Question

Please write the equation associated with (definition of) the Arrow- Pratt risk aversion coefficient, indicating the likely values (positive or negative) of the first and second derivatives of the utility function.

Answer 1

Risk aversion is the behavior of humans (especially consumers and investors), who, when exposed to uncertainty, attempt to lower that uncertainty. It is the hesitation of a person to agree to a situation with an unknown payoff rather than another situation with a more predictable payoff but possibly lower expected payoff. For example, a risk-averse investor might choose to put their money into a bank account with a low but guaranteed interest rate, rather than into a stock that may have high expected returns, but also involves a chanceof losing value. Arrow and Pratt's original measure used wealth as the argument in the Bernoulli function, so for wealth w, the Arrow-Pratt measure of risk-aversion is -u"(w)/u'(w). This has, in fact, become the traditional way in which the measure is used. In simple terms, what we are measuring above is the actual dollar amount an individual will choose to hold in risky assets, given a certain wealth level w. For this reason, the measure described above is referred to as a measure of absolute risk-aversion.Risk-aversion implies that the second derivative of utility, with respect to wealth, is negative. The assumption of risk-aversion means an investor will reject a fair gamble, because the decrease in utility caused by the loss is greater than the increase in utility of an equivalent gain.The expected utility theorem is based on a set of four axioms concerning investor behaviour. The first principle required of a utility function is that is consistent with more being preferred to less. This attribute, known as nonsatiation, states simply that the utility of more (X + 1) pounds is always higher than the utility of less (X) pounds. Thus, of a choice between alternative investments, an investor will always choose that with the largest expected payoff. Therefore, the first restriction placed on a utility function is that it has a positive first derivative.The second principle of a utility function is an assumption of an investor's taste for risk. Three assumptions are possible: the investor is either averse to risk, neutral towards risk, or seeks risk. Risk-aversion means that an investor will reject a fair gamble. For example, a certain return of £1 will be preferred to an equal chance of £2 or £0. Risk-aversion implies that the second derivative of utility, with respect to wealth, is negative.6 The assumption of risk-aversion means an investor will reject a fair gamble, because the decrease in utility caused by the loss is greater than the increase in utility of an equivalent gain. There is general agreement in the literature that most investors exhibit decreasing absolute risk aversion. However, there is doubt concerning relative risk aversion. Generally, it is assumed that investors exhibit constant relative risk aversion. However, the justification for this is one of tractability rather than a belief in its descriptive validity. If an investor can nominate the state of relative risk aversion that best describes his preferences, he can again reduce the number of portfolios to be considered, or further restrict the utility functions that may describe his behaviour.