Market risk of the united states stock market based on asymmetric distribution model

逢 甲 大 學 國際經營與貿易學系碩士班 碩 士 論 文 基於不對稱分佈模式的美國股市的市場風險 Market Risk of the United States Stock Market Based on Asymmetric Distribution Model 指導教授: 王若愚博士 研 究 生: 阮坤輝 中 華 民 國 一 百 零 七 年 五 月 Market Risk of the United States Stock Market Based on Asymmetric Distribution Model Acknowledgements “Market Risk of the United States Stock Market Based on Asymmetric Distribution Model” is the topic I chose for my graduation thesis after two years in the master's degree program of the International Business Department at Feng Chia University Having gone through all research and processes to complete my thesis, I would like to express my deepest gratitude and appreciation towards my mentor, Professor Frank Wang of the Department of International Business, who gave me a clear orientation and guided me continuously and patiently throughout my master’s program and thesis In addition, I would also like to thank the professors, teachers, and staff members of the department office and my classmates and friends who contributed their valuable opinions for the thesis Last but not least, I would like to thank my relatives for their full support and belief in me along the way I couldn’t have accomplished this without their help Sincerely, Nancy Quynh Nguyen 阮坤煇 FCU e-Thesis and Dissertation (2018) Market Risk of the United States Stock Market Based on Asymmetric Distribution Model Abstract In this paper, the Value at Risk (VaR) approach is performed to analyze the investment risk of Dow Jones Industrial Average (DJIA) and its constituent stocks Data on stock index and stock prices are collected from Thomson Reuters Datastream, and the study is divided into the financial crisis period and tranquil period By applying the following three distribution models: Skewed-T Distribution (ST), Generalized Hyperbolic Distribution (GH), and Normal Inverse Gaussian Distribution (NIG) on daily stock returns, empirical evidence shows that the VaR of the skewed distributions are better than that of the normal distribution Furthermore, as expected, the distribution of the return on stock value during the financial crisis has a fatter tail compared to the tranquil period, and out of the three models, the NIG Distribution provides the most satisfactory result Keywords: value at risk, financial crisis, skewed distribution i FCU e-Thesis and Dissertation (2018) Market Risk of the United States Stock Market Based on Asymmetric Distribution Model 摘要 在本文中，將採用風險價值（VaR）方法來分析美國道瓊股價指數及成分股之市 場投資風險。研究中股價指數及股票價格由 Datastream 資料庫取得，並將研究期 間區分為金融危機期和平靜期。本研究利用三種不同的一般化厚尾分配配使本研 究 之 資 料 ， 三 種 分 配 模 型 為 正 態 偏 差 t 分 佈 （ ST ） ， 一 般 化 Hyperbolic Distribution（GH）和 Normal Inverse Gaussian Distribution（NIG）。實證結果基 本上說明這些偏態分佈的風險值衡量績效優於常態分配的結果。 其次，研究結果也如同我們期望，與非金融風暴期間相比，在金融風暴期間股價 報 酬 率 具 有 較 厚 的 尾 部 分 配 ， 三 中 模 型 分 配 又 以 Normal Inverse Gaussian Distribution 的結果最令人滿意。 關鍵字: 風險價值，金融風暴，偏態分佈 ii FCU e-Thesis and Dissertation (2018) Market Risk of the United States Stock Market Based on Asymmetric Distribution Model Table of Contents Abstract i 摘要 ii Table of Contents iii List of Tables iv List of figures v Chapter Introduction 1.1 World financial background 1.2 Risk, Value at Risk and Asymmetrical Distribution 1.3 Research purpose 1.4 Structure of this thesis Chapter Literature Review 2.1 Previous research about VaR 2.2 Previous research about asymmetrical distributions 2.3 Differences with previous studies Chapter Research Methodology And Data .9 3.1 Research methodology 3.2 Value at Risk, Expected Shortfall and Skewness 3.2.1 Introduction to Value at Risk (VaR) 3.2.2 Expected Shortfall 10 3.2.3 Skewness 11 3.2.4 Kurtosis 12 3.3 VaR distribution models 12 3.3.1 The Generalized Hyperbolic Distributions 12 3.3.2 Skewed T distribution 13 3.3.3 The Normal-Inverse Gaussian (NIG) Distribution 14 3.4 VaR back-testing procedures .16 3.5 Data and background information 16 Chapter Empirical Results .20 4.1 Descriptive statistics .20 4.2 Pairwise correlations in daily returns 24 4.3 Model fitting distribution .24 4.4 VaR calculation 27 4.5 Backtesting procedure 31 4.5.1 Statistics 32 4.5.2 Result discussion 33 4.5.3 Violations summary 36 Chapter Conclusion 40 References .41 iii FCU e-Thesis and Dissertation (2018) Market Risk of the United States Stock Market Based on Asymmetric Distribution Model List of Tables Table Dow Jones Industrial Average components stocks 17 Table Descriptive statistic of stock returns .22 Table Pairwise correlation of daily returns .23 Table Model fitting parameters 24 Table VAR Statistics – Min value and date of stock components .33 Table Descriptive statistic of excessed VaR .33 Table Violence and ratios 34 Table Number of violations of DJIA, Apple, JS, TRVC .35 Table Expected number of violations 36 Table 10 Distribution models’ performance at confidence levels 39 iv FCU e-Thesis and Dissertation (2018) Market Risk of the United States Stock Market Based on Asymmetric Distribution Model List of figures Figure DAX price from 1993 to 1997 .3 Figure Skewness in positive and negative directions 11 Figure Kurtosis in different points of view 12 Figure Daily Stock Prices Returns of DJIA and stock components 19 Figure Fitting model results 27 Figure VaR calculation results .29 Figure Expected Shortfall calculation results .31 Figure Dow Jones 30 historical price and VAR 37 Figure VAR of Johnson & Johnson and Traveler Cos 38 v FCU e-Thesis and Dissertation (2018) Market Risk of the United States Stock Market Based on Asymmetric Distribution Model Chapter Introduction 1.1 World financial background The global financial crisis, which began in the United States in 2007, is known as the worst financial crisis in history It heavily affected global stock markets, leading to many bankruptcies and even collapses of stock markets at that time To this day, the number of people who are still willing to invest their money in stocks is a lot fewer than the number prior to the crisis, due to the high risk and low return However, the main economic variables, including growth rates, exchange rates, inflation rates and interest rates, show a significantly higher volatility (Butler and Okada, 2007) In short, financial markets have become more complex, uncertain and riskier In response to this problem, new derivative financial instruments (futures, options, swaps, etc.) were suggested to enhance risk analysis and management Currently, the derivatives market is huge because of its potential role in investment and risk management Yet, the economic changes and financial market reacted vigorously, reflecting the complexity and opacity of the financial system Despite bringing opportunities into the market, new derivatives with non-linear returns also come with vulnerability, risks and complexity That is to say, due to the increase in complexity and connectivity, financial organizations are more vulnerable to risks from price fluctuations in the financial market As a result, organizations began searching for alternative solutions in risk management In addition, with the recent financial and economic turmoil, financial institutions have proposed a plethora of rules and regulations to force organizations to take preventive measures in order to improve risk management and modeling systems 1.2 Risk, Value at Risk and Asymmetrical Distribution According to Malz (2001), risk is defined as the possibility of occurrence of the low rate of income or when consumption or losses are occurred In other words, ‘risk’ is the possibility of unexpected results or quantities deviating from its original expectations However, this deviation has a negative effect on the economy Positive deviations in economics are profitable and there are no risk problems Risk modeling uses a variety of techniques such as market risk, Value at Risk (VaR), or Extreme Value Theory (EVT) to analyze the portfolio and measure risks These risks are usually classified into credit risk, liquidity risk, market risk, and operational risk Many large financial service companies use risk modeling to help portfolio managers assess the amount of capital reserves to maintain and purchase financial assets FCU e-Thesis and Dissertation (2018) Market Risk of the United States Stock Market Based on Asymmetric Distribution Model Managers use formal risk modeling under the Basel II proposal (Chorafas, 2004) in all major international banking systems In the past, analysts often used qualitative methods to measure risk, which took up much effort, but