GARCH and VaR | 1
GARCH models applied on
Swedish Stock Exchange Indices
Bachelor’s Thesis 15 hp Department of Statistics Uppsala University June 2019
Wiktor Blad Vilim Nedic
Supervisor: Lars Forsberg
GARCH and VaR | 2
GARCH models applied on Swedish Stock Exchange Indices
By
Wiktor Blad and Vilim Nedic Supervisor: Lars Forsberg
Department of Statistics Uppsala University, SWE
June 3, 2019
Abstract
In the financial industry, it has been increasingly popular to measure risk.
One of the most common quantitative measures for assessing risk is Value- at-Risk (VaR). VaR helps to measure extreme risks that an investor is exposed to. In addition to acquiring information of the expected loss, VaR was introduced in the regulatory frameworks of Basel I and II as a standardized measure of market risk. Due to necessity of measuring VaR accurately, this thesis aims to be a contribution to the research field of applying GARCH-models to financial time series in order to forecast the conditional variance and find accurate VaR-estimations. The findings in this thesis is that GARCH-models which incorporate the asymmetric effect of positive and negative returns perform better than a standard GARCH.
Further on, leptokurtic distributions have been found to outperform normal distribution. In addition to various models and distributions, various rolling windows have been used to examine how the forecasts differ given window lengths.
Keywords: Value-at-Risk, GARCH, GJR-GARCH, EGARCH, student’s t distribution, generalized error distribution, Kupiec’s test, Christoffersen’s test, forecast
Contents
1. Introduction 4
1.1 Previous research 5
1.2 Purpose & Request Question 5
1.3 Structure of this paper 6
2. Theoretical Framework 7
2.1 VaR – Value at Risk 7
2.2 Models from the GARCH-family 8
2.2.1 ARCH – Autoregressive Conditional heteroscedasticity 8 2.2.2 GARCH – Generalized Autoregressive Conditional heteroscedasticity 9 2.2.3 GJR-GARCH – Glosten, Jagannathan Runkle - GARCH 10 2.2.4 EGARCH –Exponential Generalized Autoregressive Conditional
Heteroscedasticity 10
2.3 Statistical Distributions 11
2.3.1 Normal distribution 11
2.3.2 Student´s t-distribution 11
2.3.3 Generalized Error Distribution (GED) 12
2.4 Backtest report 12
2.4.1 Kupiec’s Unconditional Coverage test 12
2.4.2 Christoffersen’s Conditional Coverage test 13
3. Method 14
3.1 Data 14
3.2 Descriptive statistics 15
3.3 Rolling Forecast Procedure & Rolling Window Method 15
4. Results 16
4.1 Value at Risk Backtest 16
4.2 Large Cap 17
4.3 Mid Cap 19
4.4 Small Cap 20
4.5 Analysis 21
5. Discussion 22
6. Conclusion 24
6.1 Further research 24
7. References 25
Appendix A 27
Table A.1 Backtest report OMXS Large Cap 27
Table A.2 Backtest report OMXS Mid Cap 28
Table A.3 Backtest report OMXS Small Cap 29
Table A.4 Parameter estimates OMXS Large Cap 30
Table A.5 Parameter estimates OMXS Mid Cap 31
Table A.6 Parameterestimates OMXS Small Cap 32
GARCH and VaR | 4
1. Introduction
In the financial industry, it has been increasingly popular to measure risk. Investors always want to minimize their risk and at the same time get the highest possible return (Markowitz 1952). In finance, volatility is defined as the conditional standard deviation of the log return of the underlying stock. One of the most common quantitative measures for assessing risk is the Value-at-Risk (VaR). VaR helps to measure extreme risks that an investor is exposed to, i.e.
the investor obtains how much the expected loss is due to adverse price movement in the market under normal circumstances (Jorion 2007). In addition to acquiring information of the expected loss, VaR was introduced in the regulatory frameworks of Basel I and II as a standardized measure of market risk, catalyzing its popularity. To summarize VaR, which is a statistical technique, VaR measure the potential loss and the probability for a potential loss under a specific time-period and significance level (Hull 2010).
One reason why VaR has been such a useful and popular measurement tool is the simple concept and that the calculation for the model is not advanced, thus can be easily used by an everyday investor. It is known that financial instruments do not have constant conditional variance, i.e. there is issues with heteroscedasticity. When a model or a time series suffers from the condition of unequal variance of the error terms, it is assumed to be heteroscedastic and tends to violate the assumption of normality (Berry and Feldman 1985).
When estimating VaR, there are some assumptions made regarding the distribution of the asset returns which are unrealistic. For instance, the asset returns are assumed to be normally distributed. By ignoring the non-normal properties of the returns series, the normal distribution fails to capture the extreme data points of a volatile time series and therefore the assumption may in practice result in generating models that underestimates VaR (Christoffersen 2003). To summarize the unrealistic assumptions, this could result in unreliable and inaccurate VaR predictions.
One of the first methods that explicitly models the change in variance over time in a time series is the Autoregressive Conditional Heteroscedasticity (ARCH) model that was introduced by (Engle 1982). In order to be able to describe the volatility in a time series, ARCH recognized the difference between unconditional and the conditional variance and allowing the latter to change over time as a function of past errors. A second major type of model that provides the issue with volatility is an extended ARCH-model. The Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model was presented by (Bollerslev 1986). The characteristics of GARCH is that the model incorporates a moving average component in addition to the already present autoregressive component. One of the weaknesses of these two volatility models is that both models assume that positive and negative shocks have the same impact on volatility. The standard GARCH has been further developed into several variations of the model. One of these further variations of the GARCH is the Exponential GARCH (EGARCH) that was introduced by (Nelson 1991). This model contains a parameter that successfully captures the asymmetric effects of positive and negative returns on conditional variance. In addition to the EGARCH,
GARCH and VaR | 5 there is another GARCH-model that is frequently used when investigating financial time series and also contains a parameter allowing to measure asymmetric effects. It is the model introduced by Glosten Jagannathan Runkle (1993) which is known as GJR-GARCH.
The three aforementioned GARCH models, are just three of several varieties that have emerged since the introduction of ARCH and GARCH (Bollerslev 2010). There are perhaps GARCH- models that are even better than one of these, but these three GARCH-models are most commonly used when investigating variance in financial time series and therefore this paper aims to model volatility in Swedish indices using these three models.
1.1 Previous research
The financial time series are characterized by high volatility and do not have constant variance over time (Cryer and Chan 2008). Previous research of financial time series have provided compelling evidence that applying GARCH models are a suitable method when investigating volatility. Nowadays, there are several extensions to the standard GARCH that tend to perform better when modeling volatility in financial time series compared to the standard GARCH. For example, Hansen and Lunde (2005) compare a large number of models and conclude that the GARCH-models outperforms other models when forecasting is of interest. Similar conclusion is done by Köksal (2009) and Hung-Chun and Jui-Cheng (2010).
Bollerslev (1987) propose that the student´s t distribution perform better than the normal distribution that was originally used by Engle (1982) when he introduced the first GARCH- model. Vlaar (2000) tested GARCH-models under different distributions on bond portfolios and conclude that the standard GARCH under normal distribution outperform other combinations of models and distributions. In practice, a commonly used distribution when modelling volatility with GARCH-models is the Generalized Error Distribution (GED) which takes fat tails into account (Nelson 1991). Hung-Chan & Jui-Cheng (2010) conclude, after testing different distribution, that the generalized error distribution does not significantly improve volatility forecasting. Another study of Angelidis et. al (2004) found no evidence of a superior general model but concluded that leptokurtic distributions outperforms the normal distribution and that the estimation window length had an influence on the performance of VaR estimation.
