H
AAS
S
CHOOL
OF
B
USINESS
U
NIVERSITY
OF
C
ALIFORNIA
AT
B
ERKELEY
UGBA 103
A
VINASH
V
ERMA
Name:
2.
Suppose you are given that an equipment costs $150,000 to buy at t=0, and the
operating and maintenance (O&M) costs at
t
=1,2,…8 are $25,000, $30,000,
$45,000, $45,000, $60,000, $60,000, $90,000, and $100,000 respectively. The
equipment is needed on an ongoing basis, and you are asked to decide how often
we should replace one unit with another in order to minimize the total costs (sum
of replacement and O&M costs). In other words, you are asked to work out the
economic life of the equipment. Assume that the discount rate is 16% per year.
Given:
Discount rate
16.0%
Time (t)
0
1
2
3
4
5
6
7
8
DF
t
1
0.8621
0.7432
0.6407
0.5523
0.4761
0.4104
0.3538
0.3050
Costs
150
25
30
45
45
60
60
90
100
PV
0
(Costs)
150
21.552
22.295
28.83
24.853
28.5668
24.63
31.8447
30.502546
Cumulative PV
0
(Costs)
171.55
193.85
222.68
247.53
276.096
300.7
332.567
363.06983
AF(20%, t)
0.8621
1.6052
2.2459
2.7982
3.2743
3.6847
4.0386
4.3436
EAC
t
199
120.76
99.148
88.461
84.3223
81.61
82.3479
83.587482
The table above works out the EAC and shows that it is lowest for 6 years.
ALTERNATIVE
: The answer above is confirmed by the table below, which
works out the present value of the perpetuity of costs for each replacement
policy:
Time (t)
0
1
2
3
4
5
6
7
8
Costs
150
25
30
45
45
60
60
90
100
DF
t
1
0.8621
0.7432
0.6407
0.5523
0.47611
0.41
0.35383
0.3050255
PV
0
(Costs)
150
21.552
22.295
28.83
24.853
28.5668
24.63
31.8447
30.502546
Cumulative PV
0
(Costs)
171.55
193.85
222.68
247.53
276.096
300.7
332.567
363.06983
t-yearly Discount rate
16%
35%
56%
81%
110%
144%
183%
228%
PV
-t
of the Perpetuity
1072.2
560.9
397
305.35
250.919
209.4
182.107
159.35194
PV
0
of the Perpetuity
1243.8
754.75
619.68
552.88
527.015
510.1
514.674
522.42176
The t-year discount rate is worked out as r
tyr
= 1.16
t
–1. Thus, r
2yr
= 1.16
2
–1 = 1.3456 –1 = 34.56%; r
3yr
= 1.16
3
–1 = 1.560896 –1 = 56.0896%; and so
on.
Solution to Summer 2009 Take-Home Mid-term Examination
2