Paper: arXiv 2407.16878

Authors: Çağın Ararat, Zachary Feinstein

Abstract

Risk measures for random vectors have been considered in multi-asset markets with transaction costs and financial networks in the literature. While the theory of set-valued risk measures provide an axiomatic framework for assigning to a random vector its set of all capital requirements or allocation vectors, the actual decision-making process requires an additional rule to select from this set. In this paper, we define vector-valued risk measures by an analogous list of axioms and show that, in the convex and lower semicontinuous case, such functionals always ignore the dependence structures of the input random vectors. We also show that set-valued risk measures do not have this issue as long as they do not reduce to a vector-valued functional. Finally, we demonstrate that our results also generalize to the conditional setting. These results imply that convex vector-valued risk measures are not suitable for defining capital allocation rules for a wide range of financial applications including systemic risk measures.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a strong theoretical and axiomatic proof showing that convex lower semicontinuous vector-valued risk measures degenerate to trivial separable risk measures, impacting capital allocation and systemic risk modeling. While mathematically rigorous and highly novel for risk management theory, it is purely theoretical with no empirical data or backtesting.

Research Flowchart

  flowchart TD
    A[Research Goal: Evaluate viability of Vector-Valued Risk Measures for multi-asset/systemic risk] --> B[Input Data: Random vectors representing multi-asset portfolios and financial networks]
    B --> C[Axiomatic Formulation: Define vector-valued risk measures via convex & LSC properties]
    C --> D[Mathematical Analysis: Compare Vector-Valued vs. Set-Valued Risk Measures in unconditional & conditional settings]
    D --> E[Key Finding 1: Convex vector-valued risk measures ignore dependence structures of random vectors]
    D --> F[Key Finding 2: Set-valued risk measures preserve dependence structures unless reduced to vector-valued]
    E & F --> G[Outcome: Vector-valued risk measures are unsuitable for capital allocation and systemic risk]