772  Flexibility
where
u
d
=
=
=
=
FV Favorable State
PV
FV Unfavorable Stat
(
)
$
.
$
.
.
(
50 0
30 3
1 65
e
PV
)
$
.
$
.
.
=
=
16 7
30 3
0 55
Solve by substituting:
p
p
*
.
*
.
=
−
=
0 45
1
0 55
These probabilities implicitly capture the risk premium for investments 
perfectly correlated with the twin security. We discount the future cash flows 
weighted by the risk-neutral probabilities at the risk-free rate of 5 percent, 
arriving at exactly the same value determined using the replicating portfolio:
Contingent NPV =
+
=
0 45 45
0 55 0
1 05
19 5
.
($
)
.
( )
.
$
.
It is no coincidence that the replicating portfolio and risk-neutral valuation 
lead to the same result. They are mathematically equivalent, and both rely on 
the price of the twin security to derive the value of an investment project with 
an option to defer.
Valuation Based on Decision Tree Analysis
A second method for valuing a project with flexibility is to use DTA. This leads 
to the right answer in principle, but only if we apply the correct cost of capital 
for a project’s contingent cash flows.
One DTA approach is to discount the project’s contingent payoffs net of the 
investment requirements. Unfortunately, we can only derive the correct cost of 
capital for these cash flows from the ROV results. Given the project’s contin-
gent NPV of $19.50 with equal chances of paying off $45 or $0, the implied dis-
count rate from the ROV analysis is 15.5 percent.11 This is significantly above 
the underlying asset’s 10 percent cost of capital, because the contingent cash 
flows are riskier. The contingent NPV has an equal chance of increasing by 
131 percent or decreasing by 100 percent. The value of the underlying asset 
($90.90) has a 50–50 chance of going up 65 percent (to $150) or down 45 per-
cent (to $50). If the underlying asset’s cost of capital of 10 percent were used, 
the DTA results would therefore be too high relative to the correct ROV result:
Contingent NPV =
+
=
0 5 45
0 5 0
1 10
20 5
. ($
)
. ( )
.
$
.
11 In this simplified example, there is one value for the cost of capital. In general, the cost of capital for 
the contingent cash flows is not constant. It changes with the risk of the option across time and states 
of the world.

Methods for Valuing Flexibility  773
A better DTA approach separately discounts the two components of the 
contingent cash flows. The contingent payoffs from the underlying asset are 
discounted at the cost of capital of the underlying asset. The investment re-
quirements are discounted at the risk-free rate: 
Contingent NPV =
−



+
=
0 5
150
1 10
105
1 05
0 5 0
18 2
.
$
.
$
.
. ( )
$
.
For longer-term contingent payoffs, this DTA approach generates results 
that are closer to the correct ROV value. The next section discusses how this 
second DTA approach can lead to the exact ROV outcome if the underlying 
risk is either diversifiable or nondiversifiable but is too small to influence the 
future investment decision (that is, if the project value would exceed the in-
vestment requirements even in the unfavorable state).
Comparing ROV and DTA Approaches
As summarized in Exhibit 39.7, the standard NPV approach undervalues 
our mining project at –$9.10. The ROV approach generates a correct value 
(NPV = $19.50) because it captures the value of flexibility by using a replicat-
ing portfolio or risk-neutral valuation. The DTA approach at $18.20 is quite 
close in this example, capturing almost the entire gap between the standard 
NPV valuation and the more granular ROV result. But the DTA results might 
EXHIBIT 39.7  Valuation Result: Standard vs. Contingent NPV
$
Standard NPV
Contingent NPV
Decision tree analysis1
Cash flow
150
Cash flow
150
p = 
50%
1 – p = 50%
Investment
(105)
p = 
50%
1 – p = 50%
Investment
(105)
NPV
(9.1)
Net cash flow
45
NPV
18.2
Net cash flow
45
Cash flow
50
Cash flow
50
Investment
(105)
Investment
(105)
Risk-free rate = 5%
WACC = 10%
Net cash flow
(55)
Net cash flow
–
Real-option valuation2
Cash flow
150
1 – p* 
 55%
p * 
45%
Investment
(105)
NPV
19.5
Net cash flow
45
Cash flow
50
Investment
(105)
Net cash flow
–
Note: t = time, in years; p = probability; p* = binomial (risk-neutral) probability.
1 Discounting cash flows at the project’s cost of capital of 10% and investments at the risk-free rate of 5%.
2 Using risk-neutral valuation.