Risk aversion 177700 222422399 2008-06-29T07:25:41Z 41.204.105.5 Added references list. '''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 value|expected payoff]]. The inverse of a person's risk aversion is sometimes called their '''risk tolerance''' (for a more general discussion of the concept, see [[risk]]). ==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). [[Investor]]s 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 [[Expected utility hypothesis|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 <math>U(x_i)</math> where <math>x_i</math> are amounts of goods with index <math>i</math>. From this, it is possible to derive a function <math>u(c)</math>, of utility of consumption <math>c</math> as a whole. Here, consumption <math>c</math> is equivalent to [[money]] in real terms, i.e. without [[inflation]]. The utility function <math>u(c)</math> is defined only [[modulo]] [[linear transformation]]. The graph shows this situation for the risk-averse player: The utility of the bet, :<math>E(u)=(u(0)+u(100))/2</math> is as big as that of the certainty equivalence, <math>CE</math>. The risk premium is :<math>(\$50-\$40)/\$40</math> or 25%. ==Measures of risk aversion== ===Absolute risk aversion=== The higher the curvature of <math>u(c)</math>, 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 :<math>r_u(c)=-\frac{u''(c)}{u'(c)}</math>. The following expressions relate to this term: * [[Exponential utility]] of the form <math>u(c)=-e^{-\alpha c}</math> is unique in exhibiting ''constant absolute risk aversion'' (CARA): <math>r_u(c)=\alpha</math> is constant with respect to <math>c</math>. * ''Decreasing/increasing absolute risk aversion'' (DARA/IARA) if <math>r_u(c)</math> is decreasing/increasing. An example for a DARA utility function is <math>u(c)=\ln(c), r_u(c)=1/c</math>, while <math>u(c)=c-\alpha c^2,\alpha >0, r_u(c)=2 \alpha/(1-2 \alpha c)</math> would represent a utility function exhibiting IARA. *Experimental and empirical evidence is mostly consistent with decreasing absolute risk aversion.{{Fact|date=April 2007}} * 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 <math>r_u(c)=-\frac{u''(c)}{u'(c)}</math> 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.<ref>Bellemare, Marc F. and Zachary S. Brown, ''On the (Mis)Use of Wealth as a Proxy for Risk Aversion''[http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1140668], Working Paper, Duke University.</ref> ===Relative risk aversion=== The ''Arrow-Pratt measure of relative risk-aversion'' (RRA) or ''coefficient of relative risk aversion'' is defined as :<math>R_u(c) = cr_u(c)=\frac{-cu''(c)}{u'(c)}</math>. 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 <math>c</math>. 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 :<math>u(c) = \frac{c^{1-\rho}}{1-\rho}</math> exhibits constant relative risk aversion with <math>R_{u(c)} = \rho </math> and the elasticity of intertemporal substitution <math>\varepsilon_{u(c)} = 1/\rho</math>. When <math>\rho = 1</math> this simplifies to the case of ''log utility,'' and the [[income effect]] and [[substitution effect]] on saving exactly offset. ===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 [[Nth root|radical]] of the n-th [[central moment]]. The symbol used for risk aversion is A or A<sub>n</sub>. :<math>A = \frac{dE(r)}{d\sigma}</math> :<math>A_n = \frac{dE(r)}{d\sqrt[n]{\mu_n}} = \frac{1}{n} \frac{dE(r)}{d\mu_n} </math> ==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 wealth. ==Other categories== See "[[Harm reduction|Harm Reduction]]". Risk aversion theory can be applied to many aspects of life and its challenges, for example: *[[Bribery]] and [[Political corruption|corruption]] - whether the risk of being implicated or caught outweighs the potential personal or professional rewards *[[Drugs]] - whether the risk of having a [[bad trip]] outweighs the benefits of possible [[Spiritual transformation|transformative one]]; whether the risk of defying [[social bans]] is worth the experience of alteration. *[[Sex]] - judgement whether an experience that goes against [[social convention]], [[ethics|ethical mores]] or common health [[prescription]]s is worth the risk. *[[Extreme sports]] - having the ability to go against [[Instinct|biological predepositions]] like the [[Acrophobia|fear of height]]. ==See also== * [[Optimism bias]] * [[Utility]] * [[Risk premium]] * [[Equity premium puzzle]] * [[Investor profile]] * [[St. Petersburg paradox]] * [[Compulsive gambling]], a contrary behavior ==External links== * [http://cepa.newschool.edu/het/essays/uncert/aversion.htm#pratt More thorough introduction] * [http://emlab.berkeley.edu/users/rabin/ Prof. Rabin's homepage] * [http://arielrubinstein.tau.ac.il/papers/rabin3.pdf A response (2001) by Ariel Rubinstein] * [http://repositories.cdlib.org/cgi/viewcontent.cgi?article=1025&context=iber/econ Paper about problems with risk aversion] * [http://www.economist.com/science/displayStory.cfm?story_id=4102350 Economist article on monkey experiments showing behaviours resembling risk aversion] ==References== {{Reflist}} [[Category:Actuarial science]] [[Category:Behavioral finance]] [[Category:Economics of uncertainty]] [[Category:Risk]] [[Category:Risk in finance]] [[da:Risikoaversion]] [[de:Risikoaversion]] [[fr:Aversion au risque]] [[it:Avversione al rischio]] [[no:Risikoaversjon]] [[fi:Riskiaversio]] [[sv:Riskaversion]] [[zh:风险厌恶]]