Introduction
World financial background
The global financial crisis that started in the United States in 2007 is regarded as the worst in history, significantly impacting stock markets and resulting in numerous bankruptcies The aftermath has led to a decrease in investor confidence, with fewer individuals willing to invest in stocks due to perceived high risks and low returns Economic indicators such as growth rates, exchange rates, inflation rates, and interest rates have exhibited increased volatility, making financial markets more complex and uncertain In response, new derivative financial instruments like futures, options, and swaps have emerged to improve risk analysis and management, contributing to the growth of the derivatives market However, these derivatives also introduce vulnerabilities and risks due to their non-linear returns, heightening the exposure of financial organizations to market fluctuations Consequently, there has been a shift towards seeking alternative risk management solutions, alongside the introduction of new regulations aimed at enhancing preventive measures and improving risk management systems within financial institutions.
Risk, Value at Risk and Asymmetrical Distribution
Risk, as defined by Malz (2001), refers to the likelihood of experiencing low income rates or incurring losses, reflecting the potential for unexpected outcomes that deviate from initial expectations Such deviations negatively impact the economy, while positive deviations yield profits without risk concerns Risk modeling employs various techniques, including market risk assessments, Value at Risk (VaR), and Extreme Value Theory (EVT), to evaluate portfolios and measure different types of risks, which are categorized into credit risk, liquidity risk, market risk, and operational risk Major financial service firms utilize risk modeling to assist portfolio managers in determining necessary capital reserves and making informed decisions on financial asset purchases.
Managers in major international banking systems utilize formal risk modeling as outlined in the Basel II proposal, transitioning from traditional qualitative methods to advanced quantitative techniques This evolution allows for quick and efficient risk evaluation, with Value at Risk (VaR) serving as the primary measurement tool VaR represents the maximum potential loss at a specified confidence level over a defined time period Pérignon and Smith (2010) approached risk measurement through volatility and percentile methods, highlighting that while volatility indicates a likelihood of loss, it also presents opportunities for higher profits Accurate VaR estimation requires an understanding of financial data characteristics, including heavy tails, time-varying fluctuations, and asymmetric responses to news Ignoring these factors can result in significant underestimations of VaR, leading to detrimental outcomes for companies, banks, and investors.
In recent years, the focus on skewness has intensified, raising the question of whether skewness is constant or exhibits significant variability that impacts Value at Risk (VaR) estimation While a normal distribution, characterized by a Gaussian bell curve, is symmetric, real-world data often displays asymmetry, indicating an imbalance where values are more concentrated on one side Histograms, typically used to analyze risk and return curves, reveal that market return curves do not conform to perfect symmetry as suggested by modern finance theory Instead, stock return curves are irregular and deviate from the Gaussian distribution, exhibiting a "fat tail" that distorts their asymmetry Furthermore, market return graphs indicate substantial losses, which frequently appear to align with the normal distribution's equilibrium symmetry.
When applying a normal distribution, Value at Risk (VaR) may exhibit undesirable characteristics, such as a lack of sub-additivity, where the VaR of a portfolio can exceed that of a single asset The tail of the generalized hyperbolic distribution, along with the normal Gaussian inverse distribution, is heavier than that of the normal distribution, indicating that the VaR derived from these distributions is more aligned with empirical risk observations Consequently, using an asymmetric distribution for VaR calculations yields more accurate results compared to classical distributions This is a key reason for selecting the asymmetric distribution model to assess market risk in the United States stock market.
Research purpose
This research aims to explore previously overlooked aspects of market risk in the U.S stock market, focusing on asymmetric distribution By applying one-day Value at Risk (VaR) forecasts at both 95% and 99% confidence levels, the study compares various VaR distribution models, including skewed distributions.
This research focuses on the T distribution (ST), Generalized Hyperbolic distribution (GH), and Normal Inverse Gaussian distribution (NIG) for model selection through back-testing analysis It aims to illustrate that risk measurement relies on asymmetric distribution, while also offering detailed insights into the U.S economy and stock markets The study addresses key problems related to these distributions and their implications for financial risk assessment.
1 What are the causes behind risk factors of the stock market in this modern world?
2 The current performance of the U.S stock market after recent crises and how their top stocks are doing?
3 Can asymmetric distribution analysis be a consideration for stock investors?
This study analyzes risk measurement methodologies by examining the stock components of the Dow Jones Industrial Average in the U.S stock market Closing prices for these stocks were sourced from Datastream, covering the period from 1995 to 2016 This timeframe was chosen due to its inclusion of significant historical economic events.
Structure of this thesis
The thesis comprises five chapters, beginning with an introduction that outlines the global economic landscape, financial risks, Value at Risk (VaR), and the research objectives The second chapter conducts a literature review and examines current risk management practices in the U.S stock market Chapter 3 details the data and methodologies employed, focusing on advanced distribution models and VaR calculation techniques In Chapter 4, the results and analysis are presented, utilizing empirical data to assess the compatibility of the applied models and methods, supplemented by back-testing procedures for validation Finally, Chapter 5 summarizes the implications of the findings, discusses the limitations of the study, and suggests avenues for future research.
Literature Review
Previous research about VaR
The popularity of financial risk models has led to extensive research on risk valuation and methodologies Beder (1995) conducted early evaluations of eight prevalent Value at Risk (VaR) models, revealing significant fluctuations in VaR estimates for identical portfolios across different models He concluded that these variations largely stem from differences in model specifications and underlying assumptions.
Value at Risk (VaR) is a technique used to estimate the tail of the empirical distribution of asset returns, often assuming a normal distribution However, research indicates that asset returns are typically scarce and exhibit fat tails, which can result in the underestimation or overestimation of actual VaR Consequently, numerous studies advocate for the adoption of Skewed-Return models to improve accuracy in risk assessment.
The T Distribution and Generalized Error Distribution (GED) are fat-tailed models used to assess risk in extreme events, but they often overlook the skewness of returns Research by Brooks and Persand (2003) highlights that asymmetric models are essential to accurately estimate Value at Risk (VaR), regardless of income distribution or volatility specifications Giot and Laurent (2003) further argue that the normal APARCH model yields inadequate VaR predictions due to the importance of skewness and kurtosis in asset returns, which are not only leptokurtic but also asymmetric Despite this, previous studies have seldom utilized asymmetric returns for VaR analysis Notably, Giot and Laurent (2003) and Angelidis and Degiannakis (2005) have recently employed the Skewed Student’s T Distribution to enhance VaR estimations in financial markets.
In 2007, a conditional technique was proposed for estimating Value at Risk (VaR) and expected under-measures using the Skewed Generalized T (SGT) Distribution, demonstrating that a VaR model accommodating the asymmetric distribution of conditional returns is significant While most VaR literature relies on normal, Student’s T, or Generalized Error Distribution (GED), these distributions fail to fully address risk underestimation issues, as they inadequately handle fat-tail and high-frequency problems, leading to potential model risk Theodossiou (2000) introduced the Skewed Generalized Error Distribution to further tackle these challenges.
