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

Fama-French and Carhart Factor Models Explained

Updated 7 October 2026 · Fact-checked

The Fama-French three-factor model explains a stock's expected return using market excess return, size (SMB) and value (HML) factors. Carhart adds momentum (WML). To solve questions, multiply each factor's sensitivity by its factor premium, add them to the risk-free rate, and interpret signs of the loadings.

Understand Fama-French and Carhart Factor Models

CAPM says one thing drives expected return: sensitivity to the market. Researchers found that this left a lot unexplained. Small-cap stocks and stocks with low price relative to book value (high book-to-market) earned more than CAPM predicted. Factor models add extra sources of systematic risk to fix this.

The Fama-French three-factor model has three factors. The first is the market factor, the market return minus the risk-free rate. The second is SMB (small minus big): the return of small-cap stocks minus the return of large-cap stocks. The third is HML (high minus low): the return of high book-to-market (value) stocks minus low book-to-market (growth) stocks.

The Fama-French five-factor model adds two more. RMW (robust minus weak) is the return of firms with robust profitability minus firms with weak profitability. CMA (conservative minus aggressive) is the return of firms that invest conservatively minus firms that invest aggressively. The idea is that profitable, low-investment firms have earned higher returns.

The Carhart four-factor model takes the three Fama-French factors and adds momentum, often called WML (winners minus losers) or UMD (up minus down). It is the return of recent winners minus recent losers, typically measured over the past several months. Momentum is a pure return pattern, not a company characteristic.

Each factor has a sensitivity (loading, or beta) for a stock. A positive SMB loading means the stock behaves like small caps. A positive HML loading means it behaves like value stocks. A negative loading means it behaves like the opposite group, such as large caps or growth stocks. Factor values are premiums, so they can be negative in some periods.

Key formulas to remember

Fama-French three-factor model
E(Ri) − Rf = βmkt × [E(Rm) − Rf] + βSMB × E(SMB) + βHML × E(HML)
Add Rf to the right side to get expected return. Factor premiums are the expected returns of the long-short portfolios.
Fama-French five-factor model
E(Ri) − Rf = βmkt × [E(Rm) − Rf] + βSMB × E(SMB) + βHML × E(HML) + βRMW × E(RMW) + βCMA × E(CMA)
RMW is profitability (robust minus weak). CMA is investment (conservative minus aggressive).
Carhart four-factor model
E(Ri) − Rf = βmkt × [E(Rm) − Rf] + βSMB × E(SMB) + βHML × E(HML) + βWML × E(WML)
WML is winners minus losers, the momentum factor.
Factor definitions
SMB = small-cap return − large-cap return; HML = high B/M return − low B/M return
High book-to-market means value stocks; low means growth stocks.
CAPM as a special case
E(Ri) = Rf + βi × [E(Rm) − Rf]
Single factor. Multifactor models reduce to this if all other loadings are zero.

How to solve Fama-French and Carhart Factor Models questions

Use this approach for any question on Fama-French, five-factor or Carhart models.

  1. 1Identify which model the question uses: three-factor, five-factor or Carhart. Note every factor given.
  2. 2Write down the risk-free rate and each factor premium. Check whether the market figure is a total return or an excess return.
  3. 3If the market premium is not given directly, compute market return minus the risk-free rate.
  4. 4Multiply each sensitivity by its matching factor premium. Keep signs; negative loadings reduce expected return.
  5. 5Add all the products to get the expected excess return, then add the risk-free rate for expected return.
  6. 6For interpretation questions, read signs: positive SMB means small-cap tilt, positive HML means value tilt, positive WML means momentum tilt.
  7. 7Check the answer is sensible, then pick the option that matches. With three options, drop any that ignore the risk-free rate or a factor.

Quickest way: Premium-times-loading shortcut

When to use it: Use for any expected return calculation with two or more factors, where you have about 90 seconds.

  1. Write the list of loading × premium products in one line, converting percentages carefully.
  2. Add them with the risk-free rate in a single pass on your BA II Plus: enter each product and press + as you go.
  3. Eliminate options that equal the CAPM answer if extra factors clearly matter, or that omit Rf.
  4. For concept questions, remember: S = size, H = value, W = momentum, R = profitability, C = investment.

