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FRM Part I · FRM Exam Part I

The Arbitrage Pricing Theory and Multifactor Models of Risk and Return

The Arbitrage Pricing Theory (APT) says an asset's expected return is the risk-free rate plus the sum of each factor beta times that factor's risk premium. Multifactor models supply the factors. To solve questions, identify the betas, multiply by the premiums, add the risk-free rate, then compare with the quoted return to find alpha.

What this chapter covers

This chapter extends the single-factor CAPM to models where several systematic risk sources drive returns. You learn how to write a multifactor model, read factor betas, and use the Arbitrage Pricing Theory (APT) to turn those betas into an expected return. You also see why a well-diversified portfolio carries only factor risk, and how the APT differs from the CAPM in its assumptions.

The second half moves to applied models. The Fama-French three-factor model adds size and value factors to the market factor, and extensions add factors such as momentum. You then use factor exposures to hedge a portfolio and to measure alpha, the return left after factor exposure is accounted for.

This chapter builds directly on the CAPM and on regression and portfolio theory from the quantitative material. It also feeds into performance attribution, risk decomposition and the valuation and risk models topic. Most questions are short calculations, so the chapter rewards clean arithmetic and clear definitions.

Factor models appear in several parts of the paper, so the ideas pay off beyond this chapter. Questions are usually quick to solve once you know the formula: compute an expected return, find a hedge position, or identify alpha. They also test concepts, such as what APT assumes and how it differs from the CAPM. With 100 questions in 4 hours, these are marks you can collect fast if your method is automatic. GARP does not publish a pass mark, so you cannot afford to leave easy calculation questions to chance.

The Arbitrage Pricing Theory and Multifactor Models of Risk and Return: topics in the order to study them

  1. 1Multifactor Models and Factor BetasEverything else uses the factor equation and the meaning of a beta, so start here.
  2. 2Arbitrage Pricing Theory (APT)APT turns betas and factor premiums into an expected-return formula and rests on the no-arbitrage argument.
  3. 3Well-Diversified Portfolios and Risk PremiumsIt explains why only factor risk is priced, which is the logic behind the APT.
  4. 4APT versus CAPMYou can compare the two models only after you know each one's assumptions and formula.
  5. 5Fama-French Three-Factor Model and ExtensionsIt is the best-known applied multifactor model, so it makes most sense once the theory is clear.
  6. 6Hedging with Factor Exposures and AlphaIt combines betas, premiums and portfolio weights, so it works as the final application and revision of the whole chapter.

How to prepare The Arbitrage Pricing Theory and Multifactor Models of Risk and Return

Treat this chapter as one formula applied in several ways. Master the formula first, then practise variations.

  1. Write the multifactor equation from memory: R = a + β1F1 + β2F2 + … + e, and learn what each symbol means.
  2. Learn the APT expected-return formula: E(R) = Rf + β1λ1 + β2λ2 + …, where each λ is a factor risk premium. Check whether a question gives premiums or full factor returns.
  3. Work five or six calculations on expected return, then compute alpha as actual or forecast return minus the model return.
  4. Practise hedging: to neutralise a factor exposure, take an offsetting position sized by beta times portfolio value. Solve one problem for each factor, then for two factors together.
  5. Make a short table comparing APT and CAPM on assumptions, number of factors and role of the market portfolio.
  6. Finish with timed mixed questions, about one to two minutes each, and review every error by its cause: formula, arithmetic or concept.

Common mistakes in The Arbitrage Pricing Theory and Multifactor Models of Risk and Return

  • Forgetting to add the risk-free rate in the APT formula.

    Fix: Write E(R) = Rf + Σ βλ first, then fill in the numbers. Check whether the question gives premiums or total factor returns.

  • Using a factor's total return where the formula needs its risk premium.

    Fix: Check the units. If the factor figure is already an excess return or premium, do not subtract Rf again. Otherwise subtract it first.

  • Claiming APT requires the market portfolio or that it names the factors.

    Fix: Remember that APT assumes no arbitrage and a factor structure, and leaves the factor choice to the modeller.

  • Treating alpha as total return or as the residual risk.

    Fix: Define alpha as return in excess of the model-implied return given factor exposures, then compute it by subtraction.

  • Getting the sign wrong in a hedge.

    Fix: Ask what position gives a beta of the opposite sign. For a positive beta to a factor, the hedge is a short position in that exposure.

  • Confusing the size and value factors in the Fama-French model.

    Fix: Learn SMB as small minus big and HML as high minus low book-to-market, so positive exposure means tilt toward small or value stocks.

Last-day revision: The Arbitrage Pricing Theory and Multifactor Models of Risk and Return

  • A multifactor model splits return into factor-driven parts and a firm-specific residual.
  • A factor beta measures the sensitivity of an asset's return to a one-unit move in that factor.
  • APT: E(R) = Rf + Σ βi × λi, where λi is the risk premium for factor i.
  • APT rests on no-arbitrage: two assets with the same factor exposures must offer the same expected return.
  • In a well-diversified portfolio, firm-specific risk is diversified away, so only factor risk is priced.
  • The CAPM is a single-factor special case, with the market as the factor.
  • The APT does not name its factors, while the CAPM specifies the market portfolio.
  • Fama-French three factors: market, size (small minus big) and value (high minus low book-to-market).
  • Momentum is a common extension, adding a winners-minus-losers factor.
  • Alpha = actual return minus the return expected from factor exposures.
  • To hedge a factor, take a position that offsets the portfolio's beta to it, sized by beta times portfolio value.
  • Portfolio factor beta is the weighted average of the individual asset betas.

The Arbitrage Pricing Theory and Multifactor Models of Risk and Return practice questions

The Arbitrage Pricing Theory and Multifactor Models of Risk and Return in other exams

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

The Arbitrage Pricing Theory and Multifactor Models of Risk and Return: frequently asked questions

What is the difference between APT and CAPM?

The CAPM uses one factor, the market portfolio, and derives expected return from market beta. The APT allows several factors and relies on a no-arbitrage argument without specifying the factors. The CAPM can be viewed as a single-factor special case.

How do I calculate expected return under APT?

Add the risk-free rate to the sum of each factor beta multiplied by its factor risk premium. For example, with Rf of 3%, beta 1.2 on a 5% premium and beta 0.5 on a 2% premium, the expected return is 3% + 6% + 1% = 10%.

What are the three Fama-French factors?

They are the market factor, size (SMB, small minus big) and value (HML, high minus low book-to-market). Extensions add factors such as momentum.

Is this chapter calculation-heavy?

Mostly, yes, but the calculations are short. Expect expected-return, alpha and hedge-sizing problems alongside a few conceptual questions, so practise both.