Value-at-Risk Estimation using Multivariate
Volatility and Copulas
Master Thesis Actuarial Studies August 2019
Value-at-Risk Estimation using Multivariate Volatility and
Copulas
Zakiatul Wildani August 29, 2019
Abstract
Contents
1 Introduction 3
2 Methodology 8
2.1 The univariate GARCH model . . . 8
2.1.1 Extension of the GARCH Model . . . 11
2.2 Copulas . . . 12
2.2.1 The Elliptical copulas . . . 13
2.2.2 The Archimedean Copulas . . . 14
2.3 The copula-GARCH model . . . 16
2.3.1 Parameter Estimation . . . 17
2.4 The Dynamic Conditional Correlation (DCC) model . . . 18
2.5 Summary of the approach . . . 20
3 Data 23 3.1 Preliminary analysis . . . 23
4 Empirical Results 28 4.1 The univariate GARCH model . . . 28
4.2 The copula-GARCH model . . . 29
4.2.1 Rank correlations . . . 30
4.2.2 Estimation of copula parameters . . . 31
4.2.3 Tail dependence . . . 33
4.3 The DCC model . . . 35
4.4 Estimating VaR and back-testing . . . 39
4.5 Implications . . . 46
1
Introduction
Measuring the risk associated with financial assets nowadays is considered as important as measuring the return. This is possibly due to the recurring crisis that the market wit-nessed in recent decades. The financial crisis of 2008 is considered to be the most severe financial crisis since the 1930s. It began as a consequence of the massive failures in the U.S. subprime mortgage and bankruptcy of the traditional investment banks, Lehman Brothers and Merrill Lynch. Many of the financial disasters in this period were caused by an unsatisfactory risk management system. Therefore, the BCBS (Basel Committee on Banking Supervision) requires financial institutions to hold a minimum amount of capital to cover potential losses in the future. In the 1990s, J. P. Morgan Bank in-troduced Value-at-Risk (VaR) as a risk measure that can summarise risk in a single number. Currently, this risk measure has become a worldwide benchmark concerning risk estimation (Hull, 2015). VaR estimation offers considerable information regarding the maximum of loss under given probability over a certain period. Therefore, by using this information, financial institutions can decide how much capital should be reserved to avoid default in the future. Furthermore, VaR offers information on the performance of the investment and gives a better insight into financial planning (Jorion (2001); Hull (2015); McNeil et al. (2015)). Despite the large amount of shortcomings compared to other risk measures, the majority of investors and financial institutions still consider it to be one of the benchmarks in managing risk because of its simplicity.
Jorion (2001), provided an introduction into VaR and its estimation. The first classi-cal works in VaR methodology identify three estimation concepts; specificlassi-cally, historiclassi-cal simulation, variance-covariance approach and Monte-Carlo simulation. However, all of these methods cannot describe the volatile behaviour of the time series. Financial mar-kets exist with high (low) volatility accompanied by high (low) volatility, which means volatility clustering (Sheikh and Qiao (2009); McNeil et al. (2015)). In that case, es-timating VaR with the standard approach will lead to under or overestimation. An overestimation of VaR implies that the financial institution has to hold a relatively large amount of capital aside, which could lead to lower profit. Conversely, an underestima-tion of VaR indicates that the financial instituunderestima-tion has significant exposure to the risk, which could result in default. That is how important it is for the financial institution to use an accurate model to estimate VaR.
a better decision in various areas, such as portfolio selection and hedging.
The VaR estimation of portfolio is extremely challenging due to the complexity of the joint distribution and the dependence structure (Huang et al., 2009). The traditional approach for estimating VaR in the multivariate context assumes that the joint distribu-tion is known, such as the most commonly used normality joint distribudistribu-tion. However, such multivariate distribution does not provide a satisfactory result due to the presence of skewness and is heavy-tailed in the financial data. Besides, the linear correlation is insufficient in describing more complex dependence structure for the reason that stock index returns are more highly correlated during volatile markets and market downturns (Longin and Solnik (2001); Ang and Chen (2002)). Furthermore, Embrechts et al. (2002) identified the limitation of linear correlation to model the dependence structure across financial markets. The problem originated from the normality of the joint distribution and linear correlation of the portfolio could cause inadequate VaR estimate. Therefore, it is essential to find an accurate model to estimate VaR of portfolios and verify the predictive ability prior to making a decision.
Problem formulation
This study focuses on comparing the predictive ability of different VaR models for port-folio returns in purpose for finding the ”best” model to estimate VaR of portport-folios. Furthermore, the comparison is based on the back-testing tests proposed by Kupiec (1995) and Christoffersen (1998).
Literature reviews
Estimating VaR as a risk indicator has been applied in risk management by corporate treasurers and fund managers as well as by financial institutions for around a decade. VaR has a prominent role in the Basel regulatory framework and has also been influen-tial in Solvency II (McNeil et al., 2015). Simply, VaR is the maximum losses that an investment can reach in a given period and certain confidence level.
All of the VaR estimation methods that we have presented so far assume that the volatil-ity is constant, while in fact, it varies over time. Modelling the volatilvolatil-ity of returns has attracted considerable attention ever since the introduction of the Autoregressive Condi-tional Heteroskedasticity (ARCH) and Generalized ARCH (GARCH) models developed by Engle (1982) and Bollerslev (1986). The (G)ARCH models have become famous in the analysis of time series data, especially when there is an apparent clustering in the volatility. This situation is often referred to as conditional heteroscedasticity given that the overall series is assumed to be stationary, while the variance may be time-dependent.
The article by Abad and Benito (2013), provided a detailed comparison of VaR meth-ods. Particularly, parametric, historical simulation, Monte Carlo simulation, as well as extreme value theory. The findings showed that the best model to estimate VaR is the asymmetric GARCH model. Orhan and K¨oksal (2012), compared a comprehensive list in relation to 16 specifications of GARCH models in estimating VaR. They gathered data pertaining to stock market indices from both emerging (Brazil and Turkey) and devel-oped (Germany and the US) markets throughout the global financial crisis. The result shows that the GARCH model outperforms other models in estimating the volatility of out-of-sample forecasts.
Furthermore, as the financial volatilities move across time and markets, it is often a great interest to consider several markets when measuring the risk. One can consider the multivariate GARCH models as an extension of the univariate GARCH models when involving a portfolio returns. The most obvious application of the multivariate GARCH models is a study of the relationship between the volatilities of several markets. Bauwens et al. (2006), presented several reviews on existing multivariate GARCH models, such as generalisations of the univariate standard GARCH model (e.g., Vector Error Correction (VEC) and BEKK), linear combinations of univariate GARCH models (e.g., orthogonal GARCH (O-GARCH) and Generalized Orthogonal GARCH (GO-GARCH)), nonlin-ear combinations of univariate GARCH models (e.g., Constant Conditional Correlation (CCC) model, Dynamic Conditional Correlation (DCC) model and General Dynamic Covariance (GDC) model). (see Silvennoinen and Ter¨asvirta (2009); Tsay (2013); Mc-Neil et al. (2015) for more reviews on the multivariate GARCH models).
multivari-ate modelling and describing the dependence structure between several stock indices in financial risk management (see Li (2000); Patton (2001); Patton (2006); Jondeau and Rockinger (2006); Cherubini et al. (2004); McNeil et al. (2015)). In a sense, every joint distribution function for a random vector can be divided into two parts: the marginal distributions and the dependence structure. The copula approach represents the depen-dence part, which connects the joint distribution and the marginal distributions.
The use of copula theory in finance application has become well known due to Li (2000) who employed the Gaussian copula in Collateralized Debt Obligation (CDO) pricing. It is termed the Gaussian copula because it communicates the dependence structure in precisely the same way as the Gaussian distribution does. Furthermore, A growing literature on copula has attracted researchers to combine copula with volatility models for economic and financial time series in order to present a more adequate model to re-place classical models. Patton (2001) and Jondeau and Rockinger (2006) have proposed the copula-GARCH models. These models are specified by the GARCH models for the conditional variances, marginal distributions for each series (e.g., t-distributions) and a conditional copula.
