11 the basics of capital budgeting

CHAPTER

11

The Basics of Capital
Budgeting

SOURCE: © George Hall/CORBIS

04


BOEING RECOVERS
FROM ITS
FINANCIAL
TA I L S P I N

$

BOEING COMPANY

B

oeing Co. had been struggling in recent years. In

that the company’s accounting and financial practices

1997, the 82-year-old company suffered its first

were outmoded, making it hard to determine whether

loss in 50 years, and even though the overall

products were profitable. So, she immediately set out to

market was rising, its stock fell from $60 per share to

devise better procedures for measuring and controlling

just under $30.

costs.

Boeing’s troubles were not declining sales. In fact,

After reviewing the company’s $13 billion capital

sales doubled, from $23 billion in 1996 to more than

budget, Hopkins concluded that $2 billion of projects

$56 billion in 1998, as Boeing aggressively outbid its

had little chance of ever being profitable, and another

archrival Airbus for new business. However, Boeing’s

$1.6 billion were likely to only break even or generate

costs increased even faster than sales, and as a result,

modest profits at best. She developed a “value

the company lost $178 million in 1997 and barely broke

scorecard” and used it to help kill value-reducing

even in 1998.

projects and increase investments in profitable areas.

Boeing has always had a strong engineering culture,

While everyone recognizes that it is difficult to

but historically it paid little attention to financial

improve overnight, analysts believe that Boeing is

performance. Managements could get away with such

moving in the right direction. The company is once

behavior in the past, but stockholders won’t tolerate it

again profitable, and both its cash flow and operating

today, as top-level firings at GM, IBM, and others attest.

margins have improved. Most importantly, the stock

So, to turn things around, Boeing hired Deborah

price has rebounded sharply, and the stock is once again

Hopkins in December 1998 as the company’s chief

trading above $60 per share.

financial officer (CFO). She had previously starred as a

Hopkins received considerable praise for her work at

vice-president at Unisys and as CFO for General Motors

Boeing. For this reason, the markets were surprised and

Europe.

concerned when Hopkins announced in April 2000 that

Hopkins, who was hired just after her 44th birthday,

she was leaving Boeing to take a similar position at

was Boeing’s youngest senior executive and the

Lucent Technologies Inc. Indeed, Boeing’s stock fell

company’s highest-ranking woman. Known for her

more than 5 percent the day of the announcement.

dynamic energy and strong communications skills,

Despite this setback, Boeing has continued to vastly

Hopkins quickly made her presence felt. She discovered

outperform the market in the months following Hopkins’

505


departure. At the same time, Hopkins has stepped into a

based on a careful capital budgeting analysis. Hopkins

tough situation at Lucent, where the once high-flying

will certainly try to impose the same type of discipline

company has recently seen sharp declines in its stock

at Lucent. With this in mind as you read this chapter,

price.

think about how companies such as Boeing and Lucent

Boeing’s new CFO, Michael Sears, appears to be

continuing the policies that Hopkins put in place. In the

use capital budgeting analysis to make better
investment decisions. ■

future, each of Boeing’s investment decisions will be

In the last chapter, we discussed the cost of capital. Now we turn to investment
decisions involving fixed assets, or capital budgeting. Here the term capital refers
to long-term assets used in production, while a budget is a plan that details projected inflows and outflows during some future period. Thus, the capital budget is
Capital Budgeting

an outline of planned investments in fixed assets, and capital budgeting is the

The process of planning
expenditures on assets whose cash
flows are expected to extend
beyond one year.

whole process of analyzing projects and deciding which ones to include in the capital budget.
Our treatment of capital budgeting is divided into two chapters. This chapter
gives an overview and explains the various techniques used in capital budgeting
analysis. Chapter 12 goes on to explain how cash flows are estimated for projects,
and it also considers techniques for estimating project risk. ■

I M P O R TA N C E O F C A P I TA L B U D G E T I N G
A number of factors combine to make capital budgeting perhaps the most important function financial managers and their staffs must perform. First, since
the results of capital budgeting decisions continue for many years, the firm
loses some of its flexibility. For example, the purchase of an asset with an economic life of 10 years “locks in” the firm for a 10-year period. Further, because
asset expansion is based on expected future sales, a decision to buy an asset that

is expected to last 10 years requires a 10-year sales forecast. Finally, a firm’s capital budgeting decisions define its strategic direction, because moves into new
products, services, or markets must be preceded by capital expenditures.
An erroneous forecast of asset requirements can have serious consequences.
If the firm invests too much, it will incur unnecessarily high depreciation and

506

CHAPTER 11

■

T H E B A S I C S O F C A P I TA L B U D G E T I N G


other expenses. On the other hand, if it does not invest enough, two problems
may arise. First, its equipment and computer software may not be sufficiently
modern to enable it to produce competitively. Second, if it has inadequate capacity, it may lose market share to rival firms, and regaining lost customers requires heavy selling expenses, price reductions, or product improvements, all of
which are costly.
Timing is also important — capital assets must be available when they are
needed. Edward Ford, executive vice-president of Western Design, a decorative
tile company, gave the authors an illustration of the importance of capital budgeting. His firm tried to operate near capacity most of the time. During a four-year
period, Western experienced intermittent spurts in the demand for its products,
which forced it to turn away orders. After these sharp increases in demand, Western would add capacity by renting an additional building, then purchasing and installing the appropriate equipment. It would take six to eight months to get the
additional capacity ready, but by then demand had dried up — other firms with
available capacity had already taken an increased share of the market. Once Western began to properly forecast demand and plan its capacity requirements a year
or so in advance, it was able to maintain and even increase its market share.
Effective capital budgeting can improve both the timing and the quality of
asset acquisitions. If a firm forecasts its needs for capital assets in advance, it
can purchase and install the assets before they are needed. Unfortunately,
many firms do not order capital goods until existing assets are approaching fullcapacity usage. If sales increase because of an increase in general market demand, all firms in the industry will tend to order capital goods at about the

same time. This results in backlogs, long waiting times for machinery, a deterioration in the quality of the capital equipment, and an increase in costs. The
firm that foresees its needs and purchases capital assets during slack periods can
avoid these problems. Note, though, that if a firm forecasts an increase in demand and then expands to meet the anticipated demand, but sales do not increase, it will be saddled with excess capacity and high costs, which can lead to
losses or even bankruptcy. Thus, an accurate sales forecast is critical.
Capital budgeting typically involves substantial expenditures, and before a
firm can spend a large amount of money, it must have the funds lined up —
large amounts of money are not available automatically. Therefore, a firm contemplating a major capital expenditure program should plan its financing far
enough in advance to be sure funds are available.

SELF-TEST QUESTIONS
Why are capital budgeting decisions so important?
Why is the sales forecast a key element in a capital budgeting decision?

G E N E R AT I N G I D E A S F O R C A P I TA L P R O J E C T S
The same general concepts that are used in security valuation are also involved
in capital budgeting. However, whereas a set of stocks and bonds exists in the
securities market, and investors select from this set, capital budgeting projects are

G E N E R AT I N G I D E A S F O R C A P I TA L P R O J E C T S

507


created by the firm. For example, a sales representative may report that customers are asking for a particular product that the company does not now produce. The sales manager then discusses the idea with the marketing research
group to determine the size of the market for the proposed product. If it appears that a significant market does exist, cost accountants and engineers will be
asked to estimate production costs. If they conclude that the product can be
produced and sold at a sufficient profit, the project will be undertaken.
A firm’s growth, and even its ability to remain competitive and to survive, depends on a constant flow of ideas for new products, for ways to make existing
products better, and for ways to operate at a lower cost. Accordingly, a wellmanaged firm will go to great lengths to develop good capital budgeting proposals. For example, the executive vice-president of one very successful corporation indicated that his company takes the following steps to generate projects:

Strategic Business Plan
A long-run plan that outlines in
broad terms the firm’s basic
strategy for the next five to ten
years.

