ChainLadder (version 0.1.2-6)

MackChainLadder: Mack-Chain-Ladder Model

Description

The Mack-chain-ladder model forecasts IBNR (Incurred But Not Reported) claims based on a cumulative claims development triangle and estimates the standard error around it.

Usage

MackChainLadder(Triangle, weights = 1/Triangle,
tail=FALSE, est.sigma="log-linear")

Arguments

Triangle

cumulative claims triangle. 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

weights

weights. Default: 1/Triangle, to use volume weighted chain ladder factors.

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 estimate via a linear extrapolation of $log(chain ladder factors - 1)$, if tail is a nu

est.sigma

defines how to estimate $sigma_{n-1}$. Default is "log-linear" for a log-linear regression, "Mack" for Mack's approximation from his 1999 paper, or if a numeric value is given it will be used instead.

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 ratios
• f.se
  estimation error of the chain-ladder ratios
• F.se
  stochastic error of the chain-ladder ratios
• sigma
  estimation of the conditional deviation of the individual chain ladder ratios
• Mack.S.E
  Mack's estimated standard error for the reserves
• Total.Mack.S.E
  Mack's estimated overall standard error for the reserves
• tail
  tail factor used. If tail was set to TRUE the output will include the linear model used to estimate the tail factor

Details

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) $k \leq n+1-i$. In order to forecast the amounts $C_{ik}$ for $k > n+1-i$ the Mack chain-ladder-model assumes: $$E[ \frac{C_{i,k+1}}{C_{ik}} | C_{i1},C_{i2},\ldots,C_{ik} ] = f_k$$ $$Var( \frac{C_{i,k+1}}{C_{ik}} | C_{i1},C_{i2}, \ldots,C_{ik} ) = \frac{\sigma_k^2}{C_{ik}}$$ $${ C_{i1},\ldots,C_{in}}, { C_{j1},\ldots,C_{jn}},\; are\; independent\; for\; origin\; period\; i \neq j$$ If these assumptions are hold, the Mack-chain-ladder-model gives an unbiased estimator for IBNR (Incurred But Not Reported) claims. The Mack-chain-ladder model can be regarded as a weighted linear regression through the origin for each development period: lm(y ~ x + 0, weights=1/x), where y is the vector of claims at development period $k+1$ and x is the vector of claims at development period $k$.

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

See Also

See also qpaid, summary.MackChainLadder, plot.MackChainLadder, residuals.MackChainLadder, MunichChainLadder, BootChainLadder,

Examples

# See the example in Mack's 1999 paper
Mortgage
MRT <- MackChainLadder(Mortgage, est.sigma="Mack", tail=1.05)
MRT
# Table 1
MRT$f
MRT$f.se
MRT$F.se[3,]
MRT$sigma
plot(MRT) # We observe trends along calendar years.

# See the Taylor/Ashe example in Mack's 1993 paper
GenIns
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)
## Include a tail factor
GNIt <- MackChainLadder(GenIns, est.sigma="Mack", tail=TRUE)
## Extract more information on the tail factor
GNIt$tail 

  
# This data set is discussed many papers, e.g. England and Verall 2000
# paper, see Table 1 just there  
RAA
R <- MackChainLadder(RAA)
R
plot(R)
# Table 12 in England and Verall 2000 paper
R$f
R$sigma^2
# Table 13 in England and Verall 2000 paper
# Please note the different indexing of sigma
MackChainLadder(RAA, est.sigma=R$sigma[7])
# Table 14 in England and Verall 2000 paper
MackChainLadder(RAA, est.sigma=R$sigma[8])

# Different weights
# Using 1/Triangle^2 as weight will use mean chain ladder ratios 
MackChainLadder(RAA, weights=1/RAA^2)$f
apply(RAA[,-1]/RAA[,-10],2, mean, na.rm=TRUE)
 
 # Let's investigate the Mack model more detail
 R[["Models"]][[1]]   # Model for first development period
 summary( R[["Models"]][[1]]) # Look at the model stats
 op <- par(mfrow=c(2,2)) # plot residuals
   plot( R[["Models"]][[1]])
 par(op)

 # Let's include an intercept in our model
 newModel <- update(R[["Models"]][[1]], y ~ x+1, 
              weights=1/R[["Triangle"]][1:9,1],
              data=data.frame(x=R[["Triangle"]][1:9,1], 
                              y=R[["Triangle"]][1:9,2])
               ) 

# View the new model
 summary(newModel)
 op <- par(mfrow=c(2,2)) 
   plot( newModel )
 par(op)

 # Change the model for dev. period one to the newModel
 R2 <- R
 R2[["Models"]][[1]] <- newModel
 predict(R2) # predict the full triangle with the new model 
 #(only the last origin year will be affected)

 R2[["FullTriangle"]] <-  predict(R2)
 R2[["FullTriangle"]] 
 R2   # Std. Errors have not been re-estimated!
 # Plot the result
 
 plot(R2, title="Changed R Model")

##