vnm utility function risk aversion

James Cox and Vjollca Sadiraj (2004, working paper) use both income and wealth as arguments for the VNM utility function. Very cool! Then u 2 = g u 1. more risk averse than Theorem: Given any two strictly increasing Bernoulli utility functions u and v, the following are equivalent (a) Au(x) ≥ Av(x) for all x (b) CEu(x) ≤ CEv(x) for all x (c) There exists a strictly increasing concave function g such that u = g v • In that case, we say that v is (weakly) more risk averse … This solution shows how to find the von Neumann-Morgenstern utility functions that displays constant measure of absolute risk-aversion (Arrow-Pratt measure) - CARA. Vickrey, William (1945): "Measuring Marginal Utility by Reactions to Risk". If a VNM utility function displays constant absolute risk aversion, so that Ra(w) = α for all w, what functional form must it have? Therefore, we can observedA dw> 0. Since, her utility function is concave, basically we can say, she is risk averse. Morgenstern (VNM) utility function in expected utility (EU) theory can only be derived either by assuming a cubic utility function or as an approx imation.2 Menezes et al. 400 18 Define expected utility (E [u (x)] X is the prize, the consumer values, and the expectation E is determined by the probabilities of the various states of nature. This has, in fact, become the traditional way in which the measure is used. L=0.25A+0.75B{\displaystyle L=0.25A+0.75B} de­notes a sce­nario where P(A) = 25% is the prob­a­bil­ity of A oc­cur­ring and P(B) = 75% (and ex­actly one of them will occur). x��V{L[U?�^ For example, a firm might, in one year, undertake a project that has particular probabilities for three possible payoffs of $10, $20, or $30; those probabilities are 20 percent, 50 percent, and 30 percent, respectively. Lecture 04 Risk Prefs & EU (34) • Risk-aversion means that the certainty equivalent is smaller than the expected prize. %PDF-1.4 %���� Theorem:More risk individuals hold less of the risky asset, other things being equal. 0000003022 00000 n (1980) seek to put skew ness preference on a firmer choice-theoretic footing by introducing the concept of increasing downside risk. Constant Absolute Risk-Aversion (CARA) Consider the Utility function U(x) = 1 e ax a for a 6= 0 Absolute Risk-Aversion A(x) = U 00(x) U0(x) = a a is called Coe cient of Constant Absolute Risk-Aversion (CARA) For a = 0, U(x) = x (meaning Risk-Neutral) If the random outcome x ˘N( ;˙2), E[U(x)] = 8 <: 1 e a + a 2˙ 2 a for a 6= 0 for a = 0 x CE = a˙2 2 (b) Pratt’s formula for the relative risk premium (p. 18, eq. The idea of John von Neumann and Oskar Mogernstern is that, if you behave a certain way, then it turns out you're maximizing the expected value of a particular function. ), thedegeneratelotterythat placesprobabilityone on the mean of Fis (weakly) preferred to the lottery Fitself. <<038594E7482D20478BCAC1275DF66F5C>]>> Vickrey, William (1961), "Counterspeculation, Auctions, and Competitive Sealed Tenders". If all the information we need about the curvature of a function is contained in its second derivative, shouldn't that be a sufficient measure of risk-aversion? A linear function has a second derivative of zero, a concave function has a negative second derivative, and a convex function has a positive second derivative. The risk aversion function can be derived from the Utility function. �gK[!�Z/�!��-J From the discussion on risk-aversion in the Basic Concepts section, we recall that a consumer with a von Neumann-Morgenstern utility function can be one of the following: Knowing this, it seems logical that the degree of risk-aversion a consumer displays would be related to the curvature of their Bernoulli utility function. 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. 0000002510 00000 n 0000006019 00000 n Cox, James C., and Sadiraj, Vjollca (2004), "Implications of Small- and Large-Stakes Risk Aversion for Decision Theory", working paper. For the utility-of-consequences function u(w) = w1/2 we have u0(w) = 1 2 In the labor supply application for VNM utility functions, we show that if the two risks are independent, the comparative statics effect of greater risk aversion on labor supply in the presence of a background non-wage income risk is determined by a monotonic relationship be- tween labor supply and the wage rate under certainty. trailer Arrow and Pratt's original measure used wealth as the argument in the Bernoulli function, so for wealth w, the Arrow-Pratt measure of risk-aversion is -u"(w)/u'(w). By definition, a quadratic utility function must exhibit increasing relative risk aversion. 