Revealing the True Meaning of the IRR via Profiling the IRR and Defining the ERR

Since the late 1950s, most textbooks and many professors have been inadvertently defining the internal rate of return (IRR) of an investment incorrectly vis--vis the reinvestment of an investment's cash flows. The genesis of this unfortunate error can be traced to an article by Renshaw (1957). Most

textbooks and many professors have since paraphrased a quotation taken from the Renshaw article that attempted to paraphrase (out of context) the words of Solomon (1956). Both the Renshaw and the Solomon quotations, in their entireties, go on to properly explain the issue of reinvestment vis--vis the IRR. However, the paraphrasing of the partial quotation of Renshaw has perpetuated what some now call the "reinvestment rate controversy." The Renshaw (1957:193) quotation states: "...the (net) present value (NPV) approach assumes reinvestment of intermediate cash receipts at the discounting rate, while the internal rate-of-return (IRR) approach assumes reinvestment at the internal rate..."

The fact is that neither approach makes any assumption whatsoever about either the reinvestment of cash flows or the rate of return to be earned if reinvestment were to be considered. The discounted cash flow (DCF) equations for the IRR and the NPV are just that: formal statements of equivalence. Equations do not make assumptions; people make assumptions. A calculation cannot assume; people assume. Hence, reinvestment has nothing to do with the calculation of the IRR. However, reinvestment may be critical to the application of the IRR as an investment decision-making tool. What follows infra, hopefully, will clarify the confusion and settle the controversy regarding the issue of reinvestment vis--vis the IRR.

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The IRR is the calculated (not chosen) discount rate (i) that makes Value equal Price, i.e., the calculated discount rate that makes the equation's right-hand side (CF^sub 1-n^), or Value, equal the equation's left-hand side (CF^sub 0^), or Price (LHS = RHS).

In other words, the IRR is the "i" in this equation that equates the cash flows (CF^sub 1-n^) on the right-hand side of the equation with the price, cost or initial investment (CF^sub 0^) on the left-hand side of the equation. In yet other words, the IRR is the discount rate that makes the present value of the cash flows (gross present value, GPV) on the right-hand side of the equation exactly equal the price, cost or initial investment (CF^sub 0^) on the left-hand side of the equation. The IRR is the periodic (usually annual) rate of return "on" investment, or the return "on" capital, not to be confused with the return "of" capital.

However, as will be demonstrated, the IRR is not necessarily a periodic return on CF^sub 0^ in this equation; rather, the IRR is the rate of return on the capital (dollars, etc.) that is invested, or better stated, that is internal to the investment from period to period. In the first time-period, the amount of capital "internally invested" is the initial investment, cost, or price (CF^sub 0^). However, this "internal" amount often changes from period to period. The set of these internal amounts of capital is herein designated the "IRR profile" of the investment. The amount of capital that remains internal to an investment at any point in time is a function of the investment's capital recovery that, in turn, is a function of the magnitude and timing of the investment's cash flows.

Perhaps, a more specific and more comprehensive applied meaning of the calculated IRR is that it is a measure of the profitability of the capital that remains internally invested from period to period, and has not yet been recaptured or recovered by the investor. Whether or not the capital remains internally invested from period to period depends on the investment's periodic return "of capital-flow. An investment's periodic return of capital-flow is separate and distinct from its return "on" capital-flow. An investment's total periodic capital-flow is equal to the sum of its periodic return on capital-flow and its periodic return of capital-flow. The most familiar analogy to demonstrate these statements is the case of a loan amortization schedule that segregates the loan's debt service cash flow into its interest portion (return on capital-flow) and its principal portion (return of capital-flow) schedules. Note that in the DCF-IRR equation, the "i" mathematically handles the return on capital or return on investment, while the "1" mathematically handles the return of capital or return of investment.

The amount of capital that remains internally invested usually changes from period to period, as determined by the magnitude and timing of the investment's cash flows. If in any given time period the cash flow received is more (less) than is needed to earn the IRR for that period, then the extra (deficient) amount of cash flow is attributed to positive (negative) capital recapture, recovery or amortization. Said amount of capital recovery effectively decreases (increases) the amount of capital that remains internally invested, such that the IRR is very seldom a return on the cost, price, or initial investment that appears as CF^sub 0^ on the left-hand side of the DCF-IRR equation.

A totally generic demonstration and explanation of these concepts is offered via the introduction of a new methodology mentioned in the title of this presentation, namely "Profiling the Internal Rate of Return." To profile the IRR, one simply calculates an investment's "amortization" or "capital recovery" schedules that, as demonstrated in Exhibit 1, are a function of the magnitude and timing of the investment's cash flows (CF^sub 1-n^).

