File: {Elsevier}Brown/Revises-II/3d/Brown-ch005.3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22.12.2004/3:23pm Page: 107/118 hierarchy. We have purposely chosen extreme alternatives to illustrate our point. One needs a mechanism for thinking about risk in more realistic settings when the alternatives may not be so obvious. For instance, how would we compare two commercial structures, one occupied by a major clothing retailer and another by a major appliance retailer, or two similar apartment buildings on different sides of the street? Many such opportu- nities present themselves. They have different risk, and while the difference may not be great, there is a difference and one must be preferred over the other. Our goal in this chapter is to discover a way of ranking risky opportunities in a rational manner. As is so often the case, ‘‘rational’’ means mathematical. THE ‘‘CERTAINTY EQUIVALENT’’ APPROACH The search for a sound way to evaluate risky alternatives leads to an inquiry into how discounts come about. We assume that nearly anything of value can be sold if the price is lowered. Risky alternatives, as ‘‘things of value,’’ become more appealing as the entry fee is reduced (because the return increases). The idea that describes this situation well is known as the certainty equivalent (CE) approach. We ask an investor to choose a point of indifference between opportunities having a certain outcome and an uncertain outcome, given that the price of the opportunity with the uncertain outcome is sufficiently discounted. Let us use a concrete example to illustrate the concept. Suppose someone has $100,000 and a chance to invest it that provides two (and only two) equiprobable outcomes, one of $150,000 (the good result) and the other of $50,000 (the unfortunate outcome). The certain alternative is to do nothing, which pays $100,000. We want to know what is necessary to entice our investor away from this certain position and into an investment with an uncertain outcome. In Figure 5-6 we see the plot of utility of these uncertain outcomes as wealth rises or falls. Note the three points of interest, constituting the original wealth and the two outcomes. Our investor must decide if the gain in utility associated with winning $50,000 is more or less than the loss of utility associated with losing $50,000. The y-axis of Figure 5-6 provides the answer. The question of how much to pay for an investment with an uncertain outcome is answered by placing a numerical value on the difference between the utility of the certain opportunity and the utility of the uncertain one. How do we do this in practice? To begin with, notice that the expectation of wealth in this fair game is zero. That is, the mathematical expectation is Beginning Wealth þ (probability of gain  winning payoff ) À(probability of Chance: Risk in General 107 File: {Elsevier}Brown/Revises-II/3d/Brown-ch005.3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22.12.2004/3:23pm Page: 108/118 loss  amount of loss). Since the outcomes are equally probable, the probability of either event is 0.5, so we have Probability Payoff($) Change($) Begin wealth ($) End wealth ($) 0.5  (100,000) ¼ (50,000) þ 100,000 ¼ 50,000 0.5  100,000 ¼ 50,000 þ 100,000 ¼ 150,000 Expectation 0 þ 100,000 ¼ 100,000 The graphic representation of this situation is, of course, linear and represents how people who are ‘‘risk neutral’’ view the world. 4 Most people, as we will see in a moment, are presumed to be risk averse. The perspective of the risk neutral party is the reference from which we start to place a value on risk bearing. When comparing the two curves in Figure 5-7 we see that, relative to the y-axis, they both pass through the same points on the x-axis representing the alternative outcomes. But when they pass through initial wealth, they generate different values on the y-axis. Following the curved utility function, note that the difference between the change in utility associated with an increase in one’s wealth, 11.9184 À11.5129 ¼ 0.4055, and the change in utility associated with an equivalent (in nominal terms) decrease in one’s wealth, 50000 100000 150000 Wealth 10.8198 11.5129 11.9184 U [Wealth] FIGURE 5-6 Plotting utility of wealth against wealth. 4 Such people are usually not people at all, but companies, namely insurance companies having unlimited life and access to capital. 108 Private Real Estate Investment File: {Elsevier}Brown/Revises-II/3d/Brown-ch005.3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22.12.2004/3:23pm Page: 109/118 11.5129 À10.8198 ¼ 0.6931, shows that the lost utility associated with losing $50,000 is greater than the utility gained by winning $50,000. 