Expectancy and Kelly criterion calculator
Turn a win rate and an average win and loss into expectancy, profit factor and the Kelly fraction, and see how fragile that fraction is.
Calculations run in your browser. Nothing you type is sent or stored. Results are estimates for information only, not investment advice.
How it works
Expectancy is the average result of one trade. With win rate p, average win W, average loss Ls and a cost c on every trade:
Expectancy = p × (W − c) − (1 − p) × (Ls + c)
Profit factor = p × (W − c) ÷ ((1 − p) × (Ls + c))
The Kelly criterion asks a different question: if you repeat the same bet many times, what share of your capital should you risk each time to make it grow fastest? With b = (W − c) ÷ (Ls + c) as the payoff ratio, the answer is:
Kelly fraction = p − (1 − p) ÷ b
Here the "risk" is the amount you lose when a trade loses. If the result is zero or negative, the bet has no edge and no size makes it grow. Half and quarter Kelly are the usual ways to cut the size, because the inputs are estimates.
Worked example
The page opens with a 55% win rate and equal average wins and losses. The expectancy is 0.55 × 1 − 0.45 × 1 = 0.10 per unit risked, the profit factor is 1.22, and the Kelly fraction is 0.55 − 0.45 ÷ 1 = 10% of capital. Half Kelly is 5% and quarter Kelly 2.5%. If the true win rate is 50% instead of 55%, the Kelly fraction drops to 0%: a five-point error removes the whole edge. Add a cost of 0.05 on every trade and the expectancy falls to 0.05, the payoff ratio to 0.90 and the Kelly fraction to 5.3%.
Kelly fraction by win rate and payoff
| Win rate | Payoff 1 | Payoff 1.5 | Payoff 2 | Payoff 3 |
|---|---|---|---|---|
| 40% | 0% | 0.0% | 10.0% | 20.0% |
| 45% | 0% | 8.3% | 17.5% | 26.7% |
| 50% | 0% | 16.7% | 25.0% | 33.3% |
| 55% | 10.0% | 25.0% | 32.5% | 40.0% |
| 60% | 20.0% | 33.3% | 40.0% | 46.7% |
Two things stand out. A high win rate with a small payoff gives a small number, and a low win rate with a large payoff can give a bigger one. And the whole table assumes you know the win rate. How long that takes is covered by the trades needed calculator, and the risks of betting full Kelly are in why full Kelly is a trap in practice.
Questions
What is the Kelly criterion?
It is a formula from the 1950s for the bet size that maximises the long-run growth rate of capital when the odds and probabilities are known. It sizes each bet as a share of the current capital.
Why do people use half Kelly or less?
Because the inputs are estimates and full Kelly is very sensitive to them. Overestimating the win rate by a few points can turn a growing bet into a losing one, and full Kelly also produces large drawdowns even when the estimates are right.
Does Kelly apply to trading?
Only loosely. It assumes independent bets with fixed odds and a known win rate. Trades are correlated, payoffs vary, and the win rate has to be estimated from a small sample. Our study of how long it takes to prove a strategy found about 34 months of live trades for a typical trend rule.
What is the difference between expectancy and the Kelly fraction?
Expectancy says whether the average trade makes money and by how much. The Kelly fraction says how large to make each trade if you want to maximise growth. A positive expectancy is necessary for a positive Kelly fraction but does not make a large size safe.