Our R&D department is constantly searching for new products and for ways to improve existing products. In addition, our executive committee, which consists of senior executives in marketing, production, and finance, identifies the products and
markets in which our company should compete, and the committee sets long-run targets for each division. These targets, which are spelled out in the corporation’s
strategic business plan, provide a general guide to the operating executives who
must meet them. The operating executives then seek new products, set expansion
plans for existing products, and look for ways to reduce production and distribution
costs. Since bonuses and promotions are based on each unit’s ability to meet or exceed its targets, these economic incentives encourage our operating executives to
seek out profitable investment opportunities.
While our senior executives are judged and rewarded on the basis of how well
their units perform, people further down the line are given bonuses and stock options
for suggestions that lead to profitable investments. Additionally, a percentage of our
corporate profit is set aside for distribution to nonexecutive employees, and we have
an Employees’ Stock Ownership Plan (ESOP) to provide further incentives. Our objective is to encourage employees at all levels to keep an eye out for good ideas, including those that lead to capital investments.

If a firm has capable and imaginative executives and employees, and if its incentive system is working properly, many ideas for capital investment will be advanced. Some ideas will be good ones, but others will not. Therefore, procedures
must be established for screening projects, the primary topic of this chapter.

SELF-TEST QUESTION
What are some ways firms get ideas for capital projects?

P R O J E C T C L A S S I F I C AT I O N S
Analyzing capital expenditure proposals is not a costless operation — benefits
can be gained, but analysis does have a cost. For certain types of projects, a relatively detailed analysis may be warranted; for others, simpler procedures

508

CHAPTER 11

■

T H E B A S I C S O F C A P I TA L B U D G E T I N G


should be used. Accordingly, firms generally categorize projects and then analyze those in each category somewhat differently:
1. Replacement: maintenance of business. One category consists of expenditures to replace worn-out or damaged equipment used in the production of profitable products. Replacement projects are necessary if the
firm is to continue in business. The only issues here are (a) should this
operation be continued and (b) should we continue to use the same production processes? The answers are usually yes, so maintenance decisions
are normally made without going through an elaborate decision process.
2. Replacement: cost reduction. This category includes expenditures to replace serviceable but obsolete equipment. The purpose here is to lower the
costs of labor, materials, and other inputs such as electricity. These decisions are discretionary, and a fairly detailed analysis is generally required.
3. Expansion of existing products or markets. Expenditures to increase
output of existing products, or to expand retail outlets or distribution facilities in markets now being served, are included here. These decisions are
more complex because they require an explicit forecast of growth in demand. Mistakes are more likely, so a more detailed analysis is required. Also,
the go/no-go decision is generally made at a higher level within the firm.
4. Expansion into new products or markets. These are investments to
produce a new product or to expand into a geographic area not currently
being served. These projects involve strategic decisions that could change
the fundamental nature of the business, and they normally require the expenditure of large sums of money with delayed paybacks. Invariably, a detailed analysis is required, and the final decision is generally made at the
very top — by the board of directors as a part of the firm’s strategic plan.
5. Safety and/or environmental projects. Expenditures necessary to comply with government orders, labor agreements, or insurance policy terms
fall into this category. These expenditures are called mandatory investments, and they often involve nonrevenue-producing projects. How they are
handled depends on their size, with small ones being treated much like
the Category 1 projects described above.
6. Other. This catch-all includes office buildings, parking lots, executive
aircraft, and so on. How they are handled varies among companies.

In general, relatively simple calculations, and only a few supporting documents, are required for replacement decisions, especially maintenance-type investments in profitable plants. A more detailed analysis is required for costreduction replacements, for expansion of existing product lines, and especially
for investments in new products or areas. Also, within each category projects
are broken down by their dollar costs: Larger investments require increasingly
detailed analysis and approval at a higher level within the firm. Thus, whereas
a plant manager may be authorized to approve maintenance expenditures up to
$10,000 on the basis of a relatively unsophisticated analysis, the full board of directors may have to approve decisions that involve either amounts over $1 million or expansions into new products or markets. Statistical data are generally
lacking for new-product decisions, so here judgments, as opposed to detailed
cost data, are especially important.

P R O J E C T C L A S S I F I C AT I O N S

509


Note that the term “assets” encompasses more than buildings and equipment.
Computer software that a firm develops to help it buy supplies and materials
more efficiently, or to communicate with customers, is also an asset. So is a customer base like that of AOL developed by sending out millions of free CDs to
potential customers. And so is the design of a new computer chip, airplane, or
movie. All of these are “intangible” as opposed to “tangible” assets, but decisions
to invest in them are analyzed in the same way as decisions related to tangible assets. Keep this in mind as you go through the remainder of the chapter.

SELF-TEST QUESTION
Identify the major project classification categories, and explain how they are
used.

S I M I L A R I T I E S B E T W E E N C A P I TA L B U D G E T I N G
A N D S E C U R I T Y VA L U AT I O N
Once a potential capital budgeting project has been identified, its evaluation involves the same steps that are used in security analysis:
1. First, the cost of the project must be determined. This is similar to finding the price that must be paid for a stock or bond.
2. Next, management estimates the expected cash flows from the project,

including the salvage value of the asset at the end of its expected life. This
is similar to estimating the future dividend or interest payment stream on
a stock or bond, along with the stock’s expected sales price or the bond’s
maturity value.
3. Third, the riskiness of the projected cash flows must be estimated. This
requires information about the probability distribution (riskiness) of the
cash flows.
4. Given the project’s riskiness, management determines the cost of capital
at which the cash flows should be discounted.
5. Next, the expected cash inflows are put on a present value basis to obtain
an estimate of the asset’s value. This is equivalent to finding the present
value of a stock’s expected future dividends.
6. Finally, the present value of the expected cash inflows is compared with
the required outlay. If the PV of the cash flows exceeds the cost, the project should be accepted. Otherwise, it should be rejected. (Alternatively, if
the expected rate of return on the project exceeds its cost of capital, the
project is accepted.)
If an individual investor identifies and invests in a stock or bond whose market
price is less than its true value, the investor’s wealth will increase. Similarly, if a
firm identifies (or creates) an investment opportunity with a present value
greater than its cost, the value of the firm will increase. Thus, there is a direct

510

CHAPTER 11

■

T H E B A S I C S O F C A P I TA L B U D G E T I N G

link between capital budgeting and stock values: The more effective the firm’s
capital budgeting procedures, the higher its stock price.

SELF-TEST QUESTION
List the six steps in the capital budgeting process, and compare them with
the steps in security valuation.

C A P I TA L B U D G E T I N G D E C I S I O N R U L E S
Five key methods are used to rank projects and to decide whether or not they
should be accepted for inclusion in the capital budget: (1) payback, (2) discounted payback, (3) net present value (NPV), (4) internal rate of return (IRR),
and (5) modified internal rate of return (MIRR). We will explain how each ranking criterion is calculated, and then we will evaluate how well each performs in
terms of identifying those projects that will maximize the firm’s stock price.
We use the cash flow data shown in Figure 11-1 for Projects S and L to illustrate each method. Also, we assume that the projects are equally risky. Note
that the cash flows, CFt, are expected values, and that they have been adjusted
to reflect taxes, depreciation, and salvage values. Further, since many projects
require an investment in both fixed assets and working capital, the investment
outlays shown as CF0 include any necessary changes in net operating working

FIGURE

Net Cash Flows for Projects S and L

11-1

E X P E C T E D A F T E R - TA X
NET CASH FLOWS, CFt
YEAR (t)

PROJECT S

PROJECT L

a

($1,000)

($1,000)

0

1

500

100

2

400

300

3

300

400

4

100

600

0

1

2

3

4

Ϫ1,000

500

400

300

100

0

1

2

3

4

Ϫ1,000

100

300

400

600

Project S:

Project L:

a

CF0 represents the net investment outlay, or initial cost.