1.2 Purpose & research Question
Due to the findings in previous studies, the authors of this paper is predicting that there will be substantial differences in amount of VaR exceedances between different models. Furthermore, the expectation is to find that various combinations of GARCH-models together with different distributions and rolling window lengths will perform differently in VaR estimation. For this reason, this paper is containing 27 models for each time series which is a total of 81 estimations.
The purpose of this thesis is to examine the volatility in financial time series by backtesting VaR using three models in the GARCH-family. The models will be evaluated by how accurately they measure Value at Risk. This will be done through backtesting using 1000
GARCH and VaR | 6 observations. Furthermore, this thesis will investigate how well the models perform using different distributions. More accurately, the three distributions used are normal distribution, student’s t-distribution, and Generalized Error Distribution, all of which have commonly been used by previous researchers when investigating VaR. To examine this, the authors of the paper, have used the rugarch–package created by Alexios Ghalanos for the statistical programming language Rstudio. This package has the purpose to provide a set of tools for modelling GARCH-processes (Ghalanos 2017).
This paper focuses on Swedish stock indices, namely large cap, mid cap, and small cap. The rationale behind the delimitation is that these indices are expected to have a different levels of volatility through the time span studied in this paper. Further on, these indices are to contain a majority of the listed companies on the Swedish stock exchange. The time period of interest is between 2009-03-27 and 2019-03-29. The out-of-sample proportion of the time period is between 2015-04-13 and 2019-03-29 which results in a total of 1000 out-of-sample estimations.
In summary, the purpose of this paper is to examine which model from the GARCH-family combined with a specific distribution and rolling window length estimates VaR most accurately given the backtesting-report containing both Kupiec’s and Christoffersen’s tests. In conclusion, using different rolling window lengths of 6, 12, and 24 months for OMXS Large Cap, OMXS Mid Cap, and OMXS Small Cap with different distributions should provide enough evidence to conclude which models estimates VaR most accurately and do not reject the null hypotheses of Kupiec’s and Christoffersen’s tests.
The question this paper aims to answer is as following:
Which GARCH-model predicts VaR with 1% alpha best given the different window lengths and distributions mentioned above.
1.3 Structure of this paper
The thesis is structured as following, the theoretical framework is presented in the second chapter. In this chapter, the theoretical framework such as VaR-measure and the different GARCH-models will be presented in detail. Chapter three contain the methodology that has been used to answer the question this paper aims to answer. In addition, the data and necessary information regarding the data is also presented in this chapter. Further on, the results of this thesis is found in chapter four and discussion regarding the results is presented in chapter five.
Lastly, the authors’ conclusion is found in chapter six.
GARCH and VaR | 7
2. Theoretical Framework
This section includes all theoretical aspects that are of interest to understand the analysis of this paper. This section starts with an introduction of the concept of VaR and prior to describing the GARCH-models, a short introduction of ARCH is included. In addition to GARCH, the different distributions used in the models are presented and also a short description of the two tests used in the backtesting report.
2.1 VaR – Value at Risk
Value at Risk is a measurement tool to analyze the risk on the financial market. VaR can be defined as the maximum loss in returns under a given time-period under normal conditions on the market. According to Tsay (2005), there is an alternative view on VaR which is that VaR is the expected minimum loss under extreme conditions on the market. Orhan and Köksal (2011) claim that if VaR is underestimated, investors will be exposed to unwanted risk and if the VaR is overestimated, investors are expected to act more carefully on the financial market.
Explicitly, the mathematical definition of VaR can be expressed as:
Pr(𝐿 > 𝑉𝑎𝑅) = α, (2.1)
where alpha (𝛼) is the significance level and L denotes the loss on a given day. VaR is therefore a quantile in the distribution of the profit and loss that is expected to be exceeded only with a certain probability. Most common significance levels (𝛼) to use is an alpha equal to 95% or 99%. VaR is the upper limit of the left tail and Figure 2.1 illustrate VaR with an unspecified significance-level.
Figure 2.1 – Illustrating VaR is the minimum amount that will be lost with an unspecified confidence level.
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2.2 Models from the GARCH-family
The ARCH model is not applied in this thesis but the GARCH-model (explained below), is a development from the ARCH-model. Therefore, it is essential for comprehension of GARCH models to explain the fundamentals of an ARCH-model and how they differ from each other.
Prior to presenting the formulas behind ARCH/GARCH-models which have empirically been applied on the logarithmic differenced returns, following is the transformation of logarithmic differenced returns and the assumptions for the time series. Logarithmic differenced returns are calculated with formula 2.2. The log return series have better statistical behavior than the actual stock prices.
𝑟𝑡= log ( 𝑝𝑡
𝑝𝑡−1), (2.2)
where 𝑟𝑡 denotes the return between time 𝑡 and 𝑡 − 1 and 𝑝𝑡 is the price of a specific asset. The logarithmic return series can be further decomposed into the following process
𝑟𝑡= 𝜇 + 𝜀𝑡, (2.3)
where 𝜇 is the unconditional mean and assumed to be equal to zero. 𝜀𝑡 is an error term and is assumed to be given as
𝜀𝑡 = 𝜎𝑡𝑍𝑡, (2.4)
where 𝜎𝑡 is the conditional standard deviation of 𝜀𝑡 and where 𝑍𝑡 ~ 𝑖𝑖𝑑(0,1). Therefore, 𝜀𝑡 is independent identically distributed with mean zero and variance 𝜎𝑡2 which can be formulated as 𝜀𝑡 ~ 𝑖𝑖𝑑(0, 𝜎𝑡2).
2.2.1 ARCH – Autoregressive Conditional heteroscedasticity
The ARCH model is a consistent time series model that was developed by Engle (1982) and describe the change in variance of a time-series. According to Cryer and Chan (2008), the ARCH model is useful when explaining the non-constant variance of a time series. Cryer and Chan (2008) explain that the ARCH-model is like a regression-model with volatility as the regressand and the squared returns from past lags regressors. By looking at financial time series, the variance in the error term is not constant over time which is resulting in that the variance is heteroscedastic and the ARCH-model is successfully taking this into account. According to Tsay (2010), Engle’s ARCH-model was the first time-series model that is able to forecast the conditional volatility of a time series.
Consequently, ARCH-models need to be applied prudently on financial time-series. For this reason, there are a lot of weaknesses with the ARCH model. Firstly, the model assume that both positive and negative shocks have the same effect on volatility because it depends on the squared previous shocks. This is known as a symmetric effect on returns. Conversely, in
GARCH and VaR | 9 practice, it is known that positive and negative shocks on financial assets responds differently.
ARCH-models do not provide any understanding for the variance and that ARCH is only a mechanical way to describe the behavior of the variance according to Tsay (2010). The model can also overestimate the volatility because of the inability to reflect sudden changes in returns.
