What Kelly actually answers
Most betting and trading math stops at "is this worth doing." Kelly starts after that question is settled and asks the one that decides whether the edge ever reaches your pocket: how much. Two people can hold the identical winning proposition, take the identical bets, and finish years apart, because one of them sized the positions and the other guessed. If you are still working out whether there is an edge at all, that is our expected value calculator. This page picks the story up from there.
Be precise about what is being maximised, because it is not what most people assume. Kelly does not maximise your expected profit. Maximising expected profit on a favourable bet tells you to put everything on it every time, and that strategy ends at zero the first time you are wrong, which on any bet worth calling a bet is soon. Kelly does not maximise risk adjusted return in the Sharpe sense either. What Kelly maximises is the long run growth rate of a bankroll that keeps betting: the average of the logarithm of your wealth. Logarithms are the right lens because money compounds, and a 50% loss needs a 100% gain to undo. Kelly is the sizing rule that a person who intends to keep playing would choose.
And notice the shape of the answer: a fraction of your bankroll, never a fixed dollar amount. That is not a formatting choice, it is the entire safety mechanism. Bet a fraction and every loss makes the next bet smaller, so in theory you can never reach zero. In practice the theory has edges: table minimums, whole shares, whole contracts, and a bank balance that cannot be divided forever. The moment a real floor exists, ruin is possible again, and sizing has to respect it.
The formula
p is the probability you win, q is 1 − p, and b is the net payoff per unit risked: win b, or lose the 1 you put up. On a trading platform b is your reward to risk ratio. On a betting line it is the decimal odds minus 1, because decimal odds include your stake in the return and b does not. f* is the share of your bankroll to put at risk.
The second line is where every honest claim on this page comes from. g(f) is the expected growth rate of the bankroll per bet at any fraction you choose, and f* is not a rule handed down from anywhere: it is simply the f that makes g as large as it can be. Because we have g, we can also compute what happens at every other fraction, which is the interesting part. The criterion is from J. L. Kelly Jr., "A New Interpretation of Information Rate," Bell System Technical Journal, 1956, and was carried into blackjack, sports betting, and the stock market by Edward O. Thorp.
Worked example
The classic: a 55% win rate at even money, on a $10,000 bankroll. The edge is 1 × 0.55 − 0.45 = 0.10, so ten cents of expected profit per dollar risked. Kelly: f* = 0.10 ÷ 1 = 10% of bankroll, or $1,000. Growth at that size is 0.502% per bet, which would double the bankroll in about 139 bets if the edge held.
Now the three numbers that matter more. Half Kelly ($500) grows at 0.376% per bet, which is 74.9% of the full growth rate while risking exactly half as much on every bet. Quarter Kelly ($250) still keeps 43.7%. And estimation error: if your true win rate is 52% rather than 55%, a miss of three points, then betting the $1,000 you calculated grows your bankroll at minus 0.101% per bet. It shrinks. Half Kelly at the same mistaken estimate still grows, at 0.075%. Full Kelly does not survive a three point error. Half Kelly does.
A fatter edge: 60% at 1.6 to 1. f* = (1.6 × 0.6 − 0.4) ÷ 1.6 = 35% of bankroll, growing at 9.91% per bet. Half Kelly, 17.5%, keeps 75.3% of that. And the cliff: growth stays positive up to 66.13% of bankroll and goes negative above it, so a bettor who "doubled up on a sure thing" at 70% of bankroll would lose money over time on a genuinely winning bet.
Why nobody serious bets full Kelly
Take the growth curve seriously for a second and it tells you something the formula alone does not. Growth rises to a peak at f* and then falls, but the two sides of that peak are nothing alike. Near the top the curve is almost flat: in the classic example above, three quarters of Kelly still keeps 93.7% of the growth and half Kelly keeps 74.9%. Underbetting is cheap. Past the peak the curve turns over and drops through zero at roughly twice the Kelly fraction, and keeps going down. Overbetting is not cheap at all, and past that crossing point a genuine, real, verified edge still loses money.
