FRM Exam Part I · The Arbitrage Pricing Theory and Multifactor Models of Risk and Return

Multifactor Models and Factor Betas for FRM Part I

Updated 11 October 2026 · Fact-checked

A multifactor model says an asset's return equals its expected return plus the sum of factor betas times unexpected factor movements, plus firm-specific noise. To find expected return, add the risk-free rate to each beta times its factor risk premium. Betas measure sensitivity to each systematic factor.

Understand Multifactor Models and Factor Betas

A single-factor model explains an asset's return with one source of systematic risk, usually the market. Real assets react to more than that. Inflation, interest rates, industrial production and oil prices also move returns. A multifactor model adds these as separate factors.

The model splits return into three parts. The first is the expected return. The second is the effect of surprises in each factor. The third is firm-specific risk, a random term that is unrelated to the factors and has an expected value of zero.

A factor beta (also called factor loading or sensitivity) tells you how much the asset's return changes for a one-unit surprise in that factor, holding the other factors constant. A beta of 1.5 on a factor means a 1% surprise in that factor moves the return by about 1.5%. Betas can be positive or negative.

Factors are defined as surprises, meaning deviations from what was expected. Only unexpected movements change returns, because expected movements are already in the expected return. Systematic risk comes from the factors. Firm-specific risk can be diversified away in a large portfolio.

Under the APT, expected return depends on the betas and the factor risk premiums. A factor with a positive premium pays you for bearing its risk. This is why the multifactor model is the base for APT and for models such as Fama-French.

Key formulas to remember

Multifactor return model: R = E(R) + β1·F1 + β2·F2 + … + βk·Fk + e; F are surprises in each factor (zero mean). e is firm-specific return with zero mean, uncorrelated with the factors.
Expected return (excess return form): E(R) = Rf + β1·λ1 + β2·λ2 + … + βk·λk; λ is the risk premium on factor portfolio j, meaning expected return on a portfolio with beta 1 on factor j and 0 on the others, minus Rf.
Single-factor model: R = E(R) + β·F + e; The market model is a special case when F is the market surprise.
Portfolio beta: βp,j = Σ wi · βi,j; Portfolio factor beta is the weighted average of asset betas. Weights sum to 1.
Systematic variance (uncorrelated factors): σ² = β1²σ1² + β2²σ2² + … + σe²; Add covariance terms 2·β1·β2·Cov(F1,F2) if factors are correlated.

How to solve Multifactor Models and Factor Betas questions

Use this order for any question on multifactor returns or betas.

1Identify what you are asked for: expected return, actual return, a beta, or a portfolio figure.
2List the risk-free rate, each factor beta and each factor risk premium. Check that the units match (all in % or all in decimals).
3For expected return, compute Rf + Σ βj·λj. Do not include the firm-specific term.
4For actual return, add the surprises: E(R) + Σ βj·Fj + e. Use the surprise, not the factor's realised level.
5Check whether the premium is given as an excess return or as a total factor return. Subtract Rf if needed.
6For a portfolio, take weighted averages of the betas first, then apply the formula once.
7Sanity check: sign of each beta times the sign of the premium should match the direction of the result.

Quickest way: Beta times premium, then sum

When to use it: Use for expected return questions where betas and premiums are given directly.

Write Rf first.
Multiply each beta by its premium in one line each.
Add everything in one pass, keeping signs.
Compare with the answer choices. Wrong options often omit Rf or use the wrong sign for a negative beta.

Common mistakes in Multifactor Models and Factor Betas

Leaving out the risk-free rate in expected return
Students remember the sum of beta times premium and stop there.
Fix: Always start from Rf unless the question asks for the excess return.
Using the factor level instead of the surprise
The return equation uses F, and students plug in the realised value of GDP or inflation.
Fix: Subtract the expected value first. Only the unexpected part enters the return equation.
Treating the firm-specific term as part of expected return
The term e appears in the equation, so it looks like it matters for the forecast.
Fix: e has an expected value of zero. It affects realised return only, and it is diversifiable.
Ignoring a negative beta
Students drop the sign when multiplying.
Fix: Keep the sign. A negative beta on a positive-premium factor lowers expected return.
Confusing single-factor beta with a multifactor beta
In a multifactor model each beta is measured holding the other factors fixed, so it can differ from a one-factor regression beta.
Fix: Read each beta as a partial sensitivity. Do not substitute a market beta unless the model has only the market factor.

Worked examples

Example 1

A stock has factor betas of 1.2 on the market, 0.5 on a size factor and −0.3 on an inflation factor. The risk-free rate is 3%. Factor risk premiums are 6%, 2% and 4% respectively. Find the expected return.

Show the solution

Formula: E(R) = Rf + β1λ1 + β2λ2 + β3λ3.
Market: 1.2 × 6% = 7.2%.
Size: 0.5 × 2% = 1.0%.
Inflation: −0.3 × 4% = −1.2%.
Sum of premiums: 7.2 + 1.0 − 1.2 = 7.0%.
Add Rf: 3% + 7.0% = 10.0%.

Answer: 10.0%

Example 2

A stock has an expected return of 8%. Its betas are 1.5 on GDP growth and 0.8 on interest rate changes. GDP growth is 1% above expectation and interest rates are 0.5% below expectation. Firm-specific return is +0.4%. Find the actual return.

Show the solution

Formula: R = E(R) + β1F1 + β2F2 + e.
GDP effect: 1.5 × 1% = 1.5%.
Interest rate effect: 0.8 × (−0.5%) = −0.4%.
Firm-specific: +0.4%.
Add: 8% + 1.5% − 0.4% + 0.4% = 9.5%.

Answer: 9.5%

Exam tips

Check whether the question gives factor risk premiums or total factor returns. If total, subtract Rf first.
When a portfolio is described, average the betas by weight before using the formula. It is faster than computing each asset's return.
Questions often test the idea that only systematic factors are priced. Firm-specific risk earns no premium in a well-diversified portfolio.
Watch for a single-factor versus multifactor comparison: the multifactor model captures more sources of systematic risk but needs more estimates.

Practice questions from The Arbitrage Pricing Theory and Multifactor Models of Risk and Return

Multifactor Models and Factor Betas 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 and Factor Betas: frequently asked questions

What is a factor beta in a multifactor model?

It is the sensitivity of an asset's return to a surprise in one systematic factor, holding other factors constant. A beta of 2 means a 1% factor surprise moves the return by about 2%.

What is the difference between a single-factor and a multifactor model?

A single-factor model uses one systematic source of risk, such as the market. A multifactor model uses several, such as the market, inflation and interest rates. This usually explains returns better, but it needs more betas and premiums to be estimated.

How do you calculate expected return with a multifactor model?

Start with the risk-free rate. Multiply each factor beta by its factor risk premium and add the results. Do not include the firm-specific term, because its expected value is zero.

Why are factors defined as surprises?

Expected movements in a factor are already reflected in the expected return. Only unexpected movements change realised return, so the model uses deviations from expectation with mean zero.