# Internal Rate of Return (IRR) – Calculation with Examples

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The Internal Rate of Return (IRR) is one of the most widely used methods for capital appraisal purposes and there is no doubt that it’s also considered as one of the most robust ones too. However, the internal rate of return has certain limitations and it’s advised that it should be used along with other methods and not as the only criteria to accept or to reject projects for reasons that are explained below.

## Definition of the Internal Rate of Return

The internal rate of return is the discount rate which makes the costs to undertake a project equal the profits that this project will generate. In other words, the internal rate of return is the discount rate that gives a zero net present value.

The idea is that if the cost of capital which is also the interest rate that a company pays to borrow money is less than the internal rate of return, then the project can be accepted.

## Internal Rate of Return Decision Rule

As noted above, if the internal rate of return is lower than the cost of capital then the project should be rejected since it costs more money for the company to invest in this project than the return that the project is expected to bring.

## Internal Rate of Return Example

We will use one example in order to illustrate how the internal rate of return can be calculated and the approach is. Let’s say that company A uses the internal rate of return to evaluate investment opportunities and make a decision regarding the profitability and viability of a project.

There is one potential project that Company A wishes to appraise with the following characteristics:

-An initial investment of $50,000 is required during the first year.

-The project will last for four years and the cash inflows during these four years will be:

- Year 1 : $15,000
- Year 2: $20,000
- Year 3: $25,000
- Year 4: $18,000

The company has a cost of capital of 15% and wishes to appraise this project and decide whether to proceed or not.

## How to Calculate the Internal Rate of Return

There are two different approaches that can be followed to calculate the internal rate of return.

### Internal Rate of Return Calculation with Trial and Error

From the IRR definition, we know that the IRR is the discount rate that makes the present value of the cash flows become $nil. We can therefore use a trial and error approach and start increasing the discount rates until we get to a present value that is $nil.

The following table illustrates the calculations.

Time | Cash Flow | Discounted Cash Flows (10%) | Discounted Cash Flows (15%) | Discounted Cash Flows (19%) | Discounted Cash Flows (20%) |
---|---|---|---|---|---|

Period 0 | -50,000.00 | -50,000.00 | -50,000.00 | -50,000.00 | -50,000.00 |

Period 1 | 15,000.00 | 13,636.36 | 13,043.48 | 12,605.04 | 12,500.00 |

Period 2 | 20,000.00 | 16,528.93 | 15,122.87 | 14,123.30 | 13,888.89 |

Period 3 | 25,000.00 | 18,782.87 | 16,437.91 | 14,835.40 | 14,467.59 |

Period 4 | 18,000.00 | 12,294.24 | 10,291.56 | 8,976.04 | 8,680.56 |

Net Present Value | 28,000.00 | 11,242.40 | 4,895.82 | 539.77 | -462.96 |

As we can see from this table, a discount rate of 19% gives around $500 NPV while a 20% discount rate gives a $-462 NPV. We can therefore understand that the IRR is somewhere in the middle or around 19.5%.

### Internal Rate of Return Calculation using two discount rates

There is another approach we can use to calculate the internal rate of return (IRR) and involves a formula (see below). The idea is that we use two different discount rates for which one will give a positive net present value and another that we believe will result in a negative net present value. It might sound complicated but follow the example below which should help you understand what are the steps you should follow.

### Internal Rate of Return Formula

IRR= Ra + (NPVa/(NPVa-NPVb))*(Rb-Ra)

where

Ra is the discount rate that gives the positive net present value, NPVa is the positive NPV, NPVb is the negative NPV and Rb is the discount rate that gives the negative NPV.

Let’s proceed with a table that can illustrate what’s written above in an easier to follow way:

Time | Cash Flow | Discounted Cash Flows (10%) | Discounted Cash Flows (20%) |
---|---|---|---|

Period 0 | -50,000.00 | -50,000.00 | -50,000.00 |

Period 1 | 15,000.00 | 13,636.36 | 12,500.00 |

Period 2 | 20,000.00 | 16,528.93 | 13,888.89 |

Period 3 | 25,000.00 | 18,782.87 | 14,467.59 |

Period 4 | 18,000.00 | 12,294.24 | 8,680.56 |

Net Present Value | 28,000.00 | 11,242.40 | -462.96 |

The table is part of the first table and we can see that a 10% discount rate gives a positive NPV and a 20% gives a negative NPV. We can therefore use the formula above and calculate IRR as:

IRR= 10+(11,242/(11,242+462))*(20-10)=19.6%

As you can see both methods will give the same IRR (more or less) but most people prefer to calculate IRR by using the second approach since it involves less calculations.

## Limitations and Disadvantages of the Internal Rate of Return

The internal rate of return is a very robust capital appraisal method but there are certain limitations. For example:

- When a project has positive cash flows, followed by negative and then followed by positive cash flows again, there will be more than one IRR (more than solution).
- IRR is not very reliable when comparing projects that have significantly different time horizons (i.e project A will last for 5 years while project B will last for 15 years).
- Due to the limitations explained above, IRR is mostly used a decision tool (accept or reject) and not as a comparison tool (project A vs project B).

## Internal Rate of Return Advantages

There are however certain advantages that IRR has which make it one of the most preferred capital appraisal methods.

- It’s an easy way to decide to accept or reject projects.
- It’s a robust method that can be used to monitor how a project is performing based on the actuals vs the budgets and how that compares to the cost of capital.
- It takes into account the time value of money compared to other methods (payback period for example) that do not.