An empirical study in risk management: estimation of Value at Risk with GARCH family models

Transcription

1 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

2 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 s to assess the quality of VaR forecasts: - Violation ratio - Kupiec s - Christoffersen s - Joint 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 s 2

3 Contents I. INTRODUCTION... 4 II. THEORETICAL FRAMEWORK UNIVARIATE GARCH MODELS GARCH model Exponential GARCH (EGARCH) model Integrated GARCH (IGARCH) model GJR-GARCH Model MULTIVARIATE GARCH MODELS GO-GARCH model DCC-GARCH model ERROR DISTRIBUTIONS Normal distribution Student s t distribution VALUE-AT-RISK (VAR) METHODOLOGIES Historical simulation Unconditional parametric methods RiskMetrics model GARCH-based models EVALUATION TESTS Violation ratio Kupiec s Christoffersen's conditional Joint III. EMPIRICAL RESULTS DATA ESTIMATION OF VAR VaR for individual assets Portfolio VaR IV. CONCLUSION V. REFERENCES VI. APPENDIX

4 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-stepahead 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) ed 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 backing 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

5 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. 5

6 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 = σ z 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 σ = w + α ɛ where w>0, α > 0. Bollerslev (1986) in his paper proposed so-called generalized ARCH models. The GARCH (q, p) is given by ɛ = σ z σ = w + α ɛ + β σ 6

7 q represents the order of ɛ and p represents the order of σ. It is necessary to impose conditions, such as w>0, α > 0, β > 0 and α + β < 1 to obtain a positive conditional variance and stationarity. We applied higher order GARCH models in VaR estimation but the results were almost the same as the results obtained by a simple GARCH. So we prefer to use the GARCH models of orders q=1 and p=1 in estimating VaR for their simplicity and reliability. GARCH(1,1) is given by σ = w + α ɛ + β σ where - ɛ are returns with zero mean and unit variance - w, α, β model coefficients - w > 0, α > 0, β > 0 and α + β < 1 - σ =, σ is the unconditional variance of ɛ - E(σ )=E(E(ɛ ɛ, j = 1,2, ))=σ 1.2 Exponential GARCH (EGARCH) model In 1991 Nelson introduced the exponential GARCH model. Nelson (1991) noted that his EGARCH model can overcome three main drawbacks of ordinary GARCH models: - negative correlation between current and future returns which is excluded by the GARCH model assumption - GARCH models imply parameter restrictions - GARCH models do not consider asymmetric variance effects EGARCH (q, p) is given by (Terasvirta, 2006): log (σ ) = w + [α ɛ In our thesis we use the EGARCH (1,1) model: + λ ( ɛ E ɛ )] + β log (σ ) log (σ ) = w + αɛ + λ ɛ E( ɛ ) + βlog (σ ) - ɛ are returns with zero mean and unit variance - w, α, β, λ, model coefficients - αɛ is a sign or asymmetry effect - λ ɛ E( ɛ ) is a magnitude effect Since the logarithm is always positive, positivity constraints are not necessary in the EGARCH model. Also asymmetry depends on the coefficient α. For instance, when α<0, log (σ ) would be greater than its mean w if ɛ < 0 and it would be smaller if ɛ > 0. This means that when α<0 negative news have greater effects than positive news. On the other hand, when α>0 positive news have larger effects on the conditional variance than negative news. This shows us typical asymmetry of the financial time series. The meaning of E( ɛ ) depends on the error distribution, since in our paper we use the normal and Student s t distributions we show E( ɛ ) for these distributions (Franco and Zakoian, 2010.): - E( ɛ ) =, when z is normally distributed 7

8 freedom. - E( ɛ ) = Г( ) ()Г( ), when z is Student s t distribution with v degrees of 1.3 Integrated GARCH (IGARCH) model In 1986 Engle and Bollerslev proposed the integrated GARCH (IGARCH) model. Many studies have shown that the sum of the parameters in GARCH models almost always is close to unity. In the IGARCH model we consider the sum of the parameters to be equal to one which means that the return series is not covariance stationary and there is a unit root in the GARCH process (Jensen and Lange, 2007). Jensen and Lange pointed out that the conditional variance of the GARCH model converges in probability to the true unobserved volatility process even when the model is misspecified and the IGARCH effect is a consequence of the mathematical structure of a GARCH model and not a property of the true data generating mechanism. The condition for IGARCH is IGARCH (1,1) model: α + β = 1 σ = w + α ɛ where - w > 0, α > 0, β > 0 and α + β = 1 + β σ 1.4 GJR-GARCH Model The GJR-GARCH was proposed by Glosten, Jagannathan and Runkle (1993) and this model assumes to reveal and take into account the asymmetry property of financial data in obtaining the conditional heteroskedasticity (see Glosten, Jagannathan and Runkle, 1993). The general form of the GJR-GARCH (q, p) is given by σ = w + (α + λ I )ɛ σ = w + (α + λ I )ɛ 8 + β σ where I is an indicator function taking the value one if the residual is smaller than zero and the value zero if the residual is not smaller than zero. I = 1, if ɛ < 0 0, otherwise As I stated above many empirical results proved that negative shocks have more significant effects on the conditional variance than that of positive and GJR-GARCH model reflects this special feature of financial data. GJR-GARCH (1,1) model: + β σ

