  A new methodology has been introduced to clean the correlation matrix of
single stocks returns based on a constrained principal component analysis using
financial data. Portfolios were introduced, namely "Fundamental Maximum
Variance Portfolios", to capture in an optimal way the risks defined by
financial criteria ("Book", "Capitalization", etc.). The constrained
eigenvectors of the correlation matrix, which are the linear combination of
these portfolios, are then analyzed. Thanks to this methodology, several
stylized patterns of the matrix were identified: i) the increase of the first
eigenvalue with a time scale from 1 minute to several months seems to follow
the same law for all the significant eigenvalues with 2 regimes; ii) a
universal law seems to govern the weights of all the "Maximum variance"
portfolios, so according to that law, the optimal weights should be
proportional to the ranking based on the financial studied criteria; iii) the
volatility of the volatility of the "Maximum Variance" portfolios, which are
not orthogonal, could be enough to explain a large part of the diffusion of the
correlation matrix; iv) the leverage effect (increase of the first eigenvalue
with the decline of the stock market) occurs only for the first mode and cannot
be generalized for other factors of risk. The leverage effect on the beta,
which is the sensitivity of stocks with the market mode, makes variable the
weights of the first eigenvector.
