Mutually Exclusive Projects with Unequal Project Lives
Topic 4. Capital Budgeting Decision Methods (part II)
In this topic we examine nonsimple projects, projects that have a negative initial cash flow, in addition to one or more negative future cash flows. Next, we explore projects that have multiple IRRs. Finally, we discuss how to compare mutually exclusive projects with unequal project lives.
NonSimple Projects
Most capital budgeting projects start with a negative cash flow – the initial investment – at t_{0} followed by positive future cash flows. Such projects are called simple projects. Nonsimple projectsare projects that have one or more negative future cash flows after the initial investment.
To illustrate a nonsimple project, consider Project N, the expected cash flows for a nuclear power plant project. The initial investment of $500 million is a negative cash flow at t_{0}, followed by positive cash flows of $25 million per year for thirty years as electric power is generated and sold. At the end of the useful life of the project, the storage of nuclear fuel and the shutdown safety procedures require cash outlays of $100 million at the end of year 31.
With a 20 percent discount rate, an initial investment of $500 million, a thirtyyear annuity of $25 million, and a shutdown cash outlay of $100 million in year 31, the NPV of Project N follows (in million):
NPV =  $500 + $25  $100 = ( $500 × 1) + ($25 × 4.979) – ($100×0.0035) =  $375.88
We find that at a discount rate of 20 percent, Project N has a negative NPV of 375.878, so the firm considering Project N should reject it.
Multiple IRRs
Some projects may have more than one internal rate of return. That is, a project may have several different discount rates that result in a net present value of zero.
Here is an example. Suppose Project Q requires an initial cash outlay of $160 000 and is expected to generate a positive cash flow of $1 000 000 in year one. In year two, the project will require an additional cash outlay in the amount of $1 million. The cash flows for Project Q are shown in the following table:
Period  t_{0}  t_{1}  t_{2} 
Cash flows   160 000  1 000 000   1 000 000 
We find the IRR of Project Q by using the trial and error procedure. When r = 25%, the NPV is zero.
0 =   $160000 = $800000  $640000  $160000 = $0


Since 25 percent causes the NPV of Project Q to be zero, the IRR of the project must be 25%. But wait! If we had tried r = 400%, the IRR calculation would look like this:
0 =   $160000 = $200000  $40000  $160000 = $0
Since 400 percent results in an NPV of zero, 400 percent must also be the IRR of the Project Q. Figure below shows the NPV profile for Project Q. By examining this graph we see how 25% and 400% both make the NPV equal to zero.
As the graph shows, Project Q’s NPV profile crosses the horizontal axis (has a zero value) in two different places, at discount rates of 25% and 400%.
Project Q had two IRRs because the project’s cash flows changed from negative to positive (at t_{1}) and then from positive to negative (at t_{2}). It turns out that a nonsimple project may have (but does not have to have) as many IRRs as there are sign changes. In this case, two sign changes resulted in two internal rates of return.
Whenever we have two or more IRRs for a project, the IRR method is not a useful decisionmaking tool. Remember the IRR accept/reject decision rule: firms should accept projects with IRRs higher than the discount rate, and reject projects with IRRs lower than the discount rate. With more than one discount rate, decision makers will not know which IRR to use for the accept/reject decision. In projects that have multiple IRRs, then,switch to the NPV method.
100  
50  
0  
50  
100  
150 
0 0.25 4
Mutually Exclusive Projects with Unequal Project Lives
When mutually exclusive projects have different expected useful lives, selecting among the projects requires more than comparing the projects’ NPVs. To illustrate, suppose you are a business manager considering a new business telephone system. One is the Cheap Talk System, which requires an initial cash outlay of $10 000 and is expected to last three years. The other is the Rolles Voice System, which requires an initial cash outlay of $50 000 and is expected to last twelve years. The Cheap Talk System is expected to generate positive cash flows of $5 800 per year for each of its three years of life. The Rolles Voice System is expected to generate positive cash flows of $8 000 per year for each of its twelve years of life.


