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Risk aversion

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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. 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 is likely to have high returns, but also has a chance of becoming worthless.

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

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[edit] Example

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.

[edit] Utility of money

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.

The risk premium is

(\$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.

[edit] Measures of risk aversion

[edit] 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 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 − αc is unique in exhibiting constant absolute risk aversion (CARA): ru(c) = α is constant with respect to c.
  • Decreasing/increasing absolute risk aversion (DARA/IARA) if ru(c) is decreasing/increasing. An example for a DARA utility function is u(c) = log(c),ru(c) = 1 / c, while u(c) = c − αc2,α > 0,ru(c) = 2α / (1 − 2αc) would represent a utility function exhibiting IARA.
  • Experimental and empirical evidence is mostly consistent with decreasing absolute risk aversion.[citation needed]
  • 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. In other words, 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 remain unidentified.[1]

[edit] 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. 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 Ru(c) = ρ and the elasticity of intertemporal substitution \varepsilon_{u(c)} = 1/\rho. When ρ = 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.

[edit] Portfolio theory

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}

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

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.

[edit] Other categories

See "Harm Reduction".

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

[edit] See also

[edit] External links

[edit] References

  1. ^ Bellemare, Marc F. and Zachary S. Brown, On the (Mis)Use of Wealth as a Proxy for Risk Aversion[1], Working Paper, Duke University.
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