The Kelly Criterion is a mathematical formula used to determine the optimal size of a series of bets or investments to maximize the long-term growth of wealth.
Developed by John Larry Kelly Jr. in 1956, the formula has applications in gambling, investment management, and other areas where probabilistic outcomes are involved.
related: How to Gain an Advantage Sports Betting
The Formula
The Kelly Criterion formula is designed to maximize the expected logarithm of wealth, which translates to maximizing the geometric growth rate of an investment or a bet. The formula for calculating the optimal bet size, 𝑓∗f∗, is:𝑓∗=𝑏𝑝−𝑞𝑏f∗=bbp−qwhere:
- 𝑓∗f∗ is the fraction of the current bankroll to wager.
- 𝑏b is the net odds received on the bet (i.e., the odds minus 1).
- 𝑝p is the probability of winning.
- 𝑞q is the probability of losing, which is 1−𝑝1−p.
Application in Betting
The Kelly Criterion is particularly useful in scenarios where the return on an investment or bet is binary—either a win or a loss. For example, in sports betting or casino games, the formula helps bettors decide how much of their bankroll to stake based on the odds and their perceived probability of winning.
Example
Consider a coin toss where the coin is biased, giving a 60% chance of landing heads and 40% for tails. If the odds for heads are even (1:1), the Kelly Criterion would suggest:𝑓∗=(1×0.60)−0.401=0.20f∗=1(1×0.60)−0.40=0.20This means you should bet 20% of your bankroll on heads to maximize long-term growth.
Application in Investment
In the financial world, the Kelly Criterion can be used to determine the proportion of a portfolio to allocate to different assets to maximize long-term growth. This approach is known as growth optimal portfolio management. Investors like Warren Buffett are known to use Kelly-style methods to manage their investments.
Advantages and Limitations
Advantages
- Maximizes Long-Term Growth: By focusing on the logarithm of wealth, the Kelly Criterion ensures that the growth rate of the bankroll is maximized over the long term.
- Disciplined Betting: It provides a systematic approach to bet sizing, which can help prevent over-betting and under-betting.
Limitations
- Accuracy of Inputs: The formula requires accurate estimates of the probabilities of winning and losing. Incorrect estimates can lead to suboptimal or even detrimental betting sizes.
- Volatility: The Kelly Criterion can suggest large bet sizes, leading to significant volatility in the bankroll. To mitigate this, many bettors use a fractional Kelly approach, betting a fraction (e.g., half or a quarter) of the Kelly amount.
Practical Use
The Kelly Criterion has been successfully applied in various domains:
- Gambling: It is widely used in sports betting, poker, and other gambling activities to manage bankroll and optimize bet sizes.
- Investing: Investors use it to allocate capital among different assets to achieve optimal growth rates.
- Statistical Testing: In modern statistical methods, Kelly-style analysis is used to replace traditional statistical power in anytime-valid tests and confidence intervals.
The Kelly Criterion is a powerful tool for anyone looking to optimize their betting or investment strategies. By carefully calculating the optimal bet size based on the odds and probabilities, it helps maximize long-term wealth growth while managing risk. However, it is crucial to use accurate inputs and consider the potential for increased volatility, possibly adopting a fractional approach to balance growth and risk.
