The three attributes of npv are that it

Net Present Value and Other Investment Rules

When a company is deciding whether to invest in a new project, large sums of money can be at stake. For example, in October 2014, Badlands NGL announced plans to build a $4 billion polyethylene plant in North Dakota, which was the largest private-sector investment made in that state’s history. Earlier in 2014, Samsung Electronics announced plans to build a $14.7 billion chip facility in South Korea. The chip plant was expected to employ 150,000 workers when it was completed. But neither of these announcements came close to the Artic LNG project, which was being developed by ExxonMobil, ConocoPhillips, BP, pipeline company TransCanada, and the state of Alaska. The Artic LNG project would build a pipeline from Alaska’s North Slope to allow natural gas to be sent from the area.

The cost of the pipeline and plant to clean the gas of impurities was expected to be $45 to $65 billion. Decisions such as these, with price tags in the billions, are obviously major undertakings, and the risks and rewards must be carefully weighed. In this chapter, we discuss the basic tools used in making such decisions.

In Chapter 1, we show that increasing the value of a company’s stock is the goal of financial management. Thus, what we need to know is how to tell whether a particular investment will achieve that purpose or not. This chapter considers a variety of techniques financial analysts routinely use. More importantly, it shows how many of these techniques can be misleading, and it explains why the net present value approach is the right one.

5.1 Why Use Net Present Value?

Find out more about capital budgeting for small businesses at www.missouribusiness.net.

This chapter, as well as the next two, focuses on capital budgeting, the decision-making process for accepting or rejecting projects. This chapter develops the basic capital budgeting methods, leaving much of the practical application to subsequent chapters. But we don’t have to develop these methods from scratch. In Chapter 4, we pointed out that a dollar received in the future is worth less than a dollar received today. The reason, of course, is that today’s dollar can be reinvested, yielding a greater amount in the future. And we showed in Chapter 4 that the exact worth of a dollar to be received in the future is its present value. Furthermore, Section 4.1 suggested calculating the net present value of any project. That is, the section suggested calculating the difference between the sum of the present values of the project’s future cash flows and the initial cost of the project.

The net present value (NPV) method is the first one to be considered in this chapter. We begin by reviewing the approach with a simple example. Then, we ask why the method leads to good decisions.

PAGE 136EXAMPLE

5.1

Net Present Value The Alpha Corporation is considering investing in a riskless project costing $100. The project receives $107 in one year and has no other cash flows. The riskless discount rate on comparable riskless investments is 2 percent.

The NPV of the project can easily be calculated as:

(5.1)

From Chapter 4, we know that the project should be accepted because its NPV is positive. This is true because the project generates $107 of future cash flows from a $100 investment whereas comparable investments only generate $102.

The basic investment rule can be generalized to:

Accept a project if the NPV is greater than zero.

Reject a project if the NPV is less than zero.

We refer to this as the NPV rule.

Why does the NPV rule lead to good decisions? Consider the following two strategies available to the managers of Alpha Corporation:

1. Use $100 of corporate cash to invest in the project. The $107 will be paid as a dividend in one year.

2. Forgo the project and pay the $100 of corporate cash to stockholders as a dividend today.

If Strategy 2 is employed, the stockholder might deposit the cash dividend in a bank for one year. With an interest rate of 2 percent, Strategy 2 would produce cash of $102 (=$100 X 1.02) at the end of the year. The stockholder would prefer Strategy 1 because Strategy 2 produces less than $107 at the end of the year.

Our basic point is:

Accepting positive NPV projects benefits the stockholders.

How do we interpret the exact NPV of $4.90? This is the increase in the value of the firm from the project. For example, imagine that the firm today has productive assets worth $V and has $100 of cash. If the firm forgoes the project, the value of the firm today would simply be:

$V + $100

If the firm accepts the project, the firm will receive $107 in one year but will have no cash today. Thus, the firm’s value today would be:

The difference between these equations is just $4.90, the net present value of Equation 5.1. Thus:

The value of the firm rises by the NPV of the project.

Note that the value of the firm is merely the sum of the values of the different projects, divisions, or other entities within the firm. This property, called value additivity, is quite important. It implies that the contribution of any project to a firm’s value is simply the Page 137NPV of the project. As we will see later, alternative methods discussed in this chapter do not generally have this nice property.

The NPV rule uses the correct discount rate.

One detail remains. We assumed that the project was riskless, a rather implausible assumption. Future cash flows of real-world projects are invariably risky. In other words, cash flows can only be estimated, rather than known. Imagine that the managers of Alpha expect the cash flow of the project to be $107 next year. That is, the cash flow could be higher, say $117, or lower, say $97. With this slight change, the project is risky. Suppose the project is about as risky as the stock market as a whole, where the expected return this year is perhaps 10 percent. Then 10 percent becomes the discount rate, implying that the NPV of the project would be:

Because the NPV is negative, the project should be rejected. This makes sense: A stockholder of Alpha receiving a $100 dividend today could invest it in the stock market, expecting a 10 percent return. Why accept a project with the same risk as the market but with an expected return of only 7 percent?

SPREADSHEET APPLICATIONS

Calculating NPVs with a Spreadsheet

Spreadsheets are commonly used to calculate NPVs. Examining the use of spreadsheets in this context also allows us to issue an important warning. Consider the following:

In our spreadsheet example, notice that we have provided two answers. The first answer is wrong even though we used the spreadsheet’s NPV formula. What happened is that the “NPV” function in our spreadsheet is actually a PV function; unfortunately, one of the original spreadsheet programs many years ago got the definition wrong, and subsequent spreadsheets have copied it! Our second answer shows how to use the formula properly.

The example here illustrates the danger of blindly using calculators or computers without understanding what is going on; we shudder to think of how many capital budgeting decisions in the real world are based on incorrect use of this particular function.

Page 138Conceptually, the discount rate on a risky project is the return that one can expect to earn on a financial asset of comparable risk. This discount rate is often referred to as an opportunity cost because corporate investment in the project takes away the stockholder’s option to invest the dividend in other opportunities. Conceptually, we should look for the expected return of investments with similar risks available in the capital markets. The calculation of the discount rate is by no means impossible. We forgo the calculation in this chapter but present it in later chapters of the text.

Having shown that NPV is a sensible approach, how can we tell whether alternative methods are as good as NPV? The key to NPV is its three attributes:

1. NPV uses cash flows. Cash flows from a project can be used for other corporate purposes (such as dividend payments, other capital budgeting projects, or payments of corporate interest). By contrast, earnings are an artificial construct. Although earnings are useful to accountants, they should not be used in capital budgeting because they do not represent cash.

