Bonus-Malus systems are largely used in non-life insurance (in particular, in automobile insurance) to implement experience rating. The distribution of the number of claims is known to be well fitted by mixed Poisson processes. In particular, the Poisson-gamma and the Poisson-Inverse Gaussian distributions are frequently used in actuarial literature. In this paper we consider a three parameters distribution encompassing the Poisson-gamma and the Poisson-Inverse Gaussian distributions. This distribution is due to Hofmann and has been proposed by Walhin and Paris in order to construct a Bonus-Malus system. According to this distribution the construction of the Bonus-Malus system is easily done using the Bayes theorem and the form of the mixed Poisson distribution. Several good properties justify the adoption of this model. Numerical results are given.
Bonus-Malus systems. Mixed Poisson processes. Hofmann distribution
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