以邊際與條件方法，對財務報酬率的機率結構探討

Abstract

隨著金融市場蓬勃的發展，投資者能以多元的方式管理自己適合的投資組合。除了風險，投資者最關注的即是投資報酬率。過去幾十年，在財務金融領域裡，其中廣為討論的研究主題之一為投資報酬率的機率結構。在二十世紀中期，財務分析師和學者們已發現，先前建立於常態分配假設之下的研究，其研究結果是錯誤的。他們也發現到，以常態分配的假設下所推論的結果會低估潛在的財務風險。儘管人們知道投資報酬率的機率分布不是常態分配，一些研究人員或實務人員仍然繼續使用常態分配假設的推論，也因此，往往忽略了投資報酬率分布的偏斜與厚尾所帶來的訊息。 根據數學家本華•曼德柏(1963) 與經濟學家尤金•法馬(1965)，兩位學者們皆強調投資報酬率的機率分布應套用穩定分配(stable distributions)。在本研究裡，吾人執行財務報酬率的實證研究，以傳統的常態模型和其他厚尾的機率模型(如：拉普拉斯分配、穩定柏拉圖分配等)予以比較分析。吾人進行實證分析的資料來自彭博社(Bloomberg)，該資料為6支財務指數 (TWSE, HSI, NKY, SPX, INDU 和DEM/US) 的前一交易日之收盤價(PX_1D_CLOSE)，並將此資料轉換成對數報酬率進行分析。本研究分成兩大面向來分析：邊際方法的觀點和條件方法的觀點。 在邊際方法方面(去除時間因素)，對數報酬率以單變量常態、甘貝爾、拉普拉斯以及對稱和非對稱的穩定柏拉圖機率模型來進行資料配適，並且以無母數適合度檢定去比較不同機率模型的配適結果。在條件方法方面，考慮到資料的時間相依關係，對數報酬率資料很自然地形成時間序列的架構，因此吾人使用自我迴歸移動平均(ARMA)和廣義自我迴歸條件異質變異(GARCH)的結合模型去配適對數報酬時間序列。根據財務金融學文獻，吾人選取ARMA(1,1)-GARCH(1,1)的模型，並將隨機誤差項(innovation) 設定服從常態、拉普拉斯和非對稱的穩定柏拉圖機率分配。並將其三種分配的估計結果以無母數適合度檢定去比較。 綜合上述的兩種方式，在其他候選模型之中(與常態分配模型做為對照)，以非對稱的穩定柏拉圖機率分配去配適對數報酬率的資料，其統計結果通常是最佳的。

With the development of financial markets, market participants manage his or her own portfolio with a great diversity. Besides the risks, what investors concern the most is the asset returns. In the past decades, one of the widely-discussed topics of financial research is the probabilistic structure on asset returns. During mid-20 century, financial analysts and researchers found that all the past research based on Gaussian assumption is fallacious and noticed that this may underestimate the potential of financial risks. In spite of the well-known fact that asset returns are not normally distributed, some researchers and practitioners still maintain the normal inferences and henceforth ignore the information on asymmetry and heavy tail. According to Mandelbrot (1963) and Fama (1965), stressing on the use of stable distributions, we would like to conduct the empirical study on financial index returns by means of comparisons with the inferences based on normal distribution and other heavy-tailed distributions, such as Laplace distribution and stable Paretian distribution. In this study, from Bloomberg, we collected the closing price of the last trading day (PX_1D_CLOSE) of 6 financial indices (TWSE, HSI, NKY, SPX, INDU and DEM/US) and transformed them into log returns. Then we carried out two-fold analyses: marginal and conditional perspective. In marginal aspect, the returns were fitted by univariate normal, Gumbel, Laplace and (asymmetric and symmetric) stable Paretian models and we compare the results by their own goodness of fits ; whilst in conditional part, reconsidering the temporal dependency into data, the returns constitute time series naturally and therefore we fit the log returns by combination of homoscedastic part (ARMA) and heteroscedastic part (GARCH).

According to finance literature, we fit ARMA(1,1)-GARCH(1,1) with normal, Laplace and stable innovations, which can be compared with goodness of fit. Above all the methods, the statistical inferences based on asymmetric stable Paretian distribution are usually better than the ones based on normality.