  According to recent findings [1,2], empirical covariance matrices deduced
from financial return series contain such a high amount of noise that, apart
from a few large eigenvalues and the corresponding eigenvectors, their
structure can essentially be regarded as random. In [1], e.g., it is reported
that about 94% of the spectrum of these matrices can be fitted by that of a
random matrix drawn from an appropriately chosen ensemble. In view of the
fundamental role of covariance matrices in the theory of portfolio optimization
as well as in industry-wide risk management practices, we analyze the possible
implications of this effect. Simulation experiments with matrices having a
structure such as described in [1,2] lead us to the conclusion that in the
context of the classical portfolio problem (minimizing the portfolio variance
under linear constraints) noise has relatively little effect. To leading order
the solutions are determined by the stable, large eigenvalues, and the
displacement of the solution (measured in variance) due to noise is rather
small: depending on the size of the portfolio and on the length of the time
series, it is of the order of 5 to 15%. The picture is completely different,
however, if we attempt to minimize the variance under non-linear constraints,
like those that arise e.g. in the problem of margin accounts or in
international capital adequacy regulation. In these problems the presence of
noise leads to a serious instability and a high degree of degeneracy of the
solutions.
