Kelly Criterion: The Smartest Way to Manage Risk & Maximize Profits

The Kelly Strategy, developed by John Kelly at Bell Laboratories, provides an optimal strategy for capital allocation in investment and gambling. It aims to maximize the logarithm of wealth over time, ensuring optimal growth rate while managing downside risk. The Kelly Criterion finds applications in investment management, portfolio optimization methods, and sports betting bankroll management.

The Kelly fraction (Kelly f) represents the percentage of capital to allocate in a given bet or investment. It is derived using natural logarithms and ensures maximum expected log wealth over repeated bets or trades. The formula for bet sizing in the simplest form is:

[ f^* = \frac{p – (1 – p)}{b} ]

Where,:

• ( p ) = probability of success (winning probability)
• ( b ) = odds received on a successful bet

For example, in a 2-to-1 odds bet, the optimal betting fraction can be determined to maximize long-term returns.

The Kelly criteria method dates back to 1956 when John L. Kelly Jr. developed it at Bell Labs. His goal wasn’t trading—it was improving long-distance telephone signals. Gamblers, however, quickly saw its potential. 

Edward Thorp, one of the most famous blackjack players and later a hedge fund manager, used it to crush casinos. From there, it made its way into trading, promising a scientific way to optimize position sizing.

On paper, it makes perfect sense. If a strategy has an edge, why not size positions according to that edge, and compound gains faster? That’s what draws traders in. Some investment legends, including Warren Buffett and Bill Gross, have been rumored to use Kelly-like principles for position sizing. While they haven’t explicitly confirmed it, their trading strategies—concentrating capital in their best opportunities—align with the core idea of allocating more capital to high-probability trades.

The appeal is obvious: a rules-based, mathematical approach to risk management. It aligns with the way systematic traders think—test, optimize, and execute based on probability, not emotion.

Here’s where things fall apart. Markets don’t behave like casinos. A blackjack table has fixed probabilities, but market conditions shift constantly. A strategy’s win rate fluctuates, correlations change, and unexpected events can turn an edge into a liability overnight.

Practical Kelly Criterion assumes a level of precision that doesn’t exist in trading. The moment a trader miscalculates win probability or underestimates risk, Kelly’s aggressive position sizing can lead to crippling drawdowns. Many professional traders use fractional Kelly (e.g., half-Kelly) to reduce risk and smooth out returns.

The Kelly Criterion is a mathematical formula that helps traders determine the optimal percentage of their capital to allocate to each trade in order to maximize long-term growth while managing risk. 

Unlike arbitrary position sizing methods, the Kelly formula is grounded in probability theory, ensuring that traders bet in proportion to their edge.

The Standard Kelly Criterion Formula

The standard Kelly Criterion formula for trading is:

Kelly Criterion Formula for Fixed Odds Betting

Alternatively, in gambling or odds-based markets, the formula is expressed as:

For trading, we generally use the first formula since we deal with expected returns rather than fixed betting odds.

• Let’s walk through a practical trading scenario.

  Hypothetical Trade Setup:

  • A stock has a 60% probability of gaining 10%.
  • If the trade fails, it has a 40% probability of losing 5%.
  • We will use these values to calculate the optimal position size using the Kelly Criterion.

  Step 1: Define the Variables

  • Win probability (W) = 0.60 (60%)
  • Loss probability = 1 – 0.60 = 0.40 (40%)
  • Average Win = 10% (0.10)
  • Average Loss = 5% (0.05)
  • Win/Loss Ratio (R) = 0.10 / 0.05 = 2.0

  Step 2: Apply the Kelly Formula

Step 3: Interpreting the Result

The Kelly Criterion suggests allocating 40% of total trading capital to this trade. However, pure Kelly sizing is often too aggressive for real-world trading due to estimation errors and market uncertainty.

Practical Adjustments:

• Full Kelly (40%) can be highly volatile.
• Many traders use Fractional Kelly, such as:
• • ½ Kelly (20%) for a balance between growth and risk.
  • ¼ Kelly (10%) for more conservative risk management.

Key takeaway: While Kelly provides a mathematical guide for position sizing, in real-world trading, most systematic traders reduce their position size to account for fluctuations in win rate and market conditions.

The Kelly Criterion is a powerful mathematical tool for determining optimal bet sizing, but in real-world trading, especially for systematic traders, it comes with significant challenges. 

While the formula is mathematically sound, applying it without adjustments can expose traders to extreme volatility, estimation errors, and impractical position sizes. Here’s why:

1. Estimation Errors: Garbage In, Garbage Out

The biggest challenge when using the Kelly Criterion in trading is accurately estimating the win probability (W) and the win/loss ratio (R).

• Systematic traders rely on historical data, but past performance is not necessarily indicative of future results.
• Small errors in estimation can lead to massive over- or under-betting. Even slight miscalculations can drastically impact capital allocation, leading to unexpected drawdowns.
• Market conditions change, and a strategy’s edge fluctuates—meaning the inputs to the formula aren’t fixed.

