Blank L., Tarquin A. Engineering Economy (McGraw-Hill Series in Industrial Engineering and Management)
Подождите немного. Документ загружается.
3. Finally, Kathy asked Bob to reevaluate the cash
flows for Homeworth at the
MARR
of
35%, but
using a reinvestment rate
of
45% to determine
if
the series is sti
ll
justified.
Case Study Exercises
1, 2, and 3. Answer the questions for Bob using
spreadsheets.
CASE
STUDY
275
4.
If
the
i'
approximating procedure Bob developed
is
not available, use the original cash flow data to
apply the basic net-investment procedure, and
answer Exercises 2 and
3, where c
is
35 and 45%,
respectively.
5. Kathy concluded from this exercise that any cash
flow series is economically
ju
stified for any rein-
vestment rate that is larger than the MARR. Is this
a correct conclusion? Explain why or why not.
LLI
I-
I
u
Rate
of
Return Analysis:
Multiple
Alternatives
This
chapter
presents
the
methods
by
which
two
or
more
alternatives
can
be
evaluated
using a rate
of
return
(ROR)
comparison
based
on
the
methods
of
the
previous
chapter.
The
ROR
evaluation
correctly
performed
will result in
the
same
selection
as
the
PW,
AW,
and
FW
analyses,
but
the
computational
procedure
is
considerably
different
for
ROR
evaluations
.
The
first
case
study
involv
es
multiple
options
for
a business
owned
for
many
years
by
one
person.
The
second
case
explores
nonconventional
cash
flow
series
with
multiple
rates
of
return
and
the
use
of
the
PW
method
in
this
situation.
LEARNING OBJECTIVES
Purpose: Select
the
best mutually exclusive alternative on
the
basis
of
rate
of
return analysis
of
incremental
cash
flows.
Why incremental analysis?
Incremental
cash
flows
Interpretation
Incremental ROR by
PW
Incremental ROR by
AW
Multiple
alternatives
Spreadsheets
This
chapter
will
help
you:
1. State
why
an incremental analysis
is
necessary
for
comparing
alternatives
by
the
ROR
method.
2.
Prepare a
tabulation
of
incremental
cash
flows
for
two
alternatives.
3.
Interpret
the
meaning
of
ROR
on
the
incremental
initial
investment.
4.
Select
the
better
of
two
alternatives using incremental
or
breakeven
ROR
analysis based on
present
worth.
5.
Select
the
better
of
two
alternatives using a
ROR
analysis
based
on
annual
worth.
6.
Select
the
best
of
multiple
alternatives using
an
in
crementa
l
ROR
analysis.
7.
Develop
spreadsheets
that
include
PW,
AW, and
ROR
evaluation
for
multiple,
different-life
alternatives.
278
Mutua
ll
y exc
lu
si
ve
Indepe
nd
ent projects
CH
APTER
8 Rate
of
Return Anal
ys
is: Multiple Alterna
ti
ve
s
8.1
WHY
INCREMENTAL ANALYSIS
IS
NECESSARY
Wh
en two or more mutually exclusive alternatives are
ev
aluated, engin
ee
ring
economy can
id
entify the one alternative that
is
the best economically. As we
have l
ea
rn
ed, the
PW
, AW, and FW techniques can be used to do so. Now the
pro
ce
dur
e for us
in
g
ROR
to identify the b
es
t is
pr
esented.
Le
t
's
assume that a
co
mpany uses a
MARR
of 16% per year, that the c
omp
any
has
$9
0,
000
available for investment, and that two alternatives (A and B) are
being eva
lu
ated. Alternative A requires an investment
of
$50,000 and has an
internal rate of return i
't.
of 35% per year. Alternative B requires $85,000 and has
an
i
;
~
of
29% per year. Intuitively we may conclude that the better alternative is
the one that has the larger return, A
in
this case.
How
ever, this is not necessarily
so. While A has the
hi
gher pro
ject
ed return, it requires an initial investment that
is much less than the total money available ($90,
000
).
Wh
at happens to the in-
vestment
ca
pital that is left
ove
r? It is
ge
nera
ll
y assumed that excess funds will
be
in
vested at
th
e compan
y's
MARR, as we learned
in
the previous
chapt
e
r.
Us
in
g t
hi
s assumption,
it
is possible to determine the consequences
of
the alter-
na
ti
ve
in
vestment
s.
If
alterna
ti
ve A is sel
ec
ted, $
50
,
000
will return 35% per yea
r.
