# Dynamical Analysis of Stock Market Instability by Cross-correlation Matrix

## TL;DR

This study analyzes stock market instability using cross-correlation matrices of stock returns, identifying volatile periods and characterizing market risk through principal components and random matrix theory.

## Contribution

It introduces a dynamical approach combining rolling window cross-correlations, principal component analysis, and random matrix theory to detect and analyze market volatility.

## Key findings

- Identified three major volatile market stages.
- Detected increased eigenvector delocalization during market volatility.
- Quantified systemic risk via cumulative risk fraction (CRF).

## Abstract

We study stock market instability by using cross-correlations constructed from the return time series of 366 stocks traded on the Tokyo Stock Exchange from January 5, 1998 to December 30, 2013. To investigate the dynamical evolution of the cross-correlations, cross-correlation matrices are calculated with a rolling window of 400 days. To quantify the volatile market stages where the potential risk is high, we apply the principal components analysis and measure the cumulative risk fraction (CRF), which is the system variance associated with the first few principal components. From the CRF, we detected three volatile market stages corresponding to the bankruptcy of Lehman Brothers, the 2011 Tohoku Region Pacific Coast Earthquake, and the FRB QE3 reduction observation in the study period. We further apply the random matrix theory for the risk analysis and find that the first eigenvector is more equally de-localized when the market is volatile.