I BASIC RATEMAKING TECHNIQUES

TABLE OF CONTENTS Volume I BASIC RATEMAKING TECHNIQUES 1. Werner 1 "Introduction" 1 2. Werner 2 "Rating Manuals" 11 3. Werner 3 "Ratemaking Data" 15 4. Werner 4 "Exposures" 25 5. Werner 5 "Premium" 43 6. Werner 6 "Losses and LAE" 93 7. Werner 7 "Other Expenses and Profits" 147 8. Werner 8 "Overall Indication" 161 9. Werner 9 "Traditional Risk Classification" 209 10. Werner 10 "Multivariate Classification" 241 11. Werner 11 "Special Classification" 253 12. Werner 12 "Credibility" 331 13. Werner 13 "Other Consideration" 355 14. Werner 14 "Implementation" 363 15. Werner 15 "Commercial Lines Rating Mechanisms" 395 16. Werner 16 "Claims-Made Ratemaking" 441 17. ISO Personal Auto Manual 469 18. ASOP 13 "Trending Procedures in Property/Casualty Insurance Ratemaking" 485 19. CAS Ratemaking Statement of Principles Regarding P&C Insurance Ratemaking 491 20, AAA Risk Classification Statement of Principles 503 21. Feldblum "Personal Automobile Premiums: An Asset Share Approach" 525 Volume II UNPAID CLAIM ESTIMATION 22. Friedland 1 Overview 555 23. Friedland 2 The Claims Process 563 24. Friedland 3 Understanding the Types of Data Used in Estimation of Unpaid Claims 567 25. Friedland 4 Meeting with Management 579 26. Friedland 5 The Development Triangle 585 27. Friedland 6 The Development Triangle as a Diagnostic Tool 591 28. Friedland 7 Development Technique 601 29. Friedland 8 Expected Claims Technique 625 30. Friedland 9 Bornhuetter-Ferguson Technique 633 31. Friedland 10 Cape Cod Technique 667 32. Friedland 11 Frequency-Severity Technique 671 33. Friedland 12 Case Outstanding Development Technique 703 34. Friedland 13 Berquist-Sherman Techniques 715 35. Friedland 14 Recoveries: Salvage and Subrogation and Reinsurance 767 36. Friedland 15 Evaluation of Techniques 777 37. Friedland 16 Estimating Unpaid Allocated Claim Adjustment Expenses 805 38. Friedland 17 Estimating Unpaid Allocated Claim Adjustment Expenses 813 39. ASOP 9 Documentation and Disclosure in Ratemaking, Reserving, and Valuations 843 40. ASOP 43 Property/Casualty Unpaid Claim Estimates 851 41. CAS Reserves Statement of Principles Regarding Insurance P&C Reserves 857

NOTES Questions and parts of some solutions have been taken from material copyrighted by the Casualty Actuarial Society. They are reproduced in this study manual with the permission of the CAS solely to aid students studying for the actuarial exams. Some editing of questions has been done. Students may also request past exams directly from the society. I am very grateful to this organization for its cooperation and permission to use this material. It is, of course, in no way responsible for the structure or accuracy of the manual. Numbers in parentheses at the end of each question identify exam questions. CAS questions have four numbers separated by hyphens: the year of the exam, the number of the exam, the number of the question, and the points assigned. SoA or joint exam questions usually lack the number for points assigned. W indicates a written answer question; for questions of this type, the number of points assigned is also given. A indicates a question from the afternoon part of an exam. MC indicates that a multiple-choice question has been converted into a true/false question. Page numbers (p.) with solutions refer to the reading to which the question has been assigned unless otherwise noted. Although I have made a conscientious effort to eliminate mistakes and incorrect answers, I am certain some remain. I am very grateful to students who discovered errors in the past and encourage those of you who find others to bring them to my attention. I would also like to thank the following who in one way or another contributed to this manual: Tammy Applegate, Ed Jordan, Katy Murdza, Laurrie Raida, and Joanne Spalla. Hanover, NH 11/30/12 PJM

