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Selecting the appropriate frequency of discounting.

Like myriad other subjective concepts in the appraisal profession, the proper selection of frequency of discounting in income capitalization analysis is an open topic for discussion. The most common types of frequency of discounting are beginning-of-period (BOP), mid-period (MP), and end-of-period (EOP). Some appraisers believe there is relatively little difference in final values computed by the range of possible frequencies, and any margin of error should be minimal. Some lender-clients will accept only EOP, regardless of the merits of the other types. Most often, the controversy of the issue is whether to use EOP discounting or MP discounting. BOP is often less favored even though paying in advance is the most common occurrence of payments.

EOP discounting is often applied despite the preponderance of evidence that MP frequency more closely resembles a particular property's net operating income (NOI) performance. On occasion, the only justification for using EOP discounting is mere client preference. Some appraisers claim that EOP discounting is more appropriate because it is more conservative than MP discounting, as though a direction of value would justify its use. Sometimes, this is implied because no one involved in the assignment wants to go on record advocating any direction of value. The unbiased viewpoint of the real estate appraiser must always prevail.

While the EOP frequency produces a deeper discount of value, BOP discounting further enriches it. If MP discounting is, in fact, the best fit for a particular property type in terms of computing present value factors to apply to NOL it follows that values derived by BOP discounting and EOP discounting are both inaccurate to the extreme. Therefore, using EOP discounting may be a form of bias if it reflects a preference for a more conservative value.


When income and expenses are received and disbursed at the beginning of the particular frequency interval, only BOP discounting would seem to apply. Given this scenario, most appraisers will agree that EOP discounting is not an acceptable alternative to BOP discounting. Yet, many appraisers are uncertain about choosing between the appropriateness of BOP discounting or EOP discounting over MP discounting.

In selecting the frequency of discounting, a comprehensive study is needed to align the pattern for discounting the NOI with the pattern of how that NOI was achieved. Research and investigation of NOI performance should be precise and show that the frequency of discounting selected is appropriate. Selecting the particular frequency of discounting may involve developing an equivalent frequency pattern, which blends actual market occurrences, such as frequency of receipts, disbursements, and compounding of interest, with physical capabilities and capacities (income capitalization and cash flow models). More simply stated, this is how the appraiser processes retrieved data.

It is true that the value difference represented by BOP versus MP discounting, or MP versus EOP discounting, may be nominal, relatively speaking. Of course, relativity and its importance depend on whose point of view is being considered. The more important question for appraisers should be, which of these methods is most appropriate to fit the circumstances of income analysis?

The following presentation makes a convincing case in favor of MP discounting as the frequency that most appropriately matches the typical performance characteristics of receipts and disbursements of most income properties. At the same time, it is necessary to acknowledge that MP discounting should be avoided when income and expense characteristics do not dictate its use.


The assertion that appraising is more art than science is true for many reasons. Value estimates are derived from data retrieved from an imperfect market. This unscientific aspect of analyzing income intervals supports the argument that MP discounting is the most accurate.

The appraiser must analyze the subject property and the market of comparable properties to determine the typical pattern of income and expense intervals in the marketplace. It is necessary to discern the frequency of rental payments as well as the pattern of intervals of disbursements for the lessor's expense obligations.

Since all properties are different, terms of leases vary from property to property and may vary from tenant to tenant within the same property. An appraiser must meticulously learn the patterns of historical occurrences in the marketplace. Because appraising is not an exact science and realizing allowances must be made for variances in the marketplace, the appraiser must consider all the data to reach what may, by necessity, be general conclusions about the subject's anticipated intervals of earning income and incurring expense.

Most often, rents are due and payable in advance. Beginning-of-the-month receivables occur throughout the year, typically for all 12 months. Thus, income is customarily not received all at once at the beginning of the year and rarely, if ever, all at once at the end of the year.

With respect to expenses, their frequencies vary because some must be paid monthly and in advance (escrow for taxes and insurance) or in arrears (maintenance). Expenses may be disbursed annually, in advance (insurance if not escrowed) or in arrears (real estate taxes if not escrowed). Some expenses may be paid on an infrequent or irregular basis (leasing commissions and tenant improvements). Within net leases, the lessor's expense obligations will ordinarily include an allocation for management fees and reserves for replacement of short-lived items. With respect to management, most firms that provide full service require payment on a monthly basis. It is also prudent to allocate a portion of income for reserves on a regular basis.

