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Alternatives
Comparisons with Rate of Return

Executives prefer the rate of return method because it expresses the worth of an investment as the annual percentage return. This is a familiar measure to most investors. For situations involving the selection of the best of alternative solutions, we must be careful when using his method. The most intuitive application of the rate of return method will lead to decision errors. For example, consider the following scenario.

Suppose that you are offered three alternatives and must choose one of them. The first alternative has a 1000% rate of return, the second alternative has a 100% rate of return and the third has a 20% rate of return. Which do you select?

You probably chose the first alternative since it earns 10 times the return of second and 50 times the third, but that isn't the whole story. For the first alternative, an investment of one cent today will receive 11 cents in one year. For the second alternative, an investment of $10 today will receive $20 in one year. For the third alternative, an investment of $100 today will receive $120 in one year. Assuming you have the capital, which do you select now?

The first earns 10 cents, the second $10 and the third $20, so given this additional information, you say the choice is obvious: Select the third because it makes the most profit.

Both the rate of return and the profit are unreliable measures when selecting among alternatives because they both neglect part of the information. The rate of return is a relative measure; that is, the ratio of profit to investment. The amount invested is lost in the ratio. The profit subtracts the investment from the income, again losing the amount of the investment.

In fact, all three answers are correct under different situations. You should be able to find the answer by the end of this lesson, which considers two decision problems that involve sets of projects. The first is project selection. Here we desire to choose a subset of projects from a set of proposed projects. The second is how to select the best alternative project from a set of two or more proposals. Although the IRR is the measure used throughout this lesson we use rate of return as the more general term.

Goals
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  • Given a set of projects and an investment budget, use the ROR method for selecting the best subset of projects.
  • Use the ROR method to compare two or more alternatives and select the best.
  • Select and use a study period when alternative lives are different
  • Do incremental analysis for comparing two or more alternatives.
  • Use the Economics add-in for ROR comparisons.
Text
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3.4.5 Internal Rate of Return Method
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Project Selection
An organization has a set of proposed projects from which some subset must be chosen. The IRR provides a measure for project selection.

A primary purpose of most corporations is to invest capital to make profit. The capital comes from investors, debt, recovery of depreciation, and profits retained in the business. An annual exercise for some organizations is to seek proposals for investment projects and then select from the proposals a subset of projects to pursue. Using finance analysis we compute the IRR of each project. These measures are used for project selection. This lesson considers a very simple version of this problem. The text has an entire chapter that addresses many other factors including multiple criteria and risk.

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5 Project Screening and Selection
Read 5.1 for background. The remainder of the chapter is not required.
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Click the QuickTime symbol to see how select projects given the MARR and how to select projects with a fixed investment budget.

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ROR: Project Selection
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Summary of Example
Each year several departments in a corporation propose projects with different investment amounts, annual net incomes and lives. The corporation must choose from the proposals. The corporation's MARR is 25%. The list is below.

With a given MARR and no budget restriction, select the projects that have IRR ≥ MARR.

To apply this rule, first compute the IRR for each project. The MARR of 25% leads to the selection of projects 2, 3, 4, 6, 7, 9 and 10. The total investment in the projects is $10,450.

A related problem arises when a budget limit is imposed. This is the usual case.

When the budget is restricted, rank the projects by IRR and in order of the ranking choose the largest subset that does not exceed the budget.

To apply this rule it is convenient to sort the alternatives by IRR and compute the cumulative investment as below. When the budget is $5,000 we select the subset of projects 7, 6 and 2 with the total investment of $3750. Since adding project 10 exceeds the budget by only $150, we select that one too. Perhaps the budget is flexible.

This solution is only approximate because of the discrete nature of the projects. We can either accept a project or not. Selecting the very best subset is an optimization problem that requires some high-powered methods for solution. The problem is addressed by the Portfolio add-in. The tables above were also obtained via this add-in. For this course we are satisfied with this simple approximation, so you are not required to use the Portfolio add-in. The link is provided for reference.

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Portfolio Add-in
For Reference - Not Required
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The sorted list is sometimes suggested as a method for determining the MARR. The MARR depends on the capital available. To find the MARR we move down the ranking as far as possible until the budget is violated by the next project. The MARR is the IRR of the last project selected. For a $5,000 budget the MARR would be 45%. For a $10,000 budget the MARR is 33%. This leads to the conclusion that the greater the capital available, the smaller the MARR. Companies with little capital will have a high MARR, while companies with plentiful capital will have a low MARR.

Mutually Exclusive Alternatives

To choose among mutually exclusive alternatives, you must use incremental analysis. The IRR of each increment of investment must satisfy the MARR requirement.

The presentation describes how the incremental analysis is used when the rate of return is the criterion. Click the icon to see the presentation.

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ROR: Mutually Exclusive
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A distinction can be drawn between problems where some alternative must be chosen or where it is allowable to choose none. We handle this for problems of the latter variety by including the null alternative in the incremental analysis.

