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Risk-Minimizing Reinsurance Protection For Multivariate Risks

Authors

  • K. C. Cheung,

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    • K. C. Cheung and K. C. J. Sung are at the Department of Statistics, and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong. S. C. P. Yam is with the Department of Statistics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong. K. C. Cheung can be contacted via e-mail: kccg@hku.hk. The authors thank Jan Dhaene, Ludger RÜschendorf, and many seminar and conference participants, for pointing out more accurate references, and for their supportive comments and inspiring suggestions. K. C. Cheung acknowledges the financial support of the Research Grants Council of HKSAR (Project No. HKU701409P). S. C. P. Yam acknowledges financial support from The Hong Kong RGC GRF 404012 with the project title “Advanced Topics in Multivariate Risk Management in Finance and Insurance.” S. C. P. Yam also expresses his sincere gratitude to the hospitality of both Hausdorff Center for Mathematics of the University of Bonn and Mathematisches Forschungsinstitut Oberwolfach (MFO) in the German Black Forest during the preparation of the present work.
  • K. C. J. Sung,

    Search for more papers by this author
    • K. C. Cheung and K. C. J. Sung are at the Department of Statistics, and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong. S. C. P. Yam is with the Department of Statistics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong. K. C. Cheung can be contacted via e-mail: kccg@hku.hk. The authors thank Jan Dhaene, Ludger RÜschendorf, and many seminar and conference participants, for pointing out more accurate references, and for their supportive comments and inspiring suggestions. K. C. Cheung acknowledges the financial support of the Research Grants Council of HKSAR (Project No. HKU701409P). S. C. P. Yam acknowledges financial support from The Hong Kong RGC GRF 404012 with the project title “Advanced Topics in Multivariate Risk Management in Finance and Insurance.” S. C. P. Yam also expresses his sincere gratitude to the hospitality of both Hausdorff Center for Mathematics of the University of Bonn and Mathematisches Forschungsinstitut Oberwolfach (MFO) in the German Black Forest during the preparation of the present work.
  • S. C. P. Yam

    Search for more papers by this author
    • K. C. Cheung and K. C. J. Sung are at the Department of Statistics, and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong. S. C. P. Yam is with the Department of Statistics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong. K. C. Cheung can be contacted via e-mail: kccg@hku.hk. The authors thank Jan Dhaene, Ludger RÜschendorf, and many seminar and conference participants, for pointing out more accurate references, and for their supportive comments and inspiring suggestions. K. C. Cheung acknowledges the financial support of the Research Grants Council of HKSAR (Project No. HKU701409P). S. C. P. Yam acknowledges financial support from The Hong Kong RGC GRF 404012 with the project title “Advanced Topics in Multivariate Risk Management in Finance and Insurance.” S. C. P. Yam also expresses his sincere gratitude to the hospitality of both Hausdorff Center for Mathematics of the University of Bonn and Mathematisches Forschungsinstitut Oberwolfach (MFO) in the German Black Forest during the preparation of the present work.

Abstract

In this article, we study the problem of optimal reinsurance policy for multivariate risks whose quantitative analysis in the realm of general law-invariant convex risk measures, to the best of our knowledge, is still absent in the literature. In reality, it is often difficult to determine the actual dependence structure of these risks. Instead of assuming any particular dependence structure, we propose the minimax optimal reinsurance decision formulation in which the worst case scenario is first identified, then we proceed to establish that the stop-loss reinsurances are optimal in the sense that they minimize a general law-invariant convex risk measure of the total retained risk. By using minimax theorem, explicit form of and sufficient condition for ordering the optimal deductibles are also obtained.

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