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

Author

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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)

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.

Suggested Citation

  • Klüppelberg Claudia & Zhang Jianing, 2016. "Time-consistency of risk measures with GARCH volatilities and their estimation," Statistics & Risk Modeling, De Gruyter, vol. 32(2), pages 103-124, March.
    • Handle: RePEc:bpj:strimo:v:32:y:2016:i:2:p:103-124:n:2
    DOI: 10.1515/strm-2015-0010
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    References listed on IDEAS

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