Abstract
The paper deals with bonus-malus systems with different claim types and varying deductibles. The premium relativities are softened for the policyholders who are in the malus zone and these policyholders are subject to per claim deductibles depending on their levels in the bonus-malus scale and the types of the reported claims. We introduce such bonus-malus systems and study their basic properties. In particular, we investigate when it is possible to introduce varying deductibles, what restrictions we have and how we can do this. Moreover, we deal with the special case where varying deductibles are applied to the claims reported by policyholders occupying the highest level in the bonus-malus scale and consider two allocation principles for the deductibles. Finally, numerical illustrations are presented.
Highlights
Introduction and motivationOne of the main tasks of an actuary is to design a tariff structure that fairly distributes the total risk of potential losses among policyholders
To eliminate the second drawback, we introduce varying deductibles for the policyholders who are in the malus zone
We consider the special case where dl,i = 0 for all s0 ≤ l ≤ s −1 and 0 ≤ i ≤ m. This case is of great interest when the premium relativity for policyholders who are at level s is high enough and varying deductibles are applied to soften the bonus–malus system for such policyholders
Summary
One of the main tasks of an actuary is to design a tariff structure that fairly distributes the total risk of potential losses among policyholders. Gómez-Déniz, Hernández-Bastida and Fernández-Sánchez [8] obtain expressions that can be used to compute bonus–malus premiums based on the distribution of the total claim amount but not on the claims which produce the amounts Another modification of traditional bonus–malus systems, which take into account only the number of claims, is considered by Gómez-Déniz in the recent paper [7]. The deductibles depend on the level of the policyholder in the bonus–malus scale and the types of the reported claims Such bonus–malus systems present a number of advantages and seem to be very attractive for policyholders.
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