MSc Finance Dissertation: Copula Theory in Estimating VaR

MSc Finance Dissertation written by: XULI XIAO (Clive)
1
, Supervised by: Dr. ALEXANDRA DIAS, (Aug 2009)
MSc Finance
Application of the Copula theory in the estimation of the Value at Risk
(VaR) of a portfolio composed by Hong Kong and Taiwan market indices
All the work contained within is my own unaided effort and conforms with the University's
guidelines on plagiarism.
Abstract
Copula-based models are capable of capturing the non-linear dependence features between
financial assets. In this research, a portfolio comprised of two market indices is measured. The
distributions of the financial returns are filtered by ARMA-GARCH models which are monitored
by the semi-parametric model: the tail dependences of the distribution are monitored by
Generalised Pareto Distribution. In turn, the centre dependence of the distribution is measured by
Empirical Distribution. Normal Mixture, Gussian, Gumbel and Clayton Copulas were fitted.
According to various selection criteria, we found that the fitted Gussian Copula outer-performs the
others. One-step-ahead Value at Risk (VaR) is estimated on the Copula-based models. When the
portfolio is composed with a ratio of 10% of Hong Kong HangSeng index to 90% of Taiwan
Capitalization Weighted Stock index, the VaR estimation had the least number of violations. But,
the back-testing results of Value at Risk showed that all our Value at Risk results underestimate the
expected VaR. This could be due to the fact that we set the Asian Crisis period as our out-of-
sample period, during which, the data exhibited evident of regime shift.
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I would like to thank Dr. Alexandra Dias for her supervision and invaluable comments. I also want
to thank my parents for their financial supports.

Warwick Business School MSc Finance: XULI XIAO (Clive) August/September 2009
1
Contents
1. Introduction….................................................................................................................2
2. Copula and IFM methodology.......................................................................................6
2.1 Copula Theory……………………..............................................................6
2.2 Sklar’s Theorem...........................................................................................7
2.3 Concordance measure of dependence.........................................................8
2.4 Copula introduction………………………………………………………..9
2.4.1 Normal Mixture and Gussian Copula......................................…...9
2.4.2 Gumbel Copula………………….………………...……………….9
2.4.3 Clayton Copula………………….………………….…………….10
2.5 Brief explanation of IFM………………..……………………………….10
3. Empirical Analysis.........................................................................................................12
3.1 Data description..........................................................................................12
3.2 Model specification.....................................................................................16
3.2.1 Marginal distributions modelling...................................................17
3.2.2 Extreme Value Theory: GPD.........................................................22
3.2.3 Copula models and the dependence structure....….……………...28
3.3 Value at Risk (VaR) estimation and back-testing…….………………….32
3.3.1 Value at Risk estimation……………………….…………………32
3.3.2 Back-testing VaR............................................................................35
3.3.3 Kupiec’s Likelihood Ratio test........................................................38
3.3.4 Regime shift discussion............…………………...………………40
4. Conclusion.....................................................................................................................42
References.....................................................................................................................44

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1. Introduction
In finance, indentifying, monitoring and quantifying interdependence among financial
assets are extremely crucial. Because investment return could be highly affected by the
time-varying risk exposure. It is essential for investment managers to understand how to
derive the joint distribution of multivariate asset returns which enables investors to
correctly model the market risk. We know that stylized facts of univariate financial data
are obvious and relatively easy to indentify and monitor. However, dependence is a key
issue in studying multivariate financial data. Traditionally, 1) linear dependence and 2)
normality assumptions are two building blocks for a large number of models. Nevertheless,
both assumptions are not correct empirically. Firstly, we know that the variance of return
on a portfolio comprised of risky assets depends on the variance of the individual risky
return and the linear correlation, which only explains the linear dependence among
individual assets. It only monitors part of the entire risk and underestimates the total risk
exposure when non-linear dependence exists among assets. (Embrechts, et al. (2002)).
Secondly, as Cont (2001)‟s empirical study shows, the stylized statistical properties of
asset returns do not follow a Gussian/normal distribution. Bangia et al., (2002) mention
that multivariate normality does not hold in market crashes and bear market. The normality
assumption underestimates the downside risks during extreme events. Similarly, Erb,
et,al.(1994), Longin and Solnik (2001) and Ang and Chen (2002), Bae, et al (2003) argue
that during crises or volatile bear markets, the assets returns are much more correlated.
And meanwhile, distribution of financial return exhibit fat-tail features. In other words,
non-normal dependence exists among asset returns. This is a very clear empirical evident,

3
especially during market crashes and financial crises. Therefore, we can conclude
empirically that joint distribution of multivariate asset returns exhibit non-linear and non-
normal dependence. To deal with these two problems, copula theory, first proposed by
Sklar (1959), indicates that any n-dimensional joint distribution function can be
decomposed into n marginal distributions and a copula function. The entire dependence
structure including linear and non-linear dependence can be captured by the copula
function. The multivariate distribution and dependence structure can be estimated and
modelled by the copula function without assuming normality Copula theory has been
applied to a number of popular areas.
First of all, in derivative markets, copula theory could be used in derivative pricing and
generates substantial impacts on trading strategy. For example, if an investor holds a
derivative with more than one underlying assets, the dependence structure between these
underlying assets would definitely affect the price of the derivative contract itself.
Cherubini, et al. (2004) presents an introduction to copulas based on option pricing.
Rosenberg (2003), Bennett and Kennedy (2004), van den Goorbergh, et al. (2005) and
Salmon and Schleicher (2006) have considered option pricing with copulas. While Talor
and Wang (2004) and Hurd, et al. (2005) consider the implied copula using historical
market derivative prices. The recent Credit Crisis was related to two core new derivative
products, namely, the Collateralised Debt Obligations (CDO) and Credit Default Swaps
(CDS). The credit risk profiles of the CDO and CDS are extremely complex, because they
are implicitly involved multiple underlying sources of risks. Default on one of the
counterparties may generate domino effect on the other parties, and therefore, the whole
contract becomes worthless. The dependence structure among these risks is unclear using

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traditional model. So Schonbucher and Schubert (2001) and Giesecke (2004) consider
application of copula theory to default risk. Li (2000), who was the first to apply copula
theory in finance, used Gussian copula to model the credit risk of the CDOs. In 2004,
rating agencies Moody‟s and Standard & Poor‟s incorporated Mr. Li‟s Gaussian copula
default function formula into its rating methodology for the CDOs. Jones (2009) wrote a
detail report on Financial Times and explained the problem: Gaussian copula failed to
anticipate the Subprime Crisis due to the fact that it only assumes the distribution of the
financial returns as Gussian/normal distribution.
Secondly, risk management is another main area. Value at Risk (VaR) quantify the risk of
a portfolio at a given confidence level. Due to the “fat tail” and excess kurtosis natures of
empirical data, copula model can help achieve better VaR measurement. Cherubini and
Luciano (2001), Embrechts, et al. (2003) and Embrechts and Hoing (2006) research VaR
of portfolios using copula methodology. Rosenberg and Schuermann (2006) study the
joint credit, market and operational risks using copula theory. Kole, et al (2006) study the
selection of copulas among Gusssian copula, Student‟s t copula and Gumbel copula for
risk management. Palaro and Hotta (2006) use conditional copula to estimate VaR.
Jondeau and Rockinger (2002) develope a new measure of conditional dependency
between financial asset returns, use GARCH Plackett copula with assumption that its
innovation follows Student‟s t distribution and model the VaR measurement in non-
gaussian environments.
Thirdly, copula theory can be applied to make portfolio decision. Markowitz (1952), the
Nobel Prize winner suggests optimal portfolio weights depend on the first two moments of

