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Strategic Financial Management · Asset Pricing Theories

Fama-French Three Factor Model and Multifactor Models

Updated 11 October 2026 · Fact-checked

The Fama-French three-factor model says a stock's expected return depends on three risks: market risk, size (SMB) and value (HML). Expected return = Rf + βm(Rm − Rf) + βs(SMB) + βh(HML). To solve it, take each factor's beta times its premium, add them, and add the risk-free rate.

Understand Fama-French and Multifactor Models

CAPM says one thing explains differences in expected returns: beta against the market. Researchers found that this is not enough. Small companies and companies with high book value relative to market value earned higher average returns than CAPM predicted.

Eugene Fama and Kenneth French added two factors to the market factor. 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.

Each factor has a factor premium (the average return of that long-short portfolio) and each stock has a factor loading (its beta to that factor). A stock with a positive SMB loading behaves like a small company. A stock with a positive HML loading behaves like a value stock. A negative HML loading means it behaves like a growth stock.

This is a multifactor model. Other extensions add momentum (past winners minus past losers), profitability and investment factors, as in the five-factor version. The idea is the same: expected return = risk-free rate + sum of (loading × factor premium). APT is the general theory behind such models, but it does not name the factors. Fama-French names them.

In the exam, you will be given the risk-free rate, the factor premiums and the loadings. Your job is to compute expected return, compare it with the CAPM answer, or judge whether a stock is fairly priced.

Key rules to remember

Fama-French three-factor model
E(Ri) = Rf + βm × (Rm − Rf) + βs × SMB + βh × HML
Rm − Rf is the market risk premium. SMB and HML are the average factor premiums. βs and βh are the stock's loadings.
General multifactor model
E(Ri) = Rf + Σ βk × λk
βk is the loading on factor k and λk is the risk premium of factor k.
CAPM (single-factor case)
E(Ri) = Rf + β × (Rm − Rf)
Use this for comparison. It is the Fama-French model with the size and value loadings set to zero.
Factor definitions
SMB = R(small) − R(big); HML = R(high B/M) − R(low B/M)
Both are long-short portfolio returns. B/M is book value to market value.
Alpha (realised version)
α = Actual return − Expected return from the model
Positive alpha suggests the stock earned more than its factor risks justify.

How to solve Fama-French and Multifactor Models questions

Use this method for any question on Fama-French or another multifactor model.

  1. 1List the data given: risk-free rate, market return or market premium, SMB, HML and any extra factor premiums, and the stock's loadings.
  2. 2Check whether the market premium is given directly (Rm − Rf) or as Rm. If Rm is given, subtract Rf.
  3. 3Multiply each loading by its factor premium. Keep the sign of every loading and premium.
  4. 4Add all the products to Rf to get the expected return.
  5. 5If asked, compute the CAPM return using only the market beta and compare the two.
  6. 6Interpret the result: say whether size or value tilts add or reduce the required return.
  7. 7If a price or actual return is given, compare it with the expected return and state whether the stock is under- or overvalued, or its alpha, and give a conclusion.

Quickest way: Premium-by-loading shortcut

When to use it: Use this for 2-mark MCQs and for the first part of a numerical.

  1. Write Rf first.
  2. Write three products in a row: βm × market premium, βs × SMB, βh × HML.
  3. Add the products with their signs, then add to Rf.
  4. For a theory MCQ, remember: SMB is small minus big and HML is high B/M minus low B/M. Positive HML loading means value stock.

Common mistakes in Fama-French and Multifactor Models

  • Using Rm instead of (Rm − Rf) as the market premium.

    The question gives Rm and students multiply it by beta directly.

    Fix: Always check the wording. If the figure is a market return, subtract Rf first.

  • Reading HML as growth minus value.

    The letters H and L are confused with high price rather than high book-to-market.

    Fix: HML is High B/M (value) minus Low B/M (growth). A high loading means value-like.

  • Ignoring the sign of a negative loading.

    Students multiply positive numbers by habit.

    Fix: A negative loading times a positive premium reduces expected return. Carry the sign through.

  • Treating SMB and HML as stock returns.

    They are written like returns.

    Fix: They are premiums on long-short portfolios. The stock's sensitivity to them is its loading.

  • Saying the model proves CAPM is wrong in all cases.

    Over-simplified notes.

    Fix: Say that the added factors improved explanation of average returns in the evidence, and CAPM is the special case with one factor.

  • Confusing Fama-French with APT.

    Both are multifactor.

    Fix: APT is a general framework with unspecified factors. Fama-French fixes three specific factors.

Worked examples

Example 1

Risk-free rate is 7%. Market return is 13%. SMB premium is 3% and HML premium is 4%. A stock has market beta 1.1, size loading 0.5 and value loading −0.2. Find its expected return under Fama-French and under CAPM.

Show the solution
  1. Market premium = 13% − 7% = 6%.
  2. Market term = 1.1 × 6% = 6.6%.
  3. Size term = 0.5 × 3% = 1.5%.
  4. Value term = −0.2 × 4% = −0.8%.
  5. Fama-French return = 7% + 6.6% + 1.5% − 0.8% = 14.3%.
  6. CAPM return = 7% + 6.6% = 13.6%.

Answer: Fama-French expected return is 14.3%; CAPM gives 13.6%. The stock needs 0.7% more under Fama-French because its small-size tilt outweighs its growth tilt.

Example 2

Rf = 6%, market premium = 5%, SMB = 2%, HML = 3%. Stock A has βm = 0.9, βs = 0.8, βh = 1.2. Last year A returned 15%. Find its expected return and alpha, and comment.

Show the solution
  1. Market term = 0.9 × 5% = 4.5%.
  2. Size term = 0.8 × 2% = 1.6%.
  3. Value term = 1.2 × 3% = 3.6%.
  4. Expected return = 6% + 4.5% + 1.6% + 3.6% = 15.7%.
  5. Alpha = 15% − 15.7% = −0.7%.

Answer: Expected return is 15.7% and alpha is −0.7%. The stock earned slightly less than its market, size and value risks justified, so it underperformed on a factor-adjusted basis.

Exam tips

  • Expect a short numerical: given loadings and premiums, compute expected return. It is quick marks if you keep signs right.
  • Be ready to define SMB and HML in one line each in MCQs and short notes.
  • When asked to compare with CAPM, show both returns and state the difference with a reason.
  • For theory answers, include the link to APT and mention extensions such as momentum and the five-factor model.

Practice questions from Asset Pricing Theories

Fama-French and Multifactor Models 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 and Multifactor Models: frequently asked questions

What do SMB and HML mean in the Fama-French model?

SMB is Small Minus Big, the return of small-cap stocks less large-cap stocks. HML is High Minus Low, the return of high book-to-market (value) stocks less low book-to-market (growth) stocks. They capture the size and value premiums.

How is a multifactor model different from CAPM?

CAPM uses one factor, the market. A multifactor model uses several factors, each with its own loading and premium. CAPM is a special case of it.

Is the Fama-French model the same as APT?

No. APT is a general theory where several factors may drive returns, without naming them. Fama-French specifies market, size and value as the factors.

Is the Fama-French model asked in CMA Final?

It sits under asset pricing theories in Paper 14. Expect it as an MCQ, a short note, or a part of a numerical on expected return.