quant research // summarizing the paperLOADING
Christian Laudagé · 2026-06-24
This is a theoretical math-finance paper about how to define risk when you measure it on returns (like log-returns) rather than on dollar payoffs. It extends 'return risk measures' to more general mathematical spaces (AM-algebras, including Euclidean and multidimensional bounded random variables) and proves properties like when they are well-defined, continuous, and how they can be represented via duality or aggregation. It also introduces new classes of systemic and vector-valued return risk measures.
Why it matters: For quants building risk models on returns rather than on monetary losses, this offers a cleaner axiomatic foundation and a way to handle multiple assets or systemic risk jointly through vector-valued measures. Its practical value is mostly in guiding how risk functionals are constructed and connected to standard monetary risk measures, not in producing a ready-to-use signal.
⚠ This is abstract mathematical theory with no empirical testing, so practical implementation and any trading benefit remain unproven.
Monetary risk measures quantify the risk of uncertain monetary payoffs (or losses), whereas in time series analysis risk is typically assessed using logarithmic returns. Return risk measures (RRMs) provide an axiomatic foundation for this latter approach, which relies crucially on the positive cone of the space of essentially bounded random variables. We extend RRMs to general ordered vector spaces and characterize positive homogeneity via the geometric epigraph. To investigate geometric convexity and establish connections with monetary risk measures, we specialize the domain to AM-algebras, encompassing Euclidean spaces and spaces of multidimensional essentially bounded random variables. The latter is novel in the context of RRMs and leads to the new classes of systemic and vector-valued RRMs. We establish results on finiteness, continuity, separability, as well as dual and aggregation-based representations.
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AI summary generated from the paper’s public abstract via arXiv; it may miss nuance — read the source before relying on it. Thank you to arXiv for its open-access interoperability; StockTools is not affiliated with arXiv, and all rights remain with the authors. Educational only, not financial advice.