Fine modeling of the covariance/correlation matrix of stocks.

Summary A new method has been implemented to de-constrain the correlation matrix of equity returns based on a constrained principal component analysis using financial data. Portfolios, named "Fundamental Maximum variance portfolios", are constructed to optimally capture a risk style defined by a financial criterion ("Book", "Capitalization", etc.). The constrained eigenvectors of the correlation matrix, which are linear combinations of these portfolios, are then studied. Thanks to this method, several stylized facts of the matrix have been highlighted among which: i) the increase of the first eigenvalues with the time scale from 1 minute to several months seems to follow the same law for all the significant eigenvalues with two regimes. ii) a _universal_ law seems to govern the composition of all the portfolios "Maximum variance". Thus, according to this law, the optimal weights would be directly proportional to the ranking according to the financial criterion studied. iii) the volatility of the "Maximum Variance_" portfolios, which are not orthogonal, would explain a large part of the diffusion of the correlation matrix. iv) the leverage effect (increase of the first eigenvalue with the fall of the market) exists only for the first mode and does not generalize to the other risk factors. The leverage effect on the beta, the sensitivity of stocks with the "market mode", makes the weights of the first eigenvector variable.