Werner 12 331 Geoff Werner and Claudine Modlin, Chapter 12: Credibility in Basic Ratemaking, 2010, pp. 216 38 OUTLINE I. OVERVIEW A. Credibility 1. Law of large numbers principle that as the volume of similar, independent exposure units increases, the observed experience will approach the true experience, and for a sufficiently large number, the observed experience will equal the true experience 2. But data may not be large enough to produce accurate and stable rates and thus additional information is needed 3. Credibility addresses the problem of how to blend the actuarial estimate and other experience 4. Credibility a measure of the predictive value in a given application that the actuary attaches to a particular body of data B. Necessary Criteria for Measures of Credibility (Z) 1. 0 Z 1 2. Z should increase as the number of risks underlying the actuarial estimate increases, all else being equal 3. Z should increase at a nonincreasing rate II. METHODS FOR DETERMINING CREDIBILITY OF AN ESTIMATE A. Classical Credibility a.k.a. Limited Fluctuation Credibility 1. Goal to limit the effect that random fluctuations in observations have on the risk estimate 2. Estimate Estimate = (Z)(Observed Estimate) + (1 Z)(Related Experience) 3. Observed estimate also known as subject experience or base statistic 4. Fully credible characteristic of observed experience when the probability (p) is high that the observed experience will not differ significantly from the expected experience by more than some arbitrary amount (k) 5. Full credibility standard based on the total amount of losses Pr[(1 k)e(s) (1 + k)e(s)] = p, where k - selected deviation S total amount of losses p - selected probability

332 Werner 12 6. Simplifying assumptions a. Exposures are homogeneous, i.e., they have the same expected number of claims b. Frequency follows a Poisson distribution, whose variance equals its mean c. Severity is constant 7. Expected number of claims for full credibility assuming constant claim size E[Y] = [z (p+1)/2 /k] 2 8. Assuming p = 95% and k =.05, full credibility standard is 1,082 9. Partial credibility Z = Y/E[Y] 10. Full credibility standard based on the number of exposures divide the number of claims for full credibility by the expected frequency 11. Expected number of claims for full credibility assuming variation in claims size E[Y] = [z (p+1)/2 /k] 2 [1 + 2 s / 2 s] 2 s / 2 s coefficient of variation squared 12. Comments on the approach a. Advantages 1) Commonly used and accepted 2) Data needed is readily available 3) Straightforward calculations b. Disadvantages simplifying assumptions B. Bühlmann Credibility a.k.a. Least-Squares Credibility 1. Goal to minimize the square of the error between the estimate and the true expected value 2. Credibility-weighted estimate Estimate = (Z)(Observed Experience) + (1 Z)(Prior Mean) 3. Prior mean actuary s a priori assumption of the risk estimate 4. Credibility formula Z = N N + K, where N number of observations K ratio of the expected value of the process (EVPV) to the variance of the hypothetical means (VHM) 5. Expected value of the process variance average risk variance 6. Variance of the hypothetical means variance between risks

Werner 12 333 7. Comparison with classical credibility a. Complement is prior mean, not related experience as for classical credibility b. Approaches 1 asymptotically, whereas equals 1 at the full credibility number for classical credibility c. Equals 0 if assume no variation in the size of losses, unlike under classical credibility where a value exists 8. Assumptions a. Complement is applied to the prior mean b. Risk parameters and risk process do not shift over time c. EVPV increases with N d. VHM increases with N 9. Comments on the approach a. Not as common as classical credibility but is accepted b. Difficulty is determination of EVPV and VHM C. Bayesian Analysis 1. Adjustment of prior estimate to reflect new information in a probabilistic manner 2. Distributional assumptions made 3. Bühlmann credibility is weighted least-squares line of the Bayesian estimate; in certain situations values are equal III. DESIRABLE QUALITIES OF A COMPLEMENT OF CREDIBILITY A. Complement of Credibility 1. Complement of credibility related experience to be blended with the observed data 2. ASOP 25 a. Related experience should have similar characteristics b. If cannot adjust experience so that it is similar, do not use 3. Complement may be more important than the observed data B. Boor s Desirable Qualities 1. Accurate complement should result in rates with a low variance 2. Unbiased a. Difference between complement and observed experience should average to zero, though each year s variation may be considerable b. Differs from the term accurate, which describes closeness but may be biased 3. Statistically independent from the base statistic otherwise, error in the base statistic can be compounded 4. Readily available 5. Easy to compute helps in justification 6. Logical relationship to base statistic helps in justification

