INTERNAL RATE OF RETURN REVISITED

Introduction

Background

Discounted cash flow (DCF) includes the present value (PV) (or net present value (NPV)) and the internal rate of return (IRR) methods of analyzing cash flows. DCF provides insight into financial management not possible using other techniques. The NPV of the time-phased costs over the economic life of an investment project is the best single-number measure of its life-cycle cost.

Internal rate of return (IRR) is rarely used in government analyses. A minor reason is that some IRR calculations require cash inflow as well as outflow. Government usually generates no revenue and there is no IRR for an all-negative cost stream. However, a strength of IRR is in comparing project cost streams directly. IRR in this case is based on the differential between, say, a baseline and alternative cost streams. The technique is explained below under mutually exclusive projects.

The major reason IRR is not used centers on the extensive criticism of IRR found in financial management textbooks.⁽¹⁾ These criticisms overstate the minor difficulties associated with IRR and understate the coexistent difficulties with NPV. As a result, IRR is not being exploited to its potential. The aim of this paper is to put the criticisms of IRR into perspective and put the two DCF measures into balance.

Net Present Value

NPV is well accepted for sound reasons, but it has limitations. For one thing, to solve for NPV, one must first calculate the "opportunity cost of capital," also called the "discount rate." This rate is used in the discounting equation to calculate NPV. While generally a given in theoretical discussions, figuring out the cost of capital can be a difficult and time-consuming process. This is especially so in large, complex organizations. The "correct" discount rate is continuously debated in government.

A second difficulty with using NPV alone is that risk is assumed to be equal among competing projects. Risk is seldom equal in practice. Portfolio diversification is an acknowledged risk reduction technique. Similarly several smaller projects should have less risk than one large one, all other things being equal. NPV favors larger projects whereas smaller ones inherently have less risk.

A final difficulty with using NPV is that it is largely limited to comparing projects within an organization, presumably with the same opportunity cost of capital (discount rate). Using different rates can change NPV rankings. NPV is therefore not very useful for comparisons between organizations—especially those of different size. For example, Anglo-American NPV comparisons can be meaningless while IRR comparisons can be illuminating. Even with these minor difficulties, NPV is the best absolute measure of value of an outflow-inflow stream. IRR is the best relative measure. Both have difficulties, but IRR is strongest where NPV is weakest.

Internal Rate of Return

The second discounted cash flow measure, IRR, has traditionally been defined as the [sic, any] discount rate at which NPV is equal zero. NPV has been applauded and IRR criticized for decades. While the focus of the criticism has been on using IRR in capital budgeting decisions, the unfavorable coverage has spilled over into other areas. Analysis of the reasons given for supposed IRR inferiority is the focus of this paper. Correcting the misperception is the result.

IRR is used extensively despite the textbook criticism. Business people often favor it. For one thing, IRR is very good for screening projects. NPV is highly sensitive to the discount rate, while IRR bypasses the problem of deciding the "correct" one. Because IRR is a rate or ratio, not an absolute amount, it is more useful for comparing unlike investments, say stocks and bonds. It also is more useful for making comparisons between different periods and different sized firms and for making international comparisons. The intention here is not to argue for either point of view, but instead to put the issue into balance. The aim is to show:

• NPV and IRR have essentially equivalent utility;
• they are complementary ways of looking at a problem or opportunity;
• IRR and NPV together give a better analysis than either alone;
• if properly viewed, NPV and IRR give identical signals, including capital budgeting decisions;
• IRR is particularly useful for comparing different sized projects, where it receives some of its greatest criticism; and
• IRR is useful alone.

We will critically examine the professed reasons for the superiority of NPV over IRR in capital budgeting.⁽²⁾

Criticism Number One: Inflows versus Outflows

IRR is the same even if the cash flows are reversed or inverted. For example, the IRR is 25 percent for both of the following:

Table 1. Positive versus Negative Cash Flows

Table 2. "Conflicting" IRR/NPV Signals

This is a good place to deviate briefly and examine the relationships between IRR and positive and negative NPVs. See Table 3.

Table 3. NPV/IRR Relationships

With net-positive cash flows, NPV decreases from maximum at a zero percent discount rate and converges on zero as it increases. This is normal. But once past zero NPV, where IRR is determined, NPV is negative at all discount rates. This latter area is of no interest.

