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The discrete-time mean-variance portfolio selection formulation, arepresentative of general dynamic mean-risk portfolio selection problems, doesnot satisfy time consistency in efficiency (TCIE) in general, i.e., a truncatedpre-committed efficient policy may become inefficient when considering thecorresponding truncated problem, thus stimulating investors' irrationalinvestment behavior. We investigate analytically effects of portfolioconstraints on time consistency of efficiency for convex cone constrainedmarkets. More specifically, we derive the semi-analytical expressions for thepre-committed efficient mean-variance policy and the minimum-variance signedsupermartingale measure (VSSM) and reveal their close relationship. Ouranalysis shows that the pre-committed discrete-time efficient mean-variancepolicy satisfies TCIE if and only if the conditional expectation of VSSM'sdensity (with respect to the original probability measure) is nonnegative, oronce the conditional expectation becomes negative, it remains at the samenegative value until the terminal time. Our findings indicate that the propertyof time consistency in efficiency only depends on the basic market setting,including portfolio constraints, and this fact motivates us to establish ageneral solution framework in constructing TCIE dynamic portfolio selectionproblem formulations by introducing suitable portfolio constraints.

Mean-Variance Policy for Discrete-time Cone Constrained Markets: The Consistency in Efficiency and Minimum-Variance Signed Supermartingale Measure