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the assets, namely, mean and variance. Therefore, the aforementioned linear correlation
provides all dependence information in making portfolio selection decisions. But the linear
correlation is inadequate in the case of mutilvariate distribution. Patton (2004) studies an
equity portfolio containing two stocks (bivariate) using copulas. Garcia and Tsafack (2007)
construct a portfolio with four assets, the stocks and bonds returns from two countries and
consider the portfolio selection problem using copulas.
The final main area of application of copulas in finance is financial contagion effect.
Financial contagion always emerges during financial crises, most notably, the Asian Crisis
during 97 and 98 and the recent Subprime Crisis during 07 and 08. We have seen non-
fundamental linkages between different markets during crises. Hence, it is worth to study
whether the dependence level increase during a period of crisis using copula model.
Rodriguez (2007) applies Markov switching copula model to study contagion effect.
This paper is going to undertake an empirical study on a bivariate portfolio containing
Hong Kong HangSeng Index and Taiwan Capitalization Weighted Stock Index using
various bivariate copula models. We will also illustrate the concepts and theory of copula
and Value at Risk estimation of the bivariate portfolio. Based on Asian Crisis period data
as our out-of-sample observations, we will estimate the VaR on the copula-based models.
Due to the fact that, most of reseachers use equally weighted portfolio, we will alter the
index weight ratio to check for whether the performance of the VaR estimation will depend
on portfolio weights. Also, Kupiec‟s test would be implemented in order to back-test the
performance of the VaR estimations.

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We first give the formal copula definition and discuss the copula dependence theory and
the IFM method in Section 2. In Section 3, we present the empirical analysis on the fitted
models and illustrate all the empirical findings we obtained. Finally, Section 4 concludes.
2. Copula and IFM methodology
We will present the Copula Theory definition, Sklar‟s Therom, and the theory of
concordance measure of dependence in Section 2. The four fitted copula models will also
be presented. At the end of this section, we will introduce the Inference Function for
Margins (IFM) method for fitting Copula models.
2.1 Copula Theory
McNeil et al., (2005) give a definition of copula by stating that a Copula C must satisfy
three properties, namely:
(1) ),....,( 1 duuC is increasing in each component ui.
(2) ii uuC )1,...,1,,1,...1( for all  ]1,0[,,...,1  iudi .
(3) For all d
dd bbaa ]1,0[),...,(),,...,( 11  with ii ba  we have
  


2
1
2
1
1
...
1
1
1
,0),...,()1(...
i i
dii
ii
d
d
d
uuC where uj1=aj and uj2=bj for all  dj ,...,1 .
Property (1) states that the copula is an increasing function. Property (2) points out that the
marginal distribution should be uniformly distributed. Property (3) says that the volume
must be positive for all d dimensional rectangles in [0,1]d
(rectangle inequality).

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2.2 Sklar’s Theorem
Sklar (1959) highlights the fact that Copula Theory plays an important role in the study of
multivariate distribution by introducing the Sklar‟s Theorem: Let F be a joint distribution
function with margins F1,…,Fd. Then there exists a copula C : [0,1]d
→[0,1] such that, for
all x1,…,xd in R=[-∞, ∞], F(x1,…, xd)=C(F1(x1),…,Fd(xd)). If F1,…,Fd are continuous
margins, then C is unique; otherwise, C is uniquely determined on RanF1×…× Ran Fn.
Ran denotes the range. Conversely, if C is a copula and marginal functions F1,…,Fd are
univariate distributed, then F underlined is a d-dimensional distribution function with
margins F1,…,Fd. (For the proof, see Schweizer and Sklar (1983), McNeil et al.,(2005,
p.187) or Nelsen(1999, p.18).
In other words, the Skla‟s Theorem indicates that a d-dimensional joint distribution
function can be decomposed into d marginal distributions and a copula function C.
Simultaneously, a copula function and the univariate margins can produce the multivariate
distribution functions. The dependence structure of the multivariate distribution can
therefore be captured (represented) entirely by the copula function. As a result, by studying
the concordance of the copula functions, we can interpret the dependence information for
the corresponding multivariate distribution functions.

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2.3 Concordance measure of dependence
Because of the fact that the concordance measures are invariant, concordance is a
promising indicator for measuring dependence. For example, if we consider two i.i.d
random variables X and Y, concordance between X and Y increases when the following
condition 1 and 2 are held at the same time: 1) If either X or Y experiences a large increase,
the other also illustrates a large increase. 2) If either of them has a small increase, the other
exhibits a small increase too.
We consider two common concordance indicators, Kendall‟s tau and Spearman‟s rho,
which are invariant to both linear and non-linear strictly increasing transformations. But,
Pearson Correlation coefficient is variant in non-linear world, which explains the bias of
the result, when investors only consider the Pearson Correlation coefficient for measuring
the dependence in practise.
Joe (1997)‟s book gives an informative description on the two concordance indicators,
Kendall‟s tau and Spearman‟s rho:
 Kendall‟s tau:   1),(),(4 vudCvuC ,
 Spearman‟s rho:    3),(123),(12 dudvvuCvuuvdCs , where, C is a
survival function to a cdf C (Copula function).
For a bivariate case, we could compute C as: ),()()(1),( vuCvCuCvuC vu  , the
higher the values of the concordance measures, the higher the degree of the monotonic
dependence.

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2.4 Copula Introductions
Four Copulas including Normal Mixture, Gussian, Gumbel and Clayton Copulas from
three different copula families were chosen to study. We took the brief explanations of the
copulas from Ziovt and Wang (2006)‟s book.
2.4.1 Normal Mixture and Gussian Copula (Elliptical family) (Zivot and Wang (2006))
),()1(),(),( 21
vuCpvupCvuCNMC
  , where 1,,0 21  p , P is a probability value.
C is a Gussian Copula with parameter (correlation coefficient) δ, hence we have:
))(),((
)1(2
2
exp
12
1
),( 11
2
22
2
)()( 11
vu
yxyx
dydxvuC
vu

















 





Where, )(1

is the quantile (inverse) function of the standard Gussian distribution;  is
the joint CDF of a standard bivariate normal distribution with parameter δ (0≤ δ≤1).
Actually, Gussian Copula is a special case of Normal Mixture Copula, i.e. when
probability p equals to 0.5 and δ1 = δ2.
2.4.2 Gumbel Copula (Extreme Value family) (Zivot and Wang (2006))
  /1
])ln(()ln([(exp),( vuvuCGC
 , δ ≥ 1
Gumbel Copula captures the upside risk (upper tail dependence). Dependence increases
from 0 to 100% dependence while parameter δ goes up from 1 to infinity (+∞).

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2.4.3 Clayton Copula (Archimedean family) (Zivot and Wang (2006))
 

0,)1(),( /1
vuvuCCC
Clayton Copula is also called Kimeldorf-Sampson Copula. It models the downside risk
(lower tail dependence), for example, during crises or bear market movements.
2.5 Brief explanation of Inference Function for Margins (IFM)
IFM, Inference Function for Margins, which was introduced by Joe and Xu (1996), is a
two-step method for estimating the copula parameters, θ.
They proposed that firstly, we could model the marginal distributions FX and FY by either
parametric models (modelled entirely with normal or student‟s t distributions) or semi-
parametric models (modelled tails of the return distribution by Generalized Pareto
Distributions and centre of the distribution by empirical distribution) using Maximum
Likelihood Estimation (MLE) method. Zivot and Wang (2006) give a detailed explanation:
Denote the modelled marginal distributions as Fx and Fy , a pseudo sample of observations
can be formed as: niyFyxFxvu iiii ,...,1)),(),((),(  , this can be regarded as the first
stage of the IFM.
Secondly, conditional on the parameters ii vu , we estimated from Step 1, a copula-related
log-likelihood estimator l can be formed by combining the parameters we have obtained,

11
with the unknown copula parameter θ. The log-likelihood estimator would be:


n
i ii vucvul 1
);,(ln),:(  . Subsequently, the unknown parameter, θ, can be calculated
by maximizing the newly formed log-likelihood estimator l using MLE method. The
parameter θ is named the IFM estimator, which suggested by Joe (1997) that the IFM
estimator is as efficient as the Maximum Likelihood estimator, to a great extent.
This two-step IFM has four major advantages as mentioned by Joe and Xu (1996):
 IFM enables one to estimate parameters from multivariate models computationally
feasible.
 One can start modelling a multi-dimensional from a lower dimensional margin.
 They believe there should be more robustness against misspecification of the
dependence structure than one-step ML method.
 Because IFM avoids the sparseness problem to some extent, it is better than one-
step ML method that does not.
IFM method would be implemented in practise in the following section.

