Probability Models for Economic Decisions: Utility Theory with Constant Risk Tolerance

PROBABILITY MODELS FOR ECONOMIC DECISIONS excerpts from Chapter 3: Utility Theory with Constant Risk Tolerance 3.1. Taking account of risk aversion: utility analysis with probabilities In the decision analysis literature, a decision-maker is called risk-neutral if he (or she) is willing to base his decisions purely on the criterion of maximizing the expected value of his monetary income. The criterion of maximizing expected monetary value is so simple to work with that it is often used as a convenient guide to decision-making even by people who are not perfectly risk neutral. But in many situations, people feel that comparing gambles only on the basis of expected monetary values would take insufficient account of their aversion to risks. For example, imagine that you had a lottery ticket that would pay you either $20,000 or $0, each with probability 1/2. If you are risk neutral, then you should be unwilling to sell this ticket for any amount of money less than its expected value, which is $10,000. But many risk averse people might be very glad to exchange this risky lottery for a certain payment of $9000. Given any such lottery or gamble that promises to pay you an amount of money that will be drawn randomly from some probability distribution, a decision-maker's certainty equivalent (abbreviated CE) of this gamble is the lowest amount of money-for-certain that the decisionmaker would be willing to accept instead of a gamble. That is, saying that $7000 is your certainty equivalent of the lottery that would pay you either $20,000 or $0, each with probability 1/2, means that you would be just indifferent between having a ticket to this lottery or having $7000 cash in hand. In these terms, a risk-neutral person is one whose certainty equivalent of any gamble is just equal to its expected monetary value (abbreviated EMV). A person is risk averse if his or her certainty equivalent of a gamble is less than the gamble's expected monetary value. The difference between the expected monetary value of a gamble and a risk-averse decision-maker's certainty equivalent of the gamble is called the decision-maker's risk premium (abbreviated RP) for the gamble. Thus, RP = EMV!CE. So if a lottery paying $20,000 or $0, each with probability 1/2, is worth $7000 to you, then your risk premium for this lottery is $10,000!$7000 = $3000. When you have a choice among various gambles, you should choose the one for which you have the highest certainty equivalent, because it is the one that is worth the most to you. But 1 when a gamble is complicated, you may find it difficult to assess your certainty equivalent for it. The great appeal of the risk-neutrality assumption is that, by identifying your certainty equivalent with the expected monetary value, it makes your certainty equivalent something that is straightforward to compute or estimate by simulation. So what we need now is to find more general formulas that risk-averse decision-makers can use to compute their certainty equivalents for complex gambles and monetary risks. A realistic way of calculating certainty equivalents must include some way of taking account of a decision-maker's personal willingness to take risks. The full diversity of formulas that a rational decision-maker might use to calculate certainty equivalents is described by a branch of economics called utility theory. Utility theory generalizes the principle of expected value maximization in a simple but very versatile way. Instead of assuming that people want to maximize their expected monetary values, utility theory instead assumes that each individual has a personal utility function that assigns a utility value to every possible monetary income level that the individual might receive, such that the individual always wants to maximize the expected value of his or her utility. For example, suppose that you have to choose among two gambles, where the random variable X denotes the amount of money that you would get from the first gamble