In the previous section, we introduced the concept of an expected utility function, and stated how people maximize their expected utility when faced with a decision involving outcomes with known probabilities. 20 thousands is 43 units to this individual. 17.3 that the utility of money income of Rs. 4500). It is assumed that the individual knows the probabilities of making or gaining money income in different situations. Expected utility is the standard framework for modeling investor choices. Disclaimer Copyright, Share Your Knowledge It is because of the attitude of risk aversion that many people insure against various kinds of risk such as burning down of a house, sudden illness of a severe nature, car accidence and also prefer jobs or occupations with stable income to jobs and occupations with uncertain income. Every utility function that is monotone decreasing with respect to the standard Rothschild-Stiglitz (or stochastic dominance) order of more risky is averse to mean- Now the expected utility from the new risky job is less than the utility of 55 from the present job with an assured income of Rs. 3,000) with certainty. In the first gamble, the degree of variability of outcome is less and therefore the risk is less and in the second gamble, the degree of variability is greater which makes it more risky. 15,000 (Note that in the risky job also, expected income is Rs. It follows from above that in case marginal utility of money income diminishes a person will avoid fair gambles. 30 thousands, his utility is 75 and with his lower income of 10 thousands his utility is 45. 30 thousands to him is 83. 1,000. This attitude of risk aversion can be explained with Neumann-Morgenstern method of measuring expected utility. Risk Aversion, Certainty Equivalent, and Risk Premium If preferences satisfy the vNM axioms, risk aversion is completely characterized by concavity of the utility index and a non-negative risk-premium. There are four axioms of the expected utility theory that define a rational decision maker. 3000. RISK AVERSION AND EXPECTED-UTILITY THEORY: A CALIBRATION THEOREM BY MATTHEW RABIN1 1. The consumer is expected to be able to rank the items or outcomes in terms of preference, but the expected value will be conditioned by their probability of occurrence. 15,000 with no uncertainty is 55 whereas the expected utility of the new job or salesman on commission basis is 60. In the various earlier theories of consumer’s behaviour we saw that in making choices among commodity bundles when there is no risk and uncertainty, the consumer maximises his utility. 17.4. 10 thousands (that is, each has a probability of 0.5). . A fair game or gamble is one in which the expected value of income from a gamble is equal to the same amount of income with certainty. Therefore, the utility curve in Figure 17.6 represents the case of a risk averter or the attitude of risk aversion. 10,000 whose utility to the individual is 40 units. Such a person is called risk averter as he prefers an income with certainty (i.e., whose variability or risk is zero) to the gamble with the same expected value (where variability or risk is greater than zero). 3,000. It may be noted that marginal utility of income of a risk-averter diminishes as his income increases. We also learn that people are risk averse, risk neutral, or risk seeking (loving). 4000) + 1 – π U (Rs. In case of risk-neutral individual marginal utility of money remains constant as he has more money. It is seen from above that in case of risk-neutral person expected utility of an uncertain income with the same expected value (Rs. For an expected-utility maximizer with a utility function u, this implies that, for any lottery z˜ and for any initial wealth w, Eu(w +˜z) u(w +Ez).˜ (1.2) An individual’s money income represents the market basket of goods that he can buy. Specifying Risk-Aversion through a Utility function We seek a \valuation formula" for the amount we’d pay that: Increases one-to-one with the Mean of the outcome Decreases as the Variance of the outcome (i.e.. Risk) increases ... To maximize Expected Utility of Wealth W = W 1 (at time t = 1) 17.5. Suppose the individual is currently employed on a fixed monthly salary basis of Rs. 1,500. Now, suppose that the individual is considering to join a new job of a salesman on a commission basis. 30 thousands or Rs. Welcome to EconomicsDiscussion.net! Certainty equivalents are defined. With money income of Rs. C. Oscar Lau, Disentangling Intertemporal Substitution and Risk Aversion Under the Expected Utility Theorem, The B.E. This website includes study notes, research papers, essays, articles and other allied information submitted by visitors like YOU. 17.7. As shall be explained below, for a risk averse individual marginal utility of money diminishes as he has more money, while for a risk-seeker marginal utility of money increases as money with him increases. This is because if he proves to be a successful salesman his income may increase to Rs. 15,000 [E(x) = 0.5 x 0 + 0.5 x 30,000 = 15000], Note again that Figure 17.3 we are considering the choice of a risk averse individual for whom marginal utility of money declines as he has more of it. 17.3 that as money income of the individual increases from 10 to 20 thousand rupees, his total utility increases from 45 units to 65 (that is, by 20 units) and when his money increases from 20 thousand to 30 thousand rupees, his total utility increases from 65 to 75 units (that is, by 10 units). To explain the attitude toward risk we will consider a single composite commodity, namely, money income. 