nowadays, risk can be evaluated quickly and easily with the help of advanced hardware and computing software In this paper, Value at Risk (VaR) is used as the main risk measurement In brief, VaR is the maximum loss at a given confidence level for a given period of time Pérignon and Smith (2010) measured risk and VaR through two ways: volatility and percentile Volatility of financial returns means that there is a great likelihood of loss, but there is also a chance of gaining greater profits Percentile describes tail behavior VaR estimation is a complex task It is essential to recognize the main features of financial data and choose the best model The range of existing documents is extremely wide and may even be controversial, but it helps draw a general picture of the problem It is commonly believed that financial data is characterized by heavy-tail, time-varying fluctuations, asymmetric responses to bad and good news, and skewness Overlooking any of these factors may lead to the underestimation of VaR and possibly end in a bad result for companies, banks, and/or investors In recent years, skewness has gained more and more attention, and giving rise to the open problem of time-varying skew detection and modeling Is skewness constant or are there any significant variabilities which in turn affects the VaR estimation? The distribution has symmetry (Berim and Ruckenstein, 2009) It is called a "normal distribution" and its pattern is recognized as a Gaussian bell curve The asymmetric data is unbalanced because the data is more heavily weighted on one side than the other; it has a larger value on the left, right or both sides rather than concentrated in the middle Risk and return curves are usually plotted in histograms, such as bell curves, for data analysis Market return curves not demonstrate a perfect balance of symmetry, according to the modern academic theory of modern finance Stock return curves are irregular, for they not fall into the Gaussian normal distribution However, the asymmetry of market return curves is distorted because they have a "fat tail" Also, the market return graph shows that the losses are immense, and they often seem to fit the normal distribution of equilibrium symmetry However, when applying a normal distribution, VaR may have undesirable characteristics (Artzner et al., 1997, 1999), as in lack of sub-addition (i.e the VaR of a portfolio with two tools may be greater than a single VaR) Figure shows that the tail of the generalized hyperbolic distribution, or even the tail of the normal Gaussian inverse distribution, is heavier FCU e-Thesis and Dissertation (2018) Market Risk of the United States Stock Market Based on Asymmetric Distribution Model than the tail of the normal distribution Thus, we can see that the VaR of the GH Distribution and its subcategories is parametrically closer to the empirical observational risk value In summary, the asymmetric distribution used to calculate the VaR provides a more reasonable result than the classical distribution (Eberlein & Karsten, 1998) This is one of the many reasons why I chose to use the asymmetric distribution model to measure the market risk of the United States’ stock market Figure DAX price from 1993 to 1997 1.3 Research purpose The purpose of this research is to investigate some unrevealed aspects of the market risk of the U.S stock market based on the asymmetric distribution Applying the one-day forecast of VaR at both 95% and 99%, the following VaR distribution models are compared: Skewed T distribution (ST), Generalized Hyperbolic distribution (GH), and Normal Inverse Gaussian distribution (NIG) When selecting the most suitable model, a back-testing analysis is carried out This research attempts to demonstrate that risk measuring is based on asymmetric distribution Additionally, this research aims to provide comprehensive information on the economy and stock markets of the United States The problems proposed in this study can be expressed as follows: FCU e-Thesis and Dissertation (2018) Market Risk of the United States Stock Market Based on Asymmetric Distribution Model Panel 3M’ stock returns Expected Shortfall calculation result Figure Expected Shortfall calculation results 4.5 Backtesting procedure In this