Speaker:

Gilles Zumbach (Riskmetrics)

Title:

Empirical properties of large matrices in finance

Abstract:

Many problems in finance are related to the covariance matrix of an investment universe, like optimal allocations, risk evaluation, or the description by multivariate processes of the price dynamics. Today's investment portfolios can be very large, of the order of 1'000 to 100'000 positions. In view of the large system size, using random matrices is a natural approach to separate the "meaningful" from the "noisy" informations. The specificity of the financial time series are presented, and in particular the role of the heteroskedasticity and its implication for the definition of the covariance matrix. An empirical analysis for the covariance and correlation matrices is presented that shows the properties of the spectrum and its dynamics, as well as the dynamics of the eigenvectors. Notably, the eigenvalues decrease exponentially fast toward zero, and there is no stable subspace spanning the leading eigenvectors. The inverse of the covariance appears in several practical applications, and the small or null eigenvalues are an important nuisance. The implications for large scale ARCH processes is discussed.