2. NPV uses all the cash flows of the project. Other approaches ignore cash flows beyond a particular date; beware of these approaches.

3. NPV discounts the cash flows properly. Other approaches may ignore the time value of money when handling cash flows. Beware of these approaches as well.

Calculating NPVs by hand can be tedious. A nearby Spreadsheet Applications box shows how to do it the easy way and also illustrates an important caveat calculator.

5.2 The Payback Period Method

DEFINING THE RULE

One of the most popular alternatives to NPV is payback. Here is how payback works: Consider a project with an initial investment of -$50,000. Cash flows are $30,000, $20,000, and $10,000 in the first three years, respectively. These flows are illustrated in Figure 5.1. A useful way of writing down investments like the preceding is with the notation:

(−$50,000, $30,000, $20,000, $10,000)

The minus sign in front of the $50,000 reminds us that this is a cash outflow for the investor, and the commas between the different numbers indicate that they are received—or if they are cash outflows, that they are paid out—at different times. In this example we are Page 139assuming that the cash flows occur one year apart, with the first one occurring the moment we decide to take on the investment.

Figure 5.1 Cash Flows of an Investment Project

The firm receives cash flows of $30,000 and $20,000 in the first two years, which add up to the $50,000 original investment. This means that the firm has recovered its investment within two years. In this case, two years is the payback period of the investment.

The payback period rule for making investment decisions is simple. A particular cutoff date, say two years, is selected. All investment projects that have payback periods of two years or less are accepted, and all of those that pay off in more than two years—if at all—are rejected.

PROBLEMS WITH THE PAYBACK METHOD

There are at least three problems with payback. To illustrate the first two problems, we consider the three projects in Table 5.1. All three projects have the same three-year payback period, so they should all be equally attractive—right?

Actually, they are not equally attractive, as can be seen by a comparison of different pairs of projects.

Problem 1: Timing of Cash Flows within the Payback Period Let us compare Project A with Project B. In Years 1 through 3, the cash flows of Project A rise from $20 to $50, while the cash flows of Project B fall from $50 to $20. Because the large cash flow of $50 comes earlier with Project B, its net present value must be higher. Nevertheless, we just saw that the payback periods of the two projects are identical. Thus, a problem with the payback method is that it does not consider the timing of the cash flows within the payback period. This example shows that the payback method is inferior to NPV because, as we pointed out earlier, the NPV method discounts the cash flows properly.

Problem 2: Payments after the Payback Period Now consider Projects B and C, which have identical cash flows within the payback period. However, Project C is clearly preferred because it has a cash flow of $100 in the fourth year. Thus, another problem with the payback method is that it ignores all cash flows occurring after the payback period. Because of the short-term orientation of the payback method, some valuable long-term projects are likely to be rejected. The NPV method does not have this flaw because, as we pointed out earlier, this method uses all the cash flows of the project.

Problem 3: Arbitrary Standard for Payback Period We do not need to refer to Table 5.1 when considering a third problem with the payback method. Capital markets help us estimate the discount rate used in the NPV method. The riskless rate, perhaps proxied by the yield on a U.S. Treasury instrument, would be the appropriate rate for a riskless investment; a higher rate should be used for risky projects. Later chapters of this Page 140textbook show how to use historical returns in the capital markets to estimate the discount rate for a risky project. However, there is no comparable guide for choosing the payback cutoff date, so the choice is somewhat arbitrary.

Table 5.1 Expected Cash Flows for Projects A through C ($)

Year

A

B

C

0

−$100

−$100

−$100

1

20

50

50

2

30

30

30

3

50

20

20

4

60

60

$100

Payback period (years)

3

3

3

NPV

21.5

26.3

53.6

To illustrate the payback period problems, consider Table 5.1. Suppose the expected return on comparable risky projects is 10 percent. Then we would use a discount rate of 10 percent for these projects. If so, the NPV would be $21.5, $26.3, and $53.6 for A, B, and C respectively. When using the payback period, these projects are equal to one another (i.e. they each have a payback period of 3 years). However, when considering all cash flows, B has a higher NPV than A because of the timing of cash flows within the payback period. And C has the highest NPV because of the $100 cash flow after the payback period.

MANAGERIAL PERSPECTIVE

The payback method is often used by large, sophisticated companies when making relatively small decisions. The decision to build a small warehouse, for example, or to pay for a tune-up for a truck is the sort of decision that is often made by lower-level management. Typically, a manager might reason that a tune-up would cost, say, $200, and if it saved $120 each year in reduced fuel costs, it would pay for itself in less than two years. On such a basis the decision would be made.

Although the treasurer of the company might not have made the decision in the same way, the company endorses such decision making. Why would upper management condone or even encourage such retrograde activity in its employees? One answer would be that it is easy to make decisions using payback. Multiply the tune-up decision into 50 such decisions a month, and the appeal of this simple method becomes clearer.

The payback method also has some desirable features for managerial control. Just as important as the investment decision itself is the company’s ability to evaluate the manager’s decision-making ability. Under the NPV method, a long time may pass before one decides whether a decision was correct. With the payback method we know in two years whether the manager’s assessment of the cash flows was correct.

It has also been suggested that firms with good investment opportunities but no available cash may justifiably use payback. For example, the payback method could be used by small, privately held firms with good growth prospects but limited access to the capital markets. Quick cash recovery increases the reinvestment possibilities for such firms.

Finally, practitioners often argue that standard academic criticisms of the payback method overstate any real-world problems with the method. For example, textbooks typically make fun of payback by positing a project with low cash inflows in the early years but a huge cash inflow right after the payback cutoff date. This project is likely to be rejected under the payback method, though its acceptance would, in truth, benefit the firm. Project C in our Table 5.1 is an example of such a project. Practitioners point out that the pattern of cash flows in these textbook examples is much too stylized to mirror the real world. In fact, a number of executives have told us that for the overwhelming majority of real-world projects, both payback and NPV lead to the same decision. In addition, these executives indicate that if an investment-like Project C were encountered in the real world, decision makers would almost certainly make ad hoc adjustments to the payback rule so that the project would be accepted.

