# The twelve Rs: an overview of capitalization rate derivation.

In current Appraisal Institute courses, students are learning ways to create and prove an overall rate, a rate necessary to convert one year of income into value, by a process known as direct capitalization. All too often appraisers get comfortable with only one way to create or prove a capitalization rate in their appraisal practice without knowing, checking, or proving their rate with other methods. Some argue that direct capitalization is a thing of the past, and that creating or attempting to convert a single year's income into value is a useless academic exercise in today's world of computers and data available for conversion of income into value by sophisticated discounted cash flow (DCF) analyses. Even with computers and yield analyses, an appraiser must be able to prove his or her rate of conversion, whether it is a capitalization rate or a yield rate, and should know the many ways to prove or disprove a rate selection. Further, there are many commercial properties that should be valued by direct capitalization. DCF analysis may be the only way to value investment-grade real estate, but for the bulk of smaller, noninvestment-grade real estate, direct capitalization simulates the process used by buyers and sellers in those types of markets.

The authors have determined that there are 12 different ways to derive and prove an overall rate for direct capitalization.(1) They are as follows:

[R.sub.o] = [I.sub.o]/[V.sub.o] (1)

This is the universal formula of rate development. It is generally market extracted when the appraiser knows the net operating income ([I.sub.o]) and the value or price ([V.sub.o]) of a property. The [R.sub.o] derived by this formula implicitly contains all of the assumptions about investor expectations inherent in that particular sale or value, and to be useful should be applied to only those properties that have the same characteristics as the sale (i.e., same land and building ratios, use, lease terms, vacancy rates, expense ratios, value or income expectations, level of risk and investor motivation). It would be inappropriate to extract an overall rate from a sale of an 80-unit, older, urban, mid-rise apartment complex and use it directly to value a newer, 150-unit, suburban garden complex. Expense ratios, appreciation in value, risk, and investor motivation between these two properties would be too great to make a direct comparison. However, once the overall rate ([R.sub.o]) is extracted, it can be useful for comparable properties, and since it is market extracted, it is very persuasive.

[R.sub.o] = SR + LR + MR + RR (2)

This old formula of developing a built-up overall rate is antiquated. Here, an overall rate is built up through a combination of a safe rate (SR), a liquidity rate (LR), a management rate (MR), and a risk rate (RR). To assume that an appraiser would have the judgment to assign a proper factor for loss of liquidity, or management, or risk in today's complex and changing economic climate, and to develop a rate in this fashion are incomprehensible. It can, however, be useful in making comparative judgments once an overall rate has been created through other means.(2)

[R.sub.o] = NIT/GIM (3)

There are only two ways direct capitalization can be employed: by dividing income by a rate, or by multiplying income by a factor. In the second scenario, multiplying income by a factor, the income can be potential gross income (PGI) or effective gross income (EGI). This approach is generally used in the sales comparison approach to value and is not considered a capitalization approach per se. However, two simplistic parts of this formula can be quickly extracted from the market, i.e., the gross income multiplier (GIM) and the net income ratio (NIR), which is the complement of the expense ratio. As an example, simple research would indicate what the gross rental would be from the market at the time a property is sold, and a gross income multiplier could be developed by dividing the sales price by the potential gross income (SP [divided by] PGI = PGIM). Further analysis of the comparable sales would indicate the typical expense ratio, and 100% less the expense ratio would equal the net income ratio. Once these two figures are known, a potential overall rate can be developed by simply applying the formula expressed above.(3)

[R.sub.o] = DCR x M x [R.sub.M] (4)

Commonly referred to as the "underwriter's method" of developing an overall rate and used by lenders with their own requirements for debt coverage ratio (DCR), mortgage or loan-to-value ratio (M), and mortgage constant ([R.sub.M]), it is a good tool when it is market derived. All appraisers should check with their mortgage sources on the type of property they are appraising and know these components of lender requirements. It is based on the premise that the lender establishes the control on various properties by controlling the amount of risk they will take (loan-to-value ratio or M), the interest rate and term they require (the mortgage constant or [R.sub.M]), and the amount of net operating income they deem safe to cover the debt (debt coverage ratio or DCR). The loan-to-value ratio and mortgage constant are used in both the band of investment and the Ellwood formula for developing overall rates, and debt coverage ratios are easily surveyed or acquired from mortgage research sources such as the American Council of Life Insurance Companies' Investment Bulletin.(4)

