Private Real Estate Investment: Data Analysis and Decision Making_6

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Private Real Estate Investment: Data Analysis and Decision Making_6 107 Chance: Risk in General 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 $50,000. The y-axis of Figure 5-6 provides the answer. The question of how much to pay for an investment with an uncertain outcome is answered by placing a numerical value on the difference between the utility of the certain opportunity and the utility of the uncertain one. How do we do this in practice? To begin with, notice that the expectation of wealth in this fair game is zero. That is, the mathematical expectation is Beginning Wealth þ (probability of gain  winning payoff ) À (probability of 108 Private Real Estate Investment U [Wealth] 11.9184 11.5129 10.8198 Wealth 50000 100000 150000 FIGURE 5-6 Plotting utility of wealth against wealth. loss  amount of loss). Since the outcomes are equally probable, the probability of either event is 0.5, so we have Probability Payoff($) Change($) Begin End wealth ($) wealth ($) 0.5 (100,000) (50,000) 100,000 50,000  ¼ þ ¼ 0.5 100,000 50,000 100,000 150,000  ¼ þ ¼ Expectation 0 100,000 100,000 þ ¼ 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 À 11.5129 ¼ 0.4055, and the change in utility associated with an equivalent (in nominal terms) decrease in one’s wealth, 4 Such people are usually not people at all, but companies, namely insurance companies having unlimited life and access to capital. 109 Chance: Risk in General 11.9184 11.5129 11.3691 10.8198 50000 100000 150000 86603 FIGURE 5-7 Risk neutral and risk averse positions for u[w] ¼ Log[w]. 11.5129 À 10.8198 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 Fi ...