How the Kelly criterion works
Start with the two numbers that define any bet: the probabilityp that it wins, and the net odds b — how many dollars you win per dollar staked (b = 1 is an even-money bet; b = 2 means you win $2 for every $1 risked). The expected value of a $1 stake is then bp − q, where q = 1 − p is the losing probability. If that edge is positive, Kelly says the growth-maximizing fraction of your bankroll to stake is the edge divided by the odds. It is a bankroll-math lesson, not gambling encouragement: the formula assumes the edge is real and known, and the honest first step is admitting that most people overestimate theirs.
The Kelly formula
f* = (bp − q) ÷ b
g(f) = p·ln(1 + f·b) + q·ln(1 − f)
where f* is the fraction of bankroll to stake, pthe win probability, q = 1 − p, and bthe net odds (payout per $1 staked). The second line is what Kelly actually maximizes: g(f), the expected logarithm of wealth per bet — the long-run compound growth rate. Setting its derivative to zero gives f*. If f* ≤ 0 the bet has no edge and the optimal stake is zero.
Worked example
Take the calculator's default bet: a 55% chance of winning at even money (b = 1), sized at half Kelly against a $1,000.00 bankroll:
| Step | Amount |
|---|---|
| Edge per $1 stakedb·p − q = 1 × 0.55 − 0.45 — the expected value of every dollar risked | $0.10 |
| Full Kelly fraction f* = edge ÷ b0.10 ÷ 1 — the growth-maximizing share of bankroll | 10.00% |
| × half-Kelly multiplierstaking half of f* — the practitioner convention for uncertain edges | × 0.5 |
| = Applied stake, 5.00% of the $1,000.00 bankrollexpected log-growth at this stake: 0.3753% per bet, vs 0.5008% at full Kelly | $50.00 |
Computed with this calculator's default settings — open the tool above and you'll see the same numbers, then swap in your own probability, odds, and bankroll.
Why maximize log wealth instead of expected value?
Maximizing plain expected value tells you to bet your entire bankroll on any positive-edge wager — and then the first loss wipes you out, ending the game a strategy with "maximum expected value" was supposed to win. The paradox is an old one: expected value treats a coin-flip chance at $2 million as strictly better than a sure $900,000, which is the St. Petersburg-flavored observation that dollar amounts and what they are worth to a compounding bettor are not the same thing. Kelly’s fix is to maximize the expected logarithm of wealth instead. Because wealth compounds multiplicatively, the log turns a sequence of bets into a sum, and maximizing its average maximizes the long-run growth rate itself. The log is also ruthless about ruin — ln(0) is negative infinity — so a log-wealth maximizer never stakes everything, no matter how good the edge looks.
Betting more than Kelly is actively harmful
The growth curve g(f) rises from zero at f = 0, peaks at f*, and then falls — crossing back through zero at roughly twice the Kelly fraction. In the default example above, full Kelly (10.00% of bankroll) compounds at 0.5008% per bet, but staking double Kelly (20.00%) drops the growth rate to -0.0138% — negative, despite every single bet still having a positive edge of 10.00% per dollar. Oversizing loses money in the long run on a winning bet: the arithmetic of drawdowns (a 50% loss needs a 100% gain to recover) means volatility costs compound faster than the edge earns. This asymmetry is the deepest practical lesson in the formula — if you must err, err small.
Fractional Kelly: the practitioner’s compromise
Full Kelly is a wild ride even when the edge is exactly right — long stretches of deep drawdown are normal, and a full-Kelly bettor has roughly a 1-in-3 chance of halving the bankroll before doubling it. Worse, f* is only optimal if p is correct, and an overestimated win probability silently pushes you past the peak into the harmful zone. Fractional Kelly buys insurance against both problems cheaply: because the growth curve is flat near its peak, half Kelly keeps about 75% of the full-Kelly growth rate — 0.3753% versus 0.5008% per bet here — while cutting the size of the swings in half. That trade is why half and quarter Kelly, not full Kelly, are the working convention among people who use the formula with real money.
