Elementary proof that mean–variance implies quadratic utility

Elementary proof that mean–variance implies quadratic utility

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Article ID: iaor20111425
Volume: 70
Issue: 2
Start Page Number: 149
End Page Number: 155
Publication Date: Feb 2011
Journal: Theory and Decision
Authors: ,
Keywords: utility
Abstract:

An extensive literature overlapping economics, statistical decision theory and finance, contrasts expected utility [EU] with the more recent framework of mean–variance (MV). A basic proposition is that MV follows from EU under the assumption of quadratic utility. A less recognized proposition, first raised by Markowitz, is that MV is fully justified under EU, if and only if utility is quadratic. The existing proof of this proposition relies on an assumption from EU, described here as ‘Buridan's axiom’ after the French philosopher's fable of the ass that starved out of indifference between two bales of hay. To satisfy this axiom, MV must represent not only ‘pure’ strategies, but also their probability mixtures, as points in the (σ, μ) plane. Markowitz and others have argued that probability mixtures are represented sufficiently by (σ, μ) only under quadratic utility, and hence that MV, interpreted as a mathematical re‐expression of EU, implies quadratic utility. We prove a stronger form of this theorem, not involving or contradicting Buridan's axiom, nor any more fundamental axiom of utility theory.

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