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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,

(1.5)

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:

Because the investor's expected utility of holding B is higher, investment B is more attractive.

EXHIBIT 1.2 Properties of Two Hedge Fund Indices

Source: HFR and authors' calculations.