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19. a. The payback period is the time that it takes for the cumulative undiscounted cash inflows to equal
the initial investment.
NP-30:
Cumulative cash flows Year 1 = $239,000 = $239,000
NX-20:
Cumulative cash flows Year 1 = $130,000 = $130,000
NP-30. Remember the payback period does not necessarily rank projects correctly.
b. The IRR is the interest rate that makes the NPV of the project equal to zero, so the IRR of each
project is:
NP-30:
0 = –$735,000 + $239,000({1 – [1/(1 + IRR)5]}/IRR)
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we
find that:
IRRNP-30 = 18.74%
c. The profitability index is the present value of all subsequent cash flows, divided by the initial
investment, so the profitability index of each project is:
PINP-30 = ($239,000{[1 – (1/1.15)5]/.15})/$735,000
PINP-30 = 1.090
d. The NPV of each project is:
NPVNP-30 = –$735,000 + $239,000{[1 – (1/1.15)5]/.15}
NPVNP-30 = $66,165.07
20. The MIRRs for the project with all three approaches is:
Discounting approach:
In the discounting approach, we find the value of all negative cash outflows at Time 0, while any
positive cash inflows remain at the time at which they occur. So, discounting the cash outflows to
Time 0, we find:
Time 0 cash flow = –$47,000 – $9,500/1.105
Time 0 cash flow = –$52,898.75
So, the MIRR using the discounting approach is:
So, the MIRR using the reinvestment approach is:
0 = –$47,000 + $95,040.59/(1 + MIRR)5
$95,040.59/$47,000 = (1 + MIRR)5
21. With different discounting and reinvestment rates, we need to make sure to use the appropriate interest
rates. The MIRRs for the project with all three approaches are:
Discounting approach:
In the discounting approach, we find the value of all cash outflows at Time 0 at the discount rate, while
any cash inflows remain at the time at which they occur. So, discounting the cash outflows to Time 0,
In the reinvestment approach, we find the future value of all cash flows except the initial cash flow at
the end of the project using the reinvestment rate. So, reinvesting the cash flows to Time 5, we find:
Time 5 cash flow = $16,900(1.084) + $20,300(1.083) + $25,800(1.082) + $19,600(1.08) – $9,500
Time 5 cash flow = $90,325.54
So, the MIRR using the discounting approach is:
Time 5 cash flow = $16,900(1.084) + $20,300(1.083) + $25,800(1.082) + $19,600(1.08)
Time 5 cash flow = $99,825.54
So, the MIRR using the discounting approach is:
0 = –$52,637.79 + $99,825.54/(1 + MIRR)5
$99,825.54/$52,637.79 = (1 + MIRR)5
22. The equation for the IRR of the project is:
0 = –$75,000 + $155,000/(1 + IRR) – $65,000/(1 + IRR)2
From Descartes’ Rule of Signs, we know there are either zero IRRs or two IRRs since the cash flows
change signs twice. We can rewrite this equation as:
0 = –$75,000 + $155,000X – $65,000X2
X =
)000,65(2
−
X =
)000,65(2
0004,525,000,000,155
−
−
X =
000,130
12.268,67000,155
−
−
To find the maximum (or minimum) of a function, we find the derivative and set it equal to zero. The
derivative of this IRR function is:
0 = –$155,000(1 + IRR)–2 + $130,000(1 + IRR)–3
–$155,000(1 + IRR)–2 = $130,000(1 + IRR)–3
–$155,000(1 + IRR)3 = $130,000(1 + IRR)2
23. Given the six-year payback, the worst case is that the payback occurs at the end of the sixth year. Thus,
the worst case:
24. The equation for the IRR of the project is:
0 = –$6,048 + $34,344/(1 + IRR) – $72,840/(1 + IRR)2 + $68,400/(1 + IRR)3
25. a. Here the cash inflows of the project go on forever, which is a perpetuity. Unlike ordinary
perpetuity cash flows, the cash flows here grow at a constant rate forever, which is a growing
perpetuity. The PV of the future cash flows from the project is:
PV of cash inflows = C1/(R – g)
b. Here we want to know the minimum growth rate in cash flows necessary to accept the project.
The minimum growth rate is the growth rate at which we would have a zero NPV. The equation
for a zero NPV, using the equation for the PV of a growing perpetuity is:
26. a. The project involves three cash flows: the initial investment, the annual cash inflows, and the
abandonment costs. The mine will generate cash inflows over its 11-year economic life. To
express the PV of the annual cash inflows, apply the growing annuity formula, discounted at the
IRR and growing at eight percent.
b. Yes. Since the mine’s IRR exceeds the required return of 13 percent, the mine should be opened.
