In economics and finance, risk aversion is the behavior of humans (especially consumers and investors), when exposed to uncertainty, in attempting to lower that uncertainty. It is the hesitation of a person to agree to a situation with an unknown payoff rather than another situation with a more predictable payoff but possibly lower expected payoff. For example, a risk-averse investor might choose to put his or her money into a bank account with a low but guaranteed interest rate, rather than into a stock that may have high expected returns, but also involves a chance of losing value. Contents 1 Example
2
3.1 Absolute risk aversion 3.2 Relative risk aversion 3.3 Implications of increasing/decreasing absolute and relative risk aversion 3.4 Portfolio theory 4 Limitations of expected utility treatment of risk aversion 5 In the brain 6 Public understanding and risk in social activities 7 Influences 8 See also 9 References 10 External links Example[edit]
CE - Certainty equivalent; E(U(W)) -
A person is given the choice between two scenarios, one with a guaranteed payoff and one without. In the guaranteed scenario, the person receives $50. In the uncertain scenario, a coin is flipped to decide whether the person receives $100 or nothing. 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. However, individuals may have different risk attitudes.[1][2][3] A person is said to be: risk-averse (or risk-avoiding) - if he or she would accept a certain payment (certainty equivalent) of less than $50 (for example, $40), rather than taking the gamble and possibly receiving nothing. risk-neutral - if he or she is indifferent between the bet and a certain $50 payment. risk-loving (or risk-seeking) - if he or she would accept the bet even when the guaranteed payment is more than $50 (for example, $60). The average payoff of the gamble, known as its expected value, is $50.
The dollar amount that the individual would accept instead of the bet
is called the certainty equivalent, and the difference between the
expected value and the certainty equivalent is called the risk
premium. For risk-averse individuals, it is positive, for risk-neutral
persons it is zero, and for risk-loving individuals their risk premium
is negative.
E ( u ) = ( u ( 0 ) + u ( 100 ) ) / 2 displaystyle 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 ( $ 50 − $ 40 ) / $ 40 displaystyle ($50-$40)/$40 or 25% (where $50 is the expected value of the risky bet: ( 1 2 0 + 1 2 100 displaystyle tfrac 1 2 0+ tfrac 1 2 100 ). 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). Measures of risk aversion under expected utility theory[edit] There are multiple measures of the risk aversion expressed by a given utility function. Several functional forms often used for utility functions are expressed in terms of these measures. Absolute risk aversion[edit] The higher the curvature of u ( c ) displaystyle 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,[4][5] also known as the coefficient of absolute risk aversion, defined as A ( c ) = − u ″ ( c ) u ′ ( c ) displaystyle A(c)=- frac u''(c) u'(c) where u ′ ( c ) displaystyle u'(c) and u ″ ( c ) displaystyle u''(c) denote the first and second derivatives with respect to c displaystyle c of u ( c ) displaystyle u(c) . The following expressions relate to this term:
u ( c ) = 1 − e − α c displaystyle u(c)=1-e^ -alpha c is unique in exhibiting constant absolute risk aversion (CARA): A ( c ) = α displaystyle A(c)=alpha is constant with respect to c.
A ( c ) = − u ″ ( c ) u ′ ( c ) = 1 a c + b displaystyle A(c)=- frac u''(c) u'(c) = frac 1 ac+b The solution to this differential equation (omitting additive and multiplicative constant terms, which do not affect the behavior implied by the utility function) is: u ( c ) = ( c − c s ) 1 − R 1 − R displaystyle u(c)= frac (c-c_ s )^ 1-R 1-R where R = 1 / a displaystyle R=1/a and c s = − b / a displaystyle c_ s =-b/a . Note that when a = 0 displaystyle a=0 , this is CARA, as A ( c ) = 1 / b = c o n s t displaystyle A(c)=1/b=const , and when b = 0 displaystyle b=0 , this is CRRA (see below), as c A ( c ) = 1 / a = c o n s t displaystyle cA(c)=1/a=const . See [6] Decreasing/increasing absolute risk aversion (DARA/IARA) is present if A ( c ) displaystyle A(c) is decreasing/increasing. Using the above definition of ARA, the following inequality holds for DARA: ∂ A ( c ) ∂ c = − u ′ ( c ) u ‴ ( c ) − [ u ″ ( c ) ] 2 [ u ′ ( c ) ] 2 < 0 displaystyle frac partial A(c) partial c =- frac u'(c)u'''(c)-[u''(c)]^ 2 [u'(c)]^ 2 <0 and this can hold only if u ‴ ( c ) > 0 displaystyle u'''(c)>0 . Therefore, DARA implies that the utility function is positively skewed; that is, u ‴ ( c ) > 0 displaystyle u'''(c)>0 .