IDEAS home Printed from https://ideas.repec.org/a/eee/insuma/v113y2023icp199-214.html
   My bibliography  Save this article

Multi-constrained optimal reinsurance model from the duality perspectives

Author

Listed:
  • Cheung, Ka Chun
  • He, Wanting
  • Wang, He

Abstract

In the presence of multiple constraints such as the risk tolerance constraint and the budget constraint, many extensively studied (Pareto-)optimal reinsurance problems based on general distortion risk measures become technically challenging and have only been solved using ad hoc methods for certain special cases. In this paper, we extend the method developed in Lo (2017a) by proposing a generalized Neyman-Pearson framework to identify the optimal forms of the solutions. We then develop a dual formulation and show that the infinite-dimensional constrained optimization problems can be reduced to finite-dimensional unconstrained ones. With the support of the Nelder-Mead algorithm, we are able to obtain optimal solutions efficiently. We illustrate the versatility of our approach by working out several detailed numerical examples, many of which in the literature were only partially resolved.

Suggested Citation

  • Cheung, Ka Chun & He, Wanting & Wang, He, 2023. "Multi-constrained optimal reinsurance model from the duality perspectives," Insurance: Mathematics and Economics, Elsevier, vol. 113(C), pages 199-214.
    • Handle: RePEc:eee:insuma:v:113:y:2023:i:c:p:199-214
    DOI: 10.1016/j.insmatheco.2023.08.003
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167668723000756
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.insmatheco.2023.08.003?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    2. Deprez, Olivier & Gerber, Hans U., 1985. "On convex principles of premium calculation," Insurance: Mathematics and Economics, Elsevier, vol. 4(3), pages 179-189, July.
    3. 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.
    4. J. David Cummins & Olivier Mahul, 2004. "The Demand for Insurance With an Upper Limit on Coverage," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 71(2), pages 253-264, June.
    5. Tan, Ken Seng & Weng, Chengguo & Zhang, Yi, 2011. "Optimality of general reinsurance contracts under CTE risk measure," Insurance: Mathematics and Economics, Elsevier, vol. 49(2), pages 175-187, September.
    6. Boonen, Tim J. & Jiang, Wenjun, 2022. "A marginal indemnity function approach to optimal reinsurance under the Vajda condition," European Journal of Operational Research, Elsevier, vol. 303(2), pages 928-944.
    7. Russell R. Barton & John S. Ivey, Jr., 1996. "Nelder-Mead Simplex Modifications for Simulation Optimization," Management Science, INFORMS, vol. 42(7), pages 954-973, July.
    8. 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.
    9. Assa, Hirbod, 2015. "On optimal reinsurance policy with distortion risk measures and premiums," Insurance: Mathematics and Economics, Elsevier, vol. 61(C), pages 70-75.
    10. Gerber, Hans U., 1978. "Pareto-Optimal Risk Exchanges and Related Decision Problems," ASTIN Bulletin, Cambridge University Press, vol. 10(1), pages 25-33, May.
    11. Dhaene, J. & Denuit, M. & Goovaerts, M. J. & Kaas, R. & Vyncke, D., 2002. "The concept of comonotonicity in actuarial science and finance: theory," Insurance: Mathematics and Economics, Elsevier, vol. 31(1), pages 3-33, August.
    12. Boonen, Tim J. & Ghossoub, Mario, 2021. "Optimal reinsurance with multiple reinsurers: Distortion risk measures, distortion premium principles, and heterogeneous beliefs," Insurance: Mathematics and Economics, Elsevier, vol. 101(PA), pages 23-37.
    13. 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.
    14. A. Y. Golubin, 2006. "Pareto‐Optimal Insurance Policies in the Models with a Premium Based on the Actuarial Value," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 73(3), pages 469-487, September.
    15. Lo, Ambrose, 2017. "A Neyman-Pearson Perspective On Optimal Reinsurance With Constraints," ASTIN Bulletin, Cambridge University Press, vol. 47(2), pages 467-499, May.
    16. Zhuang, Sheng Chao & Weng, Chengguo & Tan, Ken Seng & Assa, Hirbod, 2016. "Marginal Indemnification Function formulation for optimal reinsurance," Insurance: Mathematics and Economics, Elsevier, vol. 67(C), pages 65-76.
