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Economics

ECN 104

Vikraman Baskaran

Fall

Description

CHAPTER 9
NET PRESENT VALUE AND OTHER INVESTMENT
CRITERIA
Learning Objectives
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
1. (LO2, 3) 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, zero, or negative,
depending on whether the discount rate is less than, equal to, or greater than the IRR. The discounted
payback includes the effect of the relevant discount rate. If a project’s discounted payback period is
less than the project’s life, it must be the case that NPV is positive.
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.
3. (LO2)
a. Payback period is simply 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 some predetermined cutoff for the payback period, the decision rule is to
accept projects that payback before this cutoff, and reject projects that take longer to payback.
b. The worst problem associated with payback period is that it ignores the time value of money. In
addition, the selection of a hurdle point for payback period is an arbitrary exercise that lacks any
steadfast rule or method. The payback period is biased towards short-term projects; it fully
ignores any cash flows that occur after the cutoff point.
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 consider-
ations often warrant a payback analysis as sufficient; maintenance projects are another example
where the detailed analysis of other methods is often not needed. Since payback is biased
towards liquidity, it may be a useful and appropriate analysis method for short-term projects
where cash management is most important.
87 4. (LO3)
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.
5. (LO4)
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 the
project divided by some average balance sheet value for the project. This text computes AAR as
average net income with respect to average (total) book value. Given some predetermined cutoff
for AAR, the decision rule is to accept projects with an AAR in excess of the target measure,
and reject all other projects.
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
manager, both the reliance on accounting numbers rather than relevant market data and the
exclusion of time value of money considerations are troubling. Despite these problems, AAR
continues to be used in practice because (1) the accounting information is usually available, (2)
analysts often use accounting ratios to analyze firm performance, and (3) managerial
compensation is often tied to the attainment of certain target accounting ratio goals.
6. (LO1)
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.
88 7. (LO5)
a. 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 IRR decision rule is to accept projects with IRRs greater
than the discount rate, and to reject projects with IRRs less than the discount rate.
b. IRR is the interest 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
flows, and also ambiguously ranks some mutually exclusive projects. However, for stand-alone
projects with conventional cash flows, IRR and NPV are interchangeable techniques.
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
may be a preferred method to NPV in situations where an appropriate discount rate is unknown
are uncertain; in this situation, IRR would provide more information about the project than
would NPV.
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.
9. (LO2, 5) PB = I / C ; – I + C / r = NPV, 0 = – I + C / IRR so IRR = C / I ; thus IRR = 1 / PB
For long-lived projects with relatively constant cash flows, the sooner the project pays back, the
greater is the IRR.
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.
11. (LO1) 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 even these require revenue and cost projections. The discounted cash flow measures
(discounted payback, NPV, IRR, and profitability index) are really only slightly more difficult in
practice.
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!
89 13. (LO5) 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 the interest rate between the two remaining cash flows is calculated. As such, the MIRR is
not a true interest rate. In contrast, consider the IRR. If you take the initial investment, and calculate
the future value at the IRR, you can replicate the future cash flows of the project exactly.
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.
15. (LO5) 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:
C 0 C1 C 2 IRR
Project A –$100 $10 $110 10%
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
pizza? No. Finally, consider the yield to maturity calculation on a bond. If you think about it, the
YTM is the IRR on the bond, but no mention of a reinvestment assumption for the bond coupons is
suggested. The reason is that reinvestment is irrelevant to the YTM calculation; in the same way,