C A P I TA L B U D G E T I N G D E C I S I O N R U L E S

511


FIGURE

Payback Period for Projects S and L

11-2

0
Project S:
Net cash flow
Cumulative NCF

Ϫ1,000
Ϫ1,000

1
500
Ϫ500

0
Project L:
Net cash flow
Cumulative NCF

Ϫ1,000
Ϫ1,000

1
100
Ϫ900

2
400
Ϫ100

2
300
Ϫ600

3

4

300
200

100
300

3

4

400
Ϫ200

600
400

capital.1 Finally, we assume that all cash flows occur at the end of the designated year. Incidentally, the S stands for short and the L for long: Project S is a
short-term project in the sense that its cash inflows come in sooner than L’s.

P AY B A C K P E R I O D
Payback Period
The length of time required for an

investment’s net revenues to cover
its cost.

The payback period, defined as the expected number of years required to recover the original investment, was the first formal method used to evaluate capital budgeting projects. The payback calculation is diagrammed in Figure 11-2,
and it is explained below for Project S.
1. Enter CF0 ϭ Ϫ1000 in your calculator. (You do not need to use the cash
flow register; just have your display show Ϫ1,000.)
2. Now add CF1 ϭ 500 to find the cumulative cash flow at the end of Year
1. The result is Ϫ500.
3. Now add CF2 ϭ 400 to find the cumulative cash flow at the end of Year
2. This is Ϫ100.
4. Now add CF3 ϭ 300 to find the cumulative cash flow at the end of Year
3. This is ϩ200.
5. We see that by the end of Year 3 the cumulative inflows have more than
recovered the initial outflow. Thus, the payback occurred during the
third year. If the $300 of inflows come in evenly during Year 3, then the
exact payback period can be found as follows:
PaybackS ϭ Year before full recovery ϩ
ϭ2ϩ

Unrecovered cost at start of year
Cash flow during year

$100
ϭ 2.33 years.
$300

Applying the same procedure to Project L, we find PaybackL ϭ 3.33 years.
1

The most difficult part of the capital budgeting process is estimating the relevant cash flows. For
simplicity, the net cash flows are treated as a given in this chapter, which allows us to focus on the
capital budgeting decision rules. However, in Chapter 12 we will discuss cash flow estimation in detail. Also, note that working capital is defined as the firm’s current assets, and that net operating working capital is current assets minus non-interest-bearing liabilities.

512

CHAPTER 11

■

T H E B A S I C S O F C A P I TA L B U D G E T I N G


FIGURE

11-3

Projects S and L: Discounted Payback Period

0
Project S:
Net cash flow
Discounted NCF (at 10%)
Cumulative discounted NCF

Project L:
Net cash flow
Discounted NCF (at 10%)
Cumulative discounted NCF

Mutually Exclusive Projects
A set of projects where only one
can be accepted.

Independent Projects
Projects whose cash flows are not
affected by the acceptance or
nonacceptance of other projects.

Discounted Payback Period
The length of time required for an
investment’s cash flows,
discounted at the investment’s cost
of capital, to cover its cost.

Ϫ1,000
Ϫ1,000
Ϫ1,000

1
500
455
Ϫ545

0

1

Ϫ1,000
Ϫ1,000

Ϫ1,000

100
91
Ϫ909

2
400
331
Ϫ214
2
300
248
Ϫ661

3

4

300
225
11

100
68
79

3

4

400
301
Ϫ360

600
410
50

The shorter the payback period, the better. Therefore, if the firm required a
payback of three years or less, Project S would be accepted but Project L would
be rejected. If the projects were mutually exclusive, S would be ranked over L
because S has the shorter payback. Mutually exclusive means that if one project
is taken on, the other must be rejected. For example, the installation of a conveyor-belt system in a warehouse and the purchase of a fleet of forklifts for the
same warehouse would be mutually exclusive projects — accepting one implies
rejection of the other. Independent projects are projects whose cash flows are
independent of one another.
Some firms use a variant of the regular payback, the discounted payback
period, which is similar to the regular payback period except that the expected
cash flows are discounted by the project’s cost of capital. Thus, the discounted
payback period is defined as the number of years required to recover the investment from discounted net cash flows. Figure 11-3 contains the discounted
net cash flows for Projects S and L, assuming both projects have a cost of capital of 10 percent. To construct Figure 11-3, each cash inflow is divided by
(1 ϩ k)t ϭ (1.10)t, where t is the year in which the cash flow occurs and k is the
project’s cost of capital. After three years, Project S will have generated $1,011
in discounted cash inflows. Since the cost is $1,000, the discounted payback is
just under three years, or, to be precise, 2 ϩ ($214/$225) ϭ 2.95 years. Project
L’s discounted payback is 3.88 years:
Discounted paybackS ϭ 2.0 ϩ $214/$225 ϭ 2.95 years.
Discounted paybackL ϭ 3.0 ϩ $360/$410 ϭ 3.88 years.
For Projects S and L, the rankings are the same regardless of which payback

method is used; that is, Project S is preferred to Project L, and Project S would
still be selected if the firm were to require a discounted payback of three years
or less. Often, however, the regular and the discounted paybacks produce conflicting rankings.
Note that the payback is a type of “breakeven” calculation in the sense that if
cash flows come in at the expected rate until the payback year, then the project
will break even. However, the regular payback does not consider the cost of capital — no cost for the debt or equity used to undertake the project is reflected in
the cash flows or the calculation. The discounted payback does consider capital
costs — it shows the breakeven year after covering debt and equity costs.
C A P I TA L B U D G E T I N G D E C I S I O N R U L E S

513


An important drawback of both the payback and discounted payback methods is that they ignore cash flows that are paid or received after the payback period. For example, consider two projects, X and Y, each of which requires an
up-front cash outflow of $3,000, so CF0 ϭ Ϫ$3,000. Assume that both projects
have a cost of capital of 10 percent. Project X is expected to produce cash inflows of $1,000 each of the next four years, while Project Y will produce no
cash flows the first four years but then generate a cash inflow of $1,000,000 five
years from now. Common sense suggests that Project Y creates more value for
the firm’s shareholders, yet its payback and discounted payback make it look
worse than Project X. Consequently, both payback methods have serious deficiencies. Therefore, we will not dwell on the finer points of payback analysis.2
Although the payback method has some serious faults as a ranking criterion,
it does provide information on how long funds will be tied up in a project.
Thus, the shorter the payback period, other things held constant, the greater
the project’s liquidity. Also, since cash flows expected in the distant future are
generally riskier than near-term cash flows, the payback is often used as an indicator of a project’s riskiness.

N E T P R E S E N T V A L U E (NPV)
Net Present Value (NPV)
Method
A method of ranking investment

proposals using the NPV, which is
equal to the present value of
future net cash flows, discounted
at the marginal cost of capital.

As the flaws in the payback were recognized, people began to search for ways
to improve the effectiveness of project evaluations. One such method is the net
present value (NPV) method, which relies on discounted cash flow (DCF)
techniques. To implement this approach, we proceed as follows:
1. Find the present value of each cash flow, including both inflows and outflows, discounted at the project’s cost of capital.
2. Sum these discounted cash flows; this sum is defined as the project’s NPV.
3. If the NPV is positive, the project should be accepted, while if the NPV
is negative, it should be rejected. If two projects with positive NPVs are
mutually exclusive, the one with the higher NPV should be chosen.

Discounted Cash Flow (DCF)
Techniques
Methods for ranking investment
proposals that employ time value
of money concepts.

