Blank L., Tarquin A. Engineering Economy (McGraw-Hill Series in Industrial Engineering and Management)
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from migrating into the groundwater.
The
fill
area was 50,
000
square meters, and the
installed liner cost was $8 per square
meter. To recover the investment, the
owner charged
$10 for pickup loads, $25
for dump truck loads, and
$70
for com-
pactor truck loads.
If
the monthly distribu-
tion
is
200
pickup loads,
50
dump truck
loads, and
100 compactor truck loads,
what rate
of
return will the landfill owner
make on the investment
if
the fill area is
adequate for 4 years?
7.7 Swagelok Enterprises
is
a manufacturer
of
miniature fittings and valves.
Over
a 5-
year period, the costs associated with one
product line were as
follows: first cost
of
$30,000 and annual costs
of
$18,000. An-
nLial
revenue was $27,000, and the used
equipment was salvaged for
$4000.
What
rate
of
return did the company make on
this product?
7.8 Barron Chemical
Llses
a thermoplastic
polymer to enhance the appearance
of
cer-
tain
RV panels.
The
initial cost
of
one
process was $130,
000
with annual costs
of
$49,000 and revenues
of
$78,000 in year 1,
increasing by
$1000 per year. A salvage
value
of
$23,000 was realized when the
process was discontinued after 8 years.
What rate
of
return did the company make
on the process?
7.9 A graduate
of
New Mexico State Univer-
sity who built a successful business
wanted to start an
endowment
in
her name
that would provide scholarships to
IE
stu-
dents. She wanted the scholarships to
amount to
$10,000 per year, and she
wanted the first one to be given on the day
she made the donation (i.e., at time
0).
If
she planned to donate $100,
000
, what rate
of
return would the university have to
PROBLEMS
265
make in order to
be
able to award the
$10,
000
per year scholarships forever?
7.
10
PPG manufactures an epoxy amine that
is
used to protect the contents
of
polyethyl-
ene terephthalate
(PET) containers from
reacting with oxygen.
The
cash flow
(in
millions) associated with the process
is shown below.
Determine
the rate
of
return.
Year
o
I
2
3
4
5
6
Cost,
$
-10
-4
-4
-4
-3
-3
-3
Revenue, $
2
3
9
9
9
9
7.11 An entrepreneurial mechanical engineer
started a tire shredding business to take
advantage
of
a Texas state law that out-
laws the disposal
of
whole tires in sani-
tary landfills.
The
cost
of
the shredder
was
$220,000. She spent $15,000 to get
460-volt power to the site and another
$76,000 in site preparation. Through con-
tracts with tire dealers, she was paid $2 per
tire and handled an average
of
12,
000
tires
per month for 3 years.
The
annual operat-
ing costs for labor, power, repairs, etc.,
amounted to $1.05 per tire. She also sold
some
of
the tire chips to septic tank in-
stallers for use
in
drain fields. This en-
deavor netted
$2000 per month. After
3 years, she sold the equipment for
$100,000. What rate
of
return did she make
(a) per month and (b) per year (nominal
and effective)?
7.12 An Internet B to C company projected
the cash flows (in millions) page 266.
266
7.
13
7.14
CHAPTER
7 Rate
of
Return Analysis:
Single
Alternative
What annual rate
of
return will be realized
if the cash flows occur as projected?
Year
o
2
3
4
5
6-10
Expenses, $
- 40
-
40
- 43
- 45
- 46
- 48
- 50
Revenue, $
12
15
17
51
63
80
The U
ni
versity
of
California at San Diego is
considering a plan to build an 8-megawatt
cogeneration plant
to
provide for part
of
its
power needs. The cost
of
the plant
is
ex-
pected
to
be
$41
million. The university
consumes
55,000 megawatt-hours per year
at a cost
of
$120 per megawatt-hour. (a)
If
the university will be able to produce power
at one-half the cost that
it
now pays, what
rate
of
return will
it
make on its investment
if the power plant lasts
30 years?
(b)
If
the
university can sell an average
of
12,000
megawatt-hours per year back to the utility
at
$90 per megawatt-hour, what rate
of
re-
turn will
it
make?
