What is the Kelly criterion formula?
For a trade or bet that either wins b times the amount at risk or loses all of it, the Kelly fraction is f* = p - (1 - p) / b, where p is the win rate and b is the payoff ratio. An equivalent form is f* = (p x b - (1 - p)) / b: the edge per unit risked, divided by the payoff.
With the default inputs, a 55% win rate and a payoff of 1.5, the formula gives 0.55 - 0.45 / 1.5 = 0.25. The second form agrees: the edge is 0.55 x 1.5 - 0.45 = 0.375 per unit risked, and 0.375 / 1.5 = 0.25. So full Kelly puts 25% of capital at risk on each trade, $2,500 of $10,000. Half Kelly, the default share on this page, is 12.5%, or $1,250.
The formula comes from J. L. Kelly Jr.'s paper "A New Interpretation of Information Rate", published in the Bell System Technical Journal in July 1956 (volume 35, issue 4, pages 917 to 926). Kelly linked the growth of a gambler's capital to the rate at which information reaches him over a communication channel. The fraction that maximizes long-run growth under those assumptions is the one this page computes.
What the inputs mean for a trader
The formula needs two numbers, and both have to come from somewhere real.
- Win rate (p): the share of trades that end in profit. From a trade log, it is winning trades divided by all trades.
- Payoff ratio (b): the average winning trade divided by the average losing trade, both in money. Winners averaging $300 and losers averaging $200 give b = 1.5.
Notice that both are averages of what already happened. Kelly treats them as the true, fixed odds of every future trade, which no trading record can promise. The page therefore describes its answer as arithmetic on your estimates: change the estimates and the answer changes with them, often by a lot.
A quick check before reading any result: Kelly is zero at a win rate of 1 / (1 + b). With b = 1.5 that is 40%, so any win rate below 40% gives no positive fraction at that payoff.
Kelly fraction across win rates and payoff ratios
The table shows full Kelly for common combinations. A dash means the formula gives zero or less, which the calculator reports as no positive edge.
| Win rate | Payoff 1 | Payoff 1.5 | Payoff 2 | Payoff 3 |
|---|---|---|---|---|
| 40% | - | - | 10.00% | 20.00% |
| 45% | - | 8.33% | 17.50% | 26.67% |
| 50% | - | 16.67% | 25.00% | 33.33% |
| 55% | 10.00% | 25.00% | 32.50% | 40.00% |
| 60% | 20.00% | 33.33% | 40.00% | 46.67% |
Two things stand out. Small changes in the win rate move the answer a great deal: at a payoff of 1.5, every 5 points of win rate adds 8.33 points of Kelly. And many of the fractions are large. Putting a quarter or a third of capital at risk on one trade is what the formula says on these inputs, which is why the page also works out a chosen share of it.
Why the win rate estimate matters so much
The bar chart in the result recalculates Kelly at win rates 5 and 10 points either side of yours, with the same payoff. On the default inputs it runs from 8.33% at 45% to 41.67% at 65%.
Suppose your records show a 55% win rate but the true rate over the coming trades is 50%. Full Kelly on your estimate is 25%, while full Kelly on the true rate would be 16.67%, so you would be risking 1.5 times the formula's own answer. Overestimating the edge pushes the size above Kelly. The arithmetic behind the formula shows the cost: on the default inputs, the average log growth per trade, p x ln(1 + b x f) + (1 - p) x ln(1 - f), is about +0.046 at the Kelly fraction of 25% and about -0.004 at twice it, 50% a trade, even though the inputs still show a positive edge.
A short record makes this worse. Twenty trades with 11 winners is a 55% win rate, but so few trades cannot separate a 55% strategy from a 45% one with any confidence. The trading expectancy calculator works from your own list of wins and losses.
Fractional Kelly: half, quarter and other shares
Fractional Kelly multiplies the full fraction by a share you choose: fractional Kelly = f* x share. Half Kelly on the default inputs is 25% x 0.5 = 12.5%, and quarter Kelly is 6.25%.
The reason people use it shows in the second chart. Each loss at a fraction f leaves (1 - f) of capital, so a run of losses compounds quickly:
| Losses in a row | Full Kelly, 25% | Half Kelly, 12.5% |
|---|---|---|
| 5 | 23.73% of capital left | 51.29% left |
| 10 | 5.63% left | 26.31% left |
Choosing a share below 100% also builds in room for the estimation error described above. The calculator does not suggest a share. It shows the arithmetic for whichever you enter, from 1% to 100% of Kelly.
Kelly criterion for stocks and forex: risk is not position size
In trading, the Kelly fraction is the share of capital you stand to lose if the trade fails, which is set by your stop. It is not the value of the position. The payoff ratio already assumes a losing trade costs the full amount at risk.
On the default inputs, half Kelly risks $1,250 of $10,000. If the stop sits 5% below the entry, the position that loses $1,250 at the stop is worth $25,000, two and a half times the capital. At full Kelly, $2,500 at risk with the same stop is a $50,000 position. That conversion from money at risk to shares or lots is what the position size calculator does. Kelly sets the size of the loss; the stop sets how big a position delivers it.
If losing trades in your system lose only part of the amount at risk, or winners and losers vary widely, the simple formula no longer describes it exactly, and the result here is an approximation of your situation.
Kelly in pounds and dirhams
The fraction is the same in every currency; only the amount at risk is stated in yours.
