# Income capitalization problems.

Determining an appropriate method and yield rate to estimate the value of income-producing property can be a problem. The structure of the generally accepted appraisal approach is simple enough, expressed by the brief formula that indicates the relationship between income (I), value (V), and yield (Y):

V = I/Y

In this format Y is commonly referred to as a capitalization rate or overall capitalization rate and designated as |R.sub.o~. The derivation of |R.sub.o~, however, is not simple. For example, one recommended method for computing |R.sub.o~ is called a mortgage-equity formula.(1) It is:

|Mathematical Expression Omitted~

where

|Mathematical Expression Omitted~

The terms used in this formula are not defined or explained here because they will not be used in this article. The formulas are shown only to make the point that over the past two decades the complexity of appraisal formulas and methods has steadily increased. At this stage, confusion and consternation are rife among those concerned.

As James H. Boykin notes in his article, "Seeking the Elusive Discount Rate," "A major problem for professional real estate appraisers is the lack of agreement about what constitutes a discount rate and how it is derived, which is a source of consternation not only among appraisers, but for their clients as well."(2)

Appraisers have been provided with conflicting definitions of a discount rate. This stems partly from the fact that different yield rates, also called discount rates, are used for different purposes. Further, as financial theory and procedures have been introduced into appraisal theory and practice over the past several decades, the process of capitalizing income has become increasingly complex and often subject to question. It is hoped that this article will shed some light on the problem of finding an appropriate procedure and yield rate for income property appraisal.

FORMULAS AND DATA

A yield rate is used in a method and structure as simple as the valuation expression previously shown (V = I/Y). That is, any expression that includes a yield rate or discount rate is merely a variation, or expansion, of V = I/Y. When a yield rate is used in the following fashion it is called a discount rate because the process is commonly called discounting or a discounted cash flow (DCF) analysis. The standard discounting expression is:(3)

|Mathematical Expression Omitted~

where

|I.sub.1~, |I.sub.2~, ... |I.sub.n~ = Stream of expected income throughout n periods

Y = Discount rate--the elusive discount rate

If the stream of income is expected to be a constant amount for all periods, the expanded expression becomes the simple expression V = I/Y.(4)

The problem with any yield capitalization process is how to select, derive, and use an appropriate yield rate. To obtain valid results when using a yield rate in conjunction with an income stream, that yield rate cannot be a rate that is assumed for demonstration purposes or one that is calculated using irrelevant data.

Analysts, practitioners, and academics have produced an astounding number of formulas, proofs, and mathematical expressions of value and yield relationships in recent years. In most cases the expressions and formulas are correct, or at least irrefutable on the grounds of mathematical consistency. But the numbers required in the equations too often do not represent understandable or generally available market data.

For example, in the mortgage-equity formula for |R.sub.o~ shown earlier, there is a term identified as |Y.sub.E~. This is defined as an equity yield rate, and instructions for its derivation are that "When income is expected to change on a curvilinear basis or constant-ratio basis, formulas must be used to solve for the yield. A calculator cannot be used to solve the problem conveniently, and an iteration technique is too time consuming."(5) Many experienced appraisers seem rightly overwhelmed by the mortgage-equity formula (i.e., method) and either do not use it or else put little faith in value estimates produced by its use.

This particular mortgage-equity formula and its associated method, generally referred to as the Ellwood method, has been the subject of criticism for some time. Paul F. Wendt, in reviewing the status of appraisal practice in 1974, pointed out that "The basic problem with the use of Ellwood rates are that, first, they are useful only when level returns are applicable and, second, they bear no direct relationship to rates actually received by either the lender or the equity holder."(6)

Wendt suggested then as many do today that appraisers should rely on DCF methods.(7) But there are serious flaws inherent in the standard form of DCF analysis used for appraisal. One problem is that of determining an appropriate discount rate.

Kenneth M. Lusht suggests that to acquire data, "It is perfectly legitimate to obtain forecasts from a survey of investors."(8) Unfortunately, most who work with real estate yields know that what James Boykin tells us is true: "Relatively few investors actually know the return rates on their real estate investments."(9) Worse still, some analysts misuse yield-based procedures.

