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Outline the assumptions on which the CAPM rests.

FRM® Part I · Foundations of Risk Management · Modern Portfolio Theory (MPT) and the Capital Asset Pricing Model (CAPM) · FRM1.FR.5.c

The idea

The CAPM is an equilibrium model, and its tidy result rests on strong assumptions. Investors are rational, risk-averse mean-variance optimisers who care only about expected return and variance over a single common period. That focus is justified by normal returns or quadratic utility. They hold homogeneous expectations, so they see the same efficient frontier. They can lend and borrow unlimited amounts at one risk-free rate. Markets are frictionless: no taxes or transaction costs, assets are infinitely divisible and tradable, and everyone is a price taker with equal access to information. Together these make every investor hold the same risky portfolio. Relax one, such as a higher borrowing rate or differing expectations, and the market portfolio need not be the one tangency portfolio. On the exam, be ready to say which assumption fails in a scenario and what that does to the model.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which of the following is an assumption of the CAPM?

  1. Investors can lend and borrow any amount at the same risk-free rate - correct
  2. Investors face transaction costs and taxes that differ across assetsWhy not: The CAPM assumes frictionless markets, so costs and taxes are ignored.
  3. Investors have different forecasts of expected returns and covariancesWhy not: The CAPM assumes homogeneous expectations; differing forecasts would give different efficient frontiers.
  4. Investors choose portfolios to maximise skewness and kurtosis preferencesWhy not: Investors are assumed to care only about the mean and variance of returns.

The CAPM assumes one common risk-free rate for unlimited lending and borrowing, frictionless markets, price-taking investors, a single period, mean-variance preferences and homogeneous expectations.

A harder one

In practice an investor can borrow only at a rate above the risk-free lending rate. What does this do to the CAPM's conclusions?

  1. Nothing changes because beta alone determines the expected returnWhy not: Ignores that the derivation relies on a single tangency portfolio, which exists only with one riskless rate.
  2. The risk-free rate in the security market line is replaced by the borrowing rate for all assetsWhy not: Applying the borrowing rate to every investor misreads the problem; lenders still face the lending rate.
  3. Investors no longer all hold one risky portfolio, so the linear capital market line breaks down - correct
  4. The market portfolio becomes risk-free and betas turn to zeroWhy not: Confuses unequal rates with the removal of risk; nothing makes the market portfolio riskless.

With different borrowing and lending rates the efficient set has a kink: lenders, borrowers and unlevered investors can prefer different risky portfolios, so the market portfolio need not be the one tangency portfolio and the simple CAPM line no longer follows.

The hardest tier

A researcher regresses stock returns on a broad equity index and finds that betas do not explain average returns across stocks. According to Roll's critique, which conclusion is most defensible?

  1. The CAPM itself is not rejected, because the true market portfolio of all assets is unobservable and the index may be a poor proxy - correct
  2. The CAPM is rejected because its assumptions are known to be unrealisticWhy not: Unrealistic assumptions do not by themselves reject a model; the critique is about testability.
  3. The CAPM is confirmed because the index is mean-variance efficient by constructionWhy not: Reverses the issue: an index proxy has no guarantee of being efficient.
  4. The CAPM is rejected only for stocks, and would hold if bonds were added to the sampleWhy not: Adding a single asset class does not fix the missing human capital, real estate and other assets in the true market portfolio.

The CAPM holds exactly when the true market portfolio is mean-variance efficient. That portfolio includes every asset, such as real estate and human capital, so any test using an equity index proxy cannot cleanly reject the theory.

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