If the assumptions of loan-to-value ratio (M), mortgage constant ([R.sub.M]), and debt coverage ratio (DCR) come from a specific lender rather than comparable market data, the overall rate will reflect the lender's value rather than market value. Appraisers who use the Ellwood or band of investment method to develop an overall rate ([R.sub.o]) would do well to check this rate by using the formula:

DCR = [R.sub.o] / M x [R.sub.M]

If the debt coverage ratio (DCR) calculated using this formula is inconsistent with current bank lending criteria applicable to the subject property, then the overall rate derived through the Ellwood or band of investment method may be flawed.(5)

[R.sub.o] = M x [R.sub.M] + (1 - M)[R.sub.E] (5)

Known as the band of investment method of developing an overall rate, this is based on the presumption that a capitalization rate should amount to a weighted average of debt and equity funds, dependent on the risk quality of the property investment. Also known as the mortgage equity analysis, the formula simply states that the overall rate ([R.sub.o]) is the weighted average of the mortgage capitalization rate (the mortgage constant or [R.sub.M]), and the equity capitalization rate ([R.sub.E]). The overall rate generated by this formula must satisfy both the mortgage requirement of the lender and the pre-tax cash flow requirement of the equity participant. Since it is a composite rate, weighted in proportion to the investment represented by debt and equity, it is believed by some to be infallible because the greatest weight toward the overall rate is the debt requirement which can be easily obtained by a survey of lenders in the type of investment property to be valued. It is a popular method used by appraisers to develop an overall rate, but is often misused because the data used to develop the rate is from surveys of lenders and equity participants, not from hard market data. It is a good way, however, to test a capitalization rate developed from better market-oriented sources.(6)

[R.sub.o] = L x [R.sub.L] + B x [R.sub.B] (6)

Another band of investment technique is derived using the relationship of known physical components of property, i.e., the land and the buildings. The ratio of land value and/or building value to the overall property value must be known, as well as the capitalization rate for each physical component. Used primarily in land and/or building residual techniques where one value is known, the income for that value can be extracted from the net operating income by multiplying the appropriate capitalization rate by the known value. The residual income is then capitalized into a value estimate by its appropriate rate.

It is a good technique in highest and best use analysis where building values are known or can be reasonably estimated and the question is what is the most profitable use of the land.(7)

[R.sub.o] = [Y.sub.o] (7)

When an appraiser expects that income and value will remain unchanged during a holding period, the property is basically valued by capitalization in perpetuity, i.e., the overall capitalization rate and the yield rate are synonymous. In other words, the yield to the property investment is equal to the rate of return.

[R.sub.o] = [Y.sub.o] - [[Delta].sub.o] a (8)

When there is a change in income and/or value over a holding period, the above formula can be adjusted accordingly. This is a general purpose formula to develop an overall capitalization rate where [R.sub.o] is the cap rate, [Y.sub.o] is the property yield rate, [[Delta].sub.o] is the change in property value and a is the appropriate conversion factor At present, there are three variations of the conversion factor based on the appraiser's perception of anticipated changes in income and value over a projected holding period, each of which will change the overall rate ([R.sub.o]).(8)

[R.sub.o] = [Y.sub.o] - [[Delta].sub.o] 1/[S.sub.n] (9)

If it is forecasted that income will remain level and value will change over the projected holding period, the overall change in value ([[Delta].sub.o]) is multiplied by the sinking fund factor or SFF (calculated based on the yield rate and the holding period) to derive an appropriate capitalization rate. Typically, it would be unusual for net income to remain completely level over a holding period and still have an expected change in value. However, this may be appropriate in those situations where a property is under a long-term net lease at a flat rent, and property values are increasing or decreasing due to supply and demand factors or are in an inflationary market.

[R.sub.o] = [Y.sub.o] - [[Delta].sub.o] 1/n (10)

Where it is assumed that both income and value change on a straight-line basis during a holding period, the overall change in value ([[Delta].sub.o]) is multiplied by the inverse of the number of years in the projected holding period (1/n).

