2
$\begingroup$

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?

$\endgroup$
  • 2
    $\begingroup$ Probably I don't understand what you propose. But generally when you multiply a thing by the inverse of that thing you just get the Identity. $\endgroup$ – noob2 Mar 27 '17 at 16:00
2
$\begingroup$

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. $$ \beta = \frac{\Sigma w}{w^T\Sigma w} $$

$\endgroup$
  • $\begingroup$ This calculation is a component of the formula for Contribution to Value at Risk in Finance. If you take the derivative of weighted standard deviation, then the result is the element-wise product of this beta vector and the weights. $\endgroup$ – John Mar 27 '17 at 21:37

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.