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Minimizing the ruin probability through capital injections

Author

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  • Nie, Ciyu
  • Dickson, David C. M.
  • Li, Shuanming

Abstract

We consider an insurer who has a fixed amount of funds allocated as the initial surplus for a risk portfolio, so that the probability of ultimate ruin for this portfolio is at a known level. We consider the question of whether the insurer can reduce this ultimate ruin probability by allocating part of the initial funds to the purchase of a reinsurance contract. This reinsurance contract would restore the insurer's surplus to a positive level k every time the surplus fell between 0 and k. The insurer's objective is to choose the level k that minimizes the ultimate ruin probability. Using different examples of reinsurance premium calculation and claim size distribution we show that this objective can be achieved, often with a substantial reduction in the ultimate ruin probability from the situation when there is no reinsurance. We also show that by purchasing reinsurance the insurer can release funds for other purposes without altering its ultimate ruin probability.

Suggested Citation

  • Nie, Ciyu & Dickson, David C. M. & Li, Shuanming, 2011. "Minimizing the ruin probability through capital injections," Annals of Actuarial Science, Cambridge University Press, vol. 5(2), pages 195-209, September.
    • Handle: RePEc:cup:anacsi:v:5:y:2011:i:02:p:195-209_00
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    Cited by:

    1. Avram, F. & Badescu, A.L. & Pistorius, M.R. & Rabehasaina, L., 2016. "On a class of dependent Sparre Andersen risk models and a bailout application," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 27-39.
    2. Muhsin Tamturk & Dominic Cortis & Mark Farrell, 2020. "Examining the Effects of Gradual Catastrophes on Capital Modelling and the Solvency of Insurers: The Case of COVID-19," Risks, MDPI, vol. 8(4), pages 1-13, December.
    3. Teng, Ye & Zhang, Zhimin, 2023. "On a time-changed Lévy risk model with capital injections and periodic observation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 214(C), pages 290-314.
    4. 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.
    5. Tamturk, Muhsin & Utev, Sergey, 2018. "Ruin probability via Quantum Mechanics Approach," Insurance: Mathematics and Economics, Elsevier, vol. 79(C), pages 69-74.
    6. Ramsden, Lewis & Papaioannou, Apostolos D., 2019. "Ruin probabilities under capital constraints," Insurance: Mathematics and Economics, Elsevier, vol. 88(C), pages 273-282.
    7. Julia Eisenberg & Paul Kruhner, 2016. "The Impact of Negative Interest Rates on Optimal Capital Injections," Papers 1612.06654, arXiv.org.
    8. A. S. Dibu & M. J. Jacob & Apostolos D. Papaioannou & Lewis Ramsden, 2021. "Delayed Capital Injections for a Risk Process with Markovian Arrivals," Methodology and Computing in Applied Probability, Springer, vol. 23(3), pages 1057-1076, September.
    9. Eisenberg, Julia & Krühner, Paul, 2018. "The impact of negative interest rates on optimal capital injections," Insurance: Mathematics and Economics, Elsevier, vol. 82(C), pages 1-10.
    10. Ramsden, Lewis & Papaioannou, Apostolos D., 2019. "On the time to ruin for a dependent delayed capital injection risk model," Applied Mathematics and Computation, Elsevier, vol. 352(C), pages 119-135.
    11. Başak Bulut Karageyik & Şule Şahin, 2017. "Determination of the Optimal Retention Level Based on Different Measures," JRFM, MDPI, vol. 10(1), pages 1-21, January.
    12. Hansjoerg Albrecher & Jevgenijs Ivanovs, 2013. "Power identities for L\'evy risk models under taxation and capital injections," Papers 1310.3052, arXiv.org, revised Mar 2014.
    13. Muhsin Tamturk & Sergey Utev, 2019. "Optimal Reinsurance via Dirac-Feynman Approach," Methodology and Computing in Applied Probability, Springer, vol. 21(2), pages 647-659, June.
    14. Muhsin Tamturk, 2023. "Quantum Computing in Insurance Capital Modelling," Mathematics, MDPI, vol. 11(3), pages 1-13, January.
    15. Abouzar Bazyari, 2023. "On the Ruin Probabilities in a Discrete Time Insurance Risk Process with Capital Injections and Reinsurance," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(2), pages 1623-1650, August.

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