A risky asset has two possible outcomes. Outcome 1 pays $20 with 20% probability

Question

A risky asset has two possible outcomes. Outcome 1 pays $20 with 20% probability, and outcome 2 pays $50 with 80% probability. What is the standard deviation of payoffs of this asset? Show work

Explanation / Answer

In expected utility theory, an agent has a utility function u(x) where x represents the value that he might receive in money or goods (in the above example x could be 0 or 100). Time does not come into this calculation, so inflation does not appear. (The utility function u(x) is defined only up to positive linear affine transformation - in other words a constant factor could be added to the value of u(x) for all x, and/or u(x) could be multiplied by a positive constant factor, without affecting the conclusions.) An agent possesses risk aversion if and only if the utility function is concave. For instance u(0) could be 0, u(100) might be 10, u(40) might be 5, and for comparison u(50) might be 6. The expected utility of the above bet (with a 50% chance of receiving 100 and a 50% chance of receiving 0) is, E(u) = (u(0) + u(100)) / 2, and if the person has the utility function with u(0)=0, u(40)=5, and u(100)=10 then the expected utility of the bet equals 5, which is the same as the known utility of the amount 40. Hence the certainty equivalent is 40. The risk premium is ($50 minus $40)=$10, or in proportional terms or 25% (where $50 is the expected value of the risky bet: (). This risk premium means that the person would be willing to sacrifice as much as $10 in expected value in order to achieve perfect certainty about how much money will be received. In other words, the person would be indifferent between the bet and a guarantee of $40, and would prefer anything over $40 to the bet. In the case of a wealthier individual, the risk of losing $100 would be less significant, and for such small amounts his utility function would be likely to be almost linear, for instance if u(0) = 0 and u(100) = 10, then u(40) might be 4.0001 and u(50) might be 5.0001. The utility function for perceived gains has two key properties: an upward slope, and concavity. (i) The upward slope implies that the person feels that more is better: a larger amount received yields greater utility, and for risky bets the person would prefer a bet which is first-order stochastically dominant over an alternative bet (that is, if the probability mass of the second bet is pushed to the right to form the first bet, then the first bet is preferred). (ii) The concavity of the utility function implies that the person is risk averse: a sure amount would always be preferred over a risky bet having the same expected value; moreover, for risky bets the person would prefer a bet which is a mean-preserving contraction of an alternative bet (that is, if some of the probability mass of the first bet is spread out without altering the mean to form the second bet, then the first bet is preferred). The above is an introduction to the mathematics of risk aversion. However it assumes that the individual concerned will act entirely rationally and will not factor into his decision non-monetary, psychological considerations such as regret at having made the wrong decision. Often an individual may come to a different decision depending on how the proposition is presented, even though there may be no mathematical difference. Absolute risk aversion The higher the curvature of u(c), the higher the risk aversion. However, since expected utility functions are not uniquely defined (are defined only up to affine transformations), a measure that stays constant with respect to these transformations is needed. One such measure is the Arrow-Pratt measure of absolute risk-aversion (ARA), after the economists Kenneth Arrow and John W. Pratt,[1][2] also known as the coefficient of absolute risk aversion, defined as . The following expressions relate to this term: Exponential utility of the form u(c) = 1 - e - ac is unique in exhibiting constant absolute risk aversion (CARA): A(c) = a is constant with respect to c. Hyperbolic absolute risk aversion (HARA) is the most general class of utility functions that are usually used in practice (specifically, CRRA (constant relative risk aversion, see below), CARA (constant absolute risk aversion), and quadratic utility all exhibit HARA and are often used because of their mathematical tractability). A utility function exhibits HARA if its absolute risk aversion is a hyperbolic function, namely The solution to this differential equation (omitting additive and multiplicative constant terms, which do not affect the behavior implied by the utility function) is: where R = 1 / a and cs = - b / a. Note that when a = 0, this is CARA, as A(c) = 1 / b = const, and when b = 0, this is CRRA (see below), as cA(c) = 1 / a = const. See [3] Decreasing/increasing absolute risk aversion (DARA/IARA) is present if A(c) is decreasing/increasing. Using the above definition of ARA, the following inequality holds for DARA: and this can hold only if u'''(c) > 0. Therefore, DARA implies that the utility function is positively skewed; that is, u'''(c) > 0[4]. Analogously, IARA can be derived with the opposite directions of inequalities, which permits but does not require a negatively skewed utility function (u'''(c) < 0). An example of a DARA utility function is u(c) = log(c), with A(c) = 1 / c, while u(c) = c - ac2, a > 0, with A(c) = 2a / (1 - 2ac) would represent a quadratic utility function exhibiting IARA. Experimental and empirical evidence is mostly consistent with decreasing absolute risk aversion.[5] Contrary to what several empirical studies have assumed, wealth is not a good proxy for risk aversion when studying risk sharing in a principal-agent setting. Although is

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