CASUALTY ACTUARIAL SOCIETY. Exam 9. Advanced Ratemaking, Rate of Return, and Individual Risk Rating Plans

Transcription

1 Mary Frames Miller Vice President-Adm ssions ThomaSG.Myers Chairpenon Examih~on committee November 1,200O CASUALTY ACTUARIAL SOCIETY Exam 9 Advanced Ratemaking, Rate of Return, and Individual Risk Rating Plans Exmhstion - Beth Genel-atol97cels Jeftrey A Englander E. Fiild Larry A Haefner Glenn G. Mqers Arlene F. Woodruff Richard P. Yocius 4 HOURS INSTRUCTIONS TO CANDIDATES 1. This 100 point examination consists of 48 questions divided into two sections. Section I contains 20 multiple choice questions worth one point each. Section II contains 28 problem and essay questions worth a total of 80 points. 2. To answer the multiple choice questions, use the short-answer card provided and a number 2 or FlB pencil. Mark your short-answer card during the examination period. No additional time will be allowed for this after the exam has ended. Please make your marks dark and fill in the spaces completely. Fill in that it is Fall 2000, and the exam number 9. Darken the spaces corresponding to your Candidate ID number. Five rows are available. lf your Candidate ID number is fewer than 5 digits, include leading zeros. (For example, if your Candidate ID number is 987, consider that your Candidate ID number is 00987, enter a zero on the first row, a zero on the second row, 9 on the third row, 8 on the fourth row, and 7 on the fifth [last] row.) Please write in your Candidate ID number next to the place where you darken the spaces for your Candidate ID number. Your name, or any other identifying mark, must not appear on the shortanswer card. For each of the multiple choice questions, select the one best answer and fill in the corresnonding letter. One quarter of the noint value of the question will be subtracted for each incorrect answer. No points will be added or subtracted for resnonses left blank. 3. For the problem and essay questions, the number of points for each full question or part of a question is indicated at the beginning of the question or part. Answer these questions on the lined sheets provided in your Examination Envelope. Use dark pencil or ink. Write your Candidate ID number and the examination number, 9, at the top of each answer sheet. Your name, or any other identifying mark, must not appear. Do not answer more than one question on a single sheet of paper. Write on onlv the lined side of the paper, and be careful to give the number of the question you are answering on each sheet. The answer should be concise and confined to the question as posed. When a list of a specific size is requested, do not offer more items in your list than the number requested. For example, if you are requested to list three items, only the first three responses will be graded. CONTINUE TO NEXT PAGE OF INSTRUCTIONS Casualty Actuarial Society

2 In order to receive full credit or to maximize partial credit on mathematical and computational questions, you must clearly outline your approach in either verbal or mathematical form, showing calculations where necessary. 4. Do all problems until you reach the last page of the examination where END OF EXAMINATION is marked. 5. Your Examina tion Envelope is pre-labeled with your Candidate ID number, name, exam number, and test center. Do not remove this label. Keep a record of your Candidate ID number for future inquiries regarding this exam. 6. At the beginning of the examination, check through the exam booklet for any missing or defective pages. The supervisor has additional exams for those candidates who have defective exam booklets. 7. Candidates must remain in the examination center until two hours after the start of the examination. You may leave the examination room to use the restroom with permission from the supervisor. To avoid excessive noise during the end of the examination, candidates may not leave the exam room during the last fifteen minutes of the examination. 8. At the end of the examination, place the short-answer card and all answer sheets in the Examination Envelope. Please insert your answer pages in your envelope in question number order. Insert a numbered page for each question, even if you have not attempted to answer that question. BEFORE YOU TURN THE EXAMINATION ENVELOPE IN TO THE SUPERVISOR, BE SURE TO SIGN IT IN THE SPACE PROVIDED ABOVE THE CUT-OUT WINDOW. Anvthinn written in the examination booklet will not be graded. Only the short-answer card and the answer sheets will be graded. 9. If you have brought a self-addressed, stamped envelope, you may put the examination booklet and scrap paper inside and submit it separately to the supervisor. It will be mailed to you. (Do not put the self-addressed stamped envelope inside the Examination Envelope.) If you do not have a self-addressed, stamped envelope, please place the examination booklet in the Examination Envelope and seal the envelope. You take it with you. Do not put scrap paper in the Examination Envelope. The supervisor will collect your scrap paper. Candidates may obtain a copy of the examination by contacting the CAS Office. All extra answer sheets, scrap paper, etc., must be returned to the supervisor for disposal. 10. Candidates must not give or receive assistance of any kind during the examination. Any cheating, any attempt to cheat, assisting others to cheat, or participating therein or other improper conduct will result in the Casualty Actuarial Society disqualifying the candidate s paper, and such other disciplinary action as may be deemed appropriate within the guidelines of the CAS Policy on Examination Discipline. CONTINUE TO NEXT PAGE OF INSTRUCTIONS

3 1. An examination survey and postage-paid reply envelope are included with the examination. No postage is necessary for surveys mailed within the United States. Candidates mailing the survey outside the United States should use the courtesy reply envelope distributed by your exam supervisor. Please complete the survey and leave it with the examination supervisor, or take the survev and envelope with you when leaving the examination center. Please submit the survey to the CAS Office bv November Please do not enclose the survey in the Examination Envelope. END OF INSTIWCTIONS

4 EXAM 9 - FALL SECTION I SECTION I, QUESTIONS 1-20, MULTIPLE CHOICE QUESTIONS (1 POINT EACH) 1. You are considering three separate experience rating plans. The table below shows manual and standard loss ratios for each plan, with risks grouped based on the prior year s experience modification using that plan. Based on Venter s Experience Rating - Equity and Predictive Accuracy and the following table, which of the following are true? Based on Prior Year *Loss ratios for current year, indexed to Plan 1 is the best at identifying underlying risk differences k B. C. D. E. Plan 2 is the best at correcting for the risk differences Plan 3 is too responsive to the experience. 2 3 I,2 I,3 I, 273 it identifies. PAGE-l-