Patton (2001), discussed an extension of copula to allow for modelling of the time-varying joint distribution of the Deutschemark-U.S dollar and the Yen-U.S dollar exchange rate returns. The finding indicates that the conditional dependence is asymmetric. The Mark and the Yen are more correlated when they are depreciating against the U.S dollar than when they are appreciating. Additionally, Jondeau and Rockinger (2006) employed a normal GARCH based copula to modelling of a portfolio consisting of four major stock indices. The results suggest that the dependency is found to be more widely affected when returns move in the same direction than when they move in opposite direction for European markets. Then, Patton (2012) delivered a brief review of more copula-based models. This article explained that the copula-copula-based models allow for marginal distributions separate from the dependence structure that connects these distributions to form a joint distribution. Copula theory offers practitioners much greater flexibility in the decomposing and estimating the joint distribution, freeing from considering only existing multivariate distributions.
Hotta et al. (2008), proposed a method for estimating the VaR of portfolio based on a copula. Each returns is modelled by the ARMA-GARCH models with the joint dis-tribution of innovations modeled by a copula. Their finding showed that the proposed model outperforms other traditional models. Furthermore, Huang et al. (2009) em-ployed the combination of conditional copula and the GARCH model to estimate VaR of two assets; specifically, NASDAQ and TAIEX. The empirical results show that the copula-GARCH model captures VaR more successfully compared to the traditional mod-els (including the historical simulation method, variance-covariance method, exponential weighted moving average method). Correspondingly, Aloui et al. (2013) used the copula-GARCH approaches to investigate the comovement between daily crude oil prices and five U.S dollar exchange rates. The finding reveals that the copula-based models im-prove the accuracy of VaR forecast. In addition, Sampid and Hasim (2018) proposed a multivariate copula-GARCH model to estimate VaR of portfolio in the banking sector of selected European countries; namely, Germany, the United Kingdom (UK), Sweden, France, Italy, Spain and Greece. The results also show that the copula-based approach provides better VaR estimates than the common methods currently used.
All of the presented literature above find that VaR estimation under the copula-GARCH models outperforms the conventional models. Therefore, it generates the question con-cerning the predictive ability of these models for VaR estimation of portfolios compared to the conventional multivariate GARCH models according to the back-testing test.
Research question
The research question is formulated as follows
What are the differences in the predictive ability of the copula-GARCH VaR esti-mates for portfolio in different continents compared to the conventional multivari-ate GARCH, according to the back-testing test?
Thus, this study aims to contribute to the literature on multivariate volatility modelling by incorporating the copula theory into the GARCH models to estimate VaR of portfo-lios. Besides, this study extends the scope of the analysis to the selected stock indices from different continents. The level of interaction and interdependence between the stock index returns in each continent have an important consequence in terms of portfo-lio diversification and asset allocation. Therefore, investment funds managers, corporate treasurers, or financial institutions can use the results obtained from this study to im-prove their risk management.
2
Methodology
This thesis applies the copula-GARCH models to estimate VaR of portfolio and sub-sequently check their predictive ability based on the back-testing test. The traditional approaches for estimating VaR of portfolio assume that the joint distribution is known, such as normality of the joint distribution and the dependence structure is linear. In reality, the stock index returns are usually non-normal and linear correlation does not capture the whole dependence structure as the stock index returns are more highly cor-related during volatile markets and market downturns. As identified by Embrechts et al. (2002), the use of linear correlation to model the dependence structure shows many dis-advantages particularly when we leave the multivariate normal distribution. Therefore, restriction on the joint distribution and linear correlation of portfolio returns might de-crease the performance of VaR. In order to overcome these drawbacks, this study resorts to the copula theory. Combination of the GARCH models and copulas allow the joint distribution to be free from normality assumption. Moreover, it enable us to capture nonlinearities in the stock indices relationship as well as some empirical stylized facts of returns distribution such as heteroscedasticity and autocorrelation, while avoiding the drawbacks of linear measures of interdependence (e.g.,, Pearson correlation).
Before proceeding to the analysis, this section provides an explanation about the theoret-ical background of the time series model starting with basic properties of the univariate GARCH model. The theory in this section is adopted from McNeil et al. (2015), if it is not mentioned otherwise.
2.1 The univariate GARCH model
The univariate GARCH model is employed to modelling the volatility of financial returns. This model accounts explicitly for heteroskedasticity of the stock returns and is able to describe the volatile behaviour of the time series exceedingly well. Some concepts of time series analysis, such as stationarity and white noise are required for constructing the GARCH model. Consider (Xt)t∈Z is a discrete-time stochastic process for a single
risk factor with the first and second moments are defined by
µ(t) = E(Xt), t ∈ Z, (1)
γ(t, s) = E((Xt− µ(t))(Xs− µ(s))), t, s ∈ Z, (2)
where the first moment µ(t) is called the mean function and the second moment γ(t, s) is the autocovariance function.
Stationarity. The time series model is considered a stationary process if it fulfils one or both of the following two types of stationarity; specifically strict stationarity and covariance stationarity.
if the distribution of two time series are the same and denoted by
(Xt1, ..., Xtn)
d
= (Xt1+k, ..., Xtn+k), (3)
for all t1, ..., tn, k ∈ Z, n ∈ N, and= signifies both time series have the same distribution.d
Definition 2.2 (covariance stationarity) The process (Xt)t∈Z is a covariance stationary
if it satisfies
µ(t) = µ, t ∈ Z, (4)
γ(t, s) = γ(t + k, s + k), t, s, k ∈ Z. (5) Generally speaking, both these definitions imply that the time series behave similarly in any two equally-spaced time periods. The mean, variance or the covariances structure do not change over time. Note, it can be confirmed that a strictly stationary time series with finite variance is termed covariance stationary. The definition of covariance sta-tionarity indicates that the autocovariance function between Xt and Xs can be defined
as γ(t − s, 0) = γ(s, t) = γ(s − t, 0), which suggets that the covariance only depends on |s − t|, which is known as the lag. We can now write the autocovariance function as a function of one variable: γ(h) := γ(h, 0), for all h ∈ Z. Thus, using this notation, we can define the autocorrelation function of the covariance stationary process.
Definition 2.3 (autocorrelation function) For the covariance stationary process (Xt)t∈Z,
the autocorrelation function (ACF) denoted by ρ(h) can be defined as
ρ(h) = ρ(Xh, X0) = γ(h)/γ(0), (6)
where γ(h) is the autocovariance between Xtand Xsthat depends on the lag |s − t| and
γ(0) denotes the Var(Xt).
White noise process. The white noise process is the basic building blocks for creating a time series model. The process (Xt)t∈Z is a white noise process if it satisfies
E(Xt) = 0, (7)
E(Xt2) = σ2, (8)
E(XtXt−j) = 0, t, j ∈ Z. (9)
This process is denoted as WN(0, σ2). An example of a WN process is known as strict white noise process if it is a sequence of independent identically distributed (i.i.d) with finite variance random variables, that is denoted as SWN(0, σ2). The SWN process will be the basic building block of the GARCH models. It represents the innovations (the standardized residuals) of the model. Finally, we can define the standard GARCH model as follows
GARCH(p, q). Let (zt)t∈Z be the sequence of independent and identically distributed
(i.i.d) random variables characterised by mean zero and unit variance, denoted by SWN(0,1). The process (Xt)t∈Z is a GARCH(p, q) process if it is strictly stationary
and if the following equations hold
Xt= µt+ t, (10) t= σtzt, (11) σ2t = α0+ p X i=1 αi2t−1+ q X j=1 βjσ2t−j, (12)
where α0, αi, βj > 0 represent the GARCH parameters, i = 1, ..., p and j = 1, ..., q
de-note the number of lags. Furthermore, the mean and volatility are dede-noted by µt and
σt, respectively, whilst t represents the innovations.
Eqn (12) shows that the volatility σ2t depends on the past volatilities as well as past innovations. It means that the model utilises the previous variance prediction to obtain the next one. The most popular GARCH model in applications is the GARCH(1,1) model because this model tends to be persistent and accounts for a more parsimonious model. To make the GARCH model more realistic, we need α1+ β1 < 1. In that case,
the variance of covariance stationary process is given by α0/(1 − α1− β1).
A complete GARCH analysis requires to not only specify and estimate the model but also to validate it. One can do this by analysing the innovations (standardized residu-als). Once the model has been estimated, we can verify the assumption of the SWN(0,1) process of the standardized residuals by constructing the correlogram ACF. Beside the correlogram, we can use the Ljung-Box Q(h) test.
an extension of the GARCH-type model that can deal with both autocorrelation and het-eroscedasticity; the so-called ARMA(p1,q1)-GARCH(p2,q2) model. Besides, the GARCH
model is symmetric, i.e., past negative and positive shocks have the same impact on cur-rent volatility. In reality, the sign matters. Thus, we also propose the GARCH model with leverage, also known as the GJR-GARCH model. These two extensions will be considered in this study.
