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Risk-Constrained Kelly Gambling

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  • Enzo Busseti
  • Ernest K. Ryu
  • Stephen Boyd

Abstract

We consider the classic Kelly gambling problem with general distribution of outcomes, and an additional risk constraint that limits the probability of a drawdown of wealth to a given undesirable level. We develop a bound on the drawdown probability; using this bound instead of the original risk constraint yields a convex optimization problem that guarantees the drawdown risk constraint holds. Numerical experiments show that our bound on drawdown probability is reasonably close to the actual drawdown risk, as computed by Monte Carlo simulation. Our method is parametrized by a single parameter that has a natural interpretation as a risk-aversion parameter, allowing us to systematically trade off asymptotic growth rate and drawdown risk. Simulations show that this method yields bets that out perform fractional-Kelly bets for the same drawdown risk level or growth rate. Finally, we show that a natural quadratic approximation of our convex problem is closely connected to the classical mean-variance Markowitz portfolio selection problem.

Suggested Citation

  • Enzo Busseti & Ernest K. Ryu & Stephen Boyd, 2016. "Risk-Constrained Kelly Gambling," Papers 1603.06183, arXiv.org.
    • Handle: RePEc:arx:papers:1603.06183
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    File URL: http://arxiv.org/pdf/1603.06183
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    References listed on IDEAS

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    1. Harry Markowitz, 1952. "Portfolio Selection," Journal of Finance, American Finance Association, vol. 7(1), pages 77-91, March.
    2. Sid Browne, 2000. "Risk-Constrained Dynamic Active Portfolio Management," Management Science, INFORMS, vol. 46(9), pages 1188-1199, September.
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    Cited by:

    1. Niels Wesselhöfft & Wolfgang K. Härdle, 2020. "Risk-Constrained Kelly Portfolios Under Alpha-Stable Laws," Computational Economics, Springer;Society for Computational Economics, vol. 55(3), pages 801-826, March.
    2. Guillermo Angeris & Alex Evans & Tarun Chitra, 2020. "When does the tail wag the dog? Curvature and market making," Papers 2012.08040, arXiv.org.
    3. V'elez Jim'enez & Rom'an Alberto & Lecuanda Ontiveros & Jos'e Manuel & Edgar Possani, 2023. "Sports Betting: an application of neural networks and modern portfolio theory to the English Premier League," Papers 2307.13807, arXiv.org.
    4. Cavallero, S. & Rousselot, A. & Pugatch, R. & Dinis, L. & Lacoste, D., 2025. "The trade-off between growth and risk in Kelly’s gambling and beyond," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 659(C).
    5. Enzo Busseti & Walaa M. Moursi & Stephen Boyd, 2019. "Solution refinement at regular points of conic problems," Computational Optimization and Applications, Springer, vol. 74(3), pages 627-643, December.
    6. Hubáček, Ondřej & Šír, Gustav, 2023. "Beating the market with a bad predictive model," International Journal of Forecasting, Elsevier, vol. 39(2), pages 691-719.

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