## 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 Θ = (Φ,...,Φ[I], Ψ,...,Ψ[I-1],σ,...,σ[I-1],τ,...,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] distributed as N(Φ[j],σ^2[j]),

ζ[k,l] distributed as 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 independent 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,...,σ[J] and τ,...,τ[J-1] and

Φ[m] distributed as N(φ[m],s^2[m]),

Ψ[n] distributed as 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, 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)