and the random variable Y denotes the amount of money that you would get from the second gamble. A risk-neutral decision-maker would prefer the first gamble if E(X) > E(Y). But according to utility theory, when U(x) denotes your "utility" for getting any amount of money x, you should prefer the first gamble if E(U(X)) > E(U(Y)). (Recall from Chapter 2 that, when X has a discrete distribution, the expected value operator E(C) can be written E(U(X)) = 3x P(X=x)*U(x), where the summation range includes every number x that is a possible value of the random variable X.) Furthermore, your certainty equivalent CE of the gamble that will pay the random monetary amount X should be the amount of money that gives you the same utility as the expected utility of the gamble. Thus, we have the basic equation U(CE) = E(U(X)). Utility theory can account for risk aversion, but it also is consistent with risk neutrality or even risk-seeking behavior, depending on the shape of the utility function. In 1947, von Neumann and Morgenstern gave an ingenious argument to show that any consistent rational decision maker should choose among risky gambles according to utility theory. Since then, 2 decision analysts have developed techniques to assess individuals' utility functions. Such assessment can be difficult, because people have difficulty thinking about decisions under uncertainty, and because there are so many possible utility functions. But we may simplify the process of assessing a decision-maker's personal utility function if we can assume that the utility function is in some mathematically natural family of utility functions. For practical decision analysis, the most convenient utility functions to use are those that have a special property called constant risk tolerance. Constant risk tolerance means that, if we change a gamble by adding a fixed additional amount of money to the decision-maker's payoff in all possible outcomes of the gamble, then the certainty equivalent of the gamble should increase by this same amount. This assumption of constant risk tolerance is very convenient in practical decision analysis. One nice consequence of constant risk tolerance is that it allows us to evaluate independent gambles separately. If you have constant risk tolerance then, when you are going to earn money from two independent gambles, your certainty equivalent for the sum of the two independent gambles is just the sum of your certainty equivalent for each gamble by itself. That is, the lowest price at which you would be willing to sell a gamble is not affected by having other independent gambles. This independence property only holds if you have one of these constantrisk-tolerance utility functions. If a risk-averse decision-maker's preferences over gambles satisfy this assumption of constant risk tolerance, then the decision-maker must have a utility function U in a simple oneparameter family of functions that are defined by the mathematical formula: U(x) = !EXP(!x'J), where the parameter J is called the risk-tolerance constant. (Here EXP is a standard function in Excel.) Thus, if we can assume that a decision-maker has constant risk tolerance, then we only need to measure this one risk-tolerance parameter J for the decision-maker. By asking the decision-maker to subjectively assess his or her personal certainty equivalent for one simple gamble, we can get enough information to compute the risk tolerance that accounts for this personal assessment. Then, once we have found an appropriate risk tolerance for this decisionmaker, we will be able to use it to estimate the decision-maker's expected utility and certainty equivalent for any gamble that we can simulate, no matter how complex. 