Principles of Corporate Finance_ 12th Edition

Chapter 19 Financing and Valuation 505

bre44380_ch19_491-524.indd 505 09/30/15 12:07 PM

Step 1. Sangria’s current debt ratio is D/V = .4. So

r = .06(.4) + .124(.6) = .0984, or 9.84%

Step 2. We will assume that the debt cost stays at 6% when the debt ratio is 20%. Then

rE = .0984 + (.0984 − .06)(.25) = .108, or 10.8%

Note that the debt–equity ratio is .2/.8 = .25.

Step 3. Recalculate WACC.

WACC = .06(1 − .35)(.2) + .108(.8) = .0942, about 9.4%

Figure 19.1 enters these numbers on the plot of WACC versus the debt–equity ratio.

Unlevering and Relevering Betas

Our three-step procedure (1) unlevers and then (2) relevers the cost of equity. Some financial
managers find it convenient to (1) unlever and then (2) relever the equity beta. Given the beta
of equity at the new debt ratio, the cost of equity is determined from the capital asset pricing
model. Then WACC is recalculated.
The formula for unlevering beta was given in Section 17-2.

βA = βD(D/V) + βE(E/V)

This equation says that the beta of a firm’s assets is revealed by the beta of a portfolio of all
of the firm’s outstanding debt and equity securities. An investor who bought such a portfolio
would own the assets free and clear and absorb only business risks.
The formula for relevering beta closely resembles MM’s proposition 2, except that betas
are substituted for rates of return:

βE = βA + (βA − βD)D/E

Use this formula to recalculate βE when D/E changes.
Suppose the debt and equity betas in our example are βD = .135 and βE = 1.06.^12 The risk-
free rate is 5%, and the market risk premium is 7.0%. The cost of equity is

rE = rf + (rm − rf)βE = .05 + (.07)1.06 = .124, or 12.4%

This matches the cost of equity in our example at a 40/60 debt–equity ratio. Let’s calculate the
equity beta and cost of equity at a 20/80 ratio. The asset beta is

βA = βD(D/V) + βE(E/V) = .135(.4) + 1.06(.6) = .690

Now recalculate the equity beta and cost of equity at D/E = .2/.8 = .25:

βE = βA + (βA − βD)D/E = .690 + (.690 − .135).25 = .829

rE = rf + (rm − rf)βE = .05 + .07(.829) = .108, or 10.8%

This cost of equity gives the WACC of 9.4% calculated above and plotted in Figure 19.1.

(^12) Debt betas are generally small, and many managers simplify and assume βD = 0. Junk-debt betas can be well above zero, however.
BEYOND THE PAGE
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WACC and
changing debt
ratios