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The Mack chain-ladder model forecasts future claims developments based on a historical cumulative claims development triangle and estimates the standard error around those.

Usage

MackChainLadder(Triangle, weights = 1, alpha=1, est.sigma="log-linear",
tail=FALSE, tail.se=NULL, tail.sigma=NULL, mse.method="Mack")

Arguments

Triangle

cumulative claims triangle. Assume columns are the development period, use transpose otherwise. A (mxn)-matrix \(C_{ik}\) which is filled for \(k \leq n+1-i; i=1,\ldots,m; m\geq n \), 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).

weights

weights. Default: 1, which sets the weights for all triangle entries to 1. Otherwise specify weights as a matrix of the same dimension as Triangle with all weight entries in [0; 1]. Hence, any entry set to 0 or NA eliminates that age-to-age factor from inclusion in the model. See also 'Details'

alpha

'weighting' parameters. Default: 1 for all development periods; alpha=1 gives the historical chain-ladder age-to-age factors, alpha=0 gives the straight average of the observed individual development factors and alpha=2 is the result of an ordinary regression of \(C_{i,k+1}\) against \(C_{i,k}\) with intercept 0, see also 'Details' below, chainladder and Mack's 1999 paper

est.sigma

defines how to estimate \(sigma_{n-1}\), the variability of the individual age-to-age factors at development time \(n-1\). Default is "log-linear" for a log-linear regression, "Mack" for Mack's approximation from his 1999 paper. Alternatively the user can provide a numeric value. If the log-linear model appears to be inappropriate (p-value > 0.05) the 'Mack' method will be used instead and a warning message printed. Similarly, if Triangle is so small that log-linear regression is being attempted on a vector of only one non-NA average link ratio, the 'Mack' method will be used instead and a warning message printed.

tail

can be logical or a numeric value. If tail=FALSE no tail factor will be applied, if tail=TRUE a tail factor will be estimated via a linear extrapolation of \(log(chain-ladder factors - 1)\), if tail is a numeric value than this value will be used instead.

tail.se

defines how the standard error of the tail factor is estimated. Only needed if a tail factor > 1 is provided. Default is NULL. If tail.se is NULL, tail.se is estimated via "log-linear" regression, if tail.se is a numeric value than this value will be used instead.

tail.sigma

defines how to estimate individual tail variability. Only needed if a tail factor > 1 is provided. Default is NULL. If tail.sigma is NULL, tail.sigma is estimated via "log-linear" regression, if tail.sigma is a numeric value than this value will be used instead

mse.method

method used for the recursive estimate of the parameter risk component of the mean square error. Value "Mack" (default) coincides with Mack's formula; "Independence" includes the additional cross-product term as in Murphy and BBMW. Refer to References below.

Details

Following Mack's 1999 paper let \(C_{ik}\) denote the cumulative loss amounts of origin period (e.g. accident year) \(i=1,\ldots,m\), with losses known for development period (e.g. development year) \(k \le n+1-i\). In order to forecast the amounts \(C_{ik}\) for \(k > n+1-i\) the Mack chain-ladder-model assumes: $$\mbox{CL1: } E[ F_{ik}| C_{i1},C_{i2},\ldots,C_{ik} ] = f_k \mbox{ with } F_{ik}=\frac{C_{i,k+1}}{C_{ik}} $$

$$\mbox{CL2: } Var( \frac{C_{i,k+1}}{C_{ik}} | C_{i1},C_{i2}, \ldots,C_{ik} ) = \frac{\sigma_k^2}{w_{ik} C^\alpha_{ik}} $$ $$\mbox{CL3: } \{ C_{i1},\ldots,C_{in}\}, \{ C_{j1},\ldots,C_{jn}\},\mbox{ are independent for origin period } i \neq j $$ with \(w_{ik} \in [0;1]\), \(\alpha \in \{0,1,2\}\). If these assumptions hold, the Mack chain-ladder gives an unbiased estimator for IBNR (Incurred But Not Reported) claims.

Here \(w_{ik} are the \code{weights} from above.\)

The Mack chain-ladder model can be regarded as a special form of a weighted linear regression through the origin for each development period: lm(y ~ x + 0, weights=weights/x^(2-alpha)), where y is the vector of claims at development period \(k+1\) and x is the vector of claims at development period \(k\).

