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Risks 2017, 5(4), 59; doi:10.3390/risks5040059

A Review and Some Complements on Quantile Risk Measures and Their Domain

1
Faculty of Economics and Management, Free University of Bozen-Bolzano, 39100 Bolzano, Italy
2
Fakultät für Mathematik, Technische Universität Chemnitz, 09126 Chemnitz, Germany
3
Fachrichtung Mathematik, Technische Universität Dresden, 01062 Dresden, Germany
*
Author to whom correspondence should be addressed.
Academic Editor: Mogens Steffensen
Received: 19 September 2017 / Revised: 23 October 2017 / Accepted: 2 November 2017 / Published: 7 November 2017
View Full-Text   |   Download PDF [372 KB, uploaded 13 November 2017]

Abstract

In the present paper, we study quantile risk measures and their domain. Our starting point is that, for a probability measure Q on the open unit interval and a wide class L Q of random variables, we define the quantile risk measure ϱ Q as the map that integrates the quantile function of a random variable in L Q with respect to Q. The definition of L Q ensures that ϱ Q cannot attain the value + and cannot be extended beyond L Q without losing this property. The notion of a quantile risk measure is a natural generalization of that of a spectral risk measure and provides another view of the distortion risk measures generated by a distribution function on the unit interval. In this general setting, we prove several results on quantile or spectral risk measures and their domain with special consideration of the expected shortfall. We also present a particularly short proof of the subadditivity of expected shortfall. View Full-Text
Keywords: integrated quantile functions; quantile risk measures; spectral risk measures; subadditivity; value at risk; expected shortfall integrated quantile functions; quantile risk measures; spectral risk measures; subadditivity; value at risk; expected shortfall
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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MDPI and ACS Style

Fuchs, S.; Schlotter, R.; Schmidt, K.D. A Review and Some Complements on Quantile Risk Measures and Their Domain. Risks 2017, 5, 59.

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