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?
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$\begingroup$ It is not wrong. The two methods are equivalent. As explained below. $\endgroup$– nbbo2Apr 30, 2020 at 20:42
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$\beta_i = \frac{\text{cov} \left(X_i, M\right)}{\text{var}\left(M\right)}$. Linearity of beta is a consequence of the linearity of covariance.
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1$\begingroup$ $\text{Cov}(\lambda X_1 +(1-\lambda) X_2, M) = \lambda \text{Cov}(X_1, M) + (1-\lambda) \text{Cov}(X_2, M)$ $\endgroup$– nbbo2Apr 30, 2020 at 20:36