I have been reading Active Portfolio Management by Grinold and Khan. In the chapter about risk, they mention,
"The third elementary model relies on historical variances and covariances. This procedure is neither robust nor reasonable. Historical models rely on data from T periods to estimate the $NxN$ covariance matrix. If T is less than or equal to N, we can find active positions that will appear riskless! So the historical approach requires $T > N$. For a monthly historical covariance matrix of S&P 500 stocks, this would require more than 40 years of data."
When forming mean-variance optimal portfolio, we would need to invert the covariance matrix hence, we require a full rank covariance matrix. In this case using historical returns is not robust.
However, if the main intention is to compute an estimate of the variance of the portfolio $w'\Sigma w$ where $w$ is the weight or holdings of stock, in such a use case, we can estimate $\Sigma$ with $T<N$ right? As we are not inverting the covariance matrix, the concern of not full rank is less of an issue, right?
Any help is very much appreciated!