A simple Bayesian state-space model for the collective risk model

TL;DR

This paper introduces a Bayesian state-space model with dynamic random effects for the collective risk model, improving predictive accuracy by accounting for claim seniority and dependence between frequency and severity.

Contribution

It develops a novel Bayesian state-space CRM with dynamic random effects, enabling better longitudinal risk modeling and dependence capture.

Findings

Model achieves closed-form likelihood expression.

Demonstrates improved predictive performance on auto insurance data.

Captures dependence between frequency and severity components.

Abstract

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