The ARCH (q)- model can be formulated as below (Cryer and Chan 2008):
𝜎𝑡2 = 𝜔 + 𝛼1𝑟𝑡−12 + 𝛼2𝑟𝑡−22 + ⋯ + 𝛼𝑞𝑟𝑡−𝑞2 , (2.5)
where q denotes the ARCH-order. This results into that an ARCH (1)-model can be written as:
𝜎𝑡2 = 𝜔 + 𝛼1𝑟𝑡−12 , (2.6)
where 𝜔 and 𝛼 are unknown parameters and 𝜀𝑡 is the error term, and ARCH assume that the variance in the error term is dependent on the previous error term variances. The returns (𝑟𝑡) in an ARCH (1)-model is produced as:
𝑟𝑡= 𝜎𝑡𝜀𝑡. (2.7)
2.2.2 GARCH – Generalized Autoregressive Conditional heteroscedasticity
In addition to the ARCH-model, Bollerslev (1986) proposed a new model, named GARCH.
The advantage of the GARCH compared to ARCH-models is that GARCH-models estimate less parameters and have empirically been better in forecasting the volatility of financial time- series. The GARCH is recognized to be able to predict the conditional variance when future periods of high and low volatility cannot be distinguished from each other (Tsay 2010).
However, the model is assuming that negative and positive innovations have the same effect on volatility. Therefore, the GARCH is recognizing both positive and negative changes in the same way. This is not correct in practice because good news and bad news is affecting people’s behavior differently according to Nelson (1991).
GARCH (p,q) is specified as following according to Tsay (2005):
𝜎𝑡2 = 𝛼0+ ∑ 𝛼𝑖𝑟𝑡−𝑖2
𝑞
𝑖=1
+ ∑ 𝛽𝑗𝜎𝑡−𝑗2
𝑝
𝑗=1
. (2.8)
where 𝛼0 > 0, 𝛼𝑖 ≥ 0, 𝑖 = 1,2, … , 𝑞 and 𝛽𝑗 ≥ 0, 𝑗 = 1, 2, … , 𝑝. The GARCH-process will be covariance stationary if 𝛼𝑖+ 𝛽𝑗 < 1. The closer 𝛼𝑖 + 𝛽𝑗 is to one, the changes in variance will have a more constant effect on the model effects. 𝛼0 is a constant in the model and the parameter 𝛼𝑖 explain how fast the model reacts on changes on the market. The parameter 𝛽𝑗 explain to what degree the conditional heteroscedasticity is persistent over time and thus, larger value of
GARCH and VaR | 10 𝛽𝑗 is showing that the effects of economic news on the market have a tendency to linger. The expression of a GARCH (1,1) model with a 1-day forecast is stated below where 𝑡 − 1 is the forecast origin (Tsay 2010):
𝜎𝑡2 = 𝛼0+ 𝛼1𝑟𝑡−12 + 𝛽1𝜎𝑡−12 . (2.9)
2.2.3 GJR-GARCH – Glosten, Jagannathan Runkle - GARCH
Another commonly used variant from the GARCH-family that was specifically developed to account for asymmetric effects of volatility is the GJR-GARCH, presented by Glosten, Jagannathan and Runkle (1993). This model differs from the standard GARCH by the inclusion of an indicator, which appends a leverage factor to the GARCH-model. According to Glosten, Jagannathan and Runkle (1993), the sign of the innovation would be independent to the response variable in a standard GARCH. Instead, it would only be a function of the size of the shock. A GJR-GARCH(p,q) model is therefore specified as following:
𝜎𝑡2 = 𝜔 + ∑(𝛼𝑖 + 𝛾𝑖𝐼𝑡−1)𝑟𝑡−𝑖2
𝑞
𝑖=1
+ ∑ 𝛽𝑗𝜎𝑡−𝑗2
𝑝
𝑖=1
. (2.10)
The 𝐼𝑡−1 is an indicator that takes value one only if the previous shock are negative (𝛾𝑡−1 < 0) and otherwise zero. In GJR-GARCH, the parameter 𝛾𝑖 accounts for the leverage effect, resulting in that a negative return is expected to have a larger impact on the conditional volatility in comparison to a positive return. When the shock is negative, the GJR-GARCH takes the leverage effect into account. The 𝛽𝑗 is the coefficient for the last period’s forecasted volatility.
The GJR-GARCH(1,1) model can be stated as:
𝜎𝑡2 = 𝜔 + 𝛼1𝑟𝑡−12 + 𝛾1𝐼𝑡−1𝑟𝑡−12 + 𝛽1𝜎𝑡−12 . (2.11)
2.2.4 EGARCH –Exponential Generalized Autoregressive Conditional Heteroscedasticity
A modified version of GARCH was proposed by Nelson (1991) and is named EGARCH. The model captures the effect on variance caused by positive and negative news on the market.
EGARCH is similar to the GJR-GARCH in terms that it incorporates both the asymmetric effect on volatility and a leverage factor. The asymmetry in the volatility can be explained that the negative shocks have a higher impact on the volatility than positive shocks (Nelson 1991).
EGARCH is accounting for the asymmetric volatility problem due to the mathematical characteristic of taking the sign of the shock into consideration. Similarly to the GJR-GARCH, this is the main difference from the standard GARCH and the rationale to use GJR-GARCH and EGARCH on financial time series. In contrast to the GJR-GARCH, the EGARCH incorporates the leverage effect differently due to the fact that an EGARCH standardizes last period’s shock.
GARCH and VaR | 11 The EGARCH (p,q) model uses the natural logarithm of the dependent variable i.e. the logarithm of the conditional variance (𝜎2). The EGARCH (p,q) can be stated as:
ln(𝜎𝑡2) = 𝜔 + ∑ 𝛼𝑖
𝑞
𝑖=1 [
|𝑟𝑡−i|
√𝜎𝑡−i2
− √2 𝜋
]
+ ∑ 𝛾𝑘
𝑟
𝑘=1
𝑟𝑡−i
√𝜎𝑡−i2
It−k+ ∑ 𝛽𝑗
𝑝
𝑗=1
ln(𝜎𝑡−j2 ). (2.12)
The It−k is an indicator that, depending on whether the returns are positive or negative, takes different values, i.e. take value one if the return is less than zero. The parameter gamma, 𝛾, is restricted to less than zero and represent the increase in volatility of the last negative shock.
2.3 Statistical Distributions
Financial time series are usually not normally distributed. To estimate parameters with the right distribution it is important to achieve satisfying estimations and reasonable prediction of the volatility (Cryer and Chan 2008). In this thesis, the models are estimated using the normal distribution, student´s t distribution, and Generalized Error Distribution.
2.3.1 Normal distribution
Under the assumption for a normal distribution, it is assumed that the error term have a zero mean and variance one, i.e. 𝜀~𝑁(0,1). The normal distribution is a continuous density function where mean and variance are estimated by maximum likelihood optimization. The distribution is stated mathematically as following:
𝑓(𝑟𝑡) = 1
𝜎√2𝜋𝑒(−(𝜀𝑡
−𝜇)2 2𝜎2 )
, (2.13)
where 𝜇 is the mean and 𝜎 is the standard deviation.
2.3.2 Student´s t-distribution
Bollerslev (1987) suggested that financial time series does not match the normal distribution and therefore student’s t-distribution can be used when modelling volatility. The purpose with student’s t distribution is to allow fatter tails that are observed in stock returns due to heteroscedasticity. The density function is stated below where 𝜐 is the degrees of freedom:
𝑓(𝑟𝑡) = 𝛤(𝜐 + 1) 2 𝛤 (𝜐
2)√𝜋(𝜈 − 2)𝜎𝑡2(1 + 𝜀𝑡2 (𝜐 − 2)𝜎𝑡2)
−(𝜐+1) 2
. (2.14)
GARCH and VaR | 12
2.3.3 Generalized Error Distribution (GED)
The Generalized Error Distribution (GED) are a symmetric family of distributions usually used when errors are not normally distributed. It is similar to the normal distribution when 𝛽 = 2.
