770  Flexibility
If an investment decision were required immediately, the project would be 
declined. The standard NPV of the mining project equals the discounted ex-
pected cash flow of $90.90 minus the present value of the investment outlay of 
$105 next year. Since the level of investment is certain, it should be discounted 
at the risk-free rate of 5 percent:
Standard NPV =
−
=
−
= −
$
.
$
.
$
.
$
$ .
90 9
105
1 05
90 9
100
9 1
The answer changes if management has flexibility to defer the invest-
ment decision for one year, allowing it to make the decision after observ-
ing next year’s mineral price and the associated cash flow outcome (see 
Exhibit 39.6). The net cash flows in the favorable state are $150 – $105 = $45. 
In the unfavorable state, management would decline to invest, accepting net 
cash flows of $0.
To value this flexibility, we first use an ROV approach and then repeat the 
valuation with the DTA approach.
Real-Option Valuation
Option-pricing models use a replicating portfolio to value the project. The basic 
idea of a replicating portfolio is straightforward: if you can construct a port-
folio of priced securities that has the same payouts as an option, the portfolio 
and option should have the same price. If the securities and the option are 
traded in an open market, this identity is required; otherwise arbitrage profits 
are possible. The interesting implication is that the ROV approach lets you 
correctly value complex, contingent cash flow patterns.
Returning to our $105 investment project, assume there exists a perfectly corre-
lated security (or commodity, in this example) that trades in the market for $30.30 
EXHIBIT 39.6  Contingent Payoffs for Investment Project, Twin Security, 
and Risk-Free Bond
$
t = 0
t = 1
Project 
without 
flexibility
Project 
with 
flexibility
Twin 
security
Risk-free 
bond
Unsuccessful project
Successful project
50%
50%
p = 
1 – p =
Cash flow
150
150
Investment
(105)
(105)
NPV = ?
Net cash flow
45
45
50
1.05
Cash flow
50
50
Investment
(105)
(105)
Risk-free rate = 5%
WACC = 10%
Net cash flow
(55)
–
16.7
1.05
Note: t = time, in years 
    p = probability

Methods for Valuing Flexibility  771
per share (or unit).8 Its payouts ($50 and $16.70) equal one-third of the payouts of 
the project, and its expected return equals the underlying project’s cost of capital.
This twin security can be used to value the project, including the option 
to defer, by forming a replicating portfolio.9 Consider a portfolio consisting of 
N shares of the twin security and B risk-free bonds with a face value of $1. In 
the favorable state, the twin security pays $50 for each of the N shares, and 
each bond pays its face value plus interest, or (1 + rf). Together, these payouts 
must equal $45. Applying a similar construction to the unfavorable state, we 
can write two equations with two unknowns:
$
.
$ .
$
$
.
$ .
50 0
1 05
45
16 7
1 05
0
N
B
N
B
+
=
+
=
The solution is N = 1.35 and B = –21.43. Thus, to build a replicating port-
folio, buy 1.35 shares and short 21.43 bonds (shorting a bond is common lan-
guage for selling a bond, or borrowing money).
This position produces the same cash flow as the investment project under 
both states. Therefore, the value of the project, including the ability to defer, 
should equal the value of the replicating portfolio:
Contingent NPV
Price of Twin Security
=
−
=
−
N
B
(
)
($ )
.
($
. )
.
1
1 35 30 3
21 43
1
19 5
($ )
$
.
=
The value of the deferral option is the difference between the total contingent 
NPV of the project and its standard NPV without flexibility: $19.50 – (–$9.10) = 
$28.60 (remember, the standard NPV was negative).
Contingent NPV can also be determined with an alternative ROV approach 
called risk-neutral valuation. The name is somewhat misleading because a risk-
neutral valuation does adjust for risk, but as part of the scenario probabilities 
rather than the discount rate. To value an option, weight the future cash flows 
by risk-adjusted (or so-called risk-neutral) probabilities instead of the actual 
scenario probabilities. Then discount the probability-weighted average cash 
flow by the risk-free rate to determine current value. The risk-neutral prob-
ability of the favorable state, p∗, is defined as follows:10
p
r
d
u
d
f
*
.
=
+
−
−
=
1
0 45
8 You could also use this twin security to value the investment project without flexibility by means of a 
replicating portfolio. Because the twin security’s cash flows are always exactly one-third of the project 
cash flows, the project without flexibility should be worth three times as much as the twin security, or 
$90.90 (= 3 × $30.30). The twin security is a basic concept that is implicitly used in standard DCF as well; 
you derive the beta of a project by identifying a highly correlated traded security and use that security’s 
beta as input for the cost of capital in the DCF valuation.
9 If the project itself were traded, you would not need a twin security but would construct a replicating 
portfolio with the traded value of the project itself, as in the case of financial options on traded stocks.
10 See, for example, Trigeorgis, Real Options, 75–76.