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21. a. The NPV of each project is:
NPVNP-30 = –$940,000 + $345,000/1.12 + $335,000/1.122 + $310,000/1.123
+ $295,000/1.124 + $205,000/1.125
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:
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we
find that:
c. Incremental IRR analysis is not necessary. The NX-20 has a higher IRR and is relatively smaller
in terms of investment, with a larger NPV. Nonetheless, we will calculate the incremental IRR.
In calculating the incremental cash flows, we subtract the cash flows from the project with the
smaller initial investment from the cash flows of the project with the large initial investment, so
the incremental cash flows are:
Year
Incremental
cash flow
0
–$290,000
1
95,000
2
85,000
3
65,000
4
65,000
5
30,000
Setting the present value of these incremental cash flows equal to zero, we find the incremental
IRR is:
+ $65,000/(1 + IRR)4 + $30,000/(1 + IRR)5
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we
find that:
of 12 percent, we reject the larger project and choose the NX-20.
d. 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:
22. a. The NPV of each project is:
NPVA = –$675,000 + $330,000/1.15 + $330,000/1.152 + $260,000/1.153 + $195,000/1.154
+ $135,000/1.155
The NPV criteria implies accepting Project B.
b. The IRR is the interest rate that makes the NPV of the project equal to zero, so the IRR of each
project is:
Project A:
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we
find that:
And the IRR of Project B is:
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we
find that:
c. In calculating the incremental cash flows, we subtract the cash flows from the project with the
smaller initial investment from the cash flows of the project with the large initial investment, so
the incremental cash flows are:
Year
Incremental
cash flow
0
–$475,000
1
15,000
2
45,000
3
125,000
4
260,000
5
440,000
Setting the present value of these incremental cash flows equal to zero, we find the incremental
IRR is:
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we
find that:
For investing-type projects, accept the larger project when the incremental IRR is greater than
d. 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:
Challenge
23. 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 = $465,000(1.04) = $483,600
Year 5 cash flow = $395,000(1.04) = $410,800
So, the NPV of the project is:
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find
that:
occur for the company.
24. Given the six-year payback, the worst case is that the payback occurs at the end of the sixth year.
Thus, the worst case:
NPV = –$319,703.03
25. The equation for the IRR of the project is:
Using Descartes rule of signs, from looking at the cash flows we know there are as many as four IRRs
for this project. Even with most computer spreadsheets, we have to do some trial and error. From trial
26. a. Here the cash inflows of the project grow at a constant rate forever, which is a growing perpetuity.
So, the PV of the future cash flows from the project is:
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:
g = .0244, or 2.44%
27. 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 6 percent.
So, the IRR equation for this project is:
0 = –$2,900,000 + $525,000{[1/(IRR – .06)] – [1/(IRR – .06)] × [(1 + .06)/(1 + IRR)]11}
Using a spreadsheet or trial and error to find the root of the equation, we find that:
IRR = 18.32%
b. Yes. Since the mine’s IRR exceeds the required return of 13 percent, the mine should be opened.
the last year.
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:
PV(A) = [C3/(R – g)]/(1 + R)2
b. If we combine the cash flow streams to form Project C, we get:
Project A = [C3/(R – g)]/(1 + R)2
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we
find that:
IRR = 13.21%
c. The correct decision rule for an investing-type project is to accept the project if the discount rate
is below the IRR. Since there is one IRR, a decision can be made. At a point in the future, the
cash flows from Stream A will be greater than those from Stream B. Therefore, although there
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:
Year
Incremental
cash flows
0
–I + $1,500
1
300
2
300
3
500
Now we can find the present value of the subsequent incremental cash flows at the discount rate, 12
percent. The present value of the incremental cash flows is:
PV = $1,500 + $300/1.12 + $300/1.122 + $500/1.123
PV = $2,362.91
So, if I0 is greater than $2,362.91, the incremental cash flows will be negative. Since we are subtracting
Project Million from Project Billion, this implies that for any value over $2,362.91 the NPV of Project
Billion will be less than that of Project Million, so I0 must be less than $2,362.91.
