The dependence in financial markets before, during and after the global financial crisis

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before, during and after the Global

Financial Crisis

Andreas Bregiannis

Master’s Thesis to obtain the degree in Actuarial Science and Mathematical Finance University of Amsterdam

Faculty of Economics and Business Amsterdam School of Economics Author: Andreas Bregiannis Student nr: 11622156

Email: bregiannis.andreas@hotmail.com Date: July 12, 2018

Supervisor: Dr. Sami Umut Can Second reader: Prof. Michel Vellekoop

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Statement of Originality

This document is written by Student Andreas Bregiannis who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it.

The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

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Abstract

In this thesis, tail dependence in financial markets is analyzed. In particular, the aim of this thesis is to investigate the existence, the magnitude and the asymmetry of the tail dependence between the log returns of the AEX and the S&P 500, DAX, FTSE 100, IBEX and TOPIX stock market indices. Therefore, the AEX index from the Netherlands is used as a benchmark. Furthermore, the impact of the Global Financial Crisis (GFC) on the magnitude of this tail dependence is examined. To achieve this, a copula approach is used. The marginal log returns are modelled by AR-t-EGARCH models and the bivariate dependence structure is modelled by constant and time-varying versions of the symmetrized Joe-Clayton copula. One important finding of this thesis is that there is asymmetric tail dependence between AEX and the aforementioned stock market indices, with the lower tail dependence stronger than the upper tail dependence in all the cases. Finally, we conclude that the tail dependence was not affected by the GFC since it remains approximately the same during the pre-GFC, GFC and post-GFC periods.

Keywords Tail dependence, Copulas, Financial markets, Global financial crisis, AR-t-EGARCH models, Symmetrized Joe-Clayton copula

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Since my master’s degree studies are nearing their end, I would like, first of all, to express my sincere gratitude to my supervisor, Dr. Sami Umut Can, for his constant support, guidance and his useful comments throughout the whole process of writing my thesis. Secondly, I would like to thank the second reader, Prof. Michel Vellekoop, for taking the time to read and review my thesis. Moreover, I would like to thank all the professors from the University of Amsterdam who gave me the opportunity to gain invaluable knowledge and develop myself during this year. Furthermore, I would like to express my heartfelt thankfulness to NN and Nuffic for awarding me the NN Future Matters Scholarship and especially to my mentor, Bart Frijns, for his continued assis-tance and his valuable advices and guidance throughout this year. Last but certainly not least, I am deeply grateful to my parents, Konstantinos and Elissavet, my sister, Stevi and my girlfriend, Maria, for their endless love, support and encouragement throughout the years.

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Introduction

It is well known that the Global Financial Crisis (GFC), occurred from mid 2007 until early 2009, started in the United States but it was expanded rapidly all over the world. As a consequence, many financial markets became extremely volatile. These financial markets were, therefore, highly interdependent during extreme events in which one of them faced extremely negative results.

In general, the dependence between positive or negative extreme values of random variables is called tail dependence. It may be the case that two random variables are not correlated during regular times but they become strongly dependent when the one becomes extremely small or large, for example due to an extreme event such as a financial crisis. This tail dependence can be asymmetric. In real world, it is often seen that stocks for example, are more strongly dependent during bear markets than bull markets and hence they tend to bust together but they do not boom together.

In this thesis, we will focus on measuring the tail dependence between several stock market indices around the world. In particular, we will investigate the existence, the magnitude and the asymmetry of the tail dependence between the Amsterdam Exchange Index (AEX) from the Netherlands and the Standard & Poor’s 500 Index (S&P 500) from the United States, the German Stock Index (DAX), the Financial Times Stock Exchange 100 Index (FTSE 100) from the United Kingdom, the Spanish Exchange Index (IBEX) and the Tokyo Price Index (TOPIX) from Japan. We decided to use the AEX index from the Netherlands as a benchmark because the Netherlands is one of the largest economies in the European Union and historically it has been active in the EU from the very beginning. In addition to this, it was also, among other countries, negatively affected by the Global Financial Crisis. The choice of the rest stock market indices was motivated by the fact that we would like to gain a better insight for the tail dependence between stock indices from countries belonging to the European Union but also from other countries around the globe.

Furthermore, our aim is to investigate the impact of the Global Financial Crisis on the magnitude of the tail dependence between the AEX and the chosen stock market indices. In particular, we will measure the tail dependence before, during and after the GFC in order to check for possible differences. In this way, we will be able to conclude if the GFC affected the tail dependence, but also whether these stock markets indices, after the GFC period, are as tail dependent as they were before and hence if they face similar risks in a new potential extreme event.

In the next section, we will summarize the research questions that we wish to answer through the subsequent analysis.

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2 Andreas Bregiannis — Tail Dependence in Financial Markets

1.1 Research Questions

Based on the aforementioned goals, the research questions that will be answered in this thesis are the following

i. Is there tail dependence between the log returns of the AEX index and the S&P 500, DAX, FTSE 100, IBEX and TOPIX stock market indices?

ii. If yes, what is the magnitude of the tail dependence? Is there asymmetry in the tail dependence?

iii. Did the Global Financial Crisis affect the magnitude of this tail dependence?

1.2 Methodology

In this section, the methodology that has been followed to answer the questions above, is described. First of all, to tackle these problems, the relevant literature needed to conduct this analysis was studied. Afterwards, we gathered daily data from the relevant stock indices for a period of 15 years representing periods before, during and after the GFC. To be more precise, we collected data from January 1st, 2001 until December 31st, 2015. The beginning of August 2007 was chosen as the start of the GFC since the liquidity was worsening in the money market, following negative announcements by the investment bank BNP Paribas with the European Central Bank inserting in total e203.7 billions into the banking market trying to improve the liquidity. At the same time, the US Federal Reserve, the Bank of Canada and the Bank of Japan started intervening. The end of March 2009, was selected as the end of the GFC since April 2009 was characterized by the lack of negative news and a stock market rally. Therefore, based on Baur (2012) and on the timelines of the GFC provided by Guill´en (2009) and by the Bank for International Settlements (2010), we denoted as the pre-GFC period from January 1st, 2001 until July 31st, 2007, as the GFC period from August 1st, 2007 until March 31st, 2009 and as the post-GFC period from April 1st, 2009 until December 31st, 2015.

After collecting the corresponding data, we computed the daily log returns of the stock market indices and we conducted an exploratory data analysis. The empirical findings of this analysis helped us to get a better understanding for our data sets, their characteristics and their corresponding interdependence.

To compute the tail dependence, we decided to use a copula approach because, as we will see in the next chapter, copulas are able to capture the entire dependence structure between random variables, in contrast with the traditional dependence measures such as linear correlation that can not do this. As a result, copulas are directly used to measure the tail dependence. Thus, in our modelling process, for each chosen pair of stock indices, we modelled the marginal log returns using AR-t-EGARCH models and afterwards we applied the so-called probability integral transformation to the univariate standardized residuals to transform them and make them follow the standard uniform distribution. Finally, we used the resulting standard uniformly data as inputs to estimate the constant and time-varying Symmetrized Joe-Clayton (SJC) copula using maximum likelihood estimation. Compared to other copulas, the time-varying SJC copula proved to be the most appropriate choice for our case since it allows for asymmetry between the lower and upper tail dependence and nests symmetry as a special case. Moreover, this flexible time-varying version of the SJC copula enabled us to capture how the tail dependence evolves over time before, during and after the GFC period.

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1.3 Outline of the Thesis

The first chapter of this thesis was the Introduction, in which the topic, the research questions and the methodology were discussed.

In chapter 2, the background information and some important concepts, needed for the analysis, are described. In particular, the concepts of unconditional and condi-tional copulas as well as the tail dependence are discussed. Furthermore, some widely used copulas and their corresponding tail dependence are mentioned, concluding the appropriateness of the SJC copula for our case.