The fact that expenses are not all due and payable at precisely the same interval - thus, not precisely fixed for the duration of the lease term - is not a valid reason to forsake the process of MP discounting. In fact, this is probably the most compelling reason to select MP discounting. Since most of the lessor's expenses are incurred periodically, though perhaps irregularly, throughout each year of the projected holding period, indicates they are not all paid in advance or in arrears, and would, thus, defeat BOP or EOP as the appropriate frequency of discounting. Therefore, if holding the frequency of payment constant and assuming MP discounting is appropriate for a particular property, it would follow that values derived using BOP and EOP discounting would constitute overstatements and understatements, respectively.

Continuing the process of income analysis, an appraiser would combine rents and expenses to form NOI. Cash flows, by and large, occur on a monthly basis. The appraiser determines the basis for compounding interest and projects a realistic holding period by incorporating the data in the marketplace into the income valuation process. Projections of income and expenses must be presented on a basis that will be meaningful to a client. Typically, within yield capitalization, appraisers present income analysis using annual projections of cash flows and compound interest. An equivalent frequency pattern can be established that blends market occurrences (income and expense performance) with physical capabilities and capacities (income capitalization and cash flow models).

Compounding interest monthly is uncomplicated with the new technology available. DCF spreadsheet analysis based on monthly compounding, although easy to produce, would be too voluminous to include in an appraisal report. Because of space constraints, appraisers often perform discounted cash flow (DCF) analysis using annual compounding even though the actual market occurrences of periodic rent receipts and expense disbursements, for the most part, may include monthly intervals. Further, another compelling factor that weighs in favor of the appropriateness of MP discounting for most income properties is that most appraisers accept annual compounding as an equivalent frequency pattern for a pattern of monthly payments.

Armed with a credible working knowledge of intervals of income receipts and expense disbursements, the appraiser is better able to form an educated opinion of the appropriateness of the frequency of discounting for the particular property. MP discounting has no inherent bias toward a conservative or liberal estimate of value. Acknowledging the potential for incongruity of frequencies of income receipts and expense disbursements, MP discounting provides a reasonable, although imperfect, link between the actual frequencies of receipts and disbursements and the present value of NOI. Even if one disregarded any merits of the "relativity" argument concerning value indicators derived by MP versus BOP or EOP discounting, the variance in value conclusions between NOIs discounted at these different intervals is apparent.

The Essence of the Problem

The genesis of the problem of selecting the frequency of discounting is not with the timing of the rent payment but with selecting the frequency of compounding of interest. A review of terminology may provide clarification.

"Frequency of payment" represents the interval at which rent payments are made and thus the interval at which income is received (i.e., monthly, quarterly, or annually). This is easily discerned by way of lease analysis for the subject property and by investigations into typical occurrences for comparable properties in the marketplace. Within the process of income valuation analysis, the appraiser must select the frequency of compounding of interest in order to compute present value factors for income capitalization and discounting. The interpretations of the terms "frequency of payment" and "frequency of compounding of interest" are synonymous and are interchangeable here.

Timing of payments is regarded as the specific chronological point when rent is due and expenses are payable, and it is customarily stated in the lease contract and is discernible by market evidence from comparable properties (i.e., in advance or in arrears). The term "frequency of discounting" is, in essence, synonymous with the term "timing of payments" and the two are used interchangeably here.

Based on this understanding of terms, the conclusion of frequency of compounding of interest may be derived directly from how frequently payments are made ("frequency of payment"). In turn, selecting the frequency of discounting is a function of the "timing of payments."

Bearing in mind the objective of income valuation analysis, appraisers must ensure that their analytical results are:

* Reflective of market occurrences in terms of frequencies of payment and of discounting

* Responsive to the capabilities and capacities of spreadsheet programs

* Informative in terms of the presentation of value findings to the client

All of these allow closer identification of and focus on the problem associated with selecting the proper frequency of discounting. The problem has two parts that must be addressed: How does an appraiser select the frequency of payment and frequency of discounting? And how can the computed data be condensed for the appraisal report?


As stated, in research and examination of the market, an appraiser discerns from the subject and the market of comparable properties the frequency of rent payments (monthly, quarterly, annually) and the timing of those payments (in advance, in arrears, or another specific time of occurrence). If the valuation analysis is to mirror the market in terms of selecting the frequency of payment and frequency of discounting, there should be little question about the proper method for performing the income analysis. For example, if the income performance of a subject property operates consistent with market frequency of rent payments (say, monthly) and with market frequency of discounting (say, in advance), the discounting of income is most accurate using monthly compounding of interest and BOP frequency of discounting. Any other scenario would not reflect the actual market occurrences.

