Policy term: Six months; Proposed rates in effect from 1/1/2012 to 6/30/2013 • Selected projected premium trend: 5% Calendar Average Earned Premium Average Written Premium Year at Current Rate Level at Current Rate Level 2009 $375 $380 2010 $390 $395 5a. (2 points) Calculate the total premium trend factor for each of CYs 2009 and 2010 using two-step trending. 5b. (0.25 point) Briefly discuss when it is appropriate to use two-step trending. Question 5 - Model Solution 1 a. Two-step trending = Use Step 1 and Step 2 premium trend factors - For CY 2009 Step 1 trend = (Avg WP@CRL Latest period) / (Historical Avg EP@CRL) = 395/375 = 1.05333 AWD for CY 2010 = 7/1/10. Average written date for the period 1/1/2012 to 6/30/2013 is 10/1/2012 Step 2 trend = Starts 7/1/10, Ends 10/1/12. Step 2 trend period from 7/1/10 - 10/1/12 = 2.25 years Step 2 trend = (1.05)2.25 = 1.116 CY 2009 total premium trend factor = (1.0533)(1.052.25) = 1.1756 - For CY 2010 Step 1 trend = 395/390 = 1.0128 (see above formula) Step 2 trend = trend from 7/1/10 – 10/1/12 = 2.25 years CY 2010 Total premium trend factor = (1.0128)(1.052.25) = 1.1303 b. It is appropriate to use two step trending when the historical trend and the prospective trend are different. Question 5 - Model Solution 2 a. (1) (2) 2010 CY Avg EP Avg. WP 2009 375 395 2010 390 395 (3) = (2)/(1) 1.0533 1.0128 (4) Premium Trend 1.052.25 1.052.25 (5) (5) = (3)x(4) 1.1755 1.1303 2nd step trend period is from 7/1/2010 to 9/30/2012 which is 2.25 years. b. When the future premium trend is different from the current trend, we cannot use one-step trend, we need to use a 2- step trend instead. Exam 5, V1a Page 150 2014 by All 10, Inc. Chapter 5 – Premium BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2012 exam: 6a. (1 point) Use the two-step trending method to calculate the projected earned premium for the calendar year ending December 31, 2009. 6b. (1 point) After completing the analysis, the actuary determines that the assumed annual increase in the amount of insurance to account for inflation was materially reduced post-January 1, 2012. Discuss any necessary adjustments to the completed analysis in part a. above Question 6 – Model Solution 1 (Exam 5A Question 6) Step 1 factor = latest average written premium @ CRL (current rate level) Calendar year 2009 average earned premium @ CRL = 560/(5,000,000/10,000) = 560/500 = 1.12 Step 2 => trend from = 11/15/2011 <-midpoint of latest period. trend to = 7/1/ 2013 <-average written date in projected period = proposed effective date + ½ the time rates are expected to be in effect. →trend period = 1.625, and the Step 2 trend factor = (1.05) ^ 1.625 Projected Earned Premium for CY 2009 = EP @ CRL x Step 1 factor x Step 2 factor = 5,000,000 x (1.12) x (1.05) ^ 1.625 = $6,062,066. b. The assumed annual increase in the amount of insurance to account for inflation is an ongoing and gradual change, and is reflected in the prospective annual premium trend. So it would be necessary to adjust the prospective annual premium trend of +5% downwards to reflect this reduction, which would resultantly adjust the Step 2 factor. Note that since 2-step trending is used in part (a), it will be appropriate to only adjust the Step 2 factor since this change means trend expected in the future will be different from historical trend. Question 6 – Model Solution 2 (Exam 5A Question 6) Step 1: 560/ (5,000,000/10,000) =1.12 Step 2: from 11/15/2011 to 7/1/2013 From avg. of latest period (4Q11) to avg. written date of prospective period (7/1/2012 t0 6/30/2014) <-2 years. Thus, the step 2 trend factor is 1.05 ^ (1.625) = 1.0825 Total Projected EP = 5,000,000 x 1.12 .x 1.0825 = 6,062,065.69 b You would need to re-calculate your selected prospective trend in step 2. Step 1 can be left alone, however the step 2 trend would be less than 5%, and would lower the projected premium. Question 6 – Model Solution 3 (Exam 5A Question 6) Average written date in 4Q 11 is Nov. 15, 2011 Average written date for 2 year effective period starting July 1, 2012 is July 1, 2013. Thus, the Prospective Trend period is 1.625 years Average earned premium for CY2009 is 5,000,000 ÷ 10,000 = 500 Projected Earned Premium for CY2009 is 5,000,000 (560/500) (1.05 ^ 1.625)= 6,062,065.69 b. The 5% prospective premium trend is likely too high and should be reduced in the analysis from a Examiner’s Comments a. The majority of candidates received full credit. Those that didn’t receive full credit typically lost points for calculating the trend period incorrectly. b. Most candidates either identified both or only one of the other elements needed for full credit. Some candidates identified that the first step in two step trending would not be affected, but this was not necessary for full credit. Exam 5, V1a Page 151 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Sec 1 2 3 4 5 6 1 Description Loss Definitions Loss Data aggregation Methods Common Ratios Involving Loss Statistics Adjustments To Losses Loss Adjustment Expenses Key Concepts Pages 90 - 91 91 -93 92 - 93 93 – 121 121 – 122 122 - 123 Loss Definitions 90 - 91 The text uses the term claim to mean demand for compensation and loss to refer to the amount of compensation. Losses and LAE usually represent largest portion of premium. This chapter discusses: The different types of insurance losses How loss data is aggregated for ratemaking analysis Common metrics involving losses Adjustments made to historical loss data to make it relevant for estimating future losses in the ratemaking process. This includes adjusting data for: • Extraordinary loss events • Changes in benefit levels • Changes in the loss estimates as immature claims become mature • Changes in loss cost levels over time Treatment of LAE Definitions • Paid losses: Payments made to claimants. • Case reserve: An amount expected to be paid on a claim, based on a claims adjuster’s estimate or determined by formula. • Reported (Case Incurred) losses: Paid Losses + Case Reserves • Incurred but not enough reported (IBNER): Reported losses adjusted to account for any anticipated shortfall in the case reserves • Incurred but not reported (IBNR): Reserves for claims incurred but that have not yet been reported. • Ultimate Losses: Reported Losses + IBNER + IBNR Aggregated losses are based on statistics (e.g. paid or reported losses), a data aggregation method (e.g. calendar, accident, policy, or report month/quarter/year), and a period of time. The time period for data aggregation is defined by an accounting period and a valuation date. The accounting period for losses should be consistent with financial statement dates (e.g. month, quarter, or calendar year). The valuation date (which can be different than the end of the accounting period) is the date losses are evaluated for analysis. It is expressed as the number of months after the start of the accounting period (e.g. AY 2010 as of 18 months implies AY 2010 as of 6/30/2011). Valuation dates can occur prior to the end of the accounting period. Exam 5, V1a Page 152 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 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). Exam 5, V1a Page 153 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 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. Exam 5, V1a Page 154 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 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. Exam 5, V1a Page 155 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Excess Loss Factor Calculation In this example, individual reported losses are capped at $1,000,000 (a.k.a. non-excess losses) The long-term average ratio of excess losses (the portion of each shock loss above the $1,000,000 threshold) to non-excess losses is used to determine an excess loss provision. Excess Loss Procedure (1) Accident Year 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Total Reported Losses $118,369,707 $117,938,146 $119,887,865 $118,488,983 $122,329,298 $120,157,205 $123,633,881 $124,854,827 $125,492,840 $127,430,355 $123,245,269 $123,466,498 $129,241,078 $123,302,570 $123,408,837 $1,841,247,359 (2) Number of Excess Claims 5 1 3 0 7 3 0 1 0 6 3 0 10 0 3 42 (3) Ground –Up Excess Losses $ 6,232,939 $1,300,000 $3,923,023 $ $12,938,382 $3,824,311 $ $3,000,000 $13,466,986 $4,642,4 $ $17,038,332 $ $4,351,805 $70,718,201 (4) Losses Excess of $1,000,000 $1,232,939 $300,000 $923,023 $ $5,938,382 $824,311 $ $2,000,000 $ $7,466,986 $1,642,423 $ $7,038332 $ $1,351,805 $28,718,201 (7) Excess Loss Factor (5) (6) Non-Excess Losses $117,136,768 $117,638,146 $118,964,842 $118,488,983 $116,390,916 $119,332,894 $123,633,881 $122,854,827 $125,492,840 $119,963,369 $121,602,846 $123,466,498 $122,202,746 $123,302,570 $122,057,032 $1,812,529,158 Excess Ratio 1.1% 0.3% 0.8% 0.0% 5.1% 0.7% 0.0% 1.6% 0.0% 6.2% 1.4% 0.0% 5.8% 0.0% 1.1% 1.6% 1.016 (4)= (3) - [$1,000,000 x (2)] (5)= (1) - (4) (6)= (4) / (5) (7)= 1.0 + (Tot 6), and is applied to the non-excess losses for each year in the historical experience period. Notes: The excess loss procedure is ideally performed on reported losses that have been trended to future levels (i.e. excess losses are calculated by censoring trended ground-up losses). Alternatively, some actuaries may fit statistical distributions to empirical data and simulate claim experience in order to calculate the expected excess losses. Exam 5, V1a Page 156 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Catastrophe Losses Ratemaking data excludes losses arising from catastrophic events. Catastrophe losses: from hurricanes, tornadoes, hail storms, earthquakes, wildfires, winter storms, explosions, oil spills and certain terrorist attacks are severe and results in a significant number of claims (unlike shock losses from individual high severity claims) are defined by the Property Claims Services (PCS) unit of the Insurance Services Office (ISO) as events that cause $25 million or more in direct insured property losses and that affect a significant number of policyholders and insurers. may have alternative definitions by insurers for internal procedures. are removed from ratemaking data and replaced with an average expected catastrophe loss amount. are broken down into non-modeled catastrophe losses and modeled catastrophe losses. Non-modeled catastrophe analysis is performed on events that occur with some regularity over decades. Example: Hail storms (which occur with some multi-year on and off regularity) is the most common catastrophic loss related to private passenger auto comprehensive coverage. Without a non-modeled cat procedure, indicated rates will increase immediately after a bad storm year and decrease in years having few or no storms. The actuary can calculate the ratio of hail storm losses to non-storm losses over a longer experience period (e.g. 10-30 years). The number of years used should balance stability and responsiveness. Example: If the concentration of exposures in the most hail-prone area of a state has increased drastically over the past 20 years, then a cat procedure based on 20 years of statewide data may understate the expected catastrophe potential. Once determined, the ratio can be used to adjust the non-catastrophe losses in consideration of future expected catastrophe loss. Alternatively, the actuary can develop a pure premium (or loss ratio) for the non-modeled cat exposure. Using a pp approach, compute the long-term ratio of cat losses to exposure (or amount of insurance years) and apply that ratio to projected exposures (or projected amount of ins years). See Appendix B. The loss ratio indication would be similar except the denominator of the long-term ratio would be EP, which is inflation-sensitive and the premium would need to be brought to current rate level. Catastrophe models are used for events that are irregular and generate high severity claims (e.g. hurricanes and earthquakes). 30 years of data may not capture the expected damage these events can inflict. Stochastic models are designed by professionals from a variety of fields (e.g., insurance, meteorologists, engineers) to estimate the likelihood that events of varying magnitudes will occur and the damages that will likely result given the insured property characteristics. The modeled cat loss provision is added to the non-catastrophe loss amount to determine the aggregate expected losses to be used for pricing. Insures writing in cat prone areas: may use non-pricing actions (e.g. restrict the writing of any new business, may require higher deductibles for catastrophe-related losses, or may purchase reinsurance) in cat prone areas to control the concentration to minimize the financial impact any one event can have on the profitability. may alter the underwriting profit provision in the rates to reflect the higher cost of capital needed to support the risk caused by the higher concentration of policies. Exam 5, V1a Page 157 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Reinsurance Historically, ratemaking for primary insurance was done on a direct basis (i.e. without reinsurance consideration). Some ratemaking analyses are now performed on a net basis (i.e. with consideration of reinsurance) as reinsurance programs have become more extensive and reinsurance costs have increased substantially for some lines of business. Proportional reinsurance means the same proportion of premium and losses are transferred or “ceded” to the reinsurer (thus, proportional reinsurance may not necessarily need to be included in the pricing consideration). With non-proportional reinsurance: the reinsurer agrees to assume some % of the losses (reinsurance recoverables to the insurer) the insurer cedes a portion of the premium (the cost of the reinsurance). Examples of non-proportional reinsurance include: cat excess-of-loss reinsurance (e.g. the reinsurer covers 50% of the losses that exceed $15,000,000 up to $30,000,000 on their entire property book of business in the event of a cat) per risk excess of loss reinsurance (e.g. the reinsurer will cover the portion of any large single event that is between $1,000,000 and $5,000,000 for specified risks). Changes in Coverage or Benefit Levels An insurer may: initiate changes in coverage (e.g. expand or contract coverage with respect to the types of losses covered) or opt to increase or decrease the amount of coverage offered. Benefit levels can be impacted by a law change or court ruling (e.g. caps on punitive damages for auto liability coverage and changes in the WC statutory benefit levels). Benefit changes can have direct and indirect effects on losses. direct effects are a direct and obvious consequence of the benefit change. indirect effects arise from changes in claimant behavior that as a result of the benefit change (and are more difficult to quantify than direct effects). Example: Quantification of benefit changes. Assume an insurer reduces the maximum amount of coverage for jewelry, watches, and furs on a standard homeowners policy from $5,000 to $3,000. The direct effect: is that any claimants with jewelry, watches, and furs losses in excess of $3,000 will now only receive $3,000 rather than at most $5,000. of this change can be calculated if a distribution of historical jewelry, watches, and furs losses is available. The table below shows the how reported losses on 6 claims would be capped under the two different thresholds. Exam 5, V1a Page 158 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 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. Exam 5, V1a Page 159 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 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). Exam 5, V1a Page 160 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 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 = Exam 5, V1a Page 161 Current Loss Level Average Loss Level of Historical Period 2014 by All 10, Inc. 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 Page 162 2014 by All 10, Inc. 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) Exam 5, V1a Page 163 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 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. Exam 5, V1a Page 164 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Age-to-Age loss development factor (a-t-a LDF) selection: The ratemaking actuary selects a suitable link ratio for each maturity (since the link ratios for each development period are fairly consistent across the AYs, the all-year arithmetic average link ratios are selected). A-t-A LDFs in practice may not be as stable as outlined above: If the ratemaking actuary believes patterns may be changing over time, the actuary may prefer to rely on more recent development patterns, and select a two- or three-year average. If there is a desire to select based on the most recent data, but the line of business is to too volatile to rely solely on a two- or three-year average, calculate weighted average link ratios giving more weight to the more recent years. If A-t-A factors vary widely between AYs or there may be a strong anomaly in one or two AYs, consider adjusted averages that eliminate the highest and lowest development factors from the calculation. Loss Development: Reported losses develop upward as losses approach ultimate (due in part to the emergence of new claims as well as adverse development on known claims). In some lines of business, development may be negative: i. In auto physical damage coverages, an insurer may declare a vehicle a total loss (i.e. pay the total limit for the car), take the damaged car, and sell it as scrap or for parts. The money received is called “salvage” and is treated as a negative loss. ii. When insurers pay losses for which another party is actually liable, it can approach the responsible party for indemnification of those amounts (called subrogation). Thus, when subrogation or salvage are common, or when early case reserves are set too high, age-toage development factors can be less than 1.00. While this example assumes losses are ultimate at 75 months, for some lines of business, the historical data triangle may not reach ultimate. Here, actuaries may fit curves to historical development factors to extrapolate the development beyond the patterns in the historical data. A ‘tail factor’ accounts for additional development beyond that included in the standard chain ladder method. Adjustments to Historical Data: Remove extraordinary losses from the historical data used to measure loss development patterns. Benefit or coverage changes may also distort loss development patterns. i. Since benefit changes often affect policies prospectively, the effect of the change will first appear in a new AY row. ii. If the change impacts all claims occurring on or after a certain date, it is possible there will be a change in the absolute amount of losses even though the development pattern is unaffected. If it is not possible to restate the losses, any such distortions should be considered during the a-t-a ldf selection process. Next Step: Calculate age-to-ultimate development factors (a-t-u ldf) for each maturity. The a-t-u ldf is the product of each selected a-t-a ldf and the selected a-t-u ldf for subsequent maturities (and the tail factor, if relevant). Example, a-t-u ldf for losses at age 51 months is the product of the selected age-to-age development factors for 51-63 months and 63-75 months (1.02 x 1.00). Exam 5, V1a Page 165 2014 by All 10, Inc. Chapter 6 – Losses and LAE 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. Exam 5, V1a Page 166 2014 by All 10, Inc. Chapter 6 – Losses and LAE 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. Exam 5, V1a Page 167 2014 by All 10, Inc. Chapter 6 – Losses and LAE 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. Exam 5, V1a Page 168 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Loss Trend Periods The loss trend period is the period of time from the average loss occurrence date of each experience period (often a calendar-accident year, CAY) to the average loss occurrence date for the period in which the rates will be in effect (i.e. the forecast period, which is a policy year or years). The average loss occurrence date depends on the policy term and the duration the new rates will be in effect. Assume the following: • The losses to be trended are from AY 2011. • The company writes annual policies. • The proposed effective date is January 1, 2015. • The length of time the rates are expected to be in effect is one year. The average loss occurrence date of CAY 2011 (called the “trend from” date) is 6/30/2011. The average accident date for PY 2011 is 12/31/2011, as polices are in effect over a 24-month period. The average loss occurrence date during the forecast period (called the “trend to” date) is 12/31/2015. This is because last policy to be written will be on 12/31/2015, and losses can continue to occur until 12/31/2016, so the midpoint of that two-year time period is 12/31/2015. Thus, the trend period for CAY 2011 is 4.5 years. The pure premium trend (+1%) is applied to CAY Year 2011 losses by multiplying the historical losses by (1.01)4.5 (which is the trend factor). If the policy term were semi-annual, the “trend from” date would not change, but the “trend to” date would be different. Coverage for policies written between 1/1/2015 and 12/31/2015 would extend over an 18-months, of which the midpoint would be 9 months (i.e. 9/302/015). The trend length would be 4.25 years as shown below. Loss Trend Period for 6-month Policy Term Exam 5, V1a Page 169 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. If data were aggregated by PY: the average loss occurrence date for an annual policy term would be one year after the start of the PY, as policies are in effect over 24-months. the “trend to” date is the average loss occurrence date for the PY in which rates will be in effect. Therefore, the trend period for PY 2011 annual term policies is 4 years (1/1/2012 to 12/31/2015), as shown below. Loss Trend Period for 12-month Policy Term and PY experience period The PY2011 trend factor, applied to PY 2011 losses, is 1.0406 ( = 1.014.0). Exhibit 6.18 (below) shows the same PY scenario but with semi-annual policies. Both the “trend from” and “trend to” dates are 3 months earlier than the annual policy scenario since the average occurrence date for semi-annual policies is 9 months after the start of the PY. Thus, the trend length remains the same as in the annual policy scenario and is still 4 years. Loss Trend Period for 6-month Policy Term and PY experience period If the trend selection is based on a linear trend, the selected trend is a constant amount rather than a %. The projected dollar change = (the selected annual trend) * (the length of the trend period). Assuming the selected annual pure premium linear trend is $1.00 per year, then the dollar increase due to 4 years of trend is $4.00 (= $1.00 x 4.0). The actuary may choose to undertake a two-step trending process. This is beneficial when the trend in the historical experience period and the expected trend for the forecast period are not equal. For example, legislative changes in the trend data call for a 2-step trending process if the trend exhibited in the historical period is clearly different from that expected in the future. Exam 5, V1a Page 170 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. In the exponential trend data shown above, historical severity trend exhibits a different pattern in more recent periods than in earlier years. The losses in the experience period are trended from the average accident date in the experience period to the average accident date of the last data point in the trend data. Example: The average loss occurrence date of CAY 2011 is 6/30/2011. If the last data point in the loss trend data is the 12 months ending fourth quarter 2013, the average accident date of that period is 6/30/2013. If the selected step 1 trend is -1%, the factor to adjust CAY 2011 losses to the end of the experience period is 0.98 (= (1.0 - 1%)2 ). Next, these trended losses are projected from the average accident date of the last data point in the trend data (the “project from” date of 6/30/2013) to the average loss occurrence date for the forecast period (the “project to” date of 12/31/2015). The length of this projection period is 2.5 years. If the trend selection is 2%, step 1 trended losses are adjusted by a factor of 1.05 (= (1.0 + 2%) 2.5). Two-Step Trend Periods for 12-month Policy When using CY data to measure loss trend, it is assumed that the book of business is not significantly increasing or decreasing in size. Problems with this assumption are: claims (or losses) in any CY may have come from older AYs, but are matched to the most recent CY exposures (or claims). a change in exposure levels causing changes in the distribution of each CY’s claims by accident year. The solution is to match the risk with the appropriate exposure. 1. Use econometric techniques or generalized linear models to measure trend, which will absorb changes in the size of the portfolio as well as changes in the mix of business. 2. Measure the trend using AY data (in lieu of CY data). The AY losses (or claim counts) need to be developed to ultimate before measuring the trend, which introduces subjectivity into the trend analysis. 3. Analyze the trend in incremental CY frequencies or severities. Assume CY 2010 has paid losses on claims from AYs 2010, 2009, and 2008. i. CY 2010 frequency is the sum of all [paid claim counts in CY 2010/ CY 2010 exposures]. ii. Alternatively, CY 2010 frequency is the sum of the following three incremental CY 2010 frequencies: • [CY 2010 paid claim counts from AY 2010 / CY 2010 exposures] • [CY 2010 paid claim counts from AY 2009 / CY 2009 exposures] • [CY 2010 paid claim counts from AY 2008 / CY 2008 exposures] The alternative method properly matches older claim counts to older exposures and is valid whether the portfolio is changing or not. Exam 5, V1a Page 171 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Leveraged Effect of Limits on Severity Trend When loss experience is subject to limits, consider the leveraged effect of those limits on the severity trend. Basic limits losses are losses that have been censored at a limit referred to as a “basic limit.” Total limits losses are losses that are uncensored Excess limits losses are the portion of the losses that exceed the basic limit (or the difference between total limits and basic limits losses). It is important to understand that severity trend affects each of these differently. Consider the following simple example in which every total limits loss is subject to a 10% severity trend. Effect of Limits on Severity Trend (1) (2) (3) (4) (5) (6) (7) (8) (9) Trended Losses Total Losses Capped @ $25,000 Capped @ Claim Limits Excess Total Limits Excess Losses Number Loss $25,000 Losses Loss Trend Loss Trend Loss Trend 1 $10,000 $10,000 $ $11,000 10.0% $11,000 10.0% $N/A 2 $15,000 $15,000 $$16,500 10.0% $16,500 10.0% $N/A 3 $24,000 $24,000 $ $26,400 10.0% $25,000 4.2% $1,400 N/A 4 $30,000 $25,000 $ 5,000 $33,000 10.0% $25,000 0.0% $8,000 60.0% 5 $50,000 $25,000 $25,000 $55,000 10.0% $25,000 0.0% $30,000 20.0% Total $129,000 $99,000 $30,000 $141,900 10.0% $102,500 3.5% $39,400 31.3% (2)=min [(1), $25,000] (3) = (1) - (2) (4) = (1) x 1.10 (5) = (4) / (1) - 1.0 (6)=min [ (4) , $25,000] (7)= (6) / (2) - 1.0 (8) = (4) - (6) The 10% trend in total limits losses affects basic limits losses and excess losses differently. Basic Limits: The 10% total limit trend is reduced to 3.5% when considering the basic limits losses. The two smallest losses (Claims 1 and 2) are well below the $25,000 limit before and after the 10% increase. Claim 3 was below $25,000 before trend was applied, but above the basic limit after applying trend. Claims 4 and 5 were already in excess of $25,000, so the amount of loss under the limit is the same before and after trend. Excess Limits: The impact of positive trend on excess losses is greater than the total limits trend. Claims 1 and 2 are significantly below the limit and do not impact the trend in the excess layer. Claim 3 was below $25,000 before trend was applied, but above the basic limit after applying trend. Since claims 4 and 5 were already higher than the basic limit, the entire increase in losses associated with these claims is realized in the excess losses trend. Exam 5, V1a Page 172 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Effect of Limits on Severity Trend Initial Loss Size Basic Limits Trend Limit Loss Total Losses Trend Excess Losses Undefined Limit 1.0 Loss Trend Undefined 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. Exam 5, V1a Page 173 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 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. Exam 5, V1a Page 174 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Given the following: The historical experience period is CAY 2010. The average date of claim occurrence is 7/1/2010. Assume it is typical for claims to settle within 18 months, so this “average claim” will settle on 12/31/2011. The projection period is the policy year beginning 1/1/ 2012 (i.e. rates are expected to be in effect for annual policies written from 1/1/2012 – 12/31/2012). The average hypothetical claim in the projected period will occur on 1/1/2013, and settle 18 months later on 6/30/2014 (i.e. consistent with the settlement lag of 18 months). Key comments: Trend adjusts the average historical claim from the loss cost level that exists on 7/1/2010 to the loss cost level expected on 1/1/2013. Development adjusts the trended, undeveloped claim to the ultimate level, expected to occur by 6/30/2014. This 48 month period represents 30 months of trend to adjust the cost level to that anticipated during the forecast period and the 18 months of development to project this trended value to its ultimate settlement value. 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. Exam 5, V1a Page 175 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 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. Exam 5, V1a Page 176 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 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 Exam 5, V1a Page 177 2014 by All 10, Inc. 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 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? Exam 5, V1a Page 178 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 1997 Exam: 44. (4 points) You are given: Calendar/Accident Year Reported Loss and ALAE Earned Exposures 1993 1994 1995 1,800,000 2,275,000 1,975,000 2,500 2,900 3,400 Losses are evaluated as of 12/31/96 Loss (incl. ALAE) Development Factors: 12 months to ultimate 24 months to ultimate 36 months to ultimate 48 months to ultimate LDFs 1.500 1.250 1.050 1.000 • Annual severity trend = +4.3% (trend is exponential) • Annual frequency trend = -2.0% (trend is exponential) • Commission = 14.0% • Taxes = 3.0% • Variable portion of General and Other Acquisition = 10.0% • Total fixed expense = $30 per exposure • Profit load = 3.0% • All policies are annual • Filed rates will be in effect for one year • Proposed effective date for the rate change is 10/1/97 Using the methodology in McClenahan, "Ratemaking," of Foundations of Casualty Actuarial Science, A. (2 points) Determine the developed and trended Loss and ALAE by accident year (chapter 6) B. (1 point) Determine the indicated pure premium (chapter 8) C. (1 point) Determine the indicated gross rate (chapter 8) Questions from the 1999 exam 39. (2 points) McClenahan in "Ratemaking," chapter 2 of Foundations of Casualty Actuarial Science, discusses the effects of limits on severity trend. Use the information shown below to determine the one-year severity trend for the loss amounts in the following three layers of loss: $0-$50 $50-$100 $100-$200 Losses occur in multiples of $40, with equal probability, up to $200, i.e., if a loss occurs, it has an equal chance of being $40, $80, $120, $160, or $200. For the next year, the severity trend will uniformly increase all losses by 10%. Exam 5, V1a Page 179 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2000 exam 40. (4 points) Using the techniques described by McClenahan in "Ratemaking," chapter 2 of Foundations of Casualty Actuarial Science, and the following data, answer the questions below. You are given the following information for your company's homeowners business in a single state: Calendar/ Ultimate Loss Accident Year and ALAE Written Premium Earned Premium 1997 635,000 1,000,000 975,000 1998 595,000 1,050,000 1,000,000 Effective Date July 1, 1996 January 1, 1998 July 1, 1999 Rate Change +4.0% +1.8% +3.0% Target Loss and ALAE Ratio Proposed effective date Effective period for rates Credibility Alternative indication Policy period Severity trend Frequency trend 0.670 July 1, 2000 One year 0.60 0.0% Twelve months +3.0% +1.0% 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 Exam 5, V1a Page 180 2014 by All 10, Inc. Chapter 6 – Losses and LAE 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) Exam 5, V1a Page 181 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2003 exam: 12. Given the following data and using the loss development method as described by McClenahan in Foundations of Casualty Actuarial Science, calculate the projected ultimate accident year 2001 losses. As of December 31, 2002 Accident Year Paid Losses Case Reserves 1999 $11,000 $1,000 2000 $6,000 $2,000 2001 $3,500 $4,000 2002 $1,000 $4,000 Projected ultimate accident year 2000 losses = $9,240 12-24 case-incurred link ratio = 1.71 24-36 case-incurred link ratio = 1.20 A. < $8,700 B. $8,700, but < $9,200 D. $9,700, but < $10,200 C. $9,200, but < $9,700 E. $10,200 Questions from the 2004 exam: 7. Given the following data, calculate the trended loss ratio. Number of Insureds 20 • • • • • A. < 68% Earned Premium $50,000 Developed Incurred Losses $35,000 Years of Trend = 2.5 Annual Exposure Trend = 2.0% Annual Premium Trend = 2.9% Annual Frequency Trend = -1 .0% Annual Severity Trend = 6.0% B. > 68% but < 71% C. > 71 % but < 74% D. > 74%, but < 77% E. > 77% 8. Which of the following statements are true regarding loss trends? 1. When an exponential curve is used to approximate severity, the assumption is a constant multiplicative increase in severity. 2. This original statement no longer applies to the content in this chapter 3. Linear trends tend to underestimate future costs when inflation is increasing at a multiplicative rate. A. 1 only B. 3 only C. 1 and 2 only D. 1 and 3 only E. 2 and 3 only Exam 5, V1a Page 182 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2004 exam (continued): 37. (5 points) Given the information below, answer the following questions. Show all work. Case-Incurred Losses Accident Year Age 12 Age 24 Age 36 Age 48 2000 $1,412 $1,816 $1,993 $1,993 2001 $1,624 $2,023 $2,137 2002 $1,841 $2,271 2003 $2,421 Ultimate losses are reached at age 48. The annual frequency trend is -2%. The annual severity trend is 8%. Planned effective date of rate change is July 1, 2004. Rates are reviewed annually. Policies have a term of 12 months. a. (1 point) Calculate the age-to-ultimate development factor for accident year 2003 as of December 31, 2003. Explain your assumptions. b. (0.5 point) Calculate the ultimate loss amount for accident year 2003. c. (1 point) Calculate the trended ultimate loss amount for accident year 2003. d. (1.5 points) Briefly describe three causes of loss development. e. (1 point) Briefly explain why it is appropriate to both trend and develop losses (i.e. why there is no overlap). Questions from the 2007 exam 22. (1.5 points) The claims department of an insurance company has historically set an initial case reserve of $10,000 for each liability claim at the time the claim is opened. If the claim is not closed within 18 months, the case reserve is adjusted to an appropriate level based on the characteristics of the claim. Starting with accidents occurring January 1, 2006 and later, the initial case reserve was set at $5,000 for each liability claim. The actuarial department was not made aware of this change. Assume incurred loss data for accident year 2006, valued as of December 31, 2006, is used to derive rates effective July 1, 2007. Explain the impact of this change on incurred loss development and rate adequacy for this liability line of insurance. Questions from the 2008 exam 17. (2.0 points) Given the following payment and reserve data about 2 different claims on 2 different policies: Policy Effective Date Date of Loss Transaction Date Payment Case Reserve July 1, 2006 December 1, 2006 December 1, 2006 $0 $5,000 March 1, 2007 $500 $3,500 $2,000 October 1, 2007 $3,500 March 1, 2008 $3,000 $0 October 1, 2006 March 1, 2007 March 1, 2007 October 1, 2007 March 1, 2008 $5,000 $9,000 $1,000 $10,000 $1,000 $0 a. (0.5 point) Calculate the calendar-year incurred losses for 2006 and 2007. b. (0.5 point) Calculate the accident-year incurred losses for 2006 and 2007 evaluated as of 12/31/2008. c. (0.5 point) Calculate the policy-year incurred losses for 2006 and 2007 evaluated as of 12/31/2008. d. (0.5 point) Identify one advantage and one disadvantage associated with using policy year incurred losses for ratemaking. Exam 5, V1a Page 183 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2009 exam 22. (2 points) an insurance company started writing annual policies in 2005. Given the following information for claims associated with policies written in 2005: Accidents Occurring in 2005 Accidents Occurring in 2006 Calendar Payments Reserve @ Calendar Payments Reserve @ Year End of Year Year End of Year 2005 $ 1,000,000 $500,000 2005 $ $ 2006 $ 300,000 $300,000 2006 $ 1,500,000 $ 1,000,000 2007 $ 250,000 $100,000 2007 $ 700,000 $ 200,000 2008 $ 50,000 $ 2008 $ 100,000 $ 50,000 a. (0.5 point) Calculate the calendar year losses for 2006. b. (0.5 point) Calculate the accident year incurred losses for 2006 evaluated as of December 31, 2007. c. (0.5 point) Calculate the policy year incurred losses for 2005 evaluated as of December 31, 2008. d. (0.5 point) Provide one advantage and one disadvantage associated with using calendar year incurred losses rather than accident year incurred losses for ratemaking. 