Interpret how the CAPM is derived and what its components are.

FRM® Part I · Foundations of Risk Management · Modern Portfolio Theory (MPT) and the Capital Asset Pricing Model (CAPM) · FRM1.FR.5.b

The idea

The CAPM is derived by asking what happens when every investor follows Markowitz with the same forecasts and can lend and borrow at the risk-free rate. Each investor then holds the same risky portfolio, the tangency portfolio, mixed with the risk-free asset. Because every risky asset must be held by someone, that portfolio has to be the market portfolio. In equilibrium an asset's extra expected return per unit of covariance with the market equals the market's. This gives Expected return = Rf + β × (E(Rm) − Rf). Beta is defined by Beta = Cov(Ri, Rm) ÷ Var(Rm). The three components are the risk-free rate, which is the price of time, the market risk premium E(Rm) − Rf, which is the price of risk, and beta, which is the quantity of systematic risk. Only systematic risk is rewarded. The usual mistake is to expect pay for total volatility.

Formulas

CAPM expected return

E(Ri) = Rf + βi × [E(Rm) − Rf]

E(Ri): the expected (required) return on asset i
Rf: the risk-free rate
βi: the asset's beta to the market portfolio
E(Rm): the expected return on the market portfolio

Beta multiplies the market risk premium, not the market return. Only systematic risk is priced, so total volatility does not enter.

Beta from covariance

β = Cov(Ri, Rm) ÷ Var(Rm)

Cov(Ri, Rm): the covariance of the asset's return with the market's
Var(Rm): the variance of the market's return

Divide by the market's variance, not its standard deviation. The market's own beta is 1.

A practice question

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

Which expression is the CAPM relationship for the expected return on asset i?

E(Ri) = Rf + beta_i x E(Rm)Why not: Multiplies beta by the market return instead of the market risk premium.
E(Ri) = E(Rm) + beta_i x [Rf - E(Rm)]Why not: Reverses the premium and starts from the market return; the result is only right at a beta of zero or one by accident.
E(Ri) = Rf + beta_i x [E(Rm) - Rf] - correct
E(Ri) = beta_i x [E(Rm) - Rf]Why not: Omits the risk-free rate, leaving only the compensation for risk.

The CAPM sets expected return equal to the risk-free rate plus beta times the market risk premium, E(Rm) - Rf. The premium, not the market return, is multiplied by beta.

A harder one

Stock X has a total volatility of 40% and a beta of 0.6. Stock Y has a total volatility of 20% and a beta of 1.3. The risk-free rate is 3% and the market risk premium is 5%. Which statement is correct under the CAPM?

Y has the higher required return, 9.5% against 6.0% for X, because only systematic risk is priced - correct
X has the higher required return because its standard deviation is double that of YWhy not: Treats total volatility as the priced risk; the CAPM rewards only beta.
Both have the same required return because they share the same risk-free rate and premiumWhy not: Confuses a common premium with a common required return; each asset's beta scales it.
Y has the higher required return, 6.5% against 3.0% for XWhy not: Ranks them correctly but omits the risk-free rate, using beta x premium alone.

Required return = 3% + beta x 5%: X = 3% + 0.6 x 5% = 6.0%, Y = 3% + 1.3 x 5% = 9.5%. X's extra volatility is mostly diversifiable and earns no premium.

The hardest tier

A hedging asset has a beta of -0.2. The risk-free rate is 3% and the expected market return is 8%. Its CAPM expected return is closest to:

3.0%, because no asset can earn less than the risk-free rateWhy not: Wrongly assumes a floor at Rf; a negative beta lowers the required return because the asset hedges the market.
4.0%Why not: Drops the minus sign on beta and adds the premium instead of subtracting it.
2.0%, below the risk-free rate - correct
1.4%Why not: Multiplies beta by the market return of 8% instead of the 5% market risk premium.

E(R) = 3% + (-0.2) x (8% - 3%) = 3% - 1.0% = 2.0%. An asset that tends to rise when the market falls reduces portfolio risk, so investors accept less than Rf for it.

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