Finally, with net-negative cash flows, NPV also converges on zero NPV with an increasing discount rate. After zero, NPV increases with an increasing discount rate. This implies that although we were losing money at all discount rates below 60 percent, the project became profitable at 80 percent. Worse still, the higher the discount rate, the more attractive it becomes. This suggests we can turn around an unfavorable project by increasing our opportunity cost of capital. This is nonsensical, but incorporated then ignored in the negative-positive criticism.
 

Criticism Number Two: Mutually Exclusive Projects

IRR can supposedly give a different decision from NPV on mutually exclusive projects. See Table 4.

Table 4. Mutually Exclusive Projects

• Who would not prefer to invest in ten projects like Project E (or in Project E ten times, NPV = $8,818.18) instead of Project F?
• Who would not prefer that people developing proposals look for projects with the highest possible rate of return?
• Should the higher rate of return receive favorable consideration for being of lower absolute market risk (i.e., less money is at risk)?
• Should we eliminate Project E simply because it is the smaller of the two?
• Can we invest in both now?
• Can we invest in one now, one later?

A focus on NPV to the exclusion of IRR would build in a bias for large projects over smaller, perhaps more cost-effective ones. What Criticism Number Two instead confirms is that having both NPV and IRR gives a better picture of the problem or opportunity than either alone.

One way to overcome the supposed mixed signal is to normalize the larger project to the smaller one (or vice versa). See Table 5. Project F is 10 times larger than Project E, so divide Project F cash flows by 10. Project F is normalized (or scaled) to Project E.

Another way to overcome the supposed mixed signals is to evaluate the difference (or "deltas") between the two. This is done in S05G.TXT, Table 5, and Figure 1.

Table 5. Normalized, Differential and Realistic Comparisons

Figure 1. Mutually Exclusive Projects

The technique of evaluating differences (also called incremental flows) bypasses the problem with different size projects. It is also a practical way to analyze the difference between alternatives with cash outflows only—a government Economic Analysis (EA) for example. Figure 1 illustrates NPV at all relevant discount rates (those between zero in the IRR).

NPV and IRR can be used together when evaluating different sized projects. If there is an apparent conflict, simply understand what is causing it and present the information differently if necessary. Of course, more realistically comparable investments, such as Projects F and H, IRR and NPV give the same answer regardless. Still, both NPV and IRR give a clearer picture than either alone give. It is not a case of either-or; we can have both.

Table 6. Financing Included

Assume a 10 percent loan to finance the project. While NPV remains the same for both projects, IRRs decline to 55 percent and 30 percent respectively. The difference between the project life cycle financing costs and project revenues (a variant of the differential cash flow technique) can give a better picture. An advantage to this approach is that we can incorporate the impact financing on the project. For example, very large projects might require external financing at higher rates than that incorporated in the discount rate.⁽⁷⁾ The opportunity cost of capital is typically assumed to be constant and equivalent for both large and small projects.

Criticism Number Three: Multiple Sign Changes

More than one IRR is possible with multiple sign changes. Additional IRRs can occur if the signs of the cash flows change more than once. Not uncommonly, this criticism is combined with Criticism Number One (negative versus positive flows) and presented as the inverted project K. This unnecessarily complicates the picture. We will use project K for demonstration, with positive cash flows, a positive NPV, and IRRs of 25 and 400 percent. See Table 7 and Figure 2.

Table 7. Sign Changes

Figure 2. Conflicting Rates of Return

 1

If we accept that both IRRs are mathematically correct, one solution is simply to take the smaller, more conservative 25 percent.

A second way is to use the IRR closest to the net return on outflows. Ignoring when they occurred, outflows are $2,500 and inflows $2,900. The $400 net difference is the NPV at a zero percent discount rate. It is also 16 percent of outflows. This return on outflows does not consider the time value of money; therefore, it should be close to the relevant IRR. The closest IRR is 25 percent. On the other hand, 400 percent is 25 times larger than the return on outflows. While "mathematically correct," 400 percent is not meaningful.

A third way is to use the IRR that is consistent with NPV converging on zero as the discount rate increases. In other words, look for the same pattern we find with an annuity or bond yield. We cannot arrive at 400 percent without going through a nonsense zone where NPV increases with increasing discount rates. While mathematically correct using the widely accepted definition, this second IRR is not meaningful. The rate calculated before entering a nonsense zone is 25 percent.

Sign Change Demonstration

If you are still not convinced, see Table 8. This might be the type of analysis a credit union manager would perform on a member account, but it is simplified to make the calculations obvious.

