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CFA Level II Exam · Cost of Capital: Advanced Topics

Multifactor and Build-Up Methods for Cost of Equity

Updated 7 October 2026 · Fact-checked

Multifactor models estimate cost of equity as the risk-free rate plus several factor betas times factor premiums, such as market, size, value and momentum. The build-up method adds risk premiums to the risk-free rate without betas, and is used for private firms. Identify the inputs in the vignette, then add each term.

Understand Expected Return Models: Multifactor and Build-Up Methods

CAPM uses one factor: market risk. It says expected return depends only on beta to the market. Multifactor models say other systematic risks are also priced, so expected return is the risk-free rate plus a sum of factor sensitivities times factor premiums.

The Fama-French model adds two factors to the market premium. SMB (small minus big) is the return of small-cap stocks minus large-cap stocks. HML (high minus low) is the return of high book-to-market (value) stocks minus low book-to-market (growth) stocks. A stock with a positive HML beta behaves like a value stock and earns a premium for it. The Carhart model adds a fourth factor, WML (winners minus losers), which captures momentum. The Pastor-Stambaugh model adds a liquidity factor to the Fama-French factors (and often to momentum as well), so stocks with a high liquidity beta carry a higher required return.

A factor premium can be negative. If SMB is negative, a stock with a positive SMB beta lowers the estimate. Also, a negative factor beta lowers the estimate when the premium is positive. Keep signs straight.

Private companies have no traded shares, so you cannot regress their returns to get a beta. The build-up method starts with the risk-free rate and adds premiums: an equity risk premium, a size premium, and a company-specific premium. The build-up has no beta in its basic form. A version using beta (the expanded CAPM) multiplies beta by the equity risk premium and then adds size and company-specific premiums. Beta for the expanded CAPM is often taken from comparable public firms, adjusted for leverage differences (unlever, then relever).

The main benefit of multifactor models is a richer explanation of returns. The cost is more inputs to estimate, and the factor premiums and betas are themselves estimated with error. The build-up method is simple but the size and company-specific premiums are subjective.

Key formulas to remember

CAPM
E(Ri) = Rf + βi × [E(Rm) − Rf]
One-factor benchmark. The bracket is the market risk premium.
Fama-French model
E(Ri) = Rf + β_mkt × [E(Rm) − Rf] + β_SMB × E(SMB) + β_HML × E(HML)
E(SMB) and E(HML) are factor premiums, not returns on one stock. Use the sign of each beta and premium.
Carhart model
E(Ri) = Rf + β_mkt × MRP + β_SMB × SMB + β_HML × HML + β_WML × WML
Fama-French plus the momentum factor (winners minus losers).
Pastor-Stambaugh model
E(Ri) = Rf + β_mkt × MRP + β_SMB × SMB + β_HML × HML + β_LIQ × LIQ
Adds a liquidity factor to the Fama-French factors. Versions often also include the momentum factor, so follow the factors listed in the exhibit.
Build-up method
re = Rf + ERP + size premium + specific-company premium
No beta in the basic form. Used for private companies.
Expanded CAPM
re = Rf + β × ERP + size premium + specific-company premium
Beta is usually from comparable public firms and adjusted for leverage.

How to solve Expected Return Models: Multifactor and Build-Up Methods questions

Use this method for any multifactor or build-up cost of equity question in an item set.

  1. 1Identify the company type. Public with factor betas points to a multifactor model. Private with no beta points to build-up or expanded CAPM.
  2. 2Find the model named in the vignette or exhibit: Fama-French (3 factors), Carhart (4, with momentum), or Pastor-Stambaugh (liquidity).
  3. 3Pull each input from the exhibit: risk-free rate, each beta, and each factor premium. Check that premiums are not returns of a single stock.
  4. 4Match every beta to its own premium. Do not mix up the SMB and HML betas.
  5. 5Keep the signs. A negative beta or a negative premium reduces the result.
  6. 6Compute each term, then add them to the risk-free rate.
  7. 7If the question asks for a change, such as adding a factor, compute only the new term and add it to the old answer.
  8. 8Check the answer is reasonable and in percent. Then answer the exact question, such as which factor drives the difference.

Quickest way: Add the terms in a list

When to use it: Use when the exhibit lists betas and premiums and you need the expected return fast.

  1. Write Rf first.
  2. Under it write each beta × premium on its own line, with signs.
  3. Sum the lines in one pass on the calculator.
  4. For a build-up, write Rf, ERP, size premium, specific premium and add.
  5. For a question on which model gives a higher figure, compare only the extra terms.

Common mistakes in Expected Return Models: Multifactor and Build-Up Methods

  • Multiplying the beta by the full market return instead of the market risk premium.

    Students forget Rf is already added separately.

