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Betting Around the Clock: Time Change and Long Term Model Risk

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  • Umberto Cherubini

Abstract

We investigate the performance of the Kelly rule in a setting in which the dynamics of the return is represented by a time change process. We find that in this general semi-martingale setting the Kelly rule does not maximize the average growth rate, unless the log-return is normally distributed. Namely, the investment position proposed by the Kelly rule is too large, and the investor could achieve a higher average growth rate by investing less aggressively. The higher the variance of the stochastic clock, the more material the failure of the Kelly rule. The ruin threshold proposed by Thorp (1969) is closer, even though examples based on stochastic clock variance estimates taken from the literature show that Kelly rule investment remains safely in the ruin-free region. Finally, the goal of keeping the investment below the ruin threshold for a family of stochastic clock distributions generates a long term investment problem that parallels the "acceptable investment" theory.

Suggested Citation

  • Umberto Cherubini, 2026. "Betting Around the Clock: Time Change and Long Term Model Risk," Papers 2603.13632, arXiv.org.
    • Handle: RePEc:arx:papers:2603.13632
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    References listed on IDEAS

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    3. Harry M. Markowitz, 2011. "Investment for the Long Run: New Evidence for an Old Rule," World Scientific Book Chapters, in: Leonard C MacLean & Edward O Thorp & William T Ziemba (ed.), THE KELLY CAPITAL GROWTH INVESTMENT CRITERION THEORY and PRACTICE, chapter 35, pages 495-508, World Scientific Publishing Co. Pte. Ltd..
    4. Alexander Cherny & Dilip Madan, 2009. "New Measures for Performance Evaluation," The Review of Financial Studies, Society for Financial Studies, vol. 22(7), pages 2371-2406, July.
    5. Daniel Ellsberg, 1961. "Risk, Ambiguity, and the Savage Axioms," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 75(4), pages 643-669.
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