Risk, Leverage, and Optimal Betting in Financial Markets

Money Management and the Kelly Criterion

Author

Nam Nguyen
December 14, 2025 • Estimated Reading Time: 11 minutes

Most research in portfolio management focuses on alpha generation; however, another critical component of portfolio construction is position sizing. In this edition, we examine key considerations in position sizing, including the Kelly criterion and the martingale betting system.

In this issue:

Latest Posts

  • The Effectiveness of Collar Structures in Equity and Commodity Markets (12 min)

  • Fractal Market Hypothesis: From Theory to Practice (11 min)

  • Volatility vs. Volatility of Volatility: Conceptual and Practical Differences (13 min)

  • Modeling Gold for Prediction and Portfolio Hedging (13 min)

  • Effectiveness of Covered Call Strategy in Developed and Emerging Markets (13 min)

Does Kelly Portfolio Outperform the Market?

A method for capital allocation and position sizing is to employ the Kelly criterion. The Kelly criterion aims to optimize the expected growth rate of capital, maximizing the anticipated value of the logarithm of wealth. This strategy is rooted in John Kelly’s paper, “A New Interpretation of Information Rate.” According to Kelly, in repeated bets, a bettor should act to maximize the expected growth rate of capital, thus maximizing expected wealth at the end.

Reference [1] applies Thorp’s approach, as outlined in “The Kelly Criterion in Blackjack, Sports Betting and the Stock Market,” [2] to construct a portfolio in the Norwegian stock market. The formula computes the optimal investment fraction in a set of assets, considering the expected excess returns of the assets and the inverse of the variance-covariance matrix.

Findings

  • The study evaluates the performance of a growth-optimal Kelly portfolio in the Norwegian stock market over the period February 2003 to December 2022.

  • It assesses abnormal performance using the CAPM, Fama–French three-factor model, and Carhart four-factor model.

  • The Kelly portfolio achieves a higher compound annual growth rate (14.1%) and higher ending wealth than the benchmark index, which grows at 12%.

  • It also outperforms a Markowitz portfolio, which delivers lower growth and final wealth.

  • The Kelly portfolio and the benchmark exhibit similar Sharpe ratios (0.58), while the Kelly portfolio attains a higher Sortino ratio (0.95).

  • Factor regressions indicate an annualized alpha of 16.8% for the Kelly portfolio, statistically significant at the 1% level before transaction costs.

  • However, the factor models display very low explanatory power, suggesting that the estimated alpha may be overstated.

  • Once transaction costs are incorporated, the Kelly portfolio no longer outperforms the benchmark in terms of final wealth.

  • After costs, the alpha remains only marginally significant at the 10% level, implying limited real-world risk-adjusted excess returns.

This paper presents several interesting findings,

  • First, the correlation of the Kelly portfolio with the market is nearly zero.

  • Second, the performance is sensitive to transaction costs. We believe that with lower transaction costs, the Kelly portfolio has the potential to outperform the market and display zero correlation with it.

  • Third, the Kelly portfolio surpasses the Markowitz mean-variance portfolio in performance.

We also concur with the author that the utilization of options can further enhance the risk-adjusted return.

Reference

[2] Thorp, E. O., The Kelly Criterion in Blackjack Sports Betting and the Stock Market, in: Zenios, S.A. & Ziemba, W.T., Handbook of Asset and Liability Management, Volume 1, 387–428, 2006

Enhanced Martingale Betting System with Stop Policy

The martingale betting system is a popular gambling strategy that involves doubling one’s wager after each loss in the pursuit of recovering previous losses and securing a profit equal to the original bet. The underlying idea is that, statistically, a win will eventually occur, allowing the player to recoup losses and gain a net profit equal to the initial stake. While simple in concept, the martingale system carries inherent risks, as it assumes unlimited funds for doubling bets and disregards the fact that losing streaks can persist longer than expected. Thus, this system will eventually result in bankruptcy.

Reference [3] however argues that different perspectives exist regarding whether stock price movements adhere strictly to a random walk, often modeled as a geometric Brownian motion. This suggests a potential for enhancement in the martingale betting system. The author has subsequently introduced an enhanced martingale betting system that includes a stop policy.

Findings

  • The paper proposes an Improved Martingale Betting System (IMBS) by modifying the traditional martingale strategy with a stop policy and adapting it from casino gambling to intraday trading.

  • The IMBS is empirically tested using TAIEX (TX) futures across three intraday trading strategies.

  • Results show that the IMBS delivers strong performance and is applicable to TX intraday trading and related markets.

  • The study finds that returns increase with leverage up to a certain threshold, beyond which traditional martingale strategies face a high probability of bankruptcy.

