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Unformatted text preview: Capital Budgeting
Decision Methods
“Everything is worth what its purchaser
will pay for it.”
—Publilius Syrus Bank of America buys Merrill Lynch
On September 15, 2008 Bank of America (BOA) bought Merrill Lynch & Co.
(ML) by offering .8595 shares of BOA stock for each share of ML stock. The
purchase was valued at $50 billion. The price paid was 1.8 times Merrill’s
tangible book value.
Did BOA pay too much? Why did the U.S. Treasury have to lend BOA
$20 billion to help consummate the merger? Should the U.S. government
have let Merrill Lynch go under as it did Lehman Brothers?
This acquisition was approved by the boards of directors of both Bank of
America and Merrill Lynch and Co. Federal Reserve Chairman Ben Bernanke
admitted, however, that if the deal fell through it could trigger a “broader
systematic crisis.” Was Bank of America pressured to purchase Merrill Lynch?
This was a capital budgeting decision. Merrill Lynch brings a stable of
16,000 financial advisors to the deal. Bank of America expects to increase
the amount of money it manages, and the accompanying fees it collects for
doing so, as a result of this merger. There will also be additional underwriting
fees to be had. Is this a value adding move by Bank of America?
This is what capital budgeting is all about. An investment is made and
future benefits are expected as a result of this investment. This chapter shows
how to evaluate a capital budgeting project. Source: http://newsroom.bankofamerica.com/index.php?s=43&item=8255
http://hispanicbusiness.com/news/news_print.asp?id=158659 270 © Cara Purdy (http://www.fotolia.com/p/13851) Chapter Overview Learning Objectives We now look at the decision methods that analysts use to determine whether to
approve a given investment project and how they account for a project’s risk.
Investment projects such as DaimlerChrysler’s expansion project can fuel a firm’s
success, so effective project selection often determines a firm’s value. The decision
methods for choosing acceptable investment projects, then, are some of the most
important tools financial managers use.
We begin by looking at the capital budgeting process and four capital budgeting
decision methods: payback, net present value, internal rate of return, and modified
internal rate of return. Then we discuss how to select projects when firms limit
their budget for capital projects, a practice called capital rationing. Finally, we look
at how firms measure and compensate for the risk of capital budgeting projects. The Capital Budgeting Process
All businesses budget for capital projects—an airline that considers purchasing
new planes, a production studio that decides to buy new film cameras, or a
pharmaceutical company that invests in research and development for a new drug.
Capital budgeting is the process of evaluating proposed large, longterm investment
projects. The projects may be the purchase of fixed assets, investments in research
and development, advertising, or intellectual property. For instance, Microsoft’s
development of Windows XP™ software was a capital budgeting project. 271 After reading this chapter,
you should be able to:
1. Explain the capital
budgeting process.
2. Calculate the payback
period, net present value,
internal rate of return,
and modified internal rate
of return for a proposed
capital budgeting project.
3. Describe capital rationing
and how firms decide
which projects to select.
4. Measure the risk of a
capital budgeting project.
5. Explain riskadjusted
discount rates. 272 Part III Capital Budgeting and Business Valuation Longterm projects may tie up cash resources, time, and additional assets. They can
also affect a firm’s performance for years to come. As a result, careful capital budgeting
is vital to ensure that the proposed investment will add value to the firm.
Before we discuss the specific decision methods for selecting investment projects,
we briefly examine capital budgeting basics: capital budgeting decision practices, types
of capital budgeting projects, the cash flows associated with such projects, and the stages
of the capital budgeting process. Decision Practices
Financial managers apply two decision practices when selecting capital budgeting
projects: accept/reject and ranking. The accept/reject decision focuses on the question
of whether the proposed project would add value to the firm or earn a rate of return that
is acceptable to the company. The ranking decision lists competing projects in order of
desirability to choose the best one.
The accept/reject decision determines whether a project is acceptable in light of
the firm’s financial objectives. That is, if a project meets the firm’s basic risk and return
requirements, it will be accepted. If not, it will be rejected.
Ranking compares projects to a standard measure and orders the projects based on
how well they meet the measure. If, for instance, the standard is how quickly the project
pays off the initial investment, then the project that pays off the investment most rapidly
would be ranked first. The project that paid off most slowly would be ranked last.
Types of Projects
Firms invest in two categories of projects: independent projects and mutually exclusive
projects. Independent projects do not compete with each other. A firm may accept
none, one, some, or all from among a group of independent projects. Say, for example,
that a firm is able to raise funds for all worthwhile projects it identifies. The firm is
considering two new projects—a new telephone system and a warehouse. The telephone
system and the warehouse would be independent projects. Accepting one does not
preclude accepting the other.
In contrast, mutually exclusive projects compete against each other. The best project
from among a group of acceptable mutually exclusive projects is selected. For example,
if a company needed only one copier, one proposal to buy a XeroxTM copier and a second
proposal to buy a ToshibaTM copier would be two projects that are mutually exclusive.
Capital Budgeting Cash Flows
Although decision makers examine accounting information such as sales and expenses,
capital budgeting decision makers base their decisions on relevant cash flows associated
with a project. The relevant cash flows are incremental cash flows—cash flows that
will occur if an investment is undertaken but that won’t occur if it isn’t.
For example, imagine that you are an animator who helps create children’s films.
Your firm is considering whether to buy a computer to help you design your characters
more quickly. The computer would enable you to create two characters in the time it
now takes you to draw one, thereby increasing your productivity.
The incremental cash flows associated with the animation project are (1) the initial
investment in the computer and (2) the additional cash the firm will receive because you
will double your animation output for the life of the computer. These cash flows will Chapter 10 Capital Budgeting Decision Methods 273 occur if the new computer is purchased and will not occur if it is not purchased. These
are the cash flows, then, that affect the capital budgeting decision.
Estimating these incremental cash flows, the subject of Chapter 11, is a major
component of capital budgeting decisions, as we see in the next section. Stages in the Capital Budgeting Process
The capital budgeting process has four major stages:
1. Finding projects
2. Estimating the incremental cash flows associated with projects
3. Evaluating and selecting projects
4. Implementing and monitoring projects
This chapter focuses on stage 3—how to evaluate and choose investment projects.
We assume that the firm has found projects in which to invest and has estimated the
projects’ cash flows effectively. Capital Budgeting Decision Methods
The four formal capital budgeting decision methods we will examine are payback, net
present value, internal rate of return, and modified internal rate of return. Let’s begin
with the payback method. The Payback Method
One of the simplest capital budgeting decision methods is the payback method. To use
it, analysts find a project’s payback period—the number of time periods it will take
before the cash inflows of a proposed project equal the amount of the initial project
investment (a cash outflow). In the Daimler Chrysler example discussed at the beginning
of this chapter, the analysts would determine how many years it would take to recoup
the initial $600 million investment.
How to Calculate the Payback Period To calculate the payback period, simply add up
a project’s projected positive cash flows, one period at a time, until the sum equals the
amount of the project’s initial investment. That number of time periods it takes for the
positive cash flows to equal the amount of the initial investment is the payback period.
To illustrate, imagine that you work for a firm called the AddVenture Corporation.
AddVenture is considering two investment proposals, Projects X and Y. The initial
investment for both projects is $5,000. The finance department estimates that the projects
will generate the following cash flows:
Cash Flows
For Initial
Investment End of
Year 1 End of
Year 2 End of
Year 3 End of
Year 4 Project X –$5,000 $2,000 $3,000 $ 500 $ Project Y –$5,000 $2,000 $2,000 $1,000 $ 2,000 0 Take Note
Capital budgeting
techniques are usually
used only for projects with
large cash outlays. Small
investment decisions
are usually made by
the “seat of the pants.”
For instance, if your
office supply of pencils
is running low, you
order more. You know
that the cost of buying
pencils is justified without
undergoing capital
budgeting analysis. 274 Part III Capital Budgeting and Business Valuation By analyzing the cash flows, we see that Project X has a payback period of two
years because the project’s initial investment of $5,000 (a cash flow of –$5,000) will be
recouped (by offsetting positive cash flows) at the end of year 2. In comparison, Project
Y has a payback of three years.
Payback Method Decision Rule To apply the payback decision method, firms must
first decide what payback time period is acceptable for longterm projects. In our case, if
the firm sets two years as its required payback period, it will accept Project X but reject
Project Y. If, however, the firm allows a threeyear payback period, then both projects
will be accepted if they are independent.
Problems with the Payback Method The payback method is used in practice because
of its simplicity, but it has a big disadvantage in that it does not consider cash flows that
occur after the payback period. For example, suppose Project Y’s cash flows in year 4
were $10 million instead of $2,000. It wouldn’t make any difference. Project Y would
still be rejected under the payback method if the company’s policy were to demand a
payback of no more than two years. Failing to look beyond the payback period can lead
to poor business decisions, as shown.
Another deficiency of the payback method is that it does not consider the time value
of money. For instance, compare the cash flows of Projects A and B:
Cash Flows
For Initial
Investment End of
Year 1 End of
Year 2 Project A –$5,000 $3,000 $2,000 Project B –$5,000 $2,000 $3,000 Assuming the required payback period is two years, we see that both projects are
equally preferable—they both pay back the initial investment in two years. However,
time value of money analysis indicates that if both projects are equally risky, then Project
A is better than Project B because the firm will get more money sooner. Nevertheless,
timing of cash flows makes no difference in the payback computation.
Because the payback method does not factor in the time value of money, nor the
cash flows after the payback period, its usefulness is limited. Although it does provide
a rough measure of a project’s liquidity and can help to supplement other techniques,
financial managers should not rely on it as a primary decision method. The Net Present Value (NPV) Method
The project selection method that is most consistent with the goal of owner wealth
maximization is the net present value method. The net present value (NPV) of a capital
budgeting project is the dollar amount of the change in the value of the firm as a result
of undertaking the project. The change in firm value may be positive, negative, or zero,
depending on the NPV value.
If a project has an NPV of zero, it means that the firm’s overall value will not change
if the new project is adopted. Why? Because the new project is expected to generate
exactly the firm’s required rate of return—no more and no less. A positive NPV means
that the firm’s value will increase if the project is adopted because the new project’s Chapter 10 Capital Budgeting Decision Methods estimated return exceeds the firm’s required rate of return. Conversely, a negative NPV
means the firm’s value will decrease if the new project is adopted because the new
project’s estimated return is less than what the firm requires.
Calculating NPV To calculate the net present value of a proposed project, we add the
present value of a project’s projected cash flows and then subtract the amount of the
initial investment.1 The result is a dollar figure that represents the change in the firm’s
value if the project is undertaken.
Formula for Net Present Value (NPV), Algebraic Method CF1 CF2 CFn NPV = − Initial Investment
1 + 2 + ... + n (1 + k ) (1 + k) (1 + k) (101a) where: CF = Cash flow at the indicated times
k = Discount rate, or required rate of return for the project
n = Life of the project measured in the number of time periods To use financial tables to solve present value problems, write the NPV formula as
follows:
Formula for Net Present Value (NPV), Table Method
NPV = CF1(PVIFk, 1) + CF2(PVIFk, 2) + . . . + CFn(PVIFk, n) –
Initial Investment (101b) where: PVIF = Present Value Interest Factor
k = Discount rate, or required rate of return for the project
n = Life of the project measured in the number of time periods To use the TI BAII Plus financial calculator to solve for NPV, switch the calculator
to the spreadsheet mode, enter the cash flow values in sequence, then the discount rate,
depicted as I on the TI BAII Plus calculator, then compute the NPV.