2.1.1 Extension of the GARCH Model
A great deals of variants and extensions of the univariate GARCH models have been proposed in the literature; see Ter¨asvirta (2009). In this study, we consider two exten-sions of the GARCH model. The first extension is the ARMA-GARCH model, whilst the second extension is the GARCH with leverage effect (the GJR-GARCH) model.
ARMA(p1, q1)-GARCH(p2, q2). Consider (zt)t∈Zbe a SWN(0,1) process. The process
(Xt)t∈Z is an ARMA(p1, q1)-GARCH(p2, q2) process if it is covariance stationary and if
it satisfies Xt= µt+ t, (13) µt= µ + p1 X i=1 φit−i+ q1 X j=1 θjt−j, (14) σt2= α0+ p2 X i=1 αi2t−i+ q2 X j=1 βjσ2t−j, (15)
where α0> 0, αi, βj ≥ 0, for i = 1, ..., p2, j = 1, ..., q2, and Pi=1p2 αi+Pqj=12 βj < 1.
The advantage of the ARMA-GARCH model is the ability to describe two important stylized facts relating to financial time series, namely autocorrelation and heteroskedas-ticity. Furthermore, the low ordered of the ARMA-GARCH model is more preferred than the high order in terms of its effectiveness though it remains parsimony.
GARCH with leverage. One of the drawbacks of the standard GARCH model is that the impact of positive and negative shocks on volatility at time t are the same (symmet-ric). However, according to economic theory, negative news tends to affect volatility more than positive news (asymmetric), especially for stock index returns. This phenomenon is termed the leverage effect. Glosten et al. (1993), proposed the GJR-GARCH model which encounters positive and negative shocks on the conditional variance via the use of the indicator function I{t−i≤0}. The volatility can be written as follows
where γirepresents the leverage term and I{t−i≤0}denotes the indicator function, which
is 1 if t−i≤ 0 and 0 otherwise.
The parameters of the ARMA-GARCH and the GJR-GARCH model can be estimated by the maximum likelihood estimation (MLE). After the univariate GARCH model has been defined, we can subsequently proceed to construct the copula-GARCH model when considering the portfolio returns by combining the copula functions with a GARCH-type model. To assist with the understanding of copula theory, we provide theoretical background of copula in the section below.
2.2 Copulas
The principal feature of a copula function is its ability to construct the joint distribution by separating the marginals and the dependence structure. Bivariate distributions, as well as distributions in higher dimensions are possible. In this section, we discuss the basic properties of copula and some copula models to be used in this study.
Definition 2.4 (McNeil et al., 2015) A d-dimensional copula is a distribution function in the [0, 1]d with standard uniform marginals. The notation C(u1, ..., ud) is given to
describe for the joint distribution functions that are copulas that satisfy the following properties
1. C(u1, ..., ud) is increasing in each component ui.
2. C(1, ..., 1, ui, 1, ..., 1) = ui for all i ∈ {1, ..., d} and ui∈ [0, 1].
3. For all (a1, ..., ad), (b1, ..., bd) ∈ [0, 1]dwith ai ≤ bi we have
2 X i1=1 ... 2 X id=1 (−1)i1+...+idC(u 1i1, ..., udid) ≥ 0, (17)
where uj 1= aj and uj 2= bj for all j ∈ {1, ..., d}.
The second condition implies that the marginal distributions are transformed to standard uniform, whereas the third condition ensures that for a random vector (U1, ..., Ud)0 has
distribution function C, subsequently, we have P (a1 ≤ U1 ≤ b1, ..., ad ≤ Ud ≤ bd) ≥ 0.
Sklar (1959) defined and provided some fundamental properties of copula.
Theorem 2.1 (Sklar, 1959) For the joint distribution function F (x1, ..., xd) with the
marginals F1, ..., Fd, there exists for all real (x1, ..., xd) the d-dimensional copula C :
[0, 1]d→ [0, 1] such that
F (x1, ..., xd) = C(F1(x1), ..., Fd(xd)). (18)
marginals F1, ..., Fd. 1 The first statement of Sklar’s theorem shows us that any
multi-variate distribution function contains copula. Meanwhile, the converse statement demon-strates that a copula can be utilised in combination with arbitrary marginals to construct a joint distribution function. Both of these statements are important in this study. The Eqn. (18) can be rewritten in terms of u = (u1, ..., ud) ∈ [0, 1]das follows
C(u1, ..., ud) = F (F1−1(u1), ..., Fd−1(ud)). (19)
The density of the joint distribution function F (x1, ..., xd) denoted by f (x1, ..., xd) is
given by f (x1, ..., xd) = c(F1(x1), ..., Fd(xd)) d Y i=1 fi(xi), (20)
where fi(xi) is the marginal densities of Fi and c is the density function of the copula
given by c(u1, ..., ud) = f (F1−1(u1), ..., Fd−1(ud)) Qd i=1fi(Fi−1(ui)) , (21)
where Fi−1 is the quantile function of the marginal distribution functions. Let xt =
(x1,t, x2,t)0, then, from (20), the likelihood function of copula can be defined as
LL = T X t=1 ln f12(x1,t, x2,t; θ) = T X t=1 ln f1(x1,t; θ1) + ln f2(x2,t; θ2) + ln c(F1(x1,t; θ1), F2(x2,t; θ2); θ), (22)
where T is the number of observations. Therefore, it can be seen from Eqn. (22) that the log-likelihood function of copula can be separated into two parts, the first two terms related to the marginals and the last term related to the copula. It has the conse-quence that the joint distribution is exclusively determined by the marginal distributions, whereas the dependence structure is determined by the copula. Two copula families will be considered in this study, namely the the Elliptical copulas and the Archimedean copulas.
2.2.1 The Elliptical copulas
The Elliptical copulas are derived from the Elliptical distribution. The most common are the Gaussian and Student-t copula.
1
The Gaussian copula. The Gaussian copula is a copula of the multivariate normal distribution which is defined by the following form:
CPGa(u) = P (Φ(X1) ≤ u1, ..., Φ(Xd) ≤ ud)
= ΦP(Φ−1(u1), ..., Φ−1(ud)), (23)
where Φ denotes the standard univariate normal distribution functions and ΦP denotes
the joint distribution function with linear correlation P . The probability density function of the Gaussian copula can be represented as
cGaP (u1, ..., ud) = 1 |P |1/2 exp −1 2ζ 0(P−1− I)ζ , (24)
where ζ = (Φ−1(u1), ..., Φ−1(ud))0 is the inverse of the univariate normal distribution
functions and I is the (d × d) identity matrix.
The Student-t copula: In the same way we construct the Gaussian copula from the multivariate normal distribution, we can extract an implicit copula from any other distri-bution with continuous marginal distridistri-bution functions. For example, the d-dimensional t-copula from the multivariate t−distribution can be described as
Cv,Pt (u1, ..., ud) = tv,P(t−1v (u1), ..., t−1v (ud)), (25)
where tv is the univariate Student-t distribution functions, tv,P is the joint distribution
function where P is the correlation matrix, and v is degrees of freedom. The density function has the form
ctv,P(u1, ..., ud) = |P |−1/2 Γ(v+d2 ) Γ(v2) Γ(v2) Γ(v+12 ) !d (1 +1vζ0P−1ζ)−v+22 Qd j=1 1 + ζ2 j v !−v+12 , (26)
where ζ = (t−1v (u1), ..., t−1v (ud))0 is the inverse of univariate Student-t distribution.
The Gaussian and the Student-t copula are implied by well-known multivariate distri-butions, such that this copula family is the most popular in finance literature due to the ease with which they can be implemented. However, they do not have simple closed forms. Several copulas do have simple closed forms, namely the Archimedean copulas. The Gumbel copula, the Clayton copula, and the Frank copula belong to this copula family and has proved useful for modelling portfolio returns.
2.2.2 The Archimedean Copulas
applications are the Gumbel copula (Gumbel, 1960), the Clayton copula (Clayton, 1978), and the Frank copula (Frank, 1979) which are constructed according to
C(u1, ..., ud) = ϕ−1(ϕ(u1) + ... + ϕ(ud)), (27)
where ϕ is called the copula generator and ϕ−1 is the generator inverse. A different generator will induce a different copula in the family of the Archimedean copulas.