45. Given that the probability of success or failure as a salesman is 0.5, the expected utility of the new job is given by. expected utility questions differentiate between the following terms/concepts: prospect and probability distribution risk and uncertainty utility function and Suppose to our person with a certain income of Rs. The gain in utility from Rs. Though the individuals is risk-averse as revealed by the nature of his utility function of money income, but since the expected utility of the risky job is greater than the utility of the present job with a certain income he will choose the risky job. 17.3 we have drawn a curve OU showing utility function of money income of an individual who is risk-averse. Under expected utility, risk aversion in the Arrow-Pratt sense implies rejection of gambles with mean-independent risk. KÛ^áî٘ä3h=kßv$“ó™€Ó9Ã.®»Ž:„M“([!¤ð›ò{òí-;?ÍDË)«˜Meëé[ i§Ì A per­son is given the choice be­tween two sce­nar­ios, one with a guar­an­teed pay­off and one with­out. Suppose in this new job there is 50-50, chance of either earning Rs. Proposition Suppose % has an expected utility representation and v is the corresponding von Content Guidelines 2. If he wins the game, his income will rise to Rs. Though the expected value of his uncertain income prospect is equal to his income with certainty a risk averter will not accept the gamble. We saw earlier that in a certain world, people like to maximize utility. A per­son is said to be: 1. risk-aver… Expected Utility and Risk Aversion – Solutions First a recap from the question we considered last week (September 23), namely repre-senting in the probability triangle diagram the version of the Allais paradox we came across in the questionnaire. We are now in a position to provide a precise definition of risk-averse individual. In conventional expected utility theory, risk aversion comes solely from the concavity of a person’s utility defined over wealth levels. Since the expected utility from the new risky job is 51.5 which is greater than the utility of 43 from the present job with a certain income of Rs. Several functional forms often used for utility functions are expressed in terms of these measures. 1,000 in case he wins is less than the loss in utility from Rs. With the even chance of winning and losing the expected value of income in the second gamble will be 1/2(1500) + 1/2 (4500) = Rs. Most individuals generally prefer the less risky situation (that is, the situation with less variability in outcomes or rewards). 2,000 if he loses) can be obtained as under: Expected Utility (EU) = π U (Rs. This chapter examines individual attitudes toward risk, risk aversion, and decision making under risk and describes the expected utility theory as a model of choice under uncertainty. a risk-averse agent always prefers receiving the expected outcome of a lottery with certainty, rather than the lottery itself. 3,000 and he is offered a fair gamble in which he has a 50-50 chance of winning or losing Rs. 3 Risk-Weighted Expected Utility Theory 3.1 Risk-weighted expected utility versus expected utility 3.2 Problems with risk-weighted expected utility theory. People’s preferences toward risk greatly differ. Share Your PPT File, Risk Aversion and Insurance (Explained With Diagram). On the N- M utility curve U (I) in Figure 17.6 we draw a straight line segment GH joining point G (corresponding to income of Rs. But the outcomes or payoffs are measured in terms of utility rather than rupees. In other words, most individuals seek to minimise risk and are called risk averter or risk averse. 10 thousands if he happens to be not so efficient in the new job with the equal probability of 0.5 in these two jobs, then the expected utility from the new job is given by. 4,000 if he wins and Rs. ), thedegeneratelotterythat placesprobabilityone on the mean of Fis (weakly) preferred to the lottery Fitself. In Figure 17.6 Neumann-Morgenstern utility function curve U (I) has been drawn. It will be seen from Fig. 3,000 with certainty. 2000). Suppose there is a $50-50$ chance that a risk-averse individual with a current wealth of $\$ 20,000$ will contact a debilitating disease and suffer a loss of $\$ 10,000$ a. A risk-averse person therefore prefers the income with certainty to any gamble with the same expected money value as the income with certainty. u(ai), is the Bernoulli utility function. In a world of uncertainty, it seems intuitive that individuals would maximize expected utility A construct to explain the level of satisfaction a person gets when faced with uncertain choices. With Rs. An individual’s money income represents the market basket of goods that he can buy. It will be seen from this straight-line segment GH that the expected utility from the expected money value of Rs. Therefore. Expected utility is shown to imply second‐order risk aversion. 20 thousands, the risk-loving individual will prefer the new risky job even though the expected income in the new risky job is also Rs. Thus the person will prefer the first gamble which has lower variability to the second gamble which has a higher degree of variability of outcome. Privacy Policy3. Before publishing your Articles on this site, please read the following pages: 1. Now, if he is offered a risky job with his income of Rs. Therefore, the person will refuse to accept the gamble (that is, he will not gamble). 20,000 in the present case), is equal to utility of an assured or a certain income. 