[7] 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 displaystyle u'''(c)<0 ). An example of a DARA utility function is u ( c ) = log ( c ) displaystyle u(c)=log(c) , with A ( c ) = 1 / c displaystyle A(c)=1/c , while u ( c ) = c − α c 2 , displaystyle u(c)=c-alpha c^ 2 , α > 0 displaystyle alpha >0 , with A ( c ) = 2 α / ( 1 − 2 α c ) displaystyle A(c)=2alpha /(1-2alpha c) would represent a quadratic utility function exhibiting IARA. Experimental and empirical evidence is mostly consistent with decreasing absolute risk aversion.[8] 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 A ( c ) = − u ″ ( c ) u ′ ( c ) displaystyle A(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 absolute risk aversion are usually not identified.[9] Relative risk aversion[edit] The Arrow-Pratt measure of relative risk-aversion (RRA) or coefficient of relative risk aversion is defined as[10] R ( c ) = c A ( c ) = − c u ″ ( c ) u ′ ( c ) displaystyle R(c)=cA(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 the utility function changes from risk-averse to risk-loving as c varies, i.e. utility is not strictly convex/concave over all c. A constant RRA implies a decreasing ARA, but the reverse is not always true. As a specific example of constant relative risk aversion, the utility function u ( c ) = log ( c ) displaystyle u(c)=log(c) implies RRA = 1. In intertemporal choice problems, the elasticity of intertemporal substitution often cannot be disentangled from the coefficient of relative risk aversion. The isoelastic utility function u ( c ) = c 1 − ρ − 1 1 − ρ displaystyle u(c)= frac c^ 1-rho -1 1-rho exhibits constant relative risk aversion with R ( c ) = ρ displaystyle R(c)=rho and the elasticity of intertemporal substitution ε u ( c ) = 1 / ρ displaystyle varepsilon _ u(c) =1/rho . When ρ = 1 , displaystyle rho =1, using l'Hôpital's rule shows that this simplifies to the case of log utility, u(c) = log c, and the income effect and substitution effect on saving exactly offset. A time varying relative risk aversion can be considered.[11] Implications of increasing/decreasing absolute and relative risk aversion[edit] The most straightforward implications of increasing or decreasing absolute or relative risk aversion, and the ones that motivate a focus on these concepts, occur in the context of forming a portfolio with one risky asset and one risk-free asset.[4][5] If the person experiences an increase in wealth, he/she will choose to increase (or keep unchanged, or decrease) the number of dollars of the risky asset held in the portfolio if absolute risk aversion is decreasing (or constant, or increasing). Thus economists avoid using utility functions such as the quadratic, which exhibit increasing absolute risk aversion, because they have an unrealistic behavioral implication. Similarly, if the person experiences an increase in wealth, he/she will choose to increase (or keep unchanged, or decrease) the fraction of the portfolio held in the risky asset if relative risk aversion is decreasing (or constant, or increasing). In monetary economics, an increase in relative risk aversion increases the impact of households' money holdings on the overall economy. In other words, the more the relative risk aversion increases, the more money demand shocks will impact the economy.[12] Portfolio theory[edit] In modern portfolio theory, risk aversion is measured as the additional expected reward an investor requires to accept additional risk. Here risk is measured as the 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 measured as the n-th root of the n-th central moment. The symbol used for risk aversion is A or An. A = d E ( r ) d σ displaystyle A= frac dE(r) dsigma A n = d E ( r ) d μ n n displaystyle A_ n = frac dE(r) d sqrt[ n ] mu _ n Limitations of expected utility treatment of risk aversion[edit]
The notion of using expected utility theory to analyze risk aversion
has come under criticism from behavioral economics.
Ambiguity aversion
Compulsive gambling, a contrary behavior
Diminishing marginal utility
Equity premium puzzle
References[edit] ^ Mr Lev Virine; Mr Michael Trumper (28 October 2013). ProjectThink:
Why Good Managers Make Poor Project Choices. Gower Publishing, Ltd.
ISBN 978-1-4724-0403-9.
^ David Hillson; Ruth Murray-Webster (2007). Understanding and
Managing Risk Attitude. Gower Publishing, Ltd.
ISBN 978-0-566-08798-1.
^ Adhikari, Binay K; Agrawal, Anup (2016). "Does local religiosity
matter for bank risk-taking?". Journal of Corporate Finance. 38:
272–293. doi:10.1016/j.jcorpfin.2016.01.009.
^ a b Arrow, K. J. (1965). "Aspects of the Theory of Risk Bearing".
The Theory of Risk Aversion. Helsinki: Yrjo Jahnssonin Saatio.
Reprinted in: Essays in the Theory of Risk Bearing, Markham Publ. Co.,
Chicago, 1971, 90–109.
^ a b Pratt, J. W. (1964). "Risk Aversion in the Small and in the
Large". Econometrica. 32 (1–2): 122–136. doi:10.2307/1913738.
JSTOR 1913738.
^ "Zender's lecture notes".
^ Levy, Haim (2006). Stochastic Dominance: Investment Decision Making
under
External links[edit] Closed form solution for a consumption savings problem with CARA
utility
Paper about problems with risk aversion
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