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2 PL-1 Eric Brosius Loss Development Using Credibility Outline I. Introduction A. What loss development method do you select when there are large random fluctuations in year to year loss experience? B. Least squares development is shown to provide the best linear approximation to the Bayesian estimate and is contrasted with other standard development techniques. II. Notation A. - estimate of ultimate losses, given losses to date of and historical experience, B. Y random variable representing claims incurred C. X random variable representing number of claims reported at year end D., expected total number of claims E., expected number of claims outstanding F. MSE mean squared error G. EVPV Expected Value of the Process Variance - H. VHM Variance of the Hypothetical Means - III. Least Squares Development A. ab, where B. b C. ab IV. Special Cases of Least Squares Development A. When and are totally uncorrelated, b = 0 1. a, the budgeted loss method B. When the observed link ratios / are all equal, a = 0 1. b, the link ratio method C. When b=1, 1. a, the Bornhuetter-Ferguson method

3 PL-2 V. Hugh White s Queston A. If actual losses are higher than expected losses what do you do? 1. Reduce the bulk reserve a corresponding amount (Budgeted Loss Method) 2. Leave the bulk reserve at the same percentage level of expected losses (Bornhuetter-Ferguson Method) 3. Increase the bulk reserve in proportion to the increase in actual reported over expected reported (Link Ratio Method) B. These options are 3 points on the least squares continuum and the actual answer is likely to lie somewhere on that continuum. VI. Bayesian Development Examples A. Various examples using Bayesian estimation are used to show that the least squares estimate is superior to the link ratio, budgeted loss and Bornhuetter-Ferguson estimates: B. Simple Model 1. included to demonstrate method 2.,, based on parameters in example 3. The function does not align with any of the three special cases, but does lie on the least squares continuum. C. Poisson-Binomial Example 1. Poisson process determines ultimate claims (y) and reported claims (x) are determined by a Binomial process with the Poisson outcome y as the first parameter. 2. 2, based on the parameters given in the paper 3. This example is used to show that the link ratio method can t reproduce the Bayesian estimate Q(x), since there is no c, such that Alternative Options for c a. Unbiased Estimate - 1 b. Minimized MSE - minimize 1 c. 0 d. Salzmann s Iceberg Technique - 0, D. General Poisson-Binomial Case 1. 1, 1 2. This is consistent with the form of the Bornhuetter-Ferguson estimate. E. Negative Binomial-Binomial Case By plugging in sample parameter values it can be seen that the special cases of the least squares do not apply, but the result does lie on the least squares continuum.

4 PL-3 F. Fixed Prior Case the ultimate number of claims is known 1., 2. This is consistent with the budgeted loss method. G. Fixed Reporting Case percentage of claims reported at yearend is always d 1., 1 2. This is consistent with the link ratio method. VII. The Linear Approximation Development Formula 1 A. Pure Bayesian analysis requires significant knowledge about the loss and loss reporting process, which may not be available. A linear approximation can be used instead (Bayesian Credibility). B. Development Formula 1 gives the best linear approximation to Q: C., D. With historical experience, we can estimate the parts: E.,,,, F. Which gives the general least squares equation: G. H. Potential problems in parameter estimation: 1. Major changes in loss experience should be adjusted for: a. Inflation b. Exposure growth 2. Sampling error 3. Should substitute link ratio method when 0 4. Should substitute budgeted loss method when 0 VIII. Credibility Form of the Development Formula Development Formula 2 A. If there is a real number 0, such that for all y, then the best linear approximation to Q is given by development formula 2: B. 1, C. This is a credibility weighting of the link ratio method and the budgeted loss method. D. Special Cases: 1. Poisson-Binomial and other Bornhuetter-Ferguson Cases a. 2. Negative Binomial-Binomial Case a.

5 PL-4 IX. The Case Load Effect Development Formula 3 A. If the rate of claim reporting is a decreasing function of the number of claims and there are real numbers 0 such that, then define development formula 3: 1. 1 X. Mechanics of the Least Squares Approach A. Adjust data for exposure growth and inflation B. Develop most mature years to ultimate based on assumed tail factor C. Develop next oldest year to ultimate using least squares on the complete years D. Repeat one year at a time until all years have been developed

6 PL-5 Past CAS Examination Questions 1. Let X and Y be the random variables describing reported losses and ultimate losses, respectively. According to Brosius, in "Loss Development Using Credibility," which of the following statements are true regarding the linear approximation to the Bayesian estimate of Y? 1. If Cov(X, Y) > Var(X), a greater-than-expected reported amount should lead to a decrease in the indicated reserve. 2. If Cov(X, Y) = Var(X), the value of the reported amount should not affect the indicated reserve. 3. If Cov(X, Y) < Var(X), a greater-than-expected reported amount should lead to an increase in the indicated reserve. A. 1 B. 2 C. 1,3 D. 1,2,3 E. None of these answers are correct. (96 7B 23 1) 2. According to Brosius, in "Loss Development Using Credibility," which of the following statements are true? 1. The Bornhuetter-Ferguson estimate for unreported claims is optimal when the Poissonbinomial process is used to model claim counts. 2. The fixed reporting case is perfectly described by the link ratio method. 3. The least-squares method is identical to the link ratio estimate when the least-squares coefficient b is zero. A. 1,2 B. 1,3 C. 2,3 D. 1,2,3 E. None of these answers are correct. (96 7B 28 1) 3. You are given the following information: Accident Earned Cumulative Reported Losses as of: Year Premium 12 mos. 24 mos. 36 mos. 48 mos. 60 mos , , ,410 2,491 3, , ,547 3,274 3,598 4, , ,691 1, ,576 1,070 4,279 3, , , ,984 1,054 Assume losses will increase by an additional 8% from 60 months to ultimate. Using the leastsquares method described by Brosius in "Loss Development Using Credibility," estimate the ultimate loss ratio for Show all work. (96 7B 51 3) 4. ln "Loss Development Using Credibility," Brosius describes one situation in which it is appropriate to use least-squares development, and another in which it is inappropriate. List these situations and indicate which is appropriate for using least-squares development. (97 7B 50 1)

7 PL-6 5. Based on loss data for a small state, you have estimated the ultimate loss for a particular accident year (L(x)) using the least-squares method described by Brosius in "Loss Development Using Credibility." In the process you calculated the following statistics: y _ = 25,525 and x _ = 21,350. What is the Bornhuetter-Ferguson estimate of the ultimate loss for this accident year if the reported losses equal $20,575? A. < $21,000 B. $21,000 but < $22,500 C. $22,500 but < $24,000 D. $24,000 but < $25,500 E. $25,500 (98 7B 19 1) 6. You are given the earned premiums and incurred losses for a small book of business as follows: Accident Earned Premium Reported Losses ($000's) Year ($000's) , ,156 1, , ,140 1, , ,100 1,400 1, ,200 1,100 1,320 1, ,500 1,000 1, ,600 1,300 You believe that the changes in the book of business are accurately reflected in the earned premiums. Using the least-squares method described by Brosius in "Loss Development Using Credibility," estimate the ultimate losses for accident year Assume a 48-to-ultimate loss development factor of for all accident years. Show all work. (98 7B 42 4) 7. You are given the following information: Accident Earned Premium Reported Losses ($000) Year ($000) 12 Months 24 Months , , , , Use the least-squares method described by Brosius in "Loss Development Using Credibility" to estimate the losses for accident year 1998 as of December 31, You may assume that changes in the book of business are accurately reflected in the earned premium. Show all work. (99 7B 59 3) 8. According to Brosius, in "Loss Development Using Credibility," when using historical data to estimate ultimate losses as of a certain development point, if incurred losses are uncorrelated from one age of development to the next, then the least-squares estimate will equal the budgeted loss estimate. ( )

8 PL-7 9. You are given the information below. The tail factor from 48 months to ultimate is Incurred Losses ($000) Accident Age of Development (Months) Year a. Based on the methodology described in Brosius's "Loss Development Using Credibility," estimate the ultimate losses for accident year 1997 using the methods below. Show all work. i) Least-squares approach ii) Link ratio approach iii) Budgeted loss approach b. Using the results from a., calculate the credibility value (Z) and use it to prove that the credibility weighted average of your results from the link ratio and budgeted loss ratio approaches equals the least-squares approach. Show all work. ( ) 10. According to Brosius, in "Loss Development Using Credibility," the relationship between covariance(x, Y), where X is the reported loss and Y is the ultimate loss, and variance(x) determines which of three reserving methodologies is optimal. Assuming that reported losses at the valuation date are higher than expected, match each of the three loss reserving methods on the left with the covariance/variance relationship on the right under which the method is optimal. 1. Budgeted loss method a. Cov(X, Y) = Var(X) 2. Bornhuetter-Ferguson method b. Cov(X, Y) < Var(X) 3. Link ratio method c. Cov(X, Y) > Var(X) A. 1a, 2c, 3b B. 1b, 2a,3c C. 1b, 2c, 3a D. 1c, 2a, 3b E. 1c, 2b, 3a ( )

9 PL You are given the following information: i) A $250,000 cap on noneconomic damages in medical malpractice suits was eliminated effective with January 1, 2000 and subsequent occurrences. ii) Expected accident year 2000 losses if cap was still in effect: $25 million. iii) Expected increase in accident year 2000 losses from cap elimination is 40%. iv) Expected percentage of accident year losses reported at 12 months before cap elimination is 40%. v) Expected percentage of accident year losses reported at 12 months after cap elimination is 30%. vi) Estimated standard deviation of ultimate losses is $10 million after the elimination of the cap. vii) Estimated standard deviation of the ratio of reported loss to ultimate loss at 12 months of development is.20 after the elimination of the cap. viii) Reported accident year 2000 losses at 12 months of development are $15 million. ix) There is no loss development beyond 48 months. Calculate the ultimate loss estimate for accident year 2000 using the Bayesian credibility method as discussed in Brosius's "Loss Development Using Credibility." Show all work. ( ) 12. You are given the following information: Incurred Losses ($000) Accident Year 48 Months Ultimate Loss Assume level premium writings throughout the time period. According to Brosius, answer the following. a. Calculate a link ratio estimate and a budgeted loss estimate of the ultimate incurred loss for accident year 1997 using an all-year weighted average. Show all work. b. Calculate the least-square estimate of ultimate incurred loss for accident year Show all work. c. Display the least-square estimate in the form of a credibility-weighted average of the link ratio estimate and budgeted loss estimate calculated in a. Show all work. ( /1.5/.5) 13. Let L(x) = a + bx be the result of a line fit to accident year pairs (x, y) of reported claims from successive development periods. Let L(x) be our estimate of y, given that we have already observed x. According to Brosius, which one of the following statements is true? A. If a > 0 and b = 1, then L(x) is identical to a Bornhuetter-Ferguson estimate. B. If a > 0 and b < 0, then L(x) is identical to a budgeted loss estimate. C. If a = 1 and b > 0, then L(x) is identical to a link ratio estimate. D. If a = 0 and b > 0, then L(x) is identical to a budgeted loss estimate. E. If a < 0 and b > 0, then L(x) is identical to a link ratio estimate. ( )

10 PL You are given the following information: Earned Incurred Losses ($000) Accident Exposures Year (000) Months Months Incurred losses will increase by an additional 20% from 39 months to ultimate. Based on Brosius, calculate the accident year 2001 ultimate loss estimate using each of the following methods. Show all work. a. All-year weighted average link ratio method b. Budgeted loss method. ( ea.) 15. You are given the following information: Cumulative Losses Reported Accident (Age of Development in Months) Year $1,200 $1,800 $2, ,100 1,650 1, ,300 1, ,400 Using the least-squares method presented by Brosius, calculate the calendar year 2005 loss emergence for accident year ( ) 16. X and Y are the two random variables describing reported losses and ultimate losses, respectively. Which of the following statements are true regarding the best linear approximation to the Bayesian estimate of Y? l. If Cov(X, Y) < Var(X), a greater-than-expected reported amount should lead to an increase in the IBNR reserve. 2. If Cov(X, Y) = Var(X), a change in the reported amount should not affect the IBNR reserve. 3. If Cov(X, Y) > Var(X), a greater-than-expected reported amount should lead to an increase in the IBNR reserve. A. 1 B. 2 C. 3 D. 1, 2 E. 2,3 ( )

