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

Fama-French Three-Factor Model and Carhart Extension

Updated 11 October 2026 · Fact-checked

The Fama-French three-factor model explains a stock's expected excess return using three betas: market, size (SMB) and value (HML). Expected return = Rf + βM × (RM − Rf) + βS × SMB + βH × HML. Carhart adds a momentum factor (MOM). Multiply each beta by its factor premium, add them up, then add Rf.

Understand Fama-French Three-Factor Model and Extensions

CAPM says one thing drives expected return: exposure to the market, measured by beta. Empirical research found this was not enough. Small-cap stocks and high book-to-market (value) stocks earned higher average returns than their market betas predicted. These patterns are called anomalies.

Fama and French responded by adding two factors. SMB (small minus big) is the return on a portfolio of small-cap stocks minus the return on a portfolio of large-cap stocks. HML (high minus low) is the return on high book-to-market (value) stocks minus the return on low book-to-market (growth) stocks. Both are long-short portfolios, so they cost roughly nothing to hold and show pure exposure to the characteristic.

These are factor-mimicking portfolios. Each is built from traded securities so that its return tracks a risk factor. You estimate a stock's factor betas by regressing its excess returns on the market excess return, SMB and HML. A positive SMB beta means the stock behaves like small caps. A positive HML beta means it behaves like value stocks. A negative beta means the opposite tilt.

Carhart added a fourth factor, momentum (MOM, sometimes called WML or UMD): winners over the past several months minus losers. Past winners tended to keep outperforming in the short run. The four-factor model is the Fama-French model plus a momentum beta.

The models are multifactor and fit the APT style: expected excess return is a sum of betas times factor risk premiums. The intercept, alpha, measures return not explained by the factors. If the model is right, alpha should be zero. A fund manager's alpha under a four-factor model is a tougher test than under CAPM, because size, value and momentum tilts no longer count as skill.

Key formulas to remember

Fama-French three-factor model (expected return)
E(Ri) = Rf + βi,M × [E(RM) − Rf] + βi,S × E(SMB) + βi,H × E(HML)
Rf is the risk-free rate. Each beta multiplies its own factor premium. The market premium is RM minus Rf; SMB and HML are already spreads.
Regression form
Ri − Rf = αi + βi,M (RM − Rf) + βi,S SMB + βi,H HML + εi
Use excess returns on the left. Alpha is the intercept. Under the model, expected alpha is zero.
SMB
SMB = Return(small-cap portfolio) − Return(large-cap portfolio)
Positive when small caps outperform large caps.
HML
HML = Return(high book-to-market) − Return(low book-to-market)
Positive when value outperforms growth.
Carhart four-factor model
E(Ri) = Rf + βi,M × [E(RM) − Rf] + βi,S × E(SMB) + βi,H × E(HML) + βi,MOM × E(MOM)
MOM is winners minus losers. Same method, one more term.

How to solve Fama-French Three-Factor Model and Extensions questions

Most questions ask you to compute an expected return, interpret a beta or alpha, or explain what a factor represents. Use this order.

  1. 1Identify the model: three factors (market, SMB, HML) or four (add MOM).
  2. 2List every input: risk-free rate, each beta and each factor premium. Check units (percent versus decimal).
  3. 3Check whether the market input is the total market return or the premium. If it is a total return, subtract Rf first.
  4. 4Multiply each beta by its own premium. Keep signs, since betas can be negative.
  5. 5Add the products to get the expected excess return.
  6. 6Add Rf to get the expected return, unless the question asks for excess return.
  7. 7If alpha is asked, subtract the model's expected return from the actual return: alpha = actual − expected.
  8. 8Sanity-check: a small value stock should show positive SMB and HML betas and a higher return than a plain market-beta estimate.

Quickest way: Beta × premium, then add

When to use it: Any numerical expected-return question with given betas and premiums.

  1. Write the premiums in a column: market premium, SMB, HML (and MOM).
  2. Write the betas beside them and multiply row by row.
  3. Sum the rows and add Rf.
  4. Scan the answer options: check which options are the excess return rather than the total return, and avoid the trap that omits Rf.

Common mistakes in Fama-French Three-Factor Model and Extensions

  • Using the market return instead of the market premium.