In chapter 3, an exploratory data analysis is conducted. First, the summary statistics for each data set are presented. Secondly, the scatter plots of the log returns as well as the scatter plots of the empirical copulas for each chosen pair are displayed. Finally, the empirical copula frequency tables are given.

In chapter 4, the modelling approach is explained. In this chapter, the existence of volatility clustering is presented and then the AR-t-EGARCH models utilized for the marginal distributions, the probability integral transformation and the estimation of the SJC copula are described.

The obtained results for the tail dependence between the chosen pairs of stock market indices are presented and discussed in chapter 5. The results are mainly focused on the existence, the magnitude and the asymmetry of the tail dependence for the entire period as well as for the periods before, during and after the GFC.

The conclusion of the thesis is found in chapter 6. The studied bibliography can be found in the end of this paper.

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Chapter 2

Background Information and

Important Concepts

In this chapter, some crucial concepts of the relevant literature are reviewed. In particular, unconditional and conditional copulas, tail dependence and some widely used copulas are discussed. These concepts will be needed in our subsequent case study and hence this chapter will enable reader to gain a better insight into them.

The first concept that will be discussed is the unconditional copulas.

2.1 Unconditional Copulas

Copulas are widely used in financial applications to model the dependence structure among variables since, unlike the dependence measures, they capture the entire depen-dence structure. Therefore, they inform us not only about the degree of the dependepen-dence but also about the structure of the dependence. As a consequence, different copulas can model, for example, the asymmetric dependence but also they can capture the tail de-pendence, which is the main purpose of this thesis. In addition to this, copulas are very popular due to the fact that they hold the invariance property. So, when for example, we apply logarithmic transformation to the marginal distributions, the copula is not affected. The concept of copulas was introduced by Sklar (1959) and the name ’copula’ was chosen because a copula ’couples’ a joint distribution function to its univariate margins.

2.1.1 Definition

The copula of the random vector (X, Y ) with joint cumulative distribution function H(x, y) and marginal cdfs F (x) and G(y) is the joint cdf of the (standard uniform) ranks F (X) and G(Y ). Therefore

C(u, v) = P r[F (X) ≤ u, G(Y ) ≤ v] = H(F−1(u), G−1(v)) (2.1)

2.1.2 Sklar’s Theorem

If H is a joint distribution function with marginal distribution functions F and G, then a copula C exists such that for all x, y ∈ R,

H(x, y) = C(F (x), G(y)) (2.2)

Furthermore, if F and G are continuous, then copula C is unique.

Conversely, if C is a copula and F and G are distribution functions, then the function H defined by (2.2) is a joint distribution function with margins F and G. Also,

h(x, y) = f (x) · g(y) · c(F (x), G(y)) (2.3) 4

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where h denotes the density of H, f and g the densities of F and G respectively and c the density of the copula C.

From Sklar’s theorem, we see that a joint distribution can be decomposed into its univariate marginal distributions, which characterize the variables that we are interested in (stock log returns in our case) and a copula, which captures the dependence structure between those variables. Thus, the equation (2.2) decomposes a bivariate cumulative distribution function and the equation (2.3) decomposes a bivariate probability density function.

This theorem can easily be extended in the multivariate case:

Let X = (X1, . . . , Xn) be a random vector with distribution function H and with marginal distribution functions Fi, that is Xi ∼ Fi, 1 ≤ i ≤ n.

A distribution function C with standard uniform marginals is called the copula of X if H(x1, . . . , xn) = C(F1(x1), . . . , Fn(xn)) (2.4) However, in this thesis, only bivariate copulas will be considered.

In the next section, the conditional copulas are discussed.

2.2 Conditional Copulas

Conditional copulas, introduced by Patton (2006), enable us to analyze the time-varying conditional dependence.

First of all, lets define the notation. X and Y are the variables of interest and W is the conditioning variable, which may be a vector. FXY W is the joint distribution of (X, Y, W ), F_{XY |W} is the conditional distribution of (X, Y ) given W , and the conditional marginal distributions of X|W and Y |W are FX|W and FY |W, respectively.

2.2.1 Definition

It is also very important to mention, according to Patton, that by extending the existing results, it can easily be shown that a conditional copula has the same properties with an unconditional copula, ∀ w ∈ W .

The Sklar’s theorem is also extended for conditional distributions.

2.2.2 Theorem

FX|W(·|w) and FY |W(·|w) are the conditional distributions of X|W = w and Y |W = w respectively, FXY |W(·|w) is the joint conditional distribution of (X, Y )|W = w, and W is the support of W. We assume that F_X|W(·|w) and F_{Y |W}(·|w) are continuous in x and y for all w ∈ W . Then, there exists a unique conditional copula C(·|w) such that

FXY |W(x, y|w) = C(FX|W(x|w), FY |W(y|w)|w), (2.5) ∀(x, y) ∈ R × R and each w ∈ W .

However, we have to be careful when we extend Sklar’s theorem to conditional dis-tributions because the conditioning variable(s), W , must be the same for both marginal distributions and the copula since, in general, ˜FXY |W1,W2 will not be the joint

dis-tribution of (X, Y )|(W1, W2). This condition is only satisfied in the, not uncommon, special case when F_X|W₁(x|w1) = FX|W1,W2(x|w1, w2) ∀ (x, w1, w2) ∈ R × W1 × W2

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and F_{Y |W}₂(y|w2) = FY |W1,W2(y|w1, w2) ∀ (y, w1, w2) ∈ R × W1× W2, that is when

cer-tain variables affect the conditional distribution of one variable but not the conditional distribution of the other.

Conversely, as before, if FX|W(·|w) is the conditional distribution of X|W = w and FY |W(·|w) is the conditional distribution of Y |W = w and {C(·|w)} is a family of conditional copulas measurable in w, then the function F_{XY |W}(·|w) defined by the equation (2.5) is a conditional bivariate distribution function with conditional marginal distributions FX|W(·|w) and FY |W(·|w).

The last result is very useful especially for multivariate density modelling, because it implies that we can couple any two univariate distributions, of any type, with any copula and in this way, we will have defined a valid bivariate distribution.

Moreover, if F_X|W and F_{Y |W} are differentiable and F_{XY |W} and C are twice differ-entiable, we can easily obtain the density function by:

f_{XY |W}(x, y|w) ≡ ∂ 2_F XY |W(x, y|w) ∂x∂y = ∂FX|W(x|w) ∂x · ∂F_{Y |W}(y|w) ∂y · ∂2_C(F X|W(x|w), FY |W(y|w)|w) ∂u∂v

=⇒ f_{XY |W}(x, y|w) ≡ f_X|W(x|w) · f_{Y |W}(y|w) · c(u, v|w) (2.6) where u ≡ F_X|W(x|w) and v ≡ F_{Y |W}(y|w), ∀ (x, y, w) ∈ R × R × W .

The next concept that we will analyze is the tail dependence.

2.3 Tail Dependence

The tail dependence is the dependence between extreme values of random variables and it is measured by the upper and lower tail dependence coefficients.

Thus, according to Joe (1997), the concept of bivariate tail dependence relates to the amount of dependence in the upper-quadrant tail or lower-quadrant tail of a bivariate distribution.

The topic of tail dependence is crucial, for example, for investors because when somebody holds a portfolio of stocks, it is possible that, during regular times, the re-turns of different stocks are weakly dependent, but during extreme events such as a financial crisis, their dependence can be quite strong if any of them becomes unusually small/large. Of course, this situation is very dangerous and due to this fact the tail dependence needs to be captured correctly in order to prevent extremely large losses in portfolios that seem to be well diversified but in fact they are not.

In the following subsection, we are going to provide a definition for the upper and lower tail dependence, similar to Joe (1997) and McNeil et al. (2015).