27. First, we need to find the future value of the cash flows for the one year in which they are blocked by
the government. So, reinvesting each cash inflow for one year, we find:
Year 2 cash flow = $295,000(1.04) = $306,800
Year 3 cash flow = $325,000(1.04) = $338,000
IRR = 10.37%
While this may look like an MIRR calculation, it is not an MIRR, rather it is a standard IRR
28. a. We can apply the growing perpetuity formula to find the PV of Stream A. The perpetuity formula
values the stream as of one year before the first payment. Therefore, the growing perpetuity
formula values the stream of cash flows as of Year 2. Next, discount the PV as of the end of Year
2 back two years to find the PV as of today, Year 0. Doing so, we find:
b. If we combine the cash flow streams to form Project C, we get:
Project A = [C3/(r – g)]/(1 + r)2
Project B = [C2/r]/(1 + r)
c. The correct decision rule for an investing-type project is to accept the project if the discount rate
29. To answer this question, we need to examine the incremental cash flows. To make the projects equally
attractive, Project Billion must have a larger initial investment. We know this because the subsequent
cash flows from Project Billion are larger than the subsequent cash flows from Project Million. So,
subtracting the Project Million cash flows from the Project Billion cash flows, we find the incremental
cash flows are:
Incremental
Year cash flows
0 –Io + $1,200
30. The IRR is the interest rate that makes the NPV of the project equal to zero. So, the IRR of the project
is:
0 = $20,000 – $26,000/(1 + IRR) + $13,000/(1 + IRR)2
Even though it appears there are two IRRs, a spreadsheet, financial calculator, or trial and error will
not give an answer. The reason is that there is no real IRR for this set of cash flows. If you examine
Calculator Solutions
1. b.
Project A
CFo
–$15,000
CFo
–$19,000
C01
$10,400
C01
$12,700
5.
CFo
–$27,000
C01
$13,100
21.80%
6.
Project A
Project B
CFo
–$7,300
CFo
–$4,390
C01
$3,940
C01
$2,170
F01
1
F01
1
7.
CFo
0
10.
CFo
$8,700
C01
–$3,900
CFo
$8,700
CFo
$8,700
C01
–$3,900
C01
–$3,900
F01
1
F01
1
11. a.
Deepwater fishing
Submarine ride
CFo
–$725,000
CFo
–$1,450,000
C01
$270,000
C01
$820,000
F01
1
F01
1
b.
CFo
–$725,000
C01
$550,000
F01
1
c.
Deepwater fishing
Submarine ride
CFo
–$725,000
CFo
–$1,450,000
12.
Project I
CFo
$0
CFo
–$45,000
C01
$23,200
C01
$23,200
F01
3
F01
3
13.
CFo
–$65,000,000
CFo
–$65,000,000
C01
$92,000,000
C01
$92,000,000
14. b.
Board game
DVD
CFo
–$850
CFo
–$1,700
C01
$670
C01
$1,300
$248.20
$364.61
c.
Board game
DVD
CFo
–$850
CFo
–$1,700
30.86%
24.96%
d.
CFo
–$850
19.29%
15. a.
CDMA
G4
Wi-Fi
CFo
0
CFo
0
CFo
0
C01
$23,000,000
C01
$21,000,000
C01
$39,000,000
F01
1
F01
1
F01
1
C02
$16,000,000
C02
$51,000,000
C02
$66,000,000
b.
CDMA
G4
Wi-Fi
CFo
–$18,000,000
CFo
–$25,000,000
CFo
–$43,000,000
C01
$23,000,000
C01
$21,000,000
C01
$39,000,000
16. b.
AZM
AZF
CFo
–$575,000
CFo
–$980,000
C01
$373,000
C01
$395,000
c.
AZM
AZF
CFo
–$575,000
CFo
–$980,000
C01
$373,000
C01
$395,000
F01
1
F01
1
17. a.
Project A
Project B
Project C
CFo
0
CFo
0
CFo
0
C01
$165,000
C01
$300,000
C01
$180,000
F01
1
F01
1
F01
1
b.
Project A
Project B
Project C
CFo
–$225,000
CFo
–$450,000
CFo
–$225,000
C01
$165,000
C01
$300,000
C01
$180,000
18. b.
Dry prepeg
Solvent prepeg
CFo
–$1,500,000
CFo
–$650,000
C01
$900,000
C01
$345,000
F01
1
F01
1
c.
Dry prepeg
Solvent prepeg
CFo
–$1,500,000
CFo
–$650,000
C01
$900,000
C01
$345,000
F01
1
F01
1
d.
CFo
–$850,000
12.68%
19. b.
NP-30
NX-20
CFo
–$735,000
CFo
–$460,000
C01
$239,000
C01
$130,000
F01
5
F01
1
18.74%
19.87%
c.
NP-30
NX-20
CFo
0
CFo
0
C01
$239,000
C01
$130,000
F01
5
F01
1
C02
C02
$143,000
F02
F02
1
d.
NP-30
NX-20
CFo
–$735,000
CFo
–$460,000
30.
CFo
$20,000
