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LONG- VERSUS SHORT-LIVED EQUIPMENT

Suppose the firm is forced to choose between two machines, D and E. The two machines are designed differently but have identical capacity and do exactly the same job. Machine D costs $15,000 and will last 3 years. It costs $4,000 per year to run. Machine E is an economy model, costing only $10,000, but it will last only 2 years and costs $6,000 per year to run. Because the two machines produce exactly the same product, the only way to choose between them is on the basis of cost. Suppose we compute the present value of the costs:

Should we take machine E, the one with the lower present value of costs? Not necessarily. All we have shown is that machine E offers 2 years of service for a lower cost than 3 years of service from machine D. But is the annual cost of using E lower than that of D?

Suppose the financial manager agrees to buy machine D and pay for its operating costs out of her budget. She then charges the plant manager an annual amount for use of the machine. There will be three equal payments starting in Year 1. Obviously, the financial manager has to make sure that the present value of these payments equals the present value of the costs of machine D, $25,690. The payment stream with such a present value when the discount rate is 6 percent turns out to be $9,610 a year. In other words, the cost of buying and operating machine D is equivalent to an annual charge of $9,610 a year for 3 years. This figure is therefore termed the equivalent annual cost of machine D.

How did we know that an annual charge of $9,610 has a present value of $25,690? The annual charge is a 3-year annuity. So we calculate the value of this annuity and set it equal to $25,690:

Equivalent annual cost АГАз 3-year annuity factor = PV costs of D = $25,690 If the cost of capital is 6 percent, the 3-year annuity factor is 2.673. So

We see now that machine D is better, because its equivalent annual cost is less ($9,610 for D versus $11,450 for E). In other words, the financial manager could afford to set a lower annual charge for the use of D.

We thus have a rule for comparing assets of different lives: Select the machine that has the lowest equivalent annual cost.

Think of the equivalent annual cost as the level annual charge6 necessary to recover the present value of investment outlays and operating costs. The annual charge continues for the life of the equipment. Calculate equivalent annual cost by dividing the appropriate present value by the annuity factor.

Equivalent Annual Cost

You need a new car. You can either purchase one outright for $15,000 or lease one for 7 years for $3,000 a year. If you buy the car, it will be worth $500 to you in 7 years. The discount rate is 10 percent. Should you buy or lease? What is the maximum lease you would be willing to pay?

The present value of the cost of purchasing is

The equivalent annual cost of purchasing the car is therefore the annuity with this present value:

Therefore, the annual lease payment of $3,000 is less than the equivalent annual cost of buying the car. You should be willing to pay up to $3,028 annually to lease.

6 This introduction to equivalent annual cost is somewhat simplified. For example, equivalent annual costs should be escalated with inflation when inflation is significant and the equipment long-lived. This would require us to equate equipment cost to the present value of a growing annuity.

EQUIVALENT ANNUAL COST The cost per period with the same present value as the cost of buying and operating a machine.