3 It is natural that this numerical measure of "risk tolerance" may seem mysterious at first. The meaning of these risk-tolerance numbers will become clearer as you get practical experience using them. Utility, with RiskTol = 36.49 A B C D E F G H 1 RiskTolerance 2 36.49 * Monetary Income EU U(CE) 3 -20 -10 0 10 20 30 40 50 60 70 -0.9466642 -0.9466642 4 .0 5 -.2 $X UTIL(X,RT) UINV(U,RT) 6 -20 -20 -1.7299782 7 -.4 -15 -15 -1.5084485 8 -.6 -10 -10 -1.3152864 9 -.8 -5 -5 -1.1468593 10 -1.0 0 0 -1 11 5 5 -0.8719465 12 -1.2 10 10 -0.7602907 13 -1.4 15 15 -0.6629328 14 -1.6 20 20 -0.578042 15 -1.8 25 25 -0.5040217 16 30 30 -0.4394799 17 -2.0 35 35 -0.383203 18 40 40 -0.3341325 19 45 FORMULAS 45 -0.2913457 20 50 A2. =RISKTOL(20,-10,2) 50 -0.2540378 21 55 B4. =-EXP(-(20)/A2)*0.5+-EXP(-(-10)/A2)*0.5 55 -0.2215074 22 60 C4. =-EXP(-(2)/A2) 60 -0.1931426 23 65 B7. =-EXP(-A7/$A$2) 65 -0.16841 24 70 C7. =-$A$2*LN(-B7) 70 -0.1468445 25 B7:C7 copied to B7:C25. 26 Chart plots (A7:A25,B7:B25). 27 28 29 *Uses add-in Simtools.xla, available at http://home.uchicago.edu/~rmyerson/addins.htm 30 Figure 3.1. A utility function with constant risk tolerance. ...To avoid the confusion of trying to interpret expected utility numbers, we should always convert them back into monetary units by asking what sure amount of money would also yield this same expected utility. This amount of money is then the certainty equivalent of the lottery. Recall basic certainty-equivalent formula U(CE) = EU. With constant risk tolerance J, the utility of the certainty equivalent becomes U(CE) = !EXP(!CE'J). So the certainty equivalent satisfies !EXP(!CE'J) = EU. But the inverse of the EXP function is the natural logarithm function LN(). So with constant risk tolerance J, the certainty equivalent of a gamble can be computed from its expected utility by the formula CE = !J*LN(!E(U(X))). 4 3.4. Note on foundations of utility theory John Von Neumann and Oskar Morgenstern showed in 1947 how simple rationality assumptions could imply that any decision-makers' risk preferences should be consistent with utility theory. In this section, we discuss a simplified version of their argument, which justifies our use of utility theory. To illustrate the logic of the argument, let us begin with an example. Suppose that we want to study a decision-maker's preferences among gambles that involve possible prizes between (say) $0 and $5000. We might start by asking him whether he would prefer to get $1000 for sure or a lottery ticket that would pay $5000 with probability 0.20 and would pay $0 otherwise. If the decision-maker were risk-neutral, he would express indifference between these two alternatives (as they both have expected monetary value $1000), but suppose that our decision-maker expresses a clear preference for the sure $1000 over the lottery ticket. Next, we might ask him whether he would prefer the sure $1000 or the lottery ticket that pays either $5000 or $0 if its probability of paying $5000 were increased to 0.30. Suppose that the decision-maker now responded that he would be willing to give up a sure $1000 for such a lottery ticket. So somewhere between 0.20 and 0.30 there should be some number p such that this decision-maker would be indifferent between getting $1000 for sure and getting a lottery ticket that pays $5000 with probability p and pays $0 with probability 1!p. Suppose we ask the decision-maker to think about such lotteries and tell us what probability p would make him just indifferent, and he tells us (perhaps after some long thought) that he would be just indifferent between $1000 for sure and a lottery ticket that pays $5000 with probability 0.27 and pays $0 otherwise. Next, we might ask this decision-maker a similar question about $2000 for sure: "For what probability p would you be indifferent between getting $2000 for sure and a lottery ticket that would pay $5000 with probability p and $0 otherwise?" Suppose that, in answer to this question, the decision-maker says that he would be just indifferent between $2000 for sure and a lottery ticket that pays $5000 with probability 0.50 and pays $0 otherwise. Now consider the following two gambles Gamble 1 pays $2000 with probability 0.5 and pays $1000 otherwise. Gamble 2 pays $5000 with probability 0.4 and pays $0 otherwise. Given only the above information about this decision-maker, should we be able to predict which of these two gambles he should prefer? 