MackChainLadder: Mack Chain-Ladder Model

Description

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")

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

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.

Author

Markus Gesmann markus.gesmann@gmail.com

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.

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

Examples

Run this code

## See the Taylor/Ashe example in Mack's 1993 paper
GenIns
plot(GenIns)
plot(GenIns, lattice=TRUE)
GNI <- MackChainLadder(GenIns, est.sigma="Mack")
GNI$f
GNI$sigma^2
GNI # compare to table 2 and 3 in Mack's 1993 paper
plot(GNI)
plot(GNI, lattice=TRUE)

## Different weights
## Using alpha=0 will use straight average age-to-age factors 
MackChainLadder(GenIns, alpha=0)$f
# You get the same result via:
apply(GenIns[,-1]/GenIns[,-10],2, mean, na.rm=TRUE)

## 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)))
MackChainLadder(GenIns, weights=weights, est.sigma = "Mack")

## Tail
## See the example in Mack's 1999 paper
Mortgage
m <- MackChainLadder(Mortgage)
round(summary(m)$Totals["CV(IBNR)",], 2) ## 26% in Table 6 of paper
plot(Mortgage)
# Specifying the tail and its associated uncertainty parameters
MRT <- MackChainLadder(Mortgage, tail=1.05, tail.sigma=71, tail.se=0.02, est.sigma="Mack")
MRT
plot(MRT, lattice=TRUE)
# Specify just the tail and the uncertainty parameters will be estimated
MRT <- MackChainLadder(Mortgage, tail=1.05)
MRT$f.se[9] # close to the 0.02 specified above
MRT$sigma[9] # less than the 71 specified above
# Note that the overall CV dropped slightly
round(summary(MRT)$Totals["CV(IBNR)",], 2) ## 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%

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

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

## One year claims development results
M <- MackChainLadder(MW2014, est.sigma="Mack")
CDR(M)

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

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