To decide which project to choose, we first compute and compare their NPVs. Assume the firm’s required rate of return is 10 percent. We solve for the NPVs as follows:
NPV of Cheap Talk:
NPV = $5800  $10000 = $5800(2.487)  $10000 = $4424
NPV of Rolles Voice:
NPV = $8000  $50000 = $8000(6.814)  $50000 = $4510
We find that Project Cheap Talk has an NPV of $4 424, compared to Project Rolles’ NPV of $4 510. We might conclude based on this information that the Rolles Voice System should be selected over the Cheap Talk System because it has the higher NPV. However, before making that decision, we must assess how Project Cheap Talk’s NPV would change if its useful life were twelve years, not three years.
Comparing Projects with Unequal Lives
Two possible methods that allow financial managers to compare projects with unequal lives are the replacement chain approach and the equivalent annual annuity (EAA) approach.
The Replacement Chain Approach. The replacement chain approach assumes each of the mutually exclusive projects can be replicated, until a common time period has passed in which the projects can be compared. The NPVs for the series of replicated projects are then compared to the projects with the longer life. An example illustrates this process. Project Cheap Talk could be repeated four times in the same time span as the twelveyear Rolles Voice Project. If a business replicated project Cheap Talk four times, the cash flows would look like this:


t_{0}  t_{1}  t_{2}  t_{3}  t_{4}  t_{5}  t_{6}  t_{7}  t_{8}  t_{9}  t_{10}  t_{11}  t_{12} 
+$5.8  +$5.8  +$5.8  +$5.8  +$5.8  +$5.8  +$5.8  +$5.8  +$5.8  +$5.8  +$5.8  +$5.8  
 $10   $10   $10   $10 
The NPV of this series of cash flows, assuming the discount rate is 10 percent, is (in thousands) $12 121. Each cash flow, be it positive or negative, is discounted back the appropriate number of years to get the NPV of the four consecutive investments in the Cheap Talk Systems.
The NPV of $12 121 for Cheap Talk System is the sum of the NPVs of the four repeated Cheap Talk projects, such that the project series would have a life of twelve years, the same life as the Rolles Voice System Project. We are now comparing apples to apples. Cheap Talk’s replacement chain NPV is $12 121, while the NPV of Project Rolles Voice is $4 510 over the same twelve year period. If a firm invested in project Cheap Talk four successive times, it would create more value than investing in one project Rolles Voice.
The Equivalent Annual Annuity (EAA). The equivalent annual annuity (EAA) approach converts the NPVs of the mutually exclusive projects into their equivalent annuity values. The equivalent annual annuity is the amount of the annuity payment that would give the same present value as the actual future cash flows for that project. The EAA approach assumes that you could repeat the mutually exclusive projects indefinitely as each project came to the end of its life.
The equivalent annual annuity (EAA) is calculated by dividing the NPV of a project by the present value interest factor for an annuity (PVIFA) that applies to the project’s life span.
Formula for an equivalent annual annuity (EAA)
EAA =
The NPVs of Cheap Talk ($4 424) and Rolles Voice ($4 510) were calculated earlier, assuming required rate of return of 10 percent. With the project’s NPV and the discount rate, we calculate each project’s EAA, as follows:


EAA of Project Cheap Talk:
EAA = = $1778.96
EAA of Project Rolles:
EAA = = $661.9
The EAA approach decision rule calls for choosing whichever mutually exclusive project has the highest EAA. Our calculations show that Project Cheap Talk has an EAA of $1778.96 and Project Rolles Voice System has an EAA of $661.90. Because Project Cheap Talk’s EAA is higher than Project Rolles’, Project Cheap Talk should be chosen.
Both the replacement chain and the EAA approach assume that mutually exclusive projects can be replicated. If the projects can be replicated, then either the replacement chain or the equivalent annual annuity methods should be used because they lead to the same correct decision. Note in our case that the EAA method results in the same project selection (Project Cheap Talk) as the replacement chain method. If the projects can not be replicated, then the normal NPVs should be used as the basis for the capital budgeting decisions.
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