334 Werner 12 C. Complements Vary with Type of Ratemaking 1. First dollar ratemaking ratemaking that covers claims from the first dollar of loss (or after some small deductible) up to some limit, e.g., homeowners a. Complements described in terms of pure premium ratemaking b. Possibly can be used in loss ratio ratemaking 2. Excess ratemaking ratemaking that covers claims that exceed some high attachment point, e.g., personal umbrella a. Data usually volatile and has low volume b. Because of low volume, reliance on loss costs below the attachment point c. Losses may be slow to develop and trend is usually higher IV. METHODS FOR DEVELOPING COMPLEMENTS OF CREDIBILITY FIRST DOLLAR RATEMAKING A. Loss Costs of a Larger Group That Includes the Group Being Rated 1. Example: use experience of a region to supplement that of a state 2. Evaluation a. Since larger, likely to have a lower process variance b. But since less homogenous, likely to be biased, but adjustment may be possible c. Independent if excludes the subject experience; if included, should not dominate d. Data usually readily available e. Easy to compute f. If all risks have something in common, logical connection exists B. Loss Costs of a Larger Related Group 1. Example: use experience of houses to supplement that of condos 2. Evaluation a. Similar to using a larger group b. Generally biased but maybe can reduce; adjustment may be difficult to explain c. Independent d. Data likely readily available e. Same calculations as for the base statistic f. If close relationship, logical connection exists C. Rate Change from the Larger Group Applied to Present Rates 1. Complement C = (Current Loss Cost of Subject Experience)(Large Group Indicated Loss Cost) Larger Group Current Average Loss Cost 2. Evaluation a. Largely unbiased b. Likely to be accurate over the long term c. Independence depends on the sizes of the data d. Data likely readily available e. Easy to compute f. In many cases, logical connection exists

Werner 12 335 D. Harwayne s Method 1. Used when subject and related experiences have significantly different distributions, e.g., using regional data with overall different cost levels 2. Steps in the calculation of the complement for a state a. Calculate the state s average pure premium b. Calculate the average pure premium for other states using the first state s class distribution c. Calculate adjustment factors for each state as the ratio of the first state s average pure premium to that for the particular state d. Apply the adjustments to each state s loss cost for a particular class e. Calculate a weighted average of the adjusted loss costs for the particular class in the different states 3. Adjustment factor for state B F B LA, where ˆL B F B - adjustment factor for state B L ˆB - reweighted average pure premium for state B using state A s distribution 4. Adjusted loss cost for class 1 in state B L ˆ 1, L B 1, F B B 5. Complement for class 1 in state A using adjusted experience of states B and C C = (L^1,B )(X 1,B ) + (L^1,C )(X 1,C ) X 1,B + X 1,C, where X 1,B - exposures in class 1 in state B 6. Evaluation a. Unbiased b. Fairly accurate if enough countrywide data to minimize the process variance c. Mostly independent d. Data usually available e. Calculations time consuming and complicated f. Logical relationship but may be difficult to explain E. Trended Present Rates 1. Adjustments made to current rates a. Adjust to what was indicated, not what was implemented b. Adjust for trend from the original target effective date to the target effective date of the new rates, e.g., inflation, distributional shifts 2. Complement pure premium approach C = (Present Rate)(Loss Trend Factor)(Previously Indicated Loss Cost) Loss Cost Implemented with Last Review

336 Werner 12 3. Complement loss ratio approach C = (Loss Trend Factor)(Prior Indicated Rate Change Factor) (Premium Trend Factor )(Prior Implemented Rate Change Factor) 4. Evaluation a. Accuracy depends on PV of historical loss costs b. Unbiased c. Independence depends on whether data overlap d. Data readily available e. Easy to calculate f. Easily explainable F. Competitor s Rates 1. Used by new or small companies as larger data has less process error 2. Evaluation a. Inaccuracy may be attributable to differences in 1) Marketing considerations 2) Judgment 3) Regulatory effects 4) Underwriting 5) Claims practices b. Independent c. Data may be difficult or time consuming to obtain d. Easy to calculate e. Generally accepted by regulators V. METHODS FOR DEVELOPING COMPLEMENTS OF CREDIBILITY EXCESS RATEMAKING A. Increased Limits Factors (ILFs) 1. Requires ground-up losses through the attachment point 2. Complement C = L A ( ILF A+L ILF A ILF A ) = L A ( ILF A+L ILF A 1), where L A - losses capped at the attachment point A ILF A - increased limits factor for the attachment point A ILF A+L - increased limits factor for the sum of the attachment point A and the excess insurer s limit of liability L 3. Evaluation a. Biased if loss distribution of subject experience differs from that for the ILF but still may be the best available estimate b. Independent of base statistic as involves parameter error, not process error c. Practical if ILFs available d. Logically related to data below the attachment point rather than data above it

