Optimal Relativities in a Modified Bonus-Malus System With Long Memory Transition Rules and Frequency-Severity Dependence

Abstract

In the classical application of a bonus-malus system (BMS) to automobile insurance, the premium for the next year is adjusted according to the policyholder’s claim history (particularly frequency) in the previous year. Variations on the classical BMS have been considered, such as taking more of the driver’s claim experience into account to better assess an individual’s risk. We note that in countries such as Belgium, Italy, Korea, and Singapore it is common practice for a BMS to adopt transition rules according to the claim history for the past several years. In this paper, we revisit a modified BMS briefly introduced by Lemaire in 1995 and Pitrebois and colleagues in 2003. Specifically, such a BMS extends the number of bonus-malus (BM) levels due to an additional component in the transition rules representing the number of consecutive claim-free years. With the extended BM levels granting a more reasonable bonus to careful drivers, this paper investigates the transition rules in a more rigorous manner, and provides the optimal BM relativities under various statistical model assumptions including the frequency random effects model and the dependent collective risk model. Numerical analysis of a real data set is provided to compare the classical BMS and our proposed BMS.Address for Correspondence: jaeyahn@ewha.ac.kr