The Kelly Criterion, derived by John Kelly Jr. in 1956, gives the fraction of capital f* to allocate to a bet in order to maximize the expected growth rate of wealth over the long run:
Discrete (binary bet): f* = p − (1−p) / b
where p is the probability of winning, (1−p) is the probability of losing, and b is the net odds (how much you win per unit wagered on a winning bet).
Continuous (portfolio form): f* = μ / σ²
where μ is the expected excess return and σ² is the variance of returns. In the portfolio context, the Kelly fraction equals the expected return divided by the variance — or equivalently, the Sharpe Ratio divided by σ.
Properties and caveats
- Kelly betting maximizes the expected value of log(wealth), equivalent to maximizing median terminal wealth over many periods.
- It guarantees that ruin is impossible under continuous-time compounding.
- It can imply very large position sizes and large drawdowns in practice, which is why most practitioners use fractional Kelly.
- It is highly sensitive to estimation error in μ and σ² — real-world parameters are noisy.