Financial markets and tail dependence : the case of Iceland during the recent financial crisis

(1)

Financial Markets and Tail

Dependence: The Case of Iceland

During the Recent Financial Crisis

Ingibj¨

org J´

onsd´

ottir

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: Ingibjörg Jónsdóttir Student nr: 10671285

Email: ingibjorgjon@gmail.com Date: January 8, 2015

Supervisor: Dr. Sami Umut Can Second reader: Dr. Roger J. A. Laeven

(2)

(3)

Financial Markets and Tail Dependence — Ingibjörg Jónsdóttir iii

Abstract

This thesis focuses on the Icelandic stock market index OMX 15. The global financial crisis that started in 2008 had a great impact on OMX 15, which lost much of its value during the crisis and ended up being discontinued in June 2009. We will explore if other major stock indices were affected by the OMX 15 by looking at the tail dependence between them. These stock indices are the German Stock Exchange (DAX), the London Stock Exchange (FTSE) and the U.S. Stock Market (S&P 500). We look at the tail dependence between the log-returns of the indices by estimating the upper and lower tail dependence coefficients to conclude whether there is any relation between them. Our findings indicate that there does exist tail dependence between OMX 15 and the selected indices.

Keywords Tail dependence, Tail-dependence coefficient, Copula, Extreme value theory, Non-parametric estimation

(4)

Preface

First and foremost, I would like to thank Dr. Sami Umut Can, my supervisor, for great guidance, advice and useful comments throughout the process of writing this thesis. I would also like to thank the second reader, Dr. Roger J. A. Laeven, for reading and reviewing the thesis. Moreover, I would like to thank Nasdaq Iceland for their help with providing data. Last but not least, I am deeply grateful to Hj´almar Helgi R¨ognvaldsson for his constant encouragement, endless love and moral support.

(6)

(7)

Chapter 1

Introduction

The recent global financial crisis that started in 2008 had major effects worldwide. One of the countries that suffered severely from it was Iceland. The Icelandic stock market index OMX Iceland 15 was heavily affected and depreciated dramatically during this period, resulting in it being discontinued in July 2009. This thesis will look at the effects the falldown of OMX 15 had on other international stock indices, specifically the German Stock Exchange (DAX), the London Stock Exchange (FTSE) and the U.S. Stock Market (S&P 500). This will be done by studying the tail depencence between the daily log-returns of the OMX 15 and each of the other stock indices.

This first chapter will cover the intended research framework of the thesis, where in Section 1.1, the research motivations will be discussed. Then, Section 1.2 will cover the main research aims and objectives of the thesis. Section 1.3 will discuss the steps to be taken to achieve those goals and answer the research questions. Finally, Section 1.4 contains an outline of the thesis and the organization of the remaining chapters.

1.1 Motivation

When a small community like Iceland, with a population of only approx. 320.000, gets thrown in a situation like the recent global financial crisis in 2008, the effects quickly spread throughout the country. Almost every family in Iceland either suffered a signif-icant loss during this period, or knew someone who did, and therefore, the crisis has been a hot topic in Iceland and is still being studied and brought up regularly.

But did the those events have effects internationally as well? The global financial crisis started with the bursting of the U.S. housing bubble, which then spread to Iceland, resulting in the falldown of OMX 15 and a banking collapse involving all three of Iceland’s major banks. After the huge drop of the OMX 15, the effects of the crisis then became apparent in Europe. Therefore, it would be interesting to see if those events are related, that is, if the falldown of OMX 15 had direct effects on the stock indices of other countries.

1.2 Aims and Objectives

In this section we will discuss the main goals we wish to achieve and the research questions we wish to answer.

1.2.1 Goals

The main goal of this thesis is to study the dependence of OMX 15 and the three other stock indices; DAX, FTSE and S&P 500, to see if there is any dependence in their “extreme” behavior. This will be done by estimating the tail dependence of the daily log-returns of those pairs of stock indices and then analyzing the results.