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3. Empirical Analysis
The portfolio was composed by two Asian market indices, Hong Kong HengSeng Index
and Taiwan Capitalization Weighted Stock Index. The database contains 1421 daily
closing prices, from 1st
May 1992 to 1st
May l 1998. Prior to the Asian Crisis, we use the
data from 1st
May 1992 to 1st
July 1997 as in-sample data and construct our copula-based
models. We set the data from 2nd
July 1997 to 1st
May 1998, (Asian Crisis period) as our
out-of-sample data in order to test the accuracy of the VaR estimation using various
models.
3.1 Data Description
We transformed the daily closing prices of the two indices into daily log-returns. Holidays
and weekends were excluded in order to avoid spurious correlation. 1225 observations
were considered as in-sample data for estimating the copula models and 195 observations
were set as out-of-sample data. From Figure 1, we observed volatility clustering in both
plots of log-returns. It is essential to check for whether both univariate returns are
stationary before fitting our Copula-based models to the data. Kwiatkowski et al., (1992)
suggest a reliable stationary test, namely KPSS test, which was regarded as the most
commonly used stationary test by Zivot and Wang (2006). We found that the test statistic
values for Hong Kong and Taiwan data were 0.0554 and 0.1611 (see Table 1), respectively,
which were both smaller than 0.446 (KPSS2
test statistic at 95% confidence level).
──────────────────
2
See table shown in Page 130 of Zivot and Wang(2006)

13
Therefore, we could not reject the null that the log-return observations were stationary. We
are then ready to proceed after the KPSS stationary test. The empirical stylized facts of the
two univariate returns are presented through the following figures and tables.
Figure 2: Histograms (on the left) of Hong Kong (top) and Taiwan (bottom) market daily log-
returns and QQ-plots (on the right) of empirical quantiles against the standard normal quantiles.
Figure 1: Daily closing price (on the left) and log-returns (on the right) of Hong Kong (top) and
Taiwan (bottom) market indices.
Figure 1: Daily closing price (on the left) and log-returns (on the right) of Hong Kong (top) and
Taiwan (bottom) stock indices.

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Table 1: Descriptive statistics of daily log-returns of Hong Kong and Taiwan stock indices
Statistics Hong Kong HangSeng
Index
Taiwan Capitalization Weighted
Index
Mean
Std. Derivation
Minimum
Median
Maximum
Skewness
KPSS statistic
Excess of Kurtosis
Jarque-Bera statistic
Jarque-Bera p-value
ARCH LM statistic
ARCH LM p-value
Ljung-Box statistic
Ljung-Box p-value
0.000832
0.01474
-0.08348
0.0005237
0.05708
-0.3201
0.0554
6.064
500.1464
0.000
125.9839
0.000
37.9989
0.0089
0.0005563
0.01603
-0.07782
0.0004226
0.07694
-0.03212
0.1611
5.993
457.4827
0.000
70.9308
0.000
33.1963
0.0321
Note: Table 1 above summarize the statistics details of the univariate distributions. 1225 observations
were monitored, from 1st May 1992 to 1st July 1997. (Holidays and weekends were eliminated.)
Table 1 also exhibited the skewness/asymmetry and excess of kurtosis of the two
univariate distributions. Both of them were not symmetry and had large kurtosis. Under
Jarque-Bera test, P-values for them are 0.000, which strongly rejected the null hypothesis
that the univariate returns were normal distributed. QQ plots and histograms from Figure 2
also presented the non-Gussian features of both univariate distributions. Comparing the
two histograms, we also observed that the return of Taiwan stock index was more
symmetry than Hong Kong stock index, which was a consistent finding with the Skewness
measurements (-0.3201 and -0.03212) exhibited in Table 1.

15
Lag
ACF
0 5 10 15 20 25 30
0.00.20.40.60.81.0
Series : TWI.LnReturn.ts
Lag
ACF
0 5 10 15 20 25 30
0.00.20.40.60.81.0
Series : TWI.LnReturn.ts^2
Lag
ACF
0 5 10 15 20 25 30
0.00.20.40.60.81.0
Series : HSI.LnReturn.ts
Lag
ACF
0 5 10 15 20 25 30
0.00.20.40.60.81.0
Series : HSI.LnReturn.ts^2
Figure 3: Correlograms for the log-returns and log-returns^2 of Hong Kong and Taiwan stock indices; Top
Left and Bottom Left show the correlograms for the log-returns of Taiwan and Hong Kong indices,
respectively; Top Right and Bottom Right present the correlograms for the squares of the log-returns.
Additionally, we implemented ARCH LM test and Ljung-Box test on the log-returns in
order to test the ARCH effect and autocorrelation. P-values of ARCH LM test for both
returns were 0.000 which we strongly rejected the null hypothesis that there is no ARCH
effect. Ljung-Box test was applied to test the autocorrelation of the data. Under a
benchmark p-value 5%, we rejected the null hypothesis that there is no autocorrelation at
95% confidence interval. Because as we can see from Table 1, Ljung-Box p-value for
Hong Kong and Taiwan stock indices are 0.0089 and 0.0321 respectively, both smaller
than 0.05. From Figure 3 below, we could also observe ARCH effects on both univariate
returns.
After examining the empirical features of the univariate returns of Hong Kong HangSeng
Index and Taiwan Capitalization Weighted Stock Index, in the following section, we will
build our copula-based models using these two log-returns as the marginal distributions
(Total number of observations: 1225*2).

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3.2 Model Specification
In this section, ARMA-GARCH Copula models were constructed. Three subsections are
included.
Firstly, in Subsection 3.2.1, ARMA-GARCH family models were fitted to the univariate
returns, individually, using Maximum Likelihood Estimation method (MLE) with
assumption that the innovation is Student‟s-t distributed in order to filter the residuals of
observations to obtain standardised innovations.
Secondly, it involved applying the first step of the IFM method. We noticed that both
univariate distributions exhibited fat-tailed features and negative skewness, which led us to
consider the Generalised Pareto Distribution (GPD) that is capable of modelling the tail
dependence. Hence, the tails of distributions of standardised residuals were modelled by
GPD according to various thresholds. The other parts of the distributions were modelled by
Empirical Distribution. The modelled observations were mapped into the uniform
distributions locating within a unit square [0, 1]2
. Additionally, dependence structures
would be monitored from the copula-based models through the concordance measures
(Kendall‟s tau and Spearman‟s rho). (See Subsection 3.2.2)
Thirdly, after modelling the marginal distribution, we could proceed to step two of the IFM
method in Subsection 3.2.3. Four copula models, including Normal Mixture Copula,
Gussian Copula, Gumbel Copula and Clayton Copula, would be fitted to the standardised
bivariate uniform distributions. The best fitted model would be distinguished by selection
criteria AIC, BIC and Log-Likelihood).