The equation for the NPV is as follows:
NPV ϭ CF0 ϩ

CF1
(1 ϩ k)

1

ϩ

CF2
(1 ϩ k)

2

ϩ и и и ϩ

CFn
(1 ϩ k)n

n

CFt
t.
tϭ1 (1 ϩ k)

ϭa

(11-1)

Here CFt is the expected net cash flow at Period t, k is the project’s cost of capital, and n is its life. Cash outflows (expenditures such as the cost of buying equipment or building factories) are treated as negative cash flows. In evaluating Projects S and L, only CF0 is negative, but for many large projects such as the Alaska
2

Another capital budgeting technique that was once used widely is the accounting rate of return (ARR),
which examines a project’s contribution to the firm’s net income. Although some companies still
calculate an ARR, it really has no redeeming features, so we will not discuss it in this text. See Eugene F. Brigham and Phillip R. Daves, Intermediate Financial Management, 7th ed. (Fort Worth, TX:
Harcourt College Publishers, 2002), Chapter 11. Yet another technique that we omit here is the profitability index, or benefit/cost ratio. Brigham and Daves also discuss this method.

514

CHAPTER 11

■

T H E B A S I C S O F C A P I TA L B U D G E T I N G


Pipeline, an electric generating plant, or IBM’s laptop computer project, outflows occur for several years before operations begin and cash flows turn positive.
At a 10 percent cost of capital, Project S’s NPV is $78.82:
0

k ϭ 10%

Ϫ1,000.00

Cash Flows

1

2

3

4

500

400

300

100

454.55
330.58
225.39
68.30
Net Present Value

78.82

By a similar process, we find NPVL ϭ $49.18. On this basis, both projects
should be accepted if they are independent, but S should be chosen if they are
mutually exclusive.
It is not hard to calculate the NPV as was done in the time line by using
Equation 11-1 and a regular calculator, along with the interest rate tables.
However, it is more efficient to use a financial calculator. Different calculators
are set up somewhat differently, but they all have a section of memory called
the “cash flow register” that is used for uneven cash flows such as those in Projects S and L (as opposed to equal annuity cash flows). A solution process for
Equation 11-1 is literally programmed into financial calculators, and all you
have to do is enter the cash flows (being sure to observe the signs), along with
the value of k ϭ I. At that point, you have (in your calculator) this equation:
NPVS ϭ Ϫ1,000 ϩ

500
400
300
100
ϩ

ϩ
ϩ
.
(1.10)1
(1.10)2
(1.10)3
(1.10)4

Notice that the equation has one unknown, NPV. Now, all you need to do is to
ask the calculator to solve the equation for you, which you do by pressing the
NPV button (and, on some calculators, the “compute” button). The answer,
78.82, will appear on the screen.3

3

The Technology Supplement that accompanies this text explains this and other commonly used calculator applications. For those who do not have the Supplement, the steps for two popular calculators, the HP-10B and the HP-17B, are shown below. If you have another type of financial calculator, see its manual or the Supplement.
HP-10B:
1. Clear the memory.
2. Enter CF0 as follows: 1000 ϩ/Ϫ CFj .
3. Enter CF1 as follows: 500 CFj .
4. Repeat the process to enter the other cash flows. Note that CF 0, CF 1, and so forth, flash
on the screen as you press the CFj button. If you hold the button down, CF 0 and so
forth, will remain on the screen until you release it.
5. Once the CFs have been entered, enter k ϭ I ϭ 10%: 10 1/YR .
6. Now that all of the inputs have been entered, you can press
NPV ϭ $78.82.

NPV to get the answer,
(footnote continues)

C A P I TA L B U D G E T I N G D E C I S I O N R U L E S

515


Most projects last for more than four years, and, as you will see in Chapter
12, most projects require many calculations to develop the estimated cash
flows. Therefore, financial analysts generally use spreadsheets when dealing
with capital budgeting projects. For Project S, this spreadsheet could be used
(disregard for now the IRR on Row 6; we discuss it in the next section):
A

1

Project S

2

kϭ

3

Time

4

B

C

D

E

F

G

10%
1

2

3

4

5

Cash flow ϭ

Ϫ1000

500

400

300

100

5

NPV ϭ

$78.82

6

IRR ϭ

14.5%

7

In Excel, the formula in Cell B5 is: ‫؍‬B4؉NPV(B2,C4:F4), and it results in a
value of $78.82.4 For a simple problem such as this, setting up a spreadsheet
may not seem worth the trouble. However, in real-world problems there will be
a number of rows above our cash flow line, starting with expected sales, then
deducting various costs and taxes, and ending up with the cash flows shown on
Row 4. Moreover, once a spreadsheet has been set up, it is easy to change input
(Footnote 3 continued)

7. If a cash flow is repeated for several years, you can avoid having to enter the CFs for each
year. For example, if the $500 cash flow for Year 1 had also been the CF for Years 2 through
10, making 10 of these $500 cash flows, then after entering 500 CFj the first time, you
could enter 10

Nj . This would automatically enter 10 CFs of 500.

HP-17B:
1. Go to the cash flow (CFLO) menu, clear if FLOW(0) ϭ ? does not appear on the screen.
2. Enter CF0 as follows: 1000 ϩ/Ϫ INPUT .
3. Enter CF1 as follows: 500 INPUT .
4. Now, the calculator will ask you if the 500 is for Period 1 only or if it is also used for several following periods. Since it is only used for Period 1, press INPUT to answer “1.” Alternatively, you could press EXIT and then #T? to turn off the prompt for the remainder of the problem. For some problems, you will want to use the repeat feature.
5. Enter the remaining CFs, being sure to turn off the prompt or else to specify “1” for each
entry.
6. Once the CFs have all been entered, press EXIT and then CALC .
7. Now enter k ϭ I ϭ 10% as follows: 10 1% .
8. Now press NPV to get the answer, NPV ϭ $78.82.
4

You could click the function wizard, fx, then Financial, then NPV, and then OK. Then insert B2 as
the rate and C4:F4 as “Value 1,” which is the cash flow range. Then click OK, and edit the equation
by adding B4. Note that you cannot enter the Ϫ$1,000 cost as part of the NPV range. It occurs at
t ϭ 0, but the Excel NPV function assumes that all cash flows occur at the end of the periods.

516

CHAPTER 11

■

T H E B A S I C S O F C A P I TA L B U D G E T I N G


values to see what would happen if inputs are changed. For example, we could
see what would happen if lower sales caused all cash flows to decline by $15, or
if the cost of capital rose to 10.5 percent. It is easy to make such changes and
then see the effects on NPV. See the model for this chapter, 11MODEL.xls.

R AT I O N A L E

NPV M E T H O D

FOR THE

The rationale for the NPV method is straightforward. An NPV of zero signifies that the project’s cash flows are exactly sufficient to repay the invested capital and to provide the required rate of return on that capital. If a project has a
positive NPV, then it is generating more cash than is needed to service its debt
and to provide the required return to shareholders, and this excess cash accrues
solely to the firm’s stockholders. Therefore, if a firm takes on a project with a
positive NPV, the position of the stockholders is improved. In our example,
shareholders’ wealth would increase by $78.82 if the firm takes on Project S,
but by only $49.18 if it takes on Project L. Viewed in this manner, it is easy
to see why S is preferred to L, and it is also easy to see the logic of the NPV
approach.5
There is also a direct relationship between NPV and EVA (economic value
added) — NPV is equal to the present value of the project’s future EVAs. Therefore, accepting positive NPV projects should result in a positive EVA and a positive MVA (market value added, or the excess of the firm’s market value over its
book value). So, a reward system that compensates managers for producing positive EVA will lead to the use of NPV for making capital budgeting decisions.

I N T E R N A L R AT E

Internal Rate of Return (IRR)
Method
A method of ranking investment
proposals using the rate of return
on an investment, calculated by
finding the discount rate that
equates the present value of future
cash inflows to the project’s cost.

