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ISE 211
Engineering Economy
Rate of Return Analysis
(Chapter 7 & 8)
Chapter 7
Rate of Return
Analysis
Calculating Rate of Return
 So far, we have learned how to use the Present Worth and
Annual Cash Flow Analyses to compare alternatives.
 PW and Annual Cost Analyses depend on the interest rate
chosen.
 Rate of return analysis finds the interest rate at which
project costs = project benefits.
 This is called the Internal Rate of Return (IRR), i*.
Calculating Rate of Return (cont’d)

Now how can we use the Rate of Return analysis
to compare alternatives?
 First convert the various consequences of the
investment into a cash flow.
 Solve the cash flow for the unknown value of
i*, which is the rate of return.
Calculating Rate of Return (cont’d)
 This can be done by using one of the following two
forms of the cash flow equation:
1. PW of benefits = PW of costs
2. EUAB = EUAC
Example - 1: An $8200 investment returned $2000
per year over a five-year useful life. What was
the rate of return on the investment?
Example 2
An investment resulted in the following cash flow.
Compute the rate of return.
Year
Cash Flow
0
–$700
1
+ 100
2
+ 175
3
+ 250
4
+ 325
Example 3
An investment resulted in the following cash
flow. Compute the rate of return.
Year
Cash Flow
0
–$100
1
+ 20
2
+ 30
3
+ 20
4
+ 40
5
+ 40
Example 4 (Problem 7.5 – page 263)
For the diagram below, compute the interest
rate at which costs are equivalent to benefits.
80
80
200
80
80
200
80
80
200
Plot of NPW vs. Interest Rate i
 The plot of NPW vs. interest rate i can be a very useful source
of information.
 A cash flow representing an investment followed by benefits
from the investment would have an NPW vs. i plot – lets call
it the NPW plot for convenience.
 A typical NPW plot for an investment looks like this
diagram:
Plot of NPW vs. Interest Rate i (cont’d)
 A typical NPW plot for borrowed
money looks like this diagram:
 Now, how can we use this kind of plot
to determine the internal rate of return?
Example 1
A new corporate bond was initially sold by a stockholder to an
investor for $1000. The issuing corporation promised to pay
the bond holder $40 interest on the $1000 face value of the
bond every six months, and to repay the $1000 at the end of
ten years. After one year the bond was sold by the original
buyer for $950.
a) What rate of return did the original buyer receive on
his investment?
b) What rate of return can the new buyer (paying $950)
expect to receive if he keeps the bond for its
remaining nine-year life?
Example 2 (Problem 7.9 – page 263)
A man buys a corporate bond from a
bond brokerage house for $925. The
bond has a face value of $1000 and
pays 4% of its face value each year. If
the bond will be paid off at the end of
ten years, what rate of return will the
man receive?
Rate Of Return Analysis (ROR)
 ROR analysis is probably the most frequently used exact analysis
technique in industry.
 Although problems in computing rate of return sometimes occur, its
major advantage outweighs the occasional difficulty.
 The major advantage is that we can compute a single figure of merit
that is readily understood.
 Consider these statement:
 The net present worth on the project is $32,000.
 The equivalent uniform annual net benefit is $2800.
 The project will produce a 23% rate of return.
Rate Of Return Analysis (ROR) – (cont’d)
 In ROR analysis, find the IRR for any single project, then compare this IRR
with a preselected MARR – the minimum attractive rate of return.

The MARR is the interest rate used in PW and Annual Cost Analysis.
 When there are two alternatives, ROR analysis is performed by computing
the incremental rate of return – ROR – on the difference between the
alternatives.
 Note: we will deal with 3 or more alternatives in Chapter 8.
 Since we want to look at increments of investments, the cash flow for the
difference between alternatives is computed.
Rate Of Return Analysis (ROR) – (cont’d)
 This can be done by taking the higher initial cost alternative minus the
lower initial cost alternative.
 Then, compare ROR with MARR and make your decision based on the
following criteria:
Two Alternative situation
Decision
 IRR  MARR
Choose the higher-cost alternative
 IRR
< MARR
Choose the lower-cost alternative
 Caution – DO NOT compare IRR for each alternative. If you do so, you
may choose the wrong alternative!!!!!
Example – 1 (Problem 7.22 – page 265)
Consider the following two alternatives. If 5%
is considered the minimum attractive rate of
return, which alternative should be selected?
Year
A
B
0
-$2000
-$2800
1
+ 800
+ 1100
2
+ 800
+ 1100
3
+ 800
+ 1100
Example – 2
If an electromagnet is installed on the input conveyor of a coal
processing plant, it will pick up scrap metal in the coal. The
removal of this metal will save an estimated $1200 per year
in machinery damage being caused by metal. The
electromagnet equipment has an estimated useful life f five
years and no salvage value. Two suppliers have been
contacted: Leasseco will provide the equipment in return
for three beginning-of-year annual payments of $1000 each;
Saleco will provide the equipment for $2783. If the MARR
is 10%, which supplier should be selected?
Consideration of the Analysis Period
 The consideration of the analysis period is also important in
the ROR analysis, just like it was in the PW and Annual Cash
Flow analyses.
 The assumption that an alternative can be replaced with one of
identical costs and performance appears to be the best option.
 Just be sure that all relevant costs and benefits are included
when computing PW(costs) = PW(benefits), or EUAC =
EUAB.
Example 1
Two machines are being considered for purchase. If the
MARR (here, the minimum required interest rate) is
10%, which machine should be bought?
