CHAPTER 9
NET PRESENT VALUE AND OTHER
INVESTMENT CRITERIA
Answers to Concepts Review and Critical Thinking Questions
1. 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
2. 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
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
b. The worst problem associated with payback period is that it ignores the time value of money. In
c. Despite its shortcomings, payback is often used because (1) the analysis is straightforward and
simple and (2) accounting numbers and estimates are readily available. Materiality
4. 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
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b. The primary disadvantage to using the discounted payback method is that it ignores all cash
c. Discounted payback is an improvement on regular payback because it takes into account the
5. a. The average accounting return is interpreted as an average measure of the accounting
performance of a project over time, computed as some average profit measure attributable to
b. AAR is not a measure of cash flows and market value, but a measure of financial statement
accounts that often bear little resemblance to the relevant value of a project. In addition, the
selection of a cutoff is arbitrary, and the time value of money is ignored. For a financial
6. a. NPV is 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
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
7. a. The IRR is the discount rate that causes the NPV of a series of cash flows to be exactly zero.
b. IRR is the interest rate that causes NPV for a series of cash flows to be zero. NPV is preferred
c. IRR is frequently used because it is easier for many financial managers and analysts to rate
performance in relative terms, such as “12%”, than in absolute terms, such as “$46,000.” IRR
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8. a. The profitability index is the present value of cash inflows relative to the project cost. As such,
b. PI = (NPV + cost)/cost = 1 + (NPV/cost). If a firm has a basket of positive NPV projects and is
9. For a project with future cash flows that are an annuity:
Payback = I/C
And the IRR is:
Solving the IRR equation for IRR, we get:
Notice this is just the reciprocal of the payback. So:
10. There are a number of reasons for such investments. Two of the most important have to do with
transportation costs and exchange rates. Manufacturing in the U.S. places the finished product much
11. 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
12. Yes, they are applicable. 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
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13. 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
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 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
15. 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
C0C1C2IRR
Project A –$100 $10 $110 10%
Suppose this $100 is a deposited 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
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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. To calculate the payback period, we need to find the time that the project requires to recover its
initial investment. After three years, the project has created:
in cash flows. The project still needs to create another:
in cash flows. During the fourth year, the cash flows from the project will be $1,700. So, the payback
2. To calculate the payback period, we need to find the time that the project requires to recover its
There is a shortcut to calculate payback period when the project cash flows are an annuity. Just
divide the initial cost by the annual cash flow. For the $3,300 cost, the payback period is:
The payback period for an initial cost of $6,100 is a little trickier. Notice that the total cash inflows
after eight years will be:
If the initial cost is $6,100, the project never pays back. Notice that if you use the shortcut for
annuity cash flows, you get:
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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. Project A has total cash flows of $29,000 after Year 2, so the cash flows are short by $6,000 of
recapturing the initial investment, so the payback for Project A is:
Project B has cash flows of:
during this first three years. The cash flows are still short by $5,000 of recapturing the initial
investment, so the payback for Project B is:
Using the payback criterion and a cutoff of three years, accept Project A and reject Project B.
4. 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 = $2,800/1.11 = $2,522.52
To find the discounted payback, we use these values to find the payback period. The discounted first
For an initial cost of $6,400, the discounted payback is:
Notice the calculation of discounted payback. We know the payback period is between two and three
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Discounted payback = Regular payback = 3.62 years
6. Our definition of AAR is the average net income divided by the average book value. The average net
income for this project is:
And the average book value is:
So, the AAR for this project is:
AAR = Average net income/Average book value
7. 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:
Since the IRR is greater than the required return, we would accept the project.
8. The NPV of a project is the PV of the inflows minus the PV of the outflows. The equation for the
NPV of this project at an 11 percent required return is:
At an 11 percent required return, the NPV is positive, so we would accept the project.
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9. The NPV of a project is the PV of the inflows minus the PV of the outflows. Since the cash inflows
are an annuity, the equation for the NPV of this project at an 8 percent required return is:
At an 8 percent required return, the NPV is positive, so we would accept the project.
The equation for the NPV of the project at a 20 percent required return is:
We would be indifferent to the project if the required return was equal to the IRR of the project,
since at that required return the NPV is zero. The IRR of the project is:
10. 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:
11. The NPV of a project is the PV of the inflows minus the PV of the outflows. 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:
The NPV at a 10 percent required return is:
The NPV at a 20 percent required return is:
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And the NPV at a 30 percent required return is:
Notice that as the required return increases, the NPV of the project decreases. This will always be
true for projects with conventional cash flows. Conventional cash flows are negative at the beginning
of the project and positive throughout the rest of the project.
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