[R.sub.o] = [Y.sub.o] - CR (11)

If both income and value are expected to change at a constant ratio (compound rate) over a holding period, then the overall rate ([R.sub.o]) is simply the expected yield rate ([Y.sub.o]) adjusted by the rate of change (CR). It stands to reason that if the investor's expected yield rate is 12%, and both income and value change at +2% per year, then the overall capitalization rate is 10%, or 12% less 2%.(9)

[R.sub.o] = [Y.sub.E] - M([Y.sub.E] + P 1/[S.sub.n] - [R.sub.M]) - [[Delta].sub.o] 1/[S.sub.n] / (1 + [[Delta].sub.I] J) or (K) (12)

Most readers will recognize this as the Ellwood formula which derives an overall capitalization rate that incorporates equity yield requirements and mortgage financing terms. Developed by L. W. Ellwood and re-defined by Charles B. Akerson, it is the algebraic equivalent of DCF analysis.(10)

The Akerson format expressed algebraically is as follows:

[R.sub.o] = (M x [R.sub.M]) + (1 - M) [Y.sub.E] - M x P 1/[S.sub.n] - [[Delta].sub.o] 1/[S.sub.n]

The Akerson format as it is more commonly used is shown below. When presented in this manner, it is easier for most people to comprehend and to solve step by step than the algebraic formula.

[TABULAR DATA OMITTED]

The first part of Akerson's format closely resembles the simple band of investment method with one important difference: The equity rate used here is the equity yield rate ([Y.sub.E]), not the equity capitalization rate (equity dividend rate or [R.sub.E]) found in the band of investment. The next line adjusts the mortgage/equity-weighted average for the equity buildup that accrues to the investor as the mortgage is paid off, and this adjusted rate is known as the basic rate (r). The last line adjusts the basic rate for total anticipated appreciation or depreciation over the holding period to arrive at the overall capitalization rate ([R.sub.o]).

Originally the Ellwood formula was developed only for level income streams, and was later refined to allow for variations when different income patterns were anticipated, by the use of the Ellwood J-factor or the K-factor. However, with the use of today's financial calculators and personal computers, a simpler method, no matter what pattern of income is expected over a holding period, is to find the equivalent level income.(11)

Knowing the 12 Rs provides the appraiser with a complete bag of tools with which to test, develop, reason, and prove value. As an example, assume an appraiser is valuing a property with a net operating level income of \$300,000. This income should remain stable for the foreseeable future. This appraiser's standard operating procedure is to use only the band of investment method to develop an overall capitalization rate ([R.sub.o]). A survey of local lenders indicates that they would lend money based on a loan-to-value ratio of 70%, at a fixed interest rate of 10% with monthly payments for a term of 20 years. Further, a survey of investors indicates that they expect a 12% cash-on-cash return ([R.sub.E]). The overall capitalization rate would be derived through the band of investment method as follows:

[R.sub.o] = M x [R.sub.M] + (1 - M) [R.sub.E]

= 0.70 x 0.1158 + 0.30 x 0.12

= 0.08106 + 0.03600

= 0.11706

The overall value ([V.sub.o]) would then be:

\$300,000 [divided by] 0.11706 = \$2,562,788

(rounded) \$2,560,000

Further investigation would have also disclosed that lenders typically expect a DCR of 1.35, that investor's holding periods are generally not more than 10 years, and that the yield expectation for investors is 14%.

A check on the band of investment rate selection could be made by (a) the underwriter's method, and (b) the Ellwood formula, as follows:

[R.sub.o] = DCR x M x [R.sub.M]

= 1.35 x 0.70 x 0.1158

= 0.10943

[V.sub.o] = \$300,000 [divided by] 0.10943

= \$2,741,478

(rounded) \$2,740,000 (a)

[R.sub.o] = [Y.sub.E] - M([Y.sub.E] + P 1/[S.sub.n] - [R.sub.M]) - [[Delta].sub.o] 1/[S.sub.n] / (1 + [[Delta].sub.I] J) or (K)

= 0.14 - 0.70 (0.14 + (0.26976 x 0.05171) - 0.1158) - (-0.10 x 0.05171)

= 0.14 - 0.70 (0.03815) + 0.005171

= 0.14 - 0.02671 + 0.005171

= 0.11846 (b)

[V.sub.o] = \$300,000 [divided by] 0.11846 = \$2,532,500

(rounded) \$2,530,000

The value estimates obtained by deriving the overall capitalization rate ([R.sub.o]) through two different methods range from \$2,530,000 to \$2,740,000. This reflects a difference of 8.3% or \$210,000 - a significant divergence although both methods are based on the same set of facts. The question begs itself: Which is the correct way to develop an overall rate, and which value estimate is right? The answer lies in the type of property being valued and the most likely way an investor in the market for the subject property develops or creates his or her capitalization rate.

