Civil Engineering Reference
In-Depth Information
and compared to the benefits. This directly assumes that all stages of the NZER
will be implemented and does not provide the decision-maker with the flexibility
to determine whether to implement the next stage or abandon it. The valuation of
this flexibility is similar to that of an exchange option (Villani
2007
; McDonald
and Siegel
1985
; Margrabe
1978
). An exchange option, a special case of American
option, involves the exchange of one asset, S
1
(risky asset), for another asset S
2
(can also be risky).
In the case of NZER, at each stage of the investment evaluation, the cost of
implementing an additional refurbishment measure, I
k
, is exchanged for the
expected benefits V. The decision to invest in the each stage, k - 1, is dependent on
the present value of exchange option, NPV
mk
, at the subsequent stage k.If
NPV
mk
CI
k - 1,
then investment in stage k - 1 should be undertaken. This will
allow for subsequent investments; otherwise, the project should be abandoned, and
no further NZER are necessary as their costs exceed the expected additional ben-
efits. This is repeated at each stage to determine whether the investment should be
undertaken until all stages are completed or the condition NPV
mk
\I
k
1
is reached.
Using Villani (
2007
), McDonald and Siegel (
1985
), and Margrabe (
1978
), NPV
mk
is calculated using Eq. (
3
) below by assuming that the investment I
k
can alterna-
tively be invested at the risk-free interest rate r between time periods k - 1 and k:
NPV
mk
ð
V
;
I
k
Þ¼
Ve
dvt
N
ð
d
1
Þ
I
k
e
rt
N
ð
d
2
Þ
ð
3
Þ
where N
ðÞ¼
Probability
f
Y
y
g
Y is a standard normal random variable;
=
r
v
p
t
p
t
d
1
¼
ln
ð
Ve
dvt
=
I
k
e
rt
Þþ
r
d
v
þ
0
:
5r
2
andd
2
¼
d
1
r
v
t is the time period between k - 1 and k.
This problem is analyzed for the two-stage investment scenario where the
$10 million total investment is divided into two stages. Investment in the first-
stage I
1
depends on the value of the investment in the subsequent stage NPV
m2
.
Thus, using Eq. (
3
), the NZER was analyzed to determine the minimum cutoff
value of V at which investment should occur for a given initial investment I
1
.This
of course depends on the level of r
v
. Figure
4
shows the cutoff value V
min
for
different initial or first-stage investments I
1
for three levels of uncertainty r
v
.
Investment I
2
will only be undertaken if V [ V
min
.
Two main observations can be made from these results. First, for the same level
of initial investment I
1
, the cutoff value of V decreases with increase in uncertainty
r
v
. This indicates that the higher the uncertainty associated with the investment,
the lower the expected value of benefits V at which the NZER should be aban-
doned. Second, the higher the initial stage investment I
1
, the higher is the cutoff
value V for the same level of uncertainty. This indicates that when the initial
investment is high, the expected benefits from subsequent investments should be
high because most of the uncertainty surrounding this value would have been
resolved during the initial stage of investment. That is a higher initial stage of the
investment indicates higher confidence by the decision-maker about the expected
value of their project.