(8)

2 Ingibjörg Jónsdóttir — Financial Markets and Tail Dependence

1.2.2 Research Questions

In this thesis, we will answer the following research questions:

1. Is there tail dependence between the daily log-returns of Iceland stock index OMX 15 and the log-returns of the stock indices DAX, FTSE and S&P 500?

2. If there is tail dependence, how strong is it?

3. How can we interpret our results with respect to the financial crisis?

1.3 Methodology

This section describes the methodology used to find answers to the research questions. The following list describes the steps that will be taken:

• Review of background information about extreme value theory, tail dependence and copulas.

• Review and comparison of various estimators of the tail dependence coefficient. • Finding standard prices and correlation of the log-returns of the pairs of stock

indices.

• Estimating the tail dependence coefficients of the log-returns of the pairs of stock indices.

• Analyse our results with respect to our research questions and goals.

1.4 Thesis Organization

This section describes the outline and structure of the thesis and what will be presented in each chapter.

1.4.1 Thesis Outline

The thesis consists of five chapters including the introduction. The second chapter will cover the literature study, the third chapter will look at different ways of estimating tail dependence, the fourth chapter will contain the estimation of the tail dependence between the log-returns of the OMX 15 and DAX, FTSE and S&P 500, and finally, the fifth chapter will state the conclusions and answers to the research questions.

Below, the specifications of each chapter can be found.

Chapter 2 contains the background information we need for our study. The impor-tance of extreme value theory will be discussed with a review of the definitions of tail dependence and the tail dependence coefficient. Finally, the definition of copulas and their relation to tail dependence will be introduced.

Chapter 3 covers the estimation of the tail dependence coefficient. First, an overview of different studies of the tail dependence coefficient through the years will be discussed, following with the results of those studies presenting the best estimator of the tail de-pendence coefficient.

Chapter 4 introduces the case study, i.e., the estimation of the tail dependence co-efficient of the pairs of OMX 15 and the other stock indices. First, the standard prices

(9)

Financial Markets and Tail Dependence — Ingibjörg Jónsdóttir 3

of the stocks are found to make comparison easy. Then, the dependence relationship is studied by looking at scatter diagrams of the log-returns and finding their correlation coefficients. Finally, we will estimate the upper and lower tail dependence coefficients for each pair; OMX 15 - DAX, OMX 15 - FTSE, and OMX 15 - S&P 500.

Chapter 5 will present the conclusions and the main results. The results from Chapter 4 will be analysed and the research questions presented in Chapter 1 will be answered. Finally, a discussion of possible future research will be presented.

(10)

Chapter 2

Background information

This chapter contains the literature review needed for our case study, about extreme value theory and tail dependence. In particular, we will review some basic definitions which will lead us to the estimation of the tail dependence coefficient, which follows in Chapter 3.

2.1 Extreme Value Theory

Extreme Value Theory (EVT) has recently become an important tool in the field of finance for modeling the occurrence of extreme events. That is, quantities becoming unusually large or small compared to their historical values, for example the prices of stocks in the stock market.

Traditionally, the standard approach for measuring financial risks is using the value-at-risk (VaR), which measures the risk of loss on a specific portfolio of financial assets over a specific time period. However, the VaR approach typically uses a normal distri-bution approximation, which has been criticized in regards of the use of financial data. Then, the risk is underestimated for the heavy tails, which occur regularly in financial data. Bensalah (2000) shows that EVT provides useful risk measures in cases where we have such extreme values as in financial data, where more appropriate distributions are used to fit extreme events.

2.1.1 Extreme value cumulative distribution function

If we suppose X1, X2, . . . are i.i.d. random variables with common cumulative

distribu-tion funcion (cdf) F , the following definidistribu-tion of an extreme value cdf was introduced by Kotz and Nadarajah (2000):

Let Mn = max{X1, X2, . . . , Xn} be the maximum of the first n random variables

and let w(F ) = sup{x : F (x) < 1} denote the upper end point of F . Then, since we have

Pr(Mn≤ x) = Pr(X1 ≤ x, . . . , Xn≤ x)

= Fn(x),

then Mnconverges a.s. to w(F ), both when it is finite or infinite. The cdf of a normalized

version of Mn then converges to a nondegenerate G, that is

Pr Mn− bn an

≤ x

= Fn(anx + bn) → G(x), (2.1)

as n → ∞, where an> 0 and bnare norming constants. Then, if this holds for a suitable

choice of an and bn, we say that G is an extreme value cdf and F is in the domain of

attraction of G, i.e., F ∈ D(G).

