Civil Engineering Reference
In-Depth Information
i
=
r
I
0
¼
Z
T
0
ie
rt
dt
¼
1
e
rT
ð
4
Þ
i
=
r
I
m
¼
Z
T
m
ie
rt
dt
¼
e
rm
e
rT
ð
5
Þ
2. V
0
is the present value of expected benefits from the investment that is made
continuously over T until the whole NZER is completed. The expected value of
benefits from this investment at time k = 0 and k = m is given in Eqs. (
6
) and
(
7
), respectively:
e
lvT
¼
Ve
dv
T
V
0
¼
Ve
lvrT
ð
6
Þ
e
lv
ð
T
m
Þ
¼
Ve
dv
ð
T
m
Þ
V
m
¼
Ve
lvr
ð
T
m
Þ
ð
7
Þ
3. Thus, the value of this multistage investment in NZER of a given building at
time t = 0 is given in Eq. (
8
) below:
Þ
b
when V
0
V*
NPV
m
ð
V
;
I
Þ¼
V
I
0
Þ
V
0
=
V
ð
ð
ð
8
Þ
Þ
whenV
0
[ V
NPV
m
ð
V
;
I
Þ¼
V
0
I
0
ð
s
r
d
v
r
v
þ
2
V
I
0
b
1
; b
¼
0
:
5
r
d
v
b
¼
0
:
5
þ
2r
=
r
v
r
v
It is important to note that Eq. (
8
) has a similar structure to that of Eq. (
2
)
because they are both solutions to a perpetual American option with time to build.
The only difference in this case is that the investment expenditure is spread over a
period of time T = I/i. The project value at each stage of the investment can be
assessed based on the remaining investment I
k
at that stage by simply replacing V
0
and I
0
in Eq. (
8
)byV
m
and I
m
, respectively.
Finally, the investment problem is analyzed for the case where the decision-
maker decides to spread the $10-million investment over a period of time T with a
maximum investment per time period i. The building will only be operational after
all of the planned refurbishment are completed over the period of time T. Figure
5
shows the cutoff values of V* at each time period for the case where the investment
is divided into T = 2 years and T = 5 years, respectively.