5 The answer to this question is Yes. If his preferences are logically consistent, he should prefer Gamble 2 here. Let me explain why. Suppose we offered to exchange Gamble 1 for the following two-stage gamble: At stage 1 we toss a fair coin, and if the coin comes up Heads (which has probability 0.5) then we give him a lottery ticket that pays $5000 with probability 0.50 and pays $0 otherwise, but if the coin comes up Tails then we give him a lottery ticket that pays $5000 with probability 0.27 and pays $0 otherwise. He already told us that he would be indifferent between $2000 for sure and the first of these lottery tickets, and he would be indifferent between $1000 for sure and the second of these lottery tickets. So he should be just indifferent between Gamble 1 and this two-stage gamble. But in this two-stage gamble, his ultimate prize is either $5000 or $0, and his probability of getting $5000 is 0.5*0.50 + 0.5*0.27 = 0.385 (This calculation could be done with a probability tree, as in Section 1.6 of Chapter 1.) Gamble 2 also pays either $5000 or $0, and its probability of paying $5000 is 0.4, which is greater than 0.385. So the decision-maker should rationally prefer Gamble 2 over the two-stage gamble, which he should think is just as good as Gamble 1. Thus he should prefer Gamble 2 over Gamble 1 if he is logically consistent. This argument can be generalized to show that that logically consistent decision-makers should always satisfy utility theory. To formulate this general argument, we first need to introduce some basic notation. Let us consider a decision-maker who may have to choose among a variety of possible monetary lotteries or gambles. A gamble may be denoted here by a random variable (in boldface) that represents the unknown monetary value that the decision-maker would be paid from this gamble, if he or she accepted it. Given such a decision-maker and given any two gambles X and Y, we may write X™Y to denote proposition that "the decision-maker would strictly prefer the gamble X over the gamble Y." Similarly, we may write X~Y to denote the proposition that "the decision-maker would be indifferent between the gambles X and Y." Utility theory says that each decision-maker's preferences among all possible monetary gambles can be represented by some utility function U(C) such that, for any gambles X and Y, 6 X ™ Y when E(U(X)) > E(U(Y)), and X ~ Y when E(U(X)) = E(U(Y)). Thus, according to utility theory, once we know a decision-maker's utility function, then we can predict the decision-maker's preferences over all possible monetary gambles. According to utility theory, then, different people may have different preferences for taking and avoiding risks because they have different utility functions. To keep the argument simple, let us only consider lotteries where the outcome will be selected from some finite set of monetary prizes, which we may denote by {W1, W2, ..., Wn}. Let W1 denote the best of these possible outcomes, and let Wn denote the worst of these possible outcomes. Now, for any other possible prize Wi in this set, let us ask the decision-maker: "If you were comparing the alternatives of (1) getting Wi dollars for sure, and (2) a binary lottery in which you will get either the best prize W1 or the worst prize Wn, then what probability of getting the best prize W1 in this binary lottery would make you just indifferent between these two alternatives." Obviously, if the probability of the best prize were 1 then the binary lottery would be better; but if the probability of the best prize were 0 then the sure Wi would be better. So there should exist some probability of getting the best prize such that the decision-maker would be just indifferent between these two alternatives. Given the decision-maker's answer to this question, let Zi denote such a random variable that will be equal to either W1 or Wn with the probability distribution that the decision-maker considers just as good as Wi for sure. That is, let Zi be such that P(Zi=W1) + P(Zi=Wn) = 1, and Zi ~ Wi. Notice that we must have P(Zn=W1) = 