(11)

2.1.2 Types of extreme value distributions

Extreme value distributions consist of three different types of distribution functions (see Kotz and Nadarajah (2000)):

1. Gumbel-type distribution Pr[X ≤ x] = exp−ex−µσ (2.2) 2. Fr´echet-type distribution Pr[X ≤ x] = ( 0, if x < µ, exp n − x−µ_σ −o if x ≥ µ (2.3) 3. Weibull-type distribution Pr[X ≤ x] = ( exp n − µ−x_σ o if x ≤ µ 0, if x > µ, (2.4)

and µ, σ > 0 and > 0 are parameters.

In multivariate extreme value theory, a widely used method of studying the depen-dencies of extreme events is to use the concept of tail dependence, see Joe (1997) and Embrechts et al. (2003).

2.2 Tail Dependence

Tail dependence is frequently used in the study of dependence between extreme values. It measures the strength of dependence in the upper-right quadrant tail, or lower-left quadrant tail, of a bivariate distribution.

A common measure of tail dependence is given by the so-called tail-dependence coefficient. The tail-dependence coefficient (TDC) was first introduced in Sibuya (1960), and measures the probability of one extreme value event occurring, given that another event assumes an extreme value as well. Thus, the tail dependence of a copula function (see below) can completely capture the dependence between extreme values of stock returns, see for example Joe (1997). In Embrechts et al. (2001), tail dependence is defined as follows:

Let (X, Y ) be a vector of continuous random variables with marginal distribution functions F and G. Then, the coefficient of upper tail dependence of (X, Y ) is

λU = lim

u%1P {Y > G −1

(u)|X > F−1(u)}, (2.5) provided that the limit λU ∈ [0, 1] exists. Then, if λU ∈ (0, 1], X and Y are said to

be asymptotically dependent in the upper tail and if λ = 0, X and Y are said to be asymptotically independent in the upper tail. Similarly the coefficient of lower tail dependence of (X, Y ) is defined as

λL= lim

u&0P {Y ≤ G

−1_{(u)|X ≤ F}−1_(u)}, _(2.6)

given that the limit λL∈ [0, 1] exists.

An equivalent definition for the upper TDC, offered by Ferreira (2013), is the fol-lowing:

λU = lim

(12)

where F and G are the distribution functions of the random variables X and Y , respec-tively.

2.3 Copulas

A copula C is a multivariate cumulative distribution function whose margins are uni-formly distributed on the interval [0, 1]. In Schmidt and Stadtm¨uller (2006), the following definition can be found for a copula C:

Every n-dimensional distribution function F can be written in the form

F (x1, . . . , xn) = C(F1(x1), . . . , Fn(xn)), (2.8)

where F1, . . . , Fn are the marginal distribution functions, assumed to be continuous.

Then, the copula C is unique and has the representation

C(u1, . . . un) = F (F1−1(u1), . . . , Fn−1(un)), (2.9)

with 0 ≤ uj ≤ 1 for j ∈ {1, . . . n} and where F1−1, . . . , Fn−1 denote the generalized

inverse distribution functions of F1, . . . , Fn. That is, for every ui ∈ (0, 1], we have

F_i−1(ui) := inf{x ∈ R|Fi(x) ≥ ui},

with i = 1, . . . , n and inf{∅} = ∞.

Definition (2.8) also works in the opposite directin. That is, if C is a copula and F1, . . . , Fn are distribution functions, then the function F defined by (2.8) is an

n-dimensional distribution function with margins F1, . . . , Fn.

