Signal Correlation, IC, and PnL Dependence

Paper: arXiv 2609.09588

Authors: Marc Nunes

Abstract

Signal correlation and PnL correlation are correlations over different index sets - across assets at each date versus across dates for scalar payoffs - and practitioners often treat the first as a proxy for the second. We give an exact decomposition that shows what that proxy sees and what it discards. We recall that at each date a normalized signal’s projection onto the realized demeaned return direction is its realized cross-sectional Pearson IC, so that fixed-signal PnL is return dispersion times IC (Qian and Hua, 2004), and we identify the normalized similarity of two signals in the orthogonal complement as their partial correlation controlling for realized returns; the residual rotational freedom is an orthogonal gauge whose invariants are the transverse Gram matrix. Signal correlation therefore equals an uncentered IC cross-moment plus transverse similarity, while Pearson PnL correlation centers and dispersion-weights the IC series alone. Our main result is a non-identifiability theorem: absent constraints on transverse geometry, neither correlation bounds or orders the other, which sharpens the simulation finding of Sorensen, Qian, Schoen, and Hua (2004) into an exact statement. For normalized ensembles the transverse Gram matrix re-enters through the normalization denominator; the ex post optimal combination under Euclidean and general covariance-risk metrics is the classical multiple-correlation bound, and we show that the two rules agree for every longitudinal exposure if and only if the risk metric is isotropic on the signal span. The classical exact null law of partial correlation under isotropy, a synthetic mechanism illustration at weak-IC scales, and inference guidance under temporal dependence complete the treatment.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a highly mathematical and theoretical decomposition of signal and PnL correlation, offering significant novelty in its exact non-identifiability theorem. However, it lacks empirical validation or backtesting, focusing purely on theoretical constructs. The clarity is good for a technical audience, but the lack of empirical application limits its immediate practical rigor.

Research Flowchart

  flowchart TD
    A[Research Goal: Understand Signal vs. PnL Correlation Dependence] --> B{Key Concepts: Signal Corr., PnL Corr., IC, Orthogonal Gauge};
    B --> C{Methodology: Exact Decomposition & Non-Identifiability Theorem};
    C --> D[Inputs: Normalized Signals, Realized Returns, Risk Metrics];
    D --> E[Computational Processes: Pearson IC Calculation, Transverse Similarity, Gram Matrix Analysis];
    E --> F[Outcomes: Non-identifiability of correlations, Optimal Combination under Risk Metrics, Null Law of Partial Correlation];

Cited References (31)

1. An Introduction to Multivariate Statistical Analysis (2003)
2. Portfolio Constraints and the Fundamental Law of Active Management (2002) — link
3. The Distribution of the Partial Correlation Coefficient (1924)
4. The Fundamental Law of Active Management (1989) — link
5. Alpha Is Volatility Times IC Times Score (1994) — link
6. Active Portfolio Management: A Quantitative Approach for Producing Superior Returns and Controlling Risk (2000)
7. Resampling Methods for Dependent Data (2003) — link
8. Directional Statistics (2000) — link

0 of 31 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.