The Gumbel copula: The generator for the Gumbel copula is ϕ(u) = (− ln(u))α and the generator inverse is ϕ−1(x) = exp(−x)1/α for 1 ≤ α < ∞. The distribution function of the Gumbel d-copula takes the form
CGumbel(u1, ..., ud) = exp − " d X i=1 (− ln ui)α #1 α! . (28)
The Clayton copula: The generator for the Clayton copula is given by ϕ(u) = (u)−α−1 and the generator inverse is ϕ−1(x) = max((αx + 1)−1/α, 0) for α > 0, which yields the Clayton d-copula with the distribution function is given by
CClayton(u1, ..., ud) = " d X i=1 u−αi − d + 1 #−α1 . (29)
The Frank copula: The generator for the Frank copula is ϕ(u) = lnee−αu−α−1−1
and the generator inverse is ϕ−1(x) = −α1 ln(1 + e−x(e−α− 1)) for α ∈ R, which will generate the distribution function of the Frank d-copula represented by
CF rank(u1, ..., ud) = − 1 αln ( 1 + Qd i=1(e −αui− 1) (e−α− 1)d−1 ) . (30)
The Archimedean copulas are advantageous in relation to risk management analysis for the reason that they capture asymmetric dependence between lower and upper tail de-pendence. The primary reason why we employ the Archimedean copulas in the modelling portfolio returns is because it enables us to model a wide range of different dependence structures instead of linear correlation. Two useful dependence measures described by copula are rank correlations and the tail dependence. The most commonly used rank correlations are the Kendall’s tau and the Spearman’s rho. Both of these correlations will be considered in this study. Furthermore, the coefficient of tail dependence can be used as a measure of the tendency of markets to crash or boom together (Karmakar, 2017). Let X1 and X2 be the random variables. The coefficients of lower and upper tail
dependence denoted as λl and λu can be expressed as
where F1−1 and F2−1 are the marginal quantile functions. If λl and λu are positive, then
there is lower or upper tail dependence, otherwise it is independence. The higher the dependence value of λl or λu, the stronger the dependence in extreme events. Different
copulas exhibit different tail dependence. For instance, the Gaussian copula is asymptot-ically independent in both tails (λl = λu = 0), while the Student-t copula exhibit both
upper and lower tail dependence. For the Archimedean copulas, the Gumbel copula captures upper tail dependence and is limited to positive dependence, while the Clayton copula exhibits greater dependence in the lower tail than in the upper tail. Furthermore, the Frank copula accommodate the full range of dependence, either positive dependence or negative dependence for marginals exposed to weak tail dependence.
2.3 The copula-GARCH model
This section describes the combination of the GARCH model and copula as an alterna-tive for modelling portfolio returns. This model is defined similarly as the traditional GARCH models, however, the joint distribution function of the innovations and de-pendence structure are generated by using copula function. In such a context, the dependence structure can easily considered conditional and time varying (Jondeau and Rockinger, 2006). By doing so, this model allows for different types of univariate distri-butions for marginal distridistri-butions and nonlinear dependence structure with a rich variety of available copula families. More clearly, we first use the univariate GARCH to model the volatility of the stock index returns, and then we incorporate copula functions into the standardized residuals (innovations). The innovations from the univariate GARCH models are thus far more suitable in relation to copula estimation than the raw return series because they are approximately i.i.d (Aloui et al., 2013). Therefore, the estima-tion and testing procedure for the copula parameter will not experience statistical biases which appear when treated the raw returns as i.i.d variables which actually they are certainly not. In addition, the innovations provide more accurate marginal distribution.
Empirical evidence shows that the copula-GARCH model can be quiet robust in estimat-ing VaR of portfolio (Hotta et al. (2008), Huang et al. (2009), Aloui et al. (2013)). The structure of the copula-GARCH model in this study is adopted from Patton (2006) and Aloui et al. (2013) approach. Patton (2006) defined the conditional copula and intro-duced the extension of Sklar’s theorem, which allows us to combine the copula with the GARCH model. The copula theory is defined in the static setting. By combining with the GARCH model, we have to deal with the dynamic setting. For any d-dimensional conditional distribution function F (xit, ..., xdt|Ft−1) with conditional marginal
distribu-tions F1(x1t|Ft−1), ...., Fd(xdt|Ft−1), there exists a d-dimensional conditional copula C
such that
Ft(xit, ..., xdt|Ft−1) = Ct(F1(x1t|Ft−1), ...., Fd(xdt|Ft−1)|Ft−1). (33)
The conditional mean E(xit|Ft−1) = µit, where Ft−1 is the information that is available
defined by: 2
xit= µit+ it, (34)
it= σitzit, (35)
σit2 = ω + α12t−1+ βσ2it−1. (36)
The copula subsequently assumes the dependence structure of the marginals with con-stant shape parameter θ. The conditional density function of copula based on (21) at time t is given by ct(u1t, ..., udt|Ft−1) = ft(F1−1(u1t|Ft−1), ..., Fd−1(udt|Ft−1)|Ft−1) Qd i=1fi(Fi−1(uit|Ft−1)|Ft−1) , (37)
where uit = Fit(xit|Ft−1) is the Probability Integral Transformed (PIT) of each series
through its conditional distribution Fit estimated via the first stage GARCH process;
Fi−1(uit|Ft−1) is the quantile transformation of the uniform marginals. Then, ft(.|Ft−1)
represents the conditional multivariate density function, and fi(.|Ft−1) defines the
condi-tional marginal density function of the ithsample. The conditional joint density function of the copula-GARCH model is given by
ft(x1t, ..., xdt|Ft−1) = ct(u1t, ..., udt|Ft−1) d
Y
i=1
fi(Fi−1(uit|Ft−1)|Ft−1). (38)
Therefore, by considering all observations, t = 1, ..., T , we obtain the conditional likeli-hood function which takes the form
L(θ, α) = T X t=1 ln ct(u1t, ..., udt|Ft−1; α, θ) + T X t=1 d X i=1 ln fi(Fi−1(uit|Ft−1; α)|Ft−1) (39) = Lc(θ, α) + Lf(α),
where θ and α are the copula and the marginal parameter, respectively.
2.3.1 Parameter Estimation
The parameter of copula can be estimated by the maximum likelihood estimation (ML). However, this method may be computationally challenging for some high dimensional cases (Karmakar, 2017). The Eqn. (39) shows that the likelihood function can be divided into two parts, the marginals part and the copula part. Therefore, this separation allows us to perform the alternative method, namely the Inference Function for Margins (IFM) proposed by Joe and Xu (1996), which estimate the parameter into two stages and proceed as follows
2
1. Estimate all the marginal parameters independently by maximising the log-likelihood function of Lf(α).
2. Given the estimated parameters from step 1, maximise the log-likelihood function of the copula Lc(θ, α) to estimate the copula parameter θ.
This 2-step estimation procedure is less computationally intensive than just 1-step proce-dure. Since our models involve a large number of parameters, we adopt the IFM method for our estimation.
2.4 The Dynamic Conditional Correlation (DCC) model
In addition to the copula-GARCH model, we also perform the traditional multivariate GARCH model for modelling portfolio returns, where in this case we perform the DCC model. The advantage of the DCC model is the fact that the correlation matrices are easier to handle than the covariance matrices (Tsay, 2013). Furthermore, this model allows the correlation of stock index returns to evolve dynamically, thus, it reflects the current market condition. Engle (2002) and Tse and Tsui (2002), proposed two types of DCC models. This thesis employs the DCC model by Engle (2002) because this model satisfies the positive definite of the conditional volatility matrix of stock index returns (Tsay, 2013).
The idea of the DCC model is to divide the composition of the conditional covariance matrix into two steps. The first step is modelling the volatility part, whereas the second step is modelling the dynamic dependence of conditional correlation. The Dynamic Conditional Correlation (DCC) model generalises the Constant Conditional Correlation (CCC) model to allow conditional correlations to evolve dynamically. Most explanation for the DCC models are adopted from Tsay (2013) and Ghalanos (2015). The DCC model focuses on specifying the conditional correlation matrix while allowing volatilities to be described by the univariate GARCH models, thus simplifies the estimation process (Ghalanos, 2015). In the DCC model, consider N -dimensional innovations zt to the
stock index returns series Xt. Then, the covariance matrix Σt = [σij,t] of Xt can be
decomposed into
Σt= DtPtDt, (40)
where the conditional correlation matrix Ptis given by
Pt= D−1t ΣtDt−1, (41)
where Dt is the diagonal matrix defined by Dt = diag(
√
σ11,t, ...,
√
σN N,t) of the N
conditional volatilities of the stock index returns. The conditional variances σii,t can
models as follows 3 σt= ω + p X i=1 Ait−i t−i+ q X j=1 Bjσt−j, (42)
where ω ∈ Rn, Ai, and Biare N × N diagonal matrices, and is the Hadamar operator.
