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Chance: Risk in General 107 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 $50,000. The y-axis of Figure 5-6 provides the answer. The question of how much to pay for an investment with an uncertain outcome is answered by placing a numerical value on the difference between the utility of the certain opportunity and the utility of the uncertain one. How do we do this in practice? To begin with, notice that the expectation of wealth in this fair game is zero. That is, the mathematical expectation is Beginning Wealth þ (probability of gain winning payoff)ÿ(probability of 108 Private Real Estate Investment U [Wealth] 11.9184 11.5129 10.8198 50000 100000 150000 Wealth FIGURE 5-6 Plotting utility of wealth against wealth. loss amount of loss). Since the outcomes are equally probable, the probability of either event is 0.5, so we have Probability 0.5 0.5 Payoff($) (100,000) 100,000 Expectation Change($) ¼ (50,000) þ ¼ 50,000 þ 0 þ Begin wealth ($) 100,000 100,000 100,000 End wealth ($) ¼ 50,000 ¼ 150,000 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, 11.9184ÿ11.5129 ¼ 0.4055, and the change in utility associated with an equivalent (in nominal terms) decrease in one’s wealth, 4Such people are usually not people at all, but companies, namely insurance companies having unlimited life and access to capital. Chance: Risk in General 109 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]. 11.5129ÿ10.8198 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 E½uðwފ ¼ 0:5uð50000Þ þ 0:5uð150000Þ ¼ 11:3691 ð5-3Þ 5There 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). 110 Private Real Estate Investment 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). Log½CEŠ ¼ 11:3691 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 e11.3691, which is easily evaluated using a calculator because e is just a number, a constant approximately equal to 2.71828. eLog½CEŠ ¼ e11:3691 ¼ 2:7182811:3691 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. Chance: Risk in General 111 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 LogðE½payoffsŠÞ ¼ Logð64:25Þ ¼ 4:16278 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. U½Eðwފ > E½Uðwފ 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 65 20 80 95 Expected value 0.15 5.25 0.25 16.25 0.10 2.00 0.45 36.00 0.05 4.75 64.25 ... - tailieumienphi.vn
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