As another example, assume a similar set of facts. The subject property is expected to continue to produce a level net operating income of \$300,000. A market investigation indicates that local lenders will lend money based on a loan-to-value ratio of 70%, at a fixed interest rate of 10% with monthly payments for a term of 20 years, and an expected debt coverage ratio (DCR) of 1.25. Investor's holding periods are generally not more than 10 years, and investors' yield expectations are 14%. The market for this class of property is expected to appreciate approximately 50% over a 10-year holding period, or 4.14% per year compounded. (This was not an unusual assumption during periods of rapid inflation such as the mid-1980s, when some markets experienced increases of 15%-20% per year) A market survey indicates that investors are willing to accept a cash-on-cash return ([R.sub.E]) of 6% probably due to the anticipation of significant appreciation in property value.

The appraiser in our example consistently uses the Ellwood mortgage-equity analysis presented in the Akerson format. After all, this advanced technique accounts for every component of value in the overall capitalization rate, including the investors' required equity yield ([Y.sub.E]), mortgage financing, the equity buildup (P) accrued by making mortgage payments over the holding period, thereby reducing the principal balance, and the effects of any appreciation or depreciation in property value ([[Delta].sub.o]). The overall capitalization rate ([R.sub.o]) is derived as follows:
```Weighted Average:

70.0% Mortgage x 0.1158 Mortgage constant = 0.08106
30.0% Equity x 0.1400 Equity yield rate = 0.04200

Weighted average: 0.12306

Less Equity Buildup:

70.0% Mortgage x 0.26976 Part paid off
x 0.05171 Sinking fund factor = 0.00976
Basic rate: 0.11330

- Appreciation/+ Depreciation:

50.0% Appr. x 0.05171 Sinking fund factor = -0.02586
Overall capitalization rate 0.08744

(rounded) 8.74%
```

Capitalizing the net operating income of \$300,000 at the indicated overall rate of 8.74% results in a value estimate of \$3,432,500. The appraiser, after checking his calculations twice concludes a value of \$3,432,500.

Once more, the formulas presented here can be used to check the validity of the derived overall capitalization rate of 8.74%, including (c) the underwriter's method, and (d) the band of investment technique:

[R.sub.o] = DCR x M x [R.sub.M] (c)

and

DCR = [R.sub.o] / M x [R.sub.M]

= 0.0874 / 0.70 x 0.1158

= 1.08

Clearly, this analysis shows a debt coverage ratio significantly lower than the 1.25 DCR required by lenders as found in the appraiser's market investigation. Something is wrong and should be a red flag to the appraiser that further investigation and a review of the assumptions on which the overall rate is based may be needed.

[R.sub.o] = M x [R.sub.M] + (1 - M)[R.sub.E] (d)

and

[R.sub.E] = [R.sub.o] - (M x [R.sub.M]) / 1 - M

= 0.0874 - (0.70 x 0.1158) / 1 - 0.70

= 0.02113 or 2.11%

Similarly, this analysis indicates a cash-on-cash return (equity dividend rate, [R.sub.E]) of 2%, far below the 6% return required by market participants. Again, there is an inconsistency and this type of check can warn an appraiser to review the assumptions on which the overall rate is based.

