## Evaluating Equipment Options with Different Lives

The final problem we consider involves choosing among different possible systems, equipment setups, or procedures. Our goal is to choose the most cost-effective. The approach we consider here is only necessary when two special circumstances exist. First, the possibilities under evaluation have different economic lives. Second, and just as important, we will need whatever we buy more or less indefinitely. As a result, when it wears out, we will buy another one.

We can illustrate this problem with a simple example. Imagine we are in the business of manufacturing stamped metal subassemblies. Whenever a stamping mechanism wears out, we have to replace it with a new one to stay in business. We are considering which of two stamping mechanisms to buy.

Machine A costs \$100 to buy and \$10 per year to operate. It wears out and must be replaced every two years. Machine B costs \$140 to buy and \$8 per year to operate. It lasts for three years and must then be replaced. Ignoring taxes, which one should we go with if we use a 10 percent discount rate?

In comparing the two machines, we notice that the first is cheaper to buy, but it costs more to operate and it wears out more quickly. How can we evaluate these trade-offs? We can start by computing the present value of the costs for each:

Ross et al.: Fundamentals of Corporate Finance, Sixth Edition, Alternate Edition

IV. Capital Budgeting

10. Making Capital Investment Decisions

PART FOUR Capital Budgeting equivalent annual cost (EAC)

The present value of a project's costs calculated on an annual basis.

Machine A: PV Machine B: PV

Notice that all the numbers here are costs, so they all have negative signs. If we stopped here, it might appear that A is the more attractive because the PV of the costs is less. However, all we have really discovered so far is that A effectively provides two years' worth of stamping service for \$117.36, whereas B effectively provides three years' worth for \$159.89. These costs are not directly comparable because of the difference in service periods.

We need to somehow work out a cost per year for these two alternatives. To do this, we ask the question, What amount, paid each year over the life of the machine, has the same PV of costs? This amount is called the equivalent annual cost (EAC).

Calculating the EAC involves finding an unknown payment amount. For example, for Machine A, we need to find a two-year ordinary annuity with a PV of -\$117.36 at 10 percent. Going back to Chapter 6, we know that the two-year annuity factor is:

For Machine A, then, we have:

PV of costs = -\$117.36 = EAC X 1.7355 EAC = -\$117.36/1.7355 = -\$67.62

For Machine B, the life is three years, so we first need the three-year annuity factor:

We calculate the EAC for B just as we did for A:

PV of costs = -\$159.89 = EAC X 2.4869 EAC = -\$159.89/2.4869 = -\$64.29

Based on this analysis, we should purchase B because it effectively costs \$64.29 per year versus \$67.62 for A. In other words, all things considered, B is cheaper. In this case, the longer life and lower operating cost are more than enough to offset the higher initial purchase price.

### Equivalent Annual Costs

This extended example illustrates what happens to the EAC when we consider taxes. You are evaluating two different pollution control options. A filtration system will cost \$1.1 million to install and \$60,000 annually, before taxes, to operate. It will have to be completely replaced every five years. A precipitation system will cost \$1.9 million to install, but only \$10,000 per year to operate. The precipitation equipment has an effective operating life of eight years. Straight-line depreciation is used throughout, and neither system has any salvage value. Which option should we select if we use a 12 percent discount rate? The tax rate is 34 percent.

We need to consider the EACs for the two systems because they have different service lives and they will be replaced as they wear out. The relevant information can be summarized as follows:

Ross et al.: Fundamentals of Corporate Finance, Sixth Edition, Alternate Edition

IV. Capital Budgeting

10. Making Capital Investment Decisions