Correlation and Beyond: Positive Definite Dependence Measures for Robust Inference, Flexible Scenarios, and Stress Testing for Financial Portfolios (2504.15268v6)

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 in useable and useful ways 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, 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, NAbCs p-values and confidence intervals remain analytically consistent at both the matrix level and the pairwise cell level. Finally, NAbC maintains validity even when selected cells in the matrix are frozen for a given scenario or stress test, thus enabling flexible, granular, and realistic scenarios. 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.