One further useful property of GED is that it can transform a normal density function into a leptokurtic distribution, which mean that it can perform well in modeling the fatter tails. The GED density function can be stated as:
𝑓(𝑟𝑡) 𝛽 2𝜎𝑡𝛤 (1
𝛽)
𝑒𝑥𝑝 {− (|𝜀𝑡− 𝜇|
𝜎𝑡 )
𝛽
}. (2.15)
2.4 Backtest report
To evaluate the estimation-process of VaR, one approach is to use backtesting. This is done by comparing the estimated everyday risk with the actual outcome. Backtesting is a common method in financial analysis and risk modeling (Jorion 2007). The important properties for a VaR model is that the total exceedances match the expected exceedances, given a specific confidence level. In the backtesting procedure, the real observed returns are compared with the predicted VaR estimates. All the counted exceedances in the data-sample are compared with the theoretical alpha. VaR estimates in this thesis will be evaluated by two tests, firstly the unconditional test, Kupiec (1995) and secondly a conditional test Christoffersen (1998). The indicator variable that measures exceedances is constructed as:
ηt= {1, if rt<-VaR 0, if rt>-VaR,
where one indicates a violation and 0 means that the observed return is less than the VaR.
2.4.1 Kupiec’s Unconditional Coverage test
One of the most widespread methods of backtesting VaR is the Kupiec´s unconditional coverage test. This test use a log-likelihood ratio test which is asymptotically 𝜒2-distributed with one degree of freedom.
𝐿𝑅𝑈𝐶 = 2 ln [(1 −𝐹 𝑇)
𝑇−𝐹
(𝐹
𝑇) 𝐹] − 2 ln[(1 − 𝑝)𝑇−𝐹𝑝𝐹]. (2.16) The out-of-sample estimates with T and F is the observed number violations. F divided by T follows a binomial distribution and 𝐿𝑅𝑈𝐶, as mention above, follows the chi-square distribution.
The implication of rejecting the null hypothesis is that the number of empirical exceedances are significantly different from the nominal amount of exceedances. In this paper the null hypothesis is rejected if the test statistic 𝐿𝑅𝑈𝐶 is greater than 3.841 on a 1% significance level.
Furthermore, the Kupiec’s test does not provide any information whether VaR is under- or overestimated.
GARCH and VaR | 13
2.4.2 Christoffersen’s Conditional Coverage test
Christoffersen’s conditional coverage test is evaluating whether the exceedances occur in cluster or independently. A weakness with Kupiec´s test is its ability to detect potential clustering of exceedances. The formula for detecting two subsequent violations can be stated as:
𝑝𝑖𝑗 = 𝑃(ηt = 𝑖|ηt-1 = 𝑗), (2.17)
where η defined the independence of violations that do not occurs two days in a row.
𝐿𝑅𝑐𝑐 = −2 ln[1 − 𝑝]𝑇−𝐹𝑝𝐹] + 2ln [1 − 𝜋𝑖𝑗)𝑖𝑗𝜋𝑖𝑗𝜂𝑖𝑗(1 − 𝜋𝑖𝑗)𝜋𝑖𝑗𝜋𝑖𝑗𝜂𝑖𝑗], (2.18) under this distribution, the 𝐿𝑅𝑐𝑐 follows a chi-square distribution with two degrees of freedom, thus the critical value is 5.991 in the test. The null-hypothesis state that the violations are independently distributed and if the null is rejected, the test is inferring that the violations are clustered and consequently not independent.
GARCH and VaR | 14
3. Method 3.1 Data
The datasets in this thesis consists of the daily closing prices from the Stockholm stock exchange indices. The data is gathered from Thomson Reuters Eikon with a sample period from May 27th 2009 to March 29th 2019 with a total of 2515 observations. In the estimation process, the out-of-sample size is 1000 observations, ranging from April 1st 2015 to March 29th 2019.
Figure 3.1: Closing price and log-diff returns of the three indices, large cap, mid cap, and small cap.
In the end of year 2008 the Swedish stock market crashed, which had a major influence on the volatility on the time series from the starting date until the end. Noticeable is that in the end of year 2011 and beginning of 2016, there are visual structural breaks in the indices shown in Figure 3.1. In the log returns series, the same time periods are characterized by high volatility.
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3.2 Descriptive statistics
Large cap Mid cap Small Cap
Mean 0.0406 0.0616 0.0601
Median 0.0620 0.1292 0.1304
Minimum -8.3659 -6.0698 -5.9266
Maximum 6.2037 5.4585 5.3149
Std.Dev 1.1623 0.9457 0.8101
Skewness -0.2880 -0.6212 -0.8274
Kurtosis 6.5509 7.0345 8.7212
Table 3.1: Descriptive statistics for the indices.
To assume a sample to be normally distributed the skewness is expected to be zero and kurtosis three. In Table 3.1 the descriptive statistics are presented for the different time series used in this thesis. For these time series, the skewness is close only for large cap. Furthermore, the kurtosis is well above three indicating that the data is too heavy-tailed in order to fit the normality assumption and therefore, a leptokurtic distribution would fit the data better.
3.3 Rolling Forecast Procedure & Rolling Window Method
One way to test the predictive power of a model is to perform a forecast estimation with rolling windows, where the estimation window moves forward one step at a time. The first 1515 observations is the in-sample estimation. The remaining 1000 observations are used as an out- of-sample testing for daily VaR estimation with three different window length of 6, 12, and 24 months. To achieve flexibility in the in-sample parameters the GARCH-parameters are refitted in each step of the rolling window estimation. As a rolling window method is used, the window length is fixed to a specific time period, either 126 days for the 6 months period, 252 days for the 12 months period, and 504 days for the 24 months period. This is due to the fact that one trading year is assumed to be 252 days.
GARCH and VaR | 16
4. Results
Before the GARCH models were estimated and evaluated in the backtest, the parameters for each GARCH-model and distribution. A complete overview of the estimates is found in Appendix A, Table A.4-A.6 for the different indices. If the estimate of 𝛼1+ 𝛽1 < 1, the model can assume conditional stationarity. The estimated parameter 𝛼1 of the conditional squared residuals explain yesterdays squared residuals and 𝛽1 is the effect of the conditional variance (𝜎𝑡−12 ) of the previous day and as 𝛽1 approaches one, it is interpreted as such that the conditional variance of yesterday have a greater effect on today’s conditional variance. The parameter
“shape” which is found in Appendix A, Table A.4 through A.6 determines the kurtosis of the probability density function and this parameter is of interest for models using either a student’s t-distribution or Generalized Error Distribution.
4.1 Value at Risk Backtest
The outputs from all of the backtesting reports for different models are presented in the appendix A, Table 1-3. To analyze how the three applied GARCH-models predict the VaR one- day ahead, each model is estimated with three different rolling window lengths of 6 months, 12 months, and 24 months.
The tests in this thesis are evaluated on the 1% significance level, the null-hypothesis is rejected if the 𝐿𝑅𝑈𝐶 > 3.84 under the kupiec´s test and if 𝐿𝑅𝐶𝐶 > 5.99 under the Christoffersen´s test.
