What Is the Kelly Criterion, and What Does It Say About Position Size?

The Kelly criterion is a rule for how much of a bankroll to stake on a repeated bet with a known edge: stake the fraction that maximises the expected logarithm of capital, which for a bet paying b units per unit risked with win probability p works out to f* = (bp − q)/b, where q = 1 − p. On an even-money bet that collapses to f* = p − q.

That is the whole formula. Everything difficult about applying it to futures trading sits in two places: what you are allowed to call the bankroll, and how confident you are in p.

What is the Kelly criterion?

The Kelly criterion comes from a 1956 paper by J. L. Kelly Jr. in the Bell System Technical Journal, which asked how fast a gambler's capital can be made to grow when he receives information about the outcome. Kelly defined what he named capital's exponential growth rate, the long-run per-bet average of the log of finishing capital over starting capital, and worked out which staking fraction makes that quantity as large as it can be.

The interesting part is what he ruled out. Staking the entire bankroll on every favourable bet is the strategy that maximises the expected amount of money, and Kelly's paper notes that a gambler doing that goes broke with probability one if he keeps going. So maximising expected money is not the same problem as making money grow. Kelly's answer was to maximise the expected value of the log of capital at every bet, and the paper is explicit that this is a mechanical choice rather than a statement about how a gambler values money. Log is the function that adds up across repeated bets, which is what lets the law of large numbers say anything at all about a long sequence.

Rotando and Thorp, writing in the American Mathematical Monthly in 1992, call the resulting method fixed fractional betting: the same percentage of current capital, every trial. They credit Kelly with proposing it and Breiman with the work establishing when it holds.

What is the Kelly formula for a trade with a stop and a target?

For one bet with two outcomes, MacLean, Thorp and Ziemba give the optimum as the edge divided by the odds. Written out, with p the win probability, q = 1 − p, and b the reward-to-risk multiple (2 if a winner returns twice what a loser costs), the fraction is f* = (bp − q)/b, which is the same as p − q/b.

Rotando and Thorp derive it from the growth coefficient G(f), which is p·log(1 + bf) plus q·log(1 − f). In the even-money case they differentiate and show the maximum is unique; the uneven-payoff fraction falls out of the same calculus. Put the numbers of a trading plan into it and the output is uncomfortable:

Those are fractions of the whole bankroll, per trade. A trader who believes he wins 55% of 2:1 trades is being told by the formula to put a third of his account behind each one. The figures are what f* equals at those inputs. They demonstrate what MacLean, Thorp and Ziemba name as the criterion's main disadvantage: the stakes it suggests can be very large. Their explanation is that log utility carries an Arrow-Pratt risk aversion of one over wealth, low beside the utility functions investors usually pick, which leaves the criterion risky in the short term even where it is right about the long one.

Note also where the table says "no bet". At 1:1 a win rate at or below 50% produces f* ≤ 0, which is the formula declining the trade. That boundary is the breakeven win rate for each reward-to-risk ratio, and Kelly only has something to say on the far side of it.

What happens if you risk more than the Kelly fraction?

Staking more than the Kelly fraction costs growth, and past a second threshold the growth rate turns negative. Rotando and Thorp show that the growth coefficient is zero at f = 0, rises to its maximum at f*, and comes back down to zero at a second fraction they label f_c, which lies above f* and below 1. Their worked case is a 53% even-money bet: f* is 6%, and f_c is 11.973%.

Their theorem attaches a different fate to each side of that number. Where the growth coefficient is positive, capital grows without bound almost surely. Where it is negative, which in their example means any fixed fraction past roughly 12%, capital tends to zero almost surely. Exactly at f_c it oscillates, with a limit superior of infinity and a limit inferior of zero, both almost surely. So doubling the optimal 6% stake does not halve the growth rate. It deletes it. MacLean, Thorp and Ziemba report a matching threshold in continuous time: betting exactly double the Kelly amount leaves growth running at precisely the risk-free rate, a result their appendix derives for a geometric Wiener process under the capital asset pricing model.

Their term for strategies past f* is "growth-security dominated" — more risk and less growth than a smaller fraction would have delivered, so nothing is being bought with the extra exposure. They name Long Term Capital Management, Amaranth and the Victor Niederhoffer funds as cases where overbetting ended badly.

