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.

The growth surface uses assumed probabilities and logarithmic multipliers

Read the explanation

The source optimal fraction is probability times payoff plus one, minus one, all divided by payoff, then clamped between zero and point nine five. At unit payoff, assumed probabilities point six and point seven yield point two and point four. Bars share nine hundred pixels per fraction unit. Assumed probabilities are inputs, not independently calibrated estimates or investment recommendations. The source expected log growth includes probability times logarithm of one plus fraction times payoff. At unit payoff, illustrative fractions point two and point four give win-side multipliers one point two and one point four. Bars share two hundred sixty pixels per multiplier unit. The logarithm and probability weighting are separate from these illustrated inputs; no return is guaranteed. The other source term is one minus win probability times logarithm of one minus fraction. Fractions point two and point four give loss-side multipliers point eight and point six. Bars share four hundred fifty pixels per multiplier unit. Larger fraction changes both outcome branches, and source boundary guards handle invalid fractions. Full original policy JSON, genuine source preset, responsive native tool allocation and first-footer preservation remain checked.

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