Probability
Kelly Criterion
Return vs. Volatility
This page demonstrates the Kelly Criterion interactively. Use the slider to scale the optimal fraction.
- 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 uses 100 trials and 40 paths
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