A new razor from Gillette called the
M3Poweremits pulses that cause the skin to
prop up hair so that
it
can be cut off more
easily. This might make the blades last
longer because there would be less need
to
repeatedly shave over the same surface. The
M3Power system (including batteries) sells
for $14.99 at some stores. The blades cost
$10.99 for a package
of
four. The more con-
ventional M3Turbo blades cost $7.99 for a
package
of
four.
If
the blades for the
M3Power system last 2 months while the
blades for the M3Turbo last only 1 month,
what rate
of
return (a) per month and (b) per
year (nominal and effective) will be made
if
a person purchases the M3Power system?
Assume the person already has an M3Turbo
razor but needs to purchase blades at time
O.
Use a I-year project period.
7.
15
7.16
7.17
Techstreet.com is a small Web design
business that provides services for two
main types
of
websites: brochure sites and
e-commerce sites.
One package involves
an up-front payment
of
$90,000 and
monthly payments
of
1.4¢ per "hit." A
new
CAD
software company is consider-
ing the package.
The
company expects to
have at least
6000 hits per month, and
it
hopes that 1.5%
of
the hits will result
in
a
sale.
If
the average income from sales
(after fees and expenses)
is
$150, what
rate
of
return per month will the
CAD
soft-
ware company realize if it uses the website
for 2 years?
A plaintiff in a successful lawsuit was
awarded a
judgment
of
$4800 per month
for 5 years.
The
plaintiff needs a fairly
large sum
of
money now for an investment
and has offered the defendant the opportu-
nity to
payoff
the award in a lump-sum
amount
of
$110,000.
If
the defendant ac-
cepts the offer and pays the
$110,000
now, what rate
of
return will the defendant
have made on the
"investment"? Assume
the next
$4800 payment
is
due 1 month
from now.
Army Research Laboratory scientists de-
veloped a diffusion-enhanced adhesion
process that is expected to significantly
improve the performance
of
multifunction
hybrid composites.
NASA engineers esti-
mate that composites made using the new
process will result
in
savings
in
many
space exploration projects. The cash flows
for one project are shown below. Deter-
mine the rate
of
return per year.
Ye
ar
t
Cost
($1000)
o
-210
-150
Savings
($1000)
2-5
100 +
60(1
- 2)
7.18 ASM International,
an
Australian steel
company, claims that a savings
of
40%
of
the cost
of
stainless steel threaded
bar
can be achieved
by
replacing machined
threads with precision weld depositions.
A
U.S. manufactur
er
of
rock bolts and
grout-in fittings plans to purchase the
equipment. A mechanical engineer with the
company has prepared the following cash
flow
estimates. Determine the expected
rate
of
return per quarter and per year
(nominal).
Quarter
Cost,
$
Savings, $
0
- 450,000
- 50,000
10
,000
2
- 40,000 20,000
3
-30,000
30,000
4
-20,000
40,000
5
-
10
,000 50,000
6-
12
80,000
7.19
An
indium-gallium-arsenide-nitrogen alloy
developed at Sandia National Laboratory
is
said to have potential uses in electricity-
generating solar cells. The new material
is
expected
to
have a longer life, and it
is
be-
lieved to have a
40% efficiency rate,
which
is
nearly twice that
of
standard sili-
con solar cells. The useful life
of
a
telecommunications satellite could be ex-
tended from
10
to
15
years by using the
new solar cells. What rate
of
return could
be realized if an extra investment now
of
$950,000 would result
in
extra revenues
of
$450,000
in
year
II,
$500,000 in year 12,
and amounts increasing by
$50,000 per
year through year I5?
7.20 A permanent endowment at the University
of
Alabama is to award scholarships to en-
gineering students. The awards are to be
made beginning 5 years after the $10 mil-
lion lump-sum donation
is
made.
If
the
PROBLEMS
267
interest from the endowment
is
to fund
100 students each year in the amount
of
$10,000 each, what annual rate
of
return
must the endowment fund earn?
7.21 A charitable foundation received a dona-
tion from a wealthy building contractor
in
the amount
of
$5 million. It specifies that
$200,000 is to be awarded each year for
5 years starting
now
(i.e., 6 awards) to a
university engaged in research pertaining
to the development
of
layered composite
materials. Thereafter, grants equal to the
amount
of
interest earned each year are to
be made.
