Kelly Criterion Calculator

How the number is worked out

Two inputs carry the whole calculation: how often you win, and how much you make when you do compared with what you lose when you do not. Everything else on the page is those two numbers rearranged.

The one thing to read before you use any of it: f is the amount you stand to lose, expressed as a share of the balance. It is not the notional size of the trade. If a losing trade costs you the whole amount you put up, the two are the same number; if a losing trade costs you 10% of the position, the position is ten times this figure. That distinction is where most misreadings of Kelly come from, and it is the one this page states rather than leaves for you to guess.

The break-even win rate is worth reading alongside the answer, because it is the number that does not depend on your estimate. At a payoff ratio of 2, break-even is 1 ÷ 3 = 33.33%: below that win rate the formula returns a negative share, which is its way of saying the trade is not worth taking. The gap between your win rate and the break-even rate is your edge, and everything else is that gap multiplied out.

What the Kelly fraction field reads as

A kelly calculator that takes a fraction of Kelly has to say what unit that field is in, and most do not. A field labelled "Kelly fraction" can mean either of two numbers that differ by a factor of a hundred: type 50 meaning "half" into a field read as a multiplier and you have asked for fifty times full Kelly — 5,000% of your balance.

This page makes you choose, and then echoes your number back in both units so there is no ambiguity left in the result:

• Multiplier — 1 is full Kelly, 0.5 is half, 0.25 is a quarter. Type 50 here and you are asking for fifty times full Kelly; the page will say so in words before you act on it.
• Percent of full — 100 is full Kelly, 50 is half, 25 is a quarter.

Half Kelly and quarter Kelly are printed alongside the full figure so the three can be compared side by side. The page does not choose between them for you: full Kelly is the fraction that maximises the growth rate and nothing else, and it maximises it under the assumption that you know your win rate exactly. Sizing below it buys a smoother ride at a lower growth rate, and how much that trade is worth is a judgement about you, not about arithmetic.

Worked example: 55% at 2:1 on a 10,000 balance

You win 55% of the time and make 2 for every 1 you risk. The balance is 10,000.00 USD and the fraction is set to 0.5 — half Kelly.

1. q = 1 − 0.55 = 0.45.
2. Full Kelly: f = 0.55 − 0.45 ÷ 2 = 0.55 − 0.225 = 0.325, so 32.50% of the balance, or 3,250.00 USD at risk.
3. Half Kelly: 0.5 × 32.50% = 16.25%, or 1,625.00 USD at risk. Quarter Kelly is 8.13%, or 812.50 USD.
4. Expected value: 0.55 × 2 − 0.45 = 0.65 per 1 risked.
5. Break-even win rate: 1 ÷ (1 + 2) = 33.33%. Your 55% sits 21.67 points above it.
6. Growth at half Kelly: g = 0.55 × ln(1 + 0.1625 × 2) + 0.45 × ln(1 − 0.1625) = 0.55 × ln(1.325) + 0.45 × ln(0.8375) = 0.1548 − 0.0798 = 0.0749, and e^0.0749 − 1 = 7.79% per trade. At full Kelly the same inputs give 10.36%.

Step 6 is the honest argument for sizing below full Kelly, and it is visible rather than asserted: halving the fraction here costs about a quarter of the growth rate. What it buys is not printed, because it is not arithmetic — it is the difference between a drawdown you can sit through and one you cannot.

The same formula on decimal odds

Decimal odds and a payoff ratio are the same number wearing different clothes. Decimal odds of 2.50 mean you get 2.50 back for every 1 staked, including your stake, so the net payoff is 1.50 for every 1 at risk — and b in the formula is 1.50, not 2.50.

A 45% chance at odds of 2.50 on a 1,000.00 balance:

• b = 2.50 − 1 = 1.50; q = 0.55.
• Full Kelly: f = 0.45 − 0.55 ÷ 1.50 = 0.45 − 0.3667 = 8.33%, or 83.33 at risk.
• Break-even win rate: 1 ÷ (1 + 1.50) = 40.00%. Your 45% is five points of edge.
• Expected value: 0.45 × 1.50 − 0.55 = 0.13 per 1 risked; growth 0.52% per event.

Notice what five points of edge buys: 8.33% of the balance. That is the shape of the whole formula near break-even — the share is small, and it stays small until the edge is large. It is also why the five-points-lower row exists on the results panel: at this input it takes the answer from 8.33% to 0.00%, because 40% is exactly break-even.

