In economics and finance, risk aversion is the tendency of people to prefer outcomes with low uncertainty to those outcomes with high uncertainty, even if the average outcome of the latter is equal to or higher in monetary value than the more certain outcome.
Risk aversion explains the inclination to agree to a situation with a lower average payoff that is more predictable rather than another situation with a less predictable payoff that is higher on average. For example, a risk-averse investor might choose to put their 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.
Example
A person is given the choice between two scenarios: one with a guaranteed payoff, and one with a risky payoff with same average value. In the former 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.
A person is said to be:
risk averse (or risk avoiding) - if they 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 they are indifferent between the bet and a certain $50 payment. risk loving (or risk seeking) – if they 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 smallest guaranteed dollar amount that an individual would be indifferent to compared to an uncertain gain of a specific average predicted value is called the certainty equivalent, which is also used as a measure of risk aversion. An individual that is risk averse has a certainty equivalent that is smaller than the prediction of uncertain gains. The risk premium is the difference between the expected value and the certainty equivalent. For risk-averse individuals, risk premium is positive, for risk-neutral persons it is zero, and for risk-loving individuals their risk premium is negative.
Utility of money In expected utility theory, an agent has a utility function u(c) where c represents the value that he might receive in money or goods (in the above example c could be $0 or $40 or $100).
The utility function u(c) is defined only up to positive affine transformation – in other words, a constant could be added to the value of u(c) for all c, and/or u(c) could be multiplied by a positive constant factor, without affecting the conclusions.
An agent is risk-averse 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 4.02 and u(50) might be 5.01.
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 There are various measures of the risk aversion expressed by those given utility function. Several functional forms often used for utility functions are represented by these measures.
Absolute risk aversion 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 rather than just the second derivative of . One such measure is the Arrow–Pratt measure of absolute risk aversion (ARA), after the economists Kenneth Arrow and John W. Pratt, Reprinted in: Essays in the Theory of Risk Bearing, Markham Publ. Co., Chicago, 1971, 90–109. also known as the coefficient of absolute risk aversion, defined as
where and denote the first and second derivatives with respect to of . For example, if so and then Note how does not depend on and so affine transformations of do not change it.
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 hyperbola, 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 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, . 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. 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.
Relative risk aversion The Arrow–Pratt measure of relative risk aversion (RRA) or coefficient of relative risk aversion is defined as .
Unlike ARA whose units are in $−1, RRA is a dimensionless quantity, which allows it to be applied universally. 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 implies .
In intertemporal choice problems, the elasticity of intertemporal substitution often cannot be disentangled from the coefficient of relative risk aversion. The isoelastic utility function
exhibits constant relative risk aversion with and the elasticity of intertemporal substitution . When using l'Hôpital's rule shows that this simplifies to the case of log utility, , and the income effect and substitution effect on saving exactly offset.
A time-varying relative risk aversion can be considered.
Implications of increasing/decreasing absolute and relative risk aversion
The most straightforward implications of changing risk aversion occur in the context of forming a portfolio with one risky asset and one risk-free asset. If an investor experiences an increase in wealth, he/she will choose to decrease the total amount of wealth invested in the risky asset in proportion to absolute risk aversion and will decrease the relative fraction of the portfolio made up of the risky asset in proportion to relative risk aversion. Thus economists avoid using utility functions which exhibit increasing absolute risk aversion, because they have an unrealistic behavioral implication.
In one model 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.
Portfolio theory
In modern portfolio theory, risk aversion is measured as the additional expected reward an investor requires to accept additional risk. If an investor is risk-averse, they will invest in multiple uncertain assets, but only when the predicted return on a portfolio that is uncertain is greater than the predicted return on one that is not uncertain will the investor prefer the former. Here, the risk-return spectrum is relevant, as it results largely from this type of risk aversion. 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.
Von Neumann-Morgenstern utility theorem The von Neumann-Morgenstern utility theorem is another model used to denote how risk aversion influences an actor’s utility function. An extension of the expected utility function, the von Neumann-Morgenstern model includes risk aversion axiomatically rather than as an additional variable.
John von Neumann and Oskar Morgenstern first developed the model in their book Theory of Games and Economic Behaviour. Essentially, von Neumann and Morgenstern hypothesised that individuals seek to maximise their expected utility rather than the expected monetary value of assets. In defining expected utility in this sense, the pair developed a function based on preference relations. As such, if an individual’s preferences satisfy four key axioms, then a utility function based on how they weigh different outcomes can be deduced.