0, because the worst prize Wn would be worse than any lottery that gave any positive probability of the best prize. Similarly, we must have P(Z1=W1) = 1. Now consider any two general lotteries X and Y that have outcomes drawn from this finite set of possible prizes {W1,W2,...,Wn}. Since the decision-maker is indifferent between 7 each Wi and the corresponding lottery Zi, he should not care if we substitute a ticket to the lottery Zi wherever he would have gotten the prize Wi. Repeating this substitution argument for each i from 1 to n, we conclude that the lottery X should be just as good, for this decision-maker, as a two-stage lottery in which the first stage gives a ticket to a lottery Z1 or Z2 or ... or Zn respectively with the probabilities P(X=W1), P(X=W2), ..., P(X=Wn). In this two-stage lottery, the final outcome will be either the best prize W1 or the worst prize Wn, and the probability of getting the best prize W1 is P(X=W1)*P(Z1=W1) + P(X=W2)*P(Z2=W1) + ... + P(X=Wn)*P(Zn=W1) So the lottery X should be just as good, for this decision-maker, as a lottery that pays either the best prize W1 or the worst prize Wn, where the probability of the best prize is 3ni=1 P(X=Wi)*P(Zi=W1). By a similar argument, the decision-maker should be indifferent lottery Y should be just as good, for this decision-maker,as a lottery that pays either the best prize W1 or the worst prize Wn, where the probability of the best prize is 3ni=1 P(Y=Wi)*P(Zi=W1). Among these two binary lotteries where the only possible outcomes are W1 or Wn, the better one is obviously the one with the higher probability of the best prize W1. So the decision-maker should think that X is better than Y if 3ni=1 P(X=Wi)*P(Zi=W1) > 3ni=1 P(Y=Wi)*P(Zi=W1). The trick now is to let U(Wi) = P(Zi=W1). Then we have just shown that X ™ Y whenever 3ni=1 P(X=Wi)*U(Wi) > 3ni=1 P(Y=Wi)*U(Wi). That is, the better gamble should always be the one with the higher expected utility. Thus, we have found a way of assigning utility values to prizes such that the decision-maker, if he is logically consistent, should always prefers the lottery with the higher expected utility. You may notice that the utility numbers U(Xi) that are generated by this argument are all between 0 and 1. However, there can also be valid utility functions in which the utility numbers are negative, or bigger than 1. In fact, we can multiply a utility function by any positive constant and add any other constant, and we will still have have an equally valid utility function: 8 Fact 1. Suppose that the decision-maker's preferences over all monetary gambles can be represented by a utility function U(C). Suppose that another function V(C) is an increasing linear transformation of U(C). That is, suppose that there exist two numbers A and B such that A>0 and V(x) = A*U(x) + B, for all x. Then V(C) is also a valid utility function for representing this decision-maker's preferences over all monetary gambles. Proof of Fact 1. For any two gambles X and Y, if E(U(X)) > E(U(Y)) then E(V(X)) = E(A*U(X)+B) = A*E(U(X)) + B > A*E(U(Y)) + B = E(A*U(Y)+B) = E(V(Y)). So comparing expectations of V and U both yield the same preference ordering over gambles. 9 BASIC FACTS ABOUT RISK AVERSION Question: Consider an individual whose expresses the risk preferences [$2000] ~ 0.50[$5000]+0.50[$0] and [$1000] ~ 0.27[$5000]+0.73[$0]. If this individual is rational (consistent), which should he prefer among the lotteries 0.5[$2000]+0.5[$1000] and 0.4[$5000]+0.6[$0]? Consider an individual with a twice-differentiable utility function u(C). ˜ Suppose this individual has wealth x plus a gamble that will pay a small random amount g, ˜ such that E(g) = 0. ˜ Let * be the maximum that he would pay to insure against this gamble. So u(x!