Kelly Criterion

The two inputs, and the one the whole answer hangs on

Two numbers carry everything on this page: how often you win, and how much you make when you do compared with what you lose when you do not. Write the win rate as p, the losing share as q = 1 − p, and the payoff as b — the amount won per 1.00 put at risk. The rule is one line:

The other number on the table, the break-even win rate, is the one worth reading first, because it decides whether there is anything to size at all. It is 1 ÷ (1 + b): at 2:1 that is 33.33%, and it does not care what your win rate is. Above it the answer is positive; at it the two terms cancel and the answer is zero; below it the answer is a negative number, which is not a smaller bet but the formula telling you this is not a bet worth taking.

The row at 60% is what full Kelly asks of you

A 60% win rate at 2:1 is a very good trade, and the answer is 4,000.00 of a 10,000.00 account. That is what maximising long-run growth actually means in money: it means accepting that two losing trades in a row take the account from 10,000.00 to 3,600.00, because 0.60 × 0.60 = 0.36.

At the 55% row the same arithmetic leaves 45.56% after two losers — 0.675 × 0.675. At half Kelly, 16.25% a trade, two losers leave 70.14%. The Kelly share is not a caution and it is not a recommendation; it is the size at which an account compounding over many trades grows fastest, and the growth it buys is paid for in exactly this coin.

What the payoff does to it, holding the win rate still

The second input moves the answer just as hard, and it moves the break-even with it. Same 10,000.00 account, win rate fixed at 55.00%:

At 1:1 the same 55% win rate buys 5 points of edge and the answer is 10.00%. At 5:1 it buys 38.33 points and the answer is 46.00% — very nearly half the account, from the same win rate. Growth per trade is the expected compounding on the whole balance for one trade at that size: 0.55 × ln(1 + 2 × 0.325) + 0.45 × ln(1 − 0.325), which is 9.86% in logs and 10.36% once it is turned back into growth.

Two formulas, and where they meet

More than one Kelly formula turns up in the pages Google showed us for this word, and one of them is introduced as a correction of the other. The CFA Institute's Enterprising Investor blog, in a post dated 14 June 2018 by Alon Bochman, opens by saying most of the sources I found use the wrong formula, writes the form it is objecting to as Kelly % = W – [(1 – W) / R], says that one works only for binary bets where the downside scenario is a total loss of capital, as in -100%, and gives a second form, Kelly % = W/A – (1 – W)/B, for outcomes where a loser costs only part of the position. It names NASDAQ, Morningstar, Wiley's For Dummies series and Old School Value as places the first form turns up. That is what that page said on the day we read it; we take no position on which form anyone should use.

The form on this page is the first one, and it is stated to cover exactly the case this page describes: a loser costs the whole amount put at risk, and there is nothing in between. Where a loser costs you only a fraction A of the position, the two are the same instruction written at two different points in the chain — f is a share of the balance at risk, and the share of the portfolio sitting in the position is f divided by A. On the numbers in that post, a 60% win rate with a gain of 20% or a loss of 20%: this page's form gives 60% − 40% = 20% of the balance at risk, and 20% divided by the 20% loss is 100% of the portfolio, which is the figure that post arrives at. Two numbers, two places in the chain, one instruction.

Half Kelly, and what full Kelly costs on the way

The reason nobody recommends the number in the table at face value is the path, not the destination. Three of the pages we read say it in their own words. Wikipedia's entry: gamblers use less than full Kelly in order to reduce the chance of ruin, reduce volatility, and account for model error, and it has been found that betting more than the Kelly amount increases the risk of ruin. Albion Research's calculator page: a Kelly bettor has a 1/3 chance of halving a bankroll before doubling it, and a 1/n chance of seeing the bankroll fall to 1/n at some point, against a 1/9 chance for a half-Kelly bettor. Aldous's page at Berkeley: there is a 40% chance that at some time your wealth will drop to only 40% of what you started with, and likewise 10% to 10%, and the Kelly strategy marks the boundary between aggressive and insane investing.

All three are statements about one bet repeated over and over with the same odds, which is not what a trading account does, so this page quotes them and does not extend them. What this page can say from its own arithmetic is the short version: at the 55% row, two losing trades in a row leave 45.56% of the account and three leave 30.75%; at half Kelly the same two leave 70.14% and the same three leave 58.74%.

The other kind of Kelly calculator

One of the nine pages Google showed for this word is a calculator, at number three: Albion Research's Kelly calculator. It asks for a bankroll, the odds on offer, your probability of winning and the multiple your stake has to be a round number of, and on its own default — a 40% chance at 1.75 to 1 — it answers 5.71%, or 57.00 on 1,000.00. The odds it asks for are the profit per 1.00 staked, which is the b in the formula above, and the bankroll is the balance; the arithmetic underneath is the same arithmetic.

