Optimal relativities and transition rules of a bonus–malus system

Abstract

When a bonus–malus system with a single set of optimal relativities and a set of simple transitionrules is implemented, two inadequacy scenarios are induced because all policyholders are subject tothe same a posteriori premium relativities (level transitions) independent of their a priori characteristics(current levels occupied). In this paper we propose a new objective function in the determination ofoptimal relativities that directly incorporates the a priori expected claim frequencies to partially addressone of the inadequacy scenarios. We derive the analytical solution for the optimal relativities under afinancial equilibrium constraint. Furthermore, we introduce a metric called effectiveness of transitionrules to compare the different specifications of transition rules. We also argue that varying transitionrules which are more flexible in addressing the other inadequacy scenario may be more effective thantheir corresponding simple rules.