Risk aversion is a concept in
psychologyPsychology is the study of the mind and behavior. Its immediate goal is to understand individuals and groups by both establishing general principles and researching specific cases. For many, the ultimate goal of psychology is to benefit society...
,
economicsEconomics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
, and
finance"Finance" is often defined simply as the management of money or “funds” management Modern finance, however, is a family of business activity that includes the origination, marketing, and management of cash and money surrogates through a variety of capital accounts, instruments, and markets created...
, based on the behavior of
humanHumans are the only living species in the Homo genus...
s (especially
consumerConsumer is a broad label for any individuals or households that use goods generated within the economy. The concept of a consumer occurs in different contexts, so that the usage and significance of the term may vary.-Economics and marketing:...
s and
investorAn investor is a party that makes an investment into one or more categories of assets --- equity, debt securities, real estate, currency, commodity, derivatives such as put and call options, etc...
s) while exposed to
uncertaintyUncertainty is a term used in subtly different ways in a number of fields, including physics, philosophy, statistics, economics, finance, insurance, psychology, sociology, engineering, and information science...
.
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 payoffIn probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
. For example, a risk-averse investor might choose to put his or her money into a
bankA bank is a financial institution that serves as a financial intermediary. The term "bank" may refer to one of several related types of entities:...
account with a low but guaranteed interest rate, rather than into a
stockThe capital stock of a business entity represents the original capital paid into or invested in the business by its founders. It serves as a security for the creditors of a business since it cannot be withdrawn to the detriment of the creditors...
that may have high expected returns, but also involves a chance of losing value.
Outside the rather mathematical fields of economics and finance, people have to make choices about how they face risks every day. Some have become very cautious, preferring to minimize risks even when the potential benefit of an action is large.
Example
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. 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
In economics and finance, a risk lover is a person who has a preference for risk. While most investors are considered risk averse, one could view casino goers as risk loving...
(or risk-seeking) - if the guaranteed payment must be more than $50 (for example, $60) to induce him or her to take the guaranteed option, rather than taking the gamble and possibly winning $100.
The average payoff of the gamble, known as its
expected valueIn probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
, is $50. The dollar amount that the individual would accept instead of the bet is called the
certainty equivalentIn economics, game theory, and decision theory the expected utility hypothesis is a theory of utility in which "betting preferences" of people with regard to uncertain outcomes are represented by a function of the payouts , the probabilities of occurrence, risk aversion, and the different utility...
, and the difference between the expected value and the certainty equivalent is called the
risk premiumA risk premium is the minimum amount of money by which the expected return on a risky asset must exceed the known return on a risk-free asset, in order to induce an individual to hold the risky asset rather than the risk-free asset...
. For risk-averse individuals, it becomes positive, for risk-neutral persons it is zero, and for risk-loving individuals their risk premium becomes negative.
Utility of money
In
expected utilityIn economics, game theory, and decision theory the expected utility hypothesis is a theory of utility in which "betting preferences" of people with regard to uncertain outcomes are represented by a function of the payouts , the probabilities of occurrence, risk aversion, and the different 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 toIn mathematics, the phrase "up to x" means "disregarding a possible difference in x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
affine transformationIn geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...
- in other words a constant factor 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
concaveIn mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap or upper convex.-Definition:...
. 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 contractionIn probability and statistics, a mean-preserving spread is a change from one probability distribution A to another probability distribution B, where B is formed by spreading out one or more portions of A's probability density function while leaving the mean unchanged...
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).
The above is an introduction to the mathematics of risk aversion. However it assumes that the individual concerned will act entirely rationally and will not factor into his decision non-monetary, psychological considerations such as regret at having made the wrong decision. Often an individual may come to a different decision depending on how the proposition is presented, even though there may be no mathematical difference.
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. One such measure is the
Arrow-Pratt measure of absolute risk-aversion (ARA), after the economists
Kenneth ArrowKenneth Joseph Arrow is an American economist and joint winner of the Nobel Memorial Prize in Economics with John Hicks in 1972. To date, he is the youngest person to have received this award, at 51....
and
John W. PrattJohn W. Pratt is Emeritus William Ziegler professor business administration at Harvard University. He has made contributions to research in risk aversion theory, notably with Kenneth Arrow on measures of risk aversion....
