在本篇文章中，我們採用長期記憶模型來估計S&P500、Nasdaq100和Dow Jones Industrial Index三個股票指數期貨的日收盤價的風險值。為了更準確地計算風險值，本文採用常態分配、t分配以及偏斜t分配來做模型估計以及風險值之計算。有鑒於大多數探討風險值的文獻只考慮買入部位的風險，本研究除了估計買入部位的風險值，也估計放空部位的風險值，以期更能全面性地估算風險。實證結果顯示，ARFIMA-FIGARCH模型配合偏斜t分配較其他兩種分配更能精確地估算樣本內的風險值。基於ARFIMA-FIGARCH模型配合偏斜t分配在樣本內風險值計算的優異表現，我們利用此模型搭配來實際求算樣本外風險值。結果如同樣本內風險值一般，ARFIMA-FIGARCH模型配合偏斜t分配在樣本外也有相當好的風險預測能力。 / In this thesis, we estimate Value-at-Risk (VaR) for daily closing price of three stock index futures contracts, S&P500, Nasdaq100, and Dow Jones, using the double long memory models. Due to the existence of a long-term persistence characterized in our data, the ARFIMA-FIGARCH models are used to compute the VaR. In order to investigate better, three kinds of density distributions, normal, Student-t, and skewed Student-t distributions, are used for estimating models and computing the VaR. In addition to the VaR for the long trading positions which most researches focus on to date, the VaR for the short trading positions are calculated as well in this study. From the empirical results we show that for the three stock index futures, the ARFIMA-FIGARCH models with skewed Student-t distribution perform better in computing in-sample VaR both in long and short trading positions than symmetric models and has a quite excellent performance in forecasting out-of-sample VaR as well.

近幾年，風險值已經成為金融機構風險控管的重要工具。它的明確及簡單易懂是其讓人接受的原因，加上巴塞爾銀行監理委員會在1996提出的巴塞爾協定修正，規定銀行將市場風險因素納入考量，並允許銀行自行發展內部模型，以風險值模型衡量市場風險後，各種風險值的估算方法相繼被提出。 本篇論文是使用部分整合自回歸條件變異數（Fractional Integrated Generalized Autoregressive Conditional Heteroskedasticity，簡稱FIGARCH）計算長期利率期貨多空部位的每日風險值。選取的三支長期利率期貨是在芝加哥期貨交易所掛牌的三十年期美國政府債券期貨（TB）、十年期美國政府債券期貨（TN） 與十年期市政債券指數期貨（MNI）。 利率期貨的研究在過去文獻中，甚少被提及。但隨著利率型商品日新月異的發展，以利率期貨避險的需求也與日遽增。尤其在台灣，利率期貨更是今年新登場的期貨商品。因此，我選擇利率期貨作為研究標的，藉由以FIGARCH模型來配適波動性，提供避險者一個估算風險值的方法。 FIGARCH模型係由Baillie、Bollerslev與Mikkelsen於1996所提出，與傳統GARCH模型所不同的是，FIGARCH模型特別適用於描述具有波動性長期記憶（Long Memory）性質的資料。所謂長期記憶性，是指衝擊所造成的持續性是以緩慢的雙曲線速率衰退。而許多市場實證分析均指出，FIGARCH較適合用來描述金融市場上的波動性。此外，本研究的風險值計算，除了一般實務界常用的常態分配以外，還考慮了t分配與偏斜t分配，以捕捉財務資料常見的厚尾與偏斜的特性。 而實證結果顯示，長期利率期貨報酬率的波動性確實存在長期記憶性，所以FIGARCH(1,d,1)模型可以適切地估算長期利率期貨的每日風險值，不論在樣本內或樣本外的風險值計算均優於傳統GARCH(1,1)模型的計算結果。至於各種不同分配的比較，在樣本內的風險值計算，當α=0.05時，常態分配FIGARCH(1,d,1)模型表現較佳；當α=0.025到0.0025時，t分配與偏斜t分配FIGARCH(1,d,1)模型表現較佳，而偏斜t分配FIGARCH又稍微優於t分配FIGARCH(1,d,1)模型。 而樣本外的風險值預測，則有不同的結果，當α=0.05，t分配與偏斜t分配FIGARCH(1,d,1)模型表現較佳；而α=0.01時，常態分配FIGARCH(1,d,1)模型表現較佳。而且t分配與偏斜t分配FIGARCH(1,d,1)模型在α=0.01會出現太過保守的情形，出現失敗率（failure rate）為零，高估風險值。 / Value-at-Risk (VaR) has become the standard measure used to quantify market risk recently, and it is defined as the maximum expected loss in the value of an asset or portfolio, for a given probability α at a determined time period. This article uses the FIGARCH(1,d,1) models to calculate daily VaR for long-term interest rate futures returns for long and short trading positions based on the normal, the Student-t, and the skewed Student-t error distributions. The U.S. Treasury bonds futures, Treasury notes futures, and municipal notes index futures of daily frequency are studied. The empirical results show that returns series for three interest rate futures all have long memory in volatility, and should be modeled using fractional integrated models. Besides, the in-sample and out-of-sample VaR values generated using FIGARCH(1,d,1) models are more accurate than those generated using traditional GARCH(1,1) models. For different distributions among FIGARCH(1,d,1) models, the normal FIGARCH(1,d,1) models are preferred for in-sample VaR computing whenα=0.05, and the Student-t and skewed Student-t models perform better for in-sample VaR computing whenα=0.025-0.0025. Nonetheless, for out-of-sample VaR, the Student-t and skewed Student-t FIGARCH(1,d,1) models perform better in the case α=0.05 while the normal FIGARCH(1,d,1) models perform better in the case α=0.01. The VaR values obtained by the Student-t and skewed Student-t FIGARCH(1,d,1) models are too conservative whenα=0.01.

本篇文章將採用長期記憶模型之一的HYGARCH模型，搭配1985年廣場協議後的日圓匯率期貨資料來估計日圓期貨匯率買入和放空部位的日報酬風險值，探討控管日圓匯率期貨在使用上的風險。為了更準確地計算風險值，本文採用常態分配、學生t分配以及偏態學生t分配來作模型估計以及風險值之計算。 本文實證的結果將有兩方面的貢獻：首先，實證結果顯示當我們採用厚尾分配估計風險值時，樣本內風險值的估計誤差會與信賴水準的高低呈正比的現象，證明在極端的風險值估計上，厚尾分配均有較佳的表現。其次，與其他使用HYGARCH模型研究日圓匯率的文章相較，本文在風險控管層面上所提供的偏態學生t分配，於估計風險值時，比起只考慮厚尾的對稱學生t分配將來得更為有效，其不但在估計誤差上較小，而且根據Kupiec檢定法，其在樣本內的風險值估計也有較好的表現。此外，本文也將多方證明此資料的偏態分配屬於右偏。 / In order to manage the exposure of the dollar/yen futures returns with regarding the long memory behavior in volatility, we use the HYGARCH(1,d,1) model with the data after the Plaza Accord to compute daily Value-at-Risk (VaR) of long and short trading positions. To take into account the fat-tail situation in financial time series, we estimate the model under the normal, Student-t, and skewed Student-t distributions. The contribution of this article is twofold. First, the empirical results show that the bias of in-sample VaR increases as the confidence level increases when VaR is calculated with a fat-tail distribution. Second, we provide a better distribution, the skewed Student-t innovation, for estimating the HYGARCH model for the Japanese yen in respect of risk management because the bias under the skewed Student-t innovation is smaller than that under the Student-t distribution, and in-sample VaR of the models with a skewed Student-t distribution outperforms based on Kupiec test. In addition, we get the innovation skewed to the right through the in-sample VaR analysis.