Mean-Variance Policy for Discrete-Time Cone Constrained Markets: Time Consistency in Efficiency and Minimum-Variance Signed Supermartingale Measure
37 Pages Posted: 18 Mar 2014 Last revised: 19 Apr 2020
Date Written: March 17, 2014
Abstract
The discrete-time mean-variance portfolio selection formulation, a representative of general dynamic mean-risk portfolio selection problems, does not satisfy time consistency in efficiency (TCIE) in general, i.e., a truncated pre-committed efficient policy may become inefficient when considering the corresponding truncated problem, thus stimulating investors' irrational investment behavior. We investigate analytically effects of portfolio constraints on time consistency of efficiency for convex cone constrained markets. More specifically, we derive the semi-analytical expressions of the pre-committed efficient mean-variance policy and the minimum-variance signed supermartingale measure (VSSM) and reveal their close relationship. Our analysis shows that the pre-committed discrete-time efficient mean-variance policy satisfies TCIE if and only if the conditional expectation of VSSM's density (with respect to the original probability measure) is nonnegative, or once the conditional expectation becomes negative, it remains the same negative value until the terminal time. Our finding indicates that the property of time consistency in efficiency only depends on the basic market setting, including portfolio constraints, and motivates us to establish a general solution framework in constructing TCIE dynamic portfolio selection problem formulations by introducing suitable portfolio constraints.
Keywords: cone constrained market, discrete-time mean-variance policy, time consistency in efficiency, minimum-variance signed supermartingale measure
JEL Classification: G11
Suggested Citation: Suggested Citation
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