2.3.1 Definition

In the case of upper tail dependence we look at the probability that X2 exceeds its q-quantile, given that X1 exceeds its q-quantile, considering the limit as q goes to one. So, if X1and X2are random variables with continuous cumulative distribution functions F1 and F2 respectively, then the coefficient of upper tail dependence of X1 and X2 is given by

λu := λu(X1, X2) = lim

q→1−P (X2 > F

←

2 (q) | X1 > F1←(q)) (2.7) where, λu ∈ [0, 1], F1← and F2← denote the inverse distribution function of X1 and X2 respectively.

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Similarly, the coefficient of lower tail dependence is given by λl:= λl(X1, X2) = lim

q→0+P (X2≤ F

←

2 (q) | X1≤ F1←(q)) (2.8) The coefficients of upper and lower tail dependence can be expressed in terms of the unique copula C of the bivariate distribution of X1 and X2. Hence, the lower tail de-pendence coefficient can be written as

λl:= lim q→0+ P (X2 ≤ F2←(q), X1≤ F1←(q)) P (X1≤ F1←(q)) = lim q→0+ C(q, q) q (2.9)

and the upper tail dependence coefficient can be expressed as λu = lim

q→1−

1 − 2q + C(q, q)

1 − q (2.10)

Below, the interpretation of these coefficients is provided.

If λu = 0, then we say that X1 and X2 are asymptotically independent in the upper tail. If λu ∈ (0, 1], we say that X1 and X2 show upper tail dependence or extremal dependence in the upper tail, that is, loosely speaking, the larger the λu the larger the probability that X2 exceeds a certain high threshold given that the X1 has already exceeded that threshold.

Analogously, if λl = 0, then we say that X1 and X2 are asymptotically independent in the lower tail. If λl∈ (0, 1], then we say that X1 and X2 show lower tail dependence or extremal dependence in the lower tail. That means, roughly speaking, that the larger the λl the larger the probability that X2 does not exceed a certain low threshold given that the X1 has not exceeded that threshold.

Since these coefficients can be expressed in terms of the unique copula, this implies that we can directly calculate them if we have a known closed form copula.

In the following section, we will mention some widely used copulas and their corre-sponding tail dependence. In addition to that, we will explain our motivation for the choice of the copula in our case.

2.4 Examples of Copulas and their Tail Dependence

One well known copula is the Gaussian. For a bivariate normal random vector (X, Y )0 with correlation coefficient ρ = cor(X, Y ), its Gaussian copula is given by

CGa(u, v) = Φ(Φ−1(u), Φ−1(v)) = Z Φ−1(u) −∞ Z Φ−1(v) −∞ 1 2πp(1 − ρ2₎exp −(s2_{+ t}2_{− 2ρst)} 2(1 − ρ2₎ dsdt. (2.11) The Gaussian copula is not an appropriate choice for our case because it does not allow for tail dependence unless ρ = 1. Therefore, it assumes that λu= λl = 0

One commonly used copula is the Student’s t copula. For a bivariate Student’s t random vector (X, Y )0 with correlation coefficient ρ = cor(X, Y ) and ν degrees of freedom, its Student’s t copula is defined as

Ct(u, v) = Z t−1ν (u) −∞ Z t−1ν (v) −∞ 1 2πp(1 − ρ2₎ 1 +s 2_{+ t}2_{− 2ρst} ν(1 − ρ2₎ −ν+2₂ dsdt. (2.12)

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In contrast with the Gaussian copula, the Student’s t copula allows for tail dependence but only for symmetric.

As a consequence, λu = λl= 2tν+1 −r (ν + 1)(1 − ρ) 1 + ρ .

However, many times this is not the case in real world since for example stock returns are often asymmetric dependent in the tails, with the lower tail dependence stronger than the upper.

Two Archimedean copulas, that are often used in practice, are the Gumbel and the Clayton copulas. The bivariate Gumbel copula is given by

CGu(u, v) = exp − (−lnu)θ+ (−lnv)θ 1/θ

, 1 ≤ θ < ∞, (2.13) where θ is the parameter which represents the strength of dependence.

The bivariate Clayton copula is given by

CCl(u, v) = (u−θ+ v−θ− 1)−1/θ, 0 < θ < ∞, (2.14) The Gumbel copula does not allow for lower tail dependence but it does allow for upper tail dependence. In particular, λu= 2 − 21/θ and λl = 0.

On the other hand, the Clayton copula allows for lower tail dependence but it does not allow for upper tail dependence. Hence, λu = 0 and λl= 2−1/θ.

In our subsequent analysis, we would like to capture both lower and upper tail depen-dence and hence these copulas are not appropriate choices.

One very important copula, in general but also for our analysis, is the Joe-Clayton or BB7 copula, introduced by Joe (1997). The Joe-Clayton copula is given by

CJ C(u, v|λu, λl) = 1 −1 − ([1 − (1 − u)κ]−γ + [1 − (1 − v)κ]−γ− 1) −1 γ 1 κ _(2.15) where κ = 1 log2(2 − λu) and γ = − 1 log2(λl) .

The Joe-Clayton copula has two parameters, the λu = 2 − 21/κ and λl = 2−1/γ, which represent the coefficients of upper and lower tail dependence respectively and hence both λu and λl∈ [0, 1].

The Joe-Clayton copula allows for both upper and lower tail dependence. However, one disadvantage of this copula is that even when λu = λl, there is still some small asymmetry in the copula.

A modification of the Joe-Clayton copula is the Symmetrized Joe-Clayton (SJC) copula which allows for asymmetric tail dependence and nests symmetry as a special case when λu = λl. Patton (2006) introduced the constant as well as a time-varying version of this copula. This copula is the one that will be used in our subsequent analysis and it was chosen among the other copulas based on its properties and the AIC values. The time-varying version of the SJC copula allows us to capture how the tail dependence between the stock market indices evolves over time. As a result, it enables us to measure the tail dependence before, during and after the Global Financial Crisis. In addition to that, its functional form allows us to take into account all the possible scenarios for tail dependence, that is, asymmetric, symmetric or no tail dependence.

In the following paragraphs, the constant as well as the time-varying versions of Symmetrized Joe-Clayton (SJC) copula are described.

2.5 Symmetrized Joe-Clayton Copula

2.5.1 Constant SJC

As already mentioned, the Symmetrized Joe-Clayton copula, proposed by Patton (2006), is a small modification of the Joe-Clayton copula that allows for asymmetry

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between the upper and lower tail dependence but it considers symmetry as a special case when λu = λl.

The SJC copula is given by

CSJ C(u, v|λu, λl) = 0.5 ×CJ C(u, v|λu, λl) + CJ C(1 − u, 1 − v|λl, λu) + u + v − 1)

(2.16) Similarly to the Joe-Clayton copula, the SJC copula has the same two parameters, λu∈ [0, 1] and λl∈ [0, 1], which represent the respective upper and lower tail dependence coefficients as before.

2.5.2 Time-varying SJC

In the time-varying specification of the SJC copula, that Patton (2006) proposed, it is assumed that the formula of the copula does not change over time, while the two parameters λu and λl vary over time based on the following evolution equations. The evolution equation proposed for the upper tail dependence coefficient is

λu_t = Λωu+ βuλut−1+ αu 1 10 10 X i=1 |u_t−i− v_t−i| (2.17) Analogously, the corresponding evolution equation for the coefficient of the lower tail dependence is λl_t= Λ ωl+ βlλlt−1+ αl 1 10 10 X i=1 |ut−i− vt−i| (2.18) where Λ(x) = 1

1 + e−x is utilized to ensure that λu and λl will be always in [0, 1]. From the evolution equations above, we see that λu and λl follow a process similar to a restricted ARMA(1,10) process that includes an autoregressive term and the mean absolute difference between the previous ten values of u and v as a forcing variable. Moreover, it should be noticed that the time-varying SJC copula has six parameters, ωu, βu, αu, ωl, βl and αlto be estimated, from which the two typical parameters of upper and lower tail dependence are obtained.

