Multivariate Chain-Ladder Models

The function MultiChainLadder implements multivariate methods to forecast insurance loss payments based on several cumulative claims development triangles. These methods are multivariate extensions of the chain-ladder technique, which develop several correlated triangles simultaneously in a way that both contemporaneous correlations and structural relationships can be accounted for. The estimated conditional Mean Square Errors (MSE) are also produced.

Usage

MultiChainLadder(Triangles, fit.method = "SUR", delta = 1,
  int = NULL, restrict.regMat = NULL, extrap = TRUE, 
  mse.method = "Mack", model = "MCL", ...)
    
MultiChainLadder2(Triangles, mse.method = "Mack", last = 3, 
  type = c("MCL", "MCL+int", "GMCL-int", "GMCL"), ...)

Arguments

Triangles: a list of cumulative claims triangles of the same dimensions.
fit.method: the method used to fit the multivariate regression in each development period. The default is "SUR" - seemingly unrelated regressions. When "OLS" (Ordinary Least Squares) is used, this is the same as developing each triangle separately.
delta: parameter for controlling weights. It is used to determine the covariance structure $D(Y_{i,k}^{\delta/2})\Sigma_k D(Y_{i,k}^{\delta/2})$. The default value 1 means that the variance is proportional to the cumulative loss from the previous period.
int: a numeric vector that indicates which development periods have intercepts specified. This only takes effect for model = "GMCL". The default NULL means that no intercepts are specified.
restrict.regMat: a list of matrix specifying parameter restriction matrix for each period. This is only used for model = "GMCL". The default value NULL means no restriction is imposed on the development matrix. For example, if there are 3 triangles, there will be 9 parameters in the development matrix for each period if restrict.regMat = NULL. See systemfit for how to specify the appropriate parameter constraints.
extrap: a logical value indicating whether to use Mack's extrapolation method for the last period to get the residual variance estimation. It only takes effect for model = "MCL". If the data are trapezoids, it is set to be FALSE automatically and a warning message is given.
mse.method: method to estimate the mean square error. It can be either "Mack" or "Independence", which are the multivariate generalization of Mack's formulas and the conditional re-sampling approach, respectively.
model: the structure of the model to be fitted. It is either "MCL" or "GMCL". See details below.
last: an integer. The MultiChainLadder2 function splits the triangles into 2 parts internally (see details below), and the last argument indicates how many of the development periods in the tail go into the second part of the split. The default is 3.
type: the type of the model structure to be specified for the first part of the split model in MultiChainLadder2. "MCL"- the multivariate chain-ladder with diagonal development matrix; "MCL+int"- the multivariate chain-ladder with additional intercepts; "GMCL-int"- the general multivariate chain-ladder without intercepts; and "GMCL" - the full general multivariate chain-ladder with intercepts and non-diagonal development matrix.
...: arguments passed to systemfit.

Details

This function implements multivariate loss reserving models within the chain-ladder framework. Two major models are included. One is the Multivariate chain-ladder (MCL) model proposed by Prohl and Schmidt (2005). This is a direct multivariate generalization of the univariate chain-ladder model in that losses from different triangles are assumed to be correlated but the mean development in one triangle only depends on its past values, not on the observed values from other triangles. In contrast, the other model, the General Multivariate chain-ladder (GMCL) model outlined in Zhang (2010), extends the MCL model by allowing development dependencies among different triangles as well as the inclusion of regression intercepts. As a result, structurally related triangles, such as the paid and incurred loss triangles or the paid loss and case reserve triangles, can be developed together while still accounting for the potential contemporaneous correlations. While the MCL model is a special case of the GMCL model, it is programmed and listed separately because: a) it is an important model for its own sake; b) different MSE methods are only available for the MCL model; c) extrapolation of the residual variance estimation can be implemented for the MCL model, which is considerably difficult for the GMCL model.

