Risk Management: Position Sizing with Kelly Criterion

Risk Management: Position Sizing with Kelly Criterion

The Kelly Criterion, developed by John Kelly at Bell Labs in 1956, answers a deceptively simple question: given a bet with known odds and win probability, what fraction of your bankroll should you wager to maximize the long-run growth rate of your wealth?

The Formula

For a simple binary bet:

Where:

• ▸$f^*$ is the optimal fraction of bankroll to bet
• ▸$b$ is the net odds received (profit per unit wagered)
• ▸$p$ is the probability of winning
• ▸$q = 1 - p$ is the probability of losing

For a continuous return distribution (more relevant for trading):

Where $\mu$ is the expected return and $\sigma^2$ is the variance of returns.

Why Full Kelly Is Too Aggressive

Full Kelly maximizes the geometric mean of wealth growth, but it produces terrifying drawdowns. A strategy at full Kelly will experience drawdowns of 50%+ with non-trivial probability. Most practitioners use Half Kelly (50% of the optimal fraction) or Quarter Kelly as a practical compromise between growth and drawdown control.

The intuition: Kelly assumes your edge estimate is perfectly accurate. In practice, your edge estimate has estimation error. Fractional Kelly provides a margin of safety against this uncertainty.

Portfolio Kelly

When trading multiple strategies simultaneously, the Kelly framework extends to a portfolio context. The optimal weight vector is:

Where $\Sigma$ is the covariance matrix of strategy returns and $\boldsymbol{\mu}$ is the vector of expected returns. This is identical to the mean-variance optimal portfolio — Kelly and Markowitz are two expressions of the same underlying mathematics.

Applied Ideas

The frameworks discussed above translate directly into deployable trading logic. Here are concrete next steps for practitioners:

• ▸Backtest first: Validate any signal-generation or risk-management approach with walk-forward analysis before committing capital.
• ▸Start small: Deploy with fractional position sizing and paper-trade for at least one full market cycle.
• ▸Monitor regime shifts: Set automated alerts for when your model detects a regime change — manual review before large rebalances is prudent.
• ▸Iterate on KPIs: Track Sharpe, Sortino, max drawdown, and win rate weekly. If any metric degrades beyond your predefined threshold, pause and re-evaluate.
• ▸Combine signals: The strongest edges come from combining uncorrelated signals — pair the ideas in this post with your existing alpha sources.