Engineering Economic Analysis
Canadian Edition
Chapter 7:
Rate of Return Analysis
Chapter 7 …
 Introduces the internal rate of return (IRR)
method of evaluating project cash flows.
 Relates IRR to net present value (NPV) by
plotting NPV versus the discount rate.
 Uses an incremental technique and IRR to
evaluate mutually exclusive alternatives.
EECE 450 — Engineering Economics
7-2
Internal Rate of Return (IRR)
 Definition (borrower’s perspective): the
interest rate on the balance of a loan such
that the unpaid loan balance equals zero
when the final payment has been made.
Plan
Year
0
1
2
3
4
5
IRR
A
-$5,000.00
$1,400.00
$1,320.00
$1,240.00
$1,160.00
$1,080.00
8.00%
EECE 450 — Engineering Economics
B
-$5,000.00
$400.00
$400.00
$400.00
$400.00
$5,400.00
8.00%
C
-$5,000.00
$1,252.28
$1,252.28
$1,252.28
$1,252.28
$1,252.28
8.00%
D
-$5,000.00
$0.00
$0.00
$0.00
$0.00
$7,346.64
8.00%
7-3
Internal Rate of Return (IRR) …
 Definition (investor’s perspective): 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 Cash Flow
0 -$500,000
1
$100,000
2
$200,000
3
$250,000
4
$150,000
5
$75,000
IRR= 17.1686%
Unrecovered
Investment
(start of year)
$500,000.00
$485,843.05
$369,255.54
$182,651.58
$64,010.32
EECE 450 — Engineering Economics
Return on
Unrecovered
Investment
$85,843.05
$83,412.49
$63,396.04
$31,358.74
$10,989.68
Investment
Repayment
$14,156.95
$116,587.51
$186,603.96
$118,641.26
$64,010.32
Unrecovered
Investment
(end of year)
$500,000.00
$485,843.05
$369,255.54
$182,651.58
$64,010.32
$0.00
7-4
Calculating Internal Rate of Return
 Alternate definition: the IRR is the discount
rate that forces the NPV to zero.
 The best approach to calculating the IRR is to
use a “solver” like a preprogrammed financial
calculator or the IRR() function of Excel®.
 If NPV is plotted versus the discount rate, we
see that the IRR is at the zero-crossing (next
page).
EECE 450 — Engineering Economics
7-5
Calculating Internal Rate of Return …
Net Present Value
Year
0
1
2
3
4
5
IRR=
i
0%
1%
2%
3%
4%
5%
6%
7%
8%
9%
10%
11%
12%
13%
14%
Unrecovered
Return on
Unrecovered
Investment
Unrecovered Investment
Investment
Cash Flow (start of year)
Investment
Repayment
(end of year)
-$500,000
$500,000.00
$100,000
$500,000.00
$85,843.05
$14,156.95
$485,843.05
$200,000
$485,843.05
$83,412.49 $116,587.51
$369,255.54
$250,000
$369,255.54
$63,396.04 $186,603.96
$182,651.58
$150,000
$182,651.58
$31,358.74 $118,641.26
$64,010.32
$75,000
$64,010.32
$10,989.68
$64,010.32
$0.00
17.1686%
NPV
NPV versus Discount Rate
$275,000
$253,224
$300,000
$232,360
$212,361
$250,000
$193,179
$200,000
$174,773
$150,000
$157,102
$100,000
$140,128
$123,817
$50,000
$108,134
$0
$93,048
5%
10%
15%
20%
25%
-$50,0000%
$78,531
-$100,000
$64,554
Discount Rate
$51,092
$38,120
EECE 450 — Engineering Economics
7-6
Rate of Return Analysis
 The rate of return is the most frequently used
measure of merit in industry.
 Compare the rate of return (usually called the
Internal Rate of Return or IRR) to the
minimum acceptable rate of return (MARR).
 If IRR > MARR, accept the investment.
 If IRR < MARR, do not accept the investment.
 If IRR = MARR, the investment is marginally
acceptable.
EECE 450 — Engineering Economics
7-7
Rate of Return Analysis …
 Where two or more mutually exclusive
alternatives with the same lifetime will provide
the same utility, the IRR is calculated on the
difference between/among the alternatives.
 Analyze the incremental project: subtract the
cash flows of a project with a lower initial cost
from the one that has the highest initial cost.
 If IRR > MARR, accept the alternative with
the higher initial cost; otherwise accept the
lower cost initial cost alternative.