In order to guarantee the positivity of the covariance matrix Σt, the correlation matrix
Pt must be positive definite, and the element of ω and the diagonal elements on Ai
and Bi are positive. Let zt = (z1t, ..., zN t)0 be the marginally innovation vector, where
zt = Dt−1t. The first type of the DCC model is proposed by Engle (2002) and it is
defined as
Qt= (1 − θ1− θ2) ¯Q + θ1Qt−1+ θ2zt−1z0t−1, (43)
Pt= JtQtJt, (44)
where ¯Q is the unconditional covariance matrix of the standardized errors zt, θi are
non-negative real numbers satisfy 0 < θ1+ θ2 < 1 to ensure stationarity and positive
definiteness of Qt, and Jt= diag(q11,t−1/2, ..., q−1/2N N,t) where qii,tis the (i, i)th element of the
Qt known as the normalization matrix.
The second type of the DCC model is proposed by Tse and Tsui (2002) can be written as
Pt= (1 − θ1− θ2) ¯P + θ1Pt−1+ θ2ψt−1, (45)
where ¯P is the unconditional correlation matrix of zt, θi are non-negative real numbers
and satisfying 0 < θ1 + θ2 < 1, and ψt−1 is the local correlation matrix depending on
{zt−1, ..., zt−m} for some integer m and it can be defined as
ψii,t−1=
Pm
i=1zi,t−izi,t−i
q (Pm i=1z2i,t−i)( Pm i=1z2i,t−i) , (46)
where the choice of m can be considered as smoothing parameter.
From Eqns. (43) to (45), these two types of the DCC models are quite similar. The dif-ference is how the local information at time (t − 1) is utilised. The DCC model proposed by Engle (2002) employs zt−1only so that the correlation matrix must be re-normalised
at each time index (t − 1). Meanwhile, the DCC model of Tse and Tsui (2002) does not require re-normalization, instead, this model applies local correlation to update the conditional correlation matrix and requires the choice of m. The advantage of the DCC model employed by Engle (2002), is that this model guarantees the conditional volatility matrix Σt remains positive definite, while the model proposed by Tse and Tsui (2002)
3The GARCH models are not restricted to one specific model, so that various GARCH models can
depends on the choice of m. The choice of m > N ensures the local correlation matrix ψt−1 as well as Pt to be positive definite.
The other advantages regarding the DCC models are its simplicity and it is extremely parsimonious as they only use two parameters θ1and θ2to govern conditional correlations
regardless of the number of stock indices N involved. Ghalanos (2015) described that the log-likelihood function (LL) of the DCC model can be separated more clearly into volatility and correlation component by adding and subtracting zt0zt= 0tD
−1
t D
−1 t t, and
it has the form:
LL = 1
2(N log(2π) + 2 log |Dt| + log |Pt| + z
0 tP −1 t z 0 t) = 1 2 T X t=1 (N log(2π) + 2 log |Dt| + 0tD −1 t D −1 t t) − 1 2 T X t=1 (z0tzt+ log |Pt| + zt0P −1 t zt0) (47) = LLV(θ1) + LLP(θ1, θ2),
where LLV(θ1) is the volatility component with parameter θ1 and LLP(θ1, θ2) is the
correlation component with parameters θ1 and θ2. For the reason of parsimony of the
model and interpretability of the variables as in the univariate case, we also employ low ordered the DCC models.
2.5 Summary of the approach
Up until this point, we have discussed theoretical background of the GARCH model, the copula theory, the copula-GARCH model, and the DCC model. To sum up, the following steps describe the estimation and the back-testing procedure of VaR portfolio using the copula-GARCH models and the multivariate DCC models.
information criterions can be defined as AIC = −2LL T + 2k N, (48) BIC = −2LL T + k ln(T ) T , (49) SIC = −2LL T + ln T + 2k T , (50) HQIC = −2LL T + 2k ln(ln(T )) T , (51)
where T is the number of observations, k is the number of parameters, and LL is the value of the likelihood function. We choose the best univariate GARCH model that yields the lowest values of AIC, BIC, SIC and HQIC, and the highest value of LL. A complete univariate GARCH model requires to verify that the standardized residuals (innovations) should follow a SWN(0,1) process. The Ljung-Box test of the raw and the squared standardized residuals and the visualisation plot, for instance, the Q-Q plot and the correlogram ACF are conducted to recognise the presence of autocorrelation and volatility clustering.
2. Prior to incorporating copula functions in the fitted GARCH model for modelling portfolio returns, we need to specify the desired marginal distributions. A good glimpse at the histogram of the standardized residuals of fitted GARCH model for each stock index could provide some insights into what distribution could be a better fit.
3. The copula theory is then incorporated into the fitted GARCH model by ex-tracting and transforming the i.i.d standardized residuals into uniform (0,1) using probability-integral transformation in order to construct the pseudo-observations. Furthermore, we fit the copula function into the pseudo-observations and subse-quently estimate copula parameters by using two-stage estimation procedure (IFM) given by ˆ θ = argmaxθ T X t=1 ln c(ˆu1t, ..., ˆudt|Ft−1; θ). (52)
We choose the best copula for each copula family that best fits the data based on the highest value of LL and related model selection criteria such as the lowest value of AIC and BIC. Furthermore, via copula and the marginals, we generate the density and the distribution function for the joint distribution.
4. Furthermore, we do a simulation to construct T × N new matrix innovations from the copula
ˆ
Σ = {ζit}, i = 1, ..., N, t = 1, ..., T, (53)
5. Convert the daily simulated innovations (53) to the new risk factor returns by reintroducing the GARCH volatility and the conditional mean term observed in the original return series, that is
rit= ˆµi+ εit, εit= ˆσitζit. (54)
Generally speaking, we are modelling the innovations by a copula.
6. Besides, we also perform the traditional multivariate GARCH model, especially the DCC model. We assume the innovations for this model is the Student-t distribution to account for heavy-tailed of the data. Subsequently, we obtain the estimated parameters by MLE. By applying the DCC model, we obtain the correlation matrix Pt for t = 1, ..., T .
7. Given the risk factor returns, we compute the value of the global portfolio. If we assume the weight of each stock index is the same, we can compute the portfolio as an equally weighted portfolio. However, the major concern for financial institutions or individual investor is to minimize the risk of investment portfolio. Therefore, we demonstrate the portfolio returns on day t as a risk weighted portfolio. We assume the weight in individual stock index within a portfolio is w, where 0 ≤ wi ≤ 1 for
i = 1, ..., N and PN
i=1wi = 1. Consider ln v as the total amount invested in the
portfolio, xi is the total investment in the stock index i. The weight in the stock
index i is computed by wi= xi/ ln v. Therefore, the portfolio returns at time t is
given by Rpt= N X i=1 wirit. (55)
8. Therefore, the one day VaR at time t with confidence level (1 − α) where α ∈ (0, 1) is defined by
V aRt(α) = inf{s ∈ R : P (t > s) ≤ 1 − α} = inf{s ∈ R : Ft(s) ≥ α}, (56)
where Ft is the distribution function of the portfolio returns Rpt at time t. Thus,
we are 100(1 − α)% confident that in the worst case scenario, the losses on the portfolio in the period t will not be larger than VaR.
10. Compare the backtesting performance of the copula-GARCH models and the DCC models. The model is reliable for estimating VaR if the null hypothesis of correct exceedances for the unconditional coverage test or the null hypothesis of correct exceedances and independence of failures for the conditional coverage test are failed to reject. Furthermore, a model which is said to be the best model for estimating VaR refers to the one with the number of actual exceptions closest to the expected number of exceptions (Aloui et al., 2013). The exception is considered if the return is greater than VaR estimate at time t.