9 where - I = 1, if ɛ < 0 0, otherwise - w, α, β, λ model coefficients - w > 0, α > 0, β > 0, λ > 0 2. Multivariate GARCH models 2.1 GO-GARCH model Currently widely used multivariate GARCH models are based on three approaches (Bauwens, Laurent and Rombouts, 2006): - direct generalizations of the univariate GARCH models (VEC, BEKK and factor models) - linear combinations of univariate GARCH models (GO-GARCH and latent factor models) - nonlinear combinations of univariate GARCH models (constant and dynamic conditional correlation models, the general dynamic covariance model and copula GARCH models). For detailed information about multivariate GARCH models refer to Bauwens et al., (2006), Franco and Zakoian, (2010). For the purpose of our paper, we consider GO-GARCH and dynamic correlations GARCH models. All multivariate GARCH models suffer from an estimation of a large number of parameters, which makes their practical application very limited. Among these models the generalized orthogonal GARCH model seems very attractive as it allows diminishing higher dimension of original time series data (Franco and Zakoian, 2010). The GO-GARCH model assumes that the original data set can be obtained as a set of linearly independent components (Weide, 2002). A formal definition of the GO-GARCH model is given by (Weide, 2002): Let ɛ t = (ɛ, ɛ,, ɛ ) be multivariate time series data and assume it is governed by a linear combination of uncorrelated components {u t = (u, u,, u )}: ɛ t = Zu t New independent components are obtained through the linear transformation matrix Z and Z matrix is assumed to be invertible and constant over time. The conditional covariance matrix of ɛ t is given by V = ZH Z where H is the conditional covariance matrix of the components. Since the components are independent, H is a diagonal matrix. Each of the components is modeled as a univariate GARCH process: u u, j = 1,2, n D(0, H ) h = w + α u, 9 + β h,

10 We can use ordinary univariate GARCH models as well as other types of GARCH such as EGARCH, IGARCH and GJR-GARCH to model components. The model was first introduced by Weide (2002), as a generalization of the orthogonal GARCH model, which was first proposed by Ding in 1994 and later on popularized by Alexander in For the derivation of the linear transformation matrix Z and for more detailed information refer to (Weide, 2002, Boswijk and Weide, 2006, Johnson and Wichern, 2007). In our paper we use the GO-GARCH (1,1) model with the normal error distribution as it is simple and fits our data as well as other higher order GO-GARCH (q,p) models. The GO- GARCH (1,1) model is given by ɛ t = Zu t V = ZH Z u u, j = 1,2, n N(0, H ) h = w + α u, + β h where - ɛ t = (ɛ, ɛ,, ɛ ) - portfolio consisting of m asset returns (n x 1 vector of demeaned log returns of m assets at time t) - u t = (u, u,, u ) - uncorrelated components - Z the linear transformation matrix - V the conditional covariance matrix of a portfolio - H the conditional covariance matrix of the components, h is a conditional variance of i-th component and is modeled by a univariate GARCH (1,1) process. We can fit a univariate GARCH (1,1) model for each uncorrelated component separately and compute the one-step-ahead conditional variance forecast h, for one component and the conditional covariance forecast matrix H for all the components. Then we obtain the one-stepahead forecast of the conditional covariance matrix for the given portfolio: V = ZH Z 2.2 DCC-GARCH model The dynamic conditional correlation GARCH model was first proposed by Engle and Sheppard (2001). Actually DCC-GARCH is an extended version of conditional correlation GARCH. The core idea is that the sample covariance matrix can be decomposed into a correlation matrix and a matrix of time varying standard deviations (Orskaug, 2009). In the CCC-GARCH model a correlation matrix is assumed to be constant while in the DCC-GARCH model it is time varying and depends on the symmetric positive definite autoregressive matrix. The DCC-GARCH model can be defined as (Orskaug, 2009): ɛ t = V / z t V = D R D where - ɛ t = (ɛ, ɛ,, ɛ ) - portfolio consisting of m asset returns (n x 1 vector of demeaned log returns of m assets at time t) - z t is n x 1 vector of independent, identically distributed random variables such that E[z t ]=0 and Var[z t ] =E(z t z t ) =I (I is an identity matrix) 10

