IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2203.04053.html
   My bibliography  Save this paper

Risk sharing in equity-linked insurance products: Stackelberg equilibrium between an insurer and a reinsurer

Author

Listed:
  • Yevhen Havrylenko
  • Maria Hinken
  • Rudi Zagst

Abstract

We study the optimal investment-reinsurance problem in the context of equity-linked insurance products. Such products often have a capital guarantee, which can motivate insurers to purchase reinsurance. Since a reinsurance contract implies an interaction between the insurer and the reinsurer, we model the optimization problem as a Stackelberg game. The reinsurer is the leader in the game and maximizes its expected utility by selecting its optimal investment strategy and a safety loading in the reinsurance contract it offers to the insurer. The reinsurer can assess how the insurer will rationally react on each action of the reinsurer. The insurance company is the follower and maximizes its expected utility by choosing its investment strategy and the amount of reinsurance the company purchases at the price offered by the reinsurer. In this game, we derive the Stackelberg equilibrium for general utility functions. For power utility functions, we calculate the equilibrium explicitly and find that the reinsurer selects the largest reinsurance premium such that the insurer may still buy the maximal amount of reinsurance. Since in the equilibrium the insurer is indifferent in the amount of reinsurance, in practice, the reinsurer should consider charging a smaller reinsurance premium than the equilibrium one. Therefore, we propose several criteria for choosing such a discount rate and investigate its wealth-equivalent impact on the expected utility of each party.

Suggested Citation

  • Yevhen Havrylenko & Maria Hinken & Rudi Zagst, 2022. "Risk sharing in equity-linked insurance products: Stackelberg equilibrium between an insurer and a reinsurer," Papers 2203.04053, arXiv.org, revised Oct 2023.
    • Handle: RePEc:arx:papers:2203.04053
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2203.04053
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Larsen, Linda Sandris & Munk, Claus, 2012. "The costs of suboptimal dynamic asset allocation: General results and applications to interest rate risk, stock volatility risk, and growth/value tilts," Journal of Economic Dynamics and Control, Elsevier, vol. 36(2), pages 266-293.
    2. Desmettre, Sascha & Seifried, Frank Thomas, 2016. "Optimal asset allocation with fixed-term securities," Journal of Economic Dynamics and Control, Elsevier, vol. 66(C), pages 1-19.
    3. Martin J. Osborne & Ariel Rubinstein, 1994. "A Course in Game Theory," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262650401, December.
    4. Escobar-Anel, Marcos & Havrylenko, Yevhen & Kschonnek, Michel & Zagst, Rudi, 2022. "Decrease of capital guarantees in life insurance products: Can reinsurance stop it?," Insurance: Mathematics and Economics, Elsevier, vol. 105(C), pages 14-40.
    5. Merton, Robert C, 1969. "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case," The Review of Economics and Statistics, MIT Press, vol. 51(3), pages 247-257, August.
    6. Chen, Lv & Shen, Yang, 2019. "Stochastic Stackelberg differential reinsurance games under time-inconsistent mean–variance framework," Insurance: Mathematics and Economics, Elsevier, vol. 88(C), pages 120-137.
    7. Chen, Lv & Shen, Yang & Su, Jianxi, 2020. "A continuous-time theory of reinsurance chains," Insurance: Mathematics and Economics, Elsevier, vol. 95(C), pages 129-146.
    8. Yanfei Bai & Zhongbao Zhou & Helu Xiao & Rui Gao & Feimin Zhong, 2019. "A hybrid stochastic differential reinsurance and investment game with bounded memory," Papers 1910.09834, arXiv.org.
    9. Yanfei Bai & Zhongbao Zhou & Helu Xiao & Rui Gao & Feimin Zhong, 2021. "A stochastic Stackelberg differential reinsurance and investment game with delay in a defaultable market," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 94(3), pages 341-381, December.
    10. Søren Asmussen & Bent Jesper Christensen & Julie Thøgersen, 2019. "Stackelberg Equilibrium Premium Strategies for Push-Pull Competition in a Non-Life Insurance Market with Product Differentiation," Risks, MDPI, vol. 7(2), pages 1-23, May.
    Full references (including those not matched with items on IDEAS)

