Private Real Estate Investment: Data Analysis and Decision Making_6 pdf

Following the curved utility function, note thatthe difference between the change in utility associated with an increase inone’s wealth, 11.9184 11.5129 ¼ 0.4055, and the change in util

hierarchy We have purposely chosen extreme alternatives to illustrate ourpoint One needs a mechanism for thinking about risk in more realisticsettings when the alternatives may not be so obvious For instance, howwould we compare two commercial structures, one occupied by a majorclothing retailer and another by a major appliance retailer, or two similarapartment buildings on different sides of the street? Many such opportu-nities present themselves They have different risk, and while the differencemay not be great, there is a difference and one must be preferred over theother Our goal in this chapter is to discover a way of ranking riskyopportunities in a rational manner As is so often the case, ‘‘rational’’ meansmathematical.

THE ‘‘CERTAINTY EQUIVALENT’’ APPROACH

The search for a sound way to evaluate risky alternatives leads to an inquiryinto 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,’’ becomemore appealing as the entry fee is reduced (because the return increases) Theidea that describes this situation well is known as the certainty equivalent(CE) approach We ask an investor to choose a point of indifference betweenopportunities having a certain outcome and an uncertain outcome, given thatthe price of the opportunity with the uncertain outcome is sufficientlydiscounted

Let us use a concrete example to illustrate the concept Suppose someonehas $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 ourinvestor away from this certain position and into an investment with anuncertain outcome In Figure 5-6 we see the plot of utility of these uncertainoutcomes as wealth rises or falls Note the three points of interest,constituting the original wealth and the two outcomes Our investor mustdecide if the gain in utility associated with winning $50,000 is more or lessthan the loss of utility associated with losing $50,000 The y-axis of Figure 5-6provides the answer

The question of how much to pay for an investment with an uncertainoutcome is answered by placing a numerical value on the difference betweenthe 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 ofwealth in this fair game is zero That is, the mathematical expectation isBeginning Wealth þ (probability of gain  winning payoff )  (probability of

loss  amount of loss) Since the outcomes are equally probable, theprobability 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

When comparing the two curves in Figure 5-7 we see that, relative to they-axis, they both pass through the same points on the x-axis representing thealternative outcomes But when they pass through initial wealth, they generatedifferent values on the y-axis Following the curved utility function, note thatthe difference between the change in utility associated with an increase inone’s wealth, 11.9184  11.5129 ¼ 0.4055, and the change in utilityassociated with an equivalent (in nominal terms) decrease in one’s wealth,

10.8198

11.512911.9184

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.

11.5129  10.8198 ¼ 0.6931, shows that the lost utility associated withlosing $50,000 is greater than the utility gained by winning $50,000.5The conclusion we reach is that in order to be compensated for bearing riskour investor must be offered the opportunity to pay less than the rawexpectation ($100,000) This is reasonable Why would someone who alreadyhas $100,000 pay $100,000 for a 50/50 chance to lose some of it, knowingthat in a large number of trials he can do no better than break even? FromFigure 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 riskyopportunity is priced at $86,603 Certainty equivalent is a way of saying,that the investor is indifferent between paying $86,603 for the 50/50opportunity 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 theutility of wealth as shown on the y-axis of the plot is

Certainty Equivalent ¼ E½uðwÞ ¼ 0:5uð50000Þ þ 0:5uð150000Þ

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).

And we know that number is produced in Equation (5-4) by a function wehave chosen u ¼ Log(w) Thus, we solve for the known value of u by

‘‘exponentiating’’ both sides of Equation (5-4)

Doing this to the left side of Equation (5-4) eliminates the Log functionand leaves the certainty equivalent wealth as the unknown Doing it to theright side of Equation (5-4) leaves e11.3691, which is easily evaluated using

a calculator because e is just a number, a constant approximately equal to2.71828

eLog½CE¼e11:3691¼2:7182811:3691¼86,603 ð5-5Þ

The difference between $86,603 and $100,000, $13,397, is the discountthe investor applies to the raw expectation, given his specific preference forrisk as represented by the shape of his utility function Stated differently, thediscount is the compensation he requires to accept a prospect involving thissort of risk When a real estate broker asks his client to take money out of asavings account to buy an apartment building, it is the discount and itsassociated prospect of a higher return on the net invested funds that motivatesthe buyer to act

