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Optimal risk control for a large corporation in the presence of returns on investments

Author

Listed:
  • Bjarne Højgaard

    (Department of Mathematical Sciences, Aalborg University, Fredrik Bajers VEJ 7G, 9220 Aalborg Øst, Denmark)

  • Michael Taksar

    (Department of Appl. Math. and Stat., SUNY - Stony Brook , New York, USA Manuscript)

Abstract

This paper represents a model for the financial valuation of a firm which has control on the dividend payment stream and its risk, as well as potential profit by choosing different business activities among those available to it. Furthermore the company invests its free reserve in an asset, which may or may not contain an element of risk. The company chooses a dividend payment policy and we associate the value of the company with the expected present value of the net dividend distributions (under the optimal policy). One of the examples could be a large corporation such as an insurance company, whose liquid assets in the absence of control and investments fluctuate as a Brownian motion with a constant positive drift and a constant diffusion coefficient. We interpret the diffusion coefficient as risk exposure, while drift is understood as potential profit. At each moment of time, there is an option to reduce risk exposure, simultaneously reducing the potential profit, like using proportional reinsurance with another carrier for an insurance company. The company invests its reserve in a financial asset, whose price evolve as a geometric Brownian motion, with mean rate $r>0$ and diffusion constant $\sigma_P\geq 0$. Thus $\sigma_P=0$ corresponds to investments in a riskless bank account. The objective is to find a policy, consisting of risk control and dividend payment scheme, which maximizes the expected total discounted dividends paid out until the time of bankruptcy. We apply the theory of controlled diffusions to solve the problem. We show that if the discount rate c is less than r, the optimal return function is infinite. If $r=c$ the return function is finite for all $x

Suggested Citation

  • Bjarne Højgaard & Michael Taksar, 2001. "Optimal risk control for a large corporation in the presence of returns on investments," Finance and Stochastics, Springer, vol. 5(4), pages 527-547.
  • Handle: RePEc:spr:finsto:v:5:y:2001:i:4:p:527-547
    Note: received: October 1999; final version received: January 2001
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    Citations

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    Cited by:

    1. Zhu, Jinxia & Chen, Feng, 2013. "Dividend optimization for regime-switching general diffusions," Insurance: Mathematics and Economics, Elsevier, vol. 53(2), pages 439-456.
    2. 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.
    3. Tan, Ken Seng & Wei, Pengyu & Wei, Wei & Zhuang, Sheng Chao, 2020. "Optimal dynamic reinsurance policies under a generalized Denneberg’s absolute deviation principle," European Journal of Operational Research, Elsevier, vol. 282(1), pages 345-362.
    4. Løkka, Arne & Zervos, Mihail, 2008. "Optimal dividend and issuance of equity policies in the presence of proportional costs," Insurance: Mathematics and Economics, Elsevier, vol. 42(3), pages 954-961, June.
    5. Yan, Jia & Liu, John J. & Li, Kevin X., 2008. "Threshold control of mutual insurance with limited commitment," Insurance: Mathematics and Economics, Elsevier, vol. 43(1), pages 108-115, August.
    6. Zhou, Ming & Yuen, Kam C., 2012. "Optimal reinsurance and dividend for a diffusion model with capital injection: Variance premium principle," Economic Modelling, Elsevier, vol. 29(2), pages 198-207.
    7. Zhengjun Jiang & Martijn Pistorius, 2008. "Optimal dividend distribution under Markov-regime switching," Papers 0812.4978, arXiv.org, revised Apr 2011.
    8. Feng, Yang & Zhu, Jinxia & Siu, Tak Kuen, 2021. "Optimal risk exposure and dividend payout policies under model uncertainty," Insurance: Mathematics and Economics, Elsevier, vol. 100(C), pages 1-29.
    9. Liang, Zongxia & Huang, Jianping, 2011. "Optimal dividend and investing control of an insurance company with higher solvency constraints," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 501-511.
    10. He, Lin & Liang, Zongxia, 2008. "Optimal financing and dividend control of the insurance company with proportional reinsurance policy," Insurance: Mathematics and Economics, Elsevier, vol. 42(3), pages 976-983, June.
    11. Feng, Runhuan, 2011. "An operator-based approach to the analysis of ruin-related quantities in jump diffusion risk models," Insurance: Mathematics and Economics, Elsevier, vol. 48(2), pages 304-313, March.
    12. Sotomayor, Luz R. & Cadenillas, Abel, 2011. "Classical and singular stochastic control for the optimal dividend policy when there is regime switching," Insurance: Mathematics and Economics, Elsevier, vol. 48(3), pages 344-354, May.
    13. He, Lin & Liang, Zongxia, 2009. "Optimal financing and dividend control of the insurance company with fixed and proportional transaction costs," Insurance: Mathematics and Economics, Elsevier, vol. 44(1), pages 88-94, February.
    14. Bjarne Højgaard & Michael Taksar, 2004. "Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy," Quantitative Finance, Taylor & Francis Journals, vol. 4(3), pages 315-327.
    15. Zongxia Liang & Bin Sun, 2010. "Optimal control of a big financial company with debt liability under bankrupt probability constraints," Papers 1007.5376, arXiv.org, revised Aug 2010.
    16. Jinxia Zhu & Hailiang Yang, 2015. "Optimal financing and dividend distribution in a general diffusion model with regime switching," Papers 1506.08360, arXiv.org.
    17. Liu, Wei & Hu, Yijun, 2014. "Optimal financing and dividend control of the insurance company with excess-of-loss reinsurance policy," Statistics & Probability Letters, Elsevier, vol. 84(C), pages 121-130.
    18. He, Lin & Hou, Ping & Liang, Zongxia, 2008. "Optimal control of the insurance company with proportional reinsurance policy under solvency constraints," Insurance: Mathematics and Economics, Elsevier, vol. 43(3), pages 474-479, December.
    19. Keppo, Jussi & Kofman, Leonard & Meng, Xu, 2010. "Unintended consequences of the market risk requirement in banking regulation," Journal of Economic Dynamics and Control, Elsevier, vol. 34(10), pages 2192-2214, October.
    20. Zongxia Liang & Lin He & Jiaoling Wu, 2010. "Optimal Dividend and reinsurance strategy of a Property Insurance Company under Catastrophe Risk," Papers 1009.1269, arXiv.org.
    21. Zhengjun Jiang & Martijn Pistorius, 2012. "Optimal dividend distribution under Markov regime switching," Finance and Stochastics, Springer, vol. 16(3), pages 449-476, July.
    22. Chen, Shumin & Liu, Yanchu & Weng, Chengguo, 2019. "Dynamic risk-sharing game and reinsurance contract design," Insurance: Mathematics and Economics, Elsevier, vol. 86(C), pages 216-231.
    23. 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.

    More about this item

    Keywords

    Dividend pay-out; proportional reinsurance; diffusion models; stochastic control theory; HJB equation; singular control;
    All these keywords.

    JEL classification:

    • 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

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