OF

R E T U R N (IRR)

In Chapter 8 we presented procedures for finding the yield to maturity, or rate
of return, on a bond — if you invest in a bond, hold it to maturity, and receive
all of the promised cash flows, you will earn the YTM on the money you invested. Exactly the same concepts are employed in capital budgeting when the
internal rate of return (IRR) method is used. The IRR is defined as the discount rate that equates the present value of a project’s expected cash inflows to
the present value of the project’s costs:
PV(Inflows) ϭ PV(Investment costs),
or, equivalently, the rate that forces the NPV to equal zero:
CF0 ϩ

IRR

CF1
(1 ϩ IRR)

1

ϩ

CF2
(1 ϩ IRR)

2

ϩ и и и ϩ

CFn
ϭ0
(1 ϩ IRR)n

n
CFt
NPV ϭ a
ϭ 0.
(1
ϩ
IRR)t
tϭ0

The discount rate that forces the
PV of a project’s inflows to equal
the PV of its costs.

(11-2)

5

This description of the process is somewhat oversimplified. Both analysts and investors anticipate
that firms will identify and accept positive NPV projects, and current stock prices reflect these expectations. Thus, stock prices react to announcements of new capital projects only to the extent
that such projects were not already expected. In this sense, we may think of a firm’s value as consisting of two parts: (1) the value of its existing assets and (2) the value of its “growth opportunities,” or projects with positive NPVs.
C A P I TA L B U D G E T I N G D E C I S I O N R U L E S

517


For our Project S, here is the time line setup:

0

IRR

Cash Flows

Ϫ1,000

Sum of PVs for CF1–4

1,000

Net Present Value

0

Ϫ1,000 ϩ

1

2

3

4

500

400

300

100

500
400
300
100
ϩ
ϩ
ϩ
ϭ 0.
1
2
3
(1 ϩ IRR)
(1 ϩ IRR)
(1 ϩ IRR)
(1 ϩ IRR)4

Thus, we have an equation with one unknown, IRR, and we need to solve for IRR.
Although it is easy to find the NPV without a financial calculator, this is
not true of the IRR. If the cash flows are constant from year to year, then we
have an annuity, and we can use annuity factors as discussed in Chapter 7 to
find the IRR. However, if the cash flows are not constant, as is generally the
case in capital budgeting, then it is difficult to find the IRR without a financial calculator. Without a calculator, you must solve Equation 11-2 by trialand-error — try some discount rate (or PVIF factor) and see if the equation
solves to zero, and if it does not, try a different discount rate, and continue
until you find the rate that forces the equation to equal zero. The discount
rate that causes the equation (and the NPV) to equal zero is defined as the
IRR. For a realistic project with a fairly long life, the trial-and-error approach

is a tedious, time-consuming task.
Fortunately, it is easy to find IRRs with a financial calculator. You follow procedures almost identical to those used to find the NPV. First, you enter the cash
flows as shown on the preceding time line into the calculator’s cash flow register. In effect, you have entered the cash flows into the equation shown below
the time line. Note that we have one unknown, IRR, which is the discount rate
that forces the equation to equal zero. The calculator has been programmed to
solve for the IRR, and you activate this program by pressing the button labeled
“IRR.” Then the calculator solves for IRR and displays it on the screen. Here
are the IRRs for Projects S and L as found with a financial calculator:6
IRRS ϭ 14.5%.
IRRL ϭ 11.8%.
It is also easy to find the IRR using the same spreadsheet we used for the NPV.
With Excel, we simply enter this formula in Cell B6: ‫؍‬IRR(B4:F4). For Project S, the result is 14.5 percent.7

6

To find the IRR with an HP-10B or HP-17B, repeat the steps given in Footnote 3. Then, with
IRR/YR , and, after a pause, 14.49, Project S’s IRR, will appear. With the
an HP-10B, press
HP-17B, simply press IRR% to get the IRR. With both calculators, you would generally want to
get both the NPV and the IRR after entering the input data, before clearing the cash flow register.
The Technology Supplement explains how to find IRR with several other calculators.
7
Note that the full range can be specified with the IRR formula, because Excel’s IRR function assumes
that the first cash flow (the negative $1,000) occurs at t ϭ 0. Note too that you can use the function
wizard to find the IRR. This is convenient if you don’t have the formula committed to memory.

518

CHAPTER 11

■

T H E B A S I C S O F C A P I TA L B U D G E T I N G


Hurdle Rate
The discount rate (cost of capital)
that the IRR must exceed if a
project is to be accepted.

If both projects have a cost of capital, or hurdle rate, of 10 percent, then the
internal rate of return rule indicates that if the projects are independent, both
should be accepted — they are both expected to earn more than the cost of the
capital needed to finance them. If they are mutually exclusive, S ranks higher
and should be accepted, while L should be rejected. If the cost of capital is
above 14.5 percent, both projects should be rejected.
Notice that the internal rate of return formula, Equation 11-2, is simply the
NPV formula, Equation 11-1, solved for the particular discount rate that forces
the NPV to equal zero. Thus, the same basic equation is used for both methods, but in the NPV method the discount rate, k, is specified and the NPV is
found, whereas in the IRR method the NPV is specified to equal zero, and the
interest rate that forces this equality (the IRR) is calculated.
Mathematically, the NPV and IRR methods will always lead to the same
accept/reject decisions for independent projects. This occurs because if NPV is
positive, IRR must exceed k. However, NPV and IRR can give conflicting rankings for mutually exclusive projects. This point will be discussed in more detail
in a later section.

R AT I O N A L E

IRR M E T H O D

FOR THE

Why is the particular discount rate that equates a project’s cost with the present
value of its receipts (the IRR) so special? The reason is based on this logic: (1)
The IRR on a project is its expected rate of return. (2) If the internal rate of return exceeds the cost of the funds used to finance the project, a surplus remains
after paying for the capital, and this surplus accrues to the firm’s stockholders.
(3) Therefore, taking on a project whose IRR exceeds its cost of capital increases shareholders’ wealth. On the other hand, if the internal rate of return is
less than the cost of capital, then taking on the project imposes a cost on current stockholders. It is this “breakeven” characteristic that makes the IRR useful in evaluating capital projects.

SELF-TEST QUESTIONS
What four capital budgeting ranking methods were discussed in this section?
Describe each method, and give the rationale for its use.
What two methods always lead to the same accept/reject decision for independent projects?
What two pieces of information does the payback period convey that are not
conveyed by the other methods?

C O M PA R I S O N O F T H E N P V A N D I R R M E T H O D S
In many respects the NPV method is better than IRR, so it is tempting to explain NPV only, to state that it should be used to select projects, and to go on
to the next topic. However, the IRR method is familiar to many corporate executives, it is widely entrenched in industry, and it does have some virtues.

C O M PA R I S O N O F T H E N P V A N D I R R M E T H O D S

519


Therefore, it is important for you to understand the IRR method but also to be
able to explain why, at times, a project with a lower IRR may be preferable to
a mutually exclusive alternative with a higher IRR.

NPV P R O F I L E S

Net Present Value Profile
A graph showing the relationship
between a project’s NPV and the
firm’s cost of capital.

A graph that plots a project’s NPV against cost of capital rates is defined as the
project’s net present value profile; profiles for Projects L and S are shown in
Figure 11-4. To construct NPV profiles, first note that at a zero cost of capital,
the NPV is simply the total of the project’s undiscounted cash flows. Thus, at a
zero cost of capital NPVS ϭ $300, and NPVL ϭ $400. These values are plotted
as the vertical axis intercepts in Figure 11-4. Next, we calculate the projects’ NPVs
at three costs of capital, 5, 10, and 15 percent, and plot these values. The four points
plotted on our graph for each project are shown at the bottom of the figure.8
Recall that the IRR is defined as the discount rate at which a project’s
NPV equals zero. Therefore, the point where its net present value profile crosses
the horizontal axis indicates a project’s internal rate of return. Since we calculated
IRRS and IRRL in an earlier section, we can confirm the validity of the graph.
When we connect the data points, we have the net present value profiles.9
NPV profiles can be very useful in project analysis, and we will use them often
in the remainder of the chapter.

