Discrete dynamic systems
109
(a)
FV
=
A
(1
+
r
)
n
−
1
r
=
1000
(1
+
0
.
065)
10
−
1
0
.
065
=
£
13494
.
40
(b)
PV
=
A
1
−
(1
+
r
)
−
n
r
=
1000
1
−
(1
+
0
.
065)
−
10
0
.
065
=
£
7188
.
83
Discounting is readily used in investment appraisal and cost–benefit analysis.
Suppose
B
t
and
C
t
denote the benefits and costs, respectively, at time
t
. Then
the present value of such ﬂows are
B
t
/
(1
+
r
)
t
and
C
t
/
(1
+
r
)
t
, respectively. It
follows, then, that the
net present value
,
NPV
, of a project with financial ﬂows over
n
periods is
NPV
=
n
t
=
0
B
t
(1
+
r
)
t
−
n
t
=
0
C
t
(1
+
r
)
t
=
n
t
=
0
B
t
−
C
t
(1
+
r
)
t
Notice that for
t
=
0 the benefits
B
0
and the costs
C
0
involve no discounting. In
many projects no benefits accrue in early years only costs. If
NPV
>
0 then a
project (or investment) should be undertaken.
Example 3.9
Bramwell plc is considering buying a new welding machine to increase its output.
The machine would cost
£
40,000 but would lead to increased revenue of
£
7,500
each year for the next ten years. Half way through the machine’s lifespan, in year 5,
there is a oneoff maintenance expense of
£
5,000. Bramwell plc consider that the
appropriate discount rate is 8%. Should they buy the machine?
NPV
= −
40000
+
10
t
=
1
7500
(1
+
r
)
t
−
5000
(1
+
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 Fall '11
 Dr.Gwartney
 Economics, Time Value Of Money, Net Present Value, Bramwell plc

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