Designing an Optimal Bonus--Malus System Using the Number of Reported Claims, Steady-State Distribution, and Mixture Claim Size Distribution

TL;DR

This paper develops an optimal Bonus--Malus system using Bayesian estimators, mixture claim size distributions, and evaluates the efficiency of linear premiums compared to Bayes premiums.

Contribution

It introduces a linear relativity premium that approximates Bayesian estimators and assesses its performance with mixture claim size distributions.

Findings

Linear relativity premium closely approximates Bayes estimators.

Mixture claim size distributions improve premium accuracy.

Linear premiums show competitive Loimaranta efficiency.

Abstract

This article, in a first step, considers two Bayes estimators for the relativity premium of a given Bonus--Malus system. It then develops a linear relativity premium that closes, in the sense of weighted mean square error loss, to such Bayes estimators. In a second step, it supposes that the claim size distribution for a given Bonus--Malus system can be formulated as a finite mixture distribution. It then evaluates the base premium under a Bayesian framework for such a finite mixture distribution. The Loimaranta efficiency of such a linear relativity premium, for several Bonus--Malus systems, has been compared with two Bayes and ordinary linear relativity premiums.