Cost Comparisons

Some investment decisions involve the purchase of equipment which provides benefits that clearly outweigh the costs. For example, a sprinkler system may be required before a building can be used, a salesman may need an automobile, an office may need a copier. In such cases the issue is really that of obtaining the equipment at the lowest cost. Consequently, you need only examine those areas in which alternatives differ.

Example B.1

As an example, suppose that an office needs a heavy duty copier and has to choose between two alternatives, Model I and Model II. Both models will do a satisfactory job. The characteristics and costs of the two models follow:

Assume the operating cost is paid at the beginning of each year.

As with our earlier work, we must allow for the time value of money – amounts paid early are more expensive than amounts paid later since the funds to make later payments are assumed to be invested until they are needed. We will assume the company expects a rate of return of 12% effective.

All amounts should be discounted to find the Net Present Value of the cash flows.

For Model I, we have:

For Model II, we have:

These net present values represent the amount of money the company would have to set aside as a lump sum, which, with its earnings, would provide the payments necessary for the copiers. Clearly in this case Model I would be the cheapest.

If, however, no allowance was made for the time of payments and the amounts were simply totaled (equivalent to a zero rate of return), then the costs of the copiers would appear exactly the same.

Effects of Different Lifetimes

In the copier example above, the two copiers were expected to last for the same time. If alternatives are expected to last for different times, their costs can be compared by putting them on a per year or per month basis. To allow for the time value of money in such cases, you can use the annuity payments which would have as a present value the NPV of the costs. If this is done on an annual basis, the annuity payment is called the Equivalent Uniform Annual Cost (EUAC).

Example B.2

Suppose that there had been a third alternative copier with the following characteristics:

MODEL III

This one would have to be compared with Model I (the cheaper of the other two) and the question would be: Is the extra initial cost of Model III justified by the longer useful life?

To answer this, start by finding the NPV of Model III costs.

The NPV cannot be simply compared to that of Model I since the NPV is for a different time of use. Instead, we place it on an annual basis by finding the equivalent yearly annuity for each model.

For Model I:

[latex]n = 3,\, i = 12%, \,PV = $12,812.43, \, FV = 0[/latex]

from which you will find the payment is -$5,334.44.

This is the Equivalent Uniform Annual Cost (EUAC), i.e., the annual payment equivalent in value to the set of payments required for the given asset (in this case Model I).

For Model III:

[latex]n = 4,\, i = 12%, \,PV = $16,850.38, \,FV = 0[/latex]

from which the payment= EUAC = – $5,547.72.

Hence, this alternative would be more expensive than Model I.

Capitalized Cost

Another approach to dealing with unequal lives is to view the equipment as being replaced at the end of each cycle with an identical piece of equipment with identical costs. Then the NPV of all of the cash flows is found. Such an NPV is called the Capitalized Cost.

For Model I above this can be found by using the EUAC and the perpetuity formula:

[latex]PV = \frac{PMT}{i} = \frac{-$5,334.44}{0.12} = -$44,453.67[/latex]

Similarly for Model III:

[latex]PV = \frac{PMT}{i} = \frac{-$5,547.72}{0.12} = -$46,231.00[/latex]

again showing that Model I is cheapest.

The capitalized cost method of comparison is particularly suited to constructions such as bridges which are often viewed as never to be replaced, simply to be built and maintained.

Example B.3

Suppose a small bridge is to be built for $4,000,000 and to be maintained indefinitely for payments of $35,000 a year, starting at the end of the first year. Then the capitalized cost at a rate of return of 10% effective would be:

[latex]\text{Capitalized Cost} = $4,000,000 + \frac{$35,000}{0.10} =$7,500,000[/latex]

this being the amount of money set aside now which, with earnings, would make all payments for the bridge.

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