For simplicity, we use only the algebraic equation and the financial calculator to
calculate NPV. If you prefer using financial tables, simply replace Equation 101a with
Equation 101b.2
To show how to calculate NPV, let’s solve for the NPVs of our two earlier projects,
Projects X and Y. First, note the following cash flows of Projects X and Y:
Cash Flows
For Initial
Investment End of
Year 1 End of
Year 2 End of
Year 3 End of
Year 4 Project X –$5,000 $2,000 $3,000 $ 500 $0 Project Y –$5,000 $2,000 $2,000 $1,000 $ 2,000 We use many of the time value of money techniques learned in Chapter 8 to calculate NPV. 1 If the future cash flows are in the form of an annuity, Equations 101a and 101b can be modified to take advantage of the
present value of annuity formulas discussed in Chapter 8, Equations 84a and 84b. 2 275 276 Part III Capital Budgeting and Business Valuation Assume the required rate of return for Projects X and Y is 10 percent. Now we have
all the data we need to solve for the NPV of Projects X and Y.
Applying Equation 101a and assuming a discount rate of 10 percent, the NPV of
Project X follows: $2, 000 $3, 000 $500n NPV= − $5,000
1 + 2 + (1 + .10) (1 + .10) (1 + .10) 3 = $2,000
$3,000
$500
+
+
− $5,000
1.1
1.21
1.331 = $1,818.1818 + $2,479.3388 + $375.6574 − $5,000
= − $326.82 We may also find Project X’s NPV with the financial calculator at a 10 percent
discount rate as follows:
TI BAII PLUS Financial Calculator Solutions
Project X NPV
Keystrokes Display CF0 = old contents
CF0 = 0.00
CF0 = –5,000.00 5000
2000 C01 = 2,000.00
F01 = 1.00 3000 C02 = 3,000.00
F02 = 1.00 500 C03 = 500.00
F03 = 1.00
I = 0.00 10 I = 10.00
NPV = –326.82 The preceding calculations show that at a 10 percent discount rate, an initial cash
outlay of $5,000, and cash inflows of $2,000, $3,000, and $500 at the end of years 1,
2, and 3, the NPV for Project X is –326.82.
To find the NPV of Project Y, we apply Equation 101a as follows: $2, 000 $2, 000 $1, 000 $2, 000 NPV = 1 +
2 +
3 +
4 − $5, 000 (1 + .10) (1 + .10) (1 + .10) (1 + .10) Chapter 10 Capital Budgeting Decision Methods = $2, 000
$2, 000
$1, 000
$2, 000
+
+
+
− $5, 000
1.1
1.21
1.331
1.4641 = $1,818.1818 + $1, 652.8926 + $751.3148 + $1, 366.0269 − $5, 000
= $588.42 Using the financial calculator, we solve for Project Y’s NPV at a 10 percent discount
rate as follows:
TI BAII PLUS Financial Calculator Solutions
Project Y NPV
Keystrokes Display CF0 = old contents
CF0 = 0.00
CF0 = –5,000.00 5000
2000 C01 = 2,000.00 2 F01 = 2.00 1000 C02 = 1,000.00
F02 = 1.00 2000 C03 = 2,00.00
F03 = 1.00
I = 0.00 10 I = 10.00
NPV = 588.42 Our calculations show that with a required rate of return of 10 percent, an initial
cash outlay of $5,000, and positive cash flows in years 1 through 4 of $2,000, $2,000,
$1,000, and $2,000, respectively, the NPV for Project Y is $588.42. If we compare Project
X’s NPV of –326.82 to Project Y’s NPV of $588.42, we see that Project Y would add
value to the business and Project X would decrease the firm’s value.
NPV Decision Rules NPV is used in two ways: (1) to determine if independent projects
should be accepted or rejected (the accept/reject decision); and (2) to compare acceptable
mutually exclusive projects (the ranking decision). The rules for these two decisions are
• NPV Accept/Reject Decision—A firm should accept all independent projects having
NPVs greater than or equal to zero. Projects with positive NPVs will add to the value
of the firm if adopted. Projects with NPVs of zero will not alter the firm’s value but
(just) meet the firm’s requirements. Projects with NPVs less than zero should be
rejected because they will reduce the firm’s value if adopted. Applying this decision
rule, Project X would be rejected and Project Y would be accepted. 277 278 Part III Capital Budgeting and Business Valuation • NPV Ranking Decision—The project from the mutually exclusive list with the highest positive NPV should be ranked first, the next highest should be ranked second, and
so on. Under this decision rule, if the two projects in our previous example were
mutually exclusive, Project Y would be ranked first and Project X second. (Not only
is Project X second in rank here, but it is unacceptable because it has a negative NPV.) The NPV Profile The k value is the cost of funds used for the project. It is the discount
rate used in the NPV calculation because the cost of funds for a given project is that
project’s required rate of return. The relationship between the NPV of a project and k is
inverse—the higher the k, the lower the NPV, and the lower the k, the higher the NPV.3
Because a project’s NPV varies inversely with k, financial managers need to know
how much the value of NPV will change in response to a change in k. If k is incorrectly
specified, what appears to be a positive NPV could in fact be a negative NPV and vice
versa—a negative NPV could turn out to be positive. Mutually exclusive project rankings
could also change if an incorrect k value is used in the NPV computations.4
To see how sensitive a project’s NPV value is to changes in k, analysts often create
an NPV profile. The NPV profile is a graph that shows how a project’s NPV changes
when different discount rate values are used in the NPV computation.
Building an NPV profile is straightforward. First, the NPV of the project is calculated
at a number of different discount rates. Then the results are plotted on the graph, with k
values on one axis and the resulting NPV values on the other. If more than one project
is included on the graph, the process is repeated for each project until all are depicted.
To illustrate, we will build an NPV profile of Projects X and Y. We will plot Project X
and then Project Y on the graph.
To begin, we first calculate the NPV of Project X with a number of different discount
rates. The different k values may be chosen arbitrarily. For our purposes, let’s use 0
percent, 5 percent, 10 percent, 15 percent, and 20 percent. The results of Project X’s
NPV calculations follow:
Discount Rate
0% $ 500.00 5% $ 57.77 10%
Take Note
The NPV profile line is
curved, not straight. The
curve is steepest at low
discount rates and gets
more shallow at higher
rates. This shape occurs
because discounting
is the inverse of the
exponential compounding
phenomenon, as
described in Chapter 8. Project X NPV –$ 326.82 15% –$ 663.68 20% –$ 960.65 Now Project X’s NPV values may be plotted on the NPV profile graph. Figure
101 shows the results.
When the data points are connected in Figure 101, we see how the NPV of Project
X varies with the discount rate changes. The graph shows that with a k of about 5.7
percent, the value of the project’s NPV is zero. At that discount rate, then, Project X
would provide the firm’s required rate of return, no more and no less.
Next, we add project Y to the NPV profile graph. We calculate the NPV of Project
Y at a number of different discount rates, 0 percent, 5 percent, 10 percent, 15 percent,
and 20 percent. The results follow:
We are assuming here that the project is a typical one, meaning that it has an initial negative cash flow, the initial investment,
followed by all positive cash flows. It is possible that if a project has negative cash flows in the future the relationship between
NPV and k might not be inverse. 3 The estimation of the cost of funds used for capital budgeting projects was covered in Chapter 9. 4 Chapter 10 Capital Budgeting Decision Methods
$600 279 $500 $400
$200
$57.77 NPV $0 Pr
oj ec –$200 tX –$326.81 –$400 Figure 101
NPV Profile of Project X –$600
–$663.68
–$800
–$960.65 –$1,000
0% 5% 10% 15% 20% Discount Rate Discount Rate The NPV profile shows how
the NPV of project X varies
inversely with the discount rate,
k. Project X’s NPV is highest
($500) when the discount
rate is zero. Its NPV is lowest
(–$960.65) when the discount
rate is 20 percent. Project Y NPV 0% $2,000.00 5% $1,228.06 10% $ 588.42 15% $ 52.44 20% –$ 401.23 Figure 102 shows these NPV values plotted on the NPV profile graph.
Notice in Figure 102 that Project Y’s NPV profile falls off more steeply than Project
X’s. This indicates that Project Y is more sensitive to changes in the discount rate than
Project X. Project X’s NPV becomes negative and, thus, the project is unacceptable when
the discount rate rises above about 6 percent. Project Y’s NPV becomes negative and,
thus, the project is unacceptable when the discount rate rises above about 16 percent.
Problems with the NPV Method Although the NPV method ensures that a firm will
choose projects that add value to a firm, it suffers from two practical problems. First,
it is difficult to explain NPV to people who are not formally trained in finance. Few
nonfinance people understand phrases such as “the present value of future cash flows”
or “the change in a firm’s value given its required rate of return.” As a result, many
financial managers have difficulty using NPV analysis persuasively.
A second problem is that the NPV method results are in dollars, not percentages.
Many owners and managers prefer to work with percentages because percentages can
be easily compared with other alternatives; Project 1 has a 10 percent rate of return
compared with Project 2’s 12 percent rate of return. The next method we discuss, the
internal rate of return, provides results in percentages. Interactive Module
Go to www.textbookmedia.
com and find the free
companion material for this
book. Follow the instructions
there. See how NPV varies
with the discount rate used.
Observe the graph that
shows how NPV and IRR
relate to each. Enter your
own values and see how the
graphs change. 280 Part III Capital Budgeting and Business Valuation $2,000 $2,000
Project X
Project Y $1,500
$1,228.06 NPV $1,000 $500 $52.44 $0 Figure 102 NPV Profile
of Projects X and Y
The NPV file shows how the
NPVs of two capital budgeting
projects, Projects X and Y,
vary inversely with the discount
rate, k. $588.42 $500 $57.77
–$326.81 –$500 –$401.23
–$663.68 –$1,000 –$960.65
0% 5% 10% 15% 20% Discount Rate The Internal Rate of Return (IRR) Method
The Internal Rate of Return (IRR) is the estimated rate of return for a proposed project,
given the project’s incremental cash flows. Just like the NPV method, the IRR method
considers all cash flows for a project and adjusts for the time value of money. However,
the IRR results are expressed as a percentage, not a dollar figure.
When capital budgeting decisions are made using the IRR method, the IRR of the
proposed project is compared with the rate of return management requires for the project.
The required rate of return is often referred to as the hurdle rate. If the project’s IRR is
greater than or equal to the hurdle rate (jumps over the hurdle), the project is accepted.
Calculating Internal Rate of Return: TrialandError Method If the present value
of a project’s incremental cash flows were computed using management’s required
rate of return as the discount rate and the result exactly equaled the cost of the project,
then the NPV of the project would be zero. When NPV equals zero, the required rate of
return, or discount rate used in the NPV calculation, is the projected rate of return, IRR.
To calculate IRR, then, we reorder the terms in Equation 101a to solve for a discount
rate, k, that results in an NPV of zero.
