Generalized Linear Mixed Models for Dependent Compound Risk Models

Abstract

In ratemaking, calculation of a pure premium has traditionally been based on modeling frequency and severity in an aggregated claims model. For simplicity, it has been a standard practice to assume the independence of loss frequency and loss severity. In recent years, there is sporadic interest in the actuarial literature exploring models that departs from this independence. In this article, we extend the work of Garrido et al. (2016) which uses generalized linear models (GLMs) that account for dependence between frequency and severity and simultaneously incorporate rating factors to capture policyholder heterogeneity. In addition, we quantify and explain the contribution of the variability of claims among policyholders through the use of random effects using generalized linear mixed models (GLMMs). We calibrated our model using a portfolio of auto insurance contracts from a Singapore insurer where we observed claim counts and amounts from policyholders for a period of six years. We compared our results with the dependent GLM considered by Garrido et al. (2016), Tweedie models, and the case of independence. The dependent GLMM shows statistical evidence of positive dependence between frequency and severity. Using validation procedures, we find that the results demonstrate a more superior model when random effects are considered within a GLMM framework.