Sample Variance of Portfolio

Question

Let $w$ denote a vector of portfolio weights, $r_i$ denote the $i$th return vector, $\Sigma$ denote the Covariance matrix of $r_i$ and let $\hat{\Sigma}$ denote the sample covariance matrix of $r_i$.

The portfolio variance is given by $$ \mathbf{Var}\left( w' r_i\right) = w' \mathbf{Var}\left( r_i\right) w = w' \Sigma w. $$ Does it hold for the sample portfolio variance that $$ \widehat{\mathbf{Var}}\left( w' r_i\right) = w' \widehat{\mathbf{Var}}\left( r_i\right) w = w' \hat{\Sigma} w? $$

Answer 1

Yes, indeed. It's a simple Linear Algebra and Expectation result:

Given:

$Var(w'r) = \mathbb{E}[(w'r)^2] = \mathbb{E}[(w'rr'w)]$

With $w$ and $r$ the vectors of weights and returns. As $w$ is constant, it holds:

$\mathbb{E}[w'rr'w] = w'\mathbb{E}[rr']w$

The sample variance, $\hat{\Sigma}$, is a estimator of for $\mathbb{E}[rr']$. Therefore, it holds what you said.