# Multiple IRR Problem

**Definition**

The multiple internal rates of return problem occur when at least one future cash inflow of a project is followed by cash outflow. In other words, there is at least one negative value after a positive one, or the signs of cash flows change more than once. In this case, we say that the project has non-normal cash flows. In contrast, normal cash flows have one or more consecutive cash outflows followed by cash inflows as in the table below.

If a project has a non-normal cash flow, it can have more the one IRR. To better sort out the problem, let’s look at the example below.

**Example**

To illustrate the multiple IRR problem, let’s assume that Project Z has non-normal cash flows. The detailed information about its cash inflows and outflows is presented in the table below.

To find the IRR, we have to solve the following equation:

The NPV of Project Z is equal to zero at an IRR of 5.0699% and 82.4254%. Now we have a dilemma regarding which one is better. To illustrate the issue, let’s look at the NPV profile in the figure below.

If the cost of capital will be less than 5.0699% or more than 82.4254%, the net present value of Project Z will be negative, and it should be rejected. The cost of capital in the range of 5.0699% and 82.4254% means a positive NPV, and Project Z should be accepted.

**Conclusions**

There are two basic ways to solve the multiple IRR problem.

- The NPV method should be used for projects with non-normal cash flows. In such cases, there is no dilemma about which IRR is better.
- An alternative way is to use the
__modified internal rate of return__(MIRR) as a screening criterion. It was just developed to eliminate the multiple internal rates of return problem.