The formula for finding IRR, Equation 102, follows:
Formula for IRR (102) CF1 CF2 CFn NPV = 0 = 1 + 2 + ... + n − Initial Investment (1 + k ) (1 + k ) (1 + k ) Chapter 10 Capital Budgeting Decision Methods 281 To find the IRR of a project using Equation 102, fill in the cash flows, the n values,
and the initial investment figure. Then choose different values for k (all the other values
are known) until the result equals zero. The IRR is the k value that causes the lefthand
side of the equation, the NPV, to equal zero.
To illustrate the process, let’s calculate the IRR for Project X. Recall that the cash
flows associated with Project X were as follows:
Cash Flows
For Initial
Investment
Project X Year 1 Year 2 Year 3 Year 4 –$5,000 $2,000 $3,000 $500 $0 First, we insert Project X’s cash flows and the times they occur into Equation 102. $2, 000 $3, 000 $500 NPV = 0 = 1 + 2 + 3 − $5, 000 (1 + k ) (1 + k ) (1 + k ) Next, we try various discount rates until we find the value of k that results in an
NPV of zero. Let’s begin with a discount rate of 5 percent. $2, 000 $3, 000 $500 0=
1 + 2 + 3 − $5, 000 (1 + .05) (1 + .05) (1 + .05) $2, 000 $3, 000 $500 = − $5, 000
+
+ 1.05 1.1025 1.157625 = $1, 904.76 + $2, 721.09 + $431.92 − $5, 000
= $57.77 This is close, but not quite zero. Let’s try a second time, using a discount rate of
6 percent. $2, 000 $3, 000 $500 0=
1 + 2 + 3 − $5, 000 (1 + .06) (1 + .06) (1 + .06) $500 $2, 000 $3, 000 = − $5, 000
+
+ 1.191016 1.06 1.1236 = $1,886.79 + $2, 669.99 + $419.81 − $5, 000
= − $23.41 This is close enough for our purposes. We conclude the IRR for Project X is slightly
less than 6 percent.
Although calculating IRR by trial and error is timeconsuming, the guesses do not
have to be made entirely in the dark. Remember that the discount rate and NPV are
inversely related. When an IRR guess is too high, the resulting NPV value will be too
low. When an IRR guess is too low, the NPV will be too high. Take Note
We could estimate the
IRR more accurately
through more trial and
error. However, carrying
the IRR computation to
several decimal points
may give a false sense of
accuracy, as in the case
in which the cash flow
estimates are in error. 282 Part III Capital Budgeting and Business Valuation Calculating Internal Rate of Return: Financial Calculator Finding solutions to IRR
problems with a financial calculator is simple. After clearing previous values, enter the
initial cash outlay and other cash flows and compute the IRR value.
The TI BAII PLUS financial calculator keystrokes for finding the IRR for Project
X are shown next.
TI BAII PLUS Financial Calculator Solution IRR
Keystrokes Display CF0 = old contents
CF0 = 0.00
–5,000.00 5000
2000 C01 = 2,000.00
F01 = 1.00 3000 C02 = 3,000.00
F02 = 1.00 500 C03 = 500.00
F03 = 1.00
IRR = 0.00
IRR = 5.71 Take Note
If you are using a
financial calculator other
than the TI BAII PLUS,
the calculator procedure
will be similar but the
keystrokes will differ.
Be sure to check your
calculator’s instruction
manual. We see that with an initial cash outflow of $5,000 and Project X’s estimated cash
inflows in years 1 through 3, the IRR is 5.71 percent.
IRR and the NPV Profile Notice in Figure 102 that the point where Project X’s NPV
profile crosses the zero line is, in fact, the IRR (5.71 percent). This is no accident. When
the required rate of return, or discount rate, equals the expected rate of return, IRR, then
NPV equals zero. Project Y’s NPV profile crosses the zero line just below 16 percent. If
you use your financial calculator, you will find that the Project Y IRR is 15.54 percent.
If an NPV profile graph is available, you can always find a project’s IRR by locating
the point where a project’s NPV profile crosses the zero line.
IRR Decision Rule When evaluating proposed projects with the IRR method, those
that have IRRs equal to or greater than the required rate of return (hurdle rate) set by
management are accepted, and those projects with IRRs that are less than the required
rate of return are rejected. Acceptable, mutually exclusive projects are ranked from
highest to lowest IRR. Chapter 10 Capital Budgeting Decision Methods Benefits of the IRR Method The IRR method for selecting capital budgeting projects
is popular among financial practitioners for three primary reasons:
1. IRR focuses on all cash flows associated with the project.
2. IRR adjusts for the time value of money.
3. IRR describes projects in terms of the rate of return they earn, which makes it
easy to compare them with other investments and the firm’s hurdle rate.
Problems with the IRR Method The IRR method has several problems, however.
First, because the IRR is a percentage number, it does not show how much the value of
the firm will change if the project is selected. If a project is quite small, for instance,
it may have a high IRR but a small effect on the value of the firm. (Consider a project
that requires a $10 investment and returns $100 a year later. The project’s IRR is 900
percent, but the effect on the value of the firm if the project is adopted is negligible.)
If the primary goal for the firm is to maximize its value, then knowing the rate of
return of a project is not the primary concern. What is most important is the amount by
which the firm’s value will change if the project is adopted, which is measured by NPV.
To drive this point home, ask yourself whether you would rather earn a 100 percent
rate of return on $5 (which would be $5) or a 50 percent rate of return on $1,000 (which
would be $500). As you can see, it is not the rate of return that is important but the
dollar value. Why? Dollars, not percentages, comprise a firm’s cash flows. NPV tells
financial analysts how much value in dollars will be created. IRR does not.
A second problem with the IRR method is that, in rare cases, a project may have
more than one IRR, or no IRR. This is shown in detail in Appendix 10A.
The IRR can be a useful tool in evaluating capital budgeting projects. As with any
tool, however, knowing its limitations will enhance decision making. Conflicting Rankings between the NPV and IRR Methods
As long as proposed capital budgeting projects are independent, both the NPV and IRR
methods will produce the same accept/reject indication. That is, a project that has a
positive NPV will also have an IRR that is greater than the discount rate. As a result,
the project will be acceptable based on both the NPV and IRR values. However, when
mutually exclusive projects are considered and ranked, a conflict occasionally arises.
For instance, one project may have a higher NPV than another project, but a lower IRR.
To illustrate how conflicts between NPV and IRR can occur, imagine that the
AddVenture Company owns a piece of land that can be used in different ways. On the
one hand, this land has minerals beneath it that could be mined, so AddVenture could
invest in mining equipment and reap the benefits of retrieving and selling the minerals.
On the other hand, the land has perfect soil conditions for growing grapes that could be
used to make wine, so the company could use it to support a vineyard.
Clearly, these two uses are mutually exclusive. The mine cannot be dug if there is
a vineyard, and the vineyard cannot be planted if the mine is dug. The acceptance of
one of the projects means that the other must be rejected. 283 284 Part III Capital Budgeting and Business Valuation Now let’s suppose that AddVenture’s finance department has estimated the cash
flows associated with each use of the land. The estimates are presented next:
Time Cash Flows for
the Mining Project Cash Flows for
the Vineyard Project t0 ($ 736,369 ) ($ 736,369 ) $ 500,000 $ 0 t2 $ 300,000 $ 0 t3 $ 100,000 $ 0 t4 $ 20,000 $ 50,000 t5 $ 5,000 $ 200,000 t6 $ 5,000 $ 500,000 t7 $ 5,000 $ 500,000 t8 $ 5,000 $ 500,000 t1 Note that although the initial outlays for the two projects are the same, the
incremental cash flows associated with the projects differ in amount and timing. The
Mining Project generates its greatest positive cash flows early in the life of the project,
whereas the Vineyard Project generates its greatest positive cash flows later. The
differences in the projects’ cash flow timing have considerable effects on the NPV and
IRR for each venture.
Assume AddVenture’s required rate of return for longterm projects is 10 percent.
The NPV and IRR results for each project given its cash flows are summarized as follows:
NPV IRR Mining Project $ 65,727.39 16.05% Vineyard Project $194,035.65 14.00% These figures were obtained with our TI BAII PLUS calculator, in the same manner
as shown earlier in the chapter.
The NPV and IRR results show that the Vineyard Project has a higher NPV than the
Mining Project ($194,035.65 versus $65,727.39), but the Mining Project has a higher
IRR than the Vineyard Project (16.05 percent versus 14.00 percent). AddVenture is
faced with a conflict between NPV and IRR results. Because the projects are mutually
exclusive, the firm can only accept one.
In cases of conflict among mutually exclusive projects, the one with the highest
NPV should be chosen because NPV indicates the dollar amount of value that will be
added to the firm if the project is undertaken. In our example, then, AddVenture should
choose the Vineyard Project (the one with the higher NPV) if its primary financial goal
is to maximize firm value. The Modified Internal Rate of Return (MIRR) Method
One problem with the IRR method that was not mentioned earlier is that the only way
you can actually receive the IRR indicated by a project is if you reinvest the intervening
cash flows in the project at the IRR. If the intervening cash flows are reinvested at any rate
lower than the IRR, you will not end up with the IRR indicated at the end of the project. Chapter 10 Capital Budgeting Decision Methods For example, assume you have a project in which you invest $100 now and expect
to receive four payments of $50 at the end of each of the next four years:
t0 t1 –100 +50 t t3 t4 +50 +50 +50 The IRR of this investment, calculated using either the trialanderror or financialcalculator methods, is 34.9 percent. On the surface, this sounds like a fabulous investment.
But, as they say, don’t count your chickens before they hatch. If you can’t reinvest each
$50 payment at the IRR of 34.9 percent, forget it; you won’t end up with an overall
return of 34.9 percent. To see why, consider what would happen if, for example, you
simply put each $50 payment in your pocket. At the end of the fourth year you would
have $200 in your pocket. Now, using Equation 86, calculate the average annual rate
of return necessary to produce $200 in four years with a beginning investment of $100:
Actual return on the investment per Equation 86 1 200 4 k=
−1 100 = .189, or 18.9% (not 34.9%) Despite the fact that the IRR of the investment was 34.9 percent, you only ended
up with an 18.9 percent annual return. The only way you could have obtained an annual
return of 34.9 percent was to have reinvested each of the $50 cash flows at 34.9 percent.
This is called the IRR reinvestment assumption.
To get around the IRR reinvestment assumption, a variation on the IRR procedure
has been developed, called the Modified Internal Rate of Return (MIRR) method.
The MIRR method calls for assuming that the intervening cash flows from a project are
reinvested at a rate of return equal to the cost of capital. To find the MIRR, first calculate
how much you would end up with at the end of a project, assuming the intervening
cash flows were invested at the cost of capital. The result is called the project’s terminal
value. Next, calculate the annual rate of return it would take to produce that end amount
from the beginning investment. That rate is the MIRR.
Here is the MIRR calculation for the project in our example, in which $100 is
invested at time zero, followed by inflows of $50 at the end of each of the next four
years. Let us assume for this example that the cost of capital is 10 percent.