17
3.2.1 Marginal Distributions Modelling: ARMA-GARCH model
From previous session, we have found that the univariate log-returns exhibited significant
ARCH and autocorrelation effects. Therefore, the following univariate ARMA (p1, q1) –
GARCH (p2, q2) model with Student‟s t distributed innovation, was fitted to the log-returns
of Hong Kong index and Taiwan index, respectively.
 

  
  



2
1
2
1
22
0
2
1
1
1
1
)(
)()(
p
i
q
i jtjititit
q
i jtjtj
P
i itit
tttt
X
XX
ZX



where   

2
1
2
120 1,,...,1,0,0
p
i
q
j jii andqi  , Zt is SWN(0, 1) and
follows student‟s t distribution in our fitting process.
We determined the appropriate number of lags (p1, q1, p2, q2) for the ARMA (p1, q1) –
GARCH (p2, q2) model by comparing the statistical significance of the parameters with
various number of lags. Finally, for the univariate return of Hong Kong HangSeng Index,
we found that the AR(1)-GARCH(1,1) model provided the best fit. For Taiwan
Capitalization Weighted Stock Index, ARMA(1,1)-GARCH(1,1) was best characterized.
We also considered other ARMA models, including AR(2), AR(3), AR(4,) ARMA (2,1),
and ARMA (3,1). Nevertheless, most of the parameters from the aforementioned models
estimated were not significant at 5% level. The coefficients of the AR(1)-GARCH(1,1)
model and ARMA(1,1)-GARCH(1,1) model and the corresponding P-values are showed
in the table below.

18
Table 2: Summary of the Univariate Marginal Distribution
Coefficients of the ARMA-GARCH model for HK and TW: Results (P-value)
 1 1 0 1 1
Hong Kong:AR(1)-
GARCH(1,1)
0.0008394*
(0.015)
0.06808*
(0.018)
N.A 3.941e-006*
(0.032)
0.06674*
(0.000)
0.9167*
(0.000)
Taiwan:ARMA(1,1)-
GARCH(1,1)
9.607e-009
(0.988)
1.005*
(0.000)
-1.005*
(0.000)
0.00001069*
(0.028)
0.07972*
(0.001)
0.89*
(0.000)
Note: AR(1)-GARCH(1,1) model was fitted to Hong Kong HangSeng Index return; ARMA(1,1)-GARCH(1,1)
model was fitted to Taiwan Capitalization Weighted Index return.
“*” represents the coefficient estimated is significant at 5% (corresponding P-value < 0.05) “N.A”: the
coefficient of the MA(1) term was not available for AR(1)-GARCH(1,1) model.
As Table 2 illustrated, all P-values (in bracket) of the parameters from the two univariate
models estimated were less than 0.05 so that we rejected the null hypothesis (the parameter
is zero), except for the constant term of the conditional mean of the Taiwan index return
distribution. We are noticed that, from Table 2, the absolute values of the constant terms of
the conditional mean and variance from both distributions were very close to 0, although
three of them were statistically significant at 5% level (  of HK, 0 of HK and TW were
significant) . The GARCH models were also compared with other models from the same
family, including EGARCH, TGARCH and PGARCH. According to AIC selection
criterion and log likelihood values, we summarised that the GARCH models outer-
performed the others in this case. (see Table 3)

19
Table 3: ARMA-GARCH family model selection for margins
Hong Kong AIC Taiwan AIC
AR(1)-GARCH(1,1) -7116* ARMA(1,1)-GARCH(1,1) -6879*
AR(1)-EGARCH(1,1) -7114 ARMA(1,1)-EGARCH(1,1) -6878
AR(1)-PGARCH(1,1) -7109 ARMA(1,1)-PGARCH(1,1) -6879*
AR(1)-TGARCH(1,1) -7113 ARMA(1,1)-TGARCH(1,1) -6877
Note: AIC selection criterion was implemented.
“*” states the best fitted family models (with smallest AIC values). Because of the two identical AIC
values, two “*” were marked on the right of the table.
According to AIC, AR(1)-GARCH(1,1) contains the smallest negative AIC value that it is
claimed to be the best fit for Hong Kong index. We could also see from Table 3 that for
Taiwan index, ARMA(1,1)-GARCH(1,1) had the same AIC value as ARMA(1,1)-
PGARCH(1,1). But, we found that P-values for the parameters of the ARMA(1,1)-
PGARCH(1,1) model (not exhibited here) were relatively larger than P-values from
ARMA(1,1)-GARCH(1,1) model showed in Table 2 so that we chose the GARCH models
(with student‟s t distributed innovation), which were found to be the best fitted model in
terms of both AIC and significance of parameters.
After being filtered by the ARMA-GARCH process, we found no ARCH and
autocorrelation features from the standardised innovations. ARCH LM and Ljung Box test
results and the correlograms for the standardized innovations and its squares were showed
in Table 4 and Figure 4 below.

20
Table 4: ARCH LM test and LB test
Statistic Hong Kong Taiwan
ARCH LM test statistic
(ARCH LM P-values)
5.5648
(0.9364)
10.4736
(0.5745)
Ljung Box test statistic: residuals
(Ljung Box test P-value)
28.7104
(0.5328)
37.8132
(0.1546)
Ljung Box test statistic: residuals^2
(Ljung Box test P-value)
11.3612
(0.9992)
15.7638
(0.9847)
Note: Table 4 illusrates the test statistics and P-values from the ARCH LM for the standardised residuals
and Ljung Box test for the standardised residuals and squared residuals of the filtered returns.
Figure 4: Correlograms for the standardised residual and squared residuals of the filtered Hong Kong
and Taiwan market returns; Top Left and Bottom Left show the correlograms for the standardised
residual of the filtered Taiwan and Hong Kong indices returns, respectively; Top Right and Bottom
Right present the correlograms for the standardised residuals.

21
From Table 4, we can see that P-values of all estimates are much greater than 5%,
especially the P-values of LB test on the standardised squared residuals (99.92% and
98.47% for HK and TW, respectively), hence we can not reject the nulls that there are no
ARCH effects (P values: 93.64% for HK and 57.45% for TW) and no serial correlation on
the standardised residuals and squared residuals after being filtered by the AR(1)-
GARCH(1,1) model on univariate distribution of Hong Kong HangSeng index daily
returns and ARMA(1,1)-GARCH(1,1) model on univariate distribution of Taiwan
Capitalization Weighted index daily returns. Correlograms in Figure 4 also illustrates the
same findings that autocorrelation were eliminated after implementing the ARMA-
GARCH family models.
Because we filtered both univariate returns by assuming the innovations are from student‟s
t distribution, the quantiles of theoretical t distribution were plotted against the quantiles of
the standardised innovations, which are showed in Figure 5 below.
-5
0
5
-10 -5 0 5 10
QQplot for Taiwandailyreturn(Student's t)
TueMay5, 1992
MonMay4, 1992
Fri Jun27, 1997
Quantiles of student's t distribution
Quantilesofstandardisedresiduals
-6
-4
-2
0
2
4
-6 -4 -2 0 2 4 6
QQplot for Hong Kong dailyreturn(Student's t)
TueMay5, 1992
Fri Jun27, 1997
MonMay4, 1992
Quantiles of student's t distribution
Quantilesofstandardisedresiduals
Figure 5: On the left: QQ plot for Hong Kong distribution of daily return; on the right: QQ plot for
distribution of Taiwan daily return. X-axis is the quantile of student’s distribution; Y-axis is the
quantile of the standardised residuals after fitting AR(1)-GARCH(1,1) in Hong Kong data and
ARMA(1,1)-GARCH(1,1) in Taiwan data, respectively.

