# Engineering Economic Analysis - 9th Edition

```Engineering Economic Analysis
9th Edition
Chapter 7
RATE OF RETURN ANALYSIS
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
1
Three Major Methods of
Economic Analysis
• PW - Present Worth
• AW - Annual Worth
• IRR - Internal Rate of Return
If PW = A(P/A,i,n)
Then (P/A,i,n) = PW/A
Solve for (P/A,i,n) and look up interest in
Compound Interest Tables
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
2
Internal Rate of Return
Lender’s Viewpoint
• The interest rate on the balance of a loan
such that the unpaid loan balance equals
zero when the final payment is made.
Plan
Year
1
2
3
4
5
IRR
A
0 \$ (5,000.00)
\$ 1,400.00
\$ 1,320.00
\$ 1,240.00
\$ 1,160.00
\$ 1,080.00
8.00%
B
\$ (5,000.00)
\$ 400.00
\$ 400.00
\$ 400.00
\$ 400.00
\$ 5,400.00
C
\$ (5,000.00)
\$ 1,252.28
\$ 1,252.28
\$ 1,252.28
\$ 1,252.28
\$ 1,252.28
D
\$ (5,000.00)
\$
\$
\$
\$
\$ 7,346.64
8.00%
8.00%
8.00%
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
3
Internal Rate of Return
Investor’s Viewpoint
• The interest rate earned on the unrecovered
investment such that the unrecovered
investment equals zero at the end of the life
of the investment.
Year
0
1
2
3
4
5
IRR
Cash flow
(\$5,000.00)
\$1,252.28
\$1,252.28
\$1,252.28
\$1,252.28
\$1,252.28
8.00%
Unrecovered
investment at
beginning of
year
\$5,000.00
\$4,147.72
\$3,227.25
\$2,233.15
\$1,159.52
Total
Return on
unrecovered
investment
Investment
Unrecovered
repayment at investment at
end of year
end of year
\$400.00
\$331.81
\$258.18
\$178.65
\$92.76
\$852.28
\$920.47
\$994.10
\$1,073.63
\$1,159.52
\$1,261.40
\$5,000.00
\$4,147.72
\$3,227.25
\$2,233.15
\$1,159.52
(\$0.00)
Reset the cash flow to determine the IRR.
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
4
Calculating Rate of Return
• The IRR is the interest rate at which the
benefits equal the costs. IRR = i*
•
•
•
•
•
PW Benefit - PW Cost = 0
PW Benefit/PW Cost = 1
NPW = 0
EUAB - EUAC = 0
PW Benefit = PW Cost
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
5
Calculating IRR
• PWB/PWC = 1
• 1252.28(P/A,i,5)/5000 = 1
• (P/A,i,5) = 5000/1252.28 = 3.9927
Example 7-1
From Compound Interest Tables
Interest rate
(P/A,i,5)
7%
4.100
8%
3.993
9%
3.890
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
6
Calculating IRR
EUAB - EUAC = 0
100 + 75(A/G,i,4) - 700(A/P, i, 4) = 0
Look up and iterate, book answer is 7%
Example 7-2
Year
0
1
2
3
4
5
IRR
Cash flow
(\$700.00)
\$100.00
\$175.00
\$250.00
\$325.00
\$0.00
6.91%
Unrecovered
investment at
beginning of
year
\$700.00
\$648.37
\$518.18
\$303.99
\$0.00
Total
Return on
unrecovered
investment
Investment
Unrecovered
repayment at investment at
end of year
end of year
\$48.37
\$44.81
\$35.81
\$21.01
\$0.00
\$51.63
\$130.19
\$214.19
\$303.99
\$0.00
\$150.00
\$700.00
\$648.37
\$518.18
\$303.99
\$0.00
\$0.00
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
7
Calculating IRR
Example 7-3
The
iterations
may be
graphed
and the
true IRR
will be
indicated
at the point
where the
NPW
curve = 0.