24. (1 point) Fully discuss why it may be inappropriate to apply a basic limits loss trend to total limits losses. 27. (1 point Fully discuss the "overlap fallacy" between trend and loss development. 42. (1 point) For homeowners insurance explain two reasons that hurricane rates should be priced separately from non-hurricane rates. Questions from the 2010 exam 20. (2 points) Given the following claim activity on an annual policy effective on December 29, 2006: Claim Number 1 1 1 1 1 2 2 2 Incremental Transaction Date Payment December 31, 2006 December 31, 2006 October 5, 2007 July 5, 2008 January 25, 2009 $ 30,000 April 1, 2007 April 5, 2007 July 1, 2008 Case Reserve as Of Transaction Date $1,000 $ 10,000 $ 25,000 $$ 25,000 $- Transaction Description Claim occurred Claim reported and reserve established Case reserve increased Case reserve increased Settlement made, Payment made, Claim closed Claim occurred Claim reported and reserve established Claim closed without payment a. (0.5 point) Calculate 2008 calendar year reported losses. b. (0.5 point) Calculate 2006 accident year reported losses evaluated as of December 31, 2007. c. (0.5 point) Calculate 2006 policy year reported losses evaluated as of December 31, 2007. d. (0.5 point) Briefly describe one advantage and one disadvantage of using calendar year losses as compared to accident year losses in a ratemaking application. Exam 5, V1a Page 184 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2010 exam 21. (2 points) Identify four adjustments made to historical losses in projecting losses for a future policy period for ratemaking. Briefly describe the purpose of each. 24. (1 point) Given the following countrywide calendar year information: Calendar Year 2006 2007 2008 2009 Earned Premium $696,667 $733,333 $805,673 $907,725 Paid Loss $475,000 $500,000 $498,750 $518,700 Paid ALAE $47,500 $50,000 $24,938 $25,935 Paid Loss and ALAE $522,500 $550,000 $523,688 $544,635 Paid ULAE $26,125 $55,000 $52,369 $54,464 Select a ULAE factor to be applied to the statewide incurred losses and paid ALAE as part of calculating statewide rate indications. Explain your selection. Questions from the 2011 exam 6. (2.5 points) Given the following information for claims associated with annual homeowners policies written in 2007: Claim Accident Report Transaction Loss Case Reserve Number Year Year Date Payment Balance 1 2007 2007 April 1, 2007 $100 $300 1 2007 2007 July 1, 2008 $200 $600 1 2007 2007 June 1, 2009 $500 $0 2 2007 2008 May 1, 2008 $500 $200 2 2007 2008 July 1, 2009 $200 $0 3 2008 2008 August 1, 2008 $50 $200 3 2008 2008 March 1, 2009 $100 $50 3 2008 2008 July 1, 2010 $200 $0 a. (0.5 point) Calculate the calendar year 2008 incurred losses. b. (0.5 point) Calculate the accident year 2008 incurred losses, evaluated at December 31, 2009. c. (0.5 point) Calculate the policy year 2007 incurred losses, evaluated at December 31, 2009. d. (0.5 point) Calculate the report year 2008 incurred losses, evaluated at December 31, 2009. e. (0.5 point) Briefly describe one advantage and one disadvantage associated with using policy year losses for ratemaking. 7. (1 point) Fully explain the overlap fallacy between loss development and loss trend. 17. (1 point) Given the following data: Claim Number Loss Amount 1 $10,000 2 $15,000 3 $30,000 4 $35,000 • Basic limit = $25,000 • Total limits severity trend = 10% Calculate the excess loss trend. Exam 5, V1a Page 185 2014 by All 10, Inc. 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. Exam 5, V1a Page 186 2014 by All 10, Inc. 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% Page 187 2014 by All 10, Inc. 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. Exam 5, V1a Page 188 2014 by All 10, Inc. 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. Exam 5, V1a Page 189 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2010 exam 23. (2.5 points) Given the following workers compensation information: • The compensation rate is 80% of the worker's pre-injury wage. • The state average weekly wage (SAWW) is $1,500. • The minimum benefit is 48% of the SAWW. • The maximum benefit is changing from 128% of the SAWW to 112% of the SAWW. • The distribution of workers (and their wages) according to how their wages compare to the SAWW is as follows: Ratio to Average Weekly Wage 0 - 60% 60 - 120% 120 - 140% 140 - 160% 160 + Number of Workers 64 144 33 21 29 Total Weekly Wages $37,550 $196,200 $64,350 $47,250 $84,000 a. (2 points) Calculate the direct effect of the change in maximum benefits on losses. b. (0.5 point) Explain a potential indirect effect of the change in maximum benefits on losses. Questions from the 2012 exam 7. Develop the projected ultimate loss and LAE for accident year 2010 losses using the data above. To satisfy the state requirements, fully describe the rationale for the selections for loss development, loss trend, and ULAE. Exam 5, V1a Page 190 2014 by All 10, Inc. 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. Exam 5, V1a Page 191 2014 by All 10, Inc. 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. Exam 5, V1a Page 192 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 1999 exam Question 39. The severity trend rate = Loss Amount Before/After(x) 40/44 80/88 120/132 160/176 200/220 E[ X '] 1.0 , where X' represents losses affected by a 10% inflation rate. E[ X ] Probability of loss (f(x)) .20 .20 .20 .20 .20 Distribution of Loss by Layer 0 - 50 50 - 100 40/44 0/0 50/50 30/38 50/50 50/50 50/50 50/50 50/50 50/50 100 - 200 0/0 0/0 20/32 60/76 100/100 Loss amounts before and after the impact of uniform 10% increase Layer 0 - 50 50 - 100 100 - 200 E [ X ] x* f ( x ) x [.2*40 .80*50] 48 E[ X '] x* f ( x ) x [.2*40*(1.1) .80*50] 48.8 [.2*30 .60*50] 36 [.2*38 .60*50] 37.6 [.2*20 .20*60 .20*100] 36 [.2*32 .20*76 .20*100] 41.6 Layer 0 – 50 50 – 100 100 – 200 One year severity Trend 48.8 1.0 1.017 or 1.7% 48 37.6 1.0 1.044 or 4.4% 36 41.6 1.0 1.156 or 15.6% 36 Solutions to questions from the 2000 exam: Question 40. b. Calculate the trended projected ultimate on-level loss and ALAE ratio for the combined experience period 1997-1998. With this information, we can compute developed and trended Loss and ALAE by accident year as follows: AY 1997 1998 Total Developed Loss and ALAE (1) 635,000 595,000 Freq trend factor (2) 1.01 1.01 On-level loss and ALAE ratio Exam 5, V1a Sev trend factor (3) 1.03 1.03 Midpoint of the experience period (4) 7/1/97 7/1/98 Midpoint of the exposure period (5) 7/1/2001 7/1/2001 Trend Factor (6) (1.01*1.03)4 (1.01*1.03)3 Developed and trended Loss and ALAE (7)=(1)*(2)*(6) 743,717 669,873 1,413,590 1,413,590 Developed and Trended losses .684 On Level Earned Pr emium 1,027,2831,039,290 Page 193 2014 by All 10, Inc. 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 Exam 5, V1a Page 194 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2003 exam 12. Calculate the projected ultimate accident year 2001 losses. Step 1: Determine AY 2001 case incurred losses at 12/31/2002 projected to 36 months. Case incurred losses at 12/31/2002 = $3500 + $4,000 = $7,500. Note that at 12/31/02, AY 2001 case incurred losses are at 24 months of development. The loss development factor from 24-36 months is given as 1.20. Thus, AY 2001 case incurred losses projected to 36 months equals $9,000. Step 2: Determine AY 2001 case incurred losses at 12/31/2002 projected to ultimate. AY 2000 36-48 months case incurred loss development factor is $9,420/$8,000 = 1.155. Thus, at 12/31/02, AY 2001 cased incurred losses are at ultimate equals $9,000 * 1.155 = $10,395. Answer E. $10,200 Solutions to questions from the 2004 exam 7. Calculate the trended loss ratio. Step 1: Based on the givens of the problem, write an equation to determine the trended loss ratio. Developed Incurred Losses Freq Trend*Sev Trend Trended Loss Ratio = * Earned Premium Premium Trend Years of Trend Step 2: Using the equation in Step 1, and the data in the problem, solve for the trended loss ratio. 2.5 $35,000 .99 * 1.06 Trended Loss Ratio = * .7352 Answer C: > 71 % but < 74% $50,000 1.029 8. Which of the following statements are true regarding loss trends? 1. When an exponential curve is used to approximate severity, the assumption is a constant multiplicative increase in severity. True. “Since this data contains random fluctuations, the minimization of these fluctuations will provide a better estimate of the underlying trend. This is achieved by fitting the data to a curve. An exponential curve is selected because it assumes a constant percentage trend from year to year.” 2. Statement no longer applicable to the content within this article 3. Linear trends tend to underestimate future costs when inflation is increasing at a multiplicative rate. True. Note that the linear model will produce a model in which the projection will increase by a constant amount (a) for each unit change in x. The exponential model will produce a constant rate of change of ea - 1, with each value being ea times the prior value. Answer: D. 1 and 3 only Exam 5, V1a Page 195 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2004 exam (continued): 37.(5 points) a. (1 point) Calculate the age-to-ultimate development factor for accident year 2003 as of December 31, 2003. Explain your assumptions. Assumptions: We are told that ultimate losses are reached at age 48, and therefore our 48-ultimate loss development factor is 1.000. Selected age to age development factors are set equal to age to age link ratios computed using the given data. Age to Age link ratios are computed by dividing case-incurred losses at successive intervals (e.g. AY 2000 12-24 link ratio = 1,816/1,412 = 1.2861) Since accident year 2003 at 12/31/03 is at 12 months of maturity, a 12 to ultimate loss development factor is necessary and is computed as follows: AY 2000 2001 2002 2003 3 yr avg Factor to Ult 12-24 1.2861 1.2457 1.2336 24-36 1.0975 1.0564 36-48 1.0000 48-ULT 1.0000 1.0000 1.0000 1.0000 1.2551 1.0769 1.0000 1.0000 1.3516 1.0769 1.0000 1.0000 , where 12 to ultimate loss development factor = 1.3516 = 1.2551 * 1.0769 * 1.0000 * 1.0000 b. (0.5 point) Calculate the ultimate loss amount for accident year 2003. AY 2003 ultimate losses = AY 2003 case incurred losses12 months * 12 to ultimate loss development factor = $2,421 * 1.3516 = $3,272.22 c. (1 point) Calculate the trended ultimate loss amount for accident year 2003. Since we have computed ultimate losses for AY 2003 as $3,272.22, what remains to be computed is the annual trend factor and the trend period. The annual trend factor is computed as the product of the given annual frequency and severity trend rates. Thus, the annual trend factor equals .98 * 1.08 = 1.0548 The trend period is determined by the time between the average accident date of the experience period and the average accident date associated with the effective period of the rates. The average accident date for AY 2003 is 7/1/2003 Since the revised rates will be in effect for a one year period (7/1/2004 – 7/1/2005) and since all polices have one year terms and are written uniformly throughout the year, the average policy will run from 1/1/2005 – 12/31/2005, and the midpoint of loss occurrence under that policy will be 7/1/2005). The trend period is therefore 2 years (7/1/2003 – 7/1/2005), and the AY 2003 trended developed loss and ALAE is $3,272.22 (1.0548)2.00= $3,640.68 d. (1.5 points) Briefly describe three causes of loss development. 1. Development on known claims. This occurs when reserves are initially set too low, and then increase as more loss related information becomes known. 2. Newly reported claims. These result from the late reporting of claims. 3. Re-opening of prior closed claims. This happens when additional damages, resulting from the original loss occurrence, arise at point in time after the claim has been closed. Exam 5, V1a Page 196 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2004 exam (continued): Question 37 (continued): e. (1 point) Briefly explain why it is appropriate to both trend and develop losses (i.e., why there is no overlap). It is appropriate to both trend and develop losses because there is no double counting of severity trend and loss development factors in the ratemaking process. The trend factor reflects the severity trend from the midpoint of the experience period to the midpoint of the exposure period. The loss development factor reflects the underlying severity trend from the midpoint of the exposure period to ultimate. Solutions to questions from the 2007 exam: 22. Explain the impact of this change on incurred loss development and rate adequacy for this liability line of insurance. CAS Model Solution Incurred loss development factors are based on losses prior to accident year 2006. Since the initial case reserves were much higher, the development factors being applied to 2006 losses will be too low. Ultimate losses for 2006 will be understated therefore indicated projected loss ratios or pure premiums will be too low. This will result in an indication that will be too low. Ultimately, the rates based on accident year 2006 will be inadequate. Solutions to questions from the 2008 exam: Model Solution - Question 17 a. (0.5 point) Calculate the calendar-year incurred losses for 2006 and 2007. CY 2006 incurred losses = CY 2006 Paid losses + CY 2006 Ending Reserves – CY 2006 Beginning Reserves Note: For CY 2006, we are only concerned with transactions associated with any policies effective during CY 2006 that also have losses during CY 2006. For CY 2006, the only policy meeting this criterion is the policy effective 7/1/2006. CY 2006 Paid losses (for policy effective 7/1/2006) = $0. CY 2006 Ending Reserves (for policy effective 7/1/2006) = $5,000 and CY 2006 Beginning Reserves = $0. Thus, CY 2006 incurred losses = $0 + $5,000 - $0 = $5,000 CY 2007 incurred losses = CY 2007 Paid losses + CY 2007 Ending Reserves – CY 2007 Beginning Reserves associated with policies having CY transactions during CY 2007. Note that both the 7/1/2006 and 10/1/2006 policies have transactions (paid and case reserve activities) during CY 2007. i. For the policy effective 7/1/2006, total paid losses (based on 2007 transaction dates) = $500 + $3,500 = $4,000. In addition, beginning reserves = $5,000 and ending reserves = $2,000. Thus, CY 2007 incurred losses (for policy effective 7/1/2006) = $4,000 + $2,000 - $5,000 = $1,000. ii. For the policy effective 10/1/2006, total paid losses (based on 2007 transaction dates) = $5,000 + $9,000 = $14,000. In addition, beginning reserves = $0 and ending reserves = $1,000. Thus, CY 2007 incurred losses (for policy effective 10/1/2006) = $14,000 + $1,000 - $0 = $15,000. Thus, CY 2007 incurred losses = $1,000 + $15,000 = $16,000. Exam 5, V1a Page 197 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2008 exam: Model Solution - Question 17 (continued) b. (0.5 point) Calculate the accident-year incurred losses for 2006 and 2007 evaluated as of 12/31/2008. Note: Here we are concerned with final payments and reserves associated with accidents occurring during AY 2006 and 2007 respectively. i. For the policy effective 7/1/2006, total paid losses (on accidents occurring during 2006) as of 12/31/2008 = $500 + $3,500 + $3,000 = $7,000. Final reserves as of 12/31/2008 = $0. Thus, AY 2006 incurred losses (for policy effective 7/1/2006) = $7,000 + $0 = $7,000. ii. For the policy effective 10/1/2006, total paid losses (on accidents occurring during 2007) = $5,000 + $9,000 + $1,000 = $15,000. Again, final reserves as of 12/31/2008 = $0 Thus, AY 2007 incurred losses (for policy effective 10/1/2006) = $15,000 + $0 = $15,000. c. (0.5 point) Calculate the policy-year incurred losses for 2006 and 2007 evaluated as of 12/31/2008. Note: Both policies are effective during 2006. No policies are effective during 2007. Therefore, there will be no policy year 2007 incurred losses. i. For the policy effective 7/1/2006, total paid losses (on accidents occurring during 2006) as of 12/31/2008 = $7,000 ii. For the policy effective 10/1/2006, total paid losses (on accidents occurring during 2007) as of 12/31/2008 = $15,000 Thus, PY 2006 incurred losses = $7,000 + $15,000 = $22,000. Thus, PY 2007 incurred losses = $0 d. (0.5 point) Identify 1 advantage and 1 disadvantage associated with using PY incurred losses for ratemaking. One advantage is that premiums and losses can be matched using policy year incurred losses. One disadvantage is that policy year data is the least mature and least responsive compared to CY or AY data. Solutions to questions from the 2009 exam: Question 22 a. CY 2006 losses. The question is ambiguous with respect to whether it refers to paid or incurred losses. Assuming Paid Losses are sought, add paid losses during CY 2006 from accidents occurring in both 2005 and 2006: 300,000 + 1,500,000 = $ 1,800,000 Assuming Incurred Losses (i.e. paid + change in reserves) are sought, use the result from above and compute the change in reserves as the ending reserves – beginning reserves, for accidents occurring in both 2005 and 2006: $1,800,000 + (300,000 - 500,000) + (1,000,000 – 0) = $2,600,000 b. AY 2006 incurred losses @ 12/31/07 =(AY 06 paid through 12/31/07) + (AY 06 reserves @ 12/31/07) = (1,500,000 + 700,000) + 200,000 = $2,400,000 c. PY 2005 incurred losses @ 12/31/08. Note: Question states that all claims given in the problem arise from policies written in 2005 = (PY 05 Paid until 12/31/08) + (PY 05 reserves @ 12/31/08) = (1,000,000 + 300,000 +250,000 + 50,000) + (0) [for accidents occurring in 2005] + (1,500,000 + 700,000 + 100,000) + (50,000) [for accidents occurring in 2006] = $1,600,000 + $2,350,000 = $ 3,950,000 d. CY incurred losses are more responsive than AY since loss info is known once CY is complete. AY incurred provides a better match to premium and loss then CY basis, although not as well as PY which matches premium and loss. Exam 5, V1a Page 198 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2009 exam (continued): Question 24 Why it may be inappropriate to apply a basic limits loss trend to total limits losses. If loss costs are increasing, basic limit losses will trend at a lower rate than total losses, and thus a basic limit trend will understate the actual underlying loss trend. Basic limit losses trend at a lower rate than total losses because for losses near or at basic limits before trending, the full trend will not be realized by limiting losses. A loss that is already at or above basic limits, in fact, will observe no basic limit trends if losses are increasing. Question 27 Fully discuss the "overlap fallacy" between trend and loss development. It was believed that loss development and loss trend capture the same change in loss patterns. Therefore, using both would be “double counting”. This belief was referred to as “overlap fallacy”. It is incorrect, because loss trend projects losses from the midpoint of experience period to the midpoint of exposure period, while loss development projects losses from midpoint of the exposure period to ultimates. This can be thought graphically as possible: Successive Evaluation Periods Question 42: For homeowners insurance, explain two reasons that hurricane rates should be priced separately from non-hurricane rates. Ratemaking becomes a much easier process if premiums are split. Traditional techniques can be applied on the non-hurricane portion without having to deduce the non-hurricane portion each time. Allows appropriate classification. For example, it does not make sense to have a 25 % discount for fire protection in an area where 80 % of losses are hurricane related. Exam 5, V1a Page 199 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2010 exam: Question 20 a. CY 2008 reported losses = CY 2008 Paid losses + CY 2008 Ending Reserves – CY 2008 Beginning Reserves Note: Since two claims are given, values for each formula component above need to be aggregated. These values are shown below as (claim 1 amount + claim 2 amount) CY 2008 reported losses = ($0 + $0) + ($25,000 + $0) – (10,000 + $25,000) = -$10,000 b. AY 2006 Reported Loss as of 12/31/2007 Note: Here we are concerned with total payments and reserves as of 12/31/2007 associated with accidents occurring during AY 2006. This limits transactions to claim 1 only. i. Total paid losses (on accidents occurring during 2006) as of 12/31/2007 = $0. Final reserves as of 12/31/2007 = $10,000. Thus, AY 2006 incurred losses $0 + $10,000 = $10,000. c. PY 2006 reported loss as of 12/31/2007 Note: Here we are concerned with total payments and reserves at 12/31/2007 associated with both claims because both claims arose from a single policy issued in 2006. PY 2006 reported loss as of 12/31/2007 = ($0 + $0) + ($10,000 + $25,000) = 35,000 d. Advantage: CY losses are readily available/immediately known. No need to wait for losses to develop. Disadvantage: AY aggregation provides a better match of premiums to losses than CY aggregation. 21. (2 points) Identify four adjustments made to historical losses in projecting losses for a future policy period for ratemaking. Briefly describe the purpose of each. 1. Development – taking losses from an early state (e.g. 24 months) to their total ultimate state when all losses are paid and the claims are closed. 2. Trend – taking historical losses from the midpoint of the experience period and projecting to the midpoint of the future period (takes things such as inflation into account) 3. Benefit Level Changes – take into account anything that would change the benefits being charged to get losses to a “current benefit level” (e.g. workers comp. change in the law affecting benefits paid) 4. Catastrophes/Shock Losses/Extraordinary Events – adjust historical losses to take out any cats and load back in an amount to account for them. If cats were always just included, rates would increase years after cats and decrease after years without them to volatile. Question 24 Select a ULAE factor to be applied to the statewide incurred losses and paid ALAE as part of calculating statewide rate indications. Explain your selection. Calendar Year 2006 2007 200 8 2009 Paid Loss & ALAE (1) 522,500 550,000 523,688 544,635 Paid ULAE (2) 26,125 55,000 52,369 54,464 Paid ULAE/ Paid Loss & ALAE (3)=(2)/(1) 5% 10% 10% 10% I would select ULAE factor =10% Calendar Year 2006 has ULAE factor of 5 % but 2007– 2009 ULAE factors are all at 10% . I believe there must have been a change in operation in 2007 that caused ULAE to increase to 10%. Exam 5, V1a Page 200 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2011 exam: Question 6 a. (0.5 point) Calculate the calendar year 2008 incurred losses. b. (0.5 point) Calculate the accident year 2008 incurred losses, evaluated at December 31, 2009. c. (0.5 point) Calculate the policy year 2007 incurred losses, evaluated at December 31, 2009. d. (0.5 point) Calculate the report year 2008 incurred losses, evaluated at December 31, 2009. e. (0.5 point) Briefly describe one advantage and one disadvantage associated with using policy year losses for ratemaking. Question 6 – Model Solution a. CY 2008 incurred losses = CY 2008 Paid losses + CY 2008 Ending Reserves – CY 2008 Beginning Reserves Note: Here we consider transaction date data occurring in 2008. Such data exists for claims 1, 2 and 3. Claim 1: CY 2008 incurred losses = ($200 + $600 - $300) = $500 Claim 2: CY 2008 incurred losses = ($500 + $200 - $0) = $700 Claim 3: CY 2008 incurred losses = ($5 + $200 - $0) = $250 CY 2008 incurred losses = $500+ $700+$250=$1,450 b. AY 2008 incurred losses = AY 2008 Paid losses + AY 2008 Ending Reserves as of 12/31/2009 Note: Here we consider transaction date data occurring during AY 2008. Such data exists for claim 3 only. Claim 3: AY 2008 paid losses = ($50 + $100) = $150. AY 2008 case reserve as of 12/31/2009 = $50 CY 2008 incurred losses = $150+ $50 = $200 c. PY 2007 incurred loss as of 12/31/2009 Note: Here we are concerned with total payments and reserves at 12/31/2009 associated with all three claims these claims arose from policies issued in 2007. PY 2007 paid losses as of 12/31/2009 = 100 + 200 + 500 + 500 + 200 + 50 + 100 = 1650 PY 2007 case reserves of 12/31/2009 = 0 + 0 + 50 = 50 PY 2007 incurred losses as of 12/31/2009 = 1650 + 50 = 1700 d. RY 2008 incurred loss as of 12/31/2009 Here we are concerned with total payments and reserves as of 12/31/2009 associated with accidents reported during 2008. This limits transactions to claim 2 and claim 3. i. Total paid losses (on accidents reported during 2008) as of 12/31/2009 = $500 + 200 + 50 + 100 = 850. Case reserves as of 12/31/2009 for claims 2 and 3= $0 + $50 = $50 Thus, RY 2008 incurred losses as of 12/31/2009 $850 + $50 = $900. a. b. c. d. e. 200 + 600 - 300 + 500 + 200 + 50 + 200 = 1,450 50 + 100 + 50 = 200 100 + 200 + 500 + 500 + 200 + 50 + 100 + 50 = 1700 500 + 200 + 50 + 100 + 50 = 900 Advantage: True match between premiums and losses Disadvantage: Extended development. It takes longer to develop. Exam 5, V1a Page 201 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2011 exam: Question 7 – Model Solution 1 There is no overlap when developing loss and trending loss. Trending loss will rend loss from the midpoint of experience period to the midpoint of the exposure period. Developing loss will develop loss from the midpoint of the exposure period to the ultimate. Question 7 – Model Solution 2 The overlap fallacy between loss development and trend clarifies than there actually is no overlap, or doublecounting, between the two adjustments. Trend brings historical losses to the projected cost level/ environment of the future period, whereas development brings these losses to their ultimate settlement value. The graph below demonstrates this: Question 17. Given 5 claim amounts; • Basic limit = $25,000; • Total limits severity trend = 10% Calculate the excess loss trend. Question 17 – Model Solution When Limit Loss , Excess loss trend [ Loss *(1.0 Trend )] Limit Loss Limit Excess loss trend = Excess trended losses/Excess losses Claim # Loss XS Loss Trended Loss XS Trended Loss = loss x (1+10%) (1) (2) (3) (4) (5) 1 10,000 0 11,000 0 2 15,000 0 16,500 0 3 30,000 5,000 33,000 8,000 4 35,000 10,000 38,500 13,500 Total 15,000 21,500 (3) = (1) - 25,000, if (1) is greater than 25,000; otherwise (3) = 0 (5) = (4) - 25,000, if (4) is greater than 25,000; otherwise (5) = 0 Excess Loss trend = 21,500/15,000 – 1 = 43.33% Exam 5, V1a Page 202 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Section 2: Effects of WC Benefit Level Changes Solutions to questions from the 1995 exam: Question 37. Direct effect of a benefit change = Average benefit (after the change) . Average benefit (before the change) Current .50 Proposed .667 None .667*1.875 = 1.25 .50*1.0=.50 .667*.75 = .50 Replacement (Compensation) rate =% of the preinjury wage = Max benefit is set equal to a % state average weekly wage (SAWW) Min benefit is set equal to .50* (SAWW) = Average Benefit Computed:: (R Rate)*(% SAWW)*(Cum % of workers) The % of workers earning > (1.25 * SAWW ) receive max benefits The % of workers earning < (.50 * SAWW ) receive min benefits None .667*1.875*(1-.96) =.05 .50*1.0*.6=.30 .667*.75*.35 = .175 Workers earning between the maximum and the minimum receive benefits of equal to a % of their pre-injury wage Total (R Rate) * (cumulative % of wages) .50*(1-.38) = .31 .667*(.86-.17) = .46 .30 + .31 = .61 .05+.175+.46=.685 The direct effect of a benefit change = .685/.610 - 1.0 = 12.3. Exam 5, V1a Page 203 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to the questions from the 1999 exam Question 38. To compute the average benefit, begin by re-stating the %s in the given table as cumulative %s. Ratio to Average Wage 50% 75 1.00 1.50 2.00 2.50 Cum % Of Workers 15% 35% 60% 80% 95 100% Cum % Of Wages 6% 18% 39% 63% 89% 100% Next, determine the % of workers receiving the maximum and minimum benefit. These values are found by looking in the table above for the % of workers earning a certain percentage of the average wage such that the product of (ratio to average wage ) * (compensation rate) equals 150% and 75% of the state average wage respectively. Maximum benefit = 1.50 of average wage Minimum benefit = Note: At the maximum benefit limit, the compensation rate (.75) times the ratio to the state average wage (2.0) equals 1.50 of the state average weekly wage. 0.75 of average wage Compensation rate = Note: At the minimum benefit limit, the compensation rate (.75) times the ratio to the state average wage (1.0) equals .75 of the state average weekly wage. 0.75 of pre-injury wage Computation of the average benefit: Workers earning > 2.0 times the state average weekly wage receive max benefits Workers earning < 1.0 times the state average weekly wage receive min benefits Workers earning between the maximum and the minimum receive benefits of = a % of their pre-injury wage (R Rate) * (cumulative % of wages) Total Benefits as a % of wages .75 * 2.0 * .05 = .075 .75 * 1.0 * .60 = .45 .75 * (.89 - .39) = .375 .075 + .45 + .375 = .90 Thus, the average benefit is equal to 90% of the state average weekly wage: Exam 5, V1a Page 204 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to the questions from the 2001 exam Question 48. a. (1 point) Calculate the average benefit as a percentage of the statewide average weekly wage. Determine the % of workers receiving the maximum and minimum benefit. These values are found by looking in the given table for the % of workers earning a certain percentage of the average wage such that the product of (ratio to average wage ) * (compensation rate) equals 100% (900/900) and 40% (360 / 900) of the state average wage respectively. Maximum benefit = 1.00 of average wage Note: At the maximum benefit limit, the compensation rate (given as .667) times the ratio to the state average wage (1.50) equals 1.00 of the state average weekly wage. Minimum benefit = Compensation rate = 0.40 of average wage Note: At the minimum benefit limit, the compensation rate (.667) times the ratio to the state average wage (.60) equals .40 of the state average weekly wage. 0.667 of pre-injury wage (given) Computation of the average benefit: Workers earning > 1.50 times the state average weekly wage receive max benefits Workers earning < 0.60 times the state average weekly wage receive min benefits Benefits as a % of wages .667 * 1.5 * .10 = .10 .667 * .60 * .25 = .10 .667 * (.82 - .13) = .4602 Workers earning between the maximum and the minimum receive benefits of = a % of their pre-injury wage (R Rate) * (cumulative % of wages) Total .10 + .10 + .4602 = .6602 Thus, the average benefit is equal to 66.2% of the state average weekly wage (900) = 594.21 b. (1 point) Calculate the direct effect of changing the compensation rate from 66.7% to 80.0% of the pre-injury wage. Average benefit (after the change) . Average benefit (before the change) Benefits as a % of wages Workers earning > 1.25 times the state average weekly wage .80 * 1.25 * .20 = .20 receive max benefits (.80 * 1.25 = 1.0) .80 * .50 * .15 = .06 Workers earning < 0.50 times the state average weekly wage receive min benefits(.80 * 50 = .40) Direct effect of a benefit change = .80 * (.67 - .07) = .48 Workers earning between the maximum and the minimum receive benefits of = a % of their pre-injury wage (R Rate) * (cumulative % of wages) Total .10 + .10 + .48 = .74 Thus, the average benefit is equal to 74% of the state average weekly wage (900) = 666 Direct effect of a benefit change = 666 / 594.21 = 1.121 or 12.1% Exam 5, V1a Page 205 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to the questions from the 2007 exam 40. (2.5 points) Workers compensation law changes can produce both direct and incentive (or indirect) effects. a. (0.5 point) Explain what is meant by direct effect. b. (0.5 point) Explain what is meant by incentive effect. c. (0.75 point) Will implementation of cost of living adjustments have a direct effect, incentive effect, or both? Explain your answer. d. (0.75 point) Will changes in administrative procedures have a direct effect, incentive effect, or both? Explain your answer. CAS Model Solution a. A direct effect is the direct impact on premium or losses solely due to law change not taking into account the human response to a change. For example, if the max benefit is increased, losses will automatically go up because those already at the max will get an increase in benefits. b. An incentive effect is the impact a change has on premium and losses because of the change in human behavior. For example, if the duration of benefits is lengthened, more people that are ready to go back may malinger to get benefits longer. c. Both. Direct – Increase in indemnity payments because they will be adjusted upwards with inflation. Indirect – More people may stay out of work longer because their benefits are keeping up with inflation. Previously, they may have returned to work because their benefits were not a sufficient amount. d. Incentive effect only – Administrative procedures that make it easier to file claims may cause some to file claims they wouldn’t have in the past. Solutions to questions from the 2008: Model Solution - Question 19 Step 1: Write an equation to determine the direct benefit level effect of increasing the maximum benefit to $100. Direct effect of a benefit change = [Avg benefit (after the change)/ Avg benefit (before the change]) – 1.0 Step 2: Write an equation to determine the average benefit (effective compensation rate). The average benefit is computed as the sum of the following: 1. Benefits, as a % of wages, for the % of workers earning the minimum % of the SAWW. 2. Benefits, as a % of wages, for the % of workers earning at least the maximum % of the SAWW. 3. Benefits, as a % of wages, for the % of workers earning between the minimum % of the SAWW and the maximum % of the SAWW. Step 3: Compute the % of workers earning benefits for each of the three groups of workers identified in Step 2, before increasing the max benefit to $100. 1. The % of workers earning the minimum % of the SAWW. With a compensation rate of .667, the minimum benefit of $50 is received by a worker making $75 ($50/.667), and $75 as a % of the SAWW of $100 equals .75. Using this as the lookup value for table give in the problem, 35% of workers earn the minimum benefit. 2. The % of workers earning the maximum % of the SAWW. With a compensation rate of .667, the maximum benefit of $67 is received by workers making at least $100 ($67/.667), and $100 as a % of the SAWW of $100 equals 1.0. Using this as the lookup value for table give in the problem, 40% (1.0 - .60) of workers earn at least the maximum benefit. 