Notwithstanding all of the preceding rationale, it is not surprising to discover that as the decisions grow in importance, which is to say when firms look at bigger projects, NPV becomes the order of the day. When questions of controlling and evaluating the manager become less important than making the right investment decision, payback is used less frequently. For big-ticket decisions, such as whether or not to buy a machine, build a factory, or acquire a company, the payback method is seldom used.Page 141

SUMMARY OF PAYBACK

The payback method differs from NPV and is therefore conceptually wrong. With its arbitrary cutoff date and its blindness to cash flows after that date, it can lead to some flagrantly foolish decisions if used too literally. Nevertheless, because of its simplicity, as well as its other mentioned advantages, companies often use it as a screen for making the myriad of minor investment decisions they continually face.

Although this means that you should be wary of trying to change approaches such as the payback method when you encounter them in companies, you should probably be careful not to accept the sloppy financial thinking they represent. After this course, you would do your company a disservice if you used payback instead of NPV when you had a choice.

5.3 The Discounted Payback Period Method

Aware of the pitfalls of payback, some decision makers use a variant called the discounted payback period method . Under this approach, we first discount the cash flows. Then we ask how long it takes for the discounted cash flows to equal the initial investment.

For example, suppose that the discount rate is 10 percent and the cash flows on a project are given by:

(−$100, $50, $50, $20)

This investment has a payback period of two years because the investment is paid back in that time.

To compute the project’s discounted payback period, we first discount each of the cash flows at the 10 percent rate. These discounted cash flows are:

[−$100, $50/1.1, $50/(1.1)2, $20/(1.1)3] = (−$100, $45.45, $41.32, $15.03)

The discounted payback period of the original investment is simply the payback period for these discounted cash flows. The payback period for the discounted cash flows is slightly less than three years because the discounted cash flows over the three years are $101.80 (=$45.45 + 41.32 1 15.03). As long as the cash flows and discount rate are positive, the discounted payback period will never be smaller than the payback period because discounting reduces the value of the cash flows.

At first glance discounted payback may seem like an attractive alternative, but on closer inspection we see that it has some of the same major flaws as payback. Like payback, discounted payback first requires us to choose an arbitrary cutoff period, and then it ignores all cash flows after that date.

If we have already gone to the trouble of discounting the cash flows, we might just as well add up all the discounted cash flows and use NPV to make the decision. Although discounted payback looks a bit like NPV, it is just a poor compromise between the payback method and NPV.

5.4 The Internal Rate of Return

Now we come to the most important alternative to the NPV method: The internal rate of return, universally known as the IRR. The IRR is about as close as you can get to the NPV without actually being the NPV. The basic rationale behind the IRR method is that it provides a single number summarizing the merits of a project. That number does not depend on the interest rate prevailing in the capital market. That is why it is called the internal rate Page 142of return; the number is internal or intrinsic to the project and does not depend on anything except the cash flows of the project.

Figure 5.2 Cash Flows for a Simple Project

For example, consider the simple project (−$100, $110) in Figure 5.2. For a given rate, the net present value of this project can be described as:

where R is the discount rate. What must the discount rate be to make the NPV of the project equal to zero?

We begin by using an arbitrary discount rate of .08, which yields:

Because the NPV in this equation is positive, we now try a higher discount rate, such as .12. This yields:

Because the NPV in this equation is negative, we try lowering the discount rate to .10. This yields:

This trial-and-error procedure tells us that the NPV of the project is zero when R equals 10 percent.1 Thus, we say that 10 percent is the project’s internal rate of return (IRR). In general, the IRR is the rate that causes the NPV of the project to be zero. The implication of this exercise is very simple. The firm should be equally willing to accept or reject the project if the discount rate is 10 percent. The firm should accept the project if the discount rate is below 10 percent. The firm should reject the project if the discount rate is above 10 percent.

The general investment rule is clear:

Accept the project if the IRR is greater than the discount rate. Reject the project if the IRR is less than the discount rate.

Page 0143FIGURE 5.3 Cash Flows for a More Complex Project

We refer to this as the basic IRR rule. Now we can try the more complicated example (−$200, $100, $100, $100) in Figure 5.3.

As we did previously, let’s use trial and error to calculate the internal rate of return. We try 20 percent and 30 percent, yielding the following:

Discount Rate

NPV

20%

$10.65

30

−18.39

After much more trial and error, we find that the NPV of the project is zero when the discount rate is 23.38 percent. Thus, the IRR is 23.38 percent. With a 20 percent discount rate, the NPV is positive and we would accept it. However, if the discount rate were 30 percent, we would reject it.

Algebraically, IRR is the unknown in the following equation:2

Figure 5.4 illustrates what the IRR of a project means. The figure plots the NPV as a function of the discount rate. The curve crosses the horizontal axis at the IRR of 23.38 percent because this is where the NPV equals zero.

It should also be clear that the NPV is positive for discount rates below the IRR and negative for discount rates above the IRR. If we accept projects like this one when the discount rate is less than the IRR, we will be accepting positive NPV projects. Thus, the IRR rule coincides exactly with the NPV rule.

If this were all there were to it, the IRR rule would always coincide with the NPV rule. But the world of finance is not so kind. Unfortunately, the IRR rule and the NPV rule are consistent with each other only for examples like the one just discussed. Several problems with the IRR approach occur in more complicated situations, a topic to be examined in the next section.

The IRR in the previous example was computed through trial and error. This laborious process can be averted through spreadsheets. A nearby Spreadsheet Applications box shows how.

Page 0144Figure 5.4 Net Present Value (NPV) and Discount Rates for a More Complex Project

SPREADSHEET APPLICATIONS

Calculating IRRs with a Spreadsheet

Because IRRs are so tedious to calculate by hand, financial calculators and, especially, spreadsheets are generally used. The procedures used by various financial calculators are too different for us to illustrate here, so we will focus on using a spreadsheet. As the following example illustrates, using a spreadsheet is very easy.

Page 0145

5.5 Problems with the IRR Approach

DEFINITION OF INDEPENDENT AND MUTUALLY EXCLUSIVE PROJECTS

An independent project is one whose acceptance or rejection is independent of the acceptance or rejection of other projects. For example, imagine that McDonald’s is considering putting a hamburger outlet on a remote island. Acceptance or rejection of this unit is likely to be unrelated to the acceptance or rejection of any other restaurant in its system. The remoteness of the outlet in question ensures that it will not pull sales away from other outlets.

Now consider the other extreme, mutually exclusive investments . What does it mean for two projects, A and B, to be mutually exclusive? You can accept A or you can accept B or you can reject both of them, but you cannot accept both of them. For example, A might be a decision to build an apartment house on a corner lot that you own, and B might be a decision to build a movie theater on the same lot.

