Distortion Risk Measures of Increasing Rearrangement

Joachim Paulusch (), Thorsten Moser and Anna Sulima
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Joachim Paulusch: R+V Lebensversicherung AG, Raiffeisenplatz 1, 65189 Wiesbaden, Germany
Thorsten Moser: R+V Lebensversicherung AG, Raiffeisenplatz 1, 65189 Wiesbaden, Germany
Anna Sulima: Katedra Ekonometrii i Badan Operacyjnych, Wroclaw University of Economics and Business, ul. Komandorska 118/120, 53-345 Wroclaw, Poland

JRFM, 2024, vol. 17, issue 10, 1-14

Abstract: Increasing rearrangement is a rewarding instrument in financial risk management. In practice, risks must be managed from different perspectives. A common example is the portfolio risk, which often can be seen from at least two perspectives: market value and book value. Different perspectives with different distributions can be coupled by increasing rearrangement. One distribution is regarded as underlying, and the other distribution can be expressed as an increasing rearrangement of the underlying distribution. Then, the risk measure for the latter can be expressed in terms of the underlying distribution. Our first objective is to introduce increasing rearrangement for application in financial risk management and to apply increasing rearrangement to the class of distortion risk measures. We derive formulae to compute risk measures in terms of the underlying distribution. Afterwards, we apply our results to a series of special distortion risk measures, namely the value at risk, expected shortfall, range value at risk, conditional value at risk, and Wang’s risk measure. Finally, we present the connection of increasing rearrangement with inverse transform sampling, Monte Carlo simulation, and cost-efficient strategies. Butterfly options serve as an illustrative example of the method.

Keywords: increasing rearrangement; risk measure; distortion risk measure; expected shortfall; ES; value at risk; VaR; range value at risk; RVaR; conditional value at risk; CVaR; Wang’s risk measure; inverse transform sampling; cost-efficient strategies (search for similar items in EconPapers)
JEL-codes: C E F2 F3 G (search for similar items in EconPapers)
Date: 2024
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