An empirical study in risk management: estimation of Value at Risk with GARCH family models
Author: Askar Nyssanov
Supervisor: Anders Ågren, Professor
Master Thesis in Statistics Department of Statistics Uppsala University Sweden
2013
Abstract
In this paper the performance of classical approaches and GARCH family models are evaluated and compared in estimation one-step-ahead VaR. The classical VaR methodology includes historical simulation (HS), RiskMetrics, and unconditional approaches. The classical VaR methods, the four univariate and two multivariate GARCH models with the Student’s t and the normal error distributions have been applied to 5 stock indices and 4 portfolios to determine the best VaR method. We used four evaluation tests to assess the quality of VaR forecasts: - Violation ratio - Kupiec’s test - Christoffersen’s test - Joint test The results point out that GARCH-based models produce far more accurate forecasts for both individual and portfolio VaR. RiskMetrics gives reliable VaR predictions but it is still substantially inferior to GARCH models. The choice of an optimal GARCH model depends on the individual asset, and the best model can be different based on different empirical data.
Keywords: Value at Risk, univariate and multivariate GARCH models, classical VaR approaches, evaluation tests
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Contents I. INTRODUCTION ...... 4 II. THEORETICAL FRAMEWORK ...... 6
1. UNIVARIATE GARCH MODELS ...... 6 1.1 GARCH model ...... 6 1.2 Exponential GARCH (EGARCH) model ...... 7 1.3 Integrated GARCH (IGARCH) model ...... 8 1.4 GJR-GARCH Model ...... 8 2. MULTIVARIATE GARCH MODELS ...... 9 2.1 GO-GARCH model ...... 9 2.2 DCC-GARCH model ...... 10 3. ERROR DISTRIBUTIONS ...... 12 3.1 Normal distribution ...... 12 3.2 Student’s t distribution ...... 12 4. VALUE-AT-RISK (VAR) METHODOLOGIES ...... 13 4.1 Historical simulation...... 14 4.2 Unconditional parametric methods ...... 15 4.3 RiskMetrics model ...... 16 4.4 GARCH-based models ...... 17 5. EVALUATION TESTS ...... 18 5.1 Violation ratio ...... 18 5.2 Kupiec’s test ...... 19 5.3 Christoffersen's conditional test ...... 19 5.4 Joint test ...... 20 III. EMPIRICAL RESULTS ...... 21
1. DATA ...... 21 2. ESTIMATION OF VAR ...... 23 2.1 VaR for individual assets ...... 24 2.2 Portfolio VaR ...... 26 IV. CONCLUSION ...... 28 V. REFERENCES ...... 30 VI. APPENDIX ...... 32
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I. INTRODUCTION
Value at Risk (VaR) is one of the widely used risk measures. VaR estimates the maximum loss of the returns or a portfolio at a given risk level over a specific period. VaR was first introduced in 1994 by J.P.Morgan and since then it has become an obligatory risk measure for thousands of financial institutions, such as investment funds, banks, corporations, and so on. Classical VaR methods have several drawbacks. These methods include historical simulation (HS), RiskMetrics, and unconditional approaches. For instance, RiskMetrics method always assumes joint normality of the returns. In the unconditional approach we use a standard deviation to estimate VaR and assume that the volatility constant over time. However, in reality these assumptions do not hold in most cases. On the other hand, the basic driving principle of the historical simulation method is its assumption that the VaR forecasts can be based on historical data. In 1982, Engle, the winner of the 2003 Nobel Memorial Prize in Economic Sciences, introduced ARCH (“Autoregressive Conditional Heteroskedasticity”) models. Then Bollerslev (1986) proposed the generalization of the ARCH process calling it GARCH models. The main advantage of the GARCH models is that they are capable of capturing several major properties of financial time series. In recent years, the estimation of the VaR using GARCH models has become very popular and most widely-used approach in VaR calculation. Many research results have shown that the GARCH models outperform classical VaR methods and make more accurate VaR forecasts. Fuss, Kaiser and Adams (2007) applied three different VaR approaches: the normal, Cornish–Fisher (CF), and the GARCH-type VaR to the S&P hedge fund index series (SPHG). They showed that the GARCH-type VaR gives more accurate VaR forecasts than other VaR methods for most of the hedge fund style indices. Totić, Bulajić and Vlastelica (2011) estimated daily returns of the FTSE100 index using non-parametric, RiskMetrics and GARCH-based VaR methods with the normal and t distributions. According to their study, RiskMetrics and GARCH models performed better than non-parametric approaches. So and Yu (2006) estimated one-step- ahead VaR predictions of 12 stock market indices and four foreign exchange rates using six GARCH models and RiskMetrics. They have concluded that all GARCH models outperform RiskMetrics in estimating 1% VaR and Student’s t distribution produces more accurate VaR forecasts than the normal. Angelidis, Benos and Degiannakis (2004) used AR-GARCH, AR-EGARCH and AR- TARCH models of different orders with the normal, Student’s t and the generalized error distributions to estimate one-step-ahead VaR for five stock indices: S&P 500, NIKKEI 225, FTSE 100, CAC 40 and DAX 30. They came to the conclusion that the sample size is crucial in defining VaR accuracy, leptokurtic distributions make better VaR predictions and the GARCH model fitting the data best depends on specific stock indices. Orhan and Koksal (2012) compared 16 GARCH models in estimating one-step-ahead VaR forecasts using Student’s t and the normal