Mean–Variance and Expected Utility: The Borch Paradox

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 (µ, σ) 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.

A General Criterion of Choice, with Discussion of Borch Paradox

“…Such a criterion satisfies the first stochastic dominance, the independence condition, and allows explaining the most popular paradoxes encountered in the applications of expected utility. This same criterion provides a simple way of discussing the so-called Borch paradox, recently focused by Johnstone and Lindley [16].…”

“…This example, reported from Johnstone & Lindley [16] and extensively commented on this same paper, considered two prospects, related to two possible binary assets: Asset 1 produces payoff 1 y with probability p and payoff x with probability ( )…”

“…The defining equations for the summary statistics can be simplified by putting 0 x = (i.e. by subtracting the original x from the original values x , 1 y , 2 y ); thus one obtains ( ) For the numerical illustration taken above from Johnstone and Lindley [16], with values x , 1 y , 2 y translated by 5 c = , the system (11)- (13) Another numerical illustration, directly applied for a value 0 x = , is the following: …”

The Simplest Non-Expected Utility Model for Lottery and Portfolio Choices

“…Borch (1969) had a fundamental argument against applying mean/variance efficient frontiers with expected utility choices. He pointed out that the efficient frontier characterization of choice was inconsistent with the expected utility model unless the returns were normally distributed and/or the utility function from which EU was constructed was quadratic (Johnstone and Lindley 2013). Borch (1969) liked the elegance of the EU model and so tended to reject the efficient frontier model of consumer's portfolio possibilities.…”

The CAPM Debate and the Logic and Philosophy of Finance

“…Borch used this apparent contradiction to prove as a general proposition that a decision maker cannot logically be indifferent between two investments by reference merely to their means and variances. The inner workings of Borch's paradox and its eventual resolution by Baron () and others are pieced together by Johnstone and Lindley ().…”