Finance and Stochastics

, Volume 11, Issue 2, pp 213–236 | Cite as

Correspondence between lifetime minimum wealth and utility of consumption



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.


Optimal control Probability of ruin Utility of consumption Investment/consumption decisions 

Jel Classification

G11 C61 

Mathematics Subject Classification (2000)

91B28 91B42 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Al-Gwaiz M.A. (1992). Theory of Distributions. Monographs and Textbooks in Pure and Applied Mathematics 159. Marcel Dekker, New York Google Scholar
  2. 2.
    Barles G., Daher C. and Romano M. (1994). Optimal control on the L norm of a diffusion process. SIAM J. Cont. Optim. 32: 612–634 MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bayraktar, E., Young, V.R.: Minimizing the lifetime shortfall or shortfall at death. Working Paper, Department of Mathematics, University of Michigan (2005). http://www.math.lsa.∼erhan/shortfall.pdfGoogle Scholar
  4. 4.
    Bayraktar, E., Young, V.R.: Minimizing the probability of lifetime ruin under borrowing constraints. Insur. Math. Econ. (to appear 2006)Google Scholar
  5. 5.
    Browne S. (1995). Optimal investment policies for a firm with a random risk process: exponential utility and minimizing the probability of ruin. Math. Oper. Res. 20: 937–958 MATHMathSciNetGoogle Scholar
  6. 6.
    Browne S. (1997). Survival and growth with a liability: Optimal portfolio strategies in continuous time. Math. Oper. Res. 22: 468–493 MATHMathSciNetGoogle Scholar
  7. 7.
    Browne S. (1999). Beating a moving target: optimal portfolio strategies for outperforming a stochastic benchmark. Financ. Stoch. 3: 275–294 MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Browne S. (1999). The risk and rewards of minimizing shortfall probability. J. Portf. Manage. 25(4): 76–85 CrossRefGoogle Scholar
  9. 9.
    Dudley R.M. (2002). Real Analysis and Probability. Cambridge University Press, Cambridge MATHGoogle Scholar
  10. 10.
    Fleming W.H. and Soner H.M. (1993). Controlled Markov Processes and Viscosity Solutions. Springer, New York MATHGoogle Scholar
  11. 11.
    Fleming W.H. and Zariphopoulou T. (1991). An optimal investment/consumption model with borrowing. Math. Oper. Res. 16: 802–822 MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Heinricher A.C. and Stockbridge R.H. (1991). Optimal control of the running max. SIAM J. Cont. Optim. 29: 936–953 MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Hipp C. and Plum M. (2000). Optimal investment for insurers. Insur. Math. Econ. 27: 215–228 MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Hipp C. and Taksar M. (2000). Stochastic control for optimal new business. Insur. Math. Econ. 26: 185–192 MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Karatzas I. and Shreve S. (1998). Methods of Mathematical Finance. Springer, New York MATHGoogle Scholar
  16. 16.
    Merton R.C. (1992). Continuous-Time Finance (revised edition). Blackwell, Cambridge Google Scholar
  17. 17.
    Milevsky M.A., Ho K. and Robinson C. (1997). Asset allocation via the conditional first exit time or how to avoid outliving your money. Rev. Quant. Financ. Account. 9: 53–70 CrossRefGoogle Scholar
  18. 18.
    Milevsky M.A., Moore K.S. and Young V.R. (2006). Asset allocation and annuity-purchase strategies to minimize the probability of financial ruin. Math. Financ. 16: 647–671 MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Milevsky M.A. and Robinson C. (2000). Self-annuitization and ruin in retirement, with discussion. N. Am. Actuar. J. 4(4): 112–129 MATHMathSciNetGoogle Scholar
  20. 20.
    Pratt J.W. (1964). Risk aversion in the small and in the large. Econometrica 32: 122–136 MATHCrossRefGoogle Scholar
  21. 21.
    Schmidli H. (2001). Optimal proportional reinsurance policies in a dynamic setting. Scand. Actuar. J. 2001(1): 55–68 MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Tenenbaum M. and Pollard H. (1963). Ordinary Differential Equations. Dover, New York MATHGoogle Scholar
  23. 23.
    Young, V.R.: Optimal investment strategy to minimize the probability of lifetime ruin. N. Am. Actuar. J. 8(4), 105–126 (2004)MATHMathSciNetGoogle Scholar
  24. 24.
    Young V.R. (2004). Optimal investment strategy to minimize the probability of lifetime ruin. N. Am. Actuar. J. 8(4): 105–126 MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA

Personalised recommendations