Parameter reduction in log-normal chain-ladder models
The chain-ladder method is one of the most popular methods to calculate the reserves in actuarial sciences. A special model is considered, where the claim development factors \(C_{i,j}\) are modelled as \[ C_{i,j} = C_{i,j-1} (\exp\{\xi_{i,j}\} + 1)\,. \] Here \(C_{i,j}\) are the aggregate payments for claims with occurrence year \(i\) paid until year \(i+j\). Given the parameters \(\Theta = (\theta_0, \theta_1, \ldots, \theta_J)\) for the occurrence years \(0,1,\ldots,J\), it is assumed that \((\xi_{i,j})\) are independent normally distributed. The variances are fixed, but the mean values \(\theta_i\) are normally distributed. For a fixed \(k\), in the model \(M^{(k)}\) the parameters \(\theta_j: 0 \leq j \leq k\) are independent, but \(\theta_j = \alpha - j \beta\) for \(j > k\), where \(\alpha,\beta\) are independent normal variables. For the model \(M^{(k)}\), Bayesian statistics can be done to predict the necessary reserves. If one does not want to make the normal assumption for the unknown parameters, an empirical Bayes approach would lead to a simple way to update the reserves if more observations are known. In order to keep the model as simple as necessary, Akaike's information criterion can be used to choose the model. Because the parameters \(\theta_k\) are random, the model risk can be included in the calculation of the reserves. An insurance example illustrates the method.
scientific article; zbMATH DE number 6530525