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Dependence properties and bounds for ruin probabilities in multivariate compound risk models

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  • Cai, Jun
  • Li, Haijun

Abstract

In risk management, ignoring the dependence among various types of claims often results in over-estimating or under-estimating the ruin probabilities of a portfolio. This paper focuses on three commonly used ruin probabilities in multivariate compound risk models, and using the comparison methods shows how some ruin probabilities increase, whereas the others decrease, as the claim dependence grows. The paper also presents some computable bounds for these ruin probabilities, which can be calculated explicitly for multivariate phase-type distributed claims, and illustrates the performance of these bounds for the multivariate compound Poisson risk models with slightly or highly dependent Marshall-Olkin exponential claim sizes.

Suggested Citation

  • Cai, Jun & Li, Haijun, 2007. "Dependence properties and bounds for ruin probabilities in multivariate compound risk models," Journal of Multivariate Analysis, Elsevier, vol. 98(4), pages 757-773, April.
    • Handle: RePEc:eee:jmvana:v:98:y:2007:i:4:p:757-773
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    References listed on IDEAS

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    Cited by:

    1. Véronique Maume-Deschamps & Didier Rullière & Khalil Said, 2017. "Impact of Dependence on Some Multivariate Risk Indicators," Methodology and Computing in Applied Probability, Springer, vol. 19(2), pages 395-427, June.
    2. Gordienko, E. & Vázquez-Ortega, P., 2018. "Continuity inequalities for multidimensional renewal risk models," Insurance: Mathematics and Economics, Elsevier, vol. 82(C), pages 48-54.
    3. Shen, Xinmei & Zhang, Yi, 2013. "Ruin probabilities of a two-dimensional risk model with dependent risks of heavy tail," Statistics & Probability Letters, Elsevier, vol. 83(7), pages 1787-1799.
    4. Véronique Maume-Deschamps & Didier Rullière & Khalil Said, 2014. "On capital allocation by minimizing multivariate risk indicators," Working Papers hal-01082559, HAL.
    5. Véronique Maume-Deschamps & Didier Rullière & Khalil Said, 2015. "A risk management approach to capital allocation," Working Papers hal-01163180, HAL.
    6. repec:hal:wpaper:hal-01171395 is not listed on IDEAS
    7. G. A. Delsing & M. R. H. Mandjes & P. J. C. Spreij & E. M. M. Winands, 2020. "Asymptotics and Approximations of Ruin Probabilities for Multivariate Risk Processes in a Markovian Environment," Methodology and Computing in Applied Probability, Springer, vol. 22(3), pages 927-948, September.
    8. Albrecher, Hansjörg & Cheung, Eric C.K. & Liu, Haibo & Woo, Jae-Kyung, 2022. "A bivariate Laguerre expansions approach for joint ruin probabilities in a two-dimensional insurance risk process," Insurance: Mathematics and Economics, Elsevier, vol. 103(C), pages 96-118.
    9. Castañer, A. & Claramunt, M.M. & Lefèvre, C., 2013. "Survival probabilities in bivariate risk models, with application to reinsurance," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 632-642.
    10. He, Yue & Kawai, Reiichiro, 2022. "Moment and polynomial bounds for ruin-related quantities in risk theory," European Journal of Operational Research, Elsevier, vol. 302(3), pages 1255-1271.
    11. Xiaohu Li & Jintang Wu & Jinsen Zhuang, 2015. "Asymptotic Multivariate Finite-time Ruin Probability with Statistically Dependent Heavy-tailed Claims," Methodology and Computing in Applied Probability, Springer, vol. 17(2), pages 463-477, June.
    12. Delsing, G.A. & Mandjes, M.R.H. & Spreij, P.J.C. & Winands, E.M.M., 2019. "An optimization approach to adaptive multi-dimensional capital management," Insurance: Mathematics and Economics, Elsevier, vol. 84(C), pages 87-97.
    13. Liu, Jingchen & Woo, Jae-Kyung, 2014. "Asymptotic analysis of risk quantities conditional on ruin for multidimensional heavy-tailed random walks," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 1-9.
    14. Véronique Maume-Deschamps & Didier Rullière & Khalil Said, 2016. "On a capital allocation by minimizing multivariate risk indicators," Post-Print hal-01082559, HAL.
    15. Cheng, Ming & Konstantinides, Dimitrios G. & Wang, Dingcheng, 2022. "Uniform asymptotic estimates in a time-dependent risk model with general investment returns and multivariate regularly varying claims," Applied Mathematics and Computation, Elsevier, vol. 434(C).
    16. G. A. Delsing & M. R. H. Mandjes & P. J. C. Spreij & E. M. M. Winands, 2018. "An optimization approach to adaptive multi-dimensional capital management," Papers 1812.08435, arXiv.org.
    17. Gong, Lan & Badescu, Andrei L. & Cheung, Eric C.K., 2012. "Recursive methods for a multi-dimensional risk process with common shocks," Insurance: Mathematics and Economics, Elsevier, vol. 50(1), pages 109-120.
    18. Cénac P. & Maume-Deschamps V. & Prieur C., 2012. "Some multivariate risk indicators: Minimization by using a Kiefer–Wolfowitz approach to the mirror stochastic algorithm," Statistics & Risk Modeling, De Gruyter, vol. 29(1), pages 47-72, March.
    19. Ivanovs, Jevgenijs & Boxma, Onno, 2015. "A bivariate risk model with mutual deficit coverage," Insurance: Mathematics and Economics, Elsevier, vol. 64(C), pages 126-134.
    20. Romain Biard, 2013. "Asymptotic multivariate finite-time ruin probabilities with heavy-tailed claim amounts: Impact of dependence and optimal reserve allocation," Post-Print hal-00538571, HAL.
    21. Boxma, Onno & Frostig, Esther & Perry, David & Yosef, Rami, 2017. "A state dependent reinsurance model," Insurance: Mathematics and Economics, Elsevier, vol. 74(C), pages 170-181.

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