0000004873 00000 n William Vickrey (1945) used income as the argument of the utility function, so for income y, the Arrow-Pratt measure of risk-aversion is -u"(y)/u'(y). ÊWe conclude that a risk-averse vNM utility function u(x 1) u(E[x]) must be concave. Definition 8. As a matter of fact, the more "curved" a concave utility function is, the lower will be a consumer's certainty equivalent, and the higher their risk premium - the "flatter" the utility function is, the closer the certainty equivalent will be to the expected value of the gamble, and the smaller the risk premium. Crucially, an expected utility function is linear in the probabilities, meaning that: U(αp+(1−α)p0)=αU(p)+(1−α)U(p0). : Using these facts, Kenneth Arrow and John Pratt developed a widely-used measure of risk-aversion called, unsurprisingly, the Arrow-Pratt measure of risk-aversion. 2.23 Consider the quadratic VNM utility function U (w)= a + bw + cw 2. The easiest way to do this is to divide the second derivative by the first derivative, i.e. For a Bernoulli utility function over wealth, income, (or in fact any commodity x), u(x), we'll represent the second derivative by u"(x). (Note that any utility funtion must be increasing in its argument, i.e. For a discussion of experiments testing risk aversion, see the risk-aversion section under Experiments. Decision-Making Under Uncertainty - Advanced Topics. Particularly, risk-averse individuals present concave utility functions and the greater the concavity, the more pronounced the risk adversity. Thus, the quadratic function is consistent with investors who reduce the nominal amount invested in risky assets as their wealth increases. 0000000016 00000 n And their description of "a certain way" is very compelling: a list of four, reasonable-seeming axioms. %%EOF The Arrow-Pratt measure of risk-aversion is therefore = -u"(x)/u'(x). In simple terms, what we are measuring above is the actual dollar amount an individual will choose to hold in risky assets, given a certain wealth level w. For this reason, the measure described above is referred to as a measure of absolute risk-aversion. 1.1. In this case, wealth represents the fixed portion of an individuals assets, while income is the portion which is subject to change. Risk Aversion is a mathematical function that indicates how risk-averse a decision-maker is. Clearly, by Jensen’s inequality, which you must know by now, risk aversion corre-sponds to the concavity of the utility function: • DM is risk averse if and only if u is concave; • he is strictly risk averse if and only if u is strictly concave; • he is risk neutral if and only if u is linear, and 0000004618 00000 n We conclude that a risk-averse vNM utility function must be concave. If you haven't already, check out the Von Neumann-Morgenstern utility theorem, a mathematical result which makes their claim rigorous, and true. Given this, Arrow and Pratt had to design a measure of risk-aversion that would remain the same even after an affine transformation of the utility function. As we explained in the Utility Functionchapter that, the absolute risk aversion is and the relative risk aversion is If we apply these operations on a scaled Utility Function equation, we get, Notice that, the absolute risk aversion of an exponential utility function is a constant (1/R), that is irrespective of wealth. preference representation (needs some utility function that represents preferences). So the answer to my question seems to be that diminishing marginal utility in the vNM utility function reflects genuine diminishing marginal utility when it comes to intensity of preferences, and thus (assuming the vNM axioms are true) diminishing marginal utility really is the cause of risk aversion. Therefore, distinguishing Bernoulli from vNM utility functions enables us to examine the effects of uncertainty apart from the mere quantity of "stuff" (be it goods or money). Therefore the consumer is risk averse. E[u(x)] u(x 0) Slide 04Slide 04--2121 x 0 E[x] x 1 x u-1(E[u(x)]) For ex­am­ple, for two out­comes A and B, 1. If we want to measure the percentage of wealth held in risky assets, for a given wealth level w, we simply multiply the Arrow-pratt measure of absolute risk-aversion by the wealth w, to get a measure of relative risk-aversion, i.e. The question is, now - how do we measure the amount of curvature of a function? This is confirmed by the above relative risk aversion function. They define that there is an increase in down �Ff膃a� �(d!��fa#�ƅ��d��h�� �m {�e. 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