The
$40
,000 l
ef
t
ove
r will be
in
vested at the
MARR
of 16% per year.
The
rate
of
return on the total capital available, then, will be the weighted average. Thus, if
a
lt
e
rn
ative A is sele
ct
ed,
Ove
ra
ll
ROR =
50
,
000
(0
.3
5) +
40
,
000
(0.
16
) = 26.6%
A 90,000
If alte
rn
a
ti
ve B is selected, $85,000 will be invested at 29% per year, and the
remaining $5000 will ea
rn
16% per year.
No
w the weighted average is
Ov
era
ll
ROR
= 85,
000
(0.29) +
5000
(0.16) = 28.3%
B 90,000
These calc
ul
a
ti
ons show that even though the i* for alterna
ti
ve A is
hi
gher,
a
lt
e
rn
a
ti
ve B presents the bett
er
ove
ra
ll
ROR
f
or
the $90,000.
If
either a PW
or AW comparison is conducted using the
MARR
of 16% per y
ea
r as i, alterna-
ti
ve B will be chosen.
This simple
exa
mple illustrates a major fact about the rate
of
return me
th
od
for
co
mpa
ri
ng alternatives:
Under
some circumstances, project
ROR
values do not provide the same
ranking
of
alternatives as do PW, A
W,
and
FW
analyses. This situation
does not occur if
we
conduct an incremental cash flow
ROR
analysis (dis-
cussed in the next section).
When independent proje
ct
s are
ev
aluated, no incremental analysis is neces-
sary between projects. Each project is
ev
aluated separately from others, and
more
th
an one
ca
n be selected.
Th
e
ref
ore, the only c
omp
arison is with the do-
nothing alternative for each project.
Th
e
ROR
can be used to accept or reje
ct
each independent project.
SECTIO 8.2
Calculation
of
In
cremental Cash Flows for ROR Analysis
TABLE
8-1 Format for Incremental Cash Flow Tabulation
Yea
r
o
A
lterna
ti
ve
A
(1 )
Cash Flow
Alternativ
e B
(2)
Incr
emen
ta
l
Cash Flow
(3) = (2) - (1)
8.2 CALCULATION OF INCREMENTAL CASH FLOWS
FOR
ROR
ANALYSIS
It
is
necessary to prepare an incremental cash flow tabulation between two alter-
natives
in
preparation for an incremental
ROR
analysis. A standardized format for
the tabulation will simplify this process.
The
column headings are shown in
Table
8-1.
If
the alternatives have equal lives, the year column will go from 0 to n.
If
the alternatives have unequal liv
es,
the year column will go from 0 to the
LCM
(least common multiple)
of
the two lives.
The
use
of
the
LCM
is necessary be-
cause incremental
ROR
analysis requires equal-service comparison between
alternatives. Therefore, all the assumptions and requirements developed earlier
apply for any incremental
ROR
evaluation. When the
LCM
of
lives is used, the
salvage value and reinvestment
in
each alternative are shown at appropriate time
s.
If
a planning period is defined, the cash flow tabulation is for the specified period.
Only for the purpose
of
simplification, use the convention that between two
alternatives, the one with the
larger initial investment will
be
regarded as alter-
native
B.
Then, for each year
in
Table
8-1,
Incremental cash flow = cash
flowB
- cash flow A
[8
.
1]
The initial investment and annual cash flows for each alternative (excluding the
salvage value)
occur
in
one
of
two patterns identified in Chapter
5:
Revenue alternative, where there are both negative and positive cash flows.
Service alternative, where all cash flow estimates are negative.
In either case, Equation
[8.1]
is
used to determine the incremental cash flow series
with the sign
of
each cash flow carefully determined.
The
next two examples
illustrate incremental cash flow tabulation
of
service alternatives
of
equal and
different live
s.
Later
examp
les treat revenue alternatives.
EXAMPLE
8.1 { ,
A tool and die company
in
Pittsburgh is considering the purchase
of
a dri
ll
press with
fuzzy-logic software to improve accuracy and reduce tool wear. The company has
the opportunity
to
buy a slightly used machine for $15,000 or a new one for $21,000.
279
I Sec.5.1 I
Revenue and
se
rvi
ce
280
CHAPTER 8 Rate
of
Return Analysis:
Mu
ltiple Alterna
ti
ves
Because
th
e new machine is a more sophisticated model, its operat
in
g cost is expected
to be
$7000 per year, while the used machine is expected to require $8200 per yea
r.