The half-Kelly-keeps-three-quarters result is a standard one (it is exact in the continuous approximation, and MacLean, Ziemba and Blazenko laid out the growth versus security tradeoff formally in Management Science in 1992), but you do not have to take it on faith here. The calculator above computes it from your own inputs every time, which is why the classic example lands on 74.9% rather than a flat 75.
That gap is worth one more paragraph, because both famous numbers are approximations and almost nobody says so. Sweeping the growth function across every realistic edge (anything Kelly would size at a quarter of bankroll or less) puts the half Kelly retention between about 73% and 82%, and the zero growth crossing between about 1.84 and 2.42 times Kelly, depending on the shape of the bet. Even-money-ish bets sit near the textbook 75% and 2x; long shots, rare wins paying big, drift to the high end of both. Push all the way to an absurd edge, the kind nobody has, and the shorthand collapses entirely: at a 94% win rate the crossing point falls to about 1.1 times Kelly, so "half Kelly is safe" would be quietly wrong. The calculator prints what your numbers actually produce rather than the folklore, and it says so when your inputs fall outside the range where the folklore works.
Half the volatility, meanwhile, is not an approximation at all. It is arithmetic. Betting half the fraction means risking exactly half as much on every bet, so every swing, up and down, is exactly half the size. Roughly three quarters of the reward for half of the pain is the kind of trade you take without thinking twice, and that is the entire argument for fractional Kelly in one line.
The real reason: Kelly assumes you know p, and you do not
Everything above treats your win probability as a fact. It is an estimate, usually from a small sample, usually of a process that does not sit still. That is the crack the whole structure falls through, and it is why the estimation margin field exists on this form.
Run the classic example again with a three point error. You believed 55%; the truth is 52%. Kelly for the real bet is 4% of bankroll, but you are betting the 10% you calculated for the optimistic number. Your actual growth rate is negative. You have a real edge, four cents on the dollar, and you are losing money anyway, purely from sizing. That is not an exotic scenario. Three percentage points is a rounding error in most people's estimate of their own win rate.
The asymmetry is what settles the argument. Underbetting has a floor: the worst case is betting nothing and growing at exactly zero. Overbetting has no floor. Growth goes negative and keeps going, because each loss takes a percentage of everything you have left. When the cost of one direction is bounded and the cost of the other is not, you lean the bounded way. That is the whole case for half Kelly, and it has nothing to do with nerve.
What Kelly assumes, and where each assumption breaks
Five things are quietly taken as true by the number this page gives you. You can size continuously. Real life has table minimums, whole shares, whole contracts, and a smallest bet you can place, so at a small bankroll the "correct" fraction may be unbuyable. Outcomes are independent. Correlated positions are the classic way a portfolio of individually sensible Kelly bets turns into one enormous bet wearing several hats. The edge is stable. Markets adapt, lines move, and the strategy that produced your win rate last year is being arbitraged by someone this year. You can repeat indefinitely. Kelly is a long run rule, and the long run is doing real work in that sentence; over ten bets, sizing barely matters and luck decides. There is no ruin threshold. No margin call, no minimum balance, no rent due on the first, nothing you cannot come back from.
That last one changes the answer rather than merely qualifying it. Kelly's promise that you can never go broke depends on being allowed to keep shrinking your bets forever. Put a hard floor underneath, a point at which you are finished, and ruin becomes a real probability that has to be calculated rather than assumed away. Our risk of ruin calculator is the page for that question, and if a floor exists in your situation, it is the more important of the two.
One last honest paragraph, in the same spirit as our odds calculator. This page prices arithmetic. It is not an endorsement of betting or trading, it does not tell you that you have an edge, and it will never link you to a broker or a sportsbook. Kelly is a lever on an edge you already have; it cannot manufacture one, and applied to a losing proposition it only sets the speed you lose at. If the numbers have stopped being interesting and started being stressful, that is worth a conversation with someone you trust rather than another calculation.