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File: {Elsevier}Brown/Revises-II/3d/Brown-ch003.3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22.12.2004/3:22pm Page: 55/72 and net income in turn is a function of vacancy and expenses, Net Operating Income ¼ Gross Potential Income ÀVacancy ÀExpenses ð3-15Þ leads us to simplify the difference between gross and net income as Net Operating Income ¼ Gross Potential Income (1 ÀevrÞð3-16Þ where evr ¼ expense and vacancy rate (0 < evr < 1). Rearranging the value equation, recalling that GPI ¼ Val GRM , and substituting our simplifying assumption produces Capitalization Rate ¼ Val = GRM 1 À evrðÞ Val ð3-17Þ Given that value and gross income are accurately reported, this makes the honest cap rate we yearn for a function of the expense and vacancy rate chosen. Further rearrangement gives us an equation for evr that is dependent on only two variables, both commonly found in reported sales of investment property: the GRM and the cap rate, of which one, GRM, is more reliable than the other. Using these two rules of thumb together in Equation (3-18), we can gain some additional insight. evr ¼ 1 À cr grm ÀÁ ð3-18Þ One should be cautioned that, mathematically, it is possible for evr to be negative. However, in real estate it can never be less than zero. Should a negative evr be calculated by Equation (3-18) from observations in a dataset, it virtually must result from misreporting of either or both the capitalization rate or GRM. A moment’s thought about what it would mean for an apartment building to sell for ten times its gross income AND at a 13% cap rate will convince you that such things do not occur in nature. In Equation (3-19) we solve for GRM in terms of the other variables. GRM ¼ 1 Àevr cr ð3-19Þ GRM is always greater than 1 (all buildings in first world countries sell for more than their annual gross income), and cr, the reciprocal of a positive The ‘‘Rules of Thumb’’ 55 File: {Elsevier}Brown/Revises-II/3d/Brown-ch003.3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22.12.2004/3:22pm Page: 56/72 real number, must always be greater than 0 and less than 1. Thus, the only way for both sides of the above equation to be greater than 1 is for the numerator of the ratio on the right to be greater than its denominator. Since evr is a rate that, by definition, is a positive number between 0 and 1, 1 Àevr must be a number between 0 and 1. For the whole right side of Equation (3-19) to be greater than 1, 1 Àevr must be larger than the caprate. THE NORMAL APPROACH TO DATA Let’s look at a dataset of 1,000 actual apartment sales that took place in the San Francisco area between October 1996 and September 2001. Each observation shows the area, price, date sold, age, building size in square feet, number of units, GRM, and capitalization rate. The first five observations are displayed in Table 3-4. Using Equation (3-18), we can combine the cap rate and GRM to create a list of expense and vacancy ratios. It is useful to look at the range of the evr observations in Table 3-5 and plot them in Figure 3-6. Most practitioners in sunny California would agree that expenses of 59.18% of income for an apartment building are at least unusual, if not unlikely. Likewise at the other extreme, expenses of 10.43% are probably understated. We need to adopt a healthy suspicion about the extreme observations. The plot of an ordered list in Figure 3-6 shows, as always, a few extreme observations, but the majority of the observations is between 25 and 45%. TABLE 3-4 First Five Observations in San Francisco Data Area Price ($) Date Age SF Units GRM CR 5 880,000 09/21/01 94 2,100 6 11.24 0.0697 5 1,075,000 09/21/01 48 5,302 9 8.07 0.0918 1 920,000 09/19/01 0 5,502 6 10.6 0.0603 5 1,000,000 09/14/01 42 5,368 8 14.56 0.0546 5 1,150,000 09/07/01 50 5,200 8 8.87 0.0835 TABLE 3-5 San Francisco EVR Extreme of EVR for San Francisco data Minimum 0.1043 Maximum 0.5918 56 Private Real Estate Investment File: {Elsevier}Brown/Revises-II/3d/Brown-ch003.3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22.12.2004/3:22pm Page: 57/72 The histogram in Figure 3-7 provides a visual way to see the discrete distribution of grouped evr data. Measures of central tendency, shape, and variance, known as descriptive statistics, are shown in Table 3-6. The values in the shape statistics measure something known as ‘‘tail behavior.’’ One of these is skewness, the measure of how much the distribution is not symmetrical. The other is excess kurtosis, 0 200 400 600 800 100 0 Observations # 0.1 0.25 0.45 0.6 EVR FIGURE 3-6 Plot of ordered list of EVRs. 0.2 0.3 0.4 0.5 0.6 20 40 60 80 100 120 140 Expense and Vacancy Ratios EVRs FIGURE 3-7 San Francisco EVRs. The ‘‘Rules of Thumb’’ 57 File: {Elsevier}Brown/Revises-II/3d/Brown-ch003.3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22.12.2004/3:22pm Page: 58/72 a measure of the thickness of the tails. These are important features of the distribution that are ‘‘non-normal.’’ We will return to these later. Our data is a sample. We know its shape, but we do not know the shape of the distribution of the population from which the sample was drawn. Assuming (naively) for the moment that the sample of expense ratio observations are from a population of expense ratios that are distributed normally, we can create a probability distribution function (pdf) for such a distribution from its first two moments, the mean and the variance. We then plot these in Figure 3-8 over a range equal to three standard deviations away from the mean. TABLE 3-6 Descriptive Statistics for San Francisco evr Data Location statistics Mean 0.333921 Harmonic mean 0.322032 Median 0.334293 Shape statistics Skewness 0.221308 Excess kurtosis 1.07313 Dispersion statistics Variance 0.00370466 Standard deviation 0.0608659 Range 0.487524 Mean deviation 0.046557 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 normally distributed EVRs FIGURE 3-8 Normally distributed San Francisco EVRs. 58 Private Real Estate Investment File: {Elsevier}Brown/Revises-II/3d/Brown-ch003.3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22.12.2004/3:22pm Page: 59/72 With the information we have developed thus far, we can assess the probability that any specific reported expense ratio will occur. To do this we use the cumulative distribution function (CDF) and ask, simply, how much of the probability mass resides below some specific point. For instance, suppose we are interested in knowing the probability that the building we are investigating has an expense ratio of 25% or less. One indication of this is that portion of the data having expenses ratios at or below that figure. The answer the CDF produces is plotted on the pdf in Figure 3-9. We must keep in mind that the illustrations and computations shown below are dependent on our assumption that the evrs are normally distributed. Earlier we plotted the data as an ordered list and concluded that most of the expense ratios were in the range of 25–45%. This is confirmed by the pdf in Figure 3-10 showing most of the probability mass in that range. Often, we want to put a number to our conclusions about probability. The common phrase ‘‘What are the chances of that?’’ is vernacular for the more formal ‘‘How much of the data is below that [certain point]?’’ Table 3-7 shows several levels at which the evr on the left is matched with its respective probability on the right. Table 3-7 shows that 99.7% of our evr observations are at or below 50%, while only 8.4% of them are at or below 25%. Thus, about 91% of them are between 25 and 50%. If we are given a capitalization rate that is based on an evr of 30%, we see from Table 3-7 that only 28.9% of the buildings have an evr that low or lower. From this we can make an assessment of the reliability of the capitalization rate. But how reliable is the model we employed to make this claim? 0.25 x FIGURE 3-9 Portion of EVR observations at or below 25%. The ‘‘Rules of Thumb’’ 59 File: {Elsevier}Brown/Revises-II/3d/Brown-ch003.3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22.12.2004/3:22pm Page: 60/72 QUESTIONING THE ASSUMPTION OF NORMALITY Before continuing, it is useful to take a step back and ask a broad question: Why do we use data and statistics? Used properly, they help us peer into the unknown. The unknown may be the way a market works, something we cannot see because it is not directly observable. The unknown may be the future, something we cannot see because we have not arrived there yet. Either way, we hope that samples are representative of the population from which they are drawn, thus approximating the unknown we hope to see or predict. A frustrating reality of using data that affects practitioners in the field is that data never predict a specific outcome for a specific act like buying a particular property. If helpful at all, it enables us to more accurately predict 0.25 0.45 FIGURE 3-10 Majority of San Francisco EVR observations. TABLE 3-7 Probability of Certain EVR Observations evr P(evr) 0.25 0.0839818 0.3 0.288662 0.35 0.604179 0.4 0.861185 0.45 0.971748 0.5 0.99682 60 Private Real Estate Investment File: {Elsevier}Brown/Revises-II/3d/Brown-ch003.3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22.12.2004/3:22pm Page: 61/72 the mean of a large number of repeated trials of the same act. Recall that earlier in this chapter we counseled a ‘‘healthy suspicion about the extreme observations.’’ This advice applies both to specifics, where we believe there is something wrong with a reported data point, and, more generally, to the way extreme observations affect our ability to predict the mean. Observations are considered ‘‘extreme’’ if they lie at a point that is far[away] from the mean. Such extreme values have an affect on the mean that exceeds their relative importance to the entire distribution. Let us now address the non-normal issue. The assumption of normality is often convenient, but also often troublesome. We already suspect that the distribution of evrs is not normal because normal distributions are symmetrical (skewness ¼ 0) and have ‘‘skinny’’ tails (kurtosis ¼ 3). Neither is the case for the evr data as the reports in Table 3-6 show. Skewness and excess kurtosis are indications of non-normality which may be the result of extreme values. Our concern is that extreme values are common in this process (market) and that this sample is drawn from a population that has a similarly skewed and heavy-tailed shape. There are many tests for normality. We shall use one known as the Jarque- Bera ( JB) test. The statistic produced by his test has a chi-squared limiting distribution with two degrees of freedom. Provided the evr sample is large enough to emulate the asymptotic properties of its limiting distribution, one rejects the hypothesis of normality if the JB statistic exceeds 5.99. JB ¼ n 6 Skewness 2 þ Excess Kurtosis 2 4 ! ¼ 1000 6 0:221308 2 þ 1:07313 2 4 ! ¼ 56:1464 ð3-20Þ The result in Equation (3-20) clearly rejects the null, suggesting that the data is not normally distributed. Another important question is whether the variance is finite. That is, does the second moment exist in the limit, a requirement for normality? There is a large and growing literature on infinite variance models that goes beyond the scope of this book. Reference to some of these are at the end of this chapter. The assumption of normality imposes a set of strong conditions. Notably, because the distribution is constructed from only its first two moments (meaning that it is a two-parameter model dependent only on its mean and variance), the assumption of normality requires one to ignore skewness and heavy tails. Thus, to believe in the probability estimates in Table 3-7, one must believe that the world from which the data is drawn is symmetrical, has thin tails, and has a finite variance. In essence, the assumption of normality is The ‘‘Rules of Thumb’’ 61 File: {Elsevier}Brown/Revises-II/3d/Brown-ch003.3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22.12.2004/3:22pm Page: 62/72 like a set of blinders. It blocks the presence of outliers from view and prohibits disproportional outcomes on only one side of the distribution. Relaxing the normality assumption permits a larger view of the unknown world we wish to see. THE STABLE APPROACH TO DATA It happens that the normal distribution is a special case of a family of Stable- Paretian (SP) distributions. The wider view of Stable distributions offers a way to estimate probabilities when extreme values are allowed. 6 The assumption of normality distorts our view of the data away from its actual shape (see Figure 3-11). Comparing the expense ratio probabilities first given in Table 3-7 with those estimated under the Stable hypothesis, in Table 3-8 we get a different view of how likely a certain building expense ratio is. In this case the normal assumption about the population produced probability estimates fairly close to those if one makes a non-normal Stable assumption. This is fine when the distribution is close to normal, but many distributions are not, leaving the opportunity for considerable estimation error. 