A project may have multiple irrs when

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