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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, Open Access Journal, 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. 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: The Journal of the International Actuarial Association, Cambridge University Press, vol. 31(01), pages 1-22, May.
    2. 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: The Journal of the International Actuarial Association, Cambridge University Press, vol. 39(01), pages 307-315, May.
    3. Bonsdorff, Heikki, 1992. "On the Convergence Rate of Bonus-Malus Systems," ASTIN Bulletin: The Journal of the International Actuarial Association, Cambridge University Press, vol. 22(02), pages 217-223, November.
    4. Lemaire, Jean & Zi, Hongmin, 1994. "A Comparative Analysis of 30 Bonus-Malus Systems," ASTIN Bulletin: The Journal of the International Actuarial Association, Cambridge University Press, vol. 24(02), pages 287-309, November.
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    Citations

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    Cited by:

    1. Julie Thøgersen, 2016. "Optimal Premium as a Function of the Deductible: Customer Analysis and Portfolio Characteristics," Risks, MDPI, Open Access Journal, vol. 4(4), pages 1-19, November.
    2. 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, Open Access Journal, vol. 4(2), pages 1-23, June.

    More about this item

    Keywords

    actuarial mathematics; Bayes premium; equilibrium distribution; experience rating; insurance portfolio; Markov chain; motor insurance; Poisson claims; stationary distribution;

    JEL classification:

    • C - Mathematical and Quantitative Methods
    • G0 - Financial Economics - - General
    • G1 - Financial Economics - - General Financial Markets
    • G2 - Financial Economics - - Financial Institutions and Services
    • G3 - Financial Economics - - Corporate Finance and Governance
    • M2 - Business Administration and Business Economics; Marketing; Accounting; Personnel Economics - - Business Economics
    • M4 - Business Administration and Business Economics; Marketing; Accounting; Personnel Economics - - Accounting
    • K2 - Law and Economics - - Regulation and Business Law

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