7 LINEAR RELATIONSHIPS In the general caution at the beginning of this chapter we cast doubt on rules of thumb based on their linear nature. This admonition carries over to the analysis of data. There is a very practical reason for a natural dependence on linear models—they offer tools that are considered ‘‘tractable,’’ a term that mathematicians use to mean manageable. However, manageable does not necessarily mean realistic or even correct. There are a number of opportunities, using linear models common to most spreadsheet programs, to either dazzle a client with numbers that argue for a spurious relationship or data mine to the point of disproving a claim that is in fact true. Time and space do not allow a full elaboration of these traps, such being left to econometric texts. However, since we have a dataset at hand, let’s use it to provide several examples of suspected relationships and look a bit closer at their true nature. 6 This methodology has been developed by Prof. John P. Nolan of the Mathematics and Statistics Department of American University in Washington, DC. As of the Fall of 2004, Dr. Nolan could be reached via his Web site http://academic2.american.edu/$jpnolan/stable/stable.html. 7 The reader can explore this subject more at www.mathestate.com which includes a page allowing the user to upload data and estimate Stable parameters. 62 Private Real Estate Investment File: {Elsevier}Brown/Revises-II/3d/Brown-ch003.3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22.12.2004/3:22pm Page: 63/72 TABLE 3-8 San Francisco EVR Probabilities under Different Assumptions about the Distribution evr Normal probability Stable probability Difference 0.25 0.0839818 0.0731308 0.010851 0.3 0.288662 0.281732 0.00692941 0.35 0.604179 0.620898 À0.0167185 0.4 0.861185 0.875039 À0.0138543 0.45 0.971748 0.969291 0.00245784 0.5 0.99682 0.990765 0.00605496 0.15(b) 0.2 0.25 0.3 0.35 0.4 0.45 0.5 normally distributed EVRs FIGURE 3-11 EVR histogram of actual data (Figure 3-11a) and normal pdf from data mean and variance (Figure 3-11b). Expense and Vacancy Ratios (EVRs) 0.2(a) 0.3 0.4 0.5 0.6 20 40 60 80 100 120 140 The ‘‘Rules of Thumb’’ 63 File: {Elsevier}Brown/Revises-II/3d/Brown-ch003.3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22.12.2004/3:22pm Page: 64/72 LINEAR TRANSFORMATIONS Linear transformations involve either adding a constant to or subtracting a constant from each observation or multiplying each observation by a constant (which, if in the form of a reciprocal, can turn into division). In general these manipulations are harmless and permissible. Here is an example. Suppose that we know that vacancy in the market from which our data was drawn was 4% during the time in question. We can remove this from evr to leave us with just the expense ratio. Note that this does not change the shape of the distribution; it only moves the range down 4%. This is a ‘‘shift’’ of the distribution along the x-axis. One of the important and useful properties of the normal distribution is that it retains its shape under linear transforma- tion. 8 Subtracting a constant from each observation, as we have done for the data’s maximum and minimum values, in Table 3-9, is a form of linear transformation. As long as the value of the constant is known, such an adjustment can be made and all the conclusions reached earlier still hold (see Figures 3-12a and b). SPURIOUS RELATIONSHIPS One might claim that property characteristics influence the expense ratio. A reasonable example of this would be a claim that older buildings have higher expenses. This suggests that expense ratios increase as property age increases. Since building age is in our dataset, we can plot these for an initial indication of a relationship (see Figure 3-13). It doesn’t look like there is much in the way of a relationship in Figure 3-13. Do owners of older buildings charge higher rents to offset higher expenses? Likewise there seems to be little relationship between our two size measures, building square footage or number of units, and expense ratios (see Figures 3-14 and 3-15). TABLE 3-9 Extreme Observations for evr Data and Expense Ratio Data evr exp ratio Maximum 0.591824 0.551824 Minimum 0.1043 0.0643 8 This is also true of the larger class of Stable-Paretian distributions of which the normal is just a special case. 64 Private Real Estate Investment [...]