The Formula That Wall Street and Professional Bettors Swear By

In 1956, a physicist named John Kelly published a paper that would change how professional gamblers, investors, and hedge fund managers think about money.

His formula — the Kelly Criterion — tells you exactly how much to bet to maximize long-term wealth growth. Not "roughly how much." Not "a good rule of thumb." The mathematically optimal amount.

Warren Buffett reportedly uses a version of Kelly. Bill Gross, the legendary bond investor, used it at PIMCO. Ed Thorp, the mathematician who beat blackjack and then beat Wall Street, used it to manage billions.

And yet, most casual bettors have never heard of it. They bet random amounts, size their wagers based on gut feeling, and wonder why their bankroll doesn't grow.

This guide will teach you the Kelly Criterion from the ground up — the math, the intuition, the practical application, and the pitfalls.

The Kelly Criterion Formula

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

Where:

• $f^*$ = Fraction of your bankroll to bet
• $b$ = Decimal odds received minus 1 (the net payout)
• $p$ = Your probability of winning
• $q$ = Your probability of losing (1 - p)

Breaking It Down

The formula has two components:

Numerator (bp - q): This is your edge — the expected value of the bet. If your edge is positive, the bet is profitable. If it's negative, don't bet.

Denominator (b): This normalizes your edge by the payout. A bet with a small payout needs a larger edge to justify the same bet size.

The result ($f^*$): The fraction of your current bankroll that maximizes long-term growth.

Kelly Criterion in Action: Sports Betting

Example 1: A Simple Value Bet

You're betting on a football match. The bookmaker offers decimal odds of 3.00 on the underdog. Your analysis says they have a 40% chance of winning.

• $b$ = 3.00 - 1 = 2.00
• $p$ = 0.40
• $q$ = 0.60

$$f^* = \frac{(2.00 \times 0.40) - 0.60}{2.00} = \frac{0.80 - 0.60}{2.00} = \frac{0.20}{2.00} = 0.10$$

Kelly says: bet 10% of your bankroll.

If your bankroll is $5,000, you bet $500.

Example 2: A Heavy Favorite

You're betting on a heavy favorite at decimal odds of 1.25. You believe they have an 85% chance of winning.

• $b$ = 1.25 - 1 = 0.25
• $p$ = 0.85
• $q$ = 0.15

$$f^* = \frac{(0.25 \times 0.85) - 0.15}{0.25} = \frac{0.2125 - 0.15}{0.25} = \frac{0.0625}{0.25} = 0.25$$

Kelly says: bet 25% of your bankroll.

This seems high, but the math is correct — when you have a large edge on a favorite, Kelly recommends a large bet. However, most professionals would use fractional Kelly here to reduce variance.

Example 3: No Value

You're offered decimal odds of 2.00 on a coin flip. You estimate the probability at 50%.

• $b$ = 2.00 - 1 = 1.00
• $p$ = 0.50
• $q$ = 0.50

$$f^* = \frac{(1.00 \times 0.50) - 0.50}{1.00} = \frac{0.50 - 0.50}{1.00} = 0$$

Kelly says: don't bet. When there's no edge, Kelly correctly tells you to pass.

👉 Calculate Kelly bet sizes for your bets with our Expected Value Calculator — find +EV bets and size them correctly.

Kelly Criterion in Poker

In poker, Kelly is most useful for two decisions:

1. Bankroll Management Across Stakes

Kelly tells you how much of your bankroll to risk at each stake level. If you have $5,000 and play $1/$2 NLHE (buy-in $200):

• Your bankroll is 25 buy-ins
• Kelly suggests you should move down when you have fewer than 20 buy-ins
• This is consistent with the 20-30 buy-in rule from our Bankroll Management Guide

2. Tournament Push/Fold Decisions

In tournament poker, when stacks are short, you're often deciding between pushing all-in or fold. Kelly can help size these decisions.

Example: You have 10 big blinds in a tournament. You're considering pushing with A♠ 5♠ from the button.

• If you estimate the blinds will call 20% of the time and you have 50% equity when called:
• $b$ = 1.50 (you win 1.5x your stack when they fold)
• $p$ = 0.60 (40% fold equity + 20% × 50% when called)
• $q$ = 0.40

$$f^* = \frac{(1.50 \times 0.60) - 0.40}{1.50} = \frac{0.90 - 0.40}{1.50} = 0.33$$

Kelly says: committing 33% of your stack (3.3 big blinds) is optimal. Since you have 10 big blinds, pushing all-in is more aggressive than Kelly suggests — but in tournaments, ICM considerations often justify being more aggressive than pure Kelly.

Why Most Professionals Use Fractional Kelly

Full Kelly maximizes long-term growth. But it comes with enormous variance. Here's why most professionals use fractional Kelly:

The Problem with Full Kelly

1. Enormous drawdowns: A full-Kelly bettor can expect to see their bankroll halve before it doubles. This is psychologically devastating.

