Kelly criterion calculator: the formula's arithmetic

Win rate (%)An estimate: the share of bets or trades that win

Win/loss ratioAn estimate: the average win ÷ the average loss, such as 1.5

Kelly fraction (formula result)—

Arithmetic: p − q ÷ b—

Edge per 1 staked: p × b − q—

Break-even win rate: 1 ÷ (1 + b)—

What the sign means—

All default values are hypothetical. The Kelly fraction assumes independent bets with known odds, while a win rate and a win/loss ratio are estimates, so the result is arithmetic on those estimates and not a recommended bet size.

Kelly criterion formula

For a bet that wins b times the stake with probability p and otherwise loses the stake, the Kelly fraction is the share of capital that the formula stakes on each bet:

f* = p − q ÷ b, where q = 1 − p

The same result is the edge divided by the odds, f* = (p × b − q) ÷ b, where p × b − q is the expected result per unit staked. A zero or negative f* means the inputs show no edge, and the formula then stakes nothing. The calculator also shows the break-even win rate, 1 ÷ (1 + b), at which the edge is zero.

J. L. Kelly Jr. derived the rule in 1956 for a gambler betting a fraction of capital on repeated bets, as the fraction that maximizes the long-run growth rate of that capital. For an even-money bet, where b is 1, it reduces to p − q.

Worked example

A five-point change in the estimated win rate moves the fraction by more than eight points, and at 35% the sign turns negative.

What the Kelly formula assumes

The formula is exact only under conditions that trading records rarely meet.

• Known odds: p and b are taken as true values. From past trades they are estimates with sampling error, and the market they came from changes.
• Independent, repeated bets: each bet is the same game with the same odds, and one outcome does not affect the next.
• Two outcomes: the stake wins b or is lost. Trades have many possible results, slippage and fees.
• The goal: it maximizes the long-run growth rate of capital, which is not the same as limiting losses along the way.

Kelly's own setting was a gambler with private advance knowledge of results, betting before they became known at the original odds; the paper is about information rates, not markets.

Why the result is not a bet size

A paper on the formula's good and bad properties by MacLean, Thorp and Ziemba lists its main drawback first: the wagers it suggests may be very large, which makes it very risky in the short term.

The same paper says the theory is straightforward when the probabilities are fairly accurately known, that in investment applications this is usually not the case, and that results are very sensitive to errors in the estimated means. It also notes that a bettor staking a fixed fraction who wins and loses the same number of even-money bets ends up behind. This page suggests no fraction of the Kelly result and no stake of any size, and Fincode does not size or manage positions.

Which values to enter

• Win rate: your estimate of the share of trades that win, as a percentage from 0 to 100.
• Win/loss ratio: the average win divided by the average loss, such as 1.5 when wins average 150 and losses 100.

Both can come from your own records through the win rate and expectancy calculator. If fees are part of each trade, use the averages after fees.

Frequently asked questions

What is the Kelly criterion formula?

f* = p − q ÷ b, where p is the probability of a win, q = 1 − p and b is the win/loss ratio. With p = 0.45 and b = 1.5 it gives 0.0833, or 8.33%.

What does a negative Kelly fraction mean?

That the inputs show no edge: the win rate is below 1 ÷ (1 + b). The formula's answer is then to stake nothing.

Is the Kelly fraction how much I should bet?

No. It is arithmetic on two estimates under assumptions trading does not meet, and this page recommends no bet size.

Why does the result change so much with a small change in the win rate?

Each point of win rate moves the fraction by 1 + 1 ÷ b points, which is 1.67 at b = 1.5, so a five-point change moves it by 8.33 points. Any error in the estimate carries straight into the result.

Does the Kelly criterion work for trading?

Its conditions, known odds, independent bets and two outcomes, do not hold for trades, and researchers who study it, such as MacLean, Thorp and Ziemba, call it very risky in the short term. This page shows the arithmetic only and sizes no position.

Sources

• J. L. Kelly Jr.: A new interpretation of information rate, Bell System Technical Journal, July 1956 (copy hosted by Princeton University) (opens in a new tab)
• L. C. MacLean, E. O. Thorp and W. T. Ziemba: Good and bad properties of the Kelly criterion (1 January 2010) (opens in a new tab)
• OpenStax, Introductory Statistics: 4.2 Mean or expected value and standard deviation (opens in a new tab)

This calculator is general information, not tax, legal or investment advice. Results depend on the values you enter; fees, margin rules and tax rules differ by exchange and country and change over time, so check the official source and ask a licensed adviser about your own situation. Sources, checks and corrections: see the editorial policy.