0
1
2
3
4
5
IRR
Trial
interest
rates
0
5
10
15
20
25
30
35
40
45
50
Cash flow
(\$100.00)
\$20.00
\$30.00
\$20.00
\$40.00
\$40.00
Return on
unrecovered
investment
\$100.00
\$93.47
\$76.07
\$66.32
\$35.25
13.47%
Total
Investment
Unrecovered
repayment at investment at
end of year
end of year
\$13.47
\$12.59
\$10.25
\$8.93
\$4.75
\$6.53
\$17.41
\$9.75
\$31.07
\$35.25
\$50.00
\$100.00
\$93.47
\$76.07
\$66.32
\$35.25
(\$0.00)
\$60.00
NPW
\$50.00
\$50.00
\$26.46
\$9.24
(\$3.49)
(\$12.97)
(\$20.06)
(\$25.37)
(\$29.36)
(\$32.34)
(\$34.54)
(\$36.16)
\$40.00
Net Present Value
Year
Unrecovered
investment at
beginning of
year
\$30.00
\$20.00
\$10.00
\$0.00
(\$10.00) 0
5
10
15
20
25
30
35
40
45
50
(\$20.00)
(\$30.00)
(\$40.00)
(\$50.00)
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
8
Calculating IRR - Bond
Example 7-4a
Period rate
1000 = 40(P/A,i,2) + 950(P/F,i,2)
By lookup and interpolation i*  1.5%
Nominal rate = 2 x 1.5% = 3%
Effective rate = (1 + 1.5%)2 - 1 = 3.02%
An easier way >
7-4a & 7b
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
9
Rate of Return (ROR) Analysis
• Most frequently used measure of merit in
industry
• More accurately called Internal Rate of
Return (IRR)
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
1
0
Calculating ROR
• Where two mutually exclusive alternatives will
provide the same benefit, ROR is performed
using an incremental rate of return-DROR-on
the difference between the alternatives.
Two-alternative situation
Decision
DROR MARR
Choose higher-cost alternative
DROR <MARR
Choose lower-cost alternative
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
1
1
ROR on Alternatives with
Equivalent Benefits
Example 7-5
MARR
= 10%
0
1
2
3
4
5
IRR/period
Trial interest
rates
0
5
10
15
20
25
30
35
40
45
50
Cash flow - Cash flow alternative A alternative B
(\$1,000.00)
\$200.00
\$200.00
\$1,200.00
\$1,200.00
\$1,200.00
(\$2,783.00)
\$1,200.00
\$1,200.00
\$1,200.00
\$1,200.00
\$1,200.00
48.72%
32.60%
8.01%
NPW
\$217.00
\$72.77
(\$43.15)
(\$136.77)
(\$212.69)
(\$274.40)
(\$324.66)
(\$365.60)
(\$398.94)
(\$426.01)
(\$447.93)
Total
\$217.00
\$1,783.00
\$300.00
\$200.00
Net Present Value
Year
Cash flow on difference
Unrecovered
Return on
Investment
Unrecovered
Cash flow investment at
unrecovered repayment at investment at
alternative B-A beginning of
investment
end of year
end of year
year
(\$1,783.00)
\$1,000.00
\$1,783.00
\$142.83
\$857.17
\$925.83
\$1,000.00
\$925.83
\$74.17
\$925.83
(\$0.00)
\$0.00
(\$0.00)
(\$0.00)
\$0.00
(\$0.00)
\$0.00
(\$0.00)
(\$0.00)
\$0.00
(\$0.00)
\$0.00
(\$0.00)
(\$0.00)
\$0.00
(\$0.00)
\$100.00
\$0.00
(\$100.00)
0
5
10
15
20
25
30
35
40
45
50
(\$200.00)
(\$300.00)
(\$400.00)
(\$500.00)
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
1
2
Criterion Is to Maximize Return
Not ROR
Examples 7- 6, 7 & 8
Year
0
1
2
3
4
5
IRR/period
MARR =
NPW =
Trial interest
rates
0
5
10
15
20
25
30
35
40
45
50
Cash flow - Cash flow Cash flow alternative A alternative B alternative B-A
(\$10.00)
\$15.00
\$0.00
\$0.00
\$0.00
\$0.00
(\$20.00)
\$28.00
\$0.00
\$0.00
\$0.00
\$0.00
(\$10.00)
\$13.00
\$0.00
\$0.00
\$0.00
\$0.00
50.00%
6.00 %
\$4.15
40.00%
30.00%
\$6.42
\$2.26
NPW
\$3.00
\$2.27
\$1.65
\$1.13
\$0.69
\$0.32
\$0.00
(\$0.27)
(\$0.51)
(\$0.71)
(\$0.89)
Return on
unrecovered
investment
\$10.00
\$0.00
\$0.00
\$0.00
\$0.00
Total
Investment
repayment at
end of year
Unrecovered
investment at
end of year
\$3.00
\$0.00
\$0.00
\$0.00
\$0.00
\$10.00
\$0.00
\$0.00
\$0.00
\$0.00
\$0.00
\$0.00
\$0.00
\$0.00
\$0.00
\$3.00
\$10.00
\$3.50
\$3.00
Net Present Value
What
happens to
NPW when
MARR is
varied? Try
6, 30 &
35%.