... simple OLS Regression produces incorrect coefficient estimates in the presence of non-normal Stable data An excellent source for this, including a real estate dataset as an example, is McCulloch (1998) 70 TABLE 3-13 Regression of San Francisco Value on Number of Units REGRESSION SUMMARY OUTPUT Regression statistics Multiple R 0.040353769 R square 0.001628427 Adjusted R square 0.000628054 Standard error... services are transferred to tenants because landlords cannot recover those costs in the rent One intuitively promising idea is that expense ratios are related to clientele occupying the unit To the extent that area serves as a proxy for this, we could compare the ‘‘Area’’ field to the expense ratio data to see if there is a relationship One should remain neutral about the outcome of any such study 72 Private. .. 72 Private Real Estate Investment Recall that we postulated earlier that expense ratios should increase with age and that was not supported by the data! Current active research in the area of heavy tails deserves watching carefully Small comfort that it is, at present the only thing we can say for certain is that the assumption of normality may very often produce measurement errors Therefore, the investigation... there are economies of scale involved, more dwelling units per land unit, and more sophisticated parties Thus, one might 68 Private Real Estate Investment TABLE 3-10 Price Per Unit for Los Angeles and San Francisco Mean price per unit in Los Angeles $71,936 Mean price per unit in San Francisco $118,847 expect the price per unit to decline as property size increases These are known as ‘‘testable hypotheses.’’... suspected relationship to regression analysis 67 The ‘‘Rules of Thumb’’ GRM vs Expense Ratio Expense Ratio 0.5 0.4 0.3 0.2 0.1 0 6 8 10 GRM 12 14 FIGURE 3-16 Plot of expense ratio against GRM CASH-ON-CASH RETURN (C/C) Arguably, cash-on-cash return (C/C) is the best of the first year measures because, if we are talking about after-tax cash-on-cash return, both debt payments and taxes are considered In... ‘‘sell’’ decisions relating to individual properties REFERENCES 1 Adler, R J., Feldman, R E., and Taqqu, M S (1998) A Practical Guide to Heavy Tails: Statistical Techniques and Applications Boston, MA: Birkhauser 2 Corgel, J B., Ling, D C., and Smith, H C (2001) Real Estate Perspectives: An Introduction to Real Estate (4th ed.) New York: McGraw-Hill 3 McCulloch, J H (1998) Linear regression with stable... COMPUTATIONAL AIDS While spreadsheet technology has removed much of the tedium involved in illustrating and projecting investment performance, it should not be 73 74 Private Real Estate Investment considered a complete tool kit Technical computing software performs a different function than spreadsheet programs It can make any numerical calculation that can be done by a spreadsheet program, but it is... mean for prediction All of this is to warn the reader that when using the sort of easy data analysis provided in many popular spreadsheet packages, one must be careful to interpret the results correctly.9 If higher prices for small properties are persistent across time and other markets, we can say it is probably not because of the number of units in the 9 One should know that simple OLS Regression produces... Parameterization and software Adler, R J., Feldman, R E., and Taqqu, M S., Eds A Practical Guide to Heavy Tails: Statistical Techniques and Applications Boston, MA: Birkhauser, pp 527–533 CHAPTER 4 Fundamental Real Estate Analysis Compound interest is the eighth wonder of the world Attributed to Albert Einstein and others INTRODUCTION There are a number of excellent real estate analysis programs for... some insight into pricing trends OTHER DATA ISSUES Many excellent texts cover statistics in depth We close this chapter with some red flags that require further study One should exercise care when comparing a continuous variable such as expense ratios to a discrete variable such as age There are also problems when one variable is constrained as expense ratios are to [0,1] differently than the other such . 0.971748 0.5 0.996 82 60 Private Real Estate Investment File: {Elsevier}Brown/Revises-II/3d/Brown-ch003.3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22 . 12. 2004/3 :22 pm Page: 61/ 72 the mean. parameters. 62 Private Real Estate Investment File: {Elsevier}Brown/Revises-II/3d/Brown-ch003.3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22 . 12. 2004/3 :22 pm Page: 63/ 72 TABLE 3-8. Birkhauser, pp. 527 –533. 72 Private Real Estate Investment File: {Elsevier}Brown/Revises-II/3d/Brown-ch004.3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22 . 12. 2004/3 :22 pm Page: 73/98 CHAPTER 4 Fundamental