How the Kelly criterion works
Kelly sizing maximizes expected logarithmic wealth growth when the win probability and payoff distribution are known and repeated bets are independent. Let p be win probability, q = 1 − p, and b be net profit per unit lost. The formula returns the full-Kelly bankroll fraction. Half Kelly multiplies it by 0.5; quarter Kelly multiplies it by 0.25.
q = 1 − p
full Kelly: f* = (p × b − q) ÷ b
half Kelly = f* × 0.5; quarter Kelly = f* × 0.25
expected log growth = p ln(1 + b f) + q ln(1 − f)A negative f* is information, not a display error. It means the stated probability and payoff do not describe a positive edge. This calculator shows the negative raw value and sets the actionable stake to zero. It never turns a negative recommendation into a small positive bet.
Worked example and expected growth
With a 55% win probability and winners paying 1.5 times losers, q is 45%. Full Kelly is (0.55 × 1.5 − 0.45) ÷ 1.5 = 25% of bankroll. Half Kelly is 12.5%, and quarter Kelly is 6.25%. Under the exact repeated-bet assumptions, full Kelly has the highest expected log growth. It also produces the largest swings and is most exposed to estimation error.
Practitioners rarely know p and b with precision. A backtest can overstate both through selection bias, regime choice, leakage, optimistic fills, or too few independent trades. Parameter uncertainty makes full Kelly aggressive in practice. Fractional Kelly is a deliberate error buffer, not proof that the underlying edge is real.
Fraction, growth, and drawdown illustration
| Fraction | Stake now | Log growth/bet | Illustrative chance of ever losing 50% |
|---|---|---|---|
| Full Kelly | 25.0% | 4.57% | 50.0% |
| Half Kelly | 12.5% | 3.44% | 12.5% |
| Quarter Kelly | 6.3% | 2.02% | 0.8% |
The drawdown column is an illustrative continuous-time approximation for a stable independent edge: probability of wealth touching one half is 0.5 raised to (2/c − 1), where c is the Kelly fraction. Real trades are discrete, correlated, nonstationary, and cost-bearing, so these values are comparison bands rather than forecasts. Before sizing, use the risk/reward calculator to ground b in price geometry and the position size calculator to translate risk into units.
Correlation and portfolio Kelly
Applying single-bet Kelly independently to several simultaneous positions can create a portfolio much larger than intended. Shared market factors make nominally different trades lose together. Portfolio Kelly requires a covariance-aware optimization and still inherits estimation error. A practical process caps aggregate exposure, scales overlapping bets down, and stress-tests the result. The drawdown recovery calculator makes the capital asymmetry visible when those assumptions fail.
Frequently asked questions
Why do traders use half Kelly instead of full Kelly?
Half Kelly sacrifices some theoretical log growth while cutting the bankroll fraction in half and reducing sensitivity to input error. Since real probabilities and payoffs drift, the smaller fraction is often a more robust starting point than treating a backtest estimate as exact.
How should win probability and payoff ratio be estimated?
Use out-of-sample, costed outcomes from a stable rule set, with enough independent trades to estimate uncertainty. Avoid choosing parameters on the same sample used to report p and b. Include losses from gaps, slippage, missed fills, and strategy downtime.
Can Kelly be used for multiple simultaneous positions?
Not safely by calculating each position in isolation. Correlated bets share risk and can make the combined stake excessive. Portfolio Kelly uses expected returns and covariance, but estimation error remains; aggregate caps and fractional scaling are still prudent.
Why does a negative Kelly fraction mean do not trade?
Negative f* means the estimated payoff-weighted win probability does not compensate for losses. The growth-maximizing long stake is therefore zero. Betting a small positive amount would not be conservative Kelly; it would contradict the stated edge estimate.