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Modeling and Performance of Bonus-Malus Systems: Stationarity versus Age-Correction

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  • Søren Asmussen

    (Department of Mathematics, Aarhus University, Ny Munkegade, Aarhus C 8000, Denmark)

Abstract

In a bonus-malus system in car insurance, the bonus class of a customer is updated from one year to the next as a function of the current class and the number of claims in the year (assumed Poisson). Thus the sequence of classes of a customer in consecutive years forms a Markov chain, and most of the literature measures performance of the system in terms of the stationary characteristics of this Markov chain. However, the rate of convergence to stationarity may be slow in comparison to the typical sojourn time of a customer in the portfolio. We suggest an age-correction to the stationary distribution and present an extensive numerical study of its effects. An important feature of the modeling is a Bayesian view, where the Poisson rate according to which claims are generated for a customer is the outcome of a random variable specific to the customer.

Suggested Citation

  • Søren Asmussen, 2014. "Modeling and Performance of Bonus-Malus Systems: Stationarity versus Age-Correction," Risks, MDPI, vol. 2(1), pages 1-25, March.
    • Handle: RePEc:gam:jrisks:v:2:y:2014:i:1:p:49-73:d:33936
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    References listed on IDEAS

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    1. Loimaranta, K., 1972. "Some asymptotic properties of bonus systems," ASTIN Bulletin, Cambridge University Press, vol. 6(3), pages 233-245, May.
    2. Lemaire, Jean & Zi, Hongmin, 1994. "A Comparative Analysis of 30 Bonus-Malus Systems," ASTIN Bulletin, Cambridge University Press, vol. 24(2), pages 287-309, November.
    3. Bonsdorff, Heikki, 1992. "On the Convergence Rate of Bonus-Malus Systems," ASTIN Bulletin, Cambridge University Press, vol. 22(2), pages 217-223, November.
    4. Frangos, Nicholas E. & Vrontos, Spyridon D., 2001. "Design of Optimal Bonus-Malus Systems With a Frequency and a Severity Component On an Individual Basis in Automobile Insurance," ASTIN Bulletin, Cambridge University Press, vol. 31(1), pages 1-22, May.
    5. Mahmoudvand, Rahim & Hassani, Hossein, 2009. "Generalized Bonus-Malus Systems with a Frequency and a Severity Component on an Individual Basis in Automobile Insurance," ASTIN Bulletin, Cambridge University Press, vol. 39(1), pages 307-315, May.
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    Cited by:

    1. Martinek, László & Arató, N. Miklós, 2019. "An approach to merit rating by means of autoregressive sequences," Insurance: Mathematics and Economics, Elsevier, vol. 85(C), pages 205-217.
    2. Julie Thøgersen, 2016. "Optimal Premium as a Function of the Deductible: Customer Analysis and Portfolio Characteristics," Risks, MDPI, vol. 4(4), pages 1-19, November.
    3. Mahmoudvand Rahim & Tan Chong It & Abbasi Narges, 2017. "Adjusting the Premium Relativities in a Bonus-Malus System: An Integrated Approach Using the First Claim Time and the Number of Claims," Asia-Pacific Journal of Risk and Insurance, De Gruyter, vol. 11(2), pages 1-19, July.
    4. Corina Constantinescu & Suhang Dai & Weihong Ni & Zbigniew Palmowski, 2016. "Ruin Probabilities with Dependence on the Number of Claims within a Fixed Time Window," Risks, MDPI, vol. 4(2), pages 1-23, June.

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