3
Data
We consider data obtained from nine different financial stock indices worldwide and classify them based on their continent. We choose the top three stock indices from the selected countries on three continents; specifically, Europe, Asia, and America4(see Table 1). The data consist of daily closing prices from January 3, 2005 to April 24, 2019 in order to include the 2008 global financial crisis. All data are from Datastream and consist of 3733 observations. The daily log returns (in percentage) for each stock index at time t are calculated by
rt= 100 × " log P1,t P1,t−1 , ..., log PN,t PN,t−1 # = (r1t, ..., rN t), (57)
where Pt is the closing price on time t and N represents the stock index. The reason
why log returns (log losses) are employed instead of raw returns (losses), is because they provide more meaningful and robust results.
Table 1: Stock indices classified based on their continent.
Portfolio I Portfolio II Portfolio III
DAX (Germany) Hang Seng (Hongkong) S&P 500 (USA)
CAC (France) Shanghai SE Composite (China) Brazil Bovespa (Brazil)
FTSE (UK) Nikkei 225 (Japan) DJIA (USA)
3.1 Preliminary analysis
We start the preliminary analysis by investigating the visualisation plot of each stock index returns (see Figs. 10 to 19 in the Appendix). However, in this section, we present only one stock index returns, specifically the DAX. Firstly, we explore the overall struc-ture of the returns distribution of DAX. Normal distribution is frequently observed to be a poor model for financial time series data. This can be confirmed using a histogram or Q-Q plot (quantile-quantile plot) against a standard normal reference distribution,
4
as well as a formal numerical test such as the Jarque-Bera (JB) test. The histogram and Q-Q plot of the DAX are depicted in Fig. 1. The red line in the plot refers to an estimate of the normal distribution. These plots are important in order to obtain a clear understanding of the underlying distribution of returns. From the histogram, we observe that the daily returns are far from the estimated normal distribution. Furthermore, the returns of DAX has a long and fat tail. Moreover, there is more mass around the centre than the normal distribution suggests. Therefore, it is unlikely that the returns of DAX follows normal distribution (see Figs. 10 and 11 in the Appendix for histograms of each stock index returns).
Figure 1: Histogram (left) and Q-Q plot (right) of the daily log returns of the DAX against the normal distribution (red line).
A Q-Q plot is a standard visual tool showing the relationship between the empirical quantiles of the data and theoretical quantiles of reference distribution. In Fig. 1, we show a Q-Q plot of the daily DAX returns against the normal reference distribution. We can observe that the daily log returns portrayed the inverted S-shaped curve of the points, which suggest that the empirical quantiles of the return distribution exhibit fat-ter tails than the corresponding quantiles of the normal distribution, indicating that the normal distribution is a poor model for the DAX. The empirical quantiles should lie on the straight red line if the observation follows a normal distribution (McNeil et al., 2015) (see Figs. 12 and 13 in the Appendix for Q-Q plots of each stock index returns).
volatil-ity during the remaining periods (see Figs. 14-16 in the Appendix for similar plots).
Figure 2: Daily closing prices (left) and daily log returns (right) of the DAX from January 3, 2005 to April 24, 2019.
Furthermore, in order to recognise the presence of autocorrelation as well as volatility clustering in the returns, we observe the correlogram ACF of the raw and squared re-turns. The correlogram is a graphical display for estimates of serial correlation that plots the sample autocorrelation ρ(h) against the time lag h. The correlogram ACF of the squared returns not only detects the presence of autocorrelation but also indicates the evidence of volatility clustering. That is the reason why we investigate both the raw and squared returns. Fig. 3 shows the correlogram ACF of the raw and the squared returns of DAX where the dashed blue lines indicate a 95% confidence level. In this case, more than 5% of the estimated correlation for both the raw and squared returns lie outside the dashed blue line. Thus, there is strong evidence of autocorrelation and volatility clustering in the DAX returns (see Figs. 17-19 in the Appendix for the correlogram of each stock index returns). Thus, the presence of autocorrelation and volatility clustering support the use of the AR(MA)-GARCH model for modelling all the stock index returns.
Figure 3: Correlogram of the raw (left) and the squared returns (right) of DAX. The dashed blue lines mark the standard 95% confidence interval for the autocorrelation of the process of i.i.d finite-variance random variables.
higher return and vice versa. Furthermore, the returns are far from being normally dis-tributed, as indicated by kurtosis, are well above three (leptokurtic) and the skewness for all stock indices is not zero. The negative skewness signifies that the returns are skewed to the left suggesting more frequent negative shocks, whilst the high excess kur-tosis denotes that the returns are heavy-tailed. The non-zero skewness and high excess kurtosis clearly indicate the non-normality of the returns distribution which is confirmed by the Jarque–Bera (JB) test statistics, where the null hypothesis states that the returns follow normal distribution. The findings reject the null hypothesis of normality at the 1% significance level for all stock indices.
Table 2: Descriptive and test statistics of the daily returns (in percentages) for each stock index on the continents of Europe, Asia and America from January 3, 2005 to April 24, 2019.
Index Obs. Mean Volatility Min. Max. Skewness Kurtosis JB Q(12) Q2(12)
DAX 3732 0.03 1.3 -7.4 10.8 -0.04 9.7 7043.9* 17.8 2843.6* Europe CAC 3732 0.01 1.3 -9.5 10.6 -0.04 10.2 8119.5* 40.8* 2035.7* FTSE 3732 0.01 1.1 -9.3 9.4 -0.16 11.9 12358.0* 45.8* 3126.8* Hang Seng 3732 0.02 1.4 -13.6 13.4 -0.01 13.4 16777.0* 22.7** 3210.8* Asia Shanghai SE 3732 0.02 1.6 -9.3 9.0 -0.57 7.9 3960.1* 32.2* 865.8* Nikkei 225 3732 0.02 1.4 -12.1 13.2 -0.54 11.9 12504.0* 15.2 3339.5* S&P 500 3732 0.02 1.2 -9.5 11.0 -0.37 15.3 23509.0* 70.1* 4184.8*
America Brazil Bovespa 3732 0.04 1.7 -12.1 13.7 -0.05 8.9 5427.9* 20.6*** 3182.9*
DJIA 3732 0.02 1.1 -8.2 10.5 -0.15 14.6 20834.0* 71.6* 4032.4*
Notes: JB indicates the Jarque-Bera test of normality distribution. Q(12) and Q2(12) are the Ljung-Box test statistics for autocorrelation and
Therefore, based on the visualisation plot and descriptive statistics as well as a couple of formal numerical tests, we conclude that the AR(MA)-GARCH-type models are suitable for modelling the considered stock index returns.
4
Empirical Results
This section presents the empirical result pertaining to how we construct the copula-GARCH model and the multivariate copula-GARCH model and subsequently employ it to analyse portfolio risk by estimating VaR and back-testing VaR models as recognition of the step-by-step procedures in Section 2. In this study, we employ the rmgarch and rugarch package in R, proposed by Ghalanos (2015) and Ghalanos (2017) to estimate the volatility model and subsequently estimate the copula parameters by way of using copula package.
4.1 The univariate GARCH model
Prior to incorporating copula functions in the GARCH model, we have to choose the fitted specification for the univariate GARCH model (Aloui et al., 2013). We consider the standard GARCH and the GJR-GARCH model introduced in the previous section to fit the returns data to modelling the conditional volatility. We fit all of the stock index returns to the standard GARCH model such as the ARMA(1,0)-GARCH(1,1) model and the ARMA(1,1)-GARCH(1,1) model. We also consider the volatility model with leverage effect, the ARMA(1,0)-GJR-GARCH(1,1) model and the ARMA(1,1)-GJR-GARCH(1,1) model. The innovations are assumed the Student-t or the skewed-t distribution because these distributions provide appropriate specifications for modelling the conditional heteroscedasticity of the data. We choose the low-ordered GARCH model in terms of parsimony.
Furthermore, we perform model selection for each standard GARCH and the GJR-GARCH model by testing the in-sample fit. We use information criterion such as the AIC, BIC, SIC and HQIC, as well as the log-likelihood (LL) values to choose the best model. The fitted GARCH model is indicated by the highest value of LL and the lowest values of other criterions. Tables 14 to 16 in the Appendix represent the LL, AIC, BIC, SIC and HQIC of eight standard GARCH and GJR-GARCH models for each stock index returns on the continents of Europe, Asia and America respectively.
prior to fitting copula functions or the conventional multivariate GARCH model.
