Paper: arXiv 2312.01034

Authors: Xia Han, Ruodu Wang, Qinyu Wu

Abstract

Mean-deviation models, along with the existing theory of coherent risk measures, are well studied in the literature. In this paper, we characterize monotonic mean-deviation (risk) measures from a general mean-deviation model by applying a risk-weighting function to the deviation part. The form is a combination of the deviation-related functional and the expectation, and such measures belong to the class of consistent risk measures. The monotonic mean-deviation measures admit an axiomatic foundation via preference relations. By further assuming the convexity and linearity of the risk-weighting function, the characterizations for convex and coherent risk measures are obtained, giving rise to many new explicit examples of convex and nonconvex consistent risk measures. Further, we specialize in the convex case of the monotonic mean-deviation measure and obtain its dual representation. The worst-case values of the monotonic mean-deviation measures are analyzed under two popular settings of model uncertainty. Further, we establish asymptotic consistency and normality of the natural estimators of the monotonic mean-deviation measures.Finally, the monotonic mean-deviation measures are applied to a problem of portfolio selection using financial data.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 5.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a rigorous axiomatic and mathematical framework for monotonic mean-deviation risk measures, including dual representations and asymptotic statistical properties. It complements the deep theoretical derivations with an empirical portfolio selection application, though the work remains predominantly theoretical.

Research Flowchart

  flowchart TD
    A[Research Goal: Characterize Monotonic Mean-Deviation Risk Measures] --> B[Model Construction: Apply Risk-Weighting Function to Deviation Part]
    B --> C[Theoretical Analysis: Axiomatic Foundation, Convexity & Duality Theory]
    C --> D[Model Uncertainty & Estimation: Evaluate Worst-Case Values & Prove Asymptotic Normality]
    D --> E[Empirical Application: Financial Equity Data for Portfolio Selection]
    E --> F[Key Outcomes: New Classes of Consistent Risk Measures & Optimal Portfolio Strategies]