CHAPTER 7
NET PRESENT VALUE AND OTHER
INVESTMENT CRITERIA
Answers to Concept 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,
payback period is less than the project’s life, it must be the case that NPV is 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
discount rate R*; thus, the IRR must be greater than the required return.
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
rule or method. The payback period is biased towards short-term projects; it fully ignores any
cash flows that occur after the cutoff point.
b. 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
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 target accounting ratio goals.
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
return is less than 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
flows, and it also may ambiguously rank some mutually exclusive projects. However, for stand–
d. 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. The profitability index can be expressed as: PI = (NPV +
cost)/cost = 1 + (NPV/cost). If a firm has a basket of positive NPV projects and is subject to
e. 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
text because it has no serious flaws. The method unambiguously ranks mutually exclusive
projects, and it 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
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,
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
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 even
these require revenue and cost projections. The discounted cash flow measures (discounted payback,
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
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.
C0
C1
C2
IRR
NPV @ 0%
Project A
–$1,000,000
$0
$1,440,000
20%
$440,000
Project B
–$2,000,000
$2,400,000
$0
20%
400,000
However, in one particular case, the statement is true for equally risky projects. If the lives of the two
projects are equal and the cash flows of Project B are twice the cash flows of Project A in every time
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 Project
A, B can have a higher PI yet have a lower NPV. Only in the case of capital rationing could the
of money. Cash flows that occur further out in the future are always more sensitive to changes in the
interest rate. This sensitivity is similar to the interest rate risk of a bond.
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
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
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:
C0
C2
–$100
$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
pizza? No. Finally, consider the yield to maturity calculation on a bond. If you think about it, the YTM
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
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 = $8,700 = $8,700
Cumulative cash flows Year 2 = $8,700 + 7,400 = $16,100
So, the payback period is less than two years. 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
$15,300. 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.
Cumulative cash flows Year 2 = $5,300 + 4,300 = $9,600
Cumulative cash flows Year 3 = $5,300 + 4,300 + 4,800 = $14,400
Payback period = 2 + ($10,700 – 9,600)/$4,800
Payback period = 2.23 years
NPV = –$101.02
Project B:
NPV = –$10,700 + $5,300/1.15 + $4,300/1.152 + $4,800/1.153
NPV = $316.19
The firm should choose Project B since it has a higher NPV than Project A.
investment. The cash flows in this problem are an annuity, so the calculation is simpler. If the initial
cost is $3,400, the payback period is:
Payback = 3 + ($655/$915) = 3.72 years
There is a shortcut to calculate the payback period if the future cash flows are an annuity. Just divide
the initial cost by the annual cash flow. For the $3,400 cost, the payback period is:
For an initial cost of $4,800, the payback period is:
Payback = $4,800/$915 = 5.25 years
The payback period for an initial cost of $7,900 is a little trickier. Notice that the total cash inflows
after eight years will be:
Value today of Year 2 cash flow = $4,900/1.122 = $3,906.25
Value today of Year 3 cash flow = $5,400/1.123 = $3,843.61
Value today of Year 4 cash flow = $5,600/1.124 = $3,558.90
To find the discounted payback, we use these values to find the payback period. The discounted first
year cash flow is $3,839.29, so the discounted payback for an initial cost of $6,000 is:
Discounted payback = 1 + ($6,000 – 3,839.29)/$3,906.25 = 1.55 years
For an initial cost of $9,000, the discounted payback is:
Discounted payback = 2 + ($9,000 – 3,839.29 – 3,906.25)/$3,843.61 = 2.33 years
R = 0%: 3 + ($1,800/$3,900) = 3.46 years
Discounted payback = Regular payback = 3.46 years
R = 7%: $3,900/1.07 + $3,900/1.072 + $3,900/1.073 + $3,900/1.074 = $13,210.12
$3,900/1.075 = $2,780.65
Average book value = (Book value0 + Book value1 + Book value2 + Book value3 +
Book value4 + Book value5)/(Economic life)
Average book value = ($57,000 + 42,750 + 28,500 + 14,250 + 0)/5
Average book value = $28,500
To find the average accounting return, we divide the average project earnings by the average
book value of the machine to calculate the average accounting return. Doing so, we find:
Average accounting return = Average project earnings/Average book value
Average accounting return = $5,800/$28,500
Average accounting return = .2035, or 20.35%
b. The three flaws of the AAR are: 1) The AAR does not work with the right raw materials. It uses
would be the same if the net income in the first year occurs in the last year. 3) The AAR uses an
arbitrary interest rate as the cutoff rate and offers no guidance on what the right targeted rate of
return should be.
