使用極值理論評估具金融報酬特性的合成資料之風險值與條件風險值

Abstract

全球金融危機激起人們對黑天鵝事件的關注，黑天鵝事件代表極值發生的事件，預測極值發生的週期與程度是極具有挑戰性。使用極值理論來估算風險值與條件風險值可以計算出在極端事件發生時，在特定信心水準下的資產損失與相對應的期望資產損失。本論文使用具有金融報酬特性的合成資料作為研究資料，可以避開真實市場資料所受的變數繁雜而不利分析的缺點。使用的合成資料分為靜態與動態兩種，靜態資料是由常態分布、Student’s t 分布或對數常態分布隨機抽樣產生的資產損失，動態資料是利用自我迴歸移動平均模型、自我迴歸條件異質變異數模型與Student’s t 函數的殘差分布所產生的時間序列。利用極值理論估算風險值與條件風險值的方法區分為靜態方法與動態方法，靜態方法有區塊最大法與穿越門檻值法，動態方法利用動態模型、穿越門檻值法以及拔靴法來作估算。對於靜態資料而言，區塊最大法與穿越門檻值法對於不同的損失分布所估算出的結果，和樣本值或理論值的關係無法維持一定的趨勢。對於動態資料而言，動態方法所算出的結果，不論其變異數模型是否為定數，皆較於樣本值接近理論值但是小於理論值，具有一定的趨勢。

The global financial crisis in 2008 has raised concerns on the events so called “the Balck Swans”. Evaluating value at risk and conditional value at risk using the extreme value theory can produce the asset loss and the corresponding expected asset loss at certain confidence level under extreme circumstances. This paper uses the synthetic data characteristic of financial returns as research targets, and in this way, we can avoid the drawbacks of analyzing the real market data which are affected by ambiguous variables. There are two types of synthesized data: static data and dynamic data. Static data are asset losses which are randomly sampled from the normal distribution, the Student’s t distribution and the log normal distribution, respectively. Dynamic data are time series generated by using the autoregressive moving average model, the autoregressive conditional heteroscedasticity model. There are two methods to evaluate the value at risk and the conditional value at risk using the extreme value theory: the static method and the dynamic method. The static method includes the block maxima method and the peaks over the threshold. The dynamic method integrates the dynamic model, the peaks over the threshold and the bootstrap sampling to evaluate the risk. For static data, there is no definite relation between statistics calculated from samples, theoretical values and the results calculated by the static method. For the dynamic data, there is a definite relation between the results calculated by the dynamic method, statistics calculated from samples, theoretic values and the results calculated by the dynamic method.