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Optimal design of reinsurance contracts under adverse selection with a continuum of types

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  • Ka Chun Cheung
  • Sheung Chi Phillip Yam
  • Fei Lung Yuen
  • Yiying Zhang

Abstract

In this paper, we use the principal-agent model to study the optimal contract design in a monopolistic reinsurance market under adverse selection with a continuum of types of insurers. Instead of adopting the classical expected utility paradigm, we model the risk preference of each insurer (agent) by his Value-at-Risk at his own chosen risk tolerance level. Under information asymmetry, the reinsurer (principal) aims to maximize her expected profit by designing an optimal menu of reinsurance contracts for a continuum of insurers with hidden characteristics. The optimization problem is constrained by agents' individual compatibility and rationality constraints. By making use of the notion of indirect utility functions, the problem is completely solved for the following three commonly encountered classes of reinsurance indemnities: stop-loss, quota-share, and change loss. Some numerical examples are provided as illustrations.

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  • Ka Chun Cheung & Sheung Chi Phillip Yam & Fei Lung Yuen & Yiying Zhang, 2025. "Optimal design of reinsurance contracts under adverse selection with a continuum of types," Papers 2504.17468, arXiv.org.
    • Handle: RePEc:arx:papers:2504.17468
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    References listed on IDEAS

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    1. Liang, Zhihang & Zou, Jushen & Jiang, Wenjun, 2022. "Revisiting the optimal insurance design under adverse selection: Distortion risk measures and tail-risk overestimation," Insurance: Mathematics and Economics, Elsevier, vol. 104(C), pages 200-221.
    2. Armstrong, Mark, 1996. "Multiproduct Nonlinear Pricing," Econometrica, Econometric Society, vol. 64(1), pages 51-75, January.
    3. K. C. Cheung & S. C. P. Yam & F. L. Yuen, 2019. "Reinsurance contract design with adverse selection," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2019(9), pages 784-798, October.
    4. 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.
    5. Powell, David & Goldman, Dana, 2021. "Disentangling moral hazard and adverse selection in private health insurance," Journal of Econometrics, Elsevier, vol. 222(1), pages 141-160.
    6. Joseph E. Stiglitz, 1977. "Monopoly, Non-linear Pricing and Imperfect Information: The Insurance Market," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 44(3), pages 407-430.
    7. 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.
    8. Ailing Gu & Frederi G. Viens & Yang Shen, 2020. "Optimal excess-of-loss reinsurance contract with ambiguity aversion in the principal-agent model," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2020(4), pages 342-375, April.
    9. Zhang, Chong & Yu, Man & Chen, Jian, 2022. "Signaling quality with return insurance: Theory and empirical evidence," Other publications TiSEM 184da313-a89e-4a81-9f23-c, Tilburg University, School of Economics and Management.
    10. Yaari, Menahem E, 1987. "The Dual Theory of Choice under Risk," Econometrica, Econometric Society, vol. 55(1), pages 95-115, January.
    11. Boonen, Tim J. & Zhang, Yiying, 2021. "Optimal Reinsurance Design With Distortion Risk Measures And Asymmetric Information," ASTIN Bulletin, Cambridge University Press, vol. 51(2), pages 607-629, May.
    12. Amy Finkelstein & James Poterba, 2004. "Adverse Selection in Insurance Markets: Policyholder Evidence from the U.K. Annuity Market," Journal of Political Economy, University of Chicago Press, vol. 112(1), pages 183-208, February.
    13. J. A. Mirrlees, 1971. "An Exploration in the Theory of Optimum Income Taxation," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 38(2), pages 175-208.
    14. Alma Cohen & Peter Siegelman, 2010. "Testing for Adverse Selection in Insurance Markets," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 77(1), pages 39-84, March.
    15. Jean-Charles Rochet & Philippe Chone, 1998. "Ironing, Sweeping, and Multidimensional Screening," Econometrica, Econometric Society, vol. 66(4), pages 783-826, July.
