Skip to content
All library documents

Why Portfolio Beta Equals the Weighted Sum of Component Betas

Article Quant Q&A · Author: hartmut
Read the original

Summary

The document asks why portfolio beta is often calculated as the weighted sum of the component betas instead of being estimated directly from the portfolio’s aggregate return. It gives the definition of an asset’s beta as its covariance with the market divided by market variance. Because covariance is linear in the asset return, the beta of a weighted portfolio is the corresponding weighted sum of its component betas, assuming the weights used to form portfolio returns are applied consistently.

This explains the relationship between the two approaches: calculating beta from aggregate portfolio returns is consistent with combining individual betas by portfolio weights. The note is brief and provides no worked numerical example or discussion of changing portfolio weights, estimation windows, or differences between realized and target weights. Those practical choices can affect an empirical calculation, but they do not alter the linearity argument for fixed weights.

Key ideas

  • Beta is covariance with the market divided by market variance.
  • Covariance is linear in portfolio returns.
  • For consistent fixed weights, portfolio beta equals the weighted sum of component betas.
  • Calculating beta from aggregate portfolio returns follows the same linear relationship.

Tags

StatisticsPortfolio constructionEquities
Full text
# Beta of sum or sum of betas


# Beta of sum or sum of betas












When interested in the beta of a portfolio, I see people make a weighted sum of the portfolio components' betas. Intuitively, I would have calculated the beta of the portfolio based on its aggregate return though. Why is my approach wrong?

## Answer by siou0107 (score 2, accepted)

https://quant.stackexchange.com/a/53754

$\beta_i = \frac{\text{cov} \left(X_i, M\right)}{\text{var}\left(M\right)}$. Linearity of beta is a consequence of the linearity of covariance.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.