It is necessary, before actually applying the model, to check if the main assumptions behind the model (i.e. Calendar Year Effect and Correlation between subsequent Accident Years, see dfCorTest, cyEffTest) are verified.

Value

MackChainLadder returns a list with the following elements

call

matched call

Triangle

input triangle of cumulative claims

FullTriangle

forecasted full triangle

Models

linear regression models for each development period

f

chain-ladder age-to-age factors

f.se

standard errors of the chain-ladder age-to-age factors f (assumption CL1)

F.se

standard errors of the true chain-ladder age-to-age factors \(F_{ik}\) (square root of the variance in assumption CL2)

sigma

sigma parameter in CL2

Mack.ProcessRisk

variability in the projection of future losses not explained by the variability of the link ratio estimators (unexplained variation)

Mack.ParameterRisk

variability in the projection of future losses explained by the variability of the link-ratio estimators alone (explained variation)

Mack.S.E

total variability in the projection of future losses by the chain-ladder method; the square root of the mean square error of the chain-ladder estimate: \(\mbox{Mack.S.E.}^2 = \mbox{Mack.ProcessRisk}^2 + \mbox{Mack.ParameterRisk}^2\)

Total.Mack.S.E

total variability of projected loss for all origin years combined

Total.ProcessRisk

vector of process risk estimate of the total of projected loss for all origin years combined by development period

Total.ParameterRisk

vector of parameter risk estimate of the total of projected loss for all origin years combined by development period

weights

weights used

alpha

alphas used

tail

tail factor used. If tail was set to TRUE the output will include the linear model used to estimate the tail factor

References

Thomas Mack. Distribution-free calculation of the standard error of chain ladder reserve estimates. Astin Bulletin. Vol. 23. No 2. 1993. pp.213:225

Thomas Mack. The standard error of chain ladder reserve estimates: Recursive calculation and inclusion of a tail factor. Astin Bulletin. Vol. 29. No 2. 1999. pp.361:366

Murphy, Daniel M. Unbiased Loss Development Factors. Proceedings of the Casualty Actuarial Society Casualty Actuarial Society - Arlington, Virginia 1994: LXXXI 154-222

Buchwalder, Bühlmann, Merz, and Wüthrich. The Mean Square Error of Prediction in the Chain Ladder Reserving Method (Mack and Murphy Revisited). Astin Bulletin Vol. 36. 2006. pp.521:542

Author

Markus Gesmann markus.gesmann@gmail.com

Note

Additional references for further reading:

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

Barnett and Zehnwirth. Best estimates for reserves. Proceedings of the CAS, LXXXVI I(167), November 2000.

See also

See also qpaid for dealing with non-square triangles, chainladder for the underlying chain-ladder method, dfCorTest to check for Calendar Year Effect, cyEffTest to check for Development Factor Correlation, summary.MackChainLadder, quantile.MackChainLadder, plot.MackChainLadder and residuals.MackChainLadder displaying results, CDR.MackChainLadder for the one year claims development result.

Examples

## See the Taylor/Ashe example in Mack's 1993 paper
GenIns
#>       dev
#> origin      1       2       3       4       5       6       7       8       9
#>     1  357848 1124788 1735330 2218270 2745596 3319994 3466336 3606286 3833515
#>     2  352118 1236139 2170033 3353322 3799067 4120063 4647867 4914039 5339085
#>     3  290507 1292306 2218525 3235179 3985995 4132918 4628910 4909315      NA
#>     4  310608 1418858 2195047 3757447 4029929 4381982 4588268      NA      NA
#>     5  443160 1136350 2128333 2897821 3402672 3873311      NA      NA      NA
#>     6  396132 1333217 2180715 2985752 3691712      NA      NA      NA      NA
#>     7  440832 1288463 2419861 3483130      NA      NA      NA      NA      NA
#>     8  359480 1421128 2864498      NA      NA      NA      NA      NA      NA
#>     9  376686 1363294      NA      NA      NA      NA      NA      NA      NA
#>     10 344014      NA      NA      NA      NA      NA      NA      NA      NA
#>       dev
#> origin      10
#>     1  3901463
#>     2       NA
#>     3       NA
#>     4       NA
#>     5       NA
#>     6       NA
#>     7       NA
#>     8       NA
#>     9       NA
#>     10      NA
plot(GenIns)

plot(GenIns, lattice=TRUE)

