A characterization of the distributions that imply mean—Variance utility functions☆
If there is a riskless asset, then the distribution of every portfolio is determined by its mean and variance if and only if the random returns are a linear transformation of a spherically distributed random vector. If there is no riskless asset, then the spherically distributed random vector is replaced by a random vector in which the last n − 1 components are spherically distributed conditional on the first component, which has an arbitrary distribution. If the number of assets is infinite, then there must exist random variables m, v, y, where the distribution of y conditional on m and v is standard normal, such that every portfolio is distributed as some linear combination of m and vy. If there is a riskless asset, then m has zero variance. These distributions exhibit two-fund separability even if the utility function is not concave.