Each machine is expected
to
have a 25-year life with a 5% sa
lv
age va
lu
e.
Tabulate the
incremental cash
flow.
Solution
In
cremental cash flow is tabulated
in
Table
8-2.
Using Equation [8.1], the subtraction
perfo
rm
ed is (new - used) since the new machine has a larger initial cost. The salvage
va
lu
es
in
year 25 are separated from ordinary cash flow for clarity. When disbursements
are the same for a number
of
consecutive years, for hand solution onl
y,
it saves time to
make a single cash flow
li
sting,
as
is done for years l
to
25. However, remember
th
at
several years were combined when performing the analysis. This approach cannot be
used for spreadsheets.
TABLE
8-2
Cash Flow Tabulation for Example 8.1
Year
o
1
-2
5
25
Total
Comment
Cash Flow
Used Press
$ -
15
,000
-8,2
00
+ 750
$
-2
19
,2
50
New
Press
$ - 21,000
- 7,000
+ 1,050
$-
194,950
Incremental
Cash Flow
(New
- Used)
$ - 6,000
+ 1,200
+3
00
$+24,3
00
When
th
e cash
fl
ow columns are subtracted, the difference between the totals of the two
cash
flow
series should equal the total
of
the incremental cash
flow
column. This on
ly
provides a check of the addition and
su
btraction
in
preparing
th
e tabulation. It is not a
basis for select
in
g an alternative.
EXAMPLE
8.2
,.~
Sandersen Meat Processors has asked
it
s lead process engineer
to
evaluate two different
types
of
conveyors for the bacon curing line. Type A has
an
initial cost
of
$70,000 a
nd
a life
of
8 years. Type B has
an
initial cost
of
$95,000 and a life expectancy
of
12 years.
The annual operating cost for type A is expected
to
be
$9000, while the AOC for
ty
pe B
is
ex
pected
to
be $7000. If
th
e salvage values are $5000 and $
10
,000 for type A a
nd
type S,
re
spective
ly,
tabulate
th
e incremental cash flow us
in
g
th
e
ir
LCM.
Solution
The LCM
of
8 a
nd
12
is 24 years. In the
in
cremental cash flow tabulation for 24 years
(Ta
bl
e
8-3)
note that the reinvestment a
nd
salvage
va
lu
es are shown
in
years 8 and
16
for type A and
in
year
12
for type B.
SECTI
ON
8
.2
Calculation of Incremental Cash Flows for ROR Analysis
TABLE
8-3
Incremental Cash Flow Tabulation, Example 8.2
Cash
Fl
ow
Incremental
Cash
Flow
Ye
ar
Type A
Type 8
(8
-
A}
0
$ - 70,000
$
- 95,000
$-2
5,000
1- 7
- 9,000
- 7,000
+2
,000
C
7O
,
OOO
8
- 9,000
- 7,000
+67,0
00
+ 5,000
9-
11
- 9,000
- 7,000
+2,000
C
95
,OOO
12
- 9,000
- 7,000 - 83,000
+10
,000
13
-
15
- 9,000
- 7,000
+2
,000
C
7O
,
OOO
16
- 9,000 - 7,000
+67
,000
+ 5,000
17
- 23
- 9,000 - 7,000 +2,000
24
- 9,000
{ - 7,000
+7,000
+ 5,000
+ 10,000
$- 411,000 $- 338,000
$+73
,000
The use
of
a spreadsheet to obtain incremental cash flows requires one entry
for each year through
th
e LCM for each alternative. Therefore, some combining
of cash flows may be necessa
ry
before the entry is made for each alternative. The
incremental cash
fl
ow column results from an application
of
Equation
[8
.1]. As
an illustration,
th
e first 8 years
of
the 24 years in Table
8-3
should appear
as
fol-
lows when entered onto a spreadsheet. The incremental values in column
Dare
obtained using a subtraction relation, for example,
C4-
B4.
Column A
Column B Column C Column D
Year Type A Type B
Incremental
0 $
- 70,000
$
-
95
,000
$
- 25,000
- 9,000
- 7,000
+ 2,000
2
- 9,000
- 7,000 + 2,000
3
- 9,000 - 7,000 + 2,000
4
- 9,000 - 7,000 + 2,000
5
-
9,
000 - 7,000 + 2,000
6
- 9,000
-7
,000 + 2,000
7
- 9,000
- 7,000
+2,000
8
- 74,000
- 7,000
+ 67,000
etc.
