# Difference between IRR and PI? Lease or Buy?

What are some differences between IRR and PI? What would Lisa Cross say about IRR?

Why would an organization choose to lease an asset instead of buying it? Refer to the text and article by D.O. Burgess.

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Please see response attached (also presented below), as well as one relevant and supporting document. I hope this helps and take care.

RESPONSE:

1. What are some differences between IRR and PI?

The following information is an excerpt from the attached article concerning the differences between IRR and PI and how they are used in choosing between two options (i.e., between two projects, between lease or buy, etc.). It also discusses the pitfalls and provides excellent pointers when to use and when not to use these indicators. I make references to specific pages in the article periodically for you to consider and make reference to.

The internal rate of return

? The basic rationale behind the internal rate of return (IRR) is that it tries to find a single number that summarizes the merits of a project. That number does not depend on the interest rate that prevails in the capital market. That is why it is called internal - the number is internal or intrinsic to the project and depends only on the project's cash flows.

? The IRR is the discount rate that would make the NPV of the project equal to zero. For example, if a project has cash flows (-C0, C1, C2, C3) the IRR will be the solution for r in the equation (1) (see attached article p. 3).

? The basic investment rule based on the IRR is "accept the project if the IRR is greater than the discount rate; reject it if the IRR is less than the discount rate."

? The first problem with the basic IRR rule is that it will give an incorrect decision if the NPV function cuts the r axis from below rather than above. In that case, if the discount rate is below the IRR, the NPV will be negative at the discount rate and the project should not be undertaken. Conversely, if the discount rate is above the IRR the NPV will be positive at the discount rate and the project should be undertaken.

? The implicit assumption in the simple IRR rule is that the negative cash flows occur immediately to be followed by positive cash flows. In that case, the NPV will be decreasing in r so the NPV curve will cut the r axis from above. If a project has positive cash flows followed by negative ones, however, the NPV will be increasing in r and the NPV curve will cut the r axis from below.

? In the case where the negative cash flows occur first, the project is a substitute for lending so the project should have a return greater than the discount rate before it should be undertaken. Conversely, when the positive cash flows occur first, to be followed by negative cash flows, the project is a substitute for borrowing. The project therefore should only be undertaken if the IRR is less than the discount rate.

? The second problem with the IRR rule is that the IRR may not be unique. The NPV function may cross the r axis a number of times. In fact, we can see that equation (1) that is to be solved for r is a cubic equation. If there are T periods of future cash flows from the project, the equation to be solved will be a T-th degree polynomial. In general, a polynomial of degree T will have T roots (values of r where it equals zero).

? Fortunately, we can discard roots that are not in the [0,1] interval as not being sensible rates of return. Furthermore, as we argued above, the NPV of projects that have a negative cash flow followed by all positive ones or, vice versa, a positive cash flow followed by all negative ones, will have a monotonic NPV as a function of r and therefore just one IRR.

? Even so, there are many important types of projects that will have multiple IRR. In particular, projects where the cash flow changes sign two or more times will have this characteristic. Examples of such projects are most surface mining investments where rehabilitation expenditures are required once the mineral resource has been exploited.

? When there are multiple IRR it is less clear where the discount rate should fall to make the project worthwhile. If we graph the NPV as a function of r, however, all discount rates that would result in a positive NPV would make the project worthwhile. For example in the following case:

Figure 1: NPV for a project with several changes in cash flow sign the project would have a positive net present value for discount rates below r1or between r2 and r3 and hence should be undertaken in these ...

#### Solution Summary

This solution identifies and discusses some of the main differences between IRR and PI, and what Lisa Cross might say about IRR. Then, reasons why an organization might choose to lease an asset instead of buying it are explored. Supplemented with a highly relevant article about alternative investment rules.