11 - V the conditional covariance matrix of portfolio returns - D is a diagonal matrix of standard deviations - R is a time dependent conditional correlation matrix The conditional variance of each asset σ, can be modeled by any of the univariate GARCH with the normal error distribution and stationarity property. D is given by: σ, 0 D = 0 σ, The positive definiteness of V depends only on the correlation matrix R that is if R is a positive definite matrix then V would also be positive definite. In order to obtain a positive definite correlation matrix, R is constructed by a symmetric positive definite autoregressive matrix Q: R = Q Q Q The correlation matrix structure can be extended to the general DCC (K,L)-GARCH model (Orskaug, 2009): Q = 1 a b Q + a l l + b Q In our paper we run DCC (1,1)-GARCH (1,1) model and so I want to give some more information regarding this model. DCC (1, 1)-GARCH (1, 1) model: where σ, ɛ t = V / z V = D R D R = Q Q Q l = D ɛ σ, 0 D = 0 σ, = w + α ɛ, + β σ, Q = (1 a b)q + al l + bq - Q is an unconditional sample covariance matrix of the standard errors - Q is a diagonal matrix, elements of the matrix are square root of the diagonal elements of Q - a, b are model parameters, a>0, b>0 and a+b<1 - l is a matrix of standard errors One property of this model that makes it very practical is that the DCC (1,1)-GARCH (1,1) model can be estimated in two steps (Bauwens, Laurent and Rombouts, 2006): 1) We estimate parameters of the univariate GARCH part by maximizing the likelihood function 11

12 2) We estimate parameters of the conditional correlation structure also by maximizing the likelihood function. More details can be found in (Bauwens, Laurent and Rombouts, 2006) and (Orskaug, 2009). 3. Error distributions As I mentioned above financial data often shows fat tail property. This makes us seek alternative distribution assumptions for error terms to the normal distribution. Still the normal distribution is widely used as an error distribution in GARCH models though more complex distributions such as Student s t, negative inverse normal and generalized error distributions might be more preferable. Student s t distribution in GARCH models was first popularized by Bollerslev (1987). Nelson (1991) showed the usefulness of the generalized error distribution in modeling financial time series with GARCH models. Indeed a lot of empirical studies in GARCH modeling proved that Student s t or the generalized error distributions fit financial data better than the normal distribution. Also one of the purposes of this paper is to determine which error distribution, the normal or Student s t is better in estimating the Value at Risk (VaR) for both GARCH family models and alternative classical VaR methods. In this paper I use two distributions for error terms: - The normal distribution - Student s t distribution 3.1 Normal distribution The normal distribution is particularly important in statistics, and is the most used classical statistical distribution. The general formula for the probability density function of the normal distribution is 1 f(x) = σ 2π e (µ ) where - σ is the standard deviation of the distribution - µ is the mean The case where µ = 0 and σ = 1 is defined as the standard normal distribution. The normal distribution is symmetric around its mean. Maximum likelihood estimations for µ is the sample mean and for σ is the sample standard deviation. 3.2 Student s t distribution Student s t distribution is one of the widely used distributions in statistics science. Student s t distribution was first introduced by W. Gosset in Many return series have fatter tails and the normal distribution cannot sufficiently take into consideration this fact in modeling GARCH. That s why the t distribution is often preferred as an error distribution as compared to the normal. The probability density function is given by: 12

13 f(x) = Г v x Г v (1 + 2 vπ v ) where - v is the number of degrees of freedom, v > 0 - mean is 0 for ν > 1, otherwise undefined - variance is for v > 2, otherwise undefined The t distribution has heavier tails which means that the probability of values falling far from its mean is higher than that of the normal and it is also symmetric and bell-shaped like the normal distribution. 4. Value-at-Risk (VaR) methodologies Nowadays Value at Risk (VaR) has become one of the key methods in estimating financial risk. Definition of VaR is given by (Danielsson, 2011): VaR is a value such that the probability is p that the losses are equal to or exceed VaR in a given trading period and (1-p) that the losses are lower than VaR. The most widespread confidence levels are 90%, 95% and 99%. For example, if the asset price is 500 USD and daily VaR is 5 USD for 99% confidence level, then we can say that we assume losing 5 USD or more once every 100 days. There are three steps in VaR estimation (Danielsson, 2011): - To choose the confidence level. As stated above, common probabilities for VaR are 1%, 5% and 10%. - To determine the holding period, in most cases it is either 1-day or 10-day. However, holding periods can vary depending on the financial institutions and data. Sometimes it can be as short as 1-hour or 1-month period. - To define the probability distribution of returns or, more generally, the distribution of profit and loss. This step is usually the most challenging part of the VaR estimation. Common distributions are the normal, the Student s t, and the generalized error distributions. The general formula for VaR is (Danielsson, 2011): Or alternatively, p = Pr [r -VaR (p)] = p () f(x)dx where - p is the probability of losses exceeding or equal to VaR - r is the profit\loss - f(x) is the probability density function of r Now we explain the computation of the VaR for continuously compounded returns (Danielsson 2011): ɛ = logy logy p = Pr[Y Y VaR(p)] = Pr[Y (e ɛ 1) VaR(p)] = Pr[ ɛ VaR(p) log ( + 1) 1 σ Y σ ] 13