If the p-value is below the five percent level, the null-hypothesis is rejected. To find the most accurate models and distributions for the different equity indices, the null-hypotheses should not be rejected and the exceedances should be around the expected exceedances with no clustered exceedances. Table 4.1 present the models that yielded the lowest number of VaR exceedance for the index OMXS Large Cap. Correspondingly to Table 4.1 in this section, Appendix A, Table A.2 and Table A.3 contain a complete overview of each models performance in the backtest for OMXS Mid Cap and OMXS Small Cap respectively.
GARCH and VaR | 17
4.2 Large Cap
VaR 1 % OMXS LARGE CAP
GARCH(1,1) GJR-GARCH(1,1) EGARCH(1)
std norm GED std norm GED std norm GED Empirical alpha
6m 2.3% 3.9% 2.4% 2.3% 2.3% 2.7% 3.0% 3.9% 3.5%
12m 2.0% 2.3% 2.0% 1.7% 2.0% 2.0% 2.3% 2.3% 2.3%
24m 1.9% 2.4% 1.8% 1.8% 1.9% 1.7% 1.8% 1.8% 1.7%
Unconditional coverage (Kupiec) Test statistic (LR.uc)
6m 12.49* 14.22* 14.20* 12.86* 12.46* 19.93* 26.32* 49.01* 38.33*
(0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) 12m 7.83* 12.46* 7.83* 4.091* 7.83* 7.83* 12.46* 12.46* 12.46*
(0.01) (0.00) (0.01) (0.04) (0.01) (0.01) (0.00) (0.00) (0.00) 24m 6.47* 14.22* 5.23* 5.23* 6.47* 4.09* 5.23* 5.23* 4.09*
(0.01) (0.00) (0.02) (0.02) (0.01) (0.04) (0.02) (0.02) (0.04) Conditional Coverage (Christoffersen)
Test statistic (LR.cc)
6m 15.06* 16.53* 19.8* 12.84* 12.84* 25.28* 29.66* 65.03* 53.30*
(0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) 12m 8.64* 12.84* 8.50* 4.68 8.64* 8.64* 15.06* 15.06* 15.06*
(0.01) (0.00) (0.01) (0.10) (0.01) (0.01) (0.00) (0.00) (0.00) 24m 7.28* 16.53* 6.18* 5.87 7.21* 4.68 6.18* 6.18* 5.21 (0.03) (0.00) (0.04) (0.05) (0.03) (0.10) (0.04) (0.04) (0.07)
Notes: Theoretical alpha is 1.0% for each model. Values in parenthesis are p-values. Star indicates rejection of the null- hypothesis std = student’s t distribution, norm = normal distribution, GED = Generalized Error Distribution
Table 4.1 Backtest report OMXS Large Cap.
There is evidence that the models tend to underestimate the risk for 1% VaR on Large cap since the empirical alpha for the three best models are 1.7 %. In addition, there is no model found in this thesis that has overestimated VaR. For this index, the GJR-GARCH(1,1) with student’s t- distribution and a rolling window length of 12 months is one of the three models that have been found to underestimate VaR the least. In addition to this, two other models with generalized error distribution has the same amount of empirical exceedances. Those are GJR-GARCH(1,1) with 24 months rolling window length and EGARCH(1,1) with similarly to the previous one, a rolling window length of 24 months.
The following three figures are visual representation of the plotted return series and estimated VaR for each model.
GARCH and VaR | 18 Figure 4.1 GJR-GARCH(1,1)~std and window length of 12 months.
Figure 4.2 GJR-GARCH(1,1)~GED with window length 24 months
Figure 4.3 EGARCH(1,1)~GED with window length 24 months
The same backtest report for large cap have been applied on mid cap and small cap and can be found in Appendix A, Table A.1-A.3. A short overview of the best model is presented in the text for mid cap and small cap.
GARCH and VaR | 19
4.3 Mid Cap
Model Distr. Window length
Empirical alpha
Kupiec’s Christoffersen’s LRUC
statistic
P- value
LRCC
statistic
P- value
E std 24 m 2.1% 9.284* 0.002 10.186* 0.006
Notes: * = p-value < 0.05. std = student’s t-distribution. Theoretical alpha is equal to 1%.
Table 4.2. The model with least exceedances for OMXS Mid cap
For mid cap, none of the models failed to reject Christoffersen’s test. The model with the least amount of exceedances is an EGARCH(1,1) with a student’s t distribution and a window length of 24 months. Recall that the theoretical alpha is1.0% which is inferring that all the models, similarly to the models used in Large cap, tend to underestimate VaR. Following is a visual representation of how the model performs.
Figure 4.4 EGARCH(1,1)~std with window length 24 months
GARCH and VaR | 20
4.4 Small Cap
Model Distr. Window length
Empirical alpha
Kupiec’s Christoffersen’s LRUC
statistic
P- value
LRCC
statistic
P- value
E GED 24 m 1.6% 3.007 0.079 3.597 0.166
Notes: * = p-value < 0.05. std = student’s t-distribution. Theoretical alpha is equal to 1%.
Table 4.3. Most accurate model for OMXS Small Cap
On the contrary to previous findings, there is one model that failed to reject both of the tests applied in the backtesting report which is an EGARCH(1,1) with generalized error distribution and a window length of 24 months. Similarly to the previously mentioned models, the best model for small cap is underestimating VaR but with fewer exceedances in comparison to previous models. A graphical representation of the models performance is found in Figure 4.5.
Figure 4.5 EGARCH(1,1)~GED with window length 24 months
GARCH and VaR | 21
4.5 Analysis
Following is an overview of the most suitable models for each index given the models in this thesis.
Index Model
GJR-GARCH(1,1)~std [12m]
OMXS Large Cap GJR-GARCH(1,1)~GED [24m]
EGARCH(1,1)~GED [24m]
OMXS Mid Cap EGARCH(1,1)~std [24m]
OMXS Small Cap EGARCH(1,1)~GED [24m]
Notes: std = Student’s t distribution, GED = Generalized Error Distribution. Window length in brackets.
Table 4.4 Overview of the models
Similar to previous research, the results in this thesis further confirms the findings from previous studies. In the attempt of modelling VaR on Large cap, this thesis found that three models performed equally well. Those were GJR-GARCH(1,1) with student’s t distribution and a rolling window length of 12 months. Worth noticing is that this is the only model that employs a rolling window length of shorter than 24 months and did not underestimate VaR more times than a model employing a rolling window length of 24 months. In addition to the GJR-GARCH above, changing the distribution to generalized error distribution, compelling evidence was found that the GJR-GARCH(1,1) and the EGARCH(1,1), both using a 24 months rolling window performed equally well and failed to reject Christoffersen’s conditional coverage test.
Neither one of the three models failed to reject Kupiec’s unconditional coverage test.
Different from the large cap, the findings in volatility modeling of mid cap showed that no model failed to reject neither one of the tests employed in the backtest report. The model with the least amount of exceedances were an EGARCH(1,1) with a student’s t distribution and with a 24 months rolling window. As stated before, this model did not perform well in the framework established in this thesis as it rejected both of the null hypotheses in the coverage tests.
Furthermore, when modelling volatility on small cap, there was one model that performed particularly well as it did not reject neither one of the tests in the backtesting procedure. This was an EGARCH(1,1) employing a Generalized Error Distribution and using a 24 months window length. In addition, this was the only estimation out of the 81 estimations in this thesis that failed to reject either of the coverage tests.
