What Is the Kelly Criterion? How Much to Risk on Each Trade

How much should you risk on one trade? Most of the answers you'll hear are rules of thumb: 1% of the account, maybe 2%.

The Kelly criterion is an answer worked out with maths. John Kelly came up with it at Bell Labs in 1956, and gamblers and investors have used it ever since. It gives you the bet size that grows your money fastest.

It also shows why betting big is a mistake, even when your strategy works. Here's the formula, what happens when you bet more or less than it says, and why almost nobody bets the full amount.

The Kelly formula

Kelly needs two numbers. Your win rate is how often you win. Your payoff, R, is your average win divided by your average loss.

Kelly % = win rate − (loss rate ÷ R)

Take a strategy that wins 40% of the time at a 1:2 risk-reward ratio. Its wins are twice the size of its losses. That's 0.4 − 0.6 ÷ 2 = 0.1.

So Kelly says risk 10% of the account on each trade.

If the formula gives zero or less, the strategy has no edge, and Kelly says risk nothing. A strategy that wins 30% at 1:2 gives 0.3 − 0.7 ÷ 2 = −0.05.

Same edge, different Kelly

Kelly also works out as your edge divided by R. Your edge is what the strategy makes per trade, on average, in R.

So two strategies can make the same money per trade and still get very different Kelly sizes. Here are three that each make +0.2R a trade.

The strategy that wins least often gets the smallest bet. That makes sense, because it also has the longest losing streaks. In 100 trades, its longest run of losses is typically 15 in a row.

Risk more than Kelly and you make less

Here's the surprising part. Risking more than Kelly doesn't make more money. It makes less.

We worked out the typical result of 100 trades at 40% wins and 1:2, risking different amounts each time. The typical result is what you end with if you win exactly 40 of the 100.

At 10% a trade, the typical account grows 164%. At 20%, twice Kelly, it grows just 7%. At 30%, it loses 93%.

That's because losses hurt more than gains help. Lose 30% and you need a 43% gain to get back. Lose 50% and you need 100%. The bigger each bet, the harder each loss is to climb back from.

Half Kelly gives up surprisingly little. Risking 5%, the account grows 0.73% a trade, against 0.97% at full Kelly. That's 76% of the growth for half the risk.

The catch: big drops along the way

Kelly grows the account fastest. It doesn't grow it smoothly.

To see the ride, we ran 10,000 accounts through 100 trades each, at every risk size. Every trade had a 40% chance of winning 2R and a 60% chance of losing 1R.

At full Kelly, the typical account's worst fall from a high was 69%. 91% of accounts fell by half from a high at some point, and 37% dropped to half the money they started with.

At half Kelly, the typical worst fall was 41%, and 26% of accounts fell by half from a high. At 2% a trade, the typical worst fall was 18%. Just 3 accounts in 10,000 fell by half.

Few people can watch an account fall 69% and keep following the plan. Kelly's maths doesn't account for that.

You don't know your real win rate

Kelly needs your real win rate. All you have is an estimate.

Say you've taken 100 trades and won 40. Your real win rate could be anywhere from about 30% to 50%. That's just the luck in 100 trades. To pin it down to within 2 points either way takes about 2,300 trades.

Getting it wrong costs far more in one direction. Say you size for 40% and risk 10%, but your real win rate is 35%. The strategy still makes money: +0.05R a trade. But at 10% a trade, the typical account loses 37% over 100 trades.

At 5% a trade you'd roughly break even, and at 2% you'd still make 6%. At a 35% win rate, your real Kelly is only 2.5%. Five points off the win rate took three quarters off Kelly.

Smaller bets also need a smaller win rate to make money. Risking 10% a trade loses money below a 36.6% win rate. Risking 2% only loses below 34.0%. The strategy itself stops making money below 33.3%.

So betting too big can turn a winning strategy into a losing account. Betting too small just makes less. That's why traders who use Kelly bet well under it.

Kelly and the 1% rule

Many traders risk 1% or 2% of the account on a trade. For our 40% strategy, that's a tenth to a fifth of Kelly. It can look too careful. It isn't, for three reasons.

• You don't know your edge. Betting under Kelly costs a little growth. Betting over it can lose money.
• Kelly is worked out for one trade at a time. Five trades open at once, each risking 10%, puts half the account on the line together. Trades in the same market often win and lose together, too.
• Costs shrink your edge. Spreads and commissions come off every trade, and a smaller edge means a smaller Kelly.

So use Kelly as a ceiling, not a target. If you're risking more than half your Kelly, you're betting a lot on your estimate being right.

How to use it

• Record your trades in R. You need your real win rate and your real average win and loss, not the ones you planned.
• Work out Kelly: win rate − (loss rate ÷ R). If it's zero or less, there's no edge to size.
• Treat full Kelly as the most you'd ever risk. Half Kelly or less is the usual choice.
• Count everything that's open. Kelly applies to all your risk at once, not to each trade on its own.
• Turn the percentage into a position size: the money you're risking divided by the distance to your stop.
• Work it out again as results come in. If your win rate slips, your Kelly falls much faster.

Method, and what this cannot tell you

The Kelly fractions, the typical results and the chance of finishing behind are worked out exactly, with no simulation. Over 100 trades, where an account ends up depends only on how many trades it won, so we added up every possible number of wins. The typical result is what you end with if you win exactly your win rate's share: 40 trades out of 100 at 40%. For the drops along the way, the order of the wins and losses matters, so we simulated 10,000 accounts making 100 trades each at every risk size. We used a fixed random seed, so anyone can run it again and get the same answer.

• Every trade here either wins 2R or loses 1R. Real trades vary, and a gap can make a loss bigger than 1R.
• Each trade is independent of the last. Real results tend to come in streaks, when a strategy stops suiting the market for a while.
• There are no costs. Spreads and commissions shrink your real edge, and your real Kelly with it.
• One trade is open at a time. Several open at once, in markets that move together, add up to a bigger bet than any one of them.