## PaidIncurredChain

### Description

The Paid-incurred Chain model (Merz, Wuthrich (2010)) combines claims payments and incurred losses information to get a unified ultimate loss prediction.

### Usage

PaidIncurredChain(triangleP, triangleI)


### Arguments

 triangleP Cumulative claims payments triangle triangleI Incurred losses triangle.

### Details

The method uses some basic properties of multivariate Gaussian distributions to obtain a mathematically rigorous and consistent model for the combination of the two information channels.

We assume as usual that I=J. The model assumptions for the Log-Normal PIC Model are the following:

• Conditionally, given \Theta = (\Phi_0,...,\Phi_I, \Psi_0,...,\Psi_{I-1},\sigma_0,...,\sigma_{I-1},\tau_0,...,\tau_{I-1}) we have

• the random vector (\xi_{0,0},...,\xi_{I,I}, \zeta_{0,0},...,\zeta_{I,I-1}) has multivariate Gaussian distribution with uncorrelated components given by

\xi_{i,j} \sim N(\Phi_j,\sigma^2_j),

\zeta_{k,l} \sim N(\Psi_l,\tau^2_l);

• cumulative payments are given by the recursion

P_{i,j} = P_{i,j-1} \exp(\xi_{i,j}),

with initial value P_{i,0} = \exp (\xi_{i,0});

• incurred losses I_{i,j} are given by the backwards recursion

I_{i,j-1} = I_{i,j} \exp(-\zeta_{i,j-1}),

with initial value I_{i,I}=P_{i,I}.

• The components of \Theta are independent and \sigma_j,\tau_j > 0 for all j.

Parameters \Theta in the model are in general not known and need to be estimated from observations. They are estimated in a Bayesian framework. In the Bayesian PIC model they assume that the previous assumptions hold true with deterministic \sigma_0,...,\sigma_J and \tau_0,...,\tau_{J-1} and

\Phi_m \sim N(\phi_m,s^2_m),

\Psi_n \sim N(\psi_n,t^2_n).

This is not a full Bayesian approach but has the advantage to give analytical expressions for the posterior distributions and the prediction uncertainty.

### Value

The function returns:

• Ult.Loss.Origin Ultimate losses for different origin years.

• Ult.Loss Total ultimate loss.

• Res.Origin Claims reserves for different origin years.

• Res.Tot Total reserve.

• s.e. Square root of mean square error of prediction for the total ultimate loss.

### Note

The model is implemented in the special case of non-informative priors.

### Author(s)

Fabio Concina, fabio.concina@gmail.com

### References

Merz, M., Wuthrich, M. (2010). Paid-incurred chain claims reserving method. Insurance: Mathematics and Economics, 46(3), 568-579.

MackChainLadder,MunichChainLadder
PaidIncurredChain(USAApaid, USAAincurred)