3. The % of wages unaffected by the min and max limits for workers earning between the minimum % and maximum % of the SAWW. Workers between the limits earn 42% - 20% = 22% of state wages. Exam 5, V1a Page 206 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2008: Model Solution - Question 19 (continued) Step 4: Compute benefits, as a % of wages, for each of the three groups of workers indentified in Step 2. 1. % of workers * min wages as a % of the SAWW * the compensation rate = .35 * .75 * .667 = .1751 2 % of workers * max wages as a % of the SAWW * the compensation rate = .40 * 1.0 * .667 = .2668 3 % of workers * the compensation rate = .22 * .667 = .1467 Current effective compensation rate = .1751 + .2668 * .1467 = .5886 Step 5: Repeat Steps 3 and 4 to determine the % of workers earning benefits for each of the three groups of workers identified in Step 2, after increasing the max benefit to $100. 1. Workers earning no more than 1 half of the SAWW receive the minimum benefit. [Two thirds of 0.75 times the SAWW equals half the SAWW which equals the min benefit.] These benefits, as a percentage of wages, are 2/3 x .75 x 35% = 17.51% 2. Workers earning at least one and a half times the SAWW receive the maximum benefit. [Two thirds of 1.5 times the SAWW equals the revised maximum benefit]. These benefits, as a percentage of wages, are 2/3 x 1.5 x (100% - 91%) = 9%. 3. Workers earning between one half of the SAWW and one and a half times the SAWW receive benefits equal to two thirds of their pre-injury wages. These benefits, as a percentage of wages, are 2/3 x (81% - 20%) = 40.69%. Revised effective compensation rate = 9% + 17.51% + 40.96% = 67.47% Step 6: Using the equation in Step 1, and the results from Steps 3 and 5, compute the direct benefit level effect. Direct benefit level affect = .6747/.5886 - 1.0= .1416 b. Incentive effects are the human behavioral responses to changes in the direct effects of increasing or decreasing benefit levels, compensation rates, etc. c. Because increasing the maximum benefit increased the effective compensation rate, we might expect to see longer duration injuries, since injured workers are receiving more benefit, they have less incentive to return to work. We would also expect an increase in claims, since workers will be paid more for injuries, they will report more injuries. Solutions to questions from the 2009 exam: Question 26 a. (0.5 point) Calculate the direct effect of this benefit change. b. (0.5 point) Briefly explain two possible indirect effects of this change. a. Before the change: benefits = (.5)(pre-tax pay) After the change: benefits = (.85)(post-tax pay) = (.85)(1 - .30)(pre-tax pay) =(.595)(pr- tax pay) The direct effect of the benefits change is that benefits have increased by (.595/.5 - 1= .19 = 19% b1 .We would expect higher frequencies, since the higher benefit will provide employees with more incentive to file claims b2. We would expect employees to stay on disability longer, rather than returning to work, since they will receive higher benefits. Exam 5, V1a Page 207 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2010 exam: Question 23 a. (2 points) Calculate the direct effect of the change in maximum benefits on losses. b. (0.5 point) Explain a potential indirect effect of the change in maximum benefits on losses. Part a. The key is to calculate the benefits provided before and after the change to determine the direct effect. The minimum benefit is 48% of the SAWW ($1,500) which equals $720 (= $1,500 x 48%). The minimum benefit of $720 applies to workers who earn less than 60% of the SAWW (i.e. $720 = 80% x 60% x $1,500), given the current compensation rate of 80%. Min compensation = .48 =60% .80 The aggregate benefits for 64 employees in this category are $46,080 (= 64 x $720). The maximum benefit is 128% of the SAWW ($1,500) and thus equals $1,920 (= $1,500 x 128%). The maximum benefit of $1,920 applies to workers who earn more than 150% of the SAWW (i.e. $1,920 = 80% x 160% x $1,500), given the current compensation rate of 80%. Max compensation= 1.28 =160% .80 The aggregate benefits for the 29 employees in this category are $55,680 (= 29 x $1,920). The remaining 198 (= 144 + 33 + 21) employees fall between the minimum and maximum benefits. This means their total benefits are 80% of their actual wages or $246,240 ( = ( 80% x 196,200 ) + ( 80% x 64,350) + ( 80% x 47,250 ) ). The sum total of benefits is $348,000 (= $46,080 + $55,680 + $246,240) under the current benefit structure. Once the maximum benefit is reduced from 128% to 112% of the SAWW, more workers will be subjected to the new maximum benefit. Workers earning approximately >140% of the SAWW are subject to the maximum (i.e. $1,680 = (80% x 140% x $1,500) > $1,680). These 50 (= 21 + 29) workers will receive $84,000 (= 50 x $1,680) in benefits. New compensation = 1.12 =140% .80 Workers subject to the minimum benefit, 64, are not impacted by the change, and their benefits remain $46,080. There are now only 177 (= 144 + 33) employees that receive a benefit equal to 80% of their pre-injury wages or: $208,440 (= (80% x 196,200) + (80% x 64,350)) because more workers are now impacted by the maximum. The new sum total of benefits is $338,520 (= 84,000 + 46,080 + 208,440). The direct effect from revising the maximum benefit is -2.724 (= 338,520/348,000 – 1.0). Part b. An indirect effect of lowering the max benefit would be a change in claimant behavior. Higher wage earnings may return to work faster as their benefits would not be as favorable as they had been prior. This might compound the decrease in total compensation. Exam 5, V1a Page 208 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2012 exam: 7. Develop the projected ultimate loss and LAE for accident year 2010 losses using the data above. To satisfy the state requirements, fully describe the rationale for the selections for loss development, loss trend, and ULAE. Question 7 – Model Solution 1 (Exam 5A Question 7) Loss Development The ‘06 12-24 factor is a one-off high valve indicating a onetime event. This should be excluded from the selection. Also, the past 3 yrs. 24-36 avg. is stable and has decreased by an absolute 0.1 value from the ‘04 and 05 levels. All other periods are stable and relatively consistent. Based on this, I select the Avg. last 3 yrs. as my LDF. Loss Trend: Frequency: The frequency over the past 12 quarters has been decreasing and leveled off in the final year. I would check w/management about any initiatives they took to decrease the frequency. I would think, based on the data, a process was taken and was effective at bringing freq down to the 0.065 level, but we can expect the stable value going forward. Freq trend = 0% Severity : The book went through a shift in Pure premium, freq, and severity after March 2009. The PP is significantly less implying smaller risks were written which brought down severity. After the pure premium stabilized in June ’09 we see an increasing trend in severity. To recognize this trend, but not include the seventy values from prior ’09 June, I would use the 6pt severity trend. Sev Trend = 5.6% Trend period: 7/1/2010 -> 4/1/2014 3.75 ULAE: The book went through a shift after ’08 and saw a reduction in freq/sev of claims. I would consult the claims dept about how this is effecting their operations w/the change in the type of claims going forward. Since ’08 is considerably different than ’09 and ’10 I would take an average of the ULAE ratio for these years as they reflect the environment going forward. Selecting only ’10 would be based on the results of my conversations w/claim and could overstate the true ULAE ratio. ULAE = (15+ 15.6) / 2 = 15.3% Ult Loss & LAE = 50k x (1.65 x 1.2 x 1.05 x 1.04) Dev x (1 + 0 + .056)^3.75 trend x 1.153 ULAE Ultimate Loss and LAE = 152.907 Exam 5, V1a Page 209 2014 by All 10, Inc. Chapter 6 – Losses and LAE BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Question 7 – Model Solution 2 (Exam 5A Question 7) Loss Development: Notice that from 36-48 and onward, the link ratio are the same. So focus on 12-24 first. Notice that the all year average is high because of Accident Year 2006 in this maturity. This is likely an anomaly- due to a large loss. The other years in the maturity do not seem substantially different, so select the ex-hi/lo average. Now consider the 24-36 category. There is steady decrease in age-to-age factors here. Given this, I would select the Average 1st 3 years average. So selected link ratios are 12-24 24-36 36-48 48-60 60-72 72-ult 1.7 1.2 1.05 1.04 1. 1 Freq Loss Trend: Over the last year, frequency is very stable. However, it is declining in all other years. To balance stability of selections (represent the decreasing trend) but also be responsive (recognize that the trend has leveled off some) I would select -2% (between the 4 and 8 point fits). Sev Since June 2009, severity trend has been increasing at about +6%. The negative trends appear to be the result of the June 2008 -> March 2009 year, which has much higher severity than all other years. Therefore, adjusting or excluding the year is appropriate. Here, I choose to exclude. Since the 6-point and 4-pt fits are so similar, I feel a 6% is well supported. Pure prem Our selections imply a (1.06) * .98 = 1.0388 => 3.88% pure premium trend. Looking at the pure premium and excluding the data points from June 2008 to March 2009, we can see that a 3.88% will balance stability and reasonableness - it falls between the 6 and 4 point fits. Thus, a 3.88% pure premium trend is appropriate. ULAE No compelling reason is seen in regards to differences in paid. Loss and ALAE by year. The ULAE ratio does seem to be going, but it could be skewed by the fact that ULAE is more responsive to claim volume growth than Paid loss is (since paid loss is often from accidents occurring in prior years). So, 15.6% is not appropriate, but 14.5% would not be either without more information on the claims dept. So we select on all-year average of 15% ULAE ratio, which has the added benefit of being explainable to regulators. Avg. date of loss Avg. date of future loss Our trend paired is from 7/1/2010 -> 4/1/2014, 3.75 years Ultimate projected loss of LAE = 50,000 x 1.7 x 1.2 x 1.05 x 1.04 x 1.0388 ^ (3.75) x 1.15 = 147,745.90 Examiner’s Comments Candidates generally justified the loss development factor selections well. Some candidates did lose credit for not including justification. Occasionally candidates’ factors did not match the justification, resulting in the loss of points. Most candidates were able to identify the flat frequency trend and picked a four-point trend. The most common error was selecting a longer projection period without justification of why a decreasing trend was reasonable given the latest points. Many candidates failed to mention either the shock loss or the increasing pattern for severity in recent periods. Some candidates incorrectly calculated the trend period. Some candidates failed to provide justification for the ULAE selection. Most candidates projected ultimate loss and LAE correctly. Exam 5, V1a Page 210 2014 by All 10, Inc. Chapter 7 – Expenses and Profit BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Sec 1 2 3 4 5 6 7 8 9 10 1 Description Simple Example Underwriting Expense Categories All Variable Expense Method Premium-Based Projection Method Exposure/Policy-based Projection Methods Trending Expenses Reinsurance Costs Underwriting Profit Provision Permissible Loss Ratios Key Concepts Pages 125 – 126 126 – 127 127 – 130 130 – 133 133 – 135 135 – 137 137 – 137 138 – 138 139 – 139 139 - 139 Simple Example 125 – 126 How expenses and profit are incorporated within the fundamental insurance equation in the ratemaking process. Assume the following: __ __ The average expected loss and LAE ( L EL ) for each policy is $180. The insurer incurs $20 in expenses ( E F ) for costs associated with printing and data entry, etc. each time it writes a policy. 15% of each dollar of premium collected covers expenses that vary with the amount of premium, (V), (e.g. premium taxes). Company management has determined that the target profit provision ( QT ) should be 5% of premium. If the rates are appropriate, the premium collected will be equivalent to the sum of the expected losses, LAE, underwriting (UW) expenses (both fixed and variable), and the target underwriting profit. Using the notation below, the fundamental insurance equation can be re-written. 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 Exam 5, V1a Page 211 2014 by All 10, Inc. Chapter 7 – Expenses and Profit BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 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. Exam 5, V1a Page 212 2014 by All 10, Inc. Chapter 7 – Expenses and Profit BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 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. Exam 5, V1a Page 213 2014 by All 10, Inc. Chapter 7 – Expenses and Profit BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 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). Exam 5, V1a Page 214 2014 by All 10, Inc. Chapter 7 – Expenses and Profit BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 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. Exam 5, V1a Page 215 2014 by All 10, Inc. Chapter 7 – Expenses and Profit BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 4 Premium-Based Projection Method 130 – 133 For insurers with a significant amount of both fixed and variable u/w expenses, the premium based projection method is used since it recognizes the two types of expenses separately. Like the All Variable Expense Method, it assumes expense ratios during the projected period will be consistent with historical expense ratios The enhancement is that this approach calculates fixed and variable expense ratios separately (as opposed to a single variable expense ratio) so that each can be handled more appropriately within the indication formulae. General Expense Provisions Premium-Based Projection Method 2013 a Countrywide Expenses b Countrywide Earned Premium c Ratio [(a) / (b)] 2014 2015 $26,531,974 $28,702,771 $31,195,169 $450,000,000 $490,950,000 $530,000,000 5.9% 5.8% 5.9% d % Assumed Fixed e Fixed Expense % [(c ) x (d)] f Variable Expense % [(c ) x (1.0-(d))] 3-Year Average 5.9% Selected 5.9% 75.0% 4.4% 1.5% Step 1: Determine the % of premium attributable to each expense type by dividing historical underwriting expenses by EP or WP for each year during the historical experience period. Here, general expenses are assumed to be incurred throughout the policy period, and thus are divided by EP. Step 2: Choose a selected ratio (e.g. if the ratios are stable over time, a 3-year average may be chosen; if the ratios demonstrated a trend over time, the most recent year’s ratio or some other value may be selected). Step 3: Divide the selected expense ratio into fixed and variable ratios (using detailed expense data so that this division can be made directly, or using activity-based cost studies that help split each expense category appropriately). The example assumes 75% of the general expenses are fixed, and that percentage is used to split the selected general expense ratio of 5.9% into a fixed expense provision of 4.4% and a variable expense provision of 1.5%. Step 4: Sum the fixed and variable expense ratios across the different expense categories to determine total fixed and variable expense provisions. If the average fixed expense per exposure (required for the pure premium approach discussed in Chapter 8) is needed, the fixed expense provision can be multiplied by the projected average premium. Fixed Expense Per Exposure = Fixed Expense Ratio x Projected Average Premium Potential Distortions Using this Approach This approach assumes that historical fixed and variable expense ratios will be the same as in the projected period. . (Note: Recall that an actuary CAN select other than the historical ratios.) However, the fixed expense ratio will be distorted if the historical and projected premium levels are materially different. Exam 5, V1a Page 216 2014 by All 10, Inc. Chapter 7 – Expenses and Profit BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 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. Exam 5, V1a Page 217 2014 by All 10, Inc. Chapter 7 – Expenses and Profit BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 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. Exam 5, V1a Page 218 2014 by All 10, Inc. Chapter 7 – Expenses and Profit BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 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. Exam 5, V1a Page 219 2014 by All 10, Inc. Chapter 7 – Expenses and Profit BASIC RATEMAKING – WERNER, G. AND MODLIN, C. In the Exposure/Policy-based Projection Method: If an inflation-sensitive exposure base (e.g. payroll per $100) is used, no trending is needed if the expenses and exposure base are changing at the same rate. If a non-inflation sensitive base (e.g. car-year or house-year) or policy counts are used, average fixed expenses are expected to change over time and trending is appropriate. Data used: Some insurers use internal expense data (examining the historical change in average expenses) to select an appropriate trend. However, internal data maybe volatile and insurers may use government indices (e.g. Consumer Price Index, Employment Cost Index, etc.) and knowledge of anticipated changes in company practices to estimate an appropriate trend (see the procedure in Appendix B). Trending: The selected fixed expense ratio will be trended from the average date that expenses were incurred in the historical expense period to the average date that expenses will be incurred in the forecast period of the rates. Expenses incurred at policy inception should be trended from the average date that the policies were written in the historical period to the average written date in the projection period. Assume annual policies are sold, a steady book of business is maintained, and projected rates will be in effect for one year: Expenses Incurred at the onset of the Policy Expenses incurred evenly throughout the policy period should be trended from the average date the policies were earned in the historical period to the average earned date in the projection period. Expenses Incurred Throughout Policy Points in time: Since the experience period is a calendar year, the average date the policies are written and earned is the same. However, expenses incurred throughout the policy are trended 6 months longer than expenses incurred at inception. Exam 5, V1a Page 220 2014 by All 10, Inc. Chapter 7 – Expenses and Profit BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 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). Exam 5, V1a Page 221 2014 by All 10, Inc. Chapter 7 – Expenses and Profit BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 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). Exam 5, V1a Page 222 2014 by All 10, Inc. Chapter 7 – Expenses and Profit BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 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 Exam 5, V1a Page 223 2014 by All 10, Inc. 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 Exam 5, V1a Page 224 2014 by All 10, Inc. Chapter 7 – Expenses and Profit BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2004 exam: Question 33 b. (1 point) Expenses can be related to written or earned premium. Briefly explain why other acquisition expenses are related to written premium, while general expenses are related to earned premium. Questions from the 2005 exam 43. (4 points) Use Werner's proposed methodology in "Incorporation of Fixed Expenses" and the information below to answer the following questions for the projected annual policy period beginning July 1, 2005. Show all work. Statewide Projected Average Premium at Present Rates $850.00 Statewide Projected Loss and LAE Ratio 68.0% Profit and Contingencies Provision 5.0% Annual Fixed Expense Trend 3.0% Countrywide General Expenses Fixed General Expense as percentage of General Expenses Countrywide Earned Exposures Countrywide Written Exposures Countrywide Earned Premium Countrywide Written Premium Fixed Variable Other Acquisition $60.00 2.5% Taxes, Licenses, and Fees $ 2.50 2.0% Commissions and Brokerage None 12.0% Annual Policy Period 2003 2004 $25,000 $28,000 75% 75% 625 645 640 700 $435,000 $450,000 $460,000 $475,000 • Assume expenses are incurred evenly throughout the policy period. a. (2 points) Calculate the fixed expense provision. b. (1 point) Calculate the variable expense provision. c. (1 point) Calculate the statewide indicated rate change. Exam 5, V1a Page 225 2014 by All 10, Inc. Chapter 7 – Expenses and Profit BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2006 exam 33. (3 points) Given the following private passenger automobile ratemaking data for the past three calendar years, answer the following questions. Written Premium Earned Premium Commissions General Expenses Home Office Salaries Home Office Utilities One-Time Expense associated with Reduction in Staff All Other General Expenses Total General Expenses Other Acquisition Expenses Taxes, Licenses, and Fees Calendar Year 2003 2004 $20,000,000 $25,000,000 19,000,000 24,000,000 3,000,000 3,750,000 2005 $30,000,000 28,000,000 3,000,000 798,000 209,000 1,056,000 216,000 1,008,000 280,000 0 190,000 1,197,000 1,780,000 500,000 360,000 240,000 1,872,000 2,175,000 625,000 0 280,000 1,568,000 2,640,000 750,000 a. (1 point) Beginning on January 1, 2005 all policies written and renewed had commissions changed in order to allow the company to compete more effectively. This new commission rate is expected to continue into the future. As the actuary for this insurance company, briefly explain the commission provision you would recommend for use in the next rate revision to be effective July 1, 2006. Show all work. b. (2 points) As shown in the table above, during 2004 the company paid a one-time expense associated with a reduction in staff. This reduction was due to increases in productivity and resulted in fewer employees during 2005. This new level of staffing is expected to continue. As the actuary for this insurance company, briefly explain the general expense provision you would recommend for use in the next rate revision to be effective July 1, 2006. Show all work. Exam 5, V1a Page 226 2014 by All 10, Inc. Chapter 7 – Expenses and Profit 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. Exam 5, V1a Page 227 2014 by All 10, Inc. 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. Exam 5, V1a Page 228 2014 by All 10, Inc. Chapter 7 – Expenses and Profit 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 Exam 5, V1a Page 229 $850.00 68.0% $578.00 $96.16 18.0% 5.0% 77.0% $875.49 3.0% 2014 by All 10, Inc. 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% Exam 5, V1a Page 230 2014 by All 10, Inc. Chapter 7 – Expenses and Profit BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2008 exam Model Solution - Question 23 Initial comments: Actuaries generally divide underwriting expenses into two groups: fixed and variable. Fixed expenses are those expenses that are assumed to be the same for each exposure, regardless of the size of the premium (i.e., the expense is a constant dollar amount for each risk). Typically, overhead costs associated with the home office are considered a fixed expense. Variable expenses are those expenses that vary directly with premium; in other words, the expense is a constant percentage of the premium. Premium taxes and commissions are two good examples of variable expenses. a. A fixed expense is an expense that is incurred that does not vary with premium. A variable expense is an expense that is incurred that varies with the amount of premium. A better solution is as follows: Fixed expenses (e.g. overhead costs associated with the home office) are assumed to be the same for each risk, regardless of premium size (i.e. the expense is a constant dollar amount for each risk or policy). Variable expenses (e.g. premium taxes and commissions) vary directly with premium and thus are constant percentage of the premium. b. Calculate the indicated rate change. Step 1: Write an equation to determine the indicated rate change. Indicated Rate Change = Projected L + LAE Ratio + Fixed Expense ratio 1.0 -V - Q Step 2: Using the given expense data in the problem, compute the fixed and variable expense ratio. Note: Since other acq. and commissions & brokerage are assumed to occur at the onset of the policy, these expenses are related to written premiums, while all other expenses are related to E premium. Fixed expense ratio= .6(100k) .5(66k) .25(40k) + + =.06+.03+.01=.10 1M 1.1M 1M Variable expense ratio= .4(100k) .5(66k) 110k .75(40k) + + + =.04+.03+.10+.03=.20 1M 1.0M 1.1M 1M Step 3: Using the equation in Step 1, and the results from Step 2, compute the indicated rate change. Indicated Rate Change= .75+.10 1.0-.20-.05 -1.0=13.3% increase c. Rate changes impact the fixed expenses as a percent of premium because the premium the ratio is applied to is different than contemplated in the ratio itself. If there had been a large rate increase after the fixed ratio was calculated the estimated fixed expenses would be higher than actual Solutions to questions from the 2010 exam Question 25 – Model Solution 1 The premium based projection method could produce distorted results if: 1. Premium is not placed at the current rate level. If rates have increased (decreased) since or throughout the historical experience period, premium used in the expense ratios would be understated (overstated), resulting in an overstated (understated) expense ratio. 2. Premium is not trended to reflect shifts in average premium. If average premium is trending upward (downward) after or throughout the historical experience period, premium used in the expense ratios would be understated (overstated), resulting in an overstated (understated) expense ratio. Question 25 – Model Solution 2 – Acceptable Response 3. If we are using a nationwide expense ratio and apply it to a state that has significantly different average premium but the same fixed expense, there will be a distortion. For states with higher (lower) average premium, fixed expense will be overestimated (underestimated). Exam 5, V1a Page 231 2014 by All 10, Inc. Chapter 8 – Overall Indication BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Sec 1 2 3 4 5 Description Introduction and the Pure Premium Method Loss Ratio Method Loss Ratio Versus Pure Premium Methods Indication Examples Key Concepts Pages 141 – 143 143 – 145 145 – 147 147 – 147 147 – 148 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 Page 232 2014 by All 10, Inc. Chapter 8 – Overall Indication BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 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] Exam 5, V1a Page 233 2014 by All 10, Inc. Chapter 8 – Overall Indication BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 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. Exam 5, V1a Page 234 2014 by All 10, Inc. Chapter 8 – Overall Indication BASIC RATEMAKING – WERNER, G. AND MODLIN, C. The goal of the ratemaking: Determine whether current rates can cover the estimated losses and expenses and produce the target profit. If the expected profit % at current rates (QC) is equivalent to the target profit % (QT), then the current rates are appropriate. It is more likely case is that (QC) is not equivalent to (QT) and rates need to be adjusted. QC = 1.0 - (L + EL )+ EF -V PC The objective: How much does the premium at current rates need to be increased or decreased to achieve the target profit percentage? Determine this by substituting: (QT) for (QC) and the indicated premium (PI) for the projected premium at current rates (PC) (indicated premium is the projected premium at current rates times the indicated change factor): QT = 1.0 - Rearranging terms leads to: 1.0 -V - QT (L + EL )+ EF -V PC * Indicated Change Factor (L + EL )+ EF PC * Indicated Change Factor Rearranging terms and dividing through by PC yields: L + EL + EF Indicated Change Factor = = PC * (1.0 -V - QT ) (L + EL ) E + F PC PC , which (1.0 -V - QT ) (L + EL ) + F PC is equivalent to the loss ratio indication formula: Indicated Change Factor = [1.0 -V - QT ] A result greater than 1.0 means the current rates are inadequate and need to be adjusted upward (and vice versa). (L + EL ) + F PC - 1.0 Subtract 1.0 from both sides to produce an indicated change: Indicated Change = [1.0 -V - QT ] Example of Loss Ratio Indicated Rate Change Formula • Projected ultimate loss and LAE ratio = 65% • Projected fixed expense ratio = 6.5% • Variable expense ratio = 25% • Target profit percentage =10% ( L EL ) F PC [65% 6.5%] 1.0 1.0 10% Indicated Change = [1.0 - V QT ] [1.00 0.25 0.10] Thus, the overall average rate level is inadequate and should be increased by 10%. Exam 5, V1a Page 235 2014 by All 10, Inc. Chapter 8 – Overall Indication BASIC RATEMAKING – WERNER, G. AND MODLIN, C. New Company It is not used to price rates for a new insurer since the loss ratio approach is dependent on current premium. The LR method is only used for making rates for a company with existing rates (since the loss ratio approach is dependent on current premium). 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). Exam 5, V1a Page 236 2014 by All 10, Inc. Chapter 8 – Overall Indication BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Equivalency of Methods Both formulae can be derived from the fundamental insurance equation (thus two approaches are mathematically equivalent). (L + EL ) + F PC 1. Start with the loss ratio indication formula: Indicated Change Factor = [1.0 -V - QT ] (L + EL ) + EF PC PC Restate the formula as: Indicated Change Factor = [1.0 -V - QT ] 2. The indicated adjustment factor, the ratio of the indicated premium (PI ) to the projected premium at current (L + EL ) + EF PC PC P = rates (PC), yields the following: I PC [1.0 -V - QT ] 3. Multiplying both sides by the projected average premium at current rates ( PC / X ) results in the pure premium indication formula (proving the two methods are equivalent): 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. Exam 5, V1a Page 237 2014 by All 10, Inc. Chapter 8 – Overall Indication BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 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 Exam 5, V1a Page 238 2014 by All 10, Inc. Chapter 8 – Overall Indication BASIC RATEMAKING – WERNER, G. AND MODLIN, C. The predecessor papers to the current syllabus reading “Basic Ratemaking” by Werner, G. and Modlin, C. were numerous. While past CAS questions were drawn from prior syllabus readings, the ones shown below remain relevant to the content covered in this chapter. Questions from the 2002 exam 17. (4 points) Based on McClenahan, "Ratemaking," chapter 2 of Foundations of Casualty Actuarial Science, and the following data, answer the questions below. Show all work. Projected rates to be effective January 1, 2003 and in effect for 1 year. Permissible loss and ALAE ratio (modified) is 65%. Experience is from the accident period January 1, 2000 to June 30, 2001. Developed accident period loss and ALAE is $21,500. Annual trend factor is 3%. All policies have one-year terms and are written uniformly throughout the year. The rate on January 1, 1999 was $120 per exposure. Effective Date January 1, 2000 January 1, 2001 Year 1998 1999 2000 2001 Rate Change +10% -15% Written Exposures 200 200 200 200 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. Page 239 2014 by All 10, Inc. Chapter 8 – Overall Indication BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2004 exam: 10. Which of the following statements is false regarding the loss ratio and pure premium methods for ratemaking? A. The loss ratio and pure premium methods are identical when using consistent assumptions. B. The pure premium method is preferable when on-level premium is difficult to calculate. C. The loss ratio method produces indicated rate changes. D. The pure premium method requires well-defined, responsive exposures. E. The loss ratio method is preferable for a new line of business. 13. Given the information below, determine the indicated rate per exposure unit. • Frequency per exposure unit = 0.25 • Severity = $100 • Fixed expense per exposure unit = $10 • Variable expense factor = 20% • Profit and contingencies factor = 5% A. < $35 B. > $35 but < $40 C. > $40 but < $45 D. > $45 but < $50 E. > $50 33. (3 points) Given the following information, answer the questions below. On-Level Trended Accident Earned Ultimate Year Premium Loss & ALAE 2000 $800 $512 2001 $900 $540 2002 $1,000 $550 • • • • • • • Ratio of commissions to written premium = 14% Ratio of taxes, licenses and fees to written premium = 3 Ratio of other acquisition expenses to written premium = 2% Ratio of general expense to earned premium = 6.25% Profit and contingency provision = 5% Fixed U/W expense ratio (modified) = 5% Assume each year of historical experience receives equal weighting. a. (2 points) Determine the indicated rate change for policies to be written from January 1, 2004 to December 31, 2004. Show all work. b. (1 point) Expenses can be related to written premium or earned premium. Briefly explain why other acquisition expenses are related to written premium, while general expenses are related to earned premium. (chapter 7) Exam 5, V1a Page 240 2014 by All 10, Inc. Chapter 8 – Overall Indication BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2005 exam: 46. (5 points) Given the following data for private passenger auto bodily injury basic limits, answer the questions below. Show all work. • Policies are annual. • Proposed Effective Date = July 1, 2005 • Rates are in effect for one year. • Current Rate = 225 Experience Period Exposures and Losses Calendar Accident Earned Loss & ALAE as of Year Exposures December 31, 2004 2002 450 $52,000 2003 500 $54,000 2004 530 $40,000 • Age-to-age loss development factors 12-24 months =1.50; 24-36 months =1.15; 36-48 months= 1.05; 48 - ultimate =1.06 • Frequency trend = 2% • Severity trend = 5% • Permissible Loss Ratio (modified) = 65% a. (4 points) Calculate the indicated statewide rate level change using the loss ratio method. b. (1 point) Using your results from part a. above, illustrate the equivalency of the loss ratio method and the pure premium method. Questions from the 2006 exam: 36. (4 points) Using the methods described by McClenahan, and the following information, answer the questions below. Show all work. Experience period on-level earned premium = $500,000 Experience period trended and developed losses = $300,000 Experience period earned exposure = 10,000 Premium-related expenses factor = 23% Fixed underwriting expenses (modified) = $21,000 Profit and Contingency factor = 5% a. (1.5 points) Calculate the indicated rate level change using the loss ratio method. b. (1.5 points) Calculate the indicated rate level change using the pure premium method. c. (1.0 point) Describe one situation in which it is preferable to use the loss ratio method, and one situation in which it is preferable to use the pure premium method. Exam 5, V1a Page 241 2014 by All 10, Inc. Chapter 8 – Overall Indication BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2007 exam: 7. You are given the following information: Indicated base rate is $300 per unit of exposure. Profit and contingencies provision is 3%. Other variable expenses represent 15% of premium. What would the revised base rate be if the company changes the profit and contingencies provision to -6%? A. < $272.00 B. > $272.00 but < $285.00 C. > $285.00 but < $298.00 D. > $298.00 but < $311.00 E. > $311.00 8. You are given the following information: On-level Earned Premium: Projected Loss & ALAE: Projected Fixed Expense Ratio (modified): Variable Expense Ratio (modified): Profit and Contingencies Ratio: $100,000 $75,000 10% 25% 0% What is the indicated rate level change? A. < 6.5% B. > 6.5% but < 8.0% C. > 8.0% but < 9.5% Exam 5, V1a Page 242 D. > 9.5% but < 11.0% E. > 11.0% 2014 by All 10, Inc. Chapter 8 – Overall Indication BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2007 exam (continued): 42. (6.0 points) You are given the following information: Incurred Earned Calendar Accident Year Losses & LAE Premium 2004 $5,000,000 $10,000,000 2005 3,750,000 11,000,000 Weights for Accident Year 35% 65% Historical Rate Level Changes July 1, 2003 5.0% July 1, 2004 -1.0% July 1, 2005 10.0% July 1, 2006 0.0% Losses are valued as of June 30, 2006. Selected annual frequency trend is 4%. Selected annual severity trend is 1%. There is no premium or exposure trend. All policies are annual. Fixed expense ratio is 7%. Profit and contingencies provision is 5%. Other variable expenses are 20% of premium. The indication is considered to be 60% credible. The complement of credibility is no change. Loss Development Factors Age Age to Ult. 6 3.500 12 2.500 18 2.000 24 1.700 30 1.500 36 1.400 42 1.350 Calculate the indicated rate change for rates to be effective from July 1, 2007 through June 30, 2008. Show all work. Note: This is a chapter 5, chapter 6 and chapter 8 question. Exam 5, V1a Page 243 2014 by All 10, Inc. Chapter 8 – Overall Indication BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2007 exam (continued): 43. (3.0 points) Using Werner and Modlin’s notation: a. (2.0 points) Demonstrate the equivalence of the pure premium and loss ratio approaches, assuming identical data and consistent assumptions. b. (0.5 point) Which approach is more appropriate when pricing a new line of business? Explain. c. (0.5 point) Which approach is more appropriate when pricing a line of business for which the historical rate change history is not available? Explain. Questions from the 2008 exam: 24. (1.0 point) The indicated average rate was determined to be $300 based on the following information: Average fixed expense per exposure = $16 Variable expense provision = 15% Profit and contingencies provision = 3% Calculate the revised indicated average rate assuming the expected loss costs will be 10% higher than those assumed in the original analysis. Exam 5, V1a Page 244 2014 by All 10, Inc. Chapter 8 – Overall Indication BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2008 exam continued: 26. (5.75 points) You are given the following information: Calendar/Accident Year 2006 2007 Earned Premium $345,704 $396,714 Base Rate Underlying Premiums $100 $100 Accident Year 2002 2003 2004 2005 2006 2007 15 $164,000 $172,000 $181,000 $190,000 $200,000 $210,000 Case Incurred Loss and ALAE Evaluation Age in Months 27 39 51 $213,200 $245,180 $262,343 $223,600 $257,140 $269,997 $235,300 $258,830 $271,772 $228,000 $250,800 $240,000 63 $262,343 $269,997 Current base rate = $110 Current rating structure is purely multiplicative. Proposed rates will be effective January 1, 2009, and will be in effect for one year. All policies are annual policies. On January 1, 2005 the claims department changed case reserving practices applicable to all outstanding claims. Premium trend = 3% Frequency trend = -1% and severity trend = 2% Unallocated loss adjustment provision = 10% of ultimate incurred loss & ALAE Fixed expense ratio = 8% and variable expense ratio = 20% Profit and contingencies provision = 5% Accident year projections should be weighted 60% to accident year 2007 and 40% to accident year 2006. Overall indication is assumed to be 75% credible. Complement of credibility should be assigned to no change. a. (1.25 points) Calculate calendar/accident year 2006 and calendar/accident year 2007 projected premium at present rates. (Chapter 5, but shown here) b. (3.0 points) Calculate accident year 2006 and accident year 2007 ultimate incurred losses and loss adjustment expenses, projected to future loss cost levels. (Chapter 6, but shown here) c. (1.5 points) Calculate the indicated rate change. (Chapter 8) 27. (1.0 point) a. (0.5 point) Provide an example of where a pure premium method is more appropriate than a loss ratio method. b. (0.5 point) Provide an example of where a loss ratio method is more appropriate than a pure premium method. Exam 5, V1a Page 245 2014 by All 10, Inc. 