Optimal Growth✓ Mathematical
◆ The PatternOptimal bet size for maximum geometric growth
Kelly sizing maximizes the expected logarithm of wealth — the fastest compounding rate without risking ruin. For a simple win/loss bet:
f* = (p · b − q) / b
p = win probability, q = 1−p, b = win/loss ratio. In continuous markets, Kelly fraction = expected excess return / variance. Most practitioners use half-Kelly or less to reduce volatility.
// Interactive — win rate and Kelly fraction
Win rate %
| Fraction | Growth | Drawdown |
|---|---|---|
| Full Kelly | Max geometric | Severe |
| Half Kelly | 75 % of max | Much lower |
| Quarter Kelly | ~50 % of max | Mild |
Overbet risk. Betting more than full Kelly guarantees sub-optimal growth and eventual ruin with parameter uncertainty.
Pattern bridge: Log-normal growth connects to The Toolkit — Distributions.