Comparison with NPV investment selection criterion
As a tool applied to making an investment decision on whether a project adds value or not, comparing the IRR of a single project with the required rate of return, in isolation from any other projects, is equivalent to the NPV method. If the appropriate IRR (if such can be found correctly) is greater than the required rate of return, using the required rate of return to discount cash flows to their present value, the NPV of that project will be positive, and vice versa. However, using IRR to sort projects in order of preference does not result in the same order as using NPV.
Maximizing NPV
One possible investment objective is to maximize the total NPV of projects.
When the objective is to maximize total value, the calculated IRR should not be used to choose between mutually exclusive projects. In cases where one project has a higher initial investment than a second mutually exclusive project, the first project may have a lower IRR (expected return), but a higher NPV (increase in shareholders' wealth) and should thus be accepted over the second project (assuming no capital constraints).
When the objective is to maximize total value, IRR should not be used to compare projects of different duration. For example, the NPV added by a project with longer duration but lower IRR could be greater than that of a project of similar size, in terms of total net cash flows, but with shorter duration and higher IRR.
Practitioner preference for IRR over NPV
Despite a strong academic preference for NPV, surveys indicate that executives prefer IRR over NPV.[9] Apparently, managers prefer to compare investments of different sizes in terms of forecast investment performance, using IRR, rather than maximize value to the firm, in terms of NPV. This preference makes a difference when comparing mutually exclusive projects.
Maximizing long-term return
Maximizing total value is not the only conceivable possible investment objective. An alternative objective would for example be to maximize long-term return. Such an objective would rationally lead to accepting first those new projects within the capital budget which have the highest IRR, because adding such projects would tend to maximize overall long-term return.
Example
To see this, consider two investors, Max Value and Max Return. Max Value wishes her net worth to grow as large as possible, and will invest every last cent available to achieve this, whereas Max Return wants to maximize his rate of return over the long term, and would prefer to choose projects with smaller capital outlay but higher returns. Max Value and Max Return can each raise up to 100,000 US dollars from their bank at an annual interest rate of 10 percent paid at the end of the year.
Investors Max Value and Max Return are presented with two possible projects to invest in, called Big-Is-Best and Small-Is-Beautiful. Big-Is-Best requires a capital investment of 100,000 US dollars today, and the lucky investor will be repaid 132,000 US dollars in a year's time. Small-Is-Beautiful only requires 10,000 US dollars capital to be invested today, and will repay the investor 13,750 US dollars in a year's time.
Solution
The cost of capital for both investors is 10 percent.
Both Big-Is-Best and Small-Is-Beautiful have positive NPVs: and the IRR of each is (of course) greater than the cost of capital:
Example
To see this, consider two investors, Max Value and Max Return. Max Value wishes her net worth to grow as large as possible, and will invest every last cent available to achieve this, whereas Max Return wants to maximize his rate of return over the long term, and would prefer to choose projects with smaller capital outlay but higher returns. Max Value and Max Return can each raise up to 100,000 US dollars from their bank at an annual interest rate of 10 percent paid at the end of the year.
Investors Max Value and Max Return are presented with two possible projects to invest in, called Big-Is-Best and Small-Is-Beautiful. Big-Is-Best requires a capital investment of 100,000 US dollars today, and the lucky investor will be repaid 132,000 US dollars in a year's time. Small-Is-Beautiful only requires 10,000 US dollars capital to be invested today, and will repay the investor 13,750 US dollars in a year's time.
Solution
The cost of capital for both investors is 10 percent.
Both Big-Is-Best and Small-Is-Beautiful have positive NPVs: and the IRR of each is (of course) greater than the cost of capital: so the IRR of Big-Is-Best is 32 percent, and so the IRR of Small-Is-Beautiful is 37.5 percent.
Both investments would be acceptable to both investors, but the twist in the tale is that these are mutually exclusive projects for both investors, because their capital budget is limited to 100,000 US dollars. How will the investors choose rationally between the two?
As an example, Max Value selects Big-Is-Best, which has the higher net present value of US$20,000, while Max Return selects Small-Is-Beautiful for its higher return of 37.5 percent, compared to the 32 percent return offered by Big-Is-Best. The result is that each decision-maker chooses a different project without conflict.
Solution
The cost of capital for both investors is 10 percent.
Both Big-Is-Best and Small-Is-Beautiful have positive NPVs: and the IRR of each is (of course) greater than the cost of capital: so the IRR of Big-Is-Best is 32 percent, and so the IRR of Small-Is-Beautiful is 37.5 percent.