NPV R A N K I N G S D E P E N D

Crossover Rate
The cost of capital at which the
NPV profiles of two projects cross
and, thus, at which the projects’
NPVs are equal.

ON THE

COST

OF

C A P I TA L

Figure 11-4 shows that the NPV profiles of both Project L and Project S decline as the cost of capital increases. But notice in the figure that Project L has
the higher NPV at a low cost of capital, while Project S has the higher NPV if
the cost of capital is greater than the 7.2 percent crossover rate. Notice also
that Project L’s NPV is “more sensitive” to changes in the cost of capital than
is NPVS; that is, Project L’s net present value profile has the steeper slope, indicating that a given change in k has a larger effect on NPVL than on NPVS.
To see why L has the greater sensitivity, recall first that the cash flows from
S are received faster than those from L. In a payback sense, S is a short-term
project, while L is a long-term project. Next, recall the equation for the NPV:
NPV ϭ

CF0
(1 ϩ k)

0

ϩ

CF1
(1 ϩ k)

1

ϩ и и и ϩ

CFn
.
(1 ϩ k)n

The impact of an increase in the cost of capital is much greater on distant than
on near-term cash flows. To illustrate, consider the following:

8

To calculate the points with a financial calculator, enter the cash flows in the cash flow register,
enter I ϭ 0, and press the NPV button to find the NPV at a zero cost of capital. Then enter I ϭ
5 to override the zero, and press NPV to get the NPV at 5 percent. Repeat these steps for 10 and
15 percent. We did the calculations and made the graph with our Excel model. See 11MODEL.xls.
9
Notice that the NPV profiles are curved — they are not straight lines. NPV approaches the t ϭ 0
cash flow (the cost of the project) as the cost of capital increases without limit. The reason is that, at
an infinitely high cost of capital, the PV of the inflows would be zero, so NPV at (k ϭ ϱ) is simply
CF0, which in our example is Ϫ$1,000. We should also note that under certain conditions the NPV
profiles can cross the horizontal axis several times, or never cross it. This point is discussed later in
the chapter.

520

CHAPTER 11

■

T H E B A S I C S O F C A P I TA L B U D G E T I N G

FIGURE

11-4

Net Present Value Profiles: NPVs of Projects S and L
at Different Costs of Capital

Net Present Value
($)

400

Project L’s Net Present Value Profile

300

200
Crossover Rate = 7.2%
100

Project S’s Net Present Value Profile
IRR S = 14.5%

0

5

7.2

10

Cost of Capital (%)
15

IRR L = 11.8%
–100

COST OF CAPITAL

0%

NPVS

NPVL

$300.00

$400.00

5

180.42

206.50

10

78.82

49.18

15

(8.33)

(80.14)

PV of $100 due in 1 year @k ϭ 5%:

$100
ϭ $95.24.
(1.05)1

PV of $100 due in 1 year @ k ϭ 10%:

$100
ϭ $90.91.
(1.10)1

Percentage decline due to higher k ϭ

$95.24 Ϫ $90.91
ϭ 4.5%.
$95.24

PV of $100 due in 20 years @ k ϭ 5%:

$100

ϭ $37.69.
(1.05)20

PV of $100 due in 20 years @ k ϭ 10%:

$100
ϭ $14.86.
(1.10)20

Percentage decline due to higher k ϭ

$37.69 Ϫ $14.86
ϭ 60.6%.
$37.69

C O M PA R I S O N O F T H E N P V A N D I R R M E T H O D S

521


Thus, a doubling of the discount rate causes only a 4.5 percent decline in the
PV of a Year 1 cash flow, but the same doubling of the discount rate causes the
PV of a Year 20 cash flow to fall by more than 60 percent. Therefore, if a project
has most of its cash flows coming in the early years, its NPV will not decline very
much if the cost of capital increases, but a project whose cash flows come later
will be severely penalized by high capital costs. Accordingly, Project L, which has
its largest cash flows in the later years, is hurt badly if the cost of capital is high,
while Project S, which has relatively rapid cash flows, is affected less by high capital costs. Therefore, Project L’s NPV profile has the steeper slope.

INDEPENDENT PROJECTS

If an independent project is being evaluated, then the NPV and IRR criteria always lead to the same accept/reject decision: if NPV says accept, IRR also says
accept. To see why this is so, assume that Projects L and S are independent, and
then look back at Figure 11-4 and notice (1) that the IRR criterion for acceptance for either project is that the project’s cost of capital is less than (or to the
left of ) the IRR and (2) that whenever a project’s cost of capital is less than its
IRR, its NPV is positive. Thus, at any cost of capital less than 11.8 percent,
Project L will be acceptable by both the NPV and the IRR criteria, while both
methods reject the project if the cost of capital is greater than 11.8 percent.
Project S — and all other independent projects under consideration — could be
analyzed similarly, and it will always turn out that if the IRR method says accept, then so will the NPV method.

M U T UA L LY E X C L U S I V E P R O J E C T S 1 0
Now assume that Projects S and L are mutually exclusive rather than independent. That is, we can choose either Project S or Project L, or we can reject
both, but we cannot accept both projects. Notice in Figure 11-4 that as long as
the cost of capital is greater than the crossover rate of 7.2 percent, then (1)
NPVS is larger than NPVL and (2) IRRS exceeds IRRL. Therefore, if k is
greater than the crossover rate of 7.2 percent, the two methods both lead to the
selection of Project S. However, if the cost of capital is less than the crossover
rate, the NPV method ranks Project L higher, but the IRR method indicates
that Project S is better. Thus, a conflict exists if the cost of capital is less than the
crossover rate. NPV says choose mutually exclusive L, while IRR says take S.
Which answer is correct? Logic suggests that the NPV method is better, because it selects the project that adds the most to shareholder wealth.11
There are two basic conditions that can cause NPV profiles to cross, and
thus conflicts to arise between NPV and IRR: (1) when project size (or scale) differences exist, meaning that the cost of one project is larger than that of the
other, or (2) when timing differences exist, meaning that the timing of cash flows
10

This section is relatively technical, but it can be omitted without loss of continuity.
The crossover rate is easy to calculate. Simply go back to Figure 11-1, where we set forth the
two projects’ cash flows, and calculate the difference in those flows in each year. The differences
are CFS Ϫ CFL ϭ $0, ϩ$400, ϩ$100, Ϫ$100, and Ϫ$500, respectively. Enter these values in the

cash flow register of a financial calculator, press the IRR button, and the crossover rate, 7.17% ഠ
7.2%, appears. Be sure to enter CF0 ϭ 0 or else you will not get the correct answer.
11

522

CHAPTER 11

■

T H E B A S I C S O F C A P I TA L B U D G E T I N G


Reinvestment Rate
Assumption
The assumption that cash flows
from a project can be reinvested
(1) at the cost of capital, if using
the NPV method, or (2) at the
internal rate of return, if using the
IRR method.

from the two projects differs such that most of the cash flows from one project
come in the early years while most of the cash flows from the other project
come in the later years, as occurred with our Projects L and S.12
When either size or timing differences occur, the firm will have different
amounts of funds to invest in the various years, depending on which of the two
mutually exclusive projects it chooses. For example, if one project costs more
than the other, then the firm will have more money at t ϭ 0 to invest elsewhere
if it selects the smaller project. Similarly, for projects of equal size, the one with

the larger early cash inflows — in our example, Project S — provides more
funds for reinvestment in the early years. Given this situation, the rate of return
at which differential cash flows can be invested is a critical issue.
The key to resolving conflicts between mutually exclusive projects is this:
How useful is it to generate cash flows sooner rather than later? The value of
early cash flows depends on the return we can earn on those cash flows, that
is, the rate at which we can reinvest them. The NPV method implicitly assumes
that the rate at which cash flows can be reinvested is the cost of capital, whereas the
IRR method assumes that the firm can reinvest at the IRR. These assumptions are
inherent in the mathematics of the discounting process. The cash flows may
actually be withdrawn as dividends by the stockholders and spent on beer and
pizza, but the NPV method still assumes that cash flows can be reinvested at
the cost of capital, while the IRR method assumes reinvestment at the project’s IRR.
Which is the better assumption — that cash flows can be reinvested at the cost
of capital, or that they can be reinvested at the project’s IRR? It can be demonstrated that the best assumption is that projects’ cash flows are reinvested at the
cost of capital.13 Therefore, we conclude that the best reinvestment rate assumption is the cost of capital, which is consistent with the NPV method. This, in turn,
leads us to prefer the NPV method, at least for a firm willing and able to obtain
capital at a cost reasonably close to its current cost of capital.
We should reiterate that, when projects are independent, the NPV and IRR
methods both lead to exactly the same accept/reject decision. However, when
evaluating mutually exclusive projects, especially those that differ in scale and/or timing, the NPV method should be used.

