Roll’s Critique and the Limits of Testing the CAPM
Summary
The document introduces Roll’s critique of empirical CAPM tests. It explains that the CAPM equation follows mathematically when a portfolio is mean-variance efficient, so testing the equation with a chosen market proxy amounts to testing that proxy’s efficiency. At the same time, the true market portfolio would include a broad range of assets beyond publicly traded securities, making it unobservable and preventing a direct test of its efficiency.
The author asks whether empirical CAPM tests remain informative and whether a similar limitation applies to multifactor models. The literature notes summarized in the document suggest that some studies examine how results change across benchmark portfolios and how closely a proxy must track the actual market portfolio for rejection to carry over. The author observes that these points address proxy choice but leave the tautology issue unresolved. The text presents questions and selected references rather than a settled answer, so it does not establish that empirical tests should be abandoned or specify what conclusions researchers can safely draw.
Key ideas
- Under Roll’s critique, the CAPM equation is equivalent to mean-variance efficiency when applied to the relevant portfolio.
- The true market portfolio is unobservable because it would include many kinds of investments.
- Empirical CAPM tests therefore rely on a proxy and assess its properties.
- The document cites research on benchmark sensitivity and proxy correlation as responses to the unobservable-market problem.
- It leaves open what empirical tests establish and whether analogous issues affect multifactor models.
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Full text
# Testing asset pricing models with Roll's critique in mind
# Testing asset pricing models with Roll's critique in mind
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. (And here is a question about how to interpret this, exactly.)
- The market portfolio is unobservable: The market portfolio in practice would necessarily include every single possible available asset, including real estate, precious metals, stamp collections, jewelry, and anything with any worth. The returns on all possible investments opportunities are unobservable. From statement 1, validity of the CAPM is equivalent to the market being mean-variance efficient with respect to all investment opportunities. Without observing all investment opportunities, it is not possible to test whether this portfolio, or indeed any portfolio, is mean-variance efficient. Consequently, it is not possible to test the CAPM.
The critique sounds quite devastating, thus my questions:
- Given Roll's critique, should we drop empirical tests of the CAPM? If not, what can be concluded from an empirical test of the CAPM?
- Does the same or analogous critique apply to multifactor models?
Here is what I was able to find in the literature:
- Cochrane "Asset Pricing" (revised edition, 2005) and his lecture series on YouTube – nothing.
- Bodie, Kane and Marcus "Investments" (12th edition, 2021) present Roll's critique and then just state the following: Given the impossibility of testing the CAPM directly, we can retreat to testing the APT.... I take this as giving up on testing the CAPM because of the critique. But researchers have continued testing asset pricing models (including the CAPM, I think) also after 1977, so there must be something to that...
- Campbell, Lo & MacKinlay "The Econometrics of Financial Markets" (1996) mention on p. 214-215 that (i) Stambaugh (1982) found the results to be insensitive to alternative versions of the benchmark portfolio and (ii) Kandel and Stambaugh (1987) and Shanken (1987a) find that as long as the correlation between the actual market portfolio and the proxy is above 0.70, rejection based on the proxy implies rejection based on the actual market portfolio. These points address the second point of Roll's critique (unobservability) but not the first one (tautology).Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.