A simple Bayesian state-space approach to the collective risk models

Abstract

The collective risk model (CRM) for frequency and severity is an important tool for retail insurance ratemaking, natural disaster forecasting, as well as operational risk in banking regulation. This model, initially designed for cross-sectional data, has recently been adapted to a longitudinal context for both a priori and a posteriori ratemaking, through random effects specifications. However, the random effects are usually assumed to be static due to computational concerns, leading to predictive premiums that omit the seniority of the claims. In this paper, we propose a new CRM model with bivariate dynamic random effects processes. The model is based on Bayesian state-space models. It is associated with a simple predictive mean and closed form expression for the likelihood function, while also allowing for the dependence between the frequency and severity components. A real data application for auto insurance is proposed to show the performance of our method.