Step 1: Calculate the project’s terminal value: Time Cash FV of Cash Flow at t4 if reinvested
@ 10% Cost of Capital
Flow (per Equation 8a) t1 $50 FV = $50 x (1 + .10)3 = $ 66.55 t2 $50 FV = $50 x (1 + .10)2 = $ 60.50 t3 $50 FV = $50 x (1 + .10)1 = $ 55.00 t4 $50 FV = $50 x (1 + .10)0 = $ 50.00
Project’s Terminal Value: $ 232.05 285 286 Part III Capital Budgeting and Business Valuation Step 2: Calculate the annual rate of return that will produce the terminal value from
the initial investment:
Project’s Terminal Value: $232.05
PV of Initial Investment: $100
per Equation 86:
1 232.05 4 MIRR = −1 100 = .234, or 23.4% This is a much more realistic expectation than the 34.9 percent return from the project
indicated by the IRR method. Remember, any time you consider investing in a project,
you will not actually receive the IRR unless you can reinvest the project’s intervening
cash flows at the IRR. Calculate the MIRR to produce a more realistic indication of
the actual project outcome.
In this section we looked at four capital budgeting decision methods: the payback
method, the net present value method, the internal rate of return method, and the modified
internal rate of return method. We also investigated how to resolve conflicts between
NPV and IRR decision methods. Next, we turn to a discussion of capital rationing. Capital Rationing
Our discussion so far has shown that all independent projects with NPVs greater than
or equal to zero should be accepted. All such projects that are adopted will add value
to the firm. To act consistently with the goal of shareholder wealth maximization, then,
it seems that if a firm locates $200 billion worth of independent positive NPV projects,
it should accept all the projects. In practice, however, many firms place dollar limits on
the total amount of capital budgeting projects they will undertake. They may wish to
limit spending on new projects to keep a ceiling on business size. This practice of setting
dollar limits on capital budgeting projects is known as capital rationing.
If capital rationing is imposed, then financial managers should seek the combination
of projects that maximizes the value of the firm within the capital budget limit. For
example, suppose a firm called BeLimited does not want its capital budget to exceed
$200,000. Seven project proposals, Proposals A to G, are available, as shown in Table 101.
Note that all the projects in Table 101 have positive net present values, so they are
all acceptable. However, BeLimited cannot adopt them all because that would require
the expenditure of more than $200,000, its selfimposed capital budget limit.
Under capital rationing, BeLimited’s managers try different combinations of projects
seeking the combination that gives the highest NPV without exceeding the $200,000
limit. For example, the combination of Projects B, C, E, F, and G costs $200,000 and
yields a total NPV of $23,700. A better combination is that of Projects A, B, C, D, and F,
which costs $200,000 and has a total NPV of $30,600. In fact, this combination has the
highest total NPV, given the $200,000 capital budget limit (try a few other combinations
to see for yourself), so that combination is the one BeLimited should choose.
In this section, we explored capital rationing. In the following section, we will
examine how risk affects capital budgeting decisions. Chapter 10 Capital Budgeting Decision Methods Table 101 BeLimited Project Proposals
Initial Cash
Outlay Net Present
Value A $20,000 $8,000 B $50,000 $7,200 C $40,000 $6,500 D $60,000 $5,100 E $50,000 $4,200 F $30,000 $3,800 G $30,000 $2,000 Project Risk and Capital Budgeting
Suppose you are a financial manager for AddVenture Corporation. You are considering
two capital budgeting projects, Project X (discussed throughout the chapter) and Project
X2. Both projects have identical future cash flow values and timing, and both have the
same NPV, if discounted at the same rate, k, and the same IRR. The two projects appear
equally desirable.
Now suppose you discover that Project X’s incremental cash flows are absolutely
certain, but Project X2’s incremental cash flows, which have the same expected value,
are highly uncertain. These cash flows could be higher or lower than forecast. In other
words, Project X is a sure thing and Project X2 appears to be quite risky. As a riskaverse
person, you would prefer Project X over Project X2.
Financial managers are not indifferent to risk. Indeed, the riskier a project, the less
desirable it is to the firm. To incorporate risk into the NPV and IRR evaluation techniques,
we use the standard deviation and coefficient of variation (CV) measures discussed in
Chapter 7. We begin by examining how to measure the risk of a capital budgeting project.
Then we explain how to include the risk measurement in the NPV and IRR evaluation. Measuring Risk in Capital Budgeting
Financial managers may measure three types of risk in the capital budgeting process.
The first type is projectspecific risk, or the risk of a specific new project. The second
type is firm risk, which is the impact of adding a new project to the existing projects of
the firm. The third is market (or beta) risk, which is the effect of a new project on the
stockholders’ nondiversifiable risk.
In this chapter we focus on measuring firm risk. To measure firm risk, we compare
the coefficient of variation (CV) of the firm’s portfolio before and after a project is
adopted. The beforeandafter difference between the two CVs serves as the project’s
risk measure from a firm perspective. If the CAPM approach (discussed in Chapter 7)
is preferred, a financial manager could estimate the risk of a project by calculating the
project’s beta instead of the change in firm risk.
Computing Changes in the Coefficient of Variation Let’s reconsider our earlier
example, Project X. Recall that the IRR of Project X was 5.71 percent. Now suppose
that after careful analysis, we determine this IRR is actually the most likely value from a 287 288 Part III Capital Budgeting and Business Valuation range of possible values, each with some probability of occurrence. We find the expected
value and standard deviation of Project X’s IRR distribution using Equations 71 and
72, respectively. We find that the expected value of the IRR distribution is 5.71 percent
and the standard deviation is 2.89 percent.
To see how adding Project X to the firm’s existing portfolio changes the portfolio’s
coefficient of variation (CV), we follow a fivestep procedure.
Step 1: Find the CV of the Existing Portfolio. Suppose the expected rate of return and
standard deviation of possible returns of AddVenture’s existing portfolio are 6
percent and 2 percent, respectively. Given this information, calculate the CV of
the firm’s existing portfolio using Equation 73:
Standard Deviation
Mean, or Expected Value CV =
= .02
.06 = .3333, or 33.33% Step 2: Find the Expected Rate of Return of the New Portfolio (the Existing Portfolio
Plus the Proposed Project). Assume that the investment in Project X represents
10 percent of the value of the portfolio. In other words, the new portfolio after
adding Project X will consist of 10 percent Project X and 90 percent of the rest
of the firm’s assets. With these figures, solve for the expected rate of return of
the new portfolio using Equation 74. In the calculations, Asset A represents
Project X and Asset B represents the firm’s existing portfolio.
E(Rp) = (wa × E(Ra)) + (wb × E(Rb))
= (.10 × .0571) + (.90 × .06)
= .00571 + .05400
= .05971, or 5.971%
σp = 2 2 2 2 w a σ a + w bσ b + 2w a w b ra,bσ aσ b
2 2 2 2 = (.10 )(0.0289 ) + (.90 )(0.02 ) + (2)(.10)(.90)(0.0)(.0289)(.02) = (.01)(.000835) + (.81)(.0004) + 0 = .00000835 + .000324 = .000332352 = .0182, or 1.82% Step 3: Find the Standard Deviation of the New Portfolio (the Existing Portfolio Plus
the Proposed Project). Calculate the standard deviation of the new portfolio
using Equation 75. Assume the degree of correlation (r) between project X
(Asset A) and the firm’s existing portfolio (Asset B) is zero. Put another way,
there is no relationship between the returns from Project X and the returns from
the firm’s existing portfolio. Chapter 10 Capital Budgeting Decision Methods Step 4: Find the CV of the New Portfolio (the Existing Portfolio Plus the Proposed
Project). To solve for the CV of the new portfolio with Project X included, we
use Equation 73, as follows:
CV=
= Standard Deviation
Mean, or Expected value
.0182
.05971 = .3048, or 30.48% Step 5: Compare the CV of the Portfolio with and without the Proposed Project.
To evaluate the effect of Project X on the risk of the AddVenture portfolio, we
compare the CV of the old portfolio (without Project X) to the CV of the new
portfolio (with Project X). The coefficients follow:
CV of the Portfolio
without Project X
33.33% CV of the Portfolio
with Project X Change
in CV 30.48% –2.85 The CV of the firm’s portfolio dropped from 33.33 percent to 30.48 percent with
the addition of Project X. This 2.85 percentage point decrease in CV is the measure of
risk in Project X from the firm’s perspective. Adjusting for Risk
Most business owners and financial managers are risk averse, so they want the capital
budgeting process to reflect risk concerns. Next, we discuss how to adjust for risk in the
capital budgeting process—riskadjusted discount rates. The discount rate is the cost
of capital discussed in Chapter 9. A project’s riskadjusted discount rate is the cost of
capital for that specific project.
RiskAdjusted Discount Rates (RADRs) One way to factor risk into the capital
budgeting process is to adjust the rate of return demanded for high and lowrisk projects.
Riskadjusted discount rates (RADRs), then, are discount rates that differ according to
their effect on a firm’s risk. The higher the risk, the higher the RADR, and the lower
the risk, the lower the RADR. A project with a normal risk level would have an RADR
equal to the actual discount rate.
To find the RADR for a capital budgeting project, we first prepare a risk adjustment
table like the one in Table 102. We assume in Table 102 that the coefficient of variation,
CV, is the risk measure to be adjusted for. The CV is based on the probability distribution
of the IRR values.
Table 102 shows how the discount rates for capital budgeting projects might be
adjusted for varying degrees of risk. The discount rate of a project that decreased the
CV of the firm’s portfolio of assets by 2.5 percentage points, for example, would be
classified as a lowrisk project. That project would be evaluated using a discount rate
two percentage points lower than that used on average projects.
This has the effect of making Project X a more desirable project. That is, financial
managers would calculate the project’s NPV using an average discount rate that is two
percentage points lower, which would increase the project’s NPV. If the firm uses the 289 290 Part III Capital Budgeting and Business Valuation Table 102 RiskAdjustment Table
If Adoption of the
Proposed Capital
Budgeting Project
Would Change the CV
of the Firm’s Portfolio IRR
by the Following Amount: Then Assign the
Project to the
Following Risk
Category: And Adjust the
Discount Rate for
the Project as
Follows: More than a 1 percent increase High risk +2% Between a 1 percent decrease
and a 1 percent increase Average risk 0 Low risk –2% More than a 1 percent decrease Take Note
The data in Table 102
are only for illustration.
In practice, the actual
amount of risk adjustment
will depend on the
degree of risk aversion
of the managers and
stockholders. IRR method, financial managers would compare the project’s IRR to a hurdle rate two
percentage points lower than average, which would make it more likely that the firm
would adopt the project.
RADRs are an important part of the capital budgeting process because they
incorporate the risk–return relationship. All other things being equal, the more a project
increases risk, the less likely it is that a firm will accept the project; the more a project
decreases risk, the more likely it is that a firm will accept the project. What’s Next
In this chapter we explored the capital budgeting process and decision methods. We
examined four capital budgeting techniques—the payback method, net present value
method, internal rate of return method, and modified internal rate of return method. We
also discussed capital rationing and how to measure and adjust for risk when making
capital budgeting decisions.
