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The objective of this paper is to provide an extension of well-known models of tarification in automobile insurance. The analysis begins by introducing a regression component in the Poisson model in order to use all available information in the estimation of the distribution. In a second step, a random variable is included in the regression component of the Poisson model and a negative binomial model with a regression component is derived. We then present our main contribution by proposing a bonus-malus system which integrates a priori and a posteriori information on an individual basis. We show how net premium tables can be derived from the model. Examples of tables are presented.

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Automobile insurance is an example of a market where multi-period contracts are observed. This form of contract can be justified by asymmetrical information between the insurer and the insured. Insurers use risk classification together with bonus-malus systems. In this paper we show that the actual methodology for the integration of these two approaches can lead to inconsistencies. We develop a statistical model that adequately integrates risk classification and experience rating. For this purpose we present Poisson and negative binomial models with regression component in order to use all available information in the estimation of accident distribution. A bonus-malus system which integrates a priori and a posteriori information on an individual basis is proposed, and insurance premium tables are derived as a function of time, past accidents and the significant variables in the regression. Statistical results were obtained from a sample of 19,013 drivers. 'Another complementary explanation is learning. See Boyer, Dionne, and Kihlstrom (1989) and Palfrey and Spatt (1985) for more details.