 # Equivalent Annual Annuity, EAA By Yuriy Smirnov Ph.D.

## Definition

The equivalent annual annuity (EAA) method is used in capital budgeting to rank mutually exclusive projects with unequal life spans. The concept is based on assuming that a project is an ordinary annuity with the same life span, its net present value (NPV) is equal to the present value of this annuity, and the cost of capital is equal to the required annual rate. The next assumption is that such an ordinary annuity is an ordinary perpetuity and its present value is equal to the NPV of a project.

## Formula

As a first step, we need to calculate the NPV.

 NPV = N CFt Σ (1 + r)t t = 0

Here, CFt is the cash flow at the end of the relevant year t, r is the cost of capital, and N is the life of a project in years.

As a second step, the equivalent annual annuity of a project should be calculated. The EAA formula can be expressed as follows:

 EAA = NPV × r 1 - (1 + r)-N

Here, the NPV is the net present value of a project, r is the cost of capital, and N is the life of a project in years.

As a third step, we need to calculate the present value of an ordinary perpetuity.

 NPV of an Ordinary Perpetuity = EAA r

## Decision Rule

When several mutually exclusive projects with unequal life spans are compared, the one with the highest EAA value should be accepted.

## Example

Company C is considering two mutually exclusive projects with the same initial investment of \$20,000,000. The life span of Project L is 5 years, and the life span of Project S is 4 years. The cost of capital is 12%.

 Initial Cost, CF0 Cash flows at the end of relevant year, CFt 0 1 2 3 4 5 Project L -\$20,000,000 \$5,000,000 \$5,500,000 \$7,500,000 \$6,000,000 \$5,500,000 Project S -\$20,000,000 \$6,250,000 \$7,000,000 \$7,500,000 \$7,250,000

As the first step, we need to calculate the NPV of both projects using the equation above.

 NPV of Project L = -\$20,000,000 + \$5,000,000 + \$5,500,000 + \$7,500,000 + \$6,000,000 + \$5,500,000 = \$1,121,160.08 (1 + 0.12)1 (1 + 0.12)2 (1 + 0.12)3 (1 + 0.12)4 (1 + 0.12)5

 NPV of Project S = -\$20,000,000 + \$6,250,000 + \$7,000,000 + \$7,500,000 + \$7,250,000 = \$988,111.64 (1 + 0.12)1 (1 + 0.12)2 (1 + 0.12)3 (1 + 0.12)4

The second step involves calculating the equivalent annual annuity for both projects.

 EAA of Project L = \$1,121,160.08 × 0.12 = \$311,020.72 1 - (1 + 0.12)-5

 EAA of Project S = \$988,111.64 × 0.12 = \$325,320.38 1 - (1 + 0.12)-4

Project S has a higher EAA value than Project L, so Company C should accept it.

If both annuities are perpetuities, their present value will be as follows:

 NPVL = \$311,020.72 = \$2,591,839.30 0.12

 NPVS = \$325,320.38 = \$2,711,003.16 0.12

Thus, under the equivalent annual annuity approach, Project S has a higher net present value and should be accepted.