In the next chapter, we’ll investigate how to estimate incremental cash flows in
capital budgeting. Summary
1. Explain the capital budgeting process.
Capital budgeting is the process of evaluating proposed investment projects. Capital
budgeting projects may be independent—the projects do not compete with each other—
or mutually exclusive—accepting one project precludes the acceptance of the other(s)
in that group.
Financial managers apply two decision practices when selecting capital budgeting
projects: accept/reject decisions and ranking decisions. The accept/reject decision is
the determination of which independent projects are acceptable in light of the firm’s
financial objectives. Ranking is a process of comparing projects to a standard measure Chapter 10 Capital Budgeting Decision Methods and ordering the mutually exclusive projects based on how well they meet the measure.
The capital budgeting process is concerned only with incremental cash flows; that is,
those cash flows that will occur if an investment is undertaken but won’t occur if it isn’t.
2. Calculate the payback period, net present value, internal rate of return, and
modified internal rate of return for a proposed capital budgeting project.
The payback period is defined as the number of time periods it will take before the
cash inflows from a proposed capital budgeting project will equal the amount of the
initial investment. To find a project’s payback period, add all the project’s positive cash
flows, one period at a time, until the sum equals the amount of the initial cash outlay
for the project. The number of time periods it takes to recoup the initial cash outlay is
the payback period. A project is acceptable if it pays back the initial investment within
a time frame set by firm management.
The net present value (NPV) of a proposed capital budgeting project is the dollar
amount of the change in the value of the firm that will occur if the project is undertaken.
To calculate NPV, total the present values of all the projected incremental cash flows
associated with a project and subtract the amount of the initial cash outlay. A project
with an NPV greater than or equal to zero is acceptable. An NPV profile—a graph that
shows a project’s NPV at many different discount rates (required rates of return)— shows
how sensitive a project’s NPV is to changes in the discount rate.
The internal rate of return (IRR) is the projected percentage rate of return that a
proposed project will earn, given its incremental cash flows and required initial cash
outlay. To calculate IRR, find the discount rate that makes the project’s NPV equal to
zero. That rate of return is the IRR. A project is acceptable if its IRR is greater than or
equal to the firm’s required rate of return (the hurdle rate).
The IRR method assumes that intervening cash flows in a project are reinvested
at a rate of return equal to the IRR. This can result in overly optimistic expectations
when the IRR is high and the intervening cash flows can’t be reinvested at that rate. To
produce more realistic results, the Modified Internal Rate of Return (MIRR) method
was developed. The MIRR method calls for assuming that the intervening cash flows
from a project are reinvested at a rate of return equal to the cost of capital. To find the
MIRR, first calculate how much you would end up with at the end of a project, assuming
the intervening cash flows were invested at the cost of capital. The result is called the
project’s terminal value. Next, calculate the annual rate of return it would take to produce
that end amount from the beginning investment. That rate is the MIRR.
3. Describe capital rationing and how firms decide which projects to select.
The practice of placing dollar limits on the total size of the capital budget is called capital
rationing. Under capital rationing, the firm will select the combination of projects that
yields the highest NPV without exceeding the capital budget limit.
4. Measure the risk of a capital budgeting project.
To measure the risk of a capital budgeting project, we determine how the project would
affect the risk of the firm’s existing asset portfolio. We compare the coefficient of
variation of possible returns of the firm’s asset portfolio with the proposed project and
without it. The difference between the two coefficients of variation is the measure of
the risk of the capital budgeting project. 291 292 Part III Capital Budgeting and Business Valuation 5. Explain riskadjusted discount rates.
To compensate for the degree of risk in capital budgeting, firms may adjust the discount
rate for each project according to risk. The more risk is increased, the higher the discount
rate. The more risk is decreased, the lower the discount rate. Rates adjusted for the risk
of projects are called riskadjusted discount rates (RADRs). Equations Introduced in This Chapter
Equation 101a. NPV Formula, Algebraic Method: CF1 CF2 CFn NPV= 1 + 2 + ... + n − Initial Investment (1 + k ) (1 + k ) (1 + k ) where: CF = Cash flow at the indicated times
k = Discount rate, or required rate of return for the project
n = Life of the project measured in the number of time periods
Equation 101b. NPV Formula, Table Method:
NPV = CF1(PVIFk, 1) + CF2(PVIFk, 2) + . . . + CFn(PVIFk, n) – Initial Investment where: PVIF = Present Value Interest Factor
k = Discount rate, or required rate of return for the project
n = Life of the project measured in the number of time periods
Equation 102. Formula for IRR Using Equation 101a: CF1 CF2 CFn NPV = 0 = 1 + 2 + ... + n − Initial Investment (1 + k ) (1 + k ) (1 + k ) where: k = the IRR value Fill in cash flows (CFs) and periods (n). Then choose values for k by trial and error
until the NPV equals zero. The value of k that causes the NPV to equal zero is the IRR. Chapter 10 Capital Budgeting Decision Methods 293 SelfTest
ST1. ST6. What is the NPV of the following project? Initial investment Joe, the cutrate bond dealer, has offered to
sell you some zerocoupon bonds for $300.
(Remember, zerocoupon bonds pay off $1,000
at maturity and involve no other cash flows
other than the purchase price.) If the bonds
mature in 10 years and your required rate of
return for cutrate bonds is 20 percent, what is
the NPV of Joe’s deal? $20,000 Net cash flow at the end of year 2 ST7. $50,000 Net cash flow at the end of year 1 Assume you have decided to reinvest your
portfolio in zerocoupon bonds. You like
zerocoupon bonds because they pay off a
known amount, $1,000 at maturity, and involve
no other cash flows other than the purchase
price. Assume your required rate of return is
12 percent. If you buy some 10year zerocoupon bonds for $400 each today, will the
bonds meet your return requirements? (Hint:
Compute the IRR of the investment given the
cash flows involved.) $40,000 Discount rate 11% ST2. What is the IRR of the project in ST1? ST3. You’ve been assigned to evaluate a project for
your firm that requires an initial investment of
$200,000, is expected to last for 10 years, and
is expected to produce aftertax net cash flows
of $44,503 per year. If your firm’s required rate
of return is 14 percent, should the project be
accepted? ST4. What is the IRR of the project in ST3? ST5. What is the MIRR of the project in ST3? Review Questions
1. How do we calculate the payback period for a
proposed capital budgeting project? What are the
main criticisms of the payback method?
2. How does the net present value relate to the value
of the firm?
3. What are the advantages and disadvantages of the
internal rate of return method?
4. Provide three examples of mutually exclusive
projects.
5. What is the decision rule for accepting or rejecting
proposed projects when using net present value?
6. What is the decision rule for accepting or rejecting
proposed projects when using internal rate of
return?
7. What is capital rationing? Should a firm practice
capital rationing? Why? 8. Explain how to resolve a ranking conflict between
the net present value and the internal rate of
return. Why should the conflict be resolved as you
explained?
9. Explain how to measure the firm risk of a capital
budgeting project.
10. Why is the coefficient of variation a better risk
measure to use than the standard deviation when
evaluating the risk of capital budgeting projects?
11. Explain why we measure a project’s risk as the
change in the CV.
12. Explain how using a riskadjusted discount rate
improves capital budgeting decision making
compared with using a single discount rate for all
projects. 294 Part III Capital Budgeting and Business Valuation Build Your Communication Skills
CS1. Arrange to have a person who has no training
in finance visit the class on the next day after
the net present value (NPV) evaluation method
has been covered. Before the visitor arrives,
choose two teams who will brief the visitor
on the results of their evaluation of a capital
budgeting project. The visitor will play the
role of the CEO of the firm. The CEO must
choose one capital budgeting project for the
firm to undertake. The CEO will choose the
project based on the recommendations of
Team A and Team B.
Now have each team describe the following
projects:
Team A Initial Investment
NPV
IRR Team B Solar Energy
Satellite
(a plan to beam
energy down
from orbit) Fresh Water
Extraction Project
(a plan to extract
drinking water
from the sea) $ 40 million
$ 10 million
12% After the projects have been described, have
the CEO ask each team for its recommendation
about which project to adopt. The CEO may also
listen to input from the class at large.
When the debate is complete, have the CEO
select one of the projects (based on the class’s
recommendations). Then critique the class’s
recommendations.
CS2. Divide the class into groups of two members
each. Have each pair prepare a written
description (closed book) of how to measure
the risk of a capital budgeting project. When
the explanations are complete, select a
volunteer from each group to present their
findings to the class. $40 million
$ 8 million
18% Problems
Payback 101. Three separate projects each have an initial cash outlay of $10,000. The cash
flow for Peter’s Project is $4,000 per year for three years. The cash flow
for Paul’s Project is $2,000 in years 1 and 3, and $8,000 in year 2. Mary’s
Project has a cash flow of $10,000 in year 1, followed by $1,000 each year
for years 2 and 3.
a. Use the payback method to calculate how many years it will take for each
project to recoup the initial investment.
b. Which project would you consider most liquid? Net Present Value 102. You have just paid $20 million in the secondary market for the winning
Powerball lottery ticket. The prize is $2 million at the end of each year for
the next 25 years. If your required rate of return is 8 percent, what is the net
present value (NPV) of the deal? Internal Rate
of Return 103. What is the internal rate of return (IRR) of the Powerball deal in question 2? Modified Internal
Rate of Return 104. What is the modified internal rate of return (MIRR) of the Powerball deal in
question 2? 295 Chapter 10 Capital Budgeting Decision Methods 105. RejuveNation needs to estimate how long the payback period would be for
their new facility project. They have received two proposals and need to
decide which one is best. Project Weights will have an initial investment of
$200,000 and generate positive cash flows of $100,000 at the end of year 1,
$75,000 at the end of year 2, $50,000 at the end of year 3, and $100,000 at
the end of year 4. Project Waters will have an initial investment of $300,000
and will generate positive cash flows of $200,000 at the end of year 1 and
$150,000 at the end of years 2, 3, and 4. What is the payback period for
Project Waters? What is the payback period for Project Weights? Which
project should RejuveNation choose? Payback Period 106. Calculate the NPV of each project in problem 105 using RejuveNation’s
cash flows and a 10 percent discount rate. NPV 107. The Bedford Falls Bridge Building Company is considering the purchase
of a new crane. George Bailey, the new manager, has had some past
management experience while he was the chief financial officer of the local
savings and loan. The cost of the crane is $17,291.42, and the expected
incremental cash flows are $5,000 at the end of year 1, $8,000 at the end of
year 2, and $10,000 at the end of year 3.
a. Calculate the net present value if the required rate of return is 12 percent.
b. Calculate the internal rate of return.
c. Should Mr. Bailey purchase this crane? NPV and IRR 108. Lin McAdam and Lola Manners, managers of the Winchester Company,
do not practice capital rationing. They have three independent projects they
are evaluating for inclusion in this year’s capital budget. One is for a new
machine to make rifle stocks. The second is for a new forklift to use in the
warehouse. The third project involves the purchase of automated packaging
equipment. The Winchester Company’s required rate of return is 13 percent.
