Risk aversion is a concept in psychology, economics, and finance, based on the behavior of humans (especially consumers and investors) while exposed to uncertainty to attempt to reduce that 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. 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.
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Utility of money[edit source | editbeta]
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 offset 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,
,
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).
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Measures of risk aversion[edit source | editbeta]
Absolute risk aversion[edit source | editbeta]
The higher the curvature of , 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 theArrow-Pratt measure of absolute risk-aversion (ARA), after the economists Kenneth Arrow and John W. Pratt,[1][2] also known as thecoefficient of absolute risk aversion, defined as
.
The following expressions relate to this term: * Exponential utility of the form is unique in exhibiting constant absolute risk aversion (CARA): 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 and . Note that when , this is CARA, as , and when , this is CRRA (see below), as . See [3] * Decreasing/increasing absolute risk aversion (DARA/IARA) is present if is decreasing/increasing. Using the above definition of ARA, the following inequality holds for DARA:

and this can hold only if . Therefore, DARA implies that the utility function is positively skewed; that is, .[4]Analogously, IARA can be derived with the opposite directions of inequalities, which permits but does not require a negatively skewed utility function (). An example of a DARA utility function is , with , while , with 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 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.[6]
Relative risk aversion[edit source | editbeta]
The Arrow-Pratt-De Finetti measure of relative risk-aversion (RRA) or coefficient of relative risk aversion is defined as
.
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, the expected utility function implies RRA = 1.
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