14 where - ɛ is a series of continuously compounded returns - Y, Y, are stock indices - σ is the volatility or the standard deviation of returns Denoting the inverse of the standardized distribution of returns by F ɛ (. ), we obtain: VaR(p) = [e ɛ () 1]Y and for small values of σf ɛ (p) the VaR can be written as (Danielsson 2011): VaR(p) σf ɛ (p)y Here Y means the value of an asset. In our paper we consider the value of an asset to be one and thereafter we exclude it from the VaR formula. There are different methodologies in calculating VaR but we can define two distinct approaches: non-parametric and parametric methods. The non parametric method is based on historical data, which is a historical simulation method. On the other hand, parametric methods make assumptions on the distribution of the returns; as a result, these methods can capture the true underlying nature of the returns, and thus give more reliable VaR values (Danielsson, 2011). Parametric methods can be divided into two parts (Dowd, 2002): - In the unconditional approach, the volatility or standard deviation of the returns is time-invariant and does not depend on the time or period at which returns are observed. We calculate this volatility based on all returns and use it for measuring the financial risk. - The conditional approach is mostly based on GARCH family models. The volatility can be modeled with any univariate or multivariate GARCH model and is then used in the estimation of VaR. 4.1 Historical simulation As the name suggests, the historical simulation (HS) method uses historical data of the returns to compute VaR forecasts. As Danielsson (2011) pointed out the core idea of this methodology is to forecast future losses based on the past performance. VaR with the univariate HS method for single asset returns can be calculated in 3 steps (Danielsson, 2011): - We sort sample returns from the lowest to the highest value. For example, we have returns of Deutche bank {ɛ } for the last 2 years, thus we have approximately 500 observations. Then we sort these N=500 observations in an ascending order starting from the lowest value. Let {y } be the sorted returns of the Deutche bank. - We find the p% quantile of the {y }. For instance p = 0.01 or 1% then we calculate p% quantile by q = p N. In our case, q = = 5. - We extract q-th value from the sorted returns {y }. So the VaR formula for the univariate HS is equal to VaR = y [p N] where - y [q] means an extraction of q-th element from {y } series. In the multivariate case we estimate VaR for a portfolio that consists of several assets. At first, we need to define the vector of weights for our portfolio, and the sum of the vector 14

15 elements must be equal to 1. Multivariate HS is a straightforward extension of the univariate HS method. Similarly, there are 3 steps in calculation the VaR (Danielsson, 2011): - Calculate sample portfolio returns with the following formula: ɛ = w ɛ where ɛ - i-th asset returns w weight on asset i n number of assets - Next, sort out the sample portfolio returns and denote it y. o Calculate p% quantile of the y. q = p N is p% quantile. N is the number of observations o Extract q-th value from the sorted returnsy. Multivariate HS VaR: VaR = y [p N] HS method has its drawbacks and gains. The main advantages are (Danielsson, 2011): - there is no necessity to assume or identify distribution of the returns - HS method directly defines nonlinear dependence - fat tails of the distribution are captured if the historical data is big enough. Disadvantages of the historical simulation method are (Dowd, 2002): - Total dependence on the data set - Needs large data sets - Cannot capture structural breaks in volatility Despite being simple and easy to use, the HS method has significant shortcomings and the most important one is the total dependence on the data. Recently many financial institutions have chosen other methods for VaR estimation rather than HS. As we show in this paper, parametric methods are more reliable and make more accurate volatility forecasts for VaR values. 4.2 Unconditional parametric methods Parametric approaches to the VaR estimation are based on a distribution of returns. A parametric model estimates VaR directly from the standard deviation of the returns. In the unconditional approach, volatility or standard deviation derived from the distribution of the returns stays constant over all VaR periods (Danielsson, 2011). In our paper we derive VaR given the most common distributional assumptions: the normal and Student s t distribution. First, we need to fit the distribution of the returns and obtain the maximum likelihood estimation of its parameters. For the normal distribution, volatility estimation is a sample standard deviation, however, in the t distribution case it is quite complicated. Here I explain how to estimate volatility from the t distribution according to Danielsson (2011). The variance of the t distribution is given by: σ = v v 2 15

