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)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.
Cumulative claims payments triangle
Incurred losses triangle.
Fabio Concina, fabio.concina@gmail.com
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.
Merz, M., Wuthrich, M. (2010). Paid-incurred chain claims reserving method. Insurance: Mathematics and Economics, 46(3), 568-579.
MackChainLadder,MunichChainLadder
PaidIncurredChain(USAApaid, USAAincurred)
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