What the Kelly criterion actually optimises
Kelly maximises the expected logarithm of your bankroll, not the expected bankroll itself. That distinction is the whole idea. If you maximise expected money on a bet with a positive edge, the answer is always to stake everything, because expected money is linear in stake. Do that repeatedly and you go broke with probability approaching one, because a single loss multiplies your bankroll by zero.
John Kelly's 1956 paper in the Bell System Technical Journal reframed the problem. Because bankrolls compound multiplicatively, the quantity that governs long-run wealth is the average of the logarithm of each period's multiplier. Maximise that and you get a fraction, not a whole bankroll. Edward Thorp then carried the result from information theory into blackjack and into portfolio management, which is where most bettors first meet it.
Two properties follow that no other staking rule has. Over enough bets, a Kelly bettor's bankroll almost surely overtakes that of anyone using a systematically different fraction, and a Kelly bettor never goes bankrupt, because the stake is always a proportion of what remains. Both properties are asymptotic: they say nothing comforting about your next fifty bets.
The formula, term by term
Write b for the net odds — the profit you receive per unit staked if the bet wins. Decimal odds of 2.50 return your stake plus 1.50 profit, so b = 1.50. American odds of −110 mean you risk 110 to win 100, so b = 100/110 = 0.9091. Write p for your probability that the bet wins and q = 1 − p for the probability it loses.
The Kelly fraction is f* = (b·p − q) / b. Read the numerator as your expected profit per unit staked: you win b with probability p and lose 1 with probability q. Dividing by b converts that expected profit into a bankroll fraction, and it is the division that makes the rule cautious about long prices. The same expected profit produces a smaller stake when b is large, because a longshot loses far more often on the way to the same expectation.
There is a second form that is easier to reason about at the betting window. If d is the decimal price, the market's break-even probability is 1/d. Call the amount your estimate exceeds that your edge, e = p − 1/d. Then f* = e·d / (d − 1), which is exact, not an approximation. At even money (d = 2) the two terms cancel and the Kelly fraction is simply twice your edge; at d = 2.00 an edge of 5 percentage points means a 10% stake.
The function being maximised is g(f) = p·ln(1 + b·f) + q·ln(1 − f). Differentiate it, set the derivative to zero, and f* falls out. The chart on this page plots that function, and its shape carries the practical lesson: it is nearly flat around the peak and falls off a cliff to the right of it. Understaking costs you little. Overstaking costs you a great deal.
Worked example: a 55% shot priced at −110
You have handicapped a side at 55% and the book is offering −110, the standard price on a two-way market. Your bankroll is $5,000 and you stake half Kelly.
- Convert the price. American −110 means risking 110 to win 100, so the decimal price is 1 + 100/110 = 1.9091 and b = 0.9091.
- Find the break-even probability. 1 ÷ 1.9091 = 0.5238, so the price needs a 52.38% winner just to break even.
- Measure the edge. 55.00% − 52.38% = 2.62 percentage points.
- Apply Kelly. f* = (0.9091 × 0.55 − 0.45) / 0.9091 = (0.5000 − 0.4500) / 0.9091 = 0.0500 / 0.9091 = 0.0550, or 5.5% of bankroll. The numerator is exactly 0.05, so at this price a 55% estimate is worth 5% of a unit in expectation per unit staked.
- Scale to half Kelly. 5.5% × 0.5 = 2.75%, so the stake is 0.0275 × $5,000 = $137.50.
- Check the growth rate. At f = 0.0275, g = 0.55 × ln(1 + 0.9091 × 0.0275) + 0.45 × ln(1 − 0.0275) = 0.55 × 0.024972 + 0.45 × (−0.027885) = 0.013735 − 0.012548 = 0.001187, or about 0.119% of bankroll per bet.
Now compare the fractions on the same bet. Full Kelly at 5.5% gives g = 0.55 × ln(1.05) + 0.45 × ln(0.945) = 0.026853 − 0.025454 = 0.001399, about 0.140% per bet. So halving the stake keeps roughly 85% of the growth while halving the swing on every single result. That trade is why fractional Kelly is the default in practice.
