A simple method to study sensitivity of BMP's

Abstract

In this article we measure the local or infinitesimal sensitivity of a kind of Bayes estimates which appear in bonus-malus systems. Bonus-malus premiums can be viewed as a functional depending on the prior distribution. To measure when small changes in the prior cause large changes in the premium we compute the norm of the Fréchet derivative and propose a simple procedure to decide if a bonus-malus premium is robust. As an application, an example where the risk has a Poisson distribution and its parameter follows a Gamma prior distribution is presented under the net and variance premium principles.