Copulas model portfolios for extreme moves in correlations and, more generally, they describe dependence. As a consequence, copulas can be used to define the TDC:

Let C denote the copula of (X, Y ). Then, if C has a joint distribution function F (x1, x2) = C(F (x1), G(x2)), Ferreira (2013) observed that

λU = 2 − lim

t→∞tP (F (X) > 1 − 1/t or G(Y ) > 1 − 1/t)

= 2 − lim

t→∞t{1 − C(1 − 1/t, 1 − 1/t)}, (2.10)

where λU > 0 corresponds to upper tail dependence with its degree measured by the

value λU, and λU = 0 means tail independence. It is very important to conclude whether

(X, Y ) is tail dependent or not, since if λU = 0, but it is estimated to be positive, we

(13)

Chapter 3

Different estimators of tail

dependence coefficient

This chapter contains a discussion of different estimators for the tail dependence coef-ficient (TDC), with a conclusion of which one we will use for our study.

3.1 Estimation of Tail Dependence Coefficient

Finding the best estimator of the TDC can be tricky when we don’t know the type of model underlying data, since the use of parametric estimators involve model risk which can lead to wrong interpretations of the dependence stucture. This was confirmed by Frahm et al. (2005), who showed that (semi)parametric estimators can have disastrous performance under a wrong model assumption. However, an alternative way is by using nonparametric approach, which we will focus on, since nonparametric methods avoid model risk, even though they usually lead to larger variance.

There have been a lot of studies of various estimations of the TDC through the years, and some comparisons between them. Coles et al. (1999) presented an equivalent equation for the upper TDC as

λU = 2 − lim u%1

log ¯C(u, u)

log(u) (3.1)

with ¯C(u1, u2) = u1+ u2− 1 + C(1 − u1, 1 − u2) denoting the survival copula function.

From equation (3.1), Frahm et al. (2005) derived the estimator for the upper TDC as

λLOG_U = 2 −log ¯Cm m−h m , m−h m log m−h_m , (3.2) with 0 < h < m and Cm(u, v) = 1 m m X j=1 I(R1j/m ≤ u, R2j/m ≤ v)

is the empirical copula, I is the indicator function, R1j and R2j are the ranks of Xlj and

Ylj, respectively, where j = 1, . . . m, and h is a threshold. The second estimator Frahm

et al. (2005) considered was derived from a special case in Joe et al. (1992), which is

λSEC_U = 2 − 1 − ¯Cn n−k n , n−k n 1 −n−k_k , (3.3)

with 0 < k ≤ n. Then the third estimator Frahm et al. compared was motivated in 7

(14)

8 Ingibjörg Jónsdóttir — Financial Markets and Tail Dependence Capra et al. (1997) as λCF G= 2 − 2 exp   1 n n X i=1 log   q log_F 1 n(Xi)log 1 Gn(Yi) log_max(F 1 n(Xi),Gn(Yi)2)     (3.4)

where Fn and Gn are the empirical distribution functions of Xi and Yi, respectively.

Next, Frahm et al. (2005) performed simulations to compare the three estimators with the results that the estimator λCF G was the best among the three estimators. Another study was done by Ou et al. (2010), where the estimator λCF G was compared to three other estimators for the TDC. Those three estimators were first introduced by Schmidt and Stadtm¨uller (2006) as follows:

λ(1)_n,k = k n −1 · ¯Cn k n, k n (3.5) λ(2)_n,k= k X i=1 i n 2!−1 · k X i=1 i n · ¯Cn i n, i n (3.6) λ(3)_n,k = Pk i=1 ¯Cn ni, i n − i n 2 _i n − i n 2 Pk i=1 i n − i n 2 (3.7)

where k is determined by n, generally as k ≡ √n and ¯Cn is the empirical copula. The

results from Ou et al. (2010) showed that the estimator λCF G was also the best among those four estimators.

(15)

Chapter 4

Estimation of the tail dependence

We will now study the tail dependence between the daily log-returns of the OMX 15 and the three chosen stock indices; DAX, FTSE and S&P 500. For our estimation of the tail dependence coefficient we will assume that the log-returns of the stock prices are i.i.d. Even though this assumption is not realistic, it keeps our model simple and relatively easy to work with. In addition, we will be using the time period from January 1998 until July 2009, when the OMX 15 was discontinued.

