Asymptotic Fairness of Bonus-Malus Systems and Optimal Scales of Premiums

In this paper we try to evaluate the asymptotic fairness of bonus-malus systems, assuming the simplest case when there is no hunger for bonus. The asymptotic fairness has to be understood as the bonus-malus system ability in assessing the individual risks in the long run. Firstly we define the asymptotic fairness of a bonus-malus system following an expression that can be found in J. Lemaire [1985]: Automobile Insurance. Actuarial Models. Dordrecht: Kluwer-Nijhoff Publishing, p. 168. Secondly, we define a measure of the global asymptotic fairness considering the structure function of the risk group. Finally we try to calculate, for each set of transition rules and a given structure function, the scale of premiums that brings the global asymptotic fairness closest to the ideal situation where each insured pays in the long run a premium corresponding to its own claim frequency. This is possible thanks to the application of a multiobjective optimization technique named Goal Programming. The Geneva Papers on Risk and Insurance Theory (2002) 27, 61–82. doi:10.1023/A:1020633525146