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13. a. The equation for the NPV of the project is:
b. The equation for the IRR of the project is:
When there are multiple IRRs, the IRR decision rule is ambiguous. Both IRRs are correct; that
14. a. The payback period is the time that it takes for the cumulative undiscounted cash inflows to
equal the initial investment.
Board game:
DVD:
Cumulative cash flows Year 1 = $1,500 = $1,500
Cumulative cash flows Year 2 = $1,500 + 1,050 = $2,550
b. The NPV is the sum of the present value of the cash flows from the project, so the NPV of each
project will be:
Board game:
DVD:
c. The IRR is the interest rate that makes the NPV of a project equal to zero. So, the IRR of each
project is:
Board game:
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we
find that:
IRR = 25.09%
Since the IRR of the board game is greater than the IRR of the DVD, IRR implies we choose
the board game. Note that this is the choice when evaluating only the IRR of each project. The
d. To calculate the incremental IRR, we subtract the smaller project’s cash flows from the larger
project’s cash flows. In this case, we subtract the board game cash flows from the DVD cash
flows. The incremental IRR is the IRR of these incremental cash flows. So, the incremental
cash flows of the DVD are:
Year 0 Year 1 Year 2 Year 3
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:
15. a. The profitability index is the PV of the future cash flows divided by the initial investment. The
profitability index for each project is:
The profitability index implies we accept the G4 project. Remember this is not necessarily correct
because the profitability index does not necessarily rank projects with different initial investments
correctly.
b. The NPV of each project is:
NPV implies we accept the Wi-Fi project since it has the highest NPV. This is the correct
decision if the projects are mutually exclusive.
c. We would like to invest in all three projects since each has a positive NPV. If the budget is
limited to $40 million, we can only accept the CDMA project and the G4 project, or the Wi-Fi
project. NPV is additive across projects and the company. The total NPV of the CDMA project
and the G4 project is:
16. a. The payback period is the time that it takes for the cumulative undiscounted cash inflows to
equal the initial investment.
AZM Mini-SUV:
Payback period = 1 + $143,000 / $198,000 = 1.72 years
AZF Full-SUV:
Since the AZM has a shorter payback period than the AZF, the company should choose the
AZM. Remember the payback period does not necessarily rank projects correctly.
b. The NPV of each project is:
The NPV criteria implies we accept the AZM because it has the highest NPV.
c. The IRR is the interest rate that makes the NPV of the project equal to zero. So, the IRR of the
AZM is:
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we
find that:
The IRR criteria implies we accept the AZM because it has the highest IRR. Remember the
IRR does not necessarily rank projects correctly.
d. Incremental IRR analysis is not necessary. The AZM has the smallest initial investment, and the
17. a. The profitability index is the PV of the future cash flows divided by the initial investment. The
profitability index for each project is:
b. The NPV of each project is:
c. Accept Projects A, B, and C. Since the projects are independent, accept all three projects
d. Accept Project B. Since the Projects are mutually exclusive, choose the Project with the highest
PI, while taking into account the scale of the Project. Because Projects A and C have the same
initial investment, the problem of scale does not arise when comparing the profitability indexes.
When calculating incremental cash flows, remember to subtract the cash flows of the project
with the smaller initial cash outflow from those of the project with the larger initial cash
outflow. This procedure insures that the incremental initial cash outflow will be negative. The
incremental PI calculation is:
e. Remember that the NPV is additive across projects. Since we can spend $450,000, we could
18. a. The payback period is the time that it takes for the cumulative undiscounted cash inflows to
equal the initial investment.
Dry Prepeg:
Payback period = 1 + ($600,000 / $900,000) = 1.67 years
Solvent Prepeg:
b. The NPV of each project is:
The NPV criteria implies accepting the dry prepeg because it has the highest NPV.
c. The IRR is the interest rate that makes the NPV of the project equal to zero. So, the IRR of the
dry prepeg is:
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we
find that:
IRRDry prepeg = 30.90%
And the IRR of the solvent prepeg is:
The IRR criteria implies accepting the solvent prepeg because it has the highest IRR.
Remember the IRR does not necessarily rank projects correctly.
d. Incremental IRR analysis is necessary. The solvent prepeg has a higher IRR, but is relatively
smaller in terms of investment and NPV. 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 0 Year 1 Year 2 Year 3
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, we accept the larger project when the incremental IRR is greater
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 = $222,000 = $222,000
Payback period = 2 + ($216,000 / $222,000) = 2.97 years
NX-20:
Cumulative cash flows Year 1 = $120,000 = $120,000
Payback period = 3 + ($22,800 / $159,720) = 3.14 years
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:
IRRNX-20 = 20.34%
The IRR criteria implies accepting the NX-20.
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:
The PI criteria implies accepting the NX-20.
d. The NPV of each project is:
Challenge
20. 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:
This is a quadratic equation. We can solve for the roots of this equation with the quadratic formula:
X =
−b±
√
b2−4ac
2a
Remember that the quadratic formula is written as:
0 = aX2 + bX + c
In this case, the equation is:
IRR = .4818, or 48.18%
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:
To determine if this is a maximum or minimum, we can find the second derivative of the IRR
function. If the second derivative is positive, we have found a minimum and if the second derivative
is negative we have found a maximum. Using the reduced equation above, that is:
21. Given the six-year payback, the worst case is that the payback occurs at the end of the sixth year.
Thus, the worst case:
The best case has infinite cash flows beyond the payback point. Thus, the best-case NPV is infinite.
22. The equation for the IRR of the project is:
Using Descartes’ rule of signs, from looking at the cash flows we know there are four IRRs for this
project. Even with most computer spreadsheets, we have to do some trial and error. From trial and
23. 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:
NPV is the PV of the outflows minus by the PV of the inflows, so the NPV 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:
24. 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.
So, the IRR equation for this project is:
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we
find that:
b. Yes. Since the mine’s IRR exceeds the required return of 10 percent, the mine should be
opened. The correct decision rule for an investment-type project is to accept the project if the
