Blank L., Tarquin A. Engineering Economy (McGraw-Hill Series in Industrial Engineering and Management)
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SECTION 2.7 Determination
of
an
Unknown Interest Rate
If
Laurel can make an investment in a friend's business
of
$3000 now in order
to
receive
$5000 five years from now, determine the rate
of
return.
If
Laurel can receive 7% per
year interest on a certificate
of
deposit, which investment should
be
made?
Solution
Since only single payment amounts are involved, i can
be
determined directly from the
PI
F factor.
. 1
P = F(PIF,I,n) = F(1 + i)"
3000 = 5000 _ 1_.
-5
(I + i)
1
0.600 = (I +
i)5
(
1
)0
.2
i = - - 1 = 0.1076 (10.76%)
0.6
Alternatively, the interest rate can be found by setting
up
the standard notation
PI
F
relation, solving for the factor value, and interpolating in the tables.
P = F(PIF,i,n)
$3000 = 5000(P I F,i,5)
(
PIF
i 5) = 3000 = 0.60
" 5000
From the interest tables, a
PI
F factor
of
0.6000 for n = 5 lies between 10 and II
%.
Interpolate between these two values
to
obtain i = 10.76%.
Since 10.76% is greater than the 7% available from a certificate
of
deposit, Laurel
should make the business investment. Since the higher rate
of
return would be received
on
d1e
business investment, Laurel would probably select this option instead
of
ilie cer-
tificate
of
deposit. However, the degree
of
risk associated with the business investment
was not specified. Obviously, risk is an important parameter that may cause selection
of
the lower rate
of
return investment. Unless specified to the contrary, equal risk for all al-
ternatives
is
assumed
in
this text.
The
IRR
spreadsheet function is one
of
the most useful
of
all those available.
IRR
means internal rate
of
return, which is a topic unto itself, discussed in detail
in Chapter
7.
However, even at this early stage
of
engineering economic analy-
sis, the
IRR
function can be used beneficially to find the interest rate (or rate
of
return) for any cash flow series that is entered into a series
of
contiguous spread-
sheet cells, vertical or horizontal.
It
is very important that any years (periods)
with a zero cash flow have an entry
of
'0
' in the cell. A cell left blank is not suf-
ficient, because an incorrect value
of
i will be displayed by the
IRR
function. The
basic format
is
IRR(firsCcell:lasCcell)
75
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E-Solve
76
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CHAPTER 2 Factors: How Time and Interest Affect Money
The
firsccell
and
lasccell
are the cell references for the start and end
of
the cash
flow series. Example 2.13 illustrates the IRR function.
The RATE function, also very useful,
is
an alternative to IRR. RATE
is
a one-
cell function that displays the compound interest rate (or rate
of
return) only
when the annual cash flows, that is, A values, are the same.
Present and future
values different from the A value can be entered.
The
format is
RATE(
number
_ye
ar
s,A,P
,F)
The
F value does not include the amount A that occurs in year
n.
No entry into
spreadsheet cells
of
each cash flow
is
necessary to use RATE, so it should be
used whenever there
is
a uniform series over n years with associated P and/or F
values stated. Example 2.13 illustrates the RATE function.
EXAMPLE
2.13
<:-
Professional Engineers, Inc., requires that $500 per year be placed into a sinking fund ac-
count
to
cover any unexpected major rework on field equipment. In one case, $500 was
deposited for
IS
years and covered a rework costing $10,000
in
year IS.
What
rate
ofre-
turn did this practice provide to the company? Solve by band and by computer.
Solution
by
Hand
The
cash flow diagram is shown in Figure 2-20. Either the
AI
For
F I A factor can be used.
UsingAIF
,
A
= F(AI F,i,n)
500 =
JO,000(A
IF,i,15)
(AI
F,
i,
IS
) = 0.0500
From interest Tables 8 and 9 under the
AI
F column for
IS
years, the va
lu
e 0.0500 lies be-
tween 3 and 4%. By interpolation,
i = 3.98%. (This
is
considered a low return for an engi-
neering project.)
F = $10,000
i = ?
