Mean‐variance Policy for Discrete‐time Cone‐constrained Markets: Time Consistency in Efficiency and the Minimum‐variance Signed Supermartingale Measure

Abstract

The discrete‐time mean‐variance portfolio selection formulation, which is a representative of general dynamic mean‐risk portfolio selection problems, typically does not satisfy time consistency in efficiency (TCIE), i.e., a truncated precommitted efficient policy may become inefficient for the corresponding truncated problem. In this paper, we analytically investigate the effect of portfolio constraints on the TCIE of convex cone‐constrained markets. More specifically, we derive semi‐analytical expressions for the precommitted efficient mean‐variance policy and the minimum‐variance signed supermartingale measure (VSSM) and examine their relationship. Our analysis shows that the precommitted discrete‐time efficient mean‐variance policy satisfies TCIE if and only if the conditional expectation of the density of the VSSM (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 finding indicates that the TCIE property depends only on the basic market setting, including portfolio constraints. This motivates us to establish a general procedure for constructing TCIE dynamic portfolio selection problems by introducing suitable portfolio constraints.

Time-Consistent Strategies for the Generalized Multiperiod Mean-Variance Portfolio Optimization Considering Benchmark Orientation

“…The classical mean-variance portfolio selection theory was first introduced by Markowitz [1] and was limited to the single-period investment situation. As far as we know, the multiperiod portfolio optimization problem is deemed to be one of the most significant extensions of the pioneering work of Markowitz [1], and it has received considerable attention in recent years (e.g., Li and Ng [2], Leippold et al [3], Wei and Ye [4], Yao et al [5], Chen et al [6], Cui et al [7], Liu and Chen [8], Zhou et al [9] and so on). Most of the existing studies mainly assume that the investors only focus on their own performance and formulate the corresponding investment strategy accordingly.…”

Big Data Challenges of High‐Dimensional Continuous‐Time Mean‐Variance Portfolio Selection and a Remedy

“…For instance, Li et al (6) and Gao and Li (7) establish the theory and implementation of MVP with cardinality constraints, Chiu and Li (8) extend the continuous-time MVP strategy to asset-liability management, and Chiu and Li (9) and Chiu, Wong, and Li (10) investigate the connection between MVP and the safety-first criterion, which is closely related to the failure probability. (11) Cui et al (12) and Cui, Li, and Li (13) develop the mean-variance (MV) framework with cone and noshorting constraints. In addition, studies have found that the dynamic MV criterion is useful in pairs trading (14) and supply-chain management.…”

“…Hence, the first two claims are established. Based on Equations (13) and (14), simple calculation yields:…”

Time-consistent multiperiod mean semivariance portfolio selection with the real constraints

“…Wu and Zeng(2015) [42] discussed the time-consistent strategy for a generalized multiperiod mean variance criterion for defined-contribution pension schemes with mortality risk. Readers may refer to Lioui(2013) [ [11] and Liu and Chen(2018) [25] for the discussion of the time-consistent strategies for different multiperiod portfolio optimization models.…”