5 EXAM 9 - FALL SECTION I 2. Based on the ISO s Experience and Schedule Bating Plans Applicable to General Liability, which of the following procedures for determining an insured s basic limits loss costs are recommended when the insured s actual exposures have been subject to dramatic changes during or since the experience period? 1. Schedule Bating Method 2. Present Average Company Bate Method 3. Historical Exposures at Present Company Bate Method A. 1 B. 2 c. 1,3 D. 2,3 E. I, 233 PAGE-2-

6 EXAM 9 - FALL SECTION I 3. According to Venter s Experience Rating - Equity and Predictive Accuracy, which of the following are true? 1. Large insureds display more variation in loss ratio than would be predicted by the law of large numbers. 2. Excess losses are more predictable than total losses. 3. For a multi-split experience rating plan, the experience modification is calculated by minimizing the conditional expected value of primary and excess losses separately. A. 1 B. 3 c. 1,2 D. 2,3 E. I, 293 PAGE-3-

7 EXAM 9 - FALL SECTION I 4. Using the split plan formula from Gillam and Snader s Vundamentals of Individual Risk Batingn and the following data, calculate the experience modification. Z,.500 Z,.400 E, $2,000 E $5,000 Ap $6,000 A $7,500 A. Less than.80 B. At least.80, but less than.95 c. At least.95, but less than 1.10 D. At least 1.10, but less than 1.25 E. At least 1.25 PAGE-4-

8 EXAM 9 - FALL SECTION I 5. You are given the following information: Standard Premium Expected Losses Expected Losses Reflecting an Occurrence Loss Limitation $100,000 $60,000 $54,000 Based on Gillam and Snader s Fundamentals of Individual Risk Rating, calculate the adjusted expected losses used to determine the appropriate Table M column. A. B. C. D. E. Less than $71,500 At least $71,500, but less than $72,500 At least $72,500, but less than $73,500 At least $73,500, but less than $74,500 At least $74,500 PAGE-5-

9 EXAM 9 - FALL SECTION I 6. You are given the following information: 1 Stock Carrier Discount Table 1 1 Over $500,000 1 Based on Gillam and Snader s Fundamentals of Individual Risk Rating, calculate the guaranteed cost premium. A. B. C. D. E. Less than $275,000 At least $275,000, but less than $300,000 At least $300,000, but less than $325,000 At least $325,000, but less than $350,000 At least $350,000 PAGE-6-

10 EXAM 9 - FALL SECTION I 7. According to Gillam s Retrospective Rating: Excess Loss Factors, which of the following are true for calculating excess loss factors? A. B. C. D. E. When choosing between alternate loss distributions, how well the curve fits in the tail is less important than the fit around the mean and median. The permissible loss ratio is calculated by dividing the target cost ratio by the loss adjustment expense factor. The flat loading becomes a more significant portion of the fmal excess loss factor for higher limits. 1 3 I,2 I,3 293 PAGE-7-

11 EXAM 9 - FALL SECTION I You have been asked to evaluate three lines of business and determine which lines will produce an underwriting profit. Using the basic form of Fairley s CAPM model, as discussed in D Arcy and Dyer s &Ratemaking: A Financial Economics Approach and the following information, which line(s) require a positive underwriting profit margin? The Risk-free Rate of Return 5% The Market Risk Premium 4% 1 Workers Compensation ] 1. Medical Malpractice 2. Homeowners 3. Workers Compensation A. 2 B. 1,2 c. 1,3 D. 2,3 E. None of 1,2, or 3 require a positive underwriting profit margin. PAGE-8-

12 EXAM 9 - FALL SECTION I 9. In Determining the Proper Interest Rate for Loss Reserve Discounting: An Economic Approach, Butsic presents a Certainty Model for reserve valuation under which a hypothetical asset is selected as the basis for determining the interest rate to be used for discounting. Based on Butsic, which of the following are criteria for selecting the hypothetical asset? 1. The asset has a yield rate equal to the imbedded yield in the insurer s portfolio. 2. The asset has zero default risk. 3. The asset has the same duration as the loss payment. A. 1 B. 3 c. 1,2 D. 2,3 E. I, 293 PAGE-g-

13 EXAM 9 - FALL SECTION I 10. Based on Roth s Analysis of Surplus and Rate of Return Without Using Leverage Ratios and the following information, calculate the ratio of the required rate of return to the actual rate of return. Average Annual Increase in the Deflated Reserves 8% Increase in Demand for Insurance 5% Exnense and Claims Inflation 4% Actual Surplus Change Surplus Paid In Stockholder Dividend Percentage of Surplus 12% 2% 5% A. Less than 0.9 B. At least 0.9, but less than 1.0 C. At least 1.0, but less than 1.1 D. At least 1.1, but less than 1.2 E. At least 1.2 PAGE-lO-

14 EXAM 9 - FALL SECTION I 11. Roth, in Analysis of Surplus and Rate of Return Without Using Leverage Ratios, considers different components of the Total Economic Income. Based on Roth, what is the correct formula for the Retained Return of Capital? A. Change in Statutory Surplus minus Stockholder Dividends B. Change in Statutory Surplus minus Surplus Paid-in C. Total Economic Income minus Surplus Paid-in D. Total Economic Income minus Change in Statutory Surplus E. None of the above are true. PAGE-ll-

15 EXAM 9 - FALL SECTION I 12. According to Robbin s The Underwriting Profit Provision, which of the following are true? 1. Calendar year data plays a key role in the risk-adjusted discounting method. 2. A fair premium is obtained directly in the present value cash flow return model. 3. The internal rate of return on equity flows model is not constrained by accounting conventions. A. 1 B. 2 c. 3 D. 2,3 E. None of 1,2, or 3 are true. PAGE

