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Risk Aversion
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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%, or instead, a certain (100% probability) payment. 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%.


MEASURES OF RISK AVERSION


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 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

r_u(c)=-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)=-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.


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.''

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.

Experiments with primates suggest risk aversion is a hard-wired biological behaviour {Link without Title} .


SEE ALSO



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