Kelly Criterion in Trading: Formula, Assumptions and Limits

The Kelly criterion estimates the fraction of capital that maximizes expected long-run logarithmic growth. In a simple binary bet with known probabilities and payoffs: Kelly fraction = (b × p − q) ÷ b where (b) is the net profit per unit lost, (p) is the probability of winning, and (q = 1 − p).

Example

Assume:

• win probability (p = 0.50);
• loss probability (q = 0.50);
• average win (b = 2) times the average loss.

Kelly fraction = (2 × 0.50 − 0.50) ÷ 2 = 0.25 The model suggests allocating 25% of capital. This is a mathematical result under the stated assumptions, not a practical recommendation. Two consecutive full losses at 25% reduce equity by 43.75%. Five reduce it by more than 76%. A strategy can maximize asymptotic growth and still be intolerable or incompatible with an external drawdown rule.

Critical assumptions

The simple formula assumes:

• probabilities and payoffs are known;
• the opportunity repeats under stable conditions;
• outcomes fit the model;
• capital can be divided continuously;
• costs, gaps, liquidity, taxes, and operational failures are absent or modeled;
• the objective is long-run log growth.

Trading rarely satisfies these assumptions exactly.

Estimation error

If win probability or average payoff is overstated, the calculated fraction can be far too large. This is especially dangerous after backtest optimization, in small samples, or when a few outliers dominate the average win. A useful sensitivity analysis recalculates Kelly under less favorable inputs and includes costs. If the fraction changes sign or magnitude quickly, the estimate is not robust.

Fractional Kelly

Half Kelly or quarter Kelly multiplies the estimated full-Kelly fraction by 0.5 or 0.25. Fractional Kelly reduces growth and drawdown relative to the same model at full Kelly, but it does not correct a false edge or an invalid distribution.

Use in prop trading

A prop-account loss limit acts as a separate absorbing boundary. Even a correctly estimated Kelly fraction can create an unacceptable probability of daily or maximum-loss breach. Use Kelly, if at all, as one analytical reference. The binding size is the smallest amount permitted by strategy risk, idea concentration, liquidity, and the available loss buffer.