Calculating Asset Betas Relative to a Weighted Portfolio

Summary

The discussion explains how to calculate each asset’s regression beta relative to a portfolio formed from weighted asset returns. In a simple regression, beta is the covariance of the dependent variable with the explanatory variable divided by the explanatory variable’s variance. Here the portfolio return is the weighted sum of asset returns, so covariance linearity expresses each asset’s covariance with the portfolio as the corresponding row of the covariance matrix multiplied by the portfolio weight vector.

The denominator is the portfolio variance, calculated as the weights transposed times the covariance matrix times the weights. Thus, the vector of asset betas to the same portfolio is the covariance matrix times the weight vector, divided by portfolio variance. This gives a direct matrix calculation and clarifies why dividing by each asset’s variance would answer a different question. The result assumes the portfolio weights and covariance matrix describe the returns being analyzed; the exchange provides no data example or guidance on estimation uncertainty.

Key ideas

• A regression beta is covariance with the explanatory return divided by that return’s variance.
• An asset’s covariance with a weighted portfolio is obtained from the covariance matrix and portfolio weights.
• The common denominator for asset betas to a portfolio is the portfolio variance.
• The vector of betas is the covariance matrix multiplied by portfolio weights and scaled by inverse portfolio variance.
• Using individual asset variances in the denominator would calculate a different relationship.

Tags

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


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












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 by msitt (score 2)

https://quant.stackexchange.com/a/33280

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

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