# 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.)?

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.

• A simple idea that comes into my mind is that the $\beta$ factors can only be additive given the returns are orthogonal, or - in other words - uncorrelated. Easiest example: You invest into two assets that shield exactly the same payoff, and therefore exhibit correlation $1$. Clearly the $\beta$ of this portfolio is not twice the $\beta$ of a portfolio consisting of only one asset but exactly the same! – muffin1974 Dec 1 '15 at 9:50
• And weighted by market cap $w_i$ that's exactly what you get. – phdstudent Dec 1 '15 at 10:05

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.

• can you please elaborate further? Originally, that's what I thought was the case, i.e. in the formula that volcompt provided, we assume that cov(rj,ei) = 0. Is this true or not? Also, I don't think that method 2 is a multivariate regression since the only regressor is still the risk factor. – user18489 Dec 1 '15 at 17:46