On the spectral decomposition of empirical correlation matrices

One of the most widely used methods to build yield curve models is to use principal components analysis on the correlation matrix of the innovations. R. Litterman and J. Scheinkman found that three factors are enough to explain most of the moves in the case of the US treasury curve. These factors are level, steepness and curvature. Working in the context of commodity futures, G. Cortazar and E. Schwartz found that the spectral structure of the correlation matrices is strikingly similar to those found by R. Litterman and J. Scheinkman. We observe that in both cases the correlation between two different contracts maturing at times t and s is roughly of the form ρ^|t-s|, for a certain (fixed) 0 ≤ p ≤ 1. Assuming this correlation structure we prove that the observed factors are perturbations of cosine waves.