Cash flow on difference
Unrecovered
investment at
beginning of
year
\$2.50
\$2.00
\$1.50
\$1.00
\$0.50
\$0.00
(\$0.50) 0
5
10
15
20
25
30
35
40
45
50
(\$1.00)
(\$1.50)
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
1
3
Fixed Input and Differing Outputs
Example 7-9
Cash flow on difference
Year
0
1
2
3
4
5
IRR/period
MARR =
NPW =
Trial interest
rates
0
5
10
15
20
25
30
35
40
45
50
Cash flow - Cash flow alternative A alternative B
(\$1,000.00)
\$400.00
\$350.00
\$300.00
\$250.00
\$200.00
(\$1,000.00)
\$300.00
\$300.00
\$300.00
\$300.00
\$300.00
17.47%
15.24%
7.00 %
\$257.75
\$230.06
0.00%
NPW
\$0.00
(\$20.10)
(\$32.72)
(\$40.40)
(\$44.80)
(\$47.00)
(\$47.75)
(\$47.52)
(\$46.66)
(\$45.40)
(\$43.90)
Return on
unrecovered
investment
Investment
Unrecovered
repayment at investment at
end of year
end of year
\$0.00
(\$0.00)
(\$0.00)
(\$0.00)
(\$0.00)
Total
(\$100.00)
(\$50.00)
\$0.00
\$50.00
\$100.00
(\$0.00)
\$100.00
\$150.00
\$150.00
\$100.00
(\$0.00)
\$0.00
(\$27.69)
\$0.00
0
5
10
15
20
25
30
35
40
45
50
(\$10.00)
Net Present Value
Choose the
ranking so
that the
difference
represents
an increment
of
investment;
goes from to + over
time.
Unrecovered
Cash flow investment at
alternative B-A beginning of
year
\$0.00
(\$100.00)
\$0.00
(\$50.00)
\$100.00
\$0.00
\$150.00
\$50.00
\$150.00
\$100.00
\$100.00
(\$20.00)
(\$30.00)
(\$40.00)
(\$50.00)
(\$60.00)
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
1
4
Analysis Period
• Just as in PW and AW analysis, the analysis
period must be considered:
• Useful life of the alternative equals the analysis
period
• Alternatives have useful lives different from the
analysis period
• The analysis period is infinite, n = ¥
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
1
5
Engineering Economic Analysis
9th Edition
Chapter Appendix 7A
Difficulties Solving for an Interest Rate
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
1
6
Multiple IRR
Example 7A-1
Cash flow
0
1
2
3
4
5
\$19.00
\$10.00
(\$50.00)
(\$50.00)
\$20.00
\$60.00
IRR/period
IRR/period
10.24%
47.30%
Trial interest
rates
0
5
10
15
20
25
30
35
40
45
50
Guess
10.00%
50.00%
NPW
\$9.00
\$3.28
\$0.11
(\$1.50)
(\$2.14)
(\$2.20)
(\$1.91)
(\$1.44)
(\$0.88)
(\$0.28)
\$0.32
Return on
unrecovered
investment
(\$19.00)
(\$30.95)
\$15.88
\$67.51
\$54.43
Total
Investment
repayment at
end of year
Unrecovered
investment at
end of year
(\$1.95)
(\$3.17)
\$1.63
\$6.91
\$5.57
\$11.95
(\$46.83)
(\$51.63)
\$13.09
\$54.43
\$9.00
(\$19.00)
(\$30.95)
\$15.88
\$67.51
\$54.43
\$0.00
\$10.00
\$8.00
Net Present Value
Occurs
when a
cash flow
produces
more than
one point
at which
NPW = 0
Year
Unrecovered
investment at
beginning of
year
\$6.00
\$4.00
\$2.00
\$0.00
(\$2.00)
0
5
10
15
20
25
30
35
40
45
50
(\$4.00)
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
1
7
Cash Flow Rule of Signs
• May be converted to a polynomial
• Then, by Descartes’ rule
Number of sign
changes, m
0
1
Number of positive
values of X
0
1
2
3
2 or 0
3 or 1
4
4, 2 or 0
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
1
8
Cash Flow Rule of Signs Expands
on This Notion
• There may be as many positive values of “i”
as there are sign changes in the cash flow.
