Bayesian credibility under a bivariate prior on the frequency and the severity of claims

In this paper, as opposed to the usual assumption of independence, we propose a credibility model in which the (unobservable) risk profiles of the claim frequency and the claim severity are dependent. Given the risk profiles, the (conditional) marginal distributions of frequency and severity are assumed to belong to the exponential family. A bivariate conjugate prior is proposed for the risk profiles, where the dependency is incorporated via a factorization structure of the joint density. The bivariate posterior is derived, and in turn the Bayesian premium for the aggregate claim is given along with some results on the predictive joint and marginal distributions involving the claim number and the aggregate claim in the next period. To demonstrate the generality of our proposed model, we provide four different examples of bivariate conjugate priors in relation to mixed Erlang, gamma mixture, Farlie-Gumbel-Morgenstern (FGM) copula, and bivariate beta, where each choice has different merits. In these examples, more explicit results can be obtained, and in particular the predictive variance and Value-at-Risk (VaR) of the aggregate claim certainly provide more information on the inherent risk than the Bayesian premium which is merely the predictive mean. Finally, numerical examples will be given to illustrate the effect of dependence on the results, including the use of a real data set that further takes observable risk factors into consideration under a regression setting.