reinvestment is irrelevant in the IRR calculation. Our caveat about blocked funds applies here as well.
90 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. (LO2) To calculate the payback period, we need to find the time that the project has recovered its
initial investment. After two years, the project has created:
$1,500 + 2,600 = $4,100
in cash flows. The project still needs to create another:
$4,800 – 4,100 = $700
in cash flows. During the third year, the cash flows from the project will be $2,900. So, the payback
period will be 2 years, plus what we still need to make divided by what we will make during the third
year. The payback period is:
Payback = 2 + ($700 / $2,900) = 2.24 years
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.
91 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 = $5,701.75
2
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
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
5. (LO3)
R = 0%: 4 + ($200 / $3,700) = 4.05 years
discounted payback = regular payback = 4.05 years
R = 5%: $3,700/1.05 + $3,700/1.05 + $3,700/1.05 + $3,700/1.05 = $13,120.01
5
$3,700/1.05 = $2,899.05
discounted payback = 4 + ($15,000 – 13,120.01) / $2,899.05 = 4.65 years
R = 15%: $3,700/1.15 + $3,700/1.15 + $3,700/1.15 + $3,700/1.15 + $3,700/1.15 + $3,700/1.15 5 6
= $14,002.59; The project never pays back.
92 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%
7. (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 = – $30,000 + $13,000/(1+IRR) + $19,000/(1+IRR) + $12,000/(1+IRR) 3
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:
IRR = 22.06%
Since the IRR is greater than the required return we would accept the project.
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:
2 3
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.
9. (LO1, 5) The NPV of a project is the PV of the outflows minus the PV of the inflows. Since the cash
inflows are an annuity, the equation for the NPV of this project at an 8 percent required return is:
NPV = –$110,000 + $24,000(PVIFA 8%, 9= $39,925.31
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:
NPV = –$110,000 + $24,000(PVIFA 20%, 9= –$13,256.80
At a 20 percent required return, the NPV is negative, so we would reject the project.
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:
0 = –$110,000 + $24,000(PVIFA IRR, 9
IRR = 16.15%
93 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%
11. (LO1) The NPV of a project is the PV of the outflows minus the PV of the inflows. 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:
NPV = –$18,000 + 9,800 + 7,500 + 7,300 = $6,600
The NPV at a 10 percent required return is:
NPV = –$18,000 + $9,800/1.1 + $7,500/1.1 + $7,300/1.1 = $2,592.04
The NPV at a 20 percent required return is:
NPV = –$18,000 + $9,800/1.2 + $7,500/1.2 + $7,300/1.2 = –$400.46
And the NPV at a 30 percent required return is:
2 3
NPV = –$18,000 + $9,800/1.3 + $7,500/1.3 + $7,300/1.3 = –$2,700.96
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.
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
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find
that:
IRR = 20.30%
The equation for the IRR of Project B is:
2 3 4
0 = –$37,000 + $6,000/(1+IRR) + $12,500/(1+IRR) + $19,000/(1+IRR) + $23,000/(1+IRR)
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find
that:
IRR = 18.55%
Examining the IRRs of the projects, we see that the IRR is greaterAthan the IRR , so IRR B
decision rule implies accepting project A. This may not be a correct decision; however, because
the IRR criterion has a ranking problem for mutually exclusive projects. To see if the IRR
decision rule is correct or not, we need to evaluate the project NPVs.
94 b. The NPV of Project A is:
NPV = A$37,000 + $19,000/1.11+ $14,500/1.11 + $12,000/1.11 + $9,000/1.113 4
NPV = $6,588.52
A
And the NPV of Project B is:
NPV = B$37,000 + $6,000/1.11 + $12,500/1.11 + $19,000/1.11 + $23,000/1.11 4
NPV = $7,594.13
B
The NPV is Breater than the NPV , so wA 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
Project A. Once we find these differential cash flows, we find the IRR. The equation for the
crossover rate is:
Crossover rate: 0 = $13,000/(1+R) + $2,000/(1+R) – $7,000/(1+R) – $14,000/(1+R) 4
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find
that:
R = 14.25%
At discount rates above 14.25% choose project A; for discount rates below 14.25% choose
project B; indifferent between A and B at a discount rate of 14.25%.
13. (LO1, 5) 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:
0 = –$10,000 + $5,400/(1+IRR) + $3,400/(1+IRR) + $4,500/(1+IRR) 3
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:
IRR = 16.41%
For Project Y, the equation to find the IRR is:
0 = –$10,000 + $4,500/(1+IRR) + $3,600/(1+IRR) + $5,400/(1+IRR) 3
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:
IRR = 16.08%
To find the crossover rate, we subtract the cash flows from one project from the cash flows of the
other project, and find the IRR of the differential cash flows. We will subtract the cash flows from