The SGED distribution is utilized for modeling the empirical distribution of financial asset returns In 2003, Lehnert integrated SGED return innovations into the GARCH model, using DAX index options to evaluate both in-sample and out-of-sample option pricing performances.
(2007) proposed the GARCH specification to model non-linear dynamic short-term interest rates, fluctuations, and a set of distribution assumptions (Normal Deviation T, GED, Skewed-
T, and SGED) At the same time, the estimation of the conditional mean and return variance required to implement the parameterization technique was based on a simple GARCH (1,1) model Nasstrom (2003) used different stock series and the Stockholm Stock Exchange Index OMX to estimate VaR's different Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models (first proposed by Engle (1982)) He concluded that some of them believed that the standard GARCH (1,1) performed well in estimating VaR Schmidt and Duda (2009) researched for their master's thesis some of the best VaR models for both the three-parameter and non-parametric models In their regression analysis, based on 250 observations, they found that the Conditional Autoregression VaR (CAViaR) introduced by Engle & Manganelli (2004) yielded the most accurate estimate of one-day 95% VaR, but within one-day 99% Value at risk When volatility was modeled using the GARCH (1,1) model, they also found improvements in the estimations.
Previous research about asymmetrical distributions
Recent empirical studies have highlighted the effectiveness of Skew Normal and Skewed-T distributions as valuable models for data analysis (Vernic, 2006; Bolancé et al., 2008; Eling, 2012) Their growing popularity in research can be attributed to their straightforward interpretation and ease of implementation in financial models (De Luca et al., 2006; Adcock, 2007).
There has been a lack of analysis regarding the fit of skewed distributions, such as the Skewed-T Distribution, Generalized Hyperbolic Distribution, and Normal Gaussian Distribution, on the Dow Jones Industrial Average The Skewed-T model has shown significant theoretical implications in portfolio selection and asset pricing, extending foundational concepts like Stein’s lemma from normal to skewed distributions This suggests that the Skewed-T model could serve as a valuable tool in actuarial science, particularly in individual and collective risk models as well as asset liability management While the classical normal distribution remains the benchmark in financial modeling, recent studies indicate that the Skewed-T Distribution, alongside the NIG Distribution, is highly effective in capturing empirical market data.
Recently, the Generalized Hyperbolic Distributions family have proposed to adapt to empirical results The development of these distributions is attributed to Barndorff-Nielsen
In 1977, the Generalized Hyperbolic Distribution was introduced to model the size of continuously blown sand Since then, this distribution has found applications across various fields, including physics, biology, and agronomy Eberlein and Keller (1995) were the first to implement these models in finance, utilizing a hyperbolic subclass to align with national empirical data Subsequently, Keller (1997) advanced the field by developing expressions for derivative pricing, further enhancing the applicability of the Generalized Hyperbolic Distribution in financial contexts.
In 1999, significant advancements were made in empirical finance by extending the work of Eberlein and Keller (1995) through the use of GH and German indices This research included derivative pricing and the measurement of Value at Risk (VaR), expanding its application to various distribution scenarios Additionally, in the early 1990s, Blresild and Sonrensen (1992) introduced a computer program named ‘Hyp’ for estimating parameters of three-dimensional hyperbolic subclass distributions, while Prause (1999) developed a program for estimating GHD parameters It is important to note that these programs are not available for free.
The Normal Inverse Gaussian (NIG) Distribution, introduced by Barndorff-Nielsen in 1994, is a specific case of the Generalized Hyperbolic Distribution that offers unique characteristics advantageous for the financial sector With its four parameters, the NIG Distribution accommodates higher skewness and kurtosis compared to the Normal Distribution, allowing for slower tail-decrement This makes it an optimal choice for simulating financial derivatives, particularly when modeling stock returns as the underlying asset Utilizing NIG Distributions instead of Gaussian Distributions results in more accurate financial return modeling, facilitating realistic Value at Risk (VaR) calculations.
Differences with previous studies
The main differences compared with previous literature are as follows:
(1) We consider a plethora of skewed distributions: Skewed-T Distribution, Generalized Hyperbolic Distribution, Normal Inverse Gaussian Distribution, and Normal Distribution
(2) Risk measurement based on both Value at Risk and Expected Shortfall
(3) Statistical terms are compared using large data with daily high frequency data
Research indicates that skewness and fat-tailed distributions outperform normal distributions in modeling financial returns and predicting Value at Risk (VaR) Among the skewed distributions analyzed in this thesis, the Normal Gaussian Distribution and the Skewed Distribution are notably significant.
Research Methodology And Data
Research methodology
This study analyzes stock risk in the DJIA using Value at Risk (VaR) to measure the maximum expected loss at a specified probability It employs Generalized Skewed-T (GST) and Generalized Hyperbolic (GH) distributions, which are asymmetrical, for comparison with the Normal Inverse Gaussian (NIG) Distribution After adopting the selected distribution models, the comparison process is conducted to derive VaR results Notably, a normal skewness test fails to identify asymmetry except in the smallest decile, and skewness-based measures offer limited insights into the cross-section of stock returns (Jiang and Wu et al., 2016) Consequently, asymmetrical distribution models are utilized in this research.
Value at Risk, Expected Shortfall and Skewness
3.2.1 Introduction to Value at Risk (VaR)
Jorion (2001) defines Value at Risk (VaR) as the maximum potential loss of target options at a specified confidence level The prevalent approach to calculating VaR involves a mathematical model represented by a specific function.
The daily standard deviation ($\sigma$) is calculated by multiplying the number of standard deviations (C) corresponding to the chosen confidence level, the square root of the time range, and the currency size of the investment to determine Value at Risk (VaR) In this study, VaR is estimated on a daily basis to eliminate time components Additionally, the currency size of the investment, which represents the monetary figures at risk, can also be disregarded Consequently, the focus remains on the standard deviation and confidence level of historical returns.
Choosing a confidence level of 90% or lower for Value at Risk (VaR) can lead to inaccuracies, as different confidence levels suit various companies and their risk appetites A manager's risk aversion directly influences the required confidence level; the more risk-averse the manager, the higher the confidence level needed When applying VaR, it is essential to value assets at their market prices, which are readily available in liquid markets, as noted by Penza & Bansal (2001) Dowd (1998) emphasizes the importance of market price basis, where tradable securities are valued using current market prices and frequent position revaluations While financial assets are typically traded daily, making current valuations easier, non-financial assets may require estimation, adding uncertainty to calculations Although the assets in this study are expected to present no valuation issues, their pricing can vary significantly While some analysts claim they can forecast stock prices based on available information, predicting future oil prices remains challenging due to OPEC's significant impact on supply and demand dynamics Sadeghi & Shavvalpour (2006) highlight VaR as a valuable tool for quantifying market risk in the oil sector.
