Kelly criterion

1. Kelly Criterion

The Kelly criterion, also known as the Kelly bet size, is a formula used to determine the optimal size of a series of bets in order to maximize the rate of growth of wealth over the long term. Developed by John L. Kelly Jr., a Bell Labs researcher, in 1956, it was initially conceived for predicting horse races but has since found applications in portfolio management, investment strategies, and even gambling. This article provides a comprehensive introduction to the Kelly criterion, its mathematical foundations, practical applications, limitations, and variations. It aims to be accessible to beginners while providing sufficient detail for intermediate users.

History and Origins

John L. Kelly Jr. was a research mathematician working at Bell Labs, known for his work in information theory and cryptography. His work on long-distance telephone transmission led him to consider problems of optimal resource allocation under uncertainty. The Kelly criterion emerged from his efforts to determine the optimal fraction of capital to allocate to different investments, aiming to maximize the long-term growth rate of wealth. Initially, it was used to predict outcomes in games of chance, specifically horse racing, where Kelly himself was a successful gambler. The criterion was initially presented in a confidential paper circulated among a small group of investors and gamblers. Its popularity grew significantly after being popularized by investors like Warren Buffett and Bill Phelps. Phelps, in particular, used the criterion extensively in his gambling endeavors and wrote about it in his book *Beat the Odds*.

The Core Formula

The basic Kelly criterion formula is:

f* = (bp - q) / b

Where:

• f* represents the fraction of your current capital to bet.
• b represents the net profit received for every unit bet (the odds less one. For example, if you bet $1 and win $2 (including your initial bet), b=1).
• p represents the probability of winning.
• q represents the probability of losing (q = 1 - p).

This formula calculates the fraction of your capital that, when bet, maximizes the expected geometric growth rate of your wealth. Crucially, it doesn't guarantee a win on any single bet, but it optimizes the long-term growth of your capital *assuming* your probability assessment (p) is accurate.

Understanding the Components

Let's break down each component of the formula to understand its significance:

• Probability of Winning (p): This is arguably the most critical element. Accurate estimation of 'p' is paramount. If you overestimate your winning probability, you'll bet too much, increasing your risk of ruin. If you underestimate it, you'll bet too little, limiting your potential growth. Estimating 'p' often involves using technical analysis, fundamental analysis, or a combination of both. Tools like moving averages, Bollinger Bands, and Fibonacci retracements can assist in assessing probabilities, though none are foolproof. Elliott Wave Theory attempts to identify recurring patterns to predict future price movements, influencing 'p' estimation. Candlestick patterns are also commonly used for probability assessment.

• Odds (b): This represents the payout you receive for a winning bet, minus your initial stake. For example, if you bet $10 on a 2-to-1 payout, you win $20 (plus your $10 stake back), making 'b' equal to 1 (20/10 - 1). Understanding the odds offered by different brokers or platforms is essential. Implied volatility in options trading significantly impacts potential payouts (and thus 'b').

• Probability of Losing (q): This is simply 1 minus the probability of winning. It represents the likelihood of losing your bet.

• Fraction of Capital to Bet (f*): The result of the formula. This is the percentage of your capital you should bet to maximize long-term growth.

Example Calculation

Let's say you believe you have a 60% (p = 0.6) chance of winning a bet with odds of 1-to-1 (b = 1). Applying the Kelly criterion:

f* = (1 * 0.6 - (1 - 0.6)) / 1 f* = (0.6 - 0.4) / 1 f* = 0.2

This means you should bet 20% of your capital on this bet.

Fractional Kelly and Risk Management

While the Kelly criterion provides an optimal bet size, it can often lead to aggressive betting, especially with high-probability, low-odds scenarios. This can expose you to significant drawdown risk – the potential for substantial losses. Therefore, many investors and traders use a *fractional Kelly* strategy.

Fractional Kelly involves betting a fraction of the Kelly-recommended bet size. Common fractions include:

• Half Kelly (f = 0.5 * f*): A more conservative approach, reducing risk significantly.
• Quarter Kelly (f = 0.25 * f*): Even more conservative, suitable for risk-averse investors.

Using fractional Kelly helps to smooth out the equity curve and reduce the likelihood of ruin. Position sizing is a related concept that incorporates risk tolerance and account size into bet sizing decisions. Stop-loss orders are crucial for mitigating potential losses, regardless of the Kelly fraction used. Trailing stops can help lock in profits while minimizing risk.

