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

TL;DR

This paper introduces the Weighted Generalized Risk Measure (WGRM) and Risk Quadrangle (WRQ), providing analytical characterizations, optimization reformulations, and empirical validation for robust financial risk assessment and decision-making.

Contribution

It develops the WGRM and WRQ frameworks, extending existing risk measures, and demonstrates their tractability and effectiveness in empirical financial portfolio management.

Findings

WGRM-based portfolios outperform traditional risk measures in risk-adjusted returns.

The WRQ allows complex risk problems to be solved as linear programs.

Empirical analysis shows improved downside resilience in stock portfolios.

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.