The asymmetry is the practical point. Undershooting f* costs growth in proportion. Overshooting it costs growth and adds risk at once, and the room between the two is not generous. In the 53% example the fraction at which growth disappears sits just under twice the optimum, 11.973% against 6%, and in the continuous-time result it is exactly double the Kelly amount that pulls growth back to the risk-free rate.

What does the Kelly criterion assume?

Four conditions: profits that can be reinvested, a stake you can vary, capital that is infinitely divisible, and independent repeated trials from a distribution you know. Kelly's paper names the first two itself, and illustrates what happens without them through a gambler allowed to bet a dollar a week who cannot reinvest his winnings — that gambler should maximise expectation instead, because nothing is compounding.

Divisibility is the assumption that makes the mathematics work, and Rotando and Thorp state it outright. They are candid that real stakes come in discrete units, so ruin in the technical sense is possible after all, and they argue the theory survives as an approximation when the minimum stake is tiny next to starting capital.

The fourth condition is where trading parts company with blackjack. MacLean, Thorp and Ziemba say the criterion is only straightforward where the distributions behind it are known accurately, and that investment applications usually are not that case — so the p and b a trader puts into the formula are estimates standing in for quantities the derivation treats as given.

One further property gets quoted without its conditions. MacLean, Thorp and Ziemba list among Kelly's good properties that the expected-log bettor never risks ruin, attributing it to Hakansson and Miller. Rotando and Thorp show where that comes from in the simplest case: wager a fixed fraction strictly between zero and one of infinitely divisible capital and the probability that capital ever equals zero is itself zero, so ruin in the gambler's-ruin sense cannot occur. That conclusion rests on the divisibility assumption above: once a minimum bet exists, they say, ruin in the sense they defined becomes possible again.

Why stake half of the Kelly fraction?

Half Kelly, staking half of what the formula returns, answers a sensitivity problem: the input the criterion reacts to most sharply is the one a trader knows least precisely. MacLean, Thorp and Ziemba cite work by Chopra and Ziemba putting the relative damage of estimation errors in means, variances and covariances at about 20:2:1 for an investor with average risk aversion, and at roughly 100:3:1 under log utility. Log utility is what Kelly maximises, so the ratio is at its most lopsided exactly where the criterion lives, and the input it punishes is the estimate of the edge. That is also how the paper closes: get the mean right, and shade the size of the wagers down for safety.

Fractional Kelly is the moderation the same paper proposes: put a fraction λ of the full Kelly proportion into the risky position and hold the rest in cash, which it notes is equivalent to running full Kelly on λ of your capital and leaving the remainder uninvested. Half Kelly means λ = 0.5.

What that buys and costs is quantified in the same paper, through an example from Ziemba and Hausch. Across 700 wagers each carrying a 14% advantage, the least likely of them winning 19% of the time, full Kelly turns a $1,000 start into at least $100,000 16.6% of the time — and can also leave $18. Half Kelly's bad outcome in the same setting is $1,000 becoming $145, and its chance of clearing $100,000 drops to 0.1%. Both tails move in. That is the trade, stated in numbers, and the paper adds that no rule, Kelly or fractional, is a sure way of winning over a finite sequence.

There is one more reason to doubt your own p. The 2010 paper notes that in coin tossing, separating a 1.0% edge from a 1.1% edge takes about two million trials for the better game to be ahead with 84% probability. An edge estimated from a few hundred trades, or from a backtest tuned until it looked good, is not the known probability the formula was written for.

Which number is the bankroll on a funded account?

Kelly's fraction is a fraction of something, and on a funded account two numbers compete for the job. The arithmetic below is ours rather than a result from either paper, and it starts from the observation that what a losing run can consume before the account closes is the distance to the drawdown line, not the face value on the statement. Our drawdown article carries the amount for each size and program — $2,000 at the $50,000 size when this was written — and the line sits that far below the highest balance the account has ever closed a day on. On a $50,000 Lumen account — a simulated account whose profits pay out for real — those two candidates differ by a factor of twenty-five.

