# Equivalent Annual Annuity

Equivalent annual annuity (EAA) is an approach used in capital budgeting to choose between mutually exclusive projects with unequal useful lives. It assumes that the projects are annuities, calculates net present value for each project, and then finds annual cash flows that when discounted at the relevant discount rate for the life of the relevant project, would equal the net present value for that project.

**Decision Rule:** The project with higher equivalent annual annuity is preferred.

The main advantage of the equivalent annual annuity approach lies in that it considers the impact of time value of money and it is easier to apply in practice than the replacement chain method. It is because we do not need to lay out the complete cash flow time line till the common life of the mutually-exclusive projects. All we need is to find NPV of each project and then use the EAA formula.

However, the EAA approach suffers from the following disadvantages:

- It assumes that the projects
*can*and*will*be repeated indefinitely. It is not likely that an alternate and equally-profitable opportunity will be available exactly at the end of the project; and even if any such opportunity exists, whether the company would want to reinvest. - It ignores inflation and assumes that that the cash flows, costs and initial investment will remain the same for each iteration of the projects.

Equivalent annual cost (EAC) is a tool similar to equivalent annual annuity with is used to calculate annualized cost of different alternatives.

## Formula

Once we have determined the net present value of a project, its equivalent annual annuity can be calculated using the following formula:

EAA = | Net Present Value | ||

1 − (1 + r)^{-n} | |||

r |

Where *r* is the discount rate and *n* is the life of the project.

The above formula is simply a modification of the formula for present value of an annuity:

PV of Annuity (NPV) = Payment (EAA) × | (1 − (1 + r)^{-n} |

r |

## Example

Combustion Systems is engaged in development of car racing games. The company must choose between two games: Outback and Accelerate. Outback introduces a revolutionary technology called SRP and is expected to stay fresh for 5 years. Accelerate is more of a low-cost traditional racing game and is sort of an upgrade to Nitrous, the current best-seller. Accelerate is expected to generate sales for only 2 years.

Following is the schedule of each project's cash flows:

Year | Outback | Accelerate |
---|---|---|

0 | (15,000,000) | (8,000,000) |

1 | 5,000,000 | 8,000,000 |

2 | 6,000,000 | 2,000,000 |

3 | 8,000,000 | |

4 | 3,000,000 | |

5 | 2,000,000 |

Since Outback is revolutionary, it involves hiring external consultants to help with optimization and testing of the game, and hence the higher initial investment. Outback has higher risk which warrants use of a 10% discount rate as compared to a rate of 8% applicable to Accelerate.

*Which project the company should go ahead with?*

### Solution

The first step in the equivalent annual annuity approach is calculation of relevant net present value:

Year | Outback | Accelerate | ||||
---|---|---|---|---|---|---|

Cash Flows | 10% Discount Factor | Present Value | Cash Flows | 8% Discount Factor | Present Value | |

0 | (15,000,000) | 1.0000 | (15,000,000) | (8,000,000) | 1.0000 | (8,000,000) |

1 | 5,000,000 | 0.9091 | 4,545,455 | 8,000,000 | 0.9259 | 7,407,407 |

2 | 6,000,000 | 0.8264 | 4,958,678 | 2,000,000 | 0.8573 | 1,714,678 |

3 | 8,000,000 | 0.7513 | 6,010,518 | |||

4 | 3,000,000 | 0.6830 | 2,049,040 | |||

5 | 2,000,000 | 0.6209 | 1,241,843 | |||

3,805,534 | 1,122,085 |

The second step involves finding the cash flows occurring at each year end that would equal the relevant net present value when discounted at the relevant discount rate.

EAA_{Outback} = | $3,805,534 | = $1,003,890 |

(1 − (1 + 10%)^{-5}) ÷ 10% |

EAA_{Accelerate} = | $1,122,085 | = $629,231 |

(1 − (1 + 8%)^{-2}) ÷ 8% |

Since Outback has higher equivalent annual annuity, it is the clear winner. The company should work on Outback.

by Obaidullah Jan, ACA, CFA and last modified on