Zero-Beta Assets and Their Contribution to Market Portfolio Risk

Summary

The document asks how to express a CAPM statement about a risky asset whose return has zero covariance with the market return. The cited passage characterizes such an asset as contributing nothing to the variance of the market portfolio, while describing a covariance offset involving the asset’s relationships with other assets. The question proposes an algebraic portfolio-variance expression using asset weights and covariances.

The text contains no response validating or correcting that expression. Its proposed formula appears to mix the variance of a portfolio containing the asset and market with a double sum over assets in the market portfolio, so the intended portfolio definition and weighting need clarification before it can serve as a derivation. It gives no empirical test or further explanation of the zero-beta result.

Key ideas

• The cited CAPM passage links zero covariance with the market to zero beta.
• The passage describes the asset as adding nothing to market portfolio variance.
• The document asks whether a proposed covariance expansion captures this claim.
• No answer is supplied, and the proposed variance expression is not verified.

Tags

# Zero-beta assets and the Sharpe-Lintner CAPM


# Zero-beta assets and the Sharpe-Lintner CAPM












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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.