GARCH and VaR | 22
5. Discussion
Similarly to previous studies in financial volatility modelling, the authors of this paper has found evidence that standard GARCH with a normal distribution does not perform particularly well in the environment of modelling VaR. Generally speaking, previous researchers have found that leptokurtic distributions and in particular, student’s t distribution or Generalized Error Distribution are more appropriate volatility models as they produce more accurate forecasts. The findings in this paper corroborate the theory that leptokurtic distributions are to prefer.
A reason for the fact that normal distribution was regularly outperformed by other distributions is that leptokurtic distribution have fatter tails. This translates into this thesis that it incorporates extreme values which are characterizing a financial time series as stated in previous chapters.
The reasoning is evident when comparing the probability distributions between normal and student’s t-distribution or Generalized Error Distribution as it is constructed to include extreme values.
A more general conclusion regarding the different rolling window lengths of this thesis is that models using a 24 months rolling window performed either equally well or better than models using a shorter rolling window. All things considered, a longer rolling window length has shown a tendency of fewer exceedances. In addition to the findings regarding the rolling window length, the models showed that the two leptokurtic distributions were better than the normal distribution. This finding was in essence expected due to previous findings. Therefore, this thesis is contributing to the evidence showing that both student’s t distribution and Generalized Error Distribution is performing better in volatility modelling than normal distribution.
Furthermore, the authors of this paper expected that a standard GARCH will perform worse than other GARCH-models. This has also been confirmed in this thesis. Regarding the less volatile time series which is large cap, GJR-GARCH using either a student’s t distribution with 12 months rolling window length or a generalized error distribution with a 24 months rolling window length performed satisfyingly. In this time series, EGARCH with generalized error distribution and a long rolling window length, as stated before, performed at least equally well.
Regarding the more volatile time series which are mid cap and small cap. GJR-GARCH did not perform in any case equally well as EGARCH.
The mathematical characteristics of an EGARCH is as such that this model is expected to perform especially well in volatile time series. The EGARCH, in similarity to the GJR-GARCH contain an indicator that capture the leverage effect. However, as the leverage effect for the returns are divided by the conditional standard deviation in the EGARCH, it standardizes the last period’s shock and therefore they do not measure the asymmetric volatility the same.
Therefore, the standardized shocks of volatile time series are expected to be modelled more accurately than non-standardized of the same volatile time series. This thesis has shown compelling evidence of how this mathematical characteristic in EGARCH differ from the GJR- GARCH in modelling the conditional variance of a volatile financial time series.
GARCH and VaR | 23 In conclusion, models that incorporate the asymmetrical effects on conditional variance are performing substantially better than standard GARCH which is assuming that both positive and negative returns has equal effect on returns. Furthermore, as the discussion regarding how each model captures asymmetric shocks, the EGARCH was expected and showed that it performed better than the GJR-GARCH on more volatile time series. For the large cap, which is less volatile, both of the models have shown that they are performing well in modelling the conditional variance, given that the distribution used is leptokurtic and incorporates extreme values.
GARCH and VaR | 24
6. Conclusion
The purpose of this thesis has been to evaluate one-day-ahead VaR on a 1% level estimated by various GARCH models under three different distributions and different rolling window length.
The out-of-sample estimates are based on OMXS Large cap, OMXS Mid cap, and OMXS Small cap from March 27th, 2015 to March 29th, 2019 which corresponds to 1000 observations.
Using this data and answering the research question of this paper, leptokurtic distributions perform better in volatility modelling which was as expected. In addition, leptokurtic distributions together with GARCH-models that incorporate asymmetric effect produce more accurate VaR forecasts. Furthermore, on the conclusion regarding which model from the GARCH-family to use, the EGARCH has performed particularly well when modeling VaR of more volatile financial time series. When modelling volatility on the large cap, both GJR- GARCH and EGARCH has performed well. On the contrary, not one model of the 27 models estimated on mid cap was found to perform well. In conclusion, models taking asymmetric effects into account perform better than the standard GARCH.
Lastly, a rolling window length of 24 months have proven to estimate VaR well. Out of the five models that showed the least amount of exceedances, four out of those was using a rolling window length of 24 months and only one model was using a rolling window length of 12 months.
To summarize the findings, regarding the different distributions and models, the findings was in accordance with previous research. However, as various rolling window lengths was employed by the models, the authors intuition led them to believe that a shorter window length was supposed to outperform longer rolling window lengths due to substantial structural breaks in financial time series in the last four years. The findings was in direct contradiction to what the authors believed.
6.1 Further research
Modelling volatility of returns has been of significant interest lately, however, understanding the co-movements of financial returns is of great practical importance. As an example, asset pricing depends on the covariance of the assets in a portfolio. It is therefore important to extend and apply multivariate GARCH models which is why models such as MGARCH could maybe perform better in modelling volatility.
In addition to multivariate GARCH-models, a distribution with skewed properties would maybe fit skewed financial time series better. Two of those would be skewed student’s t distribution or skewed Generalized Error Distribution.
GARCH and VaR | 25
7. References
Angelidis, T., Benos, A. and Degiannakis, S. (2004) The use of GARCH models in VaR estimation’, Statistical Methodology, Vol. 1, No. 1:105-128.
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Bollerslev, T. (1986). Generalized autoregressive conditional heteroscedasticity. Journal of Econometrics, 31,
Brooks, C. (2014). Introductory Econometrics for Finance, third edition. New York: Cambridge University Press.
Christoffersen P. F. (2003), Elements of financial risk management, Academic Press.
Chu-Hsiung, Lin, and Shen Shan-Shan. (2006). Can the student-t distribution provide accurate value at risk? The Journal of Risk Finance, 7(3): 292–300.
Cryer, J.D. & Chan K-S. (2008). Time Series Analysis With Applications in R, 2nd Edition, Springer, New York.
Engle RF. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of U.K. inflation. Econometrica 45: 987–1007.
Glosten, L., Ravi, J., & Runkle, D. (1992). On the relation between the expected value and the volatility of the nominal excess return on stocks. Journal of Finance, 48, 1779-1801.
Hansen, P.R. & Lunde, A. 2005, "A forecast comparison of volatility models: does anything beat a GARCH (1,1)?", Journal of Applied Econometrics, vol. 20, no. 7, pp. 873-889.
Jorion P. (2007), Value at Risk: the new benchmark for managing financial risk, Third edition, The McGraw-Hill Company.
Kupiec, Paul H. (1995). Techniques for verifying the accuracy of risk measurement models, Journal of Derivatives, 3 (2), 73–84
Köksal, B. 2009, "A Comparison of Conditional Volatility Estimators for the ISE National 100 Index Returns", Journal of Economic and social research, vol. 11, no. 2, pp. 1
Liu, H. & Hung, J. 2010, "Forecasting S&P-100 stock index volatility: The role of volatility asymmetry and distributional assumption in GARCH models", Expert Systems with Applications, vol. 37, no. 7, pp. 4928-4934.
GARCH and VaR | 26 Nelson, D. B. (1991), Conditional Heteroscedasticity in Asset Returns: A New Approach, Econometrica 59:2
Orhan, M., Köksal, B., (2011). A comparison of GARCH models for VaR estimation. eswa 2012, 2582–3592.