22
From Figure 5, we notice that, around the median, observations lied on the two 45 degree
lines, which evidenced that the distributions of the two standardised residuals follow the
student‟s t distributions. However, a few outliners and modest heavy tail features were
observed around both tails. According to Extreme Value Theory (EVT), Generalised
Pareto Distribution (GPD) is capable of modelling the tail dependence during extreme
events. Consequently, we would model the tails of the distribution using Generalised
Pareto Distribution and the centre part of the distribution using Empirical Distribution in
the following section.
3.2.2 Extreme Value Theory: Generalised Pareto Distribution (GPD)
Extreme Value Theory focuses on studying the losses occurred during extreme events.
McNeil et al., (2005) suggest that threshold exceedances related models are the most useful
models for practical applications. They also state that the main distributional model for
threshold exceedances is the Generalised Pareto Distribution (GPD) function:
 0,))(/.1(1
0)),(/.(exp1)(,
/1
)( 


 


ux
uxu xG
Where 0 , and 0x when 0 and  /0  x when 0 ,  are the scale
parameters  are the shape parameters and u is a threshold value. For three different
specific values of  , distribution function of GPD, we know that there will be three
different distributions. When  is 0, it represents an exponential distribution;  > 0, it is an
ordinary Pareto distribution;  < 0, it refers to a Pareto type II distribution.

23
GDP plays an important role in EVT. Because, by assuming GDP, two key functions are
able to calculate easily, namely, the mean excess function and excess distribution function
(exceed threshold u), which are defined by McNeil et al.,(2005) as follows:
1) Function of excess distribution over threshold:
)(1
)()(
)¦()(
uF
uFuxF
uXxuXPxFu



2) Mean excess function: u)X¦u-E(Xe(u)  ; Zivot and Wang (2006) define the
Empirical mean excess function as:  
 un
i i
u
n ux
n
ue 1 )( )(
1
)( , where x(i) (i = 1,…,nu) are
the values of xi such than xi>u.
The standardised residuals terms from both returns were modelled by semi-parametric
GPD model: used empirical distribution (nonparametric) at the centre and GPD
(parametric) at both tails in order to transform the data into uniform distributions within a
unit space [0, 1]2
, before fitting copula models.
We then determined the appropriate threshold values, u, by considering the mean excess
plot which plots the empirical mean excess values en(u) against a number of possible
threshold u. Figure 6 illustrates the mean excess plots for both tails of the standardised
innovation of Hong Kong and Taiwan returns.

24
We determined the approximate upper and lower tails thresholds u by the plots from
Figure 5, which suggest that for Hong Kong, lower and upper thresholds are -1.8 and 1.7;
for Taiwan, lower and upper thresholds are -1.5 and 1.5, respectively.
We can observe from Figure 6 that the empirical findings of QQ-plots also prove our
assumption that upper and lower tails process GPD features.
Figure 5: Mean excess plot for the tails of distributions of standardised innovations of both indices;
Top left: upper tail of Hong Kong index; Top right: lower tail of Hong Kong index; Bottom left: upper
tail of Taiwan index; Bottom right: lower tail of Taiwan index.

25
Hong Kong: Taiwan:
Values of corresponding  and )(u can be estimated by Maximum Likelihood Estimate
(MLE) method. Table 5 displays the thresholds u and corresponding estimates and
standard errors.
Table 5: Thresholds (u) & the corresponding parameters
GPD parameter estimations using MLE for HK and TW: Estimates (Standard Error)
Margin Tail Threshold (u) )(u 
Hong Kong Upper 4.49 % of the data 1.7 0.4143 (0.0801) 0.1273 (0.1397)
Lower 3.347 % of the data -1.8 0.6322(0.1469) 0.1401(0.1733)
Taiwan Upper 5.388 % of the data 1.5 0.4923(0.0916) 0.1680 (0.1411)
Lower 5.061 % of the data -1.5 0.8515 (0.1587) -0.0772 (0.1367)
Note: The threshold values displayed here were inferred from Figure 5. Standard Errors are showed in
brackets.
GPD Quantiles, for xi = 0.127269004001351
Excessoverthreshold
0 1 2 3 4 5 6
0123
Upper Tail
Excessoverthreshold
0 1 2 3 4 5 6
012345
Lower Tail
Excessoverthreshold
0 2 4 6
01234
Upper Tail
GPD Quantiles, for xi = -0.0772398455214557
Excessoverthreshold
0 1 2 3 4
0123
Lower Tail
Figure 6: QQ plots of quatile of excess exceedances against quantiles of GPD, given that lower
and upper thresholds of Hong Kong (on the left) are -1.8 and 1.7; Taiwan (on the right) are -1.5
and 1.5, respectively.

26
From Table 5 above, we notice that, for Hong Kong, fat-tailed behaviours existed in both
upper tail and lower tail of the distribution, due to the positive values of : 0.1273 for upper
tail and 0.1401 for lower tail. Additionally, for Taiwan, fat-tailed feature might be found
because of the positive  value: 0.1680. Nonetheless,  value for lower tail was estimated
to be -0.0772, which is a negative  value, suggesting short-tailed densities.
Diagnostic plots on tails of the distributions were implemented to check for the fit of the
GPD to empirical returns. Figure 7 below illustrates that the fitted GPD performed well
enough with only few outliners.
Scatter plots of the pre-GPD modelled and post-GPD modelled of the bivariate empirical
standardised innovations are showed in Figure 8 below to illustrate the uniform
distribution probability-integral transforming process, namely, mapping data into the [0,1]2
unit square.
2 4 6 8 10
0.000050.000500.00500
x(on log scale)
1-F(x)(onlogscale)
2 3 4 5 6 7
0.000050.000500.005000.05000
x(on log scale)
1-F(x)(onlogscale)
2 3 4 5 6 7 8 9
0.000050.000500.005000.05000
x(on log scale)
1-F(x)(onlogscale)
2 3 4 5 6 7
0.000010.000100.001000.01000
x(on log scale)
1-F(x)(onlogscale)
Figure 7: Diagnostic plots from GDP fit to data from tails of standardised residuals of daily returns
for Hong Kong and Taiwan indices. Top left: Upper tail of Hong Kong; Bottom left: Lower tail of
Hong Kong; Top right: Upper tail of Taiwan; Bottom right: Lower tail of Taiwan.

27
Dependence indicators of the concordance measure, Kendall‟s tau and Spearman‟s rho
were estimated and reported in Table 6. The linear dependence indicator Pearson
coefficient is also showed.
Table 6: Dependence estimation: Pearson coefficient, Kendall’s tau and Spearman’s rho
Dependence Estimation
Linear Dependence Estimation Concordance (non-linear dependence) Estimation
Pearson coefficient Kendall‟s tau Spearman‟s rho
0.13163 0.07793 0.11633
Note: Linear dependence indicator Pearson coefficient (0.13163) is reported on the left of the table. On
the right, two columns show the concordance estimator, Kendall’s tau (0.07793) and Spearman’s rho
(0.1163).
We found that linear dependence indicator Pearson coefficient is greater than the Kendall‟s
tau and Spearman‟s rho. The concordance measures the degree of monotonic dependence
rather than the degree of linear correlation. The positive values of Kendall‟s tau and
Spearman‟s rho showed that the monotonic dependence existed. IFM method mentioned in
Section 2.5 would be applied to implement the Copula models in the following section.
Figure 8: Scatter plots of data: pre and post GPD modelling; Left: scatter plots of
standardised residuals of returns of Hong Kong and Taiwan indices (pre-GPD); Right:
scatter plots of standardised residuals modelled by GPD.
0.0 0.2 0.4 0.6 0.8 1.0
Hong Kong GPD
0.00.20.40.60.81.0
TaiwanGPD

