Kelly Criterion

The Kelly Criterion is the mathematically optimal formula for bet sizing, calculating exactly what percentage of your bankroll to wager based on your edge and the odds. Unlike flat staking or arbitrary unit systems, Kelly maximizes the geometric growth rate of your bankroll over time. It's the gold standard for professional bettors, investors, and anyone managing risk with positive expected value opportunities.

Understanding Kelly Criterion
The Kelly Formula
Step-by-Step Calculation
Fractional Kelly
Kelly for Multiple Bets
Common Mistakes

Understanding Kelly Criterion {#understanding}

Imagine you have a coin that lands heads 60% of the time, and someone offers you even money (2.0 odds) on heads. You have an edge—but how much should you bet?

Bet too little: You don't capitalize on your advantage
Bet too much: One bad run wipes out your bankroll
Bet optimally (Kelly): Maximum long-term growth

The Kelly Criterion answers: Given your edge and the odds, what bet size maximizes wealth over time?

Key Insight: Kelly doesn't maximize expected profit—it maximizes expected logarithmic utility, which translates to maximum geometric growth rate. This distinction is crucial for long-term wealth accumulation.

Why Kelly Works

Strategy	Short-term	Long-term
Bet everything	High variance	Bankrupt
Flat 1% stakes	Low growth	Slow accumulation
Kelly optimal	Balanced	Maximum growth

Kelly finds the perfect balance between growth and survival. Understanding your risk of ruin helps determine optimal Kelly fraction.

The Kelly Formula {#formula}

Basic Kelly Formula

f^* = \frac{bp - q}{b}

Where:

f* = Fraction of bankroll to bet
b = Decimal odds - 1 (net odds)
p = Probability of winning
q = Probability of losing (1 - p)

Simplified Formula

For decimal odds:

\text{Kelly \%} = \frac{p \times \text{Odds} - 1}{\text{Odds} - 1}

Or even simpler:

\text{Kelly \%} = \frac{\text{Edge \%}}{\text{Odds} - 1}

Where Edge % = (Probability × Odds) - 1

Kelly Formula Derivation

The formula maximizes expected log wealth:

G = p \times \log(1 + f \times b) + q \times \log(1 - f)

Taking the derivative and setting to zero yields the Kelly formula.

Step-by-Step Calculation {#calculation}

Example 1: Football Match

Scenario: You estimate Liverpool has 55% chance to beat Chelsea. Bookmaker offers odds of 2.10.

Step 1: Identify variables

Odds = 2.10
p (your probability) = 0.55
q = 1 - 0.55 = 0.45
b = 2.10 - 1 = 1.10

Step 2: Calculate edge

\text{Edge} = (0.55 \times 2.10) - 1 = 1.155 - 1 = 0.155 = 15.5\%

Step 3: Apply Kelly formula

f^* = \frac{0.55 \times 2.10 - 1}{2.10 - 1} = \frac{0.155}{1.10} = 0.141 = 14.1\%

Result: Bet 14.1% of your bankroll.

Example 2: Tennis Match (Underdog)

Scenario: You estimate underdog has 35% chance. Odds are 3.50.

Edge = (0.35 × 3.50) - 1 = 0.225 = 22.5%
Kelly = 0.225 / (3.50 - 1) = 0.225 / 2.50 = 9%

Example 3: No Edge (Negative Kelly)

Scenario: True probability 45%, odds 2.00.

Edge = (0.45 × 2.00) - 1 = -0.10 = -10%
Kelly = -0.10 / 1.00 = -10%

Negative Kelly means don't bet. If you could bet against this outcome, you would.

Kelly Calculation Table

True Prob	Odds	Edge	Kelly %
50%	2.00	0%	0%
50%	2.20	10%	8.3%
55%	2.00	10%	10%
55%	1.90	4.5%	5%
60%	1.80	8%	10%
40%	3.00	20%	10%
30%	4.00	20%	6.7%

Fractional Kelly {#fractional}

Why Use Fractional Kelly?

Full Kelly assumes you know your exact edge. In reality:

Your probability estimates have errors
Sample sizes are limited
Edge can change over time

Fractional Kelly (betting a fraction of full Kelly) addresses these issues.

Fractional Kelly Performance

\text{Fractional Kelly Growth} = f \times G_{Kelly} \times (2 - f)

Fraction	Growth vs Full Kelly	Variance Reduction
100% (Full)	100%	None
75%	93.75%	Significant
50% (Half)	75%	Major
25% (Quarter)	43.75%	Very large

Half Kelly achieves 75% of optimal growth with far less risk.

