# Chapter 8: Investment Decision Rules

## 8.1: NPV and Stand-Alone Projects

### The Net Present Value (NPV) Decision Rule

When making an investment decision, take the alternative with the highest NPV. Choosing this alternative is equivalent to receiving its NPV in cash today. In the case of a **stand-alone project**, we must choose between accepting and rejecting the project. The NPV rule then says we should compare the project’s NPV to zero (the NPV of doing nothing) and accept the project if its NPV is positive.

### Applying the NPV Rule

Researchers believe they can produce a new environmentally-friendly fertilizer. It will require a new plant that can be built immediately at a cost of $250 million. Financial managers estimate that the benefits will be $35 million per year, starting at the end of the first year and lasting forever.

The NPV of this cash flow stream, given a discount rate *r*, is

$NPV=-\$\text{250}+\frac{\$35}{r}$

The following plot show the researchers’ NPV as a function of the discount rate *r*.

The NPV is positive only for discount rates that are less than 14%, the internal rate of return (IRR). Financial managers estimate a cost of capital of 10% per year. Therefore, the NPV is $100 million, which is positive.

### Measuring Sensitivity with IRR

If you are unsure of your cost of capital estimate, it is important to determine how sensitive your analysis is to errors in this estimate. The IRR can provide this information. For researchers, if the cost of capital estimate is more than the 14% IRR, the NPV will be negative. In general, *the difference between the cost of capital and the IRR is the maximum amount of estimation error in the cost of capital estimate that can exist without altering the original decision*.

### Alternative Rules vs. The NPV Rule

It is always better to go with the NPV rule when selecting projects, when other rules are conflicting. But let’s take a look at the other possible alternative rules nonetheless.

## 8.2: The Internal Rate of Return Rule

### The Internal Rate of Return (IRR) Rule

One interpretation of the internal rate of return is that it is the average return for taking on the investment opportunity. The **internal rate of return** (IRR) **investment rule** is based on this idea: If the average return on the investment opportunity (i.e., the IRR) is greater than the return on other alternatives in the market with equivalent risk and maturity (i.e., the project’s cost of capital), then you should undertake the investment opportunity. In other words, *take any investment opportunity where the IRR exceeds the opportunity cost of capital*.* Turn down any opportunity whose IRR is less than the opportunity cost of capital*.

### IRR Rule Example

In general, *the IRR rule works for a stand-alone project if all of the project’s negative cash flows precede its positive cash flows* (basically, cash leaves your pockets before you see a return). But in other cases, the IRR rule may disagree with the NPV rule and thus be incorrect.

#### Unconventional Cash Flows

The publisher will pay a former CEO $1 million up front if he agrees to write a book. He estimates it will take him 3 years to write the book. The time that he spends writing will cause him to forgo alternative sources of income amounting to $500,000 per year. Based on the risk of alternatives and available investment opportunities, he estimates his opportunity cost of capital to be 10%.

The NPV of Star’s investment opportunity is

$NPV=\$1,000,000-\frac{\$500,000}{1+r}-\frac{\$500,000}{\left(1+r\right)^2}-\frac{\$500,000}{\left(1+r\right)^3}$

By setting the NPV equal to 0 and solving for *r*, we find the IRR. Using the annuity spreadsheet, we get IRR = 23.38%. This is larger than the 10% opportunity cost of capital. According to the IRR rule, the CEO should sign the deal. But what does the NPV rule say?

$NPV=\$1,000,000-\frac{\$500,000}{1.1}-\frac{\$500,000}{\left(1.1\right)^2}-\frac{\$500,000}{\left(1.1\right)^3}=-\$243,426$

According to the NPV, signing the deal would reduce the CEO’s wealth; he should not sign it.

The following graph plots the NPV of the investment opportunity. It shows that the IRR rule and the NPV rule will give exactly opposite recommendations. That is the NPV is positive *only when* the opportunity cost of capital is *above* 23.38% (the IRR). This means he should accept the deal only the opportunity cost of capital is greater than the IRR, the opposite of the IRR rule.

This is because the CEO gets cash first and then has to pay the costs. It is as if the CEO borrowed money, and when you borrow money you prefer as *low* a rate as possible. His optimal rule is to borrow money so long as the rate at which he borrows is *less* than the cost of capital. Thus, the normal IRR rule must be reversed for projects with **unconventional cash flows** (or borrowing-style cash flows).

#### Multiple IRRs

The publisher has agreed to make royalty payments. Star expects these payments to amount to $20,000 per year forever, starting in four years (a year after the book’s publication date).

