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Kelly Criterion CurveThe Kelly Criterion Curve indicator gives you the leverage/return tradeoff by displaying a bell curve with growth and leverage. This indicator shows where you are on the risk curve depending your allocation/leverage used and the optimal leverage to use in any asset.
What Does It Show?
The indicator plots the Kelly growth function:
g(f) = μ·f - 0.5·σ²·f²
Where:
g(f) = Expected growth rate at leverage f
μ = Annualized return
σ = Annualized volatility
f = Leverage multiplier
The curve peaks at the Optimal Kelly leverage (full Kelly) and then declines, showing that:
Too little leverage = underutilized capital
Too much leverage = volatility drag destroys returns
The curve is dynamically divided into zones based on your asset's return profile:
Underinvesting (Green) - Too conservative, underutilized capital
Optimal Sizing (Teal) - Sweet spot for position sizing
High Risk (Yellow) - Diminishing returns, high volatility drag
Never Logical (Red) - Risk outweighs reward
Suicidal (Black) - Negative expected returns
Position Markers
★ Kelly Optimal (Green/Red) - Maximum long-term log growth leverage
½ Kelly (Yellow) - Conservative sizing (recommended most times)
Settings for Kelly Calculation
Lookback Period - Historical data window for calculations (default: 252 = 1 year)
Annual Trading Days - For annualization (default: 252)
Use Log Returns - More accurate for compounding (recommended: ON)
Curve Smoothness (20-200) - Number of points on curve (default: 100)
Maximum Leverage Display (2-10x) - X-axis range
Show Short Positions - Display negative leverage for short strategies. Note the chart is not fully optimized for shorts.
Show Optimal Kelly Marker - Mark optimal leverage on curve
Show Half Kelly Marker - Mark conservative leverage
How to Use
Look at Optimal Kelly - This is the theoretical maximum for the period analyzed
Use Half Kelly for conservative sizing
Check which risk zone your position falls into
If your leverage is in the High Risk zone → Consider reducing
If you're in Never Logical or Suicidal → I wish you good luck because you will need a lot
If you're in Underinvesting → You may be too conservative
IMPORTANT
The indicator is based on past returns and volatility. It CANNOT predict:
Market crashes
Regime changes
Black swan events
If you use Optimal Kelly and suddenly there's a crash, you are toasted.
Full Kelly maximizes long-term growth but can experience large drawdowns
Most traders use ¼ to ½ Kelly for risk management
You should almost never use full Kelly, unless you are extremely confident
Remember leverage amplifies gains and losses
Notice how Max Growth isn't simply Ann. Return × Leverage
The formula accounts for volatility drag (the cost of using leverage)
Higher volatility = lower optimal leverage
The Kelly Criterion was developed by John L. Kelly Jr. in 1956 for information theory and later adapted for gambling (card counting for example, pioneered by Edward O. Thorp), and investing.
Optimal Leverage:
f* = μ / σ²
Expected Growth Function:
g(f) = μ·f - 0.5·σ²·f²
This is a quadratic function that forms the bell curve you see on the chart.
This indicator pairs perfectly with my other indicators:
Kelly Optimal Leverage Indicator
Jensen's Inequality + Kelly Leverage
Multi-Leverage VAR/VaG Indicator
For deeper insights on Kelly Criterion and optimal leverage:
Read my article: Unlock the Power of Monte Carlo
Read these papers:
Alpha Generation and Risk Smoothing using Managed
Volatility
Leverage for the Long Run - A Systematic Approach to Managing Risk and Magnifying Returns in Stocks
Trading with leverage involves substantial risk of loss.
The Kelly Criterion provides a theoretical framework - actual trading requires additional risk management, market analysis, and psychological discipline.
Some examples of using the Kelly Criterion Curve:
Russel 2000, last 500 days Kelly curve
Here's you can see that the optimal sizing over the last 500 days would have been around 1.7x leverage and that full Kelly is 3.4x leverage.
While Russel 2000 returned 16%, full Kelly would have returned 27.8%, and more that full Kelly (3.4x leverage) would lower the returns.
Berkshire Hathaway, last 1000 days Kelly curve
BRK stock optimal Kelly (full Kelly) is 2.5x for the last 1000 trading days. To reduce volatility, one could use 1/2 Kelly which is 1.25x leverage.
Bitcoin, last 2000 trading days Kelly curve
Very interestingly, the indicator tells us not to leverage Bitcoin. Even a 2x leverage can lead to ruin given its volatility, and in fact, in 2025 many traders got liquidated while leveraging Bitcoin by 2x.
Let me know if you have questions, suggestions and comments.
- Henrique Centieiro インジケーター
ETF Leverage VerificationDo leveraged ETFs really return what they promise?
Do they return the exact 2x or 3x? Or a slightly different multiple?
How much do they deviate from the promised leverage multiples?
Do these deviations impact investors in a positive or negative manner?
These are the questions that I want to answer with this indicator.
The ETF Leverage Verification indicator challenges the conventional understanding of leveraged ETFs by measuring how they actually perform versus their theoretical targets.
Instead of assuming leveraged ETFs perfectly track their target multiple, this indicator quantifies the real-world behavior by comparing the expected returns versus the actual results on every trading day.
Key Features
Measures actual versus expected performance of leveraged ETFs
Tracks deviation patterns across thousands of trading days
Identifies asymmetric behavior in up versus down markets
Quantifies beneficial "cushioning effect" during market declines
Provides statistical summary of performance patterns
Works with any leverage factor (2x, 3x, -1x, etc.)
Compatible with all leveraged ETFs (equity, bond, commodity, volatility)
How to Use the Indicator
Enter the Expected Leverage Factor (default: 2.0)
Select the Base Asset (underlying index, e.g., SPX)
Select the Leveraged Asset (leveraged ETF, e.g., SSO)
Understanding the Results
Green markers: Days when the ETF outperformed its expected multiple
Red markers: Days when the ETF underperformed its expected multiple
Data Table:
Positive Deviations: Count of days with better-than-expected performance
Negative Deviations: Count of days with worse-than-expected performance
Avg Deviation: Average magnitude of deviation from expected returns
Frequency Skew: Difference between beneficial deviations in down vs. up markets
Impact: Overall assessment of pattern benefit to investors
Summary Label:
Percentage of positive deviations in up and down markets
Total sample size for statistical significance
Key Patterns to Look For
Positive Deviation in Negative Days:
This occurs when a leveraged ETF falls less than expected during market declines. For example, if SPX falls 1% and a 2x ETF falls only 1.8% (instead of the expected 2%), this creates a +0.2% deviation. This pattern is beneficial as it provides downside protection.
Negative Deviation in Positive Days:
This happens when a leveraged ETF rises less than expected during market advances. For example, if SPX rises 1% and a 2x ETF rises only 1.9% (instead of the expected 2%), this creates a -0.1% deviation. This pattern reduces upside performance.
Frequency Skew:
The most critical metric that measures how much more frequently beneficial deviations occur in down markets compared to up markets. A higher positive skew indicates a stronger asymmetric pattern that helps long-term performance.
Mathematical Background
The indicator computes the deviation between expected and actual performance:
Deviation = Actual Return - Expected Return
Where:
Expected Return = Base Asset Return × Leverage Factor
The deviation is then categorized into four possible outcomes:
Positive deviation on positive market days
Negative deviation on positive market days
Positive deviation on negative market days
Negative deviation on negative market days
In short, more positive deviations are good for investors.
Please feel free to criticize. I'm happy to improve the indicator. インジケーター
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