# Sample Variance of Portfolio

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 $$\widehat{\mathbf{Var}}\left( w' r_i\right) = w' \widehat{\mathbf{Var}}\left( r_i\right) w = w' \hat{\Sigma} w?$$

• Yes you are right. The true variance is $w' \Sigma w$, the estimated variance (with a hat) is given by $w' \hat{\Sigma} w$ Nov 19 '21 at 15:23

## 1 Answer

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.