What is the Kelly Criterion?

1. Introduction

The Kelly Criterion is a mathematical formula used to determine the optimal size of a series of bets or trades. It helps in maximizing the long-term growth of capital by balancing risk and reward. Originally developed by John L. Kelly Jr. in 1956 for maximizing the rate of return in gambling, it has since been widely applied in trading and investing to optimize portfolio allocation.

In this guide, we will explain what the Kelly Criterion is, how it works, and how you can apply it to your trading strategies to make more informed and profitable decisions.

2. What is the Kelly Criterion?

The Kelly Criterion calculates the optimal proportion of your capital to wager (or invest) on a given trade or investment, based on the expected probability of success and the odds (return). It aims to find the “perfect” balance between risking too much (and potentially losing everything) and risking too little (and missing out on potential returns).

Formula

The Kelly Formula is given as: f∗=pb−1−p1f^* = \frac{p}{b} – \frac{1 – p}{1}

Where:

f∗f^* = The optimal fraction of your capital to bet (or invest).
pp = The probability of a win (or the probability that the trade will be successful).
bb = The net odds received on the bet (i.e., the ratio of profit to the amount invested).

Alternative Formula (for trading)

For trading applications, the Kelly Criterion is often expressed as: f∗=2×Expected ReturnRisk of the Trade−1f^* = \frac{2 \times \text{Expected Return}}{\text{Risk of the Trade}} – 1

Where:

Expected Return = The return expected from the trade (expressed as a decimal).
Risk of the Trade = The risk or the potential loss for the trade, often represented as the size of the stop loss or the potential drawdown.

3. Understanding the Components

3.1. Probability of Success (pp)

This is the likelihood that your trade will be successful. It can be based on historical performance, statistical analysis, or subjective judgment. In trading, this could be derived from backtesting a strategy, where you estimate the success rate based on past data.

3.2. Odds or Return (bb)

In gambling, the odds are often represented as the ratio of the profit relative to the stake. In trading, this corresponds to the potential return you expect from the trade compared to your risk. For example, if a trade has a potential reward of 2:1, this means you stand to gain twice as much as the amount you risk on the trade.

3.3. Risk of the Trade

Risk refers to the amount of capital you are willing to lose on a single trade. The Kelly Criterion aims to maximize growth by minimizing the risk of losing too much, which can prevent significant drawdowns in your capital.

4. How the Kelly Criterion Works

The idea behind the Kelly Criterion is to find a balance between risk and reward. If you bet too much, you may experience high volatility, risking large losses. If you bet too little, you miss out on opportunities for growth. The Kelly Criterion recommends an optimal fraction to bet, ensuring the highest possible growth rate of your capital over time.

High Kelly Fraction: When you have high confidence in the probability of success and/or high odds (reward), the Kelly Criterion suggests you bet a larger fraction of your capital.
Low Kelly Fraction: When the probability of success is low or the odds are unfavorable, the Kelly Criterion suggests you risk a smaller fraction of your capital.

5. Example of Kelly Criterion in Action

Let’s assume you have a trading strategy with the following parameters:

Probability of success (pp): 60% (or 0.60).
Odds (or reward-to-risk ratio) (bb): 2:1 (meaning you make $2 for every $1 you risk).

Now, using the Kelly Formula: f∗=0.602−1−0.601f^* = \frac{0.60}{2} – \frac{1 – 0.60}{1} f∗=0.602−0.40f^* = \frac{0.60}{2} – 0.40 f∗=0.30−0.40=−0.10f^* = 0.30 – 0.40 = -0.10

In this case, the Kelly Criterion suggests you should not take the trade, as the optimal fraction of capital to bet is negative (indicating a losing strategy).

If the odds were 3:1, the calculation would change: f∗=0.603−1−0.601f^* = \frac{0.60}{3} – \frac{1 – 0.60}{1} f∗=0.603−0.40=0.20−0.40=−0.20f^* = \frac{0.60}{3} – 0.40 = 0.20 – 0.40 = -0.20

Again, it would suggest a losing strategy.

