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(WA)] > E[U(WB)].
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(WC)] = 4.611 and E[U(WD)] = 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
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
1.5.3 Risk Aversion and the Shape of the Utility Function
We are now prepared to introduce a more precise definition of risk aversion. An investor is said to be risk averse if his utility function is concave, which in turn means that the investor requires higher expected return to bear risk. Exhibit 1.1 displays the log function for various values of wealth. We can see that the level of utility increases but at a decreasing rate.
Alternatively, a risk-averse investor avoids taking risks with zero expected payoffs. That is, for risk-averse investors,
1.5.4 Expressing Utility Functions in Terms of Expected Return and Variance
The principle of selecting investment strategies and allocations to maximize expected utility provides a very flexible way of representing the asset owner's preferences for risk and return. The representation of expected utility can be made more operational by presenting it in terms of the parameters of the probability distribution functions of investment choices. The most common form among institutional investors is to present the expected utility of an investment in terms of the mean and variance of the investment returns. That is,
Here, μ is the expected rate of return on the investment, σ2 is the variance of the rate of return, and λ is a constant that represents the asset owner's degree of risk aversion. It can be seen that the higher the value of λ, the higher the negative effect of variance on the expected value. For example, if λ is equal to zero, then the investor is said to be risk neutral and the investment is evaluated only on the basis of its expected return. A negative value of λ would indicate that the investor is a risk seeker and actually prefers more risk to less risk.
The degree of risk aversion indicates the trade-off between risk and return for a particular investor and is often indicated by a particular parameter within a utility function, such as λ in Equation 1.5. The fact that the degree of risk aversion is divided by 2 will make its interpretation much easier. It turns out that if Equation 1.5 is used to select an optimal portfolio for an investor, then the ratio of the expected rate of return on the optimal portfolio in excess of the riskless rate divided by the portfolio's variance will be equal to the degree of risk aversion.
Example: Suppose λ = 5. Calculate the expected utility of investments C and D.
In this case, the expected utility of investment C is higher than that of investment D; therefore, it is the preferred choice. It can be verified that if λ = 1, then the expected utility of investments C and D will be 0.0059 and 0.00949, respectively, meaning that D will be preferred to C.
APPLICATION 1.5.4
Suppose that an investor's expected utility, E[U(W)], from an investment can be expressed as:
where W is wealth, μ is the expected rate of return on the investment, σ2 is the variance of the rate of return, and λ is a constant that represents the asset owner's degree of risk aversion.
Use the expected utility of an investor with λ = 0.8 to determine which of the following investments is more attractive:
Investment A: μ = 0.10 and σ2 = 0.04
Investment B: μ = 0.13 and σ2 = 0.09
The expected utility of A and B are found as:
Investment A:
Investment B:
