### Description

The Munich-chain-ladder model forecasts ultimate claims based on a cumulative paid and incurred claims triangle. The model assumes that the Mack-chain-ladder model is applicable to the paid and incurred claims triangle, see MackChainLadder.

### Usage

MunichChainLadder(Paid, Incurred,
est.sigmaP = "log-linear", est.sigmaI = "log-linear",
tailP=FALSE, tailI=FALSE, weights=1)


### Arguments

 Paid cumulative paid claims triangle. Assume columns are the development period, use transpose otherwise. A (mxn)-matrix P_{ik} which is filled for k \leq n+1-i; i=1,\ldots,m; m\geq n Incurred cumulative incurred claims triangle. Assume columns are the development period, use transpose otherwise. A (mxn)-matrix I_{ik} which is filled for k \leq n+1-i; i=1,\ldots,m, m\geq n  est.sigmaP defines how sigma_{n-1} for the Paid triangle is estimated, see est.sigma in MackChainLadder for more details, as est.sigmaP gets passed on to MackChainLadder est.sigmaI defines how sigma_{n-1} for the Incurred triangle is estimated, see est.sigma in MackChainLadder for more details, as est.sigmaI is passed on to MackChainLadder tailP defines how the tail of the Paid triangle is estimated and is passed on to MackChainLadder, see tail just there. tailI defines how the tail of the Incurred triangle is estimated and is passed on to MackChainLadder, see tail just there. 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' in MackChainladder function. The weight matrix is the same for Paid and Incurred.

### Value

MunichChainLadder returns a list with the following elements

 call matched call Paid input paid triangle Incurred input incurred triangle MCLPaid Munich-chain-ladder forecasted full triangle on paid data MCLIncurred Munich-chain-ladder forecasted full triangle on incurred data MackPaid Mack-chain-ladder output of the paid triangle MackIncurred Mack-chain-ladder output of the incurred triangle PaidResiduals paid residuals IncurredResiduals incurred residuals QResiduals paid/incurred residuals QinverseResiduals incurred/paid residuals lambdaP dependency coefficient between paid chain-ladder age-to-age factors and incurred/paid age-to-age factors lambdaI dependency coefficient between incurred chain-ladder ratios and paid/incurred ratios qinverse.f chain-ladder-link age-to-age factors of the incurred/paid triangle rhoP.sigma estimated conditional deviation around the paid/incurred age-to-age factors q.f chain-ladder age-to-age factors of the paid/incurred triangle rhoI.sigma estimated conditional deviation around the incurred/paid age-to-age factors

### Author(s)

Markus Gesmann markus.gesmann@gmail.com

### References

Gerhard Quarg and Thomas Mack. Munich Chain Ladder. Blatter DGVFM 26, Munich, 2004.

See also summary.MunichChainLadder, plot.MunichChainLadder , MackChainLadder

### Examples


MCLpaid
MCLincurred
op <- par(mfrow=c(1,2))
plot(MCLpaid)
plot(MCLincurred)
par(op)

# Following the example in Quarg's (2004) paper:
MCL <- MunichChainLadder(MCLpaid, MCLincurred, est.sigmaP=0.1, est.sigmaI=0.1)
MCL
plot(MCL)
# You can access the standard chain-ladder (Mack) output via
MCL$MackPaid MCL$MackIncurred

# Input triangles section 3.3.1
MCL$Paid MCL$Incurred
# Parameters from section 3.3.2
MCL$MackPaid$f
MCL$MackIncurred$f
MCL$MackPaid$sigma
MCL$MackIncurred$sigma
# Check Mack's assumptions graphically
plot(MCL$MackPaid) plot(MCL$MackIncurred)

MCL$q.f MCL$rhoP.sigma
MCL$rhoI.sigma MCL$PaidResiduals
MCL$IncurredResiduals MCL$QinverseResiduals
MCL$QResiduals MCL$lambdaP
MCL$lambdaI # Section 3.3.3 Results MCL$MCLPaid
MCL\$MCLIncurred