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Estimation of tail-related value-at-risk measures: range-based extreme value approach. (English) Zbl 1294.91192
Summary: This study proposes a new approach for estimating value-at-risk (VaR). This approach combines quasi-maximum-likelihood fitting of asymmetric conditional autoregressive range (ACARR) models to estimate the current volatility and classical extreme value theory (EVT) to estimate the tail of the innovation distribution of the ACARR model. The proposed approach reflects two well-known phenomena found in most financial time series: stochastic volatility and the fat-tailedness of conditional distributions. This approach presents two main advantages over the approach of McNeil and Frey (2000). First, the ACARR model in this approach is an asymmetric model that treats the upward and downward movements of the asset price asymmetrically, whereas the generalized autoregressive conditional heteroskedasticity model in the McNeil and Frey approach is a symmetric model that ignores the asymmetric structure of the asset price. Second, the proposed method uses classical EVT to estimate the tail of the distribution of the residuals to avoid the threshold issue in the modern EVT model. Since the McNeil and Frey approach uses modern EVT, it may estimate the tail of the innovation distribution poorly. Back testing of historical time series data shows that our approach gives better VaR estimates than the McNeil and Frey approach.

##### MSC:
 91G70 Statistical methods; risk measures 62P05 Applications of statistics to actuarial sciences and financial mathematics 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 60G70 Extreme value theory; extremal stochastic processes 91B84 Economic time series analysis
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