Exhibit 1 consists of five separate $100,000 investments, each with its own distinct set of three-period cash flows. Each of the five investments has an IRR of 15% calculated via the DCF-IRR equation. However, the five seemingly identical 15% IRRs are not the same; rather they are 15% IRRs on different internally invested amounts in each time-period as shown by the IRR profile for each investment. Each of the five investments has had the 15% IRR profiled to determine what the IRR is a return "on" in each period. Each profile consists of a column of numbers, labeled Outstanding Internal Investment, which change from period to period by the numbers in the Return Of column. The Return Of column numbers are calculated by subtracting the numbers in the Return On column from the respective periodic cash flows (CF^sub 1-n^) on the right-hand side of each DCF-IRR equation. The Return On column numbers are always 15% times the respective Outstanding Internal Investment column numbers. This profiling process clearly demonstrates that the IRR is the periodic rate of return on only the capital that has not been recovered (recaptured or amortized), namely the numbers in the Outstanding Internal Investment columns. These internal amounts change from period to period as determined by the magnitude and timing of the investment cash flows. The profiling process, therefore, has five steps as follows:

1. Calculate the IRR;

2. Multiply the IRR Outstanding Internal Investment = Return "On";

3. Subtract Return "On" from Cash Flow = Return "Of;

4. Subtract Return "Of" from Outstanding Internal Investment = Profile Number;

5. Repeat Steps 2-4 = IRR Profile Numbers "on" which the IRR is a return for each time period (i.e., year).

For example, Investment A's profile is identical to an amortization schedule for a three-period (three year) fully-amortized level-payment loan. Note that the 15% is not a return on $100,000 in each period. In time-period one, there is $100,000 internal to the investment and $15,000 of CF^sub 1^ ($43,798) provides a 15% return on the $100,000. The excess $28,798 in CF^sub 1^ ($43,798 - $15,000 = $28,798) is attributed to positive return of capital, also called positive capital recovery, recapture, or amortization. This $28,798 lowers the amount of capital that remains internal to the investment from the original $100,000 down to $71,202 ($100,000 $28,798 = $71,202). In time-period two, the 15% IRR is on the $71,202 and amounts to $10,681. The excess $33,117 in CF^sub 2^ ($43,798 - $10,681 = $33,117) is attributed to positive return of capital. This $33,117 lowers the amount of capital that remains internal to the investment from $71,202 down to $38,085 ($71,202 - $33,117 = $38,085). In time-period three, the 15% IRR is on the $38,085 and amounts to $5,713. The excess $38,085 in CF^sub 3^ ($43,798 - $5,713 = $38,085) is attributed to positive return of capital. This $38,085 lowers the amount of capital that remains internal to the investment from $38,085 to $0 ($38,085 - $38,085 = $0).

Hence, in this example, profiling the IRR demonstrates that the IRR is a return on a decreasing internal investment, and can be likened to the creation of the amortization schedule (principal and interest) for a fully-amortized level three-year loan of $100,000 at an annual interest rate of 15%. In such a loan, the lender earns interest on the declining balance (decreasing internal investment) each year, and the amount of each payment that is not interest, is attributed to principal amortization, resulting in a zero (0) balance upon receipt of the last payment of interest (return on) and principal (return of).

The IRR of Investment B in Exhibit 1 is also 15%. However, the 15% is not a return on the original $100,000 investment (CF^sub 0^) each year (period) here either. In fact, in this example, the $5,000 cash flow (CF^sub 1^) in time-period one is not only insufficient for the provision of any capital recovery (return of), but also is $10,000 deficient for the provision of a 15% return on (IRR), the original investment of $100,000 (CF^sub 0^). In this case, negative capital recovery (return of) causes the amount of capital remaining internal to the investment to increase to $110,000, requiring the 15% return (IRR) in time-period two to be $16,500 (15% times $110,000). However, only $5,000 is received for CF^sub 2^, causing a second deficiency of $11,500, and thereby an increase from $110,000 to $121,500 in the amount remaining internal, on which a 15% IRR must be earned and recaptured in the third and final time-period. Note that 15% on $121,500 is $18,225, and that the sum of this $18,225 plus the $121,500 remaining internal (that must be fully recovered) (return of), equals $139,725 (the amount of CF^sub 3^), which provides both the 15% return on the $121,500 and recovers the outstanding internal investment of $121,500, thereby lowering said internal investment to zero (0). Hence, in this example, profiling the IRR demonstrates that the IRR is a return on an increasing internal investment, and can be likened to a three-period (year) $100,000 negatively amortized loan with a yield rate (IRR) of 15% and a payment rate of 5%, causing a balloon payment of $121,500 in the third and final period (year).

