On the bias of commercial insurance premiums

An insurance premium is defined as the product of two elements: the a priori rate, which depends on the policyholder's observable risk characteristics at the time of contract, and the a posteriori rate, which encompasses the residual component not explained by the a priori information. This paper explores the mathematical structure of the a posteriori rate and its corresponding statistical properties. As in the cases of the Bayes premium and the credibility premium, the a posteriori rate depends on both the a priori rate and the claim history in general. However, in certain cases, such as the bonus-malus premium in auto insurance, the a posteriori rate depends solely on the claim history. We refer to such insurance premiums, where the a posteriori rate is solely a function of the claim history, as commercial insurance premiums. Although the simplified structure of commercial insurance premiums enhances communication with policyholders, it can introduce a systematic bias known as the double counting problem. The insurance literature has empirically identified this bias in bonus-malus systems. Our study extends the existing empirical analysis of bonus-malus systems to a wider range of commercial insurance premiums and provides a rigorous mathematical framework demonstrating that any commercial insurance premium is susceptible to the double counting problem. Then, we propose a simple solution to mitigate this issue while retaining the structural simplicity of the commercial insurance premium. Numerical analysis, including real data analysis, demonstrates the extent of the double counting problem in commercial insurance premiums and the effectiveness of the proposed method in mitigating this issue.