What is the Kelly Criterion? Optimal Position Sizing for Quants

The Kelly Criterion is a formula used to determine the optimal size of a series of bets to maximize the logarithm of wealth over the long term.

The Kelly Criterion is a formula used to determine the optimal size of a bet or investment to maximize long-term wealth growth. Originally developed by John Kelly at Bell Labs in 1956 to handle signal noise, it was later adopted by quants like Edward Thorp to dominate blackjack and eventually the stock market. It answers the fundamental question of quantitative trading: 'How much of my capital should I risk on this trade?'

The basic formula for the Kelly fraction ($f^$) is: $f^ = \frac{bp - q}{b}$. In this equation, $b$ is the odds received on the wager (e.g., decimal odds - 1), $p$ is the probability of winning, and $q$ is the probability of losing ($1-p$). For example, if a trading algorithm has a 55% win rate and a 1:1 reward-to-risk ratio, the formula suggests risking $0.10$ or 10% of the account. This specific fraction ensures that the trader avoids the risk of ruin while maximizing the geometric growth rate of the portfolio.

Using the 'Full Kelly' fraction is mathematically optimal but practically dangerous. If your estimation of $p$ (win probability) is slightly too high, Full Kelly can lead to ruin due to high volatility and large drawdowns. Most institutional traders use 'Fractional Kelly,' such as Half-Kelly ($f^* / 2$), to provide a safety margin against model error. Over-betting leads to a significant drop in the expected growth rate, as the variance $\sigma^2$ begins to dominate the return.