Design of an Optimal Bonus-Malus System Using the Sichel Distribution as a Model of Claim Counts

Abstract

This paper presents the design of an optimal Bonus-Malus System (BMS) using the Sichel distribution to model the claim frequency distribution. This system is proposed as an alternative to the optimal BMS obtained by the traditional Negative Binomial model (Lemaire, 1995). In fact the Sichel distribution has a thicker tail than the Negative Binomial distribution and it is considered as a plausible model for highly dispersed count data. We also consider the optimal BMS provided by the Poisson-Inverse Gaussian distribution, which is a special case of the Sichel distribution. In the above setup optimality is achieved by minimizing the insurer’s risk. Furthermore we develop a generalized BMS that takes into account both the a priori and a posteriori characteristics of each policyholder. For this purpose we consider to the generalized additive models for location, scale and shape (GAMLSS) in order to use all available information in the estimation of the claim frequency distribution. The GAMLSS were introduced by Rigby and Stasinopoulos (2001, 2005) and Akantziliotou et al. (2002) as a framework for fi tting regression type models where the distribution ofthe response variable does not have to belong to the exponential family and includes highly skew and kurtotic continuous and discrete distribution. The GAMLSS allows all the parameters of the distribution of the response variable to be modeled as linear/non-linear or smooth functions of the explanatory variables. Within the framework of the GAMLSS we propose the Sichel GAMLSS for assessing claim frequency as an alternative to the Negative Binomial regression model used by Dionne and Vanasse (1989, 1992). We also consider the Negative Binomial Type I and the Poisson-Inverse Gaussian GAMLSS for assessing claim frequency. With the aim of constructing an optimal BMS by updating the posterior mean claim frequency, we adopt the parametric linear formulation of these models and we allow only their mean parameter to be modeled as a function of the significant a priori rating variables for the number of claims.