# Equivalent annual cost

In finance, the equivalent annual cost (EAC) is the cost per year of owning and operating an asset over its entire lifespan.

EAC is often used as a decision making tool in capital budgeting when comparing investment projects of unequal lifespans. For example if project A has an expected lifetime of 7 years, and project B has an expected lifetime of 11 years it would be improper to simply compare the net present values (NPVs) of the two projects, unless neither project could be repeated.

EAC is calculated by dividing the NPV of a project by the present value of an annuity factor. Equivalently, the NPV of the project may be multiplied by the loan repayment factor.

$EAC = \frac{NPV}{A_{t,r}}$

The use of the EAC method implies that the project will be replaced by an identical project.

## A practical example

A manager must decide on which machine to purchase:

Machine A
Investment cost $50,000 Expected lifetime 3 years Annual maintenance$13,000

Machine B
Investment cost $150,000 Expected lifetime 8 years Annual maintenance$7,500

The cost of capital is 5%.

The EAC for machine A is: ($50,000/$A_{3,5}$)+$13,000=$31,360 The EAC for machine B is: ($150,000/$A_{8,5}$)+$7,500=$30,708
The conclusion is to invest in machine B since it has a lower EAC.
Note: The loan repayment factors (A values) are for t years (3 or 8 years) and 5% cost of capital. $A_{3,5}$ is given by $\frac{1 - 1/(1.05)^3}{0.05}$= 2.723 and $A_{8,5}$ is given by $\frac{1 - 1/(1.05)^8}{0.05}$= 6.463. (See ordinary annuity formulae for a derivation.) The larger an A value is, the greater the present value is on a succession of future annuity payments, thus contributing to a smaller annual cost.

Alternative method:1

The manager calculates the PV of the machines:

Machine A EAC=$85,400/$A_{3,5}$=$31,360
Machine B EAC=$198,474/$A_{8,5}$=$30,708
Note: To get the numerators add the present value of the annual maintenance to the purchase price. For example, for Machine A: 50,000 + 13,000/1.05 + 13,000/(1.05)^2 + 13,000/(1.05)^3 = 85,402.

The result is the same, although the first method is easier it is essential that the annual maintenance cost is the same each year.

Alternatively the manager can use the NPV method under the assumption that the machines will be replaced with the same cost of investment each time. This is known as the chain method since 8 repetitions of machine A are chained together and 3 repetitions of machine B are chained together. Since the time horizon used in the NPV comparison must be set to 24 years (3*8=24) in order to compare projects of equal length, this method can be slightly more complicated than calculating the EAC. In addition, the assumption of the same cost of investment for each link in the chain is essentially an assumption of zero inflation, so a real interest rate rather than a nominal interest rate is commonly used in the calculations.