Random matrix theory and nested clustered portfolios on Mexican markets

ArXiv ID: 2306.05667 “View on arXiv”

Authors: Unknown

Abstract

This work aims to deal with the optimal allocation instability problem of Markowitz’s modern portfolio theory in high dimensionality. We propose a combined strategy that considers covariance matrix estimators from Random Matrix Theory~(RMT) and the machine learning allocation methodology known as Nested Clustered Optimization~(NCO). The latter methodology is modified and reformulated in terms of the spectral clustering algorithm and Minimum Spanning Tree~(MST) to solve internal problems inherent to the original proposal. Markowitz’s classical mean-variance allocation and the modified NCO machine learning approach are tested on financial instruments listed on the Mexican Stock Exchange~(BMV) in a moving window analysis from 2018 to 2022. The modified NCO algorithm achieves stable allocations by incorporating RMT covariance estimators. In particular, the allocation weights are positive, and their absolute value adds up to the total capital without considering explicit restrictions in the formulation. Our results suggest that can be avoided the risky \emph{“short position”} investment strategy by means of RMT inference and statistical learning techniques.

Keywords: Random Matrix Theory (RMT), Nested Clustered Optimization (NCO), Spectral Clustering, Minimum Spanning Tree (MST), Portfolio Optimization, Equities

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail
• Why: The paper employs advanced mathematics including Random Matrix Theory, Wishart distributions, Marchenko-Pastur law, Tracy-Widom distributions, and Painlevé differential equations, which is dense and highly advanced. Empirically, it applies a moving window backtest from 2018 to 2022 on Mexican Stock Exchange data with specific covariance estimators and portfolio optimization methods, demonstrating implementation-heavy testing.

  flowchart TD
    A["Research Goal: Solve Markowitz's allocation instability<br>in Mexican markets (high dimensionality)"] --> B{"Key Methodology"}
    
    subgraph B ["Proposed Combined Strategy"]
        B1["Random Matrix Theory<br>RMT Covariance Estimators"]
        B2["Modified Nested Clustered Optimization<br>NCO: Spectral Clustering + MST"]
    end
    
    C["Data: BMV Equities<br>Moving Window Analysis (2018–2022)"] --> D["Computational Process"]
    
    subgraph D ["Implementation Steps"]
        D1["Estimate Covariance Matrix via RMT"]
        D2["Apply Modified NCO Algorithm"]
        D3["Compare vs. Classical Markowitz MV"]
    end
    
    D --> E{"Key Findings"}
    
    subgraph E ["Outcomes"]
        E1["Stable Allocations Achieved"]
        E2["Positive Weights (No Explicit Constraints)"]
        E3["Short Position Avoidance via RMT/ML"]
    end