Calculate what an asset is expected to return by applying the CAPM.

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

The idea

To apply the CAPM, take the risk-free rate and add beta times the market risk premium: expected return = Rf + βi × [E(Rm) − Rf]. The premium is the expected market return minus the risk-free rate, not the market return itself. A beta of 1 gives the market return, and a beta of 0 gives the risk-free rate however volatile the asset is. The result is a required return: compare it with a forecast return, and the forecast minus the required return is alpha. A positive alpha means the asset looks cheap, a negative one expensive. The same required return can be used as a discount rate. A fair price is the expected payoff in one year, price plus dividend, divided by 1 plus the CAPM return. The classic slips are using E(Rm) in place of the premium, forgetting to add Rf, and mixing percentages with decimals.

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

The result is a required return. Forecast minus required return is alpha, and the same rate can discount a one-year payoff to a fair price.

Fair price from the CAPM

P0 = (P1 + D1) ÷ (1 + k)

P1: the expected price in one year
D1: the expected dividend over the year
k: the CAPM required return

A higher price than P0 means an expected return below the SML, so the stock looks expensive.

A practice question

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

The risk-free rate is 2%, the expected market return is 8% and a stock has a beta of 1.5. The CAPM expected return on the stock is:

11.0% - correct
14.0%Why not: Multiplies beta by the market return instead of the 6% premium.
12.0%Why not: Leaves out the risk-free rate and uses the market return rather than the premium.
9.0%Why not: Uses the premium correctly but forgets to add the risk-free rate.

E(R) = Rf + beta x [E(Rm) - Rf] = 2% + 1.5 x 6% = 11.0%.

A harder one

A stock has a beta of 1.3. The risk-free rate is 4% and the expected market return is 9%. An analyst forecasts a 10.0% return on the stock. What is the stock's required return and what does it imply?

Required return 10.5%, alpha +0.5%: underpricedWhy not: Subtracts in the wrong order; alpha is forecast minus required return.
Required return 15.7%, alpha -5.7%: overpricedWhy not: Multiplies beta by the market return instead of the 5% premium.
Required return 10.5%, alpha -0.5%: overpriced - correct
Required return 6.5%, alpha +3.5%: underpricedWhy not: Omits the risk-free rate, which inflates alpha and gives a false buy signal.

Required return = 4% + 1.3 x 5% = 10.5%. The forecast of 10.0% is below it, so alpha = -0.5% and the stock plots below the SML: it offers too little for its risk.

The hardest tier

A stock is expected to pay a dividend of 2.0 and to be worth 54.0 in one year. Its beta is 1.5, the risk-free rate is 3% and the market risk premium is 6%. The price consistent with the CAPM today is closest to:

50.00 - correct
51.38Why not: Discounts at the expected market return of 9% instead of the stock's CAPM return of 12%.
48.21Why not: Discounts the future price at 12% but leaves out the dividend.
54.37Why not: Discounts at the risk-free rate, ignoring the risk premium for beta.

Required return = 3% + 1.5 x 6% = 12%. Price = (54.0 + 2.0) / 1.12 = 50.00. A higher price would give an expected return below the SML.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

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