 # Replacement Chain Method By Yuriy Smirnov Ph.D.

## Definition

The replacement chain method is used in capital budgeting to rank mutually exclusive projects with unequal life spans. As the first step, it is necessary to determine the minimum common multiple life span for all projects under consideration. As the second step, each of the projects should be repeated until the minimum common multiple life span is reached. For example, if Project X has a life span of 4 years and the life span of Project Z is 6 years, the shortest common multiple life span is 12 years, which means that Project X should be repeated 3 times and Project Z 2 times.

## Assumptions

The replacement chain method is based on the following assumptions:

• any project can be repeated exactly at its replication date
• the cost of capital used as the discount rate remains constant
• all cash inflows and outflows remain the same at any iteration

## NPV Rule

The net present value is used as a screening criterion when the replacement chain method is applied, but the decision is based on the total NPV of all iteration being performed. So the project with the highest total NPV should be accepted.

## Example

Company C is considering investing in two mutually exclusive projects with the same initial cost of \$10,000,000 and different life spans. The cost of capital raised to finance the projects is 15%. Detailed information about future cash flows is shown in the table below.

 Initial Cost, CF0 Cash flows at the end of relevant year, CFt 0 1 2 3 4 5 6 Project L -\$20,000 \$2,500 \$2,750 \$3,000 \$3,250 \$3,750 \$3,500 Project S -\$10,000 \$5,250 \$4,750 \$4,500

The life span of Project L is 6 years and 3 years for Project S, so the minimum common multiple lifespan is 6 years, which means that Project S should be repeated twice while Project L only once. To rank both projects, we need to use the NPV rule.

The NPV of Project L is \$1,461.61, and the discounted cash flows are schematically presented in the figure below.

 NPVL = -\$10,000 + \$2,500 + \$2,750 + \$3,000 + \$3,250 + \$3,750 + \$3,500 = \$1,461.61 (1 + 0.15)1 (1 + 0.15)2 (1 + 0.15)3 (1 + 0.15)4 (1 + 0.15)5 (1 + 0.15)6 The NPV of Project S is \$1,115.72, and it should be repeated once again to apply the replacement chain method. After the second iteration, the net present value is \$733.61, and the discounted cash flows are as follows:

 NPVS1 = -\$10,000 + \$5,250 + \$4,750 + \$4,500 = \$1,115.72 (1 + 0.15)1 (1 + 0.15)2 (1 + 0.15)3

 NPVS2 = -\$10,000 + \$5,250 + \$4,750 + \$4,500 = \$733.61 (1 + 0.15)3 (1 + 0.15)4 (1 + 0.15)5 (1 + 0.15)6 Thus, the total NPV of Project S is \$1,849.33 and is greater than the \$1,461.61 of Project L. So, if the replacement chain method is applied, Company C should accept Project S.

Replicating projects in perpetuity

If the minimum common multiple life span is a great number, the replacement chain method becomes cumbersome. For example, the projects with life spans of 7 and 9 years have a minimum common multiple lifespan of 63 (7×9), which means that the 7-year project should be repeated 9 times, and the 9-year project should be repeated 7 times.

This problem is solved by assuming that the projects are repeating in perpetuity. In such a scenario, the formula to calculate the NPV of a project is as follows:

 NPVP = NPV1 (1 + r)N (1 + r)N - 1

Here NPV1 is the net present value of a project at the initial replication, r is the cost of capital, and N is the life span of the project in years.

For example, there are two mutually exclusive projects: Project S with an NPV of \$450,000 and a life span of 7 years and Project L with an NPV of \$475,000 and a life span of 9 years. They have a common cost of capital of 11%. If both of them are replicated in perpetuity, their NPV is as follows:

 NPVS = \$450,000 (1 + 0.11)7 = \$868,153 (1 + 0.11)7 - 1

 NPVL = \$475,000 (1 + 0.11)9 = \$779,871 (1 + 0.11)9 - 1

In such a situation, Project S should be accepted.