Assume interest either received or paid on your credit union account is "1 percent per month on the balance," i.e., $10 per month per $1,000 balance or 12 percent per year. Assume you make the withdrawals and deposits and pay or receive interest in cash the last day of each month. Note that interest payments and receipts vary directly with the IRR or rate implicit in the cash flows.
 

Figure 3. Sign Change Demonstration

At the maximum overshoot for the 840 percent credit union rate, NPV is 0.058 percent of undiscounted net cash flow ($47.47 / $81,200) and the discount rate is 763 percent.  In electronics where oscillations are studied extensively, such near solutions are called "lost in the noise."  The 840 percent rate is the responsive IRR, but this is an extreme case for both IRR and NPV.  The 697.14 percent rate before it is 58 times a nominal 12 percent interest rate and over 25 times a a very attractive 25 percent return on investment.  Rates of return become meaningless at such high multiples.  NPV is suspect as well since it is increasing with increasing discount rates, implying increasing value with increasing cost. Rates past 697.14 are nonsensical, but it serves a purpose:  It provides a useful upper bound for meaningful IRR calculations.  IRR can be bounded into a relevant range for practical use.⁽⁹⁾

IRR Algorithms

Typical IRR algorithms calculate any "mathematically correct" IRR. In doing so, developers have accepted the sign change criticism and agreed that all IRRs are the same. These algorithms can be found in, for example, spreadsheet financial functions, in function calls in high-level programming languages, and in financial programs. These algorithms may require you to provide a "seed," i.e., enter a rate. The seed is used to do a search (typically half-interval) for the closest "IRR." If you enter a seed close to 12 percent, it would more likely than not be the IRR returned. Conversely, if you provide a seed close to a second or subsequent rate, it should be the one returned. But what is "close" when you do not know what the choices are? Negative rates are also possible with many algorithms, even with positive cash flows, resulting in a nonsensical IRR. The result is that you are uncertain what the program will calculate if the cash flows contain multiple sign changes. Financial analysis programs typically warn you of the prospects, but pretty much leave it up to you to figure out what to do about it. Spreadsheets are prone to give "ERR" as the solution to IRR calculations with multiple sign changes or to lock up. When either happens, the IRR function may have encountered a problem explained in this paper? Or it could be something else?⁽¹⁰⁾
 

Criticism Number Four: IRR Incalculable

Under some circumstances IRR is incalculable. This is perhaps the most serious criticism of IRR. But IRR critics overlook that this is not much of a problem in practice and it applies to NPV as well. The only difference is that it is not as obvious with NPV as it is with IRR. IRR is incalculable in at least five circumstances. First, IRR cannot be calculated if cash flows are all positive. See Project M1 in Table 9:

Table 9. All-Positive Annual Flows

The appearance of an all-positive cash flow situation can occur though, if we lump the actual cash flows from shorter periods into longer ones. Automated report generation routines could give flows similar to those in Project M1 even if the underlying flows were not all positive. Using more frequent intervals could correct the apparent problem. For example, +$300 in year 1 of Project M1 could have been the aggregate of quarterly or semiannual flows of which one or more were negative. If this were the case, simply using data that more closely corresponds to the period over which cash flows occur would result in a solution. In this example, semiannual flows make IRR calculation possible and do a better job of calculating NPV.

A second circumstance in which IRR is incalculable is when cash flows are all negative. This, like the all-positive case, needs no further consideration: It is a giveaway and follows the same reasoning as the all-positive case.

A third circumstance in which a positive IRR is incalculable is when net cash flows are zero—say +$1,000 in year 1 and -$1,000 in year 2.⁽¹¹⁾ Intuitively we would conclude that this was a favorable circumstance in which we were given an interest-free loan for one year. There may be an opportunity implied in such an unlikely transaction; however, it is incalculable without entering the actual cash flows. At 10 percent, such an exchange would be worth $100 to the recipient, presuming the $1,000 were invested. If so, we should include the cash flow from the investment in the calculation giving net cash flows of $100, not zero. We can handle this situation by including project financing.

