Estimate the return equity holders require with both the market model and the CAPM
The idea
The capital asset pricing model estimates the required return on equity as the risk-free rate plus beta times the equity risk premium: r = Rf + beta x (E(Rm) - Rf). With Rf at 3%, beta at 1.2 and an equity risk premium of 5%, the required return is r = 3% + 1.2 x 5% = 9%. Only systematic risk, measured by beta, earns a premium, because diversifiable risk can be removed. Beta is usually estimated with the market model, a regression of the stock's returns on the market's returns, Ri = alpha + beta x Rm + error, where the slope is beta. Estimates depend on the index, period and return frequency chosen, and a historical beta may not describe the future, so analysts often adjust it or use peers' betas.
Formulas
Capital asset pricing model
E(R) = R_f + β × (E(R_m) − R_f)
- E(R)
- the required return on the share
- R_f
- the risk-free rate
- β
- the share's beta
- E(R_m)
- the expected return on the market
The expected market return less the risk-free rate is the equity risk premium. Beta is the only risk that earns a reward.
Market model
R_i = α + β × R_m + ε
- R_i
- the stock's return in a period
- α
- the return not tied to the market
- β
- the slope, the stock's sensitivity to the market
- R_m
- the market's return in the same period
- ε
- the random error, company-specific news
Beta is estimated as the slope of a regression of the stock's returns on the market's.
A practice question
The risk-free rate is 3%, the expected market return is 8% and a stock's beta is 1.2. Its required return under the CAPM is:
- 9.0% - correct
- 12.6%Why not: Multiplies beta by the market return rather than by the market risk premium.
- 6.0%Why not: Gives only the risk premium (1.2 × 5%) and forgets to add the risk-free rate.
Required return = 3% + 1.2 × (8% − 3%) = 3% + 6% = 9.0%.
A harder one
A stock's returns have a covariance with the market of 0.024, the market's standard deviation is 20%, the risk-free rate is 4% and the expected market return is 10%. The stock's CAPM required return is:
- 4.72%Why not: Divides the covariance by the market's standard deviation rather than its variance, giving a beta of 0.12.
- 10.0%Why not: Multiplies beta by the market return (10%) rather than the market risk premium (6%).
- 7.6% - correct
Beta = 0.024 / 0.20² = 0.024 / 0.04 = 0.60. Required return = 4% + 0.60 × (10% − 4%) = 7.6%.
The hardest tier
A stock's correlation with the market is 0.6, its standard deviation is 30% and the market's is 20%. With a risk-free rate of 3.5% and an expected market return of 9%, the CAPM required return is closest to:
- 11.75%Why not: Takes beta as the ratio of standard deviations (1.5), ignoring the correlation.
- 5.70%Why not: Inverts the volatility ratio (0.6 × 20% / 30%), giving a beta of 0.4.
- 8.45% - correct
Beta = ρ × σ_i / σ_m = 0.6 × 30% / 20% = 0.9. Required return = 3.5% + 0.9 × 5.5% = 8.45%.
In the app: 9 practice questions on this outcome, a narrated walkthrough with diagrams, spaced review that brings it back before you forget it.
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