4.1 Exploratory Data Analysis

First, we compute the standard prices of the stocks to observe the change of stock prices over time. Next, the log-returns of the stock indices will be obtained. Finally, we make scatter diagrams of the log-returns of the stock indices and find their correlations.

4.1.1 Standard prices

We will now look at the change of the stock prices over time. Let Ptdenote the stock price

at time t. The daily log-returns of the stock prices are then found by rt= ln(Pt/Pt−1),

for t = 1, 2, . . .. If P0 is the initial price, we assume P0 = 1, to make the comparison of

the stock prices simpler. Then, the standard price of the stocks is denoted by

Pt= Pt−1ert

= P0er1+r2+···+rt

Figure 4.1 shows the change of stock prices over the given time period.

We can see from Figure 4.1 that the return of OMX 15 gets significantly higher than the returns of the other stock indices. Furthermore, it also has the lowest return, around the time of the dramatic falldown during the recent global financial crisis. The OMX 15 is the only stock index among the four that seems to really stand out, even though there still exists reverse change of the other stock indices as well, meaning that some stocks go up while others go down, and vice versa.

(16)

Figure 4.1: Standard prices of the stocks

4.1.2 Scatter diagrams and correlations

Next, we look at the dependence relationship between OMX 15 and the other three stocks, by making scatter diagrams of the log-returns which can be seen below.

Figure 4.2: OMX 15 and DAX

(17)

Figure 4.4: OMX 15 and S&P

We can see from the figures that the scatter diagrams look somewhat similar, which is not surprising looking at the log-returns in Figure 4.1. In each scatter diagram, there seems to be a big cluster around zero, but with a few outliers as well. If we now take a closer look at the clusters in each scatter plot, by ignoring the outliers and focusing on the range between −0.1 and 0.1, the scatter diagrams of the log-returns will be as follows:

Figure 4.5: OMX 15 and DAX

(18)

Figure 4.7: OMX 15 and S&P 500

We can observe that the scatter diagrams are still a bit clustered, but seem to be slightly positively correlated. We can confirm this by finding the sample correlations for each pair of stock indices. The correlation between the log-returns of the stock indices is found using sample Pearson correlation coefficient r, given by

r = Pn i=1(Xi− ¯X)(Yi− ¯Y ) q Pn i=1(Xi− ¯X)2 q Pn i=1(Yi− ¯Y )2 , (4.1)

where Xi and Yi denote the log-returns of the two stock indices, and

¯ X = 1 n n X i=1 Xi,

and similarly, ¯Y are the sample means. Using this formula, we obtain the following sample correlations:

Pair of stocks r OMX 15 - DAX -0,059 OMX 15 - FTSE -0,036 OMX 15 - S&P 500 -0,102

We can see the log-returns of the stock indices are slightly negatively correlated. How-ever, the extreme outliers could have a disproportionate impact on the correlation coef-ficient, resulting in a negative correlation. If we compute the correlation coefficient for the “main bulk” of the data, where we have excluded the extreme outliers, as in Figures 4.5 to 4.7, we get the following results:

Pair of stocks r OMX 15 - DAX 0,165 OMX 15 - FTSE 0,224 OMX 15 - S&P 500 0,098

Thus, we see that the correlation coefficients for these restricted data sets are posi-tive. Hence, the negative overall correlation is caused by the extreme outliers, and the bulk of the data is positively correlated.

If we compare the scatter diagrams to, e.g., the scatter diagram of the log-returns of DAX and FTSE, we can see that the DAX-FTSE scatter diagram presents a much clearer correlation:

(19)

Figure 4.8: DAX and FTSE

We can see that here we have a positive correlation between DAX and FTSE, which is not surprising compared with Figure 4.1. Computing the sample correlation of DAX and FTSE we obtain 0, 777, which tells us that we have a high positive correlation. Similarly, we have the following scatter diagram of the log-returns of DAX and S&P 500:

Figure 4.9: DAX and S&P 500

We can observe that there is also a positive correlation between DAX and S&P 500, even though it seems not as strong as the DAX-FTSE correlation. Confirming this, the sample correlation we obtain for DAX-S&P 500 is 0.566. Finally, the scatter diagram of the log-returns of FTSE and S&P 500 can be seen in Figure 4.10:

(20)

We can also recognize that there is a positive correlation between FTSE and S&P 500, and by computing the correlation coefficient we have the sample correlation of 0, 487 -which shows a positive correlation for FTSE-S&P 500.

