Therefore, risk-adjust all probabilities of the upward and downward move-
ments for the drug’s value:
p
r
d
u
d
f
T
*
(
)
.
.
.
.
.
=
+
−
−
=
−
−
=
1
1 05
0 77
1 30
0 77
0 74
3
Having applied the risk-neutral probabilities, discount all contingent payoffs 
at the risk-free rate, working from right to left in the tree. Because the techno-
logical risk is fully diversifiable, there is no need to adjust the probabilities for 
success and failure in research or testing.
For example, from Exhibit 39.18, the value of the option at the end of the 
research phase showing a drop in the value of the drug is expressed as fol-
lows:
NPV
Option
Max PV Testing
Inv
Testing
3
3
3
0
(
)
[
(
)
(
), ]
=
−
In this equation, PV3(Testing) represents the value of proceeding with testing 
at this node. It equals the value of the future payoffs weighted by risk-neutral 
probabilities and discounted at the risk-free rate:
PV Testing
3
0 40 0 74
4 164
0 26
2 416
0 60 0
1 05
(
)
.
[ .
($ ,
)
.
($ ,
)]
.
( )
( .
)
=
+
+
3
1 279
= $ ,
Inv3(Testing) equals $250 million, so the value of the development project at 
this node is as follows:
NPV
Option
Max
3
1 279
250 0
1 029
(
)
[($ ,
$
), ]
$ ,
=
−
=
Solve for the other nodes in the same way. Working backward through the 
tree gives us an estimate of the contingent NPV: $120 million, the same result 
as obtained in the DTA approach without commercial risk.
This is not surprising. A closer look at the decision tree reveals that uncer-
tainty about the future value of the drug if it is marketable is not significant 
enough to influence any of the decisions in the development process. In this 
example, the commercial risk makes no difference, even if we assume volatility 
as high as 50 percent (an amount that exceeds the volatility of many high-tech 
stocks). As noted earlier, when nondiversifiable risk (the drug’s commercial 
risk as measured by its beta) does not influence investment decisions, the DTA 
and ROV results are equivalent.
Moreover, in real situations, the prevailing uncertainty in drug develop-
ment is whether the drug proves to be an effective disease treatment without 
serious side effects. The commercial risk is far less relevant, because a truly 
effective drug almost always generates attractive margins. The example illus-
trates how in such cases it is more practical to focus on the technological risk 
entirely, using a DTA approach. Explicitly modeling the nondiversifiable (e.g., 
commercial) risk requires an ROV approach that is more complex and may not 
even affect the valuation results.
Real-Option Valuation and Decision Tree Analysis  791

792  Flexibility
In general, when faced with multiple sources of underlying risk, carefully 
assess whether all of these possible risks are important or whether one pre-
vails. Sometimes you can focus the valuation approach on just one or two 
sources of uncertainty and greatly simplify the analysis.
Summary
Managerial flexibility lets executives defer or change investment decisions as 
a business or project develops. It can substantially alter the value of a busi-
ness or project. Rigidly applying standard DCF analysis fails to account for the 
impact that exercising flexibility can have on present value.
Flexibility takes many forms, such as the option to defer, expand, contract, 
or abandon projects, or to switch them on and off. This chapter has illustrated 
only a few applications. Contingent NPV analysis, in the form of decision 
tree analysis (DTA) or real-option valuation (ROV) models, correctly captures 
flexibility’s impact on value. The ROV approach is theoretically superior to 
DTA, but applying it is more complicated. So ROV is often limited to valu-
ing flexibility in commodity-based industries where prices are measurable, 
making its application more straightforward. In most other cases, a careful 
DTA approach delivers results that are reasonably solid and can provide more 
valuable insights.

793
Appendix A
Discounted Economic 
Profit Equals Discounted 
Free Cash Flow
This appendix demonstrates algebraically the equivalence between discounted 
cash flow and discounted economic profit. In the first section, we convert the 
key value driver formula presented in Chapter 3 into a value driver formula 
based on economic profit. This formula is used in Chapter 10 to estimate 
continuing value in the economic-profit valuation. The second section of this 
appendix generalizes the proof to any set of cash flows.
Proof Using Perpetuities
To convert the key value driver formula into an economic-profit-based for-
mula, start with the growing cash flow perpetuity:
V
g
t
=
−
=
FCF
WACC
1
where 
V
t
=
=
=
=
value of operations
FCF
free cash flow in year 1
WACC
weighted
1
average cost of capital
NOPAT
growth in
g =
In Chapter 3, we convert the growing perpetuity into the key value driver 
formula:
V
g
g
t
=
−




−
=
NOPAT
RONIC
WACC
1 1