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Chapter 7 Rate-of-Return Analysis
Concept of Rate of Return
Note: Symbol convention—The symbol i* represents the break–even interest rate that
makes the PW of the project equal to zero.. The symbol IRR represents the internal rate of
return of the investment. For a simple (or pure) investment, IRR = i*. For a non-simple
investment, generally i* is not equal to IRR.
7.1
7.2
$24,500 $514.55( / , ,60)P A i=
7.3
$977.64
F
=
7.4
44
$15,767,500 $1,650( / , , 44)
9,556.06 (1 )
23.16%
F Pi
i
i
=
= +
=
7.5
7.6
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7.7
Investment Classification and Calculation of i*
7.8
(a) Simple investment: Project A, D
(c)
• Project A:
*
=
*
PW( ) $32,000 $30,000( / , ,3) $10,000( / , ,3)
0
i P Ai P Gi
=−+ −
=
7.9
We may need to find the rate of return on the first cycle, which will be the
same for the entire cycles.
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7.10
(a) Classification of investment projects:
• Simple projects: B and E
• Non–simple projects: A, C and D
(b)
Project
*
i
A
47.44%, -67.44%%
B
C
D
E
7.11
(a) Classification of investment projects:
(b)
• Project A:
*
1.08%i= −
7.12
(a)
i
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(b) With the geometric expense series
(c) To maintain
i
*
=9.4208%
7.13
(a) Rate of return calculation:
(b)
$30,000.00
$40,000.00
PW Plot
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7.14
(a) Cash flow sign rules:
Projects
Number of Sign Changes
Possible Number of
*
i
A
1
0, 1
B
2
0, 1, 2
(c)
• Project A:
*
100%i=
7.15
(a) IRR = 20%
Mixed Investments, RIC and MIRR
7.16
(a)
• Project 1:
* *
1 2
110%, 60%i i= = −
C
1
0, 1
D
1
0, 1
2
0, 1, 2
3
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• Project 2: simple and pure investment,
IRR 136.22%=
• Project 3: non-simple and mixed investment,
IRR 22.14%=
(c)
• Project 1:
2
$1, 000 $840( / ,12%, 2) $1, 669.64
• Project 2:
• Project 3:
2
$1, 000 $200( / ,12%, 2) $1,159.44
$1, 400( / ,12%,1) $1, 568
$1,568 (1 MIRR )
$1,159.44
MIRR 16.29%
PF
FP
+=
=
= +
=
(d) If MARR = 12%, all projects are acceptable.
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7.17
(a)
(b)
• Simple projects: A, B, and D
(c) Apply the cash flow sign rule:
B
1
0, 1
C
2
D
1
0, 1
2
(d)
• Project B:
IRR 15.32%
B=
(e) MIRR for Project E:
Projects
Number of Sign Changes
Possible Number of
*
i
Actual
*
i
A
1
0, 1
20%
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MIRR 12.07%
=
(f) Apply the net investment test to Project E:
Project Balances
Project E (i*=12.63%)
0
$200
1
$325
2
3
4
5
7.18
7.19 Correction: Cash flow for Project 3 should be -$1,000, $7,200, -$6,000
(a)
(b) Apply the net investment test:
• Project 1:
• Project 2:
0
PB(14.64%) $5,000= −
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• Project 3:
0
PB( 3.81%) $1,000−=−
(c)
• Project 1:
IRR 82.21%=
(d)
• Project 1: MIRR = 57.93% > 12%
(e) Projects 1 & 3 are acceptable.
7.20
(a) Pure investment: B
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7.21
(a) There are two sign changes in cash flow, indicating the possibility of multiple
*
i
s.
**
12
10%, 20ii= =
Apply the net investment test:
7.22
(a)
**
12
6%, 0%ii= =
(b) Project A
7.23
(a)
• Project A:
**
12
10%, 100%= =ii
,
11 | Page
(b) Project A:
IRR 13.57%=
A
, Project B:
IRR 342.16%=
B
, Project C:
IRR 18%=
C
, Project D:
IRR 31.07%=
D
, Project E:
IRR 19.67%=
E
(c) Sample calculation:
• Project A:
(d) All projects are acceptable.
7.24
From Excel, we find two rates of return:
i
*
1
=43.47%, i
*
2
=77.67%
(a) Apply the net investment test:
PB(43.47%) $8,000= −
3
(b) Since RIC = -16.30 % < 18%, the project is not acceptable.
(c) MIRR:
• Equivalent outlay at n = 0:
7.25
(a) Apply the net investment test using
*
10%i=
:
0
PB(10%) $150,000= −
7.26 (d)
IRR Analysis
7.27
7.28
The present worth of the project cash flow is
7.29
The present worth of the project cash flow is
7.30
(a) Since IRR = 10% and PW(10%) = 0, we have,
7.31
7.32
Let X be the annual rent per apartment unit. Then the expected net cash flows
are:
N
Capital
Investment
Revenue Maintenance Manager Net Cash Flow
0
–
14,500,000
-14,500,000
1 50 X -350,000 -85,000 50X – 435,000
7.33
Let X be the annual savings in labor
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7.34
•Net cash flow table:
N
Land
Building
Equipment
Revenue
Expenses
Net Cash Flow
0
-$1.50
-$3
-$4.50
1
-$4
-$4.00
2
$3.50
-$1.40
$2.10
3
$3.68
-$1.47
$2.21
4
$3.86
-$1.54
$2.32
5
$4.05
-$1.62
$2.43
6
$4.25
-$1.70
$2.55
7
$4.47
-$1.79
$2.68
8
$4.69
-$1.88
$2.81
9
$4.92
-$1.97
$2.95
11
$5.43
-$2.17
$3.26
12
$5.43
-$2.17
$3.26
13
$5.43
-$2.17
$3.26
14
$2
$1.40
$5.43
-$2.17
$7.16
•Rate of return calculation:
PW( ) $4.5 $4( / , ,1) $2.1( / , ,2)
=−− + +
i P Fi P Fi
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PW(18%) $150,000 ( 50,000)( / ,18%,10)
=− +−
X PA
7.36
(a)
PW( ) $30 $9( / , ,1) $18( / , , 2) $20( / , ,3)
$18( / , , 4) $10( / , ,5) $5( / , ,6)
0
=−+ + +
+++
=
i P Fi P Fi P Fi
P Fi P Fi P Fi
This is a simple investment. Therefore,
*
IRR 40.20%i= =
.
Comparing Alternatives
7.37
(a) Project A:
IRR 16.01%=
Project B:
IRR 18.18%=
7.35