Expected Shortfall (ES) extends the concept of Value at Risk (VaR), a widely used risk management tool VaR is defined as the worst-case loss at a specified confidence level (1-α) within the interval (0,1), represented as VaRα(X) For instance, at a 99% confidence level, there is a 99% probability that losses will not exceed VaR0.01 by the end of the designated risk period By defining X as the portfolio's net income from time 0 to time 1, the loss at time 0 can be approximated as L = -X for short time frames Statistically, VaRα corresponds to the (1-α) distribution of loss L, and if the distribution function FL is continuous and strictly increasing, VaR can be expressed accordingly.
Value at Risk (VaR) only represents a quantile value and overlooks losses that exceed the selected probability level α, potentially concealing tail risk To address this limitation, the concept of Expected Shortfall (ES) extends VaR by incorporating these tail values and calculating the average VaR below the α level.
By combining equations (3.2) with (3.3) and using the defined loss function F L , we can see that ES is only the average of the quantile values of L:
Value at Risk (VaR) is translation invariant and monotonous, but it lacks sub-additivity, indicating that diversification does not always yield benefits In contrast, Expected Shortfall (ES) maintains translation invariance, monotonicity, and positive homogeneity, while also being sub-additive, which qualifies it as a coherent measure of risk (Hult, Lindskog, Hammarlid, and Rehn, 2012).
Figure 2 Skewness in positive and negative direction
Skewness indicates the symmetry of data distribution, with a normal distribution exhibiting a skewness of 0, centered around its mean A data set that shows positive or negative skewness deviates from this normal distribution assumption, potentially leading to an overestimation or underestimation of the Value at Risk (VaR) value, influenced by the deviation in the underlying asset returns (Lee, 2000).
Figure 3 Kurtosis in different points of view
Kurtosis quantifies the peak of data and the concentration of returns, with high kurtosis indicating that extreme deviations contribute significantly to data changes According to Lee (2000), the kurtosis of a normal distribution is three, and any deviation from this can complicate parameter methods A distribution with a higher peak, known as 'leptokurtic,' shows that asset returns have more extreme values than the normal distribution, while excessive negative kurtosis is termed 'platykurtic.' Additionally, low kurtosis results in very small Value at Risk (VaR).
VaR distribution models
Generalized Hyperbolic (GH) Distributions, introduced by Barndorff-Nielsen in 1977, are essential for modeling financial data due to their semi-heavy tails, which surpass those of the Gaussian Distribution (Ramalho, 2004) These distributions have well-established mathematical properties (Barndorff-Nielsen and Blaesild, 1981; Blổsild, 1999) and have been further developed into subcategories like the Generalized Hyperbolic Distribution and the Diffusion Limit Distribution for financial price increments (Eberlein and Keller, 1995; Rydberg, 1997a) Jaschke (1997) noted that the stable GH Distribution emerges as the weak limit of the stochastic volatility process modeled by GARCH (1,1) time series, encompassing popular distributions such as Gaussian, Student’s T, Skewed-T, Variance Gamma, Normal Inverse Gaussian, and Hyperbolic Distributions.
In many ways, we can set the parameters of GH Distribution Models as Prause (1999):
In the expression above, Kj is the modified Bessel function of the third kind of order j (Abramowitz and Stegun, 1972) and the parameters must fulfill the following conditions:
Each subclass of GHD is obtained through different assumptions of the values In this thesis, we only focus on the two above
The Skewed-T Distribution is particularly noteworthy for its effective modeling of equity return data, as highlighted by Hu and Kercheval (2010) This distribution belongs to the Generalized Hyperbolic Distributions family (McNeil et al., 2005) and can be represented as a normal mean-variance mixture involving inverse gamma distributions Empirical studies indicate that daily economic returns often exhibit characteristics of being both fat-tailed and skewed, with one tail displaying a higher probability mass than the other To accurately represent this behavior, an asymmetric probability distribution like the Skewed-T Distribution is essential The Generalized T-Distribution, introduced by Hansen (1994), provides a relevant density function for this analysis.
If X ~ GHd (λ, χ, ψ, μ, Σ, γ), λ = -1 / 2v, χ = v and ψ = 0, then X has a Skewed-T Distribution expressed as Skewed-T (ν,μ, γ, σ) as follows:
The mean and covariance of the Skewed-T Distribution random variable X is:
The covariance matrix is defined only when v > 4 In addition, when γ = 0, the Skewed-
The T Distribution simplifies to the Student's T Distribution The inverse gamma random variable is defined as the reciprocal of the gamma random variable By utilizing the mean-variance mixture definition, we can effectively create a tilted t random variable.
3.3.3 The Normal-Inverse Gaussian (NIG) Distribution
The NIG Distribution is highly versatile, making it suitable for various financial applications Forsberg and Bollerslev (2002) utilized NIG innovations to model the GARCH process and simulate the EUR/USD exchange rate Venter and de Jongh (2002) demonstrated that NIG-based Value at Risk (VaR) outperforms extreme value theory VaR Chen et al (2005) effectively calculated VaR using NIG Distribution with adaptive volatility estimates, aligning well with German bank asset portfolio data Chen and Lu (2012) confirmed the robustness and accuracy of NIG-based VaR estimation for one-day predictions The preference for NIG Distribution over Gaussian Distribution results in more realistic financial return models, enhancing VaR calculations The NIG Distribution, as detailed by Paolella (2007), is characterized by four parameters: steepness (α), asymmetry (β), scale (δ), and position (μ).
In the context of the NIG Distribution, where \( x \in \mathbb{R} \), \( 0 \leq |\beta| < \alpha \), \( \delta > 0 \), \( \mu \in \mathbb{R} \), and \( K_1 \) represents the third type of modified Bessel function with an index of 1, a smaller value of \( \alpha \) indicates a fat-tail density, while \( |\beta| \) increases with skew The center moment can be calculated based on the parameters provided by Kalemanova et al (2007).
Estimating the parameters of the NIG Distribution is typically done using maximum likelihood estimation; however, the complexity of the possibilities can make this process challenging By solving the equations (3.12) to (3.15), we derive the condition \( k - 5 \).
In the short term, the average daily income is influenced by variance, making it impossible to dismiss the zero-mean-regression hypothesis (Christoffersen, 1998) Consequently, the average value, denoted as \( m \), is established at 0.
VaR back-testing procedures
The Coverage Ratio test, as outlined by Christoffersen (1998), is a widely used method for Value at Risk (VaR) back-testing This test assesses whether the number of exceptions, or losses that surpass the VaR predictions, aligns with the expected number of exceptions over a specified back-testing period.
