結合GARCH模型與極值理論的風險值模型

Incorporating Extreme Value Theory into GARCH Model for Value-at-Risk

本文主要延續McNeil與Frey(2000)所提出的結合GARCH模型與極值理論的風險值估計方法，並針對McNeil與Frey(2000)應用極值理論估計標準化殘差項的極值分配法加以改進，以簡化風險值的估計過程並提升估計的準確性。本文主要應用Huisman, Koedijk與 Pownall(1998)及Huisman, Koedijk, Kool與Plam(2001)的VaR-x 法來修正McNeil與Frey(2000)的極值估計方法。本文以六個股價指數作為實證研究的對象，檢定本文結合GARCH模型與VaR-x估計法的準確性。經失敗率及Kupiec(1995)之條件與非條件涵蓋比率檢定結果，顯示在高信賴水準下，結合GARCH模型與極值理論相較於GARCH 模型以及指數加權移動平均法的估計更為準確。此外，本文建立的模型較McNeil與Frey(2000)的模型更具有容易以及準確估計動態風險值的特性。本文的實證結果說明了同時掌握條件異質波動數以及厚尾分配的特性，在高信賴水準下能提升風險值估計的準確性。

The purpose of this study is to construct a Value at Risk (VaR) model describing the tail of the conditional distribution of a heteroscedastic financial return series and simplify the estimation procedure of VaR. McNeil and Frey (2000) propose a method that can simultaneously tackle both extreme event and stochastic volatility for estimating VaR. They combine pseudo-maximum-likelihood fitting of GARCH models to estimate the current volatility and extreme value theory (EVT) for estimating the tail of the innovation distribution of GARCH model and show that their approach gives more accurate 1-day estimates than the others which ignore the heavy tails of the innovations or the stochastic nature of the volatility. However, Huisman, Koedijk, Kool and Palm (2001) indicate that using Hill index estimator (1975) to estimate the tail index of the innovation distribution would suffer from severe small-sample bias. As a result, the empirical applicability of the Hill index estimator is limited to cases in which a large sample is available, either in the form of high frequency data or in the form of a long sampling period, but in many practical cases this condition is not fulfilled. Moreover, even when a long sample is available, it may be interesting to split the sample and then analyze whether the tail structure of the sample has changed over time. In addition, an important part of the bias in the Hill index estimator, which itself does not provide a optimal number of tail observations, stems from the selection of the appropriate number of tail observations to include in the estimation process. In order to overcome above drawbacks, we propose an alternative approach based on the method proposed by Huisman, et al. (2001) to correct for the small-sample bias in Hill tail-index estimates of the innovation distribution. Our method does not condition its tail-index estimate on one specific number of tail observations. Instead, our method exploits information obtained from a set of Hill tail-index estimates each conditioned on a different number of tail observations. The result is a weighted average of a set of conventional Hill estimators, with weights obtained by simple least squares techniques. To improve on accuracy of VaR estimate, we use the Var-x developed by Huisman, Koedijk and Pownall (1998) to revise the conventional Hill estimator used by McNeil and Frey (2000). First, considering the stochastic volatility structure of conditional volatility of returns, we use GARCH-type model to obtain estimates of the conditional mean and volatility, and then statistical tests confirm that the error terms or innovations do form, at least approximately, iid series that exhibit heavy tails. Second, we use the modified Hill estimator developed by Huisman et al. (2001) to estimate tail-index of the innovation distribution obtained by GARCH model. Third, linking the features of tail index and the Student-t distribution, the extreme quantile of the innovation distribution will be obtained (i.e. VaR-x). Finally, taking the estimates of the conditional mean and volatility and the extreme quantile of the innovation distribution into a dynamics model of VaR, the estimate of VaR will be obtained. Our approach above has two features: one is to construct a VaR model describing the tail of the conditional distribution of a heteroscedastic financial return series; the other is to simplify the estimation procedure of VaR. To evaluate the performance of our method in terms of failure rates and Kupiec test, we apply the method to six historical series of log returns: the FTSE 100 index, the DAX index, the S&P 500 index, the NASDAQ index, the DOW index, and the Taiwan weighted stock index. The sample period is from January 1, 1991 to December 31, 2001. In addition, we also compare the relative performance of different VaR models which include EWMA model (equally weighted moving average approach), GARCH-normal model (assuming conditional normal distribution), GARCH-t model (assuming conditional t distribution), GJR-normal model (assuming conditional normal distribution and the condition volatility is asymmetry), GJR-t model (assuming conditional t distribution and the condition volatility is asymmetry), GARCH-Hill and GJR-Hill (incorporating EVT into GARCH-type model), and the GARCH-VaR-x and GJR-VaR-x proposed by this study. According to the analysis of the failure rates and Kupiec test, our empirical results present that the incorporated VaR-x into GARCH-type model does indeed give more precise estimates of the 99th and 99.5th percentile than the other models except for GARCH-Hill and GJR-Hill models. This finding indicates that considering the tail of the conditional distribution and heteroscedastic behavior simultaneously in VaR measures will obtain more accurate estimates at higher quantiles. However, our approach, employing the VaR-x to resolve the problems of selection of the appropriate number of the tail observations and small-sample bias of the Hill estimator, simplifies the estimation procedure of VaR and achieve the same accuracy as GARCH-Hill and GJR-Hill models do. Thus our approach seems to be in favor of the measure of dynamic risk management.