3000. Risk aversion and its equivalence with concavity of the utility function (Jensen’s inequality) are explained. Let us illustrate it with another example. In case of a risk-loving individual, marginal utility of income to the individual increases as his money income increases as shown by the convex total utility function curve OU in Fig. Share Your Word File • Expected utility allows people to compare gambles • Given two gambles, we assume people prefer the situation that generates the greatest expected utility – People maximize expected utility 18 Example • Job A: certain income of $50K • Job B: 50% chance of $10K and 50% chance of $90K • Expected income is the same ($50K) but in one case, 4,000 and if he loses the gamble, his income will fall to Rs. 15,000 but if he fails in his new risky job of a salesman on commission basis, his income falls to zero, then the expected utility of the risky job is given by. (Note that in the new risky job, the expected income is 20,000 which is given by E(X) = 0.5 x 10,000 + 0.5 x 30,000 = Rs. The ex­pected pay­off for both sce­nar­ios is $50, mean­ing that an in­di­vid­ual who was in­sen­si­tive to risk would not care whether they took the guar­an­teed pay­ment or the gam­ble. Thus with the present job with a fixed salary of Rs. It will be seen from this figure that the slope of total utility function OL; decreases as the money income of the individual increases. Suppose there is a $50-50$ chance that a risk-averse individual with a current wealth of $\$ 20,000$ will contact a debilitating disease and suffer a loss of $\$ 10,000$ a. 30 thousands if he happens to be highly efficient and Rs. This new job involves risk because his income in this case is not certain. 2,000 income, the person’s utility is 50 which rises to 70 when his income increases to Rs. This means, in turn, that even the It should be carefully noted that his rejection of gamble is due to diminishing marginal utility of money income for him. 4,000, his utility rises to 75. 2,000. Iftheindividualisalwaysindi fferentbetweenthesetwo lotteries, thenthenwesaytheindividualis risk neutral . It will be seen from this figure that N- M utility curve starts from the origin and has a positive slope throughout indicating that the individual prefers more income to less. In Bernoulli’s hypothesis we have seen that a person whose marginal utility of money declines will refuse to accept a fair gamble. And in case of income with certainty there is no variability of outcome and therefore involves no risk at all. There are multiple measures of the risk aversion expressed by a given utility function. Further, according to expected utility theory, risk aversion derives from the curvature of the utility of money, so such experiment would require to vary the stakes of the lotteries proposed in order to trace out the shape of the utility of money. But given the probabilities of alternative outcomes, we can calculate the expected utility. It is assumed that the individual knows the probabilities of making or gaining money income in different situations. 30 thousands if he proves to be a successful salesman, the utility of Rs. 20 while utility of Rs. In this section we focus on examining individual’s choices in the face of risk. We assume that there is equal probability of high and low income in the new risky job. The person who refuses a fair bet is said to be risk averse. Further the N-M utility curve shown in Figure 17.6 is concave which shows the marginal utility of income of a person diminishes as his income increases. As his income further increases to Rs. 1000 as before and the second a 50:50 chance of winning or losing Rs. Whether the individual will choose the new risky job or retain the present salaried job with a certain income can be known by comparing the expected utility from the new risky job with the utility of the current job. 15,000 with certainty is 55. 4 Risk Attitudes in the Jeffrey Framework 4.1 Linearity, chance neutrality, and risk aversion 4.2 Distinguishing risk attitudes 20 thousands is 80. The underlying principles of making a choice in risky and uncertain situation, namely, expected return and the degree of risk involved apply equally well to other choices. INTRODUCTION USING EXPECTED-UTILITY THEORY, economists model risk aversion as arising solely because the utility function over wealth is concave. Suppose that if the individual in his new job proves to be successful and earns Rs. The von Neumann–Morgenstern utility function can be used to explain risk-averse, risk-neutral, and risk-loving behaviour. 15000. Now, in a risky job when income increases to Rs. “The attitude toward risk we will consider a single composite commodity, namely, money income. 20,000 as (0.5 x 10,000) + 0.5 (30,000) = Rs. 15,000 (Note that in the risky job also, expected income is Rs. The comparison of risk aversion across agents is also examined. 