# Private Real Estate Investment: Data Analysis and Decision Making_4

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## Private Real Estate Investment: Data Analysis and Decision Making_4

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1. 55 The ‘‘Rules of Thumb’’ and net income in turn is a function of vacancy and expenses, Net Operating Income ¼ Gross Potential Income À Vacancy À Expenses ð3-15Þ leads us to simplify the difference between gross and net income as Net Operating Income ¼ Gross Potential Income (1 À evrÞ ð3-16Þ where evr ¼ expense and vacancy rate (0 < evr < 1). Val Rearranging the value equation, recalling that GPI ¼ GRM, and substituting our simplifying assumption produces Val=GRMð1 À evrÞ Capitalization Rate ¼ ð3-17Þ Val 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. À Á evr ¼ 1 À cr grm 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. 1 À evr GRM ¼ ð3-19Þ cr 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
2. 56 Private Real Estate Investment 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 À evr must be a number between 0 and 1. For the whole right side of Equation (3-19) to be greater than 1, 1 À evr must be larger than the caprate. THE NORMAL APPROACH DATA TO 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 3. 57 The ‘‘Rules of Thumb’’ 0.6 0.45 EVR 0.25 0.1 0 200 400 600 800 1000 Observations # FIGURE 3-6 Plot of ordered list of EVRs. Expense and Vacancy Ratios EVRs 140 120 100 80 60 40 20 0.2 0.3 0.4 0.5 0.6 FIGURE 3-7 San Francisco EVRs. 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, 4. 58 Private Real Estate Investment 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 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. normally distributed EVRs 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 FIGURE 3-8 Normally distributed San Francisco EVRs. 5. 59 The ‘‘Rules of Thumb’’ x 0.25 FIGURE 3-9 Portion of EVR observations at or below 25%. 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? 6. 60 Private Real Estate Investment 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 QUESTIONING ASSUMPTION NORMALITY THE OF 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 7. 61 The ‘‘Rules of Thumb’’ 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. ! Excess Kurtosis2 n Skewness2 þ JB ¼ 6 4 ! ð3-20Þ 1:073132 1000 0:2213082 þ ¼ 56:1464 ¼ 6 4 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 8. 62 Private Real Estate Investment 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 DATA TO 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.
9. 63 The ‘‘Rules of Thumb’’ Expense and Vacancy Ratios (EVRs) 140 120 100 80 60 40 20 (a) 0.2 0.3 0.4 0.5 0.6 normally distributed EVRs (b) 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 FIGURE 3-11 EVR histogram of actual data (Figure 3-11a) and normal pdf from data mean and variance (Figure 3-11b). 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 À 0.0167185 0.4 0.861185 0.875039 À 0.0138543 0.45 0.971748 0.969291 0.00245784 0.5 0.99682 0.990765 0.00605496
10. 64 Private Real Estate Investment 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.
11. 65 The ‘‘Rules of Thumb’’ Expense and Vacancy Ratios (EVRs) 140 120 100 80 60 40 20 (a) 0.2 0.3 0.4 0.5 0.6 Expense Ratios 140 120 100 80 60 40 20 (b) 0.1 0.2 0.3 0.4 0.5 FIGURE 3-12 Histograms for evr data (Figure 3-12a) and expense ratio data (Figure 3-12b). Age vs. Expense Ratio 0.5 0.4 Expense Ratio 0.3 0.2 0.1 0 0 20 40 60 80 100 Age FIGURE 3-13 Plot of expense ratio against building age.
12. 66 Private Real Estate Investment Building Size vs. Expense Ratio 0.5 0.4 Expense Ratio 0.3 0.2 0.1 0 0 5000 10000 15000 20000 25000 Building SF FIGURE 3-14 Plot of expense ratio against building size. Number of Units vs. Expense Ratio 0.5 0.4 Expense Ratio 0.3 0.2 0.1 0 0 5 10 15 20 25 30 35 Number of Units FIGURE 3-15 Plot of expense ratio against number of units. There might be a weak inverse relationship between GRM and expense ratios (see Figure 3-16). However, intuition and the weakness of the relationship in the plot does not recommend pursuing this. Each of the foregoing have tested and discarded a relationship based on a graph of the data. Later, while considering the price per unit (PPU) rule of thumb, we will extend the investigation of a suspected relationship to regression analysis.
