Mutually exclusive projects: making an analysis of several alternatives from which only one can be selected, such as selecting the best way to provide service or to improve an existing operation or the best way to develop a new process, product, mining operation, or oil/gas reserve.
Non-mutually exclusive projects: analyzing several alternatives from which more than one can be selected depending on capital or budget restrictions, such as ranking research, development, and exploration projects to determine the best projects to fund with available dollars.
This lesson focuses on the analysis of mutually exclusive alternatives. Valid discounted cash flow criteria such as rate of return, net present value, and benefit-cost ratio are applied in very different ways in proper analysis of mutually exclusive and non-mutually exclusive alternative investments.
At the successful completion of this lesson, students should be able to:
This lesson will take us one week to complete. Please refer to the Course Syllabus for specific time frames and due dates. Specific directions for the assignment below can be found within this lesson.
Reading | Read Chapter 4 of the textbook. |
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Assignment | Homework 4. |
If you have any questions, please post them to our discussion forum, located under the Modules tab in Canvas. I will check that discussion forum daily to respond. While you are there, feel free to post your own responses if you, too, are able to help out a classmate.
Economic analysis of projects can be divided into two categories:
1) Mutually Exclusive
2) Non-Mutually Exclusive
Mutually Exclusive type analysis is where the investor faces different investment alternatives, but only one project can be chosen for investment. Selecting one project excludes other projects from investment.
Non-Mutually Exclusive assessments are where the investor faces different alternatives, but more than one project can be selected regarding capital or budget constraint.
Example 4-1: Assume an investor has two alternatives, project A and project B, and other opportunities exist to invest at 15% ROR. The total money that investor has is 400,000 dollars.
Project A: Includes investment of 40,000 dollars at present time which yields an income of 40,000 dollars for 5 years and the salvage value at the end of the fifth year is 40,000 dollars.
C=$40,000 | I=$40,000 | I=$40,000 | I=$40,000 | I=$40,000 | I=$40,000 | L=$40,000 | |
A) |
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0 | 1 | 2 | 3 | 4 | 5 |
Project B: Includes investment of 400,000 dollars at the present time which yields the income of 200,000 dollars for 5 years and the salvage value at the end of the fifth year is 400,000 dollars.
C=$400,000 | I=$200,000 | I=$200,000 | I=$200,000 | I=$200,000 | I=$200,000 | L=$400,000 | |
B) |
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0 | 1 | 2 | 3 | 4 | 5 |
C: Cost, I:Income, L:Salvage
ROR analysis for project A:
With trial and error or using the IRR function in Excel, we can calculate . So project A is satisfactory.
ROR analysis for project B:
With trial and error or using the IRR function in Excel, we can calculate . So project B is also satisfactory.
Many people think because project A has a higher ROR, project A has to be selected over project B. But remember, we assumed 400,000 dollars is available for the investment, and the investor can only choose one of the projects. Project A takes just 10 percent of the money and gives 100% ROR, while project B takes the entire 400,000 dollars and gives 50% ROR. If the investor chooses project A and spends 40,000 dollars on this project, the rest of the money can only be invested with a 15% ROR. So, we need one more step that is called incremental analysis to be able to compare two projects and determine which project is better. The incremental analysis helps up to find a common base to compare two projects. To do so, incremental analysis breaks project B into two projects: one is similar to project A and the other is an incremental project.
Project B is equivalent to
Please note that the investing in Project B (requires $400,000) is equivalent to investing
Choosing project A with 100% ROR + investing the rest of money with 15%
Or
Choosing project B, which is equivalent to an investment in project A with 100% ROR+ investment in the incremental project (B-A)
The incremental analysis has to be done for the bigger project minus the smaller one as:
C=$360,000 | I=$160,000 | I=$160,000 | I=$160,000 | I=$160,000 | I=$160,000 | L=$360,000 | ||
B-A |
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0 | 1 | 2 | 3 | 4 | 5 |
This investment gives 44.4 % return.
So, incremental analysis shows that investment in project B is equivalent to investing in A (which gives 100% ROR) plus investing in project B-A (which gives 44%).
Thus, the second alternative, project B, is more desirable.
