# 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?

• It is not wrong. The two methods are equivalent. As explained below. Apr 30 '20 at 20:42

$$\beta_i = \frac{\text{cov} \left(X_i, M\right)}{\text{var}\left(M\right)}$$. Linearity of beta is a consequence of the linearity of covariance.
• $\text{Cov}(\lambda X_1 +(1-\lambda) X_2, M) = \lambda \text{Cov}(X_1, M) + (1-\lambda) \text{Cov}(X_2, M)$ Apr 30 '20 at 20:36