The Kelly Criterion is a rule for how much of your bankroll (your total betting capital) to stake on a wager when you believe the price is in your favor. It’s designed to maximize the long-run growth of your bankroll by:
- Betting more when your edge (the difference between your probability and the market’s implied probability) is bigger.
- Betting zero when there’s no edge.
Key Terms
- Bankroll: Your total available betting capital right now. 💰
- Probability (p): Your estimated chance the bet wins (e.g., 0.55 = 55%).
- q: The chance the bet loses, so
q = 1 - p. - Decimal odds (O): Odds like 2.10, where a winning £1 returns £2.10.
- Net odds (b):
b = O - 1(e.g., 2.10 → b = 1.10). This is used in the formula. - Implied probability: The win probability suggested by the odds, calculated as
1/O. - Break-even probability: Same as implied probability; you only have an edge if
p > 1/O. - EV (Expected Value): The average profit per bet over the long run. A positive EV indicates a good bet.
- CLV (Closing Line Value): Whether your bet beats the closing market price/odds. Consistently beating the closing line suggests a real edge.
- Vig / Overround: The bookmaker’s margin in the market.
- Exchange commission: A fee charged by betting exchanges, which reduces effective odds.
- 1X2 market: A three-way outcome in football (soccer): 1=Home, X=Draw, 2=Away.
- Parlay / Accumulator: A multi-leg bet that wins only if all legs win.
- PPP (context-specific): Not a Kelly variable, but commonly means:
- Points Per Possession (basketball efficiency) 🏀
- Power-Play Points (ice hockey) 🏒
- Points Per Play (American football analytics) 🏈
The Kelly Formula
For a single bet with decimal odds O, let b = O - 1. If your win probability is p and your loss probability is q = 1 - p, the Kelly fraction (the portion of your bankroll to stake) is:
f* = (b · p - q) / b
- If
f* ≤ 0, don’t bet (no positive edge). - If
f* > 0, stake that fraction of your current bankroll.
Edge check: You have an edge only if p > 1/O.
Odds Conversions
- American → Decimal
- +A:
O = 1 + A/100, sob = A/100.Example: +150 → O = 2.50, b = 1.50 - -A:
O = 1 + 100/A, sob = 100/A.Example: -120 → O ≈ 1.8333, b ≈ 0.8333
- +A:
- Fractional (a/b) → Decimal:
O = 1 + a/b, sob = a/b.
Worked Examples
1. Underdog at +150 (American)
- Convert: O = 2.50, so b = 1.50.
- Your model: p = 0.45 (45%), so q = 0.55.
- Break-even:
1/O = 0.40. You have an edge since 0.45 > 0.40. - Kelly:
f* = (1.50 · 0.45 - 0.55) / 1.50 = (0.675 - 0.55) / 1.50 = 0.125 / 1.50 ≈ 0.0833. - Stake ≈ 8.33% of your bankroll.
2. Favorite at -120 (American)
- Convert:
O = 1 + 100/120 ≈ 1.8333, sob ≈ 0.8333. - Your model: p = 0.58 (58%), so q = 0.42.
- Kelly:
f* = (0.8333 · 0.58 - 0.42) / 0.8333 ≈ (0.4833 - 0.42) / 0.8333 ≈ 0.076. - Stake ≈ 7.6% of your bankroll.
3. Near even-money at 2.10 (Decimal)
- Values: b = 1.10, p = 0.52, q = 0.48.
- Kelly:
f* = (1.10 · 0.52 - 0.48) / 1.10 = (0.572 - 0.48) / 1.10 = 0.092 / 1.10 ≈ 0.0836. - Stake ≈ 8.36% of your bankroll.
Full Kelly vs. Fractional Kelly
Full Kelly is optimal for growth if your p is perfectly estimated, but it’s volatile and sensitive to errors. Fractional Kelly reduces the stake for more robustness.
- Half Kelly = 50% of f*
- Quarter Kelly = 25% of f*
Practical rule: If your model is new, start with Half Kelly or smaller.
Common Mistakes
- Overconfident probabilities: Leads to stakes that are too big and large drawdowns.
Fix: Use fractional Kelly and calibrate your models.
- Ignoring correlation between bets.
Fix: Set per-event/daily caps.
- Forgetting vig/fees.
Fix: Adjust odds for overround or commission.
- Using Kelly to chase losses.
Fix: Kelly is for growth, not recovery. Stick to the math.