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. Exam 5, V1a Page 246 2014 by All 10, Inc. Chapter 8 – Overall Indication BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2011 exam continued: 10. (1.5 points) Identify whether the loss ratio or pure premium ratemaking method is preferable in each of the following scenarios. Briefly explain each answer. a. (0.5 point) A company introduced two new rating variables within the past year. b. (0.5 point) A company is entering a new line of business. c. (0.5 point) A company writes a commercial product with multiple exposure bases. Exam 5, V1a Page 247 2014 by All 10, Inc. 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 Page 248 2014 by All 10, Inc. Chapter 8 – Overall Indication BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2003 exam: Question 36. a. (1 point) Calculate the variable permissible loss ratio (VPLR) using the loss ratio method. VPLR [1.0 -V - QT ] , where V and QT are given as 0.22 and 0.03 VPLR = (1.0 - 0.22 - 0.03) = 0.75 = 75.0% b. (1 point) Calculate the indicated rate level change using the loss ratio method (LRM). (L + EL ) + F (L + EL ) + F PC PC - 1.0 , Indicated Change = - 1.0 = [1.0 -V - QT ] VPLR E 20 K (L + EL ) 400 K 0.04 ; VPLR = (1.0 - 0.22 - 0.03) = 0.75 = 75.0% 0.80 , F F PC 500 K Pc 500 K Thus, the indicated rate level change using the LRM = [0.80+0.04]/0.75 – 1.0 = .12 = 12% c. (1 point) Calculate the indicated rate level change using the pure premium method. _________ ____ ___ L EL EF Under the pure premium method, the indicated rate (R) is computed as follows: PI . 1.0 V QT _________ L EL = Indicated pure premium = ____ EF = Fixed expense Experience Period Losses $400,000 $80 Experience Period Exposures 5,000 Non premium Re lated Expensess $20,000 $4 Experience Period Exposures 5,000 ___ V = Variable expense = .22; QT = Profit load = .03; Thus, PI $80 4 $112 1 - .22 - .03 The current rate can be computed on-level earned premium/experience period earned exposures. Thus, the current rate is computed as $500,000/5,000 = $100. Therefore, indicated rate level change using the pure premium method = $112/$100 – 1.0 = .12 = 12% d. (1 point) Describe a situation where the pure premium method cannot be used. The pure premium method cannot be used if exposure information is not available. e. (1 point) Describe a situation where the loss ratio cannot be used. The loss ratio method cannot be used for a new line of business because the method requires existing rate. Solutions to questions from the 2004 exam: 10. Which statements is false regarding the loss ratio and pure premium methods for ratemaking? A. The loss ratio and pure premium methods are identical when using consistent assumptions. True. B. The pure premium method is preferable when on-level premium is difficult to calculate. True. C. The loss ratio method produces indicated rate changes. True. D. The pure premium method requires well-defined, responsive exposures. True. E. The loss ratio method is preferable for a new line of business. False. The loss ratio method cannot be used for a new line. Exam 5, V1a Page 249 2014 by All 10, Inc. Chapter 8 – Overall Indication BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2004 exam (continued): 13. Determine the indicated rate per exposure unit. Step 1: Write an equation to determine the indicated rate per exposure unit, based on the given data The given data lends itself to computing the rate per exposure unit using the pure premium method. Under the pure premium method, the indicated rate is computed as follows: _________ ____ ____ L EL EF ___ ___ Freq Sev EF * . Based on the given data, P PI I PLR 1.0 V QT Step 2: Using the equation from Step 1, and the data given in the problem, solve for the indicated rate per exposure unit. ___ PI .25*$10010 $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 Exam 5, V1a Page 250 2014 by All 10, Inc. Chapter 8 – Overall Indication BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2005 exam: 46. (5 points) a. (4 points) Calculate the indicated statewide rate level change using the loss ratio method. Step 1: Write an equation to determine the indicated rate change (IRC). (L + EL ) PC Indicated Rate Change = PI - 1.0 [1.0 - F V - QT ] ___ (L + EL ) PC - 1.0 . [PLR] Note: The problem does not mention fixed expenses, so we assume there are no fixed expenses. So the PLR is used (which, in this case, is equal to the VPLR) Step 2: Calculate the trended projected ultimate on-level loss and ALAE ratio for the combined experience period 2002 - 2004. With the given information in the problem, compute the developed and trended Loss and ALAE by accident year as follows: AY 2002 2003 2004 Total Loss and ALAE at 12/31/2004 (1) 52,000 54,000 40,000 Age to Ult LDFS (2) 1.113 1.280 1.920 Midpoint of the experience period (3) 7/1/2002 7/1/2003 7/1/2004 Midpoint of the exposure period (4) 7/1/2006 7/1/2006 7/1/2006 Trend Factor (5) (1.071)4 (1.071)3 (1.071)2 Developed and Trended Loss and ALAE (6)=(1)*(2)*(5) 76,147.63 84,912.60 88,092.75 249,152.98 Notes: (2) Age to ultimate LDF computations: (4) Avg Accident date of the exposure period is one year beyond 36 – ult = (1.05)(1.06) = 1.113 the proposed effective date of the rates. 24 – ult = (1.15)(1.113) = 1.280 12 – ult = (1.50)(1.280) = 1.920 (5) A combined frequency and severity trend is computed as (1.02)(1.05) = 1.071. Thus, (5) = 1.071t, where t is the number of years elapsed between column 3 and column 4. Step 3: Compute the Experience Loss and ALAE ratio as Developed and Trended losses $249,152.98 $249,152.98 0.748 On - Level Earned Premium $225[450 500 530] $333, 000 Step 4: Using the equation from Step 1, the results from Step 2, and the data given in the problem, solve for the indicated rate change for policies to be written from July 1, 2005 to July 1, 2006. (L + EL ) PC .748 1 0.151 Indicated Rate Change = PI - 1.0 .65 [PLR] ___ Exam 5, V1a Page 251 2014 by All 10, Inc. Chapter 8 – Overall Indication BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2005 exam (continued): b. (1 point) Using your results from part a. above, illustrate the equivalency of the loss ratio method and the pure premium method. _________ ____ ___ L EL EF Under the pure premium method, the indicated rate (R) is computed as follows: PI . 1.0 V QT In this problem, _________ L EL = Indicated pure premium = Experience Period Developed and Trended Losses $249,152 $168.35 Experience Period Exposures (450 500 530) ____ EF = Fixed expenses per exposure, V = Variable expense, and QT = Profit load. ___ $168.35 $259 . Therefore, the indicated .65 Indicated Rate Current Rate $259 $225 0.151 rate change using the pure premium method is IRC Current Rate $225 Since F, V and QT are not given, and since (1.0 – V – QT ) = PLR, PI Solutions to questions from the 2006 exam: Question 36 a. (1.5 points) Calculate the indicated rate level change using the loss ratio method. Step 1: Write an equation to determine the indicated rate change (IRC). (L + EL ) + F PC Indicated Change = - 1.0 [1.0 -V - QT ] Step2: Using the equation from Step 1, and the data given in the problem, solve for the indicated rate change using the loss ratio method. .642 300,000 IRC [ 21, 000 / 500, 000] / (1 .23 .05) 1.0 1 .108333 10.83% .72 500,000 Exam 5, V1a Page 252 2014 by All 10, Inc. 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 Page 253 2014 by All 10, Inc. 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 Exam 5, V1a Page 254 2014 by All 10, Inc. Chapter 8 – Overall Indication BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2007 exam (continued): 42. Calculate the indicated rate change for rates to be effective from July 1, 2007 through June 30, 2008. Step 1: Write an equation to determine the indicated rate change. ( L EL ) F PC Indicated Change = 1.0 [1.0 - V QT ] Note that losses will need to be adjusted by the selected annual frequency and severity trend rates, and developed to ultimate. Premiums need to be adjusted by rate level changes only, since there is no premium or exposure trend. Since we are given two years of premiums and losses, a weighted loss ratio will need to be calculated. And after computing the indicated rate change, a credibility weighted indicated rate change must be determined since the indication is considered to be 60% credible. Step 2: Determine on-level earned premium. To do so, compute on-level factors for CYs 2004 and 2005. This is the current rate level divided by the weighted average of the rate level factors in the experience period. The weights will be relative proportions of each square or triangle. First calculate the area of all triangles (area = .5 * base * height) within a unit square and then determine the remaining proportion of the square by subtracting the sum of the areas of the triangles from 1.0. Rate Level Factors: Date Rate Change Rate Level Factor 7/1/03 5% 1.05000 = 1.05 * 1.000 7/1/04 -1% 1.03950 = 1.05 * (1-.01) 7/1/05 10% 1.14345 = 1.0395 * 1.10 7/1/06 0% 1.14345 = 1.14345 * 1.00 Current Rate Level = 1.05 * (1.0 -0.01) * 1.1 * 1.0 = 1.14345 On level Earned Premium: 2004 on level EP: 1.14345/(0.125*1.00+0.75*1.05+0.125*1.0395) * 10M = 1.097 * 10M = 10,970,000 2005 on level EP: 1.14345/(0.125*1.05+1.0395*0.75+1.14345*0.125) * 11M = 1.085 * 11M = 11,935,000 Step 3: Determine ultimate losses. As of 6/30/2006, AY 2004 losses are 30 months old while AY 2005 losses are 18 months old. 2004 ultimate losses: 5,000,000 * (30-Ult Factor) = 5,000,000 * 1.5 = 7,500,000 2005 ultimate losses: 3,750,000 * (18-Ult Factor) = 3,750,000 * 2.0 = 7,500,000 Note: Losses also need to be trended to one year beyond the effective date of the rates (i.e. 7/1/2008). For AY 2004, the average accident date is 7/1/2004. Thus, four years of frequency/severity trend is applied. Step 4: Determine the projected weighted loss ratio. Ultimate CL Earned Loss Trended Loss Premium Trend Loss 4 2004 7,500,000 10,970,000 [(1.04)(1 .01)] 9,130,196 3 2005 7,500,000 11,935,000 [(1.04)(1 .01)] 8,692,114 Thus, the project weighted loss ratio = 0.35(0.8323) + 0.65(0.7283) = 0.7647 Loss Ratio 0.8323 0.7283 Indicated change = [(L+EL)/Pc +F]/[1.0 – V – QT ] – 1.0 =(0.7647+0.07)/(1 - 0.2 - 0.05) – 1.0 = .1129 Credibility weighted indicated rate change: [0.60* 1.1129 +0.4 (1.00)] - 1.0 = .0677 = +6.77% Exam 5, V1a Page 255 2014 by All 10, Inc. Chapter 8 – Overall Indication BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2007 exam (continued): Question 43 a. (2.0 points). Demonstrate the equivalence of the pure premium and loss ratio approaches, assuming identical data and consistent assumptions. b. (0.5 point) Which approach is more appropriate when pricing a new line of business? Explain. c. (0.5 point) Which approach is more appropriate when pricing a line of business for which the historical rate change history is not available? Explain. Model Solution (L + EL ) + F PC 1. Start with the loss ratio indication formula: Indicated Change Factor = [1.0 -V - QT ] (L + EL ) + EF PC PC Restate the formula as: Indicated Change Factor = [1.0 -V - QT ] 2. The indicated adjustment factor, the ratio of the indicated premium (PI ) to the projected premium at current (L + EL ) + EF 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. Exam 5, V1a Page 256 2014 by All 10, Inc. Chapter 8 – Overall Indication BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2008 exam: Model Solution - Question 24 24. (1.0 point) The indicated average rate was determined to be $300 based on the following information: Average fixed expense per exposure = $16 Variable expense provision = 15% Profit and contingencies provision = 3% Calculate the revised indicated average rate assuming the expected loss costs will be 10% higher than those assumed in the original analysis. Step 1: Write an equation to determine the revised indicated average rate. _________ ____ L EL EF and thus the revised indicated average rate equals Indicated Average Rate PI 1.0 V QT ___ _________ ____ 1.10* L EL EF 1.0 V QT Step 2: Using the equations in Step 1, solve for the revised indicated average rate. ___ ____ _________ W are given that PI = $300, EF = $16, V = .15 and QT = .03, Thus, L EL = $300(1.0-0.18)-16 = $230 ___ Thus, revised Exam 5, V1a PI 230(1.1)16 328.05 1.15.03 Page 257 2014 by All 10, Inc. Chapter 8 – Overall Indication BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2008 exam: Model Solution - Question 26 a. (1.25 points) Calculate calendar/accident year 2006 and calendar/accident year 2007 projected premium at present rates. Step 1: Write an equation to determine CAY 2006 and CAY 2007 projected premium at present rates (PPPR). PPPR = Earned Exposures * Current Base Rate * (1.0 + Premium Trend)(midpt exper period to 1 yr after proj eff date) Step 2: Determine Earned Exposures * Current Base Rate for CAY 2006 and CAY 2007 CAY 2006 Earned Exposures * Current Base Rate = $345,704/100 * $110 = $380,274.4 CAY 2007 Earned Exposures * Current Base Rate = $396,714/100 * $110 = $436,385.4 Step 3: Compute the trend period for CAY 2006 and CAY 2007 The Trend period should extend from the midpoint of the experience period to 1 year after the projected effective date of the rates. For CAY 2006, the trend period is from 7/1/06 to 1/1/2010 = 3.5 years For CAY 2007, the trend period is from 7/1/07 to 1/1/2010 = 2.5 years Step 4: Using the equation in Step 1, and the results from Steps 2 and 3, compute PPPR CAY 2006 PPPR = $380,274.4 * (1.03)3.5 = $421,723 CAY 2007 PPPR = $436,385.4 * (1.03)2.5 = $469,854 b. (3.0 points) Calculate accident year 2006 and accident year 2007 ultimate incurred losses and loss adjustment expenses, projected to future loss cost levels. Step 1: Write an equation to determine AY 2006 and AY 2007 Trended and Ultimate Incurred L+ALAE Projected Ultimate Incurred L+ALAE+ULAE = Case Incurred Losses * LDFULT * (1+ loss Trend)(midpt exper period to 1 yr after proj eff date) * (1+ULAE factor) Step 2: Using the case incurred loss triangle, compute age to age factors, select age to ultimate factors, and compute AY 2006 and AY 2007 ultimate losses. AY 2002 2003 2004 2005 2006 15-27 1.30 1.30 1.30 1.20 1.20 Case Incurred Link Ratios 27-39 39-51 1.15 1.07 1.15 1.05 1.10 1.05 1.10 51-63 1.00 1.00 We can see the change in case reserving practices from the link ratios. We will use the link ratios below the solid line. Sel A-t-A Age to Ult 1.200 1.386 1.100 1.155 1.050 1.050 1.000 1.000 AY 2006 ultimate losses = $240,000 * 1.155 = 277,200 AY 2007 ultimate losses = $210,000 * 1.386 = 291,060 Exam 5, V1a Page 258 2014 by All 10, Inc. 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 , 1V QT 1V 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. Exam 5, V1a Page 259 2014 by All 10, Inc. 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). Exam 5, V1a Page 260 2014 by All 10, Inc. 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. Exam 5, V1a Page 261 2014 by Alll 10, Inc. 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. Exam 5, V1a Page 262 2014 by All 10, Inc. 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. Exam 5, V1a Page 263 2014 by All 10, Inc. 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. Exam 5, V1a Page 264 2014 by All 10, Inc. 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. 2014 by All 10, Inc. 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? Exam 5, V1a Page 266 2014 by All 10, Inc. 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. Exam 5, V1a Page 267 2014 by All 10, Inc. 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 Exam 5, V1a Page 268 2014 by All 10, Inc. Statement of Principles Regarding P & C Insurance Ratemaking CAS COMMITTEE ON RATEMAKING PRINCIPLES Questions from the 2004 exam (continued): 38. (1.5 points) Credibility is an important consideration in ratemaking methodology. a. (0.5 point) Define credibility. b. (0.5 point) One method of increasing credibility is by increasing the size of the groupings analyzed. Briefly describe another method to increase credibility. c. (0.5 point) Explain a potential weakness in increasing credibility by the method you provided in part b. above. Questions from the 2005 exam 35. (2 points) State the four ratemaking principles of the Casualty Actuarial Society. Questions from the 2006 exam 25. (1.5 points) The ratemaking actuary for ABC Insurance Company is proposing to change the exposure base for Homeowners Insurance from number of homes to amount of Coverage A. a. (0.5 point) According to the Statement of Principles regarding P&C Insurance Ratemaking, state two desirable characteristics of an exposure base. b. (1.0 point) Determine which exposure base better satisfies each of the characteristics stated in part a. above. Explain. Questions from the 2007 exam 11. Which of the following is true based on the Statement of Principles Regarding Property and Casualty Insurance Ratemaking? A. Unallocated loss adjustment expenses are the claim settlement costs directly assignable to specific claims. B. Taxes, licenses, and fees exclude federal income taxes. C. Policyholder dividends are a return of premium not assigned as an expense. D. Allocated loss adjustment expenses include all costs associated with the settlement of claims. E. General administrative expenses are all costs, except commission and brokerage costs, associated with the acquisition of business. Questions from the 2009 exam 39. (1.75 points) a. (1 point) Identify two considerations from the "Statement of Principles Regarding Property & Casualty Ratemaking" that could apply to the concept of insurance to value. Briefly explain the relevance of each to insurance to value. b. (0.75 point) An insurance company increases the insurance to value of its book of business. Briefly describe the impact on each of the following: • Premium • Losses • Expenses Exam 5, V1a Page 269 2014 by All 10, Inc. Statement of Principles Regarding P & C Insurance Ratemaking CAS COMMITTEE ON RATEMAKING PRINCIPLES Solution to the question from the 1989 exam Question 4. 1. T. 2. F. 3. T. Answer C. Solution to the question from the 1990 exam Question 1. 1. F. 2. T. 3. F. Answer B. Solution to the question from the 1991 exam Question 18. 1. T. 2. T. 3. T. Answer E. Solution to the question from the 1993 exam Question 23. 1. T. Risk 2. F. Organization of Data. 3. T. Answer C. Solution to the question from the 1994 exam Question 39. Principle 1: A rate is an estimate of the expected value of future costs. The recommended average rate of $150 is consistent with the indicated estimate of the expected value of future costs. Principle 2: A rate provides for all costs associated with the transfer of risk. By recommending an average rate, which provides for the costs associated with the transfer of risk, equal to the indicated average rate, equity among insureds is maintained. Principle 3: A rate provides for the costs associated with an individual risk transfer. The recommended rate of $200 for class B does not provide for the costs associated with an individual risk transfer, as it is $25 below that which is indicated. Solutions to questions from the 1995 exam Question 1. 1. F. Affordability is not one of the considerations. 2. T. Reinsurance. 3. T. Operation Changes Exam 5, V1a Page 270 Answer D. 2014 by All 10, Inc. Statement of Principles Regarding P & C Insurance Ratemaking CAS COMMITTEE ON RATEMAKING PRINCIPLES Solutions to questions from the 1995 exam Question 28. 1. Other Influences: The judicial environment, residual markets, guaranty fund assessment all vary by state. 2. Trends: Consideration of past and prospective changes in frequency, severity, exposure, expenses, which can vary by state. 3. Economic variables: Costs associated with repair and replacement all vary by state. 4. Catastrophe: The types of natural catastrophe’s vary by state, and degree of frequency and severity. Solution to the question from the 1996 exam Question 1. The "Statement of Principles Regarding Property and Casualty Insurance Ratemaking," identifies 18 considerations. 1. Reinsurance is specifically listed. 2. Quality of company management is not listed. 3. Changes in claims handling practices is just one of the items mentioned under the category "Operational Changes". Answer D. Solution to the question from the 1997 exam Question 25. A. Exposure units should vary with the hazard, and be practical and be verifiable. Solution to the question from the 1998 exam Question 46. a. Principle 1: A rate is an estimate of the expected value of future costs. The recommended rate, based on affordability, and not on expected future costs, is not consistent with this principle. Principle 2: A rate provides for all costs associated with the transfer of risk. Since any deficit is made up by the state's general fund, this principle is not satisfied. Principle 3: A rate provides for the costs associated with an individual risk transfer. Since the recommended rate is fixed, this principle is not satisfied, as the costs associated with individual risk transfer are not recognized. b. Principle 2 is now satisfied since offering a rebate or imposing a surcharge provides a mechanism to target all costs associated with the transfer of risk. Exam 5, V1a Page 271 2014 by All 10, Inc. Statement of Principles Regarding P & C Insurance Ratemaking CAS COMMITTEE ON RATEMAKING PRINCIPLES Solutions to questions from the 1999 exam Question 41. The statement of principles state that "it is desirable that exposure unit: 1. be Practical 2. be Verifiable 3. vary with the level of risk The proposed exposure base is "actual miles the vehicle is driven." 1. The proposed exposure base is not practical from a number of aspects, including: Accuracy - asking insureds to provide exposure base information makes the exposure base easy to manipulate, and thus, gives rise to a moral hazard. Expense - the expense of having the odometer read by company personnel may outweigh the benefits gained from using this exposure base. 2. The proposed exposure base is verifiable (odometers can be read), but is subject to the following types of manipulation: a. odometers can malfunction b. odometers can be adjusted by individuals and automobile shops. 3. For auto liability and collision, actual miles driven (as an exposure unit) clearly varies with the level of risk. Solutions to questions from the 2000 exam Question 22. Which of the following statements is true? A. T. Subdividing the data to minimize the effects of operational or procedural changes may increase credibility. Credibility is increased by making groupings more homogeneous or by increasing the size of the group analyzed. Homogenous groups require refinement and portioning of the data. See page 3. B. F. Creating homogeneous groupings of data will tend to decrease the credibility of the data. Credibility is increased by making groupings more homogeneous or by increasing the size of the group analyzed. See page 3. C. F. Data should not be organized by calendar year for purposes of producing rates. Acceptable methods of organizing data include calendar year, accident year, report year and policy year. See page 3. D. F. When considering the trade-off between partitioning of data into homogeneous groups versus increasing the volume of ratemaking data in each grouping, preference should be given to creating the most homogeneous groupings. Each situation requires balancing homogeneity and the volume of data. See page 3. Answer A. Exam 5, V1a Page 272 2014 by All 10, Inc. Statement of Principles Regarding P & C Insurance Ratemaking CAS COMMITTEE ON RATEMAKING PRINCIPLES Solutions to questions from the 2000 exam Question 42. a. State the three principles in which ratemaking produces actuarially sound cost estimates Principle 1 Principle 2 Principle 3 A rate is an estimate of the expected value of future costs. A rate provides for all costs associated with the transfer of risk. A rate provides for the costs associated with an individual risk transfer. (When an individual risk's experience does not provide a credible basis for estimating costs, it is appropriate to consider the aggregate experience of similar risks). b. If a rate is actuarially sound, name the four criteria commonly used by actuaries. Principle 4: A rate is actuarially sound if it is: 1. Reasonable 2. NOT excessive 3. NOT inadequate 4. NOT or unfairly discriminatory if it is an actuarially sound estimate of the expected value of all future costs associated with an individual risk transfer. Solutions to questions from the 2001 exam Question 3. Which of the following statements is true? A. Unallocated loss adjustment expenses are the claim settlement costs directly assignable to specific claims. False. Allocated loss adjustment expenses are claim settlement costs directly assignable to specific claims. B. Taxes, licenses, and fees exclude federal income taxes. True. Answer B. C . Policyholder dividends are a return of premium not assigned as an expense. False. Policyholder dividends are a non-guaranteed return of premium charged to operations as an expenses. D. Allocated loss adjustment expenses include all costs associated with the settlement of claims. False. Allocated loss adjustment expenses are the claim settlement costs directly assignable to specific claims. E. General administrative expenses are all costs, except commission and brokerage costs, associated with the acquisition of business. False. General administrative expenses are all other operational and administrative costs. Exam 5, V1a Page 273 2014 by All 10, Inc. Statement of Principles Regarding P & C Insurance Ratemaking CAS COMMITTEE ON RATEMAKING PRINCIPLES Solutions to questions from the 2001 exam Question 4. According to the Statement of Principles Regarding Property and Casualty Insurance Ratemaking, which of the following statements is true? A. Consideration should be given to changes in case reserving that affect the continuity of the experience. True. Answer A. B. Consideration should be given to the determination of an appropriate exposure unit or premium basis, although it is not essential. False. The determination of an appropriate exposure unit or premium basis it is essential. C. Ratemaking is retrospective because the property and casualty insurance rate must be developed after the transfer of risk. False. Ratemaking is prospective because the property and casualty insurance rate must be developed prior to the transfer of risk. D. Credibility is generally increased by making groupings more heterogeneous due to the diversification benefit from combining uncorrelated items. False. Credibility is generally increased by making groupings more homogeneous or by increasing the size of the group analyzed. E. Changes in policy provisions, such as coordination of benefits and second injury fund recoveries, are outside the scope of ratemaking data and thus need not be considered in ratemaking methodology. False. Changes in policy provisions, such as coordination of benefits and second injury fund recoveries, need to be considered in ratemaking methodology Solutions to questions from the 2002 exam 1. Based on the Statement of Principles Regarding Property and Casualty Insurance Ratemaking, which of the following statements is false? A. A rate is an estimate of the expected value of current costs. False. A rate is an estimate of the expected value of future costs. B. A rate provides for all costs associated with the transfer of risk. True. C. A rate provides for the costs associated with an individual risk transfer. True. D. Rates that are actuarially sound comply with the following criteria: reasonable, not excessive, not inadequate, and not unfairly discriminatory. True. E. Ratemaking is prospective because the property and casualty insurance rate must be developed prior to the transfer of risk. True. Exam 5, V1a Page 274 2014 by All 10, Inc. Statement of Principles Regarding P & C Insurance Ratemaking CAS COMMITTEE ON RATEMAKING PRINCIPLES Solutions to questions from the 2003 exam 30. (3 points) The Statement of Principles Regarding Property and Casualty Insurance Ratemaking lists numerous considerations involved in the ratemaking process. State and briefly discuss three of these considerations that have been impacted by the recent rise in worldwide terrorist activity. 1. Reinsurance. Reinsurance has become more expensive because of the major losses on Sept 11. In addition, many reinsurers have become insolvent, making recoveries uncertain. Both the cost of reinsurance and the solvency of the reinsurer must be considered. 2. Catastrophe losses. Terrorist attacks were considered a catastrophe. The potential for future catastrophic losses from terrorist attacks needs to be considered in any allowance for the catastrophe exposure in the rates. 3. Legislation. There is a bill that has or is about to be passed about government involvement in losses sustained in terrorist attacks. When this bill is passed, the effect on net losses for insurers will need to be considered in ratemaking process. Solutions to questions from the 2004 exam 9. Which of the following is true regarding ratemaking expense provisions? 1. Taxes, licenses and fees do not include federal income tax. True. See Section 1: Definitions. 2. Other acquisition expenses include commission and brokerage expenses. False. Other acquisition expenses are all costs, except commission and brokerage, associated with the acquisition of business. 3. General administrative expenses represent all costs associated with the claim settlement process not directly assignable to specific claims. False. General administrative expenses are all other operational and administrative costs. Answer A. 1 only 38. (1.5 points) Credibility is an important consideration in ratemaking methodology. a. (0.5 point) Define credibility. According to the CAS Statement of Principles regarding P&C ratemaking, “credibility is a measure of the predictive value that the actuary attaches to a particular body of data.” Note: The CAS model solution from the 2004 exam reads as follows: “Credibility is determined by how much experience is expected to be a good predictor of future experience.” b. (0.5 point) One method of increasing credibility is by increasing the size of the groupings analyzed. Briefly describe another method to increase credibility. Another method would be to increase the homogeneity of groupings analyzed. The more stable and homogeneous a group, the larger the credibility. Obtaining homogeneous groupings requires refinement and partitioning of the data. See the CAS Statement of Principles regarding P&C ratemaking. c. (0.5 point) Explain a potential weakness in increasing credibility by the method you provided in part b. above. There needs to be a balance between the size of the groupings and how homogeneous you make the groupings. If groups are segregated too much in an attempt to increase homogeneity, the groups will be too small to be credible. According to the CAS statement of principles, there is a point at which partitioning divides data into groups too small to provide credible patterns. Each situation requires balancing homogeneity and the volume of data.” Exam 5, V1a Page 275 2014 by All 10, Inc. Statement of Principles Regarding P & C Insurance Ratemaking CAS COMMITTEE ON RATEMAKING PRINCIPLES Solutions to questions from the 2005 exam 35. (2 points) State the four ratemaking principles of the Casualty Actuarial Society. 1. A rate is an estimate of the expected value of future costs. 2. A rate provides for all costs associated with the transfer of risk. 3. A rate provides for the cost associated with an individual risk transfer. 4. A rate is reasonable, not inadequate, excessive, or unfairly discriminatory if it is an actuarially sound estimate of the expected value of future costs associated with an individual transfer of risk. Solutions to questions from the 2006 exam 25. (1.5 points) The ratemaking actuary for ABC Insurance Company is proposing to change the exposure base for Homeowners Insurance from number of homes to amount of Coverage A. a. (0.5 point) According to the Statement of Principles regarding P&C Insurance Ratemaking, state two desirable characteristics of an exposure base. b. (1.0 point) Determine which exposure base better satisfies each of the characteristics stated in part a. above. Explain. Initial comments: Exposure Unit—The determination of an appropriate exposure unit or premium basis is essential. It is desirable that the exposure unit vary with the hazard and be practical and verifiable. CAS Model Solution: Part a. 1 – Verifiable. 