Evaluating transition rules for enhancing fairness in bonus-malus systems : an application to the Saudi Arabian auto insurance market

A Bonus-Malus System (BMS) is a ratemaking mechanism used in insurance to adjust premiums based on a policyholder's claim history, with the goal of segmenting risk profiles more accurately. A BMS typically comprises three key components: the number of BMS levels, the transition rules dictating the movements of policyholders within the system, and the relativities used to determine premium adjustments. This paper explores the impact of modifications to these three elements on risk classification, assessed through the mean squared error. The model parameters are calibrated with real-world data from the Saudi auto insurance market. We begin the analysis by focusing on transition rules based solely on claim frequency, a framework in which most implemented BMSs work, including the current Saudi BMS. We then consider transition rules that depend on frequency and severity, in which higher penalties are given for large claim sizes. The results show that increasing the number of levels typically improves risk segmentation but requires balancing practical implementation constraints and that the adequate selection of the penalties is critical to enhancing fairness. Moreover, the study reveals that incorporating a severitybased penalty enhances risk differentiation, especially when there is a dependence between the claim frequency and severity.