Cost of Capital Formulas

We have gathered together here (and in some cases repeated) the derivations of some of the formulas used in Chapter 19. We cover five topics: 1. Deriving the WACC formula. 2. Does WACC necessarily decline when leverage increases? 3. Levering and unlevering. 4. The Miles-Ezzell formula. 5. The MM formula.

Deriving WACC We start with a firm’s expected operating cash flows C1, C2, ... , CT. With all-equity financing, these flows would be discounted at the opportunity cost of capital r. But we assume the firm is financed with debt at a constant ratio L. That is, it follows Financing Rule 2, so that D/V = L in each future period. Consider firm value in the next-to-last period: VT-1 = DT-1 + ET-1. We can write the total cash payoff to debt and equity investors in two ways: Cash payoff in T = CT + TCT rD DT-1 = VT-1 (1+rD D/V + rE E/V)
TC rD D is the interest tax shield. Equate the two expressions and solve for VT-1 as a function of CT. CT + TC D = VT-1 (1 + rD D/V + rE E/V)

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The logic repeats for T-2. Note that the cash payoff at T-1 includes VT-1:

Cash payoff in T-1 = CT-1 + TC rD DT-2 + VT-1 = VT-2 (1 + rD D/V + rE E/V)

[pic] VT-2 =[pic]

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We can continue all the way back to t = 0:

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Does WACC necessarily decline as leverage increases?

The WACC formula seems to say that the cost of capital decreases with leverage because of interest tax shields. That is not necessarily so. Suppose, for example, that the extra personal tax on debt interest exactly offsets the advantage of the corporate interest tax shield. In that case

WACC = (1-TC) rD D/V + rE E/V = r,

and MM’s proposition II becomes rE = r + (r – rD (1-TC))D/E.
Also, the security market line would have to pass