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Multivariate Collective Risk Model: Dependent Claim Numbers and Panjer’s Recursion

Author

Listed:
  • Cordelia Rudolph

    (msg life Austria Ges.m.b.H., Ausstellungsstraße 50, 1020 Vienna, Austria)

  • Uwe Schmock

    (Department of Financial and Actuarial Mathematics, Vienna University of Technology, Wiedner Hauptstraße 8–10/E105-1, 1040 Vienna, Austria)

Abstract

In this paper, we discuss a generalization of the collective risk model and of Panjer’s recursion. The model we consider consists of several business lines with dependent claim numbers. The distributions of the claim numbers are assumed to be Poisson mixture distributions. We let the claim causes have certain dependence structures and prove that Panjer’s recursion is also applicable by finding an appropriate equivalent representation of the claim numbers. These dependence structures are of a stochastic non-negative linear nature and may also produce negative correlations between the claim causes. The consideration of risk groups also includes dependence between claim sizes. Compounding the claim causes by common distributions also keeps Panjer’s recursion applicable.

Suggested Citation

  • Cordelia Rudolph & Uwe Schmock, 2020. "Multivariate Collective Risk Model: Dependent Claim Numbers and Panjer’s Recursion," Risks, MDPI, vol. 8(2), pages 1-31, May.
    • Handle: RePEc:gam:jrisks:v:8:y:2020:i:2:p:43-:d:353271
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    References listed on IDEAS

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    1. Willmot, Gordon, 1988. "Sundt and Jewell's Family of Discrete Distributions," ASTIN Bulletin, Cambridge University Press, vol. 18(1), pages 17-29, April.
    2. Sundt, Bjørn, 1999. "On Multivariate Panjer Recursions," ASTIN Bulletin, Cambridge University Press, vol. 29(1), pages 29-45, May.
    3. Xiaolin Luo & Pavel V. Shevchenko, 2007. "The t copula with Multiple Parameters of Degrees of Freedom: Bivariate Characteristics and Application to Risk Management," Papers 0710.3959, arXiv.org, revised Feb 2010.
    4. Xiaolin Luo & Pavel Shevchenko, 2010. "The t copula with multiple parameters of degrees of freedom: bivariate characteristics and application to risk management," Quantitative Finance, Taylor & Francis Journals, vol. 10(9), pages 1039-1054.
    5. Sundt, Bjørn & Jewell, William S., 1981. "Further Results on Recursive Evaluation of Compound Distributions," ASTIN Bulletin, Cambridge University Press, vol. 12(1), pages 27-39, June.
    6. Mai, Jan-Frederik & Scherer, Matthias & Shenkman, Natalia, 2013. "Multivariate geometric distributions, (logarithmically) monotone sequences, and infinitely divisible laws," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 457-480.
    7. Lindskog, Filip & McNeil, Alexander J., 2003. "Common Poisson Shock Models: Applications to Insurance and Credit Risk Modelling," ASTIN Bulletin, Cambridge University Press, vol. 33(2), pages 209-238, November.
    8. Willmot, G. E. & Panjer, H. H., 1987. "Difference equation approaches in evaluation of compound distributions," Insurance: Mathematics and Economics, Elsevier, vol. 6(1), pages 43-56, January.
    9. Panjer, Harry H., 1981. "Recursive Evaluation of a Family of Compound Distributions," ASTIN Bulletin, Cambridge University Press, vol. 12(1), pages 22-26, June.
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    Cited by:

    1. Corina Constantinescu & Julia Eisenberg, 2021. "Special Issue “Interplay between Financial and Actuarial Mathematics”," Risks, MDPI, vol. 9(8), pages 1-3, July.

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