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Risk-adjusted Bowley reinsurance under distorted probabilities

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  • Cheung, Ka Chun
  • Yam, Sheung Chi Phillip
  • Zhang, Yiying

Abstract

In the seminal work of Chan and Gerber (1985), one of the earliest game theoretical approaches was proposed to model the interaction between the reinsurer and insurer; in particular, the optimal pricing density for the reinsurer and optimal ceded loss for the insurer were determined so that their corresponding expected utilities could be maximized. Over decades, their advocated Bowley solution (could be understood as Stackelberg equilibria) concept of equilibrium reinsurance strategy has not been revisited in the modern risk management framework. In this article, we attempt to fill this gap by extending their work to the setting of general premium principle for the reinsurer and distortion risk measure for the insurer.

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  • Cheung, Ka Chun & Yam, Sheung Chi Phillip & Zhang, Yiying, 2019. "Risk-adjusted Bowley reinsurance under distorted probabilities," Insurance: Mathematics and Economics, Elsevier, vol. 86(C), pages 64-72.
    • Handle: RePEc:eee:insuma:v:86:y:2019:i:c:p:64-72
    DOI: 10.1016/j.insmatheco.2019.02.006
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    1. Chen, Lv & Shen, Yang, 2018. "On A New Paradigm Of Optimal Reinsurance: A Stochastic Stackelberg Differential Game Between An Insurer And A Reinsurer," ASTIN Bulletin, Cambridge University Press, vol. 48(2), pages 905-960, May.
    2. Neil Doherty & Kent Smetters, 2005. "Moral Hazard in Reinsurance Markets," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 72(3), pages 375-391, September.
    3. Cai, Jun & Lemieux, Christiane & Liu, Fangda, 2016. "Optimal Reinsurance From The Perspectives Of Both An Insurer And A Reinsurer," ASTIN Bulletin, Cambridge University Press, vol. 46(3), pages 815-849, September.
    4. Michael Rothschild & Joseph Stiglitz, 1976. "Equilibrium in Competitive Insurance Markets: An Essay on the Economics of Imperfect Information," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 90(4), pages 629-649.
    5. Yaari, Menahem E, 1987. "The Dual Theory of Choice under Risk," Econometrica, Econometric Society, vol. 55(1), pages 95-115, January.
    6. Cai, Jun & Tan, Ken Seng & Weng, Chengguo & Zhang, Yi, 2008. "Optimal reinsurance under VaR and CTE risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 43(1), pages 185-196, August.
    7. Xiang Lin & Chunhong Zhang & Tak Siu, 2012. "Stochastic differential portfolio games for an insurer in a jump-diffusion risk process," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 75(1), pages 83-100, February.
    8. Wu, Xianyi & Zhou, Xian, 2006. "A new characterization of distortion premiums via countable additivity for comonotonic risks," Insurance: Mathematics and Economics, Elsevier, vol. 38(2), pages 324-334, April.
    9. Cui, Wei & Yang, Jingping & Wu, Lan, 2013. "Optimal reinsurance minimizing the distortion risk measure under general reinsurance premium principles," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 74-85.
    10. d'Ursel, L. & Lauwers, M., 1985. "Chains of reinsurance: Non-cooperative equilibria and pareto optimality," Insurance: Mathematics and Economics, Elsevier, vol. 4(4), pages 279-285, October.
    11. Assa, Hirbod, 2015. "On optimal reinsurance policy with distortion risk measures and premiums," Insurance: Mathematics and Economics, Elsevier, vol. 61(C), pages 70-75.
    12. Kaluszka, Marek, 2001. "Optimal reinsurance under mean-variance premium principles," Insurance: Mathematics and Economics, Elsevier, vol. 28(1), pages 61-67, February.
