Mathematics
The Kelly criterion: how much to stake when you have an edge
In 1956 John Kelly of Bell Labs proved there is an exact fraction of capital that maximises long-run growth. Professionals have used it for decades — almost always as a fraction.
- 1956
- Kelly's paper
- f = (bp − q) / b
- The formula
- ¼ – ½
- Fraction in practice
- 0
- Stake without an edge
The formula
f* = (b × p − q) / b
where:
- f* = the fraction of the bankroll to stake
- b = the net odds (decimal odds − 1)
- p = your estimated probability of winning
- q = 1 − p
Equivalently: f* = p − q / b.
Step-by-step example
Odds 2.20 → b = 1.20. You estimate a 50 % win probability → p = 0.50, q = 0.50.
f* = (1.20 × 0.50 − 0.50) / 1.20 = (0.60 − 0.50) / 1.20 = 0.0833
Full Kelly: 8.3 % of the bankroll. With €20,000 → €1,667.
If the estimate were 45 % (the implied probability of 2.20 is 45.5 %): f* = (1.20 × 0.45 − 0.55) / 1.20 = −0.008 → no bet.
Table: full Kelly by odds and estimated probability
| Odds \ Probability | 40 % | 50 % | 55 % | 60 % | 70 % |
|---|---|---|---|---|---|
| 1.50 | — | — | 10 % | 20 % | 40 % |
| 1.80 | — | 12.5 % | 18.8 % | 25 % | 37.5 % |
| 2.00 | — | 0 % | 10 % | 20 % | 40 % |
| 2.50 | 0 % | 16.7 % | 25 % | 33.3 % | 50 % |
| 3.00 | 10 % | 25 % | 32.5 % | 40 % | 55 % |
“—” = negative (no edge). Notice how fast the fractions become huge: a 5 % error in the estimate moves the stake by 10 percentage points of the bankroll.
Why nobody plays full Kelly
- Estimation error. The formula assumes you know p. You don’t — you estimate it, and estimates are systematically over-optimistic.
- Variance. Even with the right p, full Kelly has a 50 % chance of halving the bankroll before doubling it.
- Asymmetry. Staking 2× Kelly has zero expected growth; ½ Kelly keeps 75 % of the growth with ¼ of the variance.
Fractional Kelly — what professionals use
| Fraction | Growth (% of full) | Variance (% of full) | Who |
|---|---|---|---|
| Full | 100 % | 100 % | Almost nobody |
| ½ Kelly | 75 % | 25 % | Experienced bettors with a proven model |
| ⅓ Kelly | ~56 % | ~11 % | The most common professional choice |
| ¼ Kelly | ~44 % | ~6 % | Conservative, large bankrolls |
In the 8.3 % example: ¼ Kelly = 2.1 % of the bankroll — practically the “2 % unit” of bankroll management. Not a coincidence: the rule-of-thumb unit sizes are ¼ Kelly for typical 3–5 % edges.
Several simultaneous bets
Kelly assumes one bet at a time. With 10 open positions at the weekend, the sum of full-Kelly fractions can exceed 50 % of the bankroll. Practical rule: size each position at ¼ Kelly and cap total exposure (e.g. 10–15 % of the bankroll per day).
Limits of the method
- It needs a quantitative probability estimate. If you bet by feel, Kelly does not apply — use a flat unit.
- It ignores the operator’s stake limits and your psychology.
- In markets where you have no proven edge (the closing line in the Premier League), the correct Kelly is zero.
For the other staking schemes (flat, percentage, confidence-based) see staking plans.
FAQ
What if the formula gives a negative result?▾
It means you have no edge at that price. The correct stake is zero — or, on an exchange, a lay bet.
Why not full Kelly if it is 'optimal'?▾
It is optimal only if your probability estimate is exact. In practice we overestimate our edge; full Kelly on a wrong estimate leads to 50–80 % drawdowns.
Can I apply Kelly to accumulators?▾
Mathematically yes (with the combined odds and the product of the probabilities), but estimation error compounds on every leg. Use a smaller fraction.
Invitation
Stake by the numbers
The right answer to 'how much?' is a formula, not a feeling.
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