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Exhibit 1

Profiling The IRR: "It's All Inside" The IRR Usually is Not a Return on CF^sub 0^

Investments C and D are merely combinations of the dynamics of Investments A and B. Investment C's profile indicates that the outstanding internal investment decreases and then increases due to positive capital recovery (amortization) followed by negative capital recovery (amortization), as displayed in the return "of column in Exhibit 1. Investment D's profile indicates that the outstanding internal investment increases and then decreases due to negative capital recovery (amortization) followed by positive capital recovery (amortization), as displayed in the return "of column in Exhibit 1. In both Investments C and D, the final cash flow provides both the appropriate return "on" and return "of," thereby lowering the capital remaining internal to the investment down to zero (0).

The cash flow patterns of Investments A-D are representative of the cash flow patterns of most investments. A somewhat less common investment cash flow pattern is that of Investment E in Exhibit 1. In Investment E, the 15% IRR is "on" the 100,000 initial investment, price, or cost (CF^sub 0^) in each of the three time-periods, with all of the capital recovery taking place in the third and final time period, thereby lowering the remaining internal capital of the investment down to zero (0). However, the conditions required for the IRR to be a return "on" (CF^sub 0^) in each and every time-period are only when all its periodic cash flows, except the last period's cash flow, are exactly equal to the IRR times CF^sub 0^, and the last period's cash flow is exactly equal to the sum of one of any previous period's cash flow and CF^sub 0^ itself. Hence, in Investment E, the IRR is 15% "on" CF^sub 0^ in each period since: (1) in time-periods one and two, the cash flows received are not more or less than is needed to earn the IRR on CF^sub 0^ for those periods, such that there is no (positive or negative) periodic investment amortization (recapture or recovery); and (2) the third and final time-period's cash flow ($115,000) contains its 15% on CF^sub 0^ (15,000), plus the full and complete return "of capital-flow ($100,000) for CF^sub 0^. Since no capital recovery takes place prior to the third and final time-period, there is always $100,000 internal to the investment. This set of cash flows can be likened to an interest-only non-amortized loan or bond with a balloon payment equal to the original $100,000 initial investment (CF^sub 0^) plus the $15,000 interest (return "on") for the third and final time-period.

Very few investments in the real world have such an even and perfect set of cash flows. Even interest-only bonds and mortgage loans, that likely might have such projected perfect sets of cash flows, are seldom purchased and sold at par, such that in actuality even they seldom have such even and perfect cash flows. In fact, most investment cash flows are uneven and imperfect, such that that either or both positive or negative periodic investment amortization or capital recovery is more the norm than the exception. Therefore, the amount of capital remaining internally invested (the IRR Profile) changes quite regularly, causing the typical and usual IRR to be a measure of the return "on" a "moving target," or on an everchanging internally invested amount of capital.

The ever-changing (increasing and/or decreasing) internally invested amounts of capital that are displayed in Exhibit 1 are the IRR Profile of each of the investments (A-E). These amounts are then used to calculate the respective Return "On" and Return "Of columns. Note also in Exhibit 1 that the Return "On" and Return "Of columns are added horizontally (algebraically) left-to-right to obtain the cash flows (CF^sub 1-n^) for each investment.

These same cash flows are then added vertically (arithmetically) top-to-bottom to obtain the columns that are labeled SUM OF CF^sub 1-n^ for each investment. The intent here is to explain what is meant in the part of the title of Exhibit 1, which reads "Its All Inside."

Said quote is the tagline for the J.C. Penny department store, implying that all one's needs can be found "inside" a J.C. Penny store. Herein, these words are referring to the fact all that is needed to define, calculate and understand the IRR can be found "inside" Exhibit 1. All of the amounts of capital for the provision of Return "On" and Return "Of" for each and every investment are contained inside the cash flows (CF^sub 1-n^) of each investment. There is absolutely no need to go "outside" an investment to complete the definition, calculation, or understanding of the IRR. To go outside an investment via the reinvestment of its intermediate cash flows would be to acquire additional investments external to the original investment.

Please note that for each of the five investments (A-E) in Exhibit 1, no reinvestment of the intermediate cash flows is necessary nor required to earn the 15% IRR on the periodic and usually ever-changing internally invested amounts of capital. In fact, to "reinvest" the two intermediate cash flows (CF^sub 1-2^ in these three-period investments) would be to make two additional and separate investments external to the original three-period investments. Any rate of return calculated to include these reinvested amounts, combined with the original cash flows appropriately would/should be called the External Rate of Return (ERR). Hence, profiling the IRR and defining the ERR unequivocally settles the reinvestment rate controversy. The IRR does not assume nor implicitly require the reinvestment of the intermediate cash flows of an investment, while the ERR does assume and explicitly require the reinvestment of the intermediate cash flows of an investment.

