Why Portfolio Beta Equals the Weighted Sum of Asset Betas

Summary

The document asks why two estimates of portfolio beta can differ: one sums individual asset betas weighted by portfolio weights, while the other estimates beta from the portfolio return series. It explains that beta is covariance with the market return divided by market variance. Since covariance is linear in portfolio returns, the portfolio beta equals the weighted sum when both methods use the same asset returns, weights, market benchmark, and estimation sample.

The discussion also distinguishes estimating beta against a market benchmark from running a multivariate regression of portfolio returns on the constituent asset returns. Those are different regression setups and need not produce the same quantity. The document gives an algebraic identity rather than empirical evidence, and does not explore practical sources of discrepancy such as changing weights, missing observations, or differing data treatments. Its key lesson is to compare like-for-like definitions and inputs.

Key ideas

• Beta is covariance with the market return divided by the market return's variance.
• Covariance linearity implies that portfolio beta is the weighted sum of asset betas when the same inputs and weights are used.
• Estimating beta from portfolio returns should match the weighted-beta calculation under consistent definitions.
• A multivariate regression on constituent returns is a different procedure from estimating market beta.

Tags

# Is portfolio beta additive under all return distributions?


# Is portfolio beta additive under all return distributions?












If beta is additive i.e. ${\beta}_P =\sum w_i \beta_i$, shouldn't the two methods below yield the same number?

Method 1: Estimate beta for each asset in the portfolio. Then ${\beta}_P =\sum w_i \beta_i$

Method 2: Estimate portfolio returns $r_P =\sum w_i r_i$. Then estimate beta.

The two results though close are not identical. Why is that? Is there an implicit assumption wrt the error terms in the regressions (i.e. uncorrelated, zero mean etc.)?

## Answer by phdstudent (score 3, accepted)

https://quant.stackexchange.com/a/22004

Mathematically they must be the same:

$\frac{Cov(Portfolio_{returns},r^m)}{Var(r^m)} = \frac{Cov(\sum w_i r_i,r^m)}{{Var(r^m)}} = \frac{\sum w_i Cov(r_i,r_m)}{Var(r^m)} = \sum w_i \beta_i = Portfolio_{beta}$

This is just math and has nothing to do with finance. They must yield the same.

## Answer by Richi Wa (score 1)

https://quant.stackexchange.com/a/22036

In your method 2: if you say that you regress the portfolio return $r = \sum w_i r_i$ on the asset returns $r_i$ then you do multivariate regression and all covariances between the assets will be incorporated in the solution (the vector $\beta$).

Using method 1 then you first calculate univariate regressions and weigt them - this is something different.

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