mathematics

Roll's critique

Roll's critique is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Roll's critique rather than just read about it. In short: Roll's critique is a famous analysis of the validity of empirical tests of the capital asset pricing model (CAPM) by Richard Roll. It concerns methods to formally test the statement of the CAPM, the equation E ( R i ) = R f + β i m [ E ( R m ) − R f ] . {\displaystyle E(R_{i})=R_{f}+\beta _{im}[E(R_{m})-R_{f}].\,} This equation relates an asset's expected return E ( R i ) {\displaystyle E(R_{i})} to the asset's sens…

Key takeaways

Roll's critique belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Roll's critique to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Roll's critique from memory before moving on to harder problems.

Reference excerpt

E ( R i ) = R f + β i m [ E ( R m ) − R f ] . {\displaystyle E(R_{i})=R_{f}+\beta _{im}[E(R_{m})-R_{f}].\,}

This equation relates an asset's expected return E ( R i ) {\displaystyle E(R_{i})} to the asset's sensitivity β i m {\displaystyle \beta _{im}} to the market portfolio return R m {\displaystyle R_{m}} . The market return is defined as the wealth-weighted sum of all investment returns in the economy. Roll's critique makes two statements regarding the market portfolio: 1. Mean-variance tautology: Any mean-variance efficient portfolio R p {\displaystyle R_{p}} satisfies the CAPM equation exactly:

E ( R i ) = R f + β i p [ E ( R p ) − R f ] {\displaystyle E(R_{i})=R_{f}+\beta _{ip}[E(R_{p})-R_{f}]\,} . (A portfolio is mean-variance efficient if there is no portfolio that has a higher return and lower risk than those for the efficient portfolio.) 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. 2. 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.

Relationship to the APT The mean-variance tautology argument applies to the arbitrage pricing theory and all asset-pricing models of the form

E ( R i ) = α + β 1 F 1 + . . . + β N F N . {\displaystyle E(R_{i})=\alpha +\beta _{1}F_{1}+...+\beta _{N}F_{N}.\,}

where F 1 , . . . , F N {\displaystyle F_{1},...,F_{N}\,} are unspecified factors. If the factors are returns on a mean-variance portfolio, the equation holds exactly. It is always possible to identify in-sample mean-variance efficient portfolios within a dataset of returns. Consequently, it is also always possible to construct in-sample asset pricing models that exactly satisfy the above pricing equation. This is an example of data dredging.

Discussion Roll's critique has received a large number of citations in the financial economics literature, with tens of citations per year as of 2017–2019. The majority of these citations refer to the second statement of critique; few papers address the first statement. Many researchers and practitioners interpret Roll's critique as stating only that "the market portfolio is unobservable".

References

Roll, Richard (March 1977), "A critique of the asset pricing theory's tests Part I: On past and potential testability of the theory", Journal of Financial Economics, 4 (2): 129–176, doi:10.1016/0304-405X(77)90009-5

Worked examples

Example 1 — a first encounter with Roll's critique

Start with the simplest possible case. Write down what Roll's critique claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Roll's critique before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Roll's critique ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Roll's critique

In research: Roll's critique appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Roll's critique in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Roll's critique is common in secondary-school and first-year university syllabi. It links to neighbouring topics Financial economics, Mathematical finance, so understanding it makes those chapters shorter.
In everyday life: Look for Roll's critique outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Roll's critique in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Roll's critique means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Roll's critique out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Roll's critique in simple terms?

Roll's critique is a famous analysis of the validity of empirical tests of the capital asset pricing model (CAPM) by Richard Roll. It concerns methods to formally test the statement of the CAPM, the equation E ( R i ) = R f + β i m [ E ( R m ) − R f ] . {\displaystyle E(R_{i})=R_{f}+\beta _{im}[E(R…

Why does Roll's critique matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Roll's critique?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Roll's critique.