capital budgeting 2

260

Capital Budgeting:Introduction and Techniques

Simply put, investmentdecisions have agreater impact on abusiness’s future thanany other decision it

makes. Businesses thatinvest profitably makemoney and provide a fair return for their owners. Those that fail to investprofitably are unlikely to survive in the competitive business world. Busi-nesses must invest constantly. Vail Associates, the ski resort operator,invested $1 million in 1996 in a snowmobile and horseback riding cen-ter. Beaver Creek ski area invested $20 million in a retail complex. Bothwere hoping to attract an increasingly elusive ski customer. Were thesewise investments? No matter how sophisticated the analysis, a firm is sel-dom sure. However, we can develop methods that increase the chancethat investments yield more than they cost.

The purpose of this chapter is to investigate methods for evaluatinginvestment decisions. We will use many of the tools developed so far inthis text. For example, we must adjust an investment’s cash flows to takeinto account the time value of money. Additionally, we must be able toadjust for the risk of those cash flows.

We begin this discussion by defining capital budgeting.

Chapter ObjectivesBy the end of this chapter you should be able to:

1. Introduce TVM concepts to investment analysis

2. Develop project evaluation models

3. Compare NPV to IRR

4. Select projects under capital rationing

C H A P T E R 10

Finance: Investments, Institutions, and Management, Second Edition, by Stanley G. Eakins. Copyright © 2002 by Pearson Education, Inc. Published by Addison Wesley.

WHAT DOES CAPITAL BUDGETING MEAN?Chapter 2 introduced the capital markets. The term capital referred to long-term securi-ties and investments. The term retains the same meaning in this chapter. Capital budget-ing is the process of deciding which long-term investments or projects a firm will acquireusing the long-term funds it has available. The term budgeting is appropriate because mostfirms have more ways to spend money than they have available funds. They must allocatethese limited funds in such a way as to provide the most long-term profits. Keep in mindthat the goal of the financial manager is to increase shareholder wealth. The purpose ofthis chapter is to provide techniques for selecting projects that accomplish this.

Summary of Capital BudgetingOnce a possible project has been identified, a firm’s management must evaluate whetherfirm value will be increased if the project is accepted. There are a number of steps to this evaluation.

1. All relevant cash flows must be identified. Surprisingly, this is where the firm willmake the greatest errors. Although it is often possible to estimate the initial cost ofthe project or investment accurately, estimating the cash inflows that follow is verydifficult. For example, Robert Harshaw quit his job with Texas Instruments in 1987to market a device he had invented to help pilots go through their safety checklistwithout making errors. He thought that he was going to get rich selling his productto the airlines. What he had failed to appreciate was that airlines are extremely reluc-tant to spend money on products not required by the Federal Aviation Administra-tion. Sales were poor and losses mounted. Harshaw’s company did not earn a profituntil he won a contract from Cessna in 1993 to develop a digitized voice system toalert pilots of equipment malfunctions. Because estimating cash inflows may requireestimating the success of new products, errors are almost inevitable.

2. Once all of the cash flows have been identified, they must be analyzed. We will learnone method that does not require the use of time value of money (TVM) and fourmethods that do.

• The non-TVM method is the payback period. Although this method has manyfaults, it continues to be widely reported and used. A project’s payback is simplythe number of years until the investment is recovered.

• The most widely used capital budgeting method is the net present value (NPV).The NPV is computed by subtracting the present value of cash outflows from thepresent value of cash inflows. If inflows exceed outflows on a present value basis,the project is acceptable.

• The profitability index (PI) is closely related to NPV. It converts NPV to a ratiothat is often easier to interpret than NPV.

• The fourth method we will learn is the internal rate of return (IRR). The IRR isthe average compounded annual return earned by a project. If the return exceedsthe firm’s cost of capital, the project is accepted.

CHAPTER 10 Capital Budgeting: Introduction and Techniques 261

• Finally we will learn how to compute the modified internal rate of return. Thismethod is similar to the internal rate of return, but it is modified so that it is moretheoretically sound and in some ways easier to calculate (see Extension 10.1).

3. Finally, the results of the cash flow analysis must be interpreted. We will see thatone important advantage of the TVM-based methods is that they have a clear inter-pretation. This is not the case with the payback method.

Finding Investment OpportunitiesRiches and wealth go to those who are able to see investment opportunities before others.Consider Duffy Mazan and his partners at Electric Press. In 1994 they formed a companyto set up and maintain Internet Web sites. When they first began marketing their servicesthey had to explain to their potential customers what the Internet was. Now they receivecalls from customers who are eager to get on the Web. It takes foresight, an understandingof the market in which you are interested, and some luck to pick growth areas such as this.

On the other hand, consider IBM’s investment in OS/2 Warp. Despite huge expen-ditures, the operating system remains unpopular and not widely distributed. Similarly,Steve Jobs, one of the founders of Apple Computer, has spent an estimated $130 million,including $12 million of his own money, on Next Computer (a new computer company).There has been little payoff from these investments to date.

Most firms are constantly seeking new investment ideas and opportunities. Theseideas may be as simple as replacing two low-output copiers with one high-speed unit.Alternatively, an investment may change the whole face of a firm. In this chapter weassume that management has investment ideas to evaluate. Do not lose sight of the factthat the collection of these ideas spells the success and failure of the firm.

Steps in the Capital Budgeting ProcessThe capital budgeting process is so critical to the survival of a firm that it is worth dis-cussing the full scope of the capital budgeting process, rather than simply how the eval-uation tools are computed. We can identify five steps that a firm should follow.

1. Identification of opportunities: Initially, the firm must have some method in place bywhich new opportunities are identified and brought to the attention of management.For example, when First National Bank of Fairbanks offered a $100 reward toemployees who sent in ideas that were implemented, a vault teller in a small branchsuggested a cash counting machine. Management found that this branch receivedlarge commercial deposits and that the teller was counting each bill separately. With-out the reward offer, management would never have learned of the opportunity tosave substantial amounts of teller time with a small capital investment. Managementis often removed from the factory floor or direct customer contact. Employees onthe front lines must have both the incentive and the means to communicate ideasto those who have the authority to implement them.

2. Evaluation of opportunities: Once the firm identifies an opportunity, it must be eval-uated. This requires that all of the costs and benefits be tabulated. These data are

262 PART III Foundations of Corporate Finance

then subjected to analysis. In this chapter we focus on how to analyze data once theyhave been prepared. In the next chapter we learn how to organize the cash flowsfrom an investment opportunity.

3. Selection: Often firms have more good projects than they can accept in any givenyear. This may be because of limited funds or because of human or physical con-straints facing the firm. In this chapter we look at how a firm might rank projectsto facilitate selecting among them.

4. Implementation: Once a project has been selected, it must be implemented. Themachines will be purchased, people hired, or investments made. Management mustbe vigilant at this stage to ensure that the costs reflect what was initially proposedand evaluated. For example, Twentieth-Century Fox decided to produce the movieTitanic in 1995. They projected costs to be about $100 million and decided that theproject would be profitable. Unfortunately, by 1997 cost overruns brought the totalcost to over $200 million. Total movie revenues would have to exceed $350 millionfor the project to be profitable. Although this did happen, the risk of the project wasmuch greater than originally anticipated.

