An elementary proof of the dual representation of Expected Shortfall

Paper: arXiv 2306.14506

Authors: Martin Herdegen, Cosimo Munari

Abstract

We provide an elementary proof of the dual representation of Expected Shortfall on the space of integrable random variables over a general probability space. Unlike the results in the extant literature, our proof only exploits basic properties of quantile functions and can thus be easily implemented in any graduate course on risk measures. As a byproduct, we obtain a new proof of the subadditivity of Expected Shortfall.

Complexity vs Empirical Score

• Math Complexity: 6.0/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a purely theoretical result, offering a simplified mathematical proof for the dual representation and subadditivity of Expected Shortfall. While it scores well in clarity and mathematical rigor for its pedagogic value, it entirely lacks empirical backtesting or real-world data validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Provide an elementary proof of ES dual representation] --> B[Inputs: Integrable random variables on general probability space]
    B --> C[Methodology: Apply basic properties of quantile functions]
    C --> D[Derive dual representation without advanced functional analysis]
    D --> E[Outcome 1: Accessible proof suitable for graduate courses]
    D --> F[Outcome 2: New direct proof of Expected Shortfall subadditivity]