Schur Complementary Allocation: A Unification of Hierarchical Risk Parity and Minimum Variance Portfolios

ArXiv ID: 2411.05807 “View on arXiv”

Authors: Unknown

Abstract

Despite many attempts to make optimization-based portfolio construction in the spirit of Markowitz robust and approachable, it is far from universally adopted. Meanwhile, the collection of more heuristic divide-and-conquer approaches was revitalized by Lopez de Prado where Hierarchical Risk Parity (HRP) was introduced. This paper reveals the hidden connection between these seemingly disparate approaches.

Keywords: portfolio construction, hierarchical risk parity, Markowitz optimization, covariance estimation, Portfolio Management

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper presents a novel mathematical unification between Hierarchical Risk Parity and Minimum Variance Portfolios using Schur complements and matrix algebra, requiring advanced linear algebra to follow. However, the excerpt lacks empirical backtests, datasets, or statistical performance metrics, focusing instead on theoretical connections and critique of existing methods.

  flowchart TD
    R["Research Goal:<br/>Unify HRP & MinVar Portfolios<br/>via Schur Complement"] --> D["Input: Asset Covariance Matrix Σ"]
    D --> S["Process 1: Recursive Bisection<br/>Hierarchical Clustering"]
    D --> M["Process 2: Minimum Variance<br/>Optimization"]
    S --> C["Computation:<br/>Schur Complementary Allocation<br/>Transforms HRP into MinVar"]
    M --> C
    C --> F["Key Finding 1:<br/>HRP is a specific case of MinVar<br/>with recursive conditioning"]
    C --> G["Key Finding 2:<br/>Provides robust estimator for<br/>inverse covariance matrix"]
    C --> H["Outcome:<br/>Unified framework bridging<br/>heuristic & optimization methods"]