Re Academic Press Private Real Estate Investment_5

Uncertainty: Risk in Real Estate 133 Price 300000 280000 260000 240000 220000 200000 180000 1600 1700 1800 1900 2000Square Feet FIGURE 6-5 Plot of house data. $132 per square foot. The result is that the empirical test of our theory about houses being worth $132 per square foot shows that it is less than perfect. (Most applications are.) Univariate regression is the process of finding the ‘‘conditional mean’’ in that it helps you predict the mean of one quantity conditioned on knowing some other, independent quantity. In this case the fact that we know is the size of the house. But is that fact fully determinative? Table 6-7 shows the residuals, defined as the difference between the observed price and the predicted price. This difference, in a statistical sense, is a measure of the error between what our claim is (that houses sell for $132 per square foot) and what really happens. This sort of thing, while statistically valid, is only partially helpful in guiding an individual property owner to the value of his property. Compare the output in Table 6-7 with the regression output of the circle example in Table 6-5. The R-square for the house price regression is only 66%, residuals are non-zero, standard errors are positive, and confidence intervals are positive. All of this states the obvious: relying on price per square foot as an indicator of value is less than perfect. From this we conclude that in a more complex world determinism rarely exists, and the rule is uncertainty. Diameter completely determines the circumference of all circles. Size only partially determines the price at which a house sells. The positive standard errors of the residuals are measures of how much our theory is wrong for particular houses. The 34% ‘‘unexplained’’ part encompasses the non-size characteristics that determine value. The effects of these characteristics are embedded in the error terms. This suggests that our $132 theory of house prices based on the single variable (size) and a linear relationship is simplistic, something any home buyer or seller knows.10 10Mathematicians will object with the casual use of ‘‘linear,’’ a term that has a specific and precise mathematical meaning. There is a thread of linear, however tenuous it may be, that links binary outcomes, the normal distribution and linear regression. We do take liberties here and sometimes indulge in a metaphorical use of linear as ‘‘unduly simplistic.’’ 134 TABLE 6-7 Private Real Estate Investment Regression of House Prices on Size (Square Foot) SUMMARY OUTPUT Regression statistics Multiple R R square Adjusted R square Standard error Observations 0.813090489 0.661116143 0.618755661 21450.49633 10 ANOVA df SS MS F Significance F Regression 1 7181109658 7181109658 15.6069079 0.004232704 Residual Total 8 3680990342 460123792.8 9 10862100000 Coefficients Standard error t Stat P-value Lower 95% Upper 95% Intercept ÿ79095.58434 78357.96111 ÿ1.009413507 0.342328491 ÿ259789.4835 101598.3149 SF 178.1603607 45.09751892 3.950557923 0.004232704 74.16522831 282.155493 RESIDUAL OUTPUT Observed price 195000 210000 225000 240000 275000 285000 190000 239000 249000 185000 Predicted price 188144.9567 232685.0469 205960.9928 223777.0288 259409.101 254955.0919 214869.0108 243374.6685 272771.128 197052.9747 Residuals 6855.043317 ÿ22685.04685 19039.00725 16222.97118 15590.89905 30044.90806 ÿ24869.01079 ÿ4374.668494 ÿ23771.12801 ÿ12052.97472 Despite less than perfect results, we press on. Data analysis does not always lead us directly where we want to go. The process involves many partial glimpses of the truth and the rare epiphany. Letting the numbers talk to us enlarges our understanding about how a process works. The constant reconciliation of objective outcomes based on the numbers with our subjective reasoning based on field experience is a big part of the value of the data analysis exercise. Uncertainty: Risk in Real Estate 135 DETERMINISM AND REAL ESTATE INVESTMENT The above examples include the case (a) where a definite, constant linear relationship (the diameter and circumference of a circle) surely exists between two variables or (b) where a suspected linear relationship (house size and price) may exist between two variables. We now complicate this by examining relationships between variables in an investment context. Recall the general caution of Chapter 3. The capitalization rate rule of thumb claims that value is a function of income and that the functional relationship is I 1 CR CR ð6-2Þ where V is value, I is (net operating) income, and CR is the capitalization rate. The essence of this argument is that one may compute the value of property by multiplying its net income by the reciprocal of the capitalization rate. But there is the added complication that we must choose between income and capitalization rate, both of which influence value, to decide which is the dependent variable and which is the conditioning coefficient or slope term. In the market both income and capitalization rate vary. They may vary independent of each other or the way they vary may be connected. In Table 6-8 we see examples of how value may vary depending on the choice of either