We now present two general problems with the IRR approach that affect both independent and mutually exclusive projects. Then we deal with two problems affecting mutually exclusive projects only.

TWO GENERAL PROBLEMS AFFECTING BOTH INDEPENDENT AND MUTUALLY EXCLUSIVE PROJECTS

We begin our discussion with Project A, which has the following cash flows:

(−$100, $130)

The IRR for Project A is 30 percent. Table 5.2 provides other relevant information about the project. The relationship between NPV and the discount rate is shown for this project in Figure 5.5. As you can see, the NPV declines as the discount rate rises.

Problem 1: Investing or Financing? Now consider Project B, with cash flows of:

($100, −$130)

These cash flows are exactly the reverse of the flows for Project A. In Project B, the firm receives funds first and then pays out funds later. While unusual, projects of this type do exist. For example, consider a corporation conducting a seminar where the participants pay in advance. Because large expenses are frequently incurred at the seminar date, cash inflows precede cash outflows.

Table 5.2 The Internal Rate of Return and Net Present Value

Project A

Project B

Project C

Dates:

0

1

2

0

1

2

0

1

2

Cash flows

−$100

$130

$100

−$130

−$100

$230

−$132

IRR

30%

30%

10% and 20%

NPV @10%

$ 18.2

−$ 18.2

0

Accept if market rate

>30%

<30%

>10% but <20%

Financing or investing

Investing

Financing

Mixture

Page 0146FIGURE 5.5 Net Present Value and Discount Rates for Projects A, B, and C

Consider our trial-and-error method to calculate IRR:

As with Project A, the internal rate of return is 30 percent. However, notice that the net present value is negative when the discount rate is below 30 percent. Conversely, the net present value is positive when the discount rate is above 30 percent. The decision rule is exactly the opposite of our previous result. For this type of project, the following rule applies:

Accept the project when the IRR is less than the discount rate. Reject the project when the IRR is greater than the discount rate.

This unusual decision rule follows from the graph of Project B in Figure 5.5. The curve is upward sloping, implying that NPV is positively related to the discount rate.

The graph makes intuitive sense. Suppose the firm wants to obtain $100 immediately. It can either (1) accept Project B or (2) borrow $100 from a bank. Thus, the project is actually a substitute for borrowing. In fact, because the IRR is 30 percent, taking on Project B is equivalent to borrowing at 30 percent. If the firm can borrow from a bank at, say, only 25 percent, it should reject the project. However, if a firm can borrow from a bank only at, say, 35 percent, it should accept the project. Thus Project B will be accepted if and only if the discount rate is above the IRR.3

This should be contrasted with Project A. If the firm has $100 cash to invest, it can either (1) accept Project A or (2) lend $100 to the bank. The project is actually a substitute for lending. In fact, because the IRR is 30 percent, taking on Project A is tantamount to lending at 30 percent. The firm should accept Project A if the lending rate is below 30 percent. Conversely, the firm should reject Project A if the lending rate is above 30 percent.

Page 147Because the firm initially pays out money with Project A but initially receives money with Project B, we refer to Project A as an investing type project and Project B as a financing type project. Investing type projects are the norm. Because the IRR rule is reversed for financing type projects, be careful when using it with this type of project.

Problem 2: Multiple Rates of Return Suppose the cash flows from a project are:

(−$100, $230, −$132)

Because this project has a negative cash flow, a positive cash flow, and another negative cash flow, we say that the project’s cash flows exhibit two changes of sign, or “flipflops.” Although this pattern of cash flows might look a bit strange at first, many projects require outflows of cash after some inflows. An example would be a strip-mining project. The first stage in such a project is the initial investment in excavating the mine. Profits from operating the mine are received in the second stage. The third stage involves a further investment to reclaim the land and satisfy the requirements of environmental protection legislation. Cash flows are negative at this stage.

Projects financed by lease arrangements may produce a similar pattern of cash flows. Leases often provide substantial tax subsidies, generating cash inflows after an initial investment. However, these subsidies decline over time, frequently leading to negative cash flows in later years. (The details of leasing will be discussed in a later chapter.)

It is easy to verify that this project has not one but two IRRs, 10 percent and 20 percent.4 In a case like this, the IRR does not make any sense. What IRR are we to use—10 percent or 20 percent? Because there is no good reason to use one over the other, IRR simply cannot be used here.

Why does this project have multiple rates of return? Project C generates multiple internal rates of return because both an inflow and an outflow occur after the initial investment. In general, these flip-flops or changes in sign produce multiple IRRs. In theory, a cash flow stream with K changes in sign can have up to K sensible internal rates of return (IRRs above -100 percent). Therefore, because Project C has two changes in sign, it can have as many as two IRRs. As we pointed out, projects whose cash flows change sign repeatedly can occur in the real world.

NPV Rule Of course, we should not be too worried about multiple rates of return. After all, we can always fall back on the NPV rule. Figure 5.5 plots the NPV of Project C (-$100, $230, -$132) as a function of the discount rate. As the figure shows, the NPV is zero at both 10 percent and 20 percent and negative outside the range. Thus, the NPV rule tells us to accept the project if the appropriate discount rate is between 10 percent and 20 percent. The project should be rejected if the discount rate lies outside this range.

Page 148Modified IRR As an alternative to NPV, we now introduce the modified IRR (MIRR) method, which handles the multiple IRR problem by combining cash flows until only one change in sign remains. To see how it works, consider Project C again. With a discount rate of, say, 14 percent, the value of the last cash flow, -$132, is:

−$132 /1.14 = −$115.79

as of Date 1. Because $230 is already received at that time, the “adjusted” cash flow at Date 1 is $114.21 (=$230 – 115.79). Thus, the MIRR approach produces the following two cash flows for the project:

(−$100, $114.21)

Note that by discounting and then combining cash flows, we are left with only one change in sign. The IRR rule can now be applied. The IRR of these two cash flows is 14.21 percent, implying that the project should be accepted given our assumed discount rate of 14 percent.

Of course, Project C is relatively simple to begin with: It has only three cash flows and two changes in sign. However, the same procedure can easily be applied to more complex projects—that is, just keep discounting and combining the later cash flows until only one change of sign remains.