11 PL Given the following information: Incurred Losses Age of Development in Months Accident Year ,000 25, ,000 28, ,000 27, ,500 28, ,500 According to the least-squares method, what is the expected incurred loss for accident year 2005 at 24 months? A. < $27,500 B. $27,500, but < $28,500 C. $28,500, but < $29,500 D. $29,500, but < $30,500 E. $30,500 ( ) 18. As the result of recent tort reform, general liability expected ultimate losses decreased from $60 million to $50 million for accident year Without the reform, 55% of ultimate accident year 2005 losses would have been reported within twelve months. With the reform, this percentage is expected to rise to 63%. At December 31, 2005, $35 million of losses have been reported for accident year a. What is the link ratio estimate of the ultimate loss for accident year 2005? b. What is the Bornhuetter-Ferguson estimate of the ultimate loss for accident year 2005? c. Given that Y is expected ultimate losses and X is reported losses at 12 months, and using the estimates below, what is the ultimate loss for accident year 2005, using Brosius's Bayesian credibility method? Var Y [E(X Y)] = 14.3 E Y [Var(X Y)] = 57 d. Why is it inappropriate to use the least-squares method in the situation described in this case? ( /.5/1/.5) 19. An insurer has been experiencing a deteriorating loss ratio for the last five years on its personal auto business, due to the weakening of underwriting standards. Explain why the least-squares development method may not be appropriate. ( b.5)

12 PL Given the following: Cumulative Reported Losses ($000) Accident Age of Development in Months Year ,847 12,204 14,332 17, ,280 14,650 16, ,747 14, ,077 a. Estimate the cumulative reported loss as of 24 months for accident year 2007 using the link ratio method. b. Estimate the cumulative reported loss as of 24 months for accident year 2007 using the budgeted loss method. c. Estimate the cumulative reported loss as of 24 months for accident year 2007 using the least-squares method. ( /.5/1) 21. Given the following reported loss information: Accident Year As of 60 Months As of 72 Months 2000 $40,000 $45, ,000 60, ,000 42, ,000 32, ,000 a. Use Brosius' least-squares method to calculate the expected losses for accident year 2004 at 72 months. b. Briefly explain whether least squares is an appropriate method to use in this situation. ( /.5) 22. Given the following information ($000): Accident Incurred Loss Incurred Loss Year at 12 Months at 24 Months ,000 12, ,000 20, ,000 16, ,000 Use the method of least squares development to calculate the estimated incurred loss at 24 months for the accident year ( )

13 PL Given the following information ($000) for a line of business: Accident Written Earned Cumulative Reported Losses Year Premium Premium 12 Months 24 Months 36 Months ,756 4, ,310 5, ,907 5, , ,289 6, , ,946 8, The tail factor from 36 months to ultimate is a. Use the least squares method to estimate ultimate losses for the 2009 accident year. b. Discuss the reasonability of the estimate derive in part a. above, relative to the estimate that would be produced by the link ratio method. c. Illustrate graphically the relationships between the link ratio method, budgeted loss method and least squares method in modeling the loss development process. ( /0.5/1.5) 24. Given the following information: Incurred Loss Ratio Accident Year As of 36 Months As of 48 Months a. Estimate the loss ratio for accident year 2009 as of 48 months using the least squares method. b. An alternate approach to estimating the accident year 2009 loss ratio as of 48 months is to use the arithmetic average of the link ratio method and the budgeted loss ratio method. Using the answer from part a. above, demonstrate whether this averaging approach is optimal. ( /1.5)

14 PL Given the following information: Cumulative Losses ($000,000) Accident Reported at Ultimate Year 24 Months Loss An insurer writes annual policies that incept on January 1. Exposure and coverage levels were constant for 2008 through On January 1, 2012, policy coverage was expanded and pricing actuaries estimated the following: Loss amounts will increase by 25% due to the expanded coverage. 75% of ultimate losses are expected to be reported by 24 months, with a standard deviation of 8% of estimated ultimate loss. Standard deviation of accident year 2012 ultimate loss will be $3 million. a. Calculate the projected accident year 2012 ultimate loss using Bayesian credibility methods. b. Explain why the least squares method is not appropriate for calculating the accident year 2012 loss. ( :1.5/.5)

15 PL-14 Solutions to Past CAS Examination Questions F, p. 11 Substitute "increase" for "decrease." 2. T, p F, p. 11 Substitute "decrease" for "increase." Answer: B T, p T, p F, p. 3 Substitute "a" for "b." Answer: A 3. 1) Calculate the loss ratios at 48 months and ultimate: LR 86/48 = 761/5,370 =.1417 LR 86/ult = (943)(1.08)/5,370 =.1897 LR 87/48 = 2,491/6,674 =.3732 LR 87/ult = (3,261)(1.08)/6,674 =.5277 LR 88/48 = 3,598/8,888 =.4048 LR 88/ult = (4,207)(1.08)/8,888 =.5112 LR 89/48 = 1,482/9,982 = ) Calculate values for b and c: _ x = ( )/3 =.3066 _ y = ( )/3 =.4095 xy = [(.1417)(.1897) + (.3732)(.5277) + (.4048)(.5112)]/3 =.1436 x 2 = [(.1417) 2 + (.3732) 2 + (.4048) 2 ]/3 =.1077 b = _ xy - (x)(y) x 2 - (x _ ) 2 = (.3066)(.4095) (.3066) 2 = c = y _ /x _ =.4095/.3066 = ) Estimate the ultimate loss ratio: Z = b/c = / =.9863 L(x) = (Z)(LR 89/48 )(c) + (1 - Z)y _ = (.9861)(.1485) (1.3356) + ( )(.4095) L(x) =.2013, pp. 2 3, It is appropriate "if random chance is the primary cause of fluctuations." It is inappropriate "if year to year changes are due largely to systematic shifts in the book of business," p. 12.

