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Minimizing the Probability of Lifetime Ruin under Borrowing Constraints

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  • Erhan Bayraktar
  • Virginia R. Young

Abstract

We determine the optimal investment strategy of an individual who targets a given rate of consumption and who seeks to minimize the probability of going bankrupt before she dies, also known as {\it lifetime ruin}. We impose two types of borrowing constraints: First, we do not allow the individual to borrow money to invest in the risky asset nor to sell the risky asset short. However, the latter is not a real restriction because in the unconstrained case, the individual does not sell the risky asset short. Second, we allow the individual to borrow money but only at a rate that is higher than the rate earned on the riskless asset. We consider two forms of the consumption function: (1) The individual consumes at a constant (real) dollar rate, and (2) the individual consumes a constant proportion of her wealth. The first is arguably more realistic, but the second is closely connected with Merton's model of optimal consumption and investment under power utility. We demonstrate that connection in this paper, as well as include a numerical example to illustrate our results.

Suggested Citation

  • Erhan Bayraktar & Virginia R. Young, 2007. "Minimizing the Probability of Lifetime Ruin under Borrowing Constraints," Papers math/0703850, arXiv.org.
  • Handle: RePEc:arx:papers:math/0703850
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    1. Hyeng Keun Koo, 1998. "Consumption and Portfolio Selection with Labor Income: A Continuous Time Approach," Mathematical Finance, Wiley Blackwell, vol. 8(1), pages 49-65, January.
    2. Moshe A. Milevsky & Kristen S. Moore & Virginia R. Young, 2006. "Asset Allocation And Annuity‐Purchase Strategies To Minimize The Probability Of Financial Ruin," Mathematical Finance, Wiley Blackwell, vol. 16(4), pages 647-671, October.
    3. Victor C. Pestien & William D. Sudderth, 1985. "Continuous-Time Red and Black: How to Control a Diffusion to a Goal," Mathematics of Operations Research, INFORMS, vol. 10(4), pages 599-611, November.
    4. Olivieri, Annamaria & Pitacco, Ermanno, 2003. "Solvency requirements for pension annuities," Journal of Pension Economics and Finance, Cambridge University Press, vol. 2(2), pages 127-157, July.
    5. Sid Browne, 1995. "Optimal Investment Policies for a Firm With a Random Risk Process: Exponential Utility and Minimizing the Probability of Ruin," Mathematics of Operations Research, INFORMS, vol. 20(4), pages 937-958, November.
    6. Hipp, Christian & Taksar, Michael, 2000. "Stochastic control for optimal new business," Insurance: Mathematics and Economics, Elsevier, vol. 26(2-3), pages 185-192, May.
    7. Hipp, Christian & Plum, Michael, 2000. "Optimal investment for insurers," Insurance: Mathematics and Economics, Elsevier, vol. 27(2), pages 215-228, October.
    8. Wendell H. Fleming & Thaleia Zariphopoulou, 1991. "An Optimal Investment/Consumption Model with Borrowing," Mathematics of Operations Research, INFORMS, vol. 16(4), pages 802-822, November.
    9. Sid Browne, 1999. "Beating a moving target: Optimal portfolio strategies for outperforming a stochastic benchmark," Finance and Stochastics, Springer, vol. 3(3), pages 275-294.
    10. Darrell Duffie & Thaleia Zariphopoulou, 1993. "Optimal Investment With Undiversifiable Income Risk," Mathematical Finance, Wiley Blackwell, vol. 3(2), pages 135-148, April.
    11. Duffie, Darrell & Fleming, Wendell & Soner, H. Mete & Zariphopoulou, Thaleia, 1997. "Hedging in incomplete markets with HARA utility," Journal of Economic Dynamics and Control, Elsevier, vol. 21(4-5), pages 753-782, May.
    12. Browne, S., 1995. "Optimal Investment Policies for a Firm with a Random Risk Process: Exponential Utility and Minimizing the Probability of Ruin," Papers 95-08, Columbia - Graduate School of Business.
    13. Virginia Young, 2004. "Optimal Investment Strategy to Minimize the Probability of Lifetime Ruin," North American Actuarial Journal, Taylor & Francis Journals, vol. 8(4), pages 106-126.
    14. Vanduffel, Steven & Dhaene, Jan & Goovaerts, Marc, 2005. "On the evaluation of ‘saving-consumption’ plans," Journal of Pension Economics and Finance, Cambridge University Press, vol. 4(1), pages 17-30, March.
    15. J. Dhaene & S. Vanduffel & M. J. Goovaerts & R. Kaas & D. Vyncke, 2005. "Comonotonic Approximations for Optimal Portfolio Selection Problems," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 72(2), pages 253-300, June.
    16. Milevsky, Moshe Arye & Ho, Kwok & Robinson, Chris, 1997. "Asset Allocation via the Conditional First Exit Time or How to Avoid Outliving Your Money," Review of Quantitative Finance and Accounting, Springer, vol. 9(1), pages 53-70, July.
    17. M. H. A. Davis & A. R. Norman, 1990. "Portfolio Selection with Transaction Costs," Mathematics of Operations Research, INFORMS, vol. 15(4), pages 676-713, November.
    18. Moshe Milevsky & Chris Robinson, 2000. "Self-Annuitization and Ruin in Retirement," North American Actuarial Journal, Taylor & Francis Journals, vol. 4(4), pages 112-124.
    19. Sid Browne, 1997. "Survival and Growth with a Liability: Optimal Portfolio Strategies in Continuous Time," Mathematics of Operations Research, INFORMS, vol. 22(2), pages 468-493, May.
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    Cited by:

