What the Score Means
0-20 (green): Efficient. Returns behave like a random walk. No pattern exists to exploit. Past prices tell you nothing about future prices. Go passive, save on fees. This is most liquid stocks most of the time.
20-50 (yellow): Mild predictability. Detectable patterns exist. Could be momentum, mean-reversion, or volatility clustering. The signal is real but modest. Tactical allocation or small active positions may have edge. Monitor for regime shifts.
50-100 (red): Inefficient. Strong predictable structure. Often coincides with crisis dynamics, volatility regimes, or structural breaks. Active strategies have clear edge. If tail index is also low (alpha < 2), hedge immediately.
What the Diagnostics Mean
I_pi (nats): Mutual information between current and past returns. The raw predictability signal. Higher = more predictable.
I_vol (diagnostic): Predictability in the volatility channel. High I_vol means volatility clusters (large moves follow large moves). Useful for options strategies even when returns themselves are unpredictable.
Tail Index alpha: How heavy the tails are. alpha > 2 = normal-like tails. alpha < 2 = infinite variance (extreme moves more likely than Gaussian models predict). alpha < 1.7 = danger zone.
Sharpe / Vol / Drawdown: Standard risk metrics for context. The E score is novel, these are familiar. Together they give a complete picture.
What to Do
E < 20: Index fund. Don't pay for active management. If alpha < 1.7, add tail hedges (put options).
E 20-50, I_vol dominates: Sell volatility. Covered calls, iron condors, vol-target overlays.
E 20-50, I_level dominates: Trend-follow. Momentum strategies have edge.
E > 50, alpha < 1.7: Crisis mode. Buy puts. Reduce exposure. This combination preceded March 2020, August 2015, and the GFC.
E > 50, alpha > 2: Mean-revert. Fade extremes with tight stops.
Methodology
Score is computed via nonparametric mutual information (KSG Algorithm 1, Kraskov-Stogbauer-Grassberger 2004) with Gaussian multi-lag baseline (Toeplitz determinant ratio). Tail index via Hill estimator. Serial dependence via Ljung-Box Q statistic. Score formula: E = 100(1 - exp(-K * I_pi)), calibrated for frequentist MI estimation.