28
3.2.3 Copula models and the dependence structure
We applied Joe and Xu (1996)‟s Inference Function for Margins (IFM) method to estimate
the parameters of the fitted copula models. The previous section has implemented the first
step of IFM, which was to model the parameters of the distributions of the margins using
semi-parameter method: Generalised Pareto Distribution for tails and Empirical
Distribution for the centre of the distribution. In this section, the second step of the IFM
would be employed. We would fit the copula models to the uniform distributed margins
and estimate the copula parameters using MLE based on the estimated margin parameters.
We determined to fit four copulas including the Normal Mixture, Gussian, Gumbel and
Clayton Copula from three distinct copula families, respectively. (See description and
definition of these copulas in Section 2). The estimated parameters and the selection
criterions such as AIC, BIC and Log-Likelihood values were showed in Table 6 (Top half).
Table 7: Parameters estimated from fitted copulas and related selection criteria
Fitted Bivariate Copula
Parameters
estimated
AIC BIC
Log-
Likelihood
Normal Mixture
Copula
P = 0.499999286
δ1= 0.12577758
δ2= 0.12577681
-13.544748 1.7873402 9.772374*
Gumbel Copula δ = 1.06092320 -8.940095 -3.8293987 5.470047
Clayton Copula δ = 0.130759659 -14.696099 -9.5854024 8.348049
Gussian Copula δ = 0.12577720 -17.54475* -12.43405* 9.772374*
Student’s t-Copula
Ρ= 0.12667914
d.f.=207.49325
-15.352786 N.A N.A
Note: “*” indicates the best Copula model suggested by different selection criterion. The estimated
parameters of the four copula models are showed in the second column.

29
Table 7 presents the parameters of the copulas. Parameter P from the Normal Mixture
Copula denotes the probability which is approximately 50%. Two parameters, δ1
(0.12577758) and δ2 (0.12577681) are also very close to each other, which inspired us to
consider the Gussian Copula model itself. We found that the implemented Gussian Copula
model has estimated parameter δ (0.12577720), which is showed in the bottom half of
Table 7. We also noticed that selection criterion Log-Likelihood indicated its preference on
both Gussian and Normal Mixture Copula. Moreover, the other two selection criteria AIC
and BIC all suggest that the fitted Gussian Copula model outer-performed the other models.
We even compared it with a fitted t-Copula model3
(rho=0.12667914, degree of
freedom=207.49325). The AIC suggested that Student‟s t-copula could not outer-perform
Gussian copula in our report. (AICtCopula = -15.35, AICGussian= -17.54)
Palaro and Hotta (2006) suggest that we could compare graphical representation of the
estimated copula and empirical copula. Zivot and Wang (2006) use the contour plots of the
estimated copula and fitted copula to distinguish the fitted models. We show the
comparison of contour plots from Normal Mixture, Gussian, Gumbel and Clayton Copula,
with the empirical copula, respectively. (see Figure 10)
──────────────────
3
Student‟s t Copula was obtained from Demarta and McNeil (2004). Degree of freedom is denoted
as ν; P is a correlation matrix implied by the dispersion matrix.
In a bivariate case, d is 2 and P can be replaced by rho (ρ), where ρ is the off-diagonal element. 1
vt
is the quantile function of a univariate standard Student‟s t distribution with degree of freedom, ν.

30
From comparing contour plots of empirical copula with our fitted copulas, we found that
Normal Mixture Copula, Gussian Copula and Clayton Copula exhibit less deviations from
the empirical copula and therefore, the results of graphical findings are consistent with the
selection criteria results. Figure 11 below displays the surface and contour plots (P.D.F) of
Figure 10: Copula selection based on graphical observations. First: Contour plots of Empirical
Copula and Normal Mixture Copula; Second: Contour plots of Empirical Copula and Gussian
Copula; Third: Contour plots of Empirical Copula and Gumbel Copula; Fourth: Contour plots of
Empirical Copula and Clayton Copula.
x
y
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Copula Empirical vs. Fitted CDF
x
y
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
y
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
y
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normal Mixture Copula
Gussian Copula
Gumbel Copula
Clayton Copula

31
the four copulas. The differences of the dependence structure in different copula models
can also be observed:
Figure 11: Visualisation of fitted copulas. First: Surface and contour plots of Gussian Copula;
Second: Surface and contour plots of Bivariate Normal Mixture Copula; Third: Surface and contour
plots of Bivariate Gumbel Copula; Fourth: Surface and contour plots of Bivariate Clayton Copula.
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1.0
1.01.1
1.1
1.2
1.2
1.3
1.3
1.4
1.4
1.5
1.5
1.6
1.6
x
y
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.7
0.8
0.9
1.0
1.1
1.2
1.3
x
y
PDF
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.7
0.8
0.9
1.0
1.1
1.2
1.3
x
y
PDF
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1.0
1.01.1
1.1
1.2
1.2
1.3
1.3
1.4
1.4
1.5
1.5
1.6
1.6
x
y
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
1.0
1.2
1.4
x
y
PDF
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.8
0.8
1.0
1.01.2
1.2
1.4
1.6
1.8
2.02.22.4
x
y
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.8
1.0
1.2
1.4
x
y
PDF
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.8
0.8
1.0
1.01.21.41.61.82.02.2
x
y
Normal Mixture Copula
Gussian Copula
Gumbel Copula
Clayton Copula

32
Figure 11 shows clearly that Gumbel Copula and Clayton Copula suggest asymmetric
dependence. Gumbel Copula implies higher dependence in the upper tail. On the contrary,
Clayton Copula presents higher dependence in the lower tail. Normal Mixture Copula and
Gussian do not capture the dependence in tails, but at the centre. Due to the probability
parameter (50%) of the Normal Mixture Copula, it is not surprised to see that Normal
Mixture Copula has very similar characteristic to the Gussian Copula.
After fitting the four Bivariate ARMA-GARCH Copulas, in the next section, Value at Risk
(VaR) would be estimated and back-tested in Section 3.3.
3.3 Value at Risk (VaR) estimation and back-testing
In this section, Value at Risk would be first estimated on the four ARMA-GARCH copula
models based on various bivariate portfolios with different ratio between the weight of
total wealth invested in Hong Kong index and in Taiwan index. Consecutively, the VaR
results would be back-tested to discover their performances.
3.3.1 Value at Risk estimation
VaR was first introduced to the market as „Riskmetrics‟ by banking giant JP Morgan in
1994 and gradually served as an official Basel II indicator from 1999. An investor‟s market
risk exposure can be quantified by VaR. McNeil, et al., (2005) provide a excellent
definition: “ Given some confidence level α (0, 1), the VaR of a portfolio at the confidence
level α is given by the smallest number l such that the probability that the Loss L exceeds l
is no larger than (1- α). Formally,    1)(:inf lLPRlVaR . ”

33
Value at Risk is actually a quantile function, which denotes a maximum loss would occur
with a certain high probability across a time interval (holding period) as a positive number.
Typical values of α could be 0.90, 0.95 and 0.99. For time interval, market risk managers
tend to measure VaR using 1-day or 10-days as holding period. We chose to compute the
VaR with 1-day holding period in this research. The one-day period log-return of the two
assets (indices) portfolio, R is given by:
R = log (W1•e logR(HK)
+ W2•elogR(TW)
) , W1+W2 =1, where
• W1 denotes the weight of total wealth invested in Hong Kong index
• W2 denotes the weight of total wealth invested in Taiwan index
• logR(HK) = log (Pt+1(HK)/ Pt(HK)); i.e 1-day log return of Hong Kong index
• logR(TW) = log (Pt+1(TW)/Pt(TW)); i.e 1-day log return of Taiwan index
• Pt+1 denotes the daily index on day t+1; Pt denotes the index on day t
If we say that VaRα is one-day period with α-th quantile of the return distribution, then
   1)(:inf)( lRPlRVaR , where α is the confidence level.
We calculated the VaR of the four copula-based models at three confidence levels with
different ratio of weight of total wealth invested in Hong Kong index to Taiwan index,
varying from 10%:90% to 90%:10%. All corresponding one-day ahead VaRs from Monte
Carlo simulation (with 10,000 simulated bivariate returns) were listed in Table 8.

