Frequency-Severity Experience Rating based on Latent Markovian Risk Profiles

Bonus-Malus Systems are widely employed as a commercial form of experience rating in the non-life insurance industry. However, these systems traditionally depend only on a customer's number of claims, irrespective of their size, while customers who rarely claim may, for instance, have much smaller claims than those who claim relatively often. Independence is thus implicitly assumed between the frequency and severity component, even though this assumption is usually violated in practice. We therefore propose a novel joint experience rating approach in this paper, based on latent Markovian risk profiles to account for the frequency-severity dependence. More specifically, we let the latent risk profiles follow a Hidden Markov Model to allow them to evolve over time and to account for updates in a customer's claims experience. Conditional on these risk profiles, the claim frequencies and severities are modeled as separate components to closely follow standard practices in non-life insurance. We show that the resulting risk premia can be represented as a convex combination of standard credibility premia for each risk profile, where the profile assignment probabilities account for all potential evolutions of a profile and include a customer's observed claims experience a posteriori. We additionally allow a customer's risk characteristics to affect these credibility premia and posterior weights and efficiently estimate all (prior) parameters through an empirical Bayes version of the Baum-Welch algorithm. The proposed experience rating approach is applied to a Dutch automobile insurance portfolio to identify customer risk profiles with distinctive claiming behavior.