What the Kelly criterion answers
If you have an edge, a bet or investment where you expect to win more than you lose on average, how much should you stake? Bet too little and your bankroll grows slowly; bet too much and a run of losses can wipe you out even though every bet was favorable. In 1956, John Kelly of Bell Labs found the stake that maximizes the long-term growth rate of your bankroll.
The formula
f = p − q ÷ b*
- p: your probability of winning
- q = 1 − p: the probability of losing
- b: net odds, the profit per $1 staked
If f* is zero or negative, the bet has no edge and Kelly says not to bet.
Converting odds
| Format | Example | Net odds b |
|---|---|---|
| Decimal | 2.10 | 2.10 − 1 = 1.10 |
| American, negative | −110 | 100 ÷ 110 = 0.909 |
| American, positive | +150 | 150 ÷ 100 = 1.5 |
| Fractional | 5/2 | 2.5 |
Worked example
You estimate a 55% chance of winning a bet offered at −110.
- b = 0.909, q = 0.45
- Full Kelly = 0.55 − 0.45 ÷ 0.909 = 5.5% of bankroll
- The odds imply a 52.4% chance, so your edge is 5% per dollar staked
- With half Kelly and a $1,000 bankroll, stake $27.50
- Expected bankroll growth: about 0.10% per bet
Why most people bet a fraction of Kelly
Full Kelly is mathematically optimal only if your probability estimate is exactly right. In practice:
- Estimates are noisy. Overestimating your edge by a few points can turn Kelly into overbetting, which lowers growth and raises the risk of ruin.
- Swings are large. With full Kelly, there is a 50% chance that the bankroll will at some point fall to half its starting value.
- Half Kelly keeps about three-quarters of the growth rate with roughly half the volatility; quarter Kelly is more conservative still.
Kelly in investing
The same idea applies to position sizing: with an expected excess return μ and variance σ², the Kelly fraction of capital in a risky asset is about μ ÷ σ². For stocks this often suggests leverage, which is why most investors use a small fraction of it, if any.
A word on gambling
Sportsbooks and casinos build a margin into their odds, so most bets have a negative expected value and Kelly says to stake nothing. Gambling can be addictive: if it stops being fun, call 1-800-GAMBLER for confidential help.
This calculator gives estimates for education and planning. It is not investment, tax or legal advice; for decisions about your own situation, talk to a qualified financial professional.