The Kelly Criterion formula calculates the potential sizes of wagers or investments that are the most likely to maximize the possible return while minimizing the risk of losses. This, coupled with the systematic way the formula handles bankroll management, the Kelly Criterion has been one of the top strategies used by investors, bettors, and traders for decades.Â

John L. Kelly Jr. developed the system in 1956 to try and ascertain what percentage of a bankroll should be wagered or invested based on the perceived odds. Understanding how it works and the applications that can be derived from it will give you greater security in making decisions where uncertainty and risk come into play.

## What is the Kelly Criterion?

The Kelly Criterion formula is as follows:

$fbpâqâ$

Where:

is the fraction of the bankroll to bet or invest.**$fâ$**is the odds received on the bet (i.e., the net odds minus 1).**b**is the probability of winning.**p**is the probability of losing (which is 1â p).**q**

It’s the formula that gives the best fraction of bankroll to bet, given there’s a trade-off in growth and not losing too much at one go. If the formula returns positive, then the fraction of bankroll you should wager; if it is negative, it suggests don’t bet, since EV is not advantageous.

## How the Kelly Criterion Works

The Kelly Criterion is used for maximizing the growth rate of capital over time, having in mind the proportion of bankroll to be risked in each bet or investment. It takes into consideration neither over-betting, which can perhaps lead to great losses, nor under-betting, where suboptimal growth will be attained.

### Example of the Kelly Criterion in Betting

Suppose you are betting on a sports event with the following characteristics:

- The odds of your bet are 3.00 (which means you receive $3 for every $1 wagered, including your stake).
- You estimate the probability of winning (p) at 50%.
- The probability of losing (q) is, therefore, 50%.

Using the Kelly Criterion formula:

fâ=3Ă0.5â0.53â1f* = \frac{3 \times 0.5 – 0.5}{3 – 1}fâ=3â13Ă0.5â0.5â

fâ=1.5â0.52f* = \frac{1.5 – 0.5}{2}fâ=21.5â0.5â

fâ=12=0.5f* = \frac{1}{2} = 0.5fâ=21â=0.5

Since the Kelly Criterion suggests that you wager 50% of your betting funds on this bet, if you have a $1,000 bankroll, the optimal bet size would then be $500. This approach allows you to maximize your potential growth while maintaining a buffer against potentially significant losses.

## Conditional Factors and Assumptions

Several key factors and assumptions influence the application of the Kelly Criterion:

**Accurate Probability Estimates**

The accuracy of the Kelly Criterion depends on the ability to have reasonably correct estimates for the probability of winning ‘p.’ Incorrect estimates result in a miscalculation that leads to over- or under-betting. This can include, in gambling, an understanding of the true odds of an event, which may be hard to estimate correctly.**Constant Odds**

The Kelly Criterion assumes that the odds and probability will stay the same throughout. In the real world, these may shift based on new information or market variations. Refreshing the inputs into the formula-whenever new data becomes available is really important to keep the formula working well.**Divisibility of Bankroll**

The formula assumes that the bankroll can be divided into smaller units and that it is possible to bet fractional amounts. This can be impractical in certain betting scenarios where minimum bet sizes are imposed or in markets where liquidity is an issue.**Risk Tolerance**

The Kelly Criterion seeks growth maximization, but it does not take into consideration another factor that is just as important: individual risk tolerance. Some people may want to wager less than the Kelly amount to minimize variance and prevent losing streaks.

## Over-Betting and Under-Betting

**Over-Betting**:

This occurs when a bettor wagers too large a portion of their bankroll, increasing the risk of ruin. For instance, betting more than the Kelly fraction can lead to significant losses, even with a positive expected value.**Under-Betting**:

Betting less than the Kelly fraction results in slower growth. While it reduces risk, it also means that potential profits are not maximized. Some bettors opt for a “fractional Kelly” strategy, betting a smaller fraction, such as half or quarter Kelly, to balance growth with reduced risk.

## Application Beyond Betting

The Kelly Criterion is far from a gambling-only solution and finds extensive applications in both finance and investment. For instance, in stock trading, it may answer such questions as how much capital one needs to invest in a certain stock or asset, given the expected return and volatility. Again, the aim is to maximize portfolio growth with a minimum chance of significant losses.

### Example in Investing

An investor believes a particular stock has a 60% chance of making a 20% return and a 40% chance of losing 10%.Â

The Kelly Criterion system suggests investing 40% of the available capital in this stock; this approach helps the investor allocate their resources so they can maximize growth while accounting for the associated risk. To apply the Kelly Criterion:

- p=0.6p = 0.6p=0.6
- q=0.4q = 0.4q=0.4
- Expected return = 20% (or 0.20)
- Loss = 10% (or -0.10)

The formula becomes:

fâ=(0.20Ă0.6)â(0.10Ă0.4)0.20f* = \frac{(0.20 \times 0.6) – (0.10 \times 0.4)}{0.20}fâ=0.20(0.20Ă0.6)â(0.10Ă0.4)â

fâ=0.12â0.040.20f* = \frac{0.12 – 0.04}{0.20}fâ=0.200.12â0.04â

fâ=0.080.20=0.4f* = \frac{0.08}{0.20} = 0.4fâ=0.200.08â=0.4

The Kelly Criterion suggests investing 40% of the available capital in this stock. This approach helps the investor to allocate their resources in a way that maximizes growth while accounting for the associated risk.

## Limitations of the Kelly Criterion

Despite its benefits, the Kelly Criterion has limitations that must be considered:

**Sensitivity to Input Errors**: The formula relies on accurate estimates of probability and odds. Small errors in these estimates can lead to significantly different bet sizes.**Volatility**: Following the Kelly Criterion exactly can lead to high volatility in returns, especially when bet sizes are large. To mitigate this concern, many bettors use a fractional Kelly approach, a modification of the original Kelly Criterion system.

**Practical Constraints**: In real-world scenarios, factors such as liquidity, transaction costs, and position size limits can restrict the application of the Kelly Criterion.

## Conclusion

The Kelly Criterion offers a straightforward and mathematical strategy for choosing the size of a bet in uncertain circumstances to maximize the potential for long-term capital growth. It’s an analytical way of valuing a wager, balancing potential gains against the possibility of devastating losses.Â

This requires a certain degree of probability estimation and, if the criterion were to be effectively used, acceptance of its innate volatility. The Kelly Criterion is of value where decision-making needs to be linked to risk in a wide range of fields, from gambling to investment and trading.

Mark Sullivan, the Managing Editor at the Big Blind, leverages his two decades of journalism experience to provide clear, accessible, and reader-friendly content on the gambling industry, catering to both professionals and newcomers.