, also known as the coefficient of absolute risk aversion, defined as
.
The following expressions relate to this term:
- Exponential utility
In economics exponential discounting is a specific form of the discount function, used in the analysis of choice over time...
of the form is unique in exhibiting constant absolute risk aversion (CARA): is constant with respect to c.
- Hyperbolic absolute risk aversion
In finance, economics, and decision theory, hyperbolic absolute risk aversion refers to a type of risk aversion that is particularly convenient to model mathematically and to obtain empirical predictions from...
(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 functionIn mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. The basic hyperbolic functions are the hyperbolic sine "sinh" , and the hyperbolic cosine "cosh" , from which are derived the hyperbolic tangent "tanh" and so on.Just as the points form a...
, 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
.
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 choiceIntertemporal choice is the study of the relative value people assign to two or more payoffs at different points in time. Most choices require decision-makers to trade-off costs and benefits at different points in time. These decisions maybe about savings, work effort, education, nutrition,...
problems, the
elasticity of intertemporal substitutionElasticity of intertemporal substitution is a measure of responsiveness of the growth rate of consumption to the real interest rate...
is often unable to 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
and one is subtracted in the numerator (facilitating the use of
l'Hôpital's ruleIn calculus, l'Hôpital's rule uses derivatives to help evaluate limits involving indeterminate forms. Application of the rule often converts an indeterminate form to a determinate form, allowing easy evaluation of the limit...
), this simplifies to the case of log utility, and the
income effectIn economics, the consumer's preferences, money income and prices play an important role in solving the consumer's optimization problem...
and substitution effect on saving exactly offset.
Implications of increasing/decreasing absolute and relative risk aversion
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. 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).
Portfolio theory
In
modern portfolio theoryModern portfolio theory is a theory of investment which attempts to maximize portfolio expected return for a given amount of portfolio risk, or equivalently minimize risk for a given level of expected return, by carefully choosing the proportions of various assets...
, 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 deviationStandard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...
of the return on investment, i.e. the
square rootIn mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...
of its
varianceIn probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
. 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 momentIn probability theory and statistics, central moments form one set of values by which the properties of a probability distribution can be usefully characterised...
. The symbol used for risk aversion is A or A
_{n}.
Limitations
The notion of (constant) risk aversion has come under criticism from behavioral economics. According to
Matthew RabinMatthew Joel Rabin is the Edward G. and Nancy S. Jordan Professor of Economics in the Department of Economics at the University of California, Berkeley...
of UC Berkeley, a consumer who,
from any initial wealth level [...] turns down gambles where he 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.
It is noteworthy that Rabin's article went on to criticize the whole field of expected utility and not just constant relative risk aversion. This has led to some confusion in the field. One solution to the problem observed by Rabin is that proposed by
prospect theoryProspect theory is a theory that describes decisions between alternatives that involve risk i.e. where the probabilities of outcomes are known. The model is descriptive: it tries to model real-life choices, rather than optimal decisions.-Model:...
and
cumulative prospect theoryCumulative prospect theory is a model for descriptive decisions under risk which was introduced by Amos Tversky and Daniel Kahneman in 1992 . It is a further development and variant of prospect theory...
, where outcomes are considered relative to a reference point (usually the status quo), rather than to consider only the final wealth.
Risk aversion in the brain
Attitudes towards risk have attracted the interest of the field of neuroeconomics. A study by researchers at the University of Cambridge suggested that the activity of a specific brain area (right inferior frontal gyrus) correlates with risk aversion, with more risk averse participants (i.e. those having higher risk premia) also having higher responses to safer options. This result coincides with other studies, that show that neuromodulation of the same area results in participants making more or less risk averse choices, depending on whether the modulation increases or decreases the activity of the target area.
Public understanding and risk in social activities
In the real world, many government agencies, e.g.
Health and Safety ExecutiveThe Health and Safety Executive is a non-departmental public body in the United Kingdom. It is the body responsible for the encouragement, regulation and enforcement of workplace health, safety and welfare, and for research into occupational risks in England and Wales and Scotland...