Project Y from the cash flows from Project X. It is irrelevant which cash flows we subtract from the
other. Subtracting the cash flows, the equation to calculate the IRR for these differential cash flows is:
2 3
Crossover rate: 0 = $900/(1+R) – $200/(1+R) – $900/(1+R)
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:
R = 11.73%
95 The table below shows the NPV of each project for different required returns. Notice that Project Y
always has a higher NPV for discount rates below 11.73 percent, and always has a lower NPV for
discount rates above 11.73 percent.
R $NPV X $NPV Y
0% $3,300.00 $3,500.00
5% $2,114.03 $2,215.74
10% $1,099.92 $1,123.22
15% $225.36 $185.75
20% –$534.72 –$625.00
25% –$1,200.00 –$1,331.20
14. (LO5)
a. The equation for the NPV of the project is:
2
NPV = –$27,000,000 + $46,000,000/1.1 – $6,000,000/1.1 = $9,859,504.13
The NPV is greater than 0, so we would accept the project.
b. The equation for the IRR of the project is:
0 = –$27,000,000 + $46,000,000/(1+IRR) – $6,000,000/(1+IRR)
From Descartes rule of signs, we know there are two IRRs since the cash flows change signs twice.
From trial and error, the two IRRs are:
IRR = 56.14%, –85.77%
When there are multiple IRRs, the IRR decision rule is ambiguous. Both IRRs are correct, that
is, both interest rates make the NPV of the project equal to zero. If we are evaluating whether or
not to accept this project, we would not want to use the IRR to make our decision.
96 15. (LO7) The profitability index is defined as the PV of the cash inflows divided by the PV of the cash
outflows. The equation for the profitability index at a required return of 10 percent is:
2 3
PI = [$6,200/1.1 + $5,600/1.1 + $3,900/1.1 ] / $12,000 = 1.100
The equation for the profitability index at a required return of 15 percent is:
PI = [$6,200/1.15 + $5,600/1.15 + $3,900/1.15 ] / $12,000 = 1.016
The equation for the profitability index at a required return of 22 percent is:
PI = [$6,200/1.22 + $5,600/1.22 + $3,900/1.22 ] / $12,000 = 0.916
We would accept the project if the required return were 10 percent or 15 percent since the PI is
greater than one. We would reject the project if the required return were 22 percent since the PI is less
than one.
16. (LO1, 7)
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:
PII= $18,000(PVIFA 10%,3) / $40,000 = 1.119
PIII $6,100(PVIFA 10%,3 / $12,000 = 1.264
The profitability index decision rule implies that we accept project II, since PI is greater than
II
the PI.I
b. The NPV of each project is:
NPV = I$40,000 + $18,000(PVIFA 10%,3) = $4,763.34
NPV = II12,000 + $6,100(PVIFA 10%,3 = $3,169.80
The NPV decision rule implies accepting Project I, since the NPV is greaterIthan the NPV . II
c. Using the profitability index to compare mutually exclusive projects can be ambiguous when the
magnitude of the cash flows for the two projects are of different scale. In this problem, project I
is roughly 3 times as large as project II and produces a larger NPV, yet the profit-ability index
criterion implies that project II is more acceptable.
17. (LO1, 2, 3, 5, 7)
a. The payback period for each project is:
A: 3 + ($185,000/$430,000) = 3.43 years
B: 2 + ($7,000/$17,000) = 2.41 years
The payback criterion implies accepting project B, because it pays back sooner than project A.
b. The discounted payback for each project is:
A: $25,000/1.15 + $70,000/1.15 + $70,000/1.15 = $120,695.32
$430,000/1.15 = $245,853.90
Discounted payback = 3 + ($350,000 – 120,695.32)/$245,853.90 = 3.93 years
97 B: $17,000/1.15 + $11,000/1.15 + $17,000/1.15 = $34,277.96
$11,000/1.15 = $6,289.29
Discounted payback = 3 + ($35,000 – 34,277.96)/$6,289.29 = 3.11 years
The discounted payback criterion implies accepting project B because it pays back sooner than A
c. The NPV for each project is:
A: NPV = –$350,000 + $25,000/1.15 + $70,000/1.15 + $70,000/1.15 + $430,000/1.15 4
NPV = $16,549.22
B: NPV = –$35,000 + $17,000/1.15 + $11,000/1.15 + $17,000/1.15 + $11,000/1.15 4
NPV = $5,567.25
NPV criterion implies we accept project A because project A has a higher NPV than project B.
d. The IRR for each project is:
A: $350,000 = $25,000/(1+IRR) + $70,000/(1+IRR) + $70,000/(1+IRR) + $430,000/(1+IRR) 4
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we
find that:
IRR = 16.57%
2 3 4
B: $35,000 = $17,000/(1+IRR) + $11,000/(1+IRR) + $17,000/(1+IRR) + $11,000/(1+IRR)
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we
find that:
IRR = 23.05%
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:
2 3 4
A: PI = ($25,000/1.15 + $70,000/1.15 + $70,000/1.15 + $430,000/1.15 ) / $350,000 = 1.047
B: PI = ($17,000/1.15 + $11,000/1.15 + $17,000/1.15 + $11,000/1.15 ) / $35,000 = 1.159
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
index, payback period, discounted payback and IRR imply that you should accept project B. The
final decision should be

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