A Review and Some Complements on Quantile Risk Measures and Their Domain
Faculty of Economics and Management, Free University of Bozen-Bolzano, 39100 Bolzano, Italy
Fakultät für Mathematik, Technische Universität Chemnitz, 09126 Chemnitz, Germany
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
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
of random variables, we define the quantile risk measure
as the map that integrates the quantile function of a random variable in
with respect to Q
. The definition of
cannot attain the value
and cannot be extended beyond
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.
This is an open access article distributed under the Creative Commons Attribution License
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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.
Fuchs S, Schlotter R, Schmidt KD. A Review and Some Complements on Quantile Risk Measures and Their Domain. Risks. 2017; 5(4):59.
Fuchs, Sebastian; Schlotter, Ruben; Schmidt, Klaus D. 2017. "A Review and Some Complements on Quantile Risk Measures and Their Domain." Risks 5, no. 4: 59.
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