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Chapter 3

Exploratory Data Analysis

In this chapter, we will conduct and exploratory data analysis that will help us to get a better understanding for our data sets, their characteristics and their dependence. The six data sets consist of daily stock index log returns from January 2nd, 2001 until December 31st, 2015 leading to 3913 observations.

3.1 Summary Statistics

We would like to start our statistical analysis by computing some standard summary statistics for each one of our six data sets. In particular, the sample mean, the standard deviation, the skewness, the kurtosis as well as the minimum and maximum values are computed.

In the following table we present the results for each data set.

Table 3.1: Summary Statistics

Mean St.Dev. Skewness Kurtosis Minimum Maximum AEX -0.000094 0.014816 -0.075983 9.392053 -0.095903 0.100283 S&P 500 0.000074 0.013402 -0.084921 9.095964 -0.091080 0.103675 DAX 0.000131 0.015313 -0.022333 7.628749 -0.088747 0.107975 FTSE 100 -0.000040 0.013197 -0.143744 9.464420 -0.094799 0.096472 IBEX 0.000012 0.014922 0.091083 8.294622 -0.095859 0.134836 TOPIX -0.000003 0.014021 -0.233273 6.637233 -0.088728 0.109424 From the table 3.1, we see that the average log return of each stock index is close to zero with AEX, FTSE 100 and TOPIX having a negative average log return and S&P 500, DAX and IBEX having a positive mean log return. Roughly speaking, the standard deviation of the log returns of all stock market indices are approximately between 0.013 and 0.015. Moreover, the minimum log return for all stock market indices is roughly -0.09 and the maximum is approximately 0.10 with AEX having the lower log return of -0.095903 and IBEX having the higher log return of 0.134836.

Since our main focus in this thesis is on the tail dependence, we are interested more in the tail behaviour of our datasets. Therefore, we will focus more on the results of skewness and kurtosis. The skewness of FTSE 100 and TOPIX is negative, indicating that their distributions are skewed left, that is, left tail is long relative to the right tail. The distributions of S&P 500, AEX and DAX are slightly skewed left. We say that they are slightly skewed to the left because the value of the skewness is negative but close to zero. The distribution of IBEX, in contrast, is slightly skewed right, that is, the right tail is long relative to the left tail. Regarding the kurtosis, in all the cases, we find out that the corresponding value is quite bigger than three. We remind that standard normal distribution has a kurtosis equal to three. This indicates that all of our datasets

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come from leptokurtic distributions, that is, they follow heavy-tailed distributions. Con-sequently, they have many outliers with very high or very low values. Furthermore, we found out before that the five out of six distributions are skewed left and hence their long tail is on the negative direction. Taking these results into account, we verify that it is indeed vital to determine the tail dependence between these stock indices.

In the next section, we will start investigating the dependence between our datasets.

3.2 Scatter plots of the Log Returns

In this section, we will present the scatter plots of the logarithmic stock returns of AEX with those of the other stock indices in order to provide a graphical representation of the relationship between them.

Figure 3.1: Scatter plots of the log returns

Based on the scatter plots above, we see that there is strong positive correlation (with few outliers) between the logarithmic stock returns of the pairs AEX-DAX, AEX-FTSE 100 and AEX-IBEX. That means for each of these pairs that the two stock indices generally move in the same direction with quite similar magnitude. Thus, for example, we can say that lower log returns of the stock index AEX are related to lower log returns of these three aforementioned stock market indices. For the pair AEX-S&P 500, one could say that there is a weaker positive dependence with many outliers and for the pair AEX-TOPIX it seems that, in general, there is no significant correlation.

3.3 Scatter plots of the Empirical Copulas

In this section, we would like to get a better understanding of the dependence struc-ture in our data. For this reason, we are going to construct the scatter plots of the empirical copulas. The empirical copula is an estimate of the true copula and it is

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formed by the scaled ranks of the stock index log returns. Therefore, first, we rank the log returns of each stock index from the smallest observation to the largest one. Then, we scale the ranks in the interval [0,1] by dividing them with the size of the data set and finally we construct the empirical copula of each pair by binding the corresponding scaled ranks into a 3913 × 2 matrix.

It is important to mention that both in this section as well as in section 3.4, we refer to the scaled ranks of the original log returns, even though they are not i.i.d. However, later we will fit the copulas to the scaled ranks of the standardized residuals obtained by the AR-t-EGARCH models.

The scatter plots of the empirical copulas are presented below.

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From the figure 3.2 above, we see that the data ranks of the pairs S&P 500, AEX-DAX, AEX-FTSE 100 and AEX-IBEX exhibit a thicker concentration in the upper right and lower left corners. This indicates that both upper and lower tail dependence exist with the one of the pair AEX-S&P 500 to be weaker than those of the other three pairs. Regarding the pair AEX-TOPIX there is no indication of upper or lower tail dependence since we do not see a thicker concentration of data ranks in the upper right or in the lower left corner.

3.4 Empirical Copula Frequency Tables

Now, we would like to check numerically the result found above. Therefore, in con-tinuation to the scatter plots of the empirical copulas and similarly to Knight et al. (2005), we will calculate the empirical copula frequency tables.

In order to achieve this, first of all, we take the scaled ranks as before and then we divide them evenly into 10 bins. In this way, bin 1 contains the observations with the lowest values and bin 10 includes the observations with the highest values.

In general, from such an empirical copula frequency table we can conclude the following: i. If the two log return series are (perfectly) positively correlated, then most observations will lie on the diagonal that connects the upper left corner and the lower right corner. ii. If they are independent, then we expect approximately the same amount of observa-tions in every cell.

iii. If they are (perfectly) negatively correlated, then the most observations should lie on the diagonal that connects the upper right corner and the lower left corner.

iv. If the two log return series are dependent in the lower tail, then more observations should be in the cell(1,1).

v. If they are dependent in the upper tail, then a large amount of observations will be in the cell(10,10).

The empirical copula frequency tables for the selected pairs of stock market indices are given below.

Table 3.2: Empirical Copula Frequency Table - AEX vs S&P 500

1 2 3 4 5 6 7 8 9 10 1 171 65 38 19 18 15 12 17 19 18 2 79 74 54 50 36 20 27 16 18 17 3 40 51 64 58 38 36 28 35 26 15 4 27 39 55 54 49 54 39 32 23 19 5 19 39 39 38 84 42 48 39 21 23 6 9 39 41 50 46 56 51 39 42 18 7 8 30 31 48 38 53 53 57 42 31 8 10 13 33 41 36 55 54 55 56 38 9 11 21 27 18 33 34 54 59 82 52 10 18 20 9 15 14 26 25 42 62 161

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Table 3.3: Empirical Copula Frequency Table - AEX vs DAX

1 2 3 4 5 6 7 8 9 10 1 286 77 15 3 2 2 2 2 1 2 2 78 166 80 33 10 10 8 2 2 2 3 20 82 132 70 41 23 10 6 5 2 4 3 36 82 113 59 56 21 12 5 4 5 1 15 38 86 137 54 31 21 5 4 6 0 5 20 43 71 111 86 33 18 4 7 1 4 14 23 47 76 105 79 34 8 8 1 4 1 11 15 39 85 129 84 22 9 0 2 5 7 9 18 34 88 161 67 10 2 0 4 2 1 2 9 19 76 277

