When actuaries face the problem of pricing an insurance contract that contains different types of coverage, such as a motor insurance or homeowner's insurance policy, they usually assume that types of claim are independent. However, this assumption may not be realistic:several studies have shown that there is a positive correlation between types of claim. Here we introduce different multivariate Poisson regression models in order to relax the independence assumption, including zero-inflated models to account for excess of zeros and overdispersion.These models have been largely ignored to date, mainly because of their computational difficulties. Bayesian inference based on MCMC helps to resolve this problem (and also allows us to derive, for several quantities of interest, posterior summaries to account for uncertainty).Finally, these models are applied to an automobile insurance claims database with three different types of claims. We analyse the consequences for pure and loaded premiums when the independence assumption is relaxed by using different multivariate Poisson regression models together with their zero-inflated versions.

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Bivariate Poisson regression models for ratemaking in car insurance has been previously used. They included zero-inflated models to account for the excess of zeros and the overdispersion in the data set. These models are now revisited in order to consider alternatives. A 2-finite mixture of bivariate Poisson regression models is used to demonstrate that the overdispersion in the data requires more structure if it is to be taken into account, and that a simple zero-inflated bivariate Poisson model does not suffice. At the same time, it is shown that a finite mixture of bivariate Poisson regression models embraces zero-inflated bivariate Poisson regression models as a special case. Finally, an EM algorithm is provided in order to ensure the models' ease-offit. These models are applied to an automobile insurance claims data set and it is shown that the modelling of the data set can be improved considerably.

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.

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Recently, different bivariate Poisson regression models have been used in the actuarial literature to make an a priori ratemaking taking into account the dependence between two types of claims. A natural extension for these models is to consider a posteriori ratemaking (i.e. experience rating models) that also relaxes the independence assumption. We introduce here two bivariate experience rating models that integrate the a priori ratemaking based on the bivariate Poisson regression models, extending the existing literature for the univariate case to the bivariate case. These bivariate experience rating models are applied to an automobile insurance claims data-set to analyse the consequences for posterior premiums when the independence assumption is relaxed. The main finding is that the a posteriori risk factors obtained with the bivariate experience rating models are significantly lower than those factors derived under the independence assumption.

a b s t r a c tThe paper examines the distribution function of settlements over time in an attempt to explain the time it takes to negotiate the claim compensation in the context of motor disputes. Competing risk models are applied to a Spanish motor insurance database. The empirical analysis yielded two main findings. First, in some cases the severity of temporary injuries was found to be negatively associated with the time to settlement of the claim, suggesting that more severe cases do not always take longer to settle; this supports the assumption that the time to settlement depends primarily on the degree of informational asymmetry among litigants regarding the magnitude of damages. Second, time-varying effects were observed for the explanatory factors related to the victim's age and the seriousness of injuries. The effect of these factors on the time to settlement of the claim is attenuated over time, suggesting that the asymmetries in information present a dynamic pattern during the negotiation process, since the insurer's incomplete information is enriched during the course of negotiations.

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