5. Post audit: Once the project has been completed, management must compare the costs and revenues with the original projections. This is a critical step that is often overlooked. Holding employees responsible for errors in their projectionsgives them an incentive to make more accurate future cost and revenue projections. Employees who know that they must later explain deviations fromprojections will study the results of their last estimates diligently so as to improvein the future.

Taken together, these steps can dramatically improve a firm’s ability to select wealth-increasing projects, bring them to fruition, and learn from each experience. In the nextsection we study methods of evaluating a project once it has been identified as a possi-ble candidate for capital spending.

EVALUATING THE CASH FLOWSThe financial analyst first estimates the cash inflows and outflows that an investmentwill generate. Then these cash flows are evaluated to determine whether the projectshould be accepted. In this chapter we investigate methods for evaluating the cash flows,and in Chapter 11 we will learn how to estimate the cash flows and to handle special sit-uations that arise in evaluating them.

Businesses and investors commonly use the investment evaluation methods dis-cussed earlier. Each suffers from at least one drawback. Some have many problems butcontinue to be used because they are simple. We will learn all of the common techniques.Summarized, they include the following:

• Payback period

• Net present value

• Profitability index


• Internal rate of return

• MIRR (Extension 10.1)

Payback Period MethodThe payback period method is the easiest investment evaluation method to perform,but the theoretically worst method available. The payback is simply the number of yearsit takes to recover the initial investment. The timing and riskiness of the cash flows areignored. The reason it continues to be used is that it is easy to understand and explainto others. Small businesses are especially likely to use the payback method if the own-ers or managers are not well versed in financial principles. The method is also used tosupplement more sophisticated techniques.

ComputationThe calculation of the payback is very easy if the annual cash flows are annuities (remem-ber that annuities are equal payments received at equal intervals). The payback is foundby dividing the initial investment by the annual annuity.

If the cash flows vary from year to year, they must be accumulated until the sumequals the initial investment. Partial years can be estimated. In Example 10.1 we use pay-back to evaluate an annuity.

E X A M P L E 10.1 Payback Period: Annuity

In 2000 Consumer Reports listed Lindeman’s Bin 40 Cabernet Sauvignon as a best buy in its tastetest. If Lindeman’s wants to expand production to take advantage of the increased sales this reportmay generate, it will have to expand its facilities. Assume that expansion of its winery will cost$1,000,000. If this will generate after-tax cash inflows of $235,000 for 8 years, what is the pay-back? It will take about 4 years and 3 months for the firm to recover its initial investment.

SolutionBecause the annual cash inflows are equal, simply divide them into the initial investment:

The calculation is somewhat more complicated if the cash inflows are not equal. Anaccumulation table can be constructed to compute payback in this case. We evaluatean investment with unequal cash flows in Example 10.2.

E X A M P L E 10.2 Payback Period: Unequal Cash Flows

Suppose after reviewing its cash flow estimates, Lindeman’s decides that the publicity providedby the Consumer Reports article will wear off over time. As a result, cash inflows would decline10% the first year and then 15% per year thereafter. What is the payback?

Payback = Initial investmentAnnual cash inflow

Payback =$1,000,000$235,000

= = years, 3 months4 25 4.


SolutionSet up a table, as presented here. The initial investment and cash inflow are given in the prob-lem. The next column is the sum of the cash inflows. The last column is computed by sub-tracting the accumulated inflow column from the initial investment.

The final year can be estimated by dividing the remaining balance by the cash inflow and thenmultiplying the product by 12:

It will take Lindeman’s about 5 years and 10 months to recover its initial investment if the cashflow estimates are correct (5 years+9.89 months).

Self-Test Review Question*What is the payback for an investment that requires a $10,000 initial investment

and returns $3,000 per year thereafter?

AdvantagesThe principal advantage of the payback method is its simplicity. It also provides infor-mation about how long funds will be tied up in a project. The shorter the payback, thegreater the project’s liquidity.

DisadvantagesThere are many problems with the payback method.

• No clearly defined accept/reject criteria: Is a 4-year payback good or bad? We do nothave a method to determine this. Often a payback of 2 or 3 years is required, butclearly this is arbitrary.

• No risk adjustment: Risky cash flows are treated the same way as low-risk cash flows.The required payback period could be lengthened for low-risk projects, but the exactadjustment is still arbitrary.

$ , .$ , .

. .91 028 81

110 404 320 824 12 9 89= = months¥�

Initial AccumulatedYear Investment Cash Inflow Inflow Balance

0 :$1,000,000 0 0 :$1,000,000.001 $235,000.00 $235,000.00 :765,000.002 211,500.00 446,500.00 :553,500.003 179,775.00 626,275.00 :373,725.004 152,808.75 779,083.75 :220,916.255 129,887.44 908,971.19 :91,028.816 110,404.32 1,019,375.51 ;19,375.51


*$10,000/$3,000=3.33. This means it will take 3.33 years to recover the initial investment. One-third of ayear is 4 months. The payback is then 3 years and 4 months.

• Ignores cash flows beyond the payback period: Any cash inflows that occur after thepayback period are excluded from the analysis. This is clearly a short-sighted wayto view investments.

• Ignores time value of money: Consider Table 10.1. The payback is the same, 3 years,but cash inflow A is clearly preferred because of the time value of money.

To properly evaluate investment projects we need a method that does not suffer fromthe above problems. One such method is the net present value. One reason for learningthe payback method was to demonstrate a poor method of analysis so that you will beable to appreciate a theoretically sophisticated method. Pay attention to how the net pre-sent value approach differs from the payback method.

Net Present ValueThe net present value (NPV) is the most popular and theoretically sound evaluationtool available to analysts. NPV has grown in use among corporations as more studentsare exposed to the method in their finance or MBA coursework. Its interpretationrequires a fundamental understanding of the time value of money. Surveys of largenational corporations find that over 70% now apply NPV to project evaluation, althoughmost companies continue to use other methods as well.

TheoryMost investments have some funds being spent today in the hope that greater amountswill be received in the future. Because the cash inflows and the cash outflows occur atdifferent times, they cannot be compared directly. Instead, they must be translated intoa common time period. It is usually easiest to convert all of the cash flows into currentdollars because at least some expenditure is probably made at time zero. After the con-version into present values, the cash inflows are compared with the cash outflows. Ifinflows exceed outflows, the project is acceptable. The difference between the cash out-flows and the cash inflows is the NPV.

ComputationThe formula for calculating NPV can be written several ways. Equation 10.1 uses sum-mation notation:

NPV =CF

1 +– Initial investment 10.1

=1

tt

t

n

i( )( )∑


Do not simply add togethercash flows that occur atdifferent points in time.This will never be correct.Always adjust the cashflows by computing thevalue they have at a com-mon point in time, usuallythe present. You mustalways adjust for the timevalue of money beforecombining cash flows.

Study Tip

TABLE 10.1 Cash Inflows, $1 Million Over 3 Years

Year Cash Inflow A Cash Inflow B Balance

0 0 0 –$1,000,0001 $500,000 $200,0002 300,000 300,0003 200,000 500,000 0

The first term on the right-hand side of the equation computes the present value ofthe cash inflows where i is the discount rate. This discount rate is equal to the firm’s costof capital when evaluating projects similar in risk to others in the firm’s portfolio. Theinitial investment is assumed to be paid at time zero, so no discounting is required. Ifthe initial investment is actually paid out over a period of time, the present value of theinitial investment must be found. Each year’s cash outflows would have to be discountedback to the present before subtracting from the present value of the inflows.

