Kendall Correlation Coefficients for Portfolio Optimization

ArXiv ID: 2410.17366 “View on arXiv”

Authors: Unknown

Abstract

Markowitz’s optimal portfolio relies on the accurate estimation of correlations between asset returns, a difficult problem when the number of observations is not much larger than the number of assets. Using powerful results from random matrix theory, several schemes have been developed to “clean” the eigenvalues of empirical correlation matrices. By contrast, the (in practice equally important) problem of correctly estimating the eigenvectors of the correlation matrix has received comparatively little attention. Here we discuss a class of correlation estimators generalizing Kendall’s rank correlation coefficient which improve the estimation of both eigenvalues and eigenvectors in data-poor regimes. Using both synthetic and real financial data, we show that these generalized correlation coefficients yield Markowitz portfolios with lower out-of-sample risk than those obtained with rotationally invariant estimators. Central to these results is a property shared by all Kendall-like estimators but not with classical correlation coefficients: zero eigenvalues only appear when the number of assets becomes proportional to the square of the number of data points.

Keywords: Random Matrix Theory, Kendall’s Rank Correlation, Eigenvalue/Eigenvector Cleaning, Markowitz Portfolio Optimization, Correlation Estimation, Equities/Multi-Asset

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 6.5/10
• Quadrant: Holy Grail
• Why: The paper employs advanced random matrix theory and derives non-linear correlation estimators with strong theoretical properties, indicating high mathematical density. It also presents empirical validation using both synthetic and real financial data, comparing out-of-sample portfolio risk, demonstrating substantial implementation and data-driven rigor.

  flowchart TD
    A["<b>Research Goal</b><br/>Improve correlation estimation<br/>in Markowitz portfolio optimization<br/>(data-poor regimes)"] --> B["<b>Methodology: Generalized Kendall Correlation</b><br/>Extended Kendall's tau for multivariate data<br/>Rotationally invariant estimators"]
    
    B --> C["<b>Data Inputs</b><br/>1. Synthetic Data: Controlled regimes<br/>2. Real Financial Data: Equities/Multi-asset"]
    
    C --> D["<b>Computational Process</b><br/>1. Estimate Correlation Matrix<br/>2. Eigenvalue/Eigenvector Cleaning<br/>3. Markowitz Optimization<br/>4. Out-of-sample Risk Calculation"]
    
    D --> E["<b>Key Finding 1</b><br/>Kendall-like estimators reduce<br/>portfolio risk vs. standard methods"]
    
    D --> F["<b>Key Finding 2</b><br/>Zero eigenvalues appear only when<br/><i>N</i> ∝ <i>n²</i> (unlike classical corr.)"]
    
    E --> G["<b>Outcome</b><br/>Improved correlation estimation<br/>in data-poor regimes"]
    F --> G