For certain types of income properties, if the market is to be mirrored precisely in terms of the pattern of compounding and discounting that should be applied to reflect the pattern of receipt of income and disbursement of expenses, the frequency of compounding of interest would be computed on a daily basis with BOP frequency of discounting. For example, hotel properties generate [TABULAR DATA FOR FIGURE 1 OMITTED] income receipts and expense disbursements that are transacted literally each day of the year. The underlying premise is like that of the banking industry in the accrual of interest on the basis of daily compounding, since both the hotel property and the bank account essentially operate on a daily basis.

Time and space limitations, and the unscientific processes of analysis often require the use of annual compounding even for hotel properties. Imagine how voluminous a spreadsheet presentation would be if an appraiser used daily compounding with 365 periods (tabular columns) per year for, say, a projected ten-year holding period. Compounding of interest on a daily basis is not practical.

Working toward a solution involves the following steps. A target value is presented based on a hypothetical premise using monthly compounding with payments in advance under the theory that most income properties follow this frequency pattern. Next, three equivalent frequency patterns are presented based on the premise of annual compounding in a trial-and-error format to determine which pattern produces an answer more closely resembling that of the target value. All variables of the premise are held constant in the three test patterns, with the exception of frequency of discounting. The pattern producing a test value closest to the target value is regarded as the most appropriate equivalent for computing present value factors.

The cash flow model in Figure 1 represents the target value, using monthly compounding and BOP frequency of discounting. The assumptions of market-oriented terms that match the pattern associated with the hypothetical premise are incorporated. The computed target value of $47,458 will be tested by three equivalent frequency patterns.

Valuation analysis by virtually any combination of scenarios of frequency of payment and frequency of discounting is possible from the standpoint of the capabilities and capacities (space and memory) of current computer technology combined with myriad sophisticated spreadsheet programs. But now the second part of the identified two-part problem surfaces: considering how to present the voluminous information concisely to the client.

The discounting of NOI as presented in Figure 1 is simplistic and straightforward, and often appears this way in the valuation of leased fee estates and leasehold interests. Typically, not much space is needed in the report for this part. However, in other forms of yield capitalization, such as DCF analysis, the monthly frequency of compounding of interest and BOP frequency of discounting broadens the spreadsheet, often to such enormous proportions that many columns of tabular computations are required. The presentation order of the tests of equivalent frequency patterns is BOP, EOP and finally MP.

As stated previously, a trial-and-error test of reasonableness begins with an equivalent frequency pattern employing annual compounding with BOP frequency of discounting (see Figure 2). The computed value from BOP discounting exceeds the target value of Figure 1 by $2,580, which represents a variance of approximately 5.4%, which is not inconsequential.

The second example in the comparative analysis tests value based on an equivalent frequency pattern using annual compounding [TABULAR DATA FOR FIGURE 2 OMITTED] [TABULAR DATA FOR FIGURE 3 OMITTED] and EOP frequency of discounting. The computed value from EOP discounting is $1,969 less than the target value of Figure 1, which represents about a 4.1% variance, again a variance that is not inconsequential.

If the frequency of compounding selected is, say, monthly, but the NOI is essentially transacted on the 15th of each month, what frequency of discounting would be more appropriate than MP? It seems reasonable to assume that, while the timing of the monthly payment may not actually occur on the 1st or 15th, receipts and disbursements would be made at a constant predetermined date of each month. What is an accurate portrayal of the timing of payments if the due date is the 3rd of each month, or, the 13th or the 21st? How are appraisers to account for such subtle variances?

The variance of specific date of payment within the monthly frequency of compounding should be of little consequence. The use of BOP or MP discounting should adequately account for any daily timing of payments within monthly frequency of payments because spreadsheet analysis can reasonably make the assumption that time-zero (an effective valuation date) and the actual timing of payments are synonymous. As a practical matter, since the appraisal of income properties typically requires weeks of preparation, the effective valuation date during the assignment month may be brought very close in time to the actual timing of payments. Exceptions to this might include special circumstances, such as assignment involving proposed construction (in which absorption might preclude periods of cash flow altogether) or an assignment with a given historical effective valuation date (such as date of death for an estate). Otherwise, more closely scaling this particular aspect of discounting may constitute macro-analysis that is of little consequence.

Figure 4 presents a test of the target value and is based on selecting an equivalent frequency pattern using annual compounding with MP frequency of discounting. A comparison of the test value of MP discounting presented in Figure 4 ($47,656) with the target value presented in Figure 1 ($47,458) yields a difference of $198, representing less than 0.5% variance which, by any measure, is negligible. This test provides conclusive evidence that the most appropriate equivalent frequency pattern for converting monthly compounding with payments in advance is annual compounding with MP discounting.