Table 3.4: Empirical Copula Frequency Table - AEX vs FTSE 100

1 2 3 4 5 6 7 8 9 10 1 274 64 33 12 4 2 1 0 0 2 2 71 153 74 32 21 14 10 7 5 4 3 24 82 105 77 40 26 13 16 8 0 4 9 37 81 84 77 37 24 24 14 4 5 4 25 39 56 100 83 47 21 13 4 6 3 10 33 55 55 82 73 47 26 7 7 6 9 16 43 61 61 84 62 39 10 8 1 5 7 19 24 57 77 91 87 23 9 0 2 0 9 6 24 48 94 133 75 10 0 4 3 4 4 5 14 29 66 263

Table 3.5: Empirical Copula Frequency Table - AEX vs IBEX

1 2 3 4 5 6 7 8 9 10 1 254 90 25 9 2 5 5 2 0 0 2 81 103 106 48 22 14 7 5 4 1 3 28 86 87 75 41 33 22 11 7 1 4 16 46 73 79 59 51 33 17 7 10 5 5 31 39 67 122 51 36 25 12 4 6 4 11 27 52 59 93 62 48 30 5 7 1 8 19 38 42 75 76 79 40 13 8 2 11 11 16 27 30 82 96 87 29 9 1 3 3 6 14 29 46 89 124 76 10 0 2 1 1 4 10 22 19 80 253

Table 3.6: Empirical Copula Frequency Table - AEX vs TOPIX

1 2 3 4 5 6 7 8 9 10 1 90 45 36 33 22 21 34 33 38 40 2 57 48 42 30 50 21 33 32 45 33 3 49 51 41 45 34 22 47 41 33 28 4 36 37 36 43 48 40 28 48 38 37 5 34 28 41 53 60 42 38 31 32 33 6 18 29 37 50 58 44 50 49 30 26 7 23 38 32 40 48 40 49 35 47 39 8 20 32 46 25 41 41 46 40 51 49 9 22 38 43 46 48 36 40 45 32 41 10 43 45 37 26 36 31 26 38 44 66

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From Table 3.2, we see for the AEX-S&P 500 pair, that in cell(1,1) there are 171 observations. In other words, this means that out of 3913 observations, there are 171 times when both stock index log returns are in their respective lowest 10th percentile. Similarly, we find out that in cell(10,10) there are 161 observations, which means that 161 times both stock index log returns are in their respective top 10th percentile. The number of observations in the rest of the cells is much smaller than those in these two cells. Therefore, empirically, we can conclude that both lower and upper tail dependence exist between the AEX and the S&P 500 and it seems that possibly there is a slight asymmetry in the tail dependence with the lower tail dependence to be stronger than the upper.

Similarly, for the pairs AEX-DAX, AEX-FTSE 100 and AEX-IBEX, we observe that both upper tail and lower tail dependence are present since most observations lie in the cell(1,1) and in the cell(10,10). Moreover, for the pairs AEX-DAX and AEX-FTSE 100, a higher percentage of observations lie in the cell(1,1), indicating evidence of stronger lower tail dependence, whereas for the pair AEX-IBEX, approximately the same number of observations is in the cell(1,1) and in the cell(10,10), empirically denoting symmetry in the tail dependence. Moreover, the log returns of AEX seem to be positively correlated with the log returns of DAX, FTSE 100 and IBEX, since we notice in the corresponding tables, that most observations lie on the diagonal that connects the upper left corner and the lower right corner. This is also confirmed by the scatter plots of the log returns presented in section 3.2 above.

Finally, regarding the pair AEX-TOPIX, we observe that lower tail dependence seems to exist because most observations are in the cell(1,1). Also, there is some evidence of weak upper tail dependence since the second largest percentage of observations lie in the cell(10,10) but this percentage is not significantly larger than the percentage of observations lying in the other cells.

Thus, the empirical copula frequency tables show evidence of both lower and upper tail dependence between the AEX and the other stock market indices. Furthermore, in most of the cases, there is evidence of asymmetric tail dependence with the lower tail dependence stronger than the upper. Taking these findings into consideration, we claim that a proper selection of copula is the Symmetrized Joe-Clayton (SJC) copula because its properties will allow us to capture both asymmetric as well as symmetric tail dependence.

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Chapter 4

Modelling Approach

As we have already mentioned, our main goal is to measure the upper and lower tail dependence between the AEX stock index and the other five stock indices before, during and after the Global Financial Crisis, using a copula approach.

4.1 Modelling Process

Since we want to capture the dependence structure between the stock index log returns, we have to specify the models for the marginal distribution for each of the two stock indices belonging in the pair as well as for the joint distribution of them using a copula.

This approach is the so-called Inference Functions for Margins (IFM) estimation method, according to Joe and Xu (1996) in which we divide the estimation procedure into two steps. First we estimate the parameters of the univariate distributions and secondly, given these estimated parameters, we estimate the copula parameters. Hence, the methodology used is the following: first we estimate the two marginal distributions, then we take the univariate standardized residuals and we apply the so-called probability integral transformation to them, in order to transform the univariate distributions into standard Uniform and in the end we use the obtained standard uniformly distributed variables as inputs to estimate the copula with maximum likelihood estimation.

4.2 Modelling the Marginal Distributions

In this section, the modelling approach for the marginal distributions is described. First, we check for volatility clustering in our time series and then we implement ap-propriate AR-t-EGARCH models to the marginal distributions.

4.2.1 Volatility Clustering

It is well known and often mentioned in the relevant literature that the time series of stock returns exhibit volatility clustering. That was noted by Mandelbrot (1963) and it means that large changes and small changes in the stock returns tend to cluster together. In other words, large changes tend to be followed by large changes and small changes tend to be followed by small changes of either sign.

We would like to check for volatility clustering in the log returns of our case. Thus, below we provide the corresponding plots.

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Figure 4.1: Plots of the log returns - Volatility Clustering

Obviously, from the figure 4.1 above, we conclude that in our case, volatility clustering is also present and hence we would like to properly capture this volatility clustering. For this reason, we are going to use several Generalized Auto Regressive Conditional Heteroskedasticity (GARCH) models and exponential GARCH (EGARCH) models and we will choose the most suitable model for each series of log returns.

4.2.2 AR-t-EGARCH Models

GARCH models can capture volatility clustering because volatility σt is changing over time and it depends on past values of the process. We name this processes GARCH, that is, generalized ARCH processes, because the squared volatility is allowed to depend on previous squared volatilities, but also on previous squared values of the process. Implementing several types of GARCH and EGARCH models, we find that in all the cases, EGARCH models are the most suitable. Exponential GARCH models, introduced by Nelson (1991), model some of the stock market scenarios better than the simple GARCH models. EGARCH allow for asymmetry in volatility, which is a common feature in international equity markets as, for example, Johansson and Ljungwall (2009) found. The main difference in structure between the EGARCH and GARCH models is that the first one models the log of conditional variance while the second one models simply the conditional variance.

In addition to this, autocorrelation (serial correlation) and fat tails are also common in the financial markets. Thus, in the modelling process of the marginal distributions we will take these characteristics into account by modelling the mean log returns as autoregressive processes (AR) and by using a Student’s t distribution.

As a result, we will use an AR-t-EGARCH model for modelling the marginal dis-tributions. Denoting as LRi,t the logarithmic return of stock market index i at time t and using a similar notation with Johansson (2010), the AR-t-EGARCH model can be expressed as LRi,t = µi+ N X n=1 φinLRi,t−n+ i,t, (4.1)

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18 Andreas Bregiannis — Tail Dependence in Financial Markets log(σ_i,t2 ) = ωi+ q X k=1 βiklog(σ2i,t−k) + p X l=1 αil i,t−l σi,t−l + r X m=1 γim i,t−m σi,t−m , (4.2) s νi σ_i,t2 (νi− 2) i,t ∼ iid tνi, (4.3)

where ν is the shape parameter of the Student’s t distribution.