    13. Dutang, Christophe & Albrecher, Hansjoerg & Loisel, Stéphane, 2013. "Competition among non-life insurers under solvency constraints: A game-theoretic approach," European Journal of Operational Research, Elsevier, vol. 231(3), pages 702-711.
    14. Gerber, Hans U., 1984. "Chains of reinsurance," Insurance: Mathematics and Economics, Elsevier, vol. 3(1), pages 43-48, January.
    15. Goovaerts, Marc J. & Kaas, Rob & Laeven, Roger J.A., 2010. "Decision principles derived from risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 47(3), pages 294-302, December.
    16. Wang, Shaun, 1996. "Premium Calculation by Transforming the Layer Premium Density," ASTIN Bulletin, Cambridge University Press, vol. 26(1), pages 71-92, May.
    17. Young, Virginia R., 1999. "Optimal insurance under Wang's premium principle," Insurance: Mathematics and Economics, Elsevier, vol. 25(2), pages 109-122, November.
    18. Kaluszka, Marek, 2005. "Optimal reinsurance under convex principles of premium calculation," Insurance: Mathematics and Economics, Elsevier, vol. 36(3), pages 375-398, June.
    19. Kaluszka, Marek & Krzeszowiec, Michał, 2012. "Pricing insurance contracts under Cumulative Prospect Theory," Insurance: Mathematics and Economics, Elsevier, vol. 50(1), pages 159-166.
    20. Dionne, G. & Doherty, N. & Fombaron, N., 2000. "Adverse Selection in Insurance Markets," Ecole des Hautes Etudes Commerciales de Montreal- 00-05, Ecole des Hautes Etudes Commerciales de Montreal-Chaire de gestion des risques..
    21. Chan, Fung-Yee & Gerber, Hans U., 1985. "The Reinsurer's Monopoly and the Bowley Solution," ASTIN Bulletin, Cambridge University Press, vol. 15(2), pages 141-148, November.
    22. Alejandro Balbás & José Garrido & Silvia Mayoral, 2009. "Properties of Distortion Risk Measures," Methodology and Computing in Applied Probability, Springer, vol. 11(3), pages 385-399, September.
    23. Borch, Karl, 1969. "The optimal reinsurance treaty," ASTIN Bulletin, Cambridge University Press, vol. 5(2), pages 293-297, May.
    24. Denuit, Michel & Vermandele, Catherine, 1998. "Optimal reinsurance and stop-loss order," Insurance: Mathematics and Economics, Elsevier, vol. 22(3), pages 229-233, July.
    25. Cheung, K.C. & Liu, F. & Yam, S.C.P., 2012. "Average Value-at-Risk Minimizing Reinsurance Under Wang's Premium Principle with Constraints," ASTIN Bulletin, Cambridge University Press, vol. 42(2), pages 575-600, November.
    26. Boonen, Tim J. & Tan, Ken Seng & Zhuang, Sheng Chao, 2016. "Pricing In Reinsurance Bargaining With Comonotonic Additive Utility Functions," ASTIN Bulletin, Cambridge University Press, vol. 46(2), pages 507-530, May.
    27. Raviv, Artur, 1979. "The Design of an Optimal Insurance Policy," American Economic Review, American Economic Association, vol. 69(1), pages 84-96, March.
    28. Balbás, Alejandro & Balbás, Beatriz & Heras, Antonio, 2009. "Optimal reinsurance with general risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 44(3), pages 374-384, June.
    29. Wang, Shaun S. & Young, Virginia R. & Panjer, Harry H., 1997. "Axiomatic characterization of insurance prices," Insurance: Mathematics and Economics, Elsevier, vol. 21(2), pages 173-183, November.
    30. Jun Cai & Ying Fang & Zhi Li & Gordon E. Willmot, 2013. "Optimal Reciprocal Reinsurance Treaties Under the Joint Survival Probability and the Joint Profitable Probability," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 80(1), pages 145-168, March.
    31. Cai, Jun & Liu, Haiyan & Wang, Ruodu, 2017. "Pareto-optimal reinsurance arrangements under general model settings," Insurance: Mathematics and Economics, Elsevier, vol. 77(C), pages 24-37.
    32. Chi, Yichun & Tan, Ken Seng, 2013. "Optimal reinsurance with general premium principles," Insurance: Mathematics and Economics, Elsevier, vol. 52(2), pages 180-189.
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    Cited by:

