MackChainLadder {ChainLadder}R Documentation

Mack-Chain-Ladder Model

Description

Mack-chain-ladder model to forecast IBNR claims based on a cumulative claims triangle.

Usage

MackChainLadder(Triangle, weights = 1/Triangle)

## S3 method for class 'MackChainLadder':
print(x, ...)

## S3 method for class 'MackChainLadder':
plot(x, mfrow=c(3,2), title=NULL, ...)

## S3 method for class 'MackChainLadder':
summary(object, ...)

Arguments

Triangle a cumulative claims triangle. A quadratic (nxn)-matrix C_{ik} which is filled for k <=q n+1-i, i=1,...,n
weights weights. Default: 1/Triangle
x, object an object of class "MackChainLadder"
mfrow see par
title see title
... not in use

Details

Let C_{ik} denote the cumulative loss amounts of origin year i=1,...,n, with losses know for development year k <=q 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},...,C_{ik} ] = f_k

Var( frac{C_{i,k+1}}{C_{ik}} | C_{i1},C_{i2},...,C_{ik} ) = frac{σ_k^2}{C_{ik}}

{ C_{i1},...,C_{in}}, { C_{j1},...,C_{jn}},; are; independent; for; origin; year; 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 chain-ladder model can be regarded as weighted linear regression through the origin for each development year: lm(y ~ x + 0, weights=1/x), where y is the vector of claims at development year k+1 and x is the vector of claims at development year k.

A tail factor is not yet implemented.

Value

Triangle input triangle of cumulative claims
FullTriangle forecasted full triangle
Models linear regression models for each development year
f chain-ladder ratios
f.se standard error for chain-ladder ratios
F.se standard error for individual age-to-age ratios
sigma chain-ladder ratio variance
Mack.S.E Mack's estimated standard error for the reserves
Total.Mack.S.E Mack's estimated overall standard error for the reserves

Author(s)

Markus Gesmann markus.gesmann@web.de

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 http://www.casact.org/library/astin/vol23no2/213.pdf

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 http://www.casact.org/library/astin/vol29no2/361.pdf

See Also

See also MunichChainLadder, residuals.MackChainLadder

Examples


data(Mortgage)
Mortgage

MRT <- MackChainLadder(Mortgage)
MRT
plot(MRT) # We observe trends along calendar years.

data(GenIns)
GenIns

GNI <- MackChainLadder(GenIns)
GNI
plot(GNI)
 
  
data(RAA)
RAA  
 
MCL <- MackChainLadder(RAA)
MCL
plot(MCL)
 
 
 # investigate in more detail
 MCL[["Models"]][[1]]   # Model for first development period
 summary( MCL[["Models"]][[1]]) # Look at the model stats
 op=par(mfrow=c(2,2)) # plot residuals
   plot( MCL[["Models"]][[1]])
 par(op)

 # let's include an intercept in our model
 newModel <- update(MCL[["Models"]][[1]], y ~ x+1, 
              weights=1/MCL[["Triangle"]][1:9,1],
              data=data.frame(x=MCL[["Triangle"]][1:9,1], 
                              y=MCL[["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
 MCL2 <- MCL
 MCL2[["Models"]][[1]] <- newModel
 predict(MCL2) # predict the full triangle with the new model 
 #(only the last origin year will be affected)

 MCL2[["FullTriangle"]] <-  predict(MCL2)
 MCL2[["FullTriangle"]] 
 MCL2   # Std. Errors have not been re-estimated!
 # plot the result
 
 plot(MCL2, title="Change MCL Model")


[Package ChainLadder version 0.1.1-3 Index]