An alternative equation for NPV is

NPV=PV(Cash inflows)-PV(Cash outflows) (10.2)

You can interpret a positive NPV as meaning that the current value of the incomeexceeds the current value of the expenditure, so the project should be accepted. A neg-ative NPV means the project costs more than it will bring in and so should be rejected.The decision criteria for NPV can then be summarized as follows: Accept the project ifNPV is positive or equal to 0; reject the project if NPV is negative.

E X A M P L E 10.3 Net Present Value Calculation

The owner of a Texaco gas station in Nevada is considering buying a slot machine to put in hissmall convenience store. The slot machine will sell for $6,000 and is expected to bring in about$10 per day after expenses. Slot machines in casinos have an average take of about $150 perday after expenses, so the owner believes his cash flow estimate is conservative. If the averagecost of funds to the gas station is 15%, should the slot machine be installed? The machine isexpected to last 3 years before a newer model will be needed to attract gamblers.

SolutionWe can simplify the calculations by using the annual projected cash inflow rather than the daily cash inflow ($10!365=$3,650). Putting the numbers into Equation 10.1 yieldsthe following:

Because the NPV is positive, the gas station owner should install the slot machine. The prob-lem could also have been worked using the annuity tables to find the present value of the equalcash inflows.1

How would you explain what an NPV of $2,333.77 means to someone who has nottaken an introductory finance course? One accurate interpretation is that the project hasreturned the cost of capital (15%) plus $2,333.77. In other words, the value of the firmwill increase by $2,333.77 as a result of accepting the project.

Suppose that you had completed the analysis of an investment for a very large firm.The initial investment is $1 billion and the NPV is $1. Assuming you are absolutely

NPV =CF

1 +– Initial investment

NPV =$3,650

1.15+

$3,650

1.15+

$3,650

1.15– =

=1

tt

t

n

i( )

( ) ( ) ( )

∑

1 2 36 000 2 333 77$ , $ , .


1For example, NPV=$3,650(PVIFA15%,3 yr)-$6,000=$2,333.68.

positive of all of your calculations and estimates (this will probably never be true), doyou recommend that the firm make the investment? In other words, do you invest $1billion to get an NPV of $1? The decision criterion says to accept the project if the NPVis positive. Many students want to abandon the NPV decision criteria of accepting allpositive-NPV projects when faced with this example. You should recommend acceptance.The reason is that the project is returning much more than $1. It is returning the requiredreturn (the cost of capital) plus $1. In other words, the firm is getting all that it needs tobe satisfied that it is receiving a fair return, plus a $1 bonus. The point is that a positiveNPV is the amount the investor is receiving above what is required valued at time=0.

E X A M P L E 10.4 Net Present Value Calculation

Not all investments are made in one lump sum. Sometimes the initiation of the project takesseveral years. For example, the Trans-Alaska Pipeline took 4 years to complete, at a total costof $8 billion. Suppose $1 billion was spent the first year, $1 billion the second year, $2 billionthe third year, and $4 billion the last year (assume all investments are made at the beginningof the year). If the revenues are expected to be $1 billion per year for 20 years and the discountrate is 15%, should the pipeline have been built (assume all cash inflows occur at the end ofthe year and begin at the end of year 1)?

SolutionWe will first compute the present value of the cash outflows, and then we will compute thepresent value of the cash inflows. Finally, we will compute NPV by subtracting the present valueof the outflows from the present value of the inflows.

Because the NPV is greater than zero, the pipeline should have been built.

AdvantagesThe net present value method solves the problems listed with the payback periodapproach.

• Uses time value of money concept: The cash flows are discounted back to the presentso that all cash flows are compared on an equal basis.

• Clear decision criterion: Accept the project if the NPV is zero or greater. Reject if lessthan zero.

Step 1: PV outflows = billion1

+ billion

1.15+ billion

1.15+ billion

1.15

PV outflows = billion + $0.87 billion + $1.51 billion + $2.63 billion

PV outflows = billion

Step 2: PV inflows = billion PVIFA

PV inflows = billion = billion

Step 3: NPV = PV inflows – PV outflows

1 2 3

20 yr, 15%

( )( ) ( ) ( )

( )( )( ) ( )( )

( )

$ $ $ $

$

$ .

$

$ . $ .

1 1 2 4

1

6 01

1

1 6 259 6 259

¥�

¥�

(( ) NPV = billion – $6.01 billion = $0.248 billion$ .6 259


• Discount rate adjusts for risk: By increasing or decreasing the discount rate, the firm canadjust for the riskiness of the cash flows. We will investigate how to do this in a latersection. The discount rate used to evaluate capital budgeting projects is the firm’s costof capital, which is the average cost of its debt and equity. The cost of capital reflectsthe risk of the firm and the firm’s average required rate of return on its investments. InChapter 12 we will learn how to compute the cost of capital. For now it is best describedas the return the firm must earn on its investments to satisfy investors.

DisadvantagesThe primary disadvantage to NPV is that it may be difficult for someone without a back-ground in financial theory to understand. This problem sustains the popularity of othermethods we will study.

A second problem with NPV is that it can be difficult to use when available capitalor resources are limited. If a company must select among a group of positive-NPV pro-jects, it may want to know which projects provide the highest return for the amountinvested. NPV does not provide this information. We will point out alternative methodsthat can be helpful when the firm must rank projects.

Self-Test Review Question*The investment required to obtain a new machine is $2,500. The cash flows are

estimated to be $500 per year for 8 years. If the firm’s cost of capital is 12%, should itbuy the machine? (Compute NPV)

NPV ProfileAn NPV profile graphs the NPV at a variety of discount rates. The NPV profile demon-strates how sensitive the NPV is to changes in the discount rate. We will learn in Chap-ter 12 that it is very difficult to accurately and confidently estimate the cost of capital fora firm. At best we can determine an approximate value. Before we recommend that a firmaccept or reject a project, we should determine whether a small error in our cost of cap-ital estimate is important. We can do this by preparing an NPV profile. Once the profileis prepared, we can note whether small changes in the cost of capital will result in majorchanges to the NPV.

Let us prepare an NPV profile for the cash flows given in Table 10.2.

TABLE 10.2

Year Cash Flow

0 :$1,0001 2502 2503 2504 2505 250


*No; NPV=$2,483.82-$2,500=:16.18. Because the NPV is negative, do not buy the machine.

Many students get con-fused about which dis-count rates should be usedto construct an NPV pro-file. Any interest rateworks. Simply pick onesthat are above and belowthe crossover point. Youwill not know where thisoccurs until you begincomputing some NPVs.

Study Tip

To compute the NPV profile, select a number of different discount rates and com-pute the NPV for each. You may use any discount rates you choose, although it is usu-ally easiest to begin at 0% because the NPV is found simply by summing the cash flows.Continue using increasingly larger discount rates until the NPV turns negative. TheNPVs for five interest rates using the cash flows from Table 10.2 are reported below.