2 3 4 5 6 7 8 9
10
J 1 12
13
14 15
A
= $500
Solution by
Computer
Figure
2-20
Diagram to determine the rate
of
return, Example 2.13.
Refer
to
the cash flow diagram (Figure 2- 20) while completing the spreadsheet (Fig-
ure 2-
21).
A single-cell solution using the RATE function can be applied since A = $- 500
occurs each year and the F = $10,000 value takes place in the last year
of
the series.
Ce
ll
SECTION 2.8
Determination
of
Unknown Number
of
Years
A3
contains the function RATE(l5, -
500"
I 0000), and the answer displayed
is
3.98%. The
minus sign on 500 indicates the annual deposit. The extra comma
is
necessary to indicate
that
no
P value
is
present. This function
is
fast, but
it
allows only limited sensitivity analy-
si
s;
all the A values have to change by the same amount.
The
fRR
function
is
much better
for answering
"what
if?" questions.
To
apply the
TRR
function and obtain the same answer, enter the value 0 in a
ce
ll
(for year 0), followed by - 500 for
14
years and 9500 (from 10,000 - 500)
in
year
15.
Figure 2-
21
contains these numbers
in
cells D2 through
017.
In any cell on the spread-
sheet, enter the
IRR function
IRR(02:0
17).
The
answer i = 3.98%
is
displayed in cell E3.
It
is advisable
to
enter the year numbers 0 through n (15 in this example)
in
the column im-
mediately to the left
of
the cash flow entries.
The
IRR function does not need these num-
bers, but it makes the cash flow entry activity easier and more accurate. Now any cash flow
can be changed, and a new rate will be displayed immediately via
TRR.
Figure
2-21
Spreadsheet
solution for rate
of
return using the RATE and
TRR
functions,
Example
2.13.
2.8 DETERMINATION
OF
UNKNOWN
NUMBER
OF
YEARS
It is sometimes necessary to determine the
number
of
years (periods) required for
a cash flow series to provide a stated rate
of
return.
Other
times
it
is desirable to de-
termine when specified amounts will be available from an investment. In both
cases, the unknown value is
11.
Techniques similar to those
of
the preceding section
are used to find
11.
Some
problems can be solved directly for n by manipulation
of
77
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78
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CHAPTER 2
Factors: How Time a
nd
Interest Affect Money
the s
in
gle-payment and uniform-series formulas.
In
other cases, n is found
through interpolation in the interest tables, as illustrated below.
The
spreadsheet function
NPER
can
be
used to quickly find the
number
of
years (periods)
11
for given A, P, and/or F values.
The
format is
NPER
(i%
,
A,
P
,F)
If
th
e future value F is not involved, F is omitted; however, a present worth P and
unifo
rm
amount
A
mu
st be entered. The A entry can be zero when only single
amounts
P and F are known, as
in
the next example. At least
one
of
the entries
must have a sign opposite the others to obtain an answer from
NPER.
EXAMPLE 2.14
.%".
How long will
it
take for $1000 to double
if
the interest rate is 5% per year?
Solution
The n value can be determined using either the F I P or
PI
F factor. Using the P
IF
factor,
P =
F(P
I F,i,
n)
1000 =
2000(PIF
,5%,n)
(PI
F,5%,
/!)
= 0.500
In the 5% interest table, the value 0.500 lies between
14
and IS years. By interpolation,
11
= 14.2 year
s.
Use the function NPER(5%,0
,-1000,2
000) to display an n value
of
14.2 1 years.
2.9 SPREADSHEET
APPLICATION-BASIC
SENSITIVITY ANALYSIS
We
have performed engineering economy computations with the spreadsheet
fu
nctions
PV,
FV,
PMT,
IRR
, and
NPER
that were introduced in Section 1.8.
Most
fu
nctions took o
nl
y a single spreadsheet cell to find the answer.
The
exam-
ple below illustrates h
ow
to solve a slightly more complex problem that involves
sensitivity analysis; that is, it he
lp
s answer
"what
if?" question
s.
EXAMPLE 2.15 \
".