🔹 Example: A trader backtests a strategy and finds it has a 60% win rate. If the true forward-looking win rate is only 50%, the Kelly allocation would be significantly overestimated, leading to excessive risk-taking.

2. The “Edge” Illusion: Overconfidence in a Flawed Strategy

The Kelly Criterion assumes that you have a proven edge—but many traders overestimate their strategy’s profitability. Even with rigorous backtesting, some edges are illusory due to:

• Curve fitting (over-optimizing for past data that won’t repeat).
• Data mining bias (finding patterns that appear profitable but aren’t robust).
• Selection bias (ignoring underperforming strategies in favor of overfitted ones).

💡 Reality Check: To confidently apply Kelly, a trader needs thousands of trades to validate the edge. However, many systematic strategies only generate a handful of signals per month, making statistical significance harder to achieve.

3. Volatility and Drawdowns: The Risk of Full Kelly

Full Kelly sizing is known to maximize growth, but it also introduces severe volatility.

• Kelly bets aggressively, which can lead to huge swings in equity—something most systematic traders try to avoid.
• Even if the strategy is profitable, a string of losses can cause massive drawdowns that are psychologically and financially damaging.
• Systematic traders typically prioritize risk-adjusted returns rather than just maximizing theoretical growth.

🔹 Example: If a Full Kelly approach suggests risking 40% of capital on a trade, a string of three consecutive losses would wipe out over 78% of the account.

➡ Solution: Many professional traders use Fractional Kelly (½ Kelly, ¼ Kelly, etc.) to balance growth with survivability.

4. Diversification Neglect: A Portfolio-Level Issue

The Kelly Criterion focuses on optimizing individual trades, but systematic traders manage portfolios of trading strategies.

• A portfolio of uncorrelated trades, each sized using Full Kelly, may still expose the trader to unacceptable risk.
• Kelly does not inherently account for correlations between assets or strategies.
• Without proper adjustments, traders could end up over-leveraging multiple correlated trades, increasing exposure beyond safe levels.

🔹 Better Approach: Instead of applying Kelly to each individual trade, systematic traders optimize risk allocation across an entire portfolio using volatility-based or value-at-risk (VaR) methods.

5. Practical Constraints: The Real-World Application is Messy

Even if the Kelly formula produces an “optimal” bet size, real-world trading imposes limitations:

• Transaction costs: Frequent resizing of positions to match Kelly’s sizing increases trading costs.
• Liquidity constraints: Large Kelly-calculated positions may be impractical in low-liquidity stocks or crypto assets.
• Minimum position sizes: Some markets have lot size restrictions, making precise Kelly sizing impossible.
• Slippage: Executing large positions can impact market prices, distorting expected returns.

🔹 Example: If Kelly suggests buying 3.85 contracts of an options position, but the broker only allows whole numbers, the trader is forced to round up or down, leading to suboptimal execution.

➡ Solution: Many systematic traders use practical risk controls like:

• Volatility-based position sizing (risk per trade capped at X% of equity).
• Fixed-fractional sizing (risking a constant percentage per trade).
• Portfolio-wide risk limits (position sizes adjusted for total exposure).

Systematic traders want consistency, scalability, and survivability. While Kelly is mathematically sound, blindly applying it without adjusting for real-world uncertainty is dangerous. Many traders prefer:

• Fractional Kelly (½ Kelly or ⅓ Kelly) to limit volatility.
• Fixed fractional position sizing, where risk per trade is capped at a reasonable percentage of capital.
• Volatility-adjusted position sizing, which adapts to changing market conditions.

The math is right, but the assumptions can be deadly. If you’re trading with real money, position sizing needs to prioritize risk control first—not just theoretical growth.

In capital markets investments, the Kelly Strategy is used to allocate a fraction of the portfolio to risky assets while preserving wealth. It plays a significant role in portfolio allocation, guiding investors to balance between loss of capital and positive trade amounts. By utilizing daily returns and cumulative returns, investors can optimize their portfolio of securities.

Unlike all-in investing strategies, the Kelly Criterion provides a conservative position sizing approach. While aggressive bet size can lead to large gains, it also increases the risk of negative outcomes. Investors may prefer a minimum bet size or a constant fraction strategy to mitigate risk.

Here are some of the most effective alternatives:

1. Fractional Kelly: A More Conservative Approach

Concept: Instead of betting the full Kelly percentage, traders bet only a fraction (e.g., 50% Kelly, 25% Kelly, or even 10% Kelly).

Advantages:

• Reduces volatility and drawdowns, making the strategy more sustainable.
• Lowers the risk of ruin, especially if win probabilities (W) or win/loss ratios (R) are misestimated.
• More psychologically manageable, as it smooths the equity curve.

Example:

• Suppose the full Kelly calculation suggests allocating 40% of capital to a trade.
• Using ½ Kelly, the trader would allocate 20% instead.
• Using ¼ Kelly, the trader would allocate 10%.