Applications Beyond Gambling

The Kelly criterion has applications far beyond horse racing and casinos:

• Portfolio Management:** Allocating capital across different assets based on their expected returns and correlations. For example, determining the optimal percentage of a portfolio to invest in stocks versus bonds. Modern Portfolio Theory incorporates similar principles of diversification and risk management.

• Investment Strategies:** Determining the optimal amount to invest in a particular stock or asset. Analyzing price action and identifying support and resistance levels can aid in estimating the probability of a successful investment. Value investing and growth investing both require probability assessments of future performance.

• Venture Capital:** Deciding how much to invest in different startups, based on their potential for growth and the probability of success.

• Options Trading:** Determining the optimal size of an options position based on the probability of the option expiring in the money. Understanding Greeks (Delta, Gamma, Theta, Vega) is vital in options trading and influences probability assessments. Iron Condors and Butterfly Spreads are examples of options strategies where Kelly criterion principles can be applied.

• Forex Trading:** Determining the appropriate lot size for a currency pair trade, based on the trader's analysis and risk tolerance. Using MACD, RSI, and other oscillators can help with probability assessment. Trend lines and chart patterns also contribute to trade setup analysis.

Limitations and Criticisms

Despite its theoretical elegance, the Kelly criterion has limitations:

• Sensitivity to Probability Estimates:** The formula is highly sensitive to the accuracy of the probability estimate (p). Even small errors in estimating 'p' can lead to suboptimal bet sizes and potentially ruinous outcomes. Confirmation bias can significantly distort probability assessments.

• Requires Positive Expected Value:** The Kelly criterion only works if the bet has a positive expected value (bp > q). If 'p' is less than 1/b, the formula will result in a negative bet size, indicating that you should not bet at all.

• Drawdown Risk:** The full Kelly criterion can lead to significant drawdowns, especially in volatile markets. This is why fractional Kelly is often recommended.

• Assumes Independent Trials:** The basic Kelly criterion assumes that each bet is independent of the others. This assumption is often violated in real-world scenarios, where market conditions and correlations between assets can influence outcomes. Correlation analysis is important to understand relationships between assets.

• Transaction Costs:** The formula doesn’t account for transaction costs (brokerage fees, slippage, etc.), which can reduce the overall profitability of betting.

• Practical Difficulties in Estimating 'p':** Accurately estimating the probability of winning is often challenging, especially in complex markets. Expert opinions and sentiment analysis can supplement technical analysis, but are not always reliable.

Variations and Extensions

Several variations and extensions of the Kelly criterion have been developed to address its limitations:

• Kelly-Shannon Criterion:** This variation introduces a risk aversion parameter, allowing investors to adjust the bet size based on their tolerance for risk.

• Fractional Kelly (as discussed above):** The most common modification, reducing the bet size to mitigate drawdown risk.

• Kelly Betting with Constraints:** These models incorporate constraints on bet size or maximum drawdown. Martingale strategy (while generally discouraged) is an example of a constrained betting system.

• Adaptive Kelly Criterion:** Adjusts the bet size based on the performance of previous bets.

• Generalized Kelly Criterion:** Extends the criterion to handle multiple assets and correlated outcomes.

Advanced Considerations

• **Logarithmic Utility:** The Kelly criterion is derived from maximizing the expected logarithmic utility of wealth. This means it prioritizes consistent growth over large, but infrequent, gains.

• **Bayesian Updating:** Combining the Kelly criterion with Bayesian updating allows you to continuously refine your probability estimates based on new information. Monte Carlo simulations can be used to model different scenarios and assess the impact of different probability estimates.

• **Volatility Scaling:** Adjusting the Kelly fraction based on the volatility of the asset. Higher volatility requires a smaller Kelly fraction. Average True Range (ATR) is a common indicator for measuring volatility.

Conclusion

The Kelly criterion is a powerful tool for optimizing bet sizing and maximizing long-term growth. However, it's crucial to understand its limitations and use it judiciously. Accurate probability estimation, risk management through fractional Kelly, and consideration of transaction costs are essential for successful implementation. While it's not a guaranteed path to riches, the Kelly criterion provides a rational framework for making informed investment and betting decisions. Remember to combine it with sound fundamental analysis and technical analysis to improve your probability assessments.

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