The Kelly Criterion is a precise mathematical formula — not a rule of thumb — for sizing bets or investments to maximise the long-run compound growth rate of your bankroll or portfolio. First published by John L. Kelly Jr. at Bell Labs in 1956, it has since become foundational in sports betting, poker, options trading, and quantitative portfolio management. Unlike the common instinct to bet a fixed dollar amount or a fixed percentage, Kelly dynamically scales your wager to your edge: bet more when the edge is large, bet less when it is thin, and bet nothing at all when the expected value is negative.
The formula
For a binary bet with decimal odds d (e.g. 2.00 for evens), the net odds are b = d − 1. If your
estimated win probability is p and the loss probability is q = 1 − p, the Kelly fraction is:
f* = (b·p − q) / b
The numerator b·p − q is your edge — the expected profit per unit staked. If edge is zero or
negative, Kelly tells you to bet zero. If edge is positive, Kelly gives the exact fraction of your
bankroll that maximises the expected logarithm of wealth after many rounds.
For portfolios of assets with normally distributed continuous returns, the formula becomes:
f* = (μ − r) / σ²
where μ is the expected annual return, r the risk-free rate, and σ the annual standard deviation.
This is the continuous Kelly used in quantitative finance, directly proportional to the Sharpe ratio.
How it maximises growth
Kelly does not maximise expected dollar profit per bet — it maximises E[log(wealth)], the expected logarithm of wealth. This is equivalent to maximising the geometric mean of outcomes, which is what determines your bankroll after many bets. Betting more than Kelly grows the bankroll faster in the short run but produces a lower geometric mean — and after enough bets, the over-bettor will always fall behind the Kelly bettor. Betting less than Kelly is always sub-optimal but never ruinous. The asymmetry means over-betting is much more dangerous than under-betting.
Why practitioners use fractional Kelly
Full Kelly produces theoretically optimal growth but with brutal volatility. A typical Kelly bettor can expect 50% drawdowns regularly even on a +EV strategy. The reason: the formula is only as accurate as your probability estimate. A 5% overestimate of win probability is enough to make you over-bet and meaningfully hurt long-run growth. The solution is fractional Kelly — bet a fixed fraction (commonly 25% or 50%) of the full Kelly amount. At 50% Kelly, you capture roughly 75% of the maximum growth rate while experiencing roughly half the variance. Most professional sports bettors and quantitative traders operate between 25% and 50% Kelly for this reason.
Worked example
You estimate a sports market at 60% win probability. The available odds are 2.10 (decimal), so the
net odds are b = 1.10. Loss probability is q = 0.40.
- Edge:
b·p − q = 1.10 × 0.60 − 0.40 = 0.66 − 0.40 = 0.26 - Full Kelly:
f* = 0.26 / 1.10 = 23.6% - 50% Kelly:
0.5 × 23.6% = 11.8% - Wager on a £1,000 bankroll at 50% Kelly: £118
| Win % | Decimal odds | Edge | Full Kelly | 50% Kelly |
|---|---|---|---|---|
| 55% | 2.00 | 10% | 10.0% | 5.0% |
| 60% | 2.10 | 26% | 23.6% | 11.8% |
| 65% | 1.80 | 17% | 21.3% | 10.6% |
| 70% | 1.70 | 19% | 27.1% | 13.6% |
Every figure above is computed live in your browser — nothing is sent to any server.
The SVG growth-rate chart lets you see the classic bell-shaped Kelly curve: growth rises as you approach f*, peaks exactly at the Kelly fraction, and then falls — crossing zero again at 2×f* (the “double Kelly” ruin point). Any bet beyond 2× full Kelly produces negative expected log-growth, meaning you are mathematically guaranteed to lose money over the long run.