Mean-Variance and Expected Utility: The Borch Paradox

TL;DR

This paper examines Borch's paradox, which questions the logical coherence of mean-variance utility models in finance, and reviews the historical debate on their relation to expected utility theory.

Contribution

It provides a detailed analysis of Borch's paradox and the early philosophical arguments connecting mean-variance methods with expected utility theory.

Findings

Borch's paradox challenges the coherence of mean-variance indifference curves.

Historical debate has clarified the logical issues in mean-variance utility models.

The paper reviews key arguments that set Borch's paradox aside.

Abstract

The model of rational decision-making in most of economics and statistics is expected utility theory (EU) axiomatised by von Neumann and Morgenstern, Savage and others. This is less the case, however, in financial economics and mathematical finance, where investment decisions are commonly based on the methods of mean-variance (MV) introduced in the 1950s by Markowitz. Under the MV framework, each available investment opportunity ("asset") or portfolio is represented in just two dimensions by the ex ante mean and standard deviation $(\mu,\sigma)$ of the financial return anticipated from that investment. Utility adherents consider that in general MV methods are logically incoherent. Most famously, Norwegian insurance theorist Borch presented a proof suggesting that two-dimensional MV indifference curves cannot represent the preferences of a rational investor (he claimed that MV indifference curves "do not exist"). This is known as Borch's paradox and gave rise to an important but generally little-known philosophical literature relating MV to EU. We examine the main early contributions to this literature, focussing on Borch's logic and the arguments by which it has been set aside.