Corporations Making Investments

Learning Objectives

Upon completion of Chapter 7, you will be able to:

• Understand the goal of capital budgeting.

• Be able to calculate and interpret net present value (NPV).

• Be able to calculate and interpret payback period.

• Be able to calculate and interpret internal rate of return (IRR).

• Cite the shortcomings of IRR and payback period.

Corporations Making Investments


Iain Sarjeant/Getty Images

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CHAPTER 7Introduction

Businesses are founded to make money for their owners, and in the process they provide employment for workers and products for customers. As they are also investors, owners aim to increase their wealth, so they hire and compensate man- agement to identify promising projects in which to channel the firm’s funds. Therefore, the projects pursued by a business can be viewed as investments—in fact, it’s very useful to envision a business as an investment vehicle for the company’s owners. To increase the wealth of owners, management needs to find projects that are expected to be more valuable than their cost. When Microsoft, for example, chooses to invest in the development of a new version of Windows software, the firm must have concluded that this represents a wealth-producing use of its capital. Mathematically, this criterion for investment may be represented as

Expected Value of a Project . Cost of the Project


E(Value) . Cost

By subtracting cost from both sides, we can see that this criterion may also be stated as

(7.1) E(Value) 2 Cost . 0

So if management identifies a project with an expected value (in today’s dollars) of $10,000 and a cost (also expressed in today’s dollars) of $8,000, then this project is expected to add $2,000 to the wealth of the business’s owners. Note that we say expected value because there is an element of risk in any forecast of value. In fact, to be really accurate, we should probably also use the term expected cost because there is also quite a bit of uncertainty that surrounds the cost of many projects. It is common, for example, to hear of the U.S. Depart- ment of Defense having projects whose costs far exceed their initial budgets. There is a good chance that a similar phenomenon applies to projects in the private sector, but we’re less aware of these cost overruns because they are not broadcast on the nightly news like the overruns we may hear about with Department of Defense contracts.

The left side of Equation (7.1) measures the expected value added to the business by a project. Since investors want to maximize the value of their ownership, a firm should accept all of the projects that satisfy this inequality and add value to the firm. If the busi- ness must choose only one among a group of competing projects, then the proper strategy is to pick the project that adds the most value to the firm (the project that has the highest value added). The underlying objective of this chapter is learning to choose those projects that satisfy these value-added principles. As discussed in Chapter 6, the process for allo- cating corporate funds to these promising investments is referred to as capital budgeting because the company is allocating its capital to the most worthy investment projects the firm has identified.

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CHAPTER 7Section 7.1 Capital Budgeting Methods

7.1 Capital Budgeting Methods

There are several techniques that may used to choose the best investments among the potential projects a company is considering. This chapter focuses on the three most commonly emp- loyed methods: payback period, net present value (NPV), and internal rate of return (IRR). Each technique has its own strengths and weaknesses, which will be dis- cussed and summarized later in the chapter. To begin, we will describe each of the techniques, what it is measuring, how to calculate it, and how to apply it in capital budgeting decisions.

Payback Period

The first technique, payback period, is also the simplest and the least theoretically correct. The payback for a project is the number of years’ worth of operating cash flows it takes for these flows to cumulatively equal the project’s initial cost. Let’s look at a specific example.

Suppose that Campus Pizza is a small pizza parlor near the campus of State U. The own- ers are considering an investment in a motor scooter equipped with an insulated cargo box in order to begin delivering pizzas to dorms and apartments around campus. To date, Campus Pizza has not offered delivery service because of the difficulty of parking at State U, but the motor scooter idea is a good solution to that problem and seems like a promis- ing project.

After doing some research and getting some bids, Campus Pizza’s owners estimate the project’s initial cash outlay at $6,500. This includes the cost of purchasing the scooter, the price of the insulated cargo box—complete with a eye-catching Campus Pizza logo— the cost of additional cardboard takeout box inventory necessary given the anticipated additional demand, the cost of the initial 3 months of vehicle insurance that must be pre- paid, and the cost of the advertising campaign to kick off the delivery service (flyers, an ad in the student newspaper, a band at the grand opening, and so on).

Table 7.1 shows the anticipated cash flows for the scooter project, along with the cumula- tive cash flows. Payback is the number of periods until cumulative operating cash flows equal the project’s initial cost. The decision rule for payback is arbitrarily set by company policy and generally takes the form of a statement such as accept all projects with a payback less than 4 years.

Barbara Smaller/The New Yorker Collection/

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CHAPTER 7Section 7.1 Capital Budgeting Methods

Table 7.1: Payback illustrated using the scooter delivery project

Time (years) Initial cost CF t

Cumulative CF t ’s

0 $6,500

1 $600 $600

2 $950 $1,550

3 $1,200 $2,750

4 $400 $3,150

5 $850 $4,000

6 $1,500 $5,500

7 $1,500 $7,000

8 $2,150 $9,150

In this example, the payback period is between quarters 6 and 7. In fact it is 6.67 quarters because after quarter 6, payback requires $1,000 of the $1,500 of quarter 7’s cash flows, or two-thirds of those funds. If Campus Pizza had a policy of accepting projects with a pay- back less than 4 years, the scooter project would easily satisfy that criteria.

Problems with Using the Payback Period Payback has some critical drawbacks, though. First, it does not consider all of the project’s cash flows. This biases the technique against long-lived projects. With the 4-year payback acceptance policy, for example, there would probably never be any pharmaceutical drugs that were acceptable projects. Their cash flows occur too far into the future. Second, pay- back does not consider the time value of money. A project costing $60,000 with cash flows of $10,000 in year 1, $50,000 in year 2, and $6,000 in year 3 would have the same 2-year payback as another $60,000 project with cash flows of $59,000 in year 1, $1,000 in year 2 and $6,000 in year 3. Knowing what we know about the time value of money, it is easy to see that the second project is clearly superior based on the information provided here.

