Kelly Criterion is a mathematical strategy used to determine the optimal amount of your bankroll to bet on a given wager. Developed by John L. Kelly Jr., this formula maximizes long-term growth by accurately balancing risk and reward based on your calculated edge. In short the larger your theoretical edge the more of your bankroll you can risk wagering.

Why Use the Kelly Criterion?

• Maximizes expected long-term growth.
• Reduces the probability of risk of ruin.
• Balances risk versus reward by adjusting bet size based on your edge.

Kelly Criterion Formula

Kelly Criterion = Probability of Win - (Probability of Loss / Decimal Odds - 1)

Detailed Example

Let's use the weighted coin toss example from the What Is Expected Value guide where there is a coin that has a 60% chance of landing on heads and the payout is $95 on a $100 wager (1.95 decimal odds).

• True Probability: 60% chance (0.60)
• Sportsbook Odds: 1.95 (-105 American Odds)

Calculate using the Kelly formula:

Kelly Criterion = Probability of Win - (Probability of Loss / Decimal Odds - 1)
Kelly Criterion = 0.60 - (0.40 / 1.95 - 1)
Kelly Criterion = 0.60 - (0.40 / .95)
Kelly Criterion = 0.60 - 0.42
Kelly Criterion = 0.18 or 18%

This means wagering 18% of your bankroll is the optimal amount to maximize winnings while managing risk.

Variations of the Kelly Criterion

Wagering 18% as outlined above theoretically results in the largest possible profits while reducing the risk of losing your bankroll. However, that risk is still present and for many higher than they are willing to tolerate.

Full Kelly

• Betting the exact percentage calculated.
• Maximizes growth but can result in significant volatility and greatest risk of losing money.

Fractional Kelly

• Betting a fraction (e.g., half or quarter) of the full Kelly amount.
• Reduces variance while still capturing positive long-term growth.
• Often recommended to mitigate volatility and risk.

Example of Fractional Kelly

• Full Kelly: 18%
• Three Quarter Kelly: 13.5%
• Half Kelly: 9%
• Quarter Kelly: 4.5%

Risk of Ruin

What is Ruin

When your bankroll falls below a set threshold of its initial amount. There is not a singular amount that applies to all bettors. Some people may define ruin as losing 50% of your initial bankroll. For example if you started with $100 some may consider their bankroll ruined if they reach $50. Others may consider it 25% which means they would considering their bankroll ruined if they reach $25. This is comes down to a personal preference and individual risk tolerance.

What is Risk Of Ruin

Risk of ruin is the probability of your bankroll falling below your personally defined ruin level. The risk of ruin varies based on several factors

• Theoretical edge you have when wagering
• Wager sizing, i.e. full vs quarter Kelly
• Individual threshold for what defines ruin

Coin Toss Simulations

Taking the example above of a weighted coin with a 60% chance of landing on heads and decimal odds of 1.95 we can run simulations to see how betting different amounts such as full Kelly vs quarter Kelly vs static bet amounts impacts profit and loss.

Simulation Criteria

• Probability of Win: 60%
• Decimal Odds: 1.95
• Starting Bankroll: $100
• Number of Bets: 100
• Number of Simulations: 100,000

Simulation Results

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Key Takeaways

• Bettors in this simulation have 18% edge, being profitable should be very easy.
• Betting a static amount greatly increases the chances of losing money, despits having a large statistical edge.
• Sizing your bets based on Kelly Criterion results in higher potential profitability, and higher chances of being profitable copmpared to static bet sizing.
• The Risk of Ruin, being below $50 or $25, is much smaller when using Kelly Criterion.

Sports Betting Simulations

In the coin toss example you had an 18% edge. That is not a realistic edge to consistently have in sports betting. Below we will run simulations based on having a 2% edge, much more realistic in sports betting. In the table above we proved that using Kelly Criterion for bet sizing increases profits and at the same time reduces the Risk of Ruin. Knowing this we are now only going to compare various Fractional Kelly Criterion bet sizing.

Small Sample Size

• Probability of Win: 51%
• Decimal Odds: 2.00
• Starting Bankroll: $100
• Number of Bets: 100
• Number of Simulations: 100,000

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Key Takeaways

• 100 Bets is a very small sample size, and there is not much varaition between each fractional Kelly.
• ~46% of the simulations of each fractional Kelly is not profitable due to the small sample size.

With a small sample size of only 100 bets there is not a large difference between any of the fractional Kelly Criterion bet sizing, but it is important to note that across all fractional Kelly bet sizing methods ~46% of the simulations are not profitable. Even with a proven mathematical edge and perfectly sized bets, variance across a small sample size of 100 bets results in nearly half of bettors losing money.

This is one of the most important lessons in +EV betting.  You need a large sample size before you can realistically expect to be profitable. The minimum number of bets for this should be 1,000. Depending on the frequency you place bets it may take months to place 1,000 bets. You need patience and the mental fortitude to withstand placing that many bets over a several month time span before you can truly see the results of +EV betting.

Medium Sample Size

• Probability of Win: 51%
• Decimal Odds: 2.00
• Starting Bankroll: $100
• Number of Bets: 1,000
• Number of Simulations: 100,000

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Key Takeaways

• Variation between each fractional Kelly is increasing as sample size increases.
• Full Kelly = Largest risk, largest reward.
  • Highest max, mean, median, and mode
  • Lowest min
  • 7.66% of simulations below 50% of their starting bankroll

With a 1,000 bet sample we start to see variation between the different fractional Kelly wager sizing. As expected the full Kelly simulations resulted in the highest max, mean, median, and mode bankroll. This is great for those bettors, but you can see that the full Kelly also comes with the lowest min, the largest percentage of bettors losing money, and the largest Risk of Ruin at 7.66% of bettors ending with less than 50% of their starting bankroll.

Large Sample Size

• Probability of Win: 51%
• Decimal Odds: 2.00
• Starting Bankroll: $100
• Number of Bets: 10,000
• Number of Simulations: 100,000

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Key Takeaways

• The pattern of full Kelly resulting in the largest risk for the largest reward continues.
• 96% of 1/4 Kelly bettors are profitable and only 0.8% are below 50% of their starting bankroll.
• The max for 1/2, 3/4, and full Kelly grow exponentially and extreme outliers drag up the mean (average).
• The mode (most common result) and median (result in the middle) for 1/2, 3/4, and full Kelly are above 1/4 but the difference is significantly smaller than the differnce between the means.

The pattern of full Kelly wager resulting in the largest profits continues, but so does the associated risk. With a 100,000 simulations of 10,000 bets at full Kelly the average profit is much larger, but take note that nearly 9% of bettors that bet full Kelly ended up with less than 50% of their original bankroll. Compare this to 0.08% of bettors that wager 1/4 Kelly that ended up with 50% or less of their bankroll.