# Risk aversion

Risk aversion is a concept in economics and finance theory explaining the behaviour of consumers and investors under uncertainty. Risk aversion occurs when a person is willing to accept a lower expected payoff if it means they can have a more predictable outcome. For a more general discussion see the main article risk.

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## Example

A person is given the choice between a bet of either receiving \$100 or nothing, both with a probability of 50%. Now she is risk averse if she would accept a payoff of \$40 with probability 100%, risk neutral if she would accept no less than \$50 or risk-loving (risk-proclive) for a payoff of \$60.

(The average payoff of the bet, the expected value would be \$50. The amount accepted is called the certainty equivalent, the difference between it and expected value the risk premium).

## Utility of money

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Risk_aversion.jpg
Risk aversion example

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

The graph shows this situation for the risk-averse player: The utility of the bet, [itex]E(u)=(u(0)+u(100))/2[itex] is as big as that of the certainty equivalence, [itex]CE[itex]. The risk premium is [itex](\$50-\$40)/\$40[itex] or 25%.

## Measures of risk aversion

The higher the curvature of [itex]u(c)[itex], the higher the risk aversion. However, since utility functions are not uniquely defined (only up to linear 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

[itex]r_u(c)=-u''(c)/u'(c)[itex].

The following expressions relate to this term:

• Constant absolute risk aversion (CARA) if [itex]r_u(c)[itex] is constant with respect to [itex]c[itex]
• Decreasing/increasing absolute risk aversion (DARA/IARA) if [itex]r_u(c)[itex] is decreasing/increasing.

Also used is the Arrow-Pratt measure of relative risk-aversion (RRA) or coefficient of relative risk aversion, which is defined as [itex]R_u(c) = x*r_u(c) = -x u'(c)/u''(c)[itex]. As above, 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 [itex]c[itex].

## Limitations

The notion of (constant) risk aversion has come under criticism from behavioral economics. According to Matthew Rabin of 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.

Rabin claims that this effect brings doubt upon the applicability of utility theory. However, it should be noted that examples such as this one hinge on the assumption that the consumer turns down the low-stakes gamble no matter what their initial wealth. The usual criticism of this assumption is that the existence of such a consumer is highly dubious. See  (http://arielrubinstein.tau.ac.il/papers/rabin3.pdf) for more on this.

A perhaps more believable criticism of utility theory is the difficulty by laypersons of estimating very small probabilities and the preference of gains versus losses (loss aversion).

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