13. 67 The ‘‘Rules of Thumb’’ GRM vs. Expense Ratio 0.5 0.4 Expense Ratio 0.3 0.2 0.1 0 6 8 10 12 14 GRM 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 all types and sizes of properties, market participants value the cash-on-cash measure. Some even prefer it over multi- period models because it requires no estimation of future income. Unfortunately, for all its value it is the least observable. Because terms of debt rarely appear and income taxes never appear in datasets, we have only theory on which to rely. The data to conﬁrm any theory we may construct is sparse at best. PRICE PER UNIT (PPU) Perhaps the simplest question asked is: ‘‘How much per foot?’’ or ‘‘How much per acre?’’ Many people buy certain property types based on whether its price is appropriately scaled to its size. Some people spend considerable time reviewing sales and offerings to determine what price per unit the market supports. Table 3-10 shows the average price per unit for our two cities. We have suggested that purchasers of smaller properties tend to make decisions based on rough measures of investment value. One could also argue that as one buys larger properties there are economies of scale involved, more dwelling units per land unit, and more sophisticated parties. Thus, one might
14. 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.’’ That is, if we collect data associated with such assertions we can test their validity. In Tables 3-11 and 3-12 we compute the average price per unit for each of two size classes from our two datasets, noting that prices in San Francisco are higher than those in Los Angeles. (Caution is advised as the San Francisco data was gathered over a ﬁve-year period and the Los Angeles data was gathered over approximately the last six months of that same period. Here we faced a difﬁcult choice. Because the comparison is between two markets of different size, there are fewer sale observations over the same period of time in the smaller market. Nonetheless, it is desirable to compare datasets of approximately the same size. As the data is included in the electronic ﬁles that accompany this chapter, it is left as an exercise to determine what, if any, bias the time difference introduces.) In Table 3-11 we divide the San Francisco data into approximately half using > 10 units as ‘‘large’’ to learn that people apparently do pay more per unit for smaller buildings. The same appears to hold for the Los Angeles data (see Table 3-12). On the surface, it appears that we are correct in our theory about small properties and price, but this merely suggests a relationship between two groups of data. Does this relationship hold generally? The plot of the San TABLE 3-11 San Francisco Price Per Unit by Size Category # of sales Average price Large SF 424 $103,901 Small SF 576$129,849 TABLE 3-12 Los Angeles Price Per Unit by Size Category # of sales Average price Large LA 343 $63,550 Small LA 357$79,992
15. 69 The ‘‘Rules of Thumb’’ 250000 200000 PPU 150000 100000 50000 0 0 5 10 15 20 25 30 35 Building Size FIGURE 3-17 San Francisco price per unit vs. building size. Francisco data in Figure 3-17 indicates a weak negative relationship between unit price and building size. We see fewer large buildings in the higher price per unit range. But how strong is the relationship in general? As we shall see, it is not very strong at all. Regression analysis is a process that attempts to specify a rule, known as a functional relationship, which governs the connection between two variables. Univariate analysis, which we perform here, involves a single variable, building size. Ordinary Least Squares (OLS) Regression ﬁnds the ‘‘least squares’’ line that purports to describe how price per unit ‘‘functions’’ as building size grows. Table 3-13 shows the regression of value on number of units in San Francisco. The weakness of the relationship as seen in the plot, the low T-statistic and R-squared values in the regression analysis, suggests that there is no relationship in this case. The low F-statistic suggests that the model does not apply. Another interpretation of this is that most of the model behavior is captured in the intercept, meaning that we are little better off using the conditional probability suggested by OLS than we would be using the unconditional 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 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).