- Pounds. £5,000 of capital, a 45% win rate and a payoff of 2. Kelly is 0.45 - 0.55 / 2 = 17.5%, or £875 at risk per trade. Quarter Kelly is 4.375%, or £218.75. Kelly reaches zero at a 33.33% win rate on this payoff.
- Dirhams. AED 40,000 of capital, a 52% win rate and a payoff of 1.25. Kelly is 0.52 - 0.48 / 1.25 = 13.6%, or AED 5,440. Half Kelly is AED 2,720. The zero point here is 44.44%, only 7.56 points below the win rate entered.
Choose any of 24 currencies from the selector. The calculator never converts between them.
How to calculate the Kelly criterion in Excel or Google Sheets
Put the win rate in B2 as a decimal (0.55) and the payoff ratio in C2. Full Kelly is =B2-(1-B2)/C2, which returns 0.25 for the default inputs; format it as a percentage. Half Kelly is =MAX(0,B2-(1-B2)/C2)*0.5, where MAX stops a negative result turning into a negative size. With capital in D2, the amount at risk is =MAX(0,B2-(1-B2)/C2)*D2.
To build both inputs from a trade log with results in column F, use =COUNTIF(F:F,">0")/COUNT(F:F) for the win rate and =AVERAGEIF(F:F,">0")/-AVERAGEIF(F:F,"<0") for the payoff ratio.
What the Kelly result leaves out
- Uncertainty in the inputs. The formula treats p and b as known. Yours are estimates.
- Changing conditions. A win rate measured in one market phase may not hold in the next.
- Several open trades. Kelly here is for one trade at a time. Correlated positions open together behave like one larger trade.
- Costs and slippage. Enter averages that already include them, or the edge looks larger than it is.
- Gaps through the stop. A loss bigger than the amount at risk breaks the formula's assumption about losses.
When the formula returns zero or less, the page says no positive edge on these inputs and sizes nothing. That is a statement about the numbers entered, not a judgment of any strategy.
Glossary
Kelly fraction (f*): the share of capital the formula puts at risk on one trade. Win rate (p): the share of trades that make money. Payoff ratio (b): the average win divided by the average loss. Edge per unit risked: p x b - (1 - p), the average result for each unit put at risk. Fractional Kelly: a chosen share of the Kelly fraction. Amount at risk: what a losing trade costs if it reaches the stop.
Frequently asked questions
What is the Kelly criterion formula?
f* = p - (1 - p) / b, where p is the win rate and b is the payoff ratio of average win to average loss. With p = 0.55 and b = 1.5 the result is 0.25, so the formula puts 25% of capital at risk per trade.
How do I use the Kelly criterion for trading?
Measure your win rate and the ratio of your average winning trade to your average losing trade, enter both, and read the fraction of capital the formula puts at risk. Then use your stop to turn that amount at risk into a position size.
What is half Kelly?
Half the full Kelly fraction. If full Kelly is 25%, half Kelly is 12.5%. A smaller share makes a losing run less damaging and leaves room for a win rate or payoff that turns out lower than estimated.
What does a negative Kelly result mean?
It means the inputs show no positive edge: the win rate is too low for the payoff, so the formula sizes nothing. At a 40% win rate and a payoff of 1.2 the result is -10%, and Kelly only turns positive above a 45.45% win rate.
Is the Kelly fraction my position size?
No. It is the share of capital you would lose if the trade hits its stop. A 12.5% Kelly fraction on $10,000 is $1,250 at risk; with a stop 5% from the entry, the position carrying that risk is worth $25,000.
How sensitive is Kelly to the win rate?
Very. At a payoff of 1.5, Kelly is 16.67% at a 50% win rate and 25% at 55%. Overestimating the win rate by 5 points makes you risk 1.5 times the formula's answer for the true rate.
Where does the Kelly criterion come from?
From J. L. Kelly Jr.'s 1956 paper "A New Interpretation of Information Rate" in the Bell System Technical Journal. It related the growth of a gambler's capital to the rate of information received over a channel.
How do I calculate the Kelly criterion in Excel?
With the win rate as a decimal in B2 and the payoff ratio in C2, use =B2-(1-B2)/C2. Wrap it as =MAX(0,B2-(1-B2)/C2) so a negative result reads as zero, and multiply by your share of Kelly.
What win rate makes the Kelly fraction zero?
1 / (1 + b), where b is the payoff ratio. At a payoff of 1 it is 50%, at 1.5 it is 40% and at 2 it is 33.33%. Below that win rate the formula gives no positive fraction.
Can I use this Kelly calculator in pounds or dirhams?
Yes. The fraction does not depend on currency, and the optional capital field turns it into an amount at risk in any of 24 currencies. £5,000 at a 17.5% Kelly fraction is £875 at risk per trade.
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Last reviewed 13 November 2026 by the NIFM Academy editorial team. Next review by 13 May 2027.
- J. L. Kelly Jr., A New Interpretation of Information Rate, Bell System Technical Journal, volume 35, issue 4, pages 917 to 926, July 1956, bibliographic details confirmed from the publisher's DOI record 10.1002/j.1538-7305.1956.tb03809.x. Checked 13 November 2026.
- Kelly criterion arithmetic, f* = p - (1 - p) / b = (p x b - (1 - p)) / b; fractional Kelly = f* x share; zero at a win rate of 1 / (1 + b); capital after k losses = (1 - f)^k; average log growth per trade = p x ln(1 + b x f) + (1 - p) x ln(1 - f); shown in the working under every result. Checked 13 November 2026.