An example of inappropriate practice appears in "Discount Rate Derivation," by Robert C. Mason. In this article, we are shown an example that provides an initial value, a three-year cash flow, and a selling price. An overall cap rate is computed using the initial value and the first-year income. A discount rate is computed using the data given. Mason then concludes that "We now have the knowledge of the magnitude of the discount rate required to obtain the . . . |overall rate~."(10)

Mason shows and proves the mathematical relationship between the values given. It is misleading and improper, however, to use a given cap rate (assumed to be based on market data) to obtain a discount rate, and then use the same discount rate to obtain a value in order to confirm a value computed using that cap rate. Mason also explains how to select a discount rate based on security yields. It is true that real estate yields are related to security yields, but the relationship is so complex as to make reliable direct extraction of real estate valuation yield rates from security yield rates virtually impossible. Certainly it cannot be done with the ease and validity suggested by those who endorse such procedures.

DEFINING A YIELD RATE

What yield rate is required? The answer, of course, is that yield rates differ depending on their purposes. For example, generally yield rates derived from one type of income source, say common stocks, cannot and should not be directly used to value another type of source, say bonds--or real estate. Later in this article the reasons that different types of yields are required for different valuation procedures will be explained.

When working with most yield rate problems the question of taxes arises. Although there are conflicting opinions on this subject,(11) taxes do make a significant difference in computing real estate yields. One need only reflect on the sharp drop in real estate prices that occurred after 1986 when a new tax law changed the rules on depreciation and capital gains to appreciate the impact of taxes on real estate yields.

As financial theory and terms have become more prevalent, appraisers have had to deal with the more fundamental problem of recovering asset value (i.e., the amount invested that produces the income stream).

In financial theory, a yield rate, as in the expression Y = I/V, has always been one that includes the recovery of capital--that is, the income stream under consideration has always been measured in such a way that it is presumed to provide an amount that replaces the capital asset. This is true for securities and other financial instruments (e.g., the return of principal for bonds and mortgage loans) as well as for wasting assets. Unfortunately, early appraisal theory did not accept the idea that a capitalization rate should account for the recovery of capital.

A common problem in applying financial yields generally derived from securities such as stocks and bonds is the confusion caused by depreciation and related cash flows associated with a wasting asset. This is apparently the problem Ellwood was trying to resolve when he developed his valuation expressions. He tried to explain and measure the value of a wasting asset using the yields on debt and equity claims on the asset.

Instead of using an expanded DCF format, I/(1 + Y), and recognizing that I does or should include the return of capital, Ellwood tried to adjust the Y. In doing so he found himself dealing with the tricky problem of computing or measuring an equity yield rate on leveraged real property. The result is the extremely complex mortgage-equity expression.

Each of the three assets of real property, debt, and equity has its own unique cash flow pattern and a required unique way to compute its yield. In addition, yields are related to the inherent risk of the income-producing vehicle--a major factor commonly overlooked by those who extract real estate capitalization rates from security yields.

While on the one hand using yield rates is a complex undertaking, on the other hand it is not. The process is simple when cast in the framework of a DCF analysis. All that is required for any asset, whether it is a depreciating asset such as developed real estate or a security such as a bond, mortgage loan, lease, common stock, or the equity portion of leveraged real estate, is to project the expected cash flow throughout the period the asset will be held. This, of course, includes the return of invested capital such as the commonly recognized reversion when real estate is sold, the principal recovered in loan payments, or the principal created when bonds mature. The cash flow stream is then discounted using an appropriate yield rate. This yield rate is the discount rate. It is also the yield expected to be produced by the asset if it could be purchased for the price computed--its present (i.e., market) value.

Real estate yields are complex because many, if not most, properties are financed with debt. This means that property (i.e., asset) values and yields are based both on mortgage loan yields and leveraged equity yields (on an after-tax basis). Simply put, if the stream to be discounted (i.e., value) is an after-tax cash flow to a leveraged equity holder, an after-tax leveraged equity yield rate must be used. If the stream is net operating income a special rate must be constructed using something like the mortgage-equity formulas shown earlier.