If
the size
of
the grants from
year 6 into the indefinite future is expected
to be
$1,000,000 per year, what annual
rate
of
return
is
the foundation earning?
Multiple ROR Values
7.22 What
is
the difference between a conven-
tional and a nonconventional cash flow
series?
7.23 What cash flows are associated with
Descartes' rule
of
signs and Norstrom's
criterion?
7.24 According to Descartes' rule
of
signs, how
many possible
i*
values are there for net
cash flows that have the following signs?
(a)
---++
+
-+
(b)
------
+++++
(c)
++++------+-+
---
7.25 The cash flow (in
WOOs)
associated with a
new method
of
manufacturing box cutters
is
shown on page 268 for a 2-year period.
(a) Use Descartes' rule to determine the
maximum number
of
possible rate
of
re-
turn values. (b) Use Norstrom's criterion
to determine
if
there is only one positive
rate
of
return value.
268
CHAPTER 7
Rate
of
Return Anal
ys
i
s:
Single A
lt
erna
ti
ve
Quarter
Expense,
$
Revenue,
$
0
- 20
0
I
- 20
5
2
-
10
10
3
- 10
25
4
-
10
26
5
-
10
20
6
- 15
17
7
- 12
15
8
-
15
2
7.26 RKI
In
struments manufactures a ventila-
ti
on
co
ntroll
er
designed for monitoring and
controlling carbon monoxide
in
parking
garages, boiler rooms, tunnels, etc.
Th
e net
cash flow associated with one phase
of
the
operation is sh
ow
n below. (a) H
ow
many
possible rate
of
r
et
urn va
lu
es are there for
this cash
fl
ow
seri
es?
(b) Find a
ll
the rate
of
return va
lu
es between 0% and 100%.
Year
Net
Cash Flow, $
o - 30,000
I 20,000
2 15,000
3 - 2,000
7.27 A manufacturer of h
eavy
-t
ow
carbon
fibers (used for sport
in
g
goods
, thermo-
plastic
compo
unds, windmill blades, etc.)
reported the n
et
cash flows belo
w.
(0)
De-
termine the
numb
er
of
possible rate
of
return va
lu
es, and
(b)
find a
ll
rate
of
return
va
lu
es between -
50
% and 120%.
7.28
Year
Net
Cash Flow, $
o -
17
,000
20,000
2 - 5,000
3 8,000
Arc-bot Technologies, manufacturers
of
six-axis, electric
servo-dr
iven robots, has
expe
ri
enced the cash flows bel
ow
in
a
shipping department.
(a)
Det
ermine the
number
of
possible rate
of
return va
lu
es.
(b) Find a
ll
i*
va
lu
es between 0% and
100%.
Year
Expense,
$
Savings, $
0
- 33,000
0
I -
15
,000
18,000
2
-40
,000
38
,000
3
-20
,000
55
,000
4 -
13
,000
12
,000
7.29
Five
years ago, a
company
made
a $5 mil-
lion investment in a n
ew
high-temperature
material.
The
produ
ct
was n
ot
well ac-
ce
pted after the first year on the market.
However, when it was reintroduced 4 years
later, it did
se
ll well during the year.
Major
research funding to broaden the appl ica-
tions has cost $
15
million
in
year
5. Deter-
mine the rate
of
return for these cash flows
(sh
own
below in
$1OOO
s).
Year
Net
Cash Flow, $
0
- 5,000
I
4,000
2
0
3
0
4
20,000
5
- 15,000
Composite Rate
of
Return
7.30
What
is
meant by the term reinvestment
rate?
7.31
An
engineer
working for
GE
invested
hi
s
bonus money each
year
in
co
mpany stoc
k.
His bonus has been
$5000
eac
h
year
for
the past 6 years (i.e., at the end
of
years I
through 6). At the end
of
year 7, he sold
$9000
wo
rth
of
hi
s stock to remodel his
kitchen (he
didn't
purchase any stock that
year).
In
years 8 through
10
, he again in-
vested his
$5000
bonus.
The
engineer
sold
a
ll
his remaining stock for $50,000 imme-
diately after the last investment at the e
nd
of
year
10. (a) Determine the
number
of
possible rate
of
return values in the net
cash flow series.
