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Optimal capital growth with convex shortfall penalties

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  • MacLean, Leonard C.
  • Zhao, Yonggan
  • Ziemba, William T.

Abstract

The optimal capital growth strategy or Kelly strategy, has many desirable properties such as maximizing the asympotic long run growth of capital. However, it has considerable short run risk since the utility is logarithmic, with essentially zero Arrow-Pratt risk aversion. Most investors favor a smooth wealth path with high growth. In this paper we provide a method to obtain the maximum growth while staying above a predetermined ex-ante discrete time smooth wealth path with high probability, with shortfalls below the path penalized with a convex function of the shortfall so as to force the investor to remain above the wealth path. This results in a lower investment fraction than the Kelly strategy with less risk, and lower but maximal growth rate under the assumptions. A mixture model with Markov transitions between several normally distributed market regimes is used for the dynamics of asset prices. The investment model allows the determination of the optimal constrained growth wagers at discrete points in time in an attempt to stay above the ex-ante path.

Suggested Citation

  • MacLean, Leonard C. & Zhao, Yonggan & Ziemba, William T., 2014. "Optimal capital growth with convex shortfall penalties," LSE Research Online Documents on Economics 59292, London School of Economics and Political Science, LSE Library.
  • Handle: RePEc:ehl:lserod:59292
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    File URL: http://eprints.lse.ac.uk/59292/
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    References listed on IDEAS

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    1. Leonard Maclean & Edward Thorp & William Ziemba, 2010. "Long-term capital growth: the good and bad properties of the Kelly and fractional Kelly capital growth criteria," Quantitative Finance, Taylor & Francis Journals, vol. 10(7), pages 681-687.
    2. Andrew Ang & Geert Bekaert, 2002. "International Asset Allocation With Regime Shifts," Review of Financial Studies, Society for Financial Studies, vol. 15(4), pages 1137-1187.
    3. L. C. MacLean & W. T. Ziemba & G. Blazenko, 1992. "Growth Versus Security in Dynamic Investment Analysis," Management Science, INFORMS, vol. 38(11), pages 1562-1585, November.
    4. Donald B. Hausch & William T. Ziemba & Mark Rubinstein, 1981. "Efficiency of the Market for Racetrack Betting," Management Science, INFORMS, vol. 27(12), pages 1435-1452, December.
    5. Allan Timmermann & Massimo Guidolin, 2006. "An econometric model of nonlinear dynamics in the joint distribution of stock and bond returns," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 21(1), pages 1-22.
    6. Basak, Suleyman & Shapiro, Alexander, 2001. "Value-at-Risk-Based Risk Management: Optimal Policies and Asset Prices," Review of Financial Studies, Society for Financial Studies, vol. 14(2), pages 371-405.
    7. MacLean, Leonard C. & Sanegre, Rafael & Zhao, Yonggan & Ziemba, William T., 2004. "Capital growth with security," Journal of Economic Dynamics and Control, Elsevier, vol. 28(5), pages 937-954, February.
    8. Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228.
    9. MacLean, Leonard & Zhao, Yonggan & Ziemba, William, 2006. "Dynamic portfolio selection with process control," Journal of Banking & Finance, Elsevier, vol. 30(2), pages 317-339, February.
    10. Barr Rosenberg., 1972. "The Behavior of Random Variables with Nonstationary Variance and the Distribution of Security Prices," Research Program in Finance Working Papers 11, University of California at Berkeley.
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    JEL classification:

    • J1 - Labor and Demographic Economics - - Demographic Economics

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