Especially, in our case, an AR(1)-t-EGARCH(2,1) is used for the marginal distribu-tions of AEX and DAX, an AR(1)-t-EGARCH(3,1) is used for S&P 500, IBEX and TOPIX and an AR(5)-t-EGARCH(2,1) is used for FTSE 100.

The choice for each marginal distribution was made based on the Akaike Information Criterion (AIC), on the weighted Ljung-Box test on standardized residuals as well as on standardized squared residuals and on the weighted ARCH Lagrange Multiplier test on the standardized residuals. As a consequence, in each case, the model with the lower AIC value, in combination with acceptable high p-values in the other three tests, was chosen.

The standardized residuals and the standardized squared residuals should not be autocorrelated. This condition is tested by the corresponding weighted Ljung-Box test where the null hypothesis is that there is no serial correlation up to a specified lag. In our case, we follow the same procedure as followed in the rugarch package in R. Therefore, instead of testing up to the same fixed lag for every model, we take the individual model specifications into account and we test up to lag 4(p + q) + (p + q) − 1, where p+q=degrees of freedom. For example, if we have an EGARCH(3,1), then p+q=4 and we test for serial correlation on the standardized squared residuals up to lag 19. As can be seen from tables 4.2 and 4.3 below, the respective p-values for the chosen models are significantly larger than 0.05 and hence the null hypothesis of no autocorrelation in the standardized residuals and in standardized squared residuals up to lag 4(p+q)+(p+q)−1 can not be rejected.

Moreover, we want to ensure that our chosen models adequately capture all ARCH effects in the standardized residuals. For this reason, a weighted ARCH LM test is implemented, in which the null hypothesis is that there are no remaining ARCH effects in the standardized residuals up to the determined lag. Similarly, based on the rugarch package in R, we test for remaining ARCH effects up to lag p + q + 5. The corresponding p-values that can be found in table 4.4 are significantly higher than 0.05 and therefore the null hypothesis of no remaining ARCH effects up to lag p + q + 5 can not be rejected. Thus, we conclude that our models for the marginal distributions are well specified since they satisfy all the desired conditions. In the table 4.1, the estimated parameters for the chosen AR-t-EGARCH models are presented. The corresponding standard errors are given in the parentheses.

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Table 4.1: Parameter estimates of AR-t-EGARCH models

AEX S&P 500 DAX FTSE 100 IBEX TOPIX

µi 0.000171 0.000297 0.000437 0.000197 0.000338 0.000114 (0.000135) (0.000123) (0.000148) (0.000117) (0.000180) (0.000166) φi1 0.003979 -0.074229 -0.024064 -0.035614 0.003414 -0.081750 (0.014616) (0.014780) (0.013480) (0.016421) (0.051830) (0.015611) φi2 -0.027217 (0.016390) φi3 -0.010618 (0.016607) φi4 -0.003523 (0.015863) φi5 -0.034299 (0.015711) ωi -0.139772 -0.093798 -0.161056 -0.138508 -0.133571 -0.191924 (0.002375) (0.001580) (0.002556) (0.002085) (0.002862) (0.008370) αi1 -0.245272 -0.166908 -0.268436 -0.178073 -0.245019 -0.105129 (0.015777) (0.025800) (0.027059) (0.009802) (0.027190) (0.024252) αi2 0.110230 0.032543 0.144268 0.071612 0.048325 -0.016426 (0.016787) (0.033301) (0.026548) (0.014529) (0.036426) (0.032398) αi3 0.059301 0.086133 0.068770 (0.023031) (0.023921) (0.025065) βi1 0.984717 0.989777 0.981904 0.985046 0.985090 0.978026 (0.000001) (0.000002) (0.000063) (0.000079) (0.000001) (0.001009) γi1 -0.103919 0.014658 -0.105925 0.081781 -0.034273 0.160622 (0.031627) (0.039713) (0.037819) (0.037873) (0.036020) (0.037778) γi2 0.218289 0.144823 0.236455 0.036897 0.055852 0.002803 (0.034099) (0.052373) (0.039873) (0.038765) (0.056998) (0.051725) γi3 -0.063048 0.094450 -0.024753 (0.033192) (0.037817) (0.038353) νi 12.335462 9.545357 9.048588 10.308884 8.514495 9.820796 (2.278172) (0.837367) (1.245015) (1.520034) (1.085727) (1.126788)

Table 4.2: p-values from weighted Ljung-Box test on standardized residuals AEX S&P 500 DAX FTSE 100 IBEX TOPIX Lag[1] 0.4223 0.8432 0.5066 0.6758 0.2036 0.4628 Lag[2(p+q)+(p+q)-1] 0.8882 0.7587 0.9258 1.0000 0.1360 0.8943 Lag[4(p+q)+(p+q)-1] 0.4213 0.6369 0.4834 0.6404 0.3526 0.8793

Table 4.3: p-values from weighted Ljung-Box test on standardized squared residuals AEX S&P 500 DAX FTSE 100 IBEX TOPIX Lag[1] 0.2628 0.4774 0.1643 0.7003 0.4498 0.0725 Lag[2(p+q)+(p+q)-1] 0.5709 0.3556 0.4973 0.1998 0.1615 0.3968 Lag[4(p+q)+(p+q)-1] 0.7018 0.5024 0.5905 0.1454 0.1923 0.5817

Table 4.4: p-values from weighted ARCH LM test on standardized residuals AEX S&P 500 DAX FTSE 100 IBEX TOPIX Lag[p+q+1] 0.5687 0.7007 0.3690 0.9846 0.6409 0.9723 Lag[p+q+3] 0.7010 0.4348 0.7076 0.7215 0.7581 0.4683 Lag[p+q+5] 0.7446 0.6321 0.6967 0.3367 0.8327 0.5808

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4.3 Probability Integral Transformation

Given that our models for the marginal distributions are well specified, we are now going to apply the probability integral transformation to the univariate standardized residuals.

The idea of the probability integral transformation is that if X is a random variable following a continuous distribution with cumulative distribution function FX, then the random variable Y = FX(X) has a standard uniform distribution.

In order to transform the univariate distributions of the standardized residuals into standard Uniform, we use the empirical cumulative distribution functions as described above. The obtained serially independent standard uniformly distributed variables will be used as inputs to estimate the copulas of the desired pairs.

4.4 Estimation of the SJC copula

Based on its properties and the obtained AIC values, among other copulas, we concluded that for all the desired pairs, the most suitable was the time-varying version of the Symmetrized Joe-Clayton copula.

Consequently, we will present the estimation process only for this copula, but of course the same approach can be followed for other copulas as well.

We remind that the variables u and v used as inputs in the SJC copula are the empirical cumulative distribution functions of the standardized residuals obtained from the AR-t-EGARCH models for the marginal distributions.

The first step now is to determine the probability density function (pdf) of the SJC copula. For this reason, first we need to specify the pdf of the Joe-Clayton copula. For simplicity in the resulting formula of the pdf of the Joe-Clayton copula, we set Φ = 1 − (1 − u)κ and Ψ = 1 − (1 − v)κ.

As a result, the pdf of the Joe-Clayton copula is given by

cJ C(u, v|λu, λl) = ∂2CJ C(u, v|λu, λl) ∂u∂v = (ΦΨ)−γ−1(1 − u)κ−1(1 − v)κ−1 ×[1 − (Φ−γ _{+ Ψ}−γ_{− 1)}−1_γ ]−1+κ1(Φ−γ+ Ψ−γ − 1)−2− 1 γ_{(1 + γ)κ} +[1 − (Φ−γ + Ψ−γ− 1)−1γ_]−2+κ1(Φ−γ + Ψ−γ − 1)−2− 2 γ_{(κ − 1)} _(4.4) where κ = 1 log2(2 − λu) and γ = − 1 log2(λl) .