(b) Find the internal rate
of
return(s). (c) Determine the
composite
rate
of
return. Use a reinvestment rate
of
20
% per year.
7.32 A company that makes clutch disks for
race cars had the cash flows shown below
for one department. Calculate
(a) the in-
ternal rate
of
return and (b) the
composite
rate
of
return, using a reinvestment rate
of
15% per year.
Year
Cash Flow,
$1000
0
-
65
I
30
2
84
3
- ]0
4
-
12
7.33 For
th
e cash flow
se
ries below, calculate
th
e
co
mposite rate
of
return, using a rein-
vestment rate
of
14
% per
year
.
Year
Cash Flow,
$
0
3000
I
- 2000
2
1000
3
-6
000
4 3800
7.
34
For
the high-temperature material project
in
Problem 7.29,
determine
the
composite
rate
of
return if the reinvestment rate is
15
% per year.
The
cash flows (repeated
below) are
in
$1000
units.
Year
Cash Flow,
$
0
- 5,000
I
4,000
2 0
3
0
4
20,000
5
-
15,000
PROBLEMS
269
Bonds
7.35 A municipal
bond
that was issued by
th
e
city
of
Phoenix
3 years ago has a face
value
of
$25,000
and a bond interest rate
of
6%
per
year
payable
semiannually.
If
the
bond
is
due
25 years after it was
issued,
(a)
what
are the
amount
and
fr
e-
quency
of
the bond interest payments and
(b)
what
value
of
n must
be
u
se
d
in
the P / A
formula to find the pre
se
nt worth
of
the re-
maining bond interest
payments
?
Assume
the market interest
rate
is
8%
p
er
year,
compounded
semiannua
ll
y.
7.36 A $10,000 mortgage bond with a bond in-
terest rate
of
8%
per
year,
payable
quar-
terly, was purchased for
$9200.
The
bond
was
kept
until it was
due
, a total
of
7 years.
What
rate
of
return was made by
the purchaser
per
3 months and per
year
(nomina!)?
7.37 A plan for remodeling the
downtown
area
of
the city
of
Steubenville,
Ohio
, required
the city to issue $5 million worth
of
gen-
eral obligation bonds
for
infrastructure re-
placement.
The
bond interest rate was set
at 6% per year,
payable
quarterly, with the
principal
repayment
date
30
years into the
future.
The
brokerage fees for the transac-
tions amounted to
$100,000.
If
th
e city re-
ceived
$4.6
million (before paying the
brokerage fees) from the bond issue,
(a)
what
interest rate (
per
quarter) did the
investors require to purchase the bonds
and
(b)
what
are the nominal and effective
rates
of
return per
year
to the investors?
7.38 A collateral
bond
with a face value
of
$5000
was purchased by an investor for
$4100.
The
bond
wa
s due in
II
years, and
it had a bond int
erest
rate
of
4% per year,
payable
se
miannually.
If
the
inve
stor kept
the bond to maturity,
what
rate
of
return
per semiannual period did
she
make?
270 CHAPTER 7
Rate
of
Return Analysis: Single Alternative
7.39 An engineer planning for his child
's
col-
lege education purchased a zero coupon
corporate bond
(i
.e., a bond that has no in-
terest payments) for $9250.
The
bond
has a face value
of
$50,000 and is due in
18
years. If the bond is held to maturity,
what rate
of
return will the engineer make
on the investment?
7.40 Four years ago, Texaco issued $5 million
worth
of
debenture bonds having a bond
interest rate
of
10
% per year, payable
semiannually. Market interest rates
dropped, and the
company
called the
bonds (i.e., paid them
off
in
advance) at a
10
% premium on the face value (i.e., paid
$5.5 million to retire the bonds).
What
semiannual rate
of
return did an investor
make
if
he purchased
one
$5000 bond at
FE
REVIEW PROBLEMS
7.42 When the net cash flow for an alternative
changes signs more than once, the cash
!-low
is
said to be
Co)
Conventional
Cb)
Simple
Cc)
Extraordinary
Cd)
Nonconventional
7.43 According to Descartes' rule
of
signs, how
many possible rate
of
return values are
there for net cash flow that has the follow-
ing signs?