Werner 12 337 B. Lower Limits Analysis 1. Use losses capped at a limit lower than the attachment point because of data sparseness 2. Complement C = L d ( ILF A+L ILF A ILF d ), where L d - losses cost capped at the lower limit d ILF A - increased limits factor for attachment point A ILF A - increased limits factor for the lower limit d ILF A+L - increased limits factor for the sum of the attachment point A and the excess insurer s limit of liability L 3. Evaluation a. Uncertain whether more or less accurate than using ILFs as capped losses may provide more stability b. More biased when using losses truncated at lower levels c. Generally independent d. Data at other than the base limit and attachment point may not be as available e. Calculations not more difficult than using ILFs f. Logically related to data below the lower limit rather than data above it C. Limits Analysis 1. Where different policy limits available, perform analysis separately at each coverage limit; use premium volume and ELR for each layer 2. Complement C = (LR)( P d )( ILF min(d,a+l) ILF A ILF d A d ), where LR - total limits loss ratio P d - total premium for policies with limit d 3. Evaluation a. Biased and inaccurate like the prior two methods plus it assumes ELR is constant by limit b. May be the only available method c. Time consuming but easy to calculate d. Like prior two methods, not based on actual data from layer to be priced

338 Werner 12 D. Fitted Curves 1. Reasons for use a. Smooth out volatility b. Extrapolate to higher limits 2. Percentage of total losses in the excess layer % Losses in Layer (A, A + L) = A+L A (x A) f(x) dx + x f(x) dx A+L L f(x) dx 3. Apply percentage to total limits loss costs to produce expected losses in a layer 4. Evaluation a. Less biased and more stable than other methods if fitted curve generally fits the data b. More accurate than other methods if there are few claims c. Less independent than other methods d. Data may not be readily available e. Computationally complex and may be difficult to explain f. Logically related to the data used unlike other methods VI. CREDIBILITY WHEN USING STATISTICAL METHODS A. Statistical Diagnostics of the Meaningfulness of Model Results 1. Standard errors of the parameter estimates 2. Standard deviance tests, e.g., chi-square 3. Consistency of model results over time B. Statistical Diagnostics of the Appropriateness of Model Assumptions 1. Deviance residual plots 2. Leverage plots C. Comments 1. Results of multivariate analysis usually not weighted with univariate actuarial estimates 2. May be possible to incorporate Bayesian analysis

Werner 12 339 PAST CAS EXAMINATION QUESTIONS A. General A1. What are the three essential conditions for establishing an individual risk credibility system for experience rating? (69S 8 2 6) A2. Given the following information, what credibility has been given to the loss experience in the calculation of the indicated rate level change? Experience period loss ratio.750 Expense and profit provision.400 Indicated rate level change +10.0% A..20 B..30 C..40 D..50 E..60 (73 5 9 1) A3. Werner and Modlin list three conditions that a credibility function should satisfy. a. List each of these criteria. b. Consider the following example: Risk #1 has $1,000 of expected losses, risk #2 has $10,000 of expected losses, and each risk incurred $5,000 of actual losses during the experience rating period. Illustrate the effect of each of these criteria on the risks experience-rated premium. (78 9 17 1/3) A4. According to Werner and Modlin, which of the following conditions should be satisfied with respect to a credibility formula in an experience rating plan? 1. The credibility should not be less than zero, nor greater than unity. 2. The credibility should increase as the risk size increases. 3. As the size of risk increases, the percentage charge for any loss of a given size should decrease. A. 1,2 B. 1,3 C. 2,3 D. All should be satisfied. E. None should be satisfied. (79 9 12 1) A5. a. Werner and Modlin describe three conditions that Z should satisfy. State the conditions and describe them mathematically. b. Illustrate how these conditions work by describing the relative impact of a $10,000 loss on two risks, one with $2,000 of expected losses and the second with $17,000 of expected losses. (80 9 19 3/3) A6. Werner and Modlin give three conditions that the credibility Z should satisfy. List these conditions. (84 9 27a 2) A7. List the three conditions that should always be satisfied by the credibility factor in an experience rating plan. (87 9 33 1.5) E 2 A8. Does the formula Z = E 2 meet the three criteria that, according to Werner and Modlin, all + 1000 credibility formulas must meet? Prove your answer. (88 9 15 1.5)

