  We investigate the possible drawbacks of employing the standard Pearson
estimator to measure correlation coefficients between financial stocks in the
presence of non-stationary behavior, and we provide empirical evidence against
the well-established common knowledge that using longer price time series
provides better, more accurate, correlation estimates. Then, we investigate the
possible consequences of instabilities in empirical correlation coefficient
measurements on optimal portfolio selection. We rely on previously published
works which provide a framework allowing to take into account possible risk
underestimations due to the non-optimality of the portfolio weights being used
in order to distinguish such non-optimality effects from risk underestimations
genuinely due to non-stationarities. We interpret such results in terms of
instabilities in some spectral properties of portfolio correlation matrices.