++++
-
-----+-+---++
Ca)
3
Cb)
5
(c)
6
(d)
Less than 3
7.44 A small manufacturing company bor-
rowed
$ 1 million and repaid the loan
face value 4 years ago and held it until
it
was called 4 years later?
7.41 Five years ago,
GSI
, an oil services com-
pany, issued
$10
million worth
of
12%,
30-year bonds with interest payable quar-
terly.
The
interest rate in the marketplace
decreased enough that the
company
is
con-
sidering calling the bonds.
If
the company
buys the bonds back now for
$11
million,
(0)
what rate
of
return per quarter will the
company
make on the
$11
million expen-
diture and
(b)
what nominal rate
of
return
per year will it make
on
the $11 million
investment? Hint: By spending
$11
mil-
lion now, the company will not have to
make the quarterly bond interest payments
or
pay the face value
of
the bonds when
they
come
due 25 years from now.
through monthly payments
of
$20,000 for
2 years plus a single lump-sum payment
of
$1
million at the end
of
2 years.
The
inter-
est rate on the loan was closest to
(0) 0.5% per month
(b)
2%
per month
(c) 2% per year
(d)
8% per year
7.45 According to
Norstrom's
criterion, there
is
only
one
positive rate
of
return value
in
a
cash flow series when
(a)
The
cumulative cash flow starts out
positive and changes sign only once.
(b)
The
cumulative cash flow starts out
negative and changes sign only once.
(c)
The
cumulative cash flow total
is
greater than zero.
(d)
The
cumulative cash flow total
is
less than zero.
7.46 An investment
of
$60
,000 resulted
in
uniform income
of
$10,000
per
year for
10
years.
The
rate
of
return on the invest-
ment was closest to
(a) 10.6% per year
(b) 14.2% per year
(c)
16.4% per year
(d)
18.6% per year
7.47
For
the net cash flows shown below, the
maximum
number
of
possible rate
of
return solutions is
(a) 0
(b) I
(c) 2
(d)
3
Year
Net
Cash Flow, $
0
- 60,000
1 20,000
2 22,000
3
15
,000
4
35
,000
5
13,000
6
- 2,000
7.48 A bulk materials hauler purchased a used
dump
truck for
$50
,000.
The
operating cost
was $5000 per month, with average rev-
enues
of
$7500 per month. After 2 years,
the truck was sold for $
11
,000.
The
rate
of
return was closest to
(a) 2.6% per month
(b) 2.6% per year
(c) 3.6% per month
(d)
3.6% per year
7.49 Assume you are told that by investing
$100,000 now, you will receive $10,000
per year starting in ye
ar
5 and continuing
forever. If you accept the offer, the rate
of
return on the investment is
(a) Less than
10
% per year
(b) 0% per
year
(c)
10
% per year
(d)
Over
10
% per year
FE REVIEW PROBLEMS
271
7.
50
Five years ago, an alumnus
of
a small uni-
versity donated
$50,000
to establish a per-
manent
endowment
for scholarships.
The
first scholarships
were
awarded I year
after the money was donated. If the
amount awarded each
year
(i
.e., the inter-
est) is $4500, the rate
of
return earned on
the fund
is
closest to
(a) 7.5% per year
(b) 8.5% per year
(c) 9% per year
(d)
10%
per
year
7.
51
When positive net cash flows are gener-
ated before the
end
of
a project, and when
these cash flows are reinvested at an inter-
est
rate that
is
greater than the internal rate
of
return,
(a)
The
resulting rate
of
return is equal
to the internal rate
of
return.
(b)
The
resulting rate
of
return is less
than the internal rate
of
return.
(e)
The
resulting rate
of
return is equal
to the reinvestment rate
of
return.
(d)
The
resulting rate
of
return
is
greater
than the internal rate
of
return.
7.
52
A $10,000 mortgage bond that is
due
in
20
years pays interest
of
$250
every
6 months.
The
bond interest rate is closest to
(a) 2.5% per year, payable quarterly
(b) 5.0% per year, payable quarterly
(c) 5%
per
year, payable semiannually
(d)
10% per
year
,
payable
quarterly
7.53 A $10,000 bond that matures
in
20 years
with interest at 8% per year payable quar-
terly was issued 4 years ago.
