Robust Bayesian premium principles in actuarial science

Abstract

The term premium relates to the purchase price of an insurance contract. Bayesian models in credibility theory require a complete specification of the model (basically, the prior) and it is difficult to justify any one particular choice. According to robust Bayesian methodology, uncertainty in the prior can be modelled by specifying a class Γ of priors instead of a single prior. We examine the ranges of Bayesian premiums when the priors belong to such a class. Most robust Bayesian procedures include measures of sensitivity of quantities which can be expressed in terms of a posterior expectation (e.g. the mean, variance and probability of sets). Nevertheless, a significant difference that appears in the actuarial context is considered here. The expression for some Bayes premiums in credibility theory suggests that the quantity of interest can be expressed in terms of the ratio of posterior expectations. Appropriate techniques to do this are considered here. Two models and two situations are presented for a non-compound collective model. Even though the model is very robust, a consideration of unimodality significantly reduces the sensitivity of the Bayesian premium arising from a base prior π0. Therefore, unimodality is very convenient for modelling subjective beliefs about the risk parameter.