We have already mentioned the arbitrary nature of the investment decision criteria. There is no assurance that projects with short payback will increase owners’ wealth, even if the cash flows occur just as they are forecasted. The reason is that payback does not take into consideration risk. It could be that a very risky project has a short payback, or that a very low-risk project has a long payback. With an arbitrary rule in place, a firm could be accept- ing projects that destroy firm value and tossing aside the best projects.

Despite these drawbacks, payback has its place. For example, some kinds of projects are relatively small and low risk. It may not be worth the expense and time to do a full-blown NPV or IRR analysis of such projects, and one attractive attribute of the payback method is that it is simple to use. Suppose a firm is considering replacing an old oil-burning fur- nace with a new furnace that burns natural gas. The technicians assure management that the $40,000 upgrade will save the firm in excess of $5,000 a month. Here the payback is

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CHAPTER 7Section 7.1 Capital Budgeting Methods

in the neighborhood of 8 months and should clearly be pursued, especially given that the savings will go on for many years to come. Because the result is so clear, it is hardly worthwhile to pay a newly hired business graduate to spend a couple of days producing an in-depth capital budgeting assessment of the furnace investment.

Net Present Value

We have emphasized that a business should accept projects that are expected to add value to the company. This is the key to successful capital budgeting. There are several chal- lenges to estimating the value added by an investment project. We already mentioned how difficult it might be to estimate the cost of an endeavor. For example, consider the cost of a gold mine, which could take years to develop—during that time there is a good chance the firm might uncover unforeseen, geological challenges that increase its cost. The cost for a mining development could take years to fully estimate; it would be subject to almost constant revision and is clearly a challenge to forecast. In fact, the process of estimating these costs could be quite expensive in its own right. Once the project’s costs are estimated, the analyst must estimate the benefits, which are the future cash flows it’s expected to generate.

In Chapter 6, we took a close look at how to estimate a project’s operating cash flows, including which cash flows are relevant in project evaluation and which are not relevant. We argued that incremental cash flows (those that change if a project is pursued) are rel- evant, while those cash flows that do not change (such as sunk costs or allocated overhead) are not be included in project assessment. We also explained that, because we are taking the perspective of shareholders, it is incremental after-tax cash flows that matter, as they are the cash flows available to shareholders. Another challenge is that the cash flows gener- ated by a project will occur over time, sometimes lasting decades, as in the case of the gold mine mentioned earlier. Prescription pharmaceuticals have extremely long development lives, all of which requires ongoing investment. In the United States the average time from invention to market for a new drug is 12 years (see the Web Resources at the end of the for details of this process). Drug producers must do research, conduct effectiveness and safety trials, apply to the FDA for approval and get patent protection—a long process that delays the initiation of operating cash flows that will accrue to the firm for years.

In order to compare years of forecasted operating cash flows to the project’s initial invest- ment, all these cash flows must be converted into their equivalent present values. A useful way of thinking about this is the old adage, “You can’t add apples and oranges.” If we make our initial investment with a large pile of oranges, then next year’s profits arrive as apples, profits in 2 years are bananas, profits received in 3 years are grapes, and so on, it isn’t clear (beyond having a makings for fruit salad) whether we have made a profitable investment or not. Do five bananas, four apples, a clump of grapes, etc., add up to more than 14 oranges?

For comparison purposes, we need to convert cash flows that occur at different times to an equivalent value at a particular point in time. It’s most common to convert all cash flows into their present values (their equivalent values in today’s dollars). Once this is done, it’s easy to see whether today’s cash outflow is greater or less than all the future inflows because they are all expressed in the same terms. For these present value calculations, we utilize the time value of money mathematics that was covered in Chapters 4 and 5.

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CHAPTER 7Section 7.1 Capital Budgeting Methods

Once expected cash flows and costs are converted into today’s dollars (into their pres- ent values), we can subtract the costs from the project’s present value of cash inflows to determine whether the investment creates wealth for shareholders and owners. This dif- ference between the present value of benefits and costs is the project’s NPV. NPV directly measures the expected value added to shareholder wealth if the project is implemented. To reinforce this, look over the following development of the NPV formula. Note that we start with the idea that businesses wish to take on projects that are expected to add value to the firm—and therefore create wealth for the owners of the company. If the objective is to add value to the company, then

Expected Value Added . 0


E(Value of a Project) 2 E(Cost of the Project) . 0

Putting this into present value terms (to make cash flows comparable) gives us

(7.2) {E(CF 1 )/(1 1 R)1 1 E(CF

2 )/(1 1 R)2 1 . . . 1 E(CF

N )/(1 1 R)N } 2 E(CF

0 ) . 0

where we use the CF notation as defined in Chapter 4. More succinctly, using the summa- tion symbol, we can write this as

(7.3) a N t 5 1

E1CFt2 11 1 R2 t E(CF0) 5 NPV . 0

As you can see, this development of NPV mirrors the basic intuition introduced at the beginning of this chapter: Good investments are those whose value is expected to exceed their cost. The future cash flows are discounted at R, a rate that is appropriate for the proj- ect’s risk. Determination of that discount rate is the topic of Chapter 10. For now, just keep in mind that this is the investor’s required return for the project. These return require- ments get larger for projects with greater risk because investors are risk averse (they try to avoid risk or at least demand compensation in the form of higher returns for their risk exposure).

NPV Example: The Pizza Scooter Delivery Project Revisited Let’s look again at Campus Pizza’s delivery project, this time from the perspective of NPV. Recall that the project’s initial cash outlay is estimated at $6,500. The initial phase of the project is 2 years. Quarterly operating cash flows are shown in Table 7.2. All of the cash flows in the table been adjusted for taxes following the principles covered in Chapter 6. At the end of that time, the scooter would be sold, and the company will receive $1,500 after any applicable taxes on the sale. In Table 7.2, we name the cash flows by their type (from Chapter 6) and with a more mathematical notation. So CF

0 represents the initial

investment, OCF t represents the operating cash flows for each period (t), and TCF

8 is the

project’s terminal cash flow at the end of the eighth quarter.

