# Solving the dual IRR puzzle.

Analysts often select investments by choosing, from among potential projects, those that offer the highest net present value (NPV) or internal rate of return (IRR). The two measures are derived from the same basic equation, although some analysts prefer the NPV technique, which offers a measure of profitability in dollar terms. Others prefer the IRR approach, which generates a percentage return that can be compared to projected returns on other investments.

A problem with IRR

A problem that accompanies IRR analysis is that if an investment has at least three projected cash flows and if one IRR can be computed, there is likely to be at least one additional IRR solution. (The exception involves what mathematicians call "repeated roots.")

It is fortunate that in a simple investment situation (in which all periods following the initial outlay involve inflows), one of the multiple IRRs can be ignored because its magnitude is inconsistent with the nature of the projected cash flow.

More specifically, the value of the errant IRR will be less than -100 percent, indicating a loss of more than the amount invested. (While there are investments, such as some derivatives, on which losses can exceed funds committed, this result cannot hold if every cash flow after an initial outlay is positive.)

Unfortunately, it is not always possible for the analyst to ignore one of two IRRs because each may be of a magnitude that appears consistent with the projected cash flows. This situation can occur when there is a reversal in the signs of the cash flows such that an initial outlay is followed by at least one positive and then at least one negative cash flow. How can the analyst respond to dual IRRs in such cases?

Resolving the conflict

We seek a method that addresses all cases of dual IRRs, regardless of the magnitudes of the results. The method discussed here has great intuitive appeal, in that a rate is rejected if it moves in the "wrong" direction when a cash flow changes. A rate is ignored, for example, if it falls when computed under new assumptions that show a lower negative cash flow or a higher positive cash flow in any specified period. (It should be intuitively clear that a useful technique would treat as irrelevant a rate of return measure that falls as the investor becomes "better off" or rises when the investor is "worse off.")

This method has theoretical appeal because it is applicable to all situations in which there are two solutions to the IRR equation, regardless of whether one of the rates is clearly implausible because of its magnitude.

A graphical example

Suppose that there is an investment with three projected cash flows. We expect the initial cash flow relating to an investment to be a negative amount (the initial cash flow generally would be the cost of the project) and would typically expect subsequent cash flows to be positive, although one or more could be negative.

The NPV of a project with three cash flows calculates with the equation:

1. NPV = C[F.sub.0]/[(1 + r).sup.0] + C[F.sub.1]/[(1 + r).sup.1] + C[F.sub.2]/[(1 + r).sup.2]

In Equation 1, the subscripted CFs represent the periodic cash flows. Note that the initial C[F.sub.0] is assumed to occur today (it is discounted for zero periods), and recall that its value is negative.

Variable r represents the rate by which expected cash flows are discounted in computing present values. This rate is the return that the investor would expect to earn on other available investments that would impose similar risks.

We can multiply each side by Equation 1 by [(1 + r).sup.2] to create Equation 2:

2. C[F.sub.0][(1 + r).sup.2] + C[F.sub.1](1 + r) + C[F.sub.2]

and can present the result in a form that is easier to study by replacing (1 + r) with x, so that the equation appears as:

2A. NVP [x.sup.2] = C[F.sub.0][x.sup.2] + C[F.sub.1]x + C[F.sub.2]

Equation 2A could be called net future value (NFV) in that it represents the future equivalent of a set of cash flows, jus as NPV represents the present equivalent of a cash flow series. The equation takes the form of the well-known quadratic function, the graph of which is a parabola. Figure 1 shows a graph of this parabolic function based on an assumption that C[F.sub.1] is positive, while C[F.sub.2], like C[F.sub.0], has a negative value.

The "roots" of the function are shown graphically as points where the parabola crosses the horizontal axis. At each such point, the function's value equals zero:

NFV = 0 = C[F.sub.0][x.sup.2] + C[F.sub.1]x + C[F.sub.2]

We can find the dual IRRs (r values for which NFV = 0) by subtracting 1 from each of these roots; recall that x = 1 + r. The two IRRs are shown in Figure 1 as r and [r.sub.2]. The puzzle for the analyst to solve is to select the one that is relevant.

The simplest cash flow change to consider, for both computation and graphicat purposes, is an increase or decrease in C[F.sub.2]. (Note that in Equation 2A, C[F.sub.2] is the only cash flow not multiplied by x or [x.sup.2]. In a graphical sense. a change in C[F.sub.2] merely shifts the parabola up or down.)

Suppose that the analyst were to revise the cash flow projections such that C[F.sub.2] would rise (become less negative). It should be obvious that the investor would be better off with a higher cash flow in that final period.

A graphical representation would show the parabola shifting upward, as occurs in Figure 2. Note that the root at the left would decrease ([r.sub.1][prime] has a lower value than does [r.sub.1]), while the root at the right, [r.sub.2], would take the higher [r.sub.2][prime] value shown.

Alternatively, a revised cash flow projection showing a decrease in C[F.sub.2] obviously would be unfavorable for the investor, relative to the situation depicted in Figure 1. As shown in Figure 3, the parabola shifts downward; the right-side root falls (from [r.sub.2] to [r.sub.2][prime]) while the left-side root rises (from [r.sub.1] to [r.sub.1][prime]).

The root on the left moves in the wrong direction whenever we change an assumption regarding the cash flows. Therefore, this root must be irrelevant. The principle demonstrated is that the IRR associated with the root of the function where NFV is increasing (the slope is positive) should be rejected as an indicator of investment returns. The technique is equally applicable in the more typical investment situation (one without expected cash flow reversals), in which one of the two rates has a value of less than -100 percent.

Conclusion

An IRR, regardless of its magnitude. is a relevant measure of investment return if a marginal increase in any projected cash flow results in an increase in the rate's measured value, whereas an IRR is irrelevant if a marginal increase in a cash flow causes a decrease in the rate's value. This rule is very general, in the sense that it covers all possible IRRs, including the generally forgotten case of an IRR less than -100 percent. (In fact, if IRRs that could be considered were to be limited to those greater than -100 percent, the rule would be stated somewhat differently.) Problems can arise in the use of the preceding rule, however, if there are more than two roots to the NFV function.

Reference

Colwell, Peter F., Roger E. Cannady, and Hiram Paley, "Relevant and Irrelevant Internal Rates of Return, The Engineering Economist 32 (Fall 1986): 17-38.

Peter F. Colwell is ORER Professor of Real Estate in the Department of Finance at the University of Illinois at Urbana-Champaign. He is also the editor of the Illinois Real Estate Letter and is in his third term as the director of the university's office of Real Estate Research.

Colwell serves as the 1995 president of the American Real Estate and Urban Economics Association, is a Homer Hoyt Advanced Studies Institute Fellow and a member of its Weimer School faculty. He is also a member of the editorial review boards for the Journal of the American Real Estate and Urban Economics Association and the Journal of Real Estate Research.
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