    1. Chen, Xinfu & Landriault, David & Li, Bin & Li, Dongchen, 2015. "On minimizing drawdown risks of lifetime investments," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 46-54.
    2. Haluk Yener, 2015. "Maximizing survival, growth and goal reaching under borrowing constraints," Quantitative Finance, Taylor & Francis Journals, vol. 15(12), pages 2053-2065, December.
    3. Bayraktar, Erhan & Hu, Xueying & Young, Virginia R., 2011. "Minimizing the probability of lifetime ruin under stochastic volatility," Insurance: Mathematics and Economics, Elsevier, vol. 49(2), pages 194-206, September.
    4. Arash Fahim & Lingjiong Zhu, 2015. "Optimal Investment in a Dual Risk Model," Papers 1510.04924, arXiv.org, revised Feb 2023.
    5. Angoshtari, Bahman & Bayraktar, Erhan & Young, Virginia R., 2015. "Minimizing the expected lifetime spent in drawdown under proportional consumption," Finance Research Letters, Elsevier, vol. 15(C), pages 106-114.
    6. Erhan Bayraktar & Virginia Young, 2011. "Proving regularity of the minimal probability of ruin via a game of stopping and control," Finance and Stochastics, Springer, vol. 15(4), pages 785-818, December.
    7. Zhuo Jin & Zuo Quan Xu & Bin Zou, 2023. "Optimal moral-hazard-free reinsurance under extended distortion premium principles," Papers 2304.08819, arXiv.org.
    8. Bayraktar, Erhan & Young, Virginia R., 2008. "Mutual fund theorems when minimizing the probability of lifetime ruin," Finance Research Letters, Elsevier, vol. 5(2), pages 69-78, June.
    9. Erhan Bayraktar & Yuchong Zhang, 2014. "Stochastic Perron's Method for the Probability of lifetime ruin problem under transaction costs," Papers 1404.7406, arXiv.org, revised Nov 2014.
    10. Erhan Bayraktar & Yuchong Zhang, 2014. "Minimizing the Probability of Lifetime Ruin Under Ambiguity Aversion," Papers 1402.1809, arXiv.org, revised Nov 2014.
    11. Azcue, Pablo & Muler, Nora, 2009. "Optimal investment strategy to minimize the ruin probability of an insurance company under borrowing constraints," Insurance: Mathematics and Economics, Elsevier, vol. 44(1), pages 26-34, February.
    12. Liang, Xiaoqing & Young, Virginia R., 2023. "Annuitizing at a bounded, absolutely continuous rate to minimize the probability of lifetime ruin," Insurance: Mathematics and Economics, Elsevier, vol. 112(C), pages 80-96.
    13. Wang, Ting & Young, Virginia R., 2012. "Optimal commutable annuities to minimize the probability of lifetime ruin," Insurance: Mathematics and Economics, Elsevier, vol. 50(1), pages 200-216.
    14. Cohen, Asaf & Young, Virginia R., 2016. "Minimizing lifetime poverty with a penalty for bankruptcy," Insurance: Mathematics and Economics, Elsevier, vol. 69(C), pages 156-167.
    15. Bayraktar, Erhan & Young, Virginia R., 2008. "Maximizing utility of consumption subject to a constraint on the probability of lifetime ruin," Finance Research Letters, Elsevier, vol. 5(4), pages 204-212, December.
    16. Erhan Bayraktar & David Promislow & Virginia Young, 2014. "Purchasing Term Life Insurance to Reach a Bequest Goal while Consuming," Papers 1412.2262, arXiv.org, revised Feb 2016.
    17. Erhan Bayraktar & Asaf Cohen, 2015. "Risk Sensitive Control of the Lifetime Ruin Problem," Papers 1503.05769, arXiv.org, revised Jul 2016.
    18. Claude Bergeron, 2013. "Dividend growth, stock valuation, and long-run risk," Journal of Economics and Finance, Springer;Academy of Economics and Finance, vol. 37(4), pages 547-559, October.
    19. Xiaoqing Liang & Virginia R. Young, 2020. "Minimizing the Probability of Lifetime Exponential Parisian Ruin," Journal of Optimization Theory and Applications, Springer, vol. 184(3), pages 1036-1064, March.
    20. Bayraktar, Erhan & Young, Virginia R., 2009. "Minimizing the lifetime shortfall or shortfall at death," Insurance: Mathematics and Economics, Elsevier, vol. 44(3), pages 447-458, June.
    21. Liang, Zhibin & Bayraktar, Erhan, 2014. "Optimal reinsurance and investment with unobservable claim size and intensity," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 156-166.
    22. Asaf Cohen & Virginia R. Young, 2015. "Minimizing Lifetime Poverty with a Penalty for Bankruptcy," Papers 1509.01694, arXiv.org.
    23. Junbeom Lee & Xiang Yu & Chao Zhou, 2019. "Lifetime Ruin under High-watermark Fees and Drift Uncertainty," Papers 1909.01121, arXiv.org, revised Oct 2020.

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