PaidIncurredChain: PaidIncurredChain

View source: R/PIC.R

PaidIncurredChainR Documentation

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.

See Also

MackChainLadder,MunichChainLadder

Examples

PaidIncurredChain(USAApaid, USAAincurred)

ChainLadder documentation built on Sept. 11, 2024, 8:35 p.m.