## Friday, October 21, 2011

### Comparing Projects With Unequal Lives

CFA Level 1 - Corporate Finance

As mentioned previously, NPV and IRR can sometimes lead to conflicting results in the analysis of mutually exclusive projects. One reason for this potential problem is the timing of the cash flows of the mutually exclusive projects. As a result, we need to adjust for the timing issue in order to correct this problem.
There are two methods used to make the adjustments:
1.Replacement-chain method
2.Equivalent annual annuity

Example:
Once again, assume Newco is planning to add new machinery to its current plant. There are two machines Newco is considering, with cash flows as follows:

Figure 11.8: Discounted cash flows for Machine A and Machine B

Compare the two projects with unequal lives using both the replacement-chain method and the equivalent annual annuity (EAA) approach.

1.     Replacement-Chain Method
In this example, Machine A has an operating lifespan of six years. Machine B has an operating lifespan of three years. The cash flows for each project are discounted by Newco's calculated WACC of 8.4%.
• NPV of Machine A is equal to \$2,926.
• NPV of Machine B is equal to \$1,735.

The initial analysis indicates that Machine A, with the greater NPV, should be the project chosen.
• The IRR of Machine A is equal to 8.3%.
• The IRR of Machine B is equal to 15.5%.

This analysis indicates that Machine B, with the greater IRR, should be the project chosen. The NPV analysis and the IRR analysis have given us differing results. This is most likely due to the unequal lives of the two projects. As such, we need to analyze the two projects over a common life.

For Machine A (project 1), the lifespan is six years. For Machine B (project 2), the lifespan is three years. Given that the lifespan of the longest project is six years, in order to measure both over a common life, we must adjust the lifespan of Machine B to six years.

Because the lifespan of Machine B is three years, the lifespan of this project needs to be doubled to equal the six-year lifespan of Machine A. This indicates that another Machine B would have to be purchased (to get two machines with a lifespan of three years each) to get to the six-year lifespan of Machine A - hence, the replacement-chain method.The new cash flows would be as follows:

Figure 11.9: Cash flows over a common life

• NPV of Machine A remains \$2,926.
• NPV of Machine B is now \$3,098 given the adjustment.

The initial analysis indicates that Machine B, with the greater NPV, should be the project chosen. Recall, this is different from our first analysis where Machine A was chosen given its greater NPV.
•  The IRR of Machine A remains 8.3%.
• The IRR of Machine B remains 15.5%.

 Look Out!Note, while the NPV has changed given the additional cash flows, the IRR for the projects remain the same.

This analysis indicates that Machine B, with the greater IRR, should be the project chosen. Recall, this is the same as our first analysis, where Machine B was chosen given its greater IRR.

With the cash flows adjusted with the replacement-chain method, both the NPV and the IRR arrive at the same conclusion. With this adjusted analysis, Machine B (project 2), should be the project accepted.
2.     Equivalent-Annual-Annuity Approach
While easy to understand, the replacement-chain method can be time consuming. A simpler approach is the equivalent-annual-annuity approach.
This is the procedure for determining EAA
:1) Determine the projects' NPVs.
2) Find each project's EAA, the expected payment over the project's life, where the future value of the project would equal zero.
3) Compare the EAA of each project and select the project with the highestEAA.
From our example, the NPV of each project is as follows:
-NPV of Machine A is equal to \$2,926.
-NPV of Machine B is equal to \$1,735.

To determine each project's EAA, it is best to use your financial calculator.

-
For, Machine A (project 1), our assumptions are as follows:

i = 8.4% (the company's WACC)
n = 6PV = NPV = -2,926
FV = 0
Find for PMT

For Machine A, the EAA (the calculated PMT) is \$640.64.

-
For Machine B (project 2), our assumptions are as follows:

i = 8.4% (the company's WACC)
n = 3PV = NPV = -1,735
FV = 0
Find for PMT

For Machine B, the EAA (the calculated PMT) is \$678.10.

Machine B should be the project chosen as it has the highest EAA, which is \$678.10, relative to Machine A whose EAA is \$640.64.

Inflation Effects on Capital Budgeting Analysis
Inflation exists and should not be forgotten when making capital-budgeting decisions. It is important to build inflation expectations into the analysis. If inflation expectations are left out of the capital-budgeting analysis, the NPV calculated from the biased cash flows will be incorrect.

As an example, suppose Newco unintentionally leaves out its inflation expectations when determining the plant addition. Since inflation expectations are included in the WACC, and PV of each cash flow is discounted by the WACC, the NPV will be incorrect and have a downward bias.

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