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Bankroll and Kelly staking

Kelly sizes a bet to maximise long-run growth. It is optimal only if your probability estimate is right, and unforgiving when it is not.

Read this before using the output. The Kelly criterion is optimal only if your probability estimate is correct. It has no defence against a wrong one. Overestimating your edge is the ordinary way bettors fail, not a rare one, and Kelly converts that error into oversized stakes automatically. Betting twice the correct Kelly fraction drives long-run growth to roughly zero even when your edge is genuine, and betting more than that makes it negative while the strategy still looks like a winner for long stretches. Almost everyone who uses this in practice uses a fraction of it, and the page shows half and quarter for that reason.

Your bankroll and your estimate

Bankroll

Price (American)

Your estimate of the true win probability (%)

Flat stake for comparison (% of bankroll)

Full Kelly: —
Full Kelly stake: —
Half Kelly: —
Quarter Kelly: —
Edge over break-even: —
EV per bet at full Kelly: —
Growth rate, full Kelly: —
Growth rate, half Kelly: —
Growth rate, flat stake: —
Flat stake: —
Stake at 2x Kelly: —

b = D - 1 (net decimal odds) q = 1 - p Kelly fraction f* = (p x b - q) / b = (p x D - 1) / (D - 1) If f* ≤ 0 the bet has no positive expectation: stake nothing. Stake f* x bankroll Fractional Kelly half = f*/2, quarter = f*/4 Expected log growth per bet, at any fraction f: g(f) = p x ln(1 + f x b) + q x ln(1 - f) g is maximised exactly at f = f*. At f = 2 x f* growth is at or near zero, and beyond it growth is negative. The page computes g at each fraction rather than assuming the textbook approximation.

What the number means

Kelly answers one specific question: what fraction of a bankroll maximises the long-run growth rate of that bankroll, given a known edge? The answer is (p x D - 1) / (D - 1), which is the expected value of the bet divided by the amount you can win per unit staked. If expectation is zero or negative the formula returns zero or less, which is the correct instruction to not bet.

The growth-rate rows are the useful part of this page. They show that overbetting is punished asymmetrically: the curve rises to a peak at full Kelly and then falls away, reaching about zero at twice full Kelly and going negative past it. Underbetting costs you a little growth. Overbetting can cost you all of it while the bets themselves were still good ones.

Half Kelly is a common compromise. It captures roughly three quarters of the growth rate at roughly half the volatility, and it halves the damage from an overestimated edge. Quarter Kelly is more conservative again.

Where it misleads

Every number here is downstream of a probability you estimated. If your true edge is 2% and you believe it is 5%, Kelly will size the bet as though you had 5%, and you will be overbetting by a factor of two and a half without any signal that anything is wrong. Results will not tell you either, because variance at these stake sizes swamps the difference for hundreds of bets.

The formula also assumes one bet resolving at a time, a bankroll you can actually re-stake, a price you can get in the size the formula asks for, and no correlation between bets. Simultaneous or correlated positions require smaller fractions than the single-bet answer. Limits, minimum increments and the fact that bankrolls are also rent money all argue in the same direction.

Kelly is a sizing tool, not an edge detector. It presumes the hard problem is already solved.

Bankroll and Kelly staking

Your bankroll and your estimate

What the number means

Where it misleads

Read next

Kelly sizes an edge. It cannot tell you whether you have one.