What is the Kelly Criterion? Optimal Position Sizing for Long-Term Growth

The Kelly Criterion is the mathematical formula for optimal bet sizing. Learn how to maximize the logarithmic growth of your capital while avoiding ruin.

The Kelly Criterion provides the mathematical edge that separates professional bettors from gamblers in quant finance. What is the Kelly Criterion? It is a formula used to determine the optimal size of a series of bets to maximize the long-term growth of wealth.

In its simplest form for a trade with a win probability $p$ and a win/loss ratio $b$, the Kelly fraction $f^$ is: $f^ = \frac{bp - q}{b}$. Where $q = 1 - p$. In finance, where outcomes are continuous rather than binary, we use the 'Mean-Variance' Kelly: $f^* = \frac{\mu - r}{\sigma^2}$. This formula tells us that your allocation should be proportional to the expected excess return $(\mu - r)$ and inversely proportional to the variance $(\sigma^2)$.

While the Kelly Criterion is mathematically optimal for growth, it is extremely aggressive. Following 'Full-Kelly' leads to high volatility in equity and leaves zero margin for error in your estimation of $\mu$ and $\sigma$. If you overestimate your edge by even a small margin, Full-Kelly will lead to a total drawdown.

1. Half-Kelly: Quants often use $0.5 \cdot f^$ to reduce volatility by 50% while only sacrificing a small portion of growth. 2. Risk of Ruin: If you bet more than $2 \cdot f^$, your expected long-term growth rate becomes negative, even if every individual trade has a positive expectation. 3. Diversification: Kelly scaling applies to the total portfolio; highly correlated trades must be treated as a single position.

We can simulate the difference between betting 10%, Full-Kelly, and Over-Kelly to visualize why position sizing is more important than the signal itself.