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%
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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.
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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.
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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.
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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.
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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.
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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.
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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)
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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 = ¥
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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.
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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.
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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.
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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
• Receiving a gift
• Giving your friend a loan and not being paid
back
In either case, no “i” can be computed.
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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%
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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)
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3