The internal rate of return (** IRR**) is one of the most frequently-used metrics for assessing investment opportunities. The

**is defined as the discount rate for which the**

*IRR***of a project is zero. The definition is simple, but the**

*NPV***is generally impossible to calculate without a computer.**

*IRR*If you use Excel, there is a built-in ** IRR** function that will calculate the

**for you, given a stream of costs and benefits over time. To see an example of the function's syntax, please have a look at the**

*IRR***NPV Example.xlsx**file posted in the Lesson 09 module in Canvas. (This is the same Excel file that was discussed in the

**video.)**

*NPV*Unlike the ** NPV**, which takes units of dollars, the

**is given in percentage terms (% discount rate per year such that the project**

*IRR***is zero). We call this a "yield" measure of return. This can be very convenient when comparing different types of projects. In many cases the project with the largest yield (i.e., highest**

*NPV***) will be the most desirable. The**

*IRR***can also be compared to the investor's "hurdle rate," which is the lowest return that an investor is willing to accept before putting money into a project. Energy projects that will sell their output into competitive markets often need a yield higher than, say, a 15% hurdle rate over a five year period. This means that the project's return on investment must be at least 15%, and that this yield must be realized within five years. If the**

*IRR***of a prospective project is higher than the hurdle rate, the project could be considered attractive to an investor.**

*IRR*While the ** IRR** is often an appropriate measure for determining whether an individual project is worthwhile, there are three cases where

**may not be very useful or may yield misleading information.**

*IRR*First, if a project has some years with positive cash flow and some years with negative cash flow (an example is on the power plant pro forma from the previous lesson), then it is possible that the ** IRR **may have multiple values. Mathematically, this arises because a high-order polynomial equation may have multiple (non-unique) roots. As a simple example, if the cash flows for a project over three years are -$10, $21 and -$11, then there would be two

**s for this potential project: 0% (because it just breaks even) and 10%.**

*IRR*Second, if a project has negative total undiscounted cash flows over its lifetime, then the ** IRR **is mathematically undefined.

Third, it is possible that the project with the highest ** IRR **may not be the project with the largest

**. In other words, some types of projects can see their**

*NPV**increase when the discount rate goes up, which is the opposite of what we would expect to happen. This situation occurs when projects have large cash outlays at their end-of-life. Nuclear power plants are a good example — at the end of a nuclear power plant's life (at least in most developed countries), the owner must pay a large amount to have the plant decommissioned. If the discount rate used by the plant's owner is low, this increases the contribution of that end-of-life cash outlay to the*

**NPV****. If the discount rate used by the plant's owner is high, then that end-of-life cash outlay is not very important, in present value terms.**

*NPV*Here is a simple example. Suppose that a hypothetical investment requires a $5 cash outlay in Year 0, then earns $5 per year in Years 1 through 5. In Year 6, the project requires a cash outlay of $22.50. As an exercise, calculate the ** NPV** assuming a discount rate of 10% per year and 5% per year. You should get

*NPV**s*of $1.25 and -$0.14.