16 EXAM 9 - FALL SECTION I 13. Based on Stone s A Theory of Capacity and the Insurance of Catastrophe Risks, which of the fouowing statements are true? 1. The level of stability constraints is usually set by state regulators. 2. The exposure ratio is the coefficient of variation of expected insured loss. 3. By writing deductible policies, as opposed to writing fuii coverage policies with the same limits, an insurance company can decrease its exposure ratio. A 2 B. 1,2 c. 1,3 D. 2,3 E. None of 1,2, or 3 are true. PAGE

17 EXAM 9 - FALL SECTION I 14. Based on Stone s A Theory of Capacity and the Insurance of Catastrophe Risks and the information below, what is the new exposure ratio for an insurer after five new policies have been added to the portfolio? Assume that all losses are independent. I Original Portfolio of Policies Number of Policies 25 Total Expected Losses $12,000 Exuosure Ratio for the Portfolio 10% New Policies Number of Policies Expected Loss per Policy Exposure Ratio per Policy 5 $750 40% A. B. C. D. E. Less than 0.10 At least 0.10, but less than 0.15 At least 0.15, but less than 0.20 At least 0.20, but less than 0.25 At least 0.25 PAGE

18 EXAM 9 - FALL SECTION I 15. According to Holler et al. in Something Old, Something New in Classification Ratemaking with a Novel Use of GLMs for Credit Insurance, which of the following are goals of a premium rating plan? 1. Ensure that the insurer receives premiums at a level that is expected to be adequate. 2. Ensure that rates are competitive in the marketplace. 3. A. B. C. D. E. Allocate premiums 2 3 I, 2 I,3 I, 273 fairly between insureds. PAGE

19 EXAM 9 - FALL SECTION I 16. According to the American Academy of Actuaries Committee on Risk Classification s Wsk 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. PAGE

20 EXAM 9 - FALL SECTION I 17. Based on ISO s Experience and Schedule Rating Plans Applicable to General Liability, NCCI s CLExperience Rating Plan Manual for Workers Compensation and Employers Liability Insurance, and Gillam s Workers Compensation Experience Rating: What Every Actuary Should Know, which of the following are true regarding the General Liability and the Workers Compensation experience rating plans? 1. Both plans calculate expected losses at an undeveloped level. 2. Both plans detrend the expected losses so that they are at the same level as the experience period. 3. Neither plan gives full credibility to excess losses. A. 1 B. 3 c. 1, 2 D. 2, 3 E. I, 2, 3 PAGE-17-

21 EXAM 9 - FALL SECTION I 18. According to Miccolis u On the Theory of Increased Limits and Excess of Loss Pricing, which of the following are true? 1. Given a set of increased limits factors and the average basic limits severity, one can theoretically determine the underlying loss severity distribution. 2. The risk adjustment for a set of increased limits factors can be determined from the loss severity distribution underlying the increased limits factors. 3. Given a set of risk-adjusted increased limits factors, one can determine the risk adjustment for excess of loss coverage. A 2 B. 3 c. 1,2 D. 2,3 E. 1,2,3 PAGE-18-

22 EXAM 9 - FALL SECTION I 19. Using Lee s The Mathematics of Excess of Loss Coverages and Retrospective Rating - A Graphical Approach and the following graph, answer the following question. If the expected losses are equal to $400, what is the probability exactly $300? of a loss of Entry Ratio (r) oo Cumulative Distribution Function A. 15% B. 35% C. 40% D. 55% E. 60% PAGE

23 EXAM 9 - FALL SECTION I 20. According to Gillam and Snader s Fundamentals of Individual Risk Rating, which of the following are true for calculating loss elimination ratios? A safety factor is required. For deductible insurance, a portion of loss adjustment eliminated along with the rest of the losses. expenses is 3. A B. C. D. E. When the ex-medical discount is derived, the portion of medical pure premium eliminated is determined by judgment. 1 3 I,2 I,3 273 PAGE

24 EXAM 9 - FALL SECTION II SECTION II, QUESTIONS 21-48, WRITTEN ANSWER QUESTIONS 21. (2 points) a. Using the cash flow model in D Arcy and Dyer s Watemaking: A Financial Economics Approach and the data below, determine the premium for an annual policy. Show all work. Losses and LAE Underwriting Expenses Taxes Premium Timing of Payment End of the First Year Policy Inception Expiration Policv Inceution Amount $750 $ Risk-adiusted Discount Rate 2% I Risk-free Discount Rate 6% Marginal Tax Rate 34% Equity Supporting the Policy $300 I (1 point) b. List four enhancements to the cash flow model used in part a. that would make the model more realistic. PAGE - 21-

25 EXAM 9 - FALL SECTION II 22. Based on Butsic s Determining the Proper Interest Rate for Loss Reserve Discounting: An Economic Approach, answer the following. (1 point) a. According to Butsic, should an insurer s own assets be used to determine the interest rate used for discounting the insurer s loss reserves? State two of Butsic s reasons. (0.5 point) b. Butsic states that it is reasonable that the risk adjustment for determining the proper interest rate for loss reserve discounting is negative, whereas the risk adjustment in the Capital Asset Pricing Model is positive. Briefly state his rationale. PAGE

26 EXAM 9 - FALL SECTION II 23. (3 points) You are the consulting actuary for a small insurance company attempting to determine the impact of discounting at the risk-free interest rate. Based on Butsic s UDetermining the Proper Interest Rate for Loss Reserve Discounting: An Economic Approach and the following information, calculate the difference in the actual after-tax return on equity between discounting at the risk-free interest rate and discounting at the risk-adjusted interest rate. Show all work. Premium $250 Loss Expected to Be Paid One Year from Now $255 Tax Rate 35% Required Percentage of Reserves to Be Held as Equity Required Return on Equity Expected Return on Equity Interest Rate on Government Bonds 30% 15% 15% 5% PAGE