Summary of the Cost-Only Example in the ROR:Mutually Exclusive Lecture
With only costs, the IRR values for the individual alternatives are undefined. Incremental analysis selects the best alternative.

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Cost Only Example
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The result of each analysis depends on the MARR. After the first analysis the alternatives compared may be different when a different MARR is specified.

Different Lives
When the alternatives have different lives, they must be compared over a common study period. We choose the least common multiple of the lives.

Click the QuickTime symbol to see how to use the present worth method when the alternatives have the different lives.

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ROR: Different Lives
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Summary of the Example in the Different Lives movie
This example involves incomes, but the individual rates of returns are irrelevant because we must choose one of the mutually exclusive alternatives. Again incremental analysis is required. The problem is more complex because one of the incremental analyses involves alternatives with different lives.

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Different Lives Example
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The Economics Add-in
You should be able to perform NPW or NAW analyses by hand with equivalency expressions, but for problem solving the Economics add-in can handle problems of arbitrary complexity.

The Economics add-in is easy to use and it should be handy for homework and computer-based exams.

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Economics Add-in: Compare Projects
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The figure below shows projects defined for the machine example introduced in the ROR: Mutually Exclusive movie.

To perform an incremental analysis, we choose Compare Projects from the menu. We select the First alternative, with the smaller investment, as the defender and the Second alternative, with the greater investment, as the challenger. The results for the example are below. When two projects are compared the add-in automatically prepares an incremental analysis. The IRR value of 21.41% reported in the table is the IRR of the increment of investment. Because this value is greater than the MARR, the increment of the Second alternative over the First is justified. Economically speaking, we should choose the Second alternative.

To compare more than two alternatives with the rate of return method, more than one of these 2-way comparisons is required. Follow the incremental method. Starting with the alternative with the smallest investment as the initial defender and incumbent, we must test each increment of investment to see if the IRR of the increment exceeds the MARR. An alternative whose incremental investment is justified becomes the incumbent. The process continues in order of increasing investment until all alternatives have been considered. The final incumbent is the selected alternative.

Division

The Answer to
the Original Question

This discussion should have provided a method to answer the original question.

"You are offered three alternatives and you must choose one of them. The first alternative has a 1000% rate of return, the second alternative has a 100% rate of return and the third has a 20% rate of return. Which do you select?"

You should know at this point in the course that the answer depends on your MARR. From the material presented here, you should have learned the additional lesson that the individual values of the ROR don't tell the whole tale. You must use incremental analysis.

The three alternatives are listed below in order of increasing investment. We have added the Null alternative to make available the do-nothing option.

Alternative

Investment

Return after one year

Null

0

0

A

$0.01

$0.11

B

$10

$20

C

$100

$120

For example, let's say your MARR is 10%. The incremental analysis shows that C is the best decision.

Increment

Challenger

Defender

Investment
(chal - def)

Return
(chal - def)

IRR
(chal - def)

Decision

1

A

Null

0.01

0.11

1000%

Accept A

2

B

A

9.99

19.89

99.1%

Accept B

3

C

B

90.00

100.00

11.1%

Accept C

Decision: Choose C

Let's say your MARR is 15%. The incremental analysis shows that B is the best decision.

Increment

Challenger

Defender

Investment
(chal - def)

Return
(chal - def)

IRR
(chal - def)

Decision

1

A

Null

0.01

0.11

1000%

Accept A

2

B

A

9.99

19.89

99.1%

Accept B

3

C

B

90.00

100.00

11.1%

Reject C

Decision: Choose B

Let's say your MARR is 100%. The incremental analysis shows that A is the best decision.

Increment

Challenger

Defender

Investment
(chal - def)

Return
(chal - def)

IRR
(chal - def)

Decision

1

A

Null

0.01

0.11

1000%

Accept A

2

B

A

9.99

19.89

99.1%

Reject B

3

C

A

99.99

119.89

19.9%

Reject C

Decision: Choose A

Let's say your MARR is 1000%. Boy you sure are greedy! The analysis shows that you should select none.

Increment

Challenger

Defender

Investment
(chal - def)

Return
(chal - def)

IRR
(chal - def)

Decision

1

A

Null

0.01

0.11

1000%

Reject A

2

B

Null

10

20

100%

Reject B

3

C

Null

100

120

20%

Reject C

Decision: Choose Null

So unless you know your MARR there is no correct solution. You should answer:

"The answer could be anything. It depends on my MARR."

The important lesson to learn is that the rate of return for any alternative makes little difference when deciding between mutually exclusive alternatives. Each increment of investment must make the MARR.

Division

Summary

 

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ROR Comparison Summary
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Problems

 

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ROR Comparison Problems
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Division

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Engineering Finance
by Paul A. Jensen
Copyright 2005 - All rights reserved

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