The Economic Properties of Utility Functions

In this section we discuss how knowledge of the properties of utility functions coupled with partial information on an investor's preferences can provide an insight into the process of rational choice. 

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. 

A risk-neutral investor is indifferent as to whether a fair gamble is undertaken or not, and thus implies a zero second derivative. 

Risk-seeking means that an investor would select a fair gamble, unlike the risk averse investor; see above. 

Functions that exhibit greater change in value for larger unit changes in an argument are functions with positive second derivatives. Thus the acceptance of a fair gamble implies a positive second derivative. These conditions may be summarised as follows: 

Condition Definition Implication
Risk aversion Reject fair gamble U''(0) <0
Risk neutrality Indifferent to fair gamble U''(0) =0
Risk preference Select a fair gamble U''(0) >0

Figure 3.8 below shows preference functions exhibiting alternative properties with respect to risk aversion. Figure 3.8a represents the shape of utility functions - in utility of wealth space - that exhibit risk aversion, risk neutrality, and risk preference. Figure 3.8b represents the shape of the indifference curves in expected return standard deviation space -that would be associated with each of these three types of utility functions.

The third principle of a utility function is an assumption as to how investor preferences change with fluctuations in wealth. If an investor increases the amount invested in risky assets as wealth increases, that investor is said to exhibit decreasing absolute risk aversion. If an investor's investment in risky assets remains the same as wealth changes,

that investor is said to exhibit constant absolute risk aversion. Finally, if an investor invests less in risky assets as wealth increases, that investor is said to exhibit increasing absolute risk aversion. As previously discussed, different degrees of risk aversion maybe associated with different derivatives of the utility function. A similar result applies to absolute risk aversion. If U'(W) and U"(W) are the first and second derivatives of the utility function at wealth (level) W, then it has been shown that7

can be used to measure an investor's absolute risk aversion. Thus, A'(W), the derivative of A(W) with respect to wealth, is a measure of how absolute risk aversion behaves with changes in wealth. These conditions may be summarised as follows:

Condition Definition
Property of A'(W )8
Increasing absolute risk aversion As wealth increases hold lesser monetary value in risky assets A'(W) >0
Constant absolute risk aversion As wealth increases hold same monetary value in risky assets A'(W)=0
Decreasing absolute risk aversion As wealth increases hold more monetary value in risky assets A'(W) <0

Therefore, if investor preferences towards absolute risk aversion can be defined, the number of possible options required to be considered can be further reduced. Furthermore, this assumption restricts the possible utility functions that can be used to describe preferences. 

The fourth and final principle used to restrict an investor's utility function is that of the percentage of wealth invested in risky assets - not nominal investment as described above - changing as wealth fluctuates. For example, an investor investing 60% of his wealth in risky assets, whether his wealth is W or 2W. The investor's behaviour is then said to be characterised by constant relative risk aversion. If as his wealth increases an investor invests a greater percentage in risky investments, he is said to exhibit decreasing relative risk aversion: if he invests a smaller percentage, he is said to exhibit increasing relative risk aversion. 

Relative risk aversion is closely related to absolute risk aversion: it refers to the change in percentage of investment in risky assets as wealth changes. Absolute risk aversion refers to the change in the absolute amount invested in risky assets as wealth changes. The measure of relative risk aversion has been shown to be7

If R'(W) is the first derivative of W, then R'(W) < 0 indicates that the utility function exhibits decreasing relative risk aversion. If R'(W) = 0, than the utility function is said to exhibit constant relative risk aversion. Finally, if R'(W) > 0, then the function is said to exhibit increasing relative risk aversion. These conditions may be summarised as follows:
Condition Definition
Property of R'(W )
Increasing relative risk aversionPercentage invested in risky assets declines as wealth increases
R'(W) >0
Constant relative risk aversionPercentage invested in risky assets is unchanged as wealth increases R'(W)=0
Decreasing relative risk aversionPercentage invested in risky assets increases as wealth increases R'(W) <0

There is general agreement in the literature that most investors exhibit decreasing absolute risk aversion. However, there is doubt concerning relative risk aversion.8 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.

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7See, for example Mossin [1973] or Elton & Gruber [1991, pp. 206-209].

8For example, see Lease, Lewellen & Schlarbaum [1974], Cohn, Lewellen, Lease & Schlarbaum [1975] and Baker, Hargrove & Haslem [1977].