Beyond Correlation: Positive Definite Dependence Measures for Robust Inference, Flexible Scenarios, and Causal Modeling for Financial Portfolios

ArXiv ID: 2504.15268 “View on arXiv”

Authors: Unknown

Abstract

We live in a multivariate world, and effective modeling of financial portfolios, including their construction, allocation, forecasting, and risk analysis, simply is not possible without explicitly modeling the dependence structure of their assets. Dependence structure can drive portfolio results more than the combined effects of other parameters in investment and risk models, but the literature provides relatively little to define the finite-sample distributions of dependence measures under challenging, real-world financial data conditions. Yet this is exactly what is needed to make valid inferences about their estimates, and to use these inferences for essential purposes such as hypothesis testing, dynamic monitoring, realistic and granular scenario and reverse scenario analyses, and mitigating the effects of correlation breakdowns during market upheavals. This work develops a new and straightforward method, Nonparametric Angles-based Correlation (NAbC), for defining the finite-sample distributions of any dependence measure whose matrix of pairwise associations is positive definite (e.g. Pearsons, Kendalls, Spearmans, the Tail Dependence Matrix, and others). The solution remains valid under marginal asset distributions characterized by notably different and varying degrees of serial correlation, non-stationarity, heavy-tailedness, and asymmetry. Importantly, it provides p-values and confidence intervals at the matrix level, even when selected cells in the matrix are frozen, thus enabling flexible, granular, and realistic scenarios, reverse scenarios, and stress tests. Finally, when applied to directional dependence measures, NAbC enables accurate DAG recovery in causal modeling. NAbC stands alone in providing all of these capabilities simultaneously, and should prove to be a very useful means by which we can better understand and manage financial portfolios in our multivariate world.

Keywords: Dependence Structure, Nonparametric Angles-based Correlation, Portfolio Risk Management, Finite-Sample Distributions, Causal Modeling, Portfolio Management / Multi-asset

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 7.5/10
• Quadrant: Holy Grail
• Why: The paper introduces a sophisticated nonparametric statistical method (NAbC) requiring advanced linear algebra and multivariate statistics (e.g., positive definite matrices, spectral distributions, eigenvalues) to handle complex dependence structures, indicating high math complexity. It emphasizes robust inference, finite-sample distributions, and practical applications like scenario analysis and causal modeling for financial portfolios, backed by theoretical derivations and empirical examples, though full backtesting code/datasets are not explicitly shown.

  flowchart TD
    A["Research Goal<br>Model Dependence Structure<br>for Financial Portfolios"] --> B{"Data & Inputs"}
    B --> C["Asset Return Series<br>with Real-World Challenges<br>e.g. Non-stationarity, Heavy Tails"]
    
    C --> D["Core Methodology<br>Nonparametric Angles-based<br>Correlation NAbC"]
    
    D --> E["Computational Process"]
    subgraph E ["Matrix-Level Inference"]
        E1["Estimate Pairwise<br>Associations"] --> E2["Construct Positive<br>Definite Matrix"]
        E2 --> E3["Derive Finite-Sample<br>Distributions"]
        E3 --> E4["Calculate P-values &<br>Confidence Intervals"]
    end
    
    E --> F["Key Findings & Outcomes"]
    subgraph F ["Flexible Financial Applications"]
        F1["Valid Hypothesis Testing<br>& Dynamic Monitoring"]
        F2["Realistic Scenarios,<br>Reverse Scenarios & Stress Tests"]
        F3["Accurate Causal DAG<br>Recovery for Portfolios"]
    end