The Kelly Criterion: Mathematical Position Sizing & Compounding
Calculating optimal risk fractions ($f^*$), avoiding the risk of ruin, and implementing Fractional Kelly in institutional trading.
1. The Mathematics of Optimal Growth
Developed by Bell Labs scientist John L. Kelly Jr. in 1956, the Kelly Criterion is a mathematical formula that calculates the optimal fraction of capital to risk on a series of positive-expected-value investments to maximize the long-term compound growth rate of wealth:
Where $f^*$ is the fraction of portfolio capital to bet, $b$ is the net odds (win/loss payoff ratio), $p$ is the probability of winning, and $q = 1 - p$ is the probability of losing.
2. Fractional Kelly: Managing Volatility & Drawdowns
While "Full Kelly" mathematically maximizes terminal wealth, it generates extreme portfolio volatility and severe drawdowns (often exceeding 50% to 80%).
Institutional quantitative hedge funds utilize Fractional Kelly (Half-Kelly or Quarter-Kelly, e.g. $0.5 imes f^*$), which captures 75% to 90% of maximum geometric compounding while slashing portfolio variance and drawdown risk by more than half.