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This paper studies a multi-period mean–variance (MV) portfolio selection problem in a market of one risk-free asset and one risky asset traded with proportional transaction costs and no-shorting constraint. A particular interest of this study is to investigate the time consistency in efficiency (TCIE) of the optimal MV portfolio in the presence of transaction costs. To this end, we derive a semi-closed-form solution of the optimal pre-committed dynamic MV policy in the no-shorting non-frictionless market with a combination of embedding and dynamic programming techniques, as well as its several analytical properties. We show that the optimal MV policy is always TCIE when no-shorting constraint is imposed, which gives right to the long-only portfolios in dynamic settings advocated in some empirical evidences. Numerically, we conduct sensitivity analyses of the efficient frontiers and the width of the no-transaction region with respect to the rates of transaction costs and initial wealth allocations. Moreover, we show the significance of the TCIE and the intuitive rationale behind it.

Optimal multi-period transaction-cost-aware long-only portfolios and time consistency in efficiency