Robustness in Bayesian Models for Bonus–Malus Systems

In implementing Bayesian models for a Bonus–Malus System (BMS) it is normal to include a parametric structure, π0(λ), in the insurer's portfolio. According to Bayesian sensitivity analysis, the structure function can be modelled by specifying a class Γ of priors instead of a single prior. In this chapter, we examine the ranges of the relativities, i.e., δπ = 𝔼 [λπ(λ | data)]/𝔼[λπ(λ)], π ∈ Γ. We combine standard and robust Bayesian tools to show how the choice of the prior can critically affect the relative premiums. Extending the recent paper of Gómez et al. (2002b), we develop our model to the Negative Binomial–Pareto model (Meng and Whitmore 1999) and the Poisson–Inverse Gaussian model (Tremblay 1992, Besson and Partrat 1992) and also extend the class of prior densities to ones that are more realistic in the actuarial setting, i.e., the classes of generalized moments conditions. We illustrate our method with data from Lemaire (1979). The chapter is mainly focused on showing that there exists an appropriate methodology to develop a Bayesian sensitivity analysis of the bonus-malus of loaded premiums.