In the auto insurance industry, a Bonus-Malus System (BMS) is commonly used as a posteriori risk classification mechanism to set the premium for the next contract period based on a policyholder's claim history. Even though the recent literature reports evidence of a significant dependence between frequency and severity, the current BMS practice is to use a frequency-based transition rule while ignoring severity information. Although Oh et al. [(2020). Bonus-Malus premiums under the dependent frequency-severity modeling. Scandinavian Actuarial Journal 2020(3): 172–195] claimed that the frequency-driven BMS transition rule can accommodate the dependence between frequency and severity, their proposal is only a partial solution, as the transition rule still completely ignores the claim severity and is unable to penalize large claims. In this study, we propose to use the BMS with a transition rule based on both frequency and size of claim, based on the bivariate random effect model, which conveniently allows dependence between frequency and severity. We analytically derive the optimal relativities under the proposed BMS framework and show that the proposed BMS outperforms the existing frequency-driven BMS. Later, numerical experiments are also provided using both hypothetical and actual datasets in order to assess the effect of various dependencies on the BMS risk classification and confirm our theoretical findings.

在汽车保险行业，Bonus-Malus System (BMS) 通常用作后验风险分类机制，根据投保人的索赔历史确定下一个合同期的保费。尽管最近的文献报道了频率和严重性之间存在显着相关性的证据，但当前的 BMS 实践是使用基于频率的转换规则而忽略严重性信息。虽然哦等人。[（2020）。依赖频率严重性模型下的 Bonus-Malus 保费。斯堪的纳维亚精算杂志2020(3): 172-195]声称频率驱动的BMS过渡规则可以适应频率和严重性之间的依赖关系，他们的提议只是部分解决方案，因为过渡规则仍然完全忽略了索赔的严重性并且无法惩罚大额索赔。在这项研究中，我们建议使用基于双变量随机效应模型的具有基于频率和索赔大小的转移规则的 BMS，该模型方便地允许频率和严重性之间的依赖关系。我们通过分析推导出所提出的 BMS 框架下的最佳相关性，并表明所提出的 BMS 优于现有的频率驱动 BMS。之后，