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

## 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.

## 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.

## References

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

## Author

Fabio Concina, fabio.concina@gmail.com

## Examples

```
PaidIncurredChain(USAApaid, USAAincurred)
#> $Ult.Loss.Origin
#> [,1]
#> [1,] 983113.3
#> [2,] 1078696.6
#> [3,] 1145761.4
#> [4,] 1245171.4
#> [5,] 1371964.4
#> [6,] 1433857.4
#> [7,] 1415963.7
#> [8,] 1410065.3
#> [9,] 1320414.5
#>
#> $Ult.Loss
#> [1] 11405008
#>
#> $Res.Origin
#> [,1]
#> [1,] 965.2681
#> [2,] 3159.6104
#> [3,] 7386.4026
#> [4,] 18521.4454
#> [5,] 47232.3846
#> [6,] 113727.4102
#> [7,] 230663.7420
#> [8,] 443903.3046
#> [9,] 778393.5002
#>
#> $Res.Tot
#> [1] 1643953
#>
#> $s.e.
#> [1] 113940.2
#>
```