**Internal rate of return, IRR**

### Table of contents

## Introduction

The simplest version of internal rate of return (IRR) determines the rate of return that will set the stream of benefits from a project equal to the initial outlay.

That is, for an initial investment of 100 DKK with annual returns of 15 DKK for 10 years and a scrap value of 0, what rate of return must be applied to make the two equal?

- 100 = 150/(1+?)

If one ignores the time value of money, the relevant rate would be 50 % (1 + 0.5). However, IRR takes into account the time value of money, such that future benefits are weighted lower than initial outlays.

The simple stream is illustrated graphically below.

Note that the investment stream in Year 1 sums to -85 (-100 + 15).

For what rate of return are the two equal in Year 1? Using the internal rate of return function in a spreadsheet reveals an IRR for the full period equal to 10.41 % (1 + 0.104) for the above example.

A different way of calculating this would be to discount all amounts to present value, income (plus) and investment (minus). The discount rate which will set the result to 0 is the internal rate of return. (The easiest way is to guess an appropriate interest rate and reiterate the calculations until you reach a value of zero).

Calculating IRR is useful for comparisons of returns from a project investment relative to, for example, returns from investment into government bonds. Assuming an annual returns of 3 percent on the government bonds, the IRR equals 3.36 percent over the full period. The investment stream is shown in the graph below.

In the above graph, a project and government bonds costing 100 DKK are purchased in year 1. From year 2 to year 10 (9 years) they provide a return of 15 % and 3 % respectively. At the end of the period, the original bond is returned to the owner whereas the scrap value of the project is 0. The IA for the project is 6.46 % and for the bonds 3.36 %. For a person investing in both government bonds and the project, the project will provide the larger payoff. (Note that this calculation assumes that the positive returns are reinvested in the project each year - which explains why government bonds with a 3 % annual return has a total return rate of more than 3 %).

## Limitations of the simple IRR

There are two key problems with the simple IRR.

- in some situations a project will require additional investment during its lifetime. If this additional investment is larger than the benefit in the period, then there will be a negative value in the stream of benefits. For an investment with an initial cost of 100 DKK, a benefit in year 2 of 420 DKK and an additional investment in year 3 of 400 DKK, there are two solutions for IRR, namely 46 percent
*and*174 percent, both of which are mathematically correct. Additional scenarios could result in more than 2 (mathematically) correct solutions.

- while the simple IRR is sufficient for evaluating single project proposals, it is not suitable for comparisons of mutually exclusive projects; i.e. it is not capable of accurately ranking different project proposals. This is because the IRR assumes that all returns are reinvested at the rate of return calculated by the IRR, and no distinction is made between a rate of interest charged for borrowing and a rate of interest earned on lending, i.e. these are assumed equal, which is rarely the case - banks frequently charge more for loans than the give on deposits. This is clear from comparisons with net present value (NPV) calculations of the same projects. Instead, a variant of IRR known as 'Incremental IRR' can be used to compare competing project proposals. An example is given below.

## Practical Example

The following example compares initial costs and costs over time of three types of energy efficient lightbulbs; compact flourescent light, light-emitting diodes (LED) and halogen light. Table 1 describes the specifications of the lightbulbs used.

Table 1. Comparison of lightbulbs |
||||
---|---|---|---|---|

Type |
. | Flourescent | Halogen | LED |

Fitting |
. | E27 | E27 | E27 |

Watt/Effect |
W | 15/60-75 | 60 | 6/60 |

Lifetime |
hours | 10 000 | 2 000 | 20 000 - 50 000 |

Price |
DKK | 45 | 79 | 168 |

Annual energy usage |
KWH | 21.9 | 87.6 | 8.8 |

*Note that all prices are wholesale estimates, and all other values are approximate only. Energy estimates are based on a lamp being in use 1 000 hours a year. A KWH is assumed to cost 2 DKK*

The graph below illustrates the investment stream over time for the above 3 lightbulbs relative to a traditional incandescent lightbulb with similar specifications. Savings are composed of electricity savings and reinvestment savings, as the traditional incandescent lightbulb has a lifetime of only 1 000 hours, app. 1 year. However, the cost of the traditional lightbulb is also substantially lower, app. 6 DKK. Given the much longer lifetime of the LED bulb, the remaining years are included as scrap value at the end of the period.

In year 1, the investment for the three lightbulbs is negative, reflecting the greater cost relative to the traditional incandescent lightbulb. The fluourescent lightbulb is 7.5 times more expensive, the halogen bulb is 13.2 times more expensive, and the LED bulb is 28 times more expensive than the traditional lightbulb.

The halogen lightbulb has a lifetime only twice that of a traditional lightbulb, uses the same power (60 W) and costs significantly more (although the lighting is said to be superior). This explains why the investment is negative every second year, and why the overall savings are quite low. The two other types of lightbulbs are obviously superior investment-wise, with a lifetime extending throughout the period, and a much lower energy consumption than the traditional lightbulb. Given that the traditional lightbulb has to be replaced each year, both the fluorescent and LED lightbulbs represent significant annual savings after the initial investment. The table below gives the simple IRR and the net present value (NPV) for the three lightbulbs:

Table 2. Investment evaluation |
|||||
---|---|---|---|---|---|

Type |
Year 1 |
Year 2 |
Year 10 |
IRR |
NPV |

Flourescent | -39 | 137.4 | 137.4 | 352 | 1 031 |

Halogen | -73 | 6 | 6 | n/a | -300 |

LED | -162 | 163.6 | 247.6 | 101 | 1 176 |

*All values are in DKK except for IRR, which are in percentages. The NPV calculations are based on a discount rate of 3 percent.*

The table above shows the inital investment of each lightbulb less the cost of the traditional bulb (6 DKK) which has to be replaced annually. For the case of the halogen bulb, which has to be replaced biannually, this results in an alternating positive/negative stream of benefits since there are no gains from energy savings. The IRR could not be calculated for the halogen bulb, as IRR calculates a solution every time the value changes from positive to negative; in this case there were 5 solutions, all of which are mathematically correct.

The IRR method clearly favours the flourescent bulb over the LED, despite the larger energy savings of the latter, and the scrap value in year 10 from the longer lifetime of the bulb. The NPV ranks the LED bulb over the flourescent bulb, with the halogen bulb having a negative value; unsurprising, given the amount of reinvestment needed over the period for this type of lighting, and the lack of other savings.

## Incremental IRR

The ranking from the NPV in Table 2 above can be replicated by using incremental IRR rather than simple IRR. Since the flourescent lightbulb (A) is a low-cost option relative to the LED bulb (B), you subtract A from B to find the incremental cash flow:

The above yields an IRR of 20 percent, which is high enough to consider the LED lightbulb a better investment than the flourescent lightbulb.