Kelly Criterion

The Kelly Formula Explained

The Kelly Criterion answers a fundamental question: what percentage of your bankroll should you bet when you have an edge? The formula is:

f* = (b·p − q) / b

Where:
f* = fraction of bankroll to bet (Kelly percentage)
b  = odds received (decimal odds − 1)
p  = probability of winning
q  = probability of losing (1 − p)

This formula was developed by John Kelly Jr. at Bell Labs in 1956, originally for managing information transmission rates. It maximizes the logarithmic growth of your bankroll over time, which means it optimizes the compound growth rate rather than just expected value.

Why It Works: The Mathematical Intuition

Kelly balances two competing forces:

• Betting too little: You fail to capitalize on your edge and grow slowly
• Betting too much: Variance can devastate your bankroll even with positive expected value

The formula finds the sweet spot by maximizing E[log(Wealth)]. Logarithmic utility naturally incorporates risk aversion—doubling your bankroll is good, but losing half is catastrophic. This creates the optimal balance between aggression and safety.

Key insight: Kelly assumes you'll make this bet repeatedly over time. It's designed for long-term growth, not one-off wagers.

Full Kelly vs Fractional Kelly

While Full Kelly maximizes long-term growth rate, most professional bettors use Fractional Kelly—typically betting 25% to 50% of the Kelly recommendation. Here's why:

Problems with Full Kelly:

• Extreme volatility: Full Kelly leads to wild bankroll swings (50%+ drawdowns are common)
• Probability estimation errors: If you overestimate your edge even slightly, Full Kelly can be disastrous
• Psychological strain: The variance is difficult to stomach emotionally

Benefits of Fractional Kelly:

• Reduced variance: Half Kelly reduces volatility by ~75% while only sacrificing ~25% of growth rate
• Error tolerance: More forgiving if your probability estimates are slightly off
• Smoother equity curve: More sustainable psychologically and practically

Recommended approach: Start with 1/4 Kelly or 1/3 Kelly until you're confident in your edge estimation accuracy. Professional sports bettors rarely exceed 1/2 Kelly.

When Kelly Fails: Risk of Ruin

The Kelly Criterion has a critical assumption: you can make fractional bets and bet continuously. It also assumes your probability estimates are accurate. When these conditions aren't met, Kelly can fail:

Scenario 1: Overestimating Your Edge

If you think you have a 55% win probability but it's actually 50%, Full Kelly will recommend betting 5% of your bankroll on a break-even proposition. This is the biggest danger in practice.

Perceived edge: 55% win at +100 odds
Actual edge: 50% win (no edge)
Full Kelly bet: 10% of bankroll
Result: Slow but steady bankroll decline

Scenario 2: Correlated Bets

Kelly assumes independent bets. If you place multiple correlated bets (e.g., several player props in the same game), you're effectively over-betting and increasing ruin risk.

Scenario 3: Limited Betting Opportunities

Kelly is optimal for infinite repeated bets. If you only have a few opportunities, the variance may not even out, and Kelly can underperform more conservative strategies.

Scenario 4: Finite Bankrolls and Discrete Bets

With a $1,000 bankroll and $10 minimum bets, you can't always bet the exact Kelly percentage. This granularity can affect performance, especially with smaller bankrolls.

Finding the Right Multiplier

Most bettors apply a fractional Kelly multiplier to balance growth and variance:

• Full Kelly: Maximum growth, highest volatility (rarely used in isolation)
• Half Kelly: ~75% less variance with only ~25% less growth
• Quarter Kelly: Popular for high-variance sports or uncertain edges
• Custom: Tailor the multiplier to bankroll comfort and confidence in projections

The calculator lets you toggle preset multipliers or dial in a custom percentage for full control.

Worked Example: NFL Point Spread

The Kansas City Chiefs are playing at home against the Denver Broncos. The line is:

Chiefs -7 at -110 (American odds)
Your analysis suggests Chiefs cover 56% of the time

Step 1: Convert American Odds to Decimal

Decimal odds = 1 + (100 / 110) = 1.909
Therefore: b = 1.909 − 1 = 0.909

Step 2: Convert Probability

p = 0.56 (56% win probability)
q = 0.44 (44% loss probability)

Step 3: Calculate Kelly Percentage

f* = (b·p − q) / b
f* = (0.909 × 0.56 − 0.44) / 0.909
f* = (0.509 − 0.44) / 0.909
f* = 0.069 / 0.909
f* = 0.076 or 7.6%

Step 4: Apply to Your Bankroll

Bankroll: $5,000
Full Kelly bet: $5,000 × 0.076 = $380
Half Kelly bet: $5,000 × 0.038 = $190
Quarter Kelly bet: $5,000 × 0.019 = $95

Expected Value Check:

EV = (p × win amount) − (q × loss amount)
EV = (0.56 × $380 × 0.909) − (0.44 × $380)
EV = $193.54 − $167.20
EV = +$26.34 per bet

Important note: The 56% win probability must be well-researched. If the true probability is 52.4% or lower, you have no edge at -110 odds, and Kelly would recommend not betting at all.

When to Use Kelly (and When Not To)

Kelly IS Appropriate When:

• You have a genuine, quantifiable edge: Based on data, modeling, or superior information
• You're making many bets over time: The law of large numbers works in your favor
• You can estimate probabilities accurately: Through backtesting, statistical models, or proven methods
• Your bankroll can handle volatility: These aren't your rent money or emergency funds
• Bets are independent: Or you're adjusting for correlation in your calculations

Kelly is NOT Appropriate When:

• You're guessing at probabilities: "I feel like this team will win" isn't sufficient
• You're making one-off bets: Kelly optimizes long-term growth, not single events
• You can't tolerate drawdowns: Even Fractional Kelly will have losing streaks
• You're betting on hunches: Kelly requires mathematical edge, not gut feelings
• The odds or probabilities can change: In-play betting or markets with shifting lines
• You're prone to tilt: Kelly requires discipline to stick to the system during downswings

Alternative Strategies:

• Flat betting: Simpler, more conservative, appropriate for casual bettors
• Unit-based systems: Betting 1-5 units based on confidence level
• Fixed percentage: Always bet 1-2% of bankroll regardless of edge
• Expected value maximization: For one-off bets where bankroll growth isn't the goal

Practical Implementation Tips

• Track your results: Monitor whether your actual win rate matches predictions
• Be honest about your edge: It's usually smaller than you think
• Update your bankroll regularly: Recalculate Kelly percentages as your bankroll changes
• Set maximum bet limits: Even if Kelly says 10%, consider capping at 5%
• Adjust for uncertainty: The less certain you are, the more fractional you should go
• Don't chase losses: Kelly is systematic, not emotional. Stick to the formula
• Consider opportunity cost: If you run out of bankroll, you can't bet on future +EV opportunities

Final Thoughts

The Kelly Criterion is a powerful tool, but it's not magic. It requires accurate probability estimation, emotional discipline, and sufficient bankroll to weather variance. Most professional bettors view it as a starting point—a way to think about bet sizing systematically—rather than a rigid rule.

If you're serious about sports betting, study the Kelly Criterion, but combine it with solid handicapping, bankroll management principles, and realistic self-assessment of your abilities. And remember: when in doubt, bet smaller. Fractional Kelly keeps you in the game longer and gives you time to refine your edge.