EECE 450 — Engineering Economics
7-8
Rate of Return Analysis …
Net Present Value
Year Machine X Machine Y
X-Y
0
-$500,000
-$400,000 -$100,000
1
$75,000
$60,000 $15,000
2
$125,000
$105,000 $20,000
3
$140,000
$120,000 $20,000
4
$150,000
$125,000 $25,000
5
$150,000
$125,000 $25,000
6
$130,000
$110,000 $20,000
7
$100,000
$85,000 $15,000
8
$50,000
$40,000 $10,000
IRR=
16.3387% 17.7188% 10.4043%
i
NPV X
NPV Y
0%
$420,000
$370,000
Comparing Machines X and Y
1%
$381,456
$337,709
2%
$345,204
$307,336
$500,000
3%
$311,077
$278,741
4%
$278,921
$251,797
$400,000
5%
$248,597
$226,385
$300,000
6%
$219,975
$202,399
7%
$192,937
$179,738
$200,000
8%
$167,375
$158,314
9%
$143,188
$138,041
$100,000
10%
$120,285
$118,843
11%
$98,580
$100,649
$0
12%
$77,996
$83,393
0%
5%
10%
15%
20%
25%
-$100,000
13%
$58,459
$67,015
14%
$39,904
$51,459
-$200,000
15%
$22,268
$36,674
16%
$5,495
$22,611
Discount Rate
17%
-$10,470
$9,226
EECE 450 — Engineering Economics
Machine X
Machine Y
7-9
Analysis Period
 The analysis period must be considered and
can be used as a way of ensuring the
lifetimes are equal for incremental analysis.
 Recall that there are three possible cases:
• Useful life of the alternative equals the analysis
period
• Alternatives have useful lives different from the
analysis period
• The analysis period is infinite, n = .
EECE 450 — Engineering Economics
7-10
Engineering Economic Analysis
Canadian Edition
Chapter Appendix 7A:
Difficulties in Solving for
an Interest Rate
Problems with the IRR
 The IRR has some drawbacks that force us to
be cautious when we use it:
• The IRR technique can not distinguish between
investing and borrowing; and the criterion for
acceptance depends on which it is.
• Multiple IRR values can occur when there are two
or more reversals in the signs of the cash flows.
• The IRR technique assumes that all cash flows
are borrowed and reinvested at the IRR rate of
return during the project (may not be realistic).
 The NPV technique always works properly.
The Modified IRR fixes some of the problems.
EECE 450 — Engineering Economics
7-12
Problems with the IRR …
 Example of multiple IRRs:
0
1
2
3
IRR=
i
0%
5%
10%
15%
20%
25%
30%
35%
40%
50%
60%
70%
80%
90%
100%
120%
Cash Flow
-$1,000
$6,200
-$12,000
$7,200
20.00%
100.00%
200.00%
NPV
$400.00
$240.04
$128.47
$51.70
$0.00
-$33.60
-$54.16
-$65.39
-$69.97
-$66.67
-$54.69
-$39.69
-$24.69
-$11.23
$0.00
$15.03
NPV versus Discount Rate
$500.00
$400.00
Net Present Value
Year
$300.00
$200.00
$100.00
$0.00
0%
-$100.00
50%
100%
150%
200%
250%
300%
-$200.00
EECE 450 — Engineering Economics
Discount Rate
7-13
Modified Internal Rate of Return
 Two of the problems with the IRR can be
solved by calculating the Modified Internal
Rate of Return (MIRR):
• Multiple IRR values
• Financing and reinvesting at the IRR rate of return
 The MIRR technique requires two external
rates of return:
• einv for investing
• efin for financing
EECE 450 — Engineering Economics
7-14
Modified Internal Rate of Return …
 Find the present value at the start of the
project of all negative cash flows using efin as
the discount rate.
 Find the future value at the end of the project
of all positive cash flows using einv as the
discount rate.
 Find the rate of return MIRR that balances
these two values at both ends of the project.
 Note: this is an approximate technique; it
produces a reliable result that can be used
appropriately in the analysis/decision.
EECE 450 — Engineering Economics
7-15
Modified Internal Rate of Return …
 Example of MIRR technique applied to the
previous multiple IRR example:
Year
0
1
2
3
IRR=
Cash Flow
-$1,000
$6,200
-$12,000
$7,200
20.00%
100.00%
200.00%
EECE 450 — Engineering Economics
External reinvestment rate:
16%
External financing rate:
10%
MIRR (formulas)= 12.4959%
MIRR (Excel function)= 12.4959%
7-16
Suggested Problems
 7-31, 36, 38, 39, 46, 52, 54.
 7A-9, 23, 26. Note the instructions at the
start of the problems for this section in the
text.
EECE 450 — Engineering Economics
7-17