Linear and nonlinear market correlations: Characterizing financial crises and portfolio optimization

Pearson correlation and mutual information-based complex networks of the day-to-day returns of U.S. S&P500 stocks between 1985 and 2015 have been constructed to investigate the mutual dependencies of the stocks and their nature. We show that both networks detect qualitative differences especially during (recent) turbulent market periods, thus indicating strongly fluctuating interconnections between the stocks of different companies in changing economic environments. A measure for the strength of nonlinear dependencies is derived using surrogate data and leads to interesting observations during periods of financial market crises. In contrast to the expectation that dependencies reduce mainly to linear correlations during crises, we show that (at least in the 2008 crisis) nonlinear effects are significantly increasing. It turns out that the concept of centrality within a network could potentially be used as some kind of an early warning indicator for abnormal market behavior as we demonstrate with the example of the 2008 subprime mortgage crisis. Finally, we apply a Markowitz mean variance portfolio optimization and integrate the measure of nonlinear dependencies to scale the investment exposure. This leads to significant outperformance as compared to a fully invested portfolio.

Figure 1

(a) Return time series, (b) autocorrelation function of returns (blue, dark gray) and absolute values of returns (red, light gray), (c) phase map of Lincoln National Corporation (LNC) stock returns. Phase map: Phases $\phi \left(l\right)$ are scattered against neighboring phases $\phi (l+1)$.

Figure 2

Left: Mean (a) and variance (c) of the coefficients of the Pearson correlation-based distance matrix calculated in each time step. Right: Mean (b) and variance (d) based on the mutual information of the original series (blue) and the average distance matrix of all surrogate realizations (red). Plus/minus one sigma error ranges (gray) are drawn for the surrogate realizations; however, errors are so small that one can barely seen them.

Figure 3

Average significance coefficient $\overline{{\chi }_{sig}\left(X_i\right)}$ of each asset—Thick red line: Global average over all individual assets.

Figure 4

Significance matrices ${\chi }_{\text{sig}}$: Lower half, Calm market environment in November 1998; Upper half, Turbulent market environment in May 2009 after the crash triggered by the Lehman collapse on September 15, 2008. Labels denote stock indices.

Figure 5

Top: Normalized Tree Length of Pearson correlation-based MST. Bottom: Same for mutual information of original data (blue) and surrogates (red).

Figure 6

(a) Dynamic mean occupation layer of Pearson correlation-based MST. (b) Mutual Information of surrogate series (red) vs. Pearson correlation-based result (blue). (c) Same for mutual information of original series (blue) and surrogates (red) with plus/minus one sigma error range in gray. Vertical lines indicate the bursting of the Dot-com bubble in March 2000 and the acceleration of the crash after the Lehman Brothers collapse on September 15, 2008.

Figure 7

Assets with the highest degree centrality. Top: Pearson correlation-based; bottom: original mutual information (blue) and surrogate mutual information-based (red).

Figure 8

Degree centrality of JPM based on Pearson correlation (a) and mutual information (b) (blue, original data; red, surrogate data) vs. network avg. (red, dashed). The vertical lines indicate the peak of the stock prices on October 15, 2007 after which they began falling as well as the acceleration of the crash after Lehman Brothers collapsed on September 15, 2008.

Figure 9

Clustering coefficient of JPM (blue) vs. network avg. (red, dashed) for Pearson correlation (a), original mutual information (b), and surrogate mutual information (c).

Figure 10

Backtest results: Portfolio value (starting at 1 at $t=0$) for fixed allocation (red, dashed), fully invested max sharp ratio portfolio (red), and NLC-scaled portfolio (blue).

Figure 11

Upper plot: Portfolio weights—S&P 500 Index (red), Merrill Lynch U.S. Corporate Index (dark blue), and Merrill Lynch U.S. Treasury Index (blue). Lower plot: Cash weights based on NLC strategy.