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May 2013 Mean–Variance and Expected Utility: The Borch Paradox
David Johnstone, Dennis Lindley
Statist. Sci. 28(2): 223-237 (May 2013). DOI: 10.1214/12-STS408


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.


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David Johnstone. Dennis Lindley. "Mean–Variance and Expected Utility: The Borch Paradox." Statist. Sci. 28 (2) 223 - 237, May 2013.


Published: May 2013
First available in Project Euclid: 21 May 2013

zbMATH: 1331.91078
MathSciNet: MR3112407
Digital Object Identifier: 10.1214/12-STS408

Keywords: Borch’s paradox , CAPM , Expected utility , Mean–variance , Portfolio theory , probability mixture

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.28 • No. 2 • May 2013
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