Kelly Fraction: 0.20, Win Rate: 0.60, Expected CAGR: 4.08%, Gradient Ascent Derivative: 0.50.

Gradient Kelly Surface

Optimization Calculus & Portfolio Sizing Simulator
Regime Presets:
Log-Growth Surface $E[\ln W]$ & Gradient Field Optimal Kelly Peak $f^*$: 20.0%
500-Step Monte Carlo Portfolio Wealth Trajectories
Optimal $f^*$
Current $f$
Over-leveraged
Expected Growth $E[\ln W]$
+4.08%
Kelly Peak $f^*$
20.00%
Gradient $\partial E/\partial f$
+0.500
Ruin Risk ($W < 0.1$)
0.0%
Calculus Mechanics & Derivative Proof Surface
Growth Surface Equation: $E[\ln W] = p \cdot \ln(1 + f \cdot b) + (1-p) \cdot \ln(1 - f)$
Steepest Gradient Vector: $\frac{\partial E}{\partial f} = \frac{p \cdot b}{1 + f \cdot b} - \frac{1 - p}{1 - f}$

Why 60% Win Rate Trades Go Broke: Holding a statistical advantage ($p = 0.60$) guarantees positive expected monetary value on single trades. However, compound wealth growth is logarithmic. As leverage exceeds the Kelly peak $f^* = \frac{p(b+1)-1}{b}$, volatility drag accelerates faster than linear gains. Past $2f^*$, expected growth becomes negative ($\partial E/\partial f < 0$), driving the portfolio to exponential ruin over time despite winning most trades.

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