Although this adjustment does correct for multiple IRRs, it appears, at least to us, to violate the “spirit” of the IRR approach. As stated earlier, the basic rationale behind the IRR method is that it provides a single number summarizing the merits of a project. That number does not depend on the discount rate. In fact, that is why it is called the internal rate of return: The number is internal, or intrinsic, to the project and does not depend on anything except the project’s cash flows. By contrast, MIRR is clearly a function of the discount rate. However, a firm using this adjustment will avoid the multiple IRR problem, just as a firm using the NPV rule will avoid it.5

The Guarantee against Multiple IRRs If the first cash flow of a project is negative (because it is the initial investment) and if all of the remaining flows are positive, there can be only a single, unique IRR, no matter how many periods the project lasts. This is easy to understand by using the concept of the time value of money. For example, it is simple to verify that Project A in Table 5.2 has an IRR of 30 percent because using a 30 percent discount rate gives:

Page 149How do we know that this is the only IRR? Suppose we were to try a discount rate greater than 30 percent. In computing the NPV, changing the discount rate does not change the value of the initial cash flow of -$100 because that cash flow is not discounted. Raising the discount rate can only lower the present value of the future cash flows. In other words, because the NPV is zero at 30 percent, any increase in the rate will push the NPV into the negative range. Similarly, if we try a discount rate of less than 30 percent, the overall NPV of the project will be positive. Though this example has only one positive flow, the above reasoning still implies a single, unique IRR if there are many inflows (but no outflows) after the initial investment.

If the initial cash flow is positive—and if all of the remaining flows are negative— there can only be a single, unique IRR. This result follows from similar reasoning. Both these cases have only one change of sign or flip-flop in the cash flows. Thus, we are safe from multiple IRRs whenever there is only one sign change in the cash flows.

General Rules The following chart summarizes our rules:

Flows

Number of IRRs

IRR Criterion

NPV Criterion

First cash flow is negative and

1

Accept if IRR > R.

Accept if NPV > 0.

all remaining cash flows are positive.

Reject if IRR < R.

Reject if NPV < 0.

First cash flow is positive and

1

Accept if IRR < R.

Accept if NPV > 0.

all remaining cash flows are negative.

Reject if IRR > R.

Reject if NPV < 0.

Some cash flows after first are

May be more

No valid IRR.

Accept if NPV > 0.

positive and some cash flows after first are negative.

than 1.

Reject if NPV < 0.

Note that the NPV criterion is the same for each of the three cases. In other words, NPV analysis is always appropriate. Conversely, the IRR can be used only in certain cases. When it comes to NPV, the preacher’s words, “You just can’t lose with the stuff I use,” clearly apply.

PROBLEMS SPECIFIC TO MUTUALLY EXCLUSIVE PROJECTS

As mentioned earlier, two or more projects are mutually exclusive if the firm can accept only one of them. We now present two problems dealing with the application of the IRR approach to mutually exclusive projects. These two problems are quite similar, though logically distinct.

The Scale Problem A professor we know motivates class discussions of this topic with this statement: “Students, I am prepared to let one of you choose between two mutually exclusive ‘business’ propositions. Opportunity 1—You give me $1 now and I’ll give you $1.50 back at the end of the class period. Opportunity 2—You give me $10 and I’ll give you $11 back at the end of the class period. You can choose only one of the two opportunities. And you cannot choose either opportunity more than once. I’ll pick the first volunteer.”

Page 150Which would you choose? The correct answer is Opportunity 2.6 To see this, look at the following chart:

Cash Flow at Beginning of Class

Cash Flow at End of Class (90 Minutes Later)

NPV 7

IRR

Opportunity 1

−$ 1

+$ 1.50

$ .50

50%

Opportunity 2

−10

+ 11.00

1.00

10

As we have stressed earlier in the text, one should choose the opportunity with the highest NPV. This is Opportunity 2 in the example. Or, as one of the professor’s students explained it, “I’m bigger than the professor, so I know I’ll get my money back. And I have $10 in my pocket right now so I can choose either opportunity. At the end of the class, I’ll be able to buy one song on iTunes with Opportunity 2 and still have my original investment, safe and sound. The profit on Opportunity 1 pays for only one half of a song.”

This business proposition illustrates a defect with the internal rate of return criterion. The basic IRR rule indicates the selection of Opportunity 1 because the IRR is 50 percent. The IRR is only 10 percent for Opportunity 2.

Where does IRR go wrong? The problem with IRR is that it ignores issues of scale. Although Opportunity 1 has a greater IRR, the investment is much smaller. In other words, the high percentage return on Opportunity 1 is more than offset by the ability to earn at least a decent return8 on a much bigger investment under Opportunity 2.

Because IRR seems to be misguided here, can we adjust or correct it? We illustrate how in the next example.

EXAMPLE

5.2

NPV versus IRR Stanley Jaffe and Sherry Lansing have just purchased the rights to Corporate Finance: The Motion Picture. They will produce this major motion picture on either a small budget or a big budget. Here are the estimated cash flows:

Cash Flow at Date 0

Cash Flow at Date 1

NPV @25%

IRR

Small budget

−$10 million

$40 million

$22 million

300%

Large budget

−25 million

65 million

27 million

160

Because of high risk, a 25 percent discount rate is considered appropriate. Sherry wants to adopt the large budget because the NPV is higher. Stanley wants to adopt the small budget because the IRR is higher. Who is right?

Page 151For the reasons espoused in the classroom example, NPV is correct. Hence Sherry is right. Howwever, Stanley is very stubborn where IRR is concerned. How can Sherry justify the large budget to Stanley using the IRR approach?

This is where incremental IRR comes in. Sherry calculates the incremental cash flows from choosing the large budget instead of the small budget as follows:

Cash Flow at Date 0 (in $ millions)

Cash Flow at Date 1 (in $ millions)

Incremental cash flows from choosing large budget instead of small budget

−$25 − (-10) = −$15

$65 − 40 = $25

This chart shows that the incremental cash flows are -$15 million at Date 0 and $25 million at Date 1. Sherry calculates incremental IRR as follows:

Formula for Calculating the Incremental IRR:

IRR equals 66.67 percent in this equation, implying that the incremental IRR is 66.67 percent. Incremental IRR is the IRR on the incremental investment from choosing the large project instead of the small project.

In addition, we can calculate the NPV of the incremental cash flows:

NPV of Incremental Cash Flows:

We know the small-budget picture would be acceptable as an independent project because its NPV is positive. We want to know whether it is beneficial to invest an additional $15 million to make the large-budget picture instead of the small-budget picture. In other words, is it beneficial to invest an additional $15 million to receive an additional $25 million next year? First, our calculations show the NPV on the incremental investment to be positive. Second, the incremental IRR of 66.67 percent is higher than the discount rate of 25 percent. For both reasons, the incremental investment can be justified, so the large-budget movie should be made. The second reason is what Stanley needed to hear to be convinced.

