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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
the other project. Here, we will subtract the cash flows for Project B from the cash flows of
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find
that:
CHAPTER 27 - 2
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
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
always has a higher NPV for discount rates below 10.19 percent, and always has a lower NPV for
discount rates above 10.19 percent.
R NPVX NPVY
0% $8,020.00 $7,860.00
14. a. The equation for the NPV of the project is:
The NPV is greater than zero, so we would accept the project.
CHAPTER 27 - 3
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:
The NPV decision rule implies accepting Project I, since the NPVI is greater than the NPVII.
CHAPTER 27 - 4
c. Using the profitability index to compare mutually exclusive projects can be ambiguous when
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:
B: $31,000 / 1.15 + $28,000 / 1.152 = $48,128.54
The discounted payback criterion implies accepting Project B because it pays back sooner than A.
c. The NPV for each project is:
A: NPV = –$455,000 + $58,000 / 1.15 + $85,000 / 1.152 + $85,000 / 1.153 + $572,000 / 1.154
B: NPV = –$65,000 + $31,000 / 1.15 + $28,000 / 1.152 + $25,500 / 1.153 + $19,000 / 1.154
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:
CHAPTER 27 - 5
B: $65,000 = $31,000 / (1+IRR) + $28,000 / (1+IRR)2 + $25,500 / (1+IRR)3
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:
A: PI = ($58,000 / 1.15 + $85,000 / 1.152 + $85,000 / 1.153 + $572,000 / 1.154) / $455,000
B: PI = ($31,000 / 1.15 + $28,000 / 1.152 + $25,500 / 1.153 + $19,000 / 1.154) / $65,000
Profitability index criterion implies accept Project B because its PI is greater than Project A’s.
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
$1 trillion, at an infinite rate of interest, the value of this cash flow today is zero. So, if the future
cash flows have no value today, the NPV of the project is simply the cash flow today, so at an
infinite interest rate:
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:
CHAPTER 27 - 6
19. The MIRR 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:
0 = –$41,000 + $88,720.77 / (1 + MIRR)5
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 = –$46,836.66 + $98,120.77 / (1 + MIRR)5
CHAPTER 27 - 7
Intermediate
20. With different discounting and reinvestment rates, we need to make sure to use the appropriate
interest rate. The MIRR for the project with all three approaches is:
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, the 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
that:
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 using the reinvestment rate. So, the reinvesting the cash flows to Time 5, we find:
So, the MIRR using the discounting approach is:
0 = –$41,000 + $84,289.61 / (1 + MIRR)5
Combination approach:
In the combination approach, we find the value of all cash outflows at Time 0 using the discount
rate, and the value of all cash inflows at the end of the project using the reinvestment rate. So, the
value of the cash flows is:
CHAPTER 27 - 8
So, the MIRR using the discounting approach is:
0 = –$46,578.44 + $93,689.61 / (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:
Challenge
23. Given the seven-year payback, the worst case is that the payback occurs at the end of the seventh
year. Thus, the worst-case:
The best case has infinite cash flows beyond the payback point. Thus, the best-case NPV is infinite.
24. 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
We would accept the project when the NPV is greater than zero. See for yourself that the NPV is
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
PV of cash inflows = C1 / (R – g)
CHAPTER 27 - 9
NPV is the PV of the inflows minus the PV of the outflows, so the NPV is:
The NPV is positive, so we would accept the project.
b. Here we want to know the minimum growth rate in cash flows necessary to accept the project.
Solving for g, we get:
26. The IRR of the project is:
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find
that:
At an interest rate of 12 percent, the NPV is:
At an interest rate of zero percent, we can add cash flows, so the NPV is:
And at an interest rate of 24 percent, the NPV is:
The cash flows for the project are unconventional. Since the initial cash flow is positive and the
remaining cash flows are negative, the decision rule for IRR is invalid in this case. The NPV profile
is upward sloping, indicating that the project is more valuable when the interest rate increases.
CHAPTER 27 - 10
27. The IRR is the interest rate that makes the NPV of the project equal to zero. So, the IRR of the
project is:
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
the IRR equation, what we are really doing is solving for the roots of the equation. Going back to
high school algebra, in this problem we are solving a quadratic equation. In case you don’t
remember, the quadratic equation is:
x =
−b±
√
b2−4ac
2a
In this case, the equation is:
x =
−(−11,000 )±
√
(−11,000 )2−4(25 ,000 )(7,000 )
2(7, 000 )
The square root term works out to be:
The square root of a negative number is a complex number, so there is no real number solution,
meaning the project has no real IRR.
28. 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 = $485,000(1.04) = $504,400
So, the NPV of the project is:
And the IRR of the project is:
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find
that:
CHAPTER 27 - 11
While this may look like a MIRR calculation, it is not a MIRR, rather it is a standard IRR