27 EXAM 9 - FALL SECTION II 24. As a consulting actuary, you have been approached by the management of OneLine Insurance Company. OneLine underwrites only one line of business. You have been provided with the following information: Expected Underwriting Profit on Premium -12.6% Investment Yield on Total Assets 6.5% Current Reserve-to-Premium Ratio 1.8 Based on Ferrari s The Relationship of Underwriting, Investment, Leverage, and Exposure to Total Return on Owner s Equity, answer the following questions. (1 point) a. After an analysis based on Ferrari s methodology, OneLine s management is about to reduce the volume of the company business. Would you agree with their decision? Briefly explain. (1 point) b. After reviewing OneLine s book of business, you believe that company reserves are about 15% inadequate. How would your answer to part a. above change, if at all, in light of this finding? Briefly explain. PAGE

28 EXAM 9 - FALL SECTION II 25. Buts& in LLDetermining the Proper Interest Rate for Loss Reserve Discounting: An Economic Approach, describes the method for calculating the risk-adjusted interest rate to discount uncertain loss reserves. (3 points) a. Using the following information and Butsic s methodology,. determine the appropriate risk-adjusted interest rate to discount loss reserves. Show all work. Uncertain Loss with an Expected Value to be Paid in Two Years Dividend Paid at End of First Year Risk-free Yield Rate Required Equity as a Percent of the Discounted Reserves $ $ % 33.3% (1 point) b. What is the required return on equity? Show all work. PAGE

29 EXAM 9 -FALL SECTION II 26. (1 point) a. You have been asked to determine a target underwriting profit provision for future automobile insurance business. Based on Robbin s The Underwriting Profit Provision and the information below, determine a target before-tax underwriting profit provision using the calendar year return on equity method. Show all work. Premium to Surplus Ratio 2.5 Equity to Surplus Ratio 1.2 After-tax Return on Invested Assets 1 8% 1 (1 point) b. (1 point) c. List two similarities and two differences between the calendar year return on equity method and the calendar year investment income offset method. Based on Roth s Analysis of Surplus and Rate of Return Without Using Leverage Ratios,n list two potential problems that apply to both methods identified in part b. above. PAGE

30 EXAM 9 - FALL SECTION II 27. Based on Feldblum s LCPricing Insurance Policies: The Internal Rate of Return Model and the information below, answer the following. Assume a portfolio of annual policies with a common effective date of January 1,200O. Premium $1 oo,ooo,ooo Required Undiscounted Reserve to Surplus Ratio Loss Ratio 60% Expense Ratio 35% Tax Rate (i.e. Ignore the Effect of Taxes) 0% Insurer s Return on Assets (Nominal Reserve Plus Surplus) per Year 5% Date of Event January 1,200o December 31,200o December 31,200l December 31,2002 Surplus Required at this Date to Support Unearned Premium Reserves Loss Reserves Loss Reserves Loss Reserves if any Items Paid or Collected at this Date Premium Expenses, l/3 of Losses, Dividends to Shareholders l/3 of Losses, Dividends to Shareholders l/3 of Losses, Dividends to Shareholders (2 points) a. Complete the following table of cash flows. Show all work. (1.5 points) b. (1 point) c. Calculate the net present value of the equity flows at January 1,200O using discount rates of 20% and 25%. Show all work. Estimate the internal rate of return of the equity flows to within one percentage point, Show all work. PAGE

31 EXAM 9 - FALL SECTION II 28. As a consulting actuary, you are approached by the Aggressive Insurance Company to calculate the required profit load for a potential General Liability policy for an insured. You are provided with the following information: Aggressive s Surplus (V) Standard Deviation of Losses for Aggressive s Current Portfolio (S) Expected Return on Aggressive s Current Portfolio (R) Standard Deviation of the Insured s Losses (0) Correlation between Aggressive s Losses and the Insured s Losses (C) $250,000,000 $125,000,000!!40,000,000 1,000,000 10% 1 The management of Aggressive Insurance Company has decided to price new policies at 15% expected return on marginal surplus. Using the method for calculating the theory of ruin iu Bault s Insurers, answer the following. the marginal surplus for a new risk based on discussion of Feldblum s Risk Loads for (1 point) a. Calculate z, the Standard Normal percentile value associated with Aggressive s probability of ruin. Show all work. (2.5 points) b. Calculate (V - V), the amount of marginal surplus needed to support a potential General Liability policy for the insured. Show all work. (1 point) c. Calculate the minimum acceptable expected return r in dollars for the insured s policy. Show all work. PAGE

32 EXAM 9 -FALL SECTION II 29. Stone, in A Theory of Capacity and the Insurance of Catastrophe Risks, describes the effect on industry capacity of three types of self-insurance: increased deductibles, complete self-insurance, and vertical participation iu the risk by the insured. Based on Stone, answer the following questions. (1 point) a. What is the likely effect of complete self-insurance on industry capacity? Briefly explain why. Under what circumstances would this have the opposite effect? (1 point) b. Under what circumstances will vertical participation in the risk by the insured decrease industry capacity? Under what circumstances will it increase industry capacity? PAGE

33 EXAM 9 - FALL SECTION II 30. Based on Feldblum s Risk Loads for Insurers: answer the following. (1 point) a. What are two problems that financial analysts perceive with the Capital Asset Pricing Model (CAPM)? (0.5 point) b. List two risk load methods other than CAPM. PAGE

34 EXAM 9 - FALL SECTION II 31. You are given the following information for a classification plan consisting of two territories and two classes. I Number of Exuosures You are asked to develop a classification plan using the multiplicative Rj = XYj, where: model X = Rating factor for Terri Yj = Rating factor for ClaSSj Rj = Relative 10~s cost for Terr, & Class1 Assume that frequency follows a Poisson distribution constant across all territory and class combinations. and that severity is Based on the method described by Holler et al. in Something Old, Something New in Classification Ratemaking with a Novel Use of GLMs for Credit Insurance,n answer the following. (0.5 point) a. Show the formula to iteratively solve for the territory rating factor X. (0.5 point) b. Calculate the grand total frequency. (2 points) c. Solve for the values of X, and X, resulting from one iteration. Use initial values of Y, = 1.0 and Y2 = Show all work PAGE - 31-