2 – Vary with hazard. - OR 3 – Be practical Part b. 1 – It is easier to verify that there is a home (# homes) rather than the value of home. Thus number of homes is better for verifiability. 2 – Coverage A amount is a better exposure base for varying with hazard. The amount of damage and loss depends on the value of the home. - OR 3 – The number of homes is more practical since Coverage A amount is subject to some judgment. Exam 5, V1a Page 276 2014 by All 10, Inc. Statement of Principles Regarding P & C Insurance Ratemaking CAS COMMITTEE ON RATEMAKING PRINCIPLES Solutions to questions from the 2007 exam 11. Which of the following is true based on the Statement of Principles Regarding Property and Casualty Insurance Ratemaking? A. Unallocated loss adjustment expenses are the claim settlement costs directly assignable to specific claims False. Unallocated loss adjustment expenses are all costs associated with the claim settlement function not directly assignable to specific claims. See Definitions. B. Taxes, licenses, and fees exclude federal income taxes. True. See Definitions. C. Policyholder dividends are a return of premium not assigned as an expense. False. Policyholder dividends are a non-guaranteed return of premium charged to operations as an expense. See Definitions. D. Allocated loss adjustment expenses include all costs associated with the settlement of claims. False. Allocated loss adjustment expenses are claims settlement costs directly assignable to specific claims. See Definitions. E. General administrative expenses are all costs, except commission and brokerage costs, associated with the acquisition of business. False. Statement E. is the definition of other acquisition expenses. General administrative expenses are all other operational and administrative costs. See Definitions. Solutions to questions from the 2009 exam Question 39 – Model Solution a. Mix of business - changing mix of ITV in the book will influence premium and loss trends. Economic/Social Social trends = if there is a movement towards lower insurance to value because people are purchasing lower amounts of coverage to save money on premium due to hard economic times, the actuary may want to evaluate the insurance to value contemplated on the current rates. b. Premium - could see higher prem. as a result of larger exposure amounts written could see lower premium if there are higher cancel/non-renews Losses – expect to see larger total and near total claim amts. from larger exposures Losses may decrease from higher cancel/non-renew Losses may decrease if reinspection also leads to loss control measures implemented by homeowners. Expenses – increased inspection/reinspection may create additional expenses, however increase relative to premium change is unclear. Exam 5, V1a Page 277 2014 by All 10, Inc. ASOP 13 Trending Procedures in Property/Casualty Insurance Sec 1 2 3 4 1 Description Purpose, Scope, Cross References, and Effective Date Section 2. Definitions Section 3. Analysis of Issues and Recommended Practices Section 4. Communications and Disclosures Pages 1-1 1-2 2-3 3-4 Purpose, Scope, Cross References, and Effective Date 1-1 1.1 Purpose—To provide guidance to actuaries when performing trending procedures to estimate future values. 1.2 Scope—This standard applies to actuaries when performing work for insurance or reinsurance companies, as well as self insurers. A trending procedure does not encompass “development,” which estimates changes over time in losses (or other items) within a given exposure period (e.g. accident year or underwriting year). If the actuary departs from the guidance in this standard to comply with applicable law (statutes, regulations, and other legally binding authority) or for any other reason the actuary deems appropriate, refer to section 4.3. 1.3 Cross References—When referring to the provisions of other documents, the reference includes the referenced documents as they may be amended or restated in the future, and any successor to them, by whatever name called. If any amended or restated document differs materially from the originally referenced document, consider the guidance in this standard to the extent it is applicable and appropriate. 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). Exam 5, V1a Page 278 2014 by All 10, Inc. ASOP 13 Trending Procedures in Property/Casualty Insurance 3 Section 3. Analysis of Issues and Recommended Practices 2-3 3.1 Purpose or Use of Trending Procedures—Trending is an important component in ratemaking, reserving, valuations, underwriting, and marketing. Where multiple purposes or uses are intended, the actuary should consider the potential conflicts arising from those multiple purposes or uses and should consider adjustments to accommodate the multiple purposes or uses to the extent that, in the actuary’s professional judgment, it is appropriate and practical to make such adjustments. The actuary may present the trend estimate resulting from the trending procedure in a variety of ways (e.g. a point estimate, a range of estimates, a point estimate with a margin for adverse deviation, or a probability distribution of the trend estimate). 3.2 Historical Insurance and Non-Insurance Data The actuary should select data (historical insurance or non-insurance information) appropriate for the trends being analyzed. When selecting data, the actuary should consider the following: 1. the credibility assigned to the data by the actuary; 2. the time period for which the data is available; 3. the relationship to the items being trended; and 4. the effect of known biases or distortions on the data relied upon (e.g. the impact of catastrophic influences, seasonality, coverage changes, nonrecurring events, claim practices, and distributional changes in deductibles, types of risks, and policy limits). 3.3 Economic and Social Influences Consider economic and social influences that can have a significant impact on trends in selecting the appropriate data to review, the trending calculation, and the trending procedure. Consider the timing of the various influences. 3.4 Selection of Trending Procedures In selecting trending procedures, the actuary may consider relevant information as follows: a. procedures established by precedent or common usage in the actuarial profession; b. procedures used in previous analyses; c. procedures that predict insurance trends based on insurance, econometric, and other noninsurance data; and d. the context in which the trend estimate is used in the overall analysis. 3.5 Criteria for Determining Trending Period The actuary should consider the following when determining the trending period: the lengths of the experience and forecast periods changes in the mix of data between the experience and forecast periods when determining the trending period. When incorporating non-insurance data in the trending procedure, the actuary should consider the timing relationships among the non-insurance data, historical insurance data, and the future values being estimated. Exam 5, V1a Page 279 2014 by All 10, Inc. ASOP 13 Trending Procedures in Property/Casualty Insurance 3.6 Evaluation of Trending Procedures—The actuary should evaluate the results produced by each selected trending procedure for reasonableness and revise the procedure where appropriate. 3.7 Reliance on Data or Other Information Supplied by Others—When relying on data or other information supplied by others, the actuary should refer to ASOP No. 23, Data Quality, for guidance. 3.8 Documentation —The actuary should prepare and retain appropriate documentation regarding the methods, assumptions, procedures, and the sources of the data used. The documentation should be in a form such that another actuary qualified in the same practice area could assess the reasonableness of the actuary’s work, and should be sufficient to comply with the disclosure requirements in section 4. 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. Exam 5, V1a Page 280 2014 by All 10, Inc. ASOP 13 Trending Procedures in Property/Casualty Insurance Question from the 1993 exam 21. Based on the "Actuarial Standard of Practice No. 13, Trending Procedures in Property/Casualty Insurance Ratemaking," which of the following are examples of biases or distortions which should be considered when examining historical insurance data for trending purposes? 1. Hurricane Andrew which struck Florida in 1992. 2. The increase in the Massachusetts automobile Personal Injury Protection coverage from $2,000 to $8,000. 3. The impact of school vacations on automobile miles driven. A. 1 only B. 2 only C. 1, 3 only D. 2, 3 only E. 1, 2, 3 Question from the 1994 exam 19. Based on "Actuarial Standard of Practice No. 13, Trending Procedures in Property/Casualty Insurance Ratemaking," which of the following items should be considered in the trending procedure used in ratemaking for Workers Compensation insurance? 1. An enacted reform that restricts the use of lump sum settlements. 2. Annual revisions in the hourly rate of compensation for union employees. 3. A decrease in attorney representation as Workers Compensation returns to a true "first party" coverage. A. 1 only B. 2 only C. 1, 2 only D. 2, 3 only E. 1, 2, 3 Question from the 1995 exam There were no questions associated with this article appearing on the 1995 exam. Question from the 1997 exam 2. Based on the "Actuarial Standard of Practice No. 13, Trending Procedures in Property/Casualty Insurance Ratemaking," which of the following are biases or distortions that could affect the selection of trending procedures? 1. Revising Homeowners policy coverage from actual cash value to replacement cost value. 2. A new underwriting requirement for percentage hurricane deductibles. 3. An automatic insurance to value program at policy renewal. A. 1 Exam 5, V1a B. 2 C. 3 D. 1, 2 Page 281 E. 1, 2, 3 2014 by All 10, Inc. ASOP 13 Trending Procedures in Property/Casualty Insurance Questions from the 2001 exam Question 14. According to “Actuarial Standard of Practice No. 13: Trending Procedures in Property/Casualty Insurance Ratemaking,” which of the following items should be considered in the trending procedure used in ratemaking for private passenger automobile insurance? A. A decrease in automobile usage due to rising gas prices B. The introduction of higher policy limits C. A recently enacted tort reform that strengthens the verbal threshold for lawsuits D. Changes in price levels in the economy as measured by external indices such as the Consumer Price Index E. All of the above should be considered. Questions from the 2007 exam 6. According to ASOP No. 13, Trending Procedures in Property/Casualty Insurance Ratemaking, which of the following should be considered when selecting trending procedures? 1. Known biases (e.g., seasonality) 2. The impact on the overall indication 3. The credibility of the data A. 1 only Exam 5, V1a B. 1 and 2 only C. 1 and 3 only Page 282 D. 2 and 3 only E. 1, 2, and 3 2014 by All 10, Inc. 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 Exam 5, V1a Page 283 Answer: C. 1 and 3 only 2014 by All 10, Inc. 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. Exam 5, V1a Page 284 2014 by All 10, Inc. 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. Exam 5, V1a Page 285 2014 by All 10, Inc. 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). Exam 5, V1a Page 286 2014 by All 10, Inc. Risk Classification Statement of Principles AMERICAN ACADEMY OF ACTUARIES COMMITTEE ON RISK CLASSIFICATION 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. Exam 5, V1a Page 287 2014 by All 10, Inc. Risk Classification Statement of Principles AMERICAN ACADEMY OF ACTUARIES COMMITTEE ON RISK CLASSIFICATION 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. Exam 5, V1a Page 288 2014 by All 10, Inc. 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 Exam 5, V1a Page 289 2014 by All 10, Inc. 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. Exam 5, V1a Page 290 2014 by All 10, Inc. Risk Classification Statement of Principles 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. Exam 5, V1a Page 291 2014 by All 10, Inc. Risk Classification Statement of Principles 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.” Exam 5, V1a Page 292 2014 by All 10, Inc. Risk Classification Statement of Principles 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 Exam 5, V1a B. 2 only C. 3 only D. 1 and 3 only Page 293 E. 2 and 3 only 2014 by All 10, Inc. 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. Exam 5, V1a Page 294 2014 by All 10, Inc. Risk Classification Statement of Principles AMERICAN ACADEMY OF ACTUARIES COMMITTEE ON RISK CLASSIFICATION 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. Exam 5, V1a Page 295 2014 by All 10, Inc. Risk Classification Statement of Principles AMERICAN ACADEMY OF ACTUARIES COMMITTEE ON RISK CLASSIFICATION 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. Exam 5, V1a Page 296 2014 by All 10, Inc. Risk Classification Statement of Principles AMERICAN ACADEMY OF ACTUARIES COMMITTEE ON RISK CLASSIFICATION 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. Exam 5, V1a Page 297 2014 by All 10, Inc. Risk Classification Statement of Principles AMERICAN ACADEMY OF ACTUARIES COMMITTEE ON RISK CLASSIFICATION 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. Exam 5, V1a Page 298 2014 by All 10, Inc. Risk Classification Statement of Principles AMERICAN ACADEMY OF ACTUARIES COMMITTEE ON RISK CLASSIFICATION 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. Exam 5, V1a Page 299 2014 by All 10, Inc. Risk Classification Statement of Principles AMERICAN ACADEMY OF ACTUARIES COMMITTEE ON RISK CLASSIFICATION Solutions to questions from the 1999 Exam (continued): Question 46. a. (1 point) List three methods for determining this price. b. (1 point) List one deficiency for each method described in part a. a. 1. Reliance on wisdom, insight, and good judgment. 2. Observation of the risk's actual losses over an extended period of time. 3. Observation of losses from groups of individual risks with similar characteristics. This is the most frequently used method. b. 1. Valuable information about expected future loss experience is lost when a risk's actual loss experience is not reviewed. 2. Gradual changes in the hazard may render past information useless. 3. Identification of similar risk characteristics (commonly determined by fact and informed judgment) and related classes before the observation period is problematic. Solutions to questions from the 2000 Exam: Question 16. Which are not operational considerations relating to classification plans? All of the operational considerations listed relate to classification plans. See pages 11 – 13. Answer E. Question 35 a. (0.5 point) Briefly describe adverse selection. Adverse selection arises when buyers (looking to secure the minimum price) are free to select among different sellers, and when sellers react by offering a similar product in order to incite the movement of buyers in an attempt to gain an economic advantage, often at a price where the seller has not matched price to cost. See page 7. b. (1.5 points) Briefly explain the two methods described for controlling adverse selection. 1. Risk classification in a voluntary market - charges each risk the appropriate rate through proper risk identification and balances the economic forces governing buyer and seller actions. This is the primary means to control instability caused by adverse selection. 2. Compulsory insurance with limited choices (e.g. group insurance) reduces the voluntary choice among competing institutions. Restriction of buyer freedom prevents movement or reduces the price incentive. See pages 8 and 12-13. Solutions to questions from the 2001 Exam: 3. In which of the following situations would a refined risk classification program be most appropriate? A. Insurance premiums are determined prior to the policy period and are not adjusted on the basis of actual experience. True. To the extent that prices are NOT adjusted based on a risk’s actual experience, MORE refined risk classifications systems are needed. B. Participation in the insurance program is entirely compulsory. In government programs, participation is usually compulsory and the benefits received by, or paid on behalf of a class, are not necessarily related to the amount paid into the plan by that class. C. Dividends are paid after the initial insurance premium has been established and are based on the risk’s actual experience. To the extent that prices are adjusted based on a risk’s actual experience, less refined risk classifications systems are needed. D. The insurance premium is paid by someone other than the individual insured. Here, the individual insured is indifferent to the classification system, and thus, broad classification systems may be appropriate. E. None of A, B, C, or D are appropriate situations for a refined risk classification program. False. A is true. Exam 5, V1a Page 300 2014 by All 10, Inc. Risk Classification Statement of Principles AMERICAN ACADEMY OF ACTUARIES COMMITTEE ON RISK CLASSIFICATION Solutions to questions from the 2001 Exam (continued): 23. (1.5 points) List and briefly describe the three primary purposes of risk classification according to the American Academy of Actuaries Committee on Risk Classification’s “Risk Classification Statement of Principles.” 1. Protect the insurance system's financial soundness. Risk classification is the primary means to control instability caused by adverse selection. 2. Be fair. A proper risk classification system produces prices which are reflective of expected costs. 3. Encourage availability of coverage through economic incentives. A proper risk classification system will allow an insurer to write and better serve both higher and lower cost risks. See pages 2 – 9. Solutions to questions from the 2002 Exam 20. Which of the following best describes a basic principle of a sound risk classification system? A. The system should be applied subjectively. False. The system should be applied objectively. See page 2. B. The system should produce prices based on the observed actual losses of each risk. False. A system that produces prices based on observed actual losses of each risk is an example of experience based pricing. Further, to the extent that prices are adjusted based on a risk’s emerging actual experience after the insurance and its initial price have been established, less refined initial risk classification systems are needed. See pages 12 and 13. C. The system should reflect expected cost differences. True. See page 2. D. The system should be based solely on public acceptability. False. Although the system should be acceptable to the public, it should not be based solely on public acceptability. See page 2. E. The system should be the same for all competitors. False. Insurers should refine their risk classification systems and thus their pricing structures to be more successful than their competitors, so that it could serve both lower cost and higher cost risks in the marketplace. See pages 9 and 10. 46. (2 points) Your company is planning to implement a new classification system. List and describe two statistical and two operational considerations in designing this new classification system. Statistical: 1. Homogeneity. Individual risks within a class should have reasonably similar expected costs. Within a class there should be no clearly identifiable subgroups with significantly different loss potential. 2. Credibility. The larger the numbers of observations, the more accurate are the statistical predictions that can be made. Each class does not have to be large enough to stand on its own, since accurate predictions can be made based on statistical analysis of the experience of broader grouping of correlative classes. Note: Candidates would also receive credit for listing and defining Predictive Stability. Operational: 1. Manipulation. The ability to manipulate or misrepresent a risk’s characteristics to affect its class assignment should be minimized. 2. Measurability – Risk characteristics should lend themselves to reliable and convenient measurement, such as age, sex, occupation or location. Note: Candidates would also receive credit for listing and defining Expense, Constancy, Availability of Coverage, Avoidance of Extreme Discontinuities, Absence of Ambiguity, and Hazardous Reduction Incentives. Exam 5, V1a Page 301 2014 by All 10, Inc. Risk Classification Statement of Principles AMERICAN ACADEMY OF ACTUARIES COMMITTEE ON RISK CLASSIFICATION Solutions to questions from the 2002 Exam (continued): 48. (4 points) Your company is planning to purchase a block of boatowner's insurance business from Zeron. Zeron has raised overall rates on this block of business for three consecutive years, but does not classify risks by age or size. Despite the rate increases, loss ratios continue to worsen and growth remains high. General information. Based on the given information, we know that Zeron does not classify risks by age or size, that their loss ratios are worsening and that their growth remains high (presumably due to writing a large proportion of poor risks). This implies that that Zeron’s competitors do classify by age and size, which impacts the types of risks they underwrite, and the rates they charge. a. (1 point) Explain how adverse selection could be impacting the seller's poor results. Apparently, Zeron’s worsening loss ratios and high growth rate are the result of writing a large proportion of poor risks at inadequate rates. A review of the given premium and exposure data indicates that Zeron charges an average rate for all risks. Assuming that Zeron’s competitors classify risks by age and size, better risks will purchase from Zeron’s competitors at an actuarially fair rate while poorer risks will purchase from Zeron. Zeron’s pricing is causing a significant shift in the types of risks it underwrites. b. (3 points) Using the information below, calculate rates to address the adverse selection problem. Briefly justify your methods in light of risk classification principles. Rates should be based on measurable risk characteristics (e.g. age and size) and not on ethnicity group (since this is not a publicly acceptable classification criteria). Therefore, the data should be configured as follows: Group Age Boat Size Premium Exposures Rates (1) (2) 1 1 1 L M S 18,000 11,000 10,000 90 55 50 2 2 2 L M S 25,000 17,000 14,000 125 85 70 Total Current Loss Losses Ratio (3)=(1)/(2) (4) (5)=(4)/(1) 200 200 200 5,700 5,000 6,850 0.3167 0.4545 0.6850 200 200 200 13,700 11,300 9,700 0.5480 0.6647 0.6929 52,250 0.5500 95,000 Given the significant variability in the loss ratios, rates should be based on differences in expected costs. This can be reflected by adjusting current rates by loss ratio relativities. Age Group 1 1 1 Proposed Rates (6) 115.15 165.29 249.09 2 199.27 2 241.71 2 251.95 (6) = (3) * [(5) ÷ (5)total] Exam 5, V1a Page 302 2014 by All 10, Inc. Risk Classification Statement of Principles AMERICAN ACADEMY OF ACTUARIES COMMITTEE ON RISK CLASSIFICATION Solutions to questions from the 2003 Exam: 1. Which of the following statements are intentions of risk classification? 1. to identify good and bad risks. False. This is not mentioned. 2. to predict the experience for an individual risk. False. This is not mentioned. 3. to group individual risks having reasonably similar expectations of loss. True. See page 121. Answer C. 3 Only. Solutions to questions from the 2004 Exam: Question 23- Model Solution 1 a. (1.5 points) Given the following information: Type of Vehicle Cars Trucks Earned Exposures 100,000 75,000 Number of Claims per year 5,000 4,000 Pure Premium $200 $300 Would a classification plan that assigns cars and trucks to different classes be statistically sound? Explain why or why not. Yes, assigning cars and trucks to different classes would be statistically sound. Both cars and trucks have large volumes of data (100,000 earned exposures for cars; 75,000 for trucks). Also, the pure premiums of cars and trucks are significantly different ($200 for cars versus $300 for trucks). b. (1.5 points) Given the following information: Type of Earned Number of Pure Vehicle Exposures Claims per year Premium Type A 99,950 4,950 $199 Type B 50 5 $2,199 Would a classification plan that assigns Type A and Type B cars to different classes be statistically sound? Explain why or why not. No, assigning Type A and Type B to different classes would not be statistically sound. Even though Type B has much higher pure premium than Type A, there are only 50 exposures for Type B, which is too small to derive statistical conclusions. The high cost of Type B may only be random loss fluctuation. Question 23 – Model Solution 2 a. There would be homogeneity within the class. There are enough exposures in each to have statistical credibility. These are mutually exclusive classes that could not be manipulated by the insureds. There are differences in severity. Yes, assigning cars and trucks to different classes would be o.k. b. No, there are not enough exposures in Type B to have statistical credibility. Question 23, part b only- Model Solution 3 b. I would say yes. While Type B has very small volume, by examining the credibility-weighted differences between the types would still bring value. Type B is significantly worse in the three types of characteristics identified in A above (frequency, severity and pure premium). Exam 5, V1a Page 303 2014 by All 10, Inc. Risk Classification Statement of Principles AMERICAN ACADEMY OF ACTUARIES COMMITTEE ON RISK CLASSIFICATION Solutions to questions from the 2004 Exam (continued): Question 24 - Model Solution 1 a. (2 points) List and describe four operational considerations in designing a risk classification plan. b. (2 points) i. ii. Compare the use of miles driven and the use of accident and violation history for auto insurance based on the following risk classification considerations: Hazard Reduction Incentives Availability of Coverage a. 1. Measurability – the variables should be easy to defined & measure. 2. Manipulation – the plan should not allow for insureds to manipulate their classifications. 3. Expense – the expenses of the classification plan should be as low as possible while maximizing company value. 4. Absence of ambiguity – the classifications should be all encompassing and mutually exclusive; each insured should fit into one and only one class. b. i. It would be difficult to significantly alter the number of miles driven since most are of necessity (work, etc.). It doesn’t provide much hazard reduction incentive. Some drivers may avoid long trips. Hazard reduction incentives would work for accident and violation history because drivers would be more cautious in order to avoid higher rates. ii. Miles driven would allow for more availability of coverage because miles driven have an impact on loss exposure. Using this as a classification would improve rate accuracy and thus encourage widespread availability. Use of accident and violation history may have the same impact as described for miles driven. However, insurers may use this information to deny coverage to drivers with more than a certain number of accidents. This would reduce availability. Question 24 - Model Solution 2 a. 1. Measurability – it should be easy to measure or quantify the value of the classification (e.g., age or sex). 2. Expense – the value added by having the classification should be greater than the expense of having it in the plan. 3. Avoidance of extreme discontinuity – we should avoid a large jump in rates between a class and the one next to it. 4. Maximize coverage availability – the plan should accurate price risks so that the availability of coverage is maximized. b. i. Hazard reduction incentive a. Use of miles driven – to the extent that an insured will avoid unnecessary road trips, this may reduce the hazard. But this does not seem like an effective way to reduce hazard because people still need to drive. b. Accident / violation – this will create an incentive for insureds to drive safely and avoid accidents. ii. Availability of coverage a. Use of miles driven – to the extent that costs are correlated with miles drive, this may more accurately price risks and thus result in more availability of coverage. b. Accident / violation – since accident / violation history is correlated with costs, having this variable will promote more accurate rates, leading to better availability. Exam 5, V1a Page 304 2014 by All 10, Inc. Risk Classification Statement of Principles AMERICAN ACADEMY OF ACTUARIES COMMITTEE ON RISK CLASSIFICATION Solutions to questions from the 2005 Exam: 1. (3 points) a. (1.5 points) Describe three statistical considerations in designing a risk classification system. b. (1.5 points) Discuss one advantage and two disadvantages of using controllability as a consideration for identifying rating variables. Question 1 – Model Solution 1 a. Statistical Considerations 1. Homogeneity: risks in the same class should have reasonably similar loss potential. 2. Credibility: the number of claims should be voluminous to warrant credibility. 3. Predictive Stability: responsive to changes in the nature of insurance yet stable in avoiding unwarranted abrupt changes. b. Advantage: If the rating variable is closely associated with the efforts to reduce hazard, then the classification will help reduce the potential loss. Disadvantages: 1. If the variable is susceptible to manipulation then the insured may misuse it. 2. If the variable is irrelevant to the predictability of the losses, then the variable may not be useful in predicting future losses and this may not be acceptable to the public. Question 1 – Model Solution 2 a. Statistical Considerations 1. Homogeneity: Risks are grouped according to their traits as homogeneously as allowed (but not forgoing credibility). 2. Credibility: Risks are grouped in volumes that are adequate for the group to be credible. 3. Predictive Stability: Risks are grouped according to traits that are responsive enough to changes; but stable enough to not allow abrupt changes. b. Advantage: It is a good way to encourage reduction in hazard; insureds will want to control how much they pay in premium. Disadvantages: 1. Manipulation: Risks may tend to manipulate their exposure to reduce premiums. 2. Impractical: Some traits may not be practical to implement in a classification system. Solutions to questions from the 2006 Exam: 1. (1.5 points) Describe three primary purposes of risk classification. Question 1 – Model Solution 1 1. Protect the insurance system’s financial soundness. This is threatened by adverse selection which can occur if insurance companies are not allowed to classify. 2. Enhance fairness. Charge insureds appropriately for their potential for loss, do not punish or reward insureds at the expense of others. 3. Provide economic incentive to make coverage available. With classification, companies will be able to charge appropriately and will be able to serve higher and lower risk insureds and will be incented to provide coverage. Exam 5, V1a Page 305 2014 by All 10, Inc. Risk Classification Statement of Principles AMERICAN ACADEMY OF ACTUARIES COMMITTEE ON RISK CLASSIFICATION Solutions to questions from the 2006 Exam: Question 1 – Model Solution 2 1. To protect insurance system’s financial soundness. Minimize potential for adverse selection by matching expected costs with price. 2. To enhance fairness. By ensuring prices valid and equitable with no subsidization between classes. Each risk is charged appropriate rate through proper risk identification. 3. To permit economic incentives to operate and thus encourage widespread availability of coverage By charging higher premiums for higher risks and lower premiums for lower risks Economies of scale by offering coverage to all at appropriate rates Financial incentive to be a better risk and thus reduce one’s premium See pages 2 – 3. Solutions to questions from the 2007 Exam: Question 1 – Model Solution 1 Credibility -> enough risks in the class to allow reasonable and credible inferences to be drawn Homogeneity -> risks in the class should be similar (i.e. no subgroups identifiable) Predictive stability -> use of the classes should be responsive to changing conditions, but avoid large swings in rates from year to year Credibility and homogeneity may be in conflict. We want the risks to be very similar, but we also want enough experience so that they are credible. Question 1 – Model Solution 2 Homogeneity ->the risks within the class should be similar (i.e. there should be little variation within the class) Credibility ->there must be enough data in the class to be able [to] rely on Predictive Stability-> should be responsive to the nature of insurance losses yet stable enough to avoid abrupt price changes Homogeneity and credibility are in conflict since making a class more homogeneous by eliminating risks comes at the expense of credibility, since there may not end up being enough risks in the class to make it credible. Solutions to questions from the 2008 Exam: Question 1 – Model Solution 1 a. State these three primary purposes of risk classification. 1. Enhance insurance system financial soundness 2. Enhance fairness 3. Permit economic incentives to operate and increase availability of insurance b. Briefly describe how each of these purposes helps to establish and maintain a viable insurance system. 1. Risk classification minimizes adverse selection which will exist when buyers are free to select who they purchase insurance from 2. Rate should be in line with their expected loss costs and there shouldn’t be any subsidy between risk classes 3. Each risk class should be priced to their expected losses so that insurers have same profit potential on all risks and are willing to write high risks and low risks, rather than just going after low risks. This increases availability. Exam 5, V1a Page 306 2014 by All 10, Inc. Risk Classification Statement of Principles AMERICAN ACADEMY OF ACTUARIES COMMITTEE ON RISK CLASSIFICATION Solutions to questions from the 2008 Exam: Question 1 – Model Solution 2 a. 1. To protect the financial soundness of the insurance system 2. Enhance fairness 3. Economic incentives to make coverage available b. 1. Risk classification protects insurers from adverse selection which could impair an insurance company 2. It would provide rates that are reflective of insured’s expected cost making them fair and not unfairly discriminatory 3. Encourages insurer to refine system to better serve both high and low risk insureds because of competition. Solutions to questions from the 2008 Exam: a. (1 point) Set the discount for Age 0 (new homes) to 15%, leaving other discounts unchanged. b. (1 point) Set the discount for Age 4 to 25%, leaving other discounts unchanged. c. (1 point) Disaggregate the Age 6+ group and implement discounts of 2% for Age 6 and Age 7 and 1% for Age 8 and Age 9, leaving discounts for Age 10+ at 0%. Question 2 – Model Solution 1 a. Yes Stat Credibility There seems to be enough data to provide a reasonable prediction. Oper Manipulation The age of the home would not be subject to manipulation since it should be well documented. b. No Stat Predictive Stability This is probably random loss fluctuation and should not be too responsive. Oper Discontinuity There would be a discontinuity of coverage changing discount from 3% to 25% back to 3%. c. No Stat Homogeneity These risks should be similar and therefore can be grouped Oper Expense Expensive to implement and change system when there is not an apparent need. Question 2 – Model Solution 2 a. Agree with making the change i. From the statistical consideration, this age group has the most exposures and thus the most credibility and their loss ratios would support this change in discount. ii. From an operational consideration, this is one that could not be manipulated by the insured. b. Disagree with making the change i. Statistical – although the discount may be supported by loss ratios, this is smallest age group category so has the least credibility. ii. Operational – This would result in Age group 3 with a 3% discount, age group 4 with a 25% discount, and then age group 5 with a 3% discount again. This is an extreme discontinuity which we want to avoid. c. Disagree i. Statistical – The credibility for making this change might be in question. ii. Operational – The expense of making this change would likely outweigh the benefits. Exam 5, V1a Page 307 2014 by All 10, Inc. 