Optimal Bonus-Malus Systems Using Generalized Additive Models for Location, Scale and Shape

Abstract

This paper presents the design of optimal Bonus-Malus Systems (BMS) using generalized additive models for location, scale and shape (GAMLSS), extending the work of Tzougas, Frangos and Vrontos (2014). Specifically, for the frequency component we employ a Negative Binomial Type I, a Poisson-Inverse Gaussian, a Sichel and a finite Poisson mixture GAMLSS model, while for the severity component we employ a Pareto and a finite Exponential mixture GAMLSS models. In the path towards actuarial relevance the Bayesian view is taken and the premiums are calculated by updating the posterior mean and posterior probability of the policyholders' classes of risk. Our analysis shows that the employment of more advanced models can provide a measure of uncertainty regarding the credibility updates of claim frequency/severity of each specific risk class and the difference in the premium that they imply can act as a cushion against adverse experience. Finally, these "tailor-made" premiums are compared to those which correspond to the 'univariate',without regression components, models.