Both investments would be acceptable to both investors, but the twist in the tale is that these are mutually exclusive projects for both investors, because their capital budget is limited to 100,000 US dollars. How will the investors choose rationally between the two?
As an example, Max Value selects Big-Is-Best, which has the higher net present value of US$20,000, while Max Return selects Small-Is-Beautiful for its higher return of 37.5 percent, compared to the 32 percent return offered by Big-Is-Best. The result is that each decision-maker chooses a different project without conflict.
How can this be rational for both investors? The answer lies in the fact that the investors do not have to invest the full 100,000 US dollars. Max Return is content to invest only 10,000 US dollars for now. After all, Max Return may rationalize the outcome by thinking that maybe tomorrow there will be new opportunities available to invest the remaining 90,000 US dollars the bank is willing to lend Max Return, at even higher IRRs. Even if only seven more projects come along which are identical to Small-Is-Beautiful, Max Return would be able to match the NPV of Big-Is-Best, on a total investment of only 80,000 US dollars, with 20,000 US dollars left in the budget to spare for truly unmissable opportunities. Max Value is also happy, because she has filled her capital budget straight away, and decides she can take the rest of the year off investing.
Multiple IRRs
When the sign of the cash flows changes more than once, for example when positive cash flows are followed by negative ones and then by positive ones (+ + − − − +), the IRR may have multiple real values. In a series of cash flows like (−10, 21, −11), one initially invests money, so a high rate of return is best, but then receives more than one possesses, so then one owes money, so now a low rate of return is best. In this case, it is not even clear whether a high or a low IRR is better.
There may even be multiple real IRRs for a single project, like in the example 0% as well as 10%. Examples of this type of project are strip mines and nuclear power plants, where there is usually a large cash outflow at the end of the project.
The IRR satisfies a polynomial equation. Sturm's theorem can be used to determine if that equation has a unique real solution. In general the IRR equation cannot be solved analytically but only by iteration.
With multiple internal rates of return, the IRR approach can still be interpreted in a way that is consistent with the present value approach if the underlying investment stream is correctly identified as net investment or net borrowing.[10]
See [11] for a way of identifying the relevant IRR from a set of multiple IRR solutions.
Limitations in the context of private equity
In the context of survivorship bias which makes the high IRR of large private equity firms a poor representation of the average, according to Ludovic Phalippou,
"...a headline figure that is often shown prominently as a rate of return in presentations and documents is, in fact, an IRR. IRRs are not rates of return. Something large PE firms have in common is that their early investments did well. These early winners have set up those firms' since-inception IRR at an artificially sticky and high level. The mathematics of IRR means that their IRRs will stay at this level forever, as long as the firms avoid major disasters. In passing, this generates some stark injustice because it is easier to game IRRs on LBOs in Western countries than in any other PE investments. That means that the rest of the PE industry (e.g. emerging market growth capital) is sentenced to look relatively bad forever, for no reason other than the use of a game-able performance metric."[12]
Also,
"Another problem with the presentation of pension fund performance is that for PE, time-weighted returns...are not the most pertinent measure of performance. Asking how much pension funds gave and got back in dollar terms from PE, i.e. MoM, would be more pertinent. I went through the largest 15 funds websites to collect information on their performance. Few of them post their PE fund returns online. In most cases, they post information on their past performance in PE, but nothing that enables any meaningful benchmarking. E.g., CalSTRS [a California public pension fund] provide only the net IRR for each fund they invest in. As IRR is often misleading and can never be aggregated or compared to stock-market returns, such information is basically useless for gauging performance."[13]
Modified internal rate of return
Modified internal rate of return (MIRR) considers cost of capital, and is intended to provide a better indication of a project's probable return. It applies a discount rate for borrowing cash, and the IRR is calculated for the investment cash flows. This applies in real life for example when a customer makes a deposit before a specific machine is built.
When a project has multiple IRRs it may be more convenient to compute the IRR of the project with the benefits reinvested.[14] Accordingly, MIRR is used, which has an assumed reinvestment rate, usually equal to the project's cost of capital.
Average internal rate of return (AIRR)
Magni (2010) introduced a new approach, named AIRR approach, based on the intuitive notion of mean, that solves the problems of the IRR.[15] However, the above-mentioned difficulties are only some of the many flaws incurred by the IRR. Magni (2013) provided a detailed list of 18 flaws of the IRR and showed how the AIRR approach does not incur the IRR problems. [16]
