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

TL;DR

This paper introduces a method to simplify the correlation matrices of stock returns by aggregating stocks into blocks, reducing complexity while preserving key correlation features, with implications for financial market analysis.

Contribution

It proposes an extreme coarse graining approach that reduces a large correlation matrix to a 2x2 matrix, maintaining average correlation, and explores different block averaging strategies.

Findings

Random block averaging preserves correlation properties.

Non-random block choices show distinct properties.

Method offers a simplified yet informative view of market correlations.

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.