# Projects with Unequal Lives

When mutually-exclusive projects have unequal useful lives, capital budgeting decision is made based on annual net present value (also called equivalent annual annuity) method or replacement chain method.

Mutually exclusive projects are projects out of which only one project must be selected for investment. Unlike independent projects, they are competing projects in that selection of one excludes the other projects from consideration. When they have equal lives, decision is simple: accept the project with highest net present value and/or highest internal rate of return (IRR) and/or payback period. However, when they have unequal lives, more elaborate analysis is required to arrive at the right decision.

There are two approaches to analyzing such a situation: the annual net present value method and the replacement chain method.

## Annual Net Present Value Method

Annual net present value method involves calculation of net present value of the project adjusted for its life, i.e. estimating net present value for each project and then dividing it by the discount factor corresponding to the hurdle rate and life of the project. The resulting figure is an estimate of the net present value added by each project per year. The project with higher annual net present value is accepted. Annual net present value method is also called the equivalent annual annuity approach.

Annual net Present Value = | Net Present Value |

Annuity Discount Factor for the Project Life |

## Replacement Chain Method

In the replacement chain method, the cash flows projections for the projects under consideration are repeated up to the least common useful life. For example, if Project A has a life of 3 years and Project B has a life of 5 years, 15 years is the least common life, i.e. if Project A is repeated 5 times and Project B is repeated 3 times, both will have equal useful lives. Net present value and internal rate of return for that common useful life are compared and the project with higher NPV and IRR is accepted. Replacement chain analysis is also called **common-life approach**.

## Example

Renewable Energy, Inc. is considering investing in two projects: Solar Park or Wind Farm. Setup of the solar park will cost $20 million and will generate $7.5 million per annum for 5 years. The wind farm will cost $35 million and will generate $8 million for 10 years. If the company’s cost of capital is 10%, determine which project should the company invest in, using the annual net present value (equivalent annual annuity) method and replacement chain (common life) method.

### Solution

The Solar Park project has a net present value of $8.43 million calculated as follows:

Net present value of Solar Park = $7.5 million × 3.7908 - $20 million = $8.43 million

Where 3.7908 is the 5-year annuity present value factor at 10% discount rate.

Similarly, the Wind Farm has net present value of $14.16 (=$8 million × 6.1446 - $35 million)

Direct comparison of the Solar Park and the Wind Farm projects cannot be made because they have difference useful lives. Further, we cannot just divide the net present value of each project by the useful life of that project because it would ignore the time value of money.

#### Equivalent Annual Annuity Approach

We need to employ the equivalent annual annuity approach to find the annual net present value and then accept the project with higher annual NPV.

Annual net present value equals net present value of the project divided by the discount factor for the project life at the given discount rate.

Annual net present value for Solar Park

= $8.43 million ÷ 3.908

= $2.24 million

Annual net present value for Wind Farm

= $14.16 million ÷ 6.1446

= $2.304 million

The company should accept the Wind Farm project because it generates more value per year of project as compared to the Solar Park project.

#### Replacement Chain Method

We will reach the same conclusion using the replacement chain method. However, the approach is different as illustrated below.

Life of Wind Farm project is twice that of the Solar Park project. A comparison between the two projects is possible if we (theoretically) repeat the cash flows of Solar Park project once to make it equivalent to the Wind Farm project. The following table summarizes the cash flow pattern of both projects after Solar Park project is repeated once:

Year | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | NPV |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Solar Park | ||||||||||||

Cash flows | 7.50 | 7.50 | 7.50 | 7.50 | 7.50 | 7.50 | 7.50 | 7.50 | 7.50 | 7.50 | ||

Investment | -20 | -20 | ||||||||||

Net cash flows | -20 | 7.50 | 7.50 | 7.50 | 7.50 | -12.5 | 7.50 | 7.50 | 7.50 | 7.50 | 7.50 | |

PV Factor | 1 | 0.91 | 0.83 | 0.75 | 0.68 | 0.62 | 0.56 | 0.51 | 0.47 | 0.42 | 0.39 | |

PV of Cash Flows | -20 | 6.82 | 6.20 | 5.63 | 5.12 | -7.76 | 4.23 | 3.85 | 3.50 | 3.18 | 2.89 | 13.67 |

Wind Farm | ||||||||||||

Cash flows | 8.00 | 8.00 | 8.00 | 8.00 | 8.00 | 8.00 | 8.00 | 8.00 | 8.00 | 8.00 | ||

Investment | -35 | |||||||||||

Net cash flows | -35 | 8.00 | 8.00 | 8.00 | 8.00 | 8.00 | 8.00 | 8.00 | 8.00 | 8.00 | 8.00 | |

PV Factor | 1 | 0.91 | 0.83 | 0.75 | 0.68 | 0.62 | 0.56 | 0.51 | 0.47 | 0.42 | 0.39 | |

PV of Cash Flows | -35 | 7.27 | 6.61 | 6.01 | 5.46 | 4.97 | 4.52 | 4.11 | 3.73 | 3.39 | 3.08 | 14.16 |

The important thing to note here is that the initial investment of $20 million required at the start of the 5-year Solar Park project is repeated at the end of Year 5 to correctly represent the repeated projection of cash flows.

Net present value of Solar Park at 10-year common life = $13.67 million

Net present value of Wind Farm = $14.16 million

Since Wind Farm has higher net present value under for the least common life, it should be the preferred project.

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