5 The conclusion we reach is that in order to be compensated for bearing risk our investor must be offered the opportunity to pay less than the raw expectation ($100,000). This is reasonable. Why would someone who already has $100,000 pay $100,000 for a 50/50 chance to lose some of it, knowing that in a large number of trials he can do no better than break even? From Figure 5-7 we note that utility for the risky prospect is the same as the utility of the certainty of $100,000 (the ‘‘do nothing’’ position) if the risky opportunity is priced at $86,603. Certainty equivalent is a way of saying, that the investor is indifferent between paying $86,603 for the 50/50 opportunity to increase or decrease his wealth $50,000 or having a certain $100,000. How is $86,603 calculated? We know that the expectation of the utility of wealth as shown on the y-axis of the plot is Certainty Equivalent ¼ E½uðwÞ ¼ 0:5uð50000Þþ0:5uð150000Þ ¼ 11:3691 ð5-3Þ 50000 86603 100000 150000 10.8198 11.5129 11.9184 11.3691 FIGURE 5-7 Risk neutral and risk averse positions for u[w] ¼ Log[w]. 5 There is an important generalization at work here: the utility of the expectation is larger than the expectation of the utility. This is no surprise to mathematicians who have long known about ‘‘Jensen’s Inequality,’’ named for Johan Ludwig William Valdemar Jensen (1859–1925). Chance: Risk in General 109 File: {Elsevier}Brown/Revises-II/3d/Brown-ch005.3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22.12.2004/3:23pm Page: 110/118 And we know that number is produced in Equation (5-4) by a function we have chosen u ¼ Log(w). Thus, we solve for the known value of u by ‘‘exponentiating’’ both sides of Equation (5-4). Log½CE¼11:3691 ð5-4Þ Doing this to the left side of Equation (5-4) eliminates the Log function and leaves the certainty equivalent wealth as the unknown. Doing it to the right side of Equation (5-4) leaves e 11.3691 , which is easily evaluated using a calculator because e is just a number, a constant approximately equal to 2.71828. e Log½CE ¼ e 11:3691 ¼ 2:71828 11:3691 ¼ 86,603 ð5-5Þ The difference between $86,603 and $100,000, $13,397, is the discount the investor applies to the raw expectation, given his specific preference for risk as represented by the shape of his utility function. Stated differently, the discount is the compensation he requires to accept a prospect involving this sort of risk. When a real estate broker asks his client to take money out of a savings account to buy an apartment building, it is the discount and its associated prospect of a higher return on the net invested funds that motivates the buyer to act. Two final points are useful before we move on. Not only is the concavity of the utility function important, but ‘‘how concave’’ it is matters, as we will see in the next section. Additionally, the discount calculated above is a function of not only the shape of the utility function, but the spread of potential returns. Above our investor requires a relatively large discount of more than 13%. If we lower the potential gain or loss to $10,000, the discount drops to about 5%. The conclusion one might reach is that risk aversion is relative to both one’s initial wealth and the portion of that wealth at stake in an uncertain situation. This mathematically supports sage advice that one should not bet more than one can afford to lose. A concave utility function means that people value different dollars differently. Various microeconomic texts consider other utility functions such as those illustrated in Figure 5-4 and develop a ‘‘coefficient of risk aversion’’ to tell us how much differently those dollars are valued by different people having different risk tolerance. This has important implications for the market for uncertain investments. Such a market commands higher prices if populated by people with low coefficients of risk aversion, as they require smaller discounts. 110 Private Real Estate Investment File: {Elsevier}Brown/Revises-II/3d/Brown-ch005.3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22.12.2004/3:23pm Page: 111/118 MULTIPLE (MORE THAN TWO) OUTCOMES Returning to our first utility function (u[w] ¼ Log[w]), we can extend this result to more than two outcomes, each with different probabilities. In Table 5-2 we define some payoffs under different conditions (numbers can represent thousands or millions of dollars to make them more realistic). We then associate a specific probability with each payoff. Note the important fact that the probabilities add up to 1. Where did these probabilities come from? Quite simply, we made them up. These are subjective probabilities, what we think or feel will happen. Objective probability comes, in part, from understanding large numbers representing what has happened. Five outcomes is certainly not a large number of possible outcomes, but we are approaching these ideas in increments. Multiplying the payoffs and the probabilities together and adding them up (the ‘‘dot product’’ of two vectors in matrix algebra), we arrive at the expectation of 64.25 in Table 5-2, making the utility of this expectation, based on our original utility function LogðE½payoffsÞ ¼ Logð64:25Þ¼4:16278 In Table 5-3 we compute the utility of each payoff and compute their expectation to be 4.07608 to conclude, not surprisingly, that the utility of the expectation is greater than the expectation of the utility. U½EðwÞ > E½UðwÞ So far we have been working with discrete outcomes matched by given probabilities. In this, we claim to know the range of possibilities represented by a discrete probability distribution. The claim that we know these