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Correspondence between Lifetime Minimum Wealth and Utility of Consumption

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  • Erhan Bayraktar
  • Virginia R. Young

Abstract

We establish when the two problems of minimizing a function of lifetime minimum wealth and of maximizing utility of lifetime consumption result in the same optimal investment strategy on a given open interval $O$ in wealth space. To answer this question, we equate the two investment strategies and show that if the individual consumes at the same rate in both problems -- the consumption rate is a control in the problem of maximizing utility -- then the investment strategies are equal only when the consumption function is linear in wealth on $O$, a rather surprising result. It, then, follows that the corresponding investment strategy is also linear in wealth and the implied utility function exhibits hyperbolic absolute risk aversion.

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File URL: http://arxiv.org/pdf/math/0703820
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Paper provided by arXiv.org in its series Papers with number math/0703820.

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Date of creation: Mar 2007
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Publication status: Published in Finance and Stochastics, 2007, Volume 11 (2), 213-236
Handle: RePEc:arx:papers:math/0703820

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  1. Browne, S., 1995. "Optimal Investment Policies for a Firm with a Random Risk Process: Exponential Utility and Minimizing the Probability of Ruin," Papers, Columbia - Graduate School of Business 95-08, Columbia - Graduate School of Business.
  2. Hipp, Christian & Plum, Michael, 2000. "Optimal investment for insurers," Insurance: Mathematics and Economics, Elsevier, vol. 27(2), pages 215-228, October.
  3. 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, Springer, vol. 9(1), pages 53-70, July.
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Cited by:
  1. Bayraktar, Erhan & Young, Virginia R., 2008. "Mutual fund theorems when minimizing the probability of lifetime ruin," Finance Research Letters, Elsevier, Elsevier, vol. 5(2), pages 69-78, June.
  2. Erhan Bayraktar, 2007. "Minimizing the Lifetime Shortfall or Shortfall at Death," Papers math/0703824, arXiv.org.
  3. Erhan Bayraktar & Virginia R. Young, 2008. "Minimizing the Probability of Ruin when Consumption is Ratcheted," Papers 0806.2358, arXiv.org.
  4. Erhan Bayraktar & Yuchong Zhang, 2014. "Minimizing the Probability of Lifetime Ruin Under Ambiguity Aversion," Papers 1402.1809, arXiv.org.
  5. Konstantin Petrichev & Susan Thorp, 2007. "The Private Value of Public Pensions," Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney 211, Quantitative Finance Research Centre, University of Technology, Sydney.
  6. Erhan Bayraktar & Xueying Hu & Virginia R. Young, 2010. "Minimizing the Probability of Lifetime Ruin under Stochastic Volatility," Papers 1003.4216, arXiv.org, revised May 2011.
  7. Nielsen, Peter Holm & Steffensen, Mogens, 2008. "Optimal investment and life insurance strategies under minimum and maximum constraints," Insurance: Mathematics and Economics, Elsevier, vol. 43(1), pages 15-28, August.
  8. Bayraktar, Erhan & Young, Virginia R., 2008. "Maximizing utility of consumption subject to a constraint on the probability of lifetime ruin," Finance Research Letters, Elsevier, Elsevier, vol. 5(4), pages 204-212, December.

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