Risk measures based on target risk profiles

Paper: arXiv 2409.17676

Authors: Jascha Alexander, Christian Laudagé, Jörn Sass

Abstract

We address the problem that classical risk measures may not detect the tail risk adequately. This can occur for instance due to averaging when calculating the Expected Shortfall. The current literature proposes the so-called adjusted Expected Shortfall as a solution. This risk measure is the supremum of Expected Shortfalls for all possible levels, adjusted with a function $g$, the so-called target risk profile. We generalize this idea by using a family of risk measures which allows for more choices than Expected Shortfalls, leading to the concept of adjusted risk measures. An adjusted risk measure quantifies the minimal amount of capital that has to added to a financial position $X$ to ensure that each risk measure out of the chosen family is smaller or equal to the target risk profile $g(p)$ for the corresponding level $p\in[0,1]$. We discuss a variety of assumptions such that desirable properties for risk measures are satisfied in this setup. From a theoretical point of view, our main contribution is the analysis of equivalent assumptions such that an adjusted risk measure is positive homogeneous and subadditive. Furthermore, we show that these conditions hold for several adjusted risk measures beyond the adjusted Expected Shortfall. In addition, we derive their dual representations. Finally, we test the performance in a case study based on the S$&$P $500$.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 5.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a rigorous mathematical generalization of adjusted risk measures beyond Expected Shortfall, complete with dual representations and theoretical conditions for coherency. While the theoretical side is highly developed, the empirical component is limited to a basic S&P 500 case study without released code.

Research Flowchart

  flowchart TD
    A[Research Goal: Address tail risk underdetection in classical risk measures like ES] --> B[Methodology: Generalize adjusted ES to 'Adjusted Risk Measures' using target profiles g]
    B --> C[Data/Inputs: Financial position X, target risk profile g, family of risk measures]
    C --> D[Theoretical Analysis: Derive conditions for positive homogeneity, subadditivity, & dual representations]
    D --> E[Empirical Testing: Case study evaluating performance on S&P 500 equity data]
    E --> F[Outcomes: Validated new class of subadditive risk measures superior at detecting tail risks]

Cited References (53)

1. Adjusted Rényi entropic Value-at-Risk (2023) — link
2. Generalized deviations in risk analysis (2006) — link
3. Convex risk measures: Basic facts, law-invariance and beyond, asymptotics for large portfolios (2013) — link
4. Mean-$\rho$ portfolio selection and $\rho$-arbitrage for coherent risk measures (2022) — link
5. Adjusted higher-order Expected Shortfall (2024) — link
6. $\rho$-Arbitrage and $\rho$-Consistent Pricing for Star-Shaped Risk Measures (2024) — link
7. Infinite Dimensional Analysis : A Hitchhiker ’s Guide (2006) — link
8. Stochastic Finance : An Introduction in Discrete Time (2016) — link

0 of 53 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.