Many appraisers and analysts recommend using the standard DCF analysis for estimating value in lieu of an Ellwood (i.e., mortgage-equity) method. The DCF format as it is commonly used, however, has serious limitations that are generally overlooked by those who use and recommend it.

PROBLEMS WITH DCF ANALYSIS

There are some general problems inherent in use of a DCF analysis. First, although made easier with the help of widely available small computers, there is the somewhat lengthy and complex process of performing the DCF analysis itself. There is also the more difficult task of obtaining the necessary and correct yield rate, as previously discussed. Finally, a DCF as commonly used in finance is not suitable for use in estimating present value--of real estate or of any other income-producing asset.

The standard form of DCF analysis that appraisers are advised to use requires the projection of an expected cash flow over a holding period that includes an expected sale at a specified price. Such an analysis is suitable for an investor but not for an appraiser. The result of such an analysis produces a yield rate called an internal rate of return (IRR)--the discount rate. This is what Mason demonstrated in his article. If a property can be bought for the asking price, produces the expected income, and then sells for the expected price it will produce the computed yield. If that yield is acceptable to the investor, the investor is expected to buy the property. But an investment analysis is not an appraisal.

Both appraisers and investors are concerned with yield, but for different purposes. An appraiser uses yield to compute the expected value of an income-producing asset. An investor uses value to compute expected yield. The two purposes and approaches are related but significantly different. Both parties make estimates of future expected income, but they each collect different kinds of data and perform different types of computations. An appraiser collects yield data for the purpose of selecting and using an appropriate rate with which to compute value. In a sense, an appraiser assumes (i.e., selects) a yield. An investor uses a given value, often an asking price, which is used to compute a yield, thus assuming (i.e., selecting) a value. The approaches are fundamentally different in terms of how values and yields are used.

The reason a standard DCF analysis cannot be used for computing the asking price of real estate (i.e., its present value) is that if the current worth of a property is unknown, it is unrealistic to include an uncertain (i.e., future) selling price in a computation that is used to derive what is put forth as a reliable present value. Again, the standard DCF analysis format is suited for using values to compute an unknown yield, not for using a yield to compute an unknown value. It is possible to set up a computer program based on certain assumptions with iterative loops of computation to arrive at an estimate of value using the structure of a standard DCF analysis. This is only possible, however, if information about the investor's required yields is provided.(12) Further, market-derived after-tax equity yields should be used, because these are the yields actually used by investors who establish financial yields and values.

DIRECT CAPITALIZATION

The valuation formula V = I/Y expresses the relationship between I and V when I is expected to remain constant. Y is the investor's expected yield rate on the property. Y is also |R.sub.o~ when I is expected to remain constant. The income from most real estate or other investments, however, is usually not expected to remain constant over a long period.

In the field of finance, any expected change (e.g., growth or decline from any cause) in income is accounted for by a simple modification of the standard valuation expression. The valuation formula that considers growth is derived from the standard DCF expression shown earlier. It is:(13)

V = I/Y - g

The factor g is the expected change in the income stream, expressed as an annual percentage rate. This formula, of course, resembles the simple expression V = I/Y or V = I/|R.sub.o~, where the overall cap rate |R.sub.o~ is represented by the expression (Y - g).

Another form of the income growth valuation expression is

Y = I/V + g

which indicates that an investor's expected yield rate, Y, is made up of a current yield, Y/V, which is |R.sub.o~, plus a factor, g, which represents any expected growth or decline in the income stream. It should be noted that because |R.sub.o~ is measured by I/V, then |R.sub.o~ = Y - g, as in the expression V = I/(Y - g). Thus we have the traditional expression V = I/|R.sub.o~, which means that a properly measured |R.sub.o~ (i.e., next year's income divided by current value) includes an investor's expectation about any future change in income.

The direct capitalization formula V = I/|R.sub.o~ and its use are fairly simple because |R.sub.o~ is measured from available market data on prices and income. Next year's income, of course, must be estimated or projected, which implicitly indicates some measure of g. A measure of Y is not required because it is included in the value computed for |R.sub.o~.