section, descriptive statistics of VaR calculation is analyzed Afterwards, backtesting processes are performed to evaluate accuracy of each distribution Back-testing processes are applied based on specific data periods in the past, just to find the VaR for the ‘day in the future’, which means today, so here we can see the accuracy of VaR estimation processes If the empirical value excesses VaR value, it is considered a violation Therefore, in this research, I chose data from January 1995 to December 2010 (15 years) for the ‘past’ data to be applied into the back-testing process with ‘today’s’ data from 3/1/2011 to 30/12/2016 Results of VaR statistics are presented in Table starting from 01/03/2011 to 31/12/2016 Here, the first part is statistics of VaR calculation; the second is discussions, where I made some analyses based on stock prices and VaR values from distributions model calculations; the last 31 FCU e-Thesis and Dissertation (2018) Market Risk of the United States Stock Market Based on Asymmetric Distribution Model part is violation summaries, classifying results of the entire section Due to the main purpose of this research being risk and loss, we will focus only on left-tails, which have negative values at 95% and 99% confidence levels on the left 4.5.1 Statistics In Table 5, firstly, the value of empirical data of each stock components are selected together with date of occurrence On the same date, the value of each distribution model is displayed to make comparisons between them and to see which one is better for supporting empirical data The (%) column stands for percentage on total stock indices Positive value of percentage means that results from VaR is lower than empirical data, where things are still in control and there will be no loss for the investor On the contrary, negative value of percentage means that results from VaR is higher than empirical data and that there will be loss for investors from the difference between empirical data and VaR For example, according to empirical data from 2011 to 2016, DJIA had the lowest return at -5.5% on Aug 8th, 2011 On that day, ST Distribution gave the lowest value -0.06%, which was 12.18% lower than the empirical data This means that if investors build their backup plan based on ST Distribution, they may prevented themselves from losing money However, on that day, GH Distribution was at -4.61% (16.8% higher than empirical data) and NIG Distribution was at -4.81% (13.35% higher than empirical) This means that ST Distribution is more suitable to provide the best results for DJIA returns on the worst date The lowest daily return of DJIA goes to Cisco System (CSCS) at -14.16% on Feb 10th, 2011 On that day, the value of ST Distribution was better than others at -12.52% (11.55% higher), NIG Distribution at -10.78% (23.83% higher), and GH Distribution at -10% (29.30% higher) Pfizer (PF) had the lowest daily return at -4.75% on Aug 8th, 2011 On that day, the GH Distribution gave the closest value at -6.16% (29.82% lower), while ST and NIG Distributions gave -6.86% (44.57% lower) and -6.42% (35.18% lower) Based on this result, we can see that the GH Distribution is better than others, fitting empirical data with the closest value 32 FCU e-Thesis and Dissertation (2018) Market Risk of the United States Stock Market Based on Asymmetric Distribution Model Table VAR Statistics – Min value and date of stock components – empirical data and distributions rDJ30 rAPPLE rBOEING rCCCL rCR rCSCS rCTPL rEDN rEM rGE rHD rINTEL rINTER rJMCC rJS rMC rMCD rMCS rNIKE rPF rPG rTHREEM rTRVC rUHG rUSEP rUTNLG rVC rWD rWMS Date 8/8/2011 1/24/2013 1/27/2016 10/21/2014 8/8/2011 2/10/2011 8/8/2011 10/23/2012 8/8/2011 8/8/2011 8/8/2011 1/15/2016 4/19/2013 8/8/2011 8/8/2011 1/13/2011 7/26/2016 7/19/2013 6/29/2012 8/8/2011 4/24/2013 10/25/2011 8/8/2011 8/8/2011 1/22/2016 9/22/2011 8/8/2011 8/5/2015 8/8/2011 Empirical data -0.0555 -0.1236 -0.0893 -0.0603 -0.0754 -0.1416 -0.0922 -0.0906 -0.0619 -0.0654 -0.0589 -0.0910 -0.0828 -0.0941 -0.0941 -0.0662 -0.0447 -0.1140 -0.0940 -0.0475 -0.0657 -0.0625 -0.0759 -0.0797 -0.1210 -0.0876 -0.0551 -0.0917 -0.0922 ST value -0.0622 -0.1171 -0.0820 -0.0706 -0.0623 -0.1252 -0.0853 -0.0782 -0.0622 -0.0935 -0.0846 -0.1224 -0.0871 -0.1359 -0.0565 -0.0766 -0.0638 -0.0950 -0.0985 -0.0686 -0.0752 -0.0676 -0.0904 -0.1088 -0.2207 -0.0748 -0.0703 -0.0893 -0.0713 (%) 12.18% -5.20% -8.11% 17.11% -17.38% -11.55% -7.52% -13.67% 0.44% 43.00% 43.68% 34.46% 5.19% 44.33% -39.97% 15.61% 42.93% -16.68% 4.75% 44.57% 14.47% 8.16% 19.08% 36.54% 82.34% -14.65% 27.68% -2.59% -22.72% GH value -0.0461 -0.1116 -0.0713 -0.0544 -0.0607 -0.1001 -0.0773 -0.0701 -0.0633 -0.0790 -0.0948 -0.0963 -0.0691 -0.1356 -0.0475 -0.0733 -0.0539 -0.0812 -0.0805 -0.0616 -0.0552 -0.0703 -0.0773 -0.1065 -0.0915 -0.1120 -0.0852 -0.0755 -0.0711 (%) -16.88% -9.68% -20.18% -9.76% -19.55% -29.30% -16.12% -22.70% 2.30% 20.70% 60.92% 5.82% -16.49% 44.02% -49.55% 10.63% 20.72% -28.79% -14.35% 29.82% -15.96% 12.34% 1.88% 33.63% -24.35% 27.78% 54.74% -17.66% -22.91% NIG value -0.0481 -0.1083 -0.0709 -0.0567 -0.0583 -0.1078 -0.0776 -0.0703 -0.0576 -0.0773 -0.0777 -0.0960 -0.0718 -0.1088 -0.0512 -0.0683 -0.0543 -0.0803 -0.0842 -0.0642 -0.0601 -0.0598 -0.0769 -0.0924 -0.1027 -0.0668 -0.0644 -0.0758 -0.0646 (%) -13.35% -12.35% -20.55% -5.97% -22.69% -23.83% -15.88% -22.39% -6.91% 18.12% 32.03% 5.51% -13.23% 15.58% -45.64% 3.14% 21.62% -29.52% -10.47% 35.18% -8.48% -4.39% 1.38% 15.96% -15.14% -23.76% 16.94% -17.37% -29.90% 33 FCU e-Thesis and Dissertation (2018) Market Risk of the United States Stock Market Based on Asymmetric Distribution Model Table shows the summary of the number of times the minimum values of empirical data are covered, or not covered, by VaR distribution The results are divided into three categories: ‘Less than 10%’ shows that the number of times VaR distribution failed to predict empirical data is less than 10% (the lower the better); ‘-10% to 10%’ shows that the number of times VaR distribution failed to predict empirical data is higher than 10% and success rate is less than 10%, which represents the closest range to empirical data (the higher the better); ‘Higher than 10%’ means that success rate of VaR distribution in predicting empirical data is higher than 10% (the higher the better) As we can see, results from ST Distribution are better in the range of ‘-10% to 10%’ (27.59%) being closest to the empirical data Following up is results from NIG Distribution (24.14%), while results from GH Distribution is only 17.24% in this range Value of ST Distribution remains the best in the ‘Less than -10%’ category at 24.14% (the lower the better), however, GH and NIG Distributions fail to show good results in this range at 48.28% and 51.72% In the ‘Higher than 10%’ category, ST Distribution (48.28%) proves to be the safest choice for risk management compared to GH (34.48%) and NIG (24.14%) Therefore, ST is currently the best method to deal with extremely low empirical data Table Descriptive statistic of excessed VaR Less than -10% From -10% to 10% Higher than 10% ST Percentage (times) 24.14% 27.59% 14 48.28% GH Percentage (times) 14 48.28% 17.24% 10 34.48% NIG Percentage (times) 15 51.72% 24.14% 24.14% 4.5.2 Result discussion In this part, empirical results are discussed VaR values from distribution models and empirical data are presented along with stock price charts and analysis As we can see in Figure 8, there are two downturns of Dow Jones 30 that occurred during mid-2011 and from mid-2015 to early 2016 The first downturn in August 2011 was the debtceiling crisis in the US, after months of wrangling and fear of government meltdown, President Obama and congressional leaders reached a deal, agreeing to raise the debt ceiling by $400 billion immediately, whilst cutting government spending by hundreds of billions of dollars over the next decade Before that, on April, the ratings agencies were developing a taste for this downgrading lark Standard & Poor's snips US debt outlook from stable to negative for the first time since the Pearl Harbor attack in 1941.The second downturn in August 2015 could have 33 FCU e-Thesis and Dissertation (2018) Market Risk of the United States Stock Market Based on Asymmetric Distribution Model been caused by the Chinese stock market crisis, which lead to the historic 588-point plunge of Dow Jones 30 At that time, global fears in China's economic slowdown shook stock markets around the world for two weeks in a row, starting with Shanghai's 8.5% drop