    Fix: Use the premium, E(Rm) − Rf, when the exhibit gives market return and Rf.

  • Ignoring the sign of SMB, HML or WML premiums.

    Students assume every premium is positive.

    Fix: Copy the sign from the exhibit and multiply carefully. A negative premium times a positive beta is a negative term.

  • Applying a beta in the basic build-up method.

    Students blend it with CAPM.

    Fix: The basic build-up adds premiums with no beta. Only the expanded CAPM multiplies beta by the ERP.

  • Confusing the factors: reading HML as growth minus value, or treating WML as size.

    Names are similar and abbreviations get mixed up.

    Fix: SMB is small minus big. HML is high book-to-market (value) minus low (growth). WML is winners minus losers (momentum).

  • Forgetting leverage differences when borrowing a comparable firm's beta for a private company.

    Students copy the beta directly.

    Fix: Unlever the comparable's beta, then relever it at the target's capital structure.

  • Claiming a multifactor model is always more accurate than CAPM.

    More factors sounds better.

    Fix: It can explain returns better, but it needs more estimated inputs, each with estimation error.

Worked examples

Example 1

Vignette: An analyst estimates the cost of equity for Orion Foods using the Fama-French model. Risk-free rate 3.0%. Market risk premium 5.0%. Expected SMB premium 2.0%. Expected HML premium 3.0%. Orion's betas: market 1.10, SMB 0.50, HML −0.20. Question 1: What is Orion's cost of equity? Question 2: How does the result change if the HML beta were +0.40 instead? Question 3: Which option best describes Orion's style from its betas: A) large-cap growth, B) small-cap growth tilt, C) large-cap value?

Show the solution
  1. Q1: Market term = 1.10 × 5.0% = 5.50%.
  2. SMB term = 0.50 × 2.0% = 1.00%.
  3. HML term = −0.20 × 3.0% = −0.60%.
  4. Sum with Rf: 3.0% + 5.50% + 1.00% − 0.60% = 8.90%.
  5. Q2: New HML term = 0.40 × 3.0% = 1.20%. Change = 1.20% − (−0.60%) = 1.80%. New cost = 8.90% + 1.80% = 10.70%.
  6. Q3: Positive SMB beta means small-cap tilt. Negative HML beta means growth tilt. So B.

Answer: Q1: 8.90%. Q2: 10.70%, up 1.80 percentage points. Q3: B, a small-cap growth tilt.

Example 2

Vignette: Kavya Textiles is a private company. An analyst uses a build-up method. Risk-free rate 4.0%. Equity risk premium 5.5%. Size premium 2.5%. Company-specific premium 1.5%. A public comparable has beta 1.20 and the analyst decides to use the expanded CAPM with that beta, keeping the same premiums. Question 1: Cost of equity by the build-up method? Question 2: Cost of equity by the expanded CAPM? Question 3: Which approach uses beta: build-up or expanded CAPM?

Show the solution
  1. Q1: re = 4.0% + 5.5% + 2.5% + 1.5% = 13.5%.
  2. Q2: Beta term = 1.20 × 5.5% = 6.6%.
  3. re = 4.0% + 6.6% + 2.5% + 1.5% = 14.6%.
  4. Q3: The basic build-up has no beta. The expanded CAPM multiplies beta by the ERP.

Answer: Q1: 13.5%. Q2: 14.6%. Q3: The expanded CAPM uses beta; the basic build-up does not.

Exam tips

  • Read the exhibit for the exact model name and whether premiums are factor premiums or stock returns.
  • Write each beta × premium on a separate line and keep signs. Most lost marks come from sign errors.
  • Know the factor labels cold: SMB size, HML value, WML momentum, liquidity for Pastor-Stambaugh.
  • For private firms expect the build-up and expanded CAPM, and watch for the unlever and relever step on comparable betas.
  • Conceptual questions may ask about a stock's style from its betas. Read the sign of each beta first.

Expected Return Models: Multifactor and Build-Up Methods in other exams

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

Expected Return Models: Multifactor and Build-Up Methods: frequently asked questions

What is the difference between CAPM and multifactor models?

CAPM says only market beta is priced. Multifactor models price several systematic risks, such as size, value, momentum or liquidity. Each factor has its own beta and premium.

What does the Carhart model add to Fama-French?

It adds a momentum factor, WML (winners minus losers). A stock with a positive WML beta tends to behave like recent winners and carries a premium for that exposure.

What is the Pastor-Stambaugh model?

It adds a liquidity factor to the Fama-French factors, and versions often also include momentum. Stocks that are more sensitive to liquidity changes have a higher liquidity beta and a higher required return.

When should I use the build-up method?

Use it for private companies with no traded shares, so no beta can be estimated directly. You add premiums to the risk-free rate, or use an expanded CAPM with a comparable's beta.