• Sign changes are counted when:
• + to • - to +
• A zero cash flow is ignored
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
1
9
Cash Flow Rule of Signs
- Possibilities Number of sign changes, Number of positive values
m
of “i”
0
1
2
3
4
0
1 or 0
2, 1 or 0
3, 2, 1 or 0
4, 3, 2, 1 or 0
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
2
0
Zero Sign Changes
• Giving your friend a loan and not being paid
back
In either case, no “i” can be computed.
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
2
1
One Sign Change
The Normal Situation
Year
0
1
2
3
4
5
Sign
change
Guess
0
10
20
30
40
50
60
70
80
90
100
A
(\$50.00)
\$20.00
\$10.00
B
(\$50.00)
\$20.00
\$20.00
\$11.00
C
(\$50.00)
\$20.00
\$20.00
\$20.00
\$12.00
D
\$50.00
(\$20.00)
(\$10.00)
E
\$50.00
(\$20.00)
(\$20.00)
(\$11.00)
F
\$50.00
(\$20.00)
(\$20.00)
(\$20.00)
(\$12.00)
1
1
1
1
1
1
IRR
NPV
NPV
NPV
NPV
NPV
NPV
NPV
NPV
NPV
NPV
NPV
IRR
1.09%
1.09%
1.09%
1.09%
1.09%
1.09%
1.09%
1.09%
1.09%
1.09%
1.09%
IRR
17.64%
17.64%
17.64%
17.64%
17.64%
17.64%
17.64%
17.64%
17.64%
17.64%
17.64%
IRR
NPV
NPV
NPV
NPV
NPV
NPV
NPV
NPV
NPV
NPV
NPV
IRR
1.09%
1.09%
1.09%
1.09%
1.09%
1.09%
1.09%
1.09%
1.09%
1.09%
1.09%
IRR
17.64%
17.64%
17.64%
17.64%
17.64%
17.64%
17.64%
17.64%
17.64%
17.64%
17.64%
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
2
2
Two or More Sign Changes
• More than one IRR may exist
• Use an NPW plot to determine how many exist
0
1
2
3
4
5
\$19.00
\$10.00
(\$50.00)
(\$50.00)
\$20.00
\$60.00
IRR/period
IRR/period
10.24%
47.30%
Trial interest
rates
0
5
10
15
20
25
30
35
40
45
50
Unrecovered
investment at
beginning of
year
Cash flow
Guess
10.00%
50.00%
NPW
\$9.00
\$3.28
\$0.11
(\$1.50)
(\$2.14)
(\$2.20)
(\$1.91)
(\$1.44)
(\$0.88)
(\$0.28)
\$0.32
Return on
unrecovered
investment
(\$19.00)
(\$30.95)
\$15.88
\$67.51
\$54.43
Total
Investment
repayment at
end of year
Unrecovered
investment at
end of year
(\$1.95)
(\$3.17)
\$1.63
\$6.91
\$5.57
\$11.95
(\$46.83)
(\$51.63)
\$13.09
\$54.43
\$9.00
(\$19.00)
(\$30.95)
\$15.88
\$67.51
\$54.43
\$0.00
\$10.00
\$8.00
Net Present Value
Year
\$6.00
\$4.00
\$2.00
\$0.00
(\$2.00)
0
5
10
15
20
25
30
35
40
45
50
(\$4.00)
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
2
3
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