~
E·Solve
An
engineer and a medical doctor have teamed up to develop a major improvement
in
lap-
aroscopic surgery for gallbladder operations. They formed a small bus
in
ess corporation to
handle
th
e financial aspects
of
their partnership.
The
company has invested $500,000
in
the
project already this year
(t
= 0), and it expects to spend $500,000 annually for the n
ext
4
years, and possibly for more years. Develop a spreadsheet that helps an
swer
the following
questions.
(a) Assume the $500,000 is expended for only 4 additional years.
If
the company seUs
th
e
rights to use the n
ew
technology at the end
of
year 5 for $5 million, what is the antic-
ipated rate
of
return?
SECTION 2.9
Spreadsheet
Application-Basic
Sensitivity Analysis
(b) The engineer and doctor estimate that they will need $500,000 per year for more
than 4 additional years. How many years from now
do
they have to finish their de-
velopment work and receive the $5 million license fee to make at least
LO%
per
year? Assume the
$500,000 per year is expended through the year immediately
prior
to
the receipt for the $5 million.
Solution by
Computer
Figure
2-22
presents the spreadsheet, with all financial values
in
$LOOO
units. The IRR
function is used throughout.
(a)
The
function
lRR(B6:Bll)
in
cell
BI5
displays i = 24.07%. Note there is a cash
flow
of
$-
500
in
year
O.
The equivalence statement is: Spending $500,000 now and
$500,000 each year for 4 more years
is
equivalent to receiving $5 million at the end
of
year 5, when the interest rate is 24.07% per year.
(b) Find the rate
of
return for an increasing number
of
years that the $500
is
expended.
Columns C and D
in
Figure 2- 22 present the results
of
lRR
functions with the
$5000 cash flow
in
different years. Cells
Cl5
and
DI5
show returns
on
opposite
sides
of
10%. Therefore, the $5 million must be received some time prior to the end
of
year 7
to
make more than the 8.93% shown
in
cell D
15
.
The
engineer and doctor
have less than 6 years to complete their development work.
X M IClosoll E
.cel
. E .ample 2 15
IlIiI
Ei
A20
A
2
3
Part (a) Part (b)
4
Find i
Fnd
n such that i > 10%
5
Year
Ge
t
$5
million
in
yr
5
Get
$5
million in r 6
Ge
t
$5
million in yr 7
6 0
$ (500) $ (500) $ (500)
7
1
$ (500) $
(SOD)
$ (500)
8 2
$ (500) $ (500) $ (500)
9 3 $ (500) $
(SOD)
$ (500)
10
4 $ (500) $ (500) $ (500)
11
5
$ 5,000 $ (500) $ (500)
12
6 $ 5,
000
$ (500)
J.l
7 $ 5,000
14
15
Rate
of returns
16
17
18
893%
[
~
IRR(D6.D
13)
~
19
I.
~A
~
I
Ready
Figure
2-22
Spr
ea
dsheet solution including
se
nsitivity
ana
lysis, Example
2.1
S.
79
80
CHAPTER 2 Factors: How Time and Interest Affect Money
ADDITIONAL EXAMPLE
EXAMPLE 2.16
",r"
P, F, AND A CALCULATIONS
Ex
pl
a
in
why the uniform series factors ca
nn
ot
be used to compute P or F directly for
any
of
the cash flows shown in Figure 2-23.
$100 $100 $
100
$100
Figure
2-23
t t t t
Sample cash flow
al
diagrams.
2 3
4
5
Example 2.16.
p=?
(a)
A = $550
t t
t
t
'l
0
2 3 4
F=?
(b)
$1300
$1200
$1100
0
2
3
4
5
pc = ?
(e)
$150 $150
$100
t
31
t
0
2
F= ?
(d)
Solution
(a) The P / A factor cannot be used
to
compute P s
in
ce the $100 p
er
year receipt does
not occur each year from 1 through
5.
PROBLEMS
(b) Since there
is
no
A = $550 in year 5, the F / A factor cannot be used.
The
relation
F = 550(F / A,i,4) would furnish the future worth
in
year 4, not year
5.
ec)
The
first gradient amount G = $100 occurs in year 3. Use
of
the relation P G =
1
OO(P
/
G,i
%,4) will compute P G
in
year I, not year
O.
