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

2026-06-25

AI Summary

This study addresses the non-aggregability of conditional risk premium attribution in multi-factor portfolios: even when individual factors and pairwise combinations satisfy compatibility conditions, triplets may still harbor forward-looking information leakage. To resolve this, the authors decompose the conditional squared Sharpe ratio functional into an intervention-invariant premium, a causal distortion term, and a non-negative information loss component, analyzing its aggregability under driver-factor filtering and price-filtration immersion. They identify, for the first time, a third-order non-aggregability barrier—akin to Bernstein-type non-mutually independent triplets—that remains undetectable in lower-order tests. Leveraging L² projection, causal inference, filtration immersion theory, permutation-based calibration tests, and synthetic experiments, they establish the estimability of the decomposition and demonstrate that their approach effectively detects third-order information leakage with controlled false positive rates.

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.

Problem

• causal attribution
• immersion
• order-three obstruction
• conditional Sharpe ratio
• information loss

Innovation

• causal attribution
• immersion failure
• order-three obstruction
• conditional Sharpe ratio
• information leakage