GNI <- MackChainLadder(GenIns, est.sigma="Mack")
GNI$f
#>  [1] 3.490607 1.747333 1.457413 1.173852 1.103824 1.086269 1.053874 1.076555
#>  [9] 1.017725 1.000000
GNI$sigma^2
#> [1] 160280.3275  37736.8550  41965.2130  15182.9027  13731.3239   8185.7716
#> [7]    446.6166   1147.3660    446.6166
GNI # compare to table 2 and 3 in Mack's 1993 paper
#> 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
plot(GNI)

plot(GNI, lattice=TRUE)


## Different weights
## Using alpha=0 will use straight average age-to-age factors 
MackChainLadder(GenIns, alpha=0)$f
#>  [1] 3.566143 1.745557 1.451961 1.180984 1.111247 1.084818 1.052739 1.074753
#>  [9] 1.017725 1.000000
# You get the same result via:
apply(GenIns[,-1]/GenIns[,-10],2, mean, na.rm=TRUE)
#>        2        3        4        5        6        7        8        9 
#> 3.566143 1.745557 1.451961 1.180984 1.111247 1.084818 1.052739 1.074753 
#>       10 
#> 1.017725 

## Only use the last 5 diagonals, i.e. the last 5 calendar years
calPeriods <- (row(GenIns) + col(GenIns) - 1)
(weights <- ifelse(calPeriods <= 5, 0, ifelse(calPeriods > 10, NA, 1)))
#>       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
#>  [1,]    0    0    0    0    0    1    1    1    1     1
#>  [2,]    0    0    0    0    1    1    1    1    1    NA
#>  [3,]    0    0    0    1    1    1    1    1   NA    NA
#>  [4,]    0    0    1    1    1    1    1   NA   NA    NA
#>  [5,]    0    1    1    1    1    1   NA   NA   NA    NA
#>  [6,]    1    1    1    1    1   NA   NA   NA   NA    NA
#>  [7,]    1    1    1    1   NA   NA   NA   NA   NA    NA
#>  [8,]    1    1    1   NA   NA   NA   NA   NA   NA    NA
#>  [9,]    1    1   NA   NA   NA   NA   NA   NA   NA    NA
#> [10,]    1   NA   NA   NA   NA   NA   NA   NA   NA    NA
MackChainLadder(GenIns, weights=weights, est.sigma = "Mack")
#> MackChainLadder(Triangle = GenIns, weights = weights, 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      0.7349 5,023,131 1,331,419   341,719    0.257
#> 7  3,483,130      0.6263 5,561,629 2,078,499   547,444    0.263
#> 8  2,864,498      0.4258 6,726,585 3,862,087   975,424    0.253
#> 9  1,363,294      0.2299 5,929,927 4,566,633 1,065,926    0.233
#> 10   344,014      0.0669 5,142,278 4,798,264 1,247,449    0.260
#> 
#>                  Totals
#> Latest:   34,358,090.00
#> Dev:               0.65
#> Ultimate: 53,253,663.06
#> IBNR:     18,895,573.06
#> Mack.S.E   2,550,023.96
#> CV(IBNR):          0.13

## Tail
## See the example in Mack's 1999 paper
Mortgage
#>       dev
#> origin     1      2       3       4       5       6       7       8       9
#>      1 58046 127970  476599 1027692 1360489 1647310 1819179 1906852 1950105
#>      2 24492 141767  984288 2142656 2961978 3683940 4048898 4115760      NA
#>      3 32848 274682 1522637 3203427 4445927 5158781 5342585      NA      NA
#>      4 21439 529828 2900301 4999019 6460112 6853904      NA      NA      NA
#>      5 40397 763394 2920745 4989572 5648563      NA      NA      NA      NA
#>      6 90748 951994 4210640 5866482      NA      NA      NA      NA      NA
#>      7 62096 868480 1954797      NA      NA      NA      NA      NA      NA
#>      8 24983 284441      NA      NA      NA      NA      NA      NA      NA
#>      9 13121     NA      NA      NA      NA      NA      NA      NA      NA
m <- MackChainLadder(Mortgage)
round(summary(m)$Totals["CV(IBNR)",], 2) ## 26% in Table 6 of paper
#> [1] 0.26
plot(Mortgage)