precise probabilities is ambitious to say the least. TABLE 5-2 Expected Value of Five Payoffs Payoffs Probabilities Products Payoffs 35 0.15 5.25 65 0.25 16.25 20 0.10 2.00 80 0.45 36.00 95 0.05 4.75 Expected value 64.25 Chance: Risk in General 111 File: {Elsevier}Brown/Revises-II/3d/Brown-ch005.3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22.12.2004/3:23pm Page: 112/118 THE CONTINUOUS NORMAL CASE The final step is to imagine a very large number of possible payoffs and an equally large number of associated specific probabilities. What can one say about those circumstances? The limit of this question is the notion of a probability distribution and the related concept of a probability density function (pdf). Recall from Chapter 3 (see Figures 3-7 and 3-8) that pdfs arise from histograms, which are merely ordered groups of outcomes. In this case we assume that we know the result of investing in many buildings (the payoffs) by many investors each with different utility functions and coefficients of risk aversion. 6 Properly arranged and processed, such data would produce a pdf. Alternatively, if we choose a convenient specific distributional form, we can postulate that a large number of payoffs resulting from an equally large number of associated probabilities would produce outcomes such as those described below. One can specify a pdf (when one exists) for a continuous variable when one knows something about the distribution. In our case, the variable of interest is the different wealth result, w, arising from undertaking different propositions with uncertain outcomes. A frequent choice for a convenient specific distributional form is the normal distribution because it can be completely described if one only knows its first two moments, its mean and its variance. 7 So we assume we know these two parameters and, therefore, its shape. (Beware: this claim is a little less ambitious than the one we made above in the discrete case, but it still requires a leap of faith.) Initially, we will assume our distribution of a very large number of wealth outcomes has a TABLE 5-3 Expected Utility of Five Payoffs U(Payoffs) Probabilities Products U(Payoffs) 3.55535 0.15 0.53330 4.17439 0.25 1.04360 2.99573 0.10 0.29957 4.38203 0.45 1.97191 4.55388 0.05 0.22769 Expected value 4.07608 6 Perhaps a better characterization is investing in the same building a large number of times. 7 Or the standard deviation, which is the square root of the variance. As the square root transformation is monotonic, it does not matter which is used. The reader is asked to tolerate the rocky motion of moving back and forth between them, something that is unfortunately too common in texts on this subject. 112 Private Real Estate Investment File: {Elsevier}Brown/Revises-II/3d/Brown-ch005.3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22.12.2004/3:23pm Page: 113/118 mean of $1,000,000 and a standard deviation of $200,000. Equation (5-6) defines the normal pdf for this distribution. e À wÀ1000000ðÞ 2 80000000000 200000 ffiffiffiffiffiffi 2p p ð5-6Þ In the case of discrete outcomes, the expectation is the result of simply multiplying the outcome times the probabilities and adding up the products as we did in Tables 5-2 and 5-3. For continuous variables, the expectation is the mean. Having the pdf we can plot this function and its mean in Figure 5-8. The amount of probability mass around the mean but away from the mean represents the variance from our expectation, loosely the probability that we are wrong. Imagine the converse, a certain outcome, something guaranteed to happen without fail, such as U.S. Treasury Bills. The outcome would ALWAYS match our expectation, there would be no variance, and the entire distribution would be the straight line in the middle of Figure 5-8, a single value. But if an opportunity has uncertain outcomes, we must allow for outcomes that do not match our expectations, some better and some worse, that aggregate around the expectation. So we begin to think about risk in terms of the shape of a function, in which we have a field of possibilities sprinkled about a line called the expectation. The distribution is shaped in a way that it ‘‘peaks’’ at one (and only one) point. The area of the field is expressed graphically in Figure 5-8 for the normal distribution as a plot of its pdf. Wealth Distribution FIGURE 5-8 Wealth distribution where m ¼ $1,000,000 and s ¼ $200,000. Chance: Risk in General 113 File: {Elsevier}Brown/Revises-II/3d/Brown-ch005.3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22.12.2004/3:23pm Page: 114/118 Alternatively, suppose we had a second game, also having normally distributed outcomes but with different parameters. The critical change is that, while both have the same mean, the second proposition has a larger standard deviation of $250,000. The normal pdf for this distribution is shown as Equation (5-7). e À wÀ1000000ðÞ 2 80000000000 250000 ffiffiffiffiffiffi 2p p ð5-7Þ Plotting both distributions together in Figure 5-9 shows that the shapes, while similar, are different. The difference in these shapes means that the spread of outcomes away from our expectation is