phase, we continue to the testing phase if the future payoffs outweigh the re-
quired investments. The value of the project at this point, after three years is:
NPV
Option
Max PV Testing
Inv
Testing
3
3
3
0
(
)
[
(
)
(
), ]
=
−
In this equation, PV3(Testing) equals the probability-weighted future payoffs 
discounted by three years at the cost of capital of 7 percent:
PV Testing
3
3
0 40
6 475
150
1 07
0 60 0
2 065
(
)
.
$ ,
$
( .
)
.
( )
$ ,
=
−








+
=
With Inv3(Testing) equal to the $250 million investment requirement for the 
testing phase, the project value prior to the testing phase amounts to:
NPV
Option
Max
3
2 065
250 0
1 815
(
)
[($ ,
$
), ]
$ ,
=
−
=
Working further from right to left in the tree, we find the contingent NPV 
for the entire project prior to the research phase:
NPV
Option
Max PV Research
Inv
Research 0
Max
0
0
0
0 15
1
(
)
[
(
)
(
), ]
.
$
=
−
=
,
.
.
( )
$
,
$
815
1 07
0 85 0
100
122
3
(
)





+
−








=
0
This value including flexibility is significantly higher than the standard NPV of 
–$169 million. Note that we discounted all contingent payoffs at the underlying 
asset’s cost of capital, so the $122 million only approximates the true contin-
gent value. But the result is close, as we show in the calculations immediately 
following, and this approach is straightforward to apply and easy to explain.
The true contingent value turns out to be $120 million and follows from 
a refined DTA approach that separately discounts the asset cash flows at the 
cost of capital of 7 percent and the investment cash flows at the risk-free rate of 
5 percent.27 The value of proceeding with testing now becomes:28
PV
Testing
3
3
3
0 40
6 475
1 07
150
1 05
0 60 0
*(
)
.
$ ,
.
$
.
.
(
=
(
)
−
(
)





+
)
$ ,
$
$ ,
=
−
=
2 114
52
2 062
The value of the option to proceed with the testing phase is then:
NPV
Option
Max
3
2 144
52
250 0
2 144
302
1 812
*(
)
[($ ,
$
)
$
, ]
$ ,
$
$ ,
=
−
−
=
−
=
27 See the example in Exhibit 39.7. The assumption to discount investment outlays at the risk-free rate 
is also implicitly made in ROV approaches.
28 In prior editions of this book, we adopted an alternative but equivalent decision tree where all values 
of asset and investment cash flows were discounted to t = 0 before deriving the contingent value by 
working from right to left in the tree. The contingent NPV results are identical.
Real-Option Valuation and Decision Tree Analysis  787

788  Flexibility
Working from right to left but now separately discounting asset and invest-
ment cash flows in each step, we obtain the contingent NPV* per t = 0:
NPV
Option
Max PV
Research
Inv
Research 0
Max
0
0
0
0 15
*
*
(
)
[
(
)
(
), ]
.
=
−
=
$ ,
.
$
.
.
( )
$
,
2 114
1 07
302
1 05
0 85 0
100
3
3
(
)
−
(
)





+
−








=
0
$120
To illustrate, we obtain the same value of $120 million with yet another 
approach: the ROV method. In this approach, project the future value of the 
underlying asset under “risk-neutral” return assumptions, and then discount 
all contingent payoffs at the risk-free rate. The risk-neutral future value of a 
successfully developed drug is its value as of today, compounded at the risk-
free rate for six years:
PV
Drug
6
6
4 314 1 05
5 781
**(
)
$ ,
.
$ ,
=
(
) =
This means that the risk-neutral value of proceeding with testing is:
PV
Testing
3
3
0 40
5 781
150
1 05
0 60 0
1 9
** (
)
.
$ ,
$
.
.
( )
$ ,
=
−
(
)





+
=
46
The risk-neutral value of the option to proceed with testing as of t = 3 is:
NPV
Option
Max
3
1 946
250 0
1 696
** (
)
[$ ,
$
, ]
$ ,
=
−
=
Working from right to left while discounting all cash flows at the risk-free rate 
gives us the contingent NPV at t = 0, which is $120 million:
NPV
Option
Max PV
Research
Inv
Research 0
Max
0
0
0
0
**
**
(
)
[
(
)
(
), ]
.
=
−
=
15
1 696
1 05
0 85 0
100
120
3
$ ,
.
.
( )
$
,
$
(
)





+
−








=
0
ROV Approach: Technological and Commercial Risk
Our analysis thus far did not include the other source of uncertainty in the 
development project: the commercial risk concerning the future cash flow po-
tential of the successfully developed and marketed drug. ROV is necessary to 
handle both technological and commercial risk.
Step 1: Estimate Present Value without Flexibility  The first step, estimating 
present value without flexibility, is identical for the DTA and ROV approaches.