But if the odds were 1:1: f∗=0.601−1−0.601=0.60−0.40=0.20f^* = \frac{0.60}{1} – \frac{1 – 0.60}{1} = 0.60 – 0.40 = 0.20

Here, the Kelly Criterion suggests that you should bet 20% of your capital on each trade, which maximizes long-term growth without risking excessive capital.

6. Benefits of the Kelly Criterion

Maximizes Long-Term Growth: The primary benefit of the Kelly Criterion is that it optimizes capital growth over time by finding the ideal bet size.
Helps with Risk Management: It inherently includes risk management, as it recommends smaller bets when the strategy has a low probability of success or low reward.
Avoids Over-Betting: By calculating the optimal fraction to bet, it helps prevent overextending your capital, reducing the likelihood of large drawdowns.
Objective Decision Making: The Kelly Criterion provides an objective framework for making investment and trading decisions, reducing emotional bias in trading.

7. Limitations of the Kelly Criterion

Estimation of Probabilities and Odds: The Kelly Criterion relies on accurate estimates of the probability of success and the potential return (odds). Inaccurate estimations can lead to incorrect betting sizes and poor performance.
Volatility: Although it maximizes long-term growth, it can result in significant short-term volatility. This may not be suitable for all traders, especially those with a low risk tolerance.
Assumes Consistent Probabilities: The formula assumes that the probability of success and the odds remain constant over time. In reality, market conditions can change, affecting the reliability of these estimates.
Overconfidence in the Formula: Over-relying on the Kelly Criterion without considering other factors like liquidity, market conditions, or diversification can lead to suboptimal performance.

8. Modifying the Kelly Criterion for Lower Risk

Many traders choose to bet a fraction of the optimal Kelly amount to reduce volatility. For example, you can use half-Kelly, where you bet only 50% of the recommended amount. This strategy reduces risk but also lowers long-term growth potential.

Half-Kelly Formula

fhalf∗=12×f∗f^*_{\text{half}} = \frac{1}{2} \times f^*

This approach helps to limit large swings in your portfolio, especially if you’re uncomfortable with the high volatility the full Kelly Criterion might generate.

9. Kelly Criterion in Python

You can easily calculate the Kelly Criterion in Python using simple arithmetic. Here’s an example:

def kelly_criterion(p, b):
    return (p * (b + 1) - 1) / b

# Example parameters
probability_of_success = 0.60  # 60% chance of success
odds = 2  # 2:1 reward-to-risk ratio

# Calculate the optimal bet size
kelly_fraction = kelly_criterion(probability_of_success, odds)
print(f"Optimal bet size (Kelly Fraction): {kelly_fraction:.2f}")

This code will output the optimal fraction of your capital to bet based on the given probability and odds.

10. Conclusion

The Kelly Criterion is a powerful tool for maximizing long-term capital growth by balancing risk and reward. It provides a clear, mathematically-based decision-making framework for traders and investors, helping them determine the optimal bet size (or position size) for each trade.

Key Takeaways:

Kelly Formula: The Kelly Criterion calculates the optimal fraction to bet based on the probability of success and the expected return.
Maximizing Growth: By following the Kelly Criterion, traders can maximize their long-term capital growth while managing risk.
Risk Management: The Kelly Criterion inherently includes risk management by adjusting bet size according to the risk-return profile of the strategy.
Limitations: The Kelly Criterion depends on accurate estimates of probabilities and odds, and can result in high volatility if used without modification.

When applied correctly, the Kelly Criterion is a valuable tool in optimizing trade sizes and improving the performance of trading strategies. However, it is important to adjust the formula and consider external factors to fit your specific trading style and risk tolerance.

*Disclaimer: The content in this post is for informational purposes only. The views expressed are those of the author and may not reflect those of any affiliated organizations. No guarantees are made regarding the accuracy or reliability of the information. Use at your own risk.