    16. Alex Gershkov & Benny Moldovanu & Philipp Strack & Mengxi Zhang, 2023. "Optimal Insurance: Dual Utility, Random losses and Adverse Selection," CRC TR 224 Discussion Paper Series crctr224_2023_399, University of Bonn and University of Mannheim, Germany.
    17. 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.
    18. Chi, Yichun & Zhuang, Sheng Chao, 2020. "Optimal insurance with belief heterogeneity and incentive compatibility," Insurance: Mathematics and Economics, Elsevier, vol. 92(C), pages 104-114.
    19. Michael Landsberger & Isaac Meilijson, 1999. "A general model of insurance under adverse selection," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 14(2), pages 331-352.
    20. Talmor, Eli, 1981. "Asymmetric Information, Signaling, and Optimal Corporate Financial Decisions," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 16(4), pages 413-435, November.
    21. Kaluszka, Marek, 2001. "Optimal reinsurance under mean-variance premium principles," Insurance: Mathematics and Economics, Elsevier, vol. 28(1), pages 61-67, February.
    22. Hu, Xiang & Yang, Hailiang & Zhang, Lianzeng, 2015. "Optimal retention for a stop-loss reinsurance with incomplete information," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 15-21.
    23. Cai, Jun & Tan, Ken Seng, 2007. "Optimal Retention for a Stop-loss Reinsurance Under the VaR and CTE Risk Measures," ASTIN Bulletin, Cambridge University Press, vol. 37(1), pages 93-112, May.
    24. Mario Ghossoub & Bin Li & Benxuan Shi, 2025. "Optimal Insurance in a Monopoly: Dual Utilities with Hidden Risk Attitudes," Papers 2504.01095, arXiv.org.
    25. Young, Virginia R., 1999. "Optimal insurance under Wang's premium principle," Insurance: Mathematics and Economics, Elsevier, vol. 25(2), pages 109-122, November.
    26. Alex Gershkov & Benny Moldovanu & Philipp Strack & Mengxi Zhang, 2023. "Optimal Insurance: Dual Utility, Random Losses and Adverse Selection," ECONtribute Discussion Papers Series 242, University of Bonn and University of Cologne, Germany.
    27. Guesnerie, Roger & Laffont, Jean-Jacques, 1984. "A complete solution to a class of principal-agent problems with an application to the control of a self-managed firm," Journal of Public Economics, Elsevier, vol. 25(3), pages 329-369, December.
    28. Alex Gershkov & Benny Moldovanu & Philipp Strack & Mengxi Zhang, 2023. "Optimal Insurance: Dual Utility, Random Losses, and Adverse Selection," American Economic Review, American Economic Association, vol. 113(10), pages 2581-2614, October.
    29. Neudeck, Werner & Podczeck, Konrad, 1996. "Adverse selection and regulation in health insurance markets," Journal of Health Economics, Elsevier, vol. 15(4), pages 387-408, August.
    30. repec:dau:papers:123456789/13348 is not listed on IDEAS
    31. Vitor Farinha Luz & Piero Gottardi & Humberto Moreira, 2023. "Risk Classification in Insurance Markets with Risk and Preference Heterogeneity," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 90(6), pages 3022-3082.
    32. Mussa, Michael & Rosen, Sherwin, 1978. "Monopoly and product quality," Journal of Economic Theory, Elsevier, vol. 18(2), pages 301-317, August.
    33. Macho-Stadler, Ines & Perez-Castrillo, J. David, 2001. "An Introduction to the Economics of Information: Incentives and Contracts," OUP Catalogue, Oxford University Press, edition 2, number 9780199243259, Decembrie.
    34. Borch, Karl, 1969. "The optimal reinsurance treaty," ASTIN Bulletin, Cambridge University Press, vol. 5(2), pages 293-297, May.
    35. Ning Wang & Tak Kuen Siu & Kun Fan, 2024. "Robust reinsurance and investment strategies under principal–agent framework," Annals of Operations Research, Springer, vol. 336(1), pages 981-1011, May.
    36. Dosis, Anastasios, 2018. "On signalling and screening in markets with asymmetric information," Journal of Mathematical Economics, Elsevier, vol. 75(C), pages 140-149.
    37. Chong Zhang & Man Yu & Jian Chen, 2022. "Signaling Quality with Return Insurance: Theory and Empirical Evidence," Management Science, INFORMS, vol. 68(8), pages 5847-5867, August.
    38. 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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