We introduce some details of the GMCL model in the following. Assume N triangles are available. Denote $Y_{i,k}=(Y^{(1)}_{i,k}, \ldots ,Y^{(N)}_{i,k})$ as an $N \times 1$ vector of cumulative losses at accident year i and development year k, where (n) refers to the n-th triangle. The GMCL model in development period k (from development year k to year k+1) is: $$Y_{i,k+1}=A_k + B_k \cdot Y_{i,k}+\epsilon_{i,k},$$ where $A_k$ is a column of intercepts and $B_k$ is the $N \times N$ development matrix. By default, MultiChainLadder sets $A_k$ to be zero. This behavior can be changed by appropriately specifying the int argument. Assumptions for this model are: $$E(\epsilon_{i,k}|Y_{i,1},\ldots,Y_{i,I+1-k})=0.$$ $$cov(\epsilon_{i,k}|Y_{i,1},\ldots,Y_{i,I+1-k})=\Sigma_{\epsilon_{i,k}}=D(Y_{i,k}^{\delta/2})\Sigma_k D(Y_{i,k}^{\delta/2}).$$ $$\mbox{losses of different accident years are independent}.$$ $$\epsilon_{i,k} \, \mbox{are symmetrically distributed}.$$

The GMCL model structure is generally over-parameterized. Parameter restrictions are usually necessary for the estimation to be feasible, which can be specified through the restrict.regMat argument. We refer the users to the documentation for systemfit for details and the demo of the present function for examples.

In particular, if one restricts the development matrix to be diagonal, the GMCL model will reduce to the MCL model. When non-diagonal development matrix is used and the GMCL model is applied to paid and incurred loss triangles, it can reflect the development relationship between the two triangles, as described in Quarg and Mack (2004). The full bivariate model is identical to the "double regression" model described by Mack (2003), which is argued by him to be very similar to the Munich chain-ladder (MuCL) model. The GMCL model with intercepts can also help improve model adequacy as described in Barnett and Zehnwirth (2000).

Currently, the GMCL model only works for trapezoid data, and only implements mse.method = "Mack". The MCL model allows an additional mse estimation method that assumes independence among the estimated parameters. Further, the MCL model using fit.method = "OLS" will be equivalent to running univariate chain-ladders separately on each triangle. Indeed, when only one triangle is specified (as a list), the MCL model is equivalent to MackChainLadder.

The GMCL model allows different model structures to be specified in each development period. This is generally achieved through the combination of the int argument, which specifies the periods that have intercepts, and the restrict.regMat argument, which imposes parameter restrictions on the development matrix.

In using the multivariate methods, we often specify separate univariate chain-ladders for the tail periods to stabilize the estimation - there are few data points in the tail and running a multivariate model often produces extremely volatile estimates or even fails. In this case, we can use the subset operator "[" defined for class triangles to split the input data into two parts. We can specify a multivariate model with rich structures on the first part to reflect the multivariate dependencies, and simply apply multiple univariate chain-ladders on the second part. The two models are subsequently joined together using the Join2Fits function. We can then invoke the predict and Mse methods to produce loss predictions and mean square error estimations. They can further be combined via the JoinFitMse function to construct an object of class MultiChainLadder. See the demo for such examples.

To facilitate such a split-and-join process for most applications, we have created the function MultiChainLadder2. This function splits the data according to the last argument (e.g., if last = 3, the last three periods go into the second part), and fits the first part according to the structure indicated in the type argument. See the 'Arguments' section for details.

Note

When MultiChainLadder or MultiChainLadder2 fails, the most possible reason is that there is little or no development in the tail periods. That is, the development factor is 1 or almost equal to 1. In this case, the systemfit function may fail even for fit.method = "OLS", because the residual covariance matrix $\Sigma_k$ is singular. The simplest solution is to remove these columns using the "[" operator and fit the model on the remaining part.

Also, we recommend the use of MultiChainLadder2 over MultiChainLadder. The function MultiChainLadder2 meets the need for most applications, is relatively easy to use and produces more stable but very similar results to MultiChainLadder. Use MultiChainLadder only when non-standard situation arises, e.g., when different parameter restrictions are needed for different periods. See the demo for such examples.