Using the annuity and perpetuity formulas, the NPV of Star’s new investment opportunity is:

$NPV=\$1,000,000-\frac{\$500,000}{1+r}-\frac{\$500,000}{\left(1+r\right)^2}-\frac{\$500,000}{\left(1+r\right)^3}+\frac{\$20,000}{\left(1+r\right)^4}+\frac{\$20,000}{\left(1+r\right)^5}+...$

$=\$1,000,000-\frac{\$500,000}{r}\left(1-\frac{1}{\left(1+r\right)^3}\right)+\frac{1}{\left(1+r\right)^3}\left(\frac{\$20,000}{r}\right)$

By setting the NPV equal to zero and solving for r, we find the IRR (using a spreadsheet software). In this case, there are two IRRs—that is, there are two values of r that set the NPV equal to 0. 4.723% and 19.619%. Because there is more than one IRR, we cannot apply the IRR rule.

According to the NPV rule, if the cost of capital is *either* below 4.723% or above 19.619%, he should undertake the opportunity. Notice that even though the IRR rule fails in this case, the two IRRs are still useful as bounds on the cost of capital.

#### Nonexistent IRR

An agent has approached the former CEO and guaranteed $1 million in each of the next three years if he will agree to give four lectures per month over that period. Star estimates that preparing and delivering the lectures would take the same amount of time as writing the book—that is, the cost would be $500,000 per year. Therefore, his *net* cash flow will be $500,000 per year.

The NPV of Star’s new investment opportunity is

$NPV=\frac{\$500,000}{1+r}+\frac{\$500,000}{\left(1+r\right)^2}+\frac{\$500,000}{\left(1+r\right)^3}$

By setting the NPV equal to zero and solving for *r*, we find the IRR. In this case, however, there is no discount rate that will set the NPV equal to 0. NPV is always positive in this case (but it can also be always negative, and still not have an IRR).

#### Summary

IRR applies (normal); cash leaves first and then there are cash inflows.*A:*Multiple IRRs.*B:*Unconventional cash flows; paid first and then there are costs.*C:*Nonexistent IRR; always negative or positive NPV.*D:*

The IRR is always useful for figuring out the **average return over the life of an investment** and **indicates the sensitivity of the NPV to estimation error** in the cost of capital.

## 8.3: The Payback Rule

### The Payback Investment Rule

The simplest investment rule is the **payback investment rule**, which is based on the notion that an opportunity that pays back its initial investment quickly is a good idea. To apply the payback rule, you first calculate the amount of time it takes to pay back the initial investment, called the **payback period**. If the payback period is less than a prespecified length of time – called the **cut-off period** – which is usually a few years, then you accept the project.

### Applying the Payback Rule

Let’s go back to the researcher example. Assume these researchers require all projects to have a payback period of 5 years or less. Would they undertake the fertilizer project?

The sum of the cash flows from year 1 to year 5 is $35 × 5 = $175 million, which will not cover the initial investment of $250 million. In fact, it will not be until year 8 that the initial investment will be paid back $35 × 8 = $280 million. Because the payback period for this project is 8 years and exceeds the 5-year cut-off period, they will reject the project. (Note: If you assume the yearly cash flows are distributed evenly over the course of each year, then the payback period for the project will be $250 million / $35 million per year = 7.1428 years. But we don’t make this assumption in this example).

However, as we saw earlier, with a cost of capital of 10%, the NPV is $100 million. Following the payback rule would be a mistake.

### Payback Rule Pitfalls in Practice

The payback rule is not as reliable as the NPV rule because it (i) *ignores the project’s cost of capital and the time value of money*, (ii) *ignores cash flows after the payback period*, and (iii) *relies on an ad hoc decision criterion* (what is the right number of years to require for the payback period?).

Firms use this rule for small investment decisions, such as buying a new copy machine. In such cases, the cost of making an incorrect decision might not be large enough to justify the time required to calculate the NPV. The payback rule also provides budgeting information regarding the length of time capital will be committed to a project. Also, if the required payback period is short (one or two years), then most projects that satisfy the payback rule will have a positive NPV.

## 8.4: Choosing Between Projects

Sometimes a firm must choose just one project from among several possible projects, that is, they are **mutually exclusive projects**.

### The NPV Rule and Mutually Exclusive Projects

When projects are mutually exclusive, we need to determine which projects have positive NPV and then rank the projects to identify the best one. In this situation, the NPV rule provides a straightforward answer: *Pick the project with the highest NPV*.

### IRR Rule and Mutually Exclusive Projects

Unfortunately, picking one project over another simply because it has a larger IRR can lead to mistakes. In particular, *when projects differ in their scale of investment, the timing of their cash flows, or their riskiness, then their IRRs cannot be meaningfully compared*.

#### Differences in Scale

Would you prefer a 500% return on $1, or a 20% return on $1 million? This comparison illustrates an important shortcoming of IRR: Because it is a return, you cannot tell how much value will actually be created without knowing the scale of the investment.

If a project has a positive NPV, and if we can double its size, its NPV will double: By the Law of One Price, doubling the cash flows of an investment opportunity must make it worth twice as much. However, the IRR rule does not have this property – it is unaffected by the scale of the investment opportunity because the IRR measures the average return of the investment. Hence the IRR rule cannot be used to compare projects of different scales.

