Near singularity

We have illustrated above that the rectangular moving average model satisfying the BIS parameters (model 2) as well as the J. P. Morgan RiskMetrics model (model 3) necessarily have rank defects if the number of risk factors d exceeds the number of historic data n from which the correlation matrix is constructed. This overt existence of the rank defect however points to a deeper problem. Even when the methodology is more complex - such as when we are using model 4 or 5 with implied covariances - while it is unlikely that $\Sigma_t$ is exactly singular (and hence admits a rank defect), it can very well be near singular. This means that while there will not exist a portfolio $\lambda$ (aside from the trivial null portfolio $\lambda_i = 0$)such that $\hbox{VaR}(\lambda,\Sigma_t,c) = 0$ (for all c) there will still exist portfolios for which risk is pathologically underestimated (as defined in section ).