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CHAPTER 7Section 7.1 Capital Budgeting Methods

Table 7.2: Cash flows for Campus Pizza’s delivery project

Period Today Year 1

Q 1 Year 1

Q 2 Year 1

Q 3 Year 1

Q 4 Year 2

Q 1 Year 2

Q 2 Year 2

Q 3 Year 2

Q 4

Time 0 1 2 3 4 5 6 7 8

Initial investment (CF

0 )


Operating cash flows (OCF

t )

$600 $950 $1,200 $400 $850 $1,500 $1,500 $650

Terminal cash flows (TCF

8 )


Note that the project will begin during the fall quarter and profitability will be relatively low for the first quarter, while the service catches on and because of the training and promotional costs of the product kickoff. In the fourth quarter, cash flows drop because of lower enrollments in summer school compared with the regular school year. As is stan- dard practice, cash flows are assumed to occur at the end of each period. Campus Pizza’s owners require a 12% annual return for the scooter project, which translates into 3% per quarter. At the end of 2 years, if the project is deemed a success, the firm may conduct another NPV analysis to evaluate renewing the project either by choosing not to sell the scooter, by replacing the scooter with a new one, or by adding more scooters.

With this information, the NPV of the project can be calculated. If the NPV is positive, then Campus should move forward with the project. To calculate the NPV we use Equa- tion (7.3):

a N t 5 1

E1CFt2 11 1 R2 t E(CF0) 5 NPV . 0

For 8 quarters we get

NPV 5 $600 1.031

1 $950 1.032

1 $1,200 1.033

1 $400 1.034

1 $850 1.035

1 $1,500 1.036

1 $1,500 1.037

1 $2,150 1.038

5 $582.52 1 $895.47 1 $1,098.17 1 $355.39 1 $733.22 1 $1,256.23 1 $1,219.64 1 $1,697.23

5 $7,837.87

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CHAPTER 7Section 7.1 Capital Budgeting Methods

Now we subtract the initial investment of $6,500 to get

5 $7,837.87 – $6,500 5 $1,337.87

We added together both the operating and terminal cash flows in the eighth quarter, but it would be just as correct (and would yield the same result) if we had separately calculated the PV of the $650 period 8 operating cash flow and the $1,500 terminal cash flow (that also occurs at t 5 8). Be sure you can replicate the mathematics because it is essential for success in this course.

The scooter project, with its positive NPV, is an acceptable project. It is expected to add $1,337.87 to the value of the firm in today’s terms. Another way to interpret this positive NPV is that the scooter is expected to provide Campus Pizza’s owners with their 12% annual required return and have additional cash flows left over. Those additional funds have a present value of $1,337.87.

A third way of understanding NPV as measuring value added is to imagine Campus Pizza as a corporation with 1,000 shares of stock outstanding. In this case, once the scooter delivery project is identified and the project is announced to the firm’s investors, then (in theory at least) the value of each share of stock would increase by about $1.33, found by taking the total value added, $1,337.87, and dividing it by the number of shares outstand- ing. Next time you hear a news report of a firm discovering oil, for example, listen for the stock price reaction and think about how you might interpret the price change in terms of NPV. This can be a bit tricky because news announcements can also be negative, so the announcement that a company is being sued will make the stock price fall because the NPV of being sued is often negative—even if you win, you have to spend a lot of money on lawyers!

Figure 7.1 illustrates the relationship between discount rates and the scooter’s NPV. As your intuition should tell you, the higher the required return for a project, the lower its value will be (future benefits will be discounted to a smaller present value) and the lower its NPV.

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CHAPTER 7Section 7.1 Capital Budgeting Methods

Figure 7.1: NPV of Campus Pizza project at various quarterly discount rates

A project’s NPV decreases as the discount rate (quarterly, in this example) increases.

Although we computed the NPV of the Campus Pizza scooter project using quarterly rates to match the time period of the cash flows, we usually discuss rates in terms of annual rates. Figure 7.2 shows the NPVs for the project (computed using quarterly rates) with the equivalent annual rates on the x axis.

N e t

P re

s e n

t V a lu


Quarterly Discount Rate











0. 0%

0. 5% 1.0

% 1.5

% 2.

0% 2.

5% 3.

0% 3.

5% 4.

0% 4.

5% 5.

0% 5.

5% 6. 0%

6. 5%

7.0 %

7.5 %

8. 0%

Campus Pizza Project NPV at Various Quarterly Discount Rates

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CHAPTER 7Section 7.1 Capital Budgeting Methods

Figure 7.2: NPV of Campus Pizza project at various annual discount rates

A project’s NPV decreases as the discount rate (annual, in this example) increases.

One reason the Federal Reserve kept interest rates low during the Great Recession and the recovery (2007–2013) was that low interest rates kept required returns low. These low required returns, in turn, made projects more attractive (they had higher NPVs because of the low discount rates). The Federal Reserve’s strategy was designed to stimulate the economy as more projects became acceptable, hopefully causing increased corporate investment.

While NPV is commonly used in the business world, in the public sector a variant called cost–benefit analysis is more common. The benefit-to-cost ratio, sometimes called the prof- itability index, is the present value of benefits divided by the present value of costs. If the benefit-to-cost ratio is greater than 1, that is equivalent to a positive NPV, meaning the investment is acceptable. To learn more about this topic, see the Web Resources at the end of the chapter.