(The present worth
of
the
base amount
of
$1000
is
not included here.)
(d) The receipt values are unequal; thus the relation F =
A(F
/ A,i,3) cannot be used
to compute
F.
CHAPTER
SUMMARY
Formulas and factors derived and applied in this chapter perform equivalence
calculations for present, future, annual, and gradient cash flows. Capability in
using these formulas and their standard notation manually and with spreadsheets
is
critical to complete an engineering economy study. Using these formulas and
spreadsheet functions, you can convert single cash flows into uniform cash
flows, gradients into present worths, and much more.
You
can solve for rate
of
return i
or
time
n.
A thorough understanding
of
how to manipulate cash flows
using the material
in
this chapter will help you address financial questions
in
pro-
fessional practice as well as
in
everyday living.
PROBLEMS
81
Use of Interest Tables
2.1
Find the correct numerical value for the
following factors from the interest tables.
I. (FIP,8%,
25
)
year, what would be the equivalent value
today
of
that $140,000 expenditure?
2.
(PIA,3%,8)
3.
(PIG,9%,20)
4.
(FIA, 15%,18)
5.
(AIP,
30
%,15)
Determination of
F,
P,
and
A
2.2
The
U.S. Border Patrol
is
considering the
purchase
of
a new helicopter for aerial
survei lIance
of
the New
Mexico-Texas
border with Mexico. A similar helicopter
was purchased 4 years ago at a cost
of
$140,
000
. At an interest rate
of
7% per
2.3 Pressure Systems, Inc., manufactures
high-accuracy liquid-level transducers.
It
is investigating whether it should update
certain equipment now
or
wait to do it
later.
If
the cost now is $200,000, what
will the equivalent amount be 3 years from
now
at
an interest rate
of
10%
per
year?
2.4 Petroleum Products, Inc., is a pipeline
company that provides petroleum prod-
ucts to wholesalers in the northern United
States and Canada. The company
is
con-
sidering purchasing insertion turbine
flowmeters to allow for better monitoring
82
CHAPTER 2 Factors: How Time and Interest Affect Money
of
pipeline integrity. If these meters would
prevent one major disruption (through
early detection
of
product loss) valued at
$600,000 four years from now, how much
could the company afford to spend now at
an
interest rate
of
12% per year?
2.5 Sensotech Inc., a maker
of
microelectro-
mechanical systems, believes it can reduce
product recalls
by
10
% if it purchases new
software for detecting faulty parts. The
cost
of
the new
sof
tware
is
$225,000.
(a) How much would the company have to
save each year for 4 years to recover its in-
vestment if it uses a minimum attractive
rate
of
return
of
15%
per
year? (b) What
was the cost
of
recalls per year before the
software was purchased
if
the company
did exactly recover its investment
In
4 years from the
10
% reduction?
2.6 Thompson Mechanical Products
is
plan-
ning to set aside
$ 150,000 now for possi-
bly replacing
it
s large synchronous refiner
motors ·whenever
it
becomes necessary.
If
the replacement isn't needed for 7 years,
how much will the company have
in
its in-
vestment set-aside account
if
it achieves a
rate
of
return
of
18
% per year?
2.
7 French car maker Renault signed a
$75 million contract with ABB
of
Zurich,
Switzerland, for automated underbody as-
sembly lines, body assembly workshops,
and line control systems.
If
ABB will be
paid
in
2 years (when the systems are
ready), what
is
the present worth
of
the
contract at
18
% per year interest?
2.8 Atlas Long-Haul Transportation
is
consid-
ering installing Valutemp temperature log-
gers
in
all
of
its refrigerated trucks for
monitoring temperatures during transit.
If
the systems will reduce insurance claims
by
$100,000 two years from now, how
much should the company be willing to
spend now if it uses an interest rate
of
12% per year?
2.9
GE
Marine Systems
is
planning to supply
a Japanese shipbuilder with aero-derivative
gas turbines to
power
11
DD-class de-
stroyers for the Japanese Self-Defense
Force.
