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Licensed Unlicensed Requires Authentication Published by De Gruyter (A) February 24, 2016

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

Claudia Klüppelberg and Jianing Zhang


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.

We are grateful to the reviewer and Marcin Pitera, who pointed out some errors in a previous version of this paper.

Received: 2015-4-2
Revised: 2016-2-7
Accepted: 2016-2-8
Published Online: 2016-2-24
Published in Print: 2016-3-1

© 2016 by De Gruyter