Two final points are useful before we move on Not only is the concavity ofthe 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 about5% The conclusion one might reach is that risk aversion is relative to bothone’s initial wealth and the portion of that wealth at stake in an uncertainsituation This mathematically supports sage advice that one should not betmore than one can afford to lose

A concave utility function means that people value different dollarsdifferently Various microeconomic texts consider other utility functionssuch as those illustrated in Figure 5-4 and develop a ‘‘coefficient of riskaversion’’ to tell us how much differently those dollars are valued by differentpeople having different risk tolerance This has important implications for themarket for uncertain investments Such a market commands higher prices ifpopulated by people with low coefficients of risk aversion, as they requiresmaller discounts

MULTIPLE (MORE THAN TWO) OUTCOMES

Returning to our first utility function (u[w] ¼ Log[w]), we can extend thisresult to more than two outcomes, each with different probabilities InTable 5-2 we define some payoffs under different conditions (numbers canrepresent thousands or millions of dollars to make them more realistic) Wethen associate a specific probability with each payoff Note the important factthat the probabilities add up to 1 Where did these probabilities come from?Quite simply, we made them up These are subjective probabilities, what wethink or feel will happen Objective probability comes, in part, fromunderstanding large numbers representing what has happened Five outcomes

is certainly not a large number of possible outcomes, but we are approachingthese 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 theexpectation of 64.25 in Table 5-2, making the utility of this expectation, based

on our original utility function

LogðE½payoffsÞ ¼ Logð64:25Þ ¼ 4:16278

In Table 5-3 we compute the utility of each payoff and compute theirexpectation to be 4.07608 to conclude, not surprisingly, that the utility of theexpectation is greater than the expectation of the utility

TABLE 5-2 Expected Value of Five Payoffs

Payoffs Probabilities Products

THE CONTINUOUS NORMAL CASE

The final step is to imagine a very large number of possible payoffs and anequally large number of associated specific probabilities What can one sayabout those circumstances? The limit of this question is the notion of aprobability distribution and the related concept of a probability densityfunction (pdf) Recall from Chapter 3 (see Figures 3-7 and 3-8) that pdfs arisefrom histograms, which are merely ordered groups of outcomes In this case

we assume that we know the result of investing in many buildings (thepayoffs) by many investors each with different utility functions andcoefficients of risk aversion.6 Properly arranged and processed, such datawould produce a pdf Alternatively, if we choose a convenient specificdistributional form, we can postulate that a large number of payoffs resultingfrom an equally large number of associated probabilities would produceoutcomes such as those described below

One can specify a pdf (when one exists) for a continuous variable whenone knows something about the distribution In our case, the variable ofinterest is the different wealth result, w, arising from undertaking differentpropositions with uncertain outcomes A frequent choice for a convenientspecific distributional form is the normal distribution because it can becompletely described if one only knows its first two moments, its mean and itsvariance.7 So we assume we know these two parameters and, therefore, itsshape (Beware: this claim is a little less ambitious than the one we madeabove in the discrete case, but it still requires a leap of faith.) Initially, we willassume 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.

mean of $1,000,000 and a standard deviation of $200,000 Equation (5-6)defines the normal pdf for this distribution.

of the shape of a function, in which we have a field of possibilities sprinkledabout 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 expressedgraphically 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.

Alternatively, suppose we had a second game, also having normallydistributed outcomes but with different parameters The critical change isthat, while both have the same mean, the second proposition has a largerstandard deviation of $250,000 The normal pdf for this distribution is shown

a probability distribution, we move closer to answering that question Wehave discussed risk tolerance and utility We now judiciously combine theseideas with the notion of the distribution’s spread, more precisely, variancefrom expectation

Suppose we have an investor whose decisions about risky alternatives arebased on a logarithmic utility function How would such an investor decidebetween the two alternatives illustrated in Figure 5-9? Examine the figure

closely and notice the differences One has a higher peak The one with thelower peak has wider ‘‘shoulders’’ and spreads out more at the base.

