By Yuriy Smirnov Ph.D.

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.

Initial Cost, CF_{0} |
Cash flows at the end of relevant year, CF_{t} |
|||||

0 | 1 | 2 | 3 | 4 | 5 | |

Normal cash flows | -$10,000 | $4,000 | $6,000 | $8,000 | $7,000 | $4,000 |

Normal cash flows | -$10,000 | -$5,000 | $7,000 | $8,000 | $11,000 | $9,000 |

Non-normal cash flows | -$10,000 | $2,000 | $5,000 | $8,000 | $7,000 | -$6,000 |

Non-normal cash flows | -$10,000 | $5,000 | $4,000 | -$2,000 | $11,000 | -$4,000 |

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.

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.

Initial Cost, CF_{0} |
Cash flows at the end of relevant year, CF_{t} |
|||||

0 | 1 | 2 | 3 | 4 | 5 | |

Project Z | -$1,000,000 | $1,000,000 | $1,100,000 | $1,300,000 | $1,000,000 | -$3,700,000 |

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

NPV = -$1,000,000 + | $1,000,000 | + | $1,100,000 | + | $1,300,000 | + | $1,000,000 | + | -$3,700,000 | = 0 |

(1 + IRR)^{1} |
(1 + IRR)^{2} |
(1 + IRR)^{3} |
(1 + IRR)^{4} |
(1 + IRR)^{5} |

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.

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.

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