Kelly for investors: f* = (μ − r) ÷ σ²
Edward Thorp — who used Kelly sizing first at the blackjack table and then at his hedge fund — showed that for a continuously-held investment the same logic gives approximately f* = (μ − r)/σ²: the expected return in excess of the risk-free rate, divided by the variance of returns. An asset expected to beat cash by 5% a year with 20% volatility gets 0.05/0.04 = 1.25, suggesting slightly leveraged full Kelly — and immediately illustrating the caveat. Expected returns are far harder to estimate than the win probability of a card game, and errors in μ move f* one-for-one, so the investing form inherits everything above about fractional sizing, only more so. Estimate the inputs with ourexpected return calculator, see the closely related reward-per-unit-of-risk framing in theSharpe ratio calculator, and for the opposite end of the spectrum, thelottery odds calculatorshows a bet Kelly says to never make — a negative edge at any stake.
Frequently asked questions
What is the Kelly criterion?
The Kelly criterion, published by Bell Labs physicist John L. Kelly Jr. in 1956, answers one question: given a repeatable bet with a known edge, what fraction of your bankroll should you stake to make it grow fastest over the long run? The answer is f* = (bp − q)/b, where p is the win probability, q = 1 − p, and b is the net odds — what you win per dollar staked. Bet less and you leave growth on the table; bet more and volatility eats your compounding. It is a position-sizing rule, not a way to find an edge — the edge has to exist first.
What does full versus half Kelly mean?
Full Kelly stakes the entire fraction f* the formula produces; half Kelly stakes half of it, quarter Kelly a quarter. Fractional Kelly trades a little growth for much smoother compounding: half Kelly keeps roughly three-quarters of the full-Kelly growth rate while cutting the swings in half. Because full Kelly is only optimal when your probability estimate is exactly right — and an overestimated edge means you are unknowingly betting past the optimum — most practitioners who use Kelly at all use half or quarter Kelly as a margin of safety.
What does a negative Kelly fraction mean?
A negative f* means the bet has a negative expected value: bp − q is less than zero, so every dollar staked loses money on average. Kelly’s answer is unambiguous — do not bet. There is no stake size that turns a negative-edge wager into growth; any fraction above zero shrinks your bankroll faster the more you bet. This calculator shows the negative fraction so you can see how far underwater the bet is, but it sets the recommended stake to $0. If you could take the other side of the bet at the same odds, the mirror-image wager would have the positive edge.
Does the Kelly criterion apply to investing?
Yes, in a modified form. For a continuous investment rather than a discrete win/lose bet, Edward Thorp showed the Kelly-optimal allocation is approximately f* = (μ − r)/σ², the expected excess return divided by the variance of returns. The practical caveat is large: μ and σ must be estimated, and small errors in the expected-return estimate move f* a lot. That estimation risk — on top of full Kelly’s inherent volatility — is why investors who use Kelly thinking at all typically apply a fraction of it rather than the full allocation.
Why not bet more than the Kelly fraction?
Because growth does not just flatten past f* — it falls, and by about twice the Kelly fraction it turns negative even though every individual bet still has positive expected value. Oversized losses do disproportionate damage to a compounding bankroll (lose 50% and you need +100% to recover), so the volatility cost eventually outruns the edge. In this page’s default example, full Kelly compounds at about half a percent per bet while double Kelly compounds at slightly below zero: a bettor with a genuine edge goes broke purely from oversizing. Betting beyond Kelly is strictly worse than betting the same distance below it.
Sources
J. L. Kelly, Jr., “A New Interpretation of Information Rate,” Bell System Technical Journal 35, no. 4 (1956): 917–926 — the original paper deriving the criterion from information theory. Edward O. Thorp, “The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market,” in Handbook of Asset and Liability Management, vol. 1 (Elsevier, 2006) — the standard practitioner treatment, including the (μ − r)/σ² investing form and the fractional-Kelly growth/variance tradeoff.
Disclaimer: This calculator is foreducation and illustration only. The Kelly formula assumes your win probability and odds are exactly right — in practice edges are estimated, usually optimistically, and an overestimated edge turns "optimal" sizing into over-betting. Nothing here is gambling, investment, or financial advice, and no position-sizing rule creates an edge where none exists.