Economic Modelling · Capital Asset Pricing Model (CAPM)

Limitations, Tests and Extensions of CAPM: Roll's Critique, APT and Multi-Factor Models

Updated 11 October 2026 · Fact-checked

CAPM says expected return depends only on beta to the market portfolio. Its limitations are unrealistic assumptions, an unobservable market portfolio (Roll's critique) and weak empirical support. Extensions such as multi-factor models and Arbitrage Pricing Theory allow several risk factors, each with its own sensitivity and risk premium.

Understand Limitations, Tests and Extensions of CAPM

CAPM gives a clean result: E(Ri) = Rf + βi[E(RM) − Rf]. Only market risk is rewarded. Specific risk can be diversified away, so it earns no premium. The model is useful, but it rests on strong assumptions, so the exam asks you to criticise it.

Assumption-based limitations. CAPM assumes investors care only about mean and variance, share the same expectations, can borrow and lend at one risk-free rate, face no taxes or transaction costs, and can trade any amount of any asset. Real markets break all of these. Returns are not exactly normal, borrowing rates exceed lending rates, and investors have different horizons and views.

Empirical tests. The usual test is a two-pass method. First estimate each beta from a time-series regression of security returns on market returns. Then run a cross-section regression of average returns on those betas. CAPM predicts a positive slope equal to the market risk premium, and an intercept equal to the risk-free rate. It also predicts that nothing else, such as size or the book-to-market ratio, explains returns. Studies often find a flatter line than predicted, and other factors that do seem to matter. Betas are also estimated with error and are not stable over time.

Roll's critique. The true market portfolio contains every risky asset, including property, human capital and unlisted assets. It cannot be observed. Any test uses a proxy such as a share index. Roll argued that CAPM is therefore not testable in a pure sense. If the proxy is mean-variance efficient, the linear beta relationship holds by mathematics. If it is not, the test can reject CAPM even when CAPM is true. A test of CAPM is really a joint test of the model and of the proxy's efficiency.

Extensions. The single index model says Ri = αi + βi RM + εi, with specific risk uncorrelated across securities. It is a statistical description of returns, not an equilibrium theory like CAPM. Multi-factor models write return as a sum of factor sensitivities times factor movements plus a specific term. Arbitrage Pricing Theory (APT) assumes returns follow a factor model and that no arbitrage is possible in a well-diversified market. It then gives expected return as a linear function of factor sensitivities. APT does not require the market portfolio or mean-variance preferences. It does not say which factors to use, so they must be chosen by judgement or statistics.

Key rules to remember

CAPM: E(Ri) = Rf + βi [E(RM) − Rf]; Beta is Cov(Ri, RM) ÷ Var(RM). Only systematic risk earns a premium.
Single index model: Ri = αi + βi RM + εi; E(εi) = 0 and εi is uncorrelated with RM and with other εj. This is a return-generating model, not an equilibrium model.
Variance under single index model: Var(Ri) = βi² σM² + σεi²; Systematic part plus specific part.
Covariance under single index model: Cov(Ri, Rj) = βi βj σM²; Holds for i ≠ j. It cuts the number of inputs needed for a portfolio.
Multi-factor model: Ri = ai + bi1 F1 + bi2 F2 + … + bik Fk + εi; The bij are factor sensitivities. Factors can be macroeconomic or based on firm characteristics.
APT expected return: E(Ri) = λ0 + bi1 λ1 + bi2 λ2 + … + bik λk; λ0 is the return on a zero-sensitivity asset, often Rf. Each λj is the risk premium for factor j.
Test regression (cross-section): R̄i = γ0 + γ1 β̂i + ηi; CAPM predicts γ0 = Rf and γ1 = E(RM) − Rf. Extra variables should have zero coefficients.

How to solve Limitations, Tests and Extensions of CAPM questions

Most questions ask you to discuss, compare or calculate. This method covers all three.

1Identify the task: discuss limitations, explain a test, compare models, or calculate an expected return.
2For discussion, split your answer into three groups: assumptions, empirical evidence and the market-portfolio problem. Give one or two points under each.
3For Roll's critique, state that the market portfolio is unobservable, that tests use a proxy, and that the result is a joint test.
4For a comparison of CAPM and APT, cover the number of factors, the basis (equilibrium versus no-arbitrage), the role of the market portfolio and the assumptions about investors.
5For a calculation, write the model first. Substitute the betas or sensitivities and the premiums, and keep units consistent as percentages or decimals.
6Check that the answer is sensible. A higher beta or sensitivity to a positively priced factor should give a higher expected return.
7End with a one-line conclusion, such as when each model is useful in practice.

Quickest way: Three-column comparison under time pressure

When to use it: Use this for short 'compare' or 'discuss' questions where you have a few minutes.