16 PL L(x) = a + x = (y _ - x _ ) + x = (25,525-21,350) + 20,575 = 24,750, pp Answer: D 6. 1) Calculate the loss ratios at 36 months and ultimate: LR 92/36 = 1,156/1,700 =.68 LR 92/ult = (1,275)(1.1)/1,700 =.825 LR 93/36 = 1,140/1,900 =.60 LR 93/ult = (1,330)(1.1)/1,900 =.77 LR 94/36 = 1,400/2,000 =.70 LR 94/ult = (1,600)(1.1)/2,000 =.88 LR 95/36 = 1,430/2,200 =.65 2) Calculate values for b and c: x = ( )/3 =.66 y = ( )/3 =.825 xy = [(.68)(.825) + (.60)(.77) + (.70)(.88)]/3 =.5463 x 2 = [(.68) 2 + (.60) 2 + (.70) 2 ]/3 =.4375 b = _ xy - (x)(y) x 2 - (x _ ) 2 = (.66)(.825) (.66) 2 =.9474 c = y _ /x _ =.825/.66 = ) Estimate ultimate losses: Z = b/c =.9474/1.25 =.7579 LR(x) = (Z)(LR 95/36 )(c) + (1 Z)y _ = (.7579)(.65)(1.25) + (1.7579)(.825) LR(x) =.8155 L(x) = (2,200)(.8155) = 1,794, pp. 3, ) Calculate the loss ratios at 36 months and ultimate: LR 95/12 = 300/1,000 =.30 LR 95/24 = 500/1,000 =.50 LR 96/12 = 600/1,200 =.50 LR 96/24 = 480/1,200 =.40 LR 97/12 = 150/1,500 =.10 LR 97/24 = 600/1,500 =.40 LR 98/12 = 400/2,000 =.20 2) Calculate values for b and c: x = ( )/3 =.30 y = ( )/3 = 13/30 xy = [(.30)(.50) + (.50)(.40) + (.10)(.40)]/3 =.13 x 2 = [(.30) 2 + (.50) 2 + (.10) 2 ]/3 =.1167 b = _ xy - (x)(y) x 2 - (x _ ) 2 = 13 - (.30)(13/30) (.30) 2 = 0 c = y_ /x _ = (13/30)/.30 = ) Estimate ultimate losses: Z = b/c = 0/.6923 = 0 LR(x) = (Z)(LR 98/12 )(c) + (1 - Z)y _ = (0)(.20)(1.4444) + (1-0)(13/30) = 13/30 L(x) = (2M _ )(13/30) = 866,667, pp. 2 3,

17 PL T, p a. i) Ultimate Incurred Losses 97 = (160)(1.0375) = 166 x = ( )/2 = 140 y = ( )/2 = xy = [(130)(145) + (150)(166)]/2 = 21,875 x 2 = [(130) 2 + (150) 2 ]/2 = 19,700 b = _ xy - (x)(y) x 2 - (x _ ) 2 = 21,875 - (140)(155.50) 19,700 - (140) 2 = 1.05 a = y _ bx _ = (1.05)(140) = 8.5 L(x) = a + bx = (1.05)(150) = 166 ii) c = y _ /x _ = 155.5/140 = L(x) = cx = (1.1107)(150) = iii) L(x) = y _ = 155.5, pp b. Z = b/c = 1.05/1.107 =.9485 L(x) = (Z)(cx) + (1 - Z)y _ = (.9485)(166.6) + ( )(155.5) = 166, pp b, 2a, 3c, pp. 4, 11. Answer: B 11. 1) Calculate link ratio and budget ratio estimates: x/d = 15M _ /.3 = 50M _ E[Y] = (25M _ )(1.4) = 35M _ 2) Calculate VHM: VHM = Var(X) = Var(.3Y) = (.3) 2 (10M _ ) 2 = 9(M _ ) 2 3) Calculate EVPV: EVPV = E[X 2 ] = E[(.2Y) 2 ] = {.2} 2 {Var(Y) + (E[Y]) 2 } EVPV = [.2] 2 [(10M _ ) 2 + (35M _ ) 2 ] = 53(M _ ) 2 4) Calculate Z: Z = VHM/(VHM + EVPV) = 9(M _ ) 2 /[9(M _ ) (M _ ) 2 ] =.145 5) Calculate the ultimate loss estimate: L(x) = Zx/d + (1 Z)E[Y] = (.145)(50M _ ) + (1.145)(35M _ ) = M _, pp

18 PL a. x _ = ( )/4 = 65 y _ = ( )/4 = 87.5 c = y _ /x _ = 87.5/65 = For a link ratio estimate, we get: L(x) = cx = (1.346)(60) = For a budgeted loss estimate, we get: L(x) = _ y = 87.5 b. xy = [(65)(90) + (50)(80) + (70)(85) + (75)(95)]/4 = 5, x 2 = [(65) 2 + (50) 2 + (70) 2 + (75) 2 ]/4 = 4,312.5 b = _ xy - (x)(y) = x 2 - (x _ ) 2 5, (65)(87.5) 4, (65) 2 =.5 a = y _ bx _ = 87.5 (.5)(65) = 55 L(x) = a + bx = 55 + (.5)(60) = 85 c. Z = b/c =.5/1.346 =.3715 L(x) = (Z)(cx) + (1 - Z)y _ = (.3715)(80.76) + ( )(87.5) = 85, pp. 2 3, A. T, pp. 3 4 B. F, p. 3 Substitute "b = 0" for "b < 0." C. F, p. 3 Substitute "a = 0" for "a = 1." D. F, p. 3 Substitute "a > 0" for "a = 0" and "b = 0" for "b > 0." E. F, p. 3 Substitute "a = 0" for "a < 0." Answer: A 14. a. Since the exposure level changes, use loss ratios rather than losses: _ x = ( )/4 =.348 _ y = (1.2)( )/4 =.527 c = y _ /x _ =.527/.348 = L(x) = cx = (1.514)(97,000) = 146,858 b. L(x) = 300y _ = (300,000)(.527) = 158,100, pp. 2,