different. Therefore, the risk is different. Recall in Chapter 4 that we had an investment which when modeled in different ways still produced substantially the same 13% IRR. We postulated that the two IRRs, while quantitatively the same, were qualitatively different. We pondered how they were different from the standpoint of risk. With the introduction of a probability distribution, we move closer to answering that question. We have discussed risk tolerance and utility. We now judiciously combine these ideas with the notion of the distribution’s spread, more precisely, variance from expectation. Suppose we have an investor whose decisions about risky alternatives are based on a logarithmic utility function. How would such an investor decide between the two alternatives illustrated in Figure 5-9? Examine the figure Wealth Distribution pdw2 (σ = 250,000) pdw1 (σ = 200,000) FIGURE 5-9 Distributions with different standard deviations. 114 Private Real Estate Investment File: {Elsevier}Brown/Revises-II/3d/Brown-ch005.3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22.12.2004/3:23pm Page: 115/118 closely and notice the differences. One has a higher peak. The one with the lower peak has wider ‘‘shoulders’’ and spreads out more at the base. Before we become too tangled in the mathematics, let us step back and remind ourselves that the notion of ‘‘shape’’ assists us in understanding risk. It seems that we are interested in both the shape of the utility function and the shape of the distribution. Specifically, we want to know how much area underneath the curve is away from the mean and on what side of the mean. It is useful to keep the simple metaphor of shape in mind as we proceed. What follows proceed on the basis that the reader has some familiarity with transformations of random variables. Transformations can make an otherwise intractable problem manageable. At a minimum, one should know that certain transformations are ‘‘shape preserving.’’ Thus, after transformation, the shape of the distribution is the same. The simplest example is a linear transformation, discussed in Chapter 3, where multiplying a set of values by a constant and/or adding a constant merely rescales and re-centers the distribution. This is how a normal distribution is ‘‘standardized’’ into ‘‘standard normal,’’ where the mean is zero and the variance is one. More generally, there are rules for transformations that must be adhered to and certain properties are essential. Numerous references (such as Hogg and Craig, 5th ed., p. 168, et seq) are available to fully elaborate this area. Returning to the importance of shapes, note that the (normal) symmetry of wealth distribution in Figure 5-9 is lost when transformed by the Log utility function. In Figure 5-10 the plot on the right shows a distinct left skew with the mode to the right of the mean. This is to be expected considering the shape of the pdf of the utility function. The question becomes: Is the investor better off with the investment having the first or second probability distribution? The same Expected Utility Hypothesis that resolved Bernoulli’s paradox provides the answer. Remember that the distributions differed only in the variance. We compute the expected Wealth Distribution Utility Distribution FIGURE 5-10 The shape of distribution of wealth and the shape of distribution of utility of wealth. Chance: Risk in General 115 File: {Elsevier}Brown/Revises-II/3d/Brown-ch005.3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22.12.2004/3:23pm Page: 116/118 utility of each by integrating the product of the utility function and its probability distribution (this integration is the definition of expectation for continuous variables). The computation of these results in values as shown in Table 5-4. The result, that the expected utility of the first distribution is slightly larger, is intuitively satisfying. One would expect, given identical means and specific form of the utility function we have chosen, that the distribution having the higher variance (risk) produces less expected utility. CONCLUSION This chapter lays the foundation upon which we must stand to begin our discussion of risk in real estate. One must appreciate how risk has been dealt with by others to understand how real estate risk differs. Most risk models in finance depend on the classical mathematics of binary probability (coin flipping) and its close cousin, the normal probability distribution. Much ground has been gained on the subject using these models. Important messages to be transferred into our thinking about real estate risk include:  Utility is a powerful way to express the consequences that arise from making choices.  By valuing different dollars differently, people make decisions on the margins. It is not average outcomes that count, but marginal outcomes.  The assumption that utility functions are concave is supported by considerable evidence. Thus, the shape of the utility function bears on the way people evaluate risk.  A closer look at the shape of utility functions discloses that different people see the same risk differently. Through a bidding process in the market, their aggregate behavior determines the price of risky assets.  