Risk aversion

Risk aversion is a concept in economics, finance, and psychology explaining 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 see the main article risk.

Example
A person is given the choice between a bet of either receiving $100 or nothing, both with a probability of 50%, or instead, receiving some amount with certainty. Now he is risk averse if he would rather accept a payoff of less than $50 (for example, $40) with probability 100% than the bet, risk neutral if he was indifferent between the bet and a certain $50 payment, risk-loving (risk-proclive) if it required that the payment be more than $50 (for example, $60) to induce him to take the certain option over the bet.

The average payoff of the bet, the expected value would be $50. The certain amount accepted instead of the bet is called the certainty equivalent, 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 utility functions are not uniquely defined (only up to monotonic 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:
 * Constant absolute risk aversion (CARA) if $$r_u(c)$$ is constant with respect to $$c$$
 * Decreasing/increasing absolute risk aversion (DARA/IARA) if $$r_u(c)$$ is decreasing/increasing.

Relative risk aversion
The Arrow-Pratt measure of relative risk-aversion (RRA) or coefficient of relative risk aversion is defined as


 * $$R_u(c) = cr_u(c)=\frac{-cu''(c)}{u'(c)}$$.

As 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 $$c$$.

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.

The major solution to this problem is the one proposed by prospect theory.