Bootstrap-Chain-Ladder Model — BootChainLadder • ChainLadder

The BootChainLadder procedure provides a predictive distribution of reserves or IBNRs for a cumulative claims development triangle.

Usage

BootChainLadder(Triangle, R = 999, process.distr=c("gamma", "od.pois"), seed = NULL)

Arguments

Triangle: cumulative claims triangle. Assume columns are the development period, use transpose otherwise. A (mxn)-matrix \(C_{ik}\) which is filled for \(k \le n+1-i; i=1,\ldots,m; m\ge n \). See qpaid for how to use (mxn)-development triangles with m<n, say higher development period frequency (e.g quarterly) than origin period frequency (e.g accident years).
R: the number of bootstrap replicates.
process.distr: character string indicating which process distribution to be assumed. One of "gamma" (default), or "od.pois" (over-dispersed Poisson), can be abbreviated
seed: optional seed for the random generator

Details

The BootChainLadder function uses a two-stage bootstrapping/simulation approach. In the first stage an ordinary chain-ladder methods is applied to the cumulative claims triangle. From this we calculate the scaled Pearson residuals which we bootstrap R times to forecast future incremental claims payments via the standard chain-ladder method. In the second stage we simulate the process error with the bootstrap value as the mean and using the process distribution assumed. The set of reserves obtained in this way forms the predictive distribution, from which summary statistics such as mean, prediction error or quantiles can be derived.

Value

BootChainLadder gives a list with the following elements back:

call: matched call
Triangle: input triangle
f: chain-ladder factors
simClaims: array of dimension c(m,n,R) with the simulated claims
IBNR.ByOrigin: array of dimension c(m,1,R) with the modeled IBNRs by origin period
IBNR.Triangles: array of dimension c(m,n,R) with the modeled IBNR development triangles
IBNR.Totals: vector of R samples of the total IBNRs
ChainLadder.Residuals: adjusted Pearson chain-ladder residuals
process.distr: assumed process distribution
R: the number of bootstrap replicates

References

England, PD and Verrall, RJ. Stochastic Claims Reserving in General Insurance (with discussion), British Actuarial Journal 8, III. 2002

Barnett and Zehnwirth. The need for diagnostic assessment of bootstrap predictive models, Insureware technical report. 2007

Author

Markus Gesmann, markus.gesmann@gmail.com

Note

The implementation of BootChainLadder follows closely the discussion of the bootstrap model in section 8 and appendix 3 of the paper by England and Verrall (2002).

Examples

# See also the example in section 8 of England & Verrall (2002) on page 55.

B <- BootChainLadder(RAA, R=999, process.distr="gamma")
B
#> BootChainLadder(Triangle = RAA, R = 999, process.distr = "gamma")
#> 
#>      Latest Mean Ultimate Mean IBNR IBNR.S.E IBNR 75% IBNR 95%
#> 1981 18,834        18,834         0        0        0        0
#> 1982 16,704        16,901       197      754      205    1,704
#> 1983 23,466        24,069       603    1,318    1,117    2,967
#> 1984 27,067        28,784     1,717    2,003    2,701    5,409
#> 1985 26,180        28,994     2,814    2,291    4,168    7,335
#> 1986 15,852        19,578     3,726    2,662    4,939    8,978
#> 1987 12,314        17,999     5,685    3,238    7,558   11,419
#> 1988 13,112        24,090    10,978    4,784   13,669   19,808
#> 1989  5,395        16,654    11,259    6,258   14,787   21,771
#> 1990  2,063        19,843    17,780   14,121   26,102   43,630
#> 
#>                  Totals
#> Latest:         160,987
#> Mean Ultimate:  215,746
#> Mean IBNR:       54,759
#> IBNR.S.E         19,331
#> Total IBNR 75%:  67,005
#> Total IBNR 95%:  89,326
plot(B)

