# 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.

## 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 + Discount Rate)^{-life of the project} | |||

discount rate |

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

PV of Annuity (NPV) = Payment (EAA) × | (1 − (1 + Discount Rate)^{-life of the project} |

Discount Rate |

## Example

Combustion Systems is engaged in development of car racing games. The company has to 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.

Equivalent Annual Annuity_{Outback} = | $3,805,534 | = $1,003,890 |

(1 − (1 + 10%)^{-5}) ÷ 10% |

Equivalent Annual Annuity_{Accelerate} = | $1,122,085 | = $629,231 |

(1 − (1 + 8%)^{-2}) ÷ 8% |

#### Decision Rule

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

Written by Obaidullah Jan, ACA, CFA and last revised on