    CAPM habit mixes up RM and RM − Rf, and the question may list RM only.

    Fix: Always compute RM − Rf first. SMB, HML and MOM are already spreads, so they are not reduced by Rf.

  • Subtracting Rf from SMB, HML or MOM.

    Students treat every factor like the market factor.

    Fix: These are long-short returns. Use them as given.

  • Forgetting to add Rf back to get the expected return.

    The sum of beta × premium is an excess return, and it is easy to stop there.

    Fix: Read the question: expected return = Rf + the sum. Excess return = the sum only.

  • Reading SMB as 'small' only or HML as 'high growth'.

    The acronyms are easy to mix up.

    Fix: SMB is small minus big. HML is high book-to-market (value) minus low (growth). A positive HML beta means a value tilt.

  • Saying Fama-French is a different idea from CAPM, not an extension.

    The models are taught as rivals.

    Fix: The market factor in Fama-French is the CAPM factor. The model adds size and value factors to explain what CAPM misses.

  • Treating momentum as part of the original three-factor model.

    Momentum is often listed alongside size and value.

    Fix: Momentum comes from Carhart. The three-factor model has market, SMB and HML only.

Worked examples

Example 1

A stock has betas of 1.10 to the market, 0.50 to SMB and 0.30 to HML. The risk-free rate is 3%, the market premium is 5%, E(SMB) = 2% and E(HML) = 4%. Find its expected return under the Fama-French three-factor model.

Show the solution
  1. Market term: 1.10 × 5% = 5.5%.
  2. SMB term: 0.50 × 2% = 1.0%.
  3. HML term: 0.30 × 4% = 1.2%.
  4. Expected excess return = 5.5% + 1.0% + 1.2% = 7.7%.
  5. Expected return = 3% + 7.7% = 10.7%.

Answer: 10.7%

Example 2

Using the Carhart model, a fund has betas of 0.90 (market), −0.20 (SMB), 0.10 (HML) and 0.40 (MOM). Rf = 2%, market premium = 6%, E(SMB) = 3%, E(HML) = 4%, E(MOM) = 5%. The fund returned 8.0% over the year. What is its alpha?

Show the solution
  1. Market term: 0.90 × 6% = 5.4%.
  2. SMB term: −0.20 × 3% = −0.6%.
  3. HML term: 0.10 × 4% = 0.4%.
  4. MOM term: 0.40 × 5% = 2.0%.
  5. Sum of factor terms = 5.4% − 0.6% + 0.4% + 2.0% = 7.2%.
  6. Expected return = 2% + 7.2% = 9.2%.
  7. Alpha = actual − expected = 8.0% − 9.2% = −1.2%.

Answer: Alpha = −1.2%. The fund underperformed what its factor exposures predicted.

Exam tips

  • Write the formula first and label each input as a premium or a total return before you multiply.
  • Check whether the question asks for excess return, expected return or alpha. Options often include all three.
  • Expect conceptual items: the meaning of SMB and HML, why factor-mimicking portfolios are long-short, and what Carhart adds.
  • Interpret signs of betas in words. A negative SMB beta means a large-cap tilt. A negative HML beta means a growth tilt.
  • Remember that alpha under a multifactor model is stricter than CAPM alpha, because factor tilts are not skill.

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

Fama-French Three-Factor Model and Extensions in other exams

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

Fama-French Three-Factor Model and Extensions: frequently asked questions

What is the difference between Fama-French and CAPM?

CAPM uses one factor, the market, to explain expected returns. Fama-French adds size (SMB) and value (HML) factors to capture return patterns CAPM does not explain. The market factor is still there.

What do SMB and HML stand for?

SMB is small minus big: the return on small-cap stocks minus large-cap stocks. HML is high minus low: the return on high book-to-market (value) stocks minus low book-to-market (growth) stocks. Both are long-short factor-mimicking portfolios.

How do I calculate expected return with the Fama-French model?

Multiply each factor beta by its expected premium, add the results, and add the risk-free rate. Use the market excess return for the market factor. SMB and HML are used as given.

What does the Carhart model add?

It adds a momentum factor: past winners minus past losers. The expected return then has four beta-times-premium terms plus Rf. Carhart alpha is often used to judge fund managers after removing size, value and momentum tilts.