distributions. The data used were stock indices from growing (Turkey, Brazil) and developed (Germany, USA) economies. The conclusion again underlined that GARCH (1,1) results were the most accurate, and Student’s t slightly outperformed the normal distribution. Wong, Cheng and Wong (2003) tested the performance of 9 GARCH models in estimating VaR results for Australia’s All Ordinary Index (AOI) series. Their result showed that GARCH-based VaR models showed poor performance and did not meet Basel’s backtesting criteria. Next I have analyzed some earlier studies on portfolio VaR estimations. Santos, Nogales and Ruiz (2013) compared the performance of three multivariate GARCH models in computing VaR forecasts for equally weighted diversified portfolios with large number of assets. The models used included DCC-GARCH, CCC-GARCH and Asymmetric DCC-GARCH. This study has showed that DCC-GARCH produced more accurate VaR forecasts compared to other models. Morimoto and Kawasaki (2008) conducted a more comprehensive study in order to define the best model in 4 forecasting portfolio VaR. They have evaluated the performance of VECH, BEKK, CCC- GARCH and DCC-GARCH models with t and normal errors and RiskMetrics. Their portfolios included a large number of assets from the Tokyo Stock Exchange. According to the study’s results, the DCC-GARCH was found to be the best model in forecasting portfolio VaR. Caporin and McAleer (2012) also tried to assess the performance of the multivariate GARCH-type VaR models. They have used BEKK, DCC, Corrected DCC (cDCC), CCC, OGARCH models and RiskMetrics in their calculation of portfolio VaR forecasts. Each model was estimated for medium and large scales. Medium scale portfolios consisted of 5-15 assets while the large ones consisted of 10, 15, 20, 25, 30, 35, 40, 45, 50, 60, 70, 80 and 89 assets from S&P100. At the end, DCC-GARCH and O-GARCH slightly outperformed other methods. In the conclusion, the authors have stated that the choice of the best model in forecasting VaR portfolio mainly depends on the sample period, portfolio type, and the selection criteria relevant to the purpose of the analysis. The underlying aim of this paper is to evaluate and compare the performance of classical and GARCH-based VaR approaches in order to define the best VaR methodology. I will also analyze the implementation of RiskMetrics and assess whether it provides adequate VaR forecasts to be the most accurate VaR approach in risk management and whether it can considerably outperform other methods. My second objective is to compare GARCH models results under different distribution assumptions and define the best one for VaR estimation. There are still many questions remaining on VaR methodologies. Is there any GARCH model that substantially outranks other GARCH models? Does Student’s t distribution fit data well and give more accurate VaR predictions than the normal distribution as implied by many empirical studies? In this paper, I attempt to offer reasonable answers to these questions. In risk management we can find many research papers where the analysis part is conducted on simulated data. Recently more researchers carry out their empirical study mainly on global indices such as NASDAQ, FTSE100, NIKKEI, etc. or corporate stock prices from different fields. The choice of the data for my paper is based on a slightly different approach. The asset returns of seven largest copper producers are used to estimate 99% and 95% VaR forecasts. The scale of the world copper market counted by billions of US dollars and the empirical results of this paper might be helpful in finding the best risk forecasting model for this market.
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II. THEORETICAL FRAMEWORK
1. Univariate GARCH models
1.1 GARCH model
Engle (1982) described ARCH as “…mean zero, serially uncorrelated processes with nonconstant variances conditional on the past, but constant unconditional variances”. The main advantage of ARCH models is that they can generate accurate models in forecasting volatility of financial time series. The behavior of the time series is driven by three statistical properties (Danielsson, 2011): - volatility clusters - fat tails - nonlinear dependence Volatility clustering occurs when a period of large returns is followed by a period of small returns (Nelson, 1991). The second property indicates that large positive or large negative observations in financial data occur more frequently as compared to the standard normal distribution. Nonlinear dependence explains the relationship between multivariate financial data. For instance, nonlinear dependence between different assets can be observed during financial crisis, where many assets are likely to move together in the same direction relevant to some market conditions (Danielsson, 2011). Usually it is more practical to separate estimation of mean from volatility estimation (Danielsson, 2011), thus in this paper all the volatility models are implemented on demeaned returns, i.e. the elimination of an unconditional mean from the returns. Let ɛ be a random variable (in this paper it is the financial time series, expressed in returns) with a zero mean and variance conditional on the past time series ɛ , … , ɛ . Engle (1982) proposed a decomposition of ɛ as:
ɛ =
where z is a sequence of independent, identically distributed random variables with zero mean and unit variance. Typically, the distribution of z is assumed to be normal or leptokurtic (Terasvirta, 2006), and the conditional variance of the ARCH model of order q is modeled by