CHAPTER 5
NET PRESENT VALUE AND OTHER
INVESTMENT RULES
Answers to Concepts Review and Critical Thinking Questions
1. Assuming conventional cash flows, a payback period less than the project’s life means that the NPV
is positive for a zero discount rate, but nothing more definitive can be said. For discount rates greater
2. Assuming conventional cash flows, if a project has a positive NPV for a certain discount rate, then it
will also have a positive NPV for a zero discount rate; thus, the payback period must be less than the
3. a. Payback period is the accounting break-even point of a series of cash flows. To actually compute
the payback period, it is assumed that any cash flow occurring during a given period is realized
continuously throughout the period, and not at a single point in time. The payback is then the
point in time for the series of cash flows when the initial cash outlays are fully recovered. Given
b. The IRR is the discount rate that causes the NPV of a series of cash flows to be identically zero.
IRR can thus be interpreted as a financial break-even rate of return; at the IRR discount rate, the
net value of the project is zero. The acceptance and rejection criteria are:
If C0 < 0 and all future cash flows are positive, accept the project if the internal rate of
return is greater than or equal to the discount rate.
If C0 < 0 and all future cash flows are positive, reject the project if the internal rate of
c. The profitability index is the present value of cash inflows relative to the project cost. As such,
it is a benefit/cost ratio, providing a measure of the relative profitability of a project. The
profitability index decision rule is to accept projects with a PI greater than one, and to reject
d. NPV is the present value of a project’s cash flows, including the initial outlay. NPV specifically
measures, after considering the time value of money, the net increase or decrease in firm wealth
due to the project. The decision rule is to accept projects that have a positive NPV, and reject
4. For a project with future cash flows that are an annuity:
Payback = I/C
And the IRR is:
5. There are a number of reasons. Two of the most important have to do with transportation costs and
exchange rates. Manufacturing in the U.S. places the finished product much closer to the point of sale,
6. The single biggest difficulty, by far, is coming up with reliable cash flow estimates. Determining an
appropriate discount rate is also not a simple task. These issues are discussed in greater depth in the
7. Yes, they are. Such entities generally need to allocate available capital efficiently, just as for-profits
do. However, it is frequently the case that the “revenues” from not-for-profit ventures are intangible.
8. The statement is false. If the cash flows of Project B occur early and the cash flows of Project A occur
late, then, for a low discount rate, the NPV of A can exceed the NPV of B. Observe the following
example.
9. Although the profitability index (PI) is higher for Project B than for Project A, Project A should be
chosen because it has the greater NPV. Confusion arises because Project B requires a smaller
10. a. Project A would have a higher IRR since the initial investment for Project A is less than that of
11. Project B’s NPV would be more sensitive to changes in the discount rate. The reason is the time value
12. The MIRR is calculated by finding the present value of all cash outflows, the future value of all cash
inflows to the end of the project, and then calculating the IRR of the two cash flows. As a result, the
13. The statement is incorrect. It is true that if you calculate the future value of all intermediate cash flows
to the end of the project at the required return, then calculate the NPV of this future value and the
initial investment, you will get the same NPV. However, NPV says nothing about reinvestment of
intermediate cash flows. The NPV is the present value of the project cash flows. What is actually done
14. The statement is incorrect. It is true that if you calculate the future value of all intermediate cash flows
to the end of the project at the IRR, then calculate the IRR of this future value and the initial
investment, you will get the same IRR. However, as in the previous question, what is done with the
cash flows once they are generated does not affect the IRR. Consider the following example:
C1
C2
Project A
$10
$110
Suppose this $100 is a deposit into a bank account. The IRR of the cash flows is 10 percent. Does the
IRR change if the Year 1 cash flow is reinvested in the account, or if it is withdrawn and spent on
Solutions to Questions and Problems
NOTE: All end-of-chapter problems were solved using a spreadsheet. Many problems require multiple
steps. Due to space and readability constraints, when these intermediate steps are included in this solutions
manual, rounding may appear to have occurred. However, the final answer for each problem is found
without rounding during any step in the problem.
Basic
1. a. The payback period is the time that it takes for the cumulative undiscounted cash inflows to equal
the initial investment.
Project A:
Cumulative cash flows Year 1 = $10,400 = $10,400
Cumulative cash flows Year 2 = $10,400 + 5,900 = $16,300
Cumulative cash flows Year 3 = $12,700 + 6,100 + 5,300 = $24,100
To calculate the fractional payback period, find the fraction of Year 3’s cash flow that is needed
for the company to have cumulative undiscounted cash flows of $19,000. Divide the difference
between the initial investment and the cumulative undiscounted cash flows as of Year 2 by the
undiscounted cash flow of Year 3.