The fourth circumstance under which IRR is incalculable is subtler. It occurs with certain other combinations of cash flows not described above. See Table 10. With a positive cash flow of $0.01 (1 cent), IRR is calculable (IRR is positive but lost in rounding).  With a net negative cash flow of 1 cent, NPV diverges from zero and IRR is incalculable. If year 0 and year 1 cash flows are the same, cash flows are zero and IRR incalculable. If we decrease year 1 revenue by 1 cent (-$0.01) to $999.99, NPV is negative at any positive discount rate and IRR is incalculable.⁽¹²⁾

Table 10. IRR Incalculable - Simple Cash Flows

Table 11. IRR Incalculable - Complex Cash Flows

While this is an IRR limitation, NPV is suspect as well because it is positive at all discount rates. Said another way; the discount rate is irrelevant. Or worse, past 66.7 percent we could increase NPV by simply increasing our opportunity cost of capital—increase value by increasing cost. This is absurd. Recommendation: If IRR is incalculable with NPV diverging from zero, look carefully at your input data. If you are certain your data are okay, try a couple of different discount rates above and below your cost of capital to determine if you are in the nonsense area of the DCF plot.⁽¹⁵⁾

Figure 4. Complex Cash Flows

Summarizing, IRR is incalculable with all-positive, all-negative, net-zero, net-negative, or complex cash flows. Implications? None for either IRR or NPV in the first three cases. But while IRR is incalculable, NPV is suspect and potentially nonsensical, with net-negative or complex cash flows.
 

Criticism Number Five: Reinvestment Assumption

The final criticism is the "reinvestment assumption." Critics correctly point out that IRR implicitly assumes that positive cash flows generated are reinvested at the IRR, not the discount rate. (Negative cash flows would be disinvested as well, but generally are not mentioned, presumably to simplify the demonstration.) They argue that the discount rate, not the IRR, represents the opportunity cost of capital. This in turn makes IRR inappropriate for appraising or ranking projects. Not uncommonly, this criticism is mixed in with one or more of the other four major criticisms addressed above.⁽¹⁶⁾

To correct for contradictions resulting from using implicit assumptions, simply make them explicit. For example, if you want to know what the IRR is with cash flows reinvested at the discount rate, calculate it twice. The second time, enter the actual cash flows generated from reinvested funds for all projects. If the funds can be used to retire debt, include the interest avoided as a positive cash flow to the project. It is of course essential to treat all projects the same in this regard if comparisons are to be meaningful. While this technique would have been cumbersome several decades ago when some of the more popular textbooks were written, modern electronic tools make it straightforward. Usually, however, such adjustments are unnecessary. Commonly, reinvestment at the IRR is an appropriate assumption. It is the one made in a bond yield problem, which is, after all, an investment decision.
 

Other Criticisms

This concludes our discussion of the major criticisms of IRR as they apply to capital budgeting. The criticisms cited above are not the only ones you will find, but most of the remaining ones are either trifling or they relate to the extent to which IRR fails to conform to (admittedly useful) guidelines for a capital budgeting decision. For example, it is useful to be able to add the estimated NPV from several projects to arrive at a total NPV for all of them. This is sometimes called the "value additivity principle." We cannot add IRRs for a meaningful number, but we can average them. We can calculate an IRR for multiple projects by simply combining the cash flows. We should expect IRRs from these combined cash flows to be different and not usable alone if the projects are different size.⁽¹⁷⁾

Conclusion

It has, I believe, been shown that, practically speaking and properly viewed, IRR yields the same decision as does NPV except under some extreme circumstances that present few limitations in practice. When IRR is incalculable, NPV is suspect.  No attempt has been made to suggest that IRR is superior to NPV. They are best used together. NPV and IRR give consistent answers if handled and viewed properly. Together they give an indication of risk as well as return. IRR is not affected by the size of the cash flows. Finally, IRR is useful alone in virtually all time-value-of-money problems.⁽¹⁸⁾

NOTES

1. See, for example, Managerial Finance, Ninth Edition by J. Fred Weston and Thomas E. Copeland (Fort Worth: The Dryden Press), 1992, pp. 309-320. The first edition was published in 1962. "Summarizing the comparison between the NPV and IRR criteria, we see that the IRR has many difficulties that invalidate it as a generally acceptable capital budgeting rule." (p. 352, emphasis in original.)"   Questions, actually polite challenges, on this paper have been posed several times in the context of a Principles of Corporate Finance text by Richard. A. Brealey and Stewart C. Myers (5^th edition, Irwin-McGraw-Hill, 1996).  My  paper, Principles of Corporate Finance Questions, addresses those challenges.  It is shorter, more direct and illustrated graphically with spreadsheets from the Olin School of Business.   For an online historical discussion, see The Development of a Theory of Corporate Investment Decision Making: An Historical Perspective with Implications for Future Development and Teaching. Search using " IRR " (note the spaces) to find the relevant passages.

