A Priori Ratemaking Using Bivariate Poisson Regression Models

Abstract

In automobile insurance, it is useful to achieve a priori ratemaking by resorting to generalized linear models, and here the Poisson regression model constitutes the most widely accepted basis. However, insurance companies distinguish between claims with or without bodily injuries, or claims with full or partial liability of the insured driver. This paper examines an a priori ratemaking procedure when including two different types of claim. When assuming independence between claim types, the premium can be obtained by summing the premiums for each type of guarantee and is dependent on the rating factors chosen. If the independence assumption is relaxed, then it is unclear as to how the tariff system might be affected. In order to answer this question, bivariate Poisson regression models, suitable for paired count data exhibiting correlation, are introduced. It is shown that the usual independence assumption is unrealistic here. These models are applied to an automobile insurance claims database containing 80,994 contracts belonging to a Spanish insurance company. Finally, the consequences for pure and loaded premiums when the independence assumption is relaxed by using a bivariate Poisson regression model are analysed.

Modelling Dependence in a Ratemaking Procedure with Multivariate Poisson Regression Models

“…Certain amounts of overdispersion can be introduced by 7 considering inflated versions of multivariate Poisson regression models, like the models described in Ntzoufras (2003, 2005) and in Bermúdez (2009) Li et al, 1999). For an actuarial application see Boucher and Denuit (2008).…”

“…For each policy, the initial information at the beginning of the period and the total number of claims from policyholders at fault were reported within this yearly period. The exogenous variables, described in Table 3, were previously used in Pinquet et al (2001), Brouhns et al (2003), Bolancé et al (2003Bolancé et al ( , 2008, Boucher et al (2007Boucher et al ( , 2009), Boucher and Denuit (2008) and Bermúdez (2009).…”

“…Five different, yet representative, profiles were selected from the portfolio and classified according to their risk level. The profiles (see Table 8) are the same as in Bermúdez (2009). The first can be classified as the best profile since it presents the lowest mean score.…”

“…In Frees and Valdez (2008) and Frees et al (2009) a hierarchical statistical model is fitted using microlevel data. In Bermúdez (2009), the interpretation of a number of bivariate Poisson models was illustrated in the context of motor insurance claims and the conclusion was that using a bivariate Poisson model leads to an a priori ratemaking that presents larger variances and, hence, larger loadings than those obtained under the independence assumption. In that study, only two types of claim were considered: claims for third-party liability or for the rest of guarantees.…”

“…In the present paper we use two such models and their zero-inflated variants (to account for the excess of zeros observed in automobile databases, see e.g. Boucher et al, 2007 andBermúdez, 2009). The first one, which we call the "common covariance model", has been defined in Tsionas (2001) and the second one, the "full covariance model", in Karlis and Meligkotsidou (2005).…”