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

Abstract

In an ideal stochastic discount factor, zero pricing errors and the maximum Sharpe ratio coincide; in a low-dimensional approximation they need not. I study this separation by decomposing an investible CRSP market portfolio into dynamic value-weighted body and tail legs that recombine to the market return. All models pass the aggregate benchmark, and the recombination identity does not require zero leg alphas. Yet the identity holds for every model, while only q5 leaves systematic offsetting leg alphas-negative in the body, positive in the tail-and falls below its market-only baseline, despite dominating on spanning. Matched random splits remove the pattern.