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Modelling uncertainty in insurance Bonus-Malus premium principles by using a Bayesian robustness approach

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  • Emilio Gomez-deniz
  • Francisco Vazquez-polo

Abstract

When Bayesian models are implemented for a Bonus-Malus System (BMS), a parametric structure, π0 (λ), is normally included in the insurer's portfolio. Following Bayesian sensitivity analysis, it is possible to model the structure function by specifying a class Γ of priors instead of a single prior. This paper examines the ranges of the relativities of the form, [image omitted] Standard and robust Bayesian tools are combined to show how the choice of the prior can affect the relative premiums. As an extension of the paper by Gomez et al. (2002b), our model is developed to the variance premium principle and the class of prior densities extended to ones that are more realistic in an actuarial setting, i.e. classes of generalized moments conditions. The proposed method is illustrated with data from Lemaire (1979). The main aim of the paper is to demonstrate an appropriate methodology to perform a Bayesian sensitivity analysis of the Bonus-Malus of loaded premiums.

Suggested Citation

  • Emilio Gomez-deniz & Francisco Vazquez-polo, 2005. "Modelling uncertainty in insurance Bonus-Malus premium principles by using a Bayesian robustness approach," Journal of Applied Statistics, Taylor & Francis Journals, vol. 32(7), pages 771-784.
    • Handle: RePEc:taf:japsta:v:32:y:2005:i:7:p:771-784
    DOI: 10.1080/02664760500079746
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    References listed on IDEAS

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    1. Lemaire, Jean, 1979. "How to Define a Bonus-Malus System with an Exponential Utility Function," ASTIN Bulletin, Cambridge University Press, vol. 10(3), pages 274-282, December.
    2. David Scollnik, 2001. "Actuarial Modeling with MCMC and BUGs," North American Actuarial Journal, Taylor & Francis Journals, vol. 5(2), pages 96-124.
    3. Makov, Udi E., 1995. "Loss robustness via Fisher-weighted squared-error loss function," Insurance: Mathematics and Economics, Elsevier, vol. 16(1), pages 1-6, April.
    4. Eichenauer, Jurgen & Lehn, Jurgen & Rettig, Stefan, 1988. "A gamma-minimax result in credibility theory," Insurance: Mathematics and Economics, Elsevier, vol. 7(1), pages 49-57, January.
    5. Heilmann, Wolf-Rudiger, 1989. "Decision theoretic foundations of credibility theory," Insurance: Mathematics and Economics, Elsevier, vol. 8(1), pages 77-95, March.
    6. 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.
    7. Tremblay, Luc, 1992. "Using the Poisson Inverse Gaussian in Bonus-Malus Systems," ASTIN Bulletin, Cambridge University Press, vol. 22(1), pages 97-106, May.
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