Risk aversion

Risk aversion is a concept in economics, finance, and psychology related to the behaviour of consumers and investors under 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.

The inverse of a person's risk aversion is sometimes called their risk tolerance (for a more general discussion of the concept, see risk).

Example
A person is given the choice between two scenarios, one certain and one not. In the uncertain scenario, the person is to make a gamble with an equal probability between receiving $100 or nothing. The alternative scenario is to receive a specific dollar amount with certainty (probability of 1).

Investors have different risk attitudes. A person is
 * risk-averse if he or she would accept a certain payoff of less than $50 (for example, $40) rather than the gamble.
 * risk neutral if he or she is indifferent between the bet and a certain $50 payment.
 * risk-seeking (or risk-loving) if the certain payment must be more than $50 (for example, $60) to induce him or her to take the certain option over the gamble.

The average payoff of the gamble, known as its expected value, is $50. The dollar amount accepted instead of the bet is called the certainty equivalent, and the difference between it and the expected value is called the risk premium.

Utility of money
In utility theory, a consumer has a utility function $$U(x_i)$$ where $$x_i$$ are amounts of goods with index $$i$$. From this, it is possible to derive a function $$u(c)$$, of utility of consumption $$c$$ as a whole. Here, consumption $$c$$ is equivalent to money in real terms, i.e. without inflation. The utility function $$u(c)$$ is defined only modulo linear transformation.

The graph shows this situation for the risk-averse player: The utility of the bet,


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

is as big as that of the certainty equivalence, $$CE$$. The risk premium is


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

or 25%.

Absolute risk aversion
The higher the curvature of $$u(c)$$, the higher the risk aversion. However, since expected utility functions are not uniquely defined (only up to linear 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 $$u(c)=1-e^{-\alpha c}$$ 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 $$u(c)=\ln(c), r_u(c)=1/c$$, 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 most consistent with decreasing absolute risk aversion.

Relative risk aversion
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 $$c$$.

In intertemporal choice problems, the elasticity of intertemporal substitution is often the same as the coefficient of relative risk aversion. The "isoelastic" utility function
 * $$u(c) = \frac{c^{1-\rho}}{1-\rho}$$

exhibits constant relative risk aversion with $$R_u(c) = \rho $$. When $$\rho = 1$$ this simplifies to the case of log utility, and the income effect and substitution effect on saving exactly offset.

Portfolio theory
In modern portfolio theory, risk aversion is measured as the marginal reward an investor wants to receive if he takes for a new amount of 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 in 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.


 * $$A = \frac{dE(r)}{d\sigma}$$


 * $$A_n = \frac{dE(r)}{d\sqrt[n]{\mu_n}} = \frac{1}{n} \frac{dE(r)}{d\mu_n} $$

Limitations
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 she 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 article has often been wrongly quoted as a justification for assuming risk neutral behavior of people in small stake gambles.

The major solution to the problem observed by Rabin is the one 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.

Other categories
See "Harm Reduction".

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


 * Bribery and corruption - whether the risk of being implicated or caught outweighs the potential personal or professional rewards
 * Drugs - whether the risk of having a bad trip outweighs the benefits of possible transformative one; whether the risk of defying social bans is worth the experience of alteration.
 * Sex - judgement whether an experience that goes against social convention, ethical mores or common health prescriptions is worth the risk.
 * Extreme sports - having the ability to go against biological predepositions like the fear of height.