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Optimal Reinsurance Problem under Fixed Cost and Exponential Preferences

Author

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  • Matteo Brachetta

    (Department of Mathematics, Politecnico of Milan, Piazza Leonardo da Vinci, 32, 20133 Milano, Italy
    These authors contributed equally to this work.)

  • Claudia Ceci

    (Department of Economics, University of Chieti-Pescara, Viale Pindaro, 42, 65127 Pescara, Italy
    These authors contributed equally to this work.)

Abstract

We investigate an optimal reinsurance problem for an insurance company taking into account subscription costs: that is, a constant fixed cost is paid when the reinsurance contract is signed. Differently from the classical reinsurance problem, where the insurer has to choose an optimal retention level according to some given criterion, in this paper, the insurer needs to optimally choose both the starting time of the reinsurance contract and the retention level to apply. The criterion is the maximization of the insurer’s expected utility of terminal wealth. This leads to a mixed optimal control/optimal stopping time problem, which is solved by a two-step procedure: first considering the pure-reinsurance stochastic control problem and next discussing a time-inhomogeneous optimal stopping problem with discontinuous reward. Using the classical Cramér–Lundberg approximation risk model, we prove that the optimal strategy is deterministic and depends on the model parameters. In particular, we show that there exists a maximum fixed cost that the insurer is willing to pay for the contract activation. Finally, we provide some economical interpretations and numerical simulations.

Suggested Citation

  • Matteo Brachetta & Claudia Ceci, 2021. "Optimal Reinsurance Problem under Fixed Cost and Exponential Preferences," Mathematics, MDPI, vol. 9(4), pages 1-20, February.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:4:p:295-:d:492139
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    References listed on IDEAS

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    1. Li, Peng & Zhou, Ming & Yin, Chuancun, 2015. "Optimal reinsurance with both proportional and fixed costs," Statistics & Probability Letters, Elsevier, vol. 106(C), pages 134-141.
    2. Matteo Brachetta & Claudia Ceci, 2019. "Optimal excess-of-loss reinsurance for stochastic factor risk models," Papers 1904.05422, arXiv.org.
    3. Matteo Brachetta & Claudia Ceci, 2019. "Optimal Excess-of-Loss Reinsurance for Stochastic Factor Risk Models," Risks, MDPI, vol. 7(2), pages 1-23, May.
    4. Egami, Masahiko & Young, Virginia R., 2009. "Optimal reinsurance strategy under fixed cost and delay," Stochastic Processes and their Applications, Elsevier, vol. 119(3), pages 1015-1034, March.
    5. Yann Braouezec, 2019. "Public versus private insurance system with (and without) transaction costs: optimal segmentation policy of an informed monopolist," Applied Economics, Taylor & Francis Journals, vol. 51(18), pages 1907-1928, April.
    6. Irgens, Christian & Paulsen, Jostein, 2004. "Optimal control of risk exposure, reinsurance and investments for insurance portfolios," Insurance: Mathematics and Economics, Elsevier, vol. 35(1), pages 21-51, August.
    7. Brachetta, M. & Ceci, C., 2019. "Optimal proportional reinsurance and investment for stochastic factor models," Insurance: Mathematics and Economics, Elsevier, vol. 87(C), pages 15-33.
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    Cited by:

    1. 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.
    2. 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.

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