Please watch the following video (11:56): Mutually exclusive projects (Rate of return analysis).
Considering a discount rate of 15% (minimum rate of return), the NPV for project A and B can be calculated as:
Since the NPV for project A and B is positive at the 15% discount rate (minimum rate of return on investment), then we can conclude that both projects are economically satisfactory. But NPV for project B is higher than A, which means B is a better choice to invest.
We can also calculate the incremental NPV as:
Note that incremental NPV is exactly equal to the difference between NPVA and NPVB:
The incremental at a 15% discount rate is positive, which means the incremental investment is economically satisfactory.
Remember the two decision alternatives that the investor faces:
1) Choosing project A + investing the rest of money with 15%
2) Choosing project B, which is equivalent to an investment in project A + investment in the incremental project (B-A)
The NPV for the first decision is:
1) NPVA + NPV (of investing the remainder of the available money somewhere else with a 15% rate of return)
If an investment return of 15%, then the NPV at a discount rate of 15% for that investment cash flow equals zero. So:
1)
The NPV for the second decision is:
2)
Therefore, it can be concluded that investment in project B is a better decision.
In summary, for net present value analysis of mutually exclusive choices, two requirements need to be tested: 1) the net value on total individual project investment must be positive, and 2) the incremental net value obtained in comparing the total investment net value to the net value of the last smaller satisfactory investment level must be positive. The largest level of investment that satisfies both criteria is the economic choice. Or simply, the project with the largest positive net present value is the best choice.
Note: You can use Microsoft Excel and the NPV function in order to calculate Net Present Value as explained in Example 3-6 in Lesson 3.
Please watch the following video (3:37): Mutually exclusive alternatives
Present value ratio (PVR) also can be applied to analyze two mutually exclusive projects, A and B:
Positive PVR for project A and B indicates that both projects are economically satisfactory. But higher PVR for project A doesn’t necessarily mean project A is better than B for investment and PVR needs to be calculated for an incremental project as well.
Accepting the incremental investment indicated accepting project B over A, even though the total investment ratio on B is less than A. Just as with ROR analysis, the mutually exclusive alternative with bigger ROR, PVR is not necessarily a better mutually exclusive investment. Incremental analysis along with total individual project investment analysis is the key to a correct analysis of mutually exclusive choices.
If mutually exclusive projects that are being analyzed don’t have the same lifetimes (for example, one investment has a length of 8 years and the other alternative example has the length of 12 years), we have to be careful using the parameters that we have learned so far.
We can continue the NPV analysis without any problem for mutually exclusive projects with different lifetimes. This is because NPV analysis considers a common point in time for all projects, which is the present time.
It is also important to know that for NPV analysis, different discount rates may cause different results and may change the ranking of the projects. Thus, the selected discount rate for such should be representative of the opportunity cost of capital for consistent economic decision-making.
For ROR analysis (and other analysis, such as future value, that require a specific point on the timeline) of mutually exclusive projects with different lifetimes, we need to find a common lifetime and analyze the alternatives based on that. This common lifetime is usually the longest lifetime between alternatives.
For ROR analysis, treat all projects as having an equal life that is equal to the longest life project with net revenues and costs of zero in the later years of shorter life projects.
Consider the following two mutually exclusive projects:
Assume a minimum rate of return of 8%
Project A
C=1000 | I=250 | I=250 | I=250 | I=250 | I=250 | I=250 | I=250 |
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0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
Project B
C=2000 | C=3000 | I=1000 | I=1000 | I=1000 | I=1000 | I=1000 | I=1000 | I=1000 | I=1000 | I=1000 |
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0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
First, we need to evaluate each project individually and see if both are economically satisfactory.
Project A evaluation:
For project A: ROR> i*=8%, NPV is positive, B/C is higher than 1, and PVR is positive. So, project A is economically satisfactory.
Project B evaluation:
For project B: ROR> i*=8%, NPV is positive, B/C is higher than 1, and PVR is positive. So, project B is also economically satisfactory.
