Kelly Criterion: Definition
The Kelly Criterion is a mathematical formula that calculates the optimal fraction of trading capital to risk on a given trade, based on the trade’s win probability and payoff ratio, in order to maximize long-run capital growth.
Developed by John Kelly Jr. in 1956 for signal-transmission theory and later adapted to gambling and trading, the Kelly formula answers a specific question: “given my edge, what fraction of my bankroll grows my account fastest over many repeated trades, without risking ruin?”
The formula
f* = (bp − q) / bwhere:
- f* = the optimal fraction of capital to allocate to the trade
- b = the payoff ratio (average win size ÷ average loss size)
- p = probability of winning the trade
- q = probability of losing the trade (q = 1 − p)
Worked example
A trader has a strategy with a 55% win rate and a 1:1 payoff ratio (average win equals average loss, so b = 1):
- p = 0.55
- q = 1 − 0.55 = 0.45
- b = 1
f* = (1 × 0.55 − 0.45) / 1 = (0.55 − 0.45) / 1 = 0.10
The Kelly formula says this trader’s optimal bet size is 10% of their bankroll per trade. If the payoff ratio improves to 2:1 (b = 2) at the same 55% win rate:
f* = (2 × 0.55 − 0.45) / 2 = (1.10 − 0.45) / 2 = 0.65 / 2 = 0.325, or 32.5% of the bankroll — a much larger optimal stake, because a better payoff ratio compensates for the same win rate.
Why almost nobody trades full Kelly
Full Kelly sizing is mathematically optimal for growth but produces extremely large drawdowns in practice — a real strategy’s win rate and payoff ratio are estimates, not certainties, and betting the full Kelly fraction on overestimated edge parameters can produce account-threatening losses. For this reason:
- Half-Kelly (betting 50% of the calculated f*) reduces the strategy’s volatility and drawdown substantially while sacrificing only about 25% of the theoretical long-run growth rate.
- Quarter-Kelly is a common further-conservative choice among discretionary and systematic traders alike, especially when the win-rate/payoff estimates come from a limited backtest sample.
Betting more than full Kelly is mathematically proven to reduce long-run growth and increase ruin risk — it is never optimal, regardless of confidence in the edge.
Kelly vs. fixed-fractional position sizing
| Kelly Criterion | Fixed-fractional (e.g., always risk 1-2%) | |
|---|---|---|
| Basis | Win rate + payoff ratio, recalculated per trade/strategy | A constant percentage, chosen independent of edge size |
| Adapts to edge size | Yes — larger calculated edge sizes up automatically | No — same risk regardless of how strong the edge is |
| Risk if inputs are wrong | High — full Kelly is very sensitive to overestimated win rate/payoff | Lower — the fixed percentage caps downside regardless of edge miscalculation |
| Common practical use | Half- or quarter-Kelly, applied cautiously | Widely used as a simple, robust default |
Applying Kelly to a backtested strategy
To use Kelly sizing on a real strategy, pull the win rate and average win/loss ratio directly from a sufficiently large backtest sample (see the Backtesting glossary entry for sample-size guidance) — small samples produce unstable Kelly estimates that can swing wildly trade to trade. Most systematic traders recalculate Kelly periodically as new trade data accumulates, rather than fixing it once. Paper trade first any position-sizing model, Kelly included, before applying it with real capital — sizing errors compound faster than entry/exit errors.
How Quantzee Uses This
Kelly Criterion optimization requires accurate input data — win rate and average win/loss ratio — that can only be reliably generated from non-repainting backtest results. A backtest that uses repainting indicators will overstate win rate and the average win amount, leading to an inflated Kelly fraction and dangerous over-sizing in live trading. Quantzee’s non-repainting signals provide the clean historical data needed to calculate honest Kelly inputs: the win rate and payoff ratio you see in the Strategy Tester are the ones available in real time.
Common mistakes
- Using full Kelly. Nearly every practitioner source recommends half- or quarter-Kelly instead — full Kelly’s drawdowns are typically too severe to psychologically or financially tolerate.
- Estimating win rate/payoff from too small a sample. A 20-trade backtest gives an unreliable Kelly input; the formula is only as good as the probability estimates feeding it.
- Applying one Kelly fraction across unrelated strategies. Each strategy has its own win rate and payoff ratio — a single blended Kelly number across a multi-strategy portfolio is a common but flawed simplification.
- Ignoring correlation between concurrent positions. Kelly sizing assumes independent bets; several correlated positions sized individually via Kelly can combine into a much larger effective risk than intended.