340 Werner 12 Solutions are based on p. 216 plus the pages cited. A1. 1) Credibility should not be less than zero and not greater than unity: 0 Z 1. 2) Credibility should increase as the size of risk increases: dz/dx > 0. 3) As the size of risk increases, the percentage charge for a loss of a given size should decrease: d(z/x)/dx < 0. A2. Assume that he complement of credibility has been given to no change. Observed Experience = (L + E L + E F )/P C 1 (V + Q T ) 1 =.75 1.400 1 =.25 Z = Estimate Observed Experience =.10/.25 =.40, pp. 143 45, 223. A3. a. See A1. A4. D. b. For each risk, the $5,000 loss will result in an increase in its experience-rated premium. 1) The first criterion requires that the increase be positive (or zero) and not greater than the equivalent of the $5,000 loss ($5,000/ELR). If credibility were greater than one, the insured would be penalized more from a premium increase than the loss itself, undermining the purpose of insurance. If credibility were less than zero, an insured would be rewarded for a loss. 2) The second criterion requires that the addition to the modification formula be larger for the risk with $10,000 of expected losses. This means that a greater percentage of the loss will be considered primary for the larger risk, reflecting the greater likelihood that he is responsible for the loss. 3) The third criterion requires that the addition be a larger percentage of the premium for the risk with $1,000 of expected losses. The smaller risk will have its modification increased more since the same percentage increase has a greater dollar effect on the larger risk. A5. a. See A1. b. See A3b. A6. See A1. A7. See A1. A8. 1) Yes. Because both the numerator and denominator are always positive, and the denominator is always greater than the numerator, 0 < Z < 1. 2) Yes. dz/de = 2,000E/(E 2 + 1,000 ) 2 > 0 because E > 0. 3) No. d(z/e)/de = (1,000 E 2 )/(E 2 + 1,000 ) 2 and this is not always less than zero.

Werner 12 341 A9. Werner and Modlin state three conditions for credibility (Z): 1) 0 < Z < 1 2) Z is positive. 3) ( Z / E) is negative. Using the following example, explain why they state that Z should generally satisfy these conditions: Two risks each have a claim of $5,000. One risk has expected losses of $10,000, whereas the other risk has expected losses of $50,000. (89 9 22 1) A10. According to Werner and Modlin, in Basic Ratemaking, the credibility used to weight the actual and expected components in the rate calculation must meet three criteria. State each of the criteria and give its mathematical expression. (92 6 46 2) A11. Based on Werner and Modlin s Basic Ratemaking, state the three criteria that credibility must meet. (99 6 35a 1.5) A12. Based on Werner and Modlin in Basic Ratemaking, which of the functions in the table below define appropriate credibility rules? Assume that full credibility for a claim set is assigned when five or more claims have been observed. Number of Claims f 1 f 2 f 3 1.20.42.04 2.40.60.16 3.60.76.36 4.80.89.64 5 1.00 1.00 1.00 A. f 1 B. f 2 C. f 1, f 2 D. f 1, f 3 E. f 1, f 2, f 3 (05 5 17-1) A13. In Basic Ratemaking, Werner and Modlin discuss three criteria for credibility. a. State the criteria. b. Assess whether F(x) = X 2, 0 X 1, meets the criteria in a. Show all work. (07 5 38.75ea.)

342 Werner 12 A9. See A3b. A10. See A1. A11. See A1. A12. Since the percentage change in credibility for any loss of a given size should decrease as the size of risk, which can be represented by the number of claims, increases, the differences in the succeeding values of each function should decrease as the number of claims increases. This occurs only in the case of f 2. A13. a. See A1. b. 1) Yes. If 0 X 1, 0 X 2 1. 2) Yes. dz/dx = 2X > 0 because X > 0. 3) No. d(z/x)/dx = 1, which is not less than zero.