would multiplying a co-variance matrix by the inverse of a variance matrix generate a beta matrix?

Question

I was multiplying the popular calculation w' * Σ * w and got the idea of generating a beta matrix. multiplying Co-variance by inverse variance. Would this work? generating the beta of each asset to the portfolio consistently? Or would this generate something useless?

Answer 1

The ordinary least squares regression estimate of beta of $y$ to $x$ is given by $$ \beta = \frac{\textrm{cov}(x, y)}{\textrm{var}(x)}. $$

In your case, you want to calculate the beta of asset $i$ to your portfolio $p=\sum_j w_j x_j$. $$ \beta_i = \frac{\textrm{cov}(x_i, p)}{\textrm{var}(p)} = \frac{\textrm{cov}(x_i, \sum_j w_j x_j)}{w^T\Sigma w} = \frac{\sum_j w_j\textrm{cov}(x_i, x_j)}{w^T\Sigma w} $$ For the last step, we use the property of covariance that $$\textrm{cov}(X,aY+bZ) = a\,\textrm{cov}(X,Y)+b\,\textrm{cov}(X,Z).$$

If you want to calculate all the betas at once, you can do so in matrix form. $$ \beta = \frac{\Sigma w}{w^T\Sigma w} $$