One Axiom to Rule Them All: A Minimalist Axiomatization of Quantiles

Tolulope Fadina, Peng Liu, Ruodu Wang

Research output: Contribution to journalArticlepeer-review

Abstract

We offer a minimalist axiomatization of quantiles among all real-valued mappings on a general set of distributions through only one axiom. This axiom is called ordinality: Quantiles are the only mappings that commute with all increasing and continuous transforms. Other convenient properties of quantiles-monotonicity, semicontinuity, comonotonic additivity, elicitability, and locality in particular-follow from this axiom. Furthermore, on the set of convexly supported distributions, the median is the only mapping that commutates with all monotone and continuous transforms. On a general set of distributions, the median interval is pinned down as the unique minimal interval-valued mapping that commutes with all monotone and continuous transforms. Finally, our main result, put in a decision-Theoretic setting, leads to a minimalist axiomatization of quantile preferences. In banking and insurance, quantiles are known as the standard regulatory risk measure Value-At-Risk (VaR), and thus an axiomatization of VaR is obtained with only one axiom among law-based risk measures.

Original languageEnglish (US)
Pages (from-to)644-662
Number of pages19
JournalSIAM Journal on Financial Mathematics
Volume14
Issue number2
DOIs
StatePublished - 2023
Externally publishedYes

Keywords

  • median
  • ordinality
  • quantile maximization
  • quantiles
  • Value-At-Risk

ASJC Scopus subject areas

  • Numerical Analysis
  • Finance
  • Applied Mathematics

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