Kelly criterion betting calculator: size a stake from odds and edge

A Kelly criterion betting calculator takes your bankroll, the odds and your estimated win probability, and returns the fraction of the bankroll to stake. At -110 and a 55 percent chance, full Kelly says 5.5 percent. Almost nobody should bet that, and the reason is the whole point of this page.

The formula

For a bet that pays b units per unit staked, with win probability p and q = 1 - p:

f = (b x p - q) / b

At American -110 the profit per unit staked is b = 100 / 110 = 0.9091. At +150, b = 1.5. (Decimal odds minus 1 gives the same number.)

Thorp's well-known survey defines the criterion as the stake that maximizes the expected value of the logarithm of wealth. In plain words: it is the bet size that grows a bankroll fastest over a long run, if your probability is right.

A worked example

Hypothetical numbers, not results. You think a team wins 55 percent of the time and the price is -110.

• b = 0.9091, p = 0.55, q = 0.45
• f = (0.9091 x 0.55 - 0.45) / 0.9091 = 0.05 / 0.9091 = 5.5 percent

With a $10,000 bankroll (the Kelly formula scales with whatever you have) that is a $550 stake at full Kelly, $275 at half Kelly and $137.50 at quarter Kelly. The expected profit is 0.55 x 0.9091 - 0.45 = +0.05 per unit staked, or +5.00 on a 100 bet. You can check that number on the expected value calculator page.

If your estimate is 52 percent or lower at -110, the formula returns a negative number. The break-even rate at -110 is 52.38 percent, and below that Kelly says bet nothing.

Why bettors use a fraction of Kelly

Full Kelly assumes you know your true probability. You never do. The Kelly criterion article on Wikipedia says the method "requires accurate probability values" and that overestimating your probability pushes the stake away from the optimal one and raises the risk of ruin. It also states that betting more than the Kelly amount increases the risk of ruin, and that one approach to unknown risk is to invest less than Kelly.

The same article explains the cost of going smaller: reduced volatility is a tradeoff, because it increases the time to reach an intended wealth or decreases the growth rate.

Here is that tradeoff for the example above, assuming the 55 percent estimate is exactly right. Average log growth per bet:

• Full Kelly (5.5 percent): 0.138 percent
• Half Kelly (2.75 percent): 0.103 percent, about 75 percent of the full-Kelly growth
• Quarter Kelly (1.375 percent): 0.060 percent, about 44 percent of the full-Kelly growth
• Double Kelly (11 percent): roughly 0, all the growth gone

Half the stake keeps three quarters of the growth, and betting twice Kelly is not better than staking nothing.

What happens when your edge is smaller than you think

Suppose the true win rate is 53 percent, not 55. The Kelly stake at 53 percent is 1.30 percent. If you size at 5.5 percent because you believed 55, you are betting more than four times Kelly, and your growth per bet is about -0.073 percent: the bankroll shrinks on average even though the bet has a real edge.

Half of your estimated Kelly (2.75 percent) comes out near zero growth in that case. Quarter Kelly (1.375 percent) stays slightly positive. That is the practical argument for a fraction: it protects you when your estimate is rich, and costs you some growth when it is right. The quant+ scanner uses a fractional Kelly with a default of one quarter.

Where the probability comes from

Kelly is only as good as p. The best-documented source for a probability is the market's own price with the margin removed. Researchers have used probabilities implied by public odds to look for mispriced bets (arXiv, 2017). Convert the posted price using the implied probability calculator, remove the margin as shown on the fair odds calculator page (the power method did best in one 2017 comparison), and compare that to your own number. If your number is just the fair price, you have no edge and Kelly returns about zero.

Several bets at once

The formula above is for one bet. When bets overlap, such as two players in the same game or several bets that settle at the same time, the stakes add up as a share of the same bankroll, and the outcomes can be correlated. Sizing each bet at its own Kelly stake overstates the total you should risk. Cap the combined exposure for the day or the slate, and treat same-game bets as one position.

How to use the output

1. Estimate p honestly, and prefer the lower of two reasonable numbers.
2. Run the formula, then take a fraction: a quarter to a half is common.
3. Put a hard cap on any single bet, so one wrong estimate can't be large.
4. Log every bet and its closing price in a bet tracker. Over a few hundred bets you can see whether your estimates were too high.

Limits

• Kelly maximizes long-run growth, not comfort. Full Kelly bankrolls swing hard.
• It needs an accurate probability, and edges in sports are small and noisy.
• It ignores limits: sportsbooks can cap your stakes, so the Kelly number may not be one you can place.
• It says nothing about whether a bet is legal where you live.

quant+ shows these numbers for information only. It doesn't place bets or connect to sportsbook accounts.

Quant+ is analytics for informational purposes. 21+. If you or someone you know has a gambling problem, call 1-800-GAMBLER.