Anatomy of the Market: A Body-Tail Test of Factor Models

In an ideal stochastic discount factor, zero pricing errors and maximum Sharpe ratio coincide; in a low-dimensional approximation they need not. I test this separation by decomposing an investible CRSP market into capitalization-ranked body and tail legs that recombine to the market return. At the daily frequency, all models pass the aggregate benchmark, but q5 alone leaves systematic offsetting leg alphas-negative body, positive tail-at all nine split ratios, despite holding the strongest spanning position. Matched random splits remove the pattern. Monthly aggregation attenuates q5's joint rejection and shifts relative weakness toward FF3, showing that internal consistency is frequency-dependent.