High-dimensional covariance matrix estimators on simulated portfolios with complex structures

ArXiv ID: 2412.08756 “View on arXiv”

Authors: Unknown

Abstract

We study the allocation of synthetic portfolios under hierarchical nested, one-factor, and diagonal structures of the population covariance matrix in a high-dimensional scenario. The noise reduction approaches for the sample realizations are based on random matrices, free probability, deterministic equivalents, and their combination with a data science hierarchical method known as two-step covariance estimators. The financial performance metrics from the simulations are compared with empirical data from companies comprising the S&P 500 index using a moving window and walk-forward analysis. The portfolio allocation strategies analyzed include the minimum variance portfolio (both with and without short-selling constraints) and the hierarchical risk parity approach. Our proposed hierarchical nested covariance model shows signatures of complex system interactions. The empirical financial data reproduces stylized portfolio facts observed in the complex and one-factor covariance models. The two-step estimators proposed here improve several financial metrics under the analyzed investment strategies. The results pave the way for new risk management and diversification approaches when the number of assets is of the same order as the number of transaction days in the investment portfolio.

Keywords: Covariance Matrix Estimation, Portfolio Allocation, Random Matrix Theory, Hierarchical Risk Parity, Minimum Variance Portfolio, Equities

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail
• Why: The paper uses advanced mathematical concepts from random matrix theory, free probability, and deterministic equivalents, justifying a high complexity score. It includes rigorous empirical validation through walk-forward analysis and performance metrics on S&P 500 data, placing it in the Holy Grail quadrant.

  flowchart TD
    A["Research Goal<br>High-dimensional covariance estimation<br>for portfolio allocation"] --> B
    
    subgraph B ["Data & Inputs"]
        B1["Synthetic Portfolios<br>Complex Structures"]
        B2["Empirical Data<br>S&P 500 Companies"]
    end
    
    B --> C["Methodology: Noise Reduction<br>Random Matrices + Free Probability<br>+ Deterministic Equivalents + Two-Step Estimators"]
    
    C --> D["Computational Process<br>Moving Window & Walk-Forward Analysis"]
    
    D --> E["Portfolio Strategies<br>Minimum Variance (with/without constraints)<br>Hierarchical Risk Parity"]
    
    E --> F["Key Findings/Outcomes"]
    
    subgraph F ["Financial Performance"]
        F1["Hierarchical Nested Model<br>captures complex system signatures"]
        F2["Empirical data reproduces<br>stylized facts from models"]
        F3["Two-step estimators improve<br>financial metrics"]
        F4["Enables risk management<br>when N ≈ T (high-dimensional)"]
    end