Roll’s Critique: CAPM Tests and Mean-Variance Efficiency

Summary

The document raises a conceptual question about Roll’s critique of the CAPM: if a portfolio is mean-variance efficient, its returns satisfy a CAPM-style relation, and testing that relation for a market proxy is equivalent to testing the proxy’s efficiency. This means the CAPM relation follows mathematically once efficiency is assumed, rather than providing an independent test of that assumption.

The author considers whether this makes the CAPM tautological, since market efficiency and the pricing equation are equivalent under the stated setup. However, the document is a question rather than a resolved explanation. It does not give a derivation, empirical evidence, or guidance on how to test competing asset-pricing models. Its useful contribution is to identify the distinction between assuming market efficiency and using a proxy portfolio in a test, while leaving that distinction open for further study.

Key ideas

• A mean-variance efficient portfolio satisfies a CAPM-style expected return relation as a mathematical consequence.
• Testing the pricing relation for a market proxy is equivalent to testing that proxy’s mean-variance efficiency.
• The critique questions whether assuming efficiency makes the CAPM relation tautological.
• The document poses the issue but does not provide a resolution or empirical test.

Tags

# Understanding mean-variance tautology from Roll's critique


# Understanding mean-variance tautology from Roll's critique












One of the points of Roll's critique (Roll, 1977) can be summarized as follows (quoting Wikipedia):

- Mean-variance tautology: Any mean-variance efficient portfolio $R_{p}$ satisfies the CAPM equation exactly: $$ E(R_{i})-R_{f}=\beta_{{ip}}[E(R_{p})-R_{f}]. $$ Mean-variance efficiency of the market portfolio is equivalent to the CAPM equation holding. This statement is a mathematical fact, requiring no model assumptions. Given a proxy for the market portfolio, testing the CAPM equation is equivalent to testing mean-variance efficiency of the portfolio. The CAPM is tautological if the market is assumed to be mean-variance efficient.

Initially I understood this point as follows:

- Statement 1: The CAPM holds.

- Statement 2: The market portfolio is mean-variance efficient.

- On the one hand, we assume Statement 2 to derive Statement 1. But it turns out the two statements are equivalent. Thus, the CAPM is empty in the sense that it does not add anything to the assumption needed to derive it. Hence the tautology. Then there is nothing to test.

Later, I started doubting whether I have interpreted this right. Does the CAPM really assume that the market is mean-variance efficient? If not, how do I understand the last sentence of Roll's point (in bold)?

Related: "Testing asset pricing models with Roll's critique in mind", "Roll Critique - CAPM and mean variance tautology?"

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