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 for 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 at the same
negative value until the terminal time. Our findings indicate that the property
of time consistency in efficiency only depends on the basic market setting,
including portfolio constraints, and this fact motivates us to establish a
general solution framework in constructing TCIE dynamic portfolio selection
problem formulations by introducing suitable portfolio constraints
