Bias due to portfolio optimization

In our entire analysis of models and measures in the earlier sections, we worked with fixed, pre-defined portfolios. During this analysis, spurious minima could have occasionally appeared in the vicinity of the fixed portfolios leading to risk underestimation but spurious maxima at other times would have largely balanced this effect in the overall performance evaluation of the model.

In typical scenarios of the application of VaR methodology, however, the actual portfolio is determined with the help of the estimated covariance matrix $\Sigma_t$. This means that, on a frequent basis, investors are likely to take advantage of rank defects, near singularities and spurious minima due to stochastic errors - and tend to select portfolios in that portion of the portfolio space which necessarily underestimates risk.