, are fundamentally risk-averse in their mandate. This often means that they demand (with the power of legal enforcement) that risks be minimized, even at the cost of losing the utility of the risky activity.
It is important to consider the
opportunity costOpportunity cost is the cost of any activity measured in terms of the value of the best alternative that is not chosen . It is the sacrifice related to the second best choice available to someone, or group, who has picked among several mutually exclusive choices. The opportunity cost is also the...
when mitigating a risk; the cost of not taking the risky action. Writing laws focused on the risk without the balance of the utility may misrepresent society's goals. The public understanding of risk, which influences political decisions, is an area which has recently been recognised as deserving focus.
David SpiegelhalterDavid John Spiegelhalter OBE, FRS, is a distinguished British statistician. In 2007 he was elected Winton Professor of the Public Understanding of Risk in the Statistical Laboratory, University of Cambridge and a Fellow of Churchill College, Cambridge...
is the
Winton Professor of the Public Understanding of RiskThe Winton Professorship of the Public Understanding of Risk is a professorship within the Statistical Laboratory of the University of Cambridge. It was established in 2007 in perpetuity by a benefaction of £3.3m from the Winton Charitable Foundation, and is the only professorship of its type in...
at Cambridge University; a role he describes as "outreach".
A
vaccineThe MMR vaccine is an immunization shot against measles, mumps, and rubella . It was first developed by Maurice Hilleman while at Merck in the late 1960s....
to protect children against the three common diseases measles, mumps and rubella was developed and recommended for all children in several countries including the UK. However, a
controversyThe MMR vaccine controversy was a case of scientific misconduct which triggered a health scare. It followed the publication in 1998 of a paper in the medical journal The Lancet which presented apparent evidence that autism spectrum disorders could be caused by the MMR vaccine, an immunization...
arose around allegations that it caused autism. This alleged causal link was thoroughly disproved, and the doctor who made the claims was expelled from the
General Medical CouncilThe General Medical Council registers and regulates doctors practising in the United Kingdom. It has the power to revoke or restrict a doctor's registration if it deems them unfit to practise...
. Even years after the claims were disproved, some parents wanted to avert the risk of causing autism in their own children. They chose to spend significant amounts of their own money on alternatives from private doctors. These alternatives carried their own risks which were not balanced fairly; most often that the children were not properly immunised against the more common diseases of measles, mumps and rubella.
Mobile phoneA mobile phone is a device which can make and receive telephone calls over a radio link whilst moving around a wide geographic area. It does so by connecting to a cellular network provided by a mobile network operator...
s may carry some small
health riskThe effect of mobile phone radiation on human health is the subject of recent interest and study, as a result of the enormous increase in mobile phone usage throughout the world . Mobile phones use electromagnetic radiation in the microwave range...
. While most people would accept that unproven risk to gain the benefit of improved communication, others remain so risk averse that they do not. (The
COSMOS cohort studyCOSMOS is a cohort study of mobile phone use and health. The study will investigate the possible health effects of long term mobile phone use. It is an international survey being conducted in five European countries – UK, Denmark, Sweden, Finland and the Netherlands...
continues to study the actual risks of mobile phones.)
Risk aversion theory can be applied to many aspects of life and its challenges, for example:
- Bribery
Bribery, a form of corruption, is an act implying money or gift giving that alters the behavior of the recipient. Bribery constitutes a crime and is defined by Black's Law Dictionary as the offering, giving, receiving, or soliciting of any item of value to influence the actions of an official or...
and corruptionPolitical corruption is the use of legislated powers by government officials for illegitimate private gain. Misuse of government power for other purposes, such as repression of political opponents and general police brutality, is not considered political corruption. Neither are illegal acts by...
- whether the risk of being implicated or caught outweighs the potential personal or professional rewards
- Drugs
Destroy Rebuild Until God Shows are an American post-hardcore band formed in 2010. They released their debut self-titled album on February 22, 2011.- Formation :...
- whether the risk of having a bad tripBad trip is a disturbing experience sometimes associated with use of a psychedelic drug such as LSD, Salvinorin A, DXM, mescaline, psilocybin, DMT and sometimes even other drugs including cannabis, alcohol and MDMA....
outweighs the benefits of possible transformative oneSpiritual transformation has a variety of overlapping meanings that carry distinct connotations:*In psychology, spiritual transformation is understood within the context of an individual's meaning system, especially in relation to concepts of the sacred or ultimate concern...
