## Info

Risk preference

Select a fair gamble

U"( 0) > 0

Figures 10.1a and 10.1b show preference functions exhibiting alternative properties with respect to risk aversion. Figure 10.1a presents the shape of utility functions in utility of wealth space that exhibit risk aversion, risk neutrality, and risk preference. Figure 10.1 b presents 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.

As discussed in earlier chapters, investors who can state their feelings toward a fair gamble can significantly reduce the set of risky investments they must consider. For example, risk-averse investors must consider only the efficient frontier when choosing among alternative portfolios. Thus, an understanding of utility theory can simplify the selection problem of investors even if they are unwilling to more formally specify their utility function.

The third property of utility functions is an assumption about how the investor's preferences change with a change in wealth. If the investor's wealth increases, will more or less of that wealth be invested in risky assets? For example, assume that an investor with $10,000 to invest puts $5000 into risky assets. Now assume the same investor's wealth increases to $20,000. Will the investor invest more than $5000, less than $5000, or $5000 in risky assets? If the investor increases the amount invested in risky assets as wealth increases, then the investor is said to exhibit decreasing absolute risk aversion. If the investor's investment in risky assets is unchanged as wealth changes, then the investor is said to exhibit constant absolute risk aversion. Finally, if the investor invests fewer dollars in risky assets as wealth increases, then the investor is said to exhibit increasing absolute

Figure 10.1 Characteristics of functions with different risk-aversion coefficients. (1) Utility function of a risk-seeking investor. (2) Utility function of a risk-neutral investor. (3) Utility function of a risk-averse investor.

risk aversion. When we discussed risk aversion, we showed that different degrees of risk aversion were associated with different derivatives of the utility function. A similar result is true for absolute risk aversion. If U'(W) and U"(W) are the first and second derivatives of the utility function at wealth level W, then we show in Appendix B, at the end of this chapter that

can be used to measure an investor's absolute risk aversion. Then A'(W), the derivative of MW) with respect to wealth, is an appropriate measure of how absolute risk aversion behaves with respect to changes in wealth. Table 10.7 summarizes the important relationship between A' (W) and changes in risk aversion and presents an example of a utility function exhibiting each type of behavior described in the table.

Most evidence would indicate that, as wealth increases, the dollar amount invested in

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