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] Exam 5, V1a Page 308 2014 by All 10, Inc. 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. Exam 5, V1a Page 309 2014 by All 10, Inc. Personal Vehicle Manual 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. Exam 5, V1a Page 310 2014 by All 10, Inc. 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. Exam 5, V1a Page 311 2014 by All 10, Inc. Personal Vehicle Manual 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. Exam 5, V1a Page 312 2014 by All 10, Inc. 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. Exam 5, V1a Page 313 2014 by All 10, Inc. 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 Page 314 2014 by All 10, Inc. 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 Page 315 2014 by All 10, Inc. 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 Exam 5, V1a Page 316 2014 by All 10, Inc. 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 Page 317 D. > $27.50 but < $30.00 2014 by All 10, Inc. 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. Exam 5, V1a Page 318 2014 by All 10, Inc. Personal Vehicle Manual 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. Exam 5, V1a Page 319 2014 by All 10, Inc. Personal Vehicle Manual 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 Exam 5, V1a Page 320 2014 by All 10, Inc. 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. Exam 5, V1a Page 321 2014 by All 10, Inc. 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). Exam 5, V1a Page 322 2014 by All 10, Inc. 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). Exam 5, V1b Page 1 2014 by All 10, Inc. 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. Exam 5, V1b Page 2 2014 by All 10, Inc. Chapter 9 – Traditional Risk Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. The risks are distributed evenly amongst the two companies and the rates are set as follows: Original Distribution, Loss Cost, and Rates (1) (2) (3) (4) (5) True Refined Insurer Simple Insurer Expected Insured Charged Insured Charged Risk Cost Risks Rate Risks Rate H $230.00 25,000 $230.00 25,000 $180.00 L $130.00 25,000 $130.00 25,000 $180.00 Total $180.00 50,000 $180.00 50,000 $180.00 As shown below, if there is no movement of risks between the insurers, aggregate premium collected by both insurers is the same. For Refined, the premium charged varies by level of the rating variable and is equitable. For Simple, H risks are not charged enough premium (the $1,250,000, shortfall is completely offset by the excess premium collected from L risks). Thus, L risks are subsidizing the H risks at Simple Insurer. Static Distribution With Results (1) (2) (3) (4) (5) (6) (7) Refined Simple True Total Total Expected Insured Charged $Excess/ Insured Charged $Excess/ Risk Cost Risks Rate ($Shortfall) Risks Rate ($Shortfall) H $230.00 25,000 $230.00 $25,000 $180.00 $(1,250,000) L $130.00 25,000 $130.00 $25,000 $180.00 $1,250,000 Total $180.00 50,000 $180.00 $50,000 $180.00 $(4)= [(3)-(1)] x (2) (7)= [(6)-(1)] x (5) Since 1 out of 10 insureds shops at renewal and makes their purchase based on price, the distribution of insureds will not remain static. 2,500 =[.10 * (25,000)] Refined H risks will buy from Simple and 2,500 Simple L risks buy from Refined. This movement results in the following distribution of risks for policy year one: Policy Year One Distribution With Results (1) (2) (3) (4) (5) (6) (7) Refined Company Simple Company True Total Total Expected Insured Charged $Excess/ Insured Charged $Excess/ Risk Cost Risks Rate ($Shortfall) Risks Rate ($Shortfall) H $230.00 $230.00 $$180.00 $(1,375,000) 22,500 27,500 L $130.00 $130.00 $$180.00 $1,125,000 27,500 22,500 Total $180.00 50,000 $175.00 $50,000 $180.00 $(250,000) [(22,500 * $230) + (27,500 * $130)]/50,000 = 175.00 (4)= [(3)-(1)] x (2) (7)= [(6)-(3)] x (5) Exam 5, V1b Page 3 2014 by All 10, Inc. Chapter 9 – Traditional Risk Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 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. Exam 5, V1b Page 4 2014 by All 10, Inc. 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. Exam 5, V1b Page 5 2014 by All 10, Inc. Chapter 9 – Traditional Risk Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 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). Exam 5, V1b Page 6 2014 by All 10, Inc. Chapter 9 – Traditional Risk Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 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. Exam 5, V1b Page 7 2014 by All 10, Inc. Chapter 9 – Traditional Risk Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 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. Exam 5, V1b Page 8 2014 by All 10, Inc. 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 Exam 5, V1b Page 9 2014 by All 10, Inc. 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. Exam 5, V1b Page 10 2014 by All 10, Inc. 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. Exam 5, V1b Page 11 2014 by All 10, Inc. 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. Exam 5, V1b Page 12 2014 by All 10, Inc. 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. Exam 5, V1b Page 13 2014 by All 10, Inc. 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. Exam 5, V1b Page 14 2014 by All 10, Inc. 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 Page 15 2014 by All 10, Inc. 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. Exam 5, V1b Page 16 2014 by All 10, Inc. 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. Exam 5, V1b Page 17 2014 by All 10, Inc. 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. Exam 5, V1b Page 18 2014 by All 10, Inc. 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 Exam 5, V1b Page 19 2014 by All 10, Inc. 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. Exam 5, V1b Page 20 2014 by All 10, Inc. 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 Exam 5, V1b Page 21 2014 by All 10, Inc. 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. Exam 5, V1b Page 22 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. 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. Exam 5, V1b Page 23 2014 by All 10, Inc. Chapter 9 – Traditional Risk Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 1997 exam 43. (3 points) You are given: Territory Prior Year Base Rates Prior Year Earned Premium Current Year Base Rates Current Year Earned Premium A 100 250,000 110 300,000 B 60 400,000 55 350,000 C 120 200,000 100 250,000 D 150 100,000 160 150,000 • Full credibility is 1,082 claims • Territory A is the base territory • Incurred losses and claim counts are developed and trended • No weighting is used to combine the two years of data Combined Years Earned Premium @ Current Rates 575,000 716,667 416,667 256,667 Combined Years Incurred Losses Combined Years Claim Counts 330,000 525,000 290,000 135,000 435 800 390 275 Based on Finger, "Risk Classification," chapter 5 of Foundations of Casualty Actuarial Science, calculate the indicated territorial relativities using the loss ratio approach. Questions from the 1999 exam 13. Based on Finger, "Risk Classification" chapter 5 of Foundations of Casualty Actuarial Science, use the loss ratio approach for setting classification relativities and the data below to determine the adjustment to class B's relativity after balancing to no overall rate change. Class A B Total A. < -10% Earned Premium $100 $200 $300 Incurred Loss $60 $90 $150 B. > -10% but < -8% Credibility 0.50 1.00 C. > -8% but < -6% D. > -6% but < -4% E. > -4% Exam 5, V1b Page 24 2014 by All 10, Inc. 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. Exam 5, V1b Page 25 2014 by All 10, Inc. 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. Exam 5, V1b Page 26 2014 by All 10, Inc. Chapter 9 – Traditional Risk Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2011 exam: 11. (2.75 points) Given the following information for State X: • Only two insurance companies write automobile policies • Total expected costs (including expenses) per policy are the same for 2010 and 2011 • All policies are annual policies effective January 1 • 10% of class 1 risks shop for new insurance every year • 20% of class 2 risks shop for new insurance every year • All insureds who shop always select the carrier with the lowest rate 2010 Policy Year Total Class Insureds 1 10,000 2 10,000 Total 20,000 # Insureds 5,000 5,000 10,000 Company A Expected 2010 Costs Rates 100 150 200 150 150 150 2011 Rates 100 200 Company B # Expected 2010 Insureds Costs Rates 5,000 100 150 5,000 200 150 10,000 150 150 2011 Rates 150 150 Company A will introduce a new rating variable effective January 1, 2011, that segments the market into two 2 classes. The 2011 rate levels will be consistent with the expected costs associated with each class of business. Company B will not be changing rates on January 1, 2011. Company B uses one rate level for all insureds. a. (1.5 points) Calculate the total profit for Company A and Company B for Policy Year 2011. b. (0.5 point) Company A's goals were to improve profitability and increase market share. Briefly explain whether the goals were achieved. c. (0.25 point) Provide one recommendation to Company A to help achieve its goals of improved profitability and increased market share. d. (0.5 point) Describe the impact of Company A's action on Company B. Exam 5, V1b Page 27 2014 by All 10, Inc. Chapter 9 – Traditional Risk Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2011 exam: 15. (3 points) Given the following information: Developed Incurred Loss and Earned ALAE Total for Accident Years Territory Exposures 2009 and 2010 A 20,000 $500,000 B 5,000 $125,000 C 15,000 $250,000 Total 40,000 $875,000 Current Relativity 1.00 0.95 1.25 • The effective date for the proposed rate change is January 1, 2012 and rates will be in effect for one year. • Average date of loss is January 1, 2010. • All policies are annual. • Full credibility standard 11,050 exposures On a statewide basis, annual pure premium trends have been holding steady at 0%. However, due to fraudulent claim behavior, pure premiums are expected to trend at different rates throughout the state as follows: Territory A B C Total Annual Pure Premium Trend -5% 0% 10% 0% This fraudulent behavior is expected to continue into the foreseeable future. a. (2.75 points) Assuming Territory A is the base territory, calculate the credibility-weighted indicated relativities to the base territory. b. (0.25 point) Briefly describe a reason multivariate classification techniques are preferred over univariate classification techniques when performing territorial relativity analyses. Exam 5, V1b Page 28 2014 by All 10, Inc. 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 Page 29 2014 by All 10, Inc. 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. Exam 5, V1b Page 30 2014 by All 10, Inc. 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.” Exam 5, V1b Page 31 2014 by All 10, Inc. 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. Exam 5, V1b Page 32 2014 by All 10, Inc. 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 Page 33 2014 by All 10, Inc. 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 Page 35 2014 by All 10, Inc. 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 Page 42 2014 by Alll 10, Inc. Chapter C 9 – Traditiional Risk Classiffication BASIC RATTEMAKING – WERNER, G G. AND MOD DLIN, C. Solution ns to questio ons from th he 2011 exa am (cont’d): Question n 15 – Model Solution 2 – part a. Note: the difference be etween modell solution 1 an nd model solu ution 2 lies in how the trended pure prem mium for all territories are calculate ed. In this solution it is calc culated as Su um [losses * p pp trend]/Sum m[exposures]= =22.161 Terr A B C Pure Prem 25 25 16.67 Pure Prem m Trend 0.953 1 1.13 Tre ended Pure P Prem 21.43 25 22.18 22.161 Ind R Rel 0.967 72 1.128 81 1.001 10 Trend period p is show wn below Terr A B C Terr A B C Credibility 1.00 0.673 1.00 Curr Rel 1.00 0.95 1.25 1.0875 Adj. Curr Rel R 0.9195 0.8739 1.1494 1.00 Cred w weight Rel. 0 0.9672 1 1.0448 1.001 2nd Rel @ Base 1.00 1.08 1.03 Question n 15 – Model Solution 2 – part b. Territories s are generallly heavily corrrelated with other o variabless. Multivariate e techniques take into acccount the effects s of other varriables, where eas univariate e techniques d do not. Exam 5, V1b Page 43 2014 by Alll 10, Inc. Chapter 9 – Traditional Risk Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2012 exam: 13. Use the loss ratio method to calculate indicated territorial relativities. Question 13 – Model Solution 1 (Exam 5A Question 13) First, calculate current premium for both territories. →Territory 1 = 250(1)(.75)(750) [prem for good drivers]+ 250(1)(1.00)(250) [prem for remaining drivers] = $203,125 →Territory 2 = 250 (1.1)(.75)(600) + 250 (1.1)(1.00)(150)= $165,000 Question 13 – Model Solution 2 (Exam 5A Question 13) Terr 1 2 Curr Var Prem 750 x 250 x 0.75 + 250 x 250 = 203,125 600 x 250 x 0.75 x 1.1 + 150 x 250 x 1.1 = 165,000 Indic Rd to Base Terr 1 2 OLEP 203,125 165,000 368,125 L+ALAE 90k + 40k = 130 80k + 20k = 100 230k LR 0.640 0.606 Indic Rd to 1 (base) 0.60606 x 1.1 = 1.04167 1.0417 0.64 Examiner’s Comments Candidates in general performed well on this question. Most frequently candidates failed to use current rate level premium, which in this question is calculated via the extension of exposures method. Candidates also frequently calculated only the indicated change factors to the current relativities, as opposed to calculating the final indicated relativity. A subset of candidates misinterpreted the class plan and used the loss ratio method to solve for 4 different relativities concurrently (each combination of territory/good driver), as opposed to solving for the requested indicated territorial relativities. A small group of candidates solved for indicated territory relativities by using a pure premium approach as opposed to the requested loss ratio approach. Some candidates made adjustments to the exposure bases to reflect the class plan relativities. Exam 5, V1b Page 44 2014 by All 10, Inc. Chapter 10 – Multivariate Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Sec 1 2 3 4 5 6 7 8 9 10 11 12 1 Description Shortcomings Of Univariate Methods Minimum Bias Procedures The Adoption of Multivariate Methods The Benefits Of Multivariate Methods GLM’S Sample GLM Output A Sample Of GLM Diagnostics Practical Considerations Data Mining Techniques Augmenting Multivariate Analysis With External Data Key Concepts Appendix F – A Multivariate Classification Example Pages 170 - 171 171 - 174 174 -174 174 -175 176 - 177 177 - 179 179 - 182 183 - 183 183 - 185 185 - 185 187 - 187 Shortcomings Of Univariate Methods 170 - 171 Class ratemaking: produces more equitable individual risk pricing by analyzing loss experience of groups of similar risks. protects the insurer against adverse selection. may provide insurers with a competitive advantage and help expand the types of risks the insurer is willing and able to write profitably. Univariate class ratemaking approaches (pure premium or loss ratio) use loss experience of the levels within each rating variable to establish rate differentials to the base level. The major shortcoming of univariate approaches: Its failure to accurately account for the effect of other rating variables. The PP approach does not consider exposure correlations with other rating variables. If a rating algorithm contained several rating variables, this shortcoming could be mitigated using a two-way analysis or by making some manual adjustments. To illustrate the distortion created when using univariate methods, consider the following: Assume a one-way PP analysis for a personal auto book of business shows that older cars have high claims experience relative to newer cars. However, in reality it can be shown that this analysis is distorted by the fact that older cars tend to be driven by younger drivers (who have higher claims experience). Therefore, although the experience for both young drivers and old cars looks unfavorable, it does so primarily because of the youthful driver effect. The LR approach uses current premium to adjust for an uneven mix of business to the extent the premium varies with risk, but premium is only an approximation since it deviates from true loss cost differentials. The adjusted pure premium approach multiples exposures by the exposure-weighted average of all other rating variables’ relativities to standardize data for the uneven mix of business before calculating the oneway relativities. But, this is an approximation to reflect all exposure correlations. Exam 5, V1b Page 45 2014 by All 10, Inc. Chapter 10 – Multivariate Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 2 Minimum Bias Procedures 171 - 174 Minimum bias procedures are iterative univariate approaches. Each procedure involves the: selection of a rating structure (e.g. additive, multiplicative or combined) and selection of a bias function (e.g. balance principle, least squares, , and maximum likelihood bias functions). The bias function compares the procedure’s observed loss statistics (e.g. loss costs) to indicated loss statistics and measures the mismatch. Both sides of this equation are weighted by the exposures in each cell to adjust for an uneven mix of business. “Minimum bias” refers to the balance principle that requires that the sum of the indicated weighted pure premiums to equal the sum of the weighted observed loss costs for every level of every rating variable (a.k.a. “minimizing the bias” along the dimensions of the class system). 2 The balance principle applied to a multiplicative personal auto rating structure is shown below. There are only two rating variables: gender and territory. Gender has values male (with a rate relativity g1) and female (g2). Territory has values urban (t1) and rural (t2). The base levels relative to multiplicative indications are female and rural (hence g2 = 1.00 and t2 = 1.00). The base rate is $100. The actual loss costs (pure premiums) are as follows: Urban Rural Total Male $650 $300 $528 Female $250 $240 $244 Total $497 $267 $400 The exposure distribution is as follows: Urban Rural Male 170 90 Female 105 110 Total 275 200 Total 260 215 475 Step 1: Write four equations with observed weighted loss costs on the left and indicated weighted loss costs (the base rate, the exposure, and the indicated relativities) on the right. Males 170 x $650 +90 x $300 = ($100 x 170 xg1 x t1 ) +( $100 x 90 x g1 x t2 ) Females 105 x $250 + 110 x $240 = $100 x 105 x g2 x t1 + $100 x 110 x g2 x t2 Urban 170 x $650+ 105 x $250 = $100 x 170 x g1 x t1 + $100 x 105 x g2 x t1 Rural 90 x $300+ 110 x $240 = $100 x 90 x g1 x xt2 + $100 x 110 x g2 x t2 Step 2: Choose initial (or seed) relativities for the levels of one of the rating variables. A sensible seed is the univariate PP relativities. The urban relativity is the total urban loss costs divided by the total rural loss costs: t1 = 1.86 = ($497.27/$267.00) t2 = 1.00. Exam 5, V1b Page 46 2014 by All 10, Inc. Chapter 10 – Multivariate Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Step 3: Substituting these seed values into the first two equations, solve for the first values of g1 and g2: 170 x $650 + 90 x $300 = ($100 x 170 x g1 x 1.86) + ($100 x 90 x g1 x 1.00) $137,500 = ($31,620 x g1) + ($9,000 x g1) $137,500 = $40,620 x g1 g1 = 3.39. 105 x $250 + 110 x $240 = ($100 x 105 x g2 x 1.86) + ($100 x 110 x g2 x 1.00) $52,650 = ($19,530 x g2) + ($11,000 x g2) $52,650 = $30,530 x g2 g2 = 1.72. Step 4: Using these seed values for gender, g1 and g2, set up equations to solve for the new intermediate values of t1 and t2: 170 x $650 + 105 x $250 = ($100 x 170 x 3.39 x t1) + ($100 x 105 x 1.72 x t1) $136,750 = ($57,630 x t1) + (18,060 x t1) $136,750 = $75,690 x t1 t1 = 1.81. 90 x $300 + 110 x $240 = ($100 x 90 x 3.39 x t2) + ($100 x 110 x 1.72 x t2) $53,400 = ($30,510 x t2) + ($18,920 x t2) $53,400 =$49,430 x t2 t2 = 1.08. This procedure is repeated (each time discarding the previous relativities and solving for new ones) until there is no material change in the values of g1, g2, t1, and t2. Step 5: Upon convergence, normalize the base class relativities to 1.00. Assuming the relativities derived above represent the final iteration, then normalizing the base class relativities to 1.00 would result in: g1 = 3.39 / 1.72 = 1.97 g2 = 1.72 /1.72 = 1.00 t1 = 1.81 /1.08 = 1.68 t2 = 1.08 /1.08 = 1.00. The initial univariate relativity for t1 was 1.86, but after one iteration, the replacement value for t1 is 1.68, (reflecting the fact that the cell for urban males has considerably more exposure than the other cells, and thus the experience in that cell is given more weight). Step 6: Adjust the base loss cost (to a normalized basis): Since the base levels are female and rural (g2 and t2), and since the base loss cost = $100, then the Adjusted base loss cost = $100 x 1.72 x 1.08 = $185.76. Exam 5, V1b Page 47 2014 by All 10, Inc. Chapter 10 – Multivariate Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. The example above only considers one minimum bias method (multiplicative structure with balance principle) using the pure premium statistic. In addition, it considers only two rating variables each with two levels. The computation required to incorporate several rating variables requires at least spreadsheet programming. Sequential analysis: is related to minimum bias analysis. is mandated as the only class ratemaking method for pricing private passenger auto insurance CA. uses an adjusted one-way PP approach on the first variable to determine the indicated relativities. exposures are adjusted using the adjusted one-way PP approach and indicated relativities are calculated for the second variable; this continues until indicated relativities for every variable have been calculated. involves making only one pass through the sequence of chosen rating variables (rather than iterating until convergence is achieved). The main criticism of the non-iterative sequential approach: since it does not have a closed form solution; the results vary depending on the order of the rating variables in the sequence. 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). Exam 5, V1b Page 48 2014 by All 10, Inc. Chapter 10 – Multivariate Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 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). Exam 5, V1b Page 49 2014 by All 10, Inc. Chapter 10 – Multivariate Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. GLMs are the standard for class ratemaking. The iterations of a GLM can be tracked, and the output is a series of multipliers that can be used in rating algorithms and rating manuals. A Mathematical Foundation for GLMs: Linear Models A good way to understand GLMs is to first review linear models (LMs). Both LMs and GLMs express the relationship between an observed response variable (Y) and a number of explanatory variables (a.k.a. predictor variables). Example: The response variable may be claim frequency for homeowners insurance, and the predictor variables may include AOI, age of home, and deductible. Observations in the data (e.g. claims on individual exposures) are realizations of the response variable. Linear models: express the response variable (Y) as the sum of its mean (µ) and a random variable ( ) (a.k.a. error term): Y assume that the mean can be written as a linear combination of the predictor variables. Example: Y ( 1 X 1 2 X 2 3 X 3 4 X 4 ) where X 1 , X 2 , X 3 , and X 4 are each predictor variables, and 1 , 2 , 3 , and 4 are the parameter estimates to be derived by the LM. 2. assume that the random variable, attempt to find the parameter estimates, which, when applied to the chosen model form, produce the observed data with the highest probability. This is achieved using the likelihood function (or the log-likelihood), as maximum likelihood relies on linear algebra to solve a system of equations. Due to the high volume of observations in class ratemaking datasets, numerical techniques such as multi-dimensional Newton-Raphson algorithms are used. These techniques find the maximum of a function by finding a zero in the function’s first derivative. The likelihood function is equivalent to minimizing the sum of squared error between actual and indicated. , is normally distributed with a mean of 0 and constant variance, Generalized Linear Models: Loosening the Restrictions GLMs: are LMs that remove the restrictions of the normality assumption and a constant variance. use a link function to define the relationship between the expected response variable (e.g. claim severity) and the linear combination of the predictor variables (e.g. age of home, amount of insurance, etc.). Choice of link functions means predictor variables do not have to relate strictly in an additive fashion (as they do with LMs). Example: GLMs fit to claims experience for ratemaking often specify a log link function which assumes the rating variables relate multiplicatively to one another. Exam 5, V1b Page 50 2014 by All 10, Inc. Chapter 10 – Multivariate Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. To solve a GLM, the modeler must: • have a dataset with a sufficient number of observations of the response variable and associated predictor variables. • select a link function defining the relationship between the systematic and random components. • specify the distribution of the underlying random process (e.g. a member of the exponential family such as normal, Poisson, gamma, binomial, inverse Gaussian); this is done by specifying the mean and the variance of the distribution, the latter being a function of the mean. The maximum likelihood approach: maximizes the logarithm of the likelihood function and computes the predicted values for each variable. 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. Exam 5, V1b Page 51 2014 by All 10, Inc. Chapter 10 – Multivariate Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Effect of Vehicle Symbol on Automobile Collision Frequency The output is from a multiplicative model. The base level (to which all other levels’ parameter estimates are expressed relatively) is vehicle symbol 4. Its multiplicative differential is 1.00, and is chosen as one with the largest volume of exposure (so that statistical diagnostics are relative to a large and stable base). Notice that the GLM indicates that vehicle symbol 10 has a 25% higher indicated collision frequency than vehicle symbol 4, all other variables being considered. The pink line with square markers represents the results of a univariate analysis. The disparity b/t the GLM and univariate lines suggest vehicle symbol is strongly correlated with another variable in the model (e.g. age of driver, prior accident experience, etc). It is important to understand the phrase “all other variables being considered.” GLM results of one variable are only meaningful if the results for all other variables are considered at the same time (a.k.a. “all other variables being constant” or “all other variables at the base level.”) Chapter 13 discusses how the insurer’s final rate relativities often deviate from the actuary’s indicated relativities for business reasons. Exam 5, V1b Page 52 2014 by All 10, Inc. Chapter 10 – Multivariate Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 7 A Sample Of GLM Diagnostics 179 - 182 Statistical significance is an important criterion for evaluating rating variables, and statistical diagnostics are a major byproduct of GLMs. Statistical diagnostics: aid the modeler in understanding the certainty of the results and the appropriateness of the model. can determine if a predictive variable has a systematic effect on losses (and be retained in the model). assess the modeler’s assumptions around the link function and error term. A common statistical diagnostic for deciding whether a variable has a systematic effect on losses is the standard errors calculation. “standard errors are an indicator of the speed with which the log-likelihood falls from the maximum given a change in parameter.” 2 standard errors from the parameter estimates are akin to a 95% confidence interval. i. the GLM parameter estimate is a point estimate ii. standard errors show the range in which the modeler can be 95% confident the true answer lies within. The following graph is identical to the graph shown previously but now includes standard error lines for the non-base levels (i.e., +/- two standard errors from the differentials indicated by the GLM). Standard Errors for Effect of Vehicle Symbol on Automobile Collision Frequency Results: The upward pattern and narrow standard errors suggest this variable is statistically significant. Wide standard errors may suggest the factor is detecting mostly noise and be eliminated from the model. Symbol 17 shows wide standard errors, but that is a function of the small volume present in that level (and thus does not invalidate the strong results for symbols 1- 16, where most of the business lies). Deviance measures (an additional diagnostic) assess the statistical significance of a predictor variable. Deviance measures of how much fitted values differ from the observations. Deviance tests are used when comparing nested models (one is a subset of the other) to assess whether the additional variable(s) in the broader model are worth including. i. The deviance of each model is scaled so that the results can be compared. ii. Chi-Square or F-test gauge the theoretical trade-off between the gain in accuracy by adding the variables versus the loss of parsimony in adding more parameter estimates to be solved. Exam 5, V1b Page 53 2014 by All 10, Inc. Chapter 10 – Multivariate Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. A practical diagnostic in modeling is to compare GLM results for individual years to gauge consistency of results from one year to the next. Consistency over time of vehicle symbol on auto collision frequency separately for the two years The two lines show some random differences but in general the patterns are the same. Model validation techniques compare the expected outcome with historical results on a hold-out sample of data (i.e. data not used in developing the model so that it could be used to test the effectiveness of the model). The following output is a validation of a frequency model. The bands of expected frequencies from the GLM (from lowest to highest) track closely to the actual weighted frequency of each band in the hold-out sample of data (for most of the sample) The volatile results for the high expected frequency bands are a result of low volume of data. Exam 5, V1b Page 54 2014 by All 10, Inc. Chapter 10 – Multivariate Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Model Validation Over-fitting and Under-fitting Models: If the modeler retains variables that reflect a non-systematic effect on the response variable (i.e. noise) or over-specifies the model with high order polynomials, the result is over-fitting. The model will replicate historical data very well (including the noise) but will not predict future outcomes reliably (the future experience will not have the same noise). If the model is missing important statistical effects (containing few explanatory variables and fits to the overall mean), the result is under-fitting. This model will hardly help the modeler explain what is driving the result. See Appendix F includes for additional examples and more details. Exam 5, V1b Page 55 2014 by All 10, Inc. Chapter 10 – Multivariate Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. When using GLMs, the actuary should focus on: • ensuring data is adequate for the level of detail of the class ratemaking analysis (avoiding the GIGO principle: Garbage In, Garbage Out) • identifying when anomalous results call for additional exploratory analysis • reviewing model results as it relates to both statistical theory and business application • developing methods to communicate model results in light of an insurer’s ratemaking objectives (e.g. policyholder dislocation, competitive position) More work can be done. Retrieving of data requires careful consideration of needed volume of data; definition of homogeneous claim types; method of organization (e.g. PY vs. CAY); treatment of midterm policy changes, large losses, U/W changes during the experience period, and the effect of inflation and loss development. Balance stability and responsiveness as it relates to experience period as well as to geographies to be included in the analysis (e.g. countrywide versus individual state analysis). Commercial considerations (e.g. IT constraints, marketing objectives, and regulatory requirements) have to be carefully incorporated into the statistical analysis before any results are implemented in practice. 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.) Exam 5, V1b Page 56 2014 by All 10, Inc. Chapter 10 – Multivariate Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 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. Exam 5, V1b Page 57 2014 by All 10, Inc. Chapter 10 – Multivariate Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 10 Augmenting Multivariate Analysis With External Data 185 - 185 Insurers using GLMs seek to augment data that has already been collected and analyzed about their own policies with external data. This includes but is not limited to information about: • geo-demographics (e.g. population density of an area, average length of home ownership of an area); • weather (e.g. average rainfall or number of days below freezing of a given area); • property characteristics (e.g. square footage of a home or business, quality of the responding fire department); • information about insured individuals or business (e.g. credit information, occupation). 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 Exam 5, V1b Page 58 2014 by All 10, Inc. Chapter 10 – Multivariate Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 12 Appendix F – A Multivariate Classification Example The appendix includes output from a GLM analysis. It includes: several tests used to evaluate the predictive power of a potential rating variable hold-out sample testing used to evaluate the overall effectiveness of a particular model. EXAMPLE - PREDICTIVE VARIABLE (a multiplicative GLM fit to homeowners water damage frequency data) The graphical output isolates the effect of the prior claim history variable as a significant predictor of water damage frequency, however The model contains other explanatory variables that must be considered in conjunction with the prior claims history effect. Parameters and Standard Errors The graph shows indicated frequency relativities for prior claims history (all other variables considered). The x-axis represents the levels of the variable (0, 1, or 2 claims), with the level for zero claims being the base level, and all other levels expressed relative to it. The bars relate to the right y-axis, which show the number of policies in each level. The line with the circle marker shows the indicated relativities, and the lines with the triangle markers represent two standard errors on either side of the indicated relativities. Main Effect Test for Prior Claim History Conclusions: The upward sloping indicated relativity line with relatively tight standard errors suggests that the expected frequency is higher for risks with prior claims. Risks with 1 or 2 prior claims have a frequency about 35% and 65% higher than risks with no prior claims. Exam 5, V1b Page 59 2014 by All 10, Inc. Chapter 10 – Multivariate Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Consistency Test The prior graph shows the indicated relativities for the whole dataset. The following shows the pattern of relativities for each of the individual years included in the analysis. The lines represent the indicated frequency relativities for prior claims history, separately for each year. Consistency Test for Prior Claim History Each year’s indicated line slopes upward with roughly the same shape suggests that the pattern is consistent over time, and provides the actuary with a test supporting the stability of this variable’s predictive power. Statistical Test The actuary can test the predictive power of a variable using deviance diagnostics Using the Chi-Square test, the actuary fits models with and without the variable being studied and analyzes the trade-off between the increased accuracy of the model with the variable versus the additional complexity in having additional parameters to estimate. The null hypothesis is that the two models are approximately the same. Calculate a Chi-Square percentage based on the results of the two models (a percentage of less than 5% suggests the actuary should reject the null hypothesis that the models are the same and should use the model with the greater number of parameters). Here, the Chi-Square percentage is 0%; the actuary rejects the null hypothesis and selects the model with the greater number of parameters (e.g. select the model with prior claims history variable in it). Exam 5, V1b Page 60 2014 by All 10, Inc. Chapter 10 – Multivariate Classification 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. Exam 5, V1b Page 61 2014 by All 10, Inc. Chapter 10 – Multivariate Classification 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). Exam 5, V1b Page 62 2014 by All 10, Inc. 