; whether the risk of defying social bans is worth the experience of alteration. See "Harm reductionHarm reduction refers to a range of public health policies designed to reduce the harmful consequences associated with recreational drug use and other high risk activities...
".
- Sex
In biology, sex is a process of combining and mixing genetic traits, often resulting in the specialization of organisms into a male or female variety . Sexual reproduction involves combining specialized cells to form offspring that inherit traits from both parents...
- judgement whether an experience that goes against social convention, ethical moresEthics, also known as moral philosophy, is a branch of philosophy that addresses questions about morality—that is, concepts such as good and evil, right and wrong, virtue and vice, justice and crime, etc.Major branches of ethics include:...
or common health prescriptionA prescription is a health-care program implemented by a physician or other medical practitioner in the form of instructions that govern the plan of care for an individual patient. Prescriptions may include orders to be performed by a patient, caretaker, nurse, pharmacist or other therapist....
s is worth the risk.
- Extreme sports - weighing the risk of physical injury or death against the adrenaline rush
An adrenaline rush is the fight or flight response of the adrenal gland, in which it releases adrenaline . When releasing adrenaline, one's body releases dopamine which can act as a natural pain killer. An adrenaline rush causes the muscles to perform respiration at an increased rate improving...
and bragging rightsBragging rights refers to an informal claim one can make to holding a certain achievement, such as a record or being the first at something. This claim is not an official title or status, but rather the ability to win an argument to holding the position....
.
- Play
Play is a term employed in ethology and psychology to describe to a range of voluntary, intrinsically motivated activities normally associated with pleasure and enjoyment...
by children in playgrounds or beyond the reach of their parents.
See also
- Ambiguity aversion
In decision theory and economics, ambiguity aversion describes an attitude of preference for known risks over unknown risks. People would rather choose an option with fewer unknown elements than with many unknown elements. It is demonstrated in the Ellsberg paradox In decision theory and...
- Optimism bias
Optimism bias is the demonstrated systematic tendency for people to be overly optimistic about the outcome of planned actions. This includes over-estimating the likelihood of positive events and under-estimating the likelihood of negative events. Along with the illusion of control and illusory...
- Utility
In economics, utility is a measure of customer satisfaction, referring to the total satisfaction received by a consumer from consuming a good or service....
- Risk premium
A risk premium is the minimum amount of money by which the expected return on a risky asset must exceed the known return on a risk-free asset, in order to induce an individual to hold the risky asset rather than the risk-free asset...
- Equity premium puzzle
The equity premium puzzle is a term coined in 1985 by economists Rajnish Mehra and Edward C. Prescott. It is based on the observation that in order to reconcile the much higher returns of stocks compared to government bonds in the United States, individuals must have implausibly high risk aversion...
- Investor profile
An investor profile or style defines an individual's preferences in investment decisions, for example:* Short term trading or long term holding * Risk averse or risk tolerant / seeker...
- Loss aversion
In economics and decision theory, loss aversion refers to people's tendency to strongly prefer avoiding losses to acquiring gains. Some studies suggest that losses are twice as powerful, psychologically, as gains....
- St. Petersburg paradox
In economics, the St. Petersburg paradox is a paradox related to probability theory and decision theory. It is based on a particular lottery game that leads to a random variable with infinite expected value, i.e., infinite expected payoff, but would nevertheless be considered to be worth only a...
- Compulsive gambling
Problem gambling is an urge to continuously gamble despite harmful negative consequences or a desire to stop. Problem gambling often is defined by whether harm is experienced by the gambler or others, rather than by the gambler's behavior. Severe problem gambling may be diagnosed as clinical...
, a contrary behavior
- Neuroeconomics
Neuroeconomics is an interdisciplinary field that seeks to explain human decision making, the ability to process multiple alternatives and to choose an optimal course of action. It studies how economic behavior can shape our understanding of the brain, and how neuroscientific discoveries can...
- Prudence in economics and finance
- Diminishing marginal utility
External links