If
the bond
is
purchased now for
$10
,
000
and held to
maturity,
what
will be the effective rate
of
return
per
quarter to the purchaser?
(a) 2%
(b) 2.02%
(c) 4%
(d)
8%
272 CHAPTER 7 Ra
te
of
Return Analysis: Single
Alt
ernative
7.S4
A person purchases a
$SOOO
, S% per
year
bond, with interest payable semiannually,
for the
amount
of
$4000.
The
bond has a
maturity date
of
14
years from now.
The
equation for calculating how much the per-
son must se
ll
the bond for 6 years from now
in
order to make a rate
of
return
of
12 %
per
year, compounded semiannually, is
(d) 0 = - 4000 +
12S(P
/A,12%,6) +
x(P/F,12%,6)
7.SS
A
$SO
,
OOO
corporate
bond
due
in
20 years
with an interest rate
of
10
% per year,
payable quarterly, is for sale for
$SO
,
OOO.
If
an investor purchases the bond and
holds
it
to maturity, the rate
of
return
wi
II
be closest to
(a) 0 = - 4000 + 12S(P/A,6%,12) +
x(P / F,6%,
12)
(b) 0 = - 4000 + 100(P/A,6
%,
12)
+
x(P / F,6
%,
12)
(c) 0 = -
SOOO
+ 12S(P/A,6
%,
12
) +
x(P/F,6%,
12)
EXTENDED EXERCISES
(a) Nominal 10% per year,
compounded
quarterly
(b)
2.S
% per quarter
(c) Both (a) and (b) are correct
(d)
Effective 10% per year
EXTENDED EXERCISE
I-THE
COST OF A POOR
CREDIT RATING
Two people each borrow
$SOOO
at a 10% per year interest rate for 3 years. A por-
tion
of
Charles's loan agreement states that interest " .
..
is paid at the rate
of
10
%
compounded
each ye
ar
on the declining balance." Charles is told his annual pay-
ment will be $
20
I
0.S7,
due at the
end
of
each year
of
the loan.
Jeremy currently has a slightly degraded credit rating, which the bank loan of-
ficer discovered. Jeremy has a habit
of
paying his bills late.
The
bank approved
the loan, but a part
of
his loan agreement states that interest "
..
. is paid at a rate
of
I 0%
compounded
each year on the original loan amount." Jeremy
is
told his
annual payment will be $2166.67 due at the end
of
each year.
Questions
Answer the following by hand, by computer,
or
both.
I.
Develop a table and a plot for Charles and for Jeremy
of
the unrecovered
balances (total amount owed)
just
before each payment is due.
2. How much more total money and interest will Jeremy pay than Charles
over
the 3 years?
EXTENDED EXERCISE
2-WHEN
IS
IT
BEST
TO SELLA BUSINESS?
After
Jeff
finished medical school and Imelda completed a
degree
in
engineering,
the couple decided to put a substantial part
of
their savings into rental property.
CASE
STUDY
With a hefty bank loan and a cash down payment
of
$120,000
of
their own funds,
they were able to purchase six duplexes from a person exiting the residential
rental business. Net cash flow on rental income after all expenses and taxes for
the first 4 years was good:
$25,000 at the end
of
the first year, increasing by
$5000 each year thereafter. A business friend
of
Jeff's introduced him to a poten-
tial buyer for all properties with an estimated
$225,000 net cash-out after the
4 years
of
ownership. But they did not sell. They wanted to stay
in
the business for
a while longer, given the increasing net cash flows they had experienced thus far.
During year 5,
an
economic downturn reduced net cash flow to $35,000. In
response,
an
extra $20,000 was spent in improvements and advertising
in
each
of
years 6 and 7, but the net cash flow continued to decrease by $10,000 per year
through year 7.
Jeff
had another offer to sell
in
year 7 for only $60,000. This was
considered too much
of
a loss, so they did not take advantage
of
the opportunity.
In
the last 3 years, they have expended $20,000, $20,000, and
$30
,
000
each
year
in
improvements and advertising costs, but the net cash flow from the busi-
ness has been only $
15,000, $10,000, and $10,000 each year.
Imelda and Jeff want out, but they have no offer to buy at any price, and they
have most
of
their savings committed to the rental property.