Data and background information
The analysis focuses on the Dow Jones Industrial Average (DJIA), which consists of 30 selected large-cap U.S stocks, commonly referred to as 'DJ30' Established in 1882 by Charles Dow, Edward Jones, and Charles Bergstresser, the DJIA is the primary U.S stock index Charles Dow initially introduced an index of 11 stocks in 1884, which originally included only transportation companies It was first known as the transport average, but on May 26, 1896, Dow Jones launched two indices, leading to the creation of the Dow Jones Industrial Average By spring 1896, the index expanded to 12 stocks, and later that year, it evolved into a railway index with 20 stocks The final number of stocks was solidified in 1928, culminating in the selection of the current 30 relevant stocks.
Over the past 30 years, the Dow Jones index experienced significant growth, particularly during the mid-1990s when it surged over 315% in a decade, largely driven by the technology boom and the Internet Revolution, reaching nearly 11,000 However, the early 2000s brought challenges as the index faced substantial fluctuations influenced by the aftermath of the Internet bubble, the Afghanistan and Iraq wars, and the political and social ramifications of the 9/11 attacks.
Currently, the Dow Jones index fluctuates between 7,000 and 9,000 However, during the mid-2000s, the index experienced significant growth, reaching an all-time high of 14,164.53 on October 9, 2007, fueled by a global economic boom and positive inflation The subsequent collapse of Lehman Brothers led to a severe credit crunch, causing the Dow Jones to plummet to a 12-year low of 6,547.05 by March 9, 2009 U.S monetary policies played a crucial role in the index's recovery, which began in late 2009 Since 2013, the Dow Jones has been particularly active, soaring from 13,500 points in 2010 to nearly double its previous peak at 26,600 points.
Data spanning from January 1995 to December 2016 were collected, utilizing daily data to accurately capture short-term interactions, as weekly or monthly data may overlook these nuances (Eun and Dal Shim, 1989) Notably, Visa's stock was listed in 2008 and Goldman Sachs' in 1999, leading to their exclusion from this research The equity indices data were sourced from Thomson Reuters and downloaded from Datastream at Feng Chia University in December 2015, with firms included in the DJIA detailed in the accompanying table.
Table 1 Dow Jones Industrial Average components stocks
Boeing BOEING Aerospace and defense
Goldman Sachs GMS Banking, Financial services
Walt Disney WD Broadcasting and entertainment
Caterpillar CTPL Construction and mining equipment
UnitedHealth Group UHG Managed health care
The Home Depot HD Retailer
Daily returns 𝑟 𝑡 were computed as logarithmic differences by using the daily closing data
Panel 1 Daily Stock Prices Return of Dow Jones Industrials Average
Panel 2 Daily Stock Prices Return of Apple Corp
Panel 3 Daily Stock Prices Return of Microsoft Corp
Panel 4 Daily Stock Prices Return of Intel Corp
DOW JONES INDUSTRIALS - RETURN INDEX
Panel 5 Daily Stock Prices Return of Exxon Mobil Corp
Panel 6 Daily Stock Prices Return of JP Morgan Corp
Panel 7 Daily Stock Prices Return of Johnson & Johnson Corp
Panel 8 Daily Stock Prices Return of Wal-Mart Corp
Figure 4 Daily Stock Prices Returns of DJIA and stock components
Empirical Results
Descriptive statistics
Table 2 presents descriptive statistics for the daily log-returns of the Dow Jones 30 Industrial Average and its component stocks from January 3, 1995, to December 30, 2016 Over this period, all stocks exhibited a positive average mean, with Apple leading at 0.12%, followed by UnitedHealth Group and Cisco Systems at 0.08% The DJIA index had a mean of 0.03%, matching the lowest values of three other component stocks Notably, the DJIA index showed a right-tailed distribution, while 35.71% of stock components had left-tailed distributions Procter & Gamble recorded the highest kurtosis at 46.97 and the lowest minimum return at -31.38%, indicating a complex distribution Apple experienced a significant return drop of -51.87% due to a failed sales campaign The September 11, 2001, terrorist attacks severely impacted Boeing and United Technologies, leading to substantial losses During the 2008 financial crisis, various stocks saw significant daily return losses, while many stocks recorded high returns during recovery phases Key events influencing stock performance included strategic investments and corporate announcements, with notable returns from companies like DuPont and Microsoft The next section will detail the fitting model test.
Table 2 Descriptive statistic of stock returns
Mean Min min-date Max max-date St Dev Skew Kurt Obs DJIA 0.03% -7.87% 10/15/2008 11.08% 10/13/2008 0.0111 0.0054 8.5878 5739
JS 0.05% -15.85% 7/19/2002 12.23% 10/13/2008 0.0129 -0.0085 9.2170 5739 EDN 0.03% -11.33% 12/1/2008 11.85% 12/9/2015 0.0180 0.0468 4.6745 5739 CTPL 0.05% -14.52% 10/20/2006 14.72% 10/13/2008 0.0205 0.0826 4.2668 5739 THREEM 0.04% -9.59% 12/17/1997 11.07% 12/5/2000 0.0148 0.1228 4.7802 5739 MCD 0.05% -12.82% 9/17/2002 10.86% 9/8/1998 0.0151 0.1557 5.7712 5739 NIKE 0.07% -19.46% 2/27/2001 14.28% 3/15/2000 0.0199 0.1671 8.4484 5739
WD 0.05% -18.36% 9/17/2001 15.97% 10/13/2008 0.0190 0.1822 8.4147 5739 CCCL 0.03% -10.48% 8/31/1998 13.88% 10/13/2008 0.0140 0.2023 7.2252 5739 MCS 0.07% -15.60% 4/24/2000 19.57% 10/19/2000 0.0200 0.2311 7.5296 5739 IBM 0.05% -15.54% 10/18/2000 13.16% 4/22/1999 0.0175 0.2368 7.9906 5739 WMS 0.05% -10.04% 10/14/2015 11.07% 10/28/2008 0.0163 0.2890 4.9322 5739
Table 3 Pairwise correlation of daily returns
CR VC PF CCCL MC INTE
WD IBM HD UHG CSCS THREE
