Fair Bonus-Malus Systems
In spite of over 50 years of existence of a bonus-malus system (BMS) many crucial problems concerning its modelling, analysis and optimisation remain unsolved. Definitions of BMS proposed in literature are so general that they include systems which could not exist and perform well in the competitive automobile insurance, market, and therefore they are not used in practice. The objective of this article is to present two definitions of fair bonus-malus systems, which differ in the criterion of distinguishing BMS, however both allow for eliminating systems with non-realistic structures. The first proposal is the definition of so called bonus-malus system fair by premium (BMSFpx). The concept of this system consists in excluding from considerations such systems, in which policyholders are penalized with the greater premium after reporting fewer losses or after claiming a given number of losses if they were in better class. The criterion for distinguishing BMS in the second definition is based on the transition rules i.e. rules governing the transition of the insured, having reported a given number of claims, from one class to another. Therefore, these systems are namedfair bonus-malus sys-tems by transition rules (BMSFTIt). In this paper it is also proved that each BMSFTR is also BMSFPR. (original abstract)
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