Correlation Dynamics of Asset Returns and Systemic Crises David Melkuev Tony Wirjanto [email protected] Department of Statistics and Actuarial Science University of Waterloo Applied Report – 5 Sponsored by Society of Actuaries Centers of Actuarial Excellence 2012 Research Grant Summary In this report, we discuss correlation dynamics of financial asset returns in relation to episodes of systemic crises. The eigenvalues, which are obtained from principal component analysis on sam- ple correlation matrix of financial asset returns, are known to be equal to the variance explained by their associated eigenvectors. Referred to as an absorption ratio, which is the proportion of total system variance captured by a relatively small number of principal components, it reflects the strength of co-rrmovement of financial asset returns in the system. As a stylized fact of financial asset returns, an increasing correlation in the these returns is known to coincide with or follow significant market drawdowns in financial markets. However, studies in recent years have suggested that contagion in financial markets is facilitated by a state of a high degree of interconnectedness in the financial system which precedes the systemic crises. In this report, we explore the association between changes in the absorption ratio and aggregate market returns in the context of three notable crisis episodes across broader classes of assets than have been studied so far. Time series of the normalized-eigenvalue estimates reveal that the structure of correlations of financial asset returns is richly dynamic across various asset groups. Importantly, we find that systemic crises are characterized by a general breakdown of the correlation structure in the financial asset returns and not by an increased comovement in these returns as has been asserted in prior studies, which conclusion is mostly based on a particular episode of systemic crisis and a particular class of assets. 1 Introduction 1 Introduction Much of the observable phenomena in the empirical sciences are of a multivariate nature. Similarly in finance, values of multiple assets are observed simultaneously and form a system with complex dynamics. A good understanding of valuations of multiple assets jointly, therefore, is seen as critical to a wide range of financial applications. For instance, in portfolio management, the notion of risk in finance has been defined by variance of asset returns and an extension to the multivariate setting of covariances between assets returns is natural due to a comovement that financial asset values exhibit over time. This comovement is referred to as correlation. Correlation describes a linear relationship and, in most cases, assuming it to be such may run into the risk of oversimplification of the true relationship between returns on different financial assets.1 Nevertheless, statistical correlation sheds an important light on features of financial markets that are fundamental and universal, and is, therefore, still a relevant tool for research. As an example of this at a basic level, that the values of different assets within the equity class are generally positively correlated points to the presence of class-specific risk factors. In the remainder of this report, unless otherwise stated, we will use the terms correlation and comovement interchangeably to refer to the extent that observable quantities fluctuate over time together. More specifically, in this report, we focus our analyses on the phenomenon of changes in the strength of comovement between financial asset returns. What can be perhaps most reliably said about this particular relationship is that correlations between financial asset returns tend to increase during episodes of downturns in financial markets. There is plenty empirical evidence in support of this contention, for example in [3, 9, 12, 42, 45, 61] and the references cited therein. According to some of these studies, the prevalence of asymmetric correlations with respect to the direction of financial asset returns warrants that it be regarded as a stylized fact in stock (and possibly other) markets. Stylized facts are properties of financial markets that are so robust { for example across a wide range of economies, instruments and time periods { that they are accepted as generally true to the extent that some market models tend to emulate them. Through an extensive research inquiry into asymmetric correlation between financial asset returns, it is commonly believed to be a coincident effect, the reasons for which tend to be behavioural, like increased linkages due to forced liquidations under leverage. Indeed, correlation asymmetry in financial asset returns has been linked to asymmetric volatility of these returns, a stylized fact also known as the leverage effect. Such conclusions are robust to market fluctuations of varying amplitudes, commensurately with the definition of stylized facts. In contrast, in this report, we study return correlations between financial asset returns in reference to more circumstantial market drawdowns. Specifically, we are interested in correlation dynamics of financial asset returns in relation to systemic shocks in financial markets. Systemic events in financial markets have a special characteristic of causing drawdowns that 1The most common use of correlation in finance is in reference to the Pearson product-moment correlation coefficient. 2 1 Introduction are contagious in that they ultimately propagate to a large number of financial assets relative to the number of financial assets immediately affected. In other words, under certain market conditions, devaluation of some financial assets can cause devaluation of some other financial assets in the financial system. The notion of financial contagion is not new and is recognized in historic accounts of financial crises dating back several centuries ago. See [56]. Viewed from the perspective of systemic crises, it is intuitive to explain correlation asymmetry in financial asset returns simply as a manifestation of financial contagion itself. However, the risk of systemic shocks in financial markets tends to change over time as the conditions that facilitate contagion are created. Therefore, an important research question posed in this report is whether an increased systemic risk in financial markets is reflected in rising correlations between financial asset returns before the onset of a systemic crisis that has occurred in financial markets. To measure the strength of comovement of financial asset returns, we take an approach based on spectral analysis in time series analysis. The idea underlying this approach is to transform coordinates of the original multivariate financial asset return data, such that the variance along each axis of the new system is maximized. The values along each axis represent linearly uncor- related factors, and have successively decreasing variances. This transformation is known as a principal component analysis (PCA), which is a technique often used in multivariate analysis in Statistics (See Appendix B). More importantly, it has also been used to gain insights into changes in correlation between financial asset returns [8, 13, 22, 35, 44, 49, 66]. The vectors of values along each axis are known as principal components (PCs). The orthogonal linear combinations for the required transformation are found as the eigenvectors of the system's correlation matrix, each of which has an associated eigenvalue. Since each eigenvalue equals the variance of the PC found through its associated eigenvector, we can define the total risk in the system as the sum of these eigenvalues. Then, the proportion of total variance attributed to a small number of PCs indicates the degree of commonality between returns on the assets in question. We estimate the proportion of variance explained by a fixed small number of factors through time and study its fluctuations over time. A statistic of this sort is coined an absorption ratio (AR) in [35], which is in reference to the proportion of variance absorbed by a few factors. Studies that have taken this approach commonly find that severe downturns coincide with and are sometimes preceded by increases in the strength of comovements between financial asset returns. The explanation provided for this phenomenon is that a higher degree of integration allows for the ripple effect that characterizes systemic financial crises. Importantly, the studies drawing such a conclusion is mostly based on their investigations that are limited to certain asset classes and over certain time periods. Specifically, the Global Financial Crisis of 2008 is by far the most common case study for this stream of studies and equities are the most common asset class examined in this regard. In this report, we expand the analysis by examining three episodes of crises of systemic nature: the Global Financial Crisis of 2008, the Eurozone Sovereign Debt Crisis and the Asian Financial Crisis of 1997. Furthermore, a greater range of asset classes is also subjected to investigation in this report, including equities, bonds, credit default swaps and currencies. 3 1 Introduction In addition we are of the view that sample eigenvalues should be treated as random variables; accordingly, in this report, we place emphasis on distributional aspects of eigenvalue-based statis- tics (see Appendix C). Results from multivariate analysis include exact expressions about both joint and marginal eigenvalue distributions of Wishart matrices (see [46], for examples.) Although Wishart matrices are useful models of correlation