Dynamic Frailty Count Process in Insurance: A Unified Framework for Estimation, Pricing, and Forecasting

We study count processes in insurance, in which the underlying risk factor is time varying and unobservable. The factor follows an autoregressive gamma process, and the resulting model generalizes the static Poisson-Gamma model and allows for closed form expression for the posterior Bayes (linear or nonlinear) premium. Moreover, the estimation and forecasting can be conducted within the same framework in a rather efficient way. An example of automobile insurance pricing illustrates the ability of the model to capture the duration dependent, nonlinear impact of past claims on future ones and the improvement of the Bayes pricing method compared to the linear credibility approach. Introduction We propose a time series model for count variables encountered in insurance, when the underlying risk factor is time varying and unobservable. We introduce the au-toregressive gamma process for the latent factor dynamics and show how it provides an integrated framework for the efficient estimation, pricing, and forecasting of the risk. One typical application of the model is automobile insurance, for which the insurer holds the history of individual yearly claim counts over several periods. This information should then be taken into account in order to update the future premium regularly, on an individual basis.