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Introduction
Traditionally, the motor insurance rate-making process consists of two separate steps. In the a priori rating (the first step), insurers make use of certain observable risk classification variables to divide a portfolio of motor vehicle drivers into a number of homogeneous tariff classes. However, these a priori variables are not able to fully capture the risk characteristics of the insured drivers, so the a posteriori rating (the second step) - under the framework of credibility premium or bonus-malus system (BMS) - is needed to tackle the residual heterogeneity. These mechanisms are based on the claims experience information because it is reasonably believed that the unobservable risk characteristics would be partially revealed through the drivers' claims history. In particular, the design of a BMS can be regarded as the commercial version of the credibility premium (see e.g. Dionne & Vanasse, 1989) framework.
Each BMS is represented by three building blocks: the number of BMS levels that the BMS is operating in with a pre-specified starting level, the transition rules which govern the transition of policyholders between BMS levels over time, and the set of optimal relativities that are multiplied with the base premium to obtain the premium amounts payable. Given the specified number of BMS levels and the chosen transition rules, Norberg (1976) first determined the optimal relativity associated with each BMS level through the maximisation of asymptotic predictive accuracy (also known as the Norberg's criterion). However, this original approach does not incorporate the heterogeneity between different tariff classes, which is equivalent to the absence of a priori rating. To address this problem, Taylor (1997) developed a simulation procedure (see also Lemaire et al., 2015), whereas Pitrebois et al. (2003) obtained the analytical expression for the set of optimal relativities.
In practice, however, the optimal relativities are largely determined commercially to ease the implementation of BMS from the drivers' perspective. For instance, in Asian countries such as Singapore and Hong Kong, all the BMS levels have relativities that are below 100%. In other words, drivers are rewarded a no-claim discount in a claim-free year and are not subject to any relativity above 100% even if they incur claims. In such...