Draw three mental columns: CAPM, single index model, APT.
For each, note the risk measure (market beta, market beta, several sensitivities) and the basis (equilibrium, statistical description, no-arbitrage).
Add the main weakness of each: unobservable market and unrealistic assumptions; no theory of prices; factors not specified.
Write the answer from these notes in short points, one idea per line.
For calculations, plug into E(Ri) = λ0 + Σ bij λj and check the arithmetic once.

Common mistakes in Limitations, Tests and Extensions of CAPM

Saying Roll's critique proves CAPM is false.
Students remember that tests are unreliable and jump to 'CAPM is wrong'.
Fix: Say it shows CAPM cannot be tested conclusively, because the true market portfolio is unobservable and any test is a joint test with the proxy.
Treating the single index model and CAPM as the same.
Both use beta and a market return.
Fix: CAPM is an equilibrium model for expected returns. The single index model describes how returns are generated and includes an alpha and a random specific term.
Saying APT needs the market portfolio or assumes mean-variance investors.
Students carry CAPM assumptions over to APT.
Fix: APT relies on a factor structure and no arbitrage in a well-diversified market. It needs neither the market portfolio nor mean-variance preferences.
Claiming APT tells you which factors to use.
It is seen as a more complete model.
Fix: APT is silent on the number and identity of factors. Mention this as its main practical weakness.
Mixing percentages and decimals in APT calculations.
Premiums are given as 4% while sensitivities are plain numbers.
Fix: Convert everything to one form first, then compute λ0 + Σ bij λj and state the answer as a percentage.
Listing assumptions without saying why they matter.
Students memorise a list.
Fix: For each assumption say what breaks. For example, different borrowing and lending rates mean the line is no longer straight beyond the market portfolio.

Worked examples

Example 1

A two-factor APT model has a risk-free rate of 6%, a premium of 3% for factor 1 and a premium of 2% for factor 2. Security X has sensitivities of 1.2 to factor 1 and 0.5 to factor 2. Find the expected return on X under APT.

Show the solution

Use E(R) = λ0 + b1 λ1 + b2 λ2, taking λ0 = 6%.
Factor 1 contribution: 1.2 × 3% = 3.6%.
Factor 2 contribution: 0.5 × 2% = 1.0%.
Add: 6% + 3.6% + 1.0% = 10.6%.

Answer: The expected return on X is 10.6%.

Example 2

Under a single index model, security A has β = 1.5 and specific variance 0.0100. Security B has β = 0.8 and specific variance 0.0064. The market variance is 0.0400. Find Var(A), Var(B) and Cov(A, B).

Show the solution

Var(A) = β² σM² + σε² = 1.5² × 0.0400 + 0.0100.
1.5² = 2.25, so 2.25 × 0.0400 = 0.0900. Add 0.0100 to get 0.1000.
Var(B) = 0.8² × 0.0400 + 0.0064 = 0.64 × 0.0400 + 0.0064 = 0.0256 + 0.0064 = 0.0320.
Cov(A, B) = βA βB σM² = 1.5 × 0.8 × 0.0400 = 1.2 × 0.0400 = 0.0480.

Answer: Var(A) = 0.1000, Var(B) = 0.0320 and Cov(A, B) = 0.0480.

Exam tips

For 'discuss limitations' questions, structure by theme: assumptions, empirical evidence, Roll's critique. Examiners reward organised answers.
In APT answers, always say what it does not need (market portfolio, mean-variance preferences) and what it does not provide (the list of factors).
In calculation questions, write the formula and the substitution. Method marks are given even if the arithmetic slips.
Be precise on Roll's critique: say 'joint test' and 'proxy'. Avoid saying CAPM is disproved.
Multiple-choice questions often test the contrast between CAPM, the single index model and APT, so learn the three-column comparison.

Practice questions from Capital Asset Pricing Model (CAPM)

Limitations, Tests and Extensions of CAPM in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

FRM Part ICAPM Limitations and Extensions

Limitations, Tests and Extensions of CAPM: frequently asked questions

What is Roll's critique of CAPM in simple words?

The true market portfolio includes every risky asset, so it cannot be observed. Tests use a proxy like a share index. Therefore any test of CAPM is also a test of whether the proxy is efficient, and CAPM cannot be cleanly tested.

What is the difference between CAPM and arbitrage pricing theory?

CAPM uses one factor, the market, and rests on investor behaviour and equilibrium. APT allows several factors and rests on no arbitrage with a factor structure of returns. APT does not need the market portfolio but does not name its factors.

Is the single index model the same as CAPM?

No. The single index model describes returns as a market-linked part plus a specific part. It is used to estimate variances and covariances. CAPM is an equilibrium statement about expected returns.

Why do tests of CAPM often find a flatter line than predicted?

Common explanations include errors in estimated betas, an imperfect market proxy, borrowing restrictions and the existence of other priced factors. Which explanation is right is debated, so state them as possible reasons.