On the White Board - November 2008

Nov 17, 2008

Are correlations constant?

When dealing with a large universe of assets, a key part of the information is contained in the variance/covariance matrix. The covariance is computed using the time series corresponding to each assets, and it summarizes the volatility of each components as well as their mutual dependencies. When moving from this heuristic level of description to actual formulas, the ques- tion of the best covariance estimator arises. This is a fairly complex question, with ramifications in the given financial objectives (portfolio estimation, risk evaluation, ...) and in the time series description of a multivariate universe. Essentially two paths can be followed. First, the covariance can be computed as the natural extension of the univariate case, essentially replacing r² in the univariate formulas by the product rα · rβ in the multivariate case. Second, the volatility and correlation can be separated, and each parts evaluated with a specific and optimal formula. The underlying idea is that the volatil- ities have a fairly fast dynamics, while the correlations are more stable. Correspondingly, the optimal estimators for both quantities are different. But is this picture correct?

In this note, we want to investigate empirically this question. The separation issue can be formulated as the identification of two different characteristic time scale in the covariance, a short one corresponding to the fast evolution of the volatility (say from a few days to a few months), and a slow time scale corresponding to the correlation (say at least of one year). By pushing the underlying argument for the separation of volatilities and correlations, we can take the correlations to be constant, taking to the limit the idea of very stable correlations.

Ultimately, the question of the optimal evaluation of the covariance should be answered by looking at empirical data. To this purpose, we investi- gated three data sets of sizes 340, 55, and 54. The dynamical covariance is computed with the long memory kernel used in the RM2006 methodology, namely following the first path above and with the best univariate volatility estimator available. Then, the spectrums of the covariance and of the corre- lation matrices are computed. The attached figure plots the time evolution over 8 years of the first 11 eigenvalues of the covariance (top plot) and cor- relation (bottom plot), for a data set containing 55 time series made of FX, IR, stock indexes and commodities from the G10 countries. Essentially, this universe spans a well diversified portfolio invested in the G10 space. The similarity of the dynamics for the largest eigenvalues is clear. The volatilities have larger moves, but the time scales involves in both graphs are similar, roughly of the order of one month. In order to go beyond this intuitive picture, a deeper and rigorous quantitative analysis is needed. Yet, the idea of ”constant” correlations that could be evaluated over a (very) long time horizon can already be invalidated.