I'm reading The Capital Asset Pricing Model: Theory and Evidence (Fama and French, 2004) and came across the following statement:

"A risky asset’s return is uncorrelated with the market return—its beta is zero—when the average of the asset’s covariances with the returns on other assets just offsets the variance of the asset’s return. Such a risky asset is riskless in the market portfolio in the sense that it contributes nothing to the variance of the market return."

To express this statement algebraically would it be sufficient to show that if $Cov(R_i, R_M) = 0$, then the variance of a portfolio made up of risky asset i and the market portfolio would be:

$$\sigma_p^2 = \sigma_i^2 + \sum_{t=1}^{n} x_{i}Cov(R_i, R_M) = \sigma_i^2 + \sum_{i=1}^{n}\sum_{j=1}^{n} x_{i}x_{j}Cov(R_i, R_j) $$, where $x_i$ is the weight of asset i in the market portfolio?


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