MEAN‐VARIANCE POLICY FOR DISCRETE‐TIME CONE‐CONSTRAINED MARKETS: TIME CONSISTENCY IN EFFICIENCY AND THE MINIMUM‐VARIANCE SIGNED SUPERMARTINGALE MEASURE |
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Authors: | Xiangyu Cui Duan Li Xun Li |
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Institution: | 1. Shanghai University of Finance and Economics, and Key Laboratory of Mathematical Economics (SUFE), Ministry of Education, China;2. The Chinese University of Hong Kong;3. The Hong Kong Polytechnic University |
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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. |
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Keywords: | cone‐constrained market discrete‐time mean‐variance policy time consistency in efficiency minimum‐variance signed supermartingale measure |
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