Run the 53% even-money case through both readings. Kelly says 6%. Six per cent of $50,000 is $3,000, which is more than the entire published drawdown allowance for that size, so the first loss sized that way would breach the line and end the account before the formula got its second trial. Six per cent of the $2,000 of actual room is $120, which is a different instruction entirely.

Kelly's own condition about reinvestment is also worth checking against the rule. While the line is still trailing, every new highest daily close drags it up behind you, so the room ahead returns to the drawdown amount instead of growing — the compounding the criterion assumes does not start until the line locks at the starting balance, which happens once profit equals the drawdown amount. MacLean, Thorp and Ziemba are not writing about accounts like this one, but their advice for anyone whose first priority is protecting capital points the same way: cut risk sharply as a drawdown deepens, and they cite work on picking the fractional-Kelly coefficient so as to stay above a stated drawdown at a chosen confidence level.

What does a whole-contract minimum do to the arithmetic?

A whole-contract minimum puts a floor under the smallest stake available, which is exactly the divisibility assumption Rotando and Thorp flagged. Positions are counted in whole contracts, and our limits are published as four minis or forty micros, not as a dollar figure. On an index the smallest unit is the micro: as our instruments page listed them on 30 October 2026, the micro S&P 500 contract moves $1.25 a tick against $12.50 for the E-mini S&P 500.

Take the $120 figure from the section above and a ten-tick stop. On the micro that is $12.50 of risk per contract, so $120 buys nine of them and leaves $7.50 unused. On the E-mini the same stop risks $125 per contract, which is more than the budget, so the whole-contract answer is zero and at that budget the trade exists only in micros.

Now watch the floor arrive. Six per cent of the room falls below $12.50, the cost of one micro behind that stop, as soon as the room is under about $208. Inside the last tenth of a $2,000 allowance, the smallest trade available is already larger than the fraction the formula asked for, so a trader at that point is overbetting by construction or not trading. There is no fractional contract to round down to.

The ceiling at the other end is fixed too. The contract limits article sets what a size may hold at once, and at $50,000 that is four minis or forty micros. Either ceiling puts $500 behind the same ten-tick stop, more than four times the $120 the formula asked for, which is the clearest sign that a cap on exposure and a sizing rule answer different questions. The arithmetic of fitting stops inside the drawdown is the trader's to do either way.

FAQ

What is the Kelly formula for trading?

For a bet with two outcomes it is f* = (bp − q)/b, where p is the win probability, q = 1 − p and b is the reward-to-risk multiple. MacLean, Thorp and Ziemba describe it as the edge divided by the odds. On an even-money bet, b = 1 and the fraction reduces to p − q. Rotando and Thorp derive it as the maximum of the growth coefficient, unique in the even-money case they work through, and show that a second fraction above it returns the growth rate to zero.

Is half Kelly better than full Kelly?

The published comparison is a trade, not an improvement. In the example MacLean, Thorp and Ziemba take from Ziemba and Hausch, 700 wagers each with a 14% advantage, full Kelly takes $1,000 to at least $100,000 in 16.6% of outcomes and can also finish at $18, while half Kelly's bad case is $145 and its chance of reaching $100,000 falls to 0.1%. Both tails contract. The same paper adds that neither rule is a sure way of winning over a finite sequence.

What happens if you risk twice the Kelly fraction?

The reward for taking the risk disappears. MacLean, Thorp and Ziemba state that under a continuous-time approximation, staking exactly twice the Kelly fraction brings the growth rate down to the risk-free rate, the same return as holding cash, for all the exposure. Rotando and Thorp's discrete version puts a second zero of the growth coefficient above f*, 11.973% against an optimal 6% in their 53% example, and shows that any fixed fraction beyond it drives capital toward zero almost surely.

Can you use the Kelly criterion on a prop firm account?

The formula still computes; the conditions behind it are what to check. Kelly names two requirements for his theory to apply: profits that can be put back to work, and a stake the bettor is able to set. Rotando and Thorp add infinitely divisible capital, and say ruin becomes possible once stakes are quantised. On a funded account a trailing line returns the available room to the published drawdown amount at each new highest daily close until it locks, and contracts are whole, with nothing below one micro. For anyone whose first priority is capital protection, MacLean, Thorp and Ziemba's own prescription is to cut risk sharply as a drawdown deepens.