Hierarchical Minimum Variance Portfolios: A Theoretical and Algorithmic Approach

ArXiv ID: 2503.12328 “View on arXiv”

Authors: Unknown

Abstract

We introduce a novel approach to portfolio optimization that leverages hierarchical graph structures and the Schur complement method to systematically reduce computational complexity while preserving full covariance information. Inspired by Lopez de Prados hierarchical risk parity and Cottons Schur complement methods, our framework models the covariance matrix as an adjacency-like structure of a hierarchical graph. We demonstrate that portfolio optimization can be recursively reduced across hierarchical levels, allowing optimal weights to be computed efficiently by inverting only small submatrices regardless of portfolio size. Moreover, we translate our results into a recursive algorithm that constructs optimal portfolio allocations. Our results reveal a transparent and mathematically rigorous connection between classical Markowitz mean-variance optimization, hierarchical clustering, and the Schur complement method.

Keywords: Hierarchical Risk Parity, Schur Complement, Graph Theory, Covariance Matrix Inversion, Markowitz Mean-Variance Optimization, Portfolio Management

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats
• Why: The paper is heavily theoretical, deriving recursive relationships between hierarchical graph structures and portfolio optimization using the Schur complement and matrix algebra, which indicates high mathematical complexity. However, the excerpt contains no backtesting results, implementation code, or empirical validation datasets, focusing instead on theoretical frameworks and algorithm descriptions.

  flowchart TD
    A["Research Goal: Efficient Portfolio Optimization with Full Covariance Information"] --> B["Key Methodology: Hierarchical Graph & Schur Complement"]
    B --> C["Input: Asset Covariance Matrix Σ"]
    C --> D["Recursive Process: Partition & Invert Submatrices"]
    D --> E["Outcome: Optimal Portfolio Weights"]
    E --> F["Key Findings: Reduced Complexity, Mathematical Rigor"]