Four Steps to Valuing Flexibility  781
Based on traditional DCF using an 8 percent cost of capital, the probability 
of an up movement is 72.82 percent, and the probability of a down movement is 
27.18 percent.22 As can be verified, the present value of any branch in the event 
tree equals the expected payout discounted at the 8 percent cost of capital. For 
example, take the uppermost branch in the fifth time period. Its present value is:
PV
PV
t
t
E
k
=
=
=
+
=
+
=
4
5
1
0 7282 211 7
0 2718
156 8
1 08
182
(
)
(
)
.
($
. )
.
($
. )
.
$
.2
A similar calculation will produce any of the values in the event tree, re-
sulting in a PV of the project of $100 at t = 0. That present value equals the 
result in step 1, so we know the tree is correct.
Step 3: Model Flexibility Using a Decision Tree  When you add decision 
points to an event tree, it becomes a decision tree. Suppose the factory can be 
expanded for an additional $15. The expansion increases the factory’s value 
at that node by 20 percent. The option can be exercised at any time during the 
next five years—but only once.
Exhibit 39.12 shows the resulting decision tree. To find the payouts at a 
given point on the tree, start with the final branches. Consider the uppermost 
22 See the previous note for the derivation of the formula for estimating the upward probability: 
 
 
 
(
)
(
%)
.
.
.
.
1
1
8
0 8607
1 1618
0 8607
0 7282
+
−
−
=
+
−
−
=
k
d
u
d
T
EXHIBIT 39.12  Decision Tree: Option to Expand Factory
$
t = 0
t = 1
t = 2
t = 3
t = 4
t = 5
108
Underlying asset values
 
PV+ 
116
 
PV– 
86
 
PV 
100
239
204
175
173
149
148
127
126
124
107
106
91
90
88
77
75
65
64
55
47
Decision to expand
Management decisions (t = 5)
 
124 = Max (116,116 × 1.2 – 15)
 
 88 = Max (86, 86 × 1.2 – 15)
Portfolio replication
 
N = (124 – 88) / (116 – 86)
 
B = (88 – 86N ) / 1.05
 
N = 1.2; B = –14.3
Value of option (t = 4)
Option 
= Max (100N + 1B, 100 × 1.2 – 15)
 
= Max (106, 105)
 
= 106
Note: t = time, in years
PV = present value
N = number of replicating securities
B = number of risk-free bonds 
Incremental investment: $15 
Incremental payoff: 20%

782  Flexibility
branch in period 5. On the upward limb, the payout absent expansion would 
be $211.70, as Exhibit 39.11 shows. But with expansion, it is 1.20 × $211.70 – 
$15 = $239.00. Since the value with expansion is higher, we would decide to 
expand. On the lower limb of that same node, the payout with expansion is 
1.20 × $156.80 – $15 = $173.20, versus $156.80 without expansion, so again we 
would expand. In this way, complete the payoff estimates for all final branches.
Step 4: Estimate Contingent Net Present Value  To determine the value of 
the project with the flexibility to expand, work backward through the deci-
sion tree, using the replicating-portfolio method at each node. For the node 
highlighted in Exhibit 39.12, you can replicate the payoffs from the option to 
expand in t = 5, using a portfolio of N units of the underlying project and B 
units of $1 risk-free bonds:23
$
.
$ .
$
.
$
.
$ .
$
.
116 2
1 05
124 4
86 1
1 05
88 3
N
B
N
B
+
=
+
=
Solving the equations, we find that N = 1.2, and B = –14.3. Therefore, a rep-
licating portfolio consists of 1.2 units of the project without flexibility (at that 
node, valued at $100 in the event tree of Exhibit 39.11), plus a short position of 
14.3 bonds worth $1. As shown in the calculations in Exhibit 39.12, the value 
of the option in the node at t = 4 is then:
PV =
+
=
$
$
$
.
100
1
105 7
N
B
Work backward from right to left, node by node, to obtain a present value 
of $108.40 for a project that has an option to expand. As a result, the net present 
value of the project increases from –$5.00 to $3.40, so the option itself is worth 
$8.40. Note that the analysis also provides the value-maximizing decision strat-
egy: management should expand the factory only after five years and only if the 
factory is worth $75 or more as indicated by the boxed nodes in Exhibit 39.12.24
If, instead, management had the option to abandon the factory at any node 
for a fixed liquidation value of $100, the valuation would be as shown in Ex-
hibit 39.13. Determine the contingent payoffs for final branches. Then work 
again from right to left through the decision tree. For the highlighted node 
at t = 4, the value of the underlying factory is $116.20 in the upward branch 
and $86.10 in the downward branch (see in the event tree of Exhibit 39.11). 
Given the ability to do so, the company would abandon the project for $100 
23 If the project itself is not traded but a traded twin security exists, we can construct the portfolio in a 
similar way with units of the twin security and risk-free bonds.
24 This is analogous to a call option on a stock that does not pay dividends: it is never exercised pre-
maturely. For example, in the node highlighted in Exhibit 39.12, the value in year 4 of deferring the 
expansion of the factory to year 5 is $105.70, as calculated in the preceding equation. The value of 
expanding in year 4 is $100 × 1.20 – $15 = $105. It is therefore optimal to defer expansion, as is the case 
for all nodes before year 5.