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CFA Level I Exam · The Capital Asset Pricing Model, Market Model, and Other Factor-Based Equity Models

Arbitrage Pricing Theory and Multifactor Models

Updated 7 October 2026 · Fact-checked

Arbitrage pricing theory (APT) says an asset's expected return equals the risk-free rate plus the sum of each factor's sensitivity multiplied by that factor's risk premium. To solve a question, list the factors, multiply each beta by its premium, add the results to the risk-free rate, and read the answer.

Understand Multifactor Models: Arbitrage Pricing Theory

CAPM uses one factor: the market. It says only market beta is priced. Real returns move with more than the market. Interest rates, inflation, GDP growth and credit spreads all push stock prices. A multifactor model lets you capture several of these sources of risk at once.

Arbitrage pricing theory (APT) is a multifactor equilibrium model. It rests on three assumptions: (1) a factor model describes asset returns, (2) there are enough assets to diversify away asset-specific risk, and (3) no arbitrage opportunities exist among well-diversified portfolios. If a mispriced portfolio offered a riskless profit, investors would trade until the mispricing vanished. That is why expected returns must line up with factor exposures.

Each asset has a factor sensitivity (also called factor beta or factor loading) for each factor. It tells you how much the asset's return changes for a one-unit change in that factor, holding the others constant. Each factor also has a risk premium: the extra expected return for bearing one unit of exposure to that factor. Because the premium is for exposure beyond the risk-free rate, a factor with a sensitivity of zero earns no premium.

A macroeconomic factor model is one type of multifactor model. Its factors are surprises in economic variables, such as unexpected changes in GDP growth or inflation. The return equation is: actual return = expected return + sum of (sensitivity × factor surprise) + asset-specific error. Only surprises matter, because expected movements are already in the expected return.

APT versus CAPM: CAPM is a single-factor model that specifies the factor (the market portfolio). APT is more general. It does not say which factors or how many, so you must supply them. APT needs only well-diversified portfolios and no arbitrage. CAPM assumes mean-variance investors and a market portfolio that all investors hold.

Key formulas to remember

APT expected return
E(Rᵢ) = Rf + λ₁βᵢ₁ + λ₂βᵢ₂ + … + λₖβᵢₖ
λ is the risk premium for each factor. β is the asset's sensitivity to that factor. If a question gives the expected return on a factor portfolio rather than a premium, subtract Rf to get λ, then compute E(R) = Rf + Σβλ, adding Rf only once.
Macroeconomic factor model return
Rᵢ = E(Rᵢ) + βᵢ₁F₁ + βᵢ₂F₂ + … + βᵢₖFₖ + εᵢ
F is the surprise in each factor (actual minus expected). ε is the asset-specific return, with mean zero.
Single-factor special case
E(Rᵢ) = Rf + βᵢ × λ
With the market as the only factor this has the same form as CAPM.
Portfolio factor sensitivity
β_p,k = Σ wᵢ × βᵢ,k
A portfolio's sensitivity to a factor is the weighted average of the asset sensitivities.

How to solve Multifactor Models: Arbitrage Pricing Theory questions

Use this method for any question on APT or a multifactor expected return.

  1. 1Identify the model type: expected return (APT) or return including surprises (macroeconomic factor model).
  2. 2Write down the risk-free rate and each factor's risk premium. Check whether premiums are given directly or as factor returns minus Rf.
  3. 3List the asset's sensitivity to each factor, in the same order as the premiums.
  4. 4For an expected return, compute each term: sensitivity × premium. Keep signs, since sensitivities can be negative.
  5. 5Add all terms to the risk-free rate.
  6. 6For a return with surprises, use the surprise (actual minus expected) for each factor, not the actual factor value, and add any asset-specific term.
  7. 7Convert percentages to decimals only if needed, and check that your answer is sensible against the options.

Quickest way: Term-by-term sum

When to use it: Use it for numerical expected-return questions with two or three factors.

  1. Start with Rf as the running total.
  2. Add beta × premium for factor 1, then factor 2, then factor 3, entering in percent to avoid decimal errors.
  3. Match your total to the closest option. Options run from smallest to largest, so a sign slip often lands you on a neighbouring option.
  4. The BA II Plus defaults to chain calculation mode, so it works strictly left to right and does not apply × before +. Typing 1.2 × 4 + 0.5 × 2 + 3 = gives 13.6, not the correct 8.8. To avoid this, compute each product separately: 1.2 × 4 = 4.8 and 0.5 × 2 = 1. Then add 4.8 + 1 + 3 = 8.8. Or switch to AOS mode through 2ND FORMAT, so the calculator does × before +.

