Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Weighted Generalized Risk Measure and Risk Quadrangle: Characterization, Optimization and Application

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.

Weighted Generalized Risk Measure and Risk Quadrangle: Characterization, Optimization and Application

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.
Paper Structure (15 sections, 8 theorems, 73 equations, 3 figures, 5 tables)

This paper contains 15 sections, 8 theorems, 73 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Connection to Other Frameworks and Contribution
  4. Weighted Generalized Risk Measures
  5. Properties and Technical Discussions
  6. WGRM under a Discrete Setting
  7. WGRM under a Continuous Setting
  8. Weighted Risk Quadrangle
  9. Fundamental Risk Quadrangle Overview
  10. Weighted Risk Quadrangle
  11. Optimization under WGRM and WRQ
  12. Empirical Study
  13. Empirical Setup
  14. Baseline Results
  15. Sensitivity Analysis

Key Result

Theorem 1

(1) The aggregation function $f: \mathbb{R}^{n} \to \mathbb{R}$ satisfies (B1)-(B4) if and only if there exists a closed convex set of weights $\mathcal{W}_{1} \subseteq \mathcal{D} \subseteq \mathbb{R}^{n}$, such that where $\Phi_{\mathcal{Q}, X}^{q} := ( \Psi(X|P_{(1)}), \Psi(X|P_{(2)}),\dots,\Psi(X|P_{(n)}) )^{T}$ is the vector of order statistics obtained by sorting the individual risk assess

Figures (3)

  • Figure 1: Fundamental Risk Quadrangle in Rockafellar2013.
  • Figure 2: A Schematic Procedure for Department Managers to Aggregate Heterogeneous Risk Assessments.
  • Figure 3: Baseline Portfolio Daily Return.

Theorems & Definitions (27)

  • Definition 1
  • Definition 2
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • Remark 4
  • Proposition 1
  • Theorem 2
  • Proposition 2
  • ...and 17 more