16 The variance of the t distribution is undefined when v 2 and it cannot be equal to one. If we use a sample standard deviation in VaR estimation, the VaR might be overestimated. Therefore, we scale volatility estimate by h. σ = v v 2 h where v the number of degrees of freedom h - variance in excess implied by the standard Student s t distribution We determine the p% quantile from the assumed theoretical distribution of the returns. If p =0.01 then we use 1% quantile of the normal or the t distribution. The VaR formula is: VaR = σ q(p) where - σ is the volatility estimated from the normal or the t distribution - q(p) is p% quantile from the assumed theoretical distribution of the returns VaR estimation for a portfolio is the direct extension of the univariate case and consists of two steps. In our paper we estimate portfolio VaR only for the normal distribution assumption. - In the first step, we estimate volatility of the returns. To do so it is necessary to obtain a sample covariance matrix of portfolio returns cov(y ) and calculate the volatility of the given portfolio: σ = w covy w where w vector of weights on assets - Next, we need to obtain the p% quantile from the assumed theoretical distribution of the returns. In our case, it is the normal distribution. The portfolio VaR formula is given by: VaR = σ q (p) where - σ is the estimated portfolio volatility - q (p) is the p% quantile from the assumed theoretical normal distribution Concluded from above, unconditional parametric methods are simple and straightforward for implementation. However, they suffer from substantial drawbacks, the most important of which is the failure to capture distinct statistical properties of the financial data in modeling volatility. Many risk managers prefer conditional parametric approaches to unconditional, and the best known of them is the GARCH-based models, which we discuss in the next sections. 4.3 RiskMetrics model In 1996, JP Morgan s RiskMetrics Technical Document was published, where the exponentially weighted moving average (EWMA) method has been used in forecasting conditional volatility of asset returns (RiskMetrics Technical Document, 1996). RiskMetrics methodology has become extremely popular in estimating VaR of financial risk, though it has some disadvantages. The formula of the model is given by (Dowd, 2002): where 0<λ<1 σ = (1 λ)ɛ + λσ 16

17 J.P.Morgan proposed that λ must be set at 0.94 and since then it became the most widely used assumption in risk management. It is clear that RiskMetrics can be considered as an IGARCH (1,1) model with the constant parameters. This model assumes that the returns are normally distributed and are conditionally normal (RiskMetrics Technical Document, 1996). Calculation of VaR is straightforward: VaR = σ q(p) where - σ is the volatility estimated from RiskMetrics - q(p) is p% quantile from the normal distribution Multivariate RiskMetrics methodology is also very simple and is the direct extension of the univariate RiskMetrics. The multivariate form of the model is given by (Dowd, 2002): Σ = λσ + (1 λ)ɛ ɛ σ = w Σ w VaR = σ q (p) where - w vector of weights on assets - σ is the estimated portfolio volatility - q (p) is p% quantile from the assumed theoretical normal distribution Despite being one of the most successful methods in VaR estimation, RiskMetrics methodology has several shortcomings. The most important of them is the assumption of normality of returns. Many financial time series including returns have fat tails, and therefore cannot be properly modeled by the normal distribution. However, as Pafka and Kondor (2001) observed the success of RiskMetrics strongly depends on the choice of the risk measure as its fat tail property is not significant at smaller confidence levels like 90% or 95%, but for larger significance levels the result from the RiskMetrics method can be misleading as a consequence of the risk underestimation. In practice, there are some other useful modified versions of EWMA methods but they have not received widespread implementation by risk managers. Pafka and Kondor (2001) have thoroughly analyzed the performance of RiskMetrics and highlighted its main disadvantages: - the model completely ignores the presence of fat tails in the distribution of returns - it provides accurate results only for short (one-period ahead) forecasting horizon - the satisfactory performance in estimating Value-at-Risk by simply multiplying volatility with a constant factor strongly depends on the choice of the particular significance level. 4.4 GARCH-based models Classical VaR methods are subject to criticism for many reasons, mostly because they do not consider time dependency in estimating volatility. Unconditional parametric methods assume that the returns are correlated and volatility is time invariant (Dowd, 2002). The historical simulation method makes an assumption that it is justified to make forecasts based on the past performance which can be misleading, particularly when structural breaks occur in volatility (Danielsson, 2011). All these models drawbacks have made the GARCH models gain much popularity and, indeed, dozens of empirical studies show the reliability and usefulness of this methodology. 17

