# Kelly Criterion: Definition, How Formula Works, History, and Goals

## What Is Kelly Criterion?

The Kelly criterion is a mathematical formula relating to the long-term growth of capital developed by John L. Kelly Jr. while working at AT&T's Bell Laboratories. It is used to determine how much to invest in a given asset, in order to maximize wealth growth over time.

### Key Takeaways

• Although used for investing and other applications, the Kelly Criterion formula was originally presented as a system for gambling.
• The Kelly Criterion was formally derived by John Kelly Jr., a scientist at AT&T's Bell Laboratories.
• The formula is used to determine the optimal amount of money to put into a single trade or bet.
• Several famous investors, including Warren Buffett and Bill Gross, are said to have used the formula for their own investment strategies.
• Some argue that an individual investor's constraints can affect the formula's usefulness.

## Understanding Kelly Criterion

The Kelly criterion is currently used by gamblers and investors for risk and money management purposes, to determine what percentage of their bankroll/capital should be used in each bet/trade to maximize long-term growth.

After being published in 1956, the Kelly criterion was picked up quickly by gamblers who were able to apply the formula to horse racing. It was not until later that the formula was applied to investing. More recently, the strategy has seen a renaissance, in response to claims that legendary investors Warren Buffett and Bill Gross use a variant of the Kelly criterion.

The formula is used by investors who want to trade with the objective of growing capital, and it assumes that the investor will reinvest profits and put them at risk for future trades. The goal of the formula is to determine the optimal amount to put into any one trade.

There are two key components to the formula for the Kelly criterion:

1. Winning probability factor (W): the probability a trade will have a positive return.
2. Win/loss ratio (R): This will be equal to the total positive trade amounts, divided by the total negative trading amounts.

The result of the formula will tell investors what percentage of their total capital they should apply to each investment.

The term is often also called the Kelly strategy, Kelly formula, or Kelly bet, and the formula is as follows:

\begin{aligned} &Kelly~\% = W - \Big[\dfrac{(1-W)}{R}\Big] \\ &\textbf{where:}\\ &\begin{aligned} Kelly~\% = &\text{ Percent of investor's capital to put into}\\ &\text{ a single trade} \end{aligned}\\ &W = \text{Historical win percentage of trading system}\\ &R = \text{Trader's historical win/loss ratio}\\ \end{aligned}

While the Kelly Criterion is useful for some investors, it is important to consider the interests of diversification as well. Many investors would be wary about putting their savings into a single asset–even if the formula suggests a high probability of success.

## Kelly Criterion Limitations

The Kelly Criterion formula is not without its share of skeptics. Although the strategy's promise of outperforming all others, in the long run, looks compelling, some economists have argued against it—primarily because an individual's specific investing constraints may override the desire for optimal growth rate.

In reality, an investor's constraints, whether self-imposed or not, are a significant factor in decision-making capability. The conventional alternative includes Expected Utility Theory, which asserts that bets should be sized to maximize the expected utility of outcomes.

## What Is the Kelly Criterion?

The Kelly Criterion is a formula used to determine the optimal size of a bet when the expected returns are known. According to the formula, the optimal bet is determined by the formula

K= W - (1 - W)/R—where K is a percentage of the bettor's bankroll, W is the probability of a favorable return, and R is the ratio of average wins to average losses.

## Who Created the Kelly Criteria?

The Kelly Criteria was originally created by John Kelly, while working at AT&T's Bell Laboratories. It was first adopted by gamblers to determine how much to bet on horse races, and later adapted by some investors.

## How Do I Find My Win Probability With the Kelly Criterion?

Unlike gambling, there is no truly objective way to calculate the probability that an investment will have a positive return. Most investors using the Kelly Criterion try to estimate this value based on their historical trades: simply check a spreadsheet of your last 50 or 60 trades (available through your broker) and count how many of them had positive returns.

## How Do You Input Odds Into the Kelly Criterion?

In order to enter odds into the Kelly Criterion, one first needs to determine W, the probability of a favorable return, and R, the size of the average win divided by the size of the average loss. For investing purposes, the easiest way to estimate these percentages is from the investor's recent investment returns. These figures are then entered into the formula

K= W- (1-W) / R—where K represents the percentage of the investor's bankroll that they should invest.

## What Is Better than the Kelly Criterion?

While there are many investors who integrate the Kelly Criterion into successful moneymaking strategies, it is not foolproof and can lead to unexpected losses. Many investors have specific investment goals, such as saving for retirement, that are not well-served by seeking optimal returns. Some economists have argued that these constraints make the formula less suitable for many investors.

## How Are the Black-Scholes Model, the Kelly Criterion, and the Kalman Filter Related?

The Black-Scholes Model, Kelly Criterion, and the Kalman Filter are all mathematical systems that can be used to estimate investment returns when some key variables depend on unknown probabilities. The Black-Scholes model is used to calculate the theoretical value of options contracts, based upon their time to maturity and other factors.

The Kelly Criterion is used to determine the optimal size of an investment, based on the probability and expected size of a win or loss.

The Kalman Filter is used to estimate the value of unknown variables in a dynamic state, where statistical noise and uncertainties make precise measurements impossible.

## What Is a Good Kelly Ratio?

While some believers in the Kelly Criterion will use the formula as described, there are also drawbacks to placing a very large portion of one's portfolio in a single asset. In the interest of diversification, an investor should think twice about investing more than 20% of their bankroll in a single investment–even if the Kelly Criterion suggests a higher percentage.