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). Exam 5, V1b Page 63 2014 by All 10, Inc. Chapter 10 – Multivariate Classification 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). Exam 5, V1b Page 64 2014 by All 10, Inc. 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. Exam 5, V1b Page 65 2014 by All 10, Inc. 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. . Exam 5, V1b Page 66 2014 by Alll 10, Inc. 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. Exam 5, V1b Page 67 2014 by All 10, Inc. 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. Exam 5, V1b Page 68 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Next: Estimate the geographic risk associated with each unit. Actual experience contains both signal and random noise. The signal is driven by non-geographic elements (e.g. age, amount of insurance, number of employees) and geographic elements (e.g. density, weather indices, crime rates). The key to accurately estimating the geographic risk is isolating the geographic signal in the data. Components of Actual Experience Step 2: Calculating the Geographic Estimator Historically, actuaries used univariate techniques (e.g. pure premium approach) to develop an estimator for each geographic unit. Two major issues with this approach. 1. The geographic estimator reflects both the signal and the noise. Since geographic units tend to be small, the data is sparse and the resulting loss ratios or pure premiums or both will be too volatile to distinguish the noise from the signal. 2. Since location is highly correlated with other non-geographic factors, the resulting estimator is biased. A better approach involves using a multivariate model (e.g. a GLM) on loss cost data using a variety of nongeographic and geographic explanatory variables. Non-geographic variables include rating variables (e.g. age of insured, claim history) as well as other explanatory variables not used in rating. Geographic variables include geo-demographic variables (e.g. population density) and geo-physical variables (e.g. average rainfall). Components of Actual Experience Further Refined By including geographic and non-geographic predictors in the Signal model, the actuary controls for nongeographic effects and isolates the signal stemming from the geographic predictors. If the actuary cannot fully explain the geographic effect via the geographic predictors, there will be some systematic variation not captured by the geographic variables (a.k.a. geographic residual variation). The parameters from each geographic predictor, including the geographic residual variation, can be combined to form one composite risk index or score that represents the geographic signal for each unit. Exam 5, V1b Page 69 2014 by All 10, Inc. 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. Exam 5, V1b Page 70 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Step 4: Clustering Units are grouped into territories to minimize within group heterogeneity and maximize between group heterogeneity. Basic types of clustering routines include: • Quantile methods: Create clusters based on equal numbers of observations (e.g. geographic units) or equal weights (e.g. exposure). • Similarity methods: Create clusters based on how close the estimators are to one another. Closeness can be based on a different statistics: i. The average linkage similarity method creates boundaries based on the overall average difference between the estimators from one cluster to the next (tends to join clusters with smaller variances). ii. The centroid similarity method creates boundaries based on the overall average difference in estimators squared (tends to be more responsive to outliers). iii. Ward’s clustering method creates boundaries that lead to the smallest within cluster sum of squares difference (tends to produce clusters that have the same number of observations). These types of clustering routines do not produce contiguous groupings (i.e. groupings that only include geographic units that are adjacent to each other). If contiguous territorial boundaries are desired, then a contiguity constraint needs to be added to the clustering routine. Since geographic risk changes gradually, a discontinuity at self created boundaries will occur. Thus, the actuary should select the number of clusters that minimizes noise without creating significant discontinuities. Many insurers have eliminated grouping units into territories and simply derive rate relativities for each geographic unit (i.e. no different than creating a large number of small territories). Rather than rating territories, insurers can geo-code every risk, and the latitude and longitude of the insured item creates a unique rate relativity that changes gradually from one location to the next. II. Calculating Territorial Relativities Rate relativities or differentials can be accomplished using the techniques described in chapters 9 and 10. Since location tends to be highly correlated with other variables (e.g. low or high-valued homes tend to be concentrated in certain areas), perform this analysis using multivariate classification techniques (e.g. a new territorial boundary could be modeled along with other explanatory variables in a GLM). 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. Exam 5, V1b Page 71 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Lines of business in which IL ratemaking is used include private passenger and commercial auto liability, umbrella, any commercial product offering liability coverage (e.g. contractor’s liability, professional liability, etc). Two types of policy limits offered: 1. Single limits: Refers to the total amount the insurer will pay for a single claim (e.g. if an umbrella policy has a limit of $1,000,000, then the policy will only pay up to $1,000,000 for any one claim). 2. Compound limits: Applies two or more limits to the covered losses. Examples: i. A split limit: includes a per claimant and a per occurrence limit (e.g. in personal auto insurance, a split limit for bodily injury liability of $15,000/$30,000 means that if the insured causes an accident, the policy will pay each injured party up to $15,000 with total payment to all injured parties not to exceed $30,000). ii. An occurrence/aggregate limit: limits the amount payable for any one occurrence and for all occurrences incurred during the policy period (e.g. if an annual professional liability policy has a limit of $1,000,000/$3,000,000, the policy will not pay more than $1,000,000 for any single occurrence and will not pay more than $3,000,000 for all occurrences incurred during the policy period). The text will focus determining indicated increased limit factors (ILFs) for a single limit (compound and split limits are more complex). Standard Approach to Computing LAS and ILFs The ILF is used to modify the base rate (B, which assumes the basic limit) if the insured selects a limit of liability (H) that is different than the basic limit: Rate at Limit H = ILF for Limit H x B. Assuming all UW expenses are variable and variable expense and profit provisions do not vary by limit, the _________ Indicated ILF ( H ) ( L EL ) H _____________ (derived in the same way as Chapter 9). ( L EL ) B Actuaries may vary the profit provision by limit: because higher limits offer coverage for claims that are less frequent and very severe, and this variability adds uncertainty which makes it difficult to price and risky for insurers. to reflect the higher cost of capital needed to support the additional risk. Assume frequency and severity are independent: Indicated ILF ( H ) FrequencyH x SeverityH FrequencyB x SeverityB Assume frequency is the same regardless of the limit chosen: Indicated ILF ( H ) SeverityH SeverityB For some lines of business, frequency may vary by the limit chosen. Personal auto insureds who select a very high limit tend to have lower accident frequencies than insureds who select low limits. Selecting higher limit tends to be a sign of risk aversion and a higher degree of overall responsibility that also applies to driving behavior. A severity limited at H is referred to as the limited average severity at H or LAS (H). Indicated ILF ( H ) LAS ( H ) LAS ( B) LAS (H) is the severity assuming every loss is capped at limit H (regardless of actual policy limit), and LAS (B) is the severity assuming every loss is capped at the basic limit. Exam 5, V1b Page 72 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Example: Given the following 5,000 reported uncensored claims categorized by the size of the loss (i.e. a $150,000 loss is slotted in the $100,000100K. 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 = Exam 5, V1b Page 75 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Other Considerations Historical losses used in ILF analysis should be adjusted for any expected trend and for loss development. Recall that loss trends have a leveraged effect on increased limits losses. Assuming a constant positive percentage trend in total losses, the following relationship holds: Basic Limits Trend < Total Limits Trend < Increased Limits Trend. (See Chapter 6 for a numeric example that demonstrates this relationship). Fitted Data Approach Actuaries may fit curves to empirical data to smooth out the random fluctuations in the data. Let f(x) represent a continuous distribution of losses of size x, and H be the limit being priced. H 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). Exam 5, V1b Page 76 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Several reasons why deductibles are popular among both insureds and insurers: • Premium reduction: A deductible reduces the rate as the insured pays a portion of the losses. • Eliminates small nuisance claims: Deductibles minimize the filing of small claims (and the expense associated with investigating and handling small claims, which is often greater than the claim amount). • Provides incentive for loss control: Since the insured is responsible for the first layer of loss, the insured has a financial incentive to avoid losses. • Controls catastrophic exposure: For insurers writing a large number of policies in cat prone areas, the use of large cat deductibles can reduce its exposure to loss. Loss Elimination Ratio (LER) Approach Deductible relativities can be determined using a LER approach. Assuming all expenses are variable and that variable expenses and profit are a constant % of premium, the indicated deductible relativity for deductible D is given by the following formula (where the base level in this _________ example assumes no deductible): Indicated Deductible Relativity ( L EL ) D _________ ( L EL ) B The indicated deductible relativity is the ratio of ultimate losses and LAE after application of the deductible to ground-up ultimate losses and LAE. In the LER approach, calculate the amount of losses that are eliminated going from full coverage to a deductible or by going from one deductible to a higher deductible: LER ( D ) Losses and LAE Eliminated by Deductible ( L EL ) B ( L EL ) D Total Ground - up Losses and LAE ( L EL ) B The formula is re-written as follows: ( L EL ) D ( L EL ) B (1.0 - LER ( D )). The indicated deductible relativity can be restated as: _______ Indicated Deductible Relativity ( L EL ) B x(1.0 - LER( D)) _________ (1.0 - LER( D)). ( L EL ) B Empirical Distribution (Discrete Case) The LER can be calculated as follows: LER ( D ) [1 AllLosses Maximum[0, ( Loss Amount D )] Loss Amount ] AllLosses (assuming the ground-up loss is known for every claim) Exam 5, V1b Page 77 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Consider the size of loss distribution of ground-up homeowners losses: Size of Loss Distribution (1) Size of Loss X <= $ 100 $ 100 < X <= $ 250 $ 250< X <= $ 500 $ 500 < X <= $ 1,000 $ 1,000 < X Total (2) Reported Claims 3,200 1,225 1,187 1,845 2,543 10,000 (3) Ground-Up Reported Losses $240,365 $207,588 $463,954 $1,551,938 $11,140,545 $13,604,390 To calculate LER ($250), compute the amount of losses in each layer that will be eliminated by the deductible. The first two rows contain losses less than $250 and are completely eliminated by the deductible. The remaining rows contain individual losses that are at least $250; thus $250 will be eliminated for each of the 5,575 claims (=1,187+1,845+2,543). The LER = losses eliminated/ total losses: LER($250) ($240,365 $207,588) $250 (1,187 1,845 2,543) 0.135 $13, 604,390 The rate credit for going from full coverage to a $250 deductible is 13.5%; the deductible relativity is 0.865. The following table shows the calculations discussed above: (1) (2) Size of Loss X <= $ 100 $ 100 < X <= $ 250 $ 250< X <= $ 500 $ 500 < X <= $ 1,000 $ 1,000 < X Total (4) Losses < 250 (4) Losses>=250 (5) LER 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) Page 78 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Other Considerations Insurers may not know the ground-up losses for every claim (e.g. insureds may not report claims that are less than the deductible on their policy). When this is the case, data from policies with deductibles greater than the deductible being priced cannot be used to calculate the LER. For example: data from policies with a $500 deductible cannot be used to determine LERs for a $250 or $100 deductible, however data from policies with deductibles less than the deductible being priced can be used to determine LERs (e.g. data from policies with a $500 deductible can be used to determine the LER associated with moving from a $750 deductible to a $1,000 deductible). Calculating the credit to change from a $250 to a $500 deductible. LER Calculation to Move from a $250 to $500 Deductible (1) (2) (3) (4) (5) Net Reported Net Reported Losses Losses Reported Net Reported Assuming Assuming Deductible Claims Losses $500 Ded $250 Ded Full Cov 500 $680,220 $524,924 $588,134 $100 680 $1,268,403 $1,049,848 $1,176,269 $250 1,394 $2,940,672 $2,624,621 $2,940,672 $500 2,194 $5,249,242 $5,249,242 Unknown Unknown Unknown $1,000 254 $859,755 Total 5022 $10,998,292 (7) Net Reported Losses for Ded <=$250 (8) Losses Eliminated <=$250 Ded (9)LER (3)= Net of the deductible (4) =(3) Adjusted to a $500 deductible (6)= (5) - (4) (7)= Sum of (5) for $0, $100, $250 Deductibles (8)=Sum of (6) for $0, $100, $250 Deductibles (9)=(8)/(7) (6) Losses Eliminated Moving from $250 to $500 $63,210 $126,421 $316,051 Unknown Unknown $4,705,075 $505,682 0.107 (5)=(3) Adjusted to a $250 deductible Each row contains data for policies with different deductible amounts. The analysis can only use policies with deductibles of $250 or less (since the goal is to determine the losses eliminated when changing from a $250 to a $500 deductible) Columns 4 and 5 contain the net reported losses in Column 3 restated to $500 and $250 deductible levels, respectively. Columns 4 and 5 are not Column 3 minus the product of Column 2 and the assumed deductible. This is because not every reported loss exceeds the assumed deductible. The losses in Columns 4 and 5 are based on an assumed distribution of losses by deductible and size of loss, and cannot be recreated given the data shown. The comments made earlier with respect to trend and development in the ILF section apply to deductible pricing, too. Exam 5, V1b Page 79 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Fitted Data Approach Let f(x) represent a continuous distribution of losses of size x, and D be the size of the deductible. This formula is very similar to the formula used in the ILF section The expected loss eliminated through the use of a deductible, D: LER( D) 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. Exam 5, V1b Page 80 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 4 Size Of Risk For Workers Compensation 204 - 206 To account for differences in expense and loss levels for larger insureds, some WC insurers vary the expense component for large risks, incorporate premium discounts or loss constants, or all of these. Expense Component Commercial lines insurers use the All Variable Approach to determine the applicable expense provisions. The assumption is that UW expenses are a constant % of the premium charged. Since some expenses are fixed, using the all variable approach will cause policies with small average premium (i.e. small risks) to be undercharged and policies with large average premium (i.e. large risks) to be overcharged. Insurers may adjust for this in a few different ways. 1. WC insurers may calculate a variable expense provision that only applies to the first $5,000 of standard premium (generally defined as premium before application of premium discounts and expense constants). 2. Insurers may charge an expense constant to all risks, which accounts for costs that are the same regardless of policy size (e.g. UW and administrative expenses). Since the expense constant is a flat dollar amount, it is a decreasing % of written premium as the size of the policy increases. 3. WC insurers apply a premium discount to policies with premium above a specified amount. The following shows the calculation of the premium discount for a policy with standard premium of $400,000. Workers Compensation Premium Discount Example (1) (2) (3) (4) (5) Premium Premium Range in Range Prod General $0 $5,000 $5,000 15.0% 10.0% $5,000 $100,000 $95,000 12.0% 8.0% $100,000 $500,000 $300,000 9.0% 6.0% $500,000 above 6.0% 4.0% Standard Premium $400,000 (6) (7) (8) Taxes 3.0% 3.0% 3.0% 3.0% Profit 5.0% 5.0% 5.0% 5.0% Total 33.0% 28.0% 23.0% 18.0% (3)= Min of [(2) - (1), Standard Premium - Sum Prior(3)] (9)= (8Row 1)-(8) (10)= (9)/[1.0 -(6) - (7)] (9) Expense Reduction 0.0% 5.0% 10.0% 15.0% (10) Discount % 0.0% 5.4% 10.9% 16.3% (11)= (11) Premium Discount $0 $5,130 $32,700 $0 $37,830 (3) x (10) Loss Constants Small WC risks tend to have less favorable loss experience (as a % of premium) than large risks for several reasons. Small companies: have less sophisticated safety programs because of the large amount of capital to implement and maintain. may lack programs to help injured workers return to work. premiums are unaffected or slightly impacted by experience rating; small insureds may not be eligible for ER and may have less incentive to prevent or control injuries than large insureds. Exam 5, V1b Page 81 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. When WC insurers charge the same rate per exposure for small and large insureds, the premium will be inadequate for small companies and excessive for large companies. A loss constant added to the premium for small risks equalizes the final expected loss ratios between small and large insureds. WC Loss Constant calculation example: (1) (2) Premium Range $1 $2,500 $2,501 above (3) (4) Policies 1,000 1,000 (6)= (5) / (4) (7) = Given (5) Reported Premium Loss $1,000,000 $750,000 $5,000,000 $3,500,000 (6) Initial Loss Ratio 75.0% 70.0% (7) Target Loss Ratio 70.0% 70.0% (8) (9) Premium Shortfall $71,429 $0 Loss Constant $71.43 $0.00 (8)= [(5)/(7)] -(4)(9)= (8) / (3) The unadjusted expected loss ratios for small (premium less than or equal to $2,500) and large (premium greater than $2,500) risks are 75% and 70% (see (6)) To achieve an expected loss ratio of 70% for both types of risks, the computations in (8) and (9) are performed. With sophisticated multivariate techniques, insurers add a rating variable to account for the size of the risk, making the loss constant no longer necessary. Exam 5, V1b Page 82 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 5 Insurance To Value (ITV) 206 - 213 Insurance to value (ITV) indicates how the level of insurance chosen relates to the overall value or replacement cost of the insured item, and thus how rates vary based on the policy limit chosen (e.g. if an item is insured to full value, then the AOI equals the total value or replacement cost). Consider the following example: • Two homes worth $250,000 and $200,000 are each insured for the full amount. • Expected claim frequency is assumed to be 1% for both homes. • Expected losses are uniformly distributed. That information yields the following expected size of loss distributions and rates for each home: Rate calculations for a $250,000 Home (1) (2) Reported Loss Loss Size of Loss ($000s) Distribution ($000s) $ - < X <= $ 25 10.0% $13 $ 25 < X <= $50 10.0% $38 $ 50 < X <= $75 10.0% $63 $ 75 < X <= $100 10.0% $88 $ 100 < X <= $125 10.0% $113 $ 125 < X <= $150 10.0% $138 $ 150 0). 2. The face amount of insurance is less than the coinsurance requirement (i.e. F < cV). 3. The loss is less than the coinsurance requirement (i.e. L < cV). The amount of the penalty is as follows: L - I , if L F e = F - I , if F L cV 0, if cV L Example 1: Assume a home valued at $500,000 is insured only for $300,000 despite a coinsurance requirement of 80% (or $400,000 in this case). Since F is $300,000 a coinsurance deficiency exists and a = 0.75 (=$300,000 / $400,000). The indemnity payments and coinsurance penalties for a $200,000 loss are: F $300, 000 $200, 000 $150, 000 cV $400, 000 e L - I $200, 000 - $150, 000 $50, 000 IL Exam 5, V1b Page 85 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Example 2: The indemnity payments and coinsurance penalties for a $300,000 loss: F $300, 000 $300, 000 $225,000 cV $400, 000 e L - I $300, 000 - $225,000 $75, 000 I L Example 3: The following are the indemnity payments and coinsurance penalties for a $350,000 loss: $300, 000 F $350, 000 $262,500 $400, 000 cV e F - I $300, 000 - $262,500 $37,500 I L Example 4: The following are the indemnity payments and coinsurance penalties for a $450,000 loss: F $300, 000 $450,000 $337,500, but $337,500 F , so I F $300, 000 cV $400, 000 e F I $300,000 - $300, 000 $0. I L The coinsurance penalty for loss values between $0 and $500,000 (i.e. the full value of the home): The magnitude of the co-insurance penalty: the dollar coinsurance penalty increases linearly between $0 and F (where the penalty is the largest). the penalty decreases for loss sizes between F and cV. there is no penalty for losses larger than the cV, but the insured suffers a penalty in that the payment does not cover the total loss. Exam 5, V1b Page 86 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Achieving Rate Equity by Varying Rates Based on ITV Level I. A coinsurance penalty corrects for inequity caused by similar homes insured to different ITV levels by adjusting the indemnity payment in the event of a loss. II. Another way to achieve equity is to calculate and use rates based on the level of ITV. Recall that the indicated rate per $1,000 of insurance was the same for the two homes insured to full value (i.e. $50 per $1,000 of insurance) and higher for the underinsured home (i.e. $60 per $1,000 of insurance). If those indicated rates were used, the premium would have been equitable and no coinsurance penalty would have been necessary. A rate can be calculated given the expected frequency, the size of loss distribution, and the full value of the property. Using the following notation: f = frequency of loss s(L) = probability of loss of a given size V = maximum possible loss (which may be unlimited for some insurance) F = face value of policy The rate is the expected indemnity payment/policy face value (AOI is often shown in $100 or $1,000 increments). Given an empirical distribution of losses, the rate is as follows: F F f Ls ( L) F (1.0 s ( L)) L 1 L1 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. Exam 5, V1b Page 87 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Insurance to Value Initiatives The HO policy settles losses based on replacement cost, subject to the policy limit. One policy feature encouraging insurance to full value is guaranteed replacement cost (GRC), allowing replacement cost to exceed F if the property is 100% insured to value and subject to annual indexation. Insurers are now using more sophisticated property estimation tools, with component indicator tools considering customized features of the home (e.g. granite countertops, hardwood floors, age of plumbing and electricity). By increasing the AOI on underinsured homes to ITV level assumed in the rates, insurers generate additional premium without increasing rates. Since homeowners loss distributions are left-skewed (i.e. small losses predominate), the increased premium is more than the additional losses generated from this action. As the insureds receive increased coverage, they are more accepting of the increased premium than if rate increases were implemented. Also, the industry has made better use of property inspections, indexation clauses, and education of insureds. Exam 5, V1b Page 88 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 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 Exam 5, V1b Page 89 2014 by All 10, Inc. 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 Exam 5, V1b Page 90 2014 by All 10, Inc. 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. Exam 5, V1b Page 91 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2008 exam 18. (1.25 points) You are given the following information: Claim A B C D E Total Loss Amount $50,000 $70,000 $90,000 $110,000 $20,000 $340,000 • Total limit trend = 10% • Basic limit = $50,000 a. (0.5 point) Calculate the basic limit trend. b. (0.5 point) Calculate the excess limit trend. c. (0.25 point) Identify a situation in which the excess limit trend will be less than the basic limit trend. Note: This is more of a chapter 6 question 34. (2.0 points) a. (1.0 point) You are given the following distribution of losses. Layer of Loss Lower Limit ($) Upper Limit ($) Total $ Loss $1 $10,000 $500,000 $10,001 $250,000 $16,000,000 $250,001 $500,000 $17,500,000 $500,001 $1,000,000 $11,500,000 Occurrences 100 80 50 20 Calculate the $500,000 increased limit factor assuming the basic limit is $250,000. b. (1.0 point) Identify and briefly explain two issues that arise when using empirical data to construct increased limit factor tables. Exam 5, V1b Page 92 2014 by All 10, Inc. 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. Exam 5, V1b Page 93 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2012 exam 12. (1.25 points) Given the following information: Paid Losses $50,000 $100,000 $300,000 $500,000 Total Claim Counts by Policy Limit $100,000 $300,000 $500,000 30 25 80 150 60 120 35 50 30 180 120 280 Calculate an indicated increased limit factor for the $300,000 policy limit, assuming a basic limit of $100,000. Exam 5, V1b Page 94 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Section 2: Deductible Pricing Questions from the 2003 exam: 38. (3 points) Given the information below, calculate the loss elimination ratio for ABC Company's collision coverage in State X at a $250 deductible. Show all work. • ABC insures 5,000 cars at a $250 deductible with the following fully credible data on the collision claims: o Paid losses are $1,000,000 per year. o The average number of claims per year is 500. • A fully credible study found that in State X: o The average number of car accidents per year involving collision damage was 10,000. o The average number of vehicles was 67,000. • Assume ABC Company's expected ground-up claims frequency is equal to that of State X. • Assume the average size of accidents that fall below the deductible is $150. Questions from the 2004 exam: 39. (3 points) Given the information below, calculate the premium for a policy with a $5,000 deductible. Show all work. Loss Distribution Frequency Loss Amount 0.45 $500 0.35 $2,500 0.15 $10,000 0.05 $25,000 • • • • • • • First dollar premium is $500,000. Ground-up expected loss ratio is 60%. Allocated Loss Adjustment Expenses (as a percentage of loss) is 10%. Fixed expense is $95,000. Variable expense is 12%. Profit and contingency provision is 3%. Assume the deductible applies to loss and ALAE. Questions from the 2005 exam: 19. Given the following information, calculate the loss elimination ratio at a $500 deductible. Loss Amount Below $500 $500 Over $500 A. < 0.4 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 Page 95 D. > 0.6, but < 0.7 E. > 0.7 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2008 exam: 32. (2.5 points) Given the following information: Ground-up Severity $100 $250 $500 $1000 $3000 $8000 Probability 20% 10% 15% 30% 20% 5% Premium for a policy with no deductible = $350 Ground-up expected loss ratio = 60.9% Fixed expenses = $31.70 Variable underwriting expense provision = 22% Profit provision = 2% Allocated loss adjustment expenses (ALAE) are 10% of loss and are the responsibility of the insurer. a. (1.0 point) Calculate the loss elimination ratio (LER) for a $500 deductible. b. (1.5 points) Calculate the premium for a policy with a $500 deductible Questions from the 2010 exam 30. (1 point) Given the following information: Net Reported Policy Deductible Full Coverage $250 $500 Losses $680,000 $2,900,000 $5,200,000 Net Reported Losses Assuming a 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. Exam 5, V1b Page 96 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2011 exam 14. (1.5 points) Given the following information: Deductible Full coverage $250 $500 $750 $1,000 Total Reported Claim Counts 990 2,770 4,360 1,350 500 9,970 Net Reported Losses $1,347,000 $5,167,000 $9,198,000 $3,230,000 $1,692,000 $20,634,000 Net Reported Losses Assuming $750 Deductible $772,000 $4,024,000 $8,244,000 $3,230,000 Unknown Net Reported Losses Assuming $1000 Deductible $605,000 $3,505,000 $7,345,000 $2,926,000 $1,692,000 a. (1 point) Use the loss elimination ratio approach to deductible pricing to calculate the credit associated with moving from a $750 deductible to a $1,000 deductible. b. (0.5 point) An assumption of the loss elimination ratio approach is that claim behavior will be the same for each deductible. Describe why this assumption may not hold in practice. Exam 5, V1b Page 97 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Section 3: Size of Risk for Workers Compensation 3a. Premium Discounts Questions from the 2000 exam 52. (2 points) Based on Schofield, "Going from a Pure Premium to a Rate," and the following information, use the Workers' Compensation Method to calculate the dollar amount of Premium Discount. Standard premium = $ 475,000 Expense Table: Expense Provisions Profit and Premium Range ($) Production General Taxes Contingencies 1 – 5,000 12.0% 10.0% 4.0% 2.5% 5,001 - 100,000 9.0% 7.5% 4.0% 2.5% 100,001 - 500,000 7.0% 5.0% 4.0% 2.5% 500,001 + 6.0% 2.5% 4.0% 2.5% Questions from the 2002 exam 29. (3 points) Based on Schofield, "Going From a Pure Premium to a Rate," and the information below, use the Worker's Compensation Method to calculate the discounted premium. Show all work. Standard Premium of 500,000 For each premium gradation of 200,000 above 10,000, commissions and general expenses decrease by 25%. For the first 10,000 of Standard Premium commissions are 15% and general expenses are 10%. All other expenses total 8% of the discounted premium. Questions from the 2011 exam 16. (1.75 points) Workers compensation insurers often offer a premium discount for large premium dollar accounts. Given the following expense information for workers compensation policies: Premium Range $0 - $7,500 $7,500 - $75,000 $75,000 - $200,000 $200,000 & above Expense Percentage by Type: Production General Taxes 14% 10% 3% 10% 8% 3% 7% 6% 3% 5% 4% 3% Profit 5% 5% 5% 5% Calculate the total amount of premium discount for a policy with premium of $180,000. Exam 5, V1b Page 98 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 3b. Loss Constants Questions from the 1995 exam 35. Feldblum, “Workers' Compensation Ratemaking,” states that loss experience for large risks tends to be better than for small risks. (a) (1 point) Give two explanations that support this observation. (b) (2 points) In 1990 the NCCI recommended application of loss constants to all risks, rather than to small risks only. Using Feldblum's methodology and the information below, calculate the appropriate loss constant to be applied to all risks. Premium Size Small Risk $0 - $2,000 Large Risk $2,001 or more Number of Risks 100 50 Earned Premium $75,000 $200,000 Incurred Losses $63,000 $144,000 Loss Ratio 84.0% 72.0% (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 Exam 5, V1b Page 99 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Section 4: Insurance to Value (ITV) Questions from the 1990 exam 49. (3 Points) a. (2 Points) A building is insured for $150,000 under an agreed value policy. Assume a 12.5% loss frequency and the following size of loss distribution. Using the methods discussed by Head “Insurance to Value," calculate the pure premium rate per $100 for the building. Size of Loss (L) 0 < L <$ 50,000 $50,000 < L < 100,000 $100,000 < L < $150,000 $150,000 < L < $200,000 $200,000 < L < $250,000 $250,000 < L TOTAL Number Of Losses 340 75 50 25 10 0 500 Dollars Of Loss $3,762,000 5 625,000 6,375,000 4,463,000 2,275,000 0 $22,500,000 b. (1 Point) Is this rate higher or lower than the rate for a comparable building insured for $200,000? Why? Question from the 1992 exam 5. According to the Study Note Reading: Head, G.L.; Insurance to Value, if losses less than the policy face are possible, which of the following are true concerning the pure premium rate as the coinsurance percentage increases? 