Can the net portfolio's beta be different from the sum of long and short betas of the portfolio?

Question

The way I calculate it is summing up the weighted beta of long and shorts but I saw a table where this wasn't the case so I am wondering if this is not the correct way.

Answer 1

You can actually show by construction that the beta of the portfolio is the weighted sum of all the underlyings betas.

Assume the return of the benchmark and some asset $a$ at time $t$ are respectively denoted $r_{b,t}$ and $r_{a,t}$, then the beta of a given asset is defined by:

$$r_{a,t} = \alpha_a + \beta_a r_{b,t} + \epsilon_{a,t}$$

Let's assume you have a portfolio of $n$ assets $(a_1, ..., a_i, ..., a_n)$ each with a weight $w_i$, then the return of the portfolio is at time $t$ is defined as:

$$r_{p,t} = \sum_{i=1}^n w_i r_{a_i,t}$$

Now, by expressing each asset's return in terms of their own beta, you get:

$$ \begin{align} r_{p,t} &= \sum_{i=1}^n w_i r_{a_i,t}\\ &= \sum_{i=1}^n w_i \left( \alpha_{a_i} + \beta_{a_i} r_{b,t} + \epsilon_{a_i,t} \right)\\ &= \underbrace{\sum_{i=1}^n w_i \alpha_{a_i}}_{\alpha_p} + \underbrace{\left(\sum_{i=1}^n w_i \beta_{a_i} \right)}_{\beta_p} r_{b,t} +\sum_{i=1}^n w_i \epsilon_{a_i,t} \end{align} $$

There might be something in the table that you missed (likely the weights as Alex C pointed out) or maybe it was wrong.