2. Edge estimation errors: Kelly assumes you know your exact edge. In reality, your edge estimate has uncertainty. If you overestimate your edge by even 10%, full Kelly can lead to ruin.

3. Discrete betting: Kelly assumes you can bet any fraction of your bankroll. In practice, you're limited to discrete bet sizes (e.g., whole dollars, minimum stakes).

The Solution: Fractional Kelly

The tradeoff: You give up 25% of growth (half Kelly) to reduce your drawdown risk by roughly 50%. For most people, this is an excellent trade.

Example: With a $10,000 bankroll and a bet where full Kelly says to bet $1,000 (10%):

• Full Kelly: Bet $1,000
• Half Kelly: Bet $500
• Quarter Kelly: Bet $250

The Dangers of Over-Betting Kelly

One of the most important properties of Kelly is what happens when you bet more than Kelly suggests:

Double Kelly is a disaster. You're taking on enormous risk with zero expected growth. Your bankroll will eventually go to zero with near certainty.

The lesson: If you're unsure about your edge, bet less than Kelly. Never bet more.

Kelly Criterion for Investing

The Kelly Criterion isn't just for gambling. It's used by professional investors to size positions in their portfolios.

Warren Buffett's "Kelly Approach"

Buffett has described his investment approach as "concentrated betting" — putting large amounts of capital into his highest-conviction ideas. This is essentially Kelly thinking:

• When you have a large edge (high conviction), bet big
• When you have a small edge (low conviction), bet small
• When you have no edge, don't bet at all

Stock Market Example

You believe a stock has a 60% chance of going up 50% and a 40% chance of going down 30%.

• $b$ = 0.50 (50% upside)
• $p$ = 0.60
• $q$ = 0.40

$$f^* = \frac{(0.50 \times 0.60) - 0.40}{0.50} = \frac{0.30 - 0.40}{0.50} = \frac{-0.10}{0.50} = -0.20$$

Negative Kelly means: don't take this bet. The expected value is negative.

Now if you believe the stock has a 70% chance of going up 50%:

$$f^* = \frac{(0.50 \times 0.70) - 0.30}{0.50} = \frac{0.35 - 0.30}{0.50} = 0.10$$

Kelly says: allocate 10% of your portfolio to this stock.

Common Mistakes with Kelly Criterion

1. Overestimating Your Edge

This is the #1 mistake. If you think you have a 10% edge but actually have 5%, full Kelly will bet twice as much as it should. This leads to slower growth and higher risk of ruin.

Fix: Be conservative. If you think your edge is 10%, use 5% in the Kelly formula.

2. Not Using Fractional Kelly

Full Kelly is for theorists. Real humans can't handle 50% drawdowns. Use half Kelly or quarter Kelly.

3. Applying Kelly to Correlated Bets

Kelly assumes each bet is independent. If you're betting on multiple games involving the same team, the bets are correlated and Kelly overestimates the optimal bet size.

Fix: Reduce bet sizes when bets are correlated.

4. Ignoring Bankroll Changes

Kelly is a dynamic strategy. After every bet, you recalculate based on your new bankroll. If you win, your next bet is larger (in absolute terms). If you lose, it's smaller.

Fix: Recalculate after every bet. Don't use a fixed dollar amount.

5. Using Kelly for -EV Bets

Kelly only works for +EV bets. If the expected value is negative, Kelly correctly tells you not to bet. Don't force a Kelly calculation on a bad bet.

Key Takeaways

1. Kelly Criterion maximizes long-term growth — it's the mathematically optimal bet sizing strategy
2. The formula is simple: f* = (bp - q) / b
3. Use fractional Kelly (half or quarter) to reduce variance and account for edge uncertainty
4. Never bet more than Kelly — double Kelly has zero expected growth and near-certain ruin
5. Kelly requires accurate edge estimation — overestimating your edge is dangerous
6. Recalculate after every bet — Kelly is dynamic, not static
7. Kelly applies beyond gambling — investing, poker, tournament strategy, and more

👉 Put Kelly into practice: use our Expected Value Calculator to find +EV bets, then size them with the Kelly Criterion. Read our Bankroll Management Guide for the complete framework.

Related Tools & Resources

• Expected Value Calculator — Find +EV bets to apply Kelly sizing
• Sports Betting Calculator — Calculate payouts for Kelly-sized bets
• Bankroll Management Guide — The complete framework for managing your bankroll
• How to Read Sports Betting Odds — Master odds formats for Kelly calculations
• The Mathematics of Parlay Betting — Why parlays compound the house edge
• Glossary — Kelly Criterion, Expected Value, and other key terms