On the foundations of mean-variance analyses

Abstract

Let (μ, σ) and (μ′, σ′) be mean-standard deviation pairs of two probability distributions on the real line. Mean-variance analyses presume that the preferred distribution depends solely on these pairs, with primary preference given to larger mean and smaller variance. This presumption, in conjunction with the assumption that one distribution is better than a second distribution if the mass of the first is completely to the right of the mass of the second, implies that (μ, σ) is preferred to (μ′, σ′) if and only if either μ > μ′ or (μ = μ′ and σ < σ′), provided that the set of distributions is sufficiently rich. The latter provision fails if the outcomes of all distributions lie in a finite interval, but then it is still possible to arrive at more liberal dominance conclusions between (μ, σ) and (μ′, σ′).