Value

MultiChainLadder returns an object of class MultiChainLadder with the following slots:

model: the model structure used, either "MCL" or "GMCL"
Triangles: input triangles of cumulative claims that are converted to class triangles internally.
models: fitted models for each development period. This is the output from the call of systemfit.
coefficients: estimated regression coefficients or development parameters. They are put into the matrix format for the GMCL model.
coefCov: estimated variance-covariance matrix for the regression coefficients.
residCov: estimated residual covariance matrix.
fit.method: multivariate regression estimation method
delta: the value of delta
mse.ay: mean square error matrix for each accident year
mse.ay.est: estimation error matrix for each accident year
mse.ay.proc: process error matrix for each accident year
mse.total: mean square error matrix for all accident years combined
mse.total.est: estimation error matrix for all accident years combined
mse.total.proc: process error matrix for all accident years combined
FullTriangles: the forecasted full triangles of class triangles
int: intercept indicators

References

Buchwalder M, Bühlmann H, Merz M, Wüthrich M.V (2006). The mean square error of prediction in the chain-ladder reserving method (Mack and Murphy revisited), ASTIN Bulletin, 36(2), 521-542.

Prohl C, Schmidt K.D (2005). Multivariate chain-ladder, Dresdner Schriften zur Versicherungsmathematik.

Mack T (1993). Distribution-free calculation of the standard error, ASTIN Bulletin, 23, No.2.

Mack T (1999). The standard error of chain-ladder reserve estimates: recursive calculation and inclusion of a tail factor, ASTIN Bulletin, 29, No.2, 361-366.

Merz M, Wüthrich M (2008). Prediction error of the multivariate chain ladder reserving method, North American Actuarial Journal, 12, No.2, 175-197.

Zhang Y (2010). A general multivariate chain-ladder model.Insurance: Mathematics and Economics, 46, pp. 588-599.

Zhang Y (2010). Prediction error of the general multivariate chain ladder model.

Author

Wayne Zhang actuary_zhang@hotmail.com

Examples

# This shows that the MCL model using "OLS" is equivalent to 
# the MackChainLadder when applied to one triangle 

data(GenIns)
(U1 <- MackChainLadder(GenIns, est.sigma = "Mack"))
#> MackChainLadder(Triangle = GenIns, est.sigma = "Mack")
#> 
#>       Latest Dev.To.Date  Ultimate      IBNR  Mack.S.E CV(IBNR)
#> 1  3,901,463      1.0000 3,901,463         0         0      NaN
#> 2  5,339,085      0.9826 5,433,719    94,634    75,535    0.798
#> 3  4,909,315      0.9127 5,378,826   469,511   121,699    0.259
#> 4  4,588,268      0.8661 5,297,906   709,638   133,549    0.188
#> 5  3,873,311      0.7973 4,858,200   984,889   261,406    0.265
#> 6  3,691,712      0.7223 5,111,171 1,419,459   411,010    0.290
#> 7  3,483,130      0.6153 5,660,771 2,177,641   558,317    0.256
#> 8  2,864,498      0.4222 6,784,799 3,920,301   875,328    0.223
#> 9  1,363,294      0.2416 5,642,266 4,278,972   971,258    0.227
#> 10   344,014      0.0692 4,969,825 4,625,811 1,363,155    0.295
#> 
#>                  Totals
#> Latest:   34,358,090.00
#> Dev:               0.65
#> Ultimate: 53,038,945.61
#> IBNR:     18,680,855.61
#> Mack.S.E   2,447,094.86
#> CV(IBNR):          0.13
(U2 <- MultiChainLadder(list(GenIns), fit.method = "OLS"))
#> $`Summary Statistics for Input Triangle`
#>           Latest Dev.To.Date   Ultimate       IBNR       S.E    CV
#> 1      3,901,463      1.0000  3,901,463          0         0 0.000
#> 2      5,339,085      0.9826  5,433,719     94,634    75,535 0.798
#> 3      4,909,315      0.9127  5,378,826    469,511   121,699 0.259
#> 4      4,588,268      0.8661  5,297,906    709,638   133,549 0.188
#> 5      3,873,311      0.7973  4,858,200    984,889   261,406 0.265
#> 6      3,691,712      0.7223  5,111,171  1,419,459   411,010 0.290
#> 7      3,483,130      0.6153  5,660,771  2,177,641   558,317 0.256
#> 8      2,864,498      0.4222  6,784,799  3,920,301   875,328 0.223
#> 9      1,363,294      0.2416  5,642,266  4,278,972   971,258 0.227
#> 10       344,014      0.0692  4,969,825  4,625,811 1,363,155 0.295
#> Total 34,358,090      0.6478 53,038,946 18,680,856 2,447,095 0.131
#> 

