When actuaries face the problem of pricing an insurance contract that contains different types of coverage, such as a motor insurance or a 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 claim. 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.

Bivariate Poisson regression models for ratemaking in car insurance have 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-of-fit. These models are applied to an automobile insurance claims data set and it is shown that the modeling of the data set can be improved considerably

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.

We analyzed real telematics information for a sample of drivers with usage-based insurance policies. We examined the statistical distribution of distance driven above the posted speed limit—which presents a strong positive asymmetry—using quantile regression models. We found that, at different percentile levels, the distance driven at speeds above the posted limit depends on total distance driven and, more generally, on factors such as the percentage of urban and nighttime driving and on the driver’s gender. However, the impact of these covariates differs according to the percentile level. We stress the importance of understanding telematics information, which should not be limited to simply characterizing average drivers, but can be useful for signaling dangerous driving by predicting quantiles associated with specific driver characteristics. We conclude that the risk of driving for long distances above the speed limit is heterogeneous and, moreover, we show that prevention campaigns should target primarily male non-urban drivers, especially if they present a high percentage of nighttime driving.