α-穩定分配及其在風險值與財務預測上的應用，和t-分配做比較

Abstract

實務上，業界人士習慣將他們所要處理的財務資料假設為常態分配。然而，許多文獻指出，大多數金融資產的報酬率之分配具有厚尾的特性，也就是極端值的大小及其出現的頻率要比常態分配所估計的高，因此常態分配的假設是不適當的。而t-分配與α-穩定分配正是適合的替代方案。過去的研究顯示，t-分配完全能夠比常態分配更適合用來描述金融資產報酬率，但是，t-分配與α-穩定分配彼此之間沒有絕對的優劣。用這兩個分配來描述金融資產報酬率皆有其特色與優缺點。 本篇論文的研究主要在比較以t-分配與α-穩定分配來配適亞洲股票指數之報酬率的情形。由於過去相關研究主要著重於歐美權益指數報酬率的分析，因此本篇論文欲探討是否亞洲股市也能得到相似的結果。本篇論文的分析主要分成兩部分：分別為與時間獨立的部分，以及與時間相依的部分。在與時間獨立的部分，分別探討以無條件t-分配與α-穩定分配來配適指數報酬率所求得之風險值，並與實際資料做比較。在與時間相依的部分，以時間序列GARCH模型來分析並預測各個股票指數之報酬率，其中分別假模型之設誤差項為t-分配與α-穩定分配。 主要的發現為，無條件α-穩定分配所估算出來的風險值，在一般的風險水準之下(5%至10%)較t-分配所估算出來的風險值更為精確。而在極端情況下的風險值(小於1%)，α-穩定分配較t-分配更傾向於高估真實的風險值。這項特色使得α-穩定分配在風險管理的應用上有其獨有的價值，因為它不但能在正常情況下提供較精確的估計，還能夠在極端情況下將預期損失估計得高一點，使得風險控管者會提撥更多的準備金來預防未來可能的損失，讓風險管理更加穩健安全。 本篇論文亦發現，若從樣本外預測著眼，假設誤差項為α-穩定分配之時間序列GARCH模型會比假設誤差項為t-分配的模型擁有較小的均方根誤差(RMSE)，代表前者具有較佳的預測能力。然而，此二模型之預測能力的差異，在統計上的顯著程度大小，還因不同的資料特性而有所不同，這也提供了未來後續研究可能的方向。最後，本篇論文建構出以每個模型對股票指數報酬率做預測所對應的95%預測區間，並將之與動態風險值的概念結合。透過預測區間，我們可以在任何時間點，估算該時刻的動態風險值，使得風險值的估算不止於靜態的估計，而是動態且與時間相關，從而更貼近真實的情況。

In practice, business people used to deal with financial data as if they follow the normal distribution. However, researches have shown that most financial assets returns possess fat-tailed property, which is contradictory to that of the normal distribution. Both the t-distribution and α-stable distribution are attractive alternatives. Past study have stated that the t-distribution dominates the normal distribution, but there is no definite dominance of either the t-distribution or the α-stable distribution over the other. They both carry unique features when fitted to financial data. This paper compares the fitness of the t-distribution and the α-stable distribution to the stock indices returns in Asia, since most past researches of this kind focus on the equity indices in Europe and America. The analysis in this paper is classified into two parts, first the time independent part and followed by the time dependent part. In the first part, the Value at Risk (VaR) estimated by the unconditional t-distribution and the α-stable distribution are discussed. In the second part, the time series GARCH models with t-innovation and α-stable innovation respectively are also investigated. The main finding is that in the sense of VaR, the unconditional α-stable distribution provides better estimates of VaR at moderate levels, and extreme VaR less than 1% with α-stable distribution tends to be conservative, with comparison to t-distribution. This is a valuable feature of the application of α-stable distribution to risk management, because it allows risk managers to preserve more reservation in advance for the potential upcoming losses. Moreover, this paper also shows that the time series GARCH models with α-stable innovation always have smaller RMSE than those with t-innovation when the out-of-sample forecasting is conducted, indicating that the models with α-stable innovation may have better forecasting accuracy than those with t-innovation, though the degrees of significance are different due to the property of the data. Finally, the 95% forecasting intervals are constructed in this paper and they can be connected to the dynamic VaR, making it possible for us to estimate the VaR in accordance with time.