Recommended Fractions by Confidence

Your Edge Confidence	Recommended Fraction
Very high (verified model, 1000+ bets)	50-75%
High (good model, 500+ bets)	33-50%
Medium (reasonable estimates)	25-33%
Low (uncertain)	10-25%

Fractional Kelly Example

Full Kelly says bet 14.1%. Using half Kelly:

\text{Actual Bet} = 14.1\% \times 0.50 = 7.05\%

Kelly for Multiple Bets {#multiple-bets}

Simultaneous Independent Bets

When placing multiple bets at once, reduce each bet's Kelly fraction:

\text{Adjusted Kelly}_i = \frac{f_i^*}{\sum f_j^*} \times k

Where k is your target total exposure (often capped at 30-50% of bankroll).

Practical Approach: Diversification

Number of Bets	Max Single Bet	Max Total Exposure
1	Full Kelly	Full Kelly
2-3	2/3 Kelly each	50% total
4-5	1/2 Kelly each	50% total
6+	1/3 Kelly each	50% total

Sequential vs Simultaneous

Sequential bets (one after another resolves): Use full calculated Kelly each time—your bankroll updates.

Simultaneous bets (all pending at once): Reduce allocation to prevent over-exposure.

Common Mistakes {#mistakes}

Mistake 1: Overestimating Your Edge

The most dangerous mistake. If you think your edge is 10% but it's actually 2%, full Kelly will devastate your bankroll.

Solution:

Track your actual results over 500+ bets
Compare to closing line value (CLV)
Use fractional Kelly

Mistake 2: Ignoring Correlation

Kelly assumes independent bets. Betting on related outcomes (same game, same team) violates this assumption.

Example of correlated bets:

Liverpool to win
Liverpool over 1.5 goals
Liverpool clean sheet

These aren't independent—treat as one large bet.

Mistake 3: Not Recalculating Bankroll

Kelly percentages should apply to your current bankroll, not starting bankroll.

Bankroll	10% Kelly Bet
$1,000 (start)	$100
$1,200 (after wins)	$120
$800 (after losses)	$80

This automatic adjustment is part of Kelly's power—you bet more when winning, less when losing.

Mistake 4: Applying to Negative EV

Kelly never recommends betting on negative expected value. If your calculation gives a negative number or zero, don't bet.

Mistake 5: Ignoring Practical Constraints

Kelly might suggest betting 25% of bankroll. Practical issues:

Bookmaker limits
Liquidity (can you actually get the bet down?)
Account preservation

Kelly Criterion in Practice {#practice}

The Professional Approach

Calculate theoretical Kelly
Apply uncertainty discount (typically 25-50% of Kelly)
Cap maximum bet (usually 5-10% regardless of Kelly)
Consider correlation with other bets
Track and adjust based on results

Kelly with Bankroll Constraints

Constraint	Adjustment
Limited bankroll	Use absolute minimums ($10 regardless of %)
Bookmaker limits	May need to distribute across books
Account longevity	Sometimes underbetting preserves access

Sample Kelly Betting Log

Date	Bet	Prob	Odds	Edge	Full Kelly	Actual Bet	Result
Jan 1	Liverpool	55%	2.10	15.5%	14.1%	7%	Win
Jan 2	Man City	70%	1.50	5%	10%	5%	Win
Jan 3	Chelsea	45%	2.40	8%	5.7%	3%	Loss

Kelly Criterion Visualized {#visualization}

Growth Rate by Bet Size

For a bet with 10% edge at 2.0 odds (Kelly = 10%):

Bet Size	Expected Growth Rate
0% (no bet)	0%
5%	~0.45% per bet
10% (Kelly)	~0.50% per bet
15%	~0.45% per bet
20% (2x Kelly)	0% per bet
25%+	Negative growth

At 2x Kelly, expected growth drops to zero. Beyond that, you're mathematically expected to lose money despite having an edge.

Risk of Ruin by Strategy

Strategy	Risk of 50% Drawdown
Full Kelly	~50% eventually
Half Kelly	~11%
Quarter Kelly	~1%

Learn More

For a comprehensive guide including real-world examples, simulation results, and spreadsheet templates, read our complete Kelly Criterion guide.

Apply Kelly Criterion with these tools:

Kelly Criterion Calculator - Calculate optimal bet size
Bankroll Management - Plan your bankroll strategy
Bankroll Growth Calculator - Project growth over time

Kelly Criterion

Kelly Criterion

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