The Kelly Criterion solves a fundamental question in betting: given that you have an edge, how much should you stake?
The Kelly Formula
Kelly stake (%) = (bp - q) / b
Where:
- b = decimal odds minus 1 (net odds)
- p = your estimated probability of winning
- q = probability of losing (1 - p)
Worked Example
You estimate a 45% chance of Liverpool beating Chelsea. The bookmaker offers odds of 2.60.
- b = 2.60 - 1 = 1.60
- p = 0.45
- q = 0.55
- Kelly = (1.60 × 0.45 - 0.55) / 1.60 = (0.72 - 0.55) / 1.60 = 0.1063 (10.63%)
With a £1,000 bankroll, full Kelly recommends staking £106.30.
Full Kelly vs Fractional Kelly
| Approach | Stake (£1,000 bankroll) | Variance | Growth Rate |
|---|---|---|---|
| Full Kelly | £106.30 | Very High | Maximum |
| Half Kelly | £53.15 | High | 75% of maximum |
| Quarter Kelly | £26.58 | Moderate | 50% of maximum |
| Eighth Kelly | £13.29 | Low | 25% of maximum |
Half Kelly achieves 75% of the maximum growth rate with substantially less variance. This is why it is the most popular choice among professionals.
Why Kelly Works
Kelly automatically adjusts your stake based on:
- Edge size — bigger edge = bigger stake
- Odds — longer odds = relatively smaller stake
- Bankroll size — stakes shrink when losing, grow when winning
This means Kelly never risks total ruin. As your bankroll drops, stakes shrink proportionally.
Common Mistakes
Overestimating Your Edge
The biggest danger with Kelly is using inflated probability estimates. If you think you have a 55% edge but your true edge is 48%, Kelly will recommend oversized stakes that erode your bankroll.
Ignoring Correlation
Kelly assumes each bet is independent. If you place multiple bets on related outcomes (e.g. several Premier League matches), reduce your stakes below Kelly to account for correlation.
Using Full Kelly
Full Kelly produces extreme bankroll swings. A losing streak of 10 bets can reduce your bankroll by 50-70%. Quarter Kelly produces far smoother results.