The
buyer can pay the total contract
price
of
$1,700,000 now
or
an equivalent
amount 1 year from now (when the tur-
bines will be needed).
At
an interest rate
of
18
% per year, what is the equivalent future
amount?
2.10 What
is
the present worth
of
a future cost
of
$162,000 to Corning, Inc., 6 years from
now at an interest rate
of
12% per year?
2.
11
How much could Cryogenics Inc., a
maker
of
superconducting magnetic en-
ergy storage systems, afford to spend now
on new equipment
in
lieu
of
spending
$125,000 five years from now if the
company's rate
of
return is
14
% per year?
2
.1
2
V-
Tek Systems
is
a manufacturer
of
verti-
cal compactors, and it
is
examining its cash
flow requirements for the next 5 years.
The
company expects to replace office
machines and computer equipment at vari-
ous times over the 5-year planning period.
Specifically, the company expects to
spend
$9000 two years from now, $8000
three years from now, and $5000 five years
from now. What is the present worth
of
the
planned expenditures at an interest rate
of
10% per year?
2.
13
A proximity sensor attached to the tip
of
an endoscope could reduce risks during
eye surgery by alerting surgeons to the lo-
cation
of
critical retinal tissue.
If
a certain
eye surgeon expects that
by
using this
technology, he will avoid lawsuits
of
$1.25 and $0.5 million 2 and 5 years from
now, respectively, how much could he
afford to spend now if his out-of-pocket
costs for the lawsuits would be only 10%
of
the total amount
of
each suit? Use an in-
terest rate
of
8% per year.
2
.1
4
The
current cost
of
liabili
ty
insurance for a
certain consulting firm
is
$65,000.
[1'
the
insurance cost
is
expected to increase by
4% each year, what will be the cost 5 years
from now?
2.
15 American Gas Products manufactures a
device called a Can-Emitor that empties
the contents
of
old aerosol cans
in
2 to
3 seconds. This eliminates having to dis-
pose
of
the cans
as
hazardous wastes. If a
certain paint company can save
$75,000
per year
in
waste disposal costs, how much
could the company afford
to
spend now on
the Can-Emjtor if it wants to recover
it
s in-
vestment
in
3 years at an interest rate
of
20% per year?
2
.1
6 Atlantic Metals and Plastic uses austenitic
nickel-chromium alloys to manufacture
resistance heating wire.
The
company is
considering a new annealing-drawing
process to reduce costs.
If
the new process
will cost $1.8 million now, how much
must be saved each year to recover the in-
vestment
in
6 years at an interest rate
of
12
% per year?
2.
17 A green algae, Chlamydomonas rein-
hardtii, can produce hydrogen when tem-
porarily deprived
of
sulfur for up to 2 days
at a time. A small company needs to pur-
chase equipment costing $3.4 million to
commercialize the process.
If
the com-
pany wants to earn a rate
of
return
of
20%
per year and recover its investments in
8 years, what must be the net value
of
the
hydrogen produced each year?
2.18 How much money could RTT Environ-
mental Services borrow to finance a site
PROBLEMS
83
reclamation project if it expects revenues
of
$280,000 per year over a 5-year cleanup
period? Expenses associated with the pro-
ject
are expected to be $90,000 per year.
Assume the interest rate is 10% per year.
2.19 Western
Playland a
nd
Aquatics Park
spends $75,000 each year in consulting
services for ride inspection. New actuator
element technology enables engineers
to
simulate complex computer-controlled
movements
in
any direction.
How
much
could the amusement park afford to spend
now on the new technology if the annual
consulting services will
no
longer be
needed? Assume the park uses an interest
rate
of
15
% per year and it wants to re-
cover its investment
in
5 years.
2.20 Under an agreement with the Internet Ser-
vice Providers (ISPs) Association, SBC
Communications reduced the price it
charges
ISPs to resell its high-speed digi-
tal subscriber line
(DSL) service from
$458 to
$360 per year per customer lin
e.
