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Irreversible Reinsurance: Minimization of Capital Injections in Presence of a Fixed Cost

Author

Listed:
  • Federico, Salvatore

    (Center for Mathematical Economics, Bielefeld University)

  • Ferrari, Giorgio

    (Center for Mathematical Economics, Bielefeld University)

  • Torrente, Maria Laura

    (Center for Mathematical Economics, Bielefeld University)

Abstract

We propose a model in which, in exchange to the payment of a fixed transaction cost, an insurance company can choose the retention level as well as the time at which subscribing a perpetual reinsurance contract. The surplus process of the insurance company evolves according to the diffusive approximation of the Cramér-Lundberg model, claims arrive at a fixed constant rate, and the distribution of their sizes is general. Furthermore, we do not specify any specific functional form of the retention level. The aim of the company is to take actions in order to minimize the sum of the expected value of the total discounted flow of capital injections needed to avoid bankruptcy and of the fixed activation cost of the reinsurance contract. We provide an explicit solution to this problem, which involves the resolution of a static nonlinear optimization problem and of an optimal stopping problem for a reflected diffusion. We then illustrate the theoretical results in the case of proportional and excess-of-loss reinsurance, by providing a numerical study of the dependency of the optimal solution with respect to the model’s parameters.

Suggested Citation

  • Federico, Salvatore & Ferrari, Giorgio & Torrente, Maria Laura, 2023. "Irreversible Reinsurance: Minimization of Capital Injections in Presence of a Fixed Cost," Center for Mathematical Economics Working Papers 682, Center for Mathematical Economics, Bielefeld University.
    • Handle: RePEc:bie:wpaper:682
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    File URL: https://pub.uni-bielefeld.de/download/2983417/2983418
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    References listed on IDEAS

    as
    1. Julia Eisenberg & Hanspeter Schmidli, 2011. "Minimising expected discounted capital injections by reinsurance in a classical risk model," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2011(3), pages 155-176.
    2. Eisenberg, Julia & Fabrykowski, Lukas & Schmeck, Maren Diane, 2021. "Optimal Surplus-dependent Reinsurance under Regime-Switching in a Brownian Risk Model," Center for Mathematical Economics Working Papers 648, Center for Mathematical Economics, Bielefeld University.
    3. Yan, Tingjin & Park, Kyunghyun & Wong, Hoi Ying, 2022. "Irreversible reinsurance: A singular control approach," Insurance: Mathematics and Economics, Elsevier, vol. 107(C), pages 326-348.
    4. Hans Gerber & Elias Shiu, 1998. "On the Time Value of Ruin," North American Actuarial Journal, Taylor & Francis Journals, vol. 2(1), pages 48-72.
    5. Zhu, Jinxia & Yang, Hailiang, 2016. "Optimal capital injection and dividend distribution for growth restricted diffusion models with bankruptcy," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 259-271.
    6. Li, Peng & Zhou, Ming & Yao, Dingjun, 2022. "Optimal time for the excess of loss reinsurance with fixed costs," International Review of Economics & Finance, Elsevier, vol. 79(C), pages 466-475.
    7. Dickson, David C.M. & Waters, Howard R., 2004. "Some Optimal Dividends Problems," ASTIN Bulletin, Cambridge University Press, vol. 34(1), pages 49-74, May.
    8. Julia Eisenberg & Lukas Fabrykowski & Maren Diane Schmeck, 2021. "Optimal Surplus-Dependent Reinsurance under Regime-Switching in a Brownian Risk Model," Risks, MDPI, vol. 9(4), pages 1-25, April.
    9. Matteo Brachetta & Claudia Ceci, 2021. "Optimal Reinsurance Problem under Fixed Cost and Exponential Preferences," Mathematics, MDPI, vol. 9(4), pages 1-20, February.
    10. Matteo Brachetta & Claudia Ceci, 2021. "Optimal reinsurance problem under fixed cost and exponential preferences," Papers 2101.04975, arXiv.org.
    11. Giorgio Ferrari & Patrick Schuhmann, 2018. "An Optimal Dividend Problem with Capital Injections over a Finite Horizon," Papers 1804.04870, arXiv.org, revised May 2019.
    12. He, Yue & Kawai, Reiichiro & Shimizu, Yasutaka & Yamazaki, Kazutoshi, 2023. "The Gerber-Shiu discounted penalty function: A review from practical perspectives," Insurance: Mathematics and Economics, Elsevier, vol. 109(C), pages 1-28.
    13. Schmidli, Hanspeter, 2016. "On capital injections and dividends with tax in a classical risk model," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 138-144.
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    Cited by:

    1. Zongxia Liang & Xiaodong Luo, 2024. "Stackelberg reinsurance and premium decisions with MV criterion and irreversibility," Papers 2402.11580, arXiv.org.

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    Keywords

    reinsurance; fixed cost; capital injections; diffusive risk model; optimal stopping;
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