Replicate Meyers' Changing Settlement Rate Backtest
Feb 11, 2018

Goal

The goal of this post is to replicate the univariate results from Glenn Meyers’s paper Dependencies in Stochastic Loss Reserve Models. The stochastic reserving method used is called the Changing Settlement Ratio (CSR) model which was first described in STOCHASTIC LOSS RESERVING USING BAYESIAN MCMC MODELS.

The R code provided with the paper and from Meyers’ Actuarial Review Article were used as references.

Import data

Import the CAS loss reserve database from the reservetestr package.

Summary table shown below for full CAS database.

Table 1: Summary of CAS Loss Reserve Database
Line Number of companies Number of accident years Total number of observations
comauto 157 10 15,800
medmal 34 10 3,400
othliab 236 10 23,900
ppauto 146 10 14,600
prodliab 70 10 7,000
wkcomp 132 10 13,200

Meyers only uses a subset of the database. He chose four lines of business - comauto, ppauto, wkcomp, othliab - and selected 50 companies within each line. We can filter the full database to include only these companies by using the spreadsheet provided on the CAS website, again pulling the data from the reservetestr package.

Table 2: Summary of Data used in Meyers (2016)
Line Number of companies Number of accident years Total number of observations
comauto 50 10 5,000
othliab 50 10 5,000
ppauto 50 10 5,000
wkcomp 50 10 5,000

Now let’s replicate Table 2.1 from Meyers (2016).

(#tab:table_2.1)Group 620 - CA
acc_yr premium 1 2 3 4 5 6 7 8 9 10
1988 30,224 4,381 9,502 15,155 18,892 20,945 21,350 21,721 21,934 21,959 21,960
1989 35,778 5,456 9,887 13,338 17,505 20,180 20,977 21,855 21,877 21,912 21,981
1990 42,257 7,083 15,211 21,091 27,688 28,725 29,394 29,541 29,580 29,595 29,705
1991 47,171 9,800 17,607 23,399 29,918 32,131 33,483 33,686 34,702 34,749 34,764
1992 53,546 8,793 19,188 26,738 31,572 34,218 35,170 36,154 36,201 36,256 36,286
1993 58,004 9,586 18,297 25,998 31,635 33,760 34,785 35,653 35,779 35,837 35,852
1994 64,119 11,618 22,293 33,535 39,252 42,614 44,385 44,643 44,771 45,241 45,549
1995 68,613 12,402 27,913 39,139 45,057 47,650 50,274 50,505 50,554 50,587 50,587
1996 74,552 15,095 27,810 35,521 44,066 48,308 50,061 51,337 51,904 52,016 53,895
1997 78,855 16,361 28,545 40,940 50,449 54,212 56,722 57,658 57,734 57,883 57,906
(#tab:table_2.1)Group 620 - PA
acc_yr premium 1 2 3 4 5 6 7 8 9 10
1988 48,731 15,318 27,740 35,411 40,204 42,388 43,726 44,217 44,277 44,400 44,431
1989 49,951 15,031 30,132 37,946 42,371 43,875 44,518 44,738 45,089 45,094 45,146
1990 52,434 16,994 31,614 39,599 44,943 46,342 47,653 47,866 48,085 48,097 48,241
1991 58,191 17,717 33,767 42,741 46,881 49,117 50,419 50,641 50,787 50,942 50,980
1992 61,873 17,842 31,117 39,436 44,871 46,810 47,421 48,209 48,724 48,815 49,133
1993 63,614 20,266 37,466 45,721 50,641 52,244 53,241 53,794 54,093 54,468 54,471
1994 63,807 18,778 33,216 42,030 47,695 49,252 50,002 50,546 50,799 50,887 50,890
1995 61,157 19,900 36,442 43,585 49,177 52,052 53,150 53,420 53,488 53,649 53,659
1996 62,146 20,395 35,797 43,816 47,687 50,468 51,085 51,598 51,754 51,756 51,914
1997 68,003 20,622 36,466 44,589 50,539 52,860 53,886 54,610 54,796 55,048 55,080
(#tab:table_2.1)Group 1066 - CA
acc_yr premium 1 2 3 4 5 6 7 8 9 10
1988 5,103 1,060 3,034 4,580 5,243 4,178 4,347 4,399 4,598 4,582 4,629
1989 5,196 1,224 3,751 5,735 4,902 5,295 5,486 5,941 5,976 5,977 5,977
1990 6,947 1,252 3,568 5,265 6,102 6,607 6,315 6,343 6,370 6,445 6,419
1991 9,482 1,606 3,875 5,439 6,507 8,021 8,098 8,282 8,300 8,328 8,378
1992 10,976 1,750 4,038 5,662 6,293 6,779 7,048 7,048 7,047 7,047 7,047
1993 11,893 1,125 4,322 5,263 6,036 6,462 6,617 6,647 6,649 6,654 6,654
1994 13,029 1,403 3,746 5,800 6,737 7,078 7,110 7,225 7,346 7,366 7,366
1995 12,511 1,541 4,620 5,746 6,171 6,462 6,680 6,714 6,713 6,728 6,729
1996 14,372 1,986 4,532 4,817 5,653 5,932 5,988 6,036 6,038 6,051 6,043
1997 7,371 1,970 2,730 3,214 3,376 3,502 3,605 3,744 3,750 3,777 3,780
(#tab:table_2.1)Group 1066 - PA
acc_yr premium 1 2 3 4 5 6 7 8 9 10
1988 24,988 5,135 11,980 16,368 18,163 20,189 20,462 20,715 20,749 20,720 20,813
1989 26,082 5,655 15,108 19,498 23,097 23,819 24,296 24,622 24,735 24,736 24,741
1990 29,606 6,648 17,982 23,078 25,334 26,596 26,983 27,096 27,150 27,195 27,206
1991 33,802 5,722 14,677 19,356 21,906 22,497 22,732 23,149 23,207 23,197 23,254
1992 37,261 5,906 14,864 18,305 20,075 21,779 22,277 22,425 22,466 22,424 22,536
1993 35,849 6,439 15,146 19,187 21,576 22,539 22,941 23,037 23,029 23,135 23,174
1994 35,053 6,934 15,703 19,748 21,300 21,948 22,004 22,043 22,136 22,211 22,210
1995 33,254 6,194 12,183 15,282 17,315 18,550 18,697 18,876 19,014 19,040 19,210
1996 29,101 5,314 10,915 13,854 15,179 15,537 16,083 16,057 16,088 16,101 16,137
1997 29,149 4,301 9,758 11,914 13,216 13,740 14,098 14,427 14,448 14,491 14,513

The Changing Settlement Rate (CSR) Model

With our input data gathered, we can apply the CSR method to these 200 triangles. See the papers for details of the CSR method (Dependencies in Stochastic Loss Reserve Models and STOCHASTIC LOSS RESERVING USING BAYESIAN MCMC MODELS).

The number of iterations we ran was different from Meyers, so that increases the simulation error. We used rstan’s default assumptions of 4 chains, 2,000 iterations (1,000 warm up).

Model results comparison

The graph below compares our results with Meyers for Other Liability across four fitted parameters / model outputs. The results are reasonably close. The other lines look similar.

Figure 3.1 Uniformity Tests for the CSR Model

Very consistent results shown below when compared to figure 3.1 from the paper.

Same goes for the pp-plot and histogram with all the triangles included.

Figure 3.2 Standardized Residual Plots for the CSR Model

All the standardized residuals plots appear consistent with the paper.

Conclusion

We achieved our goal of successfully replicating the univariate CSR model results!