These numbers are graphed in Figure 10.1. We can read the point where the graphcrosses the horizontal axis. This occurs at about 8%. This is where NPV=0. To the left ofthis point NPV is positive and the project is acceptable. To the right of this point NPV isnegative and the project should be rejected. If you are confident that the cost of capital (theaverage cost of funds to the firm) is less than the crossover point, accept the project.

E X A M P L E 10.5 Preparing an NPV Profile

You are contemplating an investment in a Putt-Putt miniature golf course. If you invest $50,000today, you expect to receive annual cash flows of $15,000 for the next 5 years. You are not cer-tain of your cost of capital but expect it to be around 15%. Prepare an NPV profile and discusswhether the investment should be made.

SolutionWe will need to compute the NPV at a variety of different discount rates. We do not know whichones until we actually begin computing a few to see how the NPV profile develops. We willbegin with the discount rate equal to zero and will compute the NPV using increasingly largediscount rates until the NPV is negative. The formula for computing NPV is

NPV=$15,000!(PVIFA5 yr,i)-$50,000

Discount Rate NPV

0% $250.005 82.377.5 11.47

10 :52.3012 :98.81


Discount Rate

NP

V

$300

$200

$100

$0

($100)

($200)

0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12%

FIGURE 10.1NPV Profile

When i=0%,

NPV = $15,000(5) – $50,000

NPV = $75,000 – $50,000 = $25,000

When i=5%,

NPV = $15,000(4.329) – $50,000

NPV = $64,935 – $50,000 = $14,935

We continue computing the NPV at different discount rates until NPV is negative. The resultsare reported in this table:

We now graph the results to obtain our NPV profile:

From the NPV profile we see that we would accept the project as long as the cost of capital wasless than about 151⁄4% because the NPV is positive in that range. Alternatively, we would rejectthe project if the cost of capital was greater than about 151⁄4%. In this example, since the costof capital is 15%, you would make the investment.

In the next section we will introduce a method that converts NPV into a ratio thatis easier for some to interpret.

Profitability Index (Cost–Benefit Ratio)The profitability index (PI) uses the same inputs as the NPV, but by converting theresults to a ratio, it provides additional information. Equation 10.3 computes the PI:

PI =PV Cash inflows

PV Cash outflows

PI =PV Cash inflows

Initial investment

( )( ) ( )( )

10 3.

Discount Rate

NP

V

$30,000.00

$20,000.00

$10,000.00

$0.00

($10,000.00)

0% 5% 10% 15% 20%

Discount Rate NPV

0% $25,0005 14,935

10 6,86215 28220 (5,141)


The numerator is the present value of the benefits of taking the project. The denom-inator is the present value of the cost of taking the project. The PI is the benefit relativeto the cost, on a present value basis. An easier interpretation is that the PI is the bang forthe buck provided by the project. When NPV is zero, the PV(Cash inflows) will equal thePV(Cash outflows) and the PI will be 1. Thus, our decision criterion is to accept the pro-ject if the PI is greater than or equal to 1.

ComputationTo compute PI simply find the present value of the cash inflows and divide by the PV ofthe cash outflows. If you are also computing an NPV, these values should be readily avail-able. We can use the figures provided by Example 10.3 to illustrate the process.

E X A M P L E 10.6 Profitability Index

Suppose that a $6,000 investment will yield three cash inflows of $3,650 each. With a discountrate of 15%, what is the PI?

SolutionThe PV of the cash outflow is $6,000 because the entire investment is made today. The PV ofthe cash inflows is $3,650(PVIFA15%,3 yr), which is $3,650(2.2832)=$8,333.68. Put these fig-ures into Equation 10.3:

The profitability index is 1.39. Because it is greater than 1, we would accept the project. Noticethat this is the same decision we reached in Example 10.3. In fact, PI and NPV will alwaysprovide the same answer to the accept/reject question.

AdvantagesThe PI is useful as an aid in ranking projects from best to worst. It may be necessary torank projects if the firm does not have sufficient funds or capacity to accept all positive-NPV projects. Consider two positive-NPV projects, one small and one large. The largeone may have the largest NPV even though the smaller one has a greater return on thedollars invested. The profitability index will highlight this difference by computing thereturn per dollar invested, on a present value basis. The firm may be better off takingseveral small high-PI projects instead of one large positive-NPV project.

Self-Test Review Question*A machine will cost $2,500 to buy and is expected to yield profits of $500 per

year for 8 years. What is the profitability index? Assume a cost of capital of 12%.

PI =PV Cash inflows

PV Cash outflows

PI =$8,333.68

$6,000

PI = 1.39

( )( )


Because PI gives the returnper dollar invested, it issaid to give the “bang perbuck.”

Study Tip

*$500(PVIFA8,12%)=$2,483.80. PI=$2,483.80/$2,500=0.99352. Reject.

E X A M P L E 10.7 Using PI to Rank Projects

Suppose that you have collected the following data on four possible projects. Rank the projectsusing PI. If your capital budget was $1,000, which project(s) would you select?

SolutionBegin by computing the profitability index for each project:

Now review the PI ratios to see which projects are acceptable. Because project B has a PI lessthan 1, it is immediately rejected. Next, rank the projects in order from highest PI to lowest.Project D has the highest PI, A is next, and C is third. This analysis suggests we should acceptprojects A and D, for a total capital budget of $520. The combined NPV of these two projectsis $55, which is greater than the NPV of project C by itself.2

DisadvantagesAlthough there are no theoretical problems with PI, it should not replace NPV. Ultimately,the goal of the financial manager is to maximize shareholder wealth. PI may be used asa supplement to NPV, but not as a replacement.

In the next section we will discuss the most frequently used alternative to the netpresent value: the internal rate of return.

Internal Rate of ReturnThe internal rate of return (IRR) is the discount rate that sets the present value of thecash inflows equal to the present value of the cash outflows. Alternatively, IRR can bedefined as the discount rate that sets NPV equal to zero. If the IRR is greater than thecost of capital, the project is accepted. If the IRR is less than the cost of capital, the project is rejected.

IRR is more difficult to calculate than NPV and often requires the use of a financialcalculator or computer. However, it is far easier to interpret. For this reason it continuesto be used almost as often as NPV.

Project Net Investment PV (cash inflows) PI

A $500 $550 $550∏$500=1.1B 100 90 $90∏$100=.9C 1,000 1,052 $1,052∏$1,000=1.052D 20 25 $25∏$20=1.25

Project Net Investment PV (cash inflows) NPV

A $ 500 $ 550 $50B 100 90 :10C 1,000 1,052 52D 20 25 5


2The NPV is computed by subtracting the net investment from the PV(inflows). The NPV of A=$50,B=:$10, C=$52, and D=$5.

NPV and PI will alwaysgive the same accept/rejectdecision because all of theinputs to both models areexactly the same. The valueof PI is to help rank pro-jects by showing whichprovide the greatest returnper dollar invested.

Study Tip

TheorySuppose that your roommate offers you an opportunity to invest in his mail-order com-puter parts business. If you invest $100 today, you will receive $110 in 1 year. What isthe return on this investment? You probably answered 10% without needing paper andpencil. The return on this investment is independent of what else is happening to mar-ket returns, so we call it an internal return.