income or capitalization rate when the other value is fixed. Table 6-9 shows how different the investment situation is from our two prior examples where the value of p was p or $132 was always $132. TABLE 6-8 Investment Property Values, Income, and Capitalization Rates Income constant at $10,000 Cap rate constant at 10% Cap rate .07 .08 .09 .10 .11 .12 Value ($) 142,857 125,000 111,111 100,000 90,909 83,333 Income ($) 9,500 10,000 10,500 11,000 11,500 12,000 Value ($) 95,000 100,000 105,000 110,000 115,000 120,000 136 Private Real Estate Investment TABLE 6-9 Circles c ¼ pd Three Theories Houses p ¼ $132 sf Investments I 1 CR CR Recall that the house-price-by-square-foot function, p, was a theory based on a single number ($132). The function, V, we use for our theory about investments involves the reciprocal of a rate, 1 , by which we multiply income. The rate is not only variable, but as we will see it is a composite of several other variables. Our claim is not just that some specific constant number multiplied by income produces value, but that one of a (suitably scaled) set of different numbers multiplied by income constitutes value. Here we have three opportunities to be wrong: we may be incorrect about the shape of the function (it may not be linear); we may be incorrect about the value we pick from the set of capitalization rates we use as the multiplier; or we may incorrectly estimate the income to be collected from the property. Although these sorts of errors may also plague the house example, the opportunity for capitalization rate error goes beyond measurement error. What is different in the investment context is that the multiplier, 1 , implicitly introduces a number of other variables into the equation. Arguably, capitalization rates are a kind of interest rate, affected in many of the same ways that interest rates are affected.11 Specifically, there is a curious three-way connection between interest rates, capitalization rates, and inflation. These forces are implicitly in our thinking when we use capitalization rate. Note the composition of the interest rate, i. It is a combination of the real rate (r), the default risk (dfrt), and inflation expectations (inflexp). i ¼ r þ dfrt þ inflexp ð6-3Þ One may reasonably conclude that real estate capitalization rates (CR) are composed of the interest rate (i) and a real estate risk premium (rerp) to compensate for non-systematic, site-specific real estate risk. CR ¼ iþ rerp ð6-4Þ 11This could be challenged as overly simplistic for one could argue that the capitalization rate is more like a dividend yield rate. The purpose here is to broaden the discussion to various rates of returns that attract capital to one investment over another. Uncertainty: Risk in Real Estate 137 TABLE 6-10 Two Investment Property Regressions Regress value on cap rate Regress value on income Equation R-squared 2,777,391–19,146,005 0.0254441 –108848þ13.6603 0.962347 Substituting Equation (6-3) and Equation (6-4) into Equation (6-2), we can now decompose the value-based-on-income-capitalization function to find that, in fact, the right side of Equation (6-2) is made up of several components. V ¼ r þ dfrt þ inflexp þ rerp ð6-5Þ The curious part is that investors may discount future expected rent increases in such a way that expected inflation may, while a positive number in Equation (6-3), indirectly exert negative pressure on capitalization rates. This is reason to doubt a simple relationship between the value of investment real estate and its capitalization rate. Let’s look at a dataset of 500 actual apartment sales that took place in the Los Angeles area between May and October of 2001. Each observation shows the price sold, net operating income, and capitalization rate. The complete regression and analysis of variance in Excel format are included on the CD Rom. In the interest of space, we will summarize partial results in Table 6-10. When we claim that value is dependent on the correct selection of a capitalization rate, the capitalization rate is the ‘‘coefficient’’ of income in our theoretical relationship. The sign of the coefficient in the middle column of Table 6-10 makes sense. One expects that value would decline with increases in the capitalization rate. But the R-square of 0.025 leaves us with little confidence that the capitalization rate explains the variation in value. Hence, we should be careful about claiming that a certain capitalization rate of x will, given income, produce a correct value of y. With an R-square this low, one wonders if knowing the capitalization rate gives us very much information at all about value. What is the problem here? This is an example of model misspecification. We have assumed that the relationship between value and capitalization rate is linear. Given that the coefficient is 1 , it is clear that the relationship is non-linear, thus the use of linear regression is in error.12 12The fact that the relationship is between value and the inverse of the capitalization rate is important because it is non-linear, but in this particular case regressing value on the reciprocal of the capitalization rate does not produce meaningfully different results. ... - tailieumienphi.vn