2. To calculate the payback period, we need to find the time that the project has taken to recover its initial
investment. The cash flows in this problem are an annuity, so the calculation is simpler. If the initial
cost is $2,700, the payback period is:
Payback = 4 + ($140/$640) = 4.22 years
3. When we use discounted payback, we need to find the value of all cash flows today. The value today
of the project cash flows for the first four years is:
Value today of Year 1 cash flow = $5,000/1.11 = $4,504.50
Value today of Year 2 cash flow = $5,500/1.112 = $4,463.92
4. To calculate the discounted payback, discount all future cash flows back to the present, and use these
discounted cash flows to calculate the payback period. To find the fractional year, we divide the
amount we need to make in the last year to pay back the project by the amount we will make. Doing
5. The IRR is the interest rate that makes the NPV of the project equal to zero. So, the equation that defines
the IRR for this project is:
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:
6. The IRR is the interest rate that makes the NPV of the project equal to zero. So, the equation that defines
the IRR for Project A is:
IRR = 18.24%
And the IRR for Project B is:
IRR = 19.31%
7. The profitability index is defined as the PV of the future cash flows divided by the PV of the initial
cost. The cash flows from this project are an annuity, so the equation for the profitability index is:
8. a. The profitability index is the present value of the future cash flows divided by the initial cost. So,
for Project Alpha, the profitability index is:
PIAlpha = [$1,500/1.085 + $1,300/1.0852 + $1,100/1.0853]/$2,700 = 1.240
Intermediate
9. a. To have a payback equal to the project’s life, given C is a constant cash flow for N years:
C = I/N
10. a. The IRR is the interest rate that makes the NPV of the project equal to zero. So, the equation that
defines the IRR for this project is:
IRR = 11.21%
b. This problem differs since the initial cash flow is positive and all future cash flows are negative.
IRR = 11.21%
Discount Rate = 10%
IRR = 11.21%
Discount Rate = 20%
IRR < Discount Rate
Accept the offer when the discount rate is greater than the IRR.
d. The NPV is the sum of the present value of all cash flows, so the NPV of the project if the
discount rate is 10 percent will be:
NPV = $8,700 – $3,900/1.1 – $2,900/1.12 – $2,300/1.13 – $1,800/1.14
NPV = –$199.60
11. a. The IRR is the interest rate that makes the NPV of the project equal to zero. So, the IRR for each
project is:
IRR = 20.96%
Submarine Ride IRR:
IRR = 19.87%
Based on the IRR rule, the deepwater fishing project should be chosen because it has the higher
IRR.
b. 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 deepwater fishing cash flows from the
submarine ride cash flows. The incremental IRR is the IRR of these incremental cash flows. So,
the incremental cash flows of the submarine ride are:
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:
Incremental IRR = 18.40%
c. The NPV is the sum of the present value of the cash flows from the project, so the NPV of each
project will be:
Deepwater Fishing:
Submarine Ride:
12. 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:
b. The NPV of each project is:
c. Using the profitability index to compare mutually exclusive projects can be ambiguous when the
13. a. The equation for the NPV of the project is:
b. The equation for the IRR of the project is:
14. a. The payback period is the time that it takes for the cumulative undiscounted cash inflows to equal
the initial investment.
Board game:
Cumulative cash flows Year 1 = $670 = $670
Cumulative cash flows Year 2 = $670 + 510 = $1,180
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:
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:
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
DVD
–$1,700
$1,300
$750
$350
Board game
–850
670
510
90
DVD – Board game
–$850
$630
$240
$260
Setting the present value of these incremental cash flows equal to zero, we find the incremental
IRR is:
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:
PICDMA = [$23,000,000/1.10 + $16,000,000/1.102 + $6,000,000/1.103]/$18,000,000 = 2.15
b. The NPV of each project is:
NPVCDMA = –$18,000,000 + $23,000,000/1.10 + $16,000,000/1.102 + $6,000,000/1.103
NPVCDMA = $20,640,120.21
c. We would like to invest in all three projects since each has a positive NPV. If the budget is limited
to $43 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:
Cumulative cash flows Year 1 = $373,000 = $373,000
Cumulative cash flows Year 2 = $373,000 + 219,000 = $592,000
b. The NPV of each project is:
NPVAZM = –$575,000 + $373,000/1.10 + $219,000/1.102 + $185,000/1.103
NPVAZM = $84,075.88
c. The IRR is the interest rate that makes the NPV of the project equal to zero. So, the IRR of the
AZM is:
0 = –$575,000 + $373,000/(1 + IRR) + $219,000/(1 + IRR)2 + $185,000/(1 + IRR)3
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we
find that:
d. Incremental IRR analysis is not necessary. The AZM has the smallest initial investment, and the
largest NPV, so it should be accepted.
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:
PIA = [$165,000/1.12 + $165,000/1.122]/$225,000 = 1.24
b. The NPV of each project is:
NPVA = –$225,000 + $165,000/1.12 + $165,000/1.122
NPVA = $53,858.42
c. Accept Projects A, B, and C. Since the projects are independent, accept all three projects because
the respective profitability index of each is greater than 1.
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.
Based on the profitability index rule, Project C can be eliminated because its PI is less than the
PI of Project A. Because of the problem of scale, we cannot compare the PIs of Projects A and
B. However, we can calculate the PI of the incremental cash flows of the two projects, which are:
e. Remember that the NPV is additive across projects. Since we can spend $450,000, we could take
18. a. The payback period is the time that it takes for the cumulative undiscounted cash inflows to equal
the initial investment.
Dry Prepeg:
Cumulative cash flows Year 1 = $900,000 = $900,000
Cumulative cash flows Year 2 = $900,000 + 700,000 = $1,600,000
b. The NPV of each project is:
NPVDry prepeg = –$1,500,000 + $900,000/1.08 + $700,000/1.082 + $725,000/1.083
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:
0 = –$1,500,000 + $900,000/(1 + IRR) + $700,000/(1 + IRR)2 + $725,000/(1 + IRR)3
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we
find that:
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
Dry prepeg
–$1,500,000
$900,000
$700,000
$725,000
Solvent prepeg
Dry prepeg – Solvent prepeg
–$ 850,000
$555,000
$130,000
$365,000