Optimal design of a bonus-malus system: linear relativities revisited

In this paper, we revisit the determination of optimal relativities under the linear form of relativities that is more viable in designing a commercial bonus-malus system. We derive the analytical formulae for the optimal linear relativities subject to a financial balanced inequality constraint. We also numerically investigate the impact of different a priori risk classification towards the effectiveness of transition rules. Our results show that the a priori risk segmentation is not a sensitive factor for the effectiveness of transition rules. Furthermore, relative to the general relativities, we find that the restriction of linear relativities only produces a small amount of deterioration towards the numerical value of the optimised objective function.