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The second property of a utility function is an assumption about an investor's taste for risk. Three assumptions are possible: the investor is averse to risk, the investor is neutral toward risk, and the investor seeks risk. Risk aversion, risk neutrality, and risk seeking can all be defined in terms of a fair gamble. Consider the gambles (options) shown in Table 10.5.

The option "invest" has an expected value of (l/2)(2) + (l/2)(0) = $1. Assume that an investor would have to pay $1 to undertake this investment and obtain these outcomes. Thus, if the investor chooses not to invest, the $1 is kept. This is the alternative: do not invest. The expected value of the gamble is exactly equal to the cost. The position of the investor may be improved or hurt by undertaking the investment, but the expectation is that there will be no change in position. Because the expected value of the gamble shown in Table 10.5 is equal to its cost, it is called a fair gamble.

Risk aversion means that an investor will reject a fair gamble. In terms of Table 10.5, it means $1 for certain 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. If U(W) is the utility function and U"(W) is the second derivative, then risk aversion is usually equated with an assumption that U"(W) < 0. Let us examine why this is true.

If an investor prefers not to invest, then the expected utility of not investing must be higher than the expected utility of investing or a(l)>(l/2)C/(2) + (l/2)[/(0)

Multiplying both sides by 2 and rearranging, we have

Examine the preceding expression. The expression means that a one-unit change from 0 to I is more valuable than a one-unit change from 1 to 2. This latter change involves larger values of outcomes. A function where an additional unit increase is less valuable than the last unit increase is a function with a negative second derivative.

The assumption of risk aversion means an investor will reject a fair gamble because the disutility of the loss is greater than the utility of an equivalent gain. Functions that exhibit this property must have a negative second derivative. Therefore, the rejection of a fair gamble implies a negative second derivative.

Risk neutrality means that an investor is indifferent to whether or not a fair gamble is undertaken. In the context of Table 10.5, a risk-neutral investor would be indifferent to whether or not an investment was made. Risk neutrality implies a zero second derivative. 1 ct us examine why. For the investor to be indifferent between investing and not investing, the expected utility of investing, or not investing, must be the same, or f/(l) = (l/2)i/(2) + (l/2)i/(0) Multiplying by 2 and rearranging yields

This expression implies that the change in utility from a one-unit change in wealth is independent of whether we are moving from 0 to 1 or 1 to 2. This characteristic is associated with functions that exhibit a zero second derivative. Thus, indifference to a fair gamble implies a zero second derivative, and utility functions of risk-neutral investors should have zero second derivatives.

Risk seeking means that an investor would select a fair gamble. In the context of Table 10.5, the risk-seeking investor would choose to invest. Risk-seeking investors have utility functions with positive second derivatives. The reason why exactly parallels previous discussion. Since the risk-seeking investor chooses the investment, the expected utility of investment must be higher than the expected utility of not investing, or

Once again, multiplying by 2 and rearranging yields f/(2)-£/(!)> U(1)~ U(0)

This expression indicates that the utility of a one-unit change from 1 to 2 is greater than the utility of a one-unit change from 0 to 1. Functions that exhibit the property of greater change in value for larger unit changes in the argument are functions with positive second derivatives. Thus, the acceptance of a fair gamble implies a positive second derivative. These conditions are summarized in Table 10.6.

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