Hence, the pdf of the Symmetrized Joe-Clayton copula is cSJ C(u, v|λu, λl) = ∂2CSJ C(u, v|λu, λl) ∂u∂v = 0.5 × [∂ 2_C J C(u, v|λu, λl) ∂u∂v + ∂2CJ C(1 − u, 1 − v|λl, λu) ∂(1 − u)∂(1 − v) ] (4.5)

Note that the ∂ 2_C

J C(1 − u, 1 − v|λl, λu)

∂(1 − u)∂(1 − v) is the same as the

∂2CJ C(u, v|λu, λl) ∂u∂v

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but instead of using u and v, we use 1-u and 1-v respectively and also we take κ = 1 log2(2 − λl) and γ = − 1 log2(λu) .

Thus, knowing the probability density function of the SJC copula, we minimize the negative copula log likelihood in order to estimate the copula parameters λu and λl.

The obtained results are presented in the next chapter. It is essential to mention that the entire analysis in this thesis is performed in R or Matlab. The R and Matlab codes are available from the author upon request.

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Chapter 5

Results

In this chapter, the results obtained from the constant and time-varying Sym-metrized Joe-Clayton copula, are provided and discussed.

Particularly, first of all, the plots of the time-varying lower and upper tail depen-dence coefficients are presented. Secondly, the summary statistics for the tail dependepen-dence coefficients before, during and after the Global Financial Crisis are given. In the end, the asymmetry found in the tail dependence is discussed.

5.1 Lower and Upper Tail Dependence Coefficients

In the following graphs, after implementing the time-varying SJC copula, we see how the upper and lower tail dependence between the pairs AEX-S&P 500, AEX-DAX, AEX-FTSE 100, AEX-IBEX and AEX-TOPIX evolve over time. The constant values of upper and lower tail dependence coefficients represented by the red lines in the graphs are obtained by implementing the static SJC copula.

Figure 5.1: Lower and Upper Tail Dependence Coefficients obtained from the SJC copula

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One important finding in this analysis is, first of all, that indeed there is tail dependence between the AEX and the other stock market indices. However, the magnitude differs. In particular, the pairs AEX-DAX, AEX-FTSE 100 and AEX-IBEX which represent stock market indices from countries belonging in the European Union, exhibit very strong both upper and lower tail dependence. The strongest upper and lower tail dependence that exists between those pairs is between the AEX index from the Netherlands and the DAX index from Germany.

Relatively strong upper and lower tail dependence also exists for the pair AEX-S&P 500. However, this tail dependence between the AEX index and the S&P 500 from the United States is less stronger than the corresponding tail dependencies between the aforementioned EU countries. The pair of AEX index and the TOPIX index from Tokyo exhibits weak lower tail dependence and almost no upper tail dependence since the upper TDC is really close to zero.

Furthermore, based on the visual assessment via the produced graphs above, we also conclude that no visible important change occurs in the tail dependence coefficients for any pair as the time evolves. Therefore, we can not see significant change of upper or lower tail dependence in both switches, that is from pre-GFC period to GFC period, and from GFC to post-GFC period. Nevertheless, this finding is interesting, since it seems that the Global Financial Crisis did not significantly affect the tail dependence between these financial markets. However, in order to get a better understanding of the results, the summary statistics for the tail dependence coefficients (TDCs) before, during and after the Global Financial Crisis, will be presented in the next section.

Another important finding from this analysis, that can be seen in the graph too, is that there is asymmetry in the tail dependence. In particular, the lower tail dependence is stronger than the upper tail dependence in all of these pairs. This complies with the scenario that is believed to happen many times in real world. This scenario suggests that stocks tend to fall together as the market falls, but they do not boom together when the market rises. Therefore, this means that the stock returns are stronger dependent during bear market than bull market and our findings seem to verify and follow this scenario. We will focus more on this asymmetry later using the results from the time-varying SJC copula. However, for now it is interesting to summarize the corresponding results obtained from the constant SJC copula to see that they verify this asymmetry in tail dependence as well.

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Table 5.1: Constant TDCs obtained by static SJC copula

AEX-S&P 500 AEX-DAX AEX-FTSE 100 AEX-IBEX AEX-TOPIX

λl 0.3104 0.7313 0.6594 0.6322 0.0463

λu 0.2994 0.6733 0.5865 0.5947 0.0023

λl− λu 0.011 0.058 0.0729 0.0375 0.044

From the Table 5.1 above, we see that the lower tail dependence coefficient is larger than the upper in all the cases. The largest difference between those two coefficients is in the pair AEX-FTSE 100 where λl− λu = 0.0729 and the smallest is in the pair AEX-S&P 500 where λl− λu= 0.011.

5.2 Summary Statistics for TDCs before, during and after

GFC

In this section, a table containing the summary statistics for the upper and lower tail dependence coefficients obtained from the time-varying SJC copula for the periods before, during and after the GFC, is provided. This table will enable us to see any po-tential differences between the TDCs in those three periods.

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From the table 5.2, we see, for the pair AEX-S&P 500, that the mean of the lower TDC is 0.3132 and the mean of the upper TDC is 0.2987 for the whole period, indicating that tail dependence exists between the AEX and S&P 500. The mean of upper as well as the mean of lower TDC slightly decreases during the GFC period compared to the pre-GFC period and then it slightly increases again during the post-pre-GFC period. However, the differences between the three periods are insignificant. The minimum as well as the maximum lower TDC are found in the pre-GFC period while for the upper TDC the minimum occurs in the pre-GFC period and the maximum is in the post-GFC period. Moreover, we see that for every period, the mean lower TDC is larger than the mean upper TDC, indicating slight asymmetry in the tail dependence.

For the pair AEX-DAX, the mean lower TDC is 0.7295 and the corresponding value for the upper TDC is 0.6699 for the entire period, indicating that there is strong tail dependence between the AEX and DAX. Furthermore, there is an increase both in upper and lower TDCs during the GFC period and afterwards, during the post-GFC period, they slightly increase as well. Nevertheless, the differences between the three periods do not seem to be important. The minimum of both lower and upper TDCs is found during the pre-GFC period, whereas the maximum value of both of them is found in the post-GFC period.

Regarding the pair AEX-FTSE 100, for the whole period, the mean lower TDC is 0.6604 and the mean upper TDC is 0.5871, denoting that there is both strong lower and upper tail dependence between these stock market indices. In addition, during the GFC period, the lower TDC increases from 0.6516 to 0.6910. Similarly, the upper TDC increases from 0.5762 to 0.6240. These two increases are the most significant happened compared to all the pairs and all the periods that we analyze. After the GFC period, both lower and upper TDCs diminish again approximately to the level which were before the GFC period. Moreover, the minimum and the maximum value of lower TDC as well as the minimum of the upper TDC is found during the post-GFC period, while the maximum value of the upper TDC is found during the GFC period.

Strong tail dependence exists between the pair AEX-IBEX too, since the mean of the lower and the mean of the upper TDC for the entire period is 0.6354 and 0.5891 respec-tively. Also, the lower and upper TDCs vaguely grow during the GFC period compared to the pre-GFC period and then they decrease again during the post-GFC period. How-ever, there are not crucial differences between the three periods. The minimum value of the lower TDC is found during the post-GFC period, whereas the maximum of lower

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TDC and both the minimum and maximum value of upper TDC are found in the pre-GFC period.

The weakest tail dependence, among all these pairs, exists between the pair AEX-TOPIX in which the mean of lower TDC and the mean of upper TDC is 0.0608 and 0.0129 respectively for the whole period. In particular, these values indicate that there is weak lower tail dependence and almost no upper tail dependence. Both lower and upper TDCs faintly decrease during the GFC period and afterwards increase again in a slight extent during the post-GFC period. Finally, the minimum values of both lower and upper TDCs are found in the GFC period (no visible from the table due to the rounding), while the maximum values of both lower and upper TDCs are found during the pre-GFC period.

