Kelly Calculator: How Much to Stake When You Think the Price Is Wrong

The Kelly criterion formula

The Kelly criterion is written as:

f = (b·p − q) / b

• f is the share of the bankroll to stake.
• b is the net odds: the profit per unit staked, which is decimal odds minus one. At 2.20, b is 1.20.
• p is your probability that the bet wins.
• q is the probability that it loses, 1 − p.

The top line, b·p − q, is the edge: what one unit is expected to win at your probability. Dividing by b scales the stake to the price. A small edge at long odds gives a small stake; the same edge at short odds gives a larger one. When b·p − q is zero or negative, f is zero or below, and Kelly stakes nothing.

The edge can also be written as p × decimal odds − 1, which is how the calculator shows it.

Worked example: bankroll 1000, odds 2.20, p = 50%

You think a team wins half the time. The price is 2.20, which implies 1 ÷ 2.20 = 45.45%. Your view is more generous than the market's, so there may be an edge.

• Net odds: b = 2.20 − 1 = 1.20.
• Edge: 0.50 × 2.20 − 1 = 0.10, or 10.00% per unit staked.
• Full Kelly: (1.20 × 0.50 − 0.50) ÷ 1.20 = 0.10 ÷ 1.20 = 8.33% of the bankroll.

These are the numbers the Kelly calculator shows when it opens. A win at full Kelly adds 83.33 × 1.20 = 100.00 to the bankroll; a loss takes away 83.33. The next stake is worked out from the new bankroll, not the starting one, so stakes rise after wins and shrink after losses.

Why full Kelly swings so hard

Full Kelly maximises long-run growth, and it pays for that with large swings on the way. In the example above, five losses in a row, which happens about once in 32 attempts at an even chance, take a bankroll of 1000 down to about 647 at full Kelly. Ten losses in a row leave about 419. In the standard model of Kelly betting, the chance of the bankroll halving at some point is about one in two. Most people stop following a plan long before the maths says they should.

Betting more than Kelly is worse than betting less. In this example, staking twice full Kelly, 16.67% a bet, gives a long-run growth rate of zero: the bankroll goes nowhere on average while swinging even harder. Beyond that, growth turns negative even though every bet has a positive edge.

When your probability is too high

Kelly treats your probability as exact, and it never is. Suppose the team really wins 46% of the time, not 50%. The bet still has a small edge, 0.46 × 2.20 − 1 = 1.2%, and the correct full Kelly stake is 1% of the bankroll, or 10.00. Working from your 50% you would stake 83.33, more than eight times too much. At that size, the bankroll is expected to shrink over time despite the positive edge.

At 45% the price is no longer good at all: 0.45 × 2.20 = 0.99, a loss of 1% per unit. A four-point error in your estimate has removed the whole edge. This is why the probability field matters more than any other input. Comparing your number with the implied probability, and with prices elsewhere, is a fair check on how confident you should be. The odds calculator turns any price into its implied probability and shows the margin in a market.

Fractional Kelly

Fractional Kelly stakes a fixed share of the full Kelly figure. It is the usual answer to both problems above. In the worked example, half Kelly keeps about 75% of the long-run growth of full Kelly, and quarter Kelly keeps about 44%, while the drawdowns shrink sharply: five straight losses leave about 808 at half Kelly and about 900 at quarter Kelly, against 647 at full.

Half Kelly also forgives errors. In the 46% case, full Kelly from your 50% estimate loses money over time, while half Kelly loses about a sixth as fast, and quarter Kelly roughly breaks even. If you are not sure how good your probabilities are, a smaller fraction is the cheaper mistake. Set the fraction in the Kelly calculator and the suggested stake updates straight away.

What the calculator does not include

• Your accuracy. The Kelly calculator has no way to check your probability. A wrong input gives a confident but wrong stake.
• Several bets at once. The formula sizes one bet at a time. Bets placed together on the same day, or on linked outcomes, need smaller stakes than each would get alone.
• Stake limits and minimums. Sportsbooks set their own maximum and minimum stakes, and the suggested figure is not adjusted for them.
• Price changes. The edge is only as good as the odds you actually get. If the price drops before you bet, run the numbers again.
• Commission and fees. Any charge on winnings or deposits reduces the real edge.

Where the margins are thinnest, and so where an edge is easiest to find, is compared on the best odds betting sites page.

Staking plans and responsible gambling

Kelly decides how much to bet. It does not decide whether you should bet, and it cannot create an edge that is not there. If your probabilities are no better than the market's, every stake the formula suggests is a bet against a margin, and the bankroll will fall whatever fraction you use. A staking plan manages risk; it is not a way to win.

Treat the bankroll as money you can afford to lose, keep the fraction low, and stop if betting stops being a choice. Limits, time-outs and where to find help are on the responsible gambling page.

Which sites let you bet on this before any document check is set out in our guide to betting sites without KYC.

More in the same section: Poisson calculator, Bet calculator and all betting calculators.

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