35 EXAM 9 - FALL SECTION II 32. Based on Bailey and Simon s An Actuarial Note on the Credibility of Experience of a Single Private Passenger Car and the table below, answer the following. Private Passenger Automobile Liability Class 3 - Business Use - Non-Farmers Earned Claim Premium at Number of Frequency per Relative Merit Earned Present B Claims $1,000 of Claim Rating Car Years Rates Incurred Premium Frequency A 247,424 $25,846,000 31, X 15,868 $1,783,000 2, Y 20,369 $2,281,000 3, B 37,666 $4,129,000 7, Total 321,327 $34,039,000 45, where: Class A - Three or more years claim free Class X - Two years claim free Class Y - One year claim free Class B - Zero years claim free (1.5 points) a. Calculate the credibilities for a single private passenger car for one year, two years, and three years. Show all work. (0.5 point) b. Briefly describe the relationship that Bailey and Simon expect between the three credibilities from part a. (1 point) c. Do the credibilities calculated in part a. follow the relationship described in part b.? Briefly explain why or why not. PAGE

36 EXAM 9 - FALL SECTION II 33. (2 points) As an actuary for a personal lines insurer, you routinely apply two-way class relativity analysis as described in Holler et al. in Something Old, Something New in Classification Ratemaking with a Novel Use of GLMs for Credit Insurance. You have accurate historical loss and premium data for all class combinations, so you decide to use loss ratios instead of loss cost data. Using the procedure described by Holler and the data below, calculate the underlying loss relativities by class. Show the work for each step. Class Description Old Drivers Young Drivers Losses by Class Large Cars $1,200 $2,000 Small Cars $1,500 $2,700 Premiums by Class Class Description Large Cars Small Cars Old Drivers $1,600 $2,400 Young Drivers $4,000 $3,000 Current Relativities by Driver Age Old Drivers 1.00 Young Drivers 1.50 Current Relativities by Car Size Large Cars 1.00 Small Cars 1.20 PAGE

37 EXAM 9 - FALL SECTION II 34. Answer the following based on Mahler s An Example of Credibility and Shifting Risk Parameters. (1.5 points) a. Briefly describe three criteria used to compare the performance of credibility methods. (0.5 point) b. Mahler states that one criterion differs from the other two criteria on a conceptual level. Which criterion is that? Briefly state in what way it differs from the others. PAGE

38 EXAM 9 - FALL SECTION II 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. (0.5 point) a. Briefly describe adverse selection. (1.5 points) b. Briefly explain the two methods described for controlling adverse selection. PAGE

39 EXAM 9 - FALL SECTION II 36. (3 points) There is a group of people that would each like to purchase life insurance with a policy limit of $1,000. Each person is either high-risk or low-risk, where the probability that a high-risk person dies is twice that of a low-risk person. In Risk Classification in Life Insurance, Cummins et al. discuss different types of equilibrium. Refer to Cummins to answer the following. Consider the following three scenarios: I. II. III. There is perfect information, and insurers are permitted to classify insureds. In this case, the total aggregate premium collected from all the insureds is $1,500. A Nash separating equilibrium exists in the case in which there are independent firms and no classification. In this scenario, the total aggregate premium collected from all the insureds is $900. Life insurance is compulsory, and insurers charge the average (pooled) rate to all insureds. Each insured must purchase $1,000 of coverage. In this case, the total premium collected from all the lowrisk insureds is $1,200. In scenario II, calculate both the total premium paid by the high-risk insureds and the total premium paid by the low-risk insureds. Show all work. PAGE

40 EXAM 9 - FALL SECTION II 37. (3 points) The liability inflation rate is estimated at 13%, primarily due to increasing medical costs. In response, you file a 7% increase for your auto liability basic limit rates and a 20% increase for your auto liability excess limit rates. The insurance department has reviewed your filing. The department requires justification for the different rate increases requested for your basic limit rates and excess limit rates in light of the overall inflation rate. Based on Miccolis On the Theory of Increased Limits and Excess of Loss Pricing or Lee s The Mathematics of Excess of Loss Coverages and Retrospective Rating - A Graphical Approach, use formulas or graphs to justify your filing. Define any notation used. PAGE

41 EXAM 9 - FALL SECTION II 38. You are an actuary for XYZ Insurance. The Safe Construction Company (SCC) is interested in purchasing a large dollar deductible plan (LDD) for its workers compensation exposure, whereby SCC will reimburse XYZ for losses and allocated loss adjustment expenses (ALAE) up to $500,000 per accident, but will not reimburse more than $2,000,000 for the policy period. You are given the following information: Standard Premium 1 $2,000,000 ] 1 Expected Loss and ALAE 1 $1,400,000 1 Excess Loss Pure Premium Factor at $500,000 ULAE Loss Based Assessment, Assuming Loss Based Assessment Is to Be Paid on Total Loss, Not Just Excess Loss Insurance Charge at $2,000,000 General Overhead Expense Compensation for Credit Risk Acquisition Expense Premium Tax Rate Residual Market Assessment Rate Profit and Contingencv As a Ratio to Percentage Loss and ALAE 7.0% Loss And ALAE 8.0% Loss and ALAE 5.0% Loss and ALAE 6.0% Standard Premium Standard Premium Net Premium Net Premium Net Premium Net Premium 7.0% 4.0% 5.0% 3.0% 5.0% 3.0% Based on Teng s Pricing Workers Compensation Large Deductible and Excess Insurance and the information above, answer the following. (1.5 points) a. Calculate the indicated LDD premium. Show all work. (1 point) b. Calculate the expense ratio on the LDD premium. work Show all (1.5 points) c. Assume the excess loss coverage in the above LDD plan is offered through an excess workers compensation insurance policy (Excess WC) instead of the LDD program discussed above. Would you expect the indicated Excess WC premium to be higher or lower than the LDD premium calculated above? State three reasons. PAGE

