Multidimensional credibility: a Bayesian analysis of policyholders holding multiple policies

Property and casualty actuaries are professional experts in the economic assessment of uncertain events related to non–life insurance products (eg fire, liability or motor insurance). For the construction of a fair and reasonable tariff associated with the risks in their portfolio, actuaries have many statistical techniques in their toolbox. In this paper tools for the pricing multivariate risks are considered. Examples of situations where this problem occurs are numerous; eg workers’ compensation schemes where the insurer has information on accidents occurring ‘at work’ as well as ‘not at work’, policyholders having policies in multiple business lines (eg water, theft etc) at the same company, or policyholders holding multiple policies eg in a motor insurance context. The latter is the situation we will explore in this paper, using a data set from a European insurance company. The combination of a priori rating (through risk classification based on a priori measurable characteristics) and a posteriori rating is considered. A posteriori the claim experience of a policyholder is taken into account. In a multivariate context the pricing actuary should be able to use claim history from all business lines or risk componentsa policyholder is holding within the company. Intuitively, the number of claims a policyholder has reported on a particular business line or risk component reveals his general risk proneness and is as such relevant when pricing other lines or components. In contrast to the analytical approaches developed for multivariate experience rating in the literature, our approach is data–driven using Bayesian statistics. The intuition of positive dependence between different policies held by the same policy holder is confirmed. Focus is on applications of the multivariate Bayesian experience rating, as well as on graphical representations of the results.