  • By controlling key parameters—specifically leverage scaling (a), the number of steps (n), and total leverage—the IMBS significantly outperforms both the Equal-Weight Betting System (EWBS) and the traditional Martingale Betting System (MBS).

  • The inclusion of a stop-loss mechanism further improves performance and risk control.

  • Empirical tests indicate that IMBS performs particularly well when combined with price breakout strategies, which are identified as the most profitable approach for TX intraday trading.

In short, after testing on real data, the article concludes that

  • The conventional martingale betting system inevitably leads to bankruptcy,

  • With the integration of a stop policy, the new and improved martingale betting system demonstrates enhanced efficacy.

Reference

[3] Ting-Yuan Chen, and Szu-Lang Liao, Improved Martingale Betting System for Intraday Trading in Index Futures—Evidence of TAIEX Futures, Asian Journal of Economics and Business, Year:2023, Vol.4 (2), PP.339-366

Closing Thoughts

Taken together, the two studies highlight the trade-off between growth maximization and risk control in position sizing. The Kelly-based approach demonstrates strong theoretical and empirical growth performance, but its apparent alpha weakens once transaction costs and model limitations are accounted for, raising questions about real-world applicability. By contrast, the Improved Martingale Betting System shows that disciplined leverage control and stop policies can materially improve intraday trading outcomes relative to naive martingale schemes, especially when combined with breakout strategies. Overall, both strands of research suggest that position sizing is as critical as signal generation, and that practical constraints, parameter calibration, and market frictions ultimately determine whether theoretically attractive sizing rules translate into sustainable performance.

Educational Video

Money Management and the Kelly Criteria by S. Skiena

In this lecture, Professor Skiena emphasizes that money management and position sizing are central to long-term investment success, often more important than the accuracy of individual forecasts. He illustrates how leverage magnifies both gains and losses, and why strategies with positive expected returns can still lead to ruin if position sizes are poorly chosen. Through simple gambling examples and market analogies, he shows that maximizing expected returns is not the same as maximizing long-run wealth, especially when outcomes are repeated over time.

The lecture then introduces the Kelly criterion as a principled framework for optimal position sizing. Rather than focusing on short-term performance, the Kelly approach maximizes the expected logarithmic growth of capital, striking a balance between compounding and drawdown risk. The professor discusses the consequences of overbetting and underbetting relative to the Kelly fraction, highlighting why excessive leverage leads to volatility drag and eventual ruin.

Overall, the lecture frames the Kelly criterion as a theoretical benchmark for growth-optimal allocation, while also implicitly acknowledging the practical challenges of estimation error, uncertainty, and real-world constraints.

Volatility Weekly Recap

The figure below shows the term structures for the VIX futures (in colour) and the spot VIX (in grey).

The first half of the week was relatively quiet, as investors waited in anticipation of Wednesday’s FOMC announcement. The Fed ultimately announced a quarter-percentage-point rate cut on Wednesday, while also noting that only one additional cut may occur next year. Stocks rallied following the announcement and extended gains into Thursday. On Friday, however, weakness in the AI sector weighed on markets, leading the S&P 500 lower as investors rotated into cyclical stocks. Overall, the S&P 500 declined 0.63%, while the Nasdaq fell 1.62%.

Oil prices declined over the week, while gold moved higher. Bitcoin continued to edge upward, although Friday’s equity sell-off erased part of its gains.

On the volatility front, the first half of the week was subdued before volatility picked up modestly on Friday, yet both spot VIX and VIX futures ended the week in contango. Roll yields remained strongly positive. Although the S&P 500 declined, both VXX and VIXM posted negative returns, reflecting the low level of realized volatility.

Around the Quantosphere

  • The ex-Citi & Goldman Sachs traders of whom hedge fund Millennium will (probably) never let go (efinancialcareers-canada)

  • This French hedge fund is on a growth tear. Defying industry norms is part of its secret sauce. (businessinsider)

  • When Finance Needed More Math, It Turned to the Card Players (bloomberg)

  • Hedge funds double down using near record leverage in quest to boost returns (reuters)

  • How hedge funds performed in volatile November 2025 (reuters)

  • Inside the ‘rolling thunder’ quant crises of 2025 (ft)

  • Hedge funds are reviving appraisal arbitrage play (bloomberg)

  • Hedge fund leverage: the remedy mustn’t increase market risks (bloomberg)

  • Hedge fund managing partner Dmitry Balyasny taps AI to manage largest tail risk in 2026 (reuters)

  • Hedge funds tap US leveraged swap bets as basis trade stagnates (bloomberg)

Disclaimer

This newsletter is not investment advice. It is provided solely for entertainment and educational purposes. Always consult a financial professional before making any investment decisions.

We are not responsible for any outcomes arising from the use of the content and codes provided in the outbound links. By continuing to read this newsletter, you acknowledge and agree to this disclaimer.