Would you accept your roommate’s offer? That depends on what your required rateof return is. If your cost of capital is 12%, you would reject the proposal.

Let us continue with this example by demonstrating how we would compute theNPV. The figures are initially put into Equation 10.3:

If we know the discount rate (i), we can compute NPV. The IRR approaches theproblem from a slightly different angle. Rather than inputting a discount rate and com-puting NPV, we ask how high the discount rate can be before NPV becomes negative andthe project is unacceptable. We find this breakeven discount rate by setting NPV equalto zero and solving for i. For example,

The 10% interest rate is the value of the discount rate that sets the present value of the cash inflows equal to the present value of the cash outflows. If the 10% return is acceptable, the project should be taken. In this example, because capital cost 12%, we reject the project. Thus, the decision criterion for IRR can be summarized as follows: Accept the project if the IRR is greater than or equal to thecost of capital.3

Review Figure 10.1. We can read the IRR directly off the NPV profile. The IRR isthe discount rate where NPV=0. This is where the profile crosses the horizontal axis.

ComputationIn the preceding example we saw that the calculation of the IRR was fairly straightfor-ward when there was a single cash inflow. It becomes much more complicated whenthere are multiple cash flows. There are three methods to use depending on the natureof the cash flows and the availability of a financial calculator. They involve using finan-cial tables, trial and error, and a calculator. We will discuss each of the methods belowand illustrate them with examples.

NPV = 0 = $1101 +

–

= $1101 +

1 + = $110 =

= =

i

i

i

i

$

$

$.

. %

100

100

1001 10

0 10 10

NPV = $1101 +

–i

$100


3When used in this context, the cost of capital is often referred to as the hurdle rate. It is the rate the IRR mustexceed to be acceptable.

The first method involves using financial tables: If there is only one cash inflow orif the cash inflows are equal, the financial tables may be used to find an approximationof IRR. The steps are listed here:

1. Set up the problem as if you were solving for NPV.

2. Set NPV equal to zero.

3. Solve for PVIF or PVIFA.

4. Look up the interest rate that corresponds to the factor found in step 3 in the PVIFor PVIFA table.

E X A M P L E 10.8 Computing IRR: Factor Method

If the initial investment is $500 and the cash inflows are $200 for 3 years, what is the IRR?

Solution

NPV = 0 = $200(PVIFAIRR,3 yr) – $500

$500 = $200(PVIFAIRR,3 yr)

$500/$200 = PVIFAIRR,3 yr

2.500 = PVIFAIRR,3 yr

Look in the PVIFA table for the factor equal to 2.5 with 3 periods. We find that the interest ratefalls between 9% and 10%. We could estimate the IRR to be 9.5%.

The second method involves trial and error. This method is used if the cash flowsare not equal. The problem is again set up as if you were setting NPV equal to zero. Selectan interest rate and determine whether NPV computes to zero. If not, try another. (If thecomputed NPV was positive, try a higher interest rate; if it was negative, try a lower rate.)Keep trying interest rates until NPV is equal to zero.

E X A M P L E 10.9 IRR by Trial and Error

Use the cash flows from Example 10.8 and compute the IRR by trial and error.

SolutionTo solve this example by trial and error we would set it up using Equation 10.3:

If the discount rate is set equal to 9%, NPV=6.26. If the discount rate is set equal to 10%,NPV=:2.63. The internal rate of return is between 9% and 10%.

The third method involves using a financial calculator. Many financial calculators havebuilt-in IRR formulas. The cash flows must be entered before the IRR can be calculated.

NPV = 0 = $200

1 ++ $200

1 ++ $200

1 +–

i i i( ) ( ) ( )1 2 3500$


(Refer to the owner’s manual to find out how to do this because each brand of calculatoris different.) Solving this example using a financial calculator yields an IRR of 9.70%.

AdvantagesThe primary advantage of the IRR method of investment analysis is that it is easy to inter-pret and explain. Investors like to speak in terms of annual interest rates when evaluatinginvestment options. For this reason, many firms that use NPV also compute IRR.

DisadvantagesThere are several serious problems with IRR that must be understood. They do not nec-essarily invalidate the model, but must be considered before its application.

Reinvestment Rate Assumption The IRR assumes that the cash flows are rein-vested at the internal rate of return when they are received. In Example 10.8, three pay-ments of $200 are received. The first payment is reinvested for two periods and the secondpayment is reinvested for one period. IRR assumes that these payments earn 9.70% whenreinvested until the project is over. We consider this reinvestment rate assumption to be adisadvantage because there may not be any other investments available with returns equalto high-IRR projects, so it may not be possible to reinvest at the IRR.

The reinvestment rate assumption is a problem only when you are attempting to rankmutually exclusive projects. If you are just attempting to reach an accept/reject decisionon a project, the reinvestment rate assumption is not relevant. On the other hand, it maycause incorrect ranking of projects. If you depend on IRR to select among projects, youmay select the wrong one. Review Table 10.3. The initial investment is $1,000 for pro-jects A and B, but we get conflicting rankings from NPV and IRR. NPV is higher for pro-ject A, but IRR is greater for project B. Which project do we accept? Because the NPV iscomputed using the firm’s cost of capital, we can assume that other projects are availableat that rate. We do not know whether any more investments are available that yield 20%.For this reason, we favor NPV when ranking projects. Note that both methods gave thesame accept/reject decision. This will always be true. A project that is found acceptablewith NPV will also be acceptable with IRR.

To better understand the ranking problem, review Figure 10.2, which graphs theNPV profiles of projects A and B. Project A has the highest NPV for all discount rates


Note that we also make anassumption about reinvest-ment of periodic cashflows when computingNPV. We assume that thosecash flows are reinvested atthe firm’s cost of capital.

Study Tip

TABLE 10.3

Cash Flows

Year Project A Project B

0 :$1,000 :$1,0001 0 $1,2002 0 03 $1,500 0NPV@5% $295.76 $142.86IRR 14.47% 20%

less than 12%. Project B is superior for all discount rates greater than 12%. The pro-ject’s rank depends on the discount rate. Because the IRR method does not evaluatethe project at a particular discount rate, it cannot be used for ranking mutually exclu-sive projects.

There May Not Be a Solution to an IRR Problem In some instances thereis more than one solution to an IRR problem. Because computer programs and calcula-tors cannot tell which is correct, they return an error message. This usually happenswhen there are changing signs on the cash flows (most periods having positive cash flowsand some having negative cash flows). The multiple IRR problem can be shown graph-ically with the NPV profile.

Suppose a mining operation will spend $120 million to begin operation, will receive$310 million the second year, and will spend $200 million to clean up. The NPV profileis shown in Figure 10.3.

The NPV is initially negative, becomes positive, then becomes negative again.Because it crosses the zero NPV line twice, there are two IRRs. Because cash flows oftenalternate signs, this can be a serious problem.

Accurate Calculation Often Requires a Financial Calculator It becomesvery tedious to find the IRR by trial and error. You will probably not want to attemptmany IRR calculations without the help of a financial calculator or spreadsheet program.However, with financial calculators available for less than $30, this is less of a problemthan it used to be.


500

400

300

200

100

0

–100

–200

–300

–4000% 5%

Project AProject B

Project A hasgreatest NPV ata 5% discount rate.

