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1、Section 15, Mahler, Discussion of “Retro. Rating: 1997 Excess Loss Factors”643 644Errata for Discussion by Howard Mahler of “Retrospective Rating: 1997 Excess Loss Factors”645 At the bottom of page 320, the equation for ? R(100) is incorrect.This example of simple dispersion is an example of a mixtu
2、re with five pieces. The excess ratio of the mixture is a weighted average of individual excess ratios, with the weights the product of the means and the probabilities for each piece of the mixture.646 If the probability of each piece of a mixture is pi, pi = 1, the mean of each piece of the mixture
3、 is mi, and Ri is the excess ratio for each piece of the mixture, then ? R(L) = pi mi Ri(L).If each loss is divided by for example .75, then after development, the excess ratio at L is the same as the original excess ratio at 0.75 L.647 Ri(L) is the excess ratio when the losses have all been divided
4、 by ri.Thus Ri(L) = R(ri L). In the example on page 320, each mean is proportional to 1/divisor = 1/ri, and each probability is the same at 1/5. Thus the weights are: (1/5)(1/ri).The sum of the weights is: (1/5)(1/ri) = (1/5) (1/0.75 + 1/0.833 + 1/1 + 1/1.25 + 1/1.5) = 1.648 Thus ? R(L) = (1/5)(1/ri
5、) R(ri L) = (1/5) R(ri L) /ri.Therefore, the corrected equation at the bottom of page 320 is:? R(100) = (1/5) R(75)/0.75 + R(83.3)/0.833 + R(100)/1 + R(125)/1.25 + R(150)/1.50 = (1/5) 0.6009/0.75 + 0.5817/0.833 + 0.5582/1 + 0.5384/1.25 + 0.5191/1.50 = 0.5669.643 Appendixes B-D are for reference only
6、. Candidates do not need to memorize formulas in Appendices B-D.CAS Learning Objective B1. This reading was added to the syllabus in 2008.644 The original paper by Gillam and Couret in PCAS 1997 is not on the syllabus.The discussion by Mahler in PCAS 1998 is on the syllabus.645 Official; found by me
7、 looking over my own paper a decade later! Check the CAS webpage for any updates.646 See page 154 of “Workers Compensation Excess Ratios: An Alternate Method of Estimation” by Mahler.647 If each loss is multiplied by 1/0.75 = 1.333, this is mathematically the same as uniform inflation of 33.3%.Thus
8、we can get the excess ratio after development, by taking the original excess ratio at the deflated value of L/1.333 = 0.75 L. Increasing the sizes of loss, increases the excess ratio over a fixed limit.648 Mahler chose these loss divisors so that the total expected losses are unaffected.2014-CAS815
9、Mahler, Discussion of 1997 ELFs HCM 5/13/14, Page 923Similarly, the corrected equation at the top of page 321 is:? R(5000) = (1/5) R(3750)/0.75 + R(4165)/0.833 + R(5000)/1 + R(6250)/1.25 + R(7500)/1.50 = (1/5) 0.0157/0.75 + 0.0070/0.833 + 0/1 + 0/1.25 + 0/1.50 = 0.0059.At page 324, some of the numer
10、ical values shown in the computation of R3(2000) are mixed up, although the final value is correct at 0.384 as shown. It should have read: R3(2000) = (1.04167)(0.9999980) - (0.04167)(0.0057148)+ (0.1667)(0.0026029) - (0.8333)(0.999995) + (0.1761)(0.999987) - (0.1761)(0.0011302) = 0.384.Also, in Tabl
11、e 1 the excess ratios were computed for Gamma loss divisors with shape parameter 16.67 and inverse scale parameter 15.67. However, the text at page 323 refers to Gamma loss divisors with shape parameter s = 18.67 and inverse scale parameter l = 17.67; this distribution of loss divisors corresponds t
12、o a mean loss development of 1 and a variance of loss development of 0.060, matching the simple dispersion example. Using the intended Gamma parameters of s = 18.67 and l = 17.67 changes the excess ratios in Table 1 slightly, although the pattern remains the same.2014-CAS815 Mahler, Discussion of 19
13、97 ELFs HCM 5/13/14, Page 924The values in the simple dispersion column of Table 1 at page 320 are revised in a similar manner to that for 5000. The values in Gamma dispersion column of Table 1 at page 320 are revised based on a shape parameter of s = 18.67 and inverse scale parameter of l = 17.67.C
14、orrected Table 1Excess RatiosNo Simple Gamma LIMIT DevelopmentDispersionDispersion50 .6888 .6949 .6939100 .5582 .5669 .5673500 .3012 .3080 .30691,000 .1606 .1705 .17092,000 .0904 .0931 .09273,000 .0402 .0462 .04534,000 .0100 .0194 .01825,000 .0000 .0059 .00626,000 .0000 .0007 .00207,000 .0000 .0000
15、.00068,000 .0000 .0000 .00029,000 .0000 .0000 .0001 10,000 .0000 .0000 .0000 As can be seen in corrected Table 1, the simple dispersion effect raises the excess ratios, especially at the higher limits.649 At page 326, the formula near the bottom of page should have in place of X:R(L) = ( l/L)s-1 U(s
16、-1, s+1-, l/L).The statement in the third paragraph of page 327 of the Discussion is backwards. It should have read:As the shape parameter of the Pareto, , gets smaller, the losses have a heavier tail and the multiplicative impact of the dispersion on the excess ratios at high limits decreases.At page 331, the first equation needs parentheses around th
