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Minimizing the probability of ruin: Optimal per-loss reinsurance

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  • Liang, Xiaoqing
  • Young, Virginia R.

Abstract

We compute the optimal investment and reinsurance strategy for an insurance company that wishes to minimize its probability of ruin, when the risk process follows a compound Poisson process (CPP) and reinsurance is priced via the expected-value premium principle. We consider per-loss optimal reinsurance for the CPP after first determining optimal reinsurance for the diffusion that approximates this CPP. For both the CPP claim process and its diffusion approximation, the financial market in which the insurer invests follows the Black–Scholes model, namely, a single riskless asset that earns interest at a constant rate and a single risky asset whose price process follows a geometric Brownian motion. Under minimal assumptions about admissible forms of reinsurance, we show that optimal per-loss reinsurance is excess-of-loss. Therefore, our result extends the work of the optimality of excess-of-loss reinsurance to the problem of minimizing the probability of ruin.

Suggested Citation

  • Liang, Xiaoqing & Young, Virginia R., 2018. "Minimizing the probability of ruin: Optimal per-loss reinsurance," Insurance: Mathematics and Economics, Elsevier, vol. 82(C), pages 181-190.
  • Handle: RePEc:eee:insuma:v:82:y:2018:i:c:p:181-190
    DOI: 10.1016/j.insmatheco.2018.07.005
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    References listed on IDEAS

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    9. S. David Promislow & Virginia Young, 2005. "Minimizing the Probability of Ruin When Claims Follow Brownian Motion with Drift," North American Actuarial Journal, Taylor & Francis Journals, vol. 9(3), pages 110-128.
    10. 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.
    11. Van Heerwaarden, A. E. & Kaas, R. & Goovaerts, M. J., 1989. "Optimal reinsurance in relation to ordering of risks," Insurance: Mathematics and Economics, Elsevier, vol. 8(1), pages 11-17, March.
    12. Hipp, Christian & Taksar, Michael, 2010. "Optimal non-proportional reinsurance control," Insurance: Mathematics and Economics, Elsevier, vol. 47(2), pages 246-254, October.
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    Citations

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    Cited by:

    1. Guohui Guan & Zongxia Liang & Yilun Song, 2022. "A Stackelberg reinsurance-investment game under $\alpha$-maxmin mean-variance criterion and stochastic volatility," Papers 2212.14327, arXiv.org.
    2. Fudong Wang & Zhibin Liang, 2022. "Optimal Per-Loss Reinsurance for a Risk Model with a Thinning-Dependence Structure," Mathematics, MDPI, vol. 10(23), pages 1-23, December.
    3. Yuan, Yu & Han, Xia & Liang, Zhibin & Yuen, Kam Chuen, 2023. "Optimal reinsurance-investment strategy with thinning dependence and delay factors under mean-variance framework," European Journal of Operational Research, Elsevier, vol. 311(2), pages 581-595.
    4. Yu Yuan & Zhibin Liang & Xia Han, 2022. "Minimizing the penalized probability of drawdown for a general insurance company under ambiguity aversion," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 96(2), pages 259-290, October.
    5. Meng, Hui & Liao, Pu & Siu, Tak Kuen, 2019. "Continuous-time optimal reinsurance strategy with nontrivial curved structures," Applied Mathematics and Computation, Elsevier, vol. 363(C), pages 1-1.
    6. Kalfin & Sukono & Sudradjat Supian & Mustafa Mamat, 2023. "Model for Determining Insurance Premiums Taking into Account the Rate of Economic Growth and Cross-Subsidies in Providing Natural Disaster Management Funds in Indonesia," Sustainability, MDPI, vol. 15(24), pages 1-15, December.
    7. Bohan Li & Junyi Guo, 2021. "Optimal Investment and Reinsurance Under the Gamma Process," Methodology and Computing in Applied Probability, Springer, vol. 23(3), pages 893-923, September.
    8. Linlin Tian & Lihua Bai, 2020. "Minimizing the Ruin Probability under the Sparre Andersen Model," Papers 2004.08124, arXiv.org.
    9. Andreas Karathanasopoulos & Chia Chun Lo & Xiaorong Ma & Zhenjiang Qin, 2021. "Maintaining cost and ruin probability," Review of Quantitative Finance and Accounting, Springer, vol. 57(2), pages 759-793, August.
    10. Liang, Xiaoqing & Liang, Zhibin & Young, Virginia R., 2020. "Optimal reinsurance under the mean–variance premium principle to minimize the probability of ruin," Insurance: Mathematics and Economics, Elsevier, vol. 92(C), pages 128-146.
    11. Meng, Hui & Wei, Li & Zhou, Ming, 2023. "Multiple per-claim reinsurance based on maximizing the Lundberg exponent," Insurance: Mathematics and Economics, Elsevier, vol. 112(C), pages 33-47.

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    More about this item

    Keywords

    Probability of ruin; Optimal reinsurance; Stochastic control; Compound Poisson; Diffusion approximation;
    All these keywords.

    JEL classification:

    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics
    • 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
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

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