# show plots 
parold <- par(mfrow = c(2, 2))
plot(U2, which.plot = 1:4)

plot(U2, which.plot = 5)

par(parold)


# For mse.method = "Independence", the model is equivalent 
# to that in Buchwalder et al. (2006)

(B1 <- MultiChainLadder(list(GenIns), fit.method = "OLS", 
    mse.method = "Independence"))
#> $`Summary Statistics for Input Triangle`
#>           Latest Dev.To.Date   Ultimate       IBNR       S.E    CV
#> 1      3,901,463      1.0000  3,901,463          0         0 0.000
#> 2      5,339,085      0.9826  5,433,719     94,634    75,535 0.798
#> 3      4,909,315      0.9127  5,378,826    469,511   121,700 0.259
#> 4      4,588,268      0.8661  5,297,906    709,638   133,551 0.188
#> 5      3,873,311      0.7973  4,858,200    984,889   261,412 0.265
#> 6      3,691,712      0.7223  5,111,171  1,419,459   411,028 0.290
#> 7      3,483,130      0.6153  5,660,771  2,177,641   558,356 0.256
#> 8      2,864,498      0.4222  6,784,799  3,920,301   875,430 0.223
#> 9      1,363,294      0.2416  5,642,266  4,278,972   971,385 0.227
#> 10       344,014      0.0692  4,969,825  4,625,811 1,363,385 0.295
#> Total 34,358,090      0.6478 53,038,946 18,680,856 2,447,618 0.131
#> 

# use the unbiased residual covariance estimator 
# in Merz and Wuthrich (2008)
(W1 <- MultiChainLadder2(liab, mse.method = "Independence", 
        control = systemfit::systemfit.control(methodResidCov = "Theil"))) 
#> $`Summary Statistics for Triangle 1`
#>           Latest Dev.To.Date   Ultimate      IBNR     S.E     CV
#> 1        549,589      1.0000    549,589         0       0 0.0000
#> 2        562,795      0.9966    564,740     1,945   1,743 0.8961
#> 3        602,710      0.9911    608,104     5,394   7,354 1.3633
#> 4        784,632      0.9867    795,248    10,616   9,042 0.8518
#> 5        768,373      0.9804    783,768    15,395  11,181 0.7263
#> 6        811,100      0.9687    837,341    26,241  16,783 0.6396
#> 7        896,728      0.9542    939,740    43,012  19,706 0.4581
#> 8      1,022,241      0.9299  1,099,257    77,016  23,362 0.3033
#> 9      1,019,303      0.8817  1,156,068   136,765  29,610 0.2165
#> 10     1,141,750      0.7969  1,432,782   291,032  37,521 0.1289
#> 11     1,174,196      0.6760  1,737,032   562,836  57,669 0.1025
#> 12     1,032,684      0.4998  2,066,151 1,033,467  89,534 0.0866
#> 13       772,971      0.2907  2,659,239 1,886,268 193,211 0.1024
#> 14       204,325      0.0901  2,268,345 2,064,020 282,755 0.1370
#> Total 11,343,397      0.6483 17,497,403 6,154,006 427,041 0.0694
#> 
#> $`Summary Statistics for Triangle 2`
#>          Latest Dev.To.Date   Ultimate      IBNR     S.E      CV
#> 1       391,428       1.000    391,428         0       0  0.0000
#> 2       483,974       1.000    483,839      -135     604 -4.4825
#> 3       540,742       1.001    540,002      -740   1,436 -1.9416
#> 4       485,016       0.998    486,227     1,211   2,912  2.4043
#> 5       507,752       0.998    508,728       976   3,201  3.2801
#> 6       549,693       0.994    552,860     3,167   5,418  1.7108
#> 7       635,452       0.994    639,253     3,801   6,221  1.6367
#> 8       648,365       0.985    658,572    10,207   7,484  0.7333
#> 9       663,152       0.968    684,936    21,784   9,125  0.4189
#> 10      790,901       0.935    845,986    55,085  16,196  0.2940
#> 11      844,159       0.877    963,023   118,864  26,752  0.2251
#> 12      915,109       0.783  1,169,265   254,156  36,741  0.1446
#> 13      909,066       0.617  1,473,780   564,714  53,391  0.0945
#> 14      394,997       0.278  1,423,135 1,028,138 126,545  0.1231
#> Total 8,759,806       0.809 10,821,034 2,061,228 162,785  0.0790
#> 
#> $`Summary Statistics for Triangle 1+2`
#>           Latest Dev.To.Date   Ultimate      IBNR     S.E     CV
#> 1        941,017       1.000    941,017         0       0 0.0000
#> 2      1,046,769       0.998  1,048,579     1,810   1,845 1.0190
#> 3      1,143,452       0.996  1,148,107     4,655   7,493 1.6097
#> 4      1,269,648       0.991  1,281,475    11,827   9,499 0.8032
#> 5      1,276,125       0.987  1,292,496    16,371  12,068 0.7372
#> 6      1,360,793       0.979  1,390,201    29,408  18,888 0.6423
#> 7      1,532,180       0.970  1,578,993    46,813  22,455 0.4797
#> 8      1,670,606       0.950  1,757,828    87,222  26,019 0.2983
#> 9      1,682,455       0.914  1,841,004   158,549  33,404 0.2107
#> 10     1,932,651       0.848  2,278,768   346,117  45,438 0.1313
#> 11     2,018,355       0.748  2,700,055   681,700  72,278 0.1060
#> 12     1,947,793       0.602  3,235,416 1,287,623 112,426 0.0873
#> 13     1,682,037       0.407  4,133,019 2,450,982 223,181 0.0911
#> 14       599,322       0.162  3,691,479 3,092,157 342,288 0.1107
#> Total 20,103,203       0.710 28,318,437 8,215,234 505,376 0.0615
#> 

