Estimate the return an asset is expected to earn under a multifactor model as well as under a single-factor model.

FRM® Part I · Foundations of Risk Management · The Arbitrage Pricing Theory and Multifactor Models of Risk and Return · FRM1.FR.6.c

The idea

In a single-factor model, an asset is expected to earn the risk-free rate plus its beta times the market risk premium: Expected return = Rf + β × [E(Rm) − Rf]. In a multifactor model, each factor adds its own term, a beta times that factor's risk premium: Expected return = Rf + β1 × premium1 + β2 × premium2 + … + βk × premiumk. A factor risk premium is the expected return of a portfolio with a beta of one on that factor and zero on the others, minus the risk-free rate. Be careful with signs, because a negative beta on a factor with a positive premium lowers the expected return. In a single-factor model, beta equals the correlation times the asset's volatility divided by the factor's volatility: Beta = ρ × σi ÷ σm. When factors enter as surprises, the predicted return is the expected return plus the beta-weighted surprises. The common errors are forgetting the risk-free rate, applying a beta to the full market return rather than the premium, and using a beta with the wrong factor's premium.

Formulas

Multifactor expected return

E(R) = Rf + β1 × λ1 + β2 × λ2 + … + βk × λk

Rf: the risk-free rate
β1, β2, βk: the asset's beta to each factor
λ1, λ2, λk: each factor's risk premium, the expected return of its pure factor portfolio minus Rf

Apply each beta to its own factor's premium, not to a raw factor return, and keep the sign of a negative beta. When factors enter as surprises, add the beta-weighted surprises to E(R).

Beta from correlation

β = ρ × σi ÷ σm

ρ: the correlation between the asset's return and the market's (or the factor's)
σi: the volatility of the asset's return
σm: the volatility of the market (or factor) return

For a single-factor model this gives the beta to plug into Rf + β × premium. Use the factor's volatility, not the asset's, in the denominator.

A practice question

From the question bank, with the reason each wrong option is wrong.

Under a single-factor (CAPM-type) model, the risk-free rate is 3%, the expected return on the market is 9%, and a stock's beta is 1.2. The stock's expected return is closest to:

10.8%Why not: Multiplies beta by the market return and forgets to add the risk-free rate (and to use the premium).
10.2% - correct
13.8%Why not: Applies beta to the market return instead of the market premium of 6%.
7.2%Why not: Computes only beta times the market premium and omits the risk-free rate.

E(R) = Rf + beta x (E(Rm) - Rf) = 3% + 1.2 x 6% = 10.2%. The premium over the risk-free rate is the market return minus Rf, not the raw market return.

A harder one

The risk-free rate is 3%. A stock has a beta of 1.3 to the market factor, whose risk premium is 5.5%, and a beta of -0.4 to a second factor, whose risk premium is 3.5%. Using both factors, the stock's expected return is closest to:

11.55%Why not: Treats the negative beta as positive, so it adds rather than subtracts the second-factor compensation.
10.15%Why not: This is the single-factor answer; it ignores the second factor entirely.
7.95%Why not: Applies the market premium of 5.5% to the second factor instead of that factor's own 3.5% premium.
8.75% - correct

Single-factor: 3% + 1.3 x 5.5% = 10.15%. The second factor adds -0.4 x 3.5% = -1.40%, giving 8.75%. A negative beta lowers the required return because the stock hedges that factor.

The hardest tier

The risk-free rate is 3%. Pure factor portfolio 1 has a beta of 1 to factor 1 and 0 to factor 2, with an expected return of 8%. Pure factor portfolio 2 has a beta of 0 to factor 1 and 1 to factor 2, with an expected return of 6%. An asset has betas of 0.7 to factor 1 and 1.2 to factor 2. Its expected return under the multifactor model is closest to:

12.8%Why not: Weights the raw factor-portfolio returns by the betas; the factor risk premium is the portfolio return minus the risk-free rate.
7.1%Why not: Computes the beta-weighted premiums correctly but omits the risk-free rate.
15.8%Why not: Adds the risk-free rate to beta-weighted raw returns, counting the risk-free rate twice.
10.1% - correct

Factor premiums are 8% - 3% = 5% and 6% - 3% = 3%. E(R) = 3% + 0.7 x 5% + 1.2 x 3% = 3% + 3.5% + 3.6% = 10.1%.

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