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Necessary and Sufficient Conditions for Frequency-Based Kelly Optimal Portfolio

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  • Chung-Han Hsieh

Abstract

In this paper, we consider a discrete-time portfolio with $m \geq 2$ assets optimization problem which includes the rebalancing~frequency as an additional parameter in the maximization. The so-called Kelly Criterion is used as the performance metric; i.e., maximizing the expected logarithmic growth of a trader's account, and the portfolio obtained is called the frequency-based Kelly optimal portfolio. The focal point of this paper is to extend upon the results of our previous work to obtain various optimality characterizations on the portfolio. To be more specific, using Kelly's criterion in our frequency-based formulation, we first prove necessary and sufficient conditions for the frequency-based Kelly optimal portfolio. With the aid of these conditions, we then show several new optimality characterizations such as expected ratio optimality and asymptotic relative optimality, and a result which we call the Extended Dominant Asset Theorem. That is, we prove that the $i$th asset is dominant in the portfolio if and only if the Kelly optimal portfolio consists of that asset only. The word "extended" on the theorem comes from the fact that it was only a sufficiency result that was proved in our previous work. Hence, in this paper, we improve it to involve a proof of the necessity part. In addition, the trader's survivability issue (no bankruptcy consideration) is also studied in detail in our frequency-based trading framework. Finally, to bridge the theory and practice, we propose a simple trading algorithm using the notion called dominant asset condition to decide when should one triggers a trade. The corresponding trading performance using historical price data is reported as supporting evidence.

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  • Chung-Han Hsieh, 2020. "Necessary and Sufficient Conditions for Frequency-Based Kelly Optimal Portfolio," Papers 2004.12099, arXiv.org.
    • Handle: RePEc:arx:papers:2004.12099
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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. Daniel Kuhn & David Luenberger, 2010. "Analysis of the rebalancing frequency in log-optimal portfolio selection," Quantitative Finance, Taylor & Francis Journals, vol. 10(2), pages 221-234.
    3. Sujit R. Das & Dmitri Kaznachey & Mukul Goyal, 2014. "Computing optimal rebalance frequency for log-optimal portfolios," Quantitative Finance, Taylor & Francis Journals, vol. 14(8), pages 1489-1502, January.
    4. Chung-Han Hsieh & B. Ross Barmish & John A. Gubner, 2019. "On Positive Solutions of a Delay Equation Arising When Trading in Financial Markets," Papers 1901.02480, arXiv.org, revised Oct 2019.
    5. Chung-Han Hsieh & B. Ross Barmish & John A. Gubner, 2018. "At What Frequency Should the Kelly Bettor Bet?," Papers 1801.06737, arXiv.org, revised Aug 2018.
    6. Chung-Han Hsieh & John A. Gubner & B. Ross Barmish, 2018. "Rebalancing Frequency Considerations for Kelly-Optimal Stock Portfolios in a Control-Theoretic Framework," Papers 1807.05265, arXiv.org, revised Aug 2018.
    7. Sujit R. Das & Mukul Goyal, 2015. "Computing optimal rebalance frequency for log-optimal portfolios in linear time," Quantitative Finance, Taylor & Francis Journals, vol. 15(7), pages 1191-1204, July.
    8. Andrew W. Lo & H. Allen Orr & Ruixun Zhang, 2018. "The growth of relative wealth and the Kelly criterion," Journal of Bioeconomics, Springer, vol. 20(1), pages 49-67, April.
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    Cited by:

    1. Chung-Han Hsieh, 2022. "On Robust Optimal Linear Feedback Stock Trading," Papers 2202.02300, arXiv.org.
    2. Chung-Han Hsieh, 2022. "On Robustness of Double Linear Trading with Transaction Costs," Papers 2209.12383, arXiv.org.

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