Optimal relativities in a modified Bonus-Malus system with long memory transition rules and frequency-severity dependence

In the classical Bonus-Malus System (BMS) in automobile insurance, the premium for the next year is adjusted according to the policyholder's claim history (particularly frequency) in the previous year. Some variations of the classical BMS have been considered by taking more of driver's claim experience into account to better assess individual's risk. Nevertheless, we note that in practice it is common for a BMS to adopt transition rules according to the claim history for the past multiple years in countries such as Belgium, Italy, Korea, and Singapore. In this paper, we revisit a modified BMS which was briefly introduced in \citet{L1995} and \citet{PDW03}. 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 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 effect model and the dependent collective risk model. Also, numerical analysis of a real data set is provided to compare the classical BMS and our proposed BMS.