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CHAPTER 5
NET PRESENT VALUE AND OTHER
INVESTMENT CRITERIA
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
than zero, the payback period will still be less than the project’s life, but the NPV may be positive,
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
project life. Since discounted payback is calculated at the same discount rate as is NPV, if NPV is
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
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.
IRR is the discount rate that causes NPV for a series of cash flows to be zero. NPV is preferred
in all situations to IRR; IRR can lead to ambiguous results if there are non-conventional cash
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
projects with a negative NPV. NPV is superior to the other methods of analysis presented in the
4. For a project with future cash flows that are an annuity:
And the IRR is:
IRR = C / I
Notice this is just the reciprocal of the payback. So:
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, resulting in significant savings in transportation costs. It also reduces inventories because goods
spend less time in transit. Higher labor costs tend to offset these savings to some degree, at least
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
next several chapters. The payback approach is probably the simplest, followed by the AAR, but
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 not
tangible. For example, charitable giving has real opportunity costs, but the benefits are generally
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.
C0C1C2IRR NPV @ 0%
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
investment than Project A. Since the denominator of the PI ratio is lower for Project B than for
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
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
cash flows have been discounted or compounded by one interest rate (the required return), and then
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
One caveat: Our discussion here assumes that the cash flows are truly available once they are
generated, meaning that it is up to firm management to decide what to do with the cash flows. In
certain cases, there may be a requirement that the cash flows be reinvested. For example, in
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:
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
well.
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:
Companies can calculate a more precise value using fractional years. To calculate the fractional
payback period, find the fraction of Year 2’s cash flows that is needed for the company to have
cumulative undiscounted cash flows of $20,000. Divide the difference between the initial
investment and the cumulative undiscounted cash flows as of Year 1 by the undiscounted cash
flow of Year 2.
Project B:
To calculate the fractional payback period, find the fraction of Year 3’s cash flows that is
needed for the company to have cumulative undiscounted cash flows of $24,000. 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.
b. Discount each project’s cash flows at 15 percent. Choose the project with the highest NPV.
Project A:
The firm should choose Project B since it has a higher NPV than Project A.
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 $3,200, the payback period is:
For an initial cost of $4,800, the payback period is:
If the initial cost is $7,300, the project never pays back. Notice that if you use the shortcut for
annuity cash flows, you get:
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:
To find the discounted payback, we use these values to find the payback period. The discounted first
year cash flow is $4,464.29, so the discounted payback for an initial cost of $8,000 is:
Notice the calculation of discounted payback. We know the payback period is between two and three
years, so we subtract the discounted values of the Year 1 and Year 2 cash flows from the initial cost.
This is the numerator, which is the discounted amount we still need to make to recover our initial
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 payback the project by the amount we will make. Doing
so, we find:
r = 0%: 4 + ($300 / $4,300) = 4.07 years
Discounted payback = Regular payback = 4.07 years
r = 17%: $4,300 / 1.17 + $4,300 / 1.172 + $4,300 / 1.173 + $4,300 / 1.174 + $4,300 / 1.175
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:
Since the IRR is greater than the required return we would accept the project.
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 this Project A is:
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:
IRR = 13.40%
And the IRR for Project B is:
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:
7. The profitability index is defined as the PV of the cash inflows divided by the PV of the cash
outflows. 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:
b. According to the profitability index, you would accept Project Beta. However, remember the
profitability index rule can lead to an incorrect decision when ranking mutually exclusive
projects.
Intermediate
9. a. To have a payback equal to the project’s life, given C is a constant cash flow for N years:
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:
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we
find that:
b. This problem differs since the initial cash flow is positive and all future cash flows are
negative. In other words, this is a financing-type project. For financing projects, accept the
project when the IRR is less than the discount rate. Reject the project when the IRR is greater
than the discount rate.
Reject the offer when the discount rate is less than the IRR.
c. Using the same reason as part b., we would accept the project if the discount rate is 20 percent.
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:
When the discount rate is 20 percent, the NPV of the offer is $1,240.28. Accept the offer.
e. Yes, the decisions under the NPV rule are consistent with the choices made under the IRR rule
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:
Deepwater Fishing IRR:
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we
find that:
IRR = 17.81%
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:
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:
For investing-type projects, accept the larger project when the incremental IRR is greater than
the discount rate. Since the incremental IRR, 16.84 percent, is greater than the required rate of
return of 14 percent, choose the submarine ride project. Note that this is not the choice when
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:
Since the NPV of the submarine ride project is greater than the NPV of the deepwater fishing
project, choose the submarine ride project. The incremental IRR rule is always consistent with
the NPV rule.
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:
The profitability index decision rule implies that we accept Project II, since PIII is greater than
the PII.
b. The NPV of each project is:
The NPV decision rule implies accepting Project I, since the NPVI is greater than the NPVII.
c. Using the profitability index to compare mutually exclusive projects can be ambiguous when
the magnitudes of the cash flows for the two projects are of different scales. In this problem,