How to read the number the calculator gives you
Treat the full Kelly fraction as a ceiling, not a recommendation. Full Kelly is optimal only if your probability estimate is exactly right, and it is not. Overestimating your edge by a factor of two means you are staking double Kelly, and at double Kelly the expected log growth is exactly zero — you have converted a winning bet into a coin flip that grinds sideways forever. Anything past that is negative growth on a bet you were right about. Because estimation error only ever pushes you toward the dangerous side of a curve that is flat on the left and steep on the right, halving is not timidity, it is symmetry repair.
A full-Kelly fraction above about 10% of bankroll deserves suspicion at ordinary sports prices, because it implies an edge of several percentage points against a market that thousands of people are pricing. A fraction above 25% almost always means a data-entry error or a probability estimate that has not been calibrated against results.
The edge output is more diagnostic than the stake. If you cannot state where the edge comes from — a closing-line advantage you have measured, a model with a tracked Brier score, information the market has not absorbed — then the honest value of p is the market's own de-vigged number, and Kelly returns zero. That is the correct answer, not a failure of the calculator. To turn a two-sided market price into a fair probability, see the parlay payout calculator, which removes the overround leg by leg.
Full Kelly stake by price and edge
| Decimal price | American | Break-even p | Edge +2 pp | Edge +5 pp | Edge +10 pp |
|---|---|---|---|---|---|
| 1.50 | −200 | 66.67% | 6.00% | 15.00% | 30.00% |
| 1.91 | −110 | 52.38% | 4.21% | 10.50% | 21.00% |
| 2.00 | +100 | 50.00% | 4.00% | 10.00% | 20.00% |
| 2.50 | +150 | 40.00% | 3.33% | 8.33% | 16.67% |
| 3.00 | +200 | 33.33% | 3.00% | 7.50% | 15.00% |
| 5.00 | +400 | 20.00% | 2.50% | 6.25% | 12.50% |
The same edge in percentage points buys a smaller Kelly stake at longer prices, because the loss frequency rises faster than the payout.
Mistakes that turn Kelly into a losing system
- Using the market's implied probability as your own. If p = 1/d you have no edge by construction and f* is negative once the overround is included. Kelly needs an independent estimate.
- Forgetting that the vig is already inside the price. The break-even probability at −110 is 52.38%, not 50%. A 51% handicap is a losing bet at that price.
- Staking Kelly on several correlated bets at once. Three legs of the same game are one bet with a strange payout, not three independent Kelly opportunities. Correlated simultaneous bets need the multi-asset form of Kelly, which stakes far less in total.
- Recomputing the bankroll only after a winner. Kelly is a fraction of the current bankroll. Update it after every settlement, up or down, or you drift toward a fixed stake.
- Applying Kelly to a bet that can be partially lost. Asian handicaps that push half the stake, and anything with a partial cash-out, break the all-or-nothing assumption behind this formula.
- Betting more because the last few lost. Kelly is memoryless. The fraction depends only on the current price, the current estimate and the current bankroll.
Fractional Kelly, flat staking and when to use something else
Most professional bankroll plans run somewhere between quarter and half Kelly. The reason is visible in the growth function: near the peak it is quadratic, so scaling the stake by a factor c of full Kelly retains roughly c(2 − c) of the maximum growth. At half Kelly that is 0.75, and the worked example above shows a measured 0.85 on that particular bet; either way you are giving up a modest slice of growth to cut the standard deviation of your bankroll path roughly in half. Quarter Kelly retains about 0.44 of the theoretical maximum and makes a deep drawdown far less likely.
Flat staking — the same 1% of a fixed starting bankroll on every bet — is not irrational. It is what you should use when you cannot estimate p to better than a couple of points, because a mis-specified Kelly is worse than a conservative constant. Level-stakes plans also make record-keeping and closing-line analysis simpler, since every result carries the same weight.
Kelly transfers directly to any repeated proportional wager. In poker the analogous question is whether a single call is profitable at all, which the poker pot odds calculator answers before bankroll sizing enters the picture. For bets with more than two outcomes, such as an outright market, the two-outcome formula above does not apply and you need the general form that maximises the sum of pi·ln(1 + bifi) across all outcomes. And for wagers where the edge is structurally negative — every draw in the lottery odds calculator, for instance — Kelly's answer is always the same: stake nothing.
One more boundary worth naming. Kelly assumes the number of future bets is effectively unlimited. If you have exactly one bet to make and then walk away, the log-growth argument has nothing to compound and the right stake depends on your own utility, not on this formula. The same is true of a fixed number of attempts at a one-shot outcome.