Quadratic Utility

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.9 Quadratic utility is

Its derivatives are

As is known, if an investor displays a utility function that exhibits risk aversion, then the second derivative is negative or b must be positive. If the investor is assumed to prefer more to less, then the first derivative must be positive. No matter how small b is, as long as it is positive there is always some value of W that will make the first derivative negative. Thus, for investors who prefer more to less, the quadratic utility function may only represent their preferences over a restricted range of wealth. To be consistent with nonsatiation, the following restriction must be placed on W

1-2bW > 0

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.

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9The variance of a random variable is defined as

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, σ2w can therefore be written as 

or

Taking the expected value of a quadratic utility function provides

Rearranging the previously derived relationship to solve for E(W2) provides

Substituting this for E(W2) in the utility equation shows

Thus, expected utility can be defined in terms of means and variances when it is a quadratic function.