Portfolio Beta Is the Weight-Adjusted Sum of Asset Betas

Summary

The document explains how to calculate a portfolio’s beta from the betas of its individual assets. It represents each asset’s return as an intercept, a benchmark-linked component, and a residual, then substitutes those expressions into the weighted portfolio return. Collecting the benchmark-linked terms shows that portfolio beta equals the sum of each asset beta multiplied by its portfolio weight. Long positions and short positions are included through their signed weights.

The derivation supports the claim algebraically; it does not rely on an empirical test. If a reported portfolio beta differs from this calculation, the document suggests checking whether the weights were handled correctly or whether the table itself was wrong. The result assumes returns are combined using the stated weights and that the asset and portfolio betas use the same benchmark and return observations. It does not discuss complications such as changing weights, estimation error, or mismatched beta definitions.

Key ideas

• Portfolio beta is the weighted sum of the constituent asset betas when returns are combined using portfolio weights.
• Short positions contribute through negative weights.
• The derivation follows by substituting each asset’s benchmark regression into the portfolio return equation.
• A discrepancy may arise from mishandled weights or an incorrect table.

Tags

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


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












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 by SRKX (score 2, accepted)

https://quant.stackexchange.com/a/27912

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.

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.