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Different asset owners will have their own preferences regarding the trade-off between risk and return. Economists have developed a number of tools for expressing such preferences. Expected utility is the most common approach to specifying the preferences of an asset owner for risk and return. While a utility function is typically used to express preferences of individuals, there is nothing in the theory or application that would prevent us from applying this to institutional investors as well. Therefore, in the context of investments, we define utility as a measurement of the satisfaction that an individual receives from investment wealth or return. Expected utility is the probability weighted average value of utility over all possible outcomes. Finally, in the context of investments, a utility function is the relationship that converts an investment's financial outcome into the investor's level of utility.

Suppose the initial capital available for an investment is W and that the utility derived from W is U(W). Thus, the expected utilities associated with investments A and B can be expressed as follows:

(1.1)

(1.2)

The function U(•) is the utility function. The asset owner would prefer investment A to investment B if E[U(W_A)] > E[U(W_B)].

Suppose the utility function can be represented by the log function, and assume that the initial investment is $100. Then:

(1.3)

(1.4)

In this case the asset owner would prefer investment A to investment B because it has higher expected utility. Applying the same function to investments C and D, it can be seen that E[U(W_C)] = 4.611 and E[U(W_D)] = 4.615. In this case, the asset owner would prefer investment D to investment C.

APPLICATION 1.5.2

Suppose that an investor's utility is the following function of wealth (W):

Find the current and expected utility of the investor if the investor currently has $100 and is considering whether to speculate all the money in an investment with a 60 % chance of earning 21 % and a 40 % chance of losing 19 %. Should the investor take the speculation rather than hold the cash?

The current utility of holding the cash is 10, which can be found as . The expected utility of taking the speculation is found as:

Because the investor's expected utility of holding the cash is only 10, the investor would prefer to take the speculation, which has an expected utility of 10.2.

EXHIBIT 1.1 Logarithmic Utility Function