Campus Pizza Project NPV at Various Annual Discount Rates

0. 0%

2. 0%

4. 1%

6. 1%

8. 2%

10 .4

% 12

.6 %

14 .8

% 17

.0 %

19 .3

% 21

.6 %

23 .9

% 26

.2 %

28 .6

% 31

.1 %

33 .5

% 36

.0 %









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CHAPTER 7Section 7.1 Capital Budgeting Methods

Demonstration Problem 7.1: Calculating Cash Flows and NPV

Morris Corp. is planning to invest in some equipment. It will cost Morris $34,000 to buy the equipment, which is expected to last for 5 years. Annual cash revenues are expected to be $17,500, and annual project-related expenses are expected to be $6,500. The equipment will depreciate on a straight-line basis over the 5 years. The equipment should be sold in 5 years for $4,500. The required rate of return on this investment is 12%. The tax rate is 30%.

a. What are the cash flows for t 5 0 and t 5 1 through 5? b. Calculate the project’s NPV.


The following table shows the cash flows and NPV for the project. Annual depreciation is $34,000/5. Tax is 30% 3 EBT. Cash flow is EAT (earnings after taxes) 1 depreciation. The tax effect of the project sale is 30% 3 $4,500. NPV is the sum of the discounted cash flows, $2,898.

Project cost (34,000) 4,500

Book value 0

Gain 4,500

Tax effect (1,350)

S–E 11,000 11,000 11,000 11,000 11,000

Depreciation 6,800 6,800 6,800 6,800 6,800

EBT 4,200 4,200 4,200 4,200 4,200

Tax 1,260 1,260 1,260 1,260 1,260

EAT 2,940 2,940 2,940 2,940 2,940

Cash flow (34,000) 9,740 9,740 9,740 9,740 9,740 3,150

Discounted CF (34,000) 8,696 7,765 6,933 6,190 5,527 1,787

NPV 2,898

Internal Rate of Return (IRR)

Our final technique involves using the internal rate of return. In Figure 7.2, we can see that the scooter project is “acceptable” for all discount rates lower than about 28%. This rate, at which the NPV is equal to zero, is called the project’s internal rate of return (or IRR). Another name for the IRR is expected return because an investor who pays the initial cost of the project and receives the forecasted operating cash flows can expect to earn an annual return equal to the IRR. The IRR is the second capital budgeting technique covered in this chapter. Its decision rule is straightforward: If a project’s expected return is greater than its required return, then the project should be pursued. In finance jargon, we say that if the IRR is greater than the hurdle rate, then the project is acceptable. The hurdle rate is simply the same required rate of return used to find the NPV. In fact, using the IRR takes exactly

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CHAPTER 7Section 7.1 Capital Budgeting Methods

the same information as using the NPV for capital budgeting. For the scooter project, IRR would signal that the project is worth pursuing because its IRR of 28% is greater than the hurdle rate of 12%.

To find the IRR, one must solve for the discount rate that results in an NPV equal to zero. Unfor- tunately, solving for an IRR is frequently a trial-and-error calculation unless you have a financial calculator or a financial program built into your computer spreadsheet. If we were finding the IRR of the scooter project after having done the NPV analysis, we would know that the IRR must be above 12% per year because using the 12% discount rate yielded a positive NPV. Our next step would be to take a guess at a higher rate (we need a higher rate in order to get the NPV to equal zero). Suppose we tried 16% per year, which would result in calculating the PVs of the eight cash flows using a 4% quarterly rate. Then

NPV 5 $600/(1.04)1 1 $950/(1.04)2 1 $1,200/(1.04)3 1 $400/(1.04)4 1 $850/(1.04)5 1 $1,500/(1.04)6 1 $1,500/(1.04)7 1 $2,150/(1.04)8 – $6,500

5 $958.94

Using 16% still produces a positive NPV, so we need a much higher discount rate to bring down the PV of the cash flows even further. If we try 24% per year, that is 6% per quarter. The result is $275.05, so we’ll try a higher discount rate—32% annually or 8% per quarter. Using 32% causes us to overshoot our goal because the NPV is now negative, 2$322.80. The IRR, therefore, must be between 6% and 8% quarterly, or between 24% and 32% per year. The exact answer is 6.89% quarterly, an annual rate of 27.55%. The scooter project has an IRR (i.e., expected return) of approximately 28%, which is well above the hurdle rate (i.e., required return) of 12%. Consequently, Campus Pizza’s NPV-based decision to begin pizza deliveries is confirmed by the IRR.

It is valuable for you to understand the mechanics of finding the IRR through trial and error, but spreadsheets provide a simpler and faster way to estimate the IRR. In Excel™ we would use the IRR function, which has the form


The values are the cash flows from first to last, (i.e., time 0 to time N). The guess is exactly that, a guess of what you think the answer might be. The Excel™ function uses a trial- and-error search procedure, so including a guess helps the function search. It also helps the function distinguish between multiple possible solutions, which will be discussed in the section titled Problems With the IRR.

Figure 7.3 shows a section of the spreadsheet for the Campus Scooter project. The cash flows are in row 3. The answers are displayed in the spreadsheet, but the annotations show the function inputs and the translation from the quarterly rate (6.89%) to the annual rate (30.53%).

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CHAPTER 7Section 7.1 Capital Budgeting Methods

Figure 7.3: Spreadsheet calculation of IRR

The Excel™ IRR function only works if the values have at least one change of sign (i.e., from negative to positive). Since we usually have a large outflow for an initial investment, we sign that amount negative, and the operating and terminal cash flows naturally have positive signs. If the project loses money at some point in the future, then there might be more than one sign change, but there has to be at least one.

Problems with the IRR The relationship between NPV and IRR is such that they almost always give the manager the same, and the correct, capital budgeting decision. However, there are three problems with the IRR that can sometimes cause it to be misleading. These problems are 1. the rein- vestment assumption, 2. the multiple roots problem, and 3. mutually exclusive projects.

We begin with the reinvestment assumption. The IRR mathematically “assumes” that cash flows can be reinvested at the project’s IRR. For the scooter example, in order for Campus Pizza to actually realize the expected 30% return over the 2 years of the project, Campus Pizza’s owners must find another investment with a 30% annual return in which to place the quarterly cash flows that are generated by the scooter over its 2-year life. This may not be possible, particularly for high rates of return like this one. Thus, expecting to earn the IRR may actually be unrealistic. In this sense, the IRR overstates the expected return of a project.