EDN CTPL TRV DJ30 1 0.41 0.60 0.65 0.55 0.71 0.74 0.53 0.55 0.62 0.52 0.56 0.53 0.54 0.59 0.65 0.62 0.64 0.42 0.56 0.69 0.50 0.62 0.72 0.49 0.73 0.69 0.67 0.61 C APPLE 0.41 1 0.37 0.21 0.15 0.29 0.32 0.14 0.21 0.20 0.20 0.17 0.16 0.17 0.44 0.27 0.37 0.27 0.16 0.41 0.24 0.18 0.23 0.27 0.20 0.29 0.23 0.28 0.23 MCS 0.60 0.37 1 0.36 0.30 0.41 0.44 0.24 0.33 0.34 0.32 0.32 0.28 0.29 0.59 0.40 0.48 0.38 0.22 0.55 0.33 0.26 0.34 0.40 0.29 0.42 0.35 0.36 0.35
0.69 0.24 0.33 0.45 0.37 0.44 0.51 0.37 0.35 0.43 0.32 0.35 0.36 0.32 0.34 0.41 0.35 0.41 0.30 0.32 1 0.31 0.42 0.53 0.34 0.47 0.56 0.50 0.44 MCD 0.50 0.18 0.26 0.30 0.31 0.30 0.36 0.34 0.32 0.28 0.26 0.30 0.33 0.29 0.24 0.31 0.27 0.36 0.22 0.26 0.31 1 0.30 0.34 0.28 0.34 0.31 0.29 0.32 BOEING 0.62 0.23 0.34 0.39 0.31 0.39 0.44 0.28 0.30 0.37 0.28 0.31 0.31 0.30 0.33 0.41 0.32 0.38 0.28 0.32 0.42 0.30 1 0.54 0.32 0.43 0.43 0.42 0.37 UTNLG 0.72 0.27 0.40 0.45 0.35 0.48 0.54 0.35 0.36 0.43 0.34 0.37 0.35 0.33 0.37 0.46 0.38 0.44 0.31 0.38 0.53 0.34 0.54 1 0.35 0.50 0.52 0.52 0.44 NIKE 0.49 0.20 0.29 0.31 0.26 0.34 0.36 0.24 0.30 0.29 0.27 0.25 0.27 0.24 0.29 0.34 0.29 0.37 0.22 0.29 0.34 0.28 0.32 0.35 1 0.37 0.35 0.32 0.29 USEP 0.73 0.29 0.42 0.40 0.36 0.67 0.60 0.34 0.38 0.40 0.37 0.40 0.35 0.35 0.41 0.49 0.41 0.49 0.32 0.41 0.47 0.34 0.43 0.50 0.37 1 0.49 0.48 0.51 EDN 0.69 0.23 0.35 0.46 0.34 0.48 0.51 0.36 0.35 0.46 0.34 0.35 0.34 0.34 0.34 0.43 0.36 0.41 0.28 0.34 0.56 0.31 0.43 0.52 0.35 0.49 1 0.54 0.43 CTPL 0.67 0.28 0.36 0.43 0.30 0.44 0.49 0.29 0.31 0.44 0.29 0.31 0.30 0.28 0.36 0.41 0.36 0.41 0.26 0.34 0.50 0.29 0.42 0.52 0.32 0.48 0.54 1 0.38 TRVC 0.61 0.23 0.35 0.38 0.32 0.51 0.47 0.30 0.34 0.39 0.33 0.35 0.32 0.33 0.33 0.37 0.33 0.40 0.32 0.32 0.44 0.32 0.37 0.44 0.29 0.51 0.43 0.38 1
Pairwise correlations in daily returns
“Don’t put all your eggs in one basket!”
Effective risk management hinges on the principle of diversification However, many investors exhibit "home bias," primarily investing in domestic markets and under-allocating to foreign assets This behavior persists despite evidence that a greater allocation to international investments can mitigate risk and enhance returns (Lewis, 1999) To optimize their portfolios, investors should consider multiple stock markets and the interconnections between them As illustrated in Table 3, the pairwise correlations among DJIA stock returns are predominantly positive, yet the coefficients between individual stock components remain low, averaging at 0.37.
Model fitting distribution
Parameters Explanation λ location parameter ν shape parameter α.bar scale parameter μ mean σ standard deviation γ skewness
In this part, model fitting distribution and parameter estimation is performed using the
The analysis utilizes the 'VARES' and 'ghyp' packages in R software, focusing on 95% and 99% confidence levels Key parameters include λ (location), ν (shape), and α.bar (scale), essential for statistical inferences on population parameters, while μ, σ, and γ represent mean, standard deviation, and skewness, respectively Notably, γ is crucial, revealing that 39.29% of models in the Skewed-T Distribution are left-tailed, 50.00% in the GH distribution are left-tailed, and 32.14% in the NIG Distribution are left-tailed A comparison of model fitting distribution and parameter estimation tests with empirical descriptive statistics, particularly skewness values, is conducted to validate the findings.
The Gaussian fitting distribution test, with 32.14% left-tailed results, aligns most closely with empirical evidence at 35.71% left-tailed Following this, the Skewed-T Distribution test shows 39.29% left-tailed results, while the Generalized Hyperbolic Distribution yields the least accurate outcome at 50% left-tailed.
Next, the distribution model fitting test is performed based on the estimated parameters
This section compares the Normal Distribution with three other distributions to highlight their differences against empirical results In Figure 5, the blue lines illustrate distributions from the Generalized Hyperbolic class, while the orange line represents the Normal Distribution, the red line indicates the Skewed-T distribution, the green line shows the Normal Inverse Gaussian class, and the black line reflects the empirical results It is important to note that the negative aspects of the models are also considered.
Panel 1 of Figure 5 demonstrates that the DJIA index returns, NIG, and ST Distributions yield the most accurate results, closely aligning with empirical evidence in terms of shape, location, and scale In contrast, the GH and Normal Distribution exhibit significant discrepancies across all parameters Overall, the Generalized Hyperbolic Distribution provided acceptable results in 22 out of 29 instances, while the Normal Distribution consistently underperformed Notably, the outputs from the Normal Inverse Gaussian and Skewed-T Distribution models were deemed acceptable, as illustrated in Panels 2, 3, 4, and 5 of Figure 5.
Panel 1 DJIA’ stock returns fitting models result
Panel 2 General Electric’ stock returns fitting models result
Panel 3 Johnson & Johnson’ stock returns fitting models result
Panel 4 Apple’ stock returns fitting models result
Panel 5 Boeing’ stock returns fitting models result
The analysis of fitting models reveals that Skewed-T and NIG Distributions effectively align with empirical data, whereas Normal Distributions consistently underperform GH Distributions show occasional failures as well Despite the minor differences among Skewed-T, GH, and NIG Distributions, NIG Distributions demonstrate superior performance compared to the sub-classes Notably, Skewed-T Distributions tend to be slightly more accurate in reflecting observed returns than GH Distributions.
