# Computing covariance matrix with historical data

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 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!

• I don't see how you can estimate $\Sigma$ with $T < N$. Can you explain that ? You have an $N$ by $N$ matrix, $\Sigma$, and $T$ periods of data so it's not possible as far as I can tell. May 7 '20 at 12:34
• I think it's more difficult than that because if they say $T < N$ means that positions will appear riskless, they mean that your covariance matrix is going to have elements that are zero. So, although I'm not clear on why, I'd be careful because those two guys know what they're talking about. It may have something to do with the estimation of the covariance matrix being more complex than plugging in estimated correlations. ( maybe a risk model ? ). If you think you're way is okay, go for it. Or maybe look closer at the book to see what they mean by that statement. May 7 '20 at 20:09
• It may have something to do with independence. If stock X has T observations of returns, you probably don't want to use the same T observations for estimation of the correlations of X with the other N-1 stocks. If you do that, you'll probably end up with a matrix that's not positive definite because the resulting estimates won't be independent because of $T$ < $N$. I'm no whiz at covariance matrix estimation but it probably has something do with the rank of the resulting estimated matrix not being N. May 7 '20 at 20:19