    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. Marius Ascheberg & Nicole Branger & Holger Kraft & Frank Thomas Seifried, 2016. "When do jumps matter for portfolio optimization?," Quantitative Finance, Taylor & Francis Journals, vol. 16(8), pages 1297-1311, August.
    2. Marcos Escobar-Anel & Michel Kschonnek & Rudi Zagst, 2022. "Portfolio optimization: not necessarily concave utility and constraints on wealth and allocation," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 95(1), pages 101-140, February.
    3. Bart Diris & Franz Palm & Peter Schotman, 2015. "Long-Term Strategic Asset Allocation: An Out-of-Sample Evaluation," Management Science, INFORMS, vol. 61(9), pages 2185-2202, September.
    4. 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.
    5. Ning Bin & Huainian Zhu & Chengke Zhang, 2023. "Stochastic Differential Games on Optimal Investment and Reinsurance Strategy with Delay Under the CEV Model," Methodology and Computing in Applied Probability, Springer, vol. 25(2), pages 1-27, June.
    6. Alfonso A. Irarrazabal & Lin Ma & Juan Carlos Parra-Alvarez, 2023. "Optimal asset allocation for commodity sovereign wealth funds," Quantitative Finance, Taylor & Francis Journals, vol. 23(3), pages 471-495, March.
    7. Lu Yang & Chengke Zhang & Huainian Zhu, 2022. "Robust Stochastic Stackelberg Differential Reinsurance and Investment Games for an Insurer and a Reinsurer with Delay," Methodology and Computing in Applied Probability, Springer, vol. 24(1), pages 361-384, March.
    8. Yanfei Bai & Zhongbao Zhou & Helu Xiao & Rui Gao & Feimin Zhong, 2019. "A hybrid stochastic differential reinsurance and investment game with bounded memory," Papers 1910.09834, arXiv.org.
    9. Simon Ellersgaard & Martin Tegnér, 2018. "Stochastic volatility for utility maximizers — A martingale approach," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 5(01), pages 1-39, March.
    10. 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.
    11. Bai, Yanfei & Zhou, Zhongbao & Xiao, Helu & Gao, Rui & Zhong, Feimin, 2022. "A hybrid stochastic differential reinsurance and investment game with bounded memory," European Journal of Operational Research, Elsevier, vol. 296(2), pages 717-737.
    12. Wang, Hang & Hu, Zhijun, 2020. "Optimal consumption and portfolio decision with stochastic covariance in incomplete markets," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).
    13. Marcos Escobar-Anel & Harold A. Moreno-Franco, 2019. "Dynamic portfolio strategies under a fully correlated jump-diffusion process," Annals of Finance, Springer, vol. 15(3), pages 421-453, September.
    14. Ascheberg, Marius & Branger, Nicole & Kraft, Holger, 2013. "When do jumps matter for portfolio optimization?," SAFE Working Paper Series 16, Leibniz Institute for Financial Research SAFE.
    15. Munk, Claus, 2020. "A mean-variance benchmark for household portfolios over the life cycle," Journal of Banking & Finance, Elsevier, vol. 116(C).
    16. Masuda, Kazuto, 2024. "The Special Theory of Employment, Exchange Rate, and Money With the Focus on Inflation and Technological Progress," SocArXiv nxshd, Center for Open Science.
    17. Kar, Anirban & Mitra, Manipushpak & Mutuswami, Suresh, 2009. "On the coincidence of the prenucleolus and the Shapley value," Mathematical Social Sciences, Elsevier, vol. 57(1), pages 16-25, January.
    18. Gilli, Mario, 1999. "On Non-Nash Equilibria," Games and Economic Behavior, Elsevier, vol. 27(2), pages 184-203, May.
    19. Monica Paiella, 2007. "The Forgone Gains of Incomplete Portfolios," The Review of Financial Studies, Society for Financial Studies, vol. 20(5), pages 1623-1646, 2007 13.
    20. Francisco Gomes & Alexander Michaelides, 2003. "Portfolio Choice With Internal Habit Formation: A Life-Cycle Model With Uninsurable Labor Income Risk," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 6(4), pages 729-766, October.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    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:arx:papers:2203.04053. 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: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    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.