VaR calculation
The Basel Committee on Banking Supervision (1995) introduced a back-testing procedure to evaluate the effectiveness of Value-at-Risk (VaR) estimators This study compares empirical VaR with VaR derived from Skewed-T, GH, NIG, and Normal distributions, analyzing data from January 1, 1995, to December 30, 2016 Findings indicate that actual losses exceed the VaR, highlighting that the percentage of excess losses should align with the corresponding probability level A common method for calculating VaR involves simulating portfolio returns based on the previous 250 observed returns, known as 'historical simulation.' Alternatively, Monte Carlo methods are proposed for forecasting VaR, though both techniques are computationally demanding for large portfolios, limiting their widespread use in measuring exposure to linear market risk Figures 6 and 7 illustrate the daily VaR in relation to probability levels, with calculations performed using the 'VARES' and 'ghyp' packages in R software.
6) seem consistent with the distribution model fitting Normal Inverse Gaussian and Skewed-
T Distributions are used in model fitting, giving us 27 out of 29 precise and acceptable results
The Generalized Hyperbolic and Normal Distributions consistently fail to accurately track stock return data, as illustrated in Figure 7, where the Expected Shortfall calculation also underperforms In contrast, the NIG Distribution demonstrates superior performance, yielding acceptable results in numerous instances, while the ST Distribution often fails to align with empirical findings Overall, both the GH and Normal Distributions continue to show poor results, unable to effectively capture stock return data across most figures.
Panel 1 DJIA’ index returns VAR calculation result
Panel 2 IBM’ stock returns VAR calculation result
Panel 3 Intel’ stock returns VAR calculation result
Panel 4 Procter & Gamble’ stock returns VAR calculation result
Panel 5 Coca-Cola’ stock returns VAR calculation result
Panel 1 DJIA’ index returns Expected Shortfall calculation result
Panel 2 Apple’ stock returns Expected Shortfall calculation result
Panel 3 Microsoft’ stock returns Expected Shortfall calculation result
Panel 4 Nike’ stock returns Expected Shortfall calculation result
Panel 5 3M’ stock returns Expected Shortfall calculation result
Backtesting procedure
This section analyzes the descriptive statistics involved in Value at Risk (VaR) calculation Following this analysis, back-testing processes are conducted to assess the accuracy of each distribution These back-testing processes utilize specific historical data periods to determine the VaR.
This research examines the accuracy of Value at Risk (VaR) estimation processes by analyzing data from January 1995 to December 2010 for back-testing against contemporary data from March 1, 2011, to December 30, 2016 The findings, detailed in Table 5, include VaR statistics, discussions analyzing stock prices and VaR values derived from distribution models, and a summary of violations The primary focus is on left-tail risks, which reflect negative values at 95% and 99% confidence levels, emphasizing the study's aim of understanding risk and loss.
In Table 5, the minimum values of empirical data for each stock component, along with their occurrence dates, are presented Each distribution model's value on the same date is displayed for comparison to determine which model best supports the empirical data A positive percentage indicates that the Value at Risk (VaR) results are lower than the empirical data, suggesting no potential loss for investors, while a negative percentage signifies a higher VaR, indicating possible losses For instance, from 2011 to 2016, the DJIA recorded its lowest return of -5.5% on August 8, 2011, where the ST Distribution showed a value of -0.06%, 12.18% lower than the empirical data, indicating a protective measure for investors Conversely, GH Distribution at -4.61% and NIG Distribution at -4.81% were significantly higher than the empirical data, demonstrating that ST Distribution was the most reliable model for that date The lowest daily return for DJIA was noted at -14.16% on February 10, 2011, with ST Distribution again outperforming at -12.52%, followed by NIG Distribution at -10.78% and GH Distribution at -10% On August 8, 2011, Pfizer had a lowest return of -4.75%, where GH Distribution provided the closest estimate at -6.16%, while ST and NIG Distributions were further off at -6.86% and -6.42%, respectively, indicating that GH Distribution offers the best fit to the empirical data in this case.
Table 5 VAR Statistics – Min value and date of stock components – empirical data and distributions
The empirical data presents the values of various stocks across different dates, highlighting their respective ST, GH, and NIG percentages For instance, on August 8, 2011, the DJ30 index showed a ST value of -0.0555, with GH and NIG values of -16.88% and -13.35%, respectively Similarly, rAPPLE on January 24, 2013, recorded a ST value of -0.1236, reflecting GH and NIG percentages of -9.68% and -12.35% Other notable entries include rBOEING on January 27, 2016, with a ST of -0.0893 and GH and NIG values of -20.18% and -20.55% The dataset also reveals significant fluctuations, such as rGE on August 8, 2011, which recorded a GH value of 20.70% and an NIG of 18.12% The data underscores the volatility and performance variations in the stock market, providing insights into investment trends and stock behavior over time.
Table 6 summarizes the frequency with which the minimum values of empirical data align with the VaR distribution The findings are categorized into three groups: 'Less than 10%' indicates instances where the VaR distribution failed to predict empirical data less than 10% of the time, reflecting better predictive accuracy; '-10% to 10%' signifies cases where the failure rate exceeds 10%, with a success rate below 10%, representing the range closest to empirical data, where higher values are preferable.
The analysis indicates that the success rate of Value at Risk (VaR) distribution in predicting empirical data exceeds 10%, with the ST Distribution outperforming others in the range of ‘-10% to 10%’ at 27.59% Following closely is the NIG Distribution at 24.14%, while the GH Distribution lags behind at 17.24% In the ‘Less than -10%’ category, ST Distribution again excels with a value of 24.14%, while GH and NIG Distributions demonstrate poor performance at 48.28% and 51.72%, respectively In the ‘Higher than 10%’ category, ST Distribution remains the safest option for risk management at 48.28%, significantly better than GH at 34.48% and NIG at 24.14% Consequently, ST Distribution is identified as the most effective method for managing extremely low empirical data.
Table 6 Descriptive statistic of excessed VaR
In this part, empirical results are discussed VaR values from distribution models and empirical data are presented along with stock price charts and analysis
The Dow Jones 30 experienced significant downturns in mid-2011 and from mid-2015 to early 2016 The first downturn in August 2011 was triggered by the US debt-ceiling crisis, culminating in a deal between President Obama and congressional leaders to raise the debt ceiling by $400 billion and implement substantial government spending cuts This followed Standard & Poor's downgrade of the US debt outlook from stable to negative, the first such action since 1941 The second downturn in August 2015 was likely influenced by the Chinese stock market crisis, resulting in a historic 588-point drop in the Dow Jones 30 Global concerns over China's economic slowdown led to a two-week market turmoil, beginning with an 8.5% decline in Shanghai's stock market, and the DJIA plummeted 1,089 points within minutes of the opening bell, marking the largest single-day point loss in history, surpassing the 2010 Flash Crash.
In our analysis, we focused on empirical data that exceeded the Value at Risk (VaR) at the left-tail for both 95% and 99% confidence levels, referred to as "violations." The complete results are detailed in Table 7.