*) = E(u(x+g)). Assuming that all possible values of g˜ are near 0, Taylor series approximations yield ˜ u(x) ! * uN(x) . E[u(x) + uN(x) g˜ + uNN(x) g˜ 2'2] = u(x) + uNN(x) Var(g)'2. ˜ and then * . ![uNN(x)'uN(x)] Var(g)'2. The individual's Arrow-Pratt risk-aversion index at wealth x is !uNN(x)'uN(x). So we find that the individual's value for a small zero-expected-value gamble is approximately half of the variance of the gamble times the individual's risk-aversion index. Notice that this risk-aversion index r(x) would not change if we changed how i's utility is ˆ measured to some other equivalent scale u(x) = Au(x)+B where A>0 and B are constants. The reciprocal of the risk aversion index J(x) = 1'r(x) is called the risk tolerance index, which has the advantage of being measured in the same units as money (dollars). An individual whose risk tolerance is constant, independent of his given wealth x, must have a utility function that satisfies the differential equation !uNN(x)'uN(x) = 1'J, for some constant J. This differential equation is equivalent to d[LN(uN(x))]'dx = !1'J. For J>0, the solutions to this differential equation with uN>0 are u(x) = B!Ae!x'J, where A>0 and B are arbitary scale constants. We may use B=0 and A=1, to get the canonical constant-risk-tolerance utility function u(x) = !e!x'J, whose inverse is x = !J LN(!u). Suppose an individual with constant risk tolerance J, with utility u(x) = !e!x'J, ˜ drawn from from some given probability distribution. will get a random income Y ˜ for this individual is the sure amount of money W The certainty equivalent (CE) of this gamble Y ˜ that individual would be willing to accept instead of this lottery Y. ˜ ˜ So the certainty equivalent CE satisfies u(CE) = E(u(Y)), and so CE = !J LN(!E(u(Y))) ˜ and Y ˜ are independent Fact: Suppose u(C) is a utility function with constant risk tolerance, Y 1 2 ˜ )) for each i in {1,2}. Then u(W +W ) = E(u(Y ˜ +Y ˜ )). random variables, and u(Wi) = E(u(Y i 1 2 1 2 ˜ is drawn from a Normal distribution Fact: For an individual with constant risk tolerance J, if Y ˜ certainty equivalent is CE = µ ! (0.5'J)F2. with mean µ and standard deviation F then Y's 2 That is, !e !(µ!(0.5/J)F )/J ' +4 m!4 2 !e !y/J e !0.5((y!µ)/F) 2B F dy . (Proof: use +4 m!4 2 2 e !0.5((y![µ!F /J])/F) 2B F dy ' 1 ) 10 EFFICIENT RISK SHARING IN A SYNDICATE Question: Individuals 1 and 2 have constant risk tolerances T1 = $20,000 and T2 = $30,000. 1 has an investment paying returns drawn from a Normal distribution: µ=$35,000, F=$25,000. 1 can sell any share 2 of this investment to 2 for any price up to her CE of this share. To maximize 1's total CE, what share 2 should 1 offer to sell, for what price? (60% for $17,250) Let N denote the set of member of a syndicate or investment partnership. ˜ They hold assets which will yield returns that are a random variable Y with some given probability distribution. Each individual i in N has a given utility function for money ui(C). ˜ Let xi(y) denote the individual i's planned payoff when the syndicate earns Y=y. To be feasible, we must have 3i0N xi(y) = y, œy0ú. ˜ Let us consider an efficient allocation rule (xi(C))i0N that maximizes 3i0N 8i E(ui(xi(Y))), subject to this feasibility constraint, for some given positive utility weights (8i)i0N. To simplify the characterization of the efficient rule, we assume all appropriate differentiability. For each outcome y, the marginal weighted utility 8iuiN(xi(y)) must be equal across all i in N. That is, there must exist some function <(y) such that 8i uiN(xi(y)) = <(y), œi0N, œy0ú (1). Differentiating with respect to y, we get 8i uiNN(xi(y)) Mxi'My = =0. D19. =AVERAGE(D6:D15) I2. =SUM(D2:E2) B6 copied to B6:B15 E6. =B6-D6 E19. =AVERAGE(E6:E15) J4. =D2/I2 D20. =STDEV(D6:D15) J6. =B6*$J$4 E6 copied to E6:E15 G6. =-EXP(-D6/D$2) E20. =STDEV(E6:E15) J6 copied to J6:J15 D18. =CE(D6:D15,D2) J18. =CE(J6:J15,$D$2) H6. =-EXP(-E6/E$2) G6:H6 copied to G6:H15 E18. =CE(E6:E15,E2) L6. =J6-$J$18 D21. =D19-(0.5/D2)*D20^2 L6 copied to L6:L15 G17. =AVERAGE(G6:G15) H17. =AVERAGE(H6:H15) E21. =E19-(0.5/E2)*E20^2 L18. =CE(L6:L15,$D$2) G18. =-D2*LN(-G17) J19. =A2*$J$4 H18. =-E2*LN(-H17) J20. =A3*$J$4 G22. =-EXP(-G18/D2)-G17 J21. =J19-(0.5/D2)*J20^2 H22. =-EXP(-H18/E2)-H17 *The CE function is from Simtools.xla, available at http://home.uchicago.edu/~rmyerson/addins.htm Figure: Optimal risk sharing. 13