Within minutes after the opening bell, the DJIA plummeted 1,089 points That was the largest point loss ever within a trading day, surpassing the Flash Crash in 2010 For the analysis process, we paid close attention to any empirical data going beyond the VaR at ‘left-tail’ at both 95% and 99% confidence levels, also known as a “violation” The full result is presented in Table To conduct a deeper analysis based on periods and results, data and VaR empirical results were selected from the main index DJIA (DJ30), the most violations stock Johnson & Johnson (JS), the fewest violation stock Traveler Cos (TRVC), and the median result Apple Cos (APPLE) Based on previous analyses, data and results from these stocks are divided into sections: 2011-first downturn period, 2012 to 2014-stable and uprising period, and 2015 and 2016-second downturn period Table Violence and ratios APPL BOEING CCCL CR CSCS CTPL DJ30 EDN EM GE HD INTEL IBM JMCC JS MC MCD MCS NIKE PF PG THREEM TRVC ST01 1 1 1 0 0 0 0 0 0 (%) 0.06 0.06 0.00 0.06 0.06 0.06 0.00 0.06 0.00 0.00 0.00 0.00 0.00 0.00 0.38 0.00 0.00 0.06 0.00 0.00 0.00 0.00 0.00 ST05 8 17 11 10 12 10 48 5 4 11 (%) 0.26 0.51 0.51 1.09 0.19 0.70 0.45 0.64 0.77 0.26 0.13 0.19 0.64 0.19 3.07 0.32 0.38 0.32 0.32 0.26 0.26 0.70 0.26 GH01 10 2 4 3 20 25 16 18 (%) 0.64 0.13 0.13 0.06 0.26 0.51 0.26 0.26 0.00 0.51 0.26 0.19 0.19 1.28 1.60 0.00 0.26 1.02 0.13 0.26 1.15 0.26 0.00 GH05 13 17 17 13 12 13 12 6 12 28 28 19 22 12 (%) 0.83 0.51 0.57 1.09 0.45 1.09 0.83 0.77 0.83 0.77 0.38 0.38 0.77 1.79 1.79 0.32 0.57 1.21 0.45 0.57 1.40 0.77 0.26 NIG01 1 1 0 0 0 1 (%) 0.06 0.13 0.06 0.06 0.06 0.06 0.06 0.13 0.06 0.00 0.00 0.00 0.13 0.00 0.00 0.00 0.00 0.13 0.06 0.00 0.06 0.13 0.00 NIG05 8 17 11 10 12 3 10 7 12 34 FCU e-Thesis and Dissertation (2018) (%) 0.26 0.51 0.51 1.09 0.26 0.70 0.57 0.64 0.77 0.26 0.19 0.19 0.64 0.26 0.19 0.45 0.38 0.32 0.45 0.32 0.38 0.77 0.26 Market Risk of the United States Stock Market Based on Asymmetric Distribution Model UHG USEP UTNLG VC WD WMS Average 0 0.59 0.00 0.00 0.06 0.00 0.13 0.06 7.69 0.32 0.19 0.38 0.06 0.57 0.32 13 10 6.34 0.00 0.38 0.83 0.64 0.13 0.45 20 12 11 12.17 0.32 0.51 1.28 0.77 0.57 0.70 2 0.86 0.00 0.06 0.13 0.00 0.13 0.06 6.69 Table Number of violations of DJIA, Apple, JS, TRVC Distribution Period 2011 20122014 20152016 ST 2011 GHY 20122014 20152016 2011 NIG DJIA 20122014 20152016 APPLE JS TRVC Total by distribution 99% 95% 99% 95% 99% 95% 99% 95% 99% 95% 0 26 33 0 0 11 2 13 19 0 10 10 11 18 2 11 11 0 20 21 5 19 0 0 0 0 2 Total 99% 95% 63 39 58 20 Table is formed to summarize violations in each distributions and stocks For Dow Jones Industrial Average, ST Distribution performs the best with violations at 99% confidence in all three periods, even for 95% confidence there is only violations in two chaos periods and during the stable period Following up is the NIG Distribution with violation at 99% confidence in the first chaos period, violations at 95% confidence in the chaos period and violations in stable period These two distributions performed quite well, however, results from the GHY Distribution did not In the stable period, there were violations at both 99% and 95% confidence levels and a total of 13 violations during the chaos period For the most violation stock of DJIA-Johnson & Johnson (JS), results from ST Distribution show 55 violations, with in the chaos period and in the stable period (99% confidence); Results from 35 FCU e-Thesis and Dissertation (2018) 0.32 0.26 0.38 0.13 0.57 0.38 Market Risk of the United States Stock Market Based on Asymmetric Distribution Model GHY Distribution show a total of 53 violations, 14 in the chaos period and 11 in the stable period (99% confidence); However, results from NIG Distribution display only violations at 95% confidence and the rest during the chaos period For the least violation stock Traveler Cos (TRVC), results from all distributions are similar, with violations, all at 95% confidence, and violations in the chaos period The total in the distribution column shows a clear result that NIG and ST Distributions only have few violations at 99% confidence in the chaos period and even lesser in the stable and growth period, but results from GHY show a lot more violations in both 