Risk is a shape. Specifically, it is the shape of a probability distribution of wealth, a plot of numerous outcomes representing the realization of previously uncertain events. TABLE 5-4 Expected Utility for Two Different Wealth Distributions Mean Standard deviation Expected utility $1,000,000 $200,000 13.7937 $1,000,000 $250,000 13.7718 116 Private Real Estate Investment [...]... Genius Failed, The Rise and Fall of Long Term Capital Management New York: Random House 5 Nicholson, W (2002) Microeconomic Theory, 8th Edition New York: Thompson Learning CHAPTER 6 Uncertainty: Risk in Real Estate Real Estate is NOT Normal: A fresh look at real estate return distributions Article by Michael S Young and Richard A Graff published in the Journal of Real Estate Finance and Economics in 1995... plentiful data for real estate, we must be careful not to carelessly borrow these financial models just because they are in place and work elsewhere Anyone who has owned both real estate and stock knows that the each involve very different risks We argue that real estate risk is structurally different Whether that means we abandon classical finance theory in favor of some other remains to be seen 128 Private. .. epigram for this chapter 119 120 Private Real Estate Investment makes as much a statement about the participants in the real estate market as about the returns their investments generate Indeed, many of the ideas in this book tend toward that sentiment While it may appear unflattering at first to suggest that individual real estate investors are not normal, private real estate investors may prefer that... concepts of determinism and uncertainty, and discuss how risk fits into those ideas  Propose an enhancement to classical risk theory that fits private real estate investment  Discuss the way data now available for Tier II property may be used to empirically test the models discussed NON-NORMALITY—HOW AND WHERE DOES IT FIT? Chapter 5 ended with questions people in the private real estate investment market... solution, and accommodation of a linear view of life is not without its drawbacks If private real estate is a non-linear, dynamic world, one must be prepared to grapple with daunting mathematical complexity That is the bad news The good news far offsets the bad The real world of both real estate and finance is complex There are times when closed form, analytical solutions that are valid across the entire real. .. DETERMINISM AND HOUSE PRICES We may let a few circles convince us of the theory represented by our circumference function But things are less certain when you add people and their real estate to the equation For instance, suppose you claim that there is a functional relationship between the square footage of a house and the price 132 Private Real Estate Investment TABLE 6-6 House Sale Data Square feet... Note that a ¼ 2 and b ¼ 0 for one of the plots in Figure 6-1 The other two parameters, g and d, may be, respectively, rescaled and shifted without affecting the shape of the distribution We use 5 Under the right conditions both the normal and the stable distributions have a common parameter, the mean 124 Private Real Estate Investment a = 2, b = 0 a = 1.5, b = 1 FIGURE 6-1 The normal and the heavy right... to explain the universe Real estate analysts should not be wedded to only one, especially one as Uncertainty: Risk in Real Estate 127 FIGURE 6-3 Weibull distribution: shape parameter (a) ¼ 15, scale parameter (b) ¼ 10 restrictive and naive as the normal, merely because that has been handed to them from the world of finance The choice of which distribution best describes real estate is yet to be discovered... these ideas to real estate At times the fit is quite good At other times it is quite poor The discriminating analyst must know which times are which and when to use the right tools In this chapter we will:  Extend the discussion of classical risk to a form more relevant to the market for private (Tier II) real estate investments  Explore distributions that may be more useful for Tier II real estate  Revisit... exist in their world They have also had plentiful data with which to examine the ramifications of this Tier II real estate, lacking such data, has had to make do with the more rudimentary tools of finance 3 As distinguished from ‘‘outliers,’’ which is the name often used for extreme observations that arise from errors in the data 122 Private Real Estate Investment metaphor, this suggests that 80% . {Elsevier}Brown/Revises-II/3d/Brown-ch0 06. 3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22.12.2004 /6: 58pm Page: 119/1 56 CHAPTER 6 Uncertainty: Risk in Real Estate Real Estate is NOT Normal: A fresh look at real estate. Distributions Expectation Expected utility Expected wealth (Â10 6 ) Normal (2, 0, 1, 10) 10 2.292 26 $9,897,3 06 Stable (1.5, 1, 1, 10) 10.8858 2. 360 94 $10 ,60 0,939 1 26 Private Real Estate Investment . the market for private (Tier II) real estate investments.  Explore distributions that may be more useful for Tier II real estate.  Revisit the concepts of determinism and uncertainty, and discuss