42 EXAM 9 - FALL SECTION II 39. Using Finger s ~Estimating Pure Premiums by Layer - An Approach and the information below, which was derived from empirical data, answer the following. Attachment Point Excess Loss Distribution Estimated Coefficient of Variation of the Underlying Loss Distribution $500,000 66% $1,ooo,ooo 48% $2,000,000 29% $3,000,000 21% $4,000,000 17% The basic limit is $500,000. The unlimited expected loss for the policy is $1,000. (1 point) a. What is the pure premium for a policy with a limit of $3,000,000? Show all work. (1 point) b. What is the increased limit factor for a $3,000,000 limit? Show all work. (1 point) c. According to Finger, would this data be consistent with a lognormal distribution? Why or why not? PAGE

43 EXAM 9 - FALL SECTION II 40. In Giilam and Snader s LFundamentals of Individual Risk Bating, the authors describe three criteria for an effective credibility function in both qualitative and mathematical terms. Use Gillam and Snader to answer the following. (1 point) (1 point) (1 point) a. Briefly describe in words the authors three criteria. b. Show the three criteria in formulas. Define all notation used. C. For each of the three criteria, determine whether or not the following credibility function satisfies that criterion. If it does not, briefly explain why. Expected Losses Between 1 Credibility $0 - $5,000 0% $5,001 - $50,000 $50,001 or more 50% 100% PAGE

44 EXAM 9 - FALL SECTION II 41. In Gillam s Workers Compensation Experience Bating: What Every Actuary Should Know, the author describes the effects of the experience rating off-balance on NCCI s calculations of needed premium level. Use Gillam to answer the following. (1 point) a. If rate adequacy has deteriorated in a state for several successive years during a time of increasing costs, briefly explain how the off-balance affects the calculation of needed premium level for that state. (1 point) b. Briefly describe what may happen in that same state if adequate rates are subsequently approved. PAGE - 41-

45 EXAM 9 - FALL SECTION H 42. (3 points) Based on the NCCI s Experience Rating Plan Manual for Workers Compensation and Employers Liability Insurance and the following information, calculate the standard premium. Show all work. Payroll for the 3 Year Historical Period $4,200,000 Manual Premium for Prospective Period $100,000 State per Claim Accident Limit $95,000 Maximum Primary Value for Each Loss $5,000 Ballast Value 19,250 Expected Loss Bate Discount Ratio Weighting Value Losses from Experience Period Listed by Claim (Excluding Medical Only Losses) Claim # 1 $2,000 Claim # 2 $2,500 Claim # 3 $4,000 Claim#4 I $5.000 Claim # 5 Claim # 6 $25,000 $120,000 (One Person Injured) Medical Only Losses from Exnerience Period. Listed bv Claim Claim # 7 Claim # 8 Claim # 9 PAGE

46 EXAM 9 - FALL SECTION II 43. Use Gillam and Snader s Fundamentals of Individual Risk Rating and Venter s Experience Rating - Equity and Predictive Accuracy to answer the following. (1.5 points) a. For each of the experience rating plans listed below, identify how losses are split into primary and excess components for the purpose of experience rating: i. NCCI s 1991 Experience Rating Plan ii. NCCI s 1961 Experience Rating Plan (0.5 point) b. Briefly describe the reason for the change in the plan. PAGE

47 EXAM 9 - FALL SECTION II 44. An insured is deciding whether or not to be retrospectively rated. The insured is given the following information about a possible retrospectively rated program: Standard Premium $100,000 Basic Premium $20,000 Expected Losses $60,000 Converted Insurance Charge $1,950 Loss Conversion Factor Taxes as a Percent of Premium 2% Based on Gillam and Snader s Fundamentals of Individual Risk Rating, answer the following questions. (1.5 points) a. What premium will the insured pay if he decides not to be retrospectively rated? Show all work. (0.5 point) b. At what level of incurred losses will the insured pay the same premium whether or not he decides to be retrospectively rated? Show all work. PAGE

48 EXAM 9 - FALL SECTION II 45. (1 point) a. According to Skurnick s The California Table L, describe the problem that Table L was created to solve. (1 point) b. According to Gillam in his review of Skumick, how does the NCCI s Retrospective Rating Plan Manual for Workers Compensation and Employers Liability Insurance solve the problem in part a.? (0.5 point) c. List one advantage and one disadvantage of the Table L approach to solving the problem in part a. (0.5 point) d. List one advantage and one disadvantage of the NCCI approach to solving the problem in part a. PAGE

49 EXAM 9 - FALL SECTION II 46. (4 points) You are the pricing actuary for HIJ Reinsurance, Inc., which reinsures Primary Insurance Company s workers compensation book of business. Using the information below, calculate the rate per dollar of standard premium for the layer $500,000 excess of $500,000 for workers compensation losses and pro rata ALAE to be applied to Primary Insurance Company s workers compensation book of business. Show all work NCCI Excess Loss Factors Excess Loss and Allocated Exuense Factors Primary Insurance Company Average Experience Modification Factor Primary Insurance Company Manual Premium Hazard Group I II III Iv $50,000 $500,000 $600,000 $100,000 HIJ Reinsurance Expected Loss Ratio , ALAE 10% Expenses, Profit and Risk Load 25% PAGE

