Claims reserving models are usually based on data recorded in run-off tables, according to the origin and the development years of the payments. The amounts on the same diagonal are paid in the same calendar year and are influenced by some common effects, for example, claims inflation, that can induce dependence among payments. We introduce hierarchical generalized linear models (HGLM) with risk parameters related to the origin and the calendar years, in order to model the dependence among payments of both the same origin year and the same calendar year. Besides the random effects, the linear predictor also includes fixed effects. All the parameters are estimated within the model by the h-likelihood approach. The prediction for the outstanding claims and an approximate formula to evaluate the mean square error of prediction are obtained. Moreover, a parametric bootstrap procedure is delineated to get an estimate of the predictive distribution of the outstanding claims. A Poisson-gamma HGLM with origin and calendar year effects is studied extensively and a numerical example is provided. We find that the estimates of the correlations can be significant for payments in the same calendar year and that the inclusion of calendar effects can determine a remarkable impact on the prediction uncertainty.

This article proposes a claims reserving model for dependent lines of business with the accommodation of association among triangles by a copula function. We show that the family of elliptical copulas is a pretty convenient choice to capture the dependencies introduced by various sources, including the common calendar year effects. To quantify the associated reserving variability, we resort to parametric bootstrapping techniques for simulating the predictive distribution of outstanding liabilities and for calculating the three components of predictive uncertainty: the model error, the process error and the estimation error. Numerical analysis is performed for a portfolio of casualty insurance from a major U.S. insurer.

A central issue in claims reserving is the modelling of appropriate dependence structures. Most classical models cannot cope with this task. We define a multivariate log-normal model that allows to model both, dependence between different sub-portfolios and dependence within sub-portfolios such as claims inflation. In this model we derive closed form solutions for claims reserves and the corresponding prediction uncertainty.