Kelly Criterion

Quick Definition

The Kelly Criterion is a bankroll management formula developed by mathematician John L. Kelly Jr. in 1956. It tells you exactly what percentage of your bankroll to bet on any opportunity to maximize the long-term growth rate of your bankroll. Bet too little and you leave profit on the table. Bet too much and you risk ruin. Kelly finds the mathematically optimal point between these extremes.

The Formula

Kelly % = (bp - q) / b

Where:

• b = the net odds received (decimal odds - 1)
• p = your estimated probability of winning
• q = your estimated probability of losing (1 - p)

Simplified Version

Kelly % = Edge / Odds

Where:

• Edge = (True Win Probability × Decimal Odds) - 1 (this is your EV)
• Odds = Decimal Odds - 1

Worked Example

You estimate a team has a 55% chance of winning. The sportsbook offers +120 (2.20 decimal).

• b = 2.20 - 1 = 1.20
• p = 0.55
• q = 0.45

Kelly % = (1.20 × 0.55 - 0.45) / 1.20 = (0.66 - 0.45) / 1.20 = 0.21 / 1.20 = 17.5%

With a $10,000 bankroll, the Kelly Criterion recommends betting $1,750 on this opportunity.

Why Kelly Maximizes Long-Term Growth

Kelly is derived from information theory. It maximizes the expected logarithm of wealth, which is equivalent to maximizing the long-term geometric growth rate of your bankroll. No other fixed-fraction betting strategy grows your bankroll faster over time.

The intuition: if you bet too little, you are not extracting full value from your edge. If you bet too much, losing streaks can devastate your bankroll to the point where recovery becomes mathematically difficult. Kelly finds the exact fraction that maximizes growth.

Fractional Kelly: The Practical Standard

Full Kelly is mathematically optimal but psychologically brutal. A 17.5% bet on a single game means a losing streak of 10 games reduces your bankroll by 83%. Most professional bettors use fractional Kelly (typically half Kelly or quarter Kelly) to reduce variance while maintaining most of the growth benefit.

Half Kelly: Bet 50% of the Kelly recommendation. Reduces variance by 75% while maintaining approximately 75% of the growth rate.

Quarter Kelly: Bet 25% of the Kelly recommendation. Very conservative, suitable for bettors with high uncertainty in their probability estimates.

The trade-off: fractional Kelly grows your bankroll more slowly but with dramatically less variance. For most bettors, half Kelly is the right balance.

Kelly and Probability Estimation

Kelly is only as good as your probability estimates. If you estimate a 55% win probability but the true probability is 48%, Kelly will recommend overbetting and you will lose money. The formula amplifies both your skill and your errors.

This is why many professional bettors use conservative probability estimates or apply a discount factor to their Kelly calculations. If you are uncertain about your edge, bet less than Kelly recommends.

Practical rule: If your probability estimate has high uncertainty, use quarter Kelly. If you have strong evidence for your estimate (e.g., from a well-calibrated model), use half Kelly.

Kelly for Arbitrage and +EV Betting

For arbitrage (guaranteed profit), Kelly is straightforward: your edge is the guaranteed profit percentage, and your probability of winning is 100%. Kelly recommends betting a large fraction of your bankroll, but practical constraints (account limits, liquidity) usually cap your actual bet size.

For +EV value betting, Kelly requires accurate probability estimates. Most value betting tools (OddsJam, RebelBetting) provide Kelly recommendations based on the implied edge from sharp line comparisons.

Common Mistakes

Mistake 1: Using the sportsbook’s implied probability instead of your own estimate. The sportsbook’s implied probability already includes their margin. Using it as your “true probability” will always produce negative Kelly (recommending no bet), even on +EV opportunities.

Mistake 2: Applying Kelly to correlated bets. If you have multiple bets on the same game (e.g., spread and total), they are correlated. Standard Kelly assumes independent bets. Applying full Kelly to correlated bets leads to overbetting.

Mistake 3: Ignoring bankroll updates. Kelly is a dynamic formula. As your bankroll grows or shrinks, your bet sizes should change proportionally. Recalculate Kelly before each bet based on your current bankroll.

Mistake 4: Overconfidence in edge estimates. Beginners often overestimate their edge. A 5% edge estimate that is actually 1% leads to 5x overbetting. Start with conservative estimates and adjust as you accumulate data.

Kelly in Practice: A Day of Betting

You have a $5,000 bankroll. You find three +EV opportunities:

Bet 1: 52% estimated win probability at -105 (1.952 decimal)

• Kelly % = (0.952 × 0.52 - 0.48) / 0.952 = (0.495 - 0.48) / 0.952 = 1.6%
• Half Kelly: 0.8% = $40

Bet 2: 58% estimated win probability at +110 (2.10 decimal)

• Kelly % = (1.10 × 0.58 - 0.42) / 1.10 = (0.638 - 0.42) / 1.10 = 19.8%
• Half Kelly: 9.9% = $495

Bet 3: 45% estimated win probability at +200 (3.00 decimal)

• Kelly % = (2.00 × 0.45 - 0.55) / 2.00 = (0.90 - 0.55) / 2.00 = 17.5%
• Half Kelly: 8.75% = $437.50

Total exposure: $972.50 (19.5% of bankroll). This is reasonable for three independent +EV bets.

Risk-Free Way to Learn Kelly

Before applying Kelly to real bets, practice with paper trading. Estimate probabilities for upcoming games, calculate Kelly stakes, and track whether your estimates were accurate. After 50-100 paper bets, you will have a sense of how well-calibrated your probability estimates are.

If your paper trading results show consistent positive CLV, your probability estimates are likely accurate enough to apply Kelly to real bets. Start with quarter Kelly and scale up as your confidence grows.

Tools That Use Kelly

• OddsJam - Provides Kelly stake recommendations for each +EV opportunity
• RebelBetting - Kelly-based stake sizing in value betting mode
• Trademate Sports - Professional Kelly calculations with CLV integration
• Avo.bet - Kelly recommendations for US markets

Related Terms

• Expected Value - The edge that Kelly optimizes
• Bankroll Management - The broader framework Kelly fits into
• Closing Line Value - How to verify your probability estimates are accurate