This document discusses using independent component analysis (ICA) and locally adaptive volatility estimation (LAVE) to calculate value at risk (VaR). It introduces ICA, LAVE, GARCH, and RiskMetrics models. An empirical study applies these methods to Taiwan stock market data from 2000 to 2010. The results show that combining ICA and LAVE provides more accurate VaR estimates than RiskMetrics or GARCH alone, by better capturing changes in volatility over time. However, the document notes that more flexible settings may be needed for higher confidence levels.

1. Value at Risk Based on Independent Components Analysis 口試委員:郭維裕 教授 陳威光 教授 徐政義 教授 貿碩二 曾順延

2. Introduction Methodology 2.1.1 Independent component analysis 2.1.2 Measures of nongaussianity 2.1.3 The FastICA Algorithm 2.2.1 Locally Adaptive Volatility Estimate 2.2.2 Applying additive error terms 2.2.3 Adaptive estimation under local time homogeneity 2.3 GARCH and RiskMetrics 2.4 Back-testing Empirical Study Conclusion

3. Introduction VaR is used for risk control and management, including both credit and operational risks (Duffee and Pan, 1997) (1) RiskMetrics Independent component analysis (Hyvärinen, 2001) Using ICA to separate IC to reconstruct risk factor process would increase accuracy of forecast (CHA and CHAN, 2000), and easy to calculate VaR (Chen,2007). Locally Adaptive Volatility Estimate (Mercurio and Spokoiny, 2004, 2007) GARCH Backtesting (Kupiec ,1995)

4. Independent component analysis

5. Independent component analysis x=As (2) y=Wx (3) Central Limit Theorem

6. Independent component analysisMeasures of nongaussianity 傳統峰度估計方法，易受離群值影響 熵: 訊息論基本式子，熵愈大愈隨機，常態分配有最大的熵

7. Independent component analysis The FastICA Algorithm The FastICA is based on a fixed-point iteration scheme for finding a maximum of the nongaussianity of (Hyvärinen, 1999). 1. Choose an initial vector of unit norm, 2. Let , where g denotes the first derivative of and g’ the second derivative. In practice, the sample mean is applied for E[.] 3. Orthogonalization 4. Normalization 5. If not converged, go back to step 2. 6. Set j=j+1. For , d=number of risk factors, go back to step 2.

8. Locally Adaptive Volatility Estimate 假設在特定區間 I=[t- ,t],變異數為定值 ->

9. Locally Adaptive Volatility Estimate

10. Locally Adaptive Volatility Estimate Step 1: Enlarge the interval I from to , i.e. ,and split the new interval into two subintervals J and I. The parameters and k are integers specified according to data. In this paper, we chose m0 = 5 and k= 2. Step 2: Start homogeneity test for interval J = . If the homogeneity hypothesis isn’t rejected, enlarge J one point further to and repeat the homogeneity test. The choice of 1/3 comes from the fact that the right 1/3 part has been tested in the last homogeneous interval and the left one-thirds will be tested in the next homogeneous interval, Mercurio and Spokoiny (2004). Step 3: If (14) is violated at point s, the loop stops and the time homogeneous interval I is specified from point to point s + 1. Step 4: If time homogeneity holds for this interval, go back to Step 1.

11. GARCH and RiskMetrics The GARCH (p, q) model (where p is the order of the GARCH terms and q is the order of the ARCH terms ) is given by The RiskMetricsvariance model (also known as exponential smoother) was first established in 1989 by J.P Morgan. Lambda = 0.94

12. Back-testing Varieties of tests have been proposed to gauge the accuracy of a VaR model. Some of the earliest proposed VaRbacktests, e.g. Kupiec (1995), focused exclusively on the property of unconditional coverage. In short, these tests are concerned with whether or not the reported VaRis violated more than alpha % of the time. He assumes failure rate (n/T) is binomial distribution. The hypnosis is H0=alpha % and LR test is

13. Empirical Study 台灣 電子，金融指數 2000/05/19~2010/05/18

14. Empirical Study

15. Empirical Study LAVE of IC1 LAVE of IC2

16. Empirical Study

17. Empirical Study

18. Empirical Study

19. Conclusion Independent component analysis is a good method to simplify the calculation of value at risk, it give us a fast way to calculate distribution of portfolio. LAVE is estimator of volatility which can reflect new points more quickly than GARCH and RiskMetrics. Combine ICA and LAVE are better than RiskMetrics and GARCH when calculate value at risk. But there are two important points. First, it needs more conscientious judgment of VaR model. Second, VaR model in this thesis might be too conservative in higher alpha. The VaR model needs more flexible setting.

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