Investments: Calculating The Internal Rate of Return

By Daniel DiTieri
Return on investment (ROI) analysis has come a long way since buy and sell decisions were made strictly on the basis of the first year cash-on-cash return. All the more recent ROI techniques seek to estimate annual returns over a fairly long period – five or ten years in most cases. The trend to long-range projections reflects the fundamental change in the sources of equity capital for real estate. Formerly, equity came mostly from sophisticated real estate professionals who put up their own money and relied on their judgment in making investments. For them, the current profit and loss statement was all they needed. (It was also true that returns were often much higher than today, affording a cushion against mistakes.)

Now, however, equity capital comes primarily from large institutional investors (pension funds and equity-sharing lenders) and from the public (via syndicates and real estate investment trusts). For these investors, detailed projections substitute for the "intuitive" sense of the professional.

A projected ROI will be worth looking at only if two conditions are met: (1) the projected figures have been carefully and conservatively arrived at; and (2) the method of estimating the ROI is clearly understood by the investor. One popular method for measuring ROI is the internal rate of return (IRR). The workings (and limitations) of the IRR are detailed below, as is its close relation, the adjusted rate of return (ARR).

IRR Defined
The IRR arrives at an ROI by (1) estimating the projected cash inputs and economic benefits (cash or tax) over an assumed holding period, and (2) discounting the inputs and benefits back to their present value (the date of the initial investment). The discount rate that results in inputs and benefits balancing out to zero is the IRR.

While a schedule of cash inputs often can be prepared with some degree of accuracy, the opposite is true of future benefits, particularly cash returns. Consequently, IRRs are inherently speculative as compared to a capitalization rate based on a current income statement (but which makes no pretense of predicting the future). Nevertheless, an IRR derived from soundly based projections can be a valuable guide for the investor.

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How IRR Works
Table 1 illustrates how IRR works. Assume an initial investment of $10,000 is projected to generate a total return of $13,030 from seven annual economic benefits in the form of cash payments. By using a present value table that shows the present value of payments to be received in the future (or by using a computer program), we can determine the compounded interest rate that will permit an initial $10,000 sum to grow to $13,030 as indicated in column 2 of Table 1. The compounded interest rate is 12 percent.

This may be more clearly understood if the initial investment of $10,000 is viewed as a number of separate initial investments, each equal to the present value of each future payment (i.e., the present values listed in column 4). For example, $4,465 of the initial investment is treated as repaid to the investor at the beginning of Year 2 together with $535 interest (or a total of $5,000). This represents an annual return of 12 percent. The second segment of the initial investment, $2,391, is repaid at the beginning of the third year together with $609 of interest. This also represents a compounded return of 12 percent, and so on for the remaining five payments.

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Reinvestment Earnings Ignored
The important point to note about the process just described is that IRR does not show the compound annual growth on the entire initial investment, but only for the period of time each segment is invested. Thus, when comparing IRRs for different investments, it is necessary to make an assumption about the reinvestment rate for payments received by the investor prior to the end of the overall holding period.

To show the importance of including a reinvestment factor when comparing the total return from different investments, compare Tables 1 and 2. Table 2 shows the annual returns (including return of principal in the final year) from an investment of $10,000 in a tax-free bond yielding 8.5 percent. The total value of the annual payments is $15,950, $2,920 more than the total payments in Table 1. However, the IRR of the tax-free bond investment is only 8.5 percent, 3.5 percent points below the IRR in Table 1. The reason is obvious. Most of the annual returns in Table 1 come in the earlier years, while the opposite is true in Table 2. (That is, the average holding period is shorter in Table 1.) Thus, when the reinvestment of annual returns is ignored, Table 1 shows the higher return.

However, the conclusion may change depending on the assumption one makes about reinvestments. The lower portions of Tables 1 and 2 compare the two investments when it is assumed that annual benefits are reinvested at a 6 percent after-tax rate of return until the end of the overall holding period of seven years. It then turns out that the tax-free bond shows the higher return. The reason is that the real estate investment, which returns most of the initial investment in the early years, subjects the returns to a reinvestment rate of only six percent, significantly lower than the 12 percent return on the real estate. In the case of the tax-free bond, although the initial rate is only 8.5 percent, a larger proportion of the initial investment earns that rate for a longer period of time; consequently, less of the investment returns are reinvested at the 6 percent rate.

What IRR Really Tells Investors
As can be seen from the Tables, the IRR tells an investor what the compounded interest rate is projected to be on the equity remaining in the particular investment at any given point. It does not represent the compounded return on the total initial investment over the entire holding period. Therefore, the investor must make an assumption about a reinvestment rate whenever he is comparing two or more investments.

In the example shown in Table 1, only if annual returns could be reinvested at the same 12 percent rate the initial investment earns, could it be said that the $10,000 put up in year 1 earns 12 percent compounded annually through the end of year 8. If the reinvestment rate is more or less than 12 percent, then the total return on the investment over the holding period will be different from 12 percent.

The Adjusted Rate of Return
The adjusted rate of return (ARR) can be a better way of comparing investments because it specifically identifies a reinvestment rate to be applied to amounts repaid to the investor during the holding period. The final two lines in Tables 1 and 2 show how an ARR is calculated. When reinvestment of benefits is assumed at 6 percent after tax, the tax-free bond shows an ARR of 8 percent, versus 7.9 percent for the real estate investment. (In the case of a zero coupon bond or certificate of deposit, the ARR and the IRR are identical, since in these investments the reinvestment rate is guaranteed to be the same as the rate on the original investment.)

Dan DiTieri is a senior manager in the Real Estate practice in our New York office. He can be reached at (212) 885-8378.

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Inquiries about material in the Monitor should be directed to
Alvin L. Arnold at (212) 885-8235.