Before we become too tangled in the mathematics, let us step back andremind 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 andthe shape of the distribution Specifically, we want to know how much areaunderneath 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 withtransformations of random variables Transformations can make an otherwiseintractable problem manageable At a minimum, one should know thatcertain transformations are ‘‘shape preserving.’’ Thus, after transformation,the shape of the distribution is the same The simplest example is a lineartransformation, discussed in Chapter 3, where multiplying a set of values by aconstant and/or adding a constant merely rescales and re-centers thedistribution This is how a normal distribution is ‘‘standardized’’ into

‘‘standard normal,’’ where the mean is zero and the variance is one Moregenerally, there are rules for transformations that must be adhered to andcertain properties are essential Numerous references (such as Hogg andCraig, 5th ed., p 168, et seq) are available to fully elaborate this area.Returning to the importance of shapes, note that the (normal) symmetry ofwealth distribution in Figure 5-9 is lost when transformed by the Log utilityfunction In Figure 5-10 the plot on the right shows a distinct left skew withthe mode to the right of the mean This is to be expected considering theshape of the pdf of the utility function

The question becomes: Is the investor better off with the investment havingthe first or second probability distribution? The same Expected UtilityHypothesis that resolved Bernoulli’s paradox provides the answer Rememberthat 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.

utility of each by integrating the product of the utility function and itsprobability distribution (this integration is the definition of expectation forcontinuous variables) The computation of these results in values as shown inTable 5-4.

The result, that the expected utility of the first distribution is slightlylarger, is intuitively satisfying One would expect, given identical means andspecific form of the utility function we have chosen, that the distributionhaving the higher variance (risk) produces less expected utility

CONCLUSION

This chapter lays the foundation upon which we must stand to begin ourdiscussion of risk in real estate One must appreciate how risk has been dealtwith by others to understand how real estate risk differs Most risk models infinance depend on the classical mathematics of binary probability (coinflipping) and its close cousin, the normal probability distribution Muchground has been gained on the subject using these models Importantmessages to be transferred into our thinking about real estate risk include:

Utility is a powerful way to express the consequences that arise frommaking choices

By valuing different dollars differently, people make decisions on themargins It is not average outcomes that count, but marginal outcomes

The assumption that utility functions are concave is supported byconsiderable evidence Thus, the shape of the utility function bears onthe way people evaluate risk

A closer look at the shape of utility functions discloses that differentpeople see the same risk differently Through a bidding process in themarket, their aggregate behavior determines the price of risky assets

Risk is a shape Specifically, it is the shape of a probability distribution ofwealth, a plot of numerous outcomes representing the realization ofpreviously 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

The shape of the transformed probability distribution is related to theshape of one’s utility function through the mathematical expectationand the variance.

However, you say, empirical tests of these models have been conductedprimarily in the market for financial products In those markets one canjustify using tools based on random outcomes (stock price changes either up

or down out of control of the investor) To a good approximation theseoutcomes may be modeled as continuous A harsh position might claim thatthe stock market can be modeled with tools based on gambling because thestock market looks a lot like gambling to some people

The ‘‘some people’’ who might make this uncharitable characterization arevery often real estate investors At this point they might rightfully ask: ‘‘Whatabout us? How should we model risk? These shapes don’t look much like ourworld.’’ These people contend that their market is neither random norcontinuous It may be that it is neither linear nor static (few things are,including the stock market) Accordingly, the risk they face is a very differentkind of risk

After a long slog through the thicket of abstract utility, it is to those peopleand their questions that we now turn

Article by Michael S Young and Richard A Graff

published in the Journal of Real EstateFinance and Economics in 1995

INTRODUCTION

Having laid the foundation for thinking about risk in general terms, our task

is the adaptation of these ideas to real estate At times the fit is quite good Atother times it is quite poor The discriminating analyst must know whichtimes 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 themarket 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 howrisk fits into those ideas

Propose an enhancement to classical risk theory that fits private realestate investment

Discuss the way data now available for Tier II property may be used toempirically test the models discussed

NON-NORMALITY—HOW AND WHERE DOES IT FIT?

Chapter 5 ended with questions people in the private real estate investmentmarket might pose It is tempting to claim that the epigram for this chapter

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