# Risk aversion

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Risk aversion is a concept in psychology, economics, and finance, based on the behavior of humans (especially consumers and investors) whilst exposed to 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 returns, but also has a chance of becoming worthless.

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 minimise risks even when the potential benefit of an action is large.

## ExampleEdit

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: • risk-averse if he or she would accept a payoff of less than$50 (for example, $40), with no uncertainty, rather than taking the gamble and possibly receiving nothing. • risk neutral if he is indifferent between the bet and a certain$50 payment.
• risk-seeking (or risk-loving) if the guaranteed payment must be more than $50 (for example,$60) to induce him to take the guaranteed option, rather than taking the gamble and possibly winning $100. 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 certainty equivalent and the expected value is called the risk premium.

## Utility of moneyEdit

In utility theory, a participant 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 $u(c)$ is defined only modulo linear transformation - in other words a constant factor to be added to the value of U(x) for all x, and/or U(x) could be multiplied by a constant factor, without affecting the conclusions.)

The utility of the bet,

$E(u)=(U(0)+U(100))/2$

is as big as that of the certainty equivalence, $CE$, in this case U(40).

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.

$(\50-\40)/\40$

or 25%.

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 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. ## Measures of risk aversionEdit ### Absolute risk aversionEdit The higher the curvature of $u(c)$, 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 $r_u(c)=-\frac{u''(c)}{u'(c)}$. The following expressions relate to this term: • Exponential utility of the form $u(c)=-e^{-\alpha c}$ is unique in exhibiting constant absolute risk aversion (CARA): $r_u(c)=\alpha$ is constant with respect to $c$. • Decreasing/increasing absolute risk aversion (DARA/IARA) if $r_u(c)$ is decreasing/increasing. An example for a DARA utility function is $u(c)=\log(c), r_u(c)=1/c$, while $u(c)=c-\alpha c^2,\alpha >0, r_u(c)=2 \alpha/(1-2 \alpha c)$ would represent a utility function exhibiting IARA. • Experimental and empirical evidence is mostly consistent with decreasing absolute risk aversion.[1] • 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 $r_u(c)=-\frac{u''(c)}{u'(c)}$ 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.[2] ### Relative risk aversionEdit The Arrow-Pratt measure of relative risk-aversion (RRA) or coefficient of relative risk aversion is defined as $R_u(c) = cr_u(c)=\frac{-cu''(c)}{u'(c)}$. 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 it changes from risk-averse to risk-loving, i.e. is not strictly convex/concave over all $c$. A constant RRA implies a decreasing ARA, but the reverse is not always true. However, as a specific example, the expected utility function $u(c) = \log(c)$ does imply 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 $u(c) = \frac{c^{1-\rho}}{1-\rho}$ exhibits constant relative risk aversion with $R_u(c) = \rho$ and the elasticity of intertemporal substitution $\varepsilon_{u(c)} = 1/\rho$. When $\rho = 1$ and one is subtracted in the numerator (facilitating the use of l'Hôpital's rule), this simplifies to the case of log utility, and the income effect and substitution effect on saving exactly offset. ### Portfolio theoryEdit 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 radical of the n-th central moment. The symbol used for risk aversion is A or An. $A = \frac{dE(r)}{d\sigma}$ $A_n = \frac{dE(r)}{d\sqrt[n]{\mu_n}} = \frac{1}{n} \frac{dE(r)}{d\mu_n}$ ## LimitationsEdit 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 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.

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 Economist article went on to criticize the whole field of expected utility and not just constant absolute risk aversion. This has led to some confusion in the field. 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.

## Risk aversion in the brainEdit

Attitudes towards risk have attracted the interest of the field of neuroeconomics. A study by researchers at the University of Cambridge [3] 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 [4] [5], 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 activitiesEdit

In the real world, many government agencies, such as the British Health and Safety Executive, are fundamentally risk-averse in their constitution. 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 cost 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 Spiegelhalter is the Winton Professor of the Public Understanding of Risk at Cambridge University; a role he describes as "outreach".[6]

Children's services such as schools and playgrounds have become the focus of much risk-averse planning, meaning that children are often prevented from benefiting from activities that they would otherwise have had. Many playgrounds have been fitted with impact-absorbing matting surfaces. However, these are only designed to save children from death in the case of direct falls on their heads and do not achieve their main goals.[7] They are expensive, meaning that fewer resources are available to benefit users in other ways (such as building a playground closer to the child's home, reducing the risk of a road traffic accident on the way to it), and children are likely to attempt more dangerous acts, with confidence in the artificial surface. They grow up with a poorer understanding of risk management. Shiela Sage, an early years school advisor, observes "Children who are only ever kept in very safe places, are not the ones who are able to solve problems for themselves. Children need to be have a certain amount of risk taking ... so they'll know how to get out of situations."[8] There are also classroom courses in risk taking, for example from a business perspective.[9]

A vaccine to protect children against three common diseases was developed and recommended for all children. However, a controversy arose around allegations that it caused autism. These were thoroughly disproved,[10] but even years later, some parents chose to spend significant amounts of their own money on alternatives from private doctors. Similarly, mobile phones may carry some small[11][12] health risk. While most people would accept that unproven risk to gain the benefit of improved communication, others remain so risk averse that they do not.

Risk aversion theory can be applied to many aspects of life and its challenges, for example: