799
Appendix B
Derivation of Free Cash 
Flow, Weighted Average 
Cost of Capital, and 
Adjusted Present Value
Chapter 10 demonstrated numerically the equivalence of enterprise discounted 
cash flow (DCF), adjusted present value (APV), and the cash-flow-to-equity 
valuation when leverage (as measured by the market-based debt-to-equity 
ratio) is constant. This appendix derives the key terms in each model—namely, 
free cash flow (FCF) and the weighted average cost of capital (WACC)—and 
demonstrates their equivalence algebraically.
To simplify the analysis, we assume cash flows to equity are growing at a 
constant rate, g. This way we can use growth perpetuities to analyze the rela-
tionship between methods.1
Enterprise Discounted Cash Flow
By definition, enterprise value (V) equals the market value of debt (D) plus the 
market value of equity (E):
V
D
E
=
+
1 For an analysis that applies to more complex situations (i.e., when cash flows can follow any pat-
tern), see J. A. Miles and J. R. Ezzell, “The Weighted Average Cost of Capital, Perfect Capital Markets, 
and Project Life: A Clarification,” Journal of Financial and Quantitative Analysis 15 (1980): 719–730 (for a 
discussion of enterprise DCF and WACC); and S. C. Myers, “Interactions of Corporate Financing and 
Investment Decisions: Implications for Capital Budgeting,” Journal of Finance 29 (1974): 1–25 (for a dis-
cussion of adjusted present value).

800  Appendix  B
To examine the components of enterprise value, multiply the right side of 
the equation by a complex fraction equivalent to 1 (the numerator equals the 
denominator, an algebraic trick we will use many times):
V
D
E
D
T
k
D g
D
T
k
D g
m
d
e
m
d
e
=
+
(
)
−
(
)
+
−
( )
−
(
)
+
−
( )






1
1
CF
CF

(B.1)
where T
k
m
d
e
=
=
=
marginal tax rate
cost of debt
CF
cash flow to equity holders
g = growth in cash flow to equity holders
Over the next few steps, the fraction’s numerator will be converted to free 
cash flow (FCF). We will show later that the denominator equals the weighted 
average cost of capital. Start by defining the numerator as FCF:
FCF
CF
=
−
(
)
+
−
( )
D
T
k
D g
m
d
e
1
If the market value of debt equals the face value of debt, the cost of debt 
will equal the coupon rate, and D times kd will equal the company’s interest 
expense. Therefore,
FCF
Interest
CF
=
−
(
) +
−
( )
1
T
D g
m
e
By definition, cash flow to equity (CFe) equals earnings before interest and 
taxes (EBIT) minus interest, taxes, and net investment, plus the increase in 
debt. Assuming the ratio of debt to equity is constant, the annual increase in 
debt will equal D(g). Why? Since cash flows to equity are growing at g, the 
value of equity also grows at g. Since the ratio of debt to equity remains con-
stant (a key assumption), the value of debt must also grow at g. Substitute the 
definition of cash flow to equity into the preceding equation:
FCF
Interest
EBIT
Interest
Taxes
Net Investment
=
−
(
) +
−
−
−
+
( ) −
1
T
D g
m
D g
( )
Next, distribute the after-tax interest expression into its two components, and 
cancel D(g):
FCF
Interest
Interest
EBIT
Interest
Taxes
Net Investmen
=
−
(
) +
−
−
−
Tm
t
Simplify by canceling the interest terms and rearranging the remaining terms:
FCF
EBIT
Taxes
Interest
Net Investment
=
−
+
(
)
[
] −
Tm

Appendix  B  801
Chapter 11 defines operating taxes as the taxes a company would pay if 
the company were financed entirely with equity. Operating taxes therefore 
equal reported taxes plus the interest tax shield (as interest is eliminated, taxes 
would rise by the interest tax shield). This leads to the definition of free cash 
flow we use throughout the book:
FCF
EBIT
Operating Taxes
Net Investment
=
−
−
Next, we focus on the denominator. To derive the weighted average cost 
of capital (WACC), start with Equation B.1, and multiply CFe by 1, denoted as 
(ke - g)/(ke - g):
V
D
E
D
T
k
k
g k
g
D g
m
d
e
e
e
=
+
(
)
−
(
)
+
−
−
(
) −
( )












FCF
CF
1
where ke = cost of equity.
If equity cash flows are growing at a constant rate, the value of equity 
equals CFe divided by (ke - g). Therefore, the growing perpetuity in the de-
nominator can be replaced by the value of equity (E) and distributed:
V
D
E
D
T
k
E k
E g
D g
m
d
e
=
+
(
)
−
(
)
+ (
) −( ) −
( )






FCF
1
In the denominator, collapse E(g) and D(g) into a single term:
V
D
E
D
T
k
E k
D
E g
m
d
e
=
+
(
)
−
(
)
+ (
) −
+
(
)






FCF
1
To complete the derivation of WACC in the denominator, divide the nu-
merator and denominator by (D + E). This will eliminate the (D + E) expres-
sion on the left and place it in the denominator as a divisor. Distributing the 
term across the denominator, the result is the following equation:
V
D
D
E k
T
E
D
E k
D
E
D
E g
d
m
e
=
+
(
)
−
(
) +
+
(
) −
+
+
( )
FCF
1
The expression in the denominator is the weighted average cost of capital 
(WACC) minus the growth in cash flow (g). Therefore, Equation B.1 can be 
rewritten as:
V
g
=
−
FCF
WACC