In review, we can handle this example (or any mutually exclusive example) in one of three ways:

1. Compare the NPVs of the two choices. The NPV of the large-budget picture is greater than the NPV of the small-budget picture. That is, $27 million is greater than $22 million.

2. Calculate the incremental NPV from making the large-budget picture instead of the small-budget picture. Because the incremental NPV equals $5 million, we choose the large-budget picture.

3. Compare the incremental IRR to the discount rate. Because the incremental IRR is 66.67 percent and the discount rate is 25 percent, we take the large-budget picture.

Page 152All three approaches always give the same decision. However, we must not compare the IRRs of the two pictures. If we did, we would make the wrong choice. That is, we would accept the small-budget picture.

Although students frequently think that problems of scale are relatively unimportant, the truth is just the opposite. No real-world project comes in one clear-cut size. Rather, the firm has to determine the best size for the project. The movie budget of $25 million is not fixed in stone. Perhaps an extra $1 million to hire a bigger star or to film at a better location will increase the movie’s gross. Similarly, an industrial firm must decide whether it wants a warehouse of, say, 500,000 square feet or 600,000 square feet. And, earlier in the chapter, we imagined McDonald’s opening an outlet on a remote island. If it does this, it must decide how big the outlet should be. For almost any project, someone in the firm has to decide on its size, implying that problems of scale abound in the real world.

One final note here. Students often ask which project should be subtracted from the other in calculating incremental flows. Notice that we are subtracting the smaller project’s cash flows from the bigger project’s cash flows. This leaves an outflow at Date 0. We then use the basic IRR rule on the incremental flows.9

The Timing Problem Next we illustrate another, somewhat similar problem with the IRR approach to evaluating mutually exclusive projects.

EXAMPLE

5.3

Mutually Exclusive Investments Suppose that the Kaufold Corporation has two alternative uses for a warehouse. It can store toxic waste containers (Investment A) or electronic equipment (Investment B). The cash flows are as follows:

Cash Flow at Year

NPV

Year:

0

1

2

3

@ 0%

@10%

@15%

IRR

Investment A

−$10,000

$10,000

$1,000

$1,000

$2,000

$669

$109

16.04%

Investment B

−10,000

$1,000

$1,000

12,000

4,000

751

−484

12.94

We find that the NPV of Investment B is higher with low discount rates, and the NPV of Investment A is higher with high discount rates. This is not surprising if you look closely at the cash flow patterns. The cash flows of A occur early, whereas the cash flows of B occur later. If we assume a high discount rate, we favor Investment A because we are implicitly assuming that the early cash flow (for example, $10,000 in Year 1) can be reinvested at that rate. Because most of Investment B’s cash flows occur in Year 3, B’s value is relatively high with low discount rates.

The patterns of cash flow for both projects appear in Figure 5.6. Project A has an NPV of $2,000 at a discount rate of zero. This is calculated by simply adding up the cash flows without discounting them. Project B has an NPV of $4,000 at the zero rate. However, the NPV of Project B declines more rapidly as the discount rate increases than does the NPV of Project A. As we mentioned, this occurs because the cash flows of B occur later. Both projects have the same NPV at a discount rate of 10.55 percent. The IRR for a project is Page 153the rate at which the NPV equals zero. Because the NPV of B declines more rapidly, B actually has a lower IRR.

Figure 5.6 Net Present Value and the Internal Rate of Return for Mutually Exclusive Projects

As with the movie example, we can select the better project with one of three different methods:

1. Compare NPVs of the two projects. Figure 5.6 aids our decision. If the discount rate is below 10.55 percent, we should choose Project B because B has a higher NPV. If the rate is above 10.55 percent, we should choose Project A because A has a higher NPV.

2. Compare incremental IRR to discount rate. Method 1 employed NPV. Another way of determining whether A or B is a better project is to subtract the cash flows of A from the cash flows of B and then to calculate the IRR. This is the incremental IRR approach we spoke of earlier.

Here are the incremental cash flows:

NPV of Incremental Cash Flows

Year:

0

1

2

3

Incremental IRR

@ 0%

@10%

@15%

B−A

0

− $9,000

0

$11,000

10.55%

$2,000

$83

− $593

This chart shows that the incremental IRR is 10.55 percent. In other words, the NPV on the incremental investment is zero when the discount rate is 10.55 percent. Thus, if the relevant discount rate is below 10.55 percent, Project B is preferred to Project A. If the relevant discount rate is above 10.55 percent, Project A is preferred to Project B.

Figure 5.6 shows that the NPVs of the two projects are equal when the discount rate is 10.55 percent. In other words, the crossover rate in the figure is 10.55. The incremental cash flows chart shows that the incremental IRR is also 10.55 percent. It is not a coincidence that the crossover rate and the incremental IRR are the same; this equality must always hold. The incremental IRR is the rate that causes the incremental cash flows to have zero NPV. The incremental cash flows have zero NPV when the two projects have the same NPV.Page 154

3. Calculate NPV on incremental cash flows. Finally, we could calculate the NPV on the incremental cash flows. The chart that appears with the previous method displays these NPVs. We find that the incremental NPV is positive when the discount rate is either 0 percent or 10 percent. The incremental NPV is negative if the discount rate is 15 percent. If the NPV is positive on the incremental flows, we should choose B. If the NPV is negative, we should choose A.

In summary, the same decision is reached whether we (1) compare the NPVs of the two projects, (2) compare the incremental IRR to the relevant discount rate, or (3) examine the NPV of the incremental cash flows. However, as mentioned earlier, we should not compare the IRR of Project A with the IRR of Project B.

We suggested earlier that we should subtract the cash flows of the smaller project from the cash flows of the bigger project. What do we do here when the two projects have the same initial investment? Our suggestion in this case is to perform the subtraction so that the first nonzero cash flow is negative. In the Kaufold Corp. example we achieved this by subtracting A from B. In this way, we can still use the basic IRR rule for evaluating cash flows.

The preceding examples illustrate problems with the IRR approach in evaluating mutually exclusive projects. Both the professor-student example and the motion picture example illustrate the problem that arises when mutually exclusive projects have different initial investments. The Kaufold Corp. example illustrates the problem that arises when mutually exclusive projects have different cash flow timing. When working with mutually exclusive projects, it is not necessary to determine whether it is the scale problem or the timing problem that exists. Very likely both occur in any real-world situation. Instead, the practitioner should simply use either an incremental IRR or an NPV approach.

