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In this paper, we propose an integrated approach to adjust the premium relativities in a bonus-malus system by using the information of the first claim time (expressed in terms of sub-period in a year) and the number of claims reported by individual policyholder. We provide a formal representation for the newly proposed structure and derive the analytical expressions for the adjusted premium relativities. Other things being equal, a lower adjusted premium relativity is imposed for an earlier sub-period of the first claim made, whereas policyholders with more claims are subject to a higher adjusted premium relativity.

Adjusting the Premium Relativities in a Bonus-Malus System: An Integrated Approach Using the First Claim Time and the Number of Claims