4.1.3 Log-returns

The OMX 15 log-returns are the most extreme around October 2008 when the financial crisis hit Iceland, with the highest negative value of −1, 43. Also, we have a high negative value a couple of months later when the three major Icelandic banks had collapsed, with a log-return of −0, 51. The full OMX 15 log-return plot can be seen in Appendix A. However, we will restrict the OMX 15 plot to a smaller scale to better compare it with the extreme values of the other stock indices. Then, we can easily compare the log-returns, as can be seen from Figures 4.11 to 4.14 below:

Figure 4.11: OMX 15 log-returns

(21)

Figure 4.13: FTSE log-returns

Figure 4.14: S&P 500 log-returns

It is clear that OMX 15 has a much heavier tail than the other stock indices, which do not show such extreme negative values at the beginning of the recent global financial crisis in October 2008. Furthermore, we can also see volatility clustering in all four indices, where we have periods of persistent high or low volatility. This contradicts our assumption that the log-returns of the index prices are i.i.d., but we will nevertheless use the assumption for ease of treatment.

We will then study the dependence in extreme values between OMX 15 and the other stock indices more precisely by finding the upper and lower tail dependence coefficients.

4.2 Tail Dependence Estimations

4.2.1 OMX 15 and DAX

The first step in estimating the TDC is finding the empirical distribution functions (EDF) for the log-returns of OMX 15 and DAX, which we denote by Fn(X) and Gn(X),

(22)

Figure 4.15: Empirical distribution function for OMX 15

Figure 4.16: Empirical distribution function for DAX

We can see from the figures that OMX 15 is heavier-tailed than DAX. We then use those distribution functions to estimate the upper TDC, choosing the best estimate λCF G_U presented in section 3.1: λUOMX,DAX = 2 − 2 exp   1 n n X i=1 log   q log_F 1 n(Xi)log 1 Gn(Yi) log_max(F 1 n(Xi),Gn(Yi)2     (4.2)

Taking n = 2844 data points, this gives the upper tail dependence coefficient for the stock indices OMX 15 and DAX as

λUOMX,DAX = 0, 4895

This shows that we have a positive tail dependence estimate, and thus, there is tail dependence between OMX 15 and DAX. However, the upper tail dependence is only moderately high, or around 0.49. If we then estimate the lower TDC, by using negative log-returns, we obtain the following results:

(23)

Thus, the lower tail dependence between OMX 15 and DAX is slightly lower than the upper TDC.

4.2.2 OMX 15 and FTSE

We begin by estimating the empirical distribution function for FTSE, which can be seen below:

Figure 4.17: Empirical distribution function for FTSE

We see that the FTSE edf is not as heavy tailed as the OMX 15 edf, which can bee seen in Figure 4.15. We then estimate the upper tail dependence coefficient using equation 4.1 as before with n = 2849 data points, which gives

λUOMX,FTSE= 0, 4655.

We see there is a moderately high upper tail dependence between OMX 15 and FTSE, and the value is similar in magnitude to the OMX-DAX tail dependence. Similarly, we estimate the lower TDC using negative log-returns, which gives

λLOMX,FTSE= 0, 4239,

and hence, the lower tail dependence between OMX 15 and FTSE is not very high, and also lower than the upper TDC.

4.2.3 OMX 15 and S&P 500

As before, we begin by estimating the empirical distribution function for S&P 500, which can be observed in Figure 4.18.

