Short Communication: An Axiomatization of $\Lambda$-Quantiles

We give an axiomatic foundation to $\Lambda$-quantiles, a family of generalized quantiles introduced in [M. Frittelli, M. Maggis, and I. Peri, Math. Finance, 24 (2014), pp. 442--463] under the name Lambda Value at Risk. Under mild assumptions, we show that these functionals are characterized by a property that we call “locality,” which means that any change in the distribution of the probability mass that arises entirely above or below the value of the $\Lambda$-quantile does not modify its value. We make comparisons with a related axiomatization of the usual quantiles given by Chambers in [Math. Finance, 19 (2009), pp. 335--342], based on the stronger property of “ordinal covariance,” meaning that quantiles are covariant with respect to increasing transformations. Further, we present a systematic treatment of the properties of $\Lambda$-quantiles, refining some of the results of Frittelli, Maggis, and Peri and [M. Burzoni, I. Peri, and C. M. Ruffo, Quant. Finance, 17 (2017), pp. 1735--1743] and showing that in the case of a nonincreasing $\Lambda$ the properties of $\Lambda$-quantiles closely resemble those of the usual quantiles.