Question

13. Projects with different lives: Explain the concept of equivalent annual cost and how it is used to compare projects with different lives.

Answer 1

13. Projects with different lives: Explain the concept of equivalent annual cost and how it is used to compare projects with different lives.

Answer : The equivalent annual cost (EAC) is the annualized cost of an investment stated in nominal dollars. In other words, it is the annual payment from an annuity with a life equal to that of a project that has the same NPV as the project. Since it is a measure of the annual cost or cash inflow from a project, the EAC for one project can be compared directly with the EAC from another project, regardless of the lives of those two projects.

Equivalent annual cost (EAC) is the annual cost of owning and maintaining an asset determined by dividing the net present value of the asset purchase, operations and maintenance cost by the present value of annuity factor. It is a capital budgeting tool used by companies to compare assets with unequal useful lives. The same concept can be applied to analyse projects which have unequal useful lives.

Let’s say your company must decide between installing tube-lights or LED lights at its office. A tube light has a useful life of 2 years and an LED lights has useful life of 4 years. 200 tube-lights costs $1,000 and consume electricity of $18,000 per annum. 200 LED lights will cost $3,000 and consume electricity of $12,000 per annum.

You must be wondering how to approach it because each light has different useful life. The concept of equivalent annual cost is relevant in such situations.

Formula

We just need to find the net present value of both assets. There is no cash inflow here, so it will essentially be negative. Net present value can be calculated by discounting the future cash flows and subtracting the asset cost:

NPV =	CF1	+	CF2	+....+	CFn	− I
	(1 + r)1		(1 + r)2		(1 + r)n

Now that we have the net present value, we must convert it to annual terms which we can do by treating the net present value obtained as the present value of an ordinary annuity of duration equal to the asset life.

NPV = EAC ×	1 − (1 + r)-n
	r

Just a bit of readjustment:

Equivalent Annual Cost =	NPV × r
	1 − (1 + r)-n

Where r is the periodic discount rate, which equals annual percentage rate divided by number of periods per year, and n is the number of years.

Example

Assuming a discount rate of 10%, find out which type of lights your company must install.

Net present value of the tube-lights (NPV_TUBE) is −$26,976:

NPVTUBE = -$18,000 ×	1 − (1 + 10%)-2	− $1,000	= − $33,240
	10%

Net present value of the LED lights (NPV_LED) is -$33,491:

NPVLED = -$12,000 ×	1 − (1 + 10%)-4	− $3,000	= − $40,538
	10%

Now, we need to divide the net present value amount by the present value factor for annuity of 2 years and 4 years respectively at 10% interest rate:

EACTUBE =	− 33,240 × 10%	= − 19,152
	1 − (1 + 10%)-2

EACLED =	− 40,538 × 10%	= − 12,789
	1 − (1 + 10%)-4

The minus (−) sign indicates that it is a cost and not a benefit. Since LED lights have lower equivalent annual cost, they should be the preferred option. Equivalent annual annuity is a tool similar to equivalent annual cost but primarily used in the context of evaluation of project benefits.