281
m
E-
So
l
ve
282
CHAPTER 8 Rate
of
Return Analysis: Multiple Alternatives
8.3 INTERPRETATION
OF
RATE
OF
RETURN
ON
THE EXTRA INVESTMENT
The
incremental cash flows
in
year 0
of
Tables 8- 2 and
8-3
reflect the extra in-
vestment or cost required
if
the alternative with the larger first cost is selected.
This
is
important in an incremental ROR analysis in order to determine the
ROR
earned on the extra funds expended for the larger-investment alternative.
If
the
incremental cash flows
of
the larger investment don'tjustify it, we must select the
cheaper one. In Example 8.1 the new drill press requires an extra investment
of
$6000 (Table
8-2)
.
If
the new machine
is
purchased, there will be a "savings"
of
$1200 per year for 25 years, plus an extra $300
in
year 25.
The
decision to buy the
used or new machine can be made on the basis
of
the profitability
of
investing the
extra
$6000 in the new machine.
If
the equivalent worth
of
the savings is greater
than the equivalent worth
of
the extra investment at the MARR, the extra invest-
ment should be made (i.e., the larger first-cost proposal should be accepted).
On
the other hand, if the extra investment
is
not justified
by
the savings, select the
lower-investment proposal.
It is important to recognize that the rationale for making the selection decision
is
the
same
as if only one alternative were under consideration, that alternative
being the one represented by the incremental cash flow series. When viewed in this
manner, it
is
obvious that unless this investment yields a rate
of
return equal to or
greater than the MARR, the extra investment should not be made. As further clar-
ification
of
this extra-investment rationale, consider the following:
The
rate
of
return attainable through the incremental cash flow is an alternative to investing at
the
MARR
. Section 8.1 states that any excess funds not invested in the alternative
are assumed to be invested at the MARR.
The
conclusion
is
clear:
If
the
rate
of
return
available through the incremental cash flow equals
or
exceeds the MARR, the alternative associated with the
extra
investment
should be selected.
Not only must the return on the extra investment meet or exceed the MARR,
but also the return on the investment that
is
common
to
both alternatives must meet
or exceed the MARR. Accordingly, before starting an incremental
ROR analysis,
it
is
advisable
to
determine the internal rate
of
return
i*
for each alternative.
(Of
course, this
is
much easier with evaluation by computer than by hand.) This can be
done only for revenue alternatives, because service alternatives have only cost
(negative) cash flows and no
i*
can be determined. The guideline
is
as
follows:
For
multiple revenue alternatives, calculate the internal
rate
of
return
i*
for each alternative,
and
eliminate all alternatives
that
have
an
i*
< MARR.
Compare
the remaining alternatives incrementally.
As an illustration,
if
the
MARR
= 15% and two alternatives have
i*
values
of
12
and
21
%, the 12% alternative can be eliminated from further consideration.
With only two alternatives, it
is
obvious that the second one
is
selected.
If
both
alternatives have i*
<
MARR
, no alternative
is
justified and the do-nothing alter-
native
is
the best economically. When three or more alternatives'are evaluated, it
SECTION
8.4
Rate
of
Return Eva
lu
ation Using PW: Incremental and Bre
akeve
n
is
usually worthwhile, but not required, to calculate
i*
for each alternative for
preliminary screening. Alternatives that
cannot
meet the
MARR
may be elimi-
nated from further evaluation using this option. This option is especially useful
when performing the analysis by computer.
The
IRR
function applied to each
alternative's cash flow
est
imates can quickly indicate unacceptable alternatives,
as demonstrated later
in
Section 8.6.
When independent projects are
eva
luated, there is no comparison on the extra
investment.
Th
e ROR value
is
used to accept all projects with i*
:::::
MARR,
assum-
ing there
is
no budget limitation.
For
examp
le, assume
MARR
= 10%, and three
independent projects are available with
ROR
values of:
it = 12%
i
~
= 9% it =
23
%
Projects A and C are selected, but B is not because
i~
'
<
MARR.
Example 8.8 in
Section 8.7 on spreadsh
ee
t applications illustrates selection from independent
projects using
ROR values.
8.4
RATE
OF RETURN EVALUATION USING
PW:
INCREMENTAL
AND
BREAKEVEN
In this section
we
discuss the primary approach to making mutually exclusive
alternative selections by the incremental
ROR
method. A PW-based relation like
Equation [7.1]
is
developed for the incremental cash flows.