19 PL x _ = (1, , ,300)/3 = 1,200 y _ = (1, , ,860)/3 = 1,770 xy = [(1,200)(1,800) + (1,100)(1,650) + (1,300)(1,860)]/3 = 2,131,000 x 2 = [(1,200) 2 + (1,100) 2 + (1,300) 2 ]/3 = 1,446,667 b = _ xy - (x)(y) x 2 _ - (x) 2 = 2,131,000 - (1,200)(1,770) 1,446,667 - (1,200) 2 = 1.05 a = y _ - bx _ = 1,770 - (1.05)(1,200) = 510 L(x) = a + bx = (1.05)(1,400) = 1,980, pp F, p. 11 Substitute "decrease" for "increase." 2. T, p T, p. 11 Answer: E 17. x _ = ( )/4 = y _ = ( )/4 = 27 xy = [(10)(25) + (11)(28) + (12)(27) + (11.5)(28)]/4 = 301 x 2 = [(10) 2 + (11) 2 + (12) 2 + (11.5) 2 ]/4 = b = _ xy - (x)(y) x 2 _ - (x) 2 = (11.125)(27) (11.125) 2 = a = y _ - bx _ = 27 - ( )(11.125) = L(x) = a + bx = ( )(12.5) = , pp Answer: C 18. a. x/d = 35M _ /.63 = 55,555,556, p. 2. b. L = 35M _ + de[y] = 35M _ + (.37)(50M _ ) = 53.5M _, p. 3. c. Z = VHM/(VHM + EVPV) = 14.3/( ) =.201 L(x) = Zx/d + (1 - Z)E[Y] = (.201)(55,555,556) + ( )(50M _ ) = 51,116,667, pp d. It is inappropriate because there are significant changes in the loss history, p. 19.

20 PL It is not appropriate when "year to year changes are due largely to systematic shifts in the book of business," pp. 12, a. _ x = (8, , ,747)/3 = 10,291 _ y = (12, , ,826)/3 = 13,893 c = _ y /x _ = 13,893/10,291 = 1.35 L(x) = cx = (1.35)(12,077) = 16,304 b. L(x) = _ y = 13,893 c. xy = [(8,847)(12,204) + (10,280)(14,650) + (11,747)(14,826)]/3 = 144,243,937 x 2 = [(8,847) 2 + (10,280) 2 + (11,747) 2 ]/3 = 107,313,273 b = _ xy - (x)(y) x 2 _ - (x) 2 = 144,243,937 - (10,291)(13,893) 107,313,273 - (10,291) 2 =.902 a = y _ - bx _ = 13,893 - (.902)(10,291) = 4,611 L(x) = a + bx = 4,611 + (.902)(12,077) = 15,504, pp a. _ x = (40, , , ,000)/4 = 35,000 _ y = (45, , , ,000)/4 = 44,750 xy = [(40,000)(45,000) _ + (30,000)(60,000) + (40,000)(42,000) + (30,000)(32,000)]/4 xy = 1,560M x 2 = [(40,000) 2 + (30,000) 2 + (40,000) 2 + (30,000) 2 ]/4 = 1,250M _ b = _ xy - (x)(y) x 2 _ - (x) 2 = 1,560M_ - (35,000)(44,750) 1,250M _ - (35,000) 2 =.25 a = y _ - bx _ = 44,750 - (.25)(35,000) = 53,500 L(x) = a + bx = 53,500 + (.25)(50,000) = 41,000 b. Since b < 0, the least-squares estimate is not appropriate. Because of this the estimate produced by the budgeted loss method (y _ = 44,750) may be substituted, pp. 2 4.

21 PL ,000 16,000 10,000/3 12,000 12,000 20,000 16,000/3 16,000 10,00012,000 16,00020,000 10,00016,000/3 200,000,000 10,000 16,000 10,000 /3 152,000,000,,, 1 16,000 12,000 4,000 4,000 15,000 19, a. Ultimate losses for AY 2007 and 2008: 2007: 5, , : 1, , Loss ratios for AY 2007 and 2008 (Divide by earned premium): Year 24 Months 36 Months Ultimate % 122.3% 128.4% % 22.8% 24.0% % / / / / , , b. Since the estimate of a is less than 0 the least squares method will produce estimates of y that are less than 0 when x is small. Brosius suggests substituting the link-ratio method when a < 0. The link-ratio method will produce positive estimates of y even for small values of x. c. Y Least Squares Link Ratio X Budgeted Loss

22 PL a / / / / % b. In a credibility weighting /, where / 1.104/0.552/ Since the arithmetic average does not produce an optimal solution. 25. a. X= loss reported at 24 months Y= Ultimate losses L(x)=Z(x/d)+(1 - Z)E[Y] Z=VHM/(VHM+EVPV) VHM =(E[D].σ (y)) 2 = ((.75)(3)) 2 = EVPV=Var(D)[Var(y) +E[y] 2 ]=(0.08) 2 [3 2 +[(1.25)({ }/4)] 2 ]= Z=5.0625/( )=.606 L(x)=(.606)(21/.75)+(1.606)(22.5)= million b. The least squares method is appropriate when the distribution of loss is not changing year over year. Given the coverage expansion and change in 2012 loss distribution, we cannot use the least squares method.