This analysis focuses on the DJIA (DJ30) index and examines the Value at Risk (VaR) empirical results for selected stocks, including Johnson & Johnson (JS), which had the most violations, Traveler Cos (TRVC), with the fewest violations, and Apple Cos (APPLE), representing median results The data is categorized into three distinct periods: the first downturn period from 2011, a stable and rising phase from 2012 to 2014, and the second downturn period spanning 2015 and 2016.
ST01 (%) ST05 (%) GH01 (%) GH05 (%) NIG01 (%) NIG05 (%)
Table 8 Number of violations of DJIA, Apple, JS, TRVC
DJIA APPLE JS TRVC Total by distribution Total 99% 95% 99% 95% 99% 95% 99% 95% 99% 95% 99% 95%
Table 8 summarizes the violations across various distributions and stocks The ST Distribution stands out for the Dow Jones Industrial Average, recording zero violations at 99% confidence across all three periods, and only seven violations at 95% confidence during two chaos periods, with none in the stable period The NIG Distribution follows closely, showing one violation at 99% confidence in the first chaos period and nine violations at 95% confidence during the chaos period, but no violations in the stable period In contrast, the GHY Distribution performed poorly, with four violations at both confidence levels during the stable period and a total of 13 violations in the chaos period Notably, the stock with the highest violations in the DJIA, Johnson & Johnson (JS), exhibited 55 violations according to the ST Distribution, including five during the chaos period and one in the stable period at 99% confidence.
GHY Distribution reported a total of 53 violations, with 14 occurring during the chaos period and 11 in the stable period, at a 99% confidence level In contrast, NIG Distribution identified only 3 violations at a 95% confidence level, all occurring during the chaos period Meanwhile, the stock with the fewest violations, Traveler Cos (TRVC), showed consistent results across all distributions, with 4 violations, all at a 95% confidence level.
During the chaos period, NIG and ST Distributions exhibit minimal violations at the 99% confidence level, with even fewer in stable and growth periods In contrast, GHY shows significantly more violations at both the 95% and 99% confidence levels, with the highest violations occurring during the stable period Overall, the data indicates that NIG Distribution is the most effective option for managing risks among the four stocks and indices.
Christoffersen (1998) summarizes the back-testing procedure in Table 9, detailing the number of violations that indicate how often actual outcomes exceeded calculated values The key variables in this process are the Number of Observations (OBS), Number of Violations (NVI), Expected Number of Violations (EVI), and the Ratio of Violations (RVI), where RVI is calculated as $RVI = \frac{NVI}{EVI}$ and OBS is set at 1565 The expected number of violations is determined by the formula $EVI = OBS \times (1 - \text{confidence level})$ For a confidence level of 99%, EVI equals $15.65$, and for 95%, it equals $78.25$.
Table 9 Expected number of violations
Figure 8 Dow Jones 30 historical price and VAR
Figure 9 VAR of Johnson & Johnson and Traveler Cos
Table 7 summarizes the accuracy of the conducted back-tests, categorizing stock violations as 'acceptable' if below the EVI and 'unacceptable' if above Notably, the GH Distribution at the 99% confidence level shows violations significantly exceeding the EVI, with JP Morgan Chase & Co reporting 20 violations, Johnson & Johnson 25, Microsoft 16, and Procter & Gamble 18 The ST Distribution averages 0.59 violations at the 99% confidence level and 7.69 at 95%, while the GH Distribution permits 6.34 and 12.17 violations at 95% and 99%, respectively In contrast, the NIG Distribution allows only 0.86 and 6.69 violations at the 99% and 95% confidence levels At the 99% confidence level, both ST and NIG Distributions predominantly predict zero violations.
Table 10 illustrates the levels of violence at various confidence levels, revealing that the NIV Distribution yielded the best results with zero rejections across all confidence levels In contrast, the Skewed-T Distribution exhibited a single rejection at the 95% confidence level.
GH Distribution show 4 rejections at the 99% confidence level
Table 10 Distribution models’ performance at confidence levels
Confidence level Skew T Percentage Generalized
Conclusion
Value at Risk (VaR) quantifies the potential financial loss a company may face in its asset and derivatives portfolio It utilizes the market values of all financial products held, providing a comprehensive assessment of risk exposure.
Based on back-testing analysis using Normal Inverse Gaussian (NIG), Skewed-T, and Generalized Hyperbolic Distribution models, we find that the NIG model is the most effective for the Dow Jones Industrial Average (DJIA) and its stock components Although all three models exhibit empirical sizes that deviate from nominal sizes and show dependent back-testing exceptions, the NIG model stands out For the 99% Value at Risk (VaR), only the Generalized Hyperbolic Distribution fails the Christoffersen tests, while the other two models perform satisfactorily, indicating their utility for DJIA At the 95% VaR level, all three models provide accurate estimations with empirical sizes aligning closely to nominal levels and independent back-testing exceptions Ultimately, the choice of the best method for measuring a firm's risk is subjective, depending on the manager's preferences and the company's circumstances, yet all three models are reliable for assessing Value at Risk.
