# PaidIncurredChain: PaidIncurredChain In ChainLadder: Statistical Methods and Models for Claims Reserving in General Insurance

## Description

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

## Usage

 `1` ```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 Θ = (Φ_0,...,Φ_I, Ψ_0,...,Ψ_{I-1},σ_0,...,σ_{I-1},τ_0,...,τ_{I-1}) we have

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

ξ_{i,j} \sim N(Φ_j,σ^2_j),

ζ_{k,l} \sim N(Ψ_l,τ^2_l);

• cumulative payments are given by the recursion

P_{i,j} = P_{i,j-1} \exp(ξ_{i,j}),

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

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

I_{i,j-1} = I_{i,j} \exp(-ζ_{i,j-1}),

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

• The components of Θ are indipendent and σ_j,τ_j > 0 for all j.

Parameters Θ 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 σ_0,...,σ_J and τ_0,...,τ_{J-1} and

Φ_m \sim N(φ_m,s^2_m),

Ψ_n \sim N(ψ_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, [email protected]

## References

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

`MackChainLadder`,`MunichChainLadder`
 `1` ```PaidIncurredChain(USAApaid, USAAincurred) ```