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Statistics & Risk Modeling

with Applications in Finance and Insurance

Editor-in-Chief: Stelzer, Robert

Cite Score 2017: 0.96

SCImago Journal Rank (SJR) 2017: 0.455
Source Normalized Impact per Paper (SNIP) 2017: 0.853

Mathematical Citation Quotient (MCQ) 2017: 0.32

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Volume 32, Issue 2


Time-consistency of risk measures with GARCH volatilities and their estimation

Claudia Klüppelberg / Jianing Zhang
Published Online: 2016-02-24 | DOI: https://doi.org/10.1515/strm-2015-0010


In this paper we study time-consistent risk measures for returns that are given by a GARCH(1,1) model. We present a construction of risk measures based on their static counterparts that overcomes the lack of time-consistency. We then study in detail our construction for the risk measures Value-at-Risk (VaR) and Average Value-at-Risk (AVaR). While in the VaR case we can derive an analytical formula for its time-consistent counterpart, in the AVaR case we derive lower and upper bounds to its time-consistent version. Furthermore, we incorporate techniques from extreme value theory (EVT) to allow for a more tail-geared statistical analysis of the corresponding risk measures. We conclude with an application of our results to a data set of stock prices.

Keywords: Dynamic risk measure; time-consistency; GARCH(1,1); extreme value theory; Value-at-Risk; Average Value-at-Risk; expected shortfall; generalized Pareto distribution; aggregate returns

MSC: 60G70; 91B30; 91G80; 91G70

About the article

Received: 2015-04-02

Revised: 2016-02-07

Accepted: 2016-02-08

Published Online: 2016-02-24

Published in Print: 2016-03-01

Citation Information: Statistics & Risk Modeling, Volume 32, Issue 2, Pages 103–124, ISSN (Online) 2196-7040, ISSN (Print) 2193-1402, DOI: https://doi.org/10.1515/strm-2015-0010.

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© 2016 by De Gruyter.Get Permission

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