Please note that project A has a lifetime of seven years, while project B’s lifetime is 10 years. In this case, we chose the project with the longest lifetime (here, project B) as the base case and put zero for the years that project A doesn’t have any payment. Then we deduct the cash flow of project A from the cash flow of project B as incremental cash flow:
-1000 | -3250 | 750 | 750 | 750 | 750 | 750 | 750 | 1000 | 1000 | 1000 |
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0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
Since the incremental project (B-A) is economically satisfactory, we can conclude that project B is better than project A.
Note that
Consider this situation that a manager faces. There are three alternatives:
Year | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
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A | -200 | -350 | 100 | 100 | 150 | 150 | 150 | 150 | 150 | -- | -- |
B | -300 | -400 | 200 | 200 | 200 | 200 | 200 | 200 | 200 | 200 | 200 |
Sell | 150 | -- | -- | -- | -- | -- | -- | -- | -- | -- | -- |
Figure 4-1: cash flow for three alternatives: 1) Development plan A, 2) Development plan B, 3) Sell the property
RORA = 13.2% ROR is less than minimum rate of return of 15%, so the project is not economically satisfactory
. NPV is negative, so the project is not economically satisfactory
. PVR is negative, so the project is not economically satisfactory.
RORB=21.65% ROR is higher than the minimum rate of return of 15%, so the project is economically satisfactory
. NPV is positive, so the project is economically satisfactory
. PVR is positive, so the project is economically satisfactory.
ROR, NPV, and PVR analysis indicate that development plan B is better than investing money at a minimum rate of return of 15%.
The NPV, ROR, and PVR for selling the property:
RORsell = +∞ higher than the minimum rate of return of 15%, so the project is economically satisfactory
NPVsell = +150 is positive, so the project is economically satisfactory
PVRsell = +∞ is positive, so the project is economically satisfactory
Since NPVB is higher than NPVsell, the above analyses show that development plan B is the best economic choice among the three alternatives.
In order to compare development plan B and selling the property, we can also apply the incremental analysis as:
Year | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
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B-sell | -450 | -400 | 200 | 200 | 200 | 200 | 200 | 200 | 200 | 200 | 200 |
which is greater than 15%, so project B is economically satisfactory.
is positive, so it is economically satisfactory.
0.04 is positive, so it is economically satisfactory.
As previously explained, the incremental analysis will always lead to selecting the alternative with the largest individual NPV. Therefore, development plan B is the best economic choice. However, note that project B does not have the highest ROR or PVR.
It is negative, so the project is not economically satisfactory.
. NPV is positive, so the project is economically satisfactory
NPVsell= +150 is positive, so the project is economically satisfactory.
Therefore, at a minimum rate of return of 20%, selling the property is the best economic choice.
As calculations show, results are sensitive to discount rates (minimum rate of return)
Consider two alternatives: development plan B, and selling the property. But assume that development plan B will start in the second year. Which project is the best economic choice at a minimum rate of return of 15%?
Cash flow for these alternatives:
Year | 0 | 1 | 2 | 3 | 4 | -- | 12 |
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B | -- | -- | -300 | -400 | 200 | 200 | 200 |
Sell | -150 | -- | -- | -- | -- | -- | -- |
In this case, NPV indicates that selling the property is the best economic choice. NPV is positive, so it is economically satisfactory.
NPVsell=150 is positive, so it is economically satisfactory.
Incremental analysis can also be done as:
Year | 0 | 1 | 2 | 3 | 4 | -- | 12 |
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B-sell | -150 | -- | -300 | -400 | 200 | 200 | 200 |
RORB-sell = 11.11% which is lower than 15% minimum rate of return
which is negative, so it is not economically satisfactory.
Thus, choosing development plan B overselling the property is not economically acceptable.
So far, we have assumed that minimum rate of return is fixed over the life of the project. But there are situations where other opportunities for investment (that determine the minimum rate of return) can make different rate of returns in different time. Thus, minimum rate of return can change over time. For example, other opportunities for investment of capital can give i*=12% now; and three years from now, we might expect a project that has a return on investment of i*=15%.
For analyses with minimum rate of return that change with time, NPV and PVR are recommended as the best methods. ROR is not a reliable approach for such analyses.