Here is a very simple example illustrating the reinvestment problem. Suppose there is a project that costs $100. Its forecasted cash flows are $50 in year 1 and $150 in year 2. Given these estimates, this project has an IRR of 50%. Now, if we assume that the $50 cash flow received by the investor is deposited in an account bearing a 10% interest rate (it is rein- vested at 10% because there is no project available with a 50% promised return, which is not surprising!), then the investor’s ending wealth after 2 years is $205. This equals the $150 ending cash inflow, plus the year 1 cash flow of $50, plus the $5 interest earned in



Cash Flow









Year 1

Quarter 1




Year 1

Quarter 3


Year 1

Quarter 2


Year 1

Quarter 4


Year 2

Quarter 1


Year 2

Quarter 2


Year 2

Quarter 3


Year 2

Quarter 4



–$6,500 $600 $950

6.89% 30.53%

IRR (Quarterly) IRR (Annual)

$1,200 $400 $850 $1,500 $1,500 $2,150

0 1 2 3 4 5 6 7 8

The formula input is:


This is the quarterly rate

because we used quarterly

cash flows

Annualizing the quarterly return

using the formula

=(1+0.0689)^4 – 1

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CHAPTER 7Section 7.1 Capital Budgeting Methods

year 2 on CF 1 , yielding the total ending value of $205. This future value is actually equal

to an annual return of 43.2% given the initial investment of $100, which is certainly below the 50% IRR.

To solve the reinvestment assumption problem, another capital budgeting technique has been developed called the modified internal rate of return, or the MIRR. We have chosen not to include it in this chapter because the NPV technique will give the manager the same signal for a project as the MIRR’s signal. That is because the NPV does not suffer from a faulty reinvestment assumption. The NPV implicitly assumes that cash flows are rein- vested at investors’ required return, which is a much more achievable return.

The second weakness of the IRR is that there is a chance that there will be more than one discount rate that makes the project’s NPV equal to zero. In other words, there may not be a unique solution for finding the IRR. This is the multiple roots problem. It occurs when the cash flows of a project change signs during the life of the investment. For example, suppose that New York City is considering the purchase of a tugboat to use in its harbor, planning to earn profits from charging fees to vessels that arrive in the port. There will likely be some years during the tugboat’s life when its profitability is interrupted and the operating cash flows are forecasted to be negative. This occurs because the tugboat may need to be put into dry-dock perhaps every 5 years while it is repainted, the engines are overhauled, and so on. In this case, the initial cash flow will be negative when the boat is purchased, followed by several years of positive cash flows, then interrupted by a year with negative cash flow for maintenance, then returning to profitability, and so on. In this circumstance, there may be several interest rates that satisfy the IRR criteria. Luckily, man- agers need not be overly concerned with this problem because the NPV will always give a correct answer even with negative intermittent cash flows. These negative cash flows are simply discounted to their PVs and combined with the other positive operating cash flows to get an estimate of the project’s NPV.

The final IRR challenge is when one is analyzing mutually exclusive projects or in cases of capital rationing. In both of these cases, managers cannot pursue all projects that might satisfy the capital budgeting criteria. For mutually exclusive projects, accepting one proj- ect automatically eliminates a second project—so the manager must choose between proj- ects. An example would be to choose between using some vacant land for a Luxury Hotel project or for a refuse landfill project. It could be that both of these projects have positive NPVs and both also have IRRs above the hurdle rates established for both investments (see Table 7.3). What is the correct approach, NPV or IRR, to choosing between them?

Table 7.3: Evaluating mutually exclusive projects


Luxury hotel (hurdle rate 5 12%)

15% $6 million

Landfill (hurdle rate 5 9%) 20% $2 million

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CHAPTER 7Section 7.1 Capital Budgeting Methods

Both projects have IRRs that clear their hurdle rates, but the landfill has a much higher return at 20% compared to the hotel’s 15%. Their NPVs signal a different project prefer- ence. Which signal is correct? To answer that, ask yourself, “Would I want to measure my investment success by my rate of return or by the amount of wealth that I accumulate?” Hopefully the answer is clear—it’s wealth, not returns, that allow us to buy things, so it’s wealth that should be the basis of our choice.

Demonstration Problem 7.2: Evaluating Mutually Exclusive Projects

The Glacier Co. has two mutually exclusive projects under consideration. The required investment for each is $28,000. The required return on each is 7%. The cash flows are as follows:

Year Project A Project B

1 $3,000 $16,000

2 $6,000 $11,000

3 $10,000 $8,000

4 $14,000 $6,000

5 $18,000 $2,000

IRR 18.0% 22.8%

a. Calculate the NPV for each project. b. Which project would be selected using the NPV criterion? Which would be selected using

IRR? c. Why do NPV and IRR select different projects?


a. The following table shows the calculation of NPV for each project. NPV is the sum of the dis- counted cash flows.

Project A 0 1 2 3 4 5

Cash flow $3,000 $6,000 $10,000 $14,000 $18,000

Discounted CF $(28,000) $2,804 $5,241 $8,163 $10,681 $12,834

NPV $11,722

IRR 18.0%

Project B 0 1 2 3 4 5

Cash flow $16,000 $11,000 $8,000 $6,000 $2,000

Discounted CF $(28,000) $2,804 $5,241 $8,163 $10,681 $12,834

NPV $9,094

IRR 22.8%


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CHAPTER 7Section 7.1 Capital Budgeting Methods

Demonstration Problem 7.2: Evaluating Mutually Exclusive Projects (continued)

b. Project A has the greatest NPV, and project B has the highest IRR. NPV would select project A, and IRR would select project B.

c. IRR and NPV select different projects because the timing of project cash flows is very differ- ent. Project A’s cash flows occur for the most part in the later years, while most of project B’s cash flows occur early. Because of differences in the reinvestment rates between IRR and NPV, IRR tends to favor projects whose cash flows occur more in the early years, and NPV tends to favor those whose cash flows occur more in the later years.

