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

Klüppelberg Claudia () and Zhang Jianing ()
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Klüppelberg Claudia: Center for Mathematical Sciences, Technical University of Munich, Boltzmannstr. 3, 85748 Garching, Germany
Zhang Jianing: Center for Mathematical Sciences, Technical University of Munich, Boltzmannstr. 3, 85748 Garching, Germany

Statistics & Risk Modeling, 2016, vol. 32, issue 2, 103-124

Abstract: 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 (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1515/strm-2015-0010

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