18 In our paper we use four univariate GARCH models: GARCH, EGARCH, IGARCH and GJR-GARCH and two multivariate DCC-GARCH and GO-GARCH models in estimating VaR. VaR calculation consists of two steps: - We forecast volatility using GARCH models mentioned above - Calculate VaR based on the conditional volatility prediction VaR = σ q(p) 5. Evaluation s After using different techniques in VaR estimation we need to check their predictive accuracy using various statistical s. There are many VaR methodologies, and it is necessary to find the best model for risk forecasting. Several evaluation methods are known for model checking such as analyzing residuals, ing for normality, etc. In this paper, we use out-ofsample VaR estimates to assess risk forecasts. Out-of-sample VaR estimates are obtained based on the previous years observations and are compared with the last year s actual data. For the purposes of this paper, we explain and use four statistical techniques for evaluating the quality of VaR forecasts: - Violation ratio - Kupiec s - Christoffersen s - Joint The first method is related to the backing comparison of out-of-sample VaR values with historical observations while Christoffersen s and Kupiec s s are considered to be the formal statistical s. (Danielsson, 2011). 5.1 Violation ratio If the actual loss exceeds the VaR forecast, then the VaR is considered to have been violated. The violation ratio is the sum of actual exceedences divided by the expected number of exceedences given the forecasted period. The rate is calculated by (Danielsson, 2011): VR = 18 E p N where - E is the observed number of actual exceedences - p is the VaR probability level, in our case p=0.05 or N is the number of observations used to forecast VaR values. For instance, if VaR confidence level is at 95% and we forecast risk for one year, then the expected number of violations is equal to p N = = 12.5 per year. Most risk managers agree that the optimal value of the violation ratio must be between 0.8 and 1.2. In the case when VR>1.5 or VR<0.5, the model is imperfect. The violation ratio is good for an easy and quick check of the model but it has some serious drawbacks, one which is that it cannot show the underlying causes of the model failure. We cannot solely rely on a violation ratio as a mathematically justified method in determining the model s adequacy (Danielsson, 2011). So there exist other more powerful and commonly accepted s in checking forecast accuracy of

19 the different VaR methodologies. Most important of them are Kupiec s and Christoffersen s s, which allow making more formal conclusions about the model s adequacy. 5.2 Kupiec s We assume the number of exceedences over time follows the binomial distribution and the aim of the Kupiec s is to determine the consistency of these violations with the given confidence level. If the number of exceedences substantially differs from what is expected, then the risk model s adequacy is questionable (Danielsson, 2011). To perform the we need the number of actual violations (E), number of observations (N) and the VaR probability level (p). Assuming E is distributed by Bin(N, p) the null hypothesis is: H : p = p, p = 0.01 or 0.05 and p is estimated by. We whether the observed number of violations E is considerably different from the expected number of violations p*n. As Kupiec (1998) proposed, this is based on the likelihood ratio. E LR = 2lo g 1 E N N (1 p ) ᵪ(1) p where - N is the number of observations used to forecast VaR values - E is the observed number of actual exceedences Despite that Kupiec s does not assume the distribution of the returns, it still provides good results in the predictive accuracy of the model. On the other hand, the has some restrictions in practice, namely the sample size needs to be large and it overlooks the independence of the actual violations (Dowd, 2002). 5.3 Christoffersen's conditional As noted above, Kupiec s is a of unconditional coverage as it does not consider independence of the violations or exceedences. To check the volatility cluster in a risk model, in 1998, Christoffersen proposed a based on the conditional coverage. This Christoffersen s determines whether a violation occurred on a particular day conditionally depends on the previous day s result (Christoffersen, 1998). The procedure of carrying out the is given as in Danielsson (2011). At first we need to compute the following probabilities: p = Pr (m = i m = j) where i and j are either 0 or 1 and m means whether a VaR exceedence occurs at time t. The first-order transition matrix is given by: p П = 1 p 0 = no exceedence, 1 p p 1 = exceedence where - p is the probability of a violation or exceedence when there was no violation on a previous day 19