Tsay Ruey S. (2005) Analysis of Financial Time Series, Financial Econometrics. University of Chicago
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GARCH and VaR | 27
Appendix A
Table A.1 Backtest report OMXS Large Cap
VaR 1 % OMXS LARGE CAP
GARCH(1,1) GJR-GARCH(1,1) EGARCH(1)
std norm GED std norm GED std norm GED Empirical alpha
6m 2.3% 3.9% 2.4% 2.3% 2.3% 2.7% 3.0% 3.9% 3.5%
12m 2.0% 2.3% 2.0% 1.7% 2.0% 2.0% 2.3% 2.3% 2.3%
24m 1.9% 2.4% 1.8% 1.8% 1.9% 1.7% 1.8% 1.8% 1.7%
Unconditional coverage (Kupiec) Test statistic (LR.uc)
6m 12.49* 14.22* 14.20* 12.86* 12.46* 19.93* 26.32* 49.01* 38.33*
(0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) 12m 7.83* 12.46* 7.83* 4.091* 7.83* 7.83* 12.46* 12.46* 12.46*
(0.01) (0.00) (0.01) (0.04) (0.01) (0.01) (0.00) (0.00) (0.00) 24m 6.47* 14.22* 5.23* 5.23* 6.47* 4.09* 5.23* 5.23* 4.09*
(0.01) (0.00) (0.02) (0.02) (0.01) (0.04) (0.02) (0.02) (0.04) Conditional Coverage (Christoffersen)
Test statistic (LR.cc)
6m 15.06* 16.53* 19.8* 12.84* 12.84* 25.28* 29.66* 65.03* 53.30*
(0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) 12m 8.64* 12.84* 8.50* 4.68 8.64* 8.64* 15.06* 15.06* 15.06*
(0.01) (0.00) (0.01) (0.10) (0.01) (0.01) (0.00) (0.00) (0.00) 24m 7.28* 16.53* 6.18* 5.87 7.21* 4.68 6.18* 6.18* 5.21 (0.03) (0.00) (0.04) (0.05) (0.03) (0.10) (0.04) (0.04) (0.07)
Notes: Theoretical alpha is 1.0% for each model. Values in parenthesis are p-values. Star indicates rejection of the null- hypothesis std = student’s t distribution, norm = normal distribution, GED = Generalized Error Distribution
GARCH and VaR | 28
Table A.2 Backtest report OMXS Mid Cap
VaR 1 % OMXS MID CAP
GARCH(1,1) GJR-GARCH(1,1) EGARCH(1)
std norm GED std norm GED std norm GED Empirical alpha
6m 2.3% 3.9% 2.4% 2.3% 2.3% 2.7% 3.0% 3.9% 3.5%
12m 2.0% 2.3% 2.0% 1.7% 2.0% 2.0% 2.3% 2.3% 2.3%
24m 1.9% 2.4% 1.8% 1.8% 1.9% 1.7% 1.8% 1.8% 1.7%
Unconditional coverage (Kupiec) Test statistic (LR.uc)
6m 12.49* 14.22* 14.20* 12.86* 12.46* 19.93* 26.32* 49.01* 38.33*
(0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) 12m 7.83* 12.46* 7.83* 4.091* 7.83* 7.83* 12.46* 12.46* 12.46*
(0.01) (0.00) (0.01) (0.04) (0.01) (0.01) (0.00) (0.00) (0.00) 24m 6.47* 14.22* 5.23* 5.23* 6.47* 4.09* 5.23* 5.23* 4.09*
(0.01) (0.00) (0.02) (0.02) (0.01) (0.04) (0.02) (0.02) (0.04) Conditional Coverage (Christoffersen)
Test statistic (LR.cc)
6m 15.06* 16.53* 19.8* 12.84* 12.84* 25.28* 29.66* 65.03* 53.30*
(0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) 12m 8.64* 12.84* 8.50* 4.68 8.64* 8.64* 15.06* 15.06* 15.06*
(0.01) (0.00) (0.01) (0.10) (0.01) (0.01) (0.00) (0.00) (0.00) 24m 7.28* 16.53* 6.18* 5.87 7.21* 4.68 6.18* 6.18* 5.21 (0.03) (0.00) (0.04) (0.05) (0.03) (0.10) (0.04) (0.04) (0.07)
Notes: Theoretical alpha is 1.0% for each model. Values in parenthesis are p-values. Star indicates rejection of the null- hypothesis std = student’s t distribution, norm = normal distribution, GED = Generalized Error Distribution
GARCH and VaR | 29
Table A.3 Backtest report OMXS Small Cap
VaR 1 % OMXS SMALL CAP
GARCH(1,1) GJR-GARCH(1,1) EGARCH(1)
std norm GED std norm GED std norm GED Empirical alpha
6m 2.3% 3.9% 2.4% 2.3% 2.3% 2.7% 3.0% 3.9% 3.5%
12m 2.0% 2.3% 2.0% 1.7% 2.0% 2.0% 2.3% 2.3% 2.3%
24m 1.9% 2.4% 1.8% 1.8% 1.9% 1.7% 1.8% 1.8% 1.7%
Unconditional coverage (Kupiec) Test statistic (LR.uc)
6m 12.49* 14.22* 14.20* 12.86* 12.46* 19.93* 26.32* 49.01* 38.33*
(0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) 12m 7.83* 12.46* 7.83* 4.091* 7.83* 7.83* 12.46* 12.46* 12.46*
(0.01) (0.00) (0.01) (0.04) (0.01) (0.01) (0.00) (0.00) (0.00) 24m 6.47* 14.22* 5.23* 5.23* 6.47* 4.09* 5.23* 5.23* 4.09*
(0.01) (0.00) (0.02) (0.02) (0.01) (0.04) (0.02) (0.02) (0.04) Conditional Coverage (Christoffersen)
Test statistic (LR.cc)
6m 15.06* 16.53* 19.8* 12.84* 12.84* 25.28* 29.66* 65.03* 53.30*
(0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) 12m 8.64* 12.84* 8.50* 4.68 8.64* 8.64* 15.06* 15.06* 15.06*
(0.01) (0.00) (0.01) (0.10) (0.01) (0.01) (0.00) (0.00) (0.00) 24m 7.28* 16.53* 6.18* 5.87 7.21* 4.68 6.18* 6.18* 5.21 (0.03) (0.00) (0.04) (0.05) (0.03) (0.10) (0.04) (0.04) (0.07)
Notes: Theoretical alpha is 1.0% for each model. Values in parenthesis are p-values. Star indicates rejection of the null- hypothesis std = student’s t distribution, norm = normal distribution, GED = Generalized Error Distribution
GARCH and VaR | 30
Table A.4 Parameter estimates OMXS Large Cap
PARAMETER ESTIMATES FOR OMXS LARGE CAP
Student´s t Normal GED
GARCH (1,1) GARCH (1,1) GARCH (1,1)
Estimate Std.error t-value Pr(>|t|) Estimate Std.error t-value Pr(>|t|) Estimate Std.error t-value Pr(>|t|)
mu 0.0676 0.0173 3.920 0.00 0.0581 0.0175 3.310 0.00 0.0669 0.0174 3.850 0.00
