Signal Correlation, IC, and PnL Dependence: A partial-correlation decomposition with residual-gauge invariants and ensemble bounds under Euclidean and risk metrics

Battle Count: 58

Directly addresses a core practical question in quantitative equity: whether signal correlation can proxy for PnL correlation in portfolio construction and risk management. The non-identifiability theorem (Theorem 9) has immediate implications for how practitioners should evaluate alpha libraries, size positions, and diagnose diversification. The decomposition provides actionable diagnostics (C_sig, C_IC, C_perp) that can be computed from existing data. The distinction between Euclidean and risk metrics is critical for implementation. The paper is essentially a theoretical foundation for the Sorensen-Qian-Schoen-Hua (2004) simulation finding that IC time-series correlation matters more for diversification than cross-sectional signal correlation.

The core decomposition requires only inner products, return magnitudes, and standard partial-correlation formulas (no rotation matrices need to be computed explicitly, as noted in Technical Remark 4). The synthetic experiment is ~50 lines of NumPy. However, practical implementation requires careful handling of: (1) dependence-aware inference (HAC/block bootstrap), (2) multiplicity control across K(K-1)/2 pairs, (3) distinguishing research signals from implemented portfolios (transfer coefficient), (4) maintaining separate Euclidean and risk-metric lenses, and (5) null adjustment for multiple correlation diagnostics. The mathematical framework is elegant but the practitioner workflow involves several non-trivial decisions.

Full NumPy reproduction code provided in Appendix A with fixed random seed (20260906), explicit parameter choices (d=500, T=1000, K=12), and sensitivity analysis at four IC innovation scales. All mathematical derivations are self-contained with proofs. The paper explicitly states that the synthetic experiment should eventually be replaced by out-of-sample validation on real data.