Stability and Fairness of Premiums in Bonus–Malus Systems Based on a Premium Change Index

The bonus–malus system is a premium adjustment mechanism in insurance that links the level of premiums to the claim history of policyholders. In practice, such a system is expected to reflect risk in a fair manner while maintaining premium stability over time. However, studies addressing premium stability and fairness often remain implicit and lack clear quantitative indicators. This study aims to analyze premium stability and fairness in a bonus–malus system using a premium change index as an evaluation tool. The research adopts an evaluative and descriptive approach based on a table of relative bonus–malus premiums defined by the total number of claims, the number of large claims, and the bonus–malus period. Premium stability is assessed through inter-period premium changes, which are formalized into the Stability–Fairness Premium Index (SFPI) defined as the average absolute change in premiums over time. The results show that the index increases with the total number of claims, indicating a higher sensitivity of premiums to risk exposure. Nevertheless, for a fixed total number of claims, the index is not affected by the number of large claims, suggesting that penalties are applied proportionally without increasing premium volatility. These findings indicate that the analyzed bonus–malus system achieves a balance between premium stability and fairness. The proposed index provides a simple and transparent quantitative framework for evaluating bonus–malus systems and may support the development of fairer and more stable premium-setting policies in insurance.