The initial investment (a negative cash flow) and the expected positive net
cash flows for years 1 through 4 for each project follow: NPV Expected Net Cash Flow
Year Rifle Stock
Machine Forklift Packaging
Equipment $(12,000 ) $(18,200 ) 0 $(9,000 ) 1 2,000 5,000 0 2 5,000 4,000 5,000 3 1,000 6,000 10,000 4 4,000 2,000 12,000 a. Calculate the net present value for each project.
b. Which project(s) should be undertaken? Why? 296 Part III
NPV and IRR 109. Capital Budgeting and Business Valuation The Trask Family Lettuce Farm is located in the fertile Salinas Valley of
California. Adam Trask, the head of the family, makes all the financial
decisions that affect the farm. Because of an extended drought, the family
needs more water per acre than what the existing irrigation system can
supply. The quantity and quality of lettuce produced are expected to increase
when more water is supplied. Cal and Aron, Adam’s sons, have devised
two different solutions to their problem. Cal suggests improvements to the
existing system. Aron is in favor of a completely new system. The negative
cash flow associated with the initial investment and expected positive net
cash flows for years 1 through 7 for each project follow. Adam has no other
alternatives and will choose one of the two projects. The Trask Family
Lettuce Farm has a required rate of return of 12 percent for these projects.
Expected Net Cash Flow
Year
0 Cal’s Project
$(100,000 ) Aron’s Project
$(300,000) 1 63,655 22,611 63,655 4 22,611 63,655 5 22,611 63,655 6 22,611 63,655 7 Payback, NPV,
and IRR 22,611 3 IRR and MIRR 63,655 2 a.
b.
c.
d. 22,611 22,611 63,655 Calculate the net present value for each project.
Calculate the internal rate of return for each project.
Which project should Adam choose? Why?
Is there a conflict between the decisions indicated by the NPVs and
the IRRs? 1010. Buzz Lightyear has been offered an investment in which he expects to
receive payments of $4,000 at the end of each of the next 10 years in return
for an initial investment of $10,000 now.
a. What is the IRR of the proposed investment?
b. What is the MIRR of the proposed investment? Assume a cost capital
of 15%.
c. Why is the MIRR thought of as a more realistic indication of a project’s
potential than the IRR?
1011. Dave Hirsh publishes his own manuscripts and is unsure which of two
new printers he should purchase. He is a novelist living in Parkman,
Illinois. Having slept through most of his Finance 300 course in college,
he is unfamiliar with cash flow analysis. He enlists the help of the finance
professor at the local university, Dr. Gwen French, to assist him. Together
they estimate the following expected initial investment (a negative cash flow)
and net positive cash flows for years 1 through 3 for each machine. Dave
only needs one printer and estimates it will be worthless after three years of
heavy use. Dave’s required rate of return for this project is 10 percent. 297 Chapter 10 Capital Budgeting Decision Methods
Expected Net Cash Flow
Year
0 Cal’s Project Aron’s Project $(2,000 ) $(2,500) 1 1,500 1,100 1,300 3 a.
b.
c.
d.
e. 900 2 1,300 800 Calculate the payback period for each printer.
Calculate the net present value for each printer.
Calculate the internal rate of return for each printer.
Which printer do you think Dr. French will recommend? Why?
Suppose Dave’s required rate of return were 16 percent. Does the decision
about which printer to purchase change? 1012. Matrix.com has designed a virtualreality program that is indistinguishable
from real life to those experiencing it. The program will cost $20 million to
develop (paid up front), but the payoff is substantial: $1 million at the end of
year 1, $2 million at the end of year 2, $5 million at the end of year 3, and
$6 million at the end of each year thereafter, through year 10. Matrix.com’s
weighted average cost of capital is 15 percent. Given these conditions, what
are the NPV, IRR, and MIRR of the proposed program? NPV, IRR, and
MIRR 1013. Project A has an initial investment of $11,000, and it generates positive cash
flows of $4,000 each year for the next six years. Project B has an initial
investment of $17,000, and it generates positive cash flows of $4,500 each
year for the next six years. Assume the discount rate, or required rate of
return, is 13 percent.
a. Calculate the net present value of each project. If Project A and Project B
are mutually exclusive, which one would you select?
b. Assume Projects A and B are independent. Based on NPV, which one(s)
would you select?
c. Calculate the internal rate of return for each project.
d. Using IRR for your decision, which project would you select if Project
A and Project B were mutually exclusive? Would your answer change if
these two projects were independent instead of mutually exclusive?
e. Project C was added to the potential capital budget list at the last minute
as a mutually exclusive alternative to Project B. It has an initial cash
outlay of $17,000, and it generates its only positive cash flow of $37,500
in the sixth year. Years 1 through 5 have $0 cash flow. Calculate Project
C’s NPV. Which alternative, Project B or C, would you choose?
f. Now calculate the IRR of Project C and compare it with Project B’s IRR.
Based solely on the IRRs, which project would you select?
g. Because Projects B and C are mutually exclusive, would you recommend
that Project B or Project C be added to the capital budget for this year?
Explain your choice. Independent
and Mutually
Exclusive Projects 298 Part III
NPV, and IRR NPV Profile
Analysis Capital Budgeting and Business Valuation 1014. Joanne Crale is an independent petroleum geologist. She is taking advantage
of every opportunity to lease the mineral rights of land she thinks lies over
oil reserves. She thinks there are many reservoirs with oil reserves left
behind by major corporations when they plugged and abandoned fields
in the 1960s. Ms. Crale knows that with today’s technology, production
can be sustained at much lower reservoir pressures than was possible in
the 1960s. She would like to lease the mineral rights from George Hansen
and Ed McNeil, the landowners. Her net cost for this current oil venture
is $5 million. This includes costs for the initial leasing and three planned
development wells. The expected positive net cash flows for the project
are $1.85 million each year for four years. On depletion, she anticipates
a negative cash flow of $250,000 in year 5 because of reclamation and
disposal costs. Because of the risks involved in petroleum exploration,
Ms. Crale’s required rate of return is 16 percent.
a. Calculate the net present value for Ms. Crale’s project.
b. Calculate the internal rate of return for this project.
c. Would you recommend the project?
1015. Silkwood Power Company is considering two projects with the following
predicted cash flows:
Hydroelectric
Upgrade
Initial Investment CFs Geothermal
Upgrade ($100,000 ) ($100,000 ) End of year 1 CFs 20,000 80,000 End of year 2 CFs 30,000 40,000 End of year 3 CFs 40,000 30,000 End of year 4 CFs 90,000 10,000 a. Construct a chart showing the NPV profiles of both projects.
b. Which project(s) would be accepted if the company’s cost of capital were
6 percent?
c. Which project(s) would be accepted if the company’s cost of capital were
15 percent?
d. At what cost of capital would Silkwood value each project equally?
e. At what cost of capital would the hydroelectric project become
unacceptable?
f. At what cost of capital would the geothermal project become
unacceptable?
Challenge
Problem 1016. The product development managers at World Series Innovations are about to
recommend their final capital budget for next year. They have a selfimposed
budget limit of $100,000. Five independent projects are being considered.
Vernon Simpson, the chief scientist and CEO, has minimal financial analysis
experience and relies on his managers to recommend the projects that will
increase the value of the firm by the greatest amount. Given the following
summary of the five projects, which ones should the managers recommend? 299 Chapter 10 Capital Budgeting Decision Methods
Initial Cash
Outlay Net Present
Value Chalk Line Machine ($10,000 ) $4,000 Gel Padded Glove ($25,000 ) $3,600 Insect Repellent ($35,000 ) $3,250 Titanium Bat ($40,000 ) $2,500 Recycled Base Covers ($20,000 ) $2,100 Projects 1017. A project you are considering has an initial investment of $5,669.62. It has
positive net cash flows of $2,200 each year for three years. Your required
rate of return is 12 percent.
a. Calculate the net present value of this project.
b. Would you undertake this project if you had the cash available?
c. What would the discount rate have to be before this project’s NPV would
be positive? Construct an NPV profile with discount rates of 0 percent, 5
percent, and 10 percent to answer this question.
d. What other method might you use to determine at what discount rate the
net present value would become greater than 0? Various Capital
Budgeting Issues 1018. The internal rate of return (IRR) of a capital investment, Project B, is
expected to have the following values with the associated probabilities for an
economic life of five years. Calculate the expected value, standard deviation,
and coefficient of variation of the IRR distribution. Expected IRR IRR Probability of Occurrence 0% 0.05 1% 0.10 3% 0.20 6% 0.30 9% 0.20 11%
12% 0.10
0.05 1019. Four capital investment alternatives have the following expected IRRs and
standard deviations.
Standard
Deviation Project Expected IRR A 14% 2% B 16% 6% C
D 11%
14% 5%
4% If the firm’s existing portfolio of assets has an expected IRR of 13 percent with a
standard deviation of 3 percent, identify the lowest and the highestrisk projects
in the preceding list. You may assume the correlation coefficient of returns from
each project relative to the existing portfolio is .50 and the investment in each
project would constitute 20 percent of the firm’s total portfolio. Capital
Budgeting Risk 300
Capital Budgeting
Risk and RADRs Part III Capital Budgeting and Business Valuation 1020. Assume that a company has an existing Portfolio A, with an expected
IRR of 10 percent and standard deviation of 2 percent. The company is
considering adding a Project B, with expected IRR of 11 percent and
standard deviation of 3 percent, to its portfolio. Also assume that the amount
invested in Portfolio A is $700,000 and the amount to be invested in Project
B is $200,000.
a. If the degree of correlation between returns from Portfolio A and Project
B is .90 (r = +.9), calculate:
1. the coefficient of variation of Portfolio A
2. the expected IRR of the new combined portfolio
3. the standard deviation of the new combined portfolio
4. the coefficient of variation of the new combined portfolio
b. What is the change in the coefficient of variation as a result of adopting
Project B?
c. Assume the firm classifies projects as high risk if they raise the coefficient
of variation of the portfolio 2 percentage points or more, as low risk if
they lower the coefficient of variation of the portfolio 2 percentage points
or more, and as average risk if they change the coefficient of variation
of the portfolio by less than 2 percent in either direction. In what risk
classification would Project B be?
d. The required rate of return for averagerisk projects for the company in
this problem is 13 percent. The company policy is to adjust the required
rate of return downward by 3 percent for lowrisk projects and to raise it
3 percent for highrisk projects (averagerisk projects are evaluated at the
average required rate of return). The expected cash flows from Project B
are as follows:
CF of Initial Investment ($200,000) end of year 1 55,000 end of year 2 55,000 end of year 3 55,000 end of year 4 100,000 Calculate the NPV of Project B when its cash flows are discounted at:
1. the average required rate of return
2. the highrisk discount rate
3. the lowrisk discount rate
Risk in Capital
Budgeting 1021. Dorothy Gale is thinking of purchasing a Kansas amusement park, Glinda’s
Gulch. She already owns at least one park in each of the surrounding states
and wants to expand her operations into Kansas. She estimates the IRR from
the project may be one of a number of values, each with a certain probability
of occurrence. Her estimated IRR values and the probability of their
occurrence follow. 301 Chapter 10 Capital Budgeting Decision Methods
Glinda’s Gulch
Possible IRR Value Probability 2% .125 5%
.20 9% .35 13% .20 16% .125 Now Dorothy wants to estimate the risk of the project. Assume she has asked
you to do the following for her:
a. Calculate the mean, or expected, IRR of the project.
b. Calculate the standard deviation of the possible IRRs.
c. Assume the expected IRR of Dorothy’s existing portfolio of parks is 8
percent, with a standard deviation of 2 percent. Calculate the coefficient
of variation of Dorothy’s existing portfolio.
d. Calculate the expected IRR of Dorothy’s portfolio if the Glinda’s Gulch
project is adopted. For this calculation assume that Dorothy’s new
portfolio will consist of 80 percent existing parks and 20 percent new
Glinda’s Gulch.
e. Again assuming that Dorothy’s new portfolio will consist of 80 percent
existing parks and 20 percent new Glinda’s Gulch, calculate the standard
deviation of Dorothy’s portfolio if the Glinda’s Gulch project is added.