    17. Lu, ZhiYi & Meng, LiLi & Wang, Yujin & Shen, Qingjie, 2016. "Optimal reinsurance under VaR and TVaR risk measures in the presence of reinsurer’s risk limit," Insurance: Mathematics and Economics, Elsevier, vol. 68(C), pages 92-100.
    18. Young, Virginia R., 1999. "Optimal insurance under Wang's premium principle," Insurance: Mathematics and Economics, Elsevier, vol. 25(2), pages 109-122, November.
    19. Ka Chun Cheung & Wing Fung Chong & Ambrose Lo, 2019. "Budget-constrained optimal reinsurance design under coherent risk measures," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2019(9), pages 729-751, October.
    20. Zheng, Yanting & Cui, Wei, 2014. "Optimal reinsurance with premium constraint under distortion risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 109-120.
    21. Ghossoub, Mario, 2019. "Budget-constrained optimal insurance with belief heterogeneity," Insurance: Mathematics and Economics, Elsevier, vol. 89(C), pages 79-91.
    22. Asimit, Alexandru V. & Cheung, Ka Chun & Chong, Wing Fung & Hu, Junlei, 2020. "Pareto-optimal insurance contracts with premium budget and minimum charge constraints," Insurance: Mathematics and Economics, Elsevier, vol. 95(C), pages 17-27.
    23. Tim J. Boonen, 2016. "Optimal Reinsurance with Heterogeneous Reference Probabilities," Risks, MDPI, vol. 4(3), pages 1-11, July.
    24. Asimit, Alexandru V. & Badescu, Alexandru M. & Verdonck, Tim, 2013. "Optimal risk transfer under quantile-based risk measurers," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 252-265.
    25. Asimit, Alexandru V. & Badescu, Alexandru M. & Cheung, Ka Chun, 2013. "Optimal reinsurance in the presence of counterparty default risk," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 690-697.
    26. Yuxia Huang & Chuancun Yin, 2018. "A unifying approach to constrained and unconstrained optimal reinsurance," Papers 1807.06892, arXiv.org.
    27. Borch, Karl, 1969. "The optimal reinsurance treaty," ASTIN Bulletin, Cambridge University Press, vol. 5(2), pages 293-297, May.
    28. Chi, Yichun & Tan, Ken Seng, 2011. "Optimal Reinsurance under VaR and CVaR Risk Measures: a Simplified Approach," ASTIN Bulletin, Cambridge University Press, vol. 41(2), pages 487-509, November.
    29. 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.
    30. Jun Cai & Chengguo Weng, 2016. "Optimal reinsurance with expectile," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2016(7), pages 624-645, August.
    31. Ambrose Lo & Zhaofeng Tang, 2019. "Pareto-optimal reinsurance policies in the presence of individual risk constraints," Annals of Operations Research, Springer, vol. 274(1), pages 395-423, March.
    32. Raviv, Artur, 1979. "The Design of an Optimal Insurance Policy," American Economic Review, American Economic Association, vol. 69(1), pages 84-96, March.
    33. Ambrose Lo, 2017. "A unifying approach to risk-measure-based optimal reinsurance problems with practical constraints," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2017(7), pages 584-605, August.
    34. Boonen, Tim J. & De Waegenaere, Anja & Norde, Henk, 2017. "Redistribution of longevity risk: The effect of heterogeneous mortality beliefs," Insurance: Mathematics and Economics, Elsevier, vol. 72(C), pages 175-188.
    35. K.C. Cheung & K.C.J. Sung & S.C.P. Yam & S.P. Yung, 2014. "Optimal reinsurance under general law-invariant risk measures," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2014(1), pages 72-91.
    36. 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.
    37. Boonen, Tim J. & Ghossoub, Mario, 2019. "On the existence of a representative reinsurer under heterogeneous beliefs," Insurance: Mathematics and Economics, Elsevier, vol. 88(C), pages 209-225.
    38. Ka Chun Cheung & Ambrose Lo, 2017. "Characterizations of optimal reinsurance treaties: a cost-benefit approach," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2017(1), pages 1-28, January.