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    2. Ghossoub, Mario & Zhu, Michael B., 2024. "Stackelberg equilibria with multiple policyholders," Insurance: Mathematics and Economics, Elsevier, vol. 116(C), pages 189-201.
    3. Liu, Haiyan, 2024. "Worst-case risk with unspecified risk preferences," Insurance: Mathematics and Economics, Elsevier, vol. 116(C), pages 235-248.
    4. Anthropelos, Michail & Boonen, Tim J., 2020. "Nash equilibria in optimal reinsurance bargaining," Insurance: Mathematics and Economics, Elsevier, vol. 93(C), pages 196-205.
    5. Boonen, Tim J. & Ghossoub, Mario, 2023. "Bowley vs. Pareto optima in reinsurance contracting," European Journal of Operational Research, Elsevier, vol. 307(1), pages 382-391.
    6. Li, Danping & Young, Virginia R., 2021. "Bowley solution of a mean–variance game in insurance," Insurance: Mathematics and Economics, Elsevier, vol. 98(C), pages 35-43.
    7. Gabriela Zeller & Matthias Scherer, 2023. "Risk mitigation services in cyber insurance: optimal contract design and price structure," The Geneva Papers on Risk and Insurance - Issues and Practice, Palgrave Macmillan;The Geneva Association, vol. 48(2), pages 502-547, April.
    8. Chi, Yichun & Tan, Ken Seng & Zhuang, Sheng Chao, 2020. "A Bowley solution with limited ceded risk for a monopolistic reinsurer," Insurance: Mathematics and Economics, Elsevier, vol. 91(C), pages 188-201.
    9. Chen, Yanhong & Cheung, Ka Chun & Zhang, Yiying, 2024. "Bowley solution under the reinsurer's default risk," Insurance: Mathematics and Economics, Elsevier, vol. 115(C), pages 36-61.
    10. Zhu, Michael B. & Ghossoub, Mario & Boonen, Tim J., 2023. "Equilibria and efficiency in a reinsurance market," Insurance: Mathematics and Economics, Elsevier, vol. 113(C), pages 24-49.
    11. Cao, Jingyi & Li, Dongchen & Young, Virginia R. & Zou, Bin, 2023. "Reinsurance games with two reinsurers: Tree versus chain," European Journal of Operational Research, Elsevier, vol. 310(2), pages 928-941.
    12. Tang, Qihe & Tong, Zhiwei & Xun, Li, 2022. "Portfolio risk analysis of excess of loss reinsurance," Insurance: Mathematics and Economics, Elsevier, vol. 102(C), pages 91-110.
    13. Bensalem, Sarah & Santibáñez, Nicolás Hernández & Kazi-Tani, Nabil, 2020. "Prevention efforts, insurance demand and price incentives under coherent risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 93(C), pages 369-386.
    14. Cheung, Ka Chun & Phillip Yam, Sheung Chi & Yuen, Fei Lung & Zhang, Yiying, 2020. "Concave distortion risk minimizing reinsurance design under adverse selection," Insurance: Mathematics and Economics, Elsevier, vol. 91(C), pages 155-165.

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    More about this item

    Keywords

    Bowley solution; Stackelberg equilibria; Equilibrium reinsurance strategy; Pricing density; General premium principle; Distortion risk measure; Tail Value-at-Risk; Value-at-Risk;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • G22 - Financial Economics - - Financial Institutions and Services - - - Insurance; Insurance Companies; Actuarial Studies
    • G32 - Financial Economics - - Corporate Finance and Governance - - - Financing Policy; Financial Risk and Risk Management; Capital and Ownership Structure; Value of Firms; Goodwill

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