Common mistakes in Multifactor Models: Arbitrage Pricing Theory

  • Using the factor's actual value instead of its surprise in a macroeconomic factor model

    The wording lists a GDP growth rate and it looks like the input.

    Fix: Subtract the expected value first. Only the unexpected part moves the return away from the expected return.

  • Forgetting to add the risk-free rate

    Students stop after summing beta × premium.

    Fix: The sum of factor terms is the risk premium. Expected return = Rf + that sum.

  • Ignoring the sign of a negative sensitivity

    Negative betas feel odd, so the sign gets dropped.

    Fix: Multiply with the sign. A negative beta times a positive premium lowers the expected return.

  • Saying APT tells you which factors to use

    Students confuse it with CAPM, which names the market factor.

    Fix: APT does not identify the factors or their number. The analyst chooses them.

  • Treating the asset-specific error as part of expected return

    The error term sits in the same equation.

    Fix: Its mean is zero, so leave it out of expected return. Include it only when a question gives an asset-specific return.

Worked examples

Example 1

A stock has sensitivities of 1.2 to a market factor, 0.8 to an inflation factor and -0.5 to an interest-rate factor. The risk-free rate is 3%. The factor risk premiums are 5%, 2% and 1.5% respectively. Using APT, what is the expected return? A) 7.85% B) 9.85% C) 11.85%

Show the solution
  1. Market term: 1.2 × 5% = 6.00%.
  2. Inflation term: 0.8 × 2% = 1.60%.
  3. Interest-rate term: -0.5 × 1.5% = -0.75%.
  4. Sum the factor terms: 6.00 + 1.60 - 0.75 = 6.85%.
  5. Add the risk-free rate: 3% + 6.85% = 9.85%.

Answer: B) 9.85%

Example 2

In a macroeconomic factor model, a stock's expected return is 8%. Its sensitivity to GDP growth is 1.5 and to inflation is -0.4. GDP growth was expected at 2.0% but came in at 3.0%. Inflation was expected at 2.5% but came in at 1.5%. The asset-specific return is 0.5%. What is the stock's return? A) 9.5% B) 10.4% C) 11.4%

Show the solution
  1. GDP surprise = 3.0% - 2.0% = +1.0%. Contribution: 1.5 × 1.0% = 1.5%.
  2. Inflation surprise = 1.5% - 2.5% = -1.0%. Contribution: -0.4 × (-1.0%) = +0.4%.
  3. Add to the expected return: 8% + 1.5% + 0.4% = 9.9%.
  4. Add the asset-specific return: 9.9% + 0.5% = 10.4%.

Answer: B) 10.4%

Exam tips

  • Read whether the question gives surprises or actual values. The distractors are often built from the actual values.
  • If the question gives the expected return on a factor portfolio rather than a premium, subtract Rf to get the premium λ, then compute E(R) = Rf + Σβλ. Add Rf only once.
  • For concept questions, remember that APT assumes no arbitrage among well-diversified portfolios and does not name its factors. CAPM names one factor, the market.
  • With no penalty for wrong answers, always answer. Estimate roughly first and eliminate any option that gets the sign of the adjustment wrong.

Practice questions from The Capital Asset Pricing Model, Market Model, and Other Factor-Based Equity Models

Multifactor Models: Arbitrage Pricing Theory in other exams

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

Multifactor Models: Arbitrage Pricing Theory: frequently asked questions

What is the difference between APT and CAPM?

CAPM is a single-factor model that prices only market beta. APT allows several factors but does not say which ones. APT relies on no arbitrage among well-diversified portfolios, while CAPM relies on assumptions about investor behaviour and the market portfolio.

How do you calculate expected return with a multifactor model?

Multiply the asset's sensitivity to each factor by that factor's risk premium, add the results, then add the risk-free rate. Keep the signs of negative sensitivities.

What is a macroeconomic factor model?

It is a multifactor model whose factors are surprises in macroeconomic variables such as GDP growth, inflation or interest rates. A stock's return equals its expected return plus sensitivity times surprise for each factor, plus an asset-specific return.

What are factor sensitivities in APT?

They are the betas that show how much an asset's return changes for a one-unit move in a factor, holding other factors constant. Each asset has one sensitivity per factor, and each can be positive or negative.