If it is still not clear why NPV is the correct approach, consider a simple choice. You get to buy one of two envelopes that you will immediately open. Luckily, you know what is inside each of the envelopes. The first envelope costs $10 and has $15 inside it. If you buy this envelope, your instant return (the IRR) is 50%, and the NPV is $5 because your wealth will increase by that amount. The second envelope costs $100 and it has $120 inside it. In this case the return is only 20%, but your wealth will increase by $20. Clearly, if you can only buy only one of the two envelopes, it should be the one with the higher NPV.

Capital Rationing Capital rationing occurs when firms can raise only limited capital. Consequently these companies may not have enough funds available to invest in all the promising projects that have been identified. In this case a limited portfolio of investment projects must be identified, eliminating some investments that might otherwise be pursued. Again, the proper approach in such a circumstance is to invest in the combination of projects that provides the highest NPV (rather than the highest IRR). Table 7.4 shows the initial cost, the IRR, and the NPV of a series of projects. If the firm faces a limited capital budget, you can see what the optimal mix of projects would be depending on the amount of capital the company has available. With limited capital, the optimal capital budget is the set of investments that produce the maximum NPV.

Table 7.4: The optimal capital budgets under capital rationing

Project Cost IRR NPV $100,000 Budget

$200,000 Budget

$300,000 Budget

A $75,000 18% $14,000 A

B $30,000 45% $10,000 B

C $100,000 30% $27,000 C C C

D $130,000 17% $22,000 D

It is interesting that project A is part of the optimal capital budget when $200,000 is avail- able to invest, but when $300,000 is available, then project A drops out of the accepted set of projects for investment. It is also interesting that the highest IRR project (project B) is

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CHAPTER 7Section 7.2 Expanded Examples with Solutions

not selected in the first two budgets! Of course, if some of the projects were divisible, then the budget would change dramatically (e.g., if you could invest in half of project A for $37,500 for example).

Two things must be kept in mind in dealing with the net present value: First, NPV in theory does not suffer from any of the IRR’s weaknesses—it should always produce the correct go-versus-no-go decision for a project or set of projects. Second, even using the correct technique does not guarantee that a project will be successful; it is just the best guidance available. Investment projects are still risky, and even the most promising proj- ects sometimes fall short of expectations or even fail.

Finance Costs versus Cash Flow

In Chapter 6 we said that finance costs are not relevant cash flows to include in an NPV analysis. We will now use the NPV model and a simple example to show why this is the case. We will show that if a project’s NPV is zero, computing cash flows as we defined in Chapter 6, then all contributors of capital get their required rates of return. Suppose an investment costs $1,000 today and will return, on an after-tax incremental basis, $1,120 in exactly 1 year. Suppose further that the investment is financed with $500 of debt, which has an interest rate of 8%, and $500 of equity with a required rate of return of 16%. For now we will ignore the tax-deductibility of interest payments. The discount rate for this project is the weighted average of the two costs of capital (Chapter 10 develops this idea in more detail), or 50% 3 8% 1 50% 3 16% 5 12%. The NPV is

NPV 5 1,120 1.12

2 1,000 5 0

At the end of the year the company has $1,120. The lenders get their $500 back plus 8% interest on this investment ($40) for a total payment of $540. The equity investors who required 16% receive the residual amount, or $580 5 $1,120 2 $540. The $580 payment is a 16% return on their $500 investment, which is exactly what they required. So both groups have been paid exactly what they required, no more and no less. Had we taken the interest payments out of the cash flow, there would not have been enough money to pay the capital providers their required rate of return. The discounting process builds financ- ing costs into the NPV calculation, so including financing costs in the cash flows would be double-counting them—once in the discount rate and a second time in the cash flows.

7.2 Expanded Examples with Solutions

Example 1: The Gymerator

A small alternative energy company has just completed the R&D on a new product, the Gymerator. The Gymerator is a small flat-plane turbine that can be attached to almost any rotating device to generate electricity. The R&D efforts focused on equipment found in most gymnasiums and spas—exercise bikes, ergometers, and so on. By attaching the Gymerator to an exercise device, electricity is generate as people exercise. The electricity

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CHAPTER 7Section 7.2 Expanded Examples with Solutions

is fed from multiple exercise machines into an inverter that then feeds the electricity into either the building’s power system (reducing the electricity needed from outside sources) or to the grid, earning the gymnasium income from the local power supplier, which uses net metering, allowing an electricity meter to run backward when the customer is produc- ing electricity.

The R&D, which cost $250,000, is complete and paid for. Machinery to manufacture the Gymerator will cost $1 million. It will be depreciated over 5 years using the straight-line method to zero book value ($200,000 per year). A working capital investment of $300,000 will be needed at the beginning of the project. An additional working capital investment of $100,000 will be required at the end of year 2 to support anticipated sales growth. All working capital will be recovered at the end of the project.

Earnings before depreciation and taxes (EBDT) for the 6-year life of the project are shown in Table 7.5.

Table 7.5: Earnings ($) before depreciation and taxes for the Gymerator project

Year 1 2 3 4 5 6

EBDT 325,000 450,000 500,000 500,000 425,000 350,000

At the end of the project the machinery will be scrapped for $200,000, which is not included in Table 7.5. The company’s tax rate is 30%, and the appropriate discount rate for this proj- ect is 12%.

The Gymerator Solution

First, we find the incremental after-tax cash flows for each year of the project’s life. These are the cash flows that you will use in your NPV analysis, so be sure include all inflows and outflows.

The initial investment includes the $1 million for equipment and the $300,000 working capital investment. The R&D expense of $250,000 is not included because it has been paid, so the money is spent whether or not the project is accepted. There are no tax conse- quences for the equipment purchase or working capital investment since they are balance sheet items and have no impact on the income statement where taxes are determined. The initial investment is an outlay of $1.3 million.