Project B hasgreatest NPV ata 25% discount rate.

10% 15% 20% 25% 30%

Discount Rate

NP

VFIGURE 10.2IRR Ranking Problem

IRR Ignores Differences in Scale Suppose you had the choice of buying theKinston Indians (a small-town baseball team) or the Atlanta Braves. You can buy theIndians for $10,000. The Atlanta Braves cost $10 million, but contractual provisions limityou to owning only one baseball team of any kind. If both have an IRR of 25%, whichwould you take if you could afford either? The IRR does not give you any help becauseit converts the cash flows to percentages and ignores differences in the size or scale ofprojects being considered. Clearly anyone of sound mind would go with the Braves.

NPV Versus IRRWhich method should you use to evaluate a project? It depends on who your audienceis, whether you are ranking projects or just trying to determine which are acceptable,and whether the project has alternating signs on the cash flows.

If you are a small business owner doing calculations for your own business, you donot have to worry about the sophistication of your audience. However, most of the timeyou will be presenting your analysis to other investors. How successful would you be inconvincing your art major roommate to invest in your new mail-order pizza business ifyou spoke only of net present values, cost of capital, risk-adjusted discount rates, andthe like? Once you were convinced your numbers were correct by using NPV, a simpli-fied presentation using IRR and payback may be more successful.

The choice of analysis methodology also depends on whether you are attempting toselect among many good projects or just determining the acceptability of a single pro-ject. Remember that IRR cannot be used to rank projects.

Finally, if your project has cash flow sign changes, you may not be able to computean IRR. This will force you to focus on NPV. Never rely wholly on the payback methodbecause it leaves so much out of the analysis.

Some analysts have attempted to save the IRR method by developing an alternativecalculation that reinvests funds at the cost of capital. This method, known as the modi-fied internal rate of return, is often used in real estate analysis.


2

0

–2

–4

–6

–8

–100% 15% 25% 30% 33.30% 40%

Discount Rate

NP

V

FIGURE 10.3Multiple IRRs

E X T E N S I O N 1 0 . 1

Modified Internal Rate of Return (MIRR)Because of the problems listed above for the internal rate of return, analysts have devel-oped an alternative evaluation technique that is similar to IRR, but attempts to improve onit. The cash outflows are discounted back to the present at the cost of capital and the cashinflows are compounded at the cost of capital to the end of the project’s life. The futurevalue of the cash inflows is called the terminal value. The modified internal rate of return(MIRR) is the interest rate that sets the PV of outflows equal to the terminal value.

The calculation of MIRR, though it takes several steps, is not difficult.

1. Find the present value of all cash outflows at the firm’s cost of capital. Often the onlycash outflow is the initial investment. If any subsequent cash outflows are required,such as a future modification, compute the present value of these outflows as well.

2. Find the future value of all cash inflows at the firm’s cost of capital. All positive cashflows are compounded to the point in time at which the last cash inflow is received.

3. Compute the yield that sets the present value of the inflows equal to the presentvalue of the outflows. This yield is the modified internal rate of return.

An example will help explain this method.

E X A M P L E 10.10 Modified Internal Rate of Return

Compute the MIRR for the following cash flow stream. Assume a cost of capital of 10%. Theinitial investment is $500. The cash inflows are $300 per year for 2 years, followed by a $200expenditure and then one more $300 inflow.

SolutionPrepare a time line to better visualize the process:

1. The investment is $500+200(PVIF 3 yr, 10%)=$650.26.

2. There are three positive cash inflows that must be compounded to the end of the fourthperiod. The first $300 cash flow is compounded for three periods. The second $300 cashflow is compounded for two periods, and the last $300 earns no interest. The sum of thefuture value of the cash flows is the terminal value:

Terminal value=$300(1.103)+$300(1.102)+$300

Terminal value=$399.30+$363.00+$300

Terminal value=$1,062.30

3. In this step we compute the interest rate that will set the investment of $650.26 equal to theterminal value of $1,062.30. This is most easily done using a financial calculator.

0 1 2 3 4

$300 $300 –$200 $300–$500


Calculator solution:

PV=:650.26, FV=$1,062.30, N=4, PMT=0, compute I=13.05%

Factor table solution:

Now go to the PVIF table and find the factor closest to 0.6121 in the row corresponding to fourperiods. We find that the factor falls close to 13%, so we estimate the MIRR as about 13%.

The MIRR solves the reinvestment rate assumption problem because all cash flowsare compounded at the cost of capital. It also solves the problem of changing cash flowsigns resulting in multiple IRRs. It still suffers from scale problems. Remember that oneproblem with IRR is that it does not distinguish between large and small projects effec-tively. MIRR suffers from this same limitation. As a result, it cannot be used to rank pro-jects. Hence, it can only be used to make the accept/reject decision, which is accuratelydone by IRR. Again we reach the same conclusion: Because NPV is easy to calculate andprovides a correct wealth-maximizing decision, it is the preferred method.

Self-Test Review Question*The initial investment for a project is $706.80. It will generate cash inflows

for each of the next 3 years of $300. What is the MIRR, assuming a cost of capital of 10%?

PV =

650.26 =

PVIFn,i =

FV(PVIFn,i)1,062.30(PVIFn,i)

650.261,062.30

= 0.6121


Large corporations employ financial analystswhose primary responsibility is to evaluate cap-ital spending projects of interest to the firm. Afinancial analyst collects information fromthroughout the firm to prepare cash flow estimates. Theseestimates are then analyzed to determine whether the firmshould pursue the projects. The financial analyst is oftenalso employed in reviewing projects as they are imple-mented and post completion.

Financial analysts’ salaries range from$23,000–$27,000 for new hires by small firmsto $50,000 or $60,000 for seasoned analystsemployed by larger firms. Many financial ana-

lysts use the skills they learn analyzing individual projectsto advance into into positions of chief financial officer,where salaries can reach several hundred thousand dol-lars per year.

Careers in Finance Financial Analyst

*Terminal value is $300(1.12)+$300(1.1)+$300=$993. The initial investment is $706.80. The MIRR isfound using a financial calculator with N=3, PV=:$706.80, FV=$993, and PMT=0; we computeI=12%. Alternatively, the PVIF=$706.80/$993=0.7118, which corresponds to 12% at 3 periods.


Capital budgeting is the process of evaluating the cash flowsfrom investment opportunities and deciding which invest-ments should be accepted or rejected by the firm. The capitalbudgeting process requires two distinct steps. First, the cashflows from the project must be accurately estimated. Second,the cash flows must be evaluated to determine whether theyprovide a return sufficient to cover the firm’s cost of capital.This chapter introduced five methods to evaluate potentialinvestment opportunities. In Chapter 11 we will investigatehow cash flows are estimated.

The payback period method simply computes the num-ber of years required to recapture the initial cash outflows.This method is used primarily because of its simplicity. It canalso provide an indication of the project’s liquidity because ittells the analyst how long the firm’s funds will be tied up inthe project. However, it fails to adjust for risk or for the timevalue of money. Additionally, any cash flows that occur afterthe payback period are ignored.

The net present value (NPV) method is the preferred wayof evaluating cash flows. It adjusts for risk and for the timevalue of money by evaluating all cash flows in the present. Itis theoretically accurate and is easy to compute. Accepting allpositive-NPV projects will lead to maximizing the value of thefirm. It can also be used to rank projects if the firm is unableto accept all positive-NPV opportunities.

