PaidIncurredChain function

PaidIncurredChain

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

PaidIncurredChain(triangleP, triangleI)

Arguments

• triangleP: Cumulative claims payments triangle
• triangleI: Incurred losses triangle.

Returns

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 c("$\\Theta = (\\Phi[0],...,\\Phi[I],\n$", "$\\Psi[0],...,\\Psi[I-1],\\sigma[0],...,\\sigma[I-1],\\tau[0],...,\\tau[I-1])$")

  we have

  • the random vector c("$(\\xi[0,0],...,\\xi[I,I],\n$", "$\\zeta[0,0],...,\\zeta[I,I-1])$") has multivariate Gaussian distribution with uncorrelated components given by

$$\xi_{i,j} \sim N(\Phi_j,\sigma^2_j),\xi[i,j] distributed as N(\Phi[j],\sigma^2[j]),$$ $$\zeta_{k,l} \sim N(\Psi_l,\tau^2_l);\zeta[k,l] distributed as N(\Psi[l],\tau^2[l]);$$

* cumulative payments are given by the recursion 

$$P_{i,j} = P_{i,j-1} \exp(\xi_{i,j}),P[i,j] = P[i,j-1] * exp(\xi[i,j]),$$

  with initial value $P[i,0] = * exp (\xi[i,0])$;
* incurred losses $I[i,j]$ are given by the backwards recursion 

$$I_{i,j-1} = I_{i,j} \exp(-\zeta_{i,j-1}),I[i,j-1] = I[i,j] * exp(-\zeta[i,j-1]),$$

  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

$$\Phi_m \sim N(\phi_m,s^2_m),\Phi[m] distributed as N(\phi[m],s^2[m]),$$ $$\Psi_n \sim N(\psi_n,t^2_n).\Psi[n] distributed as N(\psi[n],t^2[n]).$$

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

Note

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

Examples

PaidIncurredChain(USAApaid, USAAincurred)

References

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

See Also

MackChainLadder,MunichChainLadder

Author(s)

Fabio Concina, fabio.concina@gmail.com