Said ERR may also be called a "portfolio IRR," because it is the geometric mean (weighted average) rate of return for more than one investment held for varying time periods, since each successive reinvested cash flow is invested for one time-period less than that of the previous reinvested cash flow.

The generic term ERR is preferred since it encompasses all of the variations on the same theme, namely the adjusted IRR, the modified IRR, the terminal value IRR, the Estate Management Rate of Return (EMRR) and even the Financial Management Rate of Return (FMRR).

Employing the ERR

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Investments A and B from Exhibit 1 both have IRRs of 15%. To demonstrate that reinvestment does matter, the cash flows of these two investments have been reinvested at three different reinvestment rates (5%, 15% and 25%) to obtain three different terminal values, which are then used with the initial investment of $100,000 to calculate the three respective ERRs. These reinvestment and ERR calculations are presented in Exhibit 2. Note that reinvesting at the 15% IRR results in an ERR equal to the IRR. However, this phenomenon does not prove (as some people think) that one must reinvest at the IRR to calculate the IRR. Rather, it simply demonstrates that a weighted average remains the same if more of the same numbers are added and averaged. For example, if a baseball player's batting average is .333 and the individual plays (reinvests in) additional games and bats 1 for 3 in each additional game that is external to the original number of games, the total overall batting average will still be .333. However, said total or overall batting average is based on the combined result of all of the games played and, therefore, more times at bat.

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Exhibit 2

Reinvestment and the ERR

In other words, the ERR of any given investment is the combined result, or weighted average, of the IRRs on more than just the original investment being analyzed, namely those additional ones made by reinvesting the original investment's cash flows. Hence, Exhibit 2 clearly demonstrates that reinvestment does matter in the application of the IRR, but has absolutely nothing to do with the calculation of the IRR. In other words, reinvestment is integral to only the ERR.

It is also worth noting that the differences in the magnitude and timing of the cash flows in Investments A and B cause the two ranges of ERRs to be quite different from one another. For Investment A, the ERR range is 7.29% (11.35%-18.64%); while for Investment B, the ERR range is only 0.83% (14.60%-15.43%). The meaning, significance and application of these numbers are the subject of another paper. However, suffice it to say that the two IRRs of 15% in Investments A and B are not as similar, let alone identical, as they might appear to be prior to the calculation of their respective ERRs.

Finally, any analyst, including the portfolio manager, must fully and completely understand both what the IRR is and what it is not, in order to properly interpret its meaning and to skillfully apply it in practice. The IRR is not exactly what many have come to think it is, and it assumes nothing regarding reinvestment. Only the analyst can make the reinvestment assumption, if any, for the application in a portfolio context. The analyst must compare the IRR profiles of seemingly similar IRRs before concluding that they are virtually identical; they may be quite different.

Conclusion

To summarize, profiling the IRR provides that if in any given time-period the cash flow received is more (less) than is needed to earn the IRR for that period, then the extra (deficient) amount of cash flow is attributed to positive (negative) capital recovery, recapture, or amortization. Said amount of capital recovery effectively decreases (increases) the amount of capital remaining internal to the investment. This process of capital recovery is analogous to positive (negative) amortization of a fully-amortized loan investment.

Profiling the IRR reveals the true meaning of the IRR as follows:

1. It is seldom a return on the initial investment (CF^sub 0^) in each period;

2. It is usually a return on varying amounts in each period;

3. There is no reinvestment in its calculation;

4. Reinvestment should be considered in the application of the IRR in order to make a fair comparison between two or more mutually-exclusive investments, especially if terminal-value wealth-maximization is a stated objective of the investor;

5. If and when reinvestment is employed, the resulting rate of return is no longer an internal rate of return (IRR), but rather an external rate of return (ERR), which is a function of the reinvestment rate chosen by the investor; and

6. Profiling the IRR unequivocally settles the reinvestment rate controversy: the IRR does not assume, nor require, reinvestment of the intermediate cash flows; whereas the ERR does assume, and require, reinvestment of the intermediate cash flows.

REFERENCE

References

Crean, M. J., Profiling the IRR and Defining the ERR, The Real Estate Appraiser and Analyst, 1989, 55, 55-61.

Dudley, C. L., Jr., A Note on Reinvestment Assumptions in Choosing Between Net Present Value and Internal Rate of Return, Journal of Finance, 1972, 27, 907-15.

Renshaw, E., A Note on the Arithmetic of Capital Budgeting Decisions, Journal of Business, 1957, 30:3, 193-201.

Solomon, E., The Arithmetic of Capital Budgeting Decisions, Journal of Business, 1956, 29:2, 124-29.

AUTHOR_AFFILIATION

by Michael J. Crean*

* University of Denver, Denver, CO 80234 or mcrean@du.edu.