23 PL-22

24 A Global Framework for Insurer Solvency Assessments, Ap. D ERM-73 International Actuarial Association A Global Framework for Insurer Solvency Assessments Appendix D Market Risk I. Definition of Market Risk Outline A. Volatility and uncertainty risk inherent in the market value of future cash flows B. Liquidity risk is related to market risk 1. Market risk could be a trigger, but other risk categories could trigger liquidity risk 2. Best handled under Pillar 2 C. Market risk can impact liabilities as well as assets 1. Asset yields impact discounting of liabilities (implicit or explicit) 2. Types of profit sharing linked to yields on assets: a. Fully based on objective indicators of capital market performance b. Related to the actual performance of the company c. Related to the actual performance of the assets that are locked-in at policyholders discretion 3. Proposed definition of market risk for insurers: a. Market risks relate to the volatility of the market values of assets and liabilities due to future changes of asset prices (/yields/returns). In this respect, the following should be taken into account: i. Market risk applies to all assets and liabilities ii. Market risk must recognize the profit sharing linkages between asset cash flows and liability cash flows iii. Market risk includes effect of changed policyholder behaviour on the liability cash flows due to changes in market yields and conditions II. Types of Market Risk A. A list of the principal sources of market risk is included in the chapter 5 section of the outline for this paper B. Determining market risk requires an accurate measure of the market value of assets and liabilities 1. Assets can be valued in the security markets 2. Liabilities are more challenging replicating asset portfolio is an option C. Typically life and health insurers match assets to liabilities, general insurers usually do not D. Insurers managing risk typically allocate their assets to one of the following: 1. Support of insurance liabilities 2. Represent economic capital 3. Represent free surplus E. Type A Market Risk Volatility in the market value of assets held and the replicating portfolio 1. Type B Market Risk Reinvestment risk when matching long term liabilities i.e. risk resulting from the inexistence of assets to match long term liabilities

25 ERM-74 A Global Framework for Insurer Solvency Assessments, Ap. D III. Time Horizon A. One year is considered appropriate for an insurer 1. Less active trading environment of insurers relative to banks 2. Time required for supervisor to assume control of struggling insurer 3. Time required to rebalance a mismatched portfolio of assets and liabilities B. Type A risk is diversifiable since the portfolio can be rebalanced C. Type B risk has some non-diversifiable risk since parts of the replicating portfolio are unknown 1. Time horizon should assess risk for the full remaining term of the liabilities IV. Confidence Level A. Market risk capital required should be aligned with the confidence level laid out in the Pillar I requirements V. Advanced Approach Type A Risks A. Type A risk can be present in any of the asset or liability cash flows of an insurer B. Advanced approach would involve risk models satisfying supervisor requirements 1. Future liability cash flows volatility can be approximated through market value techniques or with a replicating portfolio C. Market risk should include both specific risk and general market risk D. Stochastic modeling over a range of economic scenarios can be used to determine market risk 1. One year at a high confidence level E. Modeling of type A risk should reflect: 1. Actual asset allocation 2. Reinvestment policy 3. One year time horizon with sufficient assets at the end to handle remaining liabilities with prudent confidence level F. Practical approximations: 1. Deterministic liability basis at the end of the 1-year time horizon 2. Replace stochastic modeling with series of deterministic scenario shocks at 99% confidence a. Then requires stochastic modeling of liabilities at 1-year VI. Advanced Approach Type B Risks A. Type B risk can be present in any of the asset or liability cash flows of an insurer B. Modeling requirements are similar to type A risks C. Time horizon is the entire duration of liability cash flows D. Confidence level should be greater of 2 options: 1. Very high confidence level (e.g. 99%) that assets will be sufficient in one year to provide for policy liabilities determined at a moderate level (e.g. 75%) at that time 2. Fairly high confidence level (e.g. 90%) that assets will be sufficient to provide for all future liabilities

26 A Global Framework for Insurer Solvency Assessments, Ap. D ERM-75 E. Modeling Process 1. Identify assets and liabilities 2. Define impact of primary market risk factors on policyholder and company behaviour 3. Model as an integrated set of economic scenarios a. If liabilities separate from assets use replicating portfolio b. If liabilities and assets together consider: i. Asset/liability linkages ii. Pass-through of risks to policyholders iii. Reinvestment strategy iv. Policyholder behaviour F. Replicating Portfolios 1. Liability cash flows which extend beyond the longest available debt instruments are subject to both type A and B risk 2. Investment practices of the insurer need to be modeled as accurately as possible to minimize difference in market risk determination for cash flow streams shorter and longer than the replicating portfolio 3. Replicating portfolio provides a hedge against liability risks a. Theoretical since other risks can impact the liabilities G. Embedded Options 1. Replicating portfolio should include financial instruments that hedge embedded options 2. Policyholder guarantees always increase the market value of liabilities a. Value of the guarantee is value of instrument needed to hedge it b. Instruments value and volatility (and that of the guarantee) can be determined using Black-Scholes option-price formulas 3. Company options will reduce the market value of liabilities H. Incompleteness of the Capital Market 1. For some complex insurance products there are limited or no replicating financial instruments a. Option pricing theory or stochastic simulations can be used instead 2. Some embedded options are not guaranteed, but just intended a. These options still have value to policyholders (and can be granted by courts) 3. Liabilities which extend beyond 30 years typically don t have fixed interest securities to match I. Economic Scenarios 1. Desirable characteristics of constructed scenarios: a. Interest Rates i. Nominal yields positive and not indefinitely increasing ii. Subject to non-constant mean reversion iii. Volatility decreases with maturity iv. High correlation between maturities v. Distinctive yield curve shapes

27 ERM-76 A Global Framework for Insurer Solvency Assessments, Ap. D b. Equity Returns i. Negative skewness ii. Fat tails over short periods iii. Volatility clustering iv. Exogenous shocks v. Markov property vi. Market correlations increase under extreme condtions vii. Price appreciation versus dividend income c. Inflation i. Non-persistence of abnormal values ii. Realized may equal expected plus exogenous shock iii. Non-constant mean reversion iv. Volatility clustering v. Various forms of inflation vi. Relationship to other economic factors J. Discount Rates 1. Value of the replicating portfolio is determined by discounting the cash flows 2. If the cash flows are adjusted for systematic non-financial risks the discount rate can be set to the risk-free spot rates 3. Supervisors should define the spot rates so all companies use the same yields 4. Procedure is required for estimating the spot yield curve periodically a. Nelson-Siegel approach: i. spot t r 121exp t ii. With the parameters 0, 1, 2, estimated 0 2exp t t VII. Standardized Approaches Type A Risks A. Methodology 1. Required ingredients: a. Projected future cash flows b. Nature of embedded options c. Time horizon d. Confidence level e. Current economic scenario f. Series of adverse scenarios 2. Potential approximations a. Use of option adjusted durations to represent price sensitivity of cash flows and investment return shocks