    39. Chelouah, Rachid & Siarry, Patrick, 2005. "A hybrid method combining continuous tabu search and Nelder-Mead simplex algorithms for the global optimization of multiminima functions," European Journal of Operational Research, Elsevier, vol. 161(3), pages 636-654, March.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Chi, Yichun & Zhou, Xun Yu & Zhuang, Sheng Chao, 2024. "Variance insurance contracts," Insurance: Mathematics and Economics, Elsevier, vol. 115(C), pages 62-82.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Asimit, Alexandru V. & Cheung, Ka Chun & Chong, Wing Fung & Hu, Junlei, 2020. "Pareto-optimal insurance contracts with premium budget and minimum charge constraints," Insurance: Mathematics and Economics, Elsevier, vol. 95(C), pages 17-27.
    2. Jiang, Wenjun & Hong, Hanping & Ren, Jiandong, 2021. "Pareto-optimal reinsurance policies with maximal synergy," Insurance: Mathematics and Economics, Elsevier, vol. 96(C), pages 185-198.
    3. Ambrose Lo & Zhaofeng Tang, 2019. "Pareto-optimal reinsurance policies in the presence of individual risk constraints," Annals of Operations Research, Springer, vol. 274(1), pages 395-423, March.
    4. Boonen, Tim J. & Ghossoub, Mario, 2021. "Optimal reinsurance with multiple reinsurers: Distortion risk measures, distortion premium principles, and heterogeneous beliefs," Insurance: Mathematics and Economics, Elsevier, vol. 101(PA), pages 23-37.
    5. Birghila, Corina & Pflug, Georg Ch., 2019. "Optimal XL-insurance under Wasserstein-type ambiguity," Insurance: Mathematics and Economics, Elsevier, vol. 88(C), pages 30-43.
    6. Boonen, Tim J. & Jiang, Wenjun, 2022. "A marginal indemnity function approach to optimal reinsurance under the Vajda condition," European Journal of Operational Research, Elsevier, vol. 303(2), pages 928-944.
    7. Yuxia Huang & Chuancun Yin, 2018. "A unifying approach to constrained and unconstrained optimal reinsurance," Papers 1807.06892, arXiv.org.
    8. Boonen, Tim J. & Jiang, Wenjun, 2024. "Robust insurance design with distortion risk measures," European Journal of Operational Research, Elsevier, vol. 316(2), pages 694-706.
    9. 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.
    10. Wenjun Jiang & Jiandong Ren & Ričardas Zitikis, 2017. "Optimal Reinsurance Policies under the VaR Risk Measure When the Interests of Both the Cedent and the Reinsurer Are Taken into Account," Risks, MDPI, vol. 5(1), pages 1-22, February.
    11. Lu, ZhiYi & Meng, LiLi & Wang, Yujin & Shen, Qingjie, 2016. "Optimal reinsurance under VaR and TVaR risk measures in the presence of reinsurer’s risk limit," Insurance: Mathematics and Economics, Elsevier, vol. 68(C), pages 92-100.
    12. 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.
    13. Mi Chen & Wenyuan Wang & Ruixing Ming, 2016. "Optimal Reinsurance Under General Law-Invariant Convex Risk Measure and TVaR Premium Principle," Risks, MDPI, vol. 4(4), pages 1-12, December.
    14. 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.
    15. Khreshna Syuhada & Arief Hakim & Suci Sari, 2021. "The Combined Stop-Loss and Quota-Share Reinsurance: Conditional Tail Expectation-Based Optimization from the Joint Perspective of Insurer and Reinsurer," Risks, MDPI, vol. 9(7), pages 1-21, July.
    16. 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.
    17. 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.
    18. Boonen, Tim J. & Tan, Ken Seng & Zhuang, Sheng Chao, 2016. "The role of a representative reinsurer in optimal reinsurance," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 196-204.
    19. Asimit, Alexandru V. & Boonen, Tim J. & Chi, Yichun & Chong, Wing Fung, 2021. "Risk sharing with multiple indemnity environments," European Journal of Operational Research, Elsevier, vol. 295(2), pages 587-603.
    20. Chi, Yichun & Liu, Fangda, 2017. "Optimal insurance design in the presence of exclusion clauses," Insurance: Mathematics and Economics, Elsevier, vol. 76(C), pages 185-195.

    More about this item

    Keywords

    Optimal reinsurance; Generalized Neyman-Pearson lemma; Distortion risk measure; Duality;
    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

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:insuma:v:113:y:2023:i:c:p:199-214. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/inca/505554 .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.