Abramowitz, M and I A Stegun (1972) Handbook of mathematical functions Inc., New York
Adcock, C J (2007) Extensions of Stein's lemma for the skew-normal distribution Communications in Statistics— Theory and Methods, 36(9), 1661-1671
Adcock, C J (2010) Asset pricing and portfolio selection based on the multivariate extended skew-Student-t distribution Annals of Operations Research, 176(1), 221-234
Adcock, C J (2014) Mean–variance–skewness efficient surfaces, Stein’s lemma and the multivariate extended skew-Student distribution European Journal of Operational Research, 234(2), 392-401
Amu, F and M Millegard (2009) Markowitz portfolio theory Citeseer Retrieved on 12(15): 1
Ané, T (2006) An analysis of the flexibility of asymmetric power GARCH models Computational Statistics & Data Analysis, 51(2), 1293-1311
Angelidis, T., Benos, A., & Degiannakis, S (2004) The use of GARCH models in VaR estimation Statistical methodology, 1(1-2), 105-128
Angelidis, T., & Degiannakis, S (2005) Modeling risk for long and short trading positions The Journal of Risk Finance, 6(3), 226-238
Artzner, P., Delbaen, F., Eber, J M., & Heath, D (1999) Coherent measures of risk Mathematical finance, 9(3),
Aas, K., & Haff, I H (2006) The generalized hyperbolic skew student’st-distribution Journal of financial econometrics, 4(2), 275-309
B Taylor, J and A Weerapana (2010) Principles of Microeconomics
Bali, T G., & Theodossiou, P (2007) A conditional-SGT-VaR approach with alternative GARCH models Annals of Operations Research, 151(1), 241-267
Barndorff-Nielsen, O (1977) Exponentially decreasing distributions for the logarithm of particle size
Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, The Royal Society
Barndorff-Nielsen, O and P Blaesild (1981) Hyperbolic distributions and ramifications: Contributions to theory and application Statistical distributions in scientific work, Springer: 19-44
Barone-Adesi, G., et al (2008) A GARCH Option Pricing Model with Filtered Historical Simulation The Review of Financial Studies 21(3): 1223-1258
Beder, T S (1995) VAR: Seductive but dangerous Financial Analysts Journal, 51(5), 12-24
Berim, G O and E Ruckenstein (2009) Symmetry breaking of the density distribution of a quantum fluid in a nanoslit The Journal of chemical physics 131(18): 184707
Best, P (2001) Distribution and population separation of Bryde's whale Balaenoptera edeni off southern Africa Marine Ecology Progress Series 220: 277-289
Blasi, F., & Scarlatti, S (2012) From normal vs skew-normal portfolios: FSD and SSD rules Journal of
Bolance, C., Guillen, M., Pelican, E., & Vernic, R (2008) Skewed bivariate models and nonparametric estimation for the CTE risk measure Insurance: Mathematics and Economics, 43(3), 386-393
Brooks, C., & Persand, G (2003) The effect of asymmetries on stock index return Value-at-Risk estimates The Journal of Risk Finance, 4(2), 29-42
Bump, W M (1991) The Normal Curve Takes Many Forms: A Review of Skewness and Kurtosis
Butler, K and K Okada (2007) Bivariate and higher-order terms in models of international equity returns Applied Financial Economics 17(9): 725-737
Campbell, S D (2006) A review of back-testing and back-testing procedures The Journal of Risk 9(2): 1 Chen, Y., & Lu, J (2012) Value at risk estimation In Handbook of computational finance (pp 307-333) Springer, Berlin, Heidelberg
Chorafas, D N (2004) Economic capital allocation with Basel II: Cost, benefit and implementation procedures Butterworth-Heinemann
Christoffersen, P F (1998) Evaluating interval forecasts International economic review, 841-862
De Luca, G., Genton, M G., & Loperfido, N (2006) A multivariate skew-garch model In Econometric Analysis of Financial and Economic Time Series (pp 33-57) Emerald Group Publishing Limited
Dowd, K (1998) Beyond value at risk: the new science of risk management
Duffie, D and J Pan (1997) An overview of value at risk The Journal of derivatives 4(3): 7-49
Eberlein, E., Keller, U., & Prause, Karsten (1998) New insights into smile, mispricing, and value at risk: The hyperbolic model The Journal of Business, 71(3), 371-405
Eling, M (2012) Fitting insurance claims to skewed distributions: Are the skew-normal and skew-student good models? Insurance: Mathematics and Economics, 51(2), 239-248
Engle, R F (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United
Kingdom inflation Econometrica: Journal of the Econometric Society, 987-1007
Eriksson, A., Ghysels, E., & Wang, F (2009) The normal inverse gaussian distribution and the pricing of derivatives Journal of Derivatives, 16(3), 23
Eun, C S., & Dal Shim, S (1989) Macroeconomic Shocks and Exchange Rate Dynamics (No 8928) Korea Development Institute
Forsberg, L., & Bollerslev, T (2002) Bridging the gap between the distribution of realized (ECU) volatility and ARCH modelling (of the Euro): the GARCH‐NIG model Journal of Applied Econometrics, 17(5), 535-548
Lillestol, J (2000) Risk analysis and the NIG distribution Journal of Risk, 2, 41-56
Manganelli, S., & Engle, R F (2004) A Comparison of Value-at-Risk Models in Finance In G Szego (Ed.), Risk Measures for the 21st Century (pp 123-143) Chichester: John Wiley
Eun, C S and S Dal Shim (1989) Macroeconomic Shocks and Exchange Rate Dynamics Korea Development
Fama, E F and K R French (1992) The cross‐section of expected stock returns The Journal of Finance 47(2):
Giot, P., & Laurent, S (2003) Value‐at‐risk for long and short trading positions Journal of Applied Econometrics, 18(6), 641-663
Hansen, B E (1994) Autoregressive conditional density estimation International Economic Review, 705-730 Holton, G A (2002) History of Value-at-Risk Citeseer
Hu, W and A N Kercheval (2010) Portfolio optimization for student t and skewed t returns Quantitative Finance
Huang, Y C and Lin, B J (2004) Value-at-Risk analysis for Taiwan stock index futures: fat tails and conditional asymmetries in return innovations, Review of Quantitative Finance and Accounting, 22, 79–95
Hult, H., Lindskog, F., Hammarlid, O., & Rehn, C J (2012) Risk and portfolio analysis: Principles and methods
Jaschke, S R (1997) A note on stochastic volatility, GARCH models, and hyperbolic distributions Humboldt- Universitọt zu Berlin, Wirtschaftswissenschaftliche Fakultọt
Jiang, L., et al (2016) Stock return asymmetry: beyond skewness
Jorion, P (2001) Value at Risk, 2 Aufl., New York
Kalemanova, A., Schmid, B., & Werner, R (2007) The normal inverse Gaussian distribution for synthetic CDO pricing Journal of derivatives, 14(3), 80
Lambert, P., & Laurent, S (2001) Modelling financial time series using GARCH-type models with a skewed Student distribution for the innovations (No UCL-Université Catholique de Louvain)
Lee, C F., Lee, J C., & Lee, A C (2000) Statistics for business and financial economics (Vol 1, p 712) Singapore: World Scientific
Lehnert, T (2003) Explaining smiles: GARCH option pricing with conditional leptokurtosis and skewness The Journal of Derivatives, 10(3), 27-39
Lewis, K K (1999) Trying to explain home bias in equities and consumption Journal of economic literature, 37(2), 571-608
Linsmeier, T J and N D Pearson (1996) Risk measurement: An introduction to value at risk
Malz, A M (2011) Financial risk management: models, history, and institutions (Vol 538) John Wiley
Mao, J C T., et al (1966) A Decision Theory Approach to Portfolio Selection Manage Sci 12(8): B-323-B-333 Markowitz, H M (1991) Foundations of portfolio theory The Journal of Finance 46(2): 469-477
Maxwell, J and R Russo (1999) A Student’s Guide to Analysis of Variance London, UK: Routledge
McNeil, A J (1999) Extreme value theory for risk managers Departement Mathematik ETH Zentrum
Nọsstrửm, J (2003) Volatility modelling of asset prices using garch models
Paolella, M S (2007) Intermediate probability: A computational approach John Wiley & Sons
Penza, P and V K Bansal (2001) Measuring market risk with value at risk John Wiley & Sons
Pérignon, C and D R Berim (2010) The level and quality of Value-at-Risk disclosure by commercial banks
Philippe, J (2001) Value at risk: the new benchmark for managing financial risk NY: McGraw-Hill Professional Prause, K (1999) The Generalized Hyperbolic Model: Estimation Financial Derivatives, and Risk Measures,