28 A Global Framework for Insurer Solvency Assessments, Ap. D ERM-77 b. Grouping of future cash flows into various term buckets buckets multiplied by factors to produce capital requirement c. Multiply balance sheet value of assets and liabilities by a table of factors 3. Supervisor needs to determine whether the approximations used are reasonable a. Conservatism in standardized approach should incent insurers to build their own models b. Risk of insurer specific models is the possibility of manipulation B. Fixed Interest Securities and Liabilities 1. Risk of fixed income securities are related to: a. Duration b. Rating c. Currency 2. Interest rate increases drive bond prices down and thus the market risk of bonds is primarily the risk that interest rates will increase a. Price change = -1 x duration x change in yield b. Duration can be calculated explicitly or estimated as 80% of mean time to maturity 3. Standardized approach to calculating a mismatch position is through Macaulay duration: a. Disadvantages: i. First derivative approximation isn t good for large interest rate changes a) Employing second derivative (convexity) helps ii. Assumes parallel yield curve shifts, but non-parallel are common iii. Still requires complex modeling 4. Market risk determination requires probability distribution of interest rate changes a. Approximation: 100 b. Should include prudence factor when using approximation c. When aggregating bonds should assume dependence and add standard deviations rather than variances 5. Changes in spreads should be considered as part of credit risk 6. Foreign exchange risk can be estimated with a factor multiplied by the asset value i

29 ERM-78 A Global Framework for Insurer Solvency Assessments, Ap. D 8. The bond standard deviations can be used in a multivariate normal model a. With asset-liability matching true risk may be smaller C. Equity and Property 1. Subject to Type A market risk when used to fund policy liabilities 2. Time horizon to consider is 1 year 3. Equity returns have much higher volatilities than bonds When aggregating equity should assume dependence and add standard deviations rather than variances 6. Real estate can be handled the same as equity D. Derivatives and Embedded Options 1. Embedded options can be valued with the assets that are need to hedge them a. If a suitable asset does not exist, the solvency requirement should be a conservative estimate of the possible change in difference between the market values b. Typically not an easy task c. Black-Scholes or stochastic simulations can help E. Other Types of Assets 1. There are many types of assets that are not used for replicating portfolios and hedges a. The market risks are typically asset-only risks and the mismatch provision can be calculated similar to equity and property b. If the assets are illiquid, their market values and volatility should be determined conservatively F. Currency Risk 1. Valued similar to equity and property risks G. Dependencies 1. Dependencies between asset classes are low 2. Correlations between yields assets in different currencies fluctuates widely and can be considered to be 0 3. In determining total risk can assume independence 4. Valuation of derivatives should be handled in the same way as the underlying asset 5. Correlations between assets within the same class are typically high 6. Spot yields for different maturities of fixed income securities are generally high, but not perfectly correlated

30 A Global Framework for Insurer Solvency Assessments, Ap. D ERM-79 VIII. Standardized Approaches Type B Risks A. A standardized approach is difficult for Type B risks 1. Supervisors should encourage/require advanced approaches B. Possible standardized approaches: 1. Assume long term reinvestment rates return to conservative historical averages 2. Complex options should use conservative factors that have been based on industry stochastic modeling

31 ERM-80 A Global Framework for Insurer Solvency Assessments, Ap. D

32 A Global Framework for Insurer Solvency Assessments, Ap. E ERM-81 International Actuarial Association A Global Framework for Insurer Solvency Assessments Appendix E Credit Risk I. Definition of Credit Risk Outline A. Inability or unwillingness of a counterparty to fully meet its financial obligations B. Credit risk can exist with respect to any set of projected future cash flows C. Credit risk can be included in the present value of cash flows using a credit spread on the discount rate or through modeling of the cash flows 1. Market value reflects the market view of the credit risk of the provider of cash flows D. Recommendation that insurers use a similar determination of capital requirements as banks II. Types of Credit Risk A. A list of the principal sources of credit risk is included in the chapter 5 section of the outline for this paper B. Insurers managing risk typically allocate their assets to one of the following: 1. Support of insurance liabilities 2. Represent economic capital 3. Represent free surplus C. If asset performance is shared with policyholders, credit can be given in the determination of the capital requirement for credit risk D. Type A Credit Risk credit risk on the actual assets held to replicate the liability portfolio (portion of the liability portfolio for which replicating assets exist) E. Type B Credit Risk credit risk associated with future reinvested assets (portion of the liability portfolio for which no replicating assets exist) III. Key Drivers of Credit Risk A. Credit quality i.e. rating of issuer B. Maturity i.e. longer term = greater risk C. Concentration by industry i.e. correlation within a sector of the economy D. Concentration by geography i.e. correlation within a geographical region E. Size of expected loss i.e. wide variation in size of losses, includes possible associated loss if payment was need to match a scheduled outflow

33 ERM-82 A Global Framework for Insurer Solvency Assessments, Ap. E IV. Controls and Hedging Strategies A. Credit strategies used to offset credit risk: 1. Letters of credit 2. Contingency deposits 3. Securitization of mortgages 4. Securitization of other assets 5. Credit Derivatives a. Credit default swaps b. Total return swaps c. Collateralized debt obligations d. Credit-linked notes e. Credit spread options f. Basket derivatives V. General Modeling Approaches A. Default models model rates of default and recovery explicitly 1. Probabilities of cash flows under default/non-default scenarios are considered and discounted at the risk free rate 2. Only two options: default or non-default B. Credit migration models model risk of default as well as risk of value changes due to changes in credit ratings 1. Uses a transition matrix which considers probability that bond will move from one credit rating to another C. Asset models model a firms debt as an option against its assets 1. Can be combined with a model of correlations between obligors to produce a portfolio level risk management model VI. Degree of Protection A. Credit risk capital required should be aligned with the confidence level laid out in the Pillar I requirements VII. Time Horizon A. One year is considered appropriate for an insurer 1. Less active trading environment of insurers relative to banks 2. Time required for supervisor to assume control of struggling insurer 3. Time required for insurer to address credit risk in its assets B. Still need to consider the full term of all assets and liabilities of the insurer