In year 1 the EBDT is $325,000. Using the tax shield method from Chapter 6, we find that this is $227,500 5 $325,000 3 (1 2 0.30) after tax. The depreciation tax shield is the tax rate times the depreciation expense, so $60,000 5 $200,000 3 0.30. The total incremental after- tax cash flow for year 1 is $287,500 5 $227,500 1 $60,000.

The year 2 cash flows are similar to year 1, but there is an additional working capital investment of $100,000. The after-tax EBDT is $450,000 3 (1 2 0.30) 5 $315,000. The depreciation tax shield is as in year 1, $60,000. The incremental after-tax cash flow net of the working capital investment is $315,000 1 $60,000 2 $100,000 5 $275,000.

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CHAPTER 7Section 7.2 Expanded Examples with Solutions

For years 3 through 5 the cash flows are the after-tax EBDT plus the depreciation tax shield. The incremental after-tax cash flows for years 3 through 5 are $410,000 for years 3 and 4 and $357,500 for year 5.

At the end of year 6 we assume that the project ends. There are operating cash flows ($350,000 before tax), the salvage value of the equipment, and recovery of working capi- tal to consider. There is no depreciation expense because the equipment was completely depreciated to zero book value at the end of year 5. After-tax EBDT is $245,000, and the total working capital recovered is $400,000. There is no tax treatment for working capital investments. The sale of the equipment has tax consequences. Since the sales price is greater than the book value but less than the original cost, some depreciation must be recaptured. The company overdepreciated the equipment by $200,000 (the sales price of the used equipment), so it must pay taxes on that amount. The tax on $200,000 at 30% is $60,000. The after-tax income from selling the used equipment is $140,000 5 $200,000 2 $60,000. Combining all the cash flows, the incremental after-tax cash flow in year 6 is $785,000 5 $245,000 1 $400,000 1 $140,000.

We compute the NPV of this project by discount the incremental after-tax cash flows by 12% as shown in Table 7.6.

Table 7.6: NPV analysis of the Gymerator project

Year 0 1 2 3 4 5 6

Incremental after-tax cash flows

(1,300,000) 287,500 275,000 410,000 410,000 357,500 785,000

PV at 12% (1,300,000) 256,696 219,228 291,829 260,562 202,855 397,705

NPV (sum PVs) 328,877

IRR (using IRR function)


Payback period 3.80 years

What should the company do? Since the NPV is positive and the IRR is greater than 12%, the Gymerator project will increase shareholder wealth, so it should be accepted. The payback period is 3.80 years, but this doesn’t help us make a decision, as there is no accep- tance criterion that links payback and shareholder wealth.

Example 2: Lathe Replacement Proposal

This example will test your understanding of the incremental concept related to cash flows. Try to develop the cash flows yourself before looking at our solution, remembering as you do that the relevant cash flows are the change that occurs if the project is accepted and implemented.

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CHAPTER 7Section 7.2 Expanded Examples with Solutions

Western Fabricating is considering replacing one of its existing computer-assisted lathes with a newer model. The old machine was purchased 3 years ago for $42,000. It is being depreciated over 6 years to zero salvage value using the straight-line method. Its current book value is $21,000. It can be sold today for $30,000.

The new lathe will cost $80,000 and will be depreciated over 5 years to zero salvage value using the straight-line method. It is estimated to have a 10-year life, after which it can be sold for $10,000. Because it can handle bigger projects and complete them more quickly, it will increase EBDT (earnings before depreciation and taxes) by $8,000 per year. The com- pany’s tax rate is 30%, and the discount rate for projects of this type is 12%.

Lathe Replacement Solution

We will go through the years and identify the incremental after-tax cash flows, then com- pute the NPV, IRR, and payback period for this proposal.

The initial investment (CF 0 ) has three components. The company will sell the old lathe

($30,000 inflow). There are tax consequences on this sale because of depreciation recap- ture. Then the new lathe is purchased ($80,000 outflow). The company gets a check for $30,000 for the old lathe, but because the sales price exceeds the current book value ($21,000), the company must pay tax on the excessive depreciation expense. The tax is 30% of $9,000 5 $30,000 2 $21,000, or $2,700. We can now compute the initial investment as the outlay for the new lathe and the income from the sale of the old lathe less taxes on the sale. This totals

Initial Investment 5 2$52,700 5 (2$80,000 1 $30,000 2 $2,700)

Years 1 through 3 have identical cash flows. One component is the net operating income or EBDT of $8,000 per year, which has an after-tax value of $5,600 5 ($8,000 3 (1 2 Tax Rate)). Before purchasing the new lathe, the company had a depreciation expense of $7,000 per year, and this would have continued for 3 more years. If the new lathe is purchased, then the depreciation expense will be $16,000 per year for the next 5 years. Using the with-and- without principle, we credit the new lathe with an additional $9,000 (5 $16,000 2 $7,000) of depreciation expense for the next 3 years. The incremental increase in depreciation expense, and thus its value as a tax shield, is $9,000. Not until the old machine would have been fully depreciated (3 years from today) will the new lathe be assigned the full $16,000 of depreciation expense.

Combining the after-tax value of the EBDT and the incremental depreciation tax shields ($9,000 3 0.30 5 $2,700) gives us incremental after-tax cash flows for years 1 through 3 of $8,300 (5 $5,600 1 $2,700).

In years 4 and 5 there is no old depreciation expense to consider, so the cash flow com- ponents are the EBDT and the full $16,000 of depreciation expense on the new lathe. The incremental after-tax cash flows for years 4 and 5 are $10,400 (5 $5,600 1 $4,800). The $4,800 is the value of the $16,000 depreciation tax shield at a tax rate of 30%.

In years 6 through 9 the only cash flow is the EBDT of $8,000, which has an after-tax value of $5,600. The new lathe has a book value of zero, so there is no additional depreciation expense to consider.

