What is the Kelly criterion — and why does the “optimal” bet size ruin most traders?
Almost every serious discussion of position sizing ends up at the Kelly criterion, because it answers the question every trader secretly wants answered: not “how little can I risk?” but “how much should I risk to make the most money?” It gives a precise number. This page shows where that number comes from, what happens when you actually bet it, and why the people who understand it best almost never use it at full strength.
KEY TAKEAWAYS
- Kelly for a trade that wins with probability W and pays R times the risk is W − (1 − W) ÷ R. For 50% wins at 1.2:1 that is 8.3% of the account per trade.
- Full Kelly maximises long-run growth and still hit a 50% drawdown in 89.5% of 200-trade runs. Maximum growth and a tolerable ride are different goals.
- Half Kelly keeps 75% of the growth rate while cutting that drawdown risk to 22.8%. Quarter Kelly keeps 44% of the growth at 0.4%.
- Win rate three points lower than you think (47% instead of 50%) and full Kelly shrinks the median account to 0.77× — on a system that still has a positive edge. Half Kelly still grows it.
- After 100 trades your measured Kelly could honestly be anywhere from −9.6% to +26.3%. The formula is exact; its inputs are not.
What is the Kelly criterion, in one paragraph?
The Kelly criterion is a formula, published by John L. Kelly Jr. of Bell Labs in 1956, for how much of your capital to put at risk on a repeated bet with a known edge so that your capital compounds as fast as possible over the long run. It sits in a narrow gap between two ways to fail. Bet too little and the edge compounds slowly. Bet too much and the losses, which cost more in percentage terms than equal wins make back, drag the account down even though every individual bet is favourable. Kelly is the exact point between the two.
The key word is compounds. Kelly does not maximise your average profit per trade — that is maximised by betting everything, which also guarantees you eventually lose everything. It maximises the growth rate of the account: the typical result of a long sequence, where each trade’s gain or loss is applied to whatever you have at the time.
How do you calculate it?
For trading, where a winning trade makes R times what a losing trade costs, the formula is short:
W = win rate · R = average win ÷ average loss · result = fraction of the account to risk per trade
Three systems, worked through:
| System | Win rate | Reward : risk | Expectancy per trade | Kelly: risk per trade |
|---|---|---|---|---|
| Thin but real edge | 50% | 1.2 : 1 | +0.10R | 0.50 − 0.50 ÷ 1.2 = 8.3% |
| Strong edge | 45% | 2 : 1 | +0.35R | 0.45 − 0.55 ÷ 2 = 17.5% |
| Break-even | 40% | 1.5 : 1 | 0.00R | 0.40 − 0.60 ÷ 1.5 = 0% |
Two things are worth noticing before going further. First, when the expectancy is zero, Kelly is zero — the formula refuses to bet on a system with no edge, and goes negative for a losing one. That is a genuinely useful sanity check. Second, the numbers it produces are large. Anyone who has read our 1–2% rule will see 8.3% and 17.5% and flinch. The rest of this page is about whether that flinch is right.
Behind the formula is one quantity, the growth per trade: W × ln(1 + f×R) + (1 − W) × ln(1 − f), where f is the fraction risked. The Kelly fraction is simply the f at which that curve peaks. Plot it and you get the hill at the top of this page.
What happens if you actually bet full Kelly?
You get the fastest growth — and a ride almost nobody can sit through. Here is the thin-edge system (50% wins, 1.2:1) at full Kelly and at fractions of it. Each row is 200,000 simulated sequences of 200 trades, compounding on current equity. “Ruin” means a 50% drawdown from the running peak at any point, the same definition used in our risk of ruin page and simulator.
| Bet size | Risk per trade | Share of max growth | Median ending account | Chance of a 50% drawdown | Chance of ending below start |
|---|---|---|---|---|---|
| 2× Kelly | 16.7% | 0% | 1.00× | 100% | 48.3% |
| 1.5× Kelly | 12.5% | 75% | 1.87× | 99.6% | 35.9% |
| Full Kelly | 8.3% | 100% | 2.29× | 89.5% | 26.3% |
| ¾ Kelly | 6.3% | 94% | 2.18× | 64.6% | 21.7% |
| ½ Kelly | 4.2% | 75% | 1.87× | 22.8% | 17.9% |
| ¼ Kelly | 2.1% | 44% | 1.44× | 0.43% | 14.4% |
| ⅛ Kelly | 1.0% | 24% | 1.22× | 0.00% | 11.4% |
At full Kelly the typical account more than doubles over 200 trades. It is also the setting at which nine accounts in ten lose half their value at some point along the way, and more than a quarter finish below where they started despite trading a system with a real edge the entire time. That is not a flaw in the simulation. It is what maximum growth looks like from inside.