20 - p is the probability of two consecutive violations Under the null hypothesis H, we assume that there is no volatility cluster which means all the violations occurred are independent that is p = p = p and the transition matrix is: П = 1 p p 1 p p v + v p = v + v + v + v The likelihood function is: L = (1 p ) p where - v - number of cases where j follows i, i=1 means violation and i=0 means no violation For the alternative hypothesis H, maximum likelihood estimation of transition matrix and likelihood function are given by: v v П = + v v v + v v v + v v v + v The likelihood ratio is defined as: 5.4 Joint L = (1 p ) p (1 p ) p LR = 2 log(l ) ᵪ(1) log(l ) We can also combine Kupiec s and Christoffersen s s to create a new and we call it a joint : LR = LR + LR ᵪ(2) One disadvantage of the joint is that it is difficult to find the true cause of the model rejection. It might be either violation cluster or inconsistency of the violation proportion with VaR confidence level from binomial distribution. 20

21 III. EMPIRICAL RESULTS 1. Data The data consists of seven stock returns of the world s largest copper producers. In this paper, I use continuously compounded returns which are obtained by taking logarithms in price changes. Continuously compounded return at time t is defined by: ɛ = log P = log(p P ) log (P ) ɛ (n) = ɛ + ɛ + + ɛ where - P is a daily closing price of a stock at day t - ɛ is a return at day t - n - is a period of returns Freeport-McMoRan BHP Billiton Xstrata Anglo American Plc Rio Tinto Group Kazakhmys Producer Period 1 million metric Production, tons Codelco Share in the world production Country % Chile % USA % Australia % Switzerland % UK % Australia % Kazakhstan Table 1. World s largest copper producers in 2012 The length of these time series data is different since they released their stocks at various times. For the purpose of the paper, daily closing prices for the last seven years are used. Copper is one of the most widely used metals in the world, it is used almost in all spheres of industry such as electronics, automobile industry, space industry and so on. The world's seven largest copper producers accounted for over 6 million metric tons of copper production (Table 1). I use daily stock prices of seven top 10 corporations in copper 1 Period of stock prices used in the paper 21

22 production: Codelco, Anglo American Plc, Kazakhmys, Rio Tinto Group, Xstrata Plc, BHP Billiton Ltd., and Freeport-McMoRan Copper & Gold Inc. 2 At first I analyze and perform descriptive statistics on returns of each company. Codelco Freeport- McMoRan BHP Billiton Xstrata Anglo American Plc Rio Tinto Group Kazakhmys Mean Median Max Min St. dev Table 2. Descriptive statistics of stock returns As seen in Table 2, the means of the stock returns are very close to zero and BHP Billiton has the largest mean. Xstrata has the highest minimum and Kazakhmys has the largest maximum among corporations returns. Codelco Freeport- McMoRan BHP Billiton Xstrata Anglo American Plc Rio Tinto Group Kazakhmys Skewness Kurtosis Jarque Bera statistic (0.000) 3 (0.000) 7333 (0.000) (0.000) 4888 (0.000) (0.000) 5517 (0.000) Table 3. Characteristics of the returns series When observing standard deviations, Kazakhmys has the highest value, which is not good as the higher the standard deviation of return the greater is the risk of the stock. Stock prices of these corporations have significantly decreased during the world financial crisis in 2008 and immediately started growing after 2009 as can be seen in Figure 1 (see Appendix). All the stock prices show similar trends over time, which is obvious as they are all dependent to some extent on the same copper market price. We can perform visual inspection from Figure 2 to observe how the returns volatility changes over time. All the return series have volatility clusters and their volatilities are significant in 2008, which is due to the financial crisis. Xstrata has the largest outlier with Codelco and Rio Tinto Group also having big outliers. It is important to define properties of empirical distributions of given returns, such as kurtosis, skewness, symmetry, etc. As shown in Table 3 all the return series are far from being normally distributed. They all have negative skewness and large kurtosis. Xstrata has the highest kurtosis and skewness, which might indicate the existence of the big outliers. The results of Jarque Bera also show that 2 The data is available on 3 P-values of the s 22