Initial Cost
Uniform annual benefit
End-of-useful-life salvage value
Useful life (years)
Machine X
$200
95
50
6
Machine Y
$700
120
150
12
Spreadsheets and ROR Analysis
 If a cash flow diagram can be reduced to at most one P, one A,
and/or one F, then RATE investment function can be used.
i = RATE(N,A,P,F,type,guess)
 Otherwise the IRR block function is used with a cash flow in
each period.
i = IRR(selected cash flow)
Example 1
Consider the following cash flow:
Cash flow amount
Initial cost
Annual benefit
Salvage value
Useful life
-$8200
2000
0
6 years
What is the rate of return on this investment?
Example 2
Find the rate of return for the following cash
flow using Excel:
1
2
A
Year
Cash flow
B
0
-700
C
1
100
D
2
175
E
3
250
F
4
325
Chapter 8
Incremental
Analysis
Overview
 So far, we have learned how to:
 determine whether a single-project is desirable
 compare between two alternatives
 This chapter involves comparing between two or more alternatives using
incremental rate of return analysis (IROR).
 Incremental analysis can be defined as the examination of the differences
between alternatives.
 By emphasizing alternatives, we are really deciding whether or not
differential costs are justified by differential benefits.
Overview (cont’d)
 Although incremental analysis can be examined either graphically and
numerically, we will focus on the numerical solutions of ROR analysis.
 We can solve multiple-alternative problems by PW and ACFA without
any difficulties.
 ROR requires that, for two alternatives, the difference between them
must be examined to see whether or not they are desirable.
 Now if we can choose between two alternatives, then by a successive
examination we can choose from multiple alternatives.
The following figure illustrates the method of comparing multiple
alternatives:
Overview (cont’d)
Figure 1. Solving
MultipleAlternative
Problems
by successive
two-alternative
analyses.
Elements in Incremental Rate of Return Analysis
1) Check to see that all the alternatives in the problem are identified (including
the null alternative “do-nothing”, if applicable).
2) (optional!) Compute the ROR for each alternative: any alternative that have a
ROR less than the MARR may be immediately rejected.
3) Arrange the remaining alternatives in ascending order of investment.
4) Make a two-alternative analysis of the first two alternatives.
5) Take the preferred alternative from step 4, and the next alternative from the list
created in step 3 – proceed with another two-alternative comparison.
6) Continue until all alternatives have been examined and the best of the multiple
alternatives has been identified.
Decision Criteria
Decision Criteria for Increments of Investments
 If ROR  MARR, retain the higher-cost alternative.
 If ROR  MARR, retain the lower-cost alternative.
 Reject the other alternative used in the analysis.
Decision Criteria for Increments of Borrowing
 If ROR  MARR, the increment is acceptable.
 If ROR  MARR, the increment is not acceptable.
 Reject the other alternative used in the analysis.
Example 1
Using a 6% MARR, what is the preferred
alternative using a twenty-year life and no
salvage value?
Initial Cost
Uniform Annual Benefit
Alternative
A
B
C
$2,000
$5,000
$4,000
410
700
639
Example 2
The following information is provided for five mutually
exclusive alternatives that have twenty-year lives. If the
MARR 6%, which alternative should be selected?
Initial Cost
Uniform Annual Benefit
Rate of Return
A
$2,000
410
20%
Alternative
B
C
D
$6,000
$4,000
$1,000
761
639
117
11%
15%
10%
E
$9,000
785
6%
Example 3
Two mutually exclusive alternatives are being
considered. Both have a four year useful life.
Alternatives A and B have initial costs of $1000
and $500, respectively. The uniform annual
benefits from the alternatives are $350 and $165,
respectively. If the MARR is 16% and “donothing” is an option, determine which alternative
should be selected using rate of return analysis?
Example 4
Owing to perennial complaints by students and faculty about the lack of
parking spaces on campus, a parking garage on university property is
being considered. Since there are no university funds available for the
project, it will have to pay for itself from parking fees over a 15-year
period. A 10% MARR is deemed reasonable for consideration of the
question of how may levels should be constructed. Based on the cost data
shown below, determine whether a parking garage should be constructed.
If a parking building should be built, how many levels should be built?
# of levels
Construction costs
Operating Costs
Yearly Income
1
$600,000
$35,000
$100,000
2
$2,200,000
$60,000
$350,000
3
4
$3,600,000 $4,800,000
$80,000
$95,000
$570,000
$810,000
Spreadsheets and Incremental Analysis
 An incremental analysis of two alternatives is easily done with the
RATE or IRR functions when lives of the alternatives are the same.
 However, when the lives are different the problem is more difficult.
 As discussed earlier, when comparing alternatives with different length
lives, the usual approach is to assume that the alternatives are repeated
until the next least common multiple of their lives.
 This can be done easier with a spreadsheet – Excel supports an easier
approach.
 Excel has a tool called GOAL SEEK that identifies a formula cell, a
target value, and a variable cell.
Spreadsheets and Incremental Analysis
 Using this tool the variable cell is changed automatically by the
computer until the formula cell equals the target value.
 To find an IRR for an incremental analysis, the formula cell
can be the difference between two equivalent annual worth with a
target value of 0.
 Then if the variable cell is the interest rate, GOAL SEEK will
find the IRR.
Example
Two different asphalt mixes can be used on
a highway. The good mix will last 6 years,
and it will cost $600,000 to buy and lay
down. The better mix will last 10 years, and
it will cost $800,000 to buy and lay down.
Find the incremental IRR for using the
more expensive mix.
Homework # 7/8 (Chapters 7 & 8)
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