omega 0.0204 0.0062 3.270 0.00 0.0213 0.0055 3.860 0.00 0.0211 0.0064 3.360 0.00
alpha1 0.1009 0.0148 6.840 0.00 0.1009 0.0122 8.270 0.00 0.1013 0.0144 7.040 0.00
beta1 0.8850 0.0161 55.070 0.00 0.8838 0.0137 64.730 0.00 0.8833 0.0160 55.200 0.00
shape 9.4831 1.6880 5.620 0.00 1.5226 0.0604 25.230 0.00
GJR- GARCH (1,1) GJR- GARCH (1,1) GJR- GARCH (1,1)
mu 0.0353 0.0171 2.070 0.04 0.0200 0.0175 1.140 0.25 0.0359 0.0172 2.080 0.04
omega 0.0235 0.0056 4.190 0.00 0.0242 0.0049 4.900 0.00 0.0240 0.0056 4.280 0.00
alpha1 0.0028 0.0098 0.280 0.78 0.0136 0.0090 1.520 0.13 0.0087 0.0100 0.880 0.38
beta1 0.8867 0.0143 62.230 0.00 0.8869 0.0123 72.220 0.00 0.8858 0.1414 62.660 0.00
gamma1 0.1869 0.0266 7.020 0.00 0.1650 0.0226 7.290 0.00 0.1746 0.0257 0.030 0.00
shape 10.1364 1.8041 1.800 0.00 1.5668 0.0611 25.650 0.00
EGARCH (1,1) EGARCH (1,1) EGARCH (1,1)
mu 0.0299 0.0117 2.560 0.01 0.0108 0.0170 0.630 0.53 0.0302 0.0111 2.710 0.01
omega -0.0022 0.0003 -8.490 0.00 0.0046 0.0032 1.460 0.14 -0.0022 0.0031 -0.690 0.49 alpha1 -0.1376 0.0140 -9.830 0.00 -0.1145 0.0118 -9.710 0.00 -0.1244 0.0137 -9.080 0.00 beta1 0.9762 0.0008 1174.8 0.00 0.9767 0.0000 21025.0 0.00 0.9767 0.0000 23568.4 0.00 gamma1 0.1614 0.0183 8.830 0.00 0.1639 0.0003 550.500 0.00 0.1622 0.0005 303.670 0.00
shape 9.6210 1.6157 5.950 0.00 1.5529 0.0598 25.950 0.00
GARCH and VaR | 31
Table A.5 Parameter estimates OMXS Mid Cap
PARAMETER ESTIMATES FOR OMXS MID CAP
Student´s t Normal GED
GARCH (1,1) GARCH (1,1) GARCH (1,1)
Estimate Std.error t-value Pr(>|t|) Estimate Std.error t-value Pr(>|t|) Estimate Std.error t-value Pr(>|t|)
mu 0.1111 0.0143 7.490 0.00 0.0845 0.0155 5.470 0.00 0.1158 0.0151 7.663 0.00
omega 0.0350 0.0084 4.140 0.00 0.0311 0.0071 4.370 0.00 0.0337 0.0084 3.990 0.00
alpha1 0.1491 0.0217 6.860 0.00 0.1369 0.0177 7.740 0.00 0.1452 0.0215 6.750 0.00
beta1 0.8146 0.0253 32.200 0.00 0.8296 0.0218 38.061 0.00 0.8183 0.0258 31.670 0.00
shape 7.2670 1.0087 7.200 0.00 1.4317 0.0556 25.740 0.00
GJR- GARCH (1,1) GJR- GARCH (1,1) GJR- GARCH (1,1)
mu 0.0991 0.0149 6.640 0.00 0.0671 0.0155 4.330 0.00 0.1024 0.0154 6.670 0.00
omega 0.0409 0.0090 4.530 0.00 0.0344 0.0076 4.500 0.00 0.0386 0.0091 4.260 0.00
alpha1 0.0473 0.0167 2.840 0.01 0.0463 0.0138 3.350 0.00 0.0485 0.0167 2.900 0.00
beta1 0.8157 0.0261 31.240 0.00 0.8412 0.0231 36.500 0.00 0.8227 0.0227 30.350 0.00
gamma1 0.1615 0.0299 5.299 0.00 0.1321 0.0224 5.890 0.00 0.1477 0.0284 5.190 0.00
shape 7.7224 1.1167 6.920 0.00 1.4648 0.0571 26.650 0.00
EGARCH (1,1) EGARCH (1,1) EGARCH (1,1)
mu 0.0971 0.0171 5.680 0.00 0.0628 0.0148 4.230 0.00 0.1002 0.0165 6.060 0.00
omega -0.0248 0.0091 -2.710 0.00 -0.0107 0.0050 -2.150 0.03 -0.0239 0.0029 -8.390 0.00 alpha1 -0.1001 0.0224 -4.470 0.00 -0.0831 0.0119 -6.700 0.00 -0.0910 0.0115 -7.950 0.00 beta1 0.9461 0.0160 59.030 0.00 0.9562 0.0093 102.490 0.00 0.9499 0.1059 89.680 0.00
gamma1 0.2332 0.0331 7.050 0.00 0.2114 0.0234 9.030 0.00 0.2265 0.0248 9.140 0.00
shape 7.5850 1.1207 6.770 0.00 1.4575 0.0567 25.700 0.00
GARCH and VaR | 32
Table A.6 Parameterestimates OMXS Small Cap
PARAMETER ESTIMATES FOR OMXS SMALL CAP
Student´s t Normal GED
GARCH (1,1) GARCH (1,1) GARCH (1,1)
Estimate Std.error t-value Pr(>|t|) Estimate Std.error t-value Pr(>|t|) Estimate Std.error t-value Pr(>|t|)
mu 0.1052 0.0130 8.110 0.00 0.0807 0.0140 5.760 0.00 0.1108 0.0136 8.130 0.00
omega 0.0883 0.0183 4.814 0.00 0.0726 0.0126 5.770 0.00 0.0804 0.0167 4.800 0.00
alpha1 0.2125 0.0334 6.350 0.00 0.1876 0.0237 7.910 0.00 0.1972 0.0306 6.450 0.00
beta1 0.6528 0.0505 12.940 0.00 0.6984 0.0364 19.180 0.00 0.6733 0.0479 14.060 0.00
shape 5.5480 0.6058 9.160 0.00 1.3172 0.0485 27.160 0.00
GJR- GARCH (1,1) GJR- GARCH (1,1) GJR- GARCH (1,1)
mu 0.1009 0.0131 7.725 0.00 0.0719 0.0141 5.090 0.00 0.1058 0.0135 7.830 0.00
omega 0.0979 0.0197 4.970 0.00 0.0772 0.0124 6.240 0.00 0.0864 0.0171 5.040 0.00
alpha1 0.1140 0.0314 3.630 0.00 0.0933 0.0220 4.240 0.00 0.1013 0.0293 3.460 0.00
beta1 0.6398 0.0534 11.970 0.00 0.7054 0.0350 20.150 0.00 0.6731 0.0482 13.970 0.00
gamma1 0.1597 0.0445 3.580 0.00 0.1362 0.0293 4.650 0.00 0.1421 0.0390 3.650 0.00
shape 5.7816 0.6532 8.850 0.00 1.3337 0.0493 27.070 0.00
EGARCH (1,1) EGARCH (1,1) EGARCH (1,1)
mu 0.0976 0.0197 4.950 0.00
omega -0.0847 0.0260 -3.254 0.00 -0.0464 0.0120 -3.890 0.00 -0.0637 0.0161 -3.950 0.00 alpha1 -0.0795 0.0238 -3.340 0.00 -0.0855 0.0140 -6.110 0.00 -0.0859 0.0175 -4.920 0.00 beta1 0.8719 0.0363 23.990 0.00 0.8973 0.0183 49.030 0.00 0.8894 0.0234 38.090 0.00
gamma1 0.3214 0.0487 6.600 0.00 0.2759 0.0300 9.200 0.00 0.2945 0.0371 7.940 0.00
shape 5.7543 0.6004 9.580 0.00 1.4187 0.0518 27.370 0.00
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