Cash flows for two mutually exclusive investment projects A and B are given as:
C=$40 | I=$20 | I=$20 | I=$20 | I=$20 | I=$20 | L=$40 | |
A) |
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0 | 1 | 2 | 3 | ... | 10 |
C=$50 | I=$25 | I=$25 | I=$25 | I=$25 | I=$25 | L=$50 | |
B) |
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0 | 1 | 2 | 3 | ... | 10 |
C: Cost, I:Income, L:Salvage
Analyze these alternatives, assuming the minimum rate of return for the first and second years is 25% and for third to tenth year it is 15%.
Results indicate that project B is a better economic investment.
After year 2, minimum rate of return changes from 25% to 15%. In order to calculate the NPV of the cash flow, we have to separate the payments that happened at and before year 2 from payments that occurred after year 2.
Payments at year 2 and before that are not going to be affected by the change:
PV of payments from year 0 to year 2:
Project A: Present value of year 0 to year 2 payments
Project B: Present value of year 0 to year 2 payments
But payments after year 2 will be affected by the change.
To calculate the NPV of those payments and apply the change in i, first, we need to discount all the payments occurred after year 2 to this year (we set the year 2 as the base year) by i* = 15% and we calculate value of all payments at year 2:
Project A: Value of year 3 to year 10 payments at year 2
Project B: Value of year 3 to year 10 payments at year 2
Second, we discount the year 2 values for 2 years by i* = 25%to get the present value (value at year 0) of the payments:
Project A: Present Value of year 3 to year 10 payments
Project B: Present Value of year 3 to year 10 payments
In the end, we add all the values together:
Another Method:
You can also treat each payment separately. This method is especially helpful when payments are not equal or when you are using spreadsheet to calculate the NPV.
We separate the payments that happened at and before year 2 from payments after year 2. Payments at and before year 2 will be discounted just by 25%:
PV of payments from year 0 to year 2:
Project A: PV year 0 to year 2
Project B: PV year 0 to year 2
For payments after year 2, first we calculate their value at year 2:
Project A: Value of year 3 to year 10 payments at year 2
Project B: Value of year 3 to year 10 payments at year 2
Second step, we discount the year 2 values for 2 years by i* = 25% to get the present value (value at year 0) of the payments:
Project A: Present Value of year 3 to year 10 payments
Project B: Present Value of year 3 to year 10 payments
In the end we add all the values together:
If you are using Microsoft Excel or another spreadsheet to calculate the Net Present Value for the cash flow that has different discount rates over the life of project, be careful! You can not use the NPV function. However, you can calculate the Net Present Value by making a summation over calculated discounted cash flow. Figure 4-3 displays how Net Present Value for Project A cash flow with a changing minimum rate of return can be calculated. Note the formula in the cell D3 to D12.
The Rate of Return or Growth Rate of Return: With either regular ROR or Growth ROR analysis of mutually exclusive alternatives, you must evaluate both total investment ROR and incremental investment ROR, selecting the largest investment for which both are satisfactory. Use a common evaluation life for Growth ROR analysis of unequal life alternatives, normally the life of the longest life alternative assuming net revenues and costs are zero in the later years of shorter life alternatives.
Net Value Analysis: With NPV analysis, you want the mutually exclusive alternative with the largest net value – because this is the alternative with the largest investment that has both a positive total investment net value and a positive incremental net value compared to the last satisfactory smaller investment.
The Rate of Return or Growth Rate of Return: Regular ROR analysis cannot be used to consistently rank non-mutually-exclusive alternatives. Use Growth ROR, and rank the alternatives in the order of decreasing Growth ROR. This will maximize profit from available investment capital. Use a common evaluation life for Growth ROR analysis for unequal life alternatives, normally the life of the longest alternative.
Net Value Analysis: With NPV analysis of non-mutually exclusive projects, select the group of projects that will maximize cumulative net value for the dollars available to invest. This does not necessarily involve selecting the project with the largest net value on individual project investment.
You have reached the end of Lesson 4! Double-check the to-do list on the Lesson 4 Overview page [1] to make sure you have completed all of the activities listed there before you begin Lesson 5.