A
particular
ISP, which has 20,000 cus-
tomers, plans to pass
90%
of
the sav
in
gs
along to its customers. What is the total fu-
ture worth
of
these savings over
as-year
horizon at an interest rate
of
8% per year?
2.21 To improve crack detection
in
aircraft, the
U.S. Air Force combined ultrasonic in-
spection procedures with laser heating to
identify fatigue cracks. Early detection
of
cracks may reduce repair costs by as much
as
$200,000 per year. What is the present
worth
of
these savings over a 5-year pe-
riod at an interest rate
of
10% per year?
2.22 A recent engineering graduate passed the
FE
exam and was given a raise (beginn
in
g
in year
1)
of
$2000. At an interest rate
of
8% per year, what
is
the present value
of
the $2000 per year over her expected
35-year career?
84
CHAPTER 2 Factors: How Time and Interest Affect Money
2.
23
Southwestern Moving and Storage wants
to have enough money to purchase a new
tractor-trailer
in
3 years.
If
the unit will
cost
$250,000, how much should the com-
pany
set
aside each year if the account
earns
9%
per year?
2.24 Vision Technologies, Inc., is a small com-
pany that uses ultra-wideband technology
to develop devices that can detect objects
(including people) inside buildings, be-
hind walls,
or
below ground.
The
com-
pany expects to spend
$100,000 per year
for labor and
$125,000 per year for sup-
plies before a product can be marketed. At
an
interest rate
of
15
% per year, what
is
the total equivalent future amount
of
the
company's
expenses at the end
of
3 years?
Factor Values
2.25 Find the numerical value
of
the following
factors by
(a)
interpolation and
(b)
using
the appropriate formula.
l.
(PIF,18%,33)
2.
(AIG,12%,54)
2.26 Find the numerical value
of
the following
factors by
(a) interpolation and (b) using
the appropriate formula.
1.
(FIA,19%,20)
2.
(PIA,26%,15)
Arithmetic Gradient
2.27 A cash flow sequence starts
in
year 1 at
$3000 and decreases by
$200
each year
through year
10. (a) Determine the value
of
the gradient G;
(b)
determine the amount
of
cash flow
in
year 8; and
(c)
determine
the value
of
n for the gradient.
2.28 Cisco
Systems expects sales to be de-
scribed by the cash flow sequence
(6000 +
5k), where k
is
in
years and cash flow is
in
million
s.
Determine
(a)
the value
of
the
gradient G;
(b) the amount
of
cash flow in
year 6; and
(c) the value
of
n for the gradi-
ent
if
the cash flow ends
in
year
12.
2.29
For
the cash flow sequence that starts
in
year I and is described by 900 -
lOOk
,
where k represents years I through 5,
(a) determine the value
of
the gradient G
and
(b) determine the cash flow in year 5.
2.30 Omega Instruments has budgeted
$300,000 per year to pay for certain ce-
ramic parts over the next 5 years.
If
the
company expects the cost
of
the parts to
increase uniformly according to an arith-
metic gradient
of
$1
0,
000
per year, what is
it expecting the cost to be
in
year
1,
if the
interest rate is
10% per year?
2.31 Chevron-Texaco expects receipts from a
group
of
stripper wells (wells that produce
less than
10 barrels per day) to decline
according to an arithmetic gradient
of
$50,000 per year. This year's receipts
are expected to be
$280,000 (i.e
.,
end
of
year
1)
, and the company expects the use-
fullife
of
the wells to be 5 years. (a)
What
is the amount
of
the cash flow in year 3,
and
(b)
what is the equivalent uniform an-
nual worth in years
I through 5
of
the
in
-
come
from the wells at an interest rate
of
12% per year?
2.32 Income from cardboard recycling at Fort
Bliss has been increasing at a constant
rate
of
$1000
in
each
of
the last 3 years.
If
this year's income (i.e., end
of
year
1)
is
expected to be
$4000 and the increased
income trend continues through year 5,
(a) what will the income be 3 years from
now (i.e., end
of
year 3) and
(b)
what is
the present worth
of
the income over
that 5-year period at an interest rate
of
10% per year?
2.33 Amazon is considering purchasing a so-
phisticated computer system to
"cube"
a