To sum up, regarding the effects of the GFC on the tail dependence, we found that no significant changes occurred in the mean and the standard deviation of the tail dependence coefficients for all the pairs neither from pre-GFC to GFC nor from GFC to post-GFC period. As we saw above, the pair with the largest change was the AEX-FTSE 100 pair, in which both upper and lower TDCs increased during the GFC period and then they decreased again to the pre-GFC’s levels.

As already mentioned in section 5.1, it seems that there is asymmetry in the tail dependence, with the lower tail dependence stronger than the upper in all the cases. In order to verify this finding, in the next section we will plot the difference between the lower and upper tail dependence coefficients over time.

5.3 Asymmetry in the Tail Dependence

The plots of the conditional lower TDC minus the conditional upper TDC are pre-sented below.

Figure 5.2: Plots of the conditional lower TDC minus the conditional upper TDC

From the Figure 5.2, it is clearly visible that there is asymmetric tail dependence between AEX and DAX, AEX and FTSE 100 and AEX and TOPIX since we observe that the difference between the lower and upper TDCs stays consistently above zero over time, indicating that indeed the lower tail dependence is stronger than the upper tail dependence.

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For the pair AEX-IBEX, we see that there are both positive and negative differences over time. However, in general the positive differences are more and larger than the negative differences, denoting that extremes in the lower tail are stronger dependent than extremes in the upper tail. This is also verified by the difference in the means mentioned in the Table 5.2 above. Consequently, for the pair AEX-IBEX, there is also asymmetry in the tail dependence, with the lower to be stronger than the upper.

Regarding the pair AEX-S&P 500, we observe both large positive and negative dif-ferences, and hence it is not completely clear if there is stronger lower tail dependence. For this pair, this result was quite expected since we found that the mean of the time-varying lower TDC is 0.3132, the mean of the time-time-varying upper TDC is 0.2987 and hence the mean difference is 0.0145. Furthermore, the constant lower TDC is 0.3104 while the constant upper TDC is 0.2994 and the difference between them is 0.011. Therefore, in both constant and time-varying cases, we have found that the upper and lower TDCs are very close to each other. To conclude, for the pair AEX-S&P 500, it seems that the lower tail dependence is slightly stronger than the upper tail dependence but in general, this asymmetry is weaker than in the other four cases.

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Chapter 6

Conclusion

In this paper, our goal was to examine the existence, the degree, the asymmetry and the evolution of the tail dependence between the Dutch financial market and other spe-cific financial markets before, during and after the Global Financial Crisis that occurred from mid 2007 to early 2009. To achieve this, we gathered data from the Amsterdam Exchange Index (AEX) from the Netherlands, the Standard & Poor’s 500 Index (S&P 500) from the United States, the German Stock Index (DAX), the Financial Times Stock Exchange 100 Index (FTSE 100) from the United Kingdom, the Spanish Exchange In-dex (IBEX) and the Tokyo Price InIn-dex (TOPIX) from Japan and we worked with the stock index log returns. In particular, we collected data from January 1st, 2001 until December 31st, 2015 and we denoted as the pre-GFC period from January 1st, 2001 until July 31st, 2007, as the GFC period from August 1st, 2007 until March 31st, 2009 and as the post-GFC period from April 1st, 2009 until December 31st, 2015.

In our modelling process, for each chosen pair of stock indices, first we modelled the marginal distributions using AR-t-EGARCH models, then we applied the probability integral transformation to the univariate standardized residuals and in the end we used the obtained standard uniformly distributed variables as inputs to estimate the constant and time-varying Symmetrized Joe-Clayton copula using maximum likelihood estima-tion. After doing this, we obtained the constant and time-varying upper and lower tail dependence coefficients.

One of the most important findings in this thesis is that there is very strong both lower and upper tail dependence between the AEX index and the DAX, FTSE 100 and IBEX indices during the entire period with the strongest one to be between the AEX and DAX indices. Therefore, all the examined stock market indices that represent indices from EU countries exhibit very strong upper and lower tail dependence with the AEX index which is also a stock index from an EU country. Regarding the pair AEX-S&P 500, it was found that there is strong both lower and upper tail dependence as well, but the magnitude was less compared to those of the EU countries. The financial markets that seem to be less tail dependent is the Dutch and the Japanese since for the pair AEX-TOPIX, weak lower and almost no upper tail dependence was found.

Another important finding is that almost in all the cases, there is significant asym-metry in the tail dependence with the lower stronger than the upper tail dependence. As a consequence, this finding verifies the widely believed and often reported scenario that stocks tend to fall together but they do not boom together. In particular, significant asymmetry in the tail dependence was found for the pairs AEX-DAX, AEX-FTSE 100, AEX-IBEX and AEX-TOPIX, whereas for the pair AEX-S&P 500 the asymmetry is less significant.

Last but not least, for all the examined pairs of stock indices, it seems that the Global Financial Crisis did not affect significantly the magnitude of the lower and upper tail dependence. In particular, non-significant changes occurred in the lower and upper TDCs for both the pre-GFC to GFC switch and the GFC to post-GFC switch. The

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largest change happened for the pair AEX-FTSE 100 in which the average lower and upper TDCs increased by 6.05% and 8.30% respectively during the GFC period and afterwards they decreased again to reach approximately the level which were before the GFC period. As a result, this was the only pair that was affected quite importantly by the GFC.

To sum up, the aforementioned results can be very helpful for investors who want to diversify internationally their portfolios. It is well known that the Global Financial Crisis began in the United States but it expanded quickly all over the world. One very important reason for this is that the financial markets are highly dependent and many times are proven to be even more dependent during extreme events. This paper indicated the existence, the magnitude and the asymmetry of the tail dependence between the AEX and the chosen stock market indices and showed that the GFC did not significantly affect the magnitude of this dependence. Therefore, since the examined financial markets seem to be as dependent in the extreme events as they were before the GFC, investors should understand the risks involved and be very careful with the construction of their portfolios.

This paper can be a stimulus for future research in this topic since it would be very interesting to investigate the tail dependence among several financial markets, including for example more Asian and BRICS countries, as well as the impact of the Global Financial Crisis or of other extreme events on this tail dependence. A possible avenue for further research could also be to investigate structural breaks in these time series, using the tests proposed by Brown R.L., Durbin J. and Evans J.M. (1975) in the paper ’Techniques for testing the constancy of regression relationships over time’ or by Gregory C. Chow (1960) in the paper ’Tests of Equality Between Sets of Coefficients in Two Linear Regressions’.

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The dependence in financial markets before, during and after the global financial crisis

before, during and after the Global

Financial Crisis

Andreas Bregiannis

Contents

Introduction

1.1

Research Questions

1.2

Methodology

1.3

Outline of the Thesis

Chapter 2

Background Information and

Important Concepts

2.1

Unconditional Copulas

2.2

Conditional Copulas

2.3

Tail Dependence

2.4

Examples of Copulas and their Tail Dependence

2.5

Symmetrized Joe-Clayton Copula

Chapter 3

Exploratory Data Analysis

3.1

Summary Statistics

3.2

Scatter plots of the Log Returns

3.3

Scatter plots of the Empirical Copulas

3.4

Empirical Copula Frequency Tables

Chapter 4

Modelling Approach

4.1

Modelling Process

4.2

Modelling the Marginal Distributions

4.3

Probability Integral Transformation

4.4

Estimation of the SJC copula

Chapter 5

Results

5.1

Lower and Upper Tail Dependence Coefficients

5.2

Summary Statistics for TDCs before, during and after

GFC

5.3

Asymmetry in the Tail Dependence

Chapter 6

Conclusion

Bibliography