CFA Program Level I

CAPM cost of equity and beta: CFA Level I explained

Under the CAPM, the required return on equity is the risk-free rate plus beta times the market risk premium. Beta is the slope from regressing the stock's excess returns on the market's excess returns. A peer's beta is unlevered to an asset beta and relevered at the target company's own debt-to-equity ratio.

The formula, worked

Required return = Rf + beta x (E[Rm] - Rf). The market risk premium is the expected market return minus the risk-free rate.

Example: risk-free rate 3.4%, expected market return 11.8%, beta 0.80. Premium = 11.8% - 3.4% = 8.4%. Required return = 3.4% + 0.80 x 8.4% = 3.4% + 6.72% = 10.12%.

A common slip is to multiply beta by the market return instead of the premium, or to forget to add the risk-free rate back.

Estimating beta

The single-factor regression is Ri - Rf = alpha + beta x (Rm - Rf) + error, run on monthly excess returns. Beta is the slope coefficient: the sensitivity of the stock's excess return to the market's excess return. Alpha is the intercept and the error term is the unexplained part.

A Federal Reserve study used up to five years of monthly returns, citing industry practice. It also showed that analyst choices matter: equally weighting its peer betas gave 0.57, while weighting by market capitalisation gave 0.82.

Unlevering and relevering

An equity beta is the asset beta plus a mark-up for leverage, adjusted for taxes: equity beta = asset beta x [1 + (1 - t) x D/E]. More leverage raises equity beta.

Example: a peer has equity beta 1.20, D/E 1.0 and a 25% tax rate. Asset beta = 1.20 / (1 + 0.75 x 1.0) = 0.686. Relevered at the target's D/E of 0.5: 0.686 x (1 + 0.75 x 0.5) = 0.943.

With a 4% risk-free rate and a 5% premium, the cost of equity is 4% + 0.943 x 5% = 8.71%. Using the peer's 1.20 directly would give 10.0%, too high for a less levered company.

Where the exam catches people

Believing beta multiplies the expected market return: it multiplies the market risk premium, E[Rm] - Rf.
Believing beta is the regression intercept: beta is the slope; the intercept is alpha.
Believing a peer's equity beta can be used unchanged for a less levered firm: unlever it and relever at the firm's own D/E, which lowers it.
Believing leverage leaves equity beta unchanged: debt reduces shareholders' residual claim and raises equity beta.

Written from CFA Institute's documentation

Try 3 questions on this

Pick an answer to see why each option is right or wrong.

An analyst estimates the required return on equity of Tasmer Utilities with the CAPM. Tasmer's beta, estimated by regressing its monthly excess returns on the market's excess returns, is 0.80. The risk-free rate is 3.40%, and the expected return on the market portfolio is 11.80%. Tasmer's required return on equity is closest to:
Right. In the game that’s Renown for your Equities building.
Not that one: B is right. In the game your Equities building would lose a little Renown, and this question would come back later.
Your answerA.6.72%6.72% is only the risk premium, 0.80 x (11.80% - 3.40%). The CAPM adds that premium to the risk-free rate: 3.40% + 6.72% = 10.12%.The answerB.10.12%Required return = risk-free rate + beta x (expected market return - risk-free rate). Market premium = 11.80% - 3.40% = 8.40%. Required return = 3.40% + 0.80 x 8.40% = 3.40% + 6.72% = 10.12%.Your answerC.12.84%12.84% = 3.40% + 0.80 x 11.80% multiplies beta by the whole market return. Beta measures sensitivity to the market's excess return, so it multiplies the premium of 11.80% - 3.40% = 8.40%, giving 10.12%.
Written from www.federalreserve.gov/boarddocs/press/other/2005/20050518/supplement.pdf
Following common industry practice, an analyst regresses five years of Corrigan Bank's monthly stock returns in excess of the one-month Treasury bill rate on the market's monthly excess returns. In this single-factor regression, Corrigan's equity beta is the:
Right. In the game that’s Renown for your Equities building.
Not that one: A is right. In the game your Equities building would lose a little Renown, and this question would come back later.
The answerA.Slope coefficientIn the single-factor CAPM regression R(i) - R(f) = alpha + beta x [R(m) - R(f)] + error, the equity beta is the slope coefficient: it measures how sensitive the stock's excess return is to the market's excess return.Your answerB.Regression interceptThe intercept is alpha, which the Federal Reserve paper describes as a mispricing term. Beta is the slope: the change in Corrigan's excess return per unit of the market's excess return.Your answerC.Error termThe error term captures the part of Corrigan's monthly excess return that the market's excess return does not explain. Beta is the coefficient on the market's excess return, the slope.
Show the full explanation and doc quotes
The intercept, alpha, is a mispricing term, and the error term is the unexplained part of the return.
Written from www.federalreserve.gov/boarddocs/press/other/2005/20050518/supplement.pdf
Two analysts estimate Brennick Bank's cost of equity with the same single-factor CAPM, the same peer banks and the same market inputs. Brennick's debt-to-equity ratio is well below the peers' median, and its tax rate matches theirs. One analyst uses the peers' average equity beta directly; the other unlevers each peer beta and relevers it at Brennick's own ratio. Compared with the first estimate, the second analyst's estimate is most likely:
Right. In the game that’s Renown for your Equities building.
Not that one: C is right. In the game your Equities building would lose a little Renown, and this question would come back later.
Your answerA.HigherMore leverage raises an equity beta; less leverage lowers it. Relevering at Brennick's below-median debt-to-equity ratio gives a smaller leverage mark-up than the peers carry, so the estimate falls.Your answerB.UnchangedUnlevering and relevering leave the beta unchanged only if the target's debt-to-equity ratio equals the peers'. Brennick's is well below theirs, so the relevered equity beta, and the cost of equity, come out lower.The answerC.LowerAn equity beta is the asset beta plus a mark-up for leverage, adjusted for taxes: equity beta = asset beta x [1 + (1 - tax rate) x D/E]. Unlevering strips out each peer's higher leverage; relevering at Brennick's lower D/E adds back a smaller mark-up.
Show the full explanation and doc quotes
The relevered beta is lower, and with the same risk-free rate and market premium, the CAPM cost of equity (risk-free rate + beta x market premium) is lower too.
Written from www.federalreserve.gov/boarddocs/press/other/2005/20050518/supplement.pdf