50 EXAM 9 - FALL SECTION II 47. You have priced the following retrospectively rated general liability program using the ISO s Retrospective Rating Plan for Automobile, General Liability, Glass and The@. Use ISO s plan to answer the following. Standard Premium $1,000,000 Basic Premium Factor Loss Conversion Factor Expected Loss Ratio Maximum Premium Factor Minimum Premium Factor Tax Multiplier (1 point) (2 points) (1 point) a. What is the standard expense ratio for this program based on IS0 advisory expense provisions? Show all work. b. The insured has questioned the basic premium factor. Briefly explain the two components of the basic premium factor and calculate how much the insured is paying for each of them. Show aii work. C. Although you have followed the IS0 retrospective rating plan, give a possible reason why both of the components in part b. might be inappropriate for this insured. PAGE

51 EXAM 9 - FALL SECTION II Based on Gillam and Snader s Fundamentals of Individual Risk Rating, and the data below, answer the following. Assume that all risks have equal standard premium. Show all work. Number of Risks Limited Loss Ratio 1 10% 4 30% 4 40% 3 50% 2 60% 1 70% 2 80% 2 90% 1 110% (0.5 point) a. Calculate the unlimited expected loss ratio and the limited expected loss ratio. (0.5 point) b. Calculate the loss elimination ratio. (1.5 points) c. (0.5 point) d. Calculate the Table M charges at loss ratios from 0% to 120% using loss ratio increments of 10%. Calculate the Table M savings for an entry ratio of END OF EXAMINATION PAGE

52 Exam 9, Fall 2000 Final Examination Answers Answers to Section I, Multiple Choice Questions 1. C 2. D 3. C 4. E 5. B 6. B 7. B 8. A 9. D 10. C 11. B 12. E 13. A 14. A 15. D 16. E 17. D 18. C 19. A 20. B and D* *updated 12/20/00 Casualty Actuarial Society, 1100 N. Glebe Road, Suite 600, Arlington, VA Phone: (703) , Fax: (703) , office@casact.org Copyright 2001 Casualty Actuarial Society. All rights reserved.

53 EXAM 9 SAMPLE ESSAY SOLUTIONS FALL 2000 Question 21 a. PV (Prem) = PV(Loss) + PV(Exp) + PV(Tax on Inv Inc) + PV(Tax on U/W) = PV(Loss) + PV(Exp) + Tax on Inv. Inc. + PV(Tax on V/W Inc.) = 750/l [(300 + P - 250)(.06)(.34)/l.06] + [(P-250)(.34)/ (.34)/l,021 P = P P P= P = $ b. 1. Vary some expenses with premium. 2. Premium collection isn t necessarily risk free - incorporate the risk. 3. Consider more than one time period. 4. Consider how equity is allocated to the policy and incorporate items such as the variability in the losses, how long the payout period is and the line of business. Question 22 a. An insurer s own assets should not be used to determine the interest rate of discounting because: 1. Don t know which assets support the liabilities therefore don t know which yield to use. 2. Two insurers having the same undiscounted liabilities can have different asset yields depending on their asset portfolio. b. CAPM is for assets; therefore to incorporate risk, the risk adjustment needs to be positive so that asset value will be less. For reserves, the adjustment is negative so that the reserves will be greater than if used risk-free to reflect risk.

54 Question 23 ri = i - e( R - i) = ( ) =.02 Initial Equity =.3(250) = 75 Risk Free (.05 1 Risk-Adiusted C.02 1 t=o t=l t=o 5 Premium Paid Loss Inc. Loss Inv. Inc VW Inc Total Inc fi.25 Taxes Equity End Assets ROE = 82.23/75-1 = 9.64% ROE = 82.31/75-1 = 9.75% Difference = 9.64% % = -.I 1% Question 24 a. Using Ferrari s formula, T/S = I/A + R/S (I/A + U/R), the underwriting process can be considered a loan, where the borrowed funds (R/S) return VA, but interest of U/R must be paid. If ] U/R ] > VA, then the interest exceeds the return, and the company should reduce the volume of business. U/R = U/P P/R = -12.6% l/l.8 = -7%. The return on assets is less than the interest rate on the loan, so I would agree with the OneLine methodology. b. If the reserve inadequacy did not impact the expected underwriting profit, U/R = -12.6% 1/[(1.8)(1.I 5)] = -6.l%, which is less than the expected return. would then change my response and disagree with the OneLine decision. I However, if reserves are found to be inadequate, it is likely that U/P is overstated (i.e., the unde r-writing loss is more than 12.6%). Without information on the reserve inadequacy, the adjusted underwriting loss can t be determined precisely. Given how close this decision is (an underwriting loss of 6.5% I 5 = % would be breakeven) and given how badly reserves were misjudged, the expected underwriting profit needs to be re-evaluated.

55 Question 25 a. Discount loss reserves at time 0 = 10,000/[(1 + ia)*] Equity = l/3 1 O,OOO/[( 1 + ia)*] At time 1: Assets = 4/3 (1 O,OOO/[ (1 + ia)*]) (1.09) = /[(1 + ia)*] Req. res = 1 O,OOO/( 1 + ia) Req. equity = l/3 + 10,000/(1 + ia) Dividend: 233 = 14,533/[(1 + ia)*] - 13,333/(1 + ia) => 233 (1 + ia)* + 13,333( 1 + ia) - 14,533 = 0, use quadratic equation to solve => 1 + ia = [-13,333 + {13, (4 233 (-14,533))} (.5) ]/[ = 1.O7 => ia = = 7% b = /3(R ) (ia = i - e(r - i) => 0.02 = 1/3(R ) => 0.06 = P => R=015 d Question 26 a. u = 1 / ( 1 - t, ) [ R (E/P) - iant{ PHSF + l/(p/s) } ] u = 1 / (1 -.28) [.I5 (1.2/2.5) -.08(.8 + l/2.5) ] u = b. Similarities: 1. Both are based on calendar year income figures from the annual statement so the data are easily obtained and verifiable. 2. Both are distorted by change in premium volume and change in reserves adequacy. Differences: 1. Calendar year return on equity method produce a return on equity comparable to GAAP return; whereas calendar investment income method is based on offset to traditional underwriting profit. 2. Calendar year return on equity method requires surplus assumptions and target return; whereas the calendar year investment income does not. c. The proper measure of income for determining profitability is the change in surplus adjusted for dividends and paid in capital (ds/ S ).

