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In this paper we study time-consistent risk measures for returns that aregiven by a GARCH(1,1) model. We present a construction of risk measures basedon their static counterparts that overcomes the lack of time-consistency. Wethen 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 ananalytical formula for its time-consistent counterpart, in the AVaR case wederive lower and upper bounds to its time-consistent version. Furthermore, weincorporate techniques from Extreme Value Theory (EVT) to allow for a moretail-geared statistical analysis of the corresponding risk measures. Weconclude with an application of our results to a data set of stock prices.

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