34
Table 8: VaR of the Copula-based models at 90%, 95% and 99% confidence level
Value at Risk estimation (Monte Carlo simulation: 10,000 samples each time)
W1:W2 10%:90% 20%:80% 30%:70% 40%:60% 50%:50% 60%:40% 70%:30% 80%:20% 90%:10%
90% confidence level
Normal
Mixture
0.01442 0.01333 0.01288 0.01301 0.01305 0.01303 0.01390 0.01457 0.01617
Gussian 0.01510 0.01361 0.01284 0.01302 0.01288 0.01265 0.01330 0.01418 0.01502
Gumbel 0.01436 0.01318 0.01316 0.01251 0.01238* 0.01262 0.01303 0.01382 0.01527
Clayton 0.01442 0.01362 0.01290 0.01275 0.01253 0.01305 0.01309 0.01403 0.01504
Normal
Mixture
0.02134 0.01953 0.01824 0.01842 0.01811 0.01812 0.01965 0.02048 0.02308
Gussian 0.02223 0.01965 0.01846 0.01804 0.01769 0.01846 0.01916 0.02031 0.02196
Gumbel 0.02171 0.01907 0.01823 0.01736* 0.01749 0.01756 0.01831 0.01978 0.02231
Clayton 0.02123 0.01955 0.01828 0.01816 0.01836 0.01835 0.01878 0.02059 0.02176
Normal
Mixture
0.03842 0.03390 0.03031 0.02970 0.03035 0.03097 0.03348 0.03736 0.04273
Gussian 0.03764 0.03314 0.03223 0.03053 0.03040 0.03057 0.03484 0.03714 0.03856
Gumbel 0.03865 0.03219 0.03216 0.02836 0.02801* 0.03054 0.03417 0.03635 0.04230
Clayton 0.03680 0.03693 0.03213 0.03165 0.03047 0.03190 0.03494 0.03566 0.04011
Note: “*” denotes the VaR estimation with the least estimated value. Gumbel Copula model with equal
weights for both indices shows the smallest VaR at confidence level 90% and 99%. At 95% confidence
level, VaR on Gumbel Copula of portfolio with index weight ratio 40%:60% reported the least VaR value.
Table 8 illustrates the VaR estimations from Monte Carlo simulation on four Bivariate
ARMA-GARCH Copula-based models (Normal Mixed, Gussian, Gumbel and Clayton
Copula) at three different confidence levels (90%, 95% and 99%), respectively. We found
that Gumbel Copula based model suggest the least VaRs in all confidence levels 90%, 95%

35
and 99%, when we hold an equally weighted bivariate portfolio composed by Hong Kong
HangSeng Index (50%) and Taiwan Capitalization Weighted Stock Index (50%).
Moreover, Gumbel Copula also reported the smallest VaR at 95% confidence level when
the ratio of Hong Kong index to Taiwan index is 40%:60%. We then would implement
VaR back-testing procedure to monitor the performance of these one-period-ahead VaR
estimations (see Table 9 and 10).
3.3.2 VaR back-testing
We used the daily observations (total: 195 observations) during Asian Crisis period (from
2nd
July 1997 to 1st
May 1998) as our out-of-sample data. In other words, we would
compare the estimated one-day ahead VaR predictions with the realized losses occurred in
the out-of-sample observations in order to measure the performances of the VaR estimation
under various models. Ideally, the probability p that the realized losses exceed the VaR
(violations occurred) should be 1-α, where α is the confidence interval. p can be calculated
as the ratio of the number of violations to the number of total out-of-sample observations
(p = x/N, where x is the number of violations; N is the sample size: 195). Therefore, we
can monitor the performance of the VaR prediction by comparing the empirical violation
probability p and the theoretical probability 1-α. We considered the performance of the
VaR estimations on the portfolios with different index weight ratios. (Results are showed
in Table 9). At the end, we would apply Kupiec‟s Unconditional Coverage test to further
monitor the accuracy of the VaR measures.

36
Table 9: Results of VaR Back-testing conditional on different portfolio index weight and confidence level
Value at Risk Back-testing: Proportion of violations/Failure rate (f) and (number of violations: X)
W1:W2 10%:90% 20%:80% 30%:70% 40%:60% 50%:50% 60%:40% 70%:30% 80%:20% 90%:10%
90% confidence level; 1-α = 0.1 (19.5)
Normal
Mixture
0.169(33) 0.179(35) 0.200(39) 0.185(36) 0.185(36) 0.205(40) 0.215(42) 0.241(47) 0.226(44)
Gussian 0.160(31) 0.17935 0.20039 0.185(36) 0.190(37) 0.210(41) 0.226(44) 0.251(49) 0.256(50)
Gumbel 0.169(33) 0.179(35) 0.190(37) 0.190(37) 0.190(37) 0.210(41) 0.236(46) 0.251(49) 0.256(50)
Clayton 0.169(33) 0.179(35) 0.200(39) 0.190(37) 0.190(37) 0.200(39) 0.231(45) 0.251(49) 0.256(50)
Normal
Mixture
0.092(18) 0.113(22) 0.138(27) 0.138(27) 0.144(28) 0.144(28) 0.154(30) 0.164(32) 0.159(31)
Gussian 0.082(16) 0.110(21) 0.133(26) 0.149(29) 0.154(30) 0.144(28) 0.154(30) 0.164(32) 0.169(33)
Gumbel 0.082(16) 0.113(22) 0.138(27) 0.154(30) 0.154(30) 0.149(29) 0.154(30) 0.169(33) 0.169(33)
Clayton 0.097(19) 0.113(22) 0.138(27) 0.138(27) 0.138(27) 0.144(28) 0.154(30) 0.164(32) 0.169(33)
Normal
Mixture
0.021(4) 0.026(5) 0.036(7) 0.056(11) 0.056(11) 0.077(15) 0.077(15) 0.067(13) 0.062(12)
Gussian 0.021(4) 0.031(6) 0.031(6) 0.056(11) 0.056(11) 0.077(15) 0.072(14) 0.067(13) 0.067(13)
Gumbel 0.021(4) 0.031(6) 0.031(6) 0.062(12) 0.072(14) 0.077(15) 0.072(14) 0.067(13) 0.062(12)
Clayton 0.021(4) 0.021(4) 0.031(6) 0.046(9) 0.056(11) 0.072(14) 0.072(14) 0.072(14) 0.067(13)
Note: The UNDERLINED observations indicates the VaR models that have the lowest failure rates and
number of violations, in three confidence level (90%, 95% and 99%), conditional on certain index weights
ratios. (Sample size N: 195)
Table 9 above illustrates the back-testing results of the one-day-ahead VaR estimations
under different index portfolio weight ratio. The values of the proportion of observations
are showed in nine columns of Table 9. The values in brackets show the number of
realized losses exceeded the expected VaR estimation from four copula-based model,

37
Normal Mixture, Gussian, Gumbel and Clayton model with GPD modelling in tails and
empirical distribution at the centre. For 90%, 95% and 99% confidence levels, the VaR
model performs better for the portfolio consists of 10% Hong Kong index and 90% Taiwan
index in all four copula models.
The Clayton Copula model with 20% of Hong Kong index and 80% of Taiwan index,
measure at 99% confidence level, is the only model which produced the best VaR
performance without holding the assets with the 1:9 ratio. Moreover, we could also
observe a trend that the larger the weight of the Hong Kong index, the higher the failure
rates and number of violations. Although this trend is less significant in the back-testing
results at 99% confidence level (bottom group in Table 9), we can still see clearly that the
performance of the VaR prediction is affected by the portfolio weight between Hong Kong
index and Taiwan index.
In order to further investigate the accuracy of the aforementioned VaR predictions,
Kupiec‟s Likelihood Ratio test would be applied in the following section.