56 Question 27 a Premium Paid Loss UEPR Loss Reserve Reg Surplus U/W Inc. Inv. Inc. Net Inc. A Req Surplus Equity CF 1/1/ /31/ /31/ /31/ ,000, ,000,000 20,000,000 20,000,000 Loss=(0.6)(1OOM) 100,000, ,000,000 20,000,000 0 Req Surplus 50,000,000 20,000,000 10,000,000 0 = % Reserves 0 5,000, ,500,000 3,000, ,500,OOO 3,000,000 1,500,000-5o,ooo,ooo 30,000,000 10,000,000 10,000,000-5o,ooo,ooo 42,500,OOO 13,000,000 11,500,000 b. (in OOO SJ NPV~=XI = -50, ,500/ ,000/(1.20)* + 11,500/(1.20)3 = NPVd=25= -50, ,500/ ,000/(1.25)* + 11,500/(1.25)3 = c. IRR is y such that: /(1 + y) + 13/(1 + y)* /(1 + Y)~ = 0 By inspection of b. above, 20~~~25 since only 1 flow reversal, only 1 solution NPVdz2,= / /1.21* /1.213 = NPVdzZ2= / /1.22*+ 11.5/1.223 = IRR = 21.5% +/- 0.5% Question 28 Given: V=250m S=125m R=40m o=lm c =.I a. V=Sz-R 250m=(125m)z-40m z=2.32 b. (VI-V)=(S'- S)z- r S'- S = o(2 S C + o)/(s'+ S)= Im [z (125m)(.l)+ Im]/(S'+ S) (S')*- S* = 2.6 (1013) = (S')*-(125m)* S' = 125!103!957 V' -v = 103,957(2.32) -.15(V'- V) l.l5(v'-v)= 241,180 VI-V= 209,721 c. r = 0.15 (Y-V) = (.15)209,721 =

57 Question 29 a. The likely effect of complete self-insurance is an exacerbation of the capacity problem. This is because those risks tending to self-insure should be those who have lower exposure ratios and also because the exit of these insureds from the market reduces the number of insureds available to achieve a proper spread. Self-insurance would have the opposite effect if large risks with high exposure ratios tended to self-insure. b. If only those insureds with low exposure ratios vertically participate in the risk, then the volume of low exposure risks in the market will decrease leading to higher total industry portfolio exposure ratios => decreased industry capacity. If insureds with high exposure ratios and large size, vertically participate in the risk, then the opposite effect will take place => increased industry capacity. Question 30 a b The empirical security market line is less steeply sloped than CAPM predicts. Different O s result when different methods or different time periods are used. Probability of Ruin Method. Utility Theory Method. Question 31 a. Xi = [ Cj Nij Rij ] / [ Cj NijYj ] b. Grand total frequency = Total Claims / Total Exposures =[ ]/[ ]=0.32 C. Frequency Relativities Cl c2 Cl c2 TI TI 1.ooo T T Xl = [ 4000( 1) ( 1.5) ] / [ 4000( 1) ( 1.25) ] = x2= [ 3000(2) + looo(3) ] / [ 3000( 1) + looo(1.25) ] = Note: This is normalizing RI, to 1.O. Answer could be stated where RI, = 0.625

58 Question 32 a. Claims / Relative EPPR/$lOOO Claims EPPR/$lOOO Claim Freq. = Mod Total 34,039 45, A 25,846 31, A+X 27,629 34, A+X+Y 29,910 38, One-yr 2-yr 3-yr t=l-mod b. If the chance of accident for an individual risk remains constant and no risks enter or leave, then the credibility should vary approximately in proportion to the number of experience years. C. 1 Year 2 Year 3 Year / = / = / = No, risks chance for accident must be changing and/or risks may be entering or leaving. Question 33 Holler specifies four steps: 1. Create matrix of class differentials 2. Create matrix of loss ratios. 3. Create matrix of relative loss ratios from (2) using base class. 4. Multiple (1) x (3) 1. m Young => Differentials calculated from Large class relativities. Small m Young => L/Rs = Losses/Premium for Large each class. Small w Young => Use Large/Old as Base Large Small QICJ Youno => Multiply (1) x (3) Large Small 1.oo 2.16

59 Question 34 a. 1. Least squares - minimize the total squared error between actual and predicted result. 2. Small chance of large error - minimize the likelihood that any one actual observation will be a certain % different from the predicted result. 3. Meyers/Dorweiler - minimize the correlation between the ratio of actual/predicted and the predicted/average actual. b. MeyerslDorweiler is different from the first two which focus on minimizing prediction error. M/D focuses on the pattern of the errors. Question 35 a. b. When buyer has choice and there is a shift (potentially dramatic relocation) of buyers away from one product to another based on an attempt to gain an economic advantage where the seller has not matched price to cost. 1. Risk classification in a voluntary market - charges each risk the appropriate rate through proper identification and balances the economic forces governing buyer 2. and seller. Compulsory insurance with limited choices - does not permit the insured to shop for a better deal. Restriction of buyer freedom prevents movement or reduces the price incentive. Question 36 Under I, all insureds buy full coverage at the fair price. NLPL+NHPH=I~OO Where PL, PH are the actuarially fair prices NL, NH the number of insureds. We are given that PH = 2 PL => (NL + ~NH) PL = 1500 Under III, all insureds buy coverage at the pooled rate P* Given that P* NL = 1200 but also have that P*(NL + NH) = 1500 => NL =.8N => 1.2 PL = 1500/N => PL = 1250/N So at full coverage NL PI_ = 1000 NN PN = Under II, a Nash separating the high risk insureds buy full coverage and the fair price. So the high risks pay 500 =>The low risks pay (aiven)