Kelly Criterion
Learn optimal bet sizing for long term compounding. Relevant for investors, entrepreneurs.
Return vs. Volatility
This page demonstrates the Kelly Criterion interactively. Use the slider to scale the optimal fraction.
Read comic below for further explanation on this topic.
- Odds: Winnings multiplier (1.0 = even odds, 2.0 = double your money)
- Edge: Probability of winning
- Sims: Simulated trials for return/volatility analysis
- Paths: Number of independent simulation runs (10-50 range)
- X: Kelly multiplier
- Y1: Expected log growth rate (blue curve)
- Y2: Standard deviation of log growth (red curve)
- Y3: Risk-adjusted growth (Green dotted)
- Green shaded refers to optimal risk-adjusted Kelly range
- Yellow shaded refers to growth maximizing range
- Red shaded refers to ruin range
- Shows 4 Kelly multiplier strategies: 0.4, 0.7, 1.0, 1.5 Kelly.
- Each line shows multiple simulation paths with average highlighted
- Uses 10% of sims for trials to conserve bandwidth
- Click "New Simulation" to generate fresh random paths
- Histogram showing distribution of final portfolio values
- Overlaid histograms for each Kelly multiplier
- X-axis shows log₁₀ of final portfolio values (linear scale)
- Shows probability density of different outcomes
Recommended bet size should be below 0.8 K.
At K greater than 1.5, your chances of ruin gets extremely high. You notice that ruin can happen even before K reaches 1.5 when your edge or odds goes high enough.
That is another way of saying Kelly sometimes gets too overconfident in certain games, and recommends you to bet extremely big (more than 60% of your portfolio), which may work out well in few iterations, but one big loss is enough to wipe you off.
Monte Carlo Growth Paths
Ending Portfolio Value Distribution
Heatmap
- Max odds: Maximum winnings multiplier to display (e.g., 2.0 = double your money)
- P steps: Number of probability intervals (more = smoother heatmap)
- Odds steps: Number of multiplier intervals (more = smootherheatmap)
- K: Kelly multiplier
- X: Assumed probability of winnings
- Y: Assumed profit from winnings
- Z: Kelly recommended bet size
What Is This About?
The Kelly Criterion is a mathematical formula for determining the optimal bet size to maximize long-term growth while avoiding ruin. This interactive demonstration shows how different bet sizing strategies affect portfolio growth, volatility, and risk-adjusted returns.
The Challenge: Optimal Bet Sizing
When you have an edge (probability of winning > implied by odds), the question becomes: How much should you bet?
- Bet too little: You leave money on the table, grow slower than optimal
- Bet too much: High volatility, risk of ruin, lower risk-adjusted returns
- Bet optimally: Maximum long-term growth with acceptable risk
The Kelly Criterion provides the mathematical answer, but in practice, you often want to bet less than full Kelly for safety.
Who Is This For?
This tool is designed for:
- Investors & Traders: Optimizing position sizing in portfolios
- Entrepreneurs: Understanding risk management and capital allocation
- Gamblers & Sports Bettors: Learning optimal bet sizing strategies
- Risk Managers: Understanding Kelly Criterion and its practical limitations
- Anyone Learning Probability: Practical application of expected value and risk management
How to Do This Properly on Your Own
Understanding the Kelly Criterion
The Kelly Criterion formula:
f = (p × b - q) / b*
Where:
f*= Optimal fraction of bankroll to betp= Probability of winningq= Probability of losing (1 - p)b= Net odds received on the wager (e.g., if you bet $1 and win $2, b = 1)
Simplified: f* = (p × odds - 1) / (odds - 1)
Key Insights
- Full Kelly (K=1.0): Maximum long-term growth, but high volatility
- Fractional Kelly (K=0.5-0.7): Lower volatility, still good growth, more practical
- Over-Kelly (K>1.0): Dangerous, high risk of ruin
- Risk-Adjusted: Consider volatility, not just expected return
Practical Applications
- Investing: Position sizing based on edge and odds
- Trading: Optimal leverage and position sizes
- Gambling: Bet sizing in favorable games
- Business: Capital allocation decisions
Implementation Notes
- Kelly assumes you can bet repeatedly with the same edge
- Real-world edges are uncertain (use conservative estimates)
- Transaction costs reduce effective edge
- Consider using fractional Kelly (0.5-0.7) for safety
Has This Helped You?
If you found this Kelly Criterion tool useful:
- Share it with your network or on social media
- Bookmark this page for future reference
- Write backlinks to this page when referencing optimal bet sizing
This tool provides insights into:
- Optimal position sizing strategies
- Risk management and volatility control
- Long-term growth optimization
- Practical limitations of theoretical models
For Those Who Want to Do It Properly Themselves
Ready to implement Kelly Criterion in your own system? The core calculations involve:
- Calculate Edge:
Edge = (Probability × Odds) - 1 - Calculate Full Kelly:
f* = Edge / (Odds - 1) - Apply Fractional Kelly:
Bet Size = Bankroll × f* × Fraction(typically 0.5-0.7) - Simulate Outcomes: Monte Carlo simulation to understand distribution
- Risk Adjustment: Consider volatility and drawdowns, not just expected return
You can reference the chart components to understand how to:
- Calculate expected log growth
- Model volatility and risk
- Visualize probability distributions
- Compare different Kelly multiplier strategies