REDEEMING QUALITIES OF IRR

IRR probably survives because it fills a need that NPV does not. People seem to want a rule that summarizes the information about a project in a single rate of return. This single rate gives people a simple way of discussing projects. For example, one manager in a firm might say to another, “Remodeling the north wing has a 20 percent IRR.”

To their credit, however, companies that employ the IRR approach seem to understand its deficiencies. For example, companies frequently restrict managerial projections of cash flows to be negative at the beginning and strictly positive later. Perhaps, then, both the ability of the IRR approach to capture a complex investment project in a single number, and the ease of communicating that number explain the survival of the IRR.

A TEST

To test your knowledge, consider the following two statements:

1. You must know the discount rate to compute the NPV of a project, but you compute the IRR without referring to the discount rate.

2. Hence, the IRR rule is easier to apply than the NPV rule because you don’t use the discount rate when applying IRR.

The first statement is true. The discount rate is needed to compute NPV. The IRR is computed by solving for the rate where the NPV is zero. No mention is made of the discount rate in the mere computation. However, the second statement is false. To apply IRR, you must compare the internal rate of return with the discount rate. Thus the discount rate is needed for making a decision under either the NPV or IRR approach.Page 155

5.6 The Profitability Index

Another method used to evaluate projects is called the profitability index. It is the ratio of the present value of the future expected cash flows after initial investment divided by the amount of the initial investment. The profitability index can be represented as:

EXAMPLE

5.4

Profitability Index Hiram Finnegan Inc. (HFI) applies a 12 percent discount rate to two investment opportunities.

Cash Flows ($000,000)

PV @ 12% of Cash Flows Subsequent to Initial Investment ($000,000)

Project

C0

C1

C2

Profitability Index

NPV @12% ($000,000)

1

−$20

$70

$10

$70.5

3.53

$50.5

2

−10

15

40

45.3

4.53

35.3

CALCULATION OF PROFITABILITY INDEX

The profitability index is calculated for Project 1 as follows. The present value of the cash flows after the initial investment is:

The profitability index is obtained by dividing this result by the initial investment of $20. This yields:

Application of the Profitability Index How do we use the profitability index? We consider three situations:

1. Independent projects: Assume that HFI’s two projects are independent. According to the NPV rule, both projects should be accepted because NPV is positive in each case. The profitability index (PI) is greater than 1 whenever the NPV is positive. Thus, the PI decision rule is:

1. • Accept an independent project if PI > 1.

2. • Reject it if PI < 1.

2. Mutually exclusive projects: Let us now assume that HFI can only accept one of its two projects. NPV analysis says accept Project 1 because this project has the bigger NPV. Because Project 2 has the higher PI, the profitability index leads to the wrong selection.

Page 156For mutually exclusive projects, the profitability index suffers from the scale problem that IRR also suffers from. Project 2 is smaller than Project 1. Because the PI is a ratio, it ignores Project 1’s larger investment. Thus, like IRR, PI ignores differences of scale for mutually exclusive projects.

However, like IRR, the flaw with the PI approach can be corrected using incremental analysis. We write the incremental cash flows after subtracting Project 2 from Project 1 as follows:

Cash Flows ($000,000)

PV @ 12% of Cash Flows Subsequent to Initial Investment ($000,000)

Project

C0

C1

C2

Profitability Index

NPV @12% ($000,000)

1–2

−$10

$55

−$30

$25.2

2.52

$15.2

Because the profitability index on the incremental cash flows is greater than 1.0, we should choose the bigger project—that is, Project 1. This is the same decision we get with the NPV approach.

3. Capital rationing: The first two cases implicitly assumed that HFI could always attract enough capital to make any profitable investments. Now consider the case when the firm does not have enough capital to fund all positive NPV projects. This is the case of capital rationing.

Imagine that the firm has a third project, as well as the first two. Project 3 has the following cash flows:

Cash Flows ($000,000)

PV @ 12% of Cash Flows Subsequent to Initial Investment ($000,000)

Project

C0

C1

C2

Profitability Index

NPV @12% ($000,000)

3

−$10

−$5

−$60

$43.4

4.34

$33.4

Further, imagine that (1) the projects of Hiram Finnegan Inc. are independent, but (2) the firm has only $20 million to invest. Because Project 1 has an initial investment of $20 million, the firm cannot select both this project and another one. Conversely, because Projects 2 and 3 have initial investments of $10 million each, both these projects can be chosen. In other words, the cash constraint forces the firm to choose either Project 1 or Projects 2 and 3.

What should the firm do? Individually, Projects 2 and 3 have lower NPVs than Project 1 has. However, when the NPVs of Projects 2 and 3 are added together, the sum is higher than the NPV of Project 1. Thus, common sense dictates that Projects 2 and 3 should be accepted.

What does our conclusion have to say about the NPV rule or the PI rule? In the case of limited funds, we cannot rank projects according to their NPVs. Instead we should rank them according to the ratio of present value to initial investment. This is the PI rule. Both Project 2 and Project 3 have higher PI ratios than does Project 1. Thus they should be ranked ahead of Project 1 when capital is rationed.

Page 157The usefulness of the profitability index under capital rationing can be explained in military terms. The Pentagon speaks highly of a weapon with a lot of “bang for the buck.” In capital budgeting, the profitability index measures the bang (the dollar return) for the buck invested. Hence it is useful for capital rationing.

It should be noted that the profitability index does not work if funds are also limited beyond the initial time period. For example, if heavy cash outflows elsewhere in the firm were to occur at Date 1, Project 3, which also has a cash outflow at Date 1, might need to be rejected. In other words, the profitability index cannot handle capital rationing over multiple time periods.

In addition, what economists term indivisibilities may reduce the effectiveness of the PI rule. Imagine that HFI has $30 million available for capital investment, not just $20 million. The firm now has enough cash for Projects 1 and 2. Because the sum of the NPVs of these two projects is greater than the sum of the NPVs of Projects 2 and 3, the firm would be better served by accepting Projects 1 and 2. But because Projects 2 and 3 still have the highest profitability indexes, the PI rule now leads to the wrong decision. Why does the PI rule lead us astray here? The key is that Projects 1 and 2 use up all of the $30 million, whereas Projects 2 and 3 have a combined initial investment of only $20 million (= $10 + 10). If Projects 2 and 3 are accepted, the remaining $10 million must be left in the bank.

