Document Type
Article
Publication Date
9-23-2014
Department
Department of Mathematical Sciences
Abstract
A spectrum of upper bounds (Qα(X ; p) αε[0,∞] on the (largest) (1-p)-quantile Q(X;p) of an arbitrary random variable X is introduced and shown to be stable and monotonic in α, p, and X , with Q0(X ;p) = Q(X;p). If p is small enough and the distribution of X is regular enough, then Qα(X ; p) is rather close to Q(X ; p). Moreover, these quantile bounds are coherent measures of risk. Furthermore, Qα(X ; p) is the optimal value in a certain minimization problem, the minimizers in which are described in detail. This allows of a comparatively easy incorporation of these bounds into more specialized optimization problems. In finance, Q0(X;p) and Q1(X ; p) are known as the value at risk (VaR) and the conditional value at risk (CVaR). The bounds Qα(X ; p) can also be used as measures of economic inequality. The spectrum parameter α plays the role of an index of sensitivity to risk. The problems of the effective computation of the bounds are considered. Various other related results are obtained.
Publication Title
Risks
Recommended Citation
Pinelis, I.
(2014).
An optimal three-way stable and monotonic spectrum of bounds on quantiles: A spectrum of coherent measures of financial risk and economic inequality.
Risks,
2(3), 349-392.
http://doi.org/10.3390/risks2030349
Retrieved from: https://digitalcommons.mtu.edu/michigantech-p/1978
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.
Version
Publisher's PDF
Publisher's Statement
© 2014 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/). Publisher’s version of record: https://doi.org/10.3390/risks2030349