38
3.3.3 Test of frequency of exceptions (Kupiec’s Likelihood Ratio test)
The Kupiec‟s (1995) test concerns about the frequency of the out-of-sample observations
that exceed the theoretical (expected) number suggesting by Value-at-Risk model, under a
specific confidence level. The null hypothesis of the Kupiec‟s test is that the performance
of the VaR model is accurate and therefore, acceptable. In other words, we examined:
H0: f=1-α VS H1:f≠1-α, where α is the confidence level, f is the failure rate, and 1- α is the
significance level.
The Likelihood Ratio Statistic )])()1log(())1([log(2 XNxXNX
ffLR 
  follows
an asymptotic )1(2
 distribution.
Where, N is the number of out-of-sample observations (N=195); X is the number of
violations; α is the confidence level. Then f, the failure rate, can be computed as the
probability of the observations that exceeds the expected number (f =X/N), given a specific
confidence level. We tested the highlighted Copula models from the portfolio that consists
of 10% Hong Kong market index and 90% Taiwan market index under different
confidence levels in Table 9. The test results were showed in Table 10 below.

39
Table 10: Kupiec’s Unconditional Coverage test
Kupiec’s LR test on the portfolio consists of 10% HK and 90% TW stock indices
Confidence level: 90%
Models Failure Rate (f) Number of violations (N) LR test statistic P-values
Normal Mixture 0.169 33 8.78 0.003*
Gussian 0.160 31 6.37 0.013*
Gumbel 0.169 33 8.78 0.003*
Clayton 0.169 33 8.78 0.003*
Normal Mixture 0.092 18 5.95 0.019*
Gussian 0.082 16 3.56 0.056
Gumbel 0.082 16 3.56 0.056
Clayton 0.097 19 7.32 0.007*
Normal Mixture 0.021 4 1.67 0.20
Gussian 0.021 4 1.67 0.20
Gumbel 0.021 4 1.67 0.20
Clayton 0.021 4 1.67 0.20
Note: Results with three pre-determined confidence levels: 90%, 95% and 99% are showed. Critical value
of x
2
(1) at 5% significance level from statistical table is 3.841. Failure rates, number of violations, test
statistic and P-values are computed.
“*” denotes the model estimated is significant at 5% (corresponding P-value < 0.05), which rejects the
null that the VaR prediction is accurate.
Results from Table 10, suggested that, at the 90% confidence level, all copula-based
models produced poor VaR estimates. Because all the P-values were smaller than

40
benchmark significance level 5%, which rejected the null that the VaR forecast is accurate.
However, at the 95% confidence level, Gussian and Gumbel copula models have P-values
0.056, greater than 5%, which suggested VaR estimations are more accurate than Normal
Mixture and Clayton models. All four copula models reported better VaR estimations at
the highest confidence level. P-values with 99% confidence level are 0.20 which is greater
than 0.05(5% significance level). Therefore, we could not reject the null that the VaR
prediction is acceptable.
3.3.4 Regime shift and underestimation of the VaR
We also noticed that, all copula models across different portfolio weight combinations we
have measured, underestimated the expected VaRs, regardless of confidence levels (see
Table 9).
There might be due to two reasons. Firstly, it might possibly be due to the size of our out-
of-sample is too small (N= 195). Secondly, it could be due to the fact that we chose the
observations from the Asian Crisis period as our out-of-sample data. The regime shift
factor should have considered when we measured the VaR. In order to monitor the regime
shift characteristics, we firstly combined the in-sample return observations (1225) and the
out of sample observations (195), totally 1420 observations. Secondly, we divided the
sample into 20 sub-samples in chronicle order, each containing 71 observations. Thirdly,
we applied VaR estimation on the 20 sub-samples. The twenty values of the VaR clearly
explained the reason for underestimating the VaRs in our research. Figure 12 below
exhibited the regime shift evidence and its implications on our Value at Risk estimations.

41
We chose the last 195 observations as our out-of-sample data. From Figure 12, we spotted
sudden increase of the VaR at the 16th
sample across all confidence levels (90%, 95% and
99%). The last 4 sub-samples contain 4*71 = 281 observations, which mostly lie in our
out-of-sample period. The free fall of the index levels of the two markets, during the Asian
Crisis period can be seen clearly from the Index Comparison graph of Figure 12.
Especially for the Hong Kong index, it dropped from the peak of 16673.27 on 7th
Aug
2007 to the bottom of 8121.06 on 12th
Jan 2008, an astonishing 51.3% decrease, in about
five month‟s time. Therefore, our underestimation of the VaR might be due to the fact that
we back-tested the VaR when regime shift existed during the Asian Crisis period.
Figure 12: Twenty VaRs are showed with different confidence levels (90%, 95% and 99%). Note: The two
market indices comparison are showed in bottom right, where we saw the price level exhibited big
decreases during the Crisis. The dash lines divide the in-sample period (left) and out-of-sample period
(right).

42
4. Conclusion
This research focused on a portfolio composed by two indices (Hong Kong HangSeng
Index and Taiwan Capitalization Weighted Stock Index). We studied four Copula-based
models (Normal Mixed, Gussian, Gumbel, and Clayton Copulas) and their related VaR
measurements. Due to the non-Gussian, volatility-clustering, autocorrelations, heavy-tails
and asymmetric dependence stylised facts of the daily log-returns, we fitted ARMA-
GARCH models to both univariate margins (daily log-returns of Hong Kong HangSeng
index AR(1)-GARCH(1,1) and Taiwan Capitalization Weighted Stock index ARMA(1,1)-
GARCH(1,1)) and modelled the fitted distributions by GPD in tails and empirical
distribution at the centre. AIC and BIC suggest that ARMA-GARCH-Gussian Copula with
GPD modelling in tails is the best fitted model. We noticed that even the Student‟s t copula
could not outer-perform Gussian copula in our research.
We also monitored the performance of the VaR estimations in different copula-based
models and used the Asian Crisis period observations as our out-of-sample data. 9
portfolios of different index weight combinations were measured. Our research showed
that, the back-testing results of the estimation of VaR on portfolio composed by 10% Hong
Kong index and 90% Taiwan index outperformed other portfolios with different index
weight combinations, in all 90%, 95% and 99% confidence levels. Kupiec‟s Likelihood
Ratio test suggested that, at confidence level 90%, all VaR predictions were not accurately
measured (p-values < benchmark significance level: 0.05). At 95% confidence level,
Gussian and Gumbel Copulas exhibited more accurate predictions, with p-values larger

43
than 5%. Moreover, at 99% confidence level, all copula models passed the Kupiec‟s test.
We could not reject the null that the VaR prediction is accurate at 5% significance level.
However, all results we obtained underestimated the expected VaR. This could be due to
the relatively small sample size of the out-of-sample observations. It could also be
explained by the regime shift characteristics during the Asian Crisis period, where we
spotted sudden VaR increases and markets crashes. As a matter of fact, the regime shift
would lead to underestimate the expected VaR. In the future research, it is worth to apply
mutil-variate Copula models to portfolio with n assets (n>2) in order to better investigate
the natures of dependence when investors hold multivariate assets portfolio in real world
circumstances.

44
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