Determination of Insurance Premiums Using The Optimal Bonus-Malus System with The Bayesian Method

Abstract

This study develops an optimal bonus–malus premium model within a Bayesian decision-theoretic framework. Claim frequency is modeled using a Poisson distribution, while the number of claims exceeding a predefined critical threshold is modeled conditionally using a Binomial distribution. The Poisson intensity parameter is assigned an Exponential prior distribution, while the Binomial probability parameter follows a Beta prior distribution. The Exponential and Beta distributions are applied to model parameters rather than to observed data, ensuring probabilistic consistency. Since the Exponential distribution is a special case of the Gamma distribution, the Bayesian updating process remains mathematically coherent. To illustrate the model, a dataset of 1,000 simulated motor vehicle policyholders is generated in R under specified distributional assumptions. The results indicate that premiums increase with the number of claims exceeding the critical value and decrease with longer claim-free duration. The proposed framework provides a coherent and flexible approach for premium determination in bonus–malus systems. However, the findings are based on simulated data and specific modeling assumptions, which may limit direct empirical generalization.