2. The text files linked from this paper, e.g.,S05A.TXT, were generated by a Discounted Cash Flow program I wrote. It calculates IRR and NPV to the specifications implicit in this paper.

3. A discounting problem to a lender is an interest problem to the borrower.  See S21C.TXT for an example of a $200,000 mortgage loan for 15 years at 8.5 percent from both points of view.  The interest rate table might just as well be called the discount rate table in such transactions.

4. Even with this aside, how do you get a positive cash flow in the first period (year) of a project? Some have suggested investment tax credit. While that might be possible, I would hate to enter an Internal Revenue Service audit with nothing spent and $1,000 claimed in deductions. Even a 100 percent tax credit on the amount spent would be zero net cash flow, not $1,000. Still, a positive cash flow in period zero might be rationalized outside capital budgeting theory—a retainer for services to be rendered, for example. I concede some plausibility for Project C and evaluate the criticism as it has traditionally been stated.

5. When using the S05?.TXT files, but look for net negative cash flows and "(outflow)" just before IRR.

6. Internal rate of either payment or outflow would be okay.

7. Increases in debt-equity ratios suggest higher risk to lenders, who in turn increase rates. This in turn would change the cost of capital.

8. Monthly cash flows are the sum of Column (A) and the appropriate Column (C) through (I).

9. The relevant range in this case is between discount rates of zero and 697.14 percent.  While this range does not include all responsive rates, those past 697.14 percent are extraordinary returns, and all rates past it are near solutions. One way to insure only a responsive IRR is to limit its size, say to 100 percent. In this case that would be less than the relevant range.  Some IRR algorithms limit IRR this way. The view here is that it is unnecessary to limit IRR to less than 500 or 1,000 percent so long as the other criticisms are taken into account.  Such higher returns are extraordinary, but plausible.

10. Some of the IRR equations can include large or small numbers that exceed precision limits or floating-point limits. Spreadsheet algorithms, for example, can appear to lock up (or actually lock up) when calculating extremely high IRRs. Since very large percentages become meaningless in a practical sense, placing a limit of, say 500 or 1,000 percent provides a satisfactory answer without lockup.

11. Use the direction of the cash flows rather than negative IRRs.  See Criticism Number One for an explanation.

12. Some IRR algorithms calculate a zero IRR for N2 and a negative IRR for N3 and N4. Negative IRRs are logically inconsistent with positive discount rates. The reason IRR is incalculable is that any positive discount rate drives NPV more negative—away from zero. It is immediately in the nonsense zone. If we invert the flows and try again, the same thing happens. This implies negative interest from either borrower or lender point of view. DCF is unnecessary; the project should be rejected.

13. An IRR solution may still be possible if NPV turns again at subsequently higher discount rates and again converges on zero. However, this subsequent rate, if it exists, is questionable.

14. IRR may be calculable with complex cash flows using other assumptions. For example, if you use either continuous or midyear discounting with S05P2.TXT and S05P3.TXT, IRR is 30.74 and 35.08 percent respectively. In doing so, you are accepting the default assumption of not discounting year zero. This is equivalent to accepting that the year zero flow occurs at the beginning of year one.

15. IRR calculations enter the nonsense zone at a $100 NPV and a 66.7 percent discount rate. The point at which NPV begins to increase with an increasing discount rate is not meaningful, other than that is where the nonsense zone begins.

16. Combining Criticism One (negative versus positive flows) and Two (mutually exclusive projects), then concluding that IRR is at fault is a favorite. But projects that will lose money do not compare well with those that will make money. IRR has nothing to do with it. Nor should smaller projects with less money at risk be expected to generate equivalent NPVs. Combining them into one demonstration should not change the expectation.

17. Although not often formally criticized, high rates deserve mention. It puts IRR in what the British call the "too difficult" category (not worth the trouble). Millions of percent IRR are possible, though not very realistic or meaningful. A data entry error would be the more likely reason for an IRR in the hundreds of percent at acceptable risk. Typical IRR algorithms appear to "lock up" when attempting to calculate very large IRRs. Other algorithms, in spreadsheets for example, typically display "ERR" and leave it up to the user to determine what causes it.

18. Others are:

Notes

Paper prepared for the Financial Economics Network (FEN), an affiliate of the Social Science Research Network (SSRN). Copyright © 1982-1997, Ray Martin.  All rights reserved.

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