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Set out the CAPM together with its underlying assumptions, and show how the model deals with factor risk.

FRM® Part II · Risk Management and Investment Management · Factor Theory · FRM2.IM.1.b

The idea

In the CAPM, rf is the risk-free rate, Rm is the expected return on the market portfolio and β is the asset's beta. The expected return on asset i, written ER, is the risk-free rate plus the market risk premium scaled by beta: ER = rf + β × (Rm − rf). Beta is the covariance of the asset with the market divided by the market variance. With ρ the correlation with the market, σi the asset's volatility and σm the market's volatility, the same beta can be written as follows: Beta = ρ × σi ÷ σm. There is a single priced factor, the return on the market portfolio, and the only reward is the market risk premium times the asset's beta. The model assumes mean-variance investors, identical expectations, one period, unlimited borrowing and lending at one risk-free rate, no taxes or transaction costs, and price-taking investors trading every asset. Total variance splits into β² × market variance (systematic, rewarded) and residual variance (idiosyncratic, unrewarded). Typical mistakes are applying beta to the market return instead of the premium, and using σi ÷ σm as beta without the correlation.

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 rewarded, so a volatile asset with a low beta is expected to earn little.

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

The ratio of volatilities alone is not beta: it must be scaled by the correlation, or a volatile but unrelated asset gets a large beta.

Total variance split into systematic and idiosyncratic parts

total variance = β² × σm² + σε²

β
the asset's beta to the market
σm
the volatility of the market's return
σε
the idiosyncratic (residual) volatility

Holds for a single-factor model with residuals uncorrelated with the market. Square the beta and the market volatility before adding; only the first term earns a premium under the CAPM.

A practice question

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

In the CAPM, the only risk factor whose exposure earns a risk premium is:

  1. firm-specific earnings surprisesWhy not: Firm-specific risk is diversifiable and unpriced in the CAPM.
  2. the size of the firmWhy not: Size is an added factor in multifactor models, not in the one-factor CAPM.
  3. the slope of the government yield curve, a macroeconomic term spreadWhy not: Term spreads are macro factors used in multifactor models; the CAPM prices only the market.
  4. the return on the market portfolio - correct

The CAPM is a one-factor model: expected excess return equals beta times the market risk premium. Nothing else is priced.

A harder one

The risk-free rate is 3% and the market risk premium is 6%. A stock has volatility of 30% and a correlation of 0.6 with the market, whose volatility is 20%. The CAPM expected return on the stock is closest to:

  1. 6.6%Why not: Uses the correlation 0.6 as beta instead of correlation x sigma_i / sigma_m.
  2. 8.4% - correct
  3. 11.1%Why not: Applies beta to the market's expected return of 9% instead of to the 6% premium.
  4. 12.0%Why not: Uses sigma_i / sigma_m = 1.5 as beta and omits the correlation.

Beta = 0.6 x 30 / 20 = 0.9. Expected return = 3% + 0.9 x 6% = 8.4%.

The hardest tier

A stock has total volatility of 25% and a CAPM beta of 0.8. The market portfolio has volatility of 20%. Its idiosyncratic volatility is closest to:

  1. 9.0%Why not: Subtracts volatilities (25% - 0.8 x 20%) instead of subtracting variances.
  2. 17.5%Why not: Removes beta x market variance instead of beta² x market variance.
  3. 15.0%Why not: Removes the whole market variance (beta taken as 1) before taking the square root.
  4. 19.2% - correct

Total variance 0.0625 = beta² x 0.04 + residual = 0.0256 + residual, so residual variance = 0.0369 and idiosyncratic volatility = 19.2%.

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