Claim dependence with common effects in credibility models

Several credibility models found in published literature have largely been single dimensional in the sense that the observable claims are derived from a single individual risk or a single group of homogeneous risks over a period of time. In the case where the additional dimension of observing different individual risks or different groups of risks are allowed for, the assumption of independence across the observable claims is often made. This is a matter of convenience and mathematical tractability, though in general, everyone agrees this may seem unrealistic. As such, dependence must be taken into account when modelling risks for assessing credibility premiums. In this paper, we introduce the notion of modelling claim dependence across individuals and simultaneously across time within individuals using common effects. The resulting model is then used to predict expected claims given the history of all observable claims. It is well-known that this conditional expectation actually gives the best predictor of the next period claim for a single individual in the mean-squared error sense. We express this conditional expectation in the form of a credibility premium. We further give illustrative examples to demonstrate the ideas.