95% and 99% confidence levels, and violations in the stable period are even higher than others The ‘Total’ column shows the clearest results that NIG Distribution is the best choice in managing risks for the four stocks and indices 4.5.3 Violations summary According to Christoffersen (1998), Table was made as the summary of the whole back-testing procedure, where the number of violations is collected from violation results to show how many times the real number went beyond the calculated number For the main concept of back-testing procedures, we focus mainly on two variables: Number of Observation (OBS); Number of Violations (NVI); Expected (acceptable) Number of Violations (EVI), and Ratio of Violations (RVI) RVI = NVI/EVI, OBS = 1565 According to Christoffersen (1998), EVI = OBS x (1-confidence level), thus, in this case, we have EVI 99% = 1565 x (1-0.99) = 15.65; EVI 95% = 1565 x (1-0.95) = 78.25 Table Expected number of violations Obs Confidence 1566 99% 95% EVI 15.65 78.25 36 FCU e-Thesis and Dissertation (2018) Market Risk of the United States Stock Market Based on Asymmetric Distribution Model Figure Dow Jones 30 historical price and VAR 37 FCU e-Thesis and Dissertation (2018) Market Risk of the United States Stock Market Based on Asymmetric Distribution Model Figure VAR of Johnson & Johnson and Traveler Cos Table shows the summary, and also the accuracy, of the performed back-tests If the number of violations of a stock is less than the EVI, it is counted as ‘acceptable’ However, if it is higher than the EVI, it is unacceptable For example, results from the GH Distribution at the 99% confidence level is times higher than the EVI of 99%: JP Morgan Chase & Co (20 violations), Johnson & Johnson (25 violations), Microsoft (16 violations), and Procter & Gamble (18 violations) On average, ST Distribution made 0.59 violations over the empirical at the 99% confidence level and 7.69 violations at the 95% confidence level; GH Distribution allows 6.34 and 12.17 violations at the 95% and 99% confidence level separately; NIG Distribution allows only 0.86 and 6.69 violations at 99% and 95% confidence levels At the 99% confidence level, results from ST and NIG Distributions have a lot of “0s”, as they predicted to reject any violations 38 FCU e-Thesis and Dissertation (2018) Market Risk of the United States Stock Market Based on Asymmetric Distribution Model Table 10 was made to show the violence at confidence levels We can see that results from NIV Distribution were best, with rejections at all confidence levels, results from Skewed-T Distribution show only one rejection at the 95% confidence level, and results from GH Distribution show rejections at the 99% confidence level Table 10 Distribution models’ performance at confidence levels Confidence level 99 95 Skew T Percentage 3.45 Generalized Hyperbolic Percentage 13.79 Normal Inverse Gaussian Percentage 0 39 FCU e-Thesis and Dissertation (2018) 0 Market Risk of the United States Stock Market Based on Asymmetric Distribution Model Chapter Conclusion Value at Risk measures the risk of a company losing money on their portfolio of assets and derivatives It makes use of market values of all the assets, derivatives and other financial products that the company holds in its portfolio From the back-testing analysis based on Normal Inverse Gaussian, Skewed-T and Generalized Hyperbolic Distribution models, we can first conclude that in this case, NIG is best to deal with Dow Jones Industrial Average index and its stock components, despite all three models being fairly blunt with empirical sizes far from the nominal sizes, and with what it seems like dependent back-testing exceptions For the 99% VaR, only the Generalized Hyperbolic Distribution model cannot pass the Christoffersen tests, but the two other models can, which motivates us to claim that they are of good use for DJIA and its stock components The most accurate 95% VaR estimations came from all the three distribution models Here, we receive empirical sizes close to the nominal levels while, at the same time, the back-testing exceptions seemed spread out and independent from each other It is hard to say which method is the “best” way to measure 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