Four Steps to Valuing Flexibility  779
project, based on discounting the cash flows in the event tree, should still 
equal the standard DCF value from the first step.
In step 3, turn the event tree into a decision tree by identifying the types of 
managerial flexibility that are available. Build the flexibility into the nodes of 
the tree. Multiple sources of flexibility are possible at a single decision node, 
such as the option to abandon or expand, but it is important to have clear 
priorities among them. Be careful in establishing the sequence of decisions 
regarding flexibility, especially when the decision tree has compound options.
Finally, step 4 entails recognizing how the exercise of flexibility alters the 
project’s risk characteristics. If the prevailing risk affecting the contingent cash 
flows is fully diversifiable, you need no special modeling; you can use DTA, 
discounting investment cash flows at the risk-free rate and the underlying 
project’s cash flows at the weighted average cost of capital, as in the pharma-
ceutical example in the upcoming section on ROV and DTA. If the prevailing 
risk is nondiversifiable and priced in the market, the appropriate risk-adjusted 
discount rate for the project’s cash flows is no longer the weighted average 
cost of capital used in step 1. In that case, apply an ROV approach for the 
project with flexibility, using risk-neutral valuation or a replicating portfolio.
Real-Option Valuation: A Numerical Example
Using the four-step process, we illustrate the ROV approach with a straight-
forward binomial lattice for valuing flexibility that is assumed to be driven 
by nondiversifiable risk. The results are identical to alternative option-pricing 
models that use more complicated mathematics such as stochastic calculus or 
Monte Carlo simulation.
Step 1: Estimate Net Present Value without Flexibility  Assume that an invest-
ment in a project to build a factory generates cash flows whose present value 
(PV) equals $100, and its expected rate of return and cost of capital (k) equal 
EXHIBIT 39.10  Four-Step Process for Valuing Flexibility
Estimate 
NPV without 
flexibility
Model
uncertainty in 
event tree
Model
flexibility in 
decision tree
Estimate 
contingent
NPV
Objectives
Compute base-case 
present value without 
flexibility
Understand how present 
value develops with 
respect to changing 
uncertainty
Analyze event tree to 
identify and incorporate 
managerial flexibility 
to respond to new 
information
Value total project using 
DTA or ROV approach
Comments
Standard NPV approach 
is used for valuation of 
underlying asset.
No flexibility modeled; 
valuation following 
event tree should equal 
standard NPV
Flexibility is incorporated 
into event tree, 
transforming it into 
decision tree
Under high uncertainty 
and managerial flexibility, 
contingent NPV will be 
significantly higher than 
standard NPV

780  Flexibility
8 percent. The risk-free rate is 5 percent per year, and the cash outflow necessary 
to undertake the project, if we invest in it immediately, is $105. Thus, the stan-
dard NPV is –$5, equal to the expected present value of $100 less the investment 
of $105, and we would not undertake the project if we had to commit today.
Step 2: Model Uncertainty Using Event Tree  The lattice that models the 
potential values of the underlying risky asset is called an event tree. It con-
tains no decision nodes and simply models the evolution of the underlying 
asset. Exhibit 39.11 illustrates potential values the factory might take for each 
of next five years, assuming a volatility of 15 percent per year.20 Defining T as 
the number of years per upward movement and σ as the annualized volatility 
of the underlying factory value, determine the up-and-down movements by 
using the following formulas:21
Up Movement
Down Movement
=
=
=
=
u
e
d
u
T
σ
1
Substitute numerical values into these formulas:
u
e
d
=
=
=
=
0 15 1
1 1618
1
1 1618
0 8607
.
.
.
.
20 The standard deviation of the rate of change of the factory value.
21 J. Cox, M. Rubinstein, and S. Ross, “Option Pricing: A Simplified Approach,” Journal of Financial Eco-
nomics 7, no. 3 (1979): 229–263. As T becomes smaller, the binomial lattice results converge to the true 
value of the option. In this example, we have chosen T = 1 for ease of illustration.
EXHIBIT 39.11  Event Tree: Factory without Flexibility
$
t = 0
t = 1
t = 2
t = 3
t = 4
t = 5
Cumulative
probability, %
100
Underlying asset
PV = 100
Volatility = 15%
Initial investment = 105
No-flexibility NPV = 100 – 105 = (5)
Assumptions
Risk-free rate = 5%
Cost of capital (k ) = 8%
20.5
38.2
28.5
10.6
2.0
0.1
212
182
157
157
135
135
116
116
116
100
100
86
86
86
74
74
64
64
55
47
Note: t = time, in years