CFA Level II Exam · Using Multifactor Models

Arbitrage Pricing Theory (APT) for CFA Level II

Updated 7 October 2026 · Fact-checked

Arbitrage pricing theory says an asset's expected return equals the risk-free rate plus the sum of each factor's sensitivity (beta) times that factor's risk premium. It rests on a no-arbitrage argument. To solve a question, read the betas and premiums from the vignette, multiply, add, and compare the result with the stated return.

Understand Arbitrage Pricing Theory (APT)

Arbitrage pricing theory (APT) is a model of asset pricing built on one idea: two portfolios with the same risk exposures must offer the same expected return. If they did not, you could buy the cheaper one, sell the dearer one, and earn a riskless profit with no net investment. Competition removes such opportunities.

APT starts from a return-generating process. Actual return on an asset is driven by a few systematic factors plus a unique part. For one factor: Ri = E(Ri) + βi,1 F1 + εi. Here F1 is the surprise in the factor (its actual value minus its expected value), βi,1 is the asset's sensitivity to that surprise, and εi is the asset-specific error with an expected value of zero. With several factors you add one term per factor.

The main assumptions are these:
- A factor model describes returns.
- Investors can form well-diversified portfolios in which asset-specific risk is diversified away.
- No arbitrage opportunities exist among well-diversified portfolios.

APT does not require the market portfolio to be mean-variance efficient. It does not need a single market factor, and it does not say which factors to use. That is the main difference from CAPM, which is a single-factor model where the only factor is the market portfolio and which does depend on investors holding mean-variance efficient portfolios.

From the no-arbitrage argument comes the pricing relationship. Expected return is the risk-free rate plus a sum of factor risk premiums, each multiplied by the asset's beta to that factor. A factor risk premium is the extra expected return for bearing one unit of exposure to that factor. If an asset's expected return is above or below the APT value, it is mispriced, and arbitrageurs trade until the gap closes.

Key formulas to remember

Single-factor return-generating equation: Ri = E(Ri) + βi,1 F1 + εi; F1 is the factor surprise (actual minus expected). εi is the asset-specific return with mean zero.
Multifactor return-generating equation: Ri = E(Ri) + βi,1 F1 + βi,2 F2 + ... + βi,k Fk + εi; One surprise term per factor. Betas are sensitivities to each factor.
APT expected return (multifactor): E(Rp) = RF + λ1 βp,1 + λ2 βp,2 + ... + λk βp,k; λj is the risk premium for factor j. It is the expected return on a portfolio with beta 1 to factor j and 0 to all others, minus RF.
APT expected return (single factor): E(Rp) = RF + λ1 βp,1; Same form as CAPM when the factor is the market and λ1 = E(RM) − RF.
Portfolio beta: βp,k = Σ wi βi,k; Weights times asset betas, taken factor by factor. It is a weighted average.
Mispricing check: Alpha = Expected (forecast) return − APT required return; Positive means underpriced (buy). Negative means overpriced (sell or short).

How to solve Arbitrage Pricing Theory (APT) questions

Use this order for most APT questions in an item set. It works for required return, mispricing and arbitrage-portfolio questions.

1Identify the factors in the exhibit and note the risk-free rate. Check whether the numbers are factor risk premiums or total expected factor returns.
2If the exhibit gives expected factor returns rather than premiums, subtract RF to get each premium.
3List the asset's or portfolio's beta to each factor. For a portfolio, compute the weighted average beta for each factor.
4Compute required return = RF + Σ (beta × premium). Keep each factor term visible so you can check signs.
5Compare required return with the forecast or stated expected return. The difference is the mispricing.
6If the question asks about arbitrage, build a portfolio with matching factor betas, long the underpriced and short the overpriced, and compute the riskless gain.
7If the question is about a return outcome (not expected return), add beta × surprise for each factor to the expected return, not beta × the full factor value.

Quickest way: Beta-times-premium table

When to use it: Use when the vignette gives two or three factors and asks for expected return or for the better-priced asset.

Write RF on the first line, then one line per factor as beta × premium.
Do the arithmetic in decimals or percent consistently.
Add the lines. Your total is the required return.
Subtract it from the given expected return to see alpha, and read the sign to pick the answer.
Before choosing, check that you used premiums (not factor returns) and that a negative beta has a negative sign.

