Risk & Money Management · Pro · 5 min
The Kelly criterion
You can calculate the Kelly fraction for a simple bet, explain why betting more than Kelly destroys growth and why traders use a fraction of it.
How much on each flip?
Suppose you had a coin that came up heads 55% of the time, and every bet paid even money. You have a real advantage. How much of your money would you bet on each flip?
Bet too little, and your money grows slowly. Bet too much, and a few unlucky flips can wipe out the advantage. There is a mathematical answer to this question, and it says a lot about risk in trading, mostly about its limits.
A peak at ten percent
In 1956, John L. Kelly Jr., a researcher at Bell Labs, published a paper on how a gambler with an edge should size bets to make the money grow as fast as possible over the long run. The result is known as the Kelly criterion.
For a bet that pays even money, the Kelly fraction is simple:
Kelly fraction = chance of winning − chance of losing.
With a 55% chance of winning: 0.55 − 0.45 = 0.10, so 10% of the current bankroll on each bet. For bets that pay b times the stake, the general form is: chance of winning − chance of losing ÷ b.
Kelly maximizes the long-term growth rate, and the growth rate falls off on both sides of that peak. For the 55% coin, calculated per bet:
- Betting 5%, half of Kelly, gives about 75% of the maximum growth, with far smaller swings.
- Betting 10%, full Kelly, gives the maximum growth, but with deep drawdowns along the way.
- Betting 20%, double Kelly, brings the growth rate to roughly zero.
- Betting more than that makes the expected growth negative, even though every single bet is in your favor.
That is why many traders who use the idea at all use fractional Kelly: a fixed share of the Kelly fraction, such as a half or a quarter.
Kelly also has a serious limit in trading. It needs known odds and known payoffs. A trader's win rate and average win are only estimates from past trades, and they change. If you overestimate your edge, full Kelly turns into overbetting. With realistic, uncertain numbers, the Kelly fraction is best seen as an upper limit, far above what most traders actually risk.
Double Kelly ends lower
Here are 40 fictional even-money bets from the 55% coin, with 22 wins and 18 losses, starting from 10,000.
- Half Kelly, 5% per bet: ends at about 11,600, with gentle swings.
- Kelly, 10% per bet: ends at about 12,200, with clearly larger swings.
- Double Kelly, 20% per bet: ends at about 9,900, below the start, after the widest swings of all.
All three won exactly the same bets. Betting more did not mean growing more. It meant larger losses on the losing bets, and the losses outweighed the extra gains.
Common mistakes
- Plugging in a backtest win rate. Past results overstate the edge more often than they understate it.
- Treating full Kelly as safe. Even with known odds, it produces deep drawdowns.
- Ignoring correlated positions. Kelly per trade can add up to far too much in total.
- Forgetting that Kelly assumes an edge. Without an actual advantage, the Kelly fraction is zero.
The takeaway
The Kelly criterion gives the bet size that maximizes long-term growth when odds are known. In trading they are not, so traders use a fraction of Kelly or less.
Check yourself
Quick quiz
4 questions. Instant feedback, nothing is saved.
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1Tap the line that bet too much and ended below the start.
Tap the chart.
Betting twice the Kelly fraction brings the long-term growth rate down to about zero, here even slightly below, despite winning 22 of 40 bets.
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2An even-money bet wins 55% of the time. What is the Kelly fraction?
For even-money bets, Kelly = p − q = 0.55 − 0.45 = 0.10.
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3What happens to long-term growth when you bet far more than the Kelly fraction?
Larger bets increase swings so much that losses outweigh the extra gains. Beyond about twice Kelly, growth turns negative.
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4Why do traders rarely bet the full Kelly fraction?
Kelly needs known odds. Trading odds are uncertain and change, so a fraction of Kelly protects against overestimating the edge.
of 4 correct.