802  Appendix  B
such that:
WACC =
+
(
)
−
(
) +
+
(
)
D
D
E k
T
E
D
E k
d
m
e
1
Note how the after-tax cost of debt and the cost of equity are weighted 
by each security’s market-based weight to enterprise value. This is why you 
should use market-based values, and not book values, to build the cost of 
capital. This is also why you should discount free cash flow at the weighted 
average cost of capital to determine enterprise value. Remember, however, 
that you can only use a constant WACC when leverage is expected to remain 
constant (i.e., debt grows as the business grows).2
Adjusted Present Value
To determine enterprise value using adjusted present value, once again start 
with V = D + E and multiply by a fraction equal to 1. This time, however, do 
not include the marginal tax rate in the fraction:
V
D
E
D k
D g
D k
D g
d
e
d
e
=
+
(
)
(
) +
−
( )
(
) +
−
( )






CF
CF
Following the same process as before, convert each cash flow in the de-
nominator to its present value times its expected return, and divide the frac-
tion by (D + E)/(D + E):
V
D k
D g
D
D
E k
E
D
E k
g
d
e
d
e
=
(
) +
−
( )
+
(
) +
+
(
) −
CF
Appendix C shows that if the company’s interest tax shields have the 
same risk as the company’s operating assets (as one would expect when the 
company maintains a constant capital structure), the fraction’s denominator 
equals ku, the unlevered cost of equity, minus the growth in cash flow (g). 
Make this substitution into the previous equation:
V
D k
D g
k
g
d
e
u
=
(
) +
−
( )
−
CF
2 To see this restriction applied in a more general setting, see Miles and Ezzell, “Weighted Average Cost 
of Capital.”

Appendix  B  803
Next, focus on the numerator. Substitute the definitions of cash flow to 
debt and cash flow to equity, as we did earlier in this appendix:
V
D g
D g
k
g
u
=
+
−
−
−
+
( ) −
( )
−
Interest
EBIT
Interest
Taxes
Net Investment
In this equation, the two interest terms cancel and the two D(g) terms cancel, 
so simplify by canceling these terms. Also insert Tm(Interest) - Tm(Interest) 
into the numerator of the expression:
V
T
T
k
g
m
m
u
=
−
+
(
) −
(
) −
−
EBIT
Taxes
Interest
Interest
Net Investment
Aggregate reported taxes and the negative expression for Tm(Interest) into 
all-equity taxes. Move the positive expression for Tm(Interest) into a separate 
fraction:
V
T
k
g
T
m
u
m
=
−
+
(
)
[
] −
−
+
(
)
EBIT
Taxes
Interest
Net Investment
Interest
k
g
u −
At this point, we once again have free cash flow in the numerator of the 
first fraction. The second fraction equals the present value of the interest tax 
shield. Thus, enterprise value equals free cash flow discounted by the unle-
vered cost of equity plus the present value of the interest tax shield:
V
k
g
u
=
−
+
(
)
FCF
PV Interest Tax Shield
This expression is commonly referred to as adjusted present value.
In this simple proof, we assumed tax shields should be discounted at the 
unlevered cost of equity. This need not be the case. Some financial analysts 
discount expected interest tax shields at the cost of debt. If you do this, how-
ever, free cash flow discounted at the traditional WACC (defined earlier) and 
adjusted present value will lead to different valuations. In this case, WACC 
must be adjusted to reflect the alternative assumption concerning the risk of 
tax shields.

805
Appendix C
Levering and Unlevering 
the Cost of Equity
This appendix derives various formulas that can be used to compute unle-
vered beta and the unlevered cost of equity under different assumptions. 
Unlevered betas are required to estimate an industry beta, as detailed in 
Chapter 15. We prefer using an industry beta rather than a company beta to 
determine the cost of capital because company betas cannot be estimated ac-
curately. As discussed in Chapter 10, the unlevered cost of equity is used to 
discount free cash flow to compute adjusted present value. For companies 
with substantial postretirement obligations, the appendix concludes by in-
corporating pensions and other postretirement benefits into the unlevering 
process.
Unlevered Cost of Equity
Franco Modigliani and Merton Miller postulated that the market value of 
a company’s economic assets, such as operating assets (Vu) and tax shields 
(Vtxa), should equal the market value of its financial claims, such as debt (D) 
and equity (E):
V
V
D
E
u
txa
+
=
=
+
Enterprise Value

(C.1)
A second result of Modigliani and Miller’s work is that the total risk of the 
company’s economic assets, operating and financial, must equal the total risk 
of the financial claims against those assets:
V
V
V
k
V
V
V
k
D
D
E k
E
D
E k
u
u
txa
u
txa
u
txa
txa
d
e
+
(
) +
+
(
) =
+
(
) +
+
(
) 
(C.2)