794  Appendix A
where NOPAT
net operating profit after taxes
RONIC
return on new inve
t= =
=
1
sted capital
The key value driver formula can be rearranged further into a formula 
based on economic profit. We do this to demonstrate that discounted cash 
flow is equivalent to the book value of invested capital plus the present value 
of future economic profit.
To begin, start with the key value driver formula, and replace NOPAT with 
invested capital times return on invested capital (ROIC):
V
g
g
=
×
×
−




−
Invested Capital
ROIC
RONIC
WACC
0
1
If we assume that the return on new invested capital (RONIC) equals the 
return on existing invested capital (ROIC), it is possible to simplify the preced-
ing equation by distributing ROIC in the numerator:1
V
g
g
=
−
−






Invested Capital
ROIC
WACC
0
To complete the transformation to economic profit, add and subtract WACC 
in the numerator:
V
g
g
=
−
+
−
−






Invested Capital
ROIC
WACC
WACC
WACC
0
Separate the fraction into two components, and then simplify:
V
g
=
−
−





+
InvestedCapital
ROIC
WACC
WACC
InvestedCapital
WA
0
0
CC
WACC
InvestedCapital
InvestedCapital
ROIC
WA
−
−






=
+
−
g
g
0
0
CC
WACC −






g
1 This equation highlights two requirements for using the key value driver formula: both WACC 
and ROIC must be greater than the rate of growth in cash flows. If WACC is less than the cash flow 
growth rate, cash flows grow faster than they can be discounted, and value approaches infinity. 
(Perpetuity-based formulas should never be used to value cash flows whose growth rates exceed 
WACC.) If ROIC is lower than the growth rate, cash flows are negative, producing a negative value. 
In actuality, this situation is unlikely; investors would not finance a company that is never expected 
to generate or enable positive cash flow.

Appendix A  795
Economic profit is defined as invested capital times the difference of ROIC 
minus WACC. Substituting this definition into the previous equation leads to 
our final equation:
V
g
=
+
−
Invested Capital
Economic Profit
WACC
0
1
According to this formula, a company’s operating value equals the book 
value of its invested capital plus the present value of all future economic prof-
its. (The final term is a growing perpetuity of economic profits.) If future eco-
nomic profits are expected to be zero, the intrinsic value of a company equals 
its book value. In addition, if future economic profits are expected to be less 
than zero, then enterprise value should trade at less than the book value of 
invested capital—an occurrence observed in practice.
Generalized Proof
The previous section limited our proof to a set of cash flows growing at a 
constant rate. This section generalizes the proof to any set of cash flows. To 
demonstrate equivalence, start by computing the present value of a periodic 
stream of cash flows:
V
t
t
t
=
+
=
∞
∑
FCF
WACC
(
)
1
1
where
 
V
t
t
=
=
=
value of operations
FCF
free cash flow in year
WACC
weighted average cost of capital
To this value, add and subtract the cumulative sum of all current and future 
amounts of invested capital (IC):
V
t
t
t
t
t
t
t
t
t
=
+
−
+
+
+
=
∞
=
∞
=
∞
∑
∑
∑
IC
WACC
IC
WACC
FCF
WACC
(
)
(
)
(
)
1
1
1
0
0
1
where ICt = invested capital for year t.
Next, adjust the preceding equation slightly to restate the same value using 
terms that can be canceled later. First, strip invested capital at time zero from 
the first cumulative sum. Then modify the second cumulative sum to t = 1 to 
infinity, by changing each t inside the second cumulative sum to t - 1. This 

796  Appendix A
new representation is identical to the original representation but will allow us 
to cancel terms later. The new representation is as follows:
V
t
t
t
t
t
t
t
=
+
+
−
+
+
+
=
∞
−
−
=
∞
∑
∑
IC
IC
WACC
IC
WACC
FCF
WACC
0
1
1
1
1
1
1
1
(
)
(
)
(
)t
t=
∞
∑
1
Multiply the second cumulative sum by (1 + WACC)/(1 + WACC). This 
action converts the exponent t - 1 in the denominator of the cumulative sum 
to t. Also substitute for free cash flow in the third cumulative sum, using its 
definition, NOPAT less the increase in invested capital:
V
t
t
t
t
t
t
t
=
+
+
−
+
(
)
+
+
=
∞
−
=
∞
∑
∑
IC
IC
WACC
WACC IC
WACC
NOPAT
0
1
1
1
1
1
1
(
)
(
)
−
−
(
)
+
−
=
∞
∑
IC
IC
WACC
t
t
t
t
1
1
1
(
)
Because there is now a consistent denominator across all three cumulative 
sums, combine them into a single cumulative sum:
V
t
t
t
t
t
t
t
=
+
−
+
(
)
+
−
+
+
−
−
=
∞
∑
IC
IC
WACC IC
NOPAT
IC
IC
WACC
0
1
1
1
1
1
(
)
In the second term of the numerator, distribute (1 + WACC)ICt-1 into its two 
components, ICt-1 and WACC(ICt-1):
V
t
t
t
t
t
t
t
t
=
+
−
−
(
) +
−
+
+
−
−
−
=
∞
∑
IC
IC
IC
WACC IC
NOPAT
IC
IC
WACC
0
1
1
1
1
1
(
)
Simplify by collecting terms:
V
t
t
t
t
=
+
−
(
)
+
−
=
∞
∑
IC
NOPAT
WACC IC
WACC
0
1
1
1
(
)
The numerator is the definition of economic profit, so the result is a valuation 
based on economic profit:
V
t
t
t
=
+
+
=
∞
∑
IC
Economic Profit
WACC
0
1
1
(
)
The enterprise value of a company equals the book value of its invested 
capital plus the present value of all future economic profits. To calculate the 
value correctly, you must calculate economic profit using last year’s (i.e., 
beginning-of-year) invested capital—a subtle but important distinction.

Appendix A  797
The interdependence of invested capital, economic profit, and free cash 
flow is not surprising. Think of discounted cash flow this way: a portion of 
future cash flows is required to cover the required return for the investor’s 
capital. The remaining cash flow is either used to grow invested capital (to 
generate additional future cash flows) or returned to investors as an extra 
bonus. This bonus is valuable, so investors are willing to pay a premium for 
cash flows above the amount required. Subsequently, companies with posi-
tive economic profits will trade at a premium to the book value of invested 
capital.