Modelling uncertainty in insurance Bonus–Malus premium principles by using a Bayesian robustness approach

Abstract

When Bayesian models are implemented for a Bonus–Malus System (BMS), a parametric structure, π₀ (λ), is normally included in the insurer's portfolio. Following Bayesian sensitivity analysis, it is possible to model the structure function by specifying a class Γ of priors instead of a single prior. This paper examines the ranges of the relativities of the form,

Standard and robust Bayesian tools are combined to show how the choice of the prior can affect the relative premiums. As an extension of the paper by Gómez et al. (2002b), our model is developed to the variance premium principle and the class of prior densities extended to ones that are more realistic in an actuarial setting, i.e. classes of generalized moments conditions. The proposed method is illustrated with data from Lemaire (1979). The main aim of the paper is to demonstrate an appropriate methodology to perform a Bayesian sensitivity analysis of the Bonus–Malus of loaded premiums.

Keywords:

• Bonus–malus
• Bayesian robustness
• ϵ–contamination class
• generalized moments conditions

Acknowledgements

Research partially supported by grant from MCyT (Ministerio de Ciencia y Tecnología, Spain, project BEC2001–3774).