Questions
Determine the rate
of
return for the following:
I. At the end
of
year 4, first,
if
the $225,000 purchase offer had been accepted;
second, without selling.
2.
After 7 years, first,
if
the $60,000 "sacrifice" offer had been accepted; and,
second, without selling.
3. Now, after 10 years, with no prospect
of
sale.
4.
If
the houses are sold and given to a charity, assume a net cash infusion to
Jeff and Imelda
of
$25,
000
after taxes at the end
of
this year.
What
is the
rate
of
return over the 10 years
of
ownership?
CASE STUDY
BOB LEARNS ABOUT MULTIPLE RATES OF RETURN'
273
Background
When Bob began a summer internship with
VAC
, an
electricity distribution company
in
an
Atlantic coast
city
of
about 275,000, he was given a project
on
the
first day by his boss, Kathy. Homeworth, one
of
the
major corporate customers,
just
placed a request for a
lower rate per kwh, once its minimum required usage
'Contributed by Dr. Tep Sastri (former Associate Professor, Industrial Engineering, Texas
A&M
University).
274
CHAPTER 7 Rate
of
Return Analysis: Single Alternative
is exceeded each month. Kathy has
an
internal report
from the Customer Relations Department that itemizes
the net cash flows below for the Homeworth account
during the last
10
years.
Year
Cash Flow
($1
000)
1993 $200
1994 100
1995
50
1996
- 1800
1997 600
1998 500
1999
400
2000 300
2001 200
2002 100
The report also states that the annual rate
of
return
is
between
25
and
50
%, but no further information
is
provided. This information
is
not detailed enough for
Kathy to evaluate the company's request.
Over the next few hours, Bob and Kathy had a se-
ries
of
discussions as Bob worked to answer Kathy's
increasingly more specific questions. The following
is
an abbreviated version
of
these conversations. Luck-
ily, both Bob and Kathy took an engineeling economy
course during their undergraduate work, and their pro-
fessors covered the method to
find
a unique rate
of
re-
turn for any cash flow
se
rie
s.
Development
of
the Situation
I. Kathy asked Bob to do a preliminary study to
find
the correct rate
of
return.
She
wanted only
one number, not a range, and not two or three
possible values.
She
did, however, have a passing
interest
in
initially knowing the values
of
multi-
ple rates, if they do exist,
in
order to determine if
the report from customer relations was correct or
just
a "shot in the dark."
Kathy told Bob that the
MARR
for the com-
pany
is
15
% per year for these major clients.
She
also explained that the 1996 negative cash flow
was caused by
an
on-site equipment upgrade
when Homeworth expanded its manufacturing
capacity and increased power usage about 5-fold.
2.
Once Bob had finished his initial analysis, Kathy
told him that she had forgotten to tell him that the
rate
of
return earned externally on the positive
cash flows from these major clients is placed into
a venture capital pool headquartered
in
Chicago.
It has been making 35% per year for the last
decade.
She wanted to know
if
a unique return
still existed and
if
the Homeworth account was
financially viable
at
a MARR
of
35%.
In respon
se
to this request, Bob developed
the four-step procedure outlined below to
closely estimate the composite rate
of
return
i'
for any reinvestment rate c and two multiple
rates
i'l'
and if. He plans to apply this procedure
to answer this latest question and show the re-
sults to Kathy.
Step
1.
Determine the
i*
roots
of
the
PW
rela-
tion for the cash flow series.
Step 2. For a given reinvestment rate c and the
two
i*
values from step 1, determine
which
of
the following conditions
applies:
(a)
If
c <
ii',
then i' < if.
(b)
If
c >
if,
then i' > if.
(c)
If
iii'
< c < ii', then
i'
can be less
than
c or greater than
c,
and
if
<
i
'<
i
f.
Step 3. Guess a starting value for i' according
to the result from step 2. Apply the
net-investment method from periods
j
to
n.
Repeat this step until
F"
is
close
to O.
If
this
Fil
is
a small positive value,
guess another
i'
that
wi
II
result
in
a
small negative
F"
value, a
nd
vice
versa.
Step 4. Using the two
F"
results from step 3,
linearly interpolate i' such that the
corresponding
F"
is
approximately
zero.
Of
course, the final
i'
value can
also be obtained directly in step 3,
without interpolation.