Some, perhaps many, appraisers have continued to recommend and use the direct capitalization method despite the introduction of more sophisticated techniques.(14) Significantly, many appraisers have noted that when inflation is prevalent, causing an increase in expected income, cap rates measured by market data tend to be low.(15) This happens because an investor's expected Y is increased by g, which in turn reduces a market-derived |R.sub.o~. Cap rates computed directly from market data--current prices and income--have been adjusted by investors' price behavior to reflect their expectations of inflation and growth.

The advantage of direct capitalization is that it uses real market data rather than numbers derived from a complex formula or a computer iteration procedure. The typical income property appraisal problem is to predict what price an investor is likely to pay for a given property. To accurately solve this problem it is necessary to use a method that relies on real market data. If use of an after-tax computer-driven valuation model is considered, Boykin's point that required after-tax yield data are not generally available should be remembered. If an appraiser has good yield data they should be used, but the most reliable method is direct capitalization using a market-based |R.sub.o~.

CONCLUSION

Some appraisers, analysts, and academics criticize direct capitalization because it requires the use of data that are not widely available. These critics generally prefer the DCF method and the use of more readily available artificial data. In this manner they simulate otherwise unknown or unknowable investor behavior.

One or more factors are often overlooked by analysts when they extract yield rates from non-real estate sources for use in real estate valuation. These factors include risk differences among yield sources, the impact of taxes, and the recovery of capital. When capitalization rates are extracted directly from real estate market data, these rates embody investors' expectations concerning risk as well as income growth and change. These rates are created by investor price behavior and are therefore valid for predicting investor price behavior.

The valuation of income property is a difficult and complex undertaking and there is no single easy and consistently accurate way to do it. A choice must often be made between using sparse data in a reliable format and using readily available but relatively unreliable data in a flawed methodological framework. The commonsense solution, of course, is to use any and all reasonably reliable analytical methods available.

The purpose of this article is to explain why the direct capitalization approach is more reliable than other income capitalization methods. While both a mortgage-equity analysis and a standard DCF method can be used, they must be employed with care and the results should be treated with considerable suspicion.

Robert Plattner, PhD, is a professor at California State University, Fullerton, where he teaches real estate and urban land development. He received an MBA from Ohio State University and a PhD from the University of Michigan. He has published textbooks on real estate principles and investment as well as a number of articles in various journals, including The Appraisal Journal.

1. American Inst. of Real Estate Appraisers, The Appraisal of Real Estate, 9th ed. (Chicago: American Inst. of Real Estate Appraisers, 1987), 522.

2. James H. Boykin, "Seeking the Elusive Discount Rate," The Appraisal Journal (July 1990): 328.

3. The Appraisal of Real Estate, 514.

4. J. Fred Weston and Eugene F. Brigham, Essentials of Financial Management, 9th ed. (Orlando, Florida: Holt, Rinehart, Winston, 1990), 211.

5. The Appraisal of Real Estate, 529.

6. Paul F. Wendt, Real Estate Appraisal: Review and Outlook (Athens: University of Georgia Press, 1974), 138.

7. Ibid., 158.

8. William N. Kinnard, Jr., ed., 1984 Real Estate Valuation Colloquium (Boston: Oelgeschlager, Gunn, & Hain, 1986), 57.

9. Boykin, 329.

10. Robert C. Mason, "Discount Rate Derivation," The Appraisal Journal (January 1989): 83.

11. The Appraisal of Real Estate, 553.

12. Robert H. Plattner, "Income Property: Factors Affecting Value," The Real Estate Appraiser and Analyst (Fall 1986): 24.

13. Weston and Brigham, 233.

14. Charles H. Peterson, "Are Capitalization Rates Obsolete?" The Appraisal Journal (April 1981): 179.

15. John B. Bailey, "Today's Real Estate Transactions Require Advanced Valuation Methods," The Appraisal Journal (April 1981): 279; and Donald T. Nolan, "The Current Yield is Below Zero in Santa Fe, Topeka, and Panama City," The Appraisal Journal (July 1980): 390.