1. If small losses outnumber large ones, pure premium rates should decrease at a decreasing rate. 2. If large losses outnumber small ones, pure premium rates should decrease at a decreasing rate. 3. If losses of all sizes are equally numerous, pure premium rates should decrease at a constant rate. A. 1 B. 3 C. 1, 3 D. 2, 3 E. 1, 2, 3 Questions from the 1994 exam 43. (a) (2 points) Using the methods described by Head in the Study Note Reading Insurance to Value, calculate the pure premium rate per $100 for 20%, 50%, and 80% coinsurance. You have the following data: The value of property insured is $200,000. Loss frequency is 2%. Coinsurance Percentage (Cn) 20% 50% 80% Conditional Probability of Losses in Interval (Cn-1, Cn] .50 .20 .05 Arithmetic Mean Loss of Losses in Interval as % of Total Value 5% 35% 60% (b) (1 point) This question no longer applies to the content covered in this chapter Exam 5, V1b Page 100 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 1995 exam 46. (4 points) You are given: Replacement Cost $100,000 Loss Frequency 10% $200,000 10% Size of Losses in Interval ($000) $ 0- 20 21- 50 51- 80 81-100 $ 0- 20 21- 50 51- 80 81-100 101-160 161-200 Conditional Probability of Losses in the Interval .80 .10 .08 .02 .70 .15 .09 .04 .01 .01 Arithmetic Mean Loss of Losses in Interval ($000) $2 3 60 95 $3 35 65 95 150 190 A client has asked you to determine the pure premium cost of insuring his house with a $200,000 replacement cost. (a) (1 point) As described in the study note reading by Head, “ Insurance To Value," determine the pure premium rate per $100 for insuring this house for $100,000. (b) (1.5 points) How does this pure premium per $100 compare to the rate for this house if it were insured for $200,000? Explain. (c) (1.5 points) Would the pure premium rate per $100 derived in (a) match that of a house with a replacement cost of $100,000 and insured for $100,000? Why or why not? Questions from the 1996 exam 44. (3 points) You are given: Coinsurance Percentage (Cn) 40% 60% 80% 100% Conditional Probability of Losses in Interval [Cn-1 ,Cn] 65% 20% 10% 5% Arithmetic Mean Loss in Interval [Cn-1 ,Cn] $100,000 $250,000 $350,000 $500,000 • Value of Property: $500,000 • Loss Frequency: 5% (a) (2 points) Using the methods described by Head, "Insurance to Value," calculate the pure premium rate per $100 for 60% coinsurance. (b) (1 point) The property is actually insured for $200,000, with a 60% coinsurance clause. A loss of $80,000 occurs. What is the total indemnity amount payable to the insured? Exam 5, V1b Page 101 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 1998 exam 5. Based on Head, Insurance to Value, calculate the pure premium rate per $100 of insurance for a $100,000 risk and a 50% coinsurance percentage. Losses At Least Less Than 0% 10% 10% 20% 20% 30% 30% 40% 40% 50% 50% 60% 60% 70% 70% 80% 80% 90% 90% 100% A. < $1.00 Unconditional Probability Of Loss .0100 .0075 .0050 .0035 .0020 .0010 .0005 .0003 .0002 .0005 Arithmetic Mean Loss 4% 14% 23% 33% 43% 53% 62% 72% 82% 98% B. > $1.00 but < $1.05 C. > $1.05 but < $1.10 D. > $1.10 but < $1.15 E. > $1.15 Questions from the 1999 exam 15. (1 point) Based on Head, "Insurance to Value," and given the information below, what is the coinsurance penalty applicable to the insured? Coinsurance Requirement: Full Value of Structure: Amount of Insurance on Structure: Amount of Loss: 80% $1,000,000 $700,000 $600,000 A. < $20,000 B. > $20,000 but < $40,000 D. > $60,000 but < $80,000 E. > $80,000 C. > $40,000 but < $60,000 Questions from the 2000 exam 24. Based on Head, Insurance to Value, and the following information, calculate the absolute difference between the pure premium rate per $100 for a 50% coinsurance clause and a 75% coinsurance clause. • The value of the insured property is $100,000. • The loss frequency is 5%. Arithmetic Mean Loss Loss, as Percentage of Conditional Probability in Interval, as a Total Property Value of a Loss in Interval Percent of Total Value Less than or equal to 10% 0.50 4% 11 % to 25% 0.25 18% 26% to 50% 0.15 40% 51 % to 75% 0.07 70% A. < 0.10 E. > 0.40 Exam 5, V1b B. > 0.10 but < 0.20 C. > 0.20 but < 0.30 Page 102 D. > 0.30 but < 0.40 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2001 exam Question 7. Based on Head, Insurance to Value, and the following information, calculate the ratio of the pure premium rate per $100 for a 60% coinsurance clause to the pure premium rate per $100 for a 40% coinsurance clause. Loss, as a Percentage of Total Property Value Unconditional Probability of a Loss in Interval Arithmetic Mean Loss in Interval as a Percent of Total Value Less than or equal to 20% 0.050 12% 21% to 40% 0.025 30% 41% to 60% 0.015 52% 61% to 80% 0.007 75% 80% to 100 0.003 95% A. < 0.65 B. > 0.65 but < 0.75 C. > 0.75 but < 0.85 D. > 0.85 but < 0.95 E. > 0.95 Questions from the 2002 exam 42. (2 points) Based on Head, Insurance to Value, and the following information, calculate the pure premium rate per $100 for a 50% coinsurance clause. The value of the insured property is $200,000. The loss frequency is 3%. Loss, as Percentage of Total Property Value Less than or equal to 25% 26% to 50% 51% to 75% Conditional Probability of a Loss in Interval 0.75 0.12 0.08 Arithmetic Mean Loss in Interval as a Percent of Total Value 9% 40% 70% Questions from the 2003 exam 40. (2.25 points) An insurer writing fire insurance uses coinsurance in its rating structure by means of an "average clause." A coinsurance percentage of 80% applies to all policies. Based on the following information, answer the questions below. Show all work Policy Amount of Loss Property Value Face Amount of Insurance 1 2 3 $50,000 $155,000 $375,000 $200,000 $160,000 $480,000 $150,000 $120,000 $400,000 a. (1.5 points) For each of the policies above, calculate the indemnity payment made by the insurer. b. (0.75 points) For each of the policies above, calculate the additional insurance, if any, that would have been required for the insurance company to indemnify the full amount of the loss. Exam 5, V1b Page 103 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2004 exam 41. (4 points) Given the following information on an individual property policy, answer the questions below. Show all work. • The property value is $200,000. • Assume no deductible applies. • The frequency of non-zero loss is 10%. • The severity of loss distribution is as follows: • 70% at 10% of value • 20% at 50% of value • 8% at 80% of value • 2% at 90% of value • Coinsurance to 80% underlies the expected rate. • Permissible loss ratio is 65%. a. (2 points) The insured purchases a policy insuring the property to 80% of value. Determine the premium charged for the policy. b. (1 point) The insured instead purchases a policy insuring the property to 70% of value. Assuming the same rate per $100 of insured value as in part a. above, determine the expected loss ratio for this policy. c. (1 point) Assume the insurer incorporates a coinsurance clause into the policy. The insured continues to insure the property to 70% of value. What is the expected loss ratio for this policy? Briefly explain your answer. Questions from the 2005 exam 51. (2 points) Using the following information, answer the questions below. Show all work. • All properties are valued at $500,000. • The company writes 1,000 policies. • Each policy has a face value equal to the value of the insured property. • Assume only one loss per policy per period is possible, and exactly 20 insureds will incur a loss of some size during any one policy period. • Assume no coinsurance clause or deductible applies Assume losses are distributed as shown: 50% at $50,000 20% at $250,000 30% at $500,000 a. (1 point) Calculate the pure premium rate per $100 of insurance for a policy face equaling $300,000. b. (1 point) Does the pure premium rate per $100 of insurance for a $500,000 policy face differ from the rate for the $300,000 policy face? Briefly explain your answer. Exam 5, V1b Page 104 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2006 exam 44. (2.5 points) You are given the following assumptions for an insured book of property business: A company writes 1,000 property policies. Each property is valued at $500,000. Exactly 20 of these properties will experience a loss during one policy period. The losses are distributed as shown in the table below: S(L) 50% 20% 10% 5% 15% L $100,000 200,000 300,000 400,000 500,000 Find the premium rate per $100 of insurance for a policy face equaling $400,000. Show all work. Questions from the 2007 exam 49. (1.0 point) A property is valued at $300,000. The coinsurance requirement for the policy is 80% of the property value. The insured chooses a $200,000 face value. Assume there is no deductible. Calculate each of the following: a. (0.25 point) Coinsurance requirement. b. (0.25 point) Coinsurance apportionment ratio. c. (0.25 point) Coinsurance deficiency. d. (0.25 point) Maximum coinsurance penalty. Show all work. Questions from the 2008 exam 36. (2.0 points) You are given the following information: Home is valued at $350,000. Coinsurance requirement = 80% of the property value Face value of policy = $275,000 a. Calculate the coinsurance deficiency. b. Calculate the coinsurance apportionment ratio. c. Calculate the maximum coinsurance penalty possible. d. Calculate the coinsurance penalty for a $300,000 loss. Questions from the 2009 exam 40. (2 points) Given the following: • Property is valued at $500,000. • Coinsurance requirement is 88% of the property value. • Policy face value is $300,000. Graph and label the coinsurance penalty function. Exam 5, V1b Page 105 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Questions from the 2010 exam 32. (2 points) Given the following information: • Amount of loss = $200,000 • Amount of coverage = $350,000 • Replacement cost of property = $450,000 • Minimum insurance-to-value requirement = 80% a. (1 point) Calculate the coinsurance penalty. b. (0.5 point) Identify the problem with underinsurance from the insurer's perspective. c. (0.5 point) Identify the problem with underinsurance from the insured's perspective. Questions from the 2012 exam 15. (2.25 points) You are given the following information on expected claim payment distribution for properties with a replacement cost of $350,000. Claim Payment Probability $0 97.0% $10,000 1.5% $50,000 0.8% $200,000 0.5% $350,000 0.2% Assume no expenses or profit. a. (0.5 point) Assuming all homeowners purchase full coverage, calculate the pure premium per $1,000 of insurance. b. (0.75 point) Demonstrate with an example that the use of a fixed rate per $1,000 of insurance is inequitable if a subset of the insured group purchases only partial coverage. c. (1 point) Describe two insurer initiatives that would reduce the inequity from part b. above, including an explanation of how the inequity would be reduced. Exam 5, V1b Page 106 2014 by All 10, Inc. 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.” Exam 5, V1b Page 107 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2006 exam 31. (3.25 points) a. (2 points) Calculate the annual inflation rate for claims in the layer $50,000 excess of $100,000 for 2006. 2005 Ground-up Loss (1) 75,000 100,000 125,000 150,000 450,000 Ground-up 2005 Annual Losses Claims inflation 50K xs100K Rate (2) (3) 0 1.10 0 1.10 25,000 1.10 50,000 1.10 75,000 2006 Trended Ground-up Loss (4)=(1)*(3) 82,500 110,000 137,500 165,000 495,000 Trended 2006 Losses 50K xs100K (5) 0 10,000 37,500 50,000 97,500 Annual Layer Claims Inflation Rate (6)=(5)/(2)-1.0 0.3000 Col (2) and Col (5) are capped at 50,0000 b. (1.25 points) How would you expect the inflation rate in the layer $50,000 excess of $100,000 to differ from the inflation rate for claims limited to $100,000? Explain two reasons for the difference between the inflation rates. Note: This is more of a chapter 6 question The excess layer inflation rates are greater than the basic limit inflation rates for two reasons: 1. For losses already in the excess layer, inflation impacts only the portion of the loss in the excess layer. The basic limits portion does not change. 2. For losses near the basic limit, inflation causes the losses to pierce the increased limits layer, resulting in increased frequency of increased limits losses. Exam 5, V1b Page 108 2014 by All 10, Inc. 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% Exam 5, V1b Page 109 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2007 exam (continued): 47. (2.0 points) Using the methods described by Palmer in Increased Limits Ratemaking for Liability Ratemaking, calculate the following: a. (1.25 points) The $5,000,000 increased limit factor. b. (0.75 point) The limited average severity in the layer $4,000,000 in excess of $1,000,000. Show all work. Initial comments: An Increased Limit Factor (ILF) at limit L relative to basic limit B can be defined as: ILF ( L ) Expected Indemnity Cost(L) Expected Indemnity Cost(B) ILFs are developed on a per-claim or per-occurrence basis: A per-claim limit is a limit on the amount that will be paid to a single plaintiff for losses arising from a single incident. A per-occurrence limit is a limit on the total amount that will be paid to all plaintiffs for losses arising from a single incident. To evaluate an appropriate provision for indemnity costs at various limits of liability, we develop (LAS) at various limits of liability. LAS is the average size of loss when all losses have been capped at the given limit. Part A Claim A B C D E F G H I J Ground-up Loss Amount (1) 250,000 300,000 450,000 750,000 1,200,000 2,500,000 4,000,000 7,500,000 9,000,000 15,000,000 Loss at $1,000,000 Limit (2) 250,000 300,000 450,000 750,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 Loss at $5,000,000 Limit (3) 250,000 300,000 450,000 750,000 1,200,000 2,500,000 4,000,000 5,000,000 5,000,000 5,000,000 Limited Average 775,000 2,445,000 Losses in the Part B 4M x/s 1M Layer (4) 0 0 0 0 200,000 1,500,000 3,000,000 4,000,000 4,000,000 4,000,000 2,783,333 Col (2) equals Col (1) capped at 1,000,000; Col (3) equals Col (1) capped at 5,000,000 Col (4) equals Col (1) - 1,000,000, capped at 4,000,000, if (1) is greater than 1,000,000 a. The indemnity-only ILF at 5,000,00 given a basic limit of 1,000,000 equals 2,445,000/775,000 = 3.1548 b. LAS (4M xs 1M) = (200,000 + 1,500,000 + 3,000,000 + [3 x 4,000,000])/6 = 2,783,333, or LAS (4M xs 1M) = (2,445,000 – 775,000)/0.6 = 2,783,333, where .60 is equal to the probability that a loss is greater than 1M, given that a loss has occurred, or [(3 * 5,000,000 + 4,000,000 + 2,500,000 + 1,200,000]/6 - [(6 * 1,000,000]/6 = 3,783,333 -1,000,000 = 2,783,333 Exam 5, V1b Page 110 2014 by All 10, Inc. 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) x50,000 x50,000 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. Exam 5, V1b Page 111 2014 by All 10, Inc. 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. Exam 5, V1b Page 112 2014 by All 10, Inc. 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 Exam 5, V1b Page 113 2014 by All 10, Inc. 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 Exam 5, V1b Page 117 2014 by All 10, Inc. 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 Exam 5, V1b Page 118 2014 by All 10, Inc. 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 Exam 5, V1b Page 119 2014 by All 10, Inc. 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 Exam 5, V1b Page 120 2014 by All 10, Inc. 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); Exam 5, V1b Page 121 (4) = (3)/[1.0 –taxes - profit)] 2014 by All 10, Inc. 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 Exam 5, V1b Page 122 2014 by All 10, Inc. 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 $43.33 130,000 X Exam 5, V1b Page 123 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Section 4: Insurance to Value (ITV) Solutions to questions from the 1990 exam Question 49. Size of Loss (L) (1) 0 < L <$ 50,000 $50,000 < L < 100, 000 $100,000 < L < $150,000 $150,000 < L < $200,000 $200,000 < L < $250,000 $250,000 < L TOTAL Number Of Losses (2) 340 75 50 25 10 0 500 Conditional Pr[of Loss] (3)=(2) / 2(tot) .68 .15 .10 .05 .02 0 Unconditional Pr[of Loss] (4) = (3)*.125 .085 .01875 .0125 .00625 .0025 0 Dollars Of Loss (5) $ 3,762,000 5,625,000 6,375,000 4,463,000 2,275,000 0 $22,500,000 Pure premium (6)=[(5)/(2)]*(4) 941 1,406 1,594 938 375 0 5,254 Note: For L > 150,000, column (6) pure premium = $150,000 * (4) The pure premium rate per $100 for the $150,000 building = 5,254 / [150,000/100] = 3.502. (b). This rate is higher. Whenever losses < F are possible, the PP rate should decrease as F increases. Solutions to questions from the 1992 exam Question 5. 1. T. 2. F. 3. T. Answer C. Solutions to questions from the 1994 exam C C Ls(L)dL + F[1- s(L)dL] 0 R = f 0 F / 100 Question 43. (a). Co-Ins % .20 .50 .80 General Pure premium rate Equation .02* [.50(10,000) + (1-.50) * (40,000)] 40,000 / 100 02* [.50(10,000)+.2 * 70,000 + (1-.70) * (100,000)] 100,000 / 100 02* [.50 * 10,000+.2 * 70,000+.05 * 120,000 + (1-.75) * (160,000)] 160,000 / 100 Pure prem rate per $100 1.25 .98 .8125 (b). This question no longer applies to the content covered in this chapter Exam 5, V1b Page 124 2014 by All 10, Inc. 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). Exam 5, V1b Page 125 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 1996 exam Question 44. (a) The computation of % coinsurance rates. Begin with the pure premium rate equation. C C Ls(L)dL + F[1- s(L)dL] Symbol Description f frequency of loss 0 R = f 0 c coinsurance % F / 100 V property value F policy face (expressed in $'s) C cV L Loss amount Assume that L is a continuous variable, "because this assumption clarifies some relationships which might be nearly unintelligible in discrete notation." "Pure premium coinsurance rates are computed on the assumption of a policy face equal to the coinsurance requirement." Since the assumed policy face, F, = C = cV = .60 * $500,000, and using the information in the table below, we can compute the pure premium rate per $100 for 60% coinsurance as follows: Coinsurance Percentage (Cn) .40 .60 .80 1.00 Conditional Probability of Losses in Interval [Cn-1 ,Cn] 65% 20% 10% 5% Cumulative Conditional Probability of Loss > Cn .35 .15 .05 0 Arithmetic Mean Loss in Interval [Cn-1 ,Cn] $100,000 $250,000 $350,000 $500,000 C = cV $200,000 $300,000 $400,000 $500,000 $100,000*.65 + $250,000*.20 + $300,000 * (1.0 - .85) Therefore, R = .05* = 2.67. $300,000 / 100 (b) "If a policy should be less that its agreed amount, coinsurance reduces every indemnity payment proportionately." The proportion is based on the ratio of the amount of insurance purchased to the amount of insurance assumed in the pure premium coinsurance rate calculation. We are given that the insured purchased a $200,000 policy. The 60% coinsurance requirement called for the purchase of a $300,000 ($500,000 * .60) policy. $200,000 Therefore, the indemnity paid to the insured = $80,000* = $53,333.33. $300,000 Exam 5, V1b Page 126 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 1998 exam Question 5. The formula to calculate the pure premium rate per $100 of insurance: C C Ls(L)dL + F[1- s(L)dL] 0 0 R= f F / 100 At Least 0% 10% 20% 30% 40% 50% Losses Less Than 10% 20% 30% 40% 50% Unconditional Probability Of Loss .0100 .0075 .0050 .0035 .0020 .0025 (%) 4% 14% 23% 33% 43% 50% Arithmetic Mean Loss 100,000 risk 4,000 14,000 23,000 33,000 43,000 50,000 Note that the unconditional probability of a loss exceeding 50% of its value is .0010+.0005+.0003+.0002+.0005 = .0025. In addition, the policy face equals the co-insurance requirement (C = cV = .50 (100,000) = 50,000). Co-Insurance % .50 Pure premium rate per $100 [.01*4,000.0075*14,000.005*23,000.0035*33,000.002*43000.0025*50,000] = $1.17 50,000 /100 Answer E. Solutions to questions from the 1999 exam Question 15. Given: Coinsurance Requirement: Full Value of Structure: Amount of Insurance on Structure: Amount of Loss: 80% $1,000,000 $700,000 $600,000 c V F L 700,000 F Since I L 525, 000 , then I $600, 000 * cV .80*1,000,000 The coinsurance penalty equals loss amount - the indemnity payment = 600,000- 525,000 = 75,000. Answer D. Exam 5, V1b Page 127 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2000 exam Question 24. The general pure premium rate equation for percentage co-insurance is : Co-Ins % .50 .75 C C Ls(L)dL + F[1- s(L)dL] 0 0 R= f F / 100 General Pure premium rate Equation Pure prem rate per $100 [.50(.04*100,000) .25*(.18*100,000) .15*(.40*100,000) (1-.90)*(.50*100,000)] .05* 50,000 /100 1.75 .05 * [.50(.04*100,000) .25*(.18*100,000) .15*(.40*100,000) .07*(.70*100,000)] 75,000 /100 .05 * [(1-.97)*(.75*100,000)] 75,000 /100 1.31 the absolute difference between the pure premium rate per $100 for a 50% coinsurance clause and a 75% coinsurance clause is 1.75 – 1.31 = .44. Answer E. Solutions to questions from the 2001 exam Question 7. The general pure premium rate equation for percentage co-insurance is : C C Ls(L)dL + F[1- s(L)dL] 0 R = f 0 F / 100 Unlike problem 24 from the 2000 exam, we are not given in this particular problem the value of the insured property, nor the loss frequency (f). However, this information is not necessary to compute the ratio of the pure premium rate per $100 for a 60% coinsurance clause to the pure premium rate per $100 for a 40% coinsurance clause. Co-Ins % .40 General Pure premium rate Equation .60 [.05*.12 .025*.30 .015*.52 (007 .003)*.60] .60 [.05*.12 .025*.30 (.015 .007 .003)*.40] .40 Pure prem rate per $100 .0588 .0455 The ratio of the pure premium rate per $100 for a 60% coinsurance clause to the pure premium rate per $100 for a 40% coinsurance clause is .0455 ÷ .0588 = .77381 Answer E. Exam 5, V1b Page 128 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2002 exam Question 42 C C Ls(L)dL+F[1- s(L)dL] 0 : The general pure premium rate equation for percentage co-insurance is R=f 0 F/100 Using the data given in the problem, and the discrete counterpart to the continuous function above, the pure premium rate per $100 for a 50% coinsurance clause is computed as follows: General Pure premium rate Equation .03 * [.75(.09*$200,000) .12*(.40*$200,000) .08*(.50*$200,000) (1-.95)*(.50*$200,000)] (.50*$200,000) /100 Exam 5, V1b Page 129 Pure prem rate per $100 1.083 2014 by All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2003 exam 40. (2.25 points) An insurer writing fire insurance uses coinsurance in its rating structure by means of an "average clause." A coinsurance percentage of 80% applies to all policies. Based on the following information, answer the questions below. Show all work Policy Amount of Loss Property Value Face Amount of Insurance 1 2 3 $50,000 $155,000 $375,000 $200,000 $160,000 $480,000 $150,000 $120,000 $400,000 a. (1.5 points) For each of the policies above, calculate the indemnity payment made by the insurer. Note: “Insurance to value" (ITV) exists only if property is insured to the exact extent ($ amount or % value) assumed in the rate calculation. To evaluate coinsurance applications, the following formulas are given: the coinsurance requirement C = cV the coinsurance deficiency d = [cV – F] CAR = a = [F/cV] < 1.00. Compute ITV for each policy: For policy 1, ITV = $150,000/$200,000 = .75. This policy does not meet the coinsurance requirement. For policy 2, ITV = $120,000/$160,000 = .75. This policy does not meet the coinsurance requirement. For policy 3, ITV = $400,000/$480,000 = .833. This policy does meet the coinsurance requirement. Note: A standard coinsurance clause may be represented algebraically as follows: I = L*[F/cV], subject to two constraints: 1. I < L The indemnity payment cannot exceed the loss. This constraint is in concert with the principle of indemnity, which states that no insured should profit from any loss. 2. I < F The indemnity payment cannot exceed the policy face. This sets the overall limit on the amount insurance payable from a single occurrence. For policy 1, I L* FV $120,000 FV $150,000 $50,000* $46,875 . For policy 2, I L * $155, 000 * $145, 312 , .80*$160,000 cV .80*$200,000 cV but is capped at policy limits of $120,000. For policy 3, since the coinsurance requirement was met and the loss was less than policy face, indemnity equals loss amount $375,000. b. (0.75 points) For each of the policies above, calculate the additional insurance, if any, that would have been required for the insurance company to indemnify the full amount of the loss. For policy 1, the coinsurance requirement is $160,000, so an additional $10,000 is needed. For policy 2, an additional $35,000 is needed ($155,000 - $120,000). For policy 3, no additional amount is needed, since the policy limits purchased meet the coinsurance requirement and the loss is less than the policy limit. Exam 5, V1b Page 130 2014 by All 10, Inc. 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% Exam 5, V1b Page 131 2014 by All 10, Inc. 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 L1 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 Page 132 2014 by All 10, Inc. 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. Exam 5, V1b Page 133 2014 by All 10, Inc. 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 Exam 5, V1b Page 134 2014 by A All 10, Inc. Chapter 11 – Special Classification BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Solutions to questions from the 2010 exam Question 32. a. (1 point) Calculate the coinsurance penalty. We are given the following: L = 200,000 = amount of loss, V = value of property = 450,000, F= face amount = 350,000 C = Co-ins req = 80% The home is valued at $450,000 and is insured only for $350,000 despite a coinsurance requirement of 80% (or $360,000 in this case). Since F is $350,000 a coinsurance deficiency exists and a = 0.9722 (=$350,000 / $360,000), where a = apportionment ratio. The indemnity payments and coinsurance penalties for a $200,000 loss are: F $350, 000 $200, 000 $194, 444.44 cV $360, 000 e L - I $200, 000 - $194,144.44 $5,555.55 IL 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. Exam 5, V1b Page 135 2014 by All 10, Inc. 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. Exam 5, V1b Page 136 2014 by All 10, Inc. 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. Exam 5, V1b Page 137 2014 by All 10, Inc. 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 Page 138 2014 by All 10, Inc. 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. Exam 5, V1b Page 139 2014 by All 10, Inc. 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). Exam 5, V1b Page 140 2014 by All 10, Inc. 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 NK 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 Page 141 2014 by Alll 10, Inc. 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. Exam 5, V1b Page 142 2014 by Alll 10, Inc. 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. Exam 5, V1b Page 143 2014 by All 10, Inc. Chapter 12 – Credibility BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 4 Methods For Developing Complements Of Credibility 224 - 236 A variety of complements are used in practice. First dollar ratemaking is performed on products that cover claims from the first dollar of loss (or after some small deductible) up to some limit (e.g. personal auto, HO, WC, and professional liability insurance) Excess ratemaking is performed on insurance products covering claims that exceed some high attachment point (e.g. personal umbrella policies, large deductible commercial policies, and excess reinsurance). I. First Dollar Ratemaking Boor describes six commonly used methods for developing complements for first dollar ratemaking: • Loss costs of a larger group that includes the group being rated • Loss costs of a larger related group • Rate change from the larger group applied to present rates • Harwayne’s method • Trended present rates • Competitor’s rates The complements are discussed in terms of pure premium ratemaking (although some methods can be used with loss ratio methods by replacing the exposure units with earned premium). Exam 5, V1b Page 144 2014 by All 10, Inc. Chapter 12 – Credibility BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 1. Loss Costs of a Larger Group that Include the Group being Rated This complement considers a larger group’s experience to which the subject experience belongs. Examples that may apply: * A multi-state insurer using data from regional states to supplement the state experience being reviewed. * A medical malpractice insurer using experience of all primary care physicians to supplement the experience of primary care pediatricians. * An auto insurer using data of all 16-19 year old insureds to supplement the experience of 16- year-olds. * An insurer using data from a longer-term period to credibility-weight experience that is short-term. Consider the following data and possibilities for a complement of credibility to the observed experience, the latest year pure premium from Rate Group A, Class 1 ( = $50). Candidates for complement of credibility are: the 3-year pure premium for Rate Group A, Class 1; the 1 or 3-year pure premium for Rate Group A; the 1 or 3-year pure premium for the total of all experience. Another option is the total of all Class 1 experience across all rate groups (not shown). Advantages and disadvantages of complement of credibility candidates. * The 3-year pure premium of Rate Group A, Class 1 experience (i.e., $64) is problematic. i. Lack of independence (the 1-year experience comprises over 1/3rd the exposures of the 3-year experience). ii. Bias. The huge difference between the 1-year pure premium ($50) and the3-year pure premium ($64) indicates the 3-year data may be biased (i.e. changes in loss costs makes older data less relevant). * Using the total of all experience combined is: i. Better with respect to independence (Rate Group A, Class 1 is a small portion of the total experience (100 out of 4,000 exposures)). ii. Biased. The difference between the 1-year Rate Group A, Class 1 pure premium ($50) and the 1-year total pure premium ($74) implies a bias may be present. * The 1-year Rate Group A experience appears to be the best. i. The Rate Group A data should reflect risks that are more similar to Class 1. ii. The 1-year pure premium ($55) and 3-year pure premium ($57) suggests it has a low process variance. iii. The 1-year result is not too different than the 1-year Rate Group A, Class 1 result, which suggests little bias. * If the Class 1 data from all rate groups combined were available, it may be a reasonable option. Exam 5, V1b Page 145 2014 by All 10, Inc. Chapter 12 – Credibility BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Complement Evaluation 1. It has a lower process variance (because the complement is based on a greater volume of data than the subject experience). 2. The subject experience has been split out of the larger group suggests that the actuary believes the subject experience is different than the larger group. i. If so, the larger group is a biased estimator of the subject experience. ii. The actuary may be able to make an adjustment to reduce this bias. The complement can include or exclude the subject experience. i. If it excludes the subject experience, it is likely to be independent. ii. If it includes the subject experience, ensure it does not dominate the group. 3. Loss cost data of the larger group is typically available and the loss cost is easy to compute. 4. There is a logical connection between the complement and the subject experience (as long as all the risks in the larger group have something in common). 2. Loss Costs of a Larger Related Group Use loss costs of a separate but similar large group (e.g. a HO insurer may use the contents loss experience from the owners forms to supplement the contents experience for the condos form). Complement Evaluation 1. It is biased (though the magnitude and direction of bias are unknown) i. If the related experience can be adjusted to match the exposure to loss in the subject experience, the bias can be reduced. ii. In the example, consider how the exposure to loss for condos differs from owned homes and adjust the experience accordingly. 2. Independent (since the complement does not contain the subject experience) 3. The data is readily available and the loss cost is easy to compute 4. It may be difficult to explain adjustments made to the related experience to correct for bias 5. The complement will have a logical relationship to the base statistic (if the groups are closely related) 3. Rate Change from the Larger Group Applied to Present Rates This approach mitigates bias by using the rate change indicated for a larger group and applying it to the current loss cost of the subject experience (rather than using the larger group’s loss costs directly) The complement (C) can be expressed as: Larger Group Indicated Loss Cost C = Current Loss Cost of Subject Experience × Larger Group Current Average Loss Cost Assume the following: • Current loss cost of subject experience is $200. • Indicated loss cost of larger group is $330. • Current average loss cost of larger group is $300. Then the complement of credibility is calculated as follows: C = $200 x $330/300 = $220. Exam 5, V1b Page 146 2014 by All 10, Inc. Chapter 12 – Credibility BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Complement Evaluation 1. This complement is largely unbiased (even when the overall loss costs for the subject experience and the larger group are different). 2. It is likely to be accurate (assuming the rate changes are relatively small). 3. The level of independence depends on the size of the subject experience relative to the larger group. 4. The data is readily available and the calculations are very straightforward. 4. It is logical that the rate change indicated for a larger related group is indicative of the rate change for the subject experience. 4. Harwayne’s Method Is used when the subject experience and related experience have different distributions (the related experience requires adjustment before it can be blended with the subject experience). can be applied to the subject experience within a geographical area (e.g., a state), and the desired complement of credibility considers related experience in other geographical areas (e.g., other states). Other states may have distinctly different cost levels than the subject experience due to legal environment and population density. Example: The complement of credibility is determined using countrywide data (excluding the base state being reviewed), but the countrywide data is adjusted to remove overall differences between states. Steps to calculate the complement for class 1 of state A. State A 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 Exam 5, V1b Page 147 2014 by All 10, Inc. 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. Exam 5, V1b Page 148 2014 by All 10, Inc. 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. Exam 5, V1b Page 149 2014 by Alll 10, Inc. 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 Exam 5, V1b Page 150 2014 by All 10, Inc. Chapter 12 – Credibility BASIC RATEMAKING – WERNER, G. AND MODLIN, C. Complement Evaluation 1. Biased results will occur if the subject experience has a different size of loss distribution than that used to develop the ILFs (i.e. if the ILFs are based on industry data rather than the insurer’s own data). Despite the issues with accuracy, this is often the best available estimate. 2. The error is parameter error associated with the selected ILFs (the error associated with this estimate tends to be independent of the error associated with the base statistic). 3. To the extent that ILFs (preferably industry factors) and ground-up losses that have not been truncated below the attachment point is available, the procedure is practical. 4. In terms of acceptability, the estimate is more logically related to the data below the attachment point (which is used for the projection) than to the data in the layer (and this may be controversial). 2. Lower Limits Analysis Losses capped at the attachment point are used to estimate the losses in the excess layer being priced. If losses are too sparse use losses capped at a limit lower than the attachment point (i.e. the basic limit). ___ ILFA+L ILFA C Ld where ILFd • Ld is the loss cost capped at the lower limit, d; • ILFA is the ILF for the attachment point A; • ILFd is the ILF for the lower limit, d; • ILFA+L is the ILF for the sum of the attachment point A and the excess insurer’s limit of liability L (i.e. this sum is the top of the excess layer being priced). Note the first excess procedure is a special case of this procedure where d = the attachment point. Example: Calculate the complement of credibility for the layer between $500,000 and $750,000. Assume losses capped at $250,000 are $1,500,000, and the ILFs from the prior Table apply. 3.00 - 2.50 C $1,500, 000 $428,571. 1.75 Evaluation 1. It is difficult to determine whether this is more or less accurate than the previously complement. 2. It is more biased (as the differences in size of loss distributions will be exacerbated when using losses truncated at lower levels). 3. Stability of the estimate is increased when using losses capped at lower limits. 4. The error is generally independent of the error of the base statistic. 5. The data may not be available if some other lower limit is chosen, and the calculations are simple. 6. The complement is more logically related to the lower limits losses that to the losses in the layer being priced. Exam 5, V1b Page 151 2014 by All 10, Inc. Chapter 12 – Credibility BASIC RATEMAKING – WERNER, G. AND MODLIN, C. 3. Limits Analysis Insurers sell policies with a wide variety of policy limits. Some policy limits fall below the attachment point and some extend beyond the top of the excess layer. Thus, each policy’s limit and ILF needs to be considered in the calculation of the complement. i. Policies at each limit of coverage are analyzed separately. ii. Estimated losses in a layer are computed using the premium and expected loss ratio in that layer. iii. An ILF analysis on each first dollar limit’s loss costs is performed. C LR Pd ( ILFmin( d , A L ) - ILFA ) ILFd d A , where LR = Total loss ratio, Pd= Total premium for p