Best For:

• Traders who want to benefit from Kelly without extreme swings in equity.
• Those who lack high-confidence win rate estimates.

➡ Reality Check: Most professional traders use some form of fractional Kelly, rarely betting full Kelly due to the risk of overexposure.

2. Fixed Fractional Position Sizing: Simplicity Over Optimization

Concept: Allocate a fixed percentage of capital to each trade, regardless of the estimated edge.

Advantages:

• Simple and easy to implement—no need for precise probability calculations.
• Prevents overconfidence bias since it doesn’t depend on potentially flawed edge estimates.
• Maintains capital preservation, as each trade carries a fixed, known risk.

Example:

• A trader decides to risk 2% of total capital per trade.
• If they have $100,000, they risk $2,000 per trade, regardless of win probabilities.

Best For:

• Traders are looking for a structured, low-maintenance approach.
• Those who prioritize capital protection over aggressive growth.

➡ Reality Check: Many systematic traders prefer fixed fractional sizing as it is less sensitive to estimation errors than Kelly.

3. Volatility-Based Position Sizing: Keeping Risk Consistent

Concept: Position sizes are adjusted based on the asset’s volatility, ensuring a consistent risk level per trade.

Advantages:

• Prevents outsized risk exposure in volatile markets.
• Maintains risk consistency across different instruments (e.g., stocks vs. forex vs. crypto).
• Adapts to market conditions, reducing position size in high-volatility periods.

Example (ATR-Based Sizing):

• A trader uses the Average True Range (ATR) to determine stop-loss levels.
• If a stock has a high ATR (high volatility), the position size is reduced to keep risk in check.
• If a stock has a low ATR (low volatility), a larger position is taken.

Best For:

• Trend-following traders and mean-reversion traders who deal with varying volatility.
• Systematic traders aim for risk-adjusted returns rather than raw gains.

➡ Reality Check: Volatility-based sizing is widely used by hedge funds and quantitative traders.

4. Incorporating Risk Aversion: Personalizing Position Sizing

Concept: Adjust the Kelly Criterion (or any sizing method) to reflect an individual’s risk tolerance.

How It Works:

• Utility-based modifications: Instead of maximizing returns, traders weigh potential losses more heavily.
• Penalizing large bets: Some traders cap position sizes beyond a certain level, even if Kelly suggests a higher allocation.

Advantages:

• Balances personal comfort with growth potential.
• Prevents traders from making overly aggressive bets that could lead to psychological stress or forced liquidation.
• Allows for controlled risk scaling, especially when trading multiple strategies or markets.

Example:

• A trader with a low-risk tolerance might modify Kelly by using a logarithmic utility function to ensure position sizes never exceed 20% of total capital.

Best For:

• Traders prioritize psychological resilience and longevity over maximum theoretical returns.
• Those trading in leveraged or high-risk markets (e.g., options, crypto, or forex).

➡ Reality Check: Position sizing should match the trader’s risk comfort level—no formula should dictate risk beyond what’s manageable.

No single method is “best” for all traders—the ideal choice depends on your strategy, risk tolerance, and market conditions.

• The average return and distribution of returns influence the optimal Kelly fraction.
• Monthly returns and 24-month rolling portfolios help refine the strategy for investment advice.
• Stock price movements and security price variations require dynamic adjustments.

Several extensions of the Kelly Criterion include:

• Triple Kelly (aggressive betting)
• Minimum variance portfolio (balancing risk and reward)
• Growth optimal portfolios (maximizing annual growth rate)

• Edward O. Thorp, a key advocate, applied the Kelly Strategy in horse racing, sports betting, and the stock market.
• The Kelly Calculator assists in determining the proper bet size for a given probability and return structure.
• The Black-Scholes calculation and log return metrics help refine investment constraints in capital markets.

The Kelly Criterion remains a fundamental portfolio optimization method, balancing potential gain against risk-free rate. Its application to markets, including financial assets and incomplete markets, showcases its effectiveness as a profitable investment strategy. By using a mean-variance approach or an alternative approach, traders and investors can develop a growth rate function that aligns with their risk tolerance and return expectations.

Apply today to the Trader Success Program and trade confidently! are

Links to Articles About Position Sizing

Explore more insights into position sizing and improve your trading consistency! Visit our detailed article on how to calculate position sizing for all the details.

• How To Calculate Position Sizing | What Is The Best Way?
• Profile of Van Tharp
• The Dangers of Martingale Position Sizing in Trading
• Position Sizing In Amibroker With SetPositionSize Function
• Master Risk Management for Traders: Proven Strategies to Protect Your Capital and Maximize Profits
• How to become a consistent trader?
• How can I overcome my fear of taking action when trading stocks?
• Trading Psychology | 4 Causes of trading stress and anxiety PLUS how to eliminate them
• Van Tharp – Outstanding Trading Books And Resources To Improve Your Performance
• 4 things you must do to survive stock market volatility
• Is it possible to mitigate huge potential losses?
• Kelly Criterion: The Smartest Way to Manage Risk & Maximize Profits