34 A Global Framework for Insurer Solvency Assessments, Ap. E ERM-83 VIII. Advanced Approach Type A Risks A. Working party made recommendations on each of the main elements of the BIS approach: 1. Degree of protection consistency through Pillar I 2. Time horizon consistency through Pillar I 3. Diversification reflect diversification in portfolios 4. Correlation allow insurers to reflect own asset correlation 5. Cycles further study required 6. Migration supports credit migration techniques B. A General Approach: 1. Expected default probabilities are available from rating agencies 2. Commercial software is available for modeling default as well as credit spread 3. Supervisory guideline should not require commercial software but rather provide a simple formula 4. Consider a formula based on mean time to payment T and annual default probability ρ 1 a. 1 b. c. Formula has a binomial distribution and conservatively sets value to 0 after a default d. Two assumptions: i. There is a rating scale on which rating fluctuation can be described as Brownian motion ii. There is a minimum value on this scale that corresponds to a default and serves as an absorbing state of the Brownian motion e. 1, where and. is the cumulative standard normal f. 1 g. h. = 2.37 and c = C. Also need to consider diversification and correlation in the above formulas 1.,,, maximum risk assuming full dependence 2.,, assuming no dependence 3. diversification,, as a numerical approximation with b 1,, correlation of 50% and α the degree of D. Total risk of the fixed income portfolio is given by combining market and credit risk

35 ERM-84 A Global Framework for Insurer Solvency Assessments, Ap. E 1., IX. Advanced Approach Type B Risks A. Type B risk is associated with long term liabilities for which there is no replicating asset B. If valuation of liabilities uses a discount rate that is net of credit risk, the amount of the policy liabilities will automatically include a provision for credit risk for the entire term of the liability C. The credit provision can be estimated by determining the credit spread in future returns X. Standardized Approaches Type A Risks A. Recommendation of an approach similar to BIS requirements for banks XI. Standardized Approaches Type B Risks A. A standardized approach is difficult for Type B risks B. Supervisors should encourage/require advanced approaches C. Possible standardized approaches: D. Apply a factor to liabilities of long term business E. Apply a credit spread to duration and policy liabilities of long term business (assuming duration can be estimated) F. Apply a credit spread directly (assuming computation of future liability cash flows is known

36 A Global Framework for Insurer Solvency Assessments, Ap. E ERM-85 Past CAS Examination Questions 1. Given the following bond portfolio for an insurance company: Bond Issuer's Credit Risk Current Annual Years to Bond Solvency Default Maturity Rating Requirement (000) Probability Bond 1 1 BBB % Bond 3 3 BBB % Assume that the best diversification can be approximated by correlation coefficients of a. Identify and briefly describe three sources of credit risk that may be associated with this bond portfolio. b. Calculate the capital requirement due to credit risk for this bond portfolio assuming that both bonds are from the same issuer. ( :1.5/0.75) 2. A U.S.-domiciled insurance company is holding the following bonds: Bonds issued by a U.S.-based bank with credit rating AA valued at $15 million. The mean time to maturity is 5 years. Bonds issued by a technology company located in the US with credit rating of BBB and valued at $20 million. The mean time to maturity is 10 years. There are no other bonds held by the company. Identify and briefly describe three drivers of credit risk for the insurance company and briefly comment on how they relate to the insurer s bond portfolio. ( )

37 ERM-86 A Global Framework for Insurer Solvency Assessments, Ap. E Solutions to Past CAS Examination Questions 1. See pp for a description of the following acceptable risks: Direct default risk Downgrade risk Indirect credit risk Settlement risk Concentration risk b. Since both bonds are from the same issuer you can assume that they are 100% correlated: Total Capital Requirement = = Choose three (more possible): Credit risk: Credit quality of an investment or an enterprise refers to the probability that the issuer will meet all contractual obligations. This assessment normally occurs at both the initial investment and at each renewal point. One of the common measurements used in assessing credit quality is the rating assigned to the issuer. Maturity: The longer the term to maturity of an investment, the longer even a high quality issuer has to potentially deteriorate. Concentration by industry: The credit risk from concentration by industry is reduced by choosing bonds from companies in unrelated industries. Concentration by geography: There is some credit risk from concentration by geography since all of the bonds are from American companies.

38 A Global Framework for Insurer Solvency Assessments, Ap. H ERM-87 International Actuarial Association A Global Framework for Insurer Solvency Assessments Appendix H Analytic Methods Outline I. Developing a Base-Line Model A. For internal models a base-line model can be developed to model the distribution of outcome X B. Derive the cumulant generating function as:!!,, C. The Normal distribution can be viewed as first order approximation since it has: 1. If applied to all risk components the multivariate Normal distribution can be viewed as the base-line model 2. For supervisory purposes the error in the approximation can be estimated by looking at how the higher cumulants impact risk measures D. If X 1,X 2,,X n have a multivariate normal distribution with ρ i,j the correlation between the i th and j th components and σ j the standard deviation of the j-th compenent: 1., is the standard deviation of the aggregate distribution E. Major sources of error when Normal model is used as the base-line model: 1. The true probability distribution can be quite different from the Normal model, especially in the tail 2. The linear correlations used in the Normal distribution may not be well suited to combining interactions in the extreme tail of the distribution a. These are of particular importance since supervisors primary focus is typically in the tail F. Supervisor recognition of errors: 1. Require a multiple of the capital indicated by the model be used (for conservatism) 2. Directly incorporating conservatism into the assumptions of the base-line model