Kelly Criterion: The Math Behind Optimal Bet Sizing
The Kelly Criterion is a mathematical formula that tells you exactly what percentage of your capital to risk on a trade, given your win rate and your average payoff ratio. It was developed by John L. Kelly Jr. at Bell Labs in 1956 and has since been adopted by gamblers, investors, and traders worldwide.
Used correctly, it maximizes long-term growth. Used incorrectly (or with bad inputs), it can blow up your account. Here is how it works and how to apply it responsibly.
The Formula
f* = (bp - q) / b
Where:
- f* = the fraction of your capital to risk
- b = the ratio of the average win to the average loss (your payoff ratio)
- p = the probability of winning (your win rate as a decimal)
- q = the probability of losing (1 - p)
A Worked Example
Suppose your trading strategy has the following historical statistics:
- Win rate: 55% (p = 0.55)
- Average win: $600
- Average loss: $400
- Payoff ratio: $600 / $400 = 1.5 (b = 1.5)
Plugging into the formula:
f* = (1.5 x 0.55 - 0.45) / 1.5
f* = (0.825 - 0.45) / 1.5
f* = 0.375 / 1.5
f* = 0.25
The Kelly Criterion says to risk 25% of your capital on each trade. That is full Kelly. As you will see below, most practitioners use a fraction of this number.
Run your own numbers with the Kelly Criterion Calculator.
Kelly Criterion Results for Common Scenarios
| Win Rate | Payoff Ratio | Full Kelly | Half Kelly |
|---|---|---|---|
| 40% | 2.0 | 10.0% | 5.0% |
| 45% | 2.0 | 17.5% | 8.75% |
| 50% | 1.5 | 16.7% | 8.3% |
| 55% | 1.5 | 25.0% | 12.5% |
| 60% | 1.0 | 20.0% | 10.0% |
| 50% | 1.0 | 0.0% | 0.0% |
| 40% | 1.0 | -20.0% | Do not trade |
Notice the last two rows. A 50% win rate with a 1:1 payoff gives a Kelly of zero -- there is no edge. A 40% win rate with a 1:1 payoff gives a negative Kelly -- you are expected to lose money on every trade. The formula is telling you not to trade.
Full Kelly vs Half Kelly
Full Kelly maximizes long-term growth mathematically, but it comes with extreme volatility. A full Kelly bettor will frequently experience drawdowns of 50% or more. In practice, this is psychologically unbearable for most people and can lead to panic-driven mistakes.
Half Kelly (risking half the Kelly-recommended amount) reduces the long-term growth rate by only about 25% but cuts volatility in half. This is why the majority of professional traders and fund managers use Half Kelly or even Quarter Kelly:
- Full Kelly: Maximum growth, extreme volatility. Drawdowns of 50%+ are common.
- Half Kelly: 75% of the growth rate, much smoother equity curve. Most recommended for traders.
- Quarter Kelly: Very conservative, suitable for uncertain edge estimates.
If in doubt, use Half Kelly. You give up a small amount of growth in exchange for a much more survivable ride.
When to Use the Kelly Criterion
The Kelly Criterion works best when:
- You have a verified edge. At least 100 trades of historical data showing a positive expectancy. The more data, the more reliable your inputs.
- Your win rate and payoff ratio are stable. If these numbers fluctuate wildly from month to month, the Kelly output will be unreliable.
- You can handle the volatility. Even Half Kelly can produce 20-30% drawdowns. If that will make you deviate from your strategy, use Quarter Kelly or the simpler 1-2% fixed risk rule.
Limitations and Pitfalls
Garbage In, Garbage Out
The Kelly formula assumes you know your exact win rate and payoff ratio. In reality, these are estimates based on past data that may not predict the future. Overestimating your win rate by even 5% can lead to dramatically oversized positions. This is the main reason practitioners use Half Kelly -- it builds in a safety margin for estimation error.
It Assumes Independent Outcomes
Kelly assumes each trade is independent of the others. In trading, this is rarely true. Correlated positions (like holding multiple tech stocks) violate this assumption. If you have three positions that will all lose money in a tech selloff, the Kelly Criterion does not account for that correlation.
It Ignores Practical Constraints
Kelly does not account for commissions, slippage, margin requirements, or the fact that you might have multiple positions open at the same time. These real-world frictions reduce your effective edge and should push you toward a more conservative fraction.
Negative Kelly Means No Trade
If the formula gives a negative number, it means you have a negative expected value. Do not take the trade. Do not try to find a way around it. A negative Kelly means the market has an edge over you on that setup.
Practical Application for Traders
- Track at least 50-100 trades with your strategy (more is better).
- Calculate your win rate and average win/loss ratio from the data.
- Plug the numbers into the Kelly Criterion Calculator.
- Take the result and divide by 2 (Half Kelly).
- Compare with the 1-2% fixed risk rule. Use whichever is smaller.
- Recalculate every month or every 50 trades as your statistics evolve.
You can also use the Expectancy Calculator to verify that your strategy has a positive expected value before applying Kelly sizing.
Kelly Criterion vs Fixed Percentage Risk
The 1-2% fixed risk rule is simpler and safer. The Kelly Criterion is more mathematically optimal but requires accurate inputs and more discipline. For most retail traders, the fixed percentage approach is the better choice. Kelly is most valuable for systematic traders who have large sample sizes and stable statistics.
If you are just starting out, use fixed percentage risk. As your track record grows and your statistics stabilize, consider incorporating Kelly as a sanity check on your position sizes.
The Bottom Line
The Kelly Criterion gives you a mathematically grounded answer to the question "how much should I risk?" But it requires honest, accurate inputs and the discipline to use a fractional Kelly. Treat it as a guide, not a mandate. Combined with proper position sizing and drawdown management, it can significantly improve your long-term results.