Because Glinda’s Gulch is identical to Dorothy’s other parks, she
estimates the returns from Glinda’s Gulch and her existing portfolio are
perfectly positively correlated (r = +1.0).
f. Calculate the coefficient of variation of Dorothy’s new portfolio with
Glinda’s Gulch.
g. Compare the coefficient of variation of Dorothy’s existing portfolio
with and without the Glinda’s Gulch project. How does the addition of
Glinda’s Gulch affect the riskiness of the portfolio?
1022. Four capital investment alternatives have the following expected IRRs and
standard deviations.
Standard
Deviation Project Expected IRR A 18% 9% B 15% 5% C 11% 3% D 8% 1% The firm’s existing portfolio of projects has an expected IRR of 12 percent and
a standard deviation of 4 percent. Risk in Capital
Budgeting 302 Part III Capital Budgeting and Business Valuation a. Calculate the coefficient of variation of the existing portfolio.
b. Calculate the coefficient of variation of the portfolio if the firm invests
10 percent of its assets in Project A. You may assume that there is no
correlation between the returns from Project A and the returns from the
existing portfolio (r = 0). (Note: In order to calculate the coefficient
of variation, you must first calculate the expected IRR and standard
deviation of the portfolio with Project A included.)
c. Using the same procedure as in b and given the same assumptions,
calculate the coefficient of variation of the portfolio if it includes, in turn,
Projects B, C, and D.
d. Which project has the highest risk? Which has the lowest risk?
Firm Risk 1023. A firm has an existing portfolio of projects with a mean, or expected, IRR of
15 percent. The standard deviation of this estimate is 5 percent. The existing
portfolio is worth $820,000. The addition of a new project, PROJ1, is being
considered. PROJ1’s expected IRR is 18 percent with a standard deviation
of 9 percent. The initial investment for PROJ1 is expected to be $194,000.
The returns from PROJ1 and the existing portfolio are perfectly positively
correlated (r = +1.0).
a. Calculate the coefficient of variation for the existing portfolio.
b. If PROJ1 is added to the existing portfolio, calculate the weight
(proportion) of the existing portfolio in the combined portfolio.
c. Calculate the weight (proportion) of PROJ1 in the combined portfolio.
d. Calculate the standard deviation of the combined portfolio. Is the standard
deviation of the combined portfolio higher or lower than the standard
deviation of the existing portfolio?
e. Calculate the coefficient of variation of the combined portfolio.
f. If PROJ1 is added to the existing portfolio, will the firm’s risk increase or
decrease? Risk in Capital
Budgeting 1024. VOTD Pharmaceuticals is considering the mass production of a new sleeping
pill. Neely O’Hara, VOTD’s financial analyst, has gathered all the available
information from the finance, production, advertising, and marketing
departments and has estimated that the yearly net incremental cash flows
will be $298,500. She estimates the initial investment for this project will be
$2 million. The pills are expected to be marketable for 10 years, and Neely
does not expect that any of the investment costs will be recouped at the
end of the 10year period. VOTD’s required rate of return for averagerisk
projects is 8 percent. Two percent is added to the required rate of return for
highrisk projects.
a. Calculate the net present value of the project if VOTD management
considers it to be average risk.
b. A lot of uncertainty is associated with the potential competition of
new drugs that VOTD’s competitors might introduce. Because of this
uncertainty, VOTD management has changed the classification of this
project to high risk. Calculate the net present value at the riskadjusted
discount rate. Chapter 10 Capital Budgeting Decision Methods 303 c. Assuming the risk classification does not change again, should this
project be adopted?
d. What will the yearly net incremental cash flow have to be to have a
positive NPV when using the highrisk discount rate?
1025. Aluminum Building Products Company (ABPC) is considering investing in
either of the two mutually exclusive projects described as follows:
Project 1. Buying a new set of rollforming tools for its existing rollforming line to introduce a new cladding product. After its
introduction, the product will need to be promoted. This means
that cash inflows from additional production will start some time
after and will gradually pick up in subsequent periods.
Project 2. Modifying its existing rollforming line to increase productivity
of its available range of cladding products. Cash inflows from
additional production will start immediately and will reduce over
time as the products move through their life cycle.
Sarah Brown, project manager of ABPC, has requested that you do the necessary
financial analysis and give your opinion about which project ABPC should
select. The projects have the following net cash flow estimates:
Year Project 1 Project 2 0 ($200,000) ($200,000) 1 0 90,000 2 0 70,000 3 20,000 50,000 4 30,000 30,000 5 40,000 10,000 6 60,000 10,000 7 90,000 10,000 8 100,000 10,000 Both these projects have the same economic life of eight years and average
risk characteristics. ABPC’s weighted average cost of capital, or hurdle rate,
is 7.2 percent.
a. Which project would you recommend Ms. Brown accept to maximize
value of the firm? (Hint: Calculate and compare NPVs of both projects.)
b. What are the IRRs of each project? Which project should be chosen using
IRR as the selection criterion?
c. Draw the NPV profiles of both projects. What is the approximate discount
rate at which both projects would have the same NPV? What is that NPV?
d. Does the selection remain unaffected for i) WACC > 5.4 percent; ii)
WACC > 8.81 percent; and iii) WACC > 14.39 percent?
e. Further market survey research indicates that both projects have lowerthanaverage risk and, hence, the riskadjusted discount rate should be 5
percent. What happens to the ranking of the projects using NPV and IRR
as the selection criteria? Explain the conflict in ranking, if any.
f. Answer questions a, d, and e, assuming the projects are independent of
each other. Comprehensive
Problem 304 Part III Capital Budgeting and Business Valuation Answers to SelfTest
ST1. The NPV of the project can be found using Equation 101a as follows: $20, 000 $40, 000 NPV = 1 + 2 − $50, 000 (1 + .11) (1 + .11) $20, 000 $40, 000 =
+ − $50, 000 1.11 1.2321 = $18, 018 + $32, 465 − $50, 000
= $483 ST2. To find the IRR, we solve for the k value that results in an NPV of zero in
the NPV formula (Equation 101a) by trial and error:
Try using k = 11%: $20, 000 $40, 000 0=
1 + 2 − $50, 000 (1 + .11) (1 + .11) $20, 000 $40, 000 =
+ − $50, 000 1.11 1.2321 = $18, 018 + $32, 465 − $50, 000
= $483 The result does not equal zero. Try again using a slightly higher discount rate:
Second, try using k = 11.65%: $20, 000 $40, 000 0=
1 + 2 − $50, 000 (1 + .1165) (1 + .1165) $20, 000 $40, 000 =
+ − $50, 000 1.1165 1.24657225 = $17, 913 + $32, 088 − $50, 000
= $1 Close enough. The IRR of the project is almost exactly 11.65 percent.
ST3. The NPV of the project must be calculated to see if it is greater or less than
zero. Because the project in this problem is an annuity, its NPV can be found
most easily using a modified version of Equation 101b as follows:
NPV = PMT(PVIFAk, n) – Initial Investment We include this problem (and some others that follow) to show you that
Equations 101a and 101b can be modified to take advantage of the present
value of an annuity formulas covered in Chapter 8, Equations 84a and 84b. Chapter 10 Capital Budgeting Decision Methods The problem could also be solved using the basic forms of Equations 101a
and 101b, but those would take longer.
In this problem, the initial investment is $200,000, the annuity payment is
$44,503, the discount rate, k, is 14 percent, and the number of periods, n, is 10.
NPV = $44,503(PVIFA14%, 10) – $200,000 = $44,503(5.2161) – $200,000
= $232,132 – $200,000
= $32,132
Because the NPV is positive, the project should be accepted.
ST4. The IRR is found by setting the NPV formula to zero and solving for the
IRR rate, k. In this case, there are no multiple cash flows, so the equation can
be solved algebraically. For convenience we use Equation 101b modified
for an annuity:
0 = $44, 503( PVIFA k%, 10 ) − $200, 000
$200, 000 = $44, 503( PVIFA k%, 10 )
$200, 000
= PVIFA k%, 10
$44, 503
4.49408 = PVIFA k%, 10 Finding the PVIFA in Table IV in the Appendix, we see that 4.4941 (there
are only four decimal places in the table) occurs in the year 10 row in the 18
percent column. Therefore, the IRR of the project is 18 percent.
ST5. To calculate the MIRR, first calculate the project’s terminal value. Next,
calculate the annual rate of return it would take to produce that end amount
from the beginning investment. That rate is the MIRR.
For the project in ST3:
Step 1: Calculate the project’s terminal value. Note—The terminal value
could be calculated by adding up the future values of each individual cash flow, as was done on the MIRR explanation on pages
276–278. In this case, however, that would be a lengthy procedure
because there are 10 cash flows. Recognizing that the cash flows
are an annuity, we can make use of the future value of an annuity
formula, Equation 83a, to calculate the terminal value:
È (1 + .14)10  1 ˘
FV = $44,503 Í
˙
.14
Í
˙
Î
˚
FV = $44.503 [19.3372951 ]
FV = Project s Terminal Value = $860,568 305 306 Part III Capital Budgeting and Business Valuation Step 2: Calculate the annual rate of return that will produce the terminal
value from the initial investment.
Project’s Terminal Value: $860,568
PV of Initial Investment: $200,000
per Equation 86:
1 860, 568 10 MIRR = −1 200, 000 MIRR = .1571, or 15.71% ST6. There are only two cash flows in this problem, a $400 investment at time 0
and the $1,000 payoff at the end of year 10. To find the IRR, we set the NPV
formula to zero, fill in the two cash flows and periods, and solve for the IRR
rate, k, that makes the righthand side of the equation equal zero. Equation
101b is more convenient to use in this case:
0 = $1, 000( PVIFk%, 10 ) − $400
$400 = $1, 000( PVIFk%, 10 )
$400
= PVIFk%, 10
$1, 000
.4000 = PVIFk%, 10 If we find the PVIF in Table II inside the book cover, we see that .4000 occurs
in the year 10 row between the 9 percent and 10 percent columns. Therefore,
the IRR of the project is between 9 percent and 10 percent. (The exact value of
the IRR is 9.596 percent). Because your required rate of return in this problem
was 12 percent, the bonds would not meet your requirements.