For stock index returns on the continent of Asia, the returns of Shanghai SE and Nikkei provide an optimal fit under the ARMA(1,0)-GARCH(1,1) model, whereas the Hang Seng achieves a better fit with the ARMA(1,1)-GARCH(1,1) model. However, for the GJR-GARCH type model, all stock index returns can be satisfactorily modelled by the ARMA(1,0)-GJR-GARCH model. Furthermore, the fitted standard GARCH models for each stock index returns on the continent of America are in line with the stock index returns on the continent of Europe. The stock index returns provide the optimal fit under the ARMA(1,1)-GARCH(1,1) model. Meanwhile, as regards the GJR-GARCH type model, the returns of Brazil Bovespa and DJIA fit better under the ARMA(1,0)-GJR-GARCH model, whereas empirical stylized facts of the S&P 500 returns can be sufficiently modelled by the ARMA(1,1)-GJR-GARCH model.
The standardized residuals are calculated to check the adequacy of the selected uni-variate GARCH models. They should follow the SWN(0,1) process, or in other words, they should not exhibit autocorrelation and volatility clustering. The Q-Q plots and correlogram ACF of the standardized residuals for all stock index returns are depicted in Figs. 20-25 in the Appendix. We also perform the Ljung-Box Q(12) and Q2(12) test to formally examine whether or not the standardized residuals exhibit autocorrelation or volatility clustering. The Ljung–Box test statistics reported in Table 3 suggest that the standardized residuals are i.i.d and exhibit no volatility clustering. The null hypothesis of no autocorrelation in the standardized residuals of each fitted model are failed to reject except for Hang Seng and S&P 500 (for which both are statistically significant at the 10% significance level). In addition, the null hypothesis of no volatility clustering in the standardized residuals of each fitted model are also failed to reject except for the Nikkei and Hang Seng (for which both are statistically significant at the 5% significance level) and Brazil Bovespa (statistically significant at the 10% significance level). Thus, we can conclude that the fitted models are adequate in the sense that they provide serially independent and identical innovations and the volatility clustering is correctly explained by the model. After we check the i.i.d innovations of each fitted model, we can proceed to construct the copula-GARCH models and the DCC models to modelling the portfolio returns.
4.2 The copula-GARCH model
Table 3: The Ljung-Box test result of the raw and squared standardized residuals from the fitted univariate GARCH and GJR-GARCH models.
Best model The Ljung-Box test
Q(12) Q2(12) DAX ARMA(1,1)-GARCH-st 11.2 12.2 ARMA(1,0)-GJR-GARCH-st 7.6 12.6 CAC ARMA(1,1)-GARCH-st 10.7 13.8 ARMA(1,1)-GJR-GARCH-st 5.2 14.4 FTSE ARMA(1,1)-GARCH-st 10.4 10.4 ARMA(1,0)-GJR-GARCH-st 7.8 9.7
Hang Seng ARMA(1,1)-GARCH-st 18.9*** 18.1
ARMA(1,0)-GJR-GARCH-st 18.5 21.4** Shanghai SE ARMA(1,0)-GARCH-st 15.9 16.1 ARMA(1,0)-GJR-GARCH-st 15.9 16.0 Nikkei ARMA(1,0)-GARCH-st 12.1 13.3 ARMA(1,0)-GJR-GARCH-st 10.6 24.8** S&P 500 ARMA(1,1)-GARCH-st 19.1*** 16.5 ARMA(1,1)-GJR-GARCH-st 9.4 16.2 Brazil Bovespa ARMA(1,1)-GARCH-st 7.1 19.2***
ARMA(1,0)-GJR-GARCH-st 5.6 18.5
DJIA ARMA(1,1)-GARCH-st 15.3 13.2
ARMA(1,0)-GJR-GARCH-st 9.3 13.5
Notes: st is an abbreviation for the skewed-t distribution; ** and *** indicate the rejection of the null hypothesis no autocorrelation and volatility clustering in the standardized residuals at the 5% and 10% significance level.
we can construct a flexible joint distribution which allows the joint distribution of the portfolio to be free from assumption of normality and linear correlation.
4.2.1 Rank correlations
The linear correlation as a dependence measure has a number of limitations, particu-larly when we leave the Elliptical distributions (the Normal and Student-t distribution) (McNeil et al., 2015). When dealing with copulas, two common measures of dependence are Kendall’s tau and Spearman’s rho which are known as rank correlations. Unlike the linear correlation which depends both on the copula and the marginal distributions, rank correlations depend only on the copula.
strong correlation between pairs of stock indices on the continent of Europe principally because of strong political relationships between the countries on that continent. The correlation coefficient between the DAX and CAC is 0.8. It means that, an increase in the return (risk) of DAX is associated with the increase in the return (risk) of CAC and vice versa. Conversely, the correlation coefficients between the stock index on the continent of Asia are not particularly strong. By describing the correlation between the stock index, we can decide whether we should or should not take on the position of a particular stock index to hedge against other stock indices. In order to reduce portfolio risk, an investor will prefer to invest in stock index that are less or not correlated to the other stock indices. Therefore, in this case, we suggest the risk manager invests in the stock index on the continent of Asia and subsequently hedge against other stock indices on the same continent. Furthermore, there are no notable differences between the rank correlations under the copula-standard GARCH model and the copula-GJR-GARCH model. Both models show a positive relationship.
Table 4: Correlation estimates of stock indices dependence on the continents of Europe, Asia and America.
Portfolio Pair The standard GARCH The GJR-GARCH
Kendall Spearman Kendall Spearman
DAX-CAC 0.8 0.9 0.7 0.9
Europe DAX-FTSE 0.6 0.8 0.6 0.8
CAC-FTSE 0.6 0.8 0.6 0.8
Hang Seng-Shanghai SE 0.3 0.4 0.3 0.4
Asia Hang Seng-Nikkei 0.3 0.5 0.3 0.5
Shanghai SE-Nikkei 0.2 0.2 0.2 0.2
S&P 500-Brazil Bovespa 0.4 0.5 0.4 0.5
America S&P 500-DJIA 0.8 0.9 0.8 0.9
Brazil Bovespa-DJIA 0.4 0.5 0.4 0.5
Notes: The table summarize the Kendall’s tau and the Spearman’s rho correlation for each pair stock index over the overall period.
4.2.2 Estimation of copula parameters
the portfolio all together. Furthermore, we estimate the copula parameters by using In-ference Function for Marginals (IFM) as proposed by Joe and Xu (1996). The estimated copula parameters θ for five copula functions (the Gaussian, the Student-t, the Gumbel, the Clayton and the Frank copula) under the standard GARCH and the GJR-GARCH model for each continent are given in Tables 5 and Table 6, respectively.
Table 5: Estimated copula parameters from the fitted copula-standard GARCH model.
Portfolio Archimedean copulas Elliptical copulas
Gumbel Clayton Frank Gaussian Student’s-t
Europe Parameter 2.7 2.5 9.3 LL 5005.0 4607.0 4950.0 5868.0 6042.0 AIC -10007.6 -9211.8 -9898.8 -11730.7 -12075.0 BIC -10001.4 -9205.6 -9892.6 -11712.0 -12050.1 Asia Parameter 1.3 0.6 2.4 LL 634.4 786.6 711.8 1021.0 1032.0 AIC -1266.8 -1571.3 -1421.6 -2036.7 -2055.9 BIC -1260.6 -1565.0 -1415.4 -2018.0 -2030.9 America Parameter 1.9 1.4 5.5 LL 2547.0 2422.0 2493.0 5306.0 5506.0 AIC -5091.1 -4842.2 -4983.7 -10606.3 -11003.5 BIC -5084.9 -4836.0 -4977.4 -10587.6 -10978.6
Note: The bold values in the table refer to the values of LL, AIC and BIC for the best copula model.
Comparing the copula-based model is an important task after fitting the marginals and estimate copula parameters. The most widely used methods to rank the copula accord-ing to fit are the LL, AIC and BIC. These values are also reported in Tables 5 and 6, respectively. Model selection for each copula family is based on the highest value of LL and the lowest value of AIC and BIC. The same copula type has been selected. The Student-t copula appears to be the best copula that best fits all of the portfolios from the Elliptical copulas since it has the lowest values of both criterions: AIC and BIC and the highest value of LL. Therefore, the choice of the GARCH specifications (either the standard GARCH or the GJR-GARCH model) does not have a significant impact on the choice of Elliptical copulas. Both specifications highlight the relatively strong performance of the Student-t copula. Meanwhile, for the Archimedean copulas, the Clayton and Gumbel copula perform better than the Frank copula for the model under the standard GARCH specification. However, under the GJR-GARCH specifica-tion, each portfolio has different copula that provide the best fit for the data.
