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Show what follows from valuing assets with the CAPM: equilibrium and optimal holdings, factor risk exposure, how diversification gains are handled, and where the model falls short.

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

The idea

If every investor shares the CAPM assumptions, all hold the same risky portfolio, the market portfolio. They differ only in how much they lend or borrow at the risk-free rate, which is called two-fund separation. Let σm be the market's volatility and σp the volatility of the chosen mix. Choices along the capital market line then earn this expected return, written ERp: ERp = rf + [(Rm − rf) ÷ σm] × σp. Individual assets lie on the security market line. An asset's alpha is its forecast return minus the return the line requires for its beta: Alpha = forecast − [rf + β × (Rm − rf)]. Positive alpha means the asset plots above the line and is underpriced. Diversification removes idiosyncratic risk, which is therefore unrewarded, while systematic risk stays and is rewarded. The limits are Roll's critique (the true market portfolio is unobservable, so the model cannot be tested cleanly), the unrealistic assumptions, and evidence such as a flatter-than-predicted line for low-beta stocks.

Formulas

Capital market line

E(Rp) = Rf + [(E(Rm) − Rf) ÷ σm] × σp

E(Rp)
the expected return of an efficient portfolio
Rf
the risk-free rate
E(Rm)
the expected return on the market portfolio
σm
the market portfolio's volatility
σp
the efficient portfolio's volatility

Applies only to mixes of the market portfolio and the risk-free asset, where σp ÷ σm is the market weight. Individual assets are priced on the security market line by beta, not on this line.

CAPM alpha of a forecast

α = forecast return − [Rf + β × (E(Rm) − Rf)]

forecast return
the analyst's expected return for the asset
Rf
the risk-free rate
β
the asset's beta to the market
E(Rm)
the expected return on the market portfolio

Positive alpha plots above the security market line and looks underpriced. Add the risk-free rate back when the market risk premium is given rather than the market return.

A practice question

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

If the CAPM holds and markets are in equilibrium, which risky holding does every investor choose?

  1. A portfolio of the lowest-beta assets for risk-averse investors and of high-beta assets for aggressive investorsWhy not: Differences in risk tolerance are met by leverage or lending, not by changing the risky portfolio.
  2. A different tangency portfolio for each investor, reflecting personal beliefs about returnsWhy not: Homogeneous expectations make the tangency portfolio the same for everyone.
  3. The market portfolio, plus risk-free lending or borrowing - correct
  4. An equally weighted portfolio of every risky asset, held in the same proportions by all investorsWhy not: Equilibrium weights are market capitalisation weights, not equal weights.

Two-fund separation: all investors hold the market portfolio plus the risk-free asset in proportions set by risk tolerance.

A harder one

An analyst forecasts an expected return of 11% for a stock with a beta of 1.4. The risk-free rate is 2% and the market risk premium is 5%. What is the stock's CAPM alpha and what does it imply?

  1. +2.0%; the stock plots above the security market line and looks underpriced - correct
  2. -2.0%; the stock plots below the line and looks overpricedWhy not: Reverses the sign: alpha is the forecast minus the required return, 11% - 9%.
  3. +4.0%; the stock plots above the line and looks underpricedWhy not: Uses the market return of 7% as the required return and ignores the beta of 1.4.
  4. +9.0%; the stock plots above the line and looks underpricedWhy not: Subtracts only the risk-free rate (11% - 2%), which is the excess return, not the alpha.

Required return = 2% + 1.4 x 5% = 9%. Alpha = 11% - 9% = +2%. Positive alpha means the stock lies above the SML and is underpriced.

The hardest tier

A multi-decade backtest sorts stocks by beta. Realised average excess returns rise with beta but much less steeply than the security market line predicts: the lowest-beta quintile has positive alpha and the highest-beta quintile has negative alpha. Which statement is most appropriate?

  1. It confirms the CAPM, because average returns do increase with beta across the quintilesWhy not: The CAPM requires the slope to equal the market premium; a positive but flatter slope is a rejection.
  2. It shows that heterogeneous investor expectations, which the CAPM allows, are the causeWhy not: The CAPM assumes homogeneous expectations, so heterogeneity is not one of its assumptions.
  3. It proves that idiosyncratic risk is rewarded and diversification is pointlessWhy not: The result concerns beta mispricing; it does not show that diversifiable risk earns a premium.
  4. It is a known empirical shortfall: beta alone does not explain returns - correct

A flatter-than-predicted line (the low-beta anomaly) is a known failing of the model. It motivates multifactor models and explanations such as leverage constraints.

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