Elicitability and identifiability of tail risk measures

ArXiv ID: 2404.14136 “View on arXiv”

Authors: Unknown

Abstract

Tail risk measures are fully determined by the distribution of the underlying loss beyond its quantile at a certain level, with Value-at-Risk, Expected Shortfall and Range Value-at-Risk being prime examples. They are induced by law-based risk measures, called their generators, evaluated on the tail distribution. This paper establishes joint identifiability and elicitability results of tail risk measures together with the corresponding quantile, provided that their generators are identifiable and elicitable, respectively. As an example, we establish the joint identifiability and elicitability of the tail expectile together with the quantile. The corresponding consistent scores constitute a novel class of weighted scores, nesting the known class of scores of Fissler and Ziegel for the Expected Shortfall together with the quantile. For statistical purposes, our results pave the way to easier model fitting for tail risk measures via regression and the generalized method of moments, but also model comparison and model validation in terms of established backtesting procedures.

Keywords: Risk Measures, Value-at-Risk (VaR), Expected Shortfall, Elicitability, Tail Risk, General Financial Markets

Complexity vs Empirical Score

• Math Complexity: 9.2/10
• Empirical Rigor: 2.5/10
• Quadrant: Lab Rats
• Why: The paper is highly theoretical, establishing abstract results on joint identifiability and elicitability of tail risk measures using advanced measure theory and rigorous mathematical proofs, with no implementation details, datasets, or backtests provided in the excerpt.

  flowchart TD
    A["Research Goal:<br/>Establish joint identifiability and<br/>elicitability of tail risk measures<br/>with the quantile"] --> B["Methodology:<br/>Law-based risk measure generators<br/>on tail distribution"]
    B --> C["Data/Inputs:<br/>Loss distribution beyond<br/>quantile at level α"]
    C --> D["Computation:<br/>Construct weighted scores<br/>for tail expectile & quantile"]
    D --> E["Outcomes:<br/>1. Novel weighted score class<br/>2. Easier model fitting (regression, GMM)<br/>3. Model validation (backtesting)"]
    style A fill:#e1f5fe
    style E fill:#f1f8e9