Temporal evolution of financial-market correlations

We investigate financial market correlations using random matrix theory and principal component analysis. We use random matrix theory to demonstrate that correlation matrices of asset price changes contain structure that is incompatible with uncorrelated random price changes. We then identify the principal components of these correlation matrices and demonstrate that a small number of components accounts for a large proportion of the variability of the markets that we consider. We characterize the time-evolving relationships between the different assets by investigating the correlations between the asset price time series and principal components. Using this approach, we uncover notable changes that occurred in financial markets and identify the assets that were significantly affected by these changes. We show in particular that there was an increase in the strength of the relationships between several different markets following the 2007–2008 credit and liquidity crisis.

Figure 1

(Color online) Comparison of the distribution of correlation coefficients as a function of time for time windows of length $T=100$, 150, and 200. (a) Mean correlation $\mu \left(r\right)$, (b) standard deviation $\sigma \left(r\right)$, (c) skewness $y\left(r\right)$, and (d) kurtosis $\kappa \left(r\right)$.

Figure 2

(Color online) Distribution of all of the correlation coefficients $r(i,j)$ from every time window for market and random data. The shuffled and simulated data lie almost on top of each other.

Figure 3

(Color online) Distribution of eigenvalues $\rho \left(\beta \right)$ of the correlation matrices for all time windows for (a) market, (b) shuffled, and (c) simulated data. The insets show the distributions of the largest eigenvalues. In (b) and (c), we show the eigenvalue probability density functions $\rho \left(\gamma \right)$ for random matrices given by Eq. (5) for $Q\doteq 1.02$ and 1.

Figure 4

(Color online) Fraction of the variance in $\widehat{\mathbf{Z}}$ explained by the first five PCs versus time. The top curve shows the variance explained by the first PC, the next curve shows the variance explained by the second PC, and so on. The horizontal axis shows the year of the last data point in each time window.

Figure 5

(Color online) Participation ratio ${\left[I_k\right]}^{-1}$ as a function of time for the three PCs with the largest variance ($k=1,2,3$). The horizontal solid line shows the PR (averaged over 100 000 simulations) of the first PC for randomized returns, and the horizontal dashed lines show one standard deviation above and below the mean.

Figure 6

(Color online) (a) A scree plot, which gives the magnitude of the PC eigenvalues as a function of the eigenvalue index, where the eigenvalues are sorted such that ${\beta }_1\ge {\beta }_2\ge \cdots \ge {\beta }_N$. We show curves for random correlation matrices and for correlation matrices for time windows ending on 9 Mar 2001 and 27 Nov 2009. The inset zooms in on the region in which the two example curves for observed data cross the curve for random data. (b,c) Number of significant components as a function of time determined using (b) the Kaiser-Guttman criterion and (c) by comparing the scree plots of the observed and random data.

Figure 7

(Color online) The absolute correlation $\left|r\right({\widehat{\mathbf{z}}}_i,{\mathbf{y}}_k\left)\right|$ between each asset and the first six PCs ($k=1,...,6$) as a function of time. Each point on the horizontal axis represents a single time window, and each position along the vertical axis represents an asset.

Figure 8

(Color online) For individual asset classes, we show the fraction of the variance in the returns explained by the first PC versus time. The horizontal axis shows the year of the last data point in each time window.

Figure 9

Examples of the absolute correlation $\left|r\right({\widehat{\mathbf{z}}}_i,{\mathbf{y}}_k\left)\right|$ between various assets and the first five principal components ($k=1,...,5$) as a function of time. We show Danish government bonds (DEGATR), AA-rated corporate bonds (MOODCAA), Nikkei 225 (NKY), the exchange rate between Australian dollars and United States dollars (AUDUSD), gold (XAU), oil (CO1), and soybean futures (S 1). The first row shows correlations for the first PC, the second row shows them for the second PC, and so on. The horizontal red lines show the values of the 99$\mathrm{th}$ percentile of the distribution of absolute correlation coefficients for the corresponding PC for random data. As we increase $k$ from 1 to 5, the 99$\mathrm{th}$ percentile of $\left|r\right({\widehat{\mathbf{z}}}_i,{\mathbf{y}}_k\left)\right|$ for random matrices decreases from 0.47 to 0.43.

Figure 10

(Color online) Distribution of the absolute-correlation difference $|r({\widehat{\mathbf{z}}}_i,{\mathbf{y}}_k)|-|r({\widehat{\mathbf{z}}}_i,{\mathbf{w}}_k)|$ for each of the first five PCs. As explained in the text, $\left|r\right({\widehat{\mathbf{z}}}_i,{\mathbf{y}}_k\left)\right|$ denotes the absolute value of the correlation between the $i$th asset and the $k$th PC, and $\left|r\right({\widehat{\mathbf{z}}}_i,{\mathbf{w}}_k\left)\right|$ denotes the absolute value of the correlation the $i$th asset and the $k$th PC with the contribution of the $i$th asset removed from the PCs. For each component $k$, we show in a single distribution the absolute-correlation differences $|r({\widehat{\mathbf{z}}}_i,{\mathbf{y}}_k)|-|r({\widehat{\mathbf{z}}}_i,{\mathbf{w}}_k)|$ for all assets $i$(where $i=1,...,N$) over all time steps.