The preceding section discussed the construction of an opportunity set of investments and how, with some general assumptions about preferences, an investor could limit this set to the efficient frontier. If risk-free lending and borrowing can take place at the same rate in unlimited quantities, then there is only one portfolio of risky assets that will be preferred by a risk-avoiding investor. The investor's preference function determines what combination of the optimum risky portfolio and risk-free asset should be held: it also indicates how much the investor will invest and consume.
This analysis, however, assumed that the mean and variance of an asset contain all the relevant information required to solve the decision problem. In the following section no such assumption is made, with the analysis here being applicable whether the relevant decision space is formulated in terms of either the first two or higher moments of the probability distribution of returns. The following section also discusses the desirable economic features of preference functions, and the types of preference functions that possess them. An understanding of utility theory can simplify the examination of the selection problem facing investors, even if investors are unwilling or unable to formally specify their utility function. By expressing an investor's preference towards a fair gamble,5 the set of risky investments to be considered may be significantly reduced.
5A fair gamble is one where the expected value of a gamble, is equal to its cost.