What does past correlation structure tell us about the future? An answer from network filtering

We discovered that past changes in the market correlation structure are significantly related with future changes in the market volatility. By using correlationbased information filtering networks we device a new tool for forecasting the market volatility changes. In particular, we introduce a new measure, the “correlation structure persistence”, that quantifies the rate of change of the market dependence structure. This measure shows a deep interplay with changes in volatility and we demonstrate it can anticipate market risk variations. Notably, our method overcomes the curse of dimensionality that limits the applicability of traditional econometric tools to portfolios made of a large number of assets. We report on forecasting performances and statistical significance of this tool for two different equity datasets. We also identify an optimal region of parameters in terms of True Positive and False Positive trade-off, through a ROC curve analysis. We find that our forecasting method is robust and it outperforms predictors based on past volatility only. Moreover the temporal analysis indicates that our method is able to adapt to abrupt changes in the market, such as financial crises, more rapidly than methods based on past volatility. Introduction Forecasting changes in volatility is essential for risk management, asset pricing and scenario analysis. Indeed, models for describing and forecasting the evolution of volatility and covariance among financial assets are widely applied in industry [1–4]. Among the most popular approaches are worth mentioning the multivariate extensions of GARCH [5], the stochastic covariance models [6] and realized covariance [7]. However most of these econometrics tools are not able to cope with more than few assets, due to the curse of dimensionality and the increase in the number of parameters [1], limiting their insight into the volatility evolution to baskets of few assets only. This is unfortunate, since gathering insights into systemic risk and the unfolding of financial crises require modelling the evolution of entire markets which are composed by large numbers of assets [1]. We suggest to use network filtering [8–14] as a valuable tool to overcome this limitation. Correlation-based filtering networks are tools which have been widely applied to filter and reduce the complexity of covariance matrices made of large numbers of assets (of the order of hundreds), representative of entire markets. This strand of research represents an important part of the Econophysics literature and has given important insights for risk management, portfolio optimization and systemic risk regulation [15–20]. The volatility of a portfolio depends on the covariance matrix of the corresponding assets [21]. Therefore, the latter can provide insights into the former. In this work we ar X iv :1 60 5. 08 90 8v 1 [ qfi n. PM ] 2 8 M ay 2 01 6 elaborate on this connection: we show that correlation matrices can be used to predict variations of volatility, once they are analysed through the lens of network filtering. This is quite an innovative use of correlation-based networks, which have been used mostly for descriptive analyses, with the connections with risk forecasting being mostly overlooked. Some works have shown that is possible to use dimensionality reduction techniques, such as spectral methods [22], as early-warning signals for systemic risk [23,24]: however these approaches, although promising, do not provide proper forecasting tools, as they are affected by high false positive ratios and are not designed to predict a specific quantity. The approach we propose exploits network filtering to explicitly predict future volatility of markets made of hundreds of stocks. To this end, we introduce a new dynamical measure that quantifies the rate of change in the structure of the market correlation matrix: the “correlation structure persistence” 〈ES〉. This quantity is derived from the structure of network filtering from past correlations. Then we show how such measure exhibits significant predicting power on the market volatility, providing a tool to forecast it. We assess the reliability of this forecasting through out-of-sample tests on two different equity datasets. The rest of this paper is structured as follows: we first describe the two datasets we have analysed and we introduce the correlation structure persistence; then we show how our analyses point out a strong interdependence between correlation structure persistence and future changes in the market volatility; moreover, we describe how this result can be exploited to provide a forecasting tool useful for risk management, by presenting out-of-sample tests and false positive analysis; then we investigate how the forecasting performance changes in time; finally we discuss our findings and their theoretical implications. Results A measure of correlation structure persistence We have analysed two different datasets of equity data. The first set (NYSE dataset) is composed by daily closing prices of N = 342 US stocks traded in New York Stock Exchange, covering 15 years from 02/01/1997 to 31/12/2012. The second set (LSE dataset) is composed by daily closing prices of N = 214 UK stocks traded in the London Stock Exchange, covering 13 years from 05/01/2000 to 21/08/2013. All stocks have been continuously traded throughout these periods of time. These two sets of stocks have been chosen in order to provide a significant sample of the different industrial sectors in the respective markets. For each asset i (i = 1, ..., N) we have calculated the corresponding daily log-return ri(t) = log(Pi(t))− log(Pi(t− 1)), where Pi(t) is the asset i price at day t. The market return rM (t) is defined as the average of all stocks returns: rM (t) = 1/N ∑ i ri(t). In order to calculate the correlation between different assets we have then analysed the observations by using n moving time windows, Ta with a = 1, ..., n. Each time window contains θ observations of log-returns for each asset, totaling to N×n observations. The shift between adjacent time windows is fixed to dT = 5 trading days. We have calculated the correlation matrix within each time window, {ρij(Ta)}, by using an exponential smoothing method [25] that allows to assign more weight on recent observations. The smoothing factor of this scheme has been chosen equal to θ/3 according to previously established criteria [25]. From each correlation matrix {ρij(Ta)} we have then computed the corresponding Planar Maximally Filtered Graph (PMFG) [26]. The PMFG is a sparse network representation of the correlation matrix that retains only a subset of most significant entries, selected through the topological criterion of being maximally planar [9]. Such networks serve as filtering method and have been shown to provide a deep insight into the dependence structure of financial assets [9, 10,27]. Once the n PMFGs, G(Ta) with a = 1, ..., n, have been computed we have calculated two measures, a backward-looking and a forward-looking one. The first is a measure that monitors the correlation structure persistence, based on a measure of PMFG similarity. This backward-looking measure, that we call 〈ES〉(Ta), relies on past data only and indicates how slowly the correlation structure measured at time window Ta is differentiating from structures associated to previous time windows. The forward-looking