The strong-edge system makes the point more brutally. Full Kelly there is 17.5% per trade; in 200,000 runs, every single one hit a 50% drawdown. Even half Kelly (8.75%) did so 84.6% of the time. The stronger your edge looks on paper, the larger Kelly tells you to bet, and the more violent the path becomes.
Why does half Kelly keep most of the growth?
Because the top of the Kelly hill is flat. Near the peak, growth changes very little as you move away from it; the cost of stepping back rises slowly while the reduction in drawdown is immediate. Halving the bet gives up 25% of the growth rate. It cuts the chance of a 50% drawdown from 89.5% to 22.8%. Quartering it gives up 56% of the growth and brings that chance down to 0.43%.
There is a second, less obvious reason fractions win. The hill is symmetric in growth but not in danger. Betting 1.5× Kelly produces exactly the same growth rate as betting 0.5× Kelly — 75% of maximum in both cases — yet the drawdown risk is 99.6% on one side and 22.8% on the other. Overshooting the peak buys you nothing that undershooting does not, and it costs you the account. When you are unsure where the peak is, the only rational direction to err is down.
And you are always unsure where the peak is. That turns out to be the entire story.
What if your win rate is a few points off?
Kelly assumes you know your win rate and payoff exactly. Suppose you have measured 50% at 1.2:1 and sized at full Kelly, 8.3% — but the system’s true win rate is lower. Every row below is still a profitable system down to 46%; only the last is a losing one.
| True win rate | True expectancy | True Kelly | Median account at 8.3% (full) | Median account at 4.2% (half) | Ruin at full / half |
|---|---|---|---|---|---|
| 50% | +0.100R | 8.3% | 2.29× | 1.87× | 89.5% / 22.8% |
| 49% | +0.078R | 6.5% | 1.59× | 1.55× | 92.9% / 30.1% |
| 48% | +0.056R | 4.7% | 1.11× | 1.29× | 95.2% / 38.3% |
| 47% | +0.034R | 2.8% | 0.77× | 1.08× | 97.0% / 47.0% |
| 46% | +0.012R | 1.0% | 0.53× | 0.90× | 98.2% / 56.4% |
| 45% | −0.010R | none | 0.37× | 0.75× | 99.0% / 65.4% |
Read the 47% row slowly. The system still makes money on average — +0.034R a trade. A trader risking 1% on it would grow the account. But sized at the Kelly fraction for the 50% they thought they had, the typical account ends at 77% of where it started. A winning system, sized “optimally,” loses money. At 48% the full-Kelly bet is already 1.8 times the true Kelly and growth has almost vanished (1.11×); by then half Kelly is ahead.
This is the practical reason professional users of Kelly almost always bet a fraction of it. It is not timidity. The error in your inputs pushes you past the peak, and past the peak is where the growth disappears and the drawdowns take over.
How many trades before you can trust your Kelly number?
Far more than most traders ever record. A win rate measured from a sample is an estimate with a margin of error, and Kelly amplifies that error because it subtracts two uncertain numbers. Here is the range your measured win rate could plausibly sit in (95% confidence) if the truth were exactly 50%, and the Kelly fraction each end of that range would give you at 1.2:1:
| Trades recorded | Plausible win-rate range | Kelly at the low end | Kelly at the high end |
|---|---|---|---|
| 30 | 32.1% – 67.9% | −24.5% | +41.1% |
| 50 | 36.1% – 63.9% | −17.1% | +33.7% |
| 100 | 40.2% – 59.8% | −9.6% | +26.3% |
| 200 | 43.1% – 56.9% | −4.4% | +21.0% |
| 500 | 45.6% – 54.4% | +0.3% | +16.4% |
| 1,000 | 46.9% – 53.1% | +2.7% | +14.0% |
After 100 trades — more than most people have logged when they start asking about Kelly — the honest answer to “what is my Kelly fraction?” is “somewhere between do-not-trade and bet a quarter of the account.” Even after 1,000 trades the range is still wide enough to put you past the peak. Payoff ratio carries its own error on top of this; the table is the optimistic version. It is also why a trading journal is not optional: without one you do not have an estimate at all, only a feeling.
Is Kelly your position size or your risk?
Your risk — and mixing the two up is how a sensible Kelly fraction turns into a liquidation. The formula returns the fraction of the account you lose if the stop is hit. The position you open to achieve that depends on how far away the stop is.