If the appraiser uses these techniques as a check, the inconsistencies will become readily apparent. He or she reviews the assumptions and finds that the anticipated appreciation of 50% over the 10-year holding period is overly aggressive, and tempers the expected increase in property value to 25%. The resulting Ellwood/Akerson capitalization rate derivation now appears as follows:
```Weighted Average:

70% Mortgage x 0.1158 Mortgage constant = 0.08106
30% Equity x 0.1400 Equity yield rate = 0.04200

Weighted average: 0.12306

Less Equity Buildup:

70.0% Mortgage x 0.26971 Part paid off
x 0.05171 Sinking fund factor = 0.00976
Basic rate: 0.11330

- Appreciation/+ Depreciation:

25.00% Appr. x 0.05171 Sinking fund factor = -0.01293
Overall capitalization rate 0.10037

(rounded) 10.04%
```

The appraiser again applies the same formulas to check the consistency of his assumptions and finds the following:

[R.sub.o] = DCR x M x [R.sub.M] (e)

and

DCR = [R.sub.o] / M x [R.sub.M]

= 0.1004 / 0.70 x 0.1158

= 1.24

The indicated debt coverage ratio is now consistent with the DCR required by local lenders as found in the appraiser's market investigation.

[R.sub.o] = M x [R.sub.M] + (1 - M)[R.sub.E] (f)

and

[R.sub.E] = [R.sub.o] - (M x [R.sub.M]) / 1 - M

= 0.1004 - (0.70 x 0.1158) / 1 - 0.70

= 0.06447 or 6.45%

Again, using the formula as a check indicates a cash-on-cash return (equity dividend rate, [R.sub.E]) that is consistent with the 6% return required by market participants.

This process of testing capitalization rates resulted in revising the overall capitalization rate from 8.74% to 10.04%. Based on a first-year net operating income of \$300,000, the estimated value is reduced from \$3,432,000 to \$2,988,000. This reflects a decrease of 13%, which is significant in any appraisal assignment.

As has been shown in this article, it is imperative in today's real estate investment world to look at all the ways an overall capitalization rate can be developed and to use all the data available in the market to develop and test overall rates for every income valuation problem.

1. Most of the formulas discussed here are presented in Appraisal Institute courses "Basic Income Capitalization" (310) and "Advanced Income Capitalization" (510).

2. Charles B. Akerson, "Builtup and Blended Rates," Capitalization Theory and Techniques Study Guide (Chicago: American Institute of Real Estate Appraisers, 1984), 21. It should be acknowledged that these builtup rates are truly yield rates and, therefore, even if an appraiser has the judgment or expertise to develop an overall rate by this method, he or she would have to convert the yield rate to an overall capitalization rate.

3. Ibid., 18. The problem in this approach to prove an overall rate is that it may not be possible to determine the true expenses in a sale property, unless the appraiser is involved in the transaction or has direct knowledge of this information. Relying on third-party information for the actual expenses at the time of sale may be insufficient to develop an accurate overall rate. However, if the appraiser is cautious and uses typical expressed expense ratios, it still becomes a good check on other methods developed here.

4. American Council of Life Insurance Companies, Investment Bulletin (Washington, D.C.: American Council of Life Insurance Companies).

5. The Appraisal of Real Estate, 10th ed. (Chicago: Appraisal Institute, 1992): 472-473.

6. Ibid., 470-471.

7. Ibid., 472.

8. Ibid., 494-495.

9. Ibid., 498. It is unrealistic in the real world to expect both income and value to change at the same rate during an anticipated holding period. The only exception to this may be in a leased fee analysis where the lease is written to guarantee both a constant increase in income and a resale price at the same rate of growth.

10. Some argue that the Ellwood technique is obsolete with the widespread availability of easy-to-use spreadsheet programs that can perform complex DCF analyses quickly, accurately, and inexpensively. See Wayne Kelly, Donald R. Epley, and Phillip Mitchell, "A Requiem for Ellwood," The Appraisal Journal (July 1995): 284-290.

11. An excellent reference on how to calculate equivalent level income can be found in Course 510, Session 6: "Stabilizing Income and Yield Capitalization (DCF) Using an Equity Yield Rate."

Joseph H. Martin, MAI, is president of Martin Appraisal Associates, Inc., Lawrenceville, New Jersey. He received his BS in business administration at Rider College, Lawrenceville, and Is an Instructor for the Appraisal Institute. He Is an author of several textbooks on real estate licensing.

Mark W. Sussman, MAI, is vice president of John O. Lasser Associates, Inc., Livingston, New Jersey. He provides appraisal and consulting services on a wide variety of property types, with emphasis on ad valorem tax litigation and condemnation. He received his BA in psychology from Lehigh University, Bethlehem, Pennsylvania, and is an instructor for the Appraisal Institute.