23 returns do not follow normal distribution and have fat tails. According to Figure 3 (see Appendix), it is clear that all these returns have fat tails and most of them have excess kurtosis. I have constructed four portfolios from different corporate stock prices which are presented in Table 4. Portfolio Portfolio assets Weights on assets A Codelco, Kazakhmys (0.5, 0.5) B Xstrata, Rio Tinto Group (0.5, 0.5) C BHP Billiton, Freeport-McMoRan, Anglo American Plc (0.3, 0.3, 0.4) D Rio Tinto Group, Codelco, BHP Billiton (0.3, 0.3, 0.4) Table 4. Portfolios Table 4 includes 4 portfolios (A, B, C and D). A and B consist of two assets, and C and D include three assets each. In our paper we calculate the portfolio of VaR values using these four portfolios. 2. Estimation of VaR One of the main goals of this study is to compare GARCH models with classical methodologies in VaR estimation. The empirical section consists of two steps. In the first step we fit GARCH models and find parameter estimates. In the second step we compute VaR estimates for the last two years based on GARCH models obtained in the first part and compare them with VaR results from alternative methods such as historical simulation and unconditional parametric methods. We fit four types of univariate GARCH models to our data: GARCH (1,1), EGARCH (1,1), IGARCH (1,1) and GJR-GARCH (1,1) with two distributional assumptions: normal and Student s t for five assets Codelco, Anglo American Plc, Xstrata Plc, BHP Billiton Ltd. and Freeport-McMoRan Copper & Gold Inc. Then we compute VaR values based on those GARCH models and compare the results with classical VaR methods. In order to be easily understood I perform VaR analysis for individual assets separately and at the end I compare the implementation of all VaR methods to select the best one. In this paper I use out-of-sample VaR estimates to compare various VaR methodologies. I have 5 individual assets and 2300 observations for each asset which corresponds to 9 years of data. Then I take the first 1800 observations of an asset which we call estimation window and hold the remaining 500 as out-ofsample data (ing window) which includes last two years data ( ). The size of an estimation window that equals 1800 is fixed. First estimation window consists of observations from 1 to 1800 including and using this estimation window I compute 95% and 99% VaR values for 1801-st day. Then the estimation window is moved up by one day to get VaR prediction for 1802-nd day which represents the second working day of Thus I repeat this 500 times and obtain VaR estimates for 2 years. In portfolio case the length of estimation window equals 750 and ing window size includes 250 observations thus we make VaR predictions for The VaR estimation process is the same as for individual assets, at first we forecast VaR for 751-st day then for 752-nd day and so on while keeping estimation window size fixed. We run 5 VaR estimation methods and assess their forecasting accuracy. Those methods include HS, Unconditional parametric, RiskMetrics, DCC-GARCH and GO-GARCH and we assume only normal distribution. 23

24 2.1 VaR for individual assets Codelco Our first asset used to estimate VaR is Codelco. We use an unconditional parametric method with the normal distribution as it provides better results among classical methods. This method has two violations while the expected number of exceptions is no more than 5. For the results of the estimation of VaR based on GARCH models, please refer to Tables 5 and 6 located in the Appendix. Violation ratios obtained from most of the GARCH models, HS and unconditional method with Student s t distribution show good results in estimating 99% VaR. The violation ratio for 99% VaR based on GJR-GARCH model with t distribution has the perfect result equal to 1. As seen in Figures 6, 7 and 8 HS and unconditional methods do not adjust data as well as others. Their VaR lines are straight and do not consider appropriate changes in the stock return. The results of joint s are similar to those of violation ratios. The null hypothesis for all the s from all methods is not rejected. Overall, GJR-GARCH with the t distribution shows the best results in computing 99% VaR for Codelco. If we observe 95% VaR estimation, the results are a little bit different. EGARCH with the normal and IGARCH with the t distribution have the best violation ratios equal to 1. Based on the results, for Codelco GARCH-based models outperform classical methods, and the GJR-GARCH method with the t distribution provided the best VaR estimates. Freeport-McMoRan Classical methods except RiskMetrics have poor performance in VaR estimation (Tables 7 and 8 and Figures 9, 10 and 11). GJR-GARCH and EGARCH with the normal and the t distributions provided the best results. In 99% VaR estimation, GJR-GARCH and EGARCH with t distribution outperformed other methods. GJR-GARCH and EGARCH with the normal distribution have shown better results as compared to others in 95% VaR calculation. Generally, all the GJR-GARCH and EGARCH-based VaR methods showed better performance than other methods. BHP Billiton All evaluation s excluding GARCH with the normal distribution showed poor results (see Tables 9 and 10). In estimating 99% VaR HS, unconditional approaches and GARCH-based methods have only one or two violations less than allowed 5. On the other hand, the number of the violations in RiskMetrics is 11, which is inappropriately large. Only estimates from GARCH with the normal distribution have adequate number of violations and its violation ratio is equal to 0.8. The situation with 95% VaR is quite different. In this case, all the GARCH-based estimates provide good results, within them GARCH, IGARCH and GJR-GARCH methods have generated better VaR estimates. See Figures 12, 13 and 14 for the visual comparison of the methods. Xstrata It is obvious from the results (Tables 11 and 12) that GARCH-based methods significantly outperform other methods. RiskMetrics give reasonable results but it still lacks accuracy in performing VaR. Almost all the GARCH models have outstanding performance results in both 99% and 95% VaR estimates. For example, 99% VaR estimates computed from the EGARCH with the normal distribution have 5 exceedences, equal to the expected number of violations thus giving the perfect violation ratio. GARCH and IGARCH with the normal 24