From sectorial coarse graining to extreme coarse graining of S&P 500 correlation matrices

ArXiv ID: 2511.05463 “View on arXiv”

Authors: Manan Vyas, M. Mijaíl Martínez-Ramos, Parisa Majari, Thomas H. Seligman

Abstract

Starting from the Pearson Correlation Matrix of stock returns and from the desire to obtain a reduced number of parameters relevant for the dynamics of a financial market, we propose to take the idea of a sectorial matrix, which would have a large number of parameters, to the reduced picture of a real symmetric $2 \times 2$ matrix, extreme case, that still conserves the desirable feature that the average correlation can be one of the parameters. This is achieved by averaging the correlation matrix over blocks created by choosing two subsets of stocks for rows and columns and averaging over each of the resulting blocks. Averaging over these blocks, we retain the average of the correlation matrix. We shall use a random selection for two equal block sizes as well as two specific, hopefully relevant, ones that do not produce equal block sizes. The results show that one of the non-random choices has somewhat different properties, whose meaning will have to be analyzed from an economy point of view.

Keywords: Correlation Matrix, Dimensionality Reduction, Sectorial Matrix, Block Averaging, Random Matrix Theory, Equities (Stocks)

Complexity vs Empirical Score

• Math Complexity: 4.0/10
• Empirical Rigor: 6.5/10
• Quadrant: Street Traders
• Why: The mathematics is moderate, relying on standard correlation matrices and averaging, without advanced derivations. The paper is empirically rigorous, using real S&P 500 data over 4,430 days with detailed methodology, clustering (k-means), and specific numerical comparisons for different stock subset choices.

  flowchart TD
    A["Research Goal<br>Reduce Correlation Matrix<br>Dimensionality"] --> B["Input: Pearson Correlation<br>Matrix of S&P 500 Returns"]
    B --> C["Method: Block Averaging<br>Partition rows/columns into subsets"]
    C --> D{"Random Selection<br>vs. Specific Selection"}
    D -- Random --> E["Random Blocks<br>(Equal/Unequal Sizes)"]
    D -- Specific --> F["Specific Blocks<br>(Economy-Relevant Sectors)"]
    E --> G["Compute Averaged<br>2x2 Symmetric Matrix"]
    F --> G
    G --> H["Key Findings<br>Specific selection yields<br>unique dynamic properties<br>Avg. correlation retained as parameter"]