#### Differences in Timing

Even when projects have the same scale, the IRR may lead you to rank them incorrectly due to difference in the timing of the cash flows: The IRR is expressed as a return, but the dollar value of earning a given return – and therefore its NPV – depends on how long the return is earned. Earning a very high annual return is much more valuable if you earn it for several years than if you earn it for only a few days.

#### Differences in Risk

To know whether the IRR of a project is attractive, we must compare it to the project’s cost of capital, which is determined by the project’s risk. Thus, an IRR that is attractive for a safe project need not be attractive for a much riskier project. As a simple example, while you might be quite pleased to earn a 10% return on a risk-free investment opportunity, you might be much less satisfied to earn a 10% return on an investment in a risky start-up company.

### The Incremental IRR Rule

When considering a pair of mutually exclusive projects, we can avoid comparing the IRRs directly by computing the **incremental IRR**, which is the IRR of the difference between the cash flows of the two alternatives (the increment to the cash flows of one investment over the other). The incremental IRR tells us the IRR associated with switching from one project to another. Then, instead of comparing the projects, we can evaluate the decision to switch as a stand-alone decision and apply the IRR rule using the incremental IRR.

#### Example

You have two options (one minor & one major), with the following cash flows.

The cost of capital for both projects is 12%.

**What is the IRR of each proposal?**

Using annuity calculators, the minor overhaul has an of 36.3% and major one has an IRR of 23.4%.

**What is the incremental IRR?**

The IRR of both projects exceeds the cost of capital of 12%. Because the negative cash flows of each project precede the positive ones, we can apply the IRR rule to conclude that each project has positive NPV. But which project is best? Because the projects have different scales, we cannot compare their IRRs directly.

To compute the incremental IRR of switching from the minor overhaul to the major overhaul, we first compute the incremental cash flows:

These cash flows have an IRR of 20.0%, according to an annuity calculator. Because the incremental IRR exceeds the 12% cost of capital, switching to the major overhaul looks attractive (i.e., its larger scale is sufficient to make up for its lower IRR).

#### Shortcomings of the Incremental IRR Rule

It shares several problems with the regular IRR rule:

- You must keep track of which project is the incremental project and ensure that the incremental cash flows are initially negative and then become positive. Otherwise, the incremental IRR rule will have the
**unconventional cash flow**problem. - The incremental IRR
**need not exist**. - You could have
**multiple incremental IRRs**.

On top of these, there are two additional problems:

- When the incremental IRR rule indicates that one of the two projects is better, it does not imply that the better project should be accepted. It is possible that a bad project will look good when compared to a project that is even worse. This problem does not occur if you compare the NPV s of the two projects; if they are both negative, then reject both projects.
- The incremental IRR rule assumes that the riskiness of the two projects is the same. When the risks are different, the cost of capital of the incremental cash flows is not obvious, making it difficult to know whether the incremental IRR exceeds the cost of capital. In this case only the NPV rule, which allows each project to be discounted at its own cost of capital, will give a reliable answer.

## 8.5: Project Selection with Resource Constraints

In some situations, different investment opportunities demand different amounts of a particular resource. If there is a fixed supply of the resource so that you cannot undertake all possible opportunities, simply picking the highest NPV opportunity might not lead to the best decision.

### Evaluation of Projects with Different Resource Requirements

Assume you are considering the three projects in the following table, which require warehouse space.

Project A has the highest NPV, but it uses up the entire resource (the warehouse); thus it would be a mistake to take this opportunity. Projects B and C can both be undertaken (together they use all the available space), and their combined NPV exceeds the NPV of project A; thus, you should initiate both.

### Profitability Index

Practitioners often use the **profitability index** to identify the optimal combination of projects to undertake in complicated situations:

$\text{Profitability Index}\ \left(PI\right)=\frac{\text{Value Created}}{\text{Resource Consumed}}=\frac{NPV}{\text{Resource Consumed}}$

The PI measures the “bang for your buck”; the value created in terms of NPV per unit of resource consumed.

### Capital Rationing Constraints

When funds are limited, there is said to be a **capital rationing** constraint. Without the capital rationing, you would accept all positive NPV projects. When there is capital rationing, though, ranking projects by their profitability index is useful. Suppose you have a budget of $100 million to invest.

While Project I has the highest NPV, it uses up the entire budget. Projects II and III can both be under-taken (together they also take up the entire budget), and their combined NPV exceeds the NPV of Project I. The both should be initiated, for a combined NPV of $130 million.

### Shortcomings of the Profitability Index

Although the PI is simple to compute/use, 2 conditions must be satisfied for it to be completely reliable:

- The set of projects taken following the profitability index ranking completely exhausts the available resource.

There is only a single relevant resource constraint.

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