  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 Lemaire (1995) and Pitrebois et al. (2003a).
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.