Table 6: Estimated copula parameters from the fitted copula-GJR-GARCH model.
Portfolio Archimedean copulas Elliptical copulas
Gumbel Clayton Frank Gaussian Student’s-t
Europe Parameter 2.6 2.4 9.2 LL 4845.0 4440.0 4857.0 5705.0 5870.0 AIC -9688.6 -8879.0 -9712.2 -11403.2 -11732.0 BIC -9682.3 -8872.8 -9706.0 -11384.5 -11707.1 Asia Parameter 1.3 0.5 2.4 LL 622.3 764.2 704.3 996.6 1007.0 AIC -1242.7 -1526.3 -1406.5 -1987.2 -2006.2 BIC -1968.5 -1981.3 -1236.4 -1520.1 -1400.3 America Parameter 1.8 1.4 5.4 LL 2508.0 2396.0 2474.0 5146.0 5340.0 AIC -5013.5 -4789.1 -4946.8 -10285.6 -10671.8 BIC -5007.3 -4782.9 -4940.5 -10266.9 -10646.9
Note: The bold values in the table refer to the values of LL, AIC and BIC for the best copula model.
and θ → ∞ indicate independence and perfect positive dependence, respectively. From Tables 5 and 6, we can observe that estimated parameters of the Gumbel copula are not equal to one for all portfolios, which implies positive dependence. However, estimated parameters of the Clayton copula are around zero for the portfolio on the continent of Asia, which indicates independence. It denotes that the diversification among three equity markets (the Hang Seng, Shanghai SE and Nikkei) on the continent of Asia can be beneficial for international portfolio investors.
4.2.3 Tail dependence
Another alternative dependence measure derived from copula is known as tail depen-dence. Like rank correlations, the coefficient of tail dependence also depends only on the copula. The lower and upper tail dependence provide measures of extreme dependence i.e., dependence on the tails of joint distribution, which is one of the major concerns in financial risk management. The upper and lower tail dependence are particularly valuable for measuring the tendency of markets to crash or boom together. Following the best copula model, Table 7 reports estimated values of the lower and upper tail dependence coefficients of each paired stock index on the continents of Europe, Asia and America.
Table 7: Tail dependence coefficients of the best Archimedean copulas.
Portfolio Pair The standard GARCH The GJR-GARCH
λl λu λl λu
DAX-CAC 0.13 0.84 0.09 0.03
Europe DAX-FTSE 0.09 0.87 0.05 0.00
CAC-FTSE 0.31 0.84 0.03 0.04
Hang Seng-Shanghai SE 0.46 0.00 0.49 0.00
Asia Hang Seng-Nikkei 0.45 0.00 0.41 0.00
Shanghai SE-Nikkei 0.41 0.00 0.34 0.00
S&P 500-Brazil Bovespa 0.08 0.63 0.08 0.60
America S&P 500-DJIA 0.01 0.68 0.01 0.67
Brazil Bovespa-DJIA 0.19 0.66 0.19 0.64
Notes: The table presents the estimated values of the upper and lower tail dependence coefficients estimated from the copulas that provide the best fit.
and America, the upper tails dependence are higher than the lower tails dependence. Therefore, we can say that the stock indices on the continents of Europe and America are more likely to increase together than fall together. However, the dependence structure from the fitted GJR-GARCH model for each pair of stock indices on the continent of Europe are symmetric in bull and bear markets because lower and upper tail dependence do not show much difference. This is because the lower and upper tail dependence are estimated from the Frank copula, which is exposed to weak tail dependence. In contrast, the pair of stock indices on the continent of Asia show higher lower tail dependence than the upper tail dependence. Thus, there is a greater possibility that the stock index returns on the continent of Asia fall together rather than rise together.
For a more clear interpretation, take the upper tail and lower tail dependence for the DAX-CAC pair. The upper tail is estimated to be 0.84, indicating that the DAX has a price increase above a certain value and moreover, the probability that the CAC has a price increase above a corresponding value is approximately 84%. Furthermore, the estimated lower tail dependence is 0.13, meaning that given the DAX price drops below a certain value, the probability that the CAC has a price decline below a corresponding value is approximately 13%.
4.3 The DCC model
In addition to the copula-GARCH models, we also perform different classical approaches to estimate the VaR of portfolios. We construct the traditional multivariate GARCH model, where in this case, we employ the DCC models using the fitted AR(MA)-GARCH or AR(MA)-GJR-GARCH model as the marginals. The advantage of the DCC model is the fact that the correlation matrices evolve dynamically, thus, it reflects the current market conditions. The model that we will consider and the marginals for each con-tinent are presented in Table 8. The innovations are assumed to follow multivariate-t distribution to account for heavy-tailed of the returns.
Table 8: The DCC models and marginals for each continent.
Continents The DCC models The marginals
Europe DCC-GARCH ARMA(1,1)-GARCH-st ARMA(1,1)-GARCH-st ARMA(1,1)-GARCH-st DCC-GJR-GARCH ARMA(1,0)-GJR-GARCHst ARMA(1,1)-GJR-GARCHst ARMA(1,0)-GJR-GARCHst Asia DCC-GARCH ARMA(1,1)-GARCH-st ARMA(1,0)-GARCH-st ARMA(1,0)-GARCH-st DCC-GJR-GARCH ARMA(1,1)-GJR-GARCH-st ARMA(1,1)-GJR-GARCH-st ARMA(1,1)-GJR-GARCH-st America DCC-GARCH ARMA(1,1)-GARCH-st ARMA(1,1)-GARCH-st ARMA(1,1)-GARCH-st DCC-GJR-GARCH ARMA(1,1)-GJR-GARCHst ARMA(1,0)-GJR-GARCHst ARMA(1,0)-GJR-GARCHst
The estimated parameters of the DCC-GARCH and the DCC-GJR-GARCH model such as the conditional mean µ, the AR effect φ1, the MA effect θ1, the GARCH parameters
α0, α1, and β1, the leverage effect γ1 and the parameter related to the skew and shape
ξ and κ are reported in Tables 9 and 10, respectively. The result can be interpreted as follows
1. Table 9 exhibits the estimated parameters of the fitted DCC-GARCH model. The parameters of ARMA and GARCH (φ1, θ1, α0, α1, and β1) are statistically
signifi-cant at the 1% significance level except for the GARCH parameter α0 of the Hang
ex-plained by the past residuals and past conditional volatilities. Furthermore, these results also signify that the volatility is highly persistent, i.e., the extreme changes tend to be followed by other extreme changes, although not necessarily with the same sign.
2. Table 10 presents the estimated parameters of the fitted DCC-GJR-GARCH model. Certain parameters of AR and GARCH are not statistically significant indicating that the volatility clustering is not persistent. However, the leverage term γ are all statistically significant at the 1% significance level except for the Shanghai SE for which it is non-significant. This result verifies that the positive and negative shocks have a different effect on conditional volatility.
3. The positive skew (ξ) and the high excess of shape (κ) are supported by the non-normality assumption of the underlying returns distribution. The positive skew implies that positive extreme returns are higher and that they are more likely to occur than negative extreme returns. Furthermore, the high excess of shape parameters indicate that the distribution of returns are heavy-tailed. Both parameters are all statistically significant at the 1% significance level.
4. The DCC parameters a1 and b1 are both statistically significant at the 1%
sig-nificance level, which indicates that the conditional correlation between the stock index returns is persistent. Moreover, a1 and b1 are not equal to zero which means
that the DCC specification is more realistic than the conditional constant correla-tion (CCC), which assumes a1 = b1 = 0.
The estimated dynamic conditional correlation matrices Pt for each pair of stock index
returns on the continents of Europe, Asia and America with the standard GARCH specification can be shown in Figs. 4-6 below (see Fig. 30 in the Appendix for estimated dynamic conditional correlation matrices Pt for pairs of stock index under the
GJR-GARCH specification). We make the following observations regarding the dynamic conditional correlation matrix Pt.
1. All correlation coefficients are positive.
2. Each pair of stock index on the continent of Europe shows generally higher corre-lation than the correcorre-lation of stock index on other continents except for the S&P 500-DJIA on the continent of America. These results do not come as a complete surprise, because the stock indices in European countries indeed have a relatively strong political relationship.
