Peer-reviewed literature demonstrates that under expected utility theory, a mean-variance framework or preference ordering is consistent with or justified by quadratic utility functions.
THE CHOICE MODEL in which alternatives are ordered in terms of the mean and variance of their return has been utilized in a variety of fields with the best developed, both theoretically and empirically, being that of portfolio selection. Since the pioneering works of Markowitz [22] and Tobin [32], a series of results has been obtained regarding the utility theoretic foundations of this model. While these results are widely known, they continue to be the source of discussion.' For example, Markowitz demonstrated that if the ordering of alternatives is to satisfy the von Neumann-Morgenstern (NM) [35] axioms of rational behavior, only a quadratic (NM) utility function is consistent with an ordinal expected utility function that depends solely on the mean and variance of the return. Consequently, even if the return for each alternative has a normal distribution, the mean-variance framework cannot be used to rank alternatives consistently with the NM axioms unless a quadratic NM utility function is specified. The implications of this restriction are disturbing not only because of the undesirable properties of a quadratic utility function but also because, for example, the indifference curves in the mean-standard deviation plane are concentric circles with the center on the mean axis. Furthermore, a quadratic utility function leads to a rather disquieting result in portfolio theory, since it implies that in equilibrium each investor holds an equal percentage of every security (Moss
The mean-variance analysis of portfolio selection has come under heavy attacks in recent years. Borch (1969) has demonstrated that any mean standard deviation indifference curves that are drawn upward sloping can be shown to be inconsistent with the basic axiom of choice under uncertainty. Feldstein (1969) has shown that mean standard deviation indifference curves need not be convex downward, as is usually assumed to be the case, but might change from convex to concave, thus making the usual tangency solution for optimum portfolio of rather dubious value. These criticisms shake the very theoretical foundation of the prevalent portfolio theories, which are built upon the assumption of a system of upward-sloping and convex return-risk indifference curves for every investor. Tobin (1969), one of the pioneers in this field, was forced to concede that the mean-variance approach is justified only when the utility functions of investors are quadratic, or when the outcomes of all investments can be assumed to be normally distributed. Such a defensive position, however, is still vulnerable to attacks. On the one hand, it is now generally recognized that quadratic utility function has an unrealistic implication of increasing absolute risk aversion with respect to wealth, apart from its limited range of applicability. On the other hand, in view of the fact that prices of most stocks and commodities cannot become negative, it is not always appropriate to assume a normal distribution for
A new principle for choosing portfolios based on historical returns data is introduced; the optimal portfolio based on this principle is the solution to a simple linear programming problem. This principle uses minimum return rather than variance as a measure of risk. In particular, the portfolio is chosen that minimizes the maximum loss over all past observation periods, for a given level of return. This objective function avoids the logical problems of a quadratic (nonmonotone) utility function implied by mean-variance portfolio selection rules. The resulting minimax portfolios are diversifie
Portfolio diversification is a central theme in modern investment theory. We revisit the classic return–risk trade-off and propose an alternative objective Qλ(w)=μ⊤w+λ(w⊤Σw)−1/2 that balances higher expected returns with a direct penalty on portfolio volatility via the inverse standard deviation. This objective belongs to the axiomatic class of mean–variance preferences (as formalised by [1] for additively separable forms) and admits tractable solutions, including a closed-form characterisation in the two-asset case. In rolling out-of-sample backtests on standard Fama–French equity portfolios
"The requirement of positive marginal utility only makes it possible to derive a restricted twofund separation theorem for portfolio selection problems replacing the original separation theorem of Cass and Stiglitz (1970). We use our findings for a re-examination of the bias-in-beta problem in mutual funds performance evaluation and of the relevance of the standard CAPM without borrowing restrictions. We also present empirical evidence for the only limited validity of the separation theorem when explicitly recognizing positive marginal utility. Moreover, quadratic utility functions are not apt
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