# Compare to MackChainLadder
MackChainLadder(RAA)
#> MackChainLadder(Triangle = RAA)
#> 
#>      Latest Dev.To.Date Ultimate   IBNR Mack.S.E CV(IBNR)
#> 1981 18,834       1.000   18,834      0        0      NaN
#> 1982 16,704       0.991   16,858    154      143    0.928
#> 1983 23,466       0.974   24,083    617      592    0.959
#> 1984 27,067       0.943   28,703  1,636      713    0.436
#> 1985 26,180       0.905   28,927  2,747    1,452    0.529
#> 1986 15,852       0.813   19,501  3,649    1,995    0.547
#> 1987 12,314       0.694   17,749  5,435    2,204    0.405
#> 1988 13,112       0.546   24,019 10,907    5,354    0.491
#> 1989  5,395       0.336   16,045 10,650    6,332    0.595
#> 1990  2,063       0.112   18,402 16,339   24,566    1.503
#> 
#>               Totals
#> Latest:   160,987.00
#> Dev:            0.76
#> Ultimate: 213,122.23
#> IBNR:      52,135.23
#> Mack.S.E   26,880.74
#> CV(IBNR):       0.52
quantile(B, c(0.75,0.95,0.99, 0.995))
#> $ByOrigin
#>        IBNR 75%  IBNR 95%  IBNR 99% IBNR 99.5%
#> 1981     0.0000     0.000     0.000      0.000
#> 1982   204.8343  1704.318  3264.173   3545.788
#> 1983  1117.3461  2967.404  4335.517   6419.470
#> 1984  2701.4295  5408.688  7827.270   9352.420
#> 1985  4168.2175  7335.162  9190.110   9718.345
#> 1986  4938.6054  8977.874 12120.942  13083.709
#> 1987  7558.0823 11418.966 15198.655  16073.881
#> 1988 13668.6428 19808.057 24259.856  25632.204
#> 1989 14786.8103 21770.959 29016.277  35537.137
#> 1990 26101.8774 43629.720 60394.522  73357.678
#> 
#> $Totals
#>                Totals
#> IBNR 75%:    67005.02
#> IBNR 95%:    89325.58
#> IBNR 99%:   107808.51
#> IBNR 99.5%: 113249.54
#> 

# fit a distribution to the IBNR
library(MASS)
plot(ecdf(B$IBNR.Totals))
# fit a log-normal distribution 
fit <- fitdistr(B$IBNR.Totals[B$IBNR.Totals>0], "lognormal")
fit
#>      meanlog         sdlog    
#>   10.843238941    0.382203762 
#>  ( 0.012092392) ( 0.008550612)
curve(plnorm(x,fit$estimate["meanlog"], fit$estimate["sdlog"]), col="red", add=TRUE)


# See also the ABC example in  Barnett and Zehnwirth (2007) 
A <- BootChainLadder(ABC, R=999, process.distr="gamma")
A
#> BootChainLadder(Triangle = ABC, R = 999, process.distr = "gamma")
#> 
#>         Latest Mean Ultimate Mean IBNR IBNR.S.E  IBNR 75%  IBNR 95%
#> 1977   762,544       762,544         0        0         0         0
#> 1978   889,022       903,227    14,205    5,063    17,121    23,315
#> 1979 1,019,932     1,057,477    37,545    8,010    42,965    50,973
#> 1980 1,002,134     1,065,965    63,831   10,006    70,031    81,499
#> 1981 1,002,194     1,102,372   100,178   11,981   107,913   121,552
#> 1982   944,614     1,088,186   143,572   14,124   151,847   168,341
#> 1983   895,700     1,107,736   212,036   17,388   223,981   241,344
#> 1984 1,024,228     1,409,915   385,687   24,505   401,663   427,891
#> 1985 1,173,448     1,939,647   766,199   37,393   791,795   827,385
#> 1986 1,011,178     2,373,554 1,362,376   59,381 1,399,825 1,466,886
#> 1987   496,200     2,690,968 2,194,768  112,580 2,256,277 2,399,332
#> 
#>                     Totals
#> Latest:         10,221,194
#> Mean Ultimate:  15,501,592
#> Mean IBNR:       5,280,398
#> IBNR.S.E           177,892
#> Total IBNR 75%:  5,389,474
#> Total IBNR 95%:  5,586,331
plot(A, log=TRUE)


## One year claims development result
CDR(A)
#>             IBNR   IBNR.S.E  CDR(1)S.E  CDR(1)75%  CDR(1)95%
#> 1977        0.00      0.000      0.000       0.00       0.00
#> 1978    14204.79   5063.042   5063.042   17121.36   23314.51
#> 1979    37544.74   8010.141   6431.272   41495.87   48099.49
#> 1980    63831.27  10006.315   6912.543   68386.76   76139.07
#> 1981   100178.18  11980.857   7946.112  105709.99  113774.09
#> 1982   143572.48  14123.774   8721.838  149155.01  158234.96
#> 1983   212036.09  17387.774  10722.941  218889.66  230011.95
#> 1984   385686.94  24504.632  14942.826  395618.21  410498.29
#> 1985   766199.41  37392.545  25049.638  782342.36  807578.39
#> 1986  1362376.44  59380.677  42857.667 1389065.88 1431208.22
#> 1987  2194768.01 112579.822  92595.410 2245995.99 2351231.43
#> Total 5280398.35 177891.617 131914.734 5359121.03 5490246.70