# Specifying the tail and its associated uncertainty parameters
MRT <- MackChainLadder(Mortgage, tail=1.05, tail.sigma=71, tail.se=0.02, est.sigma="Mack")
MRT
#> MackChainLadder(Triangle = Mortgage, est.sigma = "Mack", tail = 1.05, 
#>     tail.se = 0.02, tail.sigma = 71)
#> 
#>      Latest Dev.To.Date   Ultimate      IBNR  Mack.S.E CV(IBNR)
#> 1 1,950,105     0.95238  2,047,610    97,505   106,544    1.093
#> 2 4,115,760     0.93126  4,419,573   303,813   179,977    0.592
#> 3 5,342,585     0.90736  5,888,041   545,456   249,708    0.458
#> 4 6,853,904     0.84904  8,072,571 1,218,667   417,857    0.343
#> 5 5,648,563     0.74548  7,577,086 1,928,523   670,156    0.347
#> 6 5,866,482     0.58427 10,040,732 4,174,250 1,127,984    0.270
#> 7 1,954,797     0.34209  5,714,195 3,759,398 1,377,496    0.366
#> 8   284,441     0.08360  3,402,595 3,118,154 1,901,740    0.610
#> 9    13,121     0.00753  1,742,908 1,729,787 2,293,437    1.326
#> 
#>                  Totals
#> Latest:   32,029,758.00
#> Dev:               0.65
#> Ultimate: 48,905,312.55
#> IBNR:     16,875,554.55
#> Mack.S.E   4,053,667.67
#> CV(IBNR):          0.24
plot(MRT, lattice=TRUE)

# Specify just the tail and the uncertainty parameters will be estimated
MRT <- MackChainLadder(Mortgage, tail=1.05)
MRT$f.se[9] # close to the 0.02 specified above
#> tail.se.tail.factor 
#>          0.02093287 
MRT$sigma[9] # less than the 71 specified above
#> tail.sigma 
#>   55.45125 
# Note that the overall CV dropped slightly
round(summary(MRT)$Totals["CV(IBNR)",], 2) ## 24%
#> [1] 0.24
# tail parameter uncertainty equal to expected value 
MRT <- MackChainLadder(Mortgage, tail=1.05, tail.se = .05)
round(summary(MRT)$Totals["CV(IBNR)",], 2) ## 27%
#> [1] 0.27

## Parameter-risk (only) estimate of the total reserve = 3142387
tail(MRT$Total.ParameterRisk, 1) # located in last (ultimate) element
#> [1] 3142387
#  Parameter-risk (only) CV is about 19%
tail(MRT$Total.ParameterRisk, 1) / summary(MRT)$Totals["IBNR", ]
#> [1] 0.1862094

## Three terms in the parameter risk estimate
## First, the default (Mack) without the tail
m <- MackChainLadder(RAA, mse.method = "Mack")
summary(m)$Totals["Mack S.E.",]
#> [1] 26880.74
## Then, with the third term
m <- MackChainLadder(RAA, mse.method = "Independence")
summary(m)$Totals["Mack S.E.",] ## Not significantly greater
#> [1] 26895.69

## One year claims development results
M <- MackChainLadder(MW2014, est.sigma="Mack")
CDR(M)
#>               IBNR   CDR(1)S.E.    Mack.S.E.
#> 1         0.000000    0.0000000    0.0000000
#> 2         1.022874    0.4083149    0.4083149
#> 3        10.085643    2.5393857    2.5652899
#> 4        21.187574   16.7232632   16.8984949
#> 5       117.662565  156.4022713  157.2756452
#> 6       223.279748  137.6522771  207.1650862
#> 7       361.808180  171.1812092  261.9266093
#> 8       469.408830   70.3161155  292.2622285
#> 9       653.504225  271.6352221  390.5874717
#> 10     1008.763182  310.1268449  502.0606072
#> 11     1011.859648  103.3834357  486.0911099
#> 12     1406.702133  632.6388191  806.9028971
#> 13     1492.903495  315.0489135  793.9381916
#> 14     1917.636398  406.1424672  891.6613403
#> 15     2458.152208  285.2076540  916.4940218
#> 16     3384.341045  668.2337878 1106.1262716
#> 17     9596.552341  733.2222786 1295.6909824
#> Total 24134.870088 1842.8507073 3233.6807352

## For more examples see:
if (FALSE) { # \dontrun{
 demo(MackChainLadder)
} # }