This situation points out that care should be exercised when using the profitability index in the real world. Nevertheless, while not perfect, the profitability index goes a long way toward handling capital rationing.

5.7 The Practice of Capital Budgeting

So far this chapter has asked “Which capital budgeting methods should companies be using?” An equally important question is this: Which methods are companies using? Table 5.3 helps answer this question. As can be seen from the table, approximately three- quarters of U.S. and Canadian companies use the IRR and NPV methods. This is not surprising, given the theoretical advantages of these approaches. Over half of these companies use the payback method, a rather surprising result given the conceptual problems with this approach. And while discounted payback represents a theoretical improvement over regular payback, the usage here is far less. Perhaps companies are attracted to the user-friendly nature of payback. In addition, the flaws of this approach, as mentioned in the current chapter, may be relatively easy to correct. For example, while the payback method ignores Page 158all cash flows after the payback period, an alert manager can make ad hoc adjustments for a project with back-loaded cash flows.

Table 5.3 Percentage of CFOs Who Always or Almost Always Use a Given Technique

% Always or Almost Always

Internal rate of return (IRR)

75.6%

Net present value (NPV)

74.9

Payback method

56.7

Discounted payback

29.5

Profitability index

11.9

SOURCE: Adapted from Figure 2 from John R. Graham and Campbell R. Harvey, “The Theory and Practice of Corporate Finance: Evidence from the Field,” Journal of Financial Economics 60 (2001). Based on a survey of 392 CFOs.

Capital expenditures by individual corporations can add up to enormous sums for the economy as a whole. For example, in 2014, ExxonMobil announced that it expected to make about $39.8 billion in capital outlays during the year, down 6 percent from the record $42.5 billion in 2013. The company further indicated that it expected to spend an average of $37 billion per year through 2017. About the same time, Chevron announced that its capital budget for 2014 would be $39.8 billion, down $2 billion from the previous year, and ConocoPhillips announced a capital expenditure budget of $16.7 billion for 2014. Other companies with large capital spending budgets in 2014 were Intel, which projected capital spending of about $11 billion, and Samsung Electronics, which projected capital spending of about $11.5 billion.

Large-scale capital spending is often an industrywide occurrence. For example, in 2014, capital spending in the semiconductor industry was expected to reach $60.9 billion. This tidy sum represented a 5.5 percent increase over industry capital spending in 2013.

According to information released by the U.S. Census Bureau in 2013, capital investment for the economy as a whole was $1.090 trillion in 2009, $1.106 trillion in 2010, and $1.226 trillion in 2011. The totals for the three years therefore equaled approximately $3.422 trillion! Given the sums at stake, it is not too surprising that successful corporations carefully analyze capital expenditures.

One might expect the capital budgeting methods of large firms to be more sophisticated than the methods of small firms. After all, large firms have the financial resources to hire more sophisticated employees. Table 5.4 provides some support for this idea. Here firms indicate frequency of use of the various capital budgeting methods on a scale of 0 (never) to 4 (always). Both the IRR and NPV methods are used more frequently, and payback less frequently, in large firms than in small firms. Conversely, large and small firms employ the last two approaches about equally.

The use of quantitative techniques in capital budgeting varies with the industry. As one would imagine, firms that are better able to estimate cash flows are more likely to use NPV. For example, estimation of cash flow in certain aspects of the oil business is quite feasible. Because of this, energy-related firms were among the first to use NPV analysis. Conversely, the cash flows in the motion picture business are very hard to project. The grosses of great hits like Spiderman, Harry Potter, and Star Wars were far, far greater than anyone imagined. The big failures like Alamo and Waterworld were unexpected as well. Because of this, NPV analysis is frowned upon in the movie business.

Table 5.4 Frequency of Use of Various Capital Budgeting Methods

Large Firms

Small Firms

Internal rate of return (IRR)

3.41

2.87

Net present value (NPV)

3.42

2.83

Payback method

2.25

2.72

Discounted payback

1.55

1.58

Profitability index

.75

.78

Firms indicate frequency of use on a scale from 0 (never) to 4 (always). Numbers in table are averages across respondents. SOURCE: Adapted from Table 2 from Graham and Harvey (2001), op. cit.

Page 159How does Hollywood perform capital budgeting? The information that a studio uses to accept or reject a movie idea comes from the pitch. An independent movie producer schedules an extremely brief meeting with a studio to pitch his or her idea for a movie. Consider the following four paragraphs of quotes concerning the pitch from the thoroughly delightful book Reel Power:10

“They [studio executives] don’t want to know too much,” says Ron Simpson. “They want to know concept… . They want to know what the three-liner is, because they want it to suggest the ad campaign. They want a title… . They don’t want to hear any esoterica. And if the meeting lasts more than five minutes, they’re probably not going to do the project.”

“A guy comes in and says this is my idea: ‘Jaws on a spaceship,’” says writer Clay Frohman ( Under Fire). “And they say, ‘Brilliant, fantastic.’ Becomes Alien. That is Jaws on a spaceship, ultimately… . And that’s it. That’s all they want to hear. Their attitude is ‘Don’t confuse us with the details of the story.’ ”

“… Some high-concept stories are more appealing to the studios than others. The ideas liked best are sufficiently original that the audience will not feel it has already seen the movie, yet similar enough to past hits to reassure executives wary of anything too far- out. Thus, the frequently used shorthand: It’s Flashdance in the country (Footloose) or High Noon in outer space (Outland ).”

“… One gambit not to use during a pitch,” says executive Barbara Boyle, “is to talk about big box-office grosses your story is sure to make. Executives know as well as anyone that it’s impossible to predict how much money a movie will make, and declarations to the contrary are considered pure malarkey.”

Summary and Conclusions

1. In this chapter, we covered different investment decision rules. We evaluated the most popular alternatives to the NPV: The payback period, the discounted payback period, the internal rate of return, and the profitability index. In doing so we learned more about the NPV.

2. While we found that the alternatives have some redeeming qualities, when all is said and done, they are not the NPV rule; for those of us in finance, that makes them decidedly second-rate.

3. Of the competitors to NPV, IRR must be ranked above payback. In fact, IRR always reaches the same decision as NPV in the normal case where the initial outflows of an independent investment project are followed only by a series of inflows.

4. We classified the flaws of IRR into two types. First, we considered the general case applying to both independent and mutually exclusive projects.