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CHAPTER 7Summary

The terminal cash flows in year 10 include EBDT for the year and the cash flow from the sale or scrapping of the new lathe. We know that the EBDT has an after-tax value of $5,600. The new lathe, now 10 years old and fully depreciated, will be scrapped for $10,000. This means that there is depreciation recapture on $10,000 at 30%, for a $3,000 tax bill. Com- bining these three cash flows we arrive at a year 10 incremental after-tax cash flow of $12,600 (5 $5,600 1 $10,000 2 $3,000).

Table 7.7 summarizes the cash flows, shows the present value of the cash flows at 12%, and displays the NPV, IRR, and payback period for the investment.

Table 7.7: Cash flows and NPV, IRR, and payback period results for the new lathe project

Year Incremental after- tax cash flow

PV , 12%

0 252,700 (52,700)

1 8,300 7,411

2 8,300 6,617

3 8,300 5,908

4 10,400 6,609

5 10,400 5,901

6 5,600 2,837

7 5,600 2,533

8 5,600 2,262

9 5,600 2,019

10 12,600 4,057

NPV (6,546)

IRR 8.80%

Payback 6.25

The negative NPV means that this is not an acceptable project for Western Fabrication to pursue. The additional EBDT doesn’t offset the cost of the new lathe, so the company should keep the old lathe for a few more years.


The firm’s objective of creating wealth for its owners requires management to evalu-ate projects based on whether they can be expected to add value to the company. This process is known as capital budgeting, and it was the subject of this chapter. Three techniques used for capital budgeting were introduced. The first method for allocat- ing capital is the payback period. Despite its many shortcomings that were discussed in the chapter, payback can be useful because it is quick and easy to do. The second, net pres- ent value or NPV, directly estimates the value added by a project by comparing the pres- ent value of the project’s forecasted cash flows to its initial cost. Projects with cash flows valued above their cost, positive NPV projects, should be acceptable to the firm because

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CHAPTER 7Web Resources

they satisfy the value-added criteria. The last technique, the internal rate of return or IRR, is an estimate of the rate of return that a project is expected to generate over its life. If the IRR exceeds the required return for the project (also known as the project’s hurdle rate), then the project should be acceptable to the firm. Generally, projects with a positive net present value also have internal rates of return that are greater than the hurdle rate. The chapter discusses several weaknesses of IRR that are not shared by NPV. These IRR prob- lems include the faulty reinvestment assumption, the evaluation of mutually exclusive projects, and capital rationing when access to capital is limited.

Key Terms

capital rationing A determinant of the capital budget in which the required rates of return on high-risk investments are raised.

hurdle rate A required rate of return, or reference point, against which to compare a project’s internal rate of return.

incremental cash flows The change in cor- porate cash flows attributable to a project.

mutually exclusive projects Investment projects that are related such that only one can be taken.

net present value (NPV) The present value of future cash flows minus the initial investment. NPV is the present value of all cash flows connected to the investment.

optimal capital budget The set of projects that maximizes the value of the firm.

payback period A measure of how many years it takes to recoup the initial invest- ment in a project.

required return The minimum return investors must expect in order to be inter- ested in investing in an asset.

Web Resources

For a description of the long (extremely long) time required to market for prescription drugs, see

An excellent description and discussion of cost–benefit analysis can be found at

Investopedia has a nice prep page for the CFA (Chartered Financial Analyst) exam that discusses the weaknesses of both NPV and IRR. If you are thinking about becoming a financial analyst, a good site to review is -cash-flow-npv-irr.asp#axzz24sa9QXgc.

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CHAPTER 7Critical Thinking and Discussion Questions

This website contains an interesting take on the payback period: -analysis.aspx.

See if you can find the flaws in logic.

We have been using the Gymerator example for several years, but someone is actually trying it. See,8599,2032281,00.html and -sweat-into-electricity.

Critical Thinking and Discussion Questions

1. Some companies spend millions of dollars each year on employee training. The cost of this training is treated as an accounting expense, but it may really be an investment. Why might training be an investment?

2. Many years ago, W. R. Grace was a steamship company operating passenger liners between the West Coast and Hawaii. Now it is a specialty chemical and energy products manufacturer. Why do you think W. R. Grace made such an abrupt change in its line of business? If you had been a shareholder of W. R. Grace, would you have welcomed the change?

3. What is the minimum rate of return that an investment should provide in order to be acceptable to the company’s investors? Why?

4. Consider the following projects for a large auto manufacturer that is considering entering the minivan market. Eliminate complementary projects by combining projects and identify mutually exclusive projects. Are there any independent projects in this list? Explain. a. a minivan assembly plant in the United States b. ships to transport minivans into the United States c. a minivan assembly plant in Korea d. a plant to produce minivan parts in Taiwan. Alternatively, parts may be pur-

chased from independent suppliers. e. a plant to produce minivan parts in Malaysia f. a distribution network for minivans in the United States—rail cars, ware-

houses, etc. 5. Explain the three main issues with using internal rate of return (IRR) for project


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CHAPTER 7Practice Problems

Practice Problems

1. We have the following information for a project:

Discount rate NPV

8% $1,700

9% $1,100

10% $300

11% 2$150

Based on this information, what is the approximate internal rate of return for this project?

2. We have the following information for these mutually exclusive projects. All have equal required rates of return.


A $3,800 11%

B $1,250 14%

C 2$560 7%

D $2,500 12%

E $0 9%

a. What is the required rate of return of these projects? b. Rank acceptable projects using NPV. c. Rank acceptable projects using IRR.

3. Suppose a project has cash flows equal to $8,500, $2,500, $1,000, and $8,000 in years 1, 2, 3 and 4, respectively. If the project costs $12,000, what is its NPV assuming a 15% required rate of return?

4. Using the project from Problem 3, what is its IRR? 5. Using the project from Problems 3 and 4, what is the project’s payback?

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