NET PRESENT VALUE AND OTHER INVESTMENT
LO1 How to compute the net present value and why it is the best decision criterion.
LO2 The payback rule and some of its shortcomings.
LO3 The discounted payback rule and some of its shortcomings.
LO4 Accounting rates of return and some of the problems with them.
LO5 The internal rate of return criterion and its strengths and weaknesses.
LO6 The modified internal rate of return.
LO7 The profitability index and its relation to net present value.
Answers to Concepts Review and Critical Thinking Questions
2. (LO2, 3, 6, 7) 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 positive, the
discounted payback period must be less than the project’s life. If NPV is positive, then the present
value of future cash inflows is greater than the initial investment cost; thus PI must be greater than 1.
If NPV is positive for a certain discount rate R, then it will be zero for some larger discount rate R*;
thus the IRR must be greater than the required return.
a. The discounted payback is calculated the same as is regular payback, with the exception that
each cash flow in the series is first converted to its present value. Thus discounted payback
provides a measure of financial/economic break-even because of this discounting; just as regular
payback provides a measure of accounting break-even because it does not discount the cash
flows. Given some predetermined cutoff for the discounted payback period, the decision rule is
to accept projects that whose discounted cash flows payback before this cutoff period, and to
reject all other projects.
b. The primary disadvantage to using the discounted payback method is that it ignores all cash
flows that occur after the cutoff date, thus biasing this criterion towards short-term projects. As
a result, the method may reject projects that in fact have positive NPVs, or it may accept
projects with large future cash outlays resulting in negative NPVs. In addition, the selection of a
cutoff point is again an arbitrary exercise.
c. Discounted payback is an improvement on regular payback because it takes into account the
time value of money. For conventional cash flows and strictly positive discount rates, the
discounted payback will always be greater than the regular payback period.
a. NPV is simply the present value of a project’s cash flows. 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.
b. NPV is superior to the other methods of analysis presented in the text because it has no serious
flaws. The method unambiguously ranks mutually exclusive projects, and can differentiate
between projects of different scale and time horizon. The only drawback to NPV is that it relies
on cash flow and discount rate values that are often estimates and not certain, but this is a
problem shared by the other performance criteria as well. A project with NPV = $2,500 implies
that the total shareholder wealth of the firm will increase by $2,500 if the project is accepted.
87 8. (LO7)
a. 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
projects with a PI less than one.
b. PI = (NPV + cost)/cost = 1 + (NPV/cost). If a firm has a basket of positive NPV projects and is
subject to capital rationing, PI may provide a good ranking measure of the projects, indicating
the “bang for the buck” of each particular project.
10. (LO1) 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
compared to other possible manufacturing locations. Of great importance is the fact that
manufacturing in the U.S. means that a much higher proportion of the costs are paid in dollars. Since
sales are in dollars, the net effect is to immunize profits to a large extent against fluctuations in
exchange rates. This issue is discussed in greater detail in the chapter on international finance.
12. (LO1, 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
hard to measure. To the extent that benefits are measurable, the question of an appropriate required
return remains. Payback rules are commonly used in such cases. Finally, realistic cost/benefit analysis
along the lines indicated should definitely be used by governments and would go a long way toward
balancing the budget!
14. (LO1, 6) 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 with those cash flows once they are generated is not relevant. Put differently,
the value of a project depends on the cash flows generated by the project, not on the future value of
those cash flows. The fact that the reinvestment “works” only if you use the required return as the
reinvestment rate is also irrelevant simply because reinvestment is not relevant in the first place to the
value of the project.
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
international investing, a company may be required to reinvest the cash flows in the country in which
they are generated and not “repatriate” the money. Such funds are said to be “blocked” and
reinvestment becomes relevant because the cash flows are not truly available.
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.