if (FALSE) {
# use the iterative residual covariance estimator
for (i in 1:5){
  W2 <- MultiChainLadder2(liab, mse.method = "Independence", 
      control = systemfit::systemfit.control(
      methodResidCov = "Theil", maxiter = i))
  print(format(summary(W2)@report.summary[[3]][15, 4:5], 
          digits = 6, big.mark = ","))    
}

# The following fits an MCL model with intercepts for years 1:7
# and separate chain-ladder models for the rest periods
f1 <- MultiChainLadder2(auto, type = "MCL+int")

# compare with the model without intercepts through residual plots
f0 <- MultiChainLadder2(auto, type = "MCL")

parold <- par(mfrow = c(2, 3), mar = c(3, 3, 2, 1))
mt <- list(c("Personal Paid", "Personal Incured", "Commercial Paid"))
plot(f0, which.plot = 3, main = mt)
plot(f1, which.plot = 3, main = mt)
par(parold)

## summary statistics
summary(f1, portfolio = "1+3")@report.summary[[4]]


# model for joint development of paid and incurred triangles 
da <- auto[1:2]
# MCL with diagonal development
M0 <- MultiChainLadder(da)
# non-diagonal development matrix with no intercepts
M1 <- MultiChainLadder2(da, type = "GMCL-int")
# Munich chain-ladder
M2 <- MunichChainLadder(da[[1]], da[[2]])
# compile results and compare projected paid to incurred ratios
r1 <- lapply(list(M0, M1), function(x){
          ult <- summary(x)@Ultimate
          ult[, 1] / ult[, 2]
      })
names(r1) <- c("MCL", "GMCL")
r2 <- summary(M2)[[1]][, 6]
r2 <- c(r2, summary(M2)[[2]][2, 3])
print(do.call(cbind, c(r1, list(MuCl = r2))) * 100, digits = 4)

}

# To reproduce results in Zhang (2010) and see more examples, use:
if (FALSE) {
 demo(MultiChainLadder)
}