Worked example: a $10,000 account, half Kelly on the thin-edge system, so 4.2% risk = $420 per trade. If the stop is 5% below entry, the position is $420 ÷ 0.05 = $8,400, 84% of the account. If the stop is 2% away, the position is $21,000 — 2.1× the account, which on a derivatives venue means leverage. Same Kelly fraction, very different exposure, and in the second case a gap through the stop costs a great deal more than 4.2%. The position size calculator does this conversion; do it every time, and never read a Kelly percentage as a leverage setting.
What the Kelly criterion is NOT
It is not a target. Full Kelly is a ceiling on sensible aggression, not a place to aim. Every point on the table above that sits at or past it is dominated by a smaller bet with less risk and similar or better growth.
It is not a measure of how good your system is. A high Kelly fraction can come from a genuine edge or from a small sample that happened to run hot. The formula cannot tell the difference, and it will size you up for either.
It is not a promise about any single path. It describes the growth rate of a very long sequence. Over the 200 trades a trader might take in a year, 26% of full-Kelly paths ended in a loss.
It is not a reason to ignore the 1–2% rule. Read properly, it is the explanation for it. One percent on the thin-edge system is exactly one-eighth Kelly: 24% of the maximum growth rate, zero 50% drawdowns in 200,000 runs, and a very large margin for the fact that you do not really know your win rate.
Where this reasoning breaks down
1. Results are assumed independent. Real trading results cluster — trend systems lose in runs when conditions change. Our risk of ruin page shows clustering alone can multiply drawdown risk by 81 at an unchanged win rate. Clustering effectively lowers your usable Kelly fraction, and nothing in the formula knows it is happening.
2. Losses are assumed to cost exactly what you planned. Slippage, gaps and failed stops make some losses bigger than 1R. Kelly is especially sensitive to the size of the worst outcome, so a fat left tail should push you further down the hill than any table here suggests.
3. It assumes one bet at a time. Three correlated positions at half Kelly each are one large bet at 1.5× Kelly on the day they move together — the far side of the peak.
4. When the advice to “bet a fraction” is itself wrong. If you have thousands of independent, well-measured outcomes and a cap on the worst loss — closer to a market-maker or a casino than a discretionary trader — betting near Kelly is defensible, and staying at one-eighth leaves real growth unclaimed. Almost no individual crypto trader is in that position. If you are not sure whether you are, you are not.
5. It says nothing about the risks you cannot model. An exchange halting withdrawals or a stablecoin depegging is not in your win rate. Those argue for staying well below any fraction the maths recommends.
Where should you go from here?
Put your own numbers in rather than ours: the risk of ruin simulator will show what a given risk setting does to drawdowns with your win rate and payoff, which is the half of the picture Kelly leaves out. Then read what risk of ruin is for the survival side of the same maths, position sizing for turning a percentage into an order, and risk-to-reward ratio for the R in the formula. If you do not yet have 100 recorded trades, the most valuable next step is not a formula at all — it is starting a journal so that one day you have inputs worth feeding it.
FAQ
What is the Kelly criterion in simple terms?
It is a formula for the share of your account to risk on each trade so that the account grows as fast as possible over many trades. For a trade that wins with probability W and pays R times what it risks, the Kelly fraction is W minus (1 minus W) divided by R. For a 50% win rate at 1.2 to 1 it gives 8.3% per trade.
Should I use full Kelly for trading?
Almost certainly not. In 200,000 simulated 200-trade runs of a system winning 50% at 1.2 to 1, full Kelly produced a 50% drawdown in 89.5% of runs. And because you never know your true win rate exactly, full Kelly on an overestimated edge can shrink the account even when the system is profitable.
What is half Kelly?
Half Kelly means risking half the fraction the formula gives. Because the growth curve is flat near its peak, half Kelly keeps about 75% of the maximum growth rate while greatly reducing drawdowns - in our simulation, from an 89.5% chance of a 50% drawdown to 22.8%. Many practitioners go further, to a quarter or less.
What if the Kelly formula gives a negative number?
A negative Kelly fraction means your measured expectancy is negative: the system loses money on average. The formula is telling you not to take the trade at all. With a small sample, it can also mean your data is too thin to say anything yet.
How is Kelly related to the 1-2% rule?
They are not in conflict. For a realistic thin edge such as 50% wins at 1.2 to 1, risking 1% per trade is about one-eighth Kelly. It keeps roughly a quarter of the maximum growth rate while making deep drawdowns extremely rare and leaving a wide margin for error in your estimated win rate.