The edf for OMX 15 in Figure 4.15 is more heavy tailed due to the huge fall in stock prices in October 2008, as can be seen in Figure 4.1. We again use equation 4.1 to find the upper tail dependence coefficient using the edf and n = 2779 data points which gives

λUOMX,S&P = 0, 5116.

We see that the upper tail dependence for OMX 15 and S&P 500 is higher than for OMX 15 and the other stock indices, even though it is still only moderately high.

(24)

Figure 4.18: Empirical distribution function for S&P 500

Then, finding the lower TDC for OMX 15 and S&P 500 we obtain λLOMX,S&P = 0, 4990.

Thus, the lower tail dependence between OMX 15 and S&P 500 is also higher comparing it to the other stock indices, but still lower than the upper tail dependence.

(25)

Chapter 5

Conclusions

In this last chapter, we will show our main conclusions of the estimation of the tail dependences of log-returns between OMX 15 and the three other stock indices, DAX, FTSE and S&P 500. In addition, we will answer the research questions presented in Chapter 1 and conclude with some possible future research.

5.1 Main Results

Below, we present our main results of the upper and lower tail dependence coefficients for each pair of stock indices:

Stocks λU λL

OMX 15 - DAX 0,4895 0,4595 OMX 15 - FTSE 0,4655 0,4239 OMX 15 - S&P 500 0,5166 0,4990

We can see that the tail dependence is moderately high for each pair of stock indices, and the lower TDC is slightly lower than the upper TDC. This was expected seeing how extreme the negative return of OMX 15 was compared to the other stock indices.

A possible explanation of why the tail dependence coefficient is not higher could be the timing of the effects of the recent global financial crisis. As said in Chapter 1, the trigger for the crisis was the bursting of the housing bubble in the U.S. From there, the effects were first seen in Iceland with the bankruptcy of the three major banks, resulting in a domino effect which spread through the rest of Europe a few months later. Therefore, there could be a better dependence between OMX 15 and the other stock indices if we would shift the timeline and compare the extreme events during the crisis when the prices of the stock indices fell, even though they were not as drastic as in Iceland. This would be an interesting example of possible future research of this topic. In addition, modeling the log-returns as a time series like GARCH (rather than an i.i.d. sample) would be another future research, which would lead to a more realistic model of the tail dependence.

(26)

Appendix A: OMX 15 log-returns

Figure 5.1: OMX 15 log-returns 20

(27)

References

Bensalah, Y. (2000) “Steps in applying extreme value theory to finance: A review”, Bank of Canada.

Capra, P., Fougeres, A.L., and Genest, C. (1997) “A nonparametric estimation proce-dure for bivariate extreme value copulas”, Biometrika, 84, 567–577.

Coles, S., Heffernan, J. and Tawn, J. (1999) “Dependence measures for extreme value analyses,” Extremes, 2, 339–365.

Embrechts, P., Lindskog, F. and McNeil, A. (2003). Modelling dependence with copulas and applications to risk management. In “Handbook of Heavy Tailed Distributions in Finance” (S. Rachev, Ed.), Elsevier, 329–384.

Ferreira, M. (2013). “Nonparametric estimation of the tail-dependence coefficient”, REVSTAT - Statistical Journal, 11, 1–16

Frahm, G., Junker, M. and Schmidt, R. (2005). “Estimating the tail-dependence coeffi-cient: properties and pitfalls”, Insurance: Mathematics & Economics, 37(1), 80–100. Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman & Hall,

Lon-don.

Joe, H., Smith, R. L. and Weissman, I. (1992). “Bivariate threshold models for ex-tremes”, Journal of the Royal Statistical Society, Series B, vol. 54, 171 – 183. Kotz, S. and Nadarajah, S. (2000). Extreme Value Distributions: Theory and

Applica-tions, London: Imperial College Press.

Schmidt, R. and Stadtm¨uller, U. (2006). “Nonparametric estimation of tail dependence”, Scandinavian Journal of Statistics, 33, 307–335.

Sibuya, M. (1960). “Bivariate extreme statistics”, Annals of the Institute of Statistical Mathematics, 11, 195–210.

Financial markets and tail dependence : the case of Iceland during the recent financial crisis