Common mistakes in Arbitrage Pricing Theory (APT)

Using the full factor return instead of the factor risk premium.
Exhibits sometimes list expected factor returns, and students multiply beta by that number directly.
Fix: Read the exhibit label. If it is a return that includes the risk-free rate, subtract RF first. If it says premium, use it as is.
Saying APT requires the market portfolio to be efficient.
Students blend APT with CAPM.
Fix: Remember that CAPM needs the market portfolio to be mean-variance efficient. APT needs only a factor model and no arbitrage among well-diversified portfolios.
Adding beta × factor value instead of beta × factor surprise in the return-generating equation.
The equation looks like a regression on the factor level.
Fix: In Ri = E(Ri) + βF + ε, F is the unexpected change. If the factor came in as expected, its contribution is zero.
Averaging betas without weights for a portfolio.
Time pressure leads to a simple average.
Fix: Multiply each asset's beta by its portfolio weight, then add. Do this separately for each factor.
Treating a mispriced asset as a guaranteed profit for any investor.
The arbitrage idea sounds riskless.
Fix: The profit is riskless only under the model's assumptions: the arbitrage portfolio has zero net investment and zero factor exposure, and asset-specific risk is fully diversified away. In practice some residual risk remains, so say this in conceptual answers.
Buying the overpriced asset because its beta is high.
Students confuse high risk with high return.
Fix: Decide by comparing forecast return with required return. Alpha above zero means underpriced and a buy.

Worked examples

Example 1

A vignette gives a risk-free rate of 3.0%. A two-factor APT model has a GDP-growth factor premium of 4.0% and an inflation-surprise factor premium of −1.5%. Stock X has betas of 1.2 to GDP growth and 0.8 to inflation. Analysts forecast a return of 7.5% for Stock X. Q1: What is the APT required return for X? Q2: Is X underpriced or overpriced? Q3: What happens to the required return if the inflation beta falls to 0.2?

Show the solution

Q1: GDP term = 1.2 × 4.0% = 4.8%.
Inflation term = 0.8 × (−1.5%) = −1.2%.
Required return = 3.0% + 4.8% − 1.2% = 6.6%.
Q2: Forecast 7.5% minus required 6.6% = +0.9%. The forecast is above the required return, so X is underpriced.
Q3: New inflation term = 0.2 × (−1.5%) = −0.3%. Required return = 3.0% + 4.8% − 0.3% = 7.5%. It rises by 0.9 percentage points because the exposure to a negatively priced factor shrinks.

Answer: Q1: 6.6%. Q2: Underpriced, alpha of +0.9%. Q3: The required return rises to 7.5%.

Example 2

A single-factor return-generating model for Stock Y is Ry = E(Ry) + 1.5 F + ε. The expected return on Y is 9%. The factor, an industrial-production surprise, comes in 2.0 percentage points above expectations, and the asset-specific return is −0.5%. Q1: What is Y's actual return? Q2: What would the return be if the factor had come in exactly as expected and ε were zero? Q3: Portfolio P is 40% Stock Y with a beta of 1.5 and 60% Stock Z with a beta of 0.5. What is the portfolio beta to the factor?

Show the solution

Q1: Factor contribution = 1.5 × 2.0% = 3.0%.
Actual return = 9% + 3.0% − 0.5% = 11.5%.
Q2: With a zero surprise and zero ε, the return equals the expected return, 9%.
Q3: Portfolio beta = 0.40 × 1.5 + 0.60 × 0.5 = 0.60 + 0.30 = 0.90.

Answer: Q1: 11.5%. Q2: 9%. Q3: 0.90.

Exam tips

Read each exhibit label carefully. Premium versus expected factor return decides whether you subtract the risk-free rate.
For conceptual questions, link each assumption to its role: factor model, diversification removing specific risk, and no arbitrage producing the pricing line.
Keep the sign of each premium. A negative premium paired with a positive beta lowers required return.
For compare-and-contrast items, remember APT allows many factors and does not name them, while CAPM has one factor, the market.
Check your arithmetic by recomputing the sum once. In an item set, an error on required return often carries into later questions.

Arbitrage Pricing Theory (APT) in other exams

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

Arbitrage Pricing Theory (APT): frequently asked questions

What is the difference between APT and CAPM?

CAPM uses a single factor, the market portfolio, and assumes investors hold mean-variance efficient portfolios. APT allows several factors and relies only on a factor model and no arbitrage among well-diversified portfolios. APT does not tell you which factors to use.

How do you calculate expected return using APT?

Add the risk-free rate to the sum of each factor beta times its risk premium. For example, E(R) = RF + β1λ1 + β2λ2. If the exhibit gives factor returns, subtract RF to get premiums first.

What is an arbitrage opportunity in APT?

It exists when an asset's expected return differs from the APT required return for its factor exposures. You buy the underpriced asset and sell a matching overpriced position with equal factor betas and no net investment. Trading pushes prices until the gap disappears.

Does APT need the market portfolio?

No. APT does not require a market portfolio or that it be efficient. The factors can be macroeconomic, fundamental or statistical, and the model only needs returns to follow the factor structure.