Measures of Risk aversion

 Risk aversion is a concept in economics, finance, and psychology related to the behaviour of consumers and investors under uncertainty. Risk aversion is the reluctance of a person to accept a bargain with an uncertain payoff rather than another bargain with a more certain, but possibly lower, expected payoff. The inverse of a person’s risk aversion is sometimes called their risk tolerance. Example A person is given the choice between two scenarios, one certain and one not. In the uncertain scenario, the person is to make a gamble with an equal probability between receiving $100 or nothing. The alternative scenario is to receive a specific dollar amount with certainty (probability of 1). Investors have different risk attitudes. A person is * risk-averse if he or she would accept a certain payoff of less than $50 (for example, $40) rather than the gamble. * risk neutral if he or she is indifferent between the bet and a certain $50 payment. * risk-seeking (or risk-loving) if the certain payment must be more than $50 (for example, $60) to induce him or her to take the certain option over the gamble. The average payoff of the gamble, known as its expected value, is $50. The dollar amount accepted instead of the bet is called the certainty equivalent, and the difference between it and the expected value is called the risk premium. Utility of money In utility theory, a consumer has a utility function U(xi) where xi are amounts of goods with index i. From this, it is possible to derive a function u(c), of utility of consumption c as a whole. Here, consumption c is equivalent to money in real terms, i.e. without inflation. The utility function u(c) is defined only modulo linear transformation. The graph shows this situation for the risk-averse player: The utility of the bet, E(u) = (u(0) + u(100)) / 2 is as big as that of the certainty equivalence, CE. The risk premium is (\$50-\$40)/\$40 or 25%. Measures of risk aversion 1) Absolute risk aversion The higher the curvature of u(c), the higher the risk aversion. However, since expected utility functions are not uniquely defined (only up to affine transformations), a measure that stays constant is needed. This measure is the Arrow-Pratt measure of absolute risk-aversion (ARA), after the economists Kenneth Arrow and John W. Pratt or coefficient of absolute risk aversion, defined as r_u(c)=-\frac{u”(c)}{u'(c)}. The following expressions relate to this term: * Exponential utility of the form u(c) = − e − αc is unique in exhibiting constant absolute risk aversion (CARA): ru(c) = α is constant with respect to c. * Decreasing/increasing absolute risk aversion (DARA/IARA) if ru(c) is decreasing/increasing. An example for a DARA utility function is u(c) = ln(c),ru(c) = 1 / c, while u(c) = c − αc2,α > 0,ru(c) = 2α / (1 − 2αc) would represent a utility function exhibiting IARA. * Experimental and empirical evidence is mostly consistent with decreasing absolute risk aversion.[citation needed] * 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. In other words, although r_u(c)=-\frac{u”(c)}{u'(c)} is monotonic in wealth under either DARA or IARA and constant in wealth under CARA, tests of contractual risk sharing relying on wealth as a proxy for risk aversion remain unidentified.[1] 2) Relative risk aversion The Arrow-Pratt measure of relative risk-aversion (RRA) or coefficient of relative risk aversion is defined as R_u(c) = cr_u(c)=\frac{-cu”(c)}{u'(c)}. Like for absolute risk aversion, the corresponding terms constant relative risk aversion (CRRA) and decreasing/increasing relative risk aversion (DRRA/IRRA) are used. This measure has the advantage that it is still a valid measure of risk aversion, even if it changes from risk-averse to risk-loving, i.e. is not strictly convex/concave over all c. In intertemporal choice problems, the elasticity of intertemporal substitution is often unable to be disentangled from the coefficient of relative risk aversion. The “isoelastic” utility function u(c) = \frac{c^{1-\rho}}{1-\rho} exhibits constant relative risk aversion with Ru(c) = ρ and the elasticity of intertemporal substitution \varepsilon_{u(c)} = 1/\rho. When ρ = 1 and one is subtracted in the numerator (facilitating the use of l’Hospital’s Rule), this simplifies to the case of log utility, and the income effect and substitution effect on saving exactly offset. 3) Portfolio theory In modern portfolio theory, risk aversion is measured as the additional marginal reward an investor requires to accept additional risk. In modern portfolio theory, risk is being measured as standard deviation of the return on investment, i.e. the square root of its variance. In advanced portfolio theory, different kinds of risk are taken into consideration. They are being measured as the n-th radical of the n-th central moment. The symbol used for risk aversion is A or An. A = \frac{dE(r)}{d\sigma} A_n = \frac{dE(r)}{d\sqrt[n]{\mu_n}} = \frac{1}{n} \frac{dE(r)}{d\mu_n Limitations The notion of (constant) risk aversion has come under criticism from behavioral economics. According to Matthew Rabin of UC Berkeley, a consumer who, from any initial wealth level […] turns down gambles where she loses $100 or gains $110, each with 50% probability […] will turn down 50-50 bets of losing $1,000 or gaining any sum of money. The point is that if we calculate the constant relative risk aversion (CRRA) from the first small-stakes gamble it will be so great that the same CRRA, applied to gambles with larger stakes, will lead to absurd predictions. The bottom line is that we cannot infer a CRRA from one gamble and expect it to scale up to larger gambles. It is noteworthy that Rabin’s article has often been wrongly quoted as a justification for assuming risk neutral behavior of people in small stake gambles. One solution to the problem observed by Rabin is that proposed by prospect theory and cumulative prospect theory, where outcomes are considered relative to a reference point (usually the status quo), rather than to consider only the final wealt How Absolute Risk-Aversion Changes with Wealth Type of Risk-Aversion Description Example of Bernoulli Function Increasing absolute risk-aversion As wealth increases, hold fewer dollars in risky assets w-cw2 Constant absolute risk-aversion As wealth increases, hold the same dollar amount in risky assets -e-cw Decreasing absolute risk-aversion As wealth increases, hold more dollars in risky assets ln(w) How Relative Risk-Aversion Changes with Wealth Type of Risk-Aversion Description Example of Bernoulli Function Increasing relative risk-aversion As wealth increases, hold a smaller percentage of wealth in risky assets w – cw2 Constant relative risk-aversion As wealth increases, hold the same percentage of wealth in risky assets ln(w) Decreasing relative risk-aversion As wealth increases, hold a larger percentage of wealth in risky assets.

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