M U LT I P L E IRR S 1 4
There is one other situation in which the IRR approach may not be usable —
this is when projects with nonnormal cash flows are involved. A project has normal cash flows if it has one or more cash outflows (costs) followed by a series of

12

Of course, it is possible for mutually exclusive projects to differ with respect to both scale and
timing. Also, if mutually exclusive projects have different lives (as opposed to different cash flow

patterns over a common life), this introduces further complications, and for meaningful comparisons, some mutually exclusive projects must be evaluated over a common life. For a discussion of
comparing projects with unequal lives refer to Eugene F. Brigham and Joel F. Houston, Fundamentals of Financial Management, 9th ed. (Fort Worth, TX: Harcourt College Publishers, 2001), Chapter 13 or Appendix 12D on the Concise web site.

13
Again, see Brigham and Daves, Intermediate Financial Management, 7th ed., Chapter 11, for a discussion of this point.
14
This section is relatively technical, but it can be omitted without loss of continuity.

C O M PA R I S O N O F T H E N P V A N D I R R M E T H O D S

523


Multiple IRRs
The situation where a project has
two or more IRRs.

cash inflows. If, however, a project calls for a large cash outflow sometime during or at the end of its life, then the project has nonnormal cash flows. Projects
with nonnormal cash flows can present unique difficulties when they are evaluated by the IRR method, with the most common problem being the existence
of multiple IRRs.
When one solves Equation 11-2 to find the IRR for a project with nonnormal cash flows,
n

CFt
a (1 ϩ IRR)t ϭ 0,

(11-2)

tϭ0

it is possible to obtain more than one solution value for IRR, which means that
multiple IRRs occur. Notice that Equation 11-2 is a polynomial of degree n, so
it has n different roots, or solutions. All except one of the roots are imaginary
numbers when investments have normal cash flows (one or more cash outflows
followed by cash inflows), so in the normal case, only one value of IRR appears.
However, the possibility of multiple real roots, hence multiple IRRs, arises
when the project has nonnormal cash flows (negative net cash flows occur during some year after the project has been placed in operation).
To illustrate this problem, suppose a firm is considering the expenditure of
$1.6 million to develop a strip mine (Project M). The mine will produce a cash
flow of $10 million at the end of Year 1. Then, at the end of Year 2, $10 million must be expended to restore the land to its original condition. Therefore,
the project’s expected net cash flows are as follows (in millions of dollars):
EXPECTED NET CASH FLOWS
YEAR 0

END OF YEAR 1

END OF YEAR 2

Ϫ$1.6

ϩ$10

Ϫ$10

These values can be substituted into Equation 11-2 to derive the IRR for the
investment:
NPV ϭ

$10 million
Ϫ$10 million

Ϫ$1.6 million
ϩ
ϩ
ϭ 0.
0
1
(1 ϩ IRR)
(1 ϩ IRR)
(1 ϩ IRR)2

When solved, we find that NPV ϭ 0 when IRR ϭ 25% and also when IRR ϭ
400%.15 Therefore, the IRR of the investment is both 25 and 400 percent. This

15

If you attempted to find the IRR of Project M with many financial calculators, you would get an
error message. This same message would be given for all projects with multiple IRRs. However,
you can still find Project M’s IRRs by first calculating NPVs using several different values for k and
then plotting the NPV profile. The intersections with the X-axis give a rough idea of the IRR values. Finally, you can use trial-and-error to find the exact values of k that force NPV ϭ 0.
Note, too, that some calculators, including the HP-10B and 17B, can find the IRR. At the error
message, key in a guess, store it, and repress the IRR key. With the HP-10B, type 10 ■ STO ■
IRR, and the answer, 25.00, appears. If you enter as your guess a cost of capital less than the one at
which NPV in Figure 11-5 is maximized (about 100%), the lower IRR, 25%, is displayed. If you
guess a high rate, say, 150, the higher IRR is shown.
The IRR function in spreadsheets also begins its trial-and-error search for a solution with an
initial guess. If you omit the initial guess, the Excel default starting point is 10 percent. Now suppose the values Ϫ1.6, ϩ10, and Ϫ10 were in Cells A1:C1. You could use this Excel formula:
‫؍‬IRR(A1:C1,10%), where 10 percent is the initial guess, and it would produce a result of 25 per(footnote continues)

524

CHAPTER 11

■

T H E B A S I C S O F C A P I TA L B U D G E T I N G


FIGURE

11-5

NPV Profile for Project M

NPV
(Millions of Dollars)
1.5
NPV = –$1.6 + $10 – $10
(1 + k) (1 + k)2

1.0

IRR2 = 400%

0.5

0

–0.5

100

200

300

400

500

Cost of Capital (%)

IRR1 = 25%

–1.0

–1.5

relationship is depicted graphically in Figure 11-5.16 Note that no dilemma
would arise if the NPV method were used; we would simply use Equation 11-1,
find the NPV, and use this to evaluate the project. If Project M’s cost of capital
were 10 percent, then its NPV would be Ϫ$0.77 million, and the project should
be rejected. If k were between 25 and 400 percent, the NPV would be positive.
One of the authors encountered another example of multiple internal rates
of return when a major California bank borrowed funds from an insurance company and then used these funds (plus an initial investment of its own) to buy a
number of jet engines, which it then leased to a major airline. The bank expected to receive positive net cash flows (lease payments plus tax savings minus
interest on the insurance company loan) for a number of years, then several
large negative cash flows as it repaid the insurance company loan, and, finally,
a large inflow from the sale of the engines when the lease expired.
The bank discovered two IRRs and wondered which was correct. It could
not ignore the IRR and use the NPV method since the lease was already on the

books, and the bank’s senior loan committee, as well as Federal Reserve bank
examiners, wanted to know the return on the lease. The bank’s solution called
for calculating and then using the “modified internal rate of return” as discussed in the next section.
(Footnote 15 continued)
cent. If you used a guess of 150 percent, you would have this formula: ‫؍‬IRR(A1:C1,150%), and
it would produce a result of 400 percent.
16
Does Figure 11-5 suggest that the firm should try to raise its cost of capital to about 100 percent
in order to maximize the NPV of the project? Certainly not. The firm should seek to minimize its
cost of capital; this will cause its stock price to be maximized. Actions taken to raise the cost of capital might make this particular project look good, but those actions would be terribly harmful to the
firm’s more numerous projects with normal cash flows. Only if the firm’s cost of capital is high in
spite of efforts to keep it down will the illustrative project have a positive NPV.
C O M PA R I S O N O F T H E N P V A N D I R R M E T H O D S

525


The examples just presented illustrate one problem, multiple IRRs, that can
arise when the IRR criterion is used with a project that has nonnormal cash
flows. Use of the IRR method on projects having nonnormal cash flows could
produce other problems such as no IRR or an IRR that leads to an incorrect
accept/reject decision. In all such cases, the NPV criterion could be easily applied, and this method leads to conceptually correct capital budgeting decisions.

