5.2 Internal Rate of Return

Introduction

The net present value presented in the previous section provides an exact monetary magnitude of a project. However, the common mindset in business focuses on profitability as a percentage and not as a dollar amount. Thus, decisions are based on the internal rate of return for a project, or [latex]IRR[/latex] for short. The [latex]IRR[/latex] is the annual percentage rate of return on the investment so that the present value of all cash inflows equals the present value of all cash outflows. That is, the [latex]IRR[/latex] is the rate of return required to make [latex]NPV=0[/latex].

To interpret the [latex]IRR[/latex] and how the [latex]IRR[/latex] can be used to make a decision about a project, consider the relationship between the value of [latex]NPV[/latex] and the [latex]IRR[/latex].

• [latex]NPV=0[/latex]. By definition, the cost of capital and the [latex]IRR[/latex] are the same value.
• [latex]NPV>0[/latex].Because the [latex]IRR[/latex] is the rate of return that makes [latex]NPV=0[/latex], this means that a higher rate of return must be used in order for the [latex]NPV>0[/latex]. In other words, the [latex]IRR[/latex] is greater than the cost of capital.
• [latex]NPV<0[/latex]. Because the [latex]IRR[/latex] is the rate of return that makes [latex]NPV=0[/latex], this means that a lower rate of return must must be used in order for the [latex]NPV<0[/latex]. In other words, the [latex]IRR[/latex] is less than the cost of capital.

In summary, here is how to interpret the [latex]IRR[/latex] and use it to make a decision about the viability of a project.

• [latex]IRR=\mbox{cost of capital}[/latex]. The project breaks even. This is the minimum financial condition to pursue the project. The business should accept the project.
• [latex]IRR>\mbox{cost of capital}[/latex].The project is profitable because the project makes enough money to cover the costs. The business should accept the project.
• [latex]IRR<\mbox{cost of capital}[/latex]. The project is not profitable because the project does not make enough money to cover the costs. The business should reject the project.

Solving for the internal rate of return is difficult without the use of a financial calculator or other technology. Similar to calculating the [latex]NPV[/latex] in the previous section, the [latex]IRR[/latex] can be calculated by using the cash flow worksheet on the financial calculator and a built-in function to calculate the internal rate of return.

Using the TI BAII Plus Cash Flow Worksheet to Calculate the Internal Rate of Return

Enter the cash flow data into the cash flow worksheet, as discussed in the previous section. Once you have entered all cash flows, you must access other functions to generate the output. For internal rate of return, press [latex]IRR[/latex] on the keypad, as illustrated in the photo, followed by the [latex]CPT[/latex] button to perform the calculation. The output is a percentage in percent format. To exit the window, press 2nd QUIT.

_{Video: Cash Flows 1: Net Present Value (NPV) and IRR Calculations by Joshua Emmanuel [2:52] Transcript Available.}

Like the net present value, the internal rate of return can be used to choose one project from several different projects. Relative to their initial investment, projects with higher [latex]IRR[/latex]s are more profitable. Consequently, [latex]IRR[/latex] can be used to rank projects. When making a choice from several options, the option with the highest [latex]IRR[/latex] should be accepted because it will be the most profitable.

Paths to Success

As mentioned above, the internal rate of return can be used to rank projects. However, a couple of things need to be considered about using the internal rate of return in this way.

1. The Cost of Capital is Ignored. The internal rate of return does not factor in the cost of capital in its computations. Recall that the fundamental concept of time value of money requires all money to be on the same date using an appropriate rate of interest—the cost of capital—before any decision can be made. Therefore, if you have not factored in the cost of capital, then your analysis is incomplete.
2. The Magnitude of the Decision is Ignored. It can easily happen that an alternative has a high [latex]IRR[/latex] but a low [latex]NPV[/latex]. For example, using a cost of capital of [latex]10\%[/latex] consider two alternatives. Alternative A invests [latex]\$1[/latex] and one year later returns [latex]\$1.50[/latex]. The [latex]IRR[/latex] is [latex]50\%[/latex], while the [latex]NPV[/latex] is [latex]\$0.36[/latex]. Alternative B invests [latex]\$1,000[/latex] and one year later returns [latex]\$1,250[/latex]. The [latex]IRR[/latex] is [latex]25\%[/latex], while the [latex]NPV[/latex] is [latex]\$136.36[/latex]. If choosing between these two options based on the [latex]IRR[/latex], you select Alternative A, resulting in a net present value that is [latex]\$136[/latex] lower than for Alternative B.

Section 5.2 Exercises

Attribution

“8.2 Internal Rate of Return” from Business and Financial Mathematics by Valerie Watts is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

“15.2: Other Measures for Making Decisions” from Business Math: A Step-by-Step Handbook (2021B) by J. Olivier and Lyryx Learning Inc. through a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License unless otherwise noted.