Use
hand
or
com-
puter means to find
~i'~
-
A'
the internal
ROR
for the series. Placing
~
(delta) be-
fore
iJ
~
- A
distinguishes it from the ROR values it and
i~.
Since incremental ROR requires equal-service comparison, the
LCM
of
lives
must be used
in
the PW formu lation. Because
of
the reinvestment requirement for
PW
analysis for different-life assets, the incremental cash flow series may con-
tain several sign changes, indicating multiple
~i
*
values. Though incorrect, this
indication
is
usually neglected in actual practice.
The
correct approach
is
to es-
tablish the reinvestment rate c and follow the approach
of
Section 7.5. This means
that the unique composite rate
of
return
(~i')
for the incremental cash flow series
is determined.
These
three required
elements-incremental
cash flow series,
LCM, and multiple root
s-a
re the primary reaso
ns
that the
ROR
method is often
applied incorrectly
in
engineering economy analyses
of
multiple alternatives. As
stated earlier,
it
is always possible, and generally advisable, to use a
PW
or
AW
analysis at an established MARR
in
lieu
of
the ROR method when multiple rates
are indicated.
Th
e complete procedure by hand
or
computer
for an incremental
ROR
analy-
sis for two alternatives is as follows:
1.
Order the alternatives by initial investment
or
cost, starting with the smaller
one
, called
A.
The
one
with the larger initial investment
is
in the column
labeled B
in
Table 8-
1.
2. Develop the cash flow and incremental cash flow series using the
LCM
of
years, assuming reinvestment
in
alternatives.
3.
Draw an incremental cash flow diagram,
if
needed.
283
Q-Solv
284
Multiple-root tests
CHAPTER
8 Rate
of
Return Analysis: Multiple Alternatives
4. Count the number
of
sign changes in the incremental cash flow series to deter-
mine
if
multiple rates
of
return may be present.
If
necessary, use Norstrom's
criterion on the cumulative incremental cash flow series to determine
if
a
si
ngle positive root exists.
5.
Set up the
PW
equation for the incremental cash flows
in
the form
of
Equa-
tion [7.1], and determine
~i~~A
using trial and error by hand
or
spreadsheet
functions.
6. Select the economically better alternative as follows:
If
~i~~A
<
MARR,
select
alternative
A.
If
~i~~A
2':
MARR,
the
extra
investment is justified; select
alterna-
tive B.
If
the incremental i* is exactly equal to
or
very near the
MARR,
noneco-
nomic considerations will most likely be used to help
in
the selection
of
the
"best" alternative.
In
step 5,
if
trial and error
is
used to calculate the rate
of
return, time may be
saved
if
the
~i'~
~ A
value is bracketed, rather than approximated by a point value
using lin
ear
interpolation, provided that a single
ROR
value is not needed.
For
example
, if the
MARR
is
15
%
per
year and you have estab
li
shed that
~i~
~ A
is in
the
15
to 20% range, an exact value
is
not necessary to accept B since you al-
ready know that
~i~
~ A
2': MARR.
The
IRR function on a spreadsheet will normally determine
one
~i*
value.
Multiple guess va
lu
es
ca
n be input to find multiple roots
in
the range - 100%
to
00
for a nonconventional series, as illustrated
in
Examples 7.4 and 7.5.
If
this
is
not the case, to be correct, the indication
of
multiple roots
in
step 4 requires
that the net-investment procedure, Equation [7.6], be applied
in
step 5 to make
~i'
=
~i*
.
If
one
of
these multiple roots is the same as the expected reinvest-
ment rate c, this root can be used as the
ROR
value, and the net-investment pro-
cedure
is
not necessary. In this
case
only,
~i'
=
~i*
,
as concluded at the end
of
Section 7.5.
Tn
2000, Bell Atlantic and
GTE
merged to form a gia
nt
telecommunications corporation
named Verizon Communications. As expected, some equipment incompatibil.ities had to be
rectified, especia
ll
y for long distance and international wireless and video services.
One
item had two
suppliers-a
U.S. firm (A) and an Asian firm (B). Approximately 3000 units
of
this
equipment
were needed. Estimates for vendors A and B are given for each uni
t.
A ·8
Initial cost, $
- 8,000
-13,0
00
Annual costs, $
- 3,500 - 1
,6
00
Salvage value, $
0
2,000
Life, years
10
5