On Partial Stochastic Comparisons Based on Tail Values at Risk

• Published in: Mathematics (Basel) Vol. 8; no. 7; p. 1181
• Main Authors: Bello, Alfonso J., Mulero, Julio, Sordo, Miguel A., Suárez-Llorens, Alfonso
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.07.2020
• Subjects: Agreements; financial risk; Random variables; Risk levels; stochastic orders; tail value at risk; value at risk
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math8071181

The tail value at risk at level p, with p ∈ ( 0 , 1 ) , is a risk measure that captures the tail risk of losses and asset return distributions beyond the p quantile. Given two distributions, it can be used to decide which is riskier. When the tail values at risk of both distributions agree, whenever the probability level p ∈ ( 0 , 1 ) , about which of them is riskier, then the distributions are ordered in terms of the increasing convex order. The price to pay for such a unanimous agreement is that it is possible that two distributions cannot be compared despite our intuition that one is less risky than the other. In this paper, we introduce a family of stochastic orders, indexed by confidence levels p 0 ∈ ( 0 , 1 ) , that require agreement of tail values at risk only for levels p > p 0 . We study its main properties and compare it with other families of stochastic orders that have been proposed in the literature to compare tail risks. We illustrate the results with a real data example.