6. First, we need to determine the average book value of the project. The book value is the gross
investment minus accumulated depreciation.
Purchase Date
Year 1
Year 2
Year 3
Gross investment
$27,000
$27,000
$27,000
$27,000
Less: Accumulated depreciation
0
8,100
20,500
27,000
Net Investment
$27,000
$18,900
$6,500
$0
Now, we can calculate the average book value as:
Average book value = ($27,000 + 18,900 + 6,500 + 0)/4
Average book value = $13,100
To calculate the average accounting return, we must remember to use the aftertax average net income
when calculating the average accounting return. So, the average aftertax net income is:
Average aftertax net income = (1 – tc)Annual pretax net income
Average aftertax net income = (1 – .25)$3,340
Average aftertax net income = $2,505
The average accounting return is the average after-tax net income divided by the average book value,
Average accounting return = $2,505/$13,100
Average accounting return = .1912, or 19.12%
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:
0 = –$16,100 + $7,800/(1 + IRR) + $9,100/(1 + IRR)2 + $5,300/(1 + IRR)3
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find
that:
IRR = 19.11%
Since the IRR is greater than the required return we would accept the project.
8. The IRR is the interest rate that makes the NPV of the project equal to zero. So, the equation that
0 = –$6,700 + $2,100/(1 + IRR) + $3,900/(1 + IRR)2 + $2,700/(1 + IRR)3
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find
that:
IRR = 13.71%
And the IRR for Project B is:
0 = C0 + C1/(1 + IRR) + C2/(1 + IRR)2 + C3/(1 + IRR)3
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:
PI = $64,000(PVIFA15%,7)/$207,000
PI = $266,266.86/$207,000
PI = 1.286
Therefore, the project should be accepted.
And for Project Beta the profitability index is:
PIBeta = [$1,700/1.10 + $2,900/1.102 + $1,400/1.103]/$3,400 = 1.469
11. a. To have a payback equal to the project’s life, given C is a constant cash flow for N years:
C = I/N
b. To have a positive NPV, I < C (PVIFAR%, N). Thus, C > I/(PVIFAR%, N).
c. Benefits = C (PVIFAR%, N) = 2 × costs = 2I
C = 2I/(PVIFAR%, N)
12. 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:
0 = C0 + C1/(1 + IRR) + C2/(1 + IRR)2 + C3/(1 + IRR)3 + C4/(1 + IRR)4
– $3,700/(1 + IRR)4
Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we
find that:
b. This problem differs from previous ones because the initial cash flow is positive and all future
cash flows are negative. In other words, this is a financing-type project, while previous projects
were investing-type projects. For financing situations, accept the project when the IRR is less
than the discount rate. Reject the project when the IRR is greater than the discount rate.
IRR = 12.41%
Discount Rate = 10%
IRR > Discount Rate
IRR < Discount Rate
Accept the offer when the discount rate is greater than the IRR.
NPV = –$710.70
When the discount rate is 10 percent, the NPV of the offer is negative, so reject the offer.
And the NPV of the project if the discount rate is 20 percent will be:
When the discount rate is 20 percent, the NPV of the offer is positive. So accept the offer.
e. Yes, the decisions under the NPV rule are consistent with the choices made under the IRR rule