16. 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 55,995.41781 Observations 1,000 ANOVA df SS MS F Signiﬁcance F Regression 1 5104009925 5,104,009,925 1.627820567 0.202301623 Residual 998 3.12922E þ 12 3,135,486,815 Total 999 3.13432E þ 12 Private Real Estate Investment Coefﬁcients Standard error t Stat P-value Lower 95% Upper 95% Intercept 119,205.3808 1,792.870938 66.48854544 0 115,687.1514 122,723.6103 Units À 20.25644089 15.87668672 À 1.275860716 0.202301623 À 51.41196116 10.89907938
17. 71 The ‘‘Rules of Thumb’’ building. This raises more questions. Do higher prices result from a larger land component? One can test for the signiﬁcance of the price per unit difference, given lot size. Separate from the small building value question, this data can be used to investigate other matters. Are there economies of scale? One can relate expense ratio to size via a closer examination of price per unit. Is there a time component? Price per unit as a function of the date sold can give 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 as age [R þ ]. These issues also cause problems in a regression context, something discussed in the Regression Primer included in the electronic ﬁles for Chapter 6. Sample size is a factor in data analysis. We arbitrarily chose a random sample of data from a larger dataset for the above illustrations. Taking the entire dataset or just a larger sample could improve our conclusions. This dataset is drawn from a large area for which we have several subsets based on location. A reductionist view would have one analyzing the data from each subset. What would your expectation be about data drawn from an area in Los Angeles near the ocean versus an area in Central Los Angeles? An expansionist view would collect data from other cities and compare them. What would your expectation be about expense ratios in Los Angeles versus Minneapolis or San Francisco versus Madrid, Spain? The San Francisco dataset covers a long period of time. There is little reason to believe that expense ratios are time dependent. On the other hand, local building code and zoning laws together with market demands can affect the level of services landlords offer tenants. The price of these services—in a free market—is impounded into the rent. Expenses as a portion of rent will change with these factors. Price-restricted markets, those with rent control such as San Francisco, may show a decline in expense ratios as the cost of 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.
18. 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 of other aspects of the distribution, however ﬂawed, has a probative value and should not be ignored. One last time we must be reminded that, regardless of how good the data and the models become in time, there is no magic bullet here. We still have to rely on a careful inspection of the property itself to be conﬁdent about a certain expense ratio or capitalization rate for that particular property. Remember that data does not supplant good ﬁeldwork at the site, it only provides context about the market in which the property competes for tenants and investors. It is the general understanding of the market that comes from data analysis. Such understanding contributes to successful ‘‘buy’’ and ‘‘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 disturbances. 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. 359–376. 4. Nolan, J. P. (1998). Maximum Likelihood Estimation and Diagnostics for Stable Distributions. [Working Paper]. 5. Nolan, J. P. (1997). Numerical calculation of stable densities and distribution functions. Communications in Statistics – Stochastic Models, 13(4), 759–774. 6. Nolan, J. P. ( June 1998). Parameterizations and modes of stable distributions. Statistics and Probability Letters, 38(2), 187–195. 7. Nolan, J. P. (1998). Univariate stable distributions: 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.
19. 4 CHAPTER 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 practitioner use in ‘‘numbers crunching.’’ Essentially, these are variant forms of spread- sheets that perform numerical analysis. An Excel version of such a program is included with the electronic ﬁles for this chapter to provide the reader with a fully elaborated example in a familiar context. Programs such as these all perform the same basic tasks, usually offering different menus of reporting and printing options to dress up the appearance of the analysis for use with clients. In this chapter we will:  List all the relevant variables required to perform a complete multi- period discounted cash flow analysis of a real estate investment.  Perform that analysis, computing various performance measures, includ- ing net present value and internal rate of return.  Demonstrate the interdependencies between the variables.  Set up a series of scenarios based on the variables for simple risk evaluation via sensitivity testing.  Introduce ways in which data may be used to further understand the process. THE ROLE COMPUTATIONAL AIDS OF While spreadsheet technology has removed much of the tedium involved in illustrating and projecting investment performance, it should not be 73