The majority of optimal Bonus-Malus Systems (BMS) presented up to now in the actuarial literature assign to each policyholder a premium based on the number of his accidents. In this way a policyholder who had an accident with a small size of loss is penalized unfairly in the same way with a policyholder who had an accident with a big size of loss. Motivated by this, we develop in this paper, the design of optimal BMS with both a frequency and a severity component. The optimal BMS designed are based both on the number of accidents of each policyholder and on the size of loss (severity) for each accident incurred. Optimality is obtained by minimizing the insurer's risk. Furthermore we incorporate in the above design of optimal BMS the important a priori information we have for each policyholder. Thus we propose a generalised BMS that takes into consideration simultaneously the individual's characteristics, the number of his accidents and the exact level of severity for each accident.

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This paper presents the design of optimal Bonus-Malus Systems (BMS) using …nite mixture models, extending the work of Lemaire (1995) and Frangos and Vrontos (2001). Speci…cally, for the frequency component we employ a …nite Poisson, Delaporte and Negative Binomial mixture, while for the severity component we employ a …nite Exponential, Gamma, Weibull and Generalized Beta Type II mixture, updating the posterior probability. We also consider the case of a …nite Negative Binomial mixture and a …nite Pareto mixture updating the posterior mean. The generalized BMS we propose, integrate risk classi…cation and experience rating by taking into account both the a priori and a posteriori characteristics of each policyholder.

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