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

Cost of Equity Using CAPM and Factor Models

Updated 7 October 2026 · Fact-checked

Cost of equity is the return shareholders require. CAPM estimates it as the risk-free rate plus beta times the equity risk premium. Multifactor models add a premium for each factor exposure. In portfolio construction, factor models split returns into factor and active parts, and guide passive, active and factor-based (smart beta) choices.

Understand Cost of Equity and Portfolio Construction Applications

Cost of equity is the return investors demand for holding a company's shares. Companies use it to discount equity cash flows and to build the weighted average cost of capital. Analysts cannot observe it, so they estimate it with a model.

The CAPM says only market risk is rewarded. You take the risk-free rate, then add beta times the market risk premium. A beta of 1 earns the market premium. A beta above 1 earns more. A beta below 1 earns less.

Multifactor models say more than one source of systematic risk is rewarded. The expected return is the risk-free rate plus the sum of each factor sensitivity times its factor risk premium. Common factors are market, size, value, momentum, profitability and investment. Examples are the Fama-French and Carhart models.

In portfolio construction, factor models split a portfolio's return into the part explained by factor exposures and the part left over, called alpha. A passive portfolio tracks a market-cap benchmark and aims for beta near the benchmark. An active portfolio tries to earn alpha. A factor-based (smart beta or rules-based) portfolio tilts on purpose to rewarded factors such as value or momentum at lower cost than traditional active management.

Active return attribution compares the portfolio with its benchmark. Active return equals portfolio return minus benchmark return. A factor model shows how much of it comes from deliberate factor tilts and how much is true security selection skill.

Key formulas to remember

CAPM cost of equity
r(e) = R(f) + β × [E(R_m) − R(f)]
E(R_m) − R(f) is the equity risk premium. Use the premium, not the market return, in the bracket.
Multifactor cost of equity
r(e) = R(f) + β1 × λ1 + β2 × λ2 + … + βk × λk
Each λ is the risk premium for that factor. Each β is the sensitivity to that factor.
Fama-French style factor set
r(e) = R(f) + β_mkt × λ_mkt + β_size × λ_size + β_value × λ_value
Size is small minus big. Value is high minus low book-to-market. Carhart adds momentum. Use the size and value premiums as given. Subtract R(f) only if a total market return is given instead of the market premium.
Active return
Active return = R(p) − R(benchmark)
Can also be split into factor-driven return and residual (selection) return.
Active return by factor exposure
Active return = Σ (β_p,i − β_b,i) × F_i + residual
Differences in factor sensitivity between portfolio and benchmark, times factor returns.

How to solve Cost of Equity and Portfolio Construction Applications questions

Use this method for any cost of equity or factor application question.

  1. 1Read what is asked: required return, active return, or which approach (passive, active, factor-based) fits.
  2. 2List the inputs: risk-free rate, beta or betas, and premiums. Check each premium is a premium, not a total return.
  3. 3If the market return is given instead, subtract the risk-free rate first to get the premium.
  4. 4Multiply each sensitivity by its factor premium and add to the risk-free rate.
  5. 5For attribution, compute the difference in exposures (portfolio minus benchmark) and multiply by the factor return.
  6. 6Treat what is left after factor effects as the residual, or selection return.
  7. 7Check the size: is the answer sensible for the beta? Then eliminate the two wrong options.

Quickest way: Premium first, then add

When to use it: Numerical cost of equity questions with three close options.

  1. Write the risk-free rate.
  2. Compute each beta × premium in your head or on the calculator.
  3. Add them in one pass. Negative betas or premiums reduce the answer.
  4. Compare with options listed smallest to largest. When the equity risk premium is positive, a beta above 1 must give a result above the market return.
  5. If a market return is given, confirm you subtracted R(f) once, not twice.

Common mistakes in Cost of Equity and Portfolio Construction Applications

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

    The formula bracket looks like a return, and the question gives E(R_m).