88 2. (LO2) To calculate the payback period, we need to find the time that the project has recovered its
initial investment. The cash flows in this problem are an annuity, so the calculation is simpler. If the
initial cost is $3,000, the payback period is:
Payback = 3 + ($420 / $860) = 3.49 years
There is a shortcut to calculate the future cash flows are an annuity. Just divide the initial cost by the
annual cash flow. For the $3,000 cost, the payback period is:
Payback = $3,000 / $860 = 3.49 years
For an initial cost of $5,000, the payback period is:
Payback = $5,000 / $860 = 5.81 years
The payback period for an initial cost of $7,000 is a little trickier. Notice that the total cash inflows
after eight years will be:
Total cash inflows = 8($860) = $6,880
If the initial cost is $7,000, the project never pays back. Notice that if you use the shortcut for annuity
cash flows, you get:
Payback = $7,000 / $860 = 8.14 years.
This answer does not make sense since the cash flows stop after eight years, so again, we must
conclude the payback period is never.
3. (LO2) Project A has cash flows of $35,000 in Year 1, so the cash flows are short by $15,000 of
recapturing the initial investment, so the payback for Project A is:
Payback = 1 + ($15,000 / $21,000) = 1.71 years
Project B has cash flows of:
Cash flows = $15,000 + 22,000 + 31,000 = $68,000
during this first three years. The cash flows are still short by $2,000 of recapturing the initial
investment, so the payback for Project B is:
B: Payback = 3 + ($2,000 / $240,000) = 3.008 years
Using the payback criterion and a cutoff of 3 years, accept project A and reject project B.
4. (LO3) 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 = $6,500/1.14 2 = $5,701.75
Value today of Year 2 cash flow = $7,000/1.14 = $5,386.27
Value today of Year 3 cash flow = $7,500/1.14 = $5,062.29
Value today of Year 4 cash flow = $8,000/1.14 = $4,736.64
To find the discounted payback, we use these values to find the payback period. The discounted first
year cash flow is $5,701.75, so the discounted payback for an $8,000 initial cost is:
Discounted payback = 1 + ($8,000 – 5,701.75)/$5,386.27 = 1.43 years
89 For an initial cost of $13,000, the discounted payback is:
Discounted payback = 2 + ($13,000 – 5,701.75 – 5,386.27)/$5,062.29 = 2.38 years
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
investment. We divide this amount by the discounted amount we will earn in Year 3 to get the
fractional portion of the discounted payback.
If the initial cost is $18,000, the discounted payback is:
Discounted payback = 3 + ($18,000 – 5,701.75 – 5,386.27 – 5,062.29) / $4,736.64 = 3.39 years
6. (LO4) Our definition of AAR is the average net income divided by the average book value. The
average net income for this project is:
Average net income = ($1,632,000 + 2,106,500 + 1,941,700 + 1,298,000) / 4 = $1,744,550
And the average book value is:
Average book value = ($18,000,000 + 0) / 2 = $9,000,000
So, the AAR for this project is:
AAR = Average net income / Average book value = $1,744,500 / $9,000,000 = .1938 or 19.38%
8. (LO1) The NPV of a project is the PV of the outflows minus the PV of the inflows. The equation for
the NPV of this project at an 11 percent required return is:
NPV = – $30,000 + $13,000/1.11 + $19,000/1.11 + $12,000/1.11 = $5,906.83
At an 11 percent required return, the NPV is positive, so we would accept the project.
The equation for the NPV of the project at a 30 percent required return is:
NPV = – $30,000 + $13,000/1.30 + $19,000/1.30 + $12,000/1.30 = –$3,295.40
At a 30 percent required return, the NPV is negative, so we would reject the project.
10. (LO5) 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:
0 = –$18,000 + $9,800/(1+IRR) + $7,500/(1+IRR) + $7,300/(1+IRR) 3
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:
IRR = 18.49%
12. (LO1, 5)
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:
0 = –$37,000 + $19,000/(1+IRR) + $14,500/(1+IRR) + $12,000/(1+IRR) + $9,000/(1+IRR) 4