### 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 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

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 markus.gesmann@gmail.com

### 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 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$f GNI$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$f.se[9] # close to the 0.02 specified above MRT$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