 # Internal Rate of Return, IRR By Yuriy Smirnov Ph.D.

## Definition

The internal rate of return (IRR) is the discount rate at which the net present value (NPV) of a project is equal to zero. In other words, the sum of a project’s expected cash flows is equal to the amount of its initial cost. The IRR method is based on the discounted cash flow technique and is widely used in capital budgeting and investment decisions as a screening criterion for accepting or rejecting projects or investment.

## Formula

To calculate the IRR of a project, we should use the NPV formula.It is necessary to set the NPV to zero and solve the equation below.

 NPV = CF0 + CF1 + CF2 + … + CFN = 0 (1 + IRR)1 (1 + IRR)2 (1 + IRR)N

or

 N CFt = 0 Σ (1 + IRR)t t = 0

## Decision rules

The decision rules used in the IRR method are as follows:

1. The project’s internal rate of return must exceed a project’s weighted average cost of capital (WACC); otherwise, the project should be rejected.
2. If several independent projects meet the rule above, all of them should be accepted. In case of several mutually exclusive projects, the project with the highest IRR must be accepted.

## Example

Let’s assume two equally risky projects with the same initial cost and the same total expected cash flows. To illustrate the influence of the time value of money concept, Project Y cash flows come in relatively sooner, and Project Z cash flows relatively later.

 Initial Cost, CF0 Cash flows at the end of relevant year, CFt 0 1 2 3 4 5 Project Y -\$20,000,000 \$10,000,000 \$8,000,000 \$6,000,000 \$4,000,000 \$2,000,000 Project Z -\$20,000,000 \$2,000,000 \$4,000,000 \$6,000,000 \$8,000,000 \$10,000,000

Let’s put all data available into the formula above.

 -\$20,000,000 + \$10,000,000 + \$8,000,000 + \$6,000,000 + \$4,000,000 + \$2,000,000 = 0 (1 + IRRY)1 (1 + IRRY)2 (1 + IRRY)3 (1 + IRRY)4 (1 + IRRY)5

 -\$20,000,000 + \$2,000,000 + \$4,000,000 + \$6,000,000 + \$8,000,000 + \$10,000,000 = 0 (1 + IRRZ)1 (1 + IRRZ)2 (1 + IRRZ)3 (1 + IRRZ)4 (1 + IRRZ)5

To solve these equations, you can use the financial calculator, or to find IRR using Excel, see the figure below. 1. Select output cell I4.
2. Click fx button, select All category, and select IRR function from the list.
3. In field Values, select the data range C4:H4, leave empty field Guess, and press the OK button.

Thus, the internal rate of return of Project Y is 20.27% and 12.01% for Project Z. The discounted cash flow of both projects is presented in the figure below. Let’s assume that the WACC for both projects is 9.5%. If the projects are independent, they should be accepted because IRR exceeds WACC. If they are mutually exclusive, Project Y should be accepted because of higher IRR than Project Z.

## Disadvantages of the IRR method

The internal rate of return method has three serious disadvantages:

1. The assumption that all positive future cash flows are reinvested at IRR. In fact, such a scenario is unlikely, especially for projects with a very high rate.
2. If at least one of future cash flows is negative, the equation can have several solutions. This situation is known as multiple IRR problem.
3. The conflict between NPV and IRR method can occur in case of mutually exclusive projects. It can happen that one project has higher IRR but lower NPV and the other has lower IRR and higher NPV. In such a situation, NPV criterion is generally better.