Traditional portfolio optimization typically relies on a single fitted return distribution, thus ignores non-normality and uncertainty across distributional states. Mean–variance–ambiguity (MVA) models, motivated by the smooth-ambiguity preferences of Klibanoff, Marinacci, and Mukerji (2005) and the KMM-to-MVA Arrow–Pratt reduction of Maccheroni, Marinacci, and Ruffino (2013), extend conventional risk penalties to account for distributional uncertainty, but they missed how state-conditional return distributions affect portfolio optimization stability. We develop Deep Mean–Variance–Ambiguity (Deep MVA) to resolve this gap by learning state-conditional return distributions and embedding the resulting conditional information loss into the ambiguity penalty. The resulting local certainty equivalent combines reduced-law mean–variance risk with a quadratic penalty for second-order disagreement. Conditional-mixture state resolution yields a Deep ambiguity covariance equal to the Traditional ambiguity covariance plus a positive-semidefinite state-feature increment. This increment is the conditional L2 loss from averaging state-specific distributions before optimization. On compact convex feasible sets, we show that the state-resolving program has stronger ambiguity-relevant curvature, lower mean-input sensitivity, and a unique global solution under restricted positive curvature or a common ridge; its projected iterations converge linearly. An LSTM-MDN learns state-conditional distribution densities from 60-minute bars, and Probability and VoV scores enter weekly MVA, with GARCH_Resid as a comparator. Tests on the SSE 50, CSI 300, and CSI 500 universes from 2017–2025 find statistically significant Sharpe out-of-sample improvements over Traditional MVA and MVO across all three universes and ambiguity specifications.

We propose a clustering-based dimensionality reduction approach to minimum variance portfolio optimization. Rather than constructing the global minimum variance (GMV) portfolio over the full stock universe, which is subject to severe estimation error due to high-dimensionality, we apply Ward’s hierarchical clustering to partition stocks into groups of similarly behaving assets, select one representative per cluster, and optimize on the resulting low-dimensional sub-universe. We show theoretically that clustering preserves the factor structure and yields a better-conditioned covariance matrix than random selection. Empirically, on the Chinese A-share market, the proposed strategies substantially outperform the full-universe benchmark, with gains robust to transaction costs.

We examine the risk-return trade-off in market anomalies within the A-share market, showing that even decaying anomalies may proxy for latent risk factors. To balance forecast bias and variance, we integrate the 1/N and mean-variance frameworks, minimizing out-of-sample forecast error. Treating anomalies as tradable assets, we construct optimized long-short portfolios with strong performance: an average annualized Sharpe ratio of 1.56 and a certainty-equivalent return of 29.4% for a mean-variance investor. These premiums persist post-publication and are largely driven by liquidity risk exposures. Our results remain robust to market frictions, including short-sale constraints and transaction costs. We conclude that even decaying market anomalies may reflect priced risk premia rather than mere mispricing. This research provides practical guidance for academics and investors in return predictability and asset allocation, especially in the unique context of the Chinese A-share market.

Abstract: It is negative risk if there is a good chance of coming out better than our reference level. This paper proposes a general risk measure: bilateral partial moment, where downside risk is supplemented with the "upside potential". Variance, mean absolute deviation, semi-variance and other downside risk definition are all incorporated in this framework. The portfolio selection problems in this general class of risk model are discussed. The portfolio optimization provides the flexibility for the selection of an appropriate target return and the weightiness of upside potentials.