Common mistakes in Fama-French and Carhart Factor Models

  • Forgetting to add the risk-free rate at the end.

    The model is written with excess return on the left, so the sum feels like the final answer.

    Fix: Ask whether the question wants expected return or excess return. If expected return, add Rf.

  • Treating HML as a high-growth factor.

    The word 'high' sounds like strong performance.

    Fix: High means high book-to-market, which is value. A positive HML loading signals a value tilt; negative signals growth.

  • Using the market return instead of market return minus Rf.

    Students copy the CAPM habit of reading the market figure directly.

    Fix: The market factor premium is Rm − Rf. Subtract Rf first if the question gives only Rm.

  • Thinking SMB and HML are company characteristics, not returns.

    Size and book-to-market are firm attributes, so the factor names get confused with them.

    Fix: The factors are long-short portfolio returns. Loadings measure how much the stock moves with them.

  • Assuming momentum is part of the Fama-French three-factor model.

    The factors are often listed together in one table.

    Fix: Momentum was added by Carhart. The three-factor model has market, SMB and HML only.

  • Ignoring signs of negative loadings or negative premiums.

    Rushing and multiplying absolute values.

    Fix: Keep the sign on each product. A negative loading times a positive premium lowers expected return.

Worked examples

Example 1

A stock has loadings of 1.10 on the market factor, 0.40 on SMB and −0.30 on HML. The risk-free rate is 3.0%, the market risk premium is 5.0%, E(SMB) is 2.0% and E(HML) is 4.0%. Using the Fama-French three-factor model, what is the expected return? (A) 7.1% (B) 8.1% (C) 9.1%

Show the solution
  1. Market term: 1.10 × 5.0% = 5.5%.
  2. SMB term: 0.40 × 2.0% = 0.8%.
  3. HML term: −0.30 × 4.0% = −1.2%.
  4. Expected excess return = 5.5% + 0.8% − 1.2% = 5.1%.
  5. Expected return = 3.0% + 5.1% = 8.1%.

Answer: B. Expected return is 8.1%.

Example 2

A portfolio has a Carhart regression with a positive SMB loading, a negative HML loading and a positive WML loading. Which description fits best? (A) Large-cap value stocks with negative momentum (B) Small-cap growth stocks with recent winners (C) Small-cap value stocks with recent losers

Show the solution
  1. Positive SMB loading: the portfolio moves with small caps, so it tilts small.
  2. Negative HML loading: it moves opposite to value, so it tilts growth.
  3. Positive WML loading: it moves with recent winners, so it has positive momentum.
  4. Combine: small-cap, growth, recent winners. Option A is wrong on all three characteristics (large cap, value, negative momentum). Option C has small-cap right but is wrong on value and losers.

Answer: B. Small-cap growth stocks with recent winners.

Exam tips

  • Memorize the factor letters: SMB size, HML value, RMW profitability, CMA investment, WML momentum.
  • Expected return questions are plug-in calculations. Check the units and whether the premium is excess return.
  • Interpretation questions test signs of loadings. Translate each sign into a style tilt before reading the options.
  • Know the contrast with CAPM: CAPM has one factor; these models add factors that capture size, value, profitability, investment and momentum.
  • If options differ by exactly the risk-free rate, the question is testing whether you added Rf.

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

Fama-French and Carhart Factor 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 Carhart Factor Models: frequently asked questions

What is the difference between CAPM and the Fama-French model?

CAPM uses one factor, sensitivity to the market. Fama-French adds size (SMB) and value (HML) factors, so expected return depends on three sources of systematic risk. This helps explain return patterns CAPM could not.

What do SMB and HML mean?

SMB is small minus big: the return of small-cap stocks minus large-cap stocks. HML is high minus low: the return of high book-to-market (value) stocks minus low book-to-market (growth) stocks.

What does the Carhart model add?

Carhart adds a momentum factor to the three Fama-French factors. Momentum is the return of recent winners minus recent losers. It gives a four-factor model.

What are RMW and CMA in the five-factor model?

RMW is robust minus weak profitability, the return of highly profitable firms minus weakly profitable firms. CMA is conservative minus aggressive investment, the return of firms that invest little minus those that invest heavily.