Mean–variance Portfolio Choice: Quadratic Partial Hedging

Abstract

In this paper we investigate the problem of mean-variance portfolio choice with bankruptcy prohibition. For incomplete markets with continuous assets' price processes and for complete markets, it is shown that the mean-variance efficient portfolios can be expressed as the optimal strategies of partial hedging for quadratic loss function. Thus, mean-variance portfolio choice, in these cases, can be viewed as expected utility maximization with non-negative marginal utility.

A Note on the Mean-Variance Criteria for Discrete Time Financial Markets

“…For incomplete markets with continuous assets' price processes and for complete markets, Xia [3] showed that Problem (1.2) is equivalent to the following optimization problem…”

“…Isn't the Markowitz mean-variance portfolio selection theory reasonable? Xia [3] showed that for incomplete markets with continuous assets' price processes and for complete markets, the mean-variance efficient portfolios can be viewed as those of expected utility maximization with non-negative marginal utility. Now we present the main results of Xia [3] .…”

“…Xia [3] showed that for incomplete markets with continuous assets' price processes and for complete markets, the mean-variance efficient portfolios can be viewed as those of expected utility maximization with non-negative marginal utility. Now we present the main results of Xia [3] . Let (Ω, F, P) be a complete probability space, and (F t ) 0≤t≤T be a filtration satisfying the usual conditions, where the positive number T is a fixed and finite time horizon, F 0 is generated by all P-null set and F T = F. We consider a financial market which consists of d + 1 assets which price processes S j , j = 0, 1, · · · , d, are assumed to be strictly positive semi-martingales with càdlàg paths (i.e., the paths that are right continuous and have left limits).…”

“…

It was shown in Xia [3] that for incomplete markets with continuous assets' price processes and for complete markets the mean-variance portfolio selection can be viewed as expected utility maximization with non-negative marginal utility. In this paper we show that for discrete time incomplete markets this result is not true.

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A Unified Approach to Dynamic Mean-Variance Analysis in Discrete and Continuous Time

“…It is well-known that optimizing expected quadratic utility yields a solution to the MV problem, see e.g. Xia (2005). However, quadratic utility generates a negative marginal utility for relatively large levels of wealth and thereby violates the classical assumption of strict monotonicity of the utility function.…”