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Designing a Bonus-Malus system reflecting the claim size under the dependent frequency–severity model

Published online by Cambridge University Press:  07 June 2021

Rosy Oh ,
Joseph H.T. Kim  and
Jae Youn Ahn Open the ORCID record for Jae Youn Ahn [Opens in a new window]
Rosy Oh
Affiliation:
Department of Industrial & Systems Engineering, Korea Advanced Institute of Science and Technology, Daejeon, Korea. E-mail: rosy.oh5@gmail.com
Joseph H.T. Kim
Affiliation:
Department of Applied Statistics, Department of Statistics and Data Science, College of Business and Economics, Yonsei University, Seoul, Korea. E-mail: jhtkim@yonsei.ac.kr
Jae Youn Ahn
Affiliation:
Department of Statistics, Ewha Womans University, Seoul, Korea. E-mail: jaeyahn@ewha.ac.kr
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Abstract

In the auto insurance industry, a Bonus-Malus System (BMS) is commonly used as a posteriori risk classification mechanism to set the premium for the next contract period based on a policyholder's claim history. Even though the recent literature reports evidence of a significant dependence between frequency and severity, the current BMS practice is to use a frequency-based transition rule while ignoring severity information. Although Oh et al. [(2020). Bonus-Malus premiums under the dependent frequency-severity modeling. Scandinavian Actuarial Journal 2020(3): 172–195] claimed that the frequency-driven BMS transition rule can accommodate the dependence between frequency and severity, their proposal is only a partial solution, as the transition rule still completely ignores the claim severity and is unable to penalize large claims. In this study, we propose to use the BMS with a transition rule based on both frequency and size of claim, based on the bivariate random effect model, which conveniently allows dependence between frequency and severity. We analytically derive the optimal relativities under the proposed BMS framework and show that the proposed BMS outperforms the existing frequency-driven BMS. Later, numerical experiments are also provided using both hypothetical and actual datasets in order to assess the effect of various dependencies on the BMS risk classification and confirm our theoretical findings.

Keywords

Bivariate random effects modelBonus-Malus systemDependenceGeneralized linear modelRate making
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 36 , Issue 4 , October 2022 , pp. 963 - 987
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press

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