Kelly Criterion

What is the Kelly Criterion?

The Kelly criterion is a formula for the fraction of capital to commit to a favorable bet so that the long-run growth rate of the account is maximized. It was derived by John L. Kelly Jr. at Bell Labs in 1956, originally in information-theoretic terms, and was later adopted by gamblers and investors as a sizing rule. For a simple bet with known win probability and payoff odds, Kelly gives the unique fraction that maximizes the expected logarithm of wealth: bet more and long-run growth falls while risk rises; bet less and growth is merely slower.

Traders care because it draws a hard ceiling. Any sizing above full Kelly is irrational on the strategy's own statistics, producing more volatility and a lower expected long-run growth rate simultaneously. The formula also formalizes intuitions that expectancy alone does not: a high win rate with small payoffs and a low win rate with large payoffs can justify very different fractions, and a strategy with zero edge has a Kelly fraction of zero.

In practice almost nobody trades full Kelly, and the standard qualifier matters: Kelly is optimal only if the inputs are exactly right. Trading edges are estimated from limited samples, drift over time, and are usually overestimated. Because the growth penalty for betting above the true optimum is severe while the penalty for betting below it is mild, practitioners use fractional Kelly, commonly a half or a quarter of the computed fraction. Fractional Kelly sacrifices a modest amount of theoretical growth for a large reduction in drawdown and in the damage done by estimation error. Full Kelly drawdowns are brutal even with perfect inputs: in the idealized model, the probability of the account eventually halving is roughly one half.

How it's calculated

For continuous returns a common approximation is f_star = mu / sigma^2, with mu the expected excess return and sigma^2 the return variance.

The two-outcome formula assumes each trade risks the full committed fraction; applying it to stop-based trading requires mapping the stop distance to the fraction actually at risk.

How traders use it

• As a sizing ceiling rather than a target: traders compute full Kelly from backtest statistics and then size well below it, treating any position that exceeds full Kelly as a definitional error regardless of conviction.
• As fractional Kelly in live sizing: a quarter to a half of the computed fraction is a common operating range, chosen because estimated edges decay and the cost of oversizing is asymmetric.
• As a strategy comparison lens: computing implied Kelly fractions across systems reveals which edges actually support meaningful size, complementing raw expectancy and drawdown statistics.
• As a sanity check against ruin dynamics: because Kelly assumes accurate inputs and infinitely divisible bets, prudent implementations pair it with independent risk-of-ruin analysis using pessimistic parameter estimates.
• With honest limits: Kelly says nothing about parameter uncertainty, correlated simultaneous positions, or fat-tailed loss distributions, all of which push the true optimal fraction below the formula's output.

Kelly criterion vs other sizing frameworks