The difference is the shape of the question. That page assumes a stake you either win whole or lose whole, and answers in the size of the stake. This page assumes you name what a loser costs you first and then let the rule set the size, which is why it can print a bracketed table, a half-Kelly column and a break-even win rate — none of which the stake-size question produces. Both are legitimate readings of the same formula; this one is the one a position is sized in.

Run your own numbers

Every figure on this page is one payoff and five win rates on one balance. Your win rate, your payoff and your own account give you different rows, and yours are the only ones that matter. The Kelly criterion calculator takes the same two inputs — quoted either as a payoff ratio or as decimal odds — lets you take any fraction of the answer, and prints the money, the break-even win rate, the answer five points of win rate lower, and what a run of consecutive losers leaves behind.

Size from your own win rate

Questions people ask

It is a rule for how much of your money to put at risk when the odds are in your favour. It answers with a share of the balance: at a 55% win rate with a 2:1 payoff it says 32.50%. Risk that share every time and the account compounds faster over many trades than it would at any other fixed share. It says nothing about which trades to take — it only sizes the ones you have already decided are worth taking.

Take your win rate as a decimal, take the losing share, divide the losing share by your payoff, and subtract. At a 55% win rate with a payoff of 2.00 won per 1.00 risked: 0.55 - 0.45 / 2 = 0.325, which is 32.50% of the balance, or 3,250.00 on 10,000.00. The payoff is net — 3.00 in decimal odds is a payoff of 2.00, because the stake comes back to you.

The break-even win rate, which is 1 divided by 1 plus the payoff. At 2:1 that is 33.33%, and it is the same in every row of the table above because it does not depend on your win rate at all. Above it the answer is positive; at it the two terms cancel and the answer is zero; below it the answer is negative, which is the formula saying the bet is not worth taking rather than saying to take a small one.

Half is the usual answer, and the reason is the path rather than the growth rate. At the 55% row, full Kelly is 32.50% a trade and two losers in a row leave 45.56% of the account; at half Kelly, 16.25% a trade, the same two losers leave 70.14%. Wikipedia's entry gives the reasons as reducing the chance of ruin, reducing volatility and allowing for the fact that your estimated win rate is an estimate; Albion Research's page puts the odds of halving a bankroll before doubling it at 1 in 3 at full Kelly and 1 in 9 at half.

Of the money at risk, not of the position. If a losing trade costs you the whole amount you put up then 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 reason the answer on this page looks so large to people who are used to sizing in notional.

Because the whole gap between your win rate and the break-even lands in the answer, and nothing damps it. At 2:1 the break-even is 33.33%, so a win rate of 35% carries 1.67 points of edge and an answer of 2.50%, while 40% carries 6.67 points and an answer of 10.00% — five points of win rate, four times the size. The same is true of the payoff: at a 55% win rate, going from 1:1 to 5:1 takes the answer from 10.00% to 46.00%.

This page uses f = p - q / b and states plainly what that form assumes: a loser costs the whole amount put at risk. The CFA Institute's Enterprising Investor blog, in its 14 June 2018 post, gives the other form for outcomes where a loser costs only a fraction of the position, and says the popular form works only for binary bets with a total loss of capital. The two are the same instruction at two points in the chain — the share of the portfolio in the position is the share of the balance at risk divided by the loss fraction — and on that post's own numbers this page's form gives 20% at risk, which divided by the 20% loss is the 100% the post arrives at. This page does not call either form wrong.

Where this page stops

Two outcomes only: a winner pays b for every 1.00 at risk and a loser costs the whole amount at risk. No partial losses, no slippage, no gaps through a stop, no trades that neither win the full payoff nor lose the full stake. The win rate and the payoff are treated as known and exact; the formula has no way to hold an estimate, which is why the calculator shows the answer five points lower as well. The bets are assumed independent, identical and repeatable, with no correlation between them, no edge that decays and no other positions compounding alongside.

No taxes, commissions, spreads, financing or margin. Nothing here is a portfolio: no diversification, no correlation with what else you hold, and no continuous version of the criterion of the kind that sets a fraction from an expected return and a variance. The drawdown figures are quoted from the three pages named above, which state them about one bet repeated; this page does not derive a ruin probability of its own.

Of the ten results Google showed for this word, two — the Investopedia entry at number two and the discussion thread at number five — could not be opened on any of the three routes we tried. We make no claim about what they say or do not say. The seven we did read were read on 22 October 2026, and pages change. Nothing on this page is advice.

Related guides

• Kelly criterion calculator — the same two inputs at any fraction of Kelly, with the break-even win rate, the answer five points lower, and what a run of losers leaves behind.
• Win rate calculator — the win rate itself, from your own closed trades, before it goes into any sizing rule.
• Risk of ruin calculator — the chance a given risk per trade and a given win rate empties the account before it grows.
• Risk reward calculator — the payoff side of the pair, worked out from your entry, stop and target.
• Stock position size — turning the money figure into a number of shares once you have decided what to risk.