Kelly Criterion Position Sizing Calculator

The Kelly Criterion is a mathematical formula that determines the optimal size of a bet or investment position to maximize long-term growth while minimizing risk. Developed by John L. Kelly Jr., this criterion balances the probability of winning and losing to find the fraction of the bankroll that should be wagered on each bet.

What is the Kelly Criterion?

The Kelly Criterion provides a mathematical framework for determining the optimal position size in gambling, trading, or investment. It calculates the fraction of the current bankroll that should be wagered on each bet to maximize the expected growth of the bankroll over time.

Key concepts in the Kelly Criterion include:

Probability of winning (p): The chance that a bet will be successful
Probability of losing (q): The chance that a bet will fail (1 - p)
Odds of winning (b): The amount won if the bet succeeds (e.g., 1.5 for a 50% profit)
Fraction (f): The optimal fraction of the bankroll to wager

The Kelly Criterion is based on the assumption that the bankroll grows exponentially over time. In practice, this means that the optimal position size is smaller than the theoretical maximum to account for the risk of prolonged losing streaks.

How to Use the Kelly Criterion

To apply the Kelly Criterion:

Determine the probability of winning (p) for your bet or investment
Calculate the odds of winning (b) - this is the amount you win if successful (e.g., 1.5 for a 50% profit)
Use the Kelly Criterion formula to calculate the optimal fraction (f) of your bankroll to wager
Adjust the position size based on your risk tolerance and the volatility of your bets

For example, if you have a 60% chance of winning a bet with 2:1 odds, the Kelly Criterion would recommend wagering approximately 27.3% of your bankroll on each bet.

The Kelly Criterion Formula

The basic Kelly Criterion formula is:

f = (bp - q) / b

Where:

f = optimal fraction of bankroll to wager
b = odds of winning (amount won if successful)
p = probability of winning
q = probability of losing (1 - p)

For fractional Kelly, which accounts for the risk of prolonged losing streaks, the formula becomes:

f = (bp - q) / b * k

Where k is a fraction between 0 and 1 (typically 0.5 to 0.75) representing the fraction of the full Kelly bet you're willing to risk.

Worked Example

Suppose you're trading a stock with these characteristics:

Probability of winning (p) = 60% (0.6)
Odds of winning (b) = 1.5 (50% profit)

Using the Kelly Criterion formula:

f = (1.5 * 0.6 - 0.4) / 1.5 f = (0.9 - 0.4) / 1.5 f = 0.5 / 1.5 f ≈ 0.333 or 33.3%

This means you should wager approximately 33.3% of your bankroll on each trade. For fractional Kelly with k = 0.5:

f = 0.333 * 0.5 ≈ 0.167 or 16.7%

This more conservative position size accounts for the risk of prolonged losing streaks.

Limitations of the Kelly Criterion

While the Kelly Criterion provides a useful framework, it has several limitations:

It assumes independent bets with the same probability and odds
It doesn't account for emotional factors or psychological biases
It may lead to large position sizes that are impractical for most traders
It doesn't consider the time value of money or opportunity costs

In practice, traders often use fractional Kelly (typically 20-50% of the full Kelly bet) to account for these limitations and reduce risk.

FAQ

What is the difference between full Kelly and fractional Kelly?: Full Kelly represents the theoretical maximum position size that maximizes long-term growth. Fractional Kelly is a smaller portion of the full Kelly bet that accounts for the risk of prolonged losing streaks.
How do I determine the probability of winning for the Kelly Criterion?: The probability of winning depends on your specific trading or investment strategy. You can estimate it based on historical performance, backtesting, or expert analysis.
Can I use the Kelly Criterion for stocks or other investments?: Yes, the Kelly Criterion can be applied to stocks, options, futures, and other financial instruments. The key is to accurately estimate the probability of winning and the odds of winning for your specific strategy.
What should I do if my Kelly Criterion position size is too large?: If the Kelly Criterion suggests a position size that's too large for your risk tolerance or capital, you can use fractional Kelly or adjust your strategy to improve your probability of winning.
Is the Kelly Criterion suitable for all types of trading?: The Kelly Criterion works best for independent bets with stable probabilities and odds. It may not be suitable for strategies with strong momentum or autocorrelation between bets.