Predictive compound risk models with dependence

The two-part regression model, which subdivides aggregate claims into frequency and severity components, is a widespread tool in actuarial practice for predicting pure premium. The assumption of independence between frequency and severity is conventional but there is increased interest in advancing models to capture the possible dependence as is done in Garrido et al. (2016). This paper extends the work of Garrido et al. (2016) and explores the benefits of using random effects for predicting insurance claims observed longitudinally, or over a period of time, within a two-part framework relaxing the assumption of independence. More specifically, we introduce a generalized formula for credibility premium of compound sum with dependence, which extends and integrates previous work in both credibility premium of compound sums and dependent two-part compound risk models. In this generalized formula of credibility premium of compound sum, we are able to derive a dependence function, DN(γ), that offers an informative measure of the strength and direction of the association between frequency and severity. This function is easy to interpret and allows for practical implementation useful for actuarial ratemaking. Our model calibration, based on longitudinal claims from a Singapore automobile insurance company, shows that there is a strong negative dependence between frequency and severity.