70 which is the utility of income of Rs. Let us now slightly change the data. It will be seen from the utility function curve OU in Fig. 15,000 [E(x) = 0.5 x 0 + 0.5 x 30,000 = 15000], Note again that Figure 17.3 we are considering the choice of a risk averse individual for whom marginal utility of money declines as he has more of it. 4,000 is 75 (point B on the utility curve and utility from 2000 is 50 (point A in Figure 17.6), the expected utility from this uncertain prospect will be: In the N-M utility curve U (I) in Figure 17.6 the expected utility can be found by joining point A (corresponding to Rs. Further, in case of new risky job if he is proved to be a successful salesman and his income increases to Rs. In the un­cer­tain sce­nario, a coin is flipped to de­cide whether the per­son re­ceives $100 or noth­ing. When there is uncertainty, the individual does not know the actual utility from taking a particular action. Risk Lover On the other hand, a person is risk-preferred or risk-loving who prefers a risky outcome with the same expected income as a certain income. But it is important to note that these different preferences toward risk depend on whether for an individual marginal utility of money diminishes or increases or remains constant. Prudence coefficient and precautionary savingsPrudence coefficient and precautionary savings [DD5] 6.6. It will be seen from this figure that utility of a certain income of Rs. The expected money value of his income in this situation of uncertain outcome is given by: E (V) = 1/2 x 4000 + 1/2 x 2000 = Rs. It is risk-loving individuals who indulge in gambling, buy lotteries, engage in criminal activities such as robberies, big frauds even at risk of getting heavy punishment if caught. 17.4 that the utility of Rs. The total utility function of a risk neutral person is shown in Fig. Thus in this concave utility function depicted in Fig. So an expected utility function over a gamble g takes the form: u(g) = p1u(a1) + p2u(a2) + ... + pnu(an) where the utility function over the outcomes, i.e. In Fig. As mentioned above, most of the individuals are risk averse but there is a good deal of evidence of people who are risk seekers. a fundamental rule in statistics relating to conditional and marginal associated with each outcome 3 On the other hand, if in a new risky job, he proves to be a bad salesman, his income goes down to Rs. 3,000, the expected value of the utility is M2D (= 62.5) which is less than M2C or Rs. Risk aversion is equivalent to concavity of utility function if the expected utility theory holds. In the guar­an­teed sce­nario, the per­son re­ceives $50. An individual will be risk neutral if his marginal utility of money income remains constant with the increase in his money. 3,000, two fair gambles are offered to him. Journal of Theoretical … Risk aversion is the most common attitude toward risk. This is because as he acts on the basis of expected utility of his income in the uncertain situation (that is, Rs. As will be seen from Figure 17.6 the utility of the person from Rs. It will be seen from Fig. 20,000). How­ever, in­di­vid­u­als may have dif­fer­ent risk attitudes. 3.3. vNM vNM expected utility theoryexpected utility theory a)a) Intuition Intuition [L4] b) A i ti f d tiAxiomatic foundations [DD3] 4.4. 3,000 from the second gamble is M2L which is less than M2D of the first gamble. 30 thousands, his utility from Rs. The expected utility of the new risky job is given by. 30 thousand per month but if he does not happen to be a good salesman his income may go down to Rs. TOS4. Expected utility is introduced. 20,000). Risk-averse investors also are known as conservative investors. But the outcomes or payoffs are measured in terms of utility rather than rupees”. 30 thousands is 120 units. Ù8ؗzá’þ06ßzÍa[CÂÕ©ÀÙ. Now the expected utility from the new risky job is less than the utility of 55 from the present job with an assured income of Rs. 10 thousands to this individual is Rs. Note that we measure money income on the X-axis and utility on the Y-axis. That is, risk-neutral person is indifferent between them. The notion of local risk aversion is introduced in general and with respect to the expected utility case, where again it is equivalent to concavity of utility function. 2,000) and point B (corresponding to Rs. Thus, the probability of his winning is 1/2 or 0.5. First, a 50:50 chance of winning or losing Rs. 4,000) by a straight line segment AB and then reading a point on it corresponding to the expected value of the gamble Rs. The concepts of relative risk aversion, absolute risk aversion, and risk tolerance are introduced. 10 thousand per month. 20 thousands. Share Your PDF File Note that expected value of income in the new job with an uncertain income is 20,000 as (0.5 x 10,000 + 0.5 (30,000) = 20,000. In the questionnaire, Question 2 asked you to choose from a pair of lotteries A, B defined However, some individuals prefer risk and are therefore called risk-seekers or risk lovers. Now suppose the person’s current income is Rs. It will be seen from Fig. The decision made will also depend on the agent’s risk aversion and the utility of other agents. People differ greatly in their attitudes towards risk. 1000 if he loses the gamble. Suppose this risk-loving individual has a present job with a certain income of Rs. A person is called risk neutral, if he is indifferent between a certain given income and an uncertain income with the same expected value. Risk aversion coefficients and Risk aversion coefficients and pportfolio choice ortfolio choice [DD5,L4] 5. 30 thousands is 75, and if he fails as a good salesman, his income falls to Rs 10 thousands which yields him utility of Rs. Risk aversion is the most common attitude towards risk. 30,000, double the present assured income of Rs. That is why his expected utility from the uncertain income prospect has been found to be lower than the utility he obtains from the same income with certainty. In Bernoulli's formulation, this function was a logarithmic function, which is strictly concave, so that the decision-mak… Some other individuals are indifferent toward risk and are called risk-neutral. 1500) and H corresponding to income of Rs. 2 Consider the link between utility, risk aversion, and risk premia for particular assets. 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