Paper: arXiv 2603.10327

Authors: Yang Liu, Yunran Wei, Xintao Ye

Abstract

Various financial market scenarios may cause heterogeneous risk assessments among analysts, which motivates the usage of the Generalized Risk Measure in Fadina et al. (2024, Finance and Stochastics). Effectively synthesizing these diverse assessments avoids over-relying on a single, potentially flawed or conservative forecast and promotes more robust decision-making. Motivated by this, we establish analytical characterizations of the Weighted Generalized Risk Measure (WGRM) under both discrete and continuous settings. Building upon the WGRM, we incorporate the Fundamental Risk Quadrangle (FRQ) in Rockafellar and Uryasev (2013, Surveys in Operations Research and Management Science) into the Weighted Risk Quadrangle (WRQ) and show that the intrinsic relationships among risk, deviation, regret, error, and statistics in FRQ are preserved under weighted aggregation across scenarios. Moreover, we demonstrate that certain complex risk optimization problems under the WGRM can be reformulated as tractable linear programs through the WRQ structure, thus ensuring computational feasibility. Finally, the WGRM and WRQ framework is applied to empirical analyses using constituents of the NASDAQ 100 and S&P 500 indices across recession and expansion regimes, which validates that WGRM-based portfolios exhibit superior risk-adjusted performance and enhanced downside resilience and effectively mitigate losses arising from erroneous single-scenario judgments.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 6.0/10 (capped from 6.5: transaction costs not modelled; see penalty flags)
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a rigorous mathematical extension of Generalized Risk Measures and the Fundamental Risk Quadrangle by incorporating weighted aggregation across heterogeneous scenario assessments. The theoretical derivations are complemented by empirical evaluations using major stock index constituents across different economic regimes.

Research Flowchart

  flowchart TD
    A[Research Goal: Synthesize heterogeneous risk assessments via Weighted Generalized Risk Measure WGRM] --> B[Theory: Extend WGRM to Weighted Risk Quadrangle WRQ]
    B --> C[Optimization: Reformulate complex WGRM problems into tractable Linear Programs]
    D[Inputs: NASDAQ 100 & S&P 500 data across recession & expansion regimes] --> E[Empirical Portfolio Optimization using LP-WRQ framework]
    C --> E
    E --> F[Outcomes: Superior risk-adjusted returns, downside resilience, & mitigation of single-scenario errors]

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0 of 39 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.