In financial economics, the utility function most frequently used to describe investor behaviour is the quadratic utility function. Its popularity stems from the fact that, under the assumption of quadratic utility, mean-variance analysis is optimal. Its derivatives are As is known, if an investor displays a utility function that exhibits risk aversion, then the second derivative is negative or 1-2 The absolute and relative risk aversion measures are As shown, the absolute risk aversion function demonstrates that the quadratic utility function exhibits increasing absolute risk aversion. Thus, the quadratic function is consistent with investors who reduce the nominal amount invested in risky assets as their wealth increases. By definition, a quadratic utility function must exhibit increasing relative risk aversion. This is confirmed by the above relative risk aversion function. Thus, quadratic utility functions have characteristics that are undesirable. Accordingly, the next section will consider real estate's role within a mixed-asset portfolio without the imposition of a restriction on the distribution of returns. This analysis is applicable whether the relevant decision space is formulated in terms of the first two or higher moments of the probability distribution of returns. ___________________________
Squaring the function, provides As is known, the expected value of the sum of random variables is the sum of the expected values. Therefore As is also known, the expected value of a constant times a random variable is the constant times the expected value of the random variable, σ or Taking the expected value of a quadratic utility function provides Rearranging the previously derived relationship to solve for Substituting this for |