    Fix: Always compute E(R_m) − R(f) first, then multiply by beta.

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

    You focus on the premium terms and stop early.

    Fix: Write R(f) first on your scratch paper, then add the premium terms.

  • Ignoring a negative factor premium or sensitivity.

    You treat all numbers as positive.

    Fix: Keep the signs. A negative sensitivity times a positive premium lowers the cost of equity.

  • Calling any factor tilt 'alpha'.

    Both show up as return above the benchmark.

    Fix: Return explained by rewarded factor exposures is not skill. Alpha is the residual after factor effects.

  • Treating smart beta as the same as traditional active management.

    Both deviate from market-cap weights.

    Fix: Factor-based strategies follow transparent rules to harvest known factor premiums, usually at lower cost. Active managers use discretionary security selection.

  • Subtracting R(f) from a factor premium that is already a premium.

    Fama-French style factor premiums such as size and value are already long-short spreads, but candidates treat them like total returns and subtract R(f) again.

    Fix: Use the size and value premiums as given. Subtract R(f) only when a total market return is given instead of the market premium.

Worked examples

Example 1

The risk-free rate is 3.0%, the expected market return is 9.0% and a stock's beta is 1.2. What is the stock's cost of equity using CAPM? Options: A. 9.6%, B. 10.2%, C. 12.0%.

Show the solution
  1. Equity risk premium = 9.0% − 3.0% = 6.0%.
  2. Beta × premium = 1.2 × 6.0% = 7.2%.
  3. Cost of equity = 3.0% + 7.2% = 10.2%.
  4. Check: when the equity risk premium is positive, a beta above 1 gives a result above the market return. Here the premium is 6.0%, so 10.2% is above the 9.0% market return, which fits.

Answer: B. 10.2%

Example 2

A three-factor model has a risk-free rate of 2.0%. Factor premiums are market 5.0%, size 2.0% and value 3.0%. A stock has sensitivities of 1.1 to market, 0.5 to size and −0.4 to value. What is its cost of equity? Options: A. 6.3%, B. 7.3%, C. 8.5%.

Show the solution
  1. Market term = 1.1 × 5.0% = 5.5%.
  2. Size term = 0.5 × 2.0% = 1.0%.
  3. Value term = −0.4 × 3.0% = −1.2%.
  4. Sum of factor terms = 5.5% + 1.0% − 1.2% = 5.3%.
  5. Add the risk-free rate: 2.0% + 5.3% = 7.3%.

Answer: B. 7.3%

Exam tips

  • Questions on cost of equity are usually short calculations. Do them fast and use the saved time elsewhere.
  • Watch whether the question gives the market return or the market premium. This is the most common trap.
  • For concept questions, remember that factor-explained return is not alpha, and smart beta is rules-based.
  • Eliminate options that ignore the sign of a negative factor sensitivity.
  • Active return questions often ask you to compare exposures of portfolio and benchmark, so keep the order portfolio minus benchmark.

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

Cost of Equity and Portfolio Construction Applications in other exams

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

Cost of Equity and Portfolio Construction Applications: frequently asked questions

How do you calculate cost of equity using CAPM?

Subtract the risk-free rate from the expected market return to get the equity risk premium. Multiply it by beta and add the risk-free rate. The result is the required return on the stock.

What is smart beta or factor investing?

It is a rules-based approach that tilts a portfolio toward factors such as value, size or momentum that have been linked to higher returns. It sits between passive indexing and discretionary active management. It is usually cheaper than active management.

How do factor models help separate active and passive returns?

They explain a portfolio's return using its factor exposures. Return from those exposures can be got through low-cost factor strategies. What remains unexplained is the residual, or alpha, which reflects selection skill or luck.

Why use a multifactor model instead of CAPM for cost of equity?

CAPM uses only market risk. Multifactor models allow other systematic risks to be priced, which may describe returns better. They need more inputs, so estimates can be less stable.