The profitability index (PI) is the ratio of the present valueof the cash inflows to the present value of the cash outflows

(initial investment). If the PI is greater than 1, the projectshould be accepted. Although the PI always gives the sameaccept/reject decision as NPV, it has the advantage of providingan indication of the return per dollar invested (the bang for thebuck). This can be useful in attempting to rank projects.

The internal rate of return (IRR) is popular because itprovides a percentage return on the project that is easy tointerpret and to explain to others. It suffers from several prob-lems, however. First, there is a fundamental theoretical prob-lem in that the cash flows are assumed to be reinvested at theIRR instead of at the cost of capital. Second, when there arealternating signs in the cash flows, no single solution may beavailable. Third, the IRR is difficult to compute. It usuallyrequires a financial calculator. Finally, it cannot be used torank projects because it does not adjust for differences in scale.

The IRR and the NPV always provide the sameaccept/reject decision. This means that as long as IRR is notbeing used to rank the merits of projects, it can be used toevaluate potential investment opportunities.

The modified IRR (MIRR) attempts to solve some of theproblems of the IRR. All cash flows are assumed to be rein-vested at the cost of capital. The MIRR is the rate that sets thepresent value of the initial investment equal to the futurevalue of the periodic cash flows.

In Chapter 11 we will learn to estimate cash flows andsome refinements to capital budgeting techniques that areoften required.

CHAPTER SUMMARY

KEY WORDScapital budgeting 261cost of capital 269internal rate of return

(IRR) 273

modified internal rate ofreturn (MIRR) 279

net present value (NPV) 266

NPV profile 269payback period 264profitability index (PI) 271

DISCUSSION QUESTIONS

1. Why is capital an appropriate word to use to describe theprocess of evaluating possible investment projects?

2. What steps should a firm take to maximize its chanceof successfully identifying and implementing invest-ment projects?

3. What is the purpose of the post audit?4. What are the advantages and disadvantages of the

payback method, NPV, IRR, PI, and MIRR?5. What is the decision criterion for NPV, PI, IRR, and

MIRR?

6. What is the purpose of the NPV profile? Where is theIRR on the profile? Which region of the profile showsacceptable projects?

7. Why would you choose to use the PI over the NPV?(What does the PI tell you that the NPV does not?)

8. What is the reinvestment assumption for the NPV andfor the IRR? Which is more theoretically sound?

9. When is the IRR as good a method to use as the NPV?When should IRR not be used?

10. What problems with the IRR are fixed by the MIRR?(Extension 10.1)


PROBLEMS

1. Consider the cash flows for the following two investments:

a. What are the payback periods on these twoinvestments?

b. What are the NPV and PI for each project if therequired rate of return is 8%?

c. If these two investments were mutually exclu-sive, which would you choose?

2. Consider the following cash flows:

a. Calculate the NPV and PI for each project. There is a 10% required return.

b. Calculate the IRR for project 2.3. Using the data from problem 2,

a. Use the cash flows from project 2 to prepare anNPV profile.

b. On the graph prepared in step a, identify therange of discount rates at which the project isacceptable.

c. On the graph prepared in step a, identify therange of discount rates at which the project isnot acceptable.

d. On the graph prepared in step a, locate the IRR.4. Compute the NPV, PI, and IRR for the following projects.

Which projects should be accepted?a. The project requires an initial investment of

$1,200 and provides five annual cash inflows of$350. Assume a cost of capital of 13%.

b. The project requires an initial investment of$12,000 and provides five annual cash inflowsof $3,500. Assume a cost of capital of 13%.

c. The project requires an initial investment of$12,000 and provides 10 annual cash inflows of$1,750. Assume a cost of capital of 13%.

Year Project 1 Project 2

0 :$200 :$3001 0 1002 50 1003 100 1004 150 100

Year Investment 1 Investment 2

0 :$150 :$1501 20 302 50 403 70 1004 120 110

d. The project requires an initial investment of $12,000 and provides 10 annual cash inflows of $1,750. Assume a cost of capital of 8%.

e. The project requires an initial investment of $12,000 and provides 10 annual cash inflows of $1,750. Assume a cost of capital of 6%.

5. Project L has a cost of $40,000, and its expected net cashinflows are $9,000 per year for 8 years.

a. What is the project’s payback period?b. The cost of capital is 12%. What are the pro-

ject’s NPV and PI?c. What is the project’s IRR?

6. A factory costs $550,000. You forecast that it will pro-duce cash inflows of $100,000 in year 1, $200,000 inyear 2, and $300,000 in year 3. The cost of capital is12%. What is the NPV of the factory?

7. You are presented a proposal for a project. Project Ironcosts $5,000 and will bring in $25,000 in the first year.The next year you will have to pay out $20,000. With a10% cost of capital, calculate the NPV for the project. Doyou accept the project?

8. Rollins Supplies Company is considering an expansionproject. The cash flows are shown in the following table.The cost of capital is 20%.

a. Calculate the NPV and PI for the expansionproject.

b. What is the IRR of the project?9. You are considering building a shopping mall. The ini-

tial investment for the mall is $1 million. The cash flowsare $500,000 for year 1, $400,000 for year 2, $300,000for year 3, and $100,000 for year 4.

a. What are the NPV and PI of the project if thecost of capital is 10%?

b. Compute the IRR for the project.c. Construct an NPV profile for the project.

10. Consider the following two projects. All cash flowsshown are on an after-tax basis.

Year Cash Flow

0 :$2,5001 1,5002 1,7003 1,0004 1,0005 1,000


a. If the discount rate is 16%, what are the PI andNPV of project A?

b. If the discount rate is 16%, what are the PI andNPV of project B?

c. Find the IRR of project A.d. Find the IRR of project B.e. Which project would you prefer?f. If the cost of capital for project A is 13% and

the cost of capital for project B is still 16%,which project would you prefer?

11. A firm has a project with a cost of $65,000 that isexpected to produce benefits of $14,000 per year for 10years. Calculate the project’s payback period, NPV, PI,and IRR. Assume a cost of capital of 14%.

12. Eastern Building is considering two mutually exclusiveprojects. With a 12% cost of capital, evaluate the givennet cash flows to determine which project, if any, shouldbe accepted, and why. Comment on any differences inNPV, MIRR (Extension 10.1), and IRR (if any exist). TheIRR is 17.28% for Rivergate and 17.12% for Treywood.The initial cost of Rivergate is $445,000, whereas the costof Treywood is $1,400,000.

Rivergate TreywoodYears Net Cash Flows Net Cash Flows

1 $160,000 $275,0002 160,000 275,0003 160,000 600,0004 95,000 600,0005 95,000 600,000

Year Project A Project B

0 :$75,000 :$55,0001 30,000 22,0002 18,000 13,2003 50,000 37,000

13. The Renn project cost $55,000 and its expected net cashinflows are $12,000 per year for 8 years.

a. What is the project’s payback period?b. The cost of capital is 12%. What is the project’s

NPV?c. What is the project’s IRR?d. Calculate the project’s MIRR assuming a 12%

cost of capital. (Extension 10.1)14. Lacey Industries Co. has been evaluating a project with

a cost of $700,000. Estimated net cash flows of $180,000are expected for a 7-year period. The cost of capital is14%. Find the NPV and IRR.

