A sharp order-three obstruction to the aggregation of conditional price-of-risk attribution

Abstract

We study the squared price-of-risk premium of a portfolio -- an integrated conditional squared Sharpe-ratio functional, not an expected excess return -- and its attribution to causal drivers. Relative to a declared admissible benchmark it decomposes into intervention-stable premium, a signed causal distortion (the confounding wedge), and a nonnegative information loss; the loss is an $L^2$ projection residual, the wedge is not. The decomposition is well posed exactly when the driver filtration is immersed in the price filtration. It need not aggregate across portfolios pooling drivers: we identify an order-three obstruction that is invisible to every singleton and pairwise admissibility screen -- each one- and two-driver sub-book is immersed while the pooled triple reveals a future innovation -- the analogue of Bernstein's pairwise-but-not-mutually-independent triple, and minimal relative to such pairwise diagnostics. We separate its two ingredients, combinatorial masking and anticipative coupling. The failure is one of immersion, not of no-arbitrage. Experiments on synthetic single- and multi-driver panels show the decomposition and its causal correction are estimable, and that a permutation-calibrated screen detects planted order-three leakage with controlled false positives.