The complexion of a valuation problem may be viewed differently by various users of appraisal services. While the task of valuation analysis may be complex and subjective, objectivity must prevail. An understatement of value is just as inaccurate as an overstatement.

FIGURE 5 Summary of Values Computed by Equivalent Frequency Patterns

Target value = $47,458 (Figure 1):

Equivalent Frequency Pattern Reference Values

Annually: BOP Figure 2 $50,038
Annually: EOP Figure 3 $45,489
Annually: MP Figure 4 $47,656

With respect to market value estimates, the objective should be to reflect market occurrences by the frequency intervals of receipts and disbursements within the particular type of income property. When NOI is typically transacted monthly throughout the year (essentially without regard to whether the timing of payments is monthly, in advance, or monthly, in arrears), the frequency of discounting that most closely reflects this pattern is MP. Conversely, when market performance of income and expense intervals clearly reflects total lump sum receipts and disbursements annually in advance or in arrears, MP discounting should not be employed.

Figure 5 summarizes the results of the tests of computed values based on the three selections of equivalent frequency patterns. In a comparison of the target value and the three test patterns, the closest match is represented by MP discounting with a variance of less than one half of 1%. Based on the tests of computed values presented in this article, MP frequency of discounting more closely relates to the relationship and frequency of income receipts and expense disbursements. When viewed in this context, MP discounting is essentially a bridge between lower and upper tendencies of value. As an equivalent frequency pattern, it allows a closer match to the target value.

From the examples presented in this article, the other two equivalent frequency patterns selected (BOP and EOP) produced variances ranging from approximately 4.1% to about 5.4%. Without question, both are small fractions. Are these variances worthy of distinction from an appraisal point of view? For valuation purposes within the context of the appraisal process, any single cause or influence that impacts value by as much as 4%5% should be given careful consideration. This is the essence of the case to be made for MP discounting. The concern should be accuracy, not relativity. It is for others to determine the relative significance of a 4% or 5% variance. While such a variance may constitute a nominal dollar value for a particular property, this small percentage would represent substantial dollars for a property of great magnitude, such as the World Trade Center in New York City.

As additional support to these conclusions, it is noteworthy that when the hypothetical premise of Figure 1 is amended to include quarterly compounding with BOP discounting, the target value becomes $47,937 ($3,000/quarter x 15.978891 PV factor). Once again, when annual compounding is selected in the equivalent frequency pattern, MP discounting produces the closest match to the target value.

To real estate appraisers who claim that the choice of frequency of discounting is of relatively little importance and that computed value differentials are inconsequential, the $4,549 spread between BOP and EOP discounting represents a variance of almost 10%. This level of variance must be recognized as a substantial potential margin of error.

Once again it is noted that if the market indicates that income and expenses of the property produce NOI in a single lump sum payment at the end of each year, the most proper frequency by that scenario would include EOP discounting. Because rent is hardly ever paid in arrears, regardless of the frequency of payments, EOP discounting would rarely be selected as the most appropriate frequency of discounting. There may be at least one practical use of EOP discounting. In valuing the reversion in yield capitalization, reversion may be computed, using terminal holding period NOI capitalized at the terminal overall rate. However, this is just as correct as using BOP frequency of discounting for the following period since the end of one cash flow period and the beginning of the next subsequent period are precisely the same points in time. In this instance, the two intervals are considered a distinction without a difference.

As providers of unbiased appraisal services, a chief purpose of appraisers is to employ accuracy, using reliable methods. Appraisers are not to be influenced by a client who might favor conservative or liberal value estimates. Using EOP or BOP discounting when MP is more appropriate is like preparing a taxpayer's return without full consideration of all allowable deductions. The objective is accuracy, not catering to client preference.


Kevin L. Bruckner, "Mid-Year Versus Year-End Present Worth Factors in DCF Analysis," The Appraisal Journal (January 1991): 126.

Richard C. Sorenson, "Mid-Year Discounting Precautions," Letter to the Editor, The Appraisal Journal (July 1991): 438.

Stephen J. Albright, Sr., MAI, SRA, is president of Albright & Associates of Ocala, Inc., Ocala, Florida. He earned a BA in history and political science from Presbyterian College, Clinton, South Carolina. His work has previously been published in The Appraisal Journal.
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Author:Albright, Stephen J., Sr.
Publication:Appraisal Journal
Date:Oct 1, 1997
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