When the answer is larger than your whole balance

It can be, and this page will print it rather than quietly trimming it. A 90% win rate at a payoff ratio of 5 gives f = 0.90 − 0.10 ÷ 5 = 0.88, so 88.00% of the balance at risk, with a break-even win rate of 16.67%.

Take that same setup and say a losing trade costs you only 10% of the position — a wide, soft stop — and the position itself is 0.88 ÷ 0.10 = 880% of the balance. Both figures are the same formula. The first is what you risk, the second is what you hold, and confusing them is how a plausible-looking percentage turns into a margin call.

Nothing here caps the answer, on purpose. A cap is a second opinion wearing the costume of a calculation, and this page would rather hand you the arithmetic and say plainly what it means: at or above 100% of the balance, the figure is leverage you would have to borrow, the per-trade growth rate stops being defined because the formula would be taking the logarithm of a non-positive number, and the honest reading is that the inputs describe an opportunity you almost certainly do not have.

Rounding, precision and what is not capped

The rules, stated rather than left to be discovered from the display:

• Shares are shown to two decimal places — 8.13%, not 8.125%.
• Money is shown to two decimal places, rounded half away from zero, the way a bank statement rounds.
• Growth is shown to two decimal places, as e^g − 1, with g in natural logarithms.
• No intermediate value is rounded before it is used. The full-precision share is what goes into the money figure, the growth figure and the copy text; only the number you see is rounded.
• Nothing is capped. There is no hidden maximum, no clamp at 25% or at 100%, and no floor that turns a negative Kelly into a small positive one. If the arithmetic says 880%, it prints 880% and flags it.
• Anything within a ten-trillionth of zero is treated as zero, so the page never prints −0.00% where the answer is exactly nothing.

One thing this number is not: a lot size. Kelly returns money at risk. Turning that into units requires the instrument, the distance to your stop, the spread and the commission, and that is a different calculation with different inputs — the position size calculator does it from a risk budget, and the forex version does it from a stop distance in pips.

Where each number comes from

The win rate is the input most likely to be wrong, and the page says so with a number rather than a warning. The standard error of a win rate measured over n trades is √(p(1−p) ÷ n): at 55% over 100 trades that is about 5 percentage points, which is why the results panel carries a row showing the answer five points lower. Everything else — the payoff ratio, and whether the future resembles the past at all — is held constant in that row, and stated as such.

What this page will not tell you

• Whether your win rate is real. The formula treats the number you type as a fact. It is an estimate, and the estimate has error in it.
• Whether your trades are independent. Kelly assumes each one is. Ten positions opened on the same view are not ten independent events, and the formula has no way to know.
• Whether the payoff ratio holds. An average win ÷ average loss from a hundred trades is not a distribution. Fat tails on either side move the answer.
• Anything about margin, leverage or financing. Kelly knows nothing about margin requirements, borrow costs or overnight charges. At large fractions those dominate.
• A lot size, a stop distance or a take-profit level. Those come from the instrument and the chart, not from this formula.
• What to do. It prints the arithmetic and the meaning of each figure. The fraction is yours to choose.

What people call this

A kelly formula calculator, a kelly calculator, a kelly criterion position sizing tool and a half kelly calculator are all the same question: given how often I win and how much I make when I do, what share of the balance goes on the next one. This page answers all four in the same place, and prints full, half and quarter Kelly together so the fraction is a comparison rather than a hunt.

The two halves of the question come from two different worlds, which is why both are accepted here. Traders think in a payoff ratio — average win over average loss — and get it from their own closed trades. Bettors think in decimal odds, and get it from a price. Subtract 1 from the odds and the two become the same number, so one formula covers both.

Kelly is one answer to "how much", and it is the answer that starts from your edge. The other starts from what you can afford to lose: the risk-per-trade percentage page works that one through, and the position size calculator turns either into units.

FAQ

Related pages

• Position size calculator — turn a risk budget into units, with the spread and commission inside it.
• How much to risk per trade — the other answer to "how much", and the losing-streak arithmetic behind it.
• Forex position size calculator — size from a stop distance in pips.
• Forex profit calculator — what a closed trade actually returned after costs.
• Pip calculator — what one pip is worth at each lot size.
• Sizing on a small account — when the percentage is smaller than the minimum step.