Understanding correlations in complex systems is crucial in the face of turbulence, such as the ongoing financial crisis. However, in complex systems, such as financial systems, correlations are not constant but instead vary in time. Here we address the question of quantifying state-dependent correlations in stock markets. Reliable estimates of correlations are absolutely necessary to protect a portfolio. We analyze 72 years of daily closing prices of the 30 stocks forming the Dow Jones Industrial Average (DJIA). We find the striking result that the average correlation among these stocks scales linearly with market stress reflected by normalized DJIA index returns on various time scales. Consequently, the diversification effect which should protect a portfolio melts away in times of market losses, just when it would most urgently be needed. Our empirical analysis is consistent with the interesting possibility that one could anticipate diversification breakdowns, guiding the design of protected portfolios. 1-8 continue to have a huge impact on the world economy and the personal fortunes of millions, shedding light on the complex nature of financial and economic systems. For these systems, a truly gargantuan amount of pre-existing precise financial market data 9-11 complemented by new big data ressources 12-15 is available for analyses. The complex mechanisms of financial market moves can lead to sudden trend switches [16] [17] [18] in a number of stocks. Such sudden trend switches can occur in a synchronized fashion, in a large number of stocks simultaneously, or in an unsynchronized fashion, affecting only a few stocks at the same time. Diversification in stock markets refers to the reduction of portfolio risk caused by the investment in a variety of stocks. If stock prices do not move up and down in perfect synchrony, a diversified portfolio will have less risk than the weighted average risk of its constituent stocks 19, 20 . Hence it should be possible to reduce risk in price of individual stocks by the combination of an appropriate set of stocks. To identify such an appropriate set of stocks with anti-correlated price time series, the assumption mostly used is that the correlations among stocks are constant over time [21] [22] [23] [24] [25] [26] . This widely used assumption is also the basis for the determination of capital requirements of financial institutions that usually own a huge variety of constituents belonging to different asset classes. Recent studies building on the availability of huge and detailed data sets of financial markets have analyzed and modeled the static and dynamic behavior of this very complex system [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] , suggesting that financial markets are governed by systemic shifts and display non-equilibrium properties. A very well known stylized fact of financial markets is the leverage effect, a term coined by Black to describe the negative correlation between past price returns and future realized volatilities in stock markets. According to Reigneron et al. 40 , the index leverage effect can be decomposed into a volatility effect and a correlation effect. In the course of recent financial market crises, this effect has regained center stage, and the work of different groups has focused on uncovering its true nature [40] [41] [42] [43] [44] [45] [46] ). Reigneron et al. analyzed daily returns of six indices from 2000 to 2010 and found that a downward index trends increase the average correlation between stocks, as quantified by measurements of eigenvalues of the conditional correlation matrix. They suggest that a quadratic term should be included to the linear regressions of the dependence of mean correlation on the index return the previous day. Here, we will expand on these results utilizing 72 years of trading of the 30 Dow Jones industrial average (DJIA) components (see also 47, 48
In REF , the average correlations among stocks are found to scale linearly with the market stress reflected by normalized DJIA index returns on various time scales.
Quantifying the Behavior of Stock Correlations Under Market Stress
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