One axiom to rule them all: An axiomatization of quantiles

We offer an axiomatic characterization of quantiles through only one axiom. Among all real-valued mappings on a general set of distributions, left quantiles are the only ones satisfying left-ordinal covariance, meaning that they commute with increasing left-continuous transforms; the case of right quantiles is analogous. Other convenient properties of quantiles, monotonicity in particular, follow from this axiom. In banking and insurance, quantiles are known as Value-at-Risk (VaR), a standard regulatory risk measure. Thus, we obtain an axiomatization of VaR with only one axiom among law-based risk measures. We further show that VaR can be alternatively characterized via the axiom of locality, plus four standard axioms relevant in financial risk management, namely, monotonicity, normalization, cash additivity, and semicontinuity