Correlation-Based Distances for Dependence and Portfolio Analysis

Summary

This reference explains several ways to measure dependence among asset returns and distance between correlation structures. Distance correlation can detect nonlinear dependence and is zero exactly when variables are independent, unlike Pearson correlation, which can miss some nonlinear relationships. Three angular-distance variants transform Pearson correlation into distances: standard angular distance treats negative correlation as greater separation, while absolute and squared variants make the sign less influential. The document relates those choices to long-only and long-short portfolio construction.

It also describes Kullback–Leibler distance between multivariate Gaussian distributions represented by correlation matrices, noting its use in assessing the stability of filtering or denoising under statistical uncertainty. Norm distance is introduced as a general matrix-comparison measure. Examples show how the metrics may be applied to asset returns and to correlation matrices from different periods. This is methodological documentation rather than a trading strategy or empirical performance study; it gives formulas and references but no evidence that any one metric improves portfolio outcomes. Metric choice depends on whether correlation sign matters and what type of dependence or matrix change the analysis is intended to capture.

Key ideas

• Distance correlation can capture nonlinear dependence that Pearson correlation may not reveal.
• Standard angular distance treats negatively correlated variables as more distant, which can suit long-only diversification analysis.
• Absolute and squared angular distances reduce the influence of correlation sign for similarity comparisons.
• Kullback–Leibler distance can compare correlation structures while reflecting their statistical distributions.
• The document describes analytical metrics and examples but reports no portfolio performance results.

Tags

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.