15. The Fitness Center is considering including two piecesof equipment, a treadmill and a step machine, in this year’s capital budget. The projects are independent.The cash outlay for the treadmill is $1,700 and for the step machine it is $2,200. The firm’s cost of capitalis 14%. After-tax cash flows, including depreciation, are as shown in the following table. Calculate the NPV,the IRR, and the MIRR (Extension 10.1) for each project, and indicate the correct accept/reject decisionfor each.

Years Treadmill Step Machine

1 $510 $7502 510 7503 510 7504 510 7505 510 750

SELF-TEST PROBLEMS

1. XYZ Company wants to know the payback period fora project with an initial investment of $4 million andannual cash flows of $800,000. What is it?

2. Suppose the annual cash flows listed for problem 1start at $800,000 and then decrease by 15% each year.What is the payback period?

3. A firm is evaluating a project with an initial cost of $3.35million and annual cash flows of $1.15 million for 4years. If the cost of capital for the firm is 14%, what isthe NPV? Should the firm accept or reject the project?

4. Suppose the cost of capital for the firm in problem 3increases to 15%. What is the NPV? Should the firmaccept or reject the project?


5. Evergreen Inc. is evaluating a project with an initialcost of $6 million. Cash flows would start at $1 mil-lion and increase by $750,000 annually for the next 3 years. If the cost of the capital for the firm is 12%,what is the NPV? Should the firm accept or reject the project?

6. A project has projected cash outflows of $2 million inthe current time period. Additional cash outflows of$1 million, $1 million, and $2 million are projectedduring the next 3 years of operation. Cash inflows of$1.3 million are expected in years 4 through 13 (10 cash inflows). Should the project be accepted ifthe company has a cost of capital of 14%? (What isthe NPV?)

7. Would the accept/reject decision change for the pro-ject described in problem 6 if the costs of capital fellto 12%?

8. Suppose that a $10,000 investment will yield threeannual cash flows of $4,000 each. With a discountrate of 12%, what is the PI? What is the accept/reject decision?

9. Rank the following projects by PI:

10. If the initial investment for a project is $500 and thecash inflows are $300 for 3 years, what is the IRR?

11. If the initial investment for a project is $500 and thecash inflows are $300 for year 1, $250 for year 2, and$150 for year 3, what is the IRR?

12. If the initial investment for a project is $1,500 and the cash inflows are $300 for 4 years, what is the IRR?

13. Suppose Project A has an NPV of $350 with an IRR of12% and Project B has an NPV of $300 with an IRR of20%. If these projects are mutually exclusive, whichproject should you accept? Why?

14. Compute the MIRR for a project with a $10,000investment and cash flows of $3,000 for 4 years.Assume of the cost of capital is 12%.

15. Compute the IRR and MIRR for a project with a $5,000investment and cash flows of $1,500 for 4 years.Assume the cost of capital is 13%.

16. Compute the MIRR for a project with a $6,000 invest-ment and cash flows of $1,000 in year 1, $2,000 in year2, $3,000 in year 3, and $4,000 in year 4. Assume thecost of capital is 10%.

17. Compute the MIRR after reversing the cash flows inproblem 16. (Cash flows of $4,000 in year 1, $3,000 inyear 2, etc.) Assume the investment and cost of capitaldo not change.

18. Compute the IRR for problem 16.19. Compute the IRR for problem 17.20. What is the NPV of a project with initial investments

of $3 million at the beginning of years 1 and 2, andcash inflows of $1.5 million at the beginning of years2 through 6? Assume the cost of capital is 10%. Shouldthe project be accepted? (Hint: At the beginning of year2, the net cash flow is :1.5 million.)

Net PV (cashProject Investment inflows) NPV

A $ 400 $480 $80B 700 735 35C 150 225 75D 1,000 950 (50)E 250 275 25

WEB EXPLORATION

1. The concepts behind NPV and IRR apply equally toinvesting in capital projects or in securities. Go towww.financenter.com/calculate/all_calculate.fcs andchoose the calculator titled What Selling Price ProvidesMy Desired Return? This site allows you to input a vari-ety of variables and to look at the rate of return theinvestment provides. It also allows you to view graphs ofthe answers. Relate the results you find using this calcu-lator to IRR and NPV.

2. There are many instructional sites on the Web that aresponsored by educational institutions. One particularlygood one sponsored by the University of South Carolinais available at hadm.sph.sc.edu/courses/econ/invest/invest.html. Review this site and investigate any of thelinks that you feel may improve your understanding ofthe concepts behind NPV and IRR. The calculatorsattached can be very helpful.


You have recently gone to work for a development/con-struction firm. This company does contract and bid

construction work as well as real estate development. Youwork on the development side helping to select projects thatwill be profitable. The development company is organizedas a separate entity from the construction firm. This requiresthat both firms be independently profitable.

The company founder, Jerry Hammer, is primarilyresponsible for identifying development opportunities. Oncean opportunity is identified, Hammer turns it over to his stafffor analysis. Jerry began as a carpenter and has built the firminto a multi–million dollar enterprise mostly based on goodintuition and street smarts. He has no college education.

Last week Jerry called the staff of the development firmtogether to discuss his latest idea. He would like to build anew strip mall on a corner of property near the university.He visualizes a group of tenets that would service the needsof college students. He directed you to let him know if theproject was feasible.

Your first step was to collect cost and revenue esti-mates. The proposed mall is a duplicate of one built last yearfor $3,750,000 with minor cosmetic changes. The mall willhave 30,000 square feet, all of which can be leased. You con-tact the owner of the property and find it can be purchasedfor $500,000.

The revenues are more difficult to estimate. You decidethe most practical approach is to assume the mall will leasefor $2.75 per square foot per month. The mall will take about10 months to build and you think the mall will be 10%leased by the end of the first year. You will get 10% of the

possible revenues for 2 months. Thus, the first year revenueswill be computed as $16,500 (30,000!$2.75!.1!2).You project to get about 60% of the possible revenues dur-ing the second year (i.e., 30,000!$2.75!12!.6). Thiswill rise to 90% by the end of the third year. During thefourth year you project the mall to be fully leased. It is theintent of the company to sell the property once it is fullyleased. You think it reasonable to project a sale by the end ofthe fifth year for $150 per square foot.

You decide a 15% discount accurately reflects the firm’scost of capital.

a. Project the annual cash flows. Remember that the leaserevenues will be received until the property is sold.Assume the initial cash flow occurs at the beginning ofthe first year and that all other cash flows occur at theend of the year. Ignore taxes and depreciation.

b. Put the cash flows onto a time line.c. Compute the payback period. Is this any help in mak-

ing an accept/reject decision?d. Compute the NPV.e. Prepare an NPV profile. Identify the IRR on the graph.

Identify on the graph the discount rates that make theproject acceptable and those that make it unacceptable.

f. Compute the IRR.g. Compute the PI.h. Compute the MIRR. (Extension 10.1)i. Do all of the methods (except payback) give the same

accept/reject result? Is this surprising?j. Discuss how you would present your results to Jerry

Hammer.

MINI CASE

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