### 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)
```

### 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 ≤q n+1-i; i=1,…,m; m≥q 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 ≤q n+1-i; i=1,…,m, m≥q 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.

### 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

```