ST7. Solve for the NPV using Equation 101a or b. Equation 101a would be
the more convenient version for this problem. The cash flows are a $300
investment at time 0 and a $1,000 future value (FV) in 10 years. The
discount rate is 20 percent. $1, 000 NPV = 10 − $300 (1 + .20) $1, 000 = − $300 6.191736 = $161.51 − $300
= − $138.49 Chapter 10 Capital Budgeting Decision Methods 307 Appendix 10A
Wrinkles in Capital Budgeting
In this appendix we discuss three situations that change the capital budgeting decision
process. First, we examine nonsimple projects, projects that have a negative initial cash
flow, in addition to one or more negative future cash flows. Next, we explore projects
that have multiple IRRs. Finally, we discuss how to compare mutually exclusive projects
with unequal project lives. Nonsimple Projects
Most capital budgeting projects start with a negative cash flow—the initial investment—
at t0, followed by positive future cash flows. Such projects are called simple projects.
Nonsimple projects are projects that have one or more negative future cash flows after
the initial investment.
To illustrate a nonsimple project, consider Project N, the expected cash flows for
a nuclear power plant project. The initial investment of $500 million is a negative cash
flow at t0, followed by positive cash flows of $25 million per year for 30 years as electric
power is generated and sold. At the end of the useful life of the project, the storage of
nuclear fuel and the shutdown safety procedures require cash outlays of $100 million at
the end of year 31. The timeline depicted in Figure 10A1 shows the cash flow pattern:
With a 20 percent discount rate, an initial investment of $500 million, a 30year
annuity of $25 million, and a shutdown cash outlay of $100 million in year 31, the NPV
of Project N follows (in millions), per Equation 101a:
1 − NPVn = $25 1
(1 + .20)30
.20 $100 −
31 − $500 (1 + .20) $100 = ($25 × 4.9789364) − − $500 284.8515766 = $124.47341 − $.35106 − $500
= − $375.8776 We find that at a discount rate of 20 percent, Project N has a negative net present
value of minus $375.88, so the firm considering Project N should reject it. –$500 25 25 25 25 –100 t0 t1 t2 t3 t30 t31 Figure 10A1 Cash
Flow Timeline for Project N
(in millions) 308 Take Note
The $25 million annual
payment for 30 years
constitutes an annuity,
so we were able to
adapt Equation 101a
by using the present
value of annuity formula,
Equation 84a. Part III Capital Budgeting and Business Valuation Multiple IRRs
Some projects may have more than one internal rate of return. That is, a project may
have several different discount rates that result in a net present value of zero.
Here is an example. Suppose Project Q requires an initial cash outlay of $160,000
and is expected to generate a positive cash flow of $1 million in year 1. In year 2, the
project will require an additional cash outlay in the amount of $1 million. The cash
flows for Project Q are shown on the timeline in Figure 10A2.
We find the IRR of Project Q by using the trialanderror procedure, Equation 102.
When k = 25 percent, the NPV is zero. $1, 000, 000 $1, 000, 000 0=
1 −
2 − $160, 000 (1 + .25) (1 + .25) $1, 000, 000 $1, 000, 000 =
− − $160, 000 1.5625 1.25
= $800, 000 − $640, 000 − $160, 000
= $0 Because 25 percent causes the NPV of Project Q to be zero, the IRR of the project
must be 25 percent. But wait! If we had tried k = 400%, the IRR calculation would
look like this: $1, 000, 000 $1, 000, 000 0=
1 − 2 − $160, 000 (1 + 4.00) (1 + 4.00) $1, 000, 000 $1, 000, 000 =
− − $160, 000 25.00 5.00
= $200, 000 − $40, 000 − $160, 000
= $0 Because 400 percent results in an NPV of zero, 400 percent must also be the IRR
of the Project Q. Figure 10A3 shows the net present value profile for Project Q. By
examining this graph, we see how 25 percent and 400 percent both make the net present
value equal to zero.
As the graph shows, Project Q’s net present value profile crosses the horizontal axis
(has a zero value) in two different places, at discount rates of 25 percent and 400 percent.
Project Q had two IRRs because the project’s cash flows changed from negative to
positive (from t0 to t1) and then from positive to negative at (from t1 to t2). It turns out
that a nonsimple project may have (but does not have to have) as many IRRs as there
are sign changes. In this case, two sign changes resulted in two internal rates of return.
t0 Figure 10A2 Cash
Flow Timeline for Project Q t1 t2 –$160,000 $1,000,000 –$1,000,000 Cash flows: Chapter 10 Capital Budgeting Decision Methods 309 Net Present Value (in thousands of dollars) 100 50 0 –50 –100 Figure 10A3
Net Present Value Profile
for Project Q –150 –200 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25 4.5 4.75 5
Discount Rate Whenever we have two or more IRRs for a project, the IRR method is not a useful
decisionmaking tool. Remember the IRR accept/reject decision rule: Firms should
accept projects with IRRs higher than the discount rate and reject projects with IRRs
lower than the discount rate. With more than one discount rate, decision makers will
not know which IRR to use for the accept/reject decision. In projects that have multiple
IRRs, then, switch to the NPV method. Mutually Exclusive Projects with Unequal
Project Lives
When mutually exclusive projects have different expected useful lives, selecting among
the projects requires more than comparing the projects’ NPVs. To illustrate, suppose
you are a business manager considering a new business telephone system. One is the
Cheap Talk System, which requires an initial cash outlay of $10,000 and is expected
to last three years. The other is the Rolles Voice System, which requires an initial cash
outlay of $50,000 and is expected to last 12 years. The Cheap Talk System is expected
to generate positive cash flows of $5,800 per year for each of its three years of life. The
Rolles Voice System is expected to generate positive cash flows of $8,000 per year for
each of its 12 years of life. The cash flows associated with each project are summarized
in Figure10A4. The net present value profile
for project Q crosses the zero
line twice, showing that two
IRRs exist. 310 Part III Capital Budgeting and Business Valuation Cheap Talk
t0
t1 Figure 10A4 Cash
Flows for the Cheap
Talk and Rolles Voice
Communications System
(in thousands)
This figure shows the cash flows
for two projects with unequal
lives, Project Cheap Talk and
Project Rolles Voice. –$10 t2 t3 +$5.8 +$5.8 +$5.8 Rolles
t0 t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 –$50 +$8 +$8 +$8 +$8 +$8 +$8 +$8 +$8 +$8 +$8 +$8 +$8 To decide which project to choose, we first compute and compare their NPVs.
Assume the firm’s required rate of return is 10 percent. We solve for the NPVs as follows:
1. NPV of Cheap Talk. NPVCT 1 − = $5,800 1
(1 + .10)3
.10 − $10, 000 = ($5,800 × 2.48685) − $10, 000
= $14.424 − $10, 000
= $4, 424 2. NPV of Rolles Voice. NPVRolles 1 − = $8, 000 1
(1 + .10)12
.10 − $50, 000 = ($8, 000 × 6.81369) − $50, 000
= $54, 510 − $50, 000
= $4, 510 We find that Project Cheap Talk has an NPV of $4,424, compared with Project
Rolles’ NPV of $4,510. We might conclude based on this information that the Rolles
Voice System should be selected over the Cheap Talk system because it has the higher
NPV. However, before making that decision, we must assess how Project Cheap Talk’s
NPV would change if its useful life were 12 years, not three years. Chapter 10 Capital Budgeting Decision Methods 311 Comparing Projects with Unequal Lives
Two possible methods that allow financial managers to compare projects with unequal
lives are the replacement chain approach and the equivalent annual annuity (EAA)
approach. The Replacement Chain Approach
The replacement chain approach assumes each of the mutually exclusive projects can be
replicated, until a common time period has passed in which the projects can be compared.
The NPVs for the series of replicated projects are then compared to the project with the
longer life. An example illustrates this process. Project Cheap Talk could be repeated four
times in the same time span as the 12year Rolles Voice project. If a business replicated
project Cheap Talk four times, the cash flows would look as depicted in Figure 10A5.
The NPV of this series of cash flows, assuming the discount rate is 10 percent, is
(in thousands) $12,121. Each cash flow, be it positive or negative, is discounted back
the appropriate number of years to get the NPV of the four consecutive investments in
the Cheap Talk system.
The NPV of $12,121 for Cheap Talk system is the sum of the NPVs of the four
repeated Cheap Talk projects, such that the project series would have a life of 12 years,
the same life as the Rolles Voice System Project. We are now comparing apples to apples.
Cheap Talk’s replacement chain NPV is $12,121, whereas the NPV of Project Rolles
Voice is $4,510 over the same 12year period. If a firm invested in project Cheap Talk four
successive times, it would create more value than investing in one project Rolles Voice.
The Equivalent Annual Annuity (EAA)
The equivalent annual annuity (EAA) approach converts the NPVs of the mutually
exclusive projects into their equivalent annuity values. The equivalent annual annuity is
the amount of the annuity payment that would give the same present value as the actual
future cash flows for that project. The EAA approach assumes that you could repeat
the mutually exclusive projects indefinitely as each project came to the end of its life.
The equivalent annuity value (EAA) is calculated by dividing the NPV of a project
by the present value interest factor for an annuity (PVIFA) that applies to the project’s
life span.
Formula for an Equivalent Annual Annuity (EAA)
EAA = NPV
PVIFA k, n (10A1) where: k = Discount rate used to calculate the NPV
n = Life span of the project Cheap Talk
t0
t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 Figure 10A5 Cash
Flows for the Cheap Talk
+$5.8 +$5.8 +$5.8 +$5.8 +$5.8 +$5.8 +$5.8 +$5.8 +$5.8 +$5.8 +$5.8 +$5.8 System Repeated 4 times
(in thousands)
–$10
–$10
–$10
–$10 312 Part III Capital Budgeting and Business Valuation The NPVs of Cheap Talk ($4,424) and Rolles Voice ($4,510) were calculated earlier,
assuming a required rate of return of 10 percent. With the project’s NPV and the discount
rate, we calculate each project’s EAA, per Equation 10A1, as follows:
1. EAA of Project Cheap Talk:
$4, 424
2.48685 EAA CT = = $1, 778.96 2. EAA of Project Rolles
EAA Rolles = $4, 510
6.81369 = $661.90 Take Note
Note that the unequal
lives problem is only an
issue when the projects
under consideration
are mutually exclusive.
If the projects were
independent, we would
adopt both. The EAA approach decision rule calls for choosing whichever mutually exclusive
project has the highest EAA. Our calculations show that Project Cheap Talk has an EAA
of $1,778.96 and Project Rolles Voice System has an EAA of $661.90. Because Project
Cheap Talk’s EAA is higher than Project Rolles’s, Project Cheap Talk should be chosen.
Both the replacement chain and the EAA approach assume that mutually exclusive
projects can be replicated. If the projects can be replicated, then either the replacement
chain or the equivalent annual annuity methods should be used because they lead to
the same correct decision. Note in our case that the EAA method results in the same
project selection (Project Cheap Talk) as the replacement chain method. If the projects
cannot be replicated, then the normal NPVs should be used as the basis for the capital
budgeting decision. Equations Introduced in This Appendix
Equation 10A1. The Formula for an Equivalent Annual Annuity:
EAA = NPV
PVIFA k, n where: k = Discount rate used to calculate the NPV
n = Life span of the project ...
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This note was uploaded on 11/17/2011 for the course BUS 330 taught by Professor Nugent during the Fall '08 term at SUNY Stony Brook.
 Fall '08
 Nugent
 Internal Rate Of Return (IRR)

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