Conditional value-at-risk under reward-penalty mechanism with applications to robust portfolio management

Paper: arXiv 2610.09246

Authors: Jun Cai, Tiantian Mao, Zhiqiao Song

Abstract

In this paper, we present robust portfolio selection models by incorporating a reward and penalty mechanism into portfolio management. We assume that the joint distribution of the losses of the underlying risky assets in a portfolio is uncertain but lies within a multivariate distribution set. Our goal is to identify optimal portfolio allocations by minimizing the worst-case conditional value-at-risk (CVaR) of portfolio loss under the reward and penalty mechanism and distribution uncertainty. Our models can also be used to investigate the problem of how to balance portfolio losses with their associated downside risk in portfolio management. We first derive an explicit closed-form expression for the worst-case CVaR under the reward-penalty mechanism, which generalizes several existing models and results regarding the worst-case CVaR, such as those studied in Jagannathan (1977), Chen et al. (2011), and Cai et al. (2024). We then apply this expression to obtain optimal portfolio allocations that minimize the worst-case CVaR under both a classical mean-covariance-based multivariate distribution set and a generalized mean-covariance-based multivariate distribution set introduced in Kang et al. (2019). Additionally, we utilize real market data to illustrate the application of the proposed models and the corresponding optimal portfolio allocations in portfolio management. Our empirical experiments show that portfolios based on the proposed models have the potential to outperform those based on several existing related models. Furthermore, the results demonstrate that incorporating downside risk into portfolio loss helps better manage risk and can achieve higher investment returns than considering either downside risk or portfolio loss alone. Moreover, our experiments reveal the trade-off between improving expected portfolio return and controlling the worst-case portfolio CVaR.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 6.0/10 (capped from 7.0: transaction costs not modelled; see penalty flags)
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper presents a mathematically sophisticated approach to robust portfolio optimization by introducing a novel reward-penalty mechanism within the CVaR framework. It demonstrates strong empirical application with real market data, showing potential outperformance and clear trade-offs. The theoretical derivations are rigorous, generalizing existing models.

Research Flowchart

  flowchart TD
    A[Research Goal: Robust Portfolio Selection with Reward-Penalty & Uncertainty] --> B{Key Methodology: Minimize Worst-Case CVaR};
    B --> C{Derive Closed-Form for Worst-Case CVaR (R-P)};
    C --> D[Data/Inputs: Real Market Data];
    D --> E{Computational Process: Apply Expression to Optimize Allocations};
    E --> F[Outcomes: Outperformance, Better Risk Management, Higher Returns, Trade-off Analysis];

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0 of 41 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.