MackChainLadder: Mack Chain-Ladder Model

In ChainLadder: Statistical Methods and Models for Claims Reserving in General Insurance

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")

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

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.

Author(s)

Markus Gesmann [e-mail]

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

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
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:
## Not run: 
 demo(MackChainLadder)

## End(Not run)

ChainLadder documentation built on Sept. 11, 2024, 8:35 p.m.