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12. a. The IRR is the interest rate that makes the NPV of the project equal to zero. The equation for the IRR
of Project A is:
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find
that:
The equation for 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:
Examining the IRRs of the projects, we see that IRRA is greater than IRRB, so the IRR decision
b. The NPV of Project A is:
And the NPV of Project B is:
The NPVB is greater than the NPVA, so we should accept Project B.
c. To find the crossover rate, we subtract the cash flows from one project from the cash flows of
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find
that:
13. The IRR is the interest rate that makes the NPV of the project equal to zero. The equation to
calculate the IRR of Project X is:
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:
For Project Y, the equation to find the IRR is:
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:
To find the crossover rate, we subtract the cash flows from one project from the cash flows of the
other project, and find the IRR of the differential cash flows. We will subtract the cash flows from
Project Y from the cash flows from Project X. It is irrelevant which cash flows we subtract from the
other. Subtracting the cash flows, the equation to calculate the IRR for these differential cash flows
is:
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:
The table below shows the NPV of each project for different required returns. Notice that Project X
R NPVX NPVY
0% $8,890.00 $8,760.00
14. a. The equation for the NPV of the project is:
The NPV is greater than zero, so we would accept the project.
b. The equation for the IRR of the project is:
From Descartes rule of signs, we know there are potentially two IRRs since the cash flows
change signs twice. From trial and error, the two IRRs are:
When there are multiple IRRs, the IRR decision rule is ambiguous. Both IRRs are correct, that
15. The profitability index is defined as the PV of the future cash flows divided by the initial investment.
The equation for the profitability index at a required return of 10 percent is:
The equation for the profitability index at a required return of 15 percent is:
The equation for the profitability index at a required return of 22 percent is:
We would accept the project if the required return were 10 percent or 15 percent since the PI is
16. a. The profitability index is the PV of the future cash flows divided by the initial investment. The
cash flows for both projects are an annuity, so:
The profitability index decision rule implies that we accept Project II, since PIII is greater than
PII.
b. The NPV of each project is:
c. Using the profitability index to compare mutually exclusive projects can be ambiguous when
the magnitude of the cash flows for the two projects are of different scale. In this problem,
17. a. The payback period for each project is:
The payback criterion implies accepting Project B, because it pays back sooner than Project A.
b. The discounted payback for each project is:
The discounted payback criterion implies accepting Project B because it pays back sooner than A.
c. The NPV for each project is:
NPV criterion implies we accept Project A because Project A has a higher NPV than Project B.
d. The IRR for each project is:
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we
find that:
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation,
we find that:
IRR decision rule implies we accept Project B because IRR for B is greater than IRR for A.
e. The profitability index for each project is:
Profitability index criterion implies we accept Project B because the PI for B is greater than the
PI for A.
f. In this instance, the NPV criteria implies that you should accept Project A, while profitability
18. At a zero discount rate (and only at a zero discount rate), the cash flows can be added together across
time. So, the NPV of the project at a zero percent required return is:
If the required return is infinite, future cash flows have no value. Even if the cash flow in one year is
The interest rate that makes the NPV of a project equal to zero is the IRR. The equation for the IRR
of this project is:
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find
that:
19. 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:
So, the MIRR using the discounting approach is:
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find:
Reinvestment approach:
In the reinvestment approach, we find the future value of all cash except the initial cash flow at the
end of the project. So, reinvesting the cash flows to Time 5, we find:
So, the MIRR using the reinvestment approach is:
Combination approach:
In the combination approach, we find the value of all cash outflows at Time 0, and the value of all
cash inflows at the end of the project. So, the value of the cash flows is:
So, the MIRR using the combination approach is:
0 = –$52,898.75 + $104,540.59/(1 + MIRR)5
Intermediate
20. 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,
So, the MIRR using the discounting approach is:
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find
that:
Reinvestment approach:
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:
So, the MIRR using the discounting approach is:
0 = –$47,000 + $90,325.54/(1 + MIRR)5
Combination approach:
In the combination approach, we find the value of all cash outflows at Time 0 using the discount
So, the MIRR using the discounting approach is:
0 = –$52,637.79 + $99,825.54/(1 + MIRR)5
22. a. To have a payback equal to the project’s life, given C is a constant cash flow for N years:
C = I/N
Challenge