SELF-TEST QUESTIONS
Describe how NPV profiles are constructed.
What is the crossover rate, and how does it affect the choice between mutually exclusive projects?
What two basic conditions can lead to conflicts between the NPV and IRR
methods?
Why is the “reinvestment rate” considered to be the underlying cause of
conflicts between the NPV and IRR methods?

If a conflict exists, should the capital budgeting decision be made on the
basis of the NPV or the IRR ranking? Why?
Explain the difference between normal and nonnormal cash flows.
What is the “multiple IRR problem,” and what condition is necessary for it
to occur?

M O D I F I E D I N T E R N A L R AT E
OF RETURN (MIRR)17
In spite of a strong academic preference for NPV, surveys indicate that executives
prefer IRR over NPV. Apparently, managers find it intuitively more appealing to
evaluate investments in terms of percentage rates of return than dollars of NPV.
Given this fact, can we devise a percentage evaluator that is better than the regular IRR? The answer is yes — we can modify the IRR and make it a better indicator of relative profitability, hence better for use in capital budgeting. The new
measure is called the modified IRR, or MIRR, and it is defined as follows:

Modified IRR (MIRR)
The discount rate at which the
present value of a project’s cost is
equal to the present value of its
terminal value, where the terminal
value is found as the sum of the
future values of the cash inflows,
compounded at the firm’s cost of
capital.

PV costs ϭ PV terminal value
n

a CIFt(1 ϩ k)

nϪt

n

COFt
a (1 ϩ k)t ϭ
tϭ0
PV costs ϭ

tϭ0

(1 ϩ MIRR)n
TV
.
(1 ϩ MIRR)n

(11-2a)

Here COF refers to cash outflows (negative numbers), or the cost of the project, and CIF refers to cash inflows (positive numbers). The left term is simply

17

526

CHAPTER 11

■

Again, this section is relatively technical, but it can be omitted without loss of continuity.

T H E B A S I C S O F C A P I TA L B U D G E T I N G

the PV of the investment outlays when discounted at the cost of capital, and
the numerator of the right term is the compounded value of the inflows, assuming that the cash inflows are reinvested at the cost of capital. The compounded value of the cash inflows is also called the terminal value, or TV. The
discount rate that forces the PV of the TV to equal the PV of the costs is defined as the MIRR.18
If the investment costs are all incurred at t ϭ 0, and if the first operating inflow occurs at t ϭ 1, as is true for the illustrative Projects S and L that we first
presented in Figure 11-1, then this equation may be used:
n
nϪt
a CIFt(1 ϩ k)
TV
tϭ1
Cost ϭ
ϭ
.
(1 ϩ MIRR)n
(1 ϩ MIRR)n

(11-2b)

We can illustrate the calculation with Project S:

Cash Flows

0

1

2

3

4

Ϫ1,000

500

400

300

100.00

k ϭ 10%
k ϭ 10%
k ϭ 10%
Terminal Value (TV)
PV of TV

1,000

NPV

0

330.00
484.00
665.50
1,579.50

MIRR ϭ 12.1%

Using the cash flows as set out on the time line, first find the terminal value by
compounding each cash inflow at the 10 percent cost of capital. Then enter
N ϭ 4, PV ϭ Ϫ1000, PMT ϭ 0, FV ϭ 1579.5, and then press the I button to
find MIRRS ϭ 12.1%. Similarly, we find MIRRL ϭ 11.3%.19
18

There are several alternative definitions for the MIRR. The differences primarily relate to
whether negative cash flows that occur after positive cash flows begin should be compounded and
treated as part of the TV or discounted and treated as a cost. A related issue is whether negative
and positive flows in a given year should be netted or treated separately. For a complete discussion,
see William R. McDaniel, Daniel E. McCarty, and Kenneth A. Jessell, “Discounted Cash Flow
with Explicit Reinvestment Rates: Tutorial and Extension,” The Financial Review, August 1988,
369–385, and David M. Shull, “Interpreting Rates of Return: A Modified Rate of Return Approach,” Financial Practice and Education, Fall 1993, 67–71.

19

With some calculators, including the HP-17B, you could enter the cash inflows in the cash flow
register (being sure to enter CF0 ϭ 0), enter I ϭ 10, and then press the NFV key to find TVS ϭ
1,579.50. The HP-10B does not have an NFV key, but you can still use the cash flow register to
find TV. Enter the cash inflows in the cash flow register (with CF0 ϭ 0), then enter I ϭ 10, then
press ■ NPV to find the PV of the inflows, which is 1,078.82. Now, with the regular time value
keys, enter N ϭ 4, I ϭ 10, PV ϭ Ϫ1078.82, PMT ϭ 0, and press FV to find TVS ϭ 1,579.50. Similar procedures can be used with other financial calculators.
Most spreadsheets have a function for finding the MIRR. Refer back to our spreadsheet for
Project S, with cash flows of Ϫ1,000, 500, 400, 300, and 100 in Cells B4:F4. You could use the Excel
function wizard to set up the following formula: ‫؍‬MIRR(B4:F4,10%,10%). Here the first 10 percent is the cost of capital used for discounting, and the second one is the rate used for compounding, or the reinvestment rate. In our definition of the MIRR, we assume that reinvestment is at the
cost of capital, so we enter 10 percent twice. The result is an MIRR of 12.1 percent.

M O D I F I E D I N T E R N A L R AT E O F R E T U R N ( M I R R )

527


The modified IRR has a significant advantage over the regular IRR. MIRR
assumes that cash flows from all projects are reinvested at the cost of capital,
while the regular IRR assumes that the cash flows from each project are reinvested at the project’s own IRR. Since reinvestment at the cost of capital is generally more correct, the modified IRR is a better indicator of a project’s true
profitability. The MIRR also solves the multiple IRR problem. To illustrate,
with k ϭ 10%, Project M (the strip mine project) has MIRR ϭ 5.6% versus its
10 percent cost of capital, so it should be rejected. This is consistent with the decision based on the NPV method, because at k ϭ 10%, NPV ϭ Ϫ$0.77 million.
Is MIRR as good as NPV for choosing between mutually exclusive projects?
If two projects are of equal size and have the same life, then NPV and MIRR
will always lead to the same decision. Thus, for any set of projects like our Projects S and L, if NPVS Ͼ NPVL, then MIRRS Ͼ MIRRL, and the kinds of conflicts we encountered between NPV and the regular IRR will not occur. Also,
if the projects are of equal size, but differ in lives, the MIRR will always lead to
the same decision as the NPV if the MIRRs are both calculated using as the
terminal year the life of the longer project. (Just fill in zeros for the shorter project’s missing cash flows.) However, if the projects differ in size, then conflicts
can still occur. For example, if we were choosing between a large project and a
small mutually exclusive one, then we might find NPVL Ͼ NPVS, but MIRRS Ͼ
MIRRL.
Our conclusion is that the MIRR is superior to the regular IRR as an indicator of a project’s “true” rate of return, or “expected long-term rate of return,”
but the NPV method is still the best way to choose among competing projects
because it provides the best indication of how much each project will increase
the value of the firm.

SELF-TEST QUESTIONS
Describe how the modified IRR (MIRR) is calculated.
What is the primary difference between the MIRR and the regular IRR?
What advantages does the MIRR have over the regular IRR for making capital budgeting decisions?
What condition can cause the MIRR and NPV methods to produce conflicting

rankings?

C O N C L U S I O N S O N C A P I TA L
BUDGETING METHODS
We have discussed five capital budgeting decision methods, comparing the
methods with one another, and highlighting their relative strengths and weaknesses. In the process, we probably created the impression that “sophisticated”
firms should use only one method in the decision process, NPV. However, virtually all capital budgeting decisions are analyzed by computer, so it is easy to
calculate and list all the decision measures: payback and discounted payback,
NPV, IRR, and modified IRR (MIRR). In making the accept/reject decision,

528

CHAPTER 11

■

T H E B A S I C S O F C A P I TA L B U D G E T I N G