Estimating the value-at-risk of European stock portfolio using Copula-GARCH model

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Estimating the Value-at-Risk of

European Stock Portfolio Using

Copula-GARCH Model

Bangchi Wang

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: Bangchi Wang Student nr: 11803800

Email: Bangchiwang217@gmail.com Date: August 15, 2018

Supervisor: Dr. Lu Yang Second reader: Dr. S. Umut Can

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Statement of Originality

This document is written by Bangchi Wang who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this docu-ment are original and that no sources other than those docu-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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Estimating the value-at-risk of European Stock Portfolio Using

Copula-GARCH Model — Bangchi Wang v

Abstract

As financial markets change everyday, measuring risk accurately has become the basis for risk management and for investment decision making. In order to effectively hedge the risks caused by price fluctuations of financial assets, it is very important to establish a reliable and accurate mathematical model to measure the risk of financial portfolios. Nowadays, the GARCH model is a popular choice for describing and modeling the volatility of financial time series, but other statistical tools are needed to account for the dependencies between different assets in a portfolio. The copula functions can solve this problem, as they provide an analysis of the correlation structures between financial assets, and also deal with the construction of multivariate joint distribution.

In this paper the international standard measurement of risk management, Value-at-Risk (VaR), is estimated under a copula-GARCH model. We apply the Monte Carlo simulation method to estimate the VaR of an equally-weighted European stock portfolio of the CAC 40 index and FTSE 100 index, with the level of confidence at 95% , 99% and 99.9%. We found that both assets show non-normal distributions with the characteristic of leptokurtosis, and can be well fitted by ARMA(1,1)-GARCH(1,1) with conditional student-t distribution, while a t copula offers an effective compromise and enables a good fit of the correlation structure between the two indices. Compared with the forecasting performances of VaR which is estimated by the univariate GARCH model, we conclude that tcopula-GARCH model provided a better fit for VaR, with a higher level of confidence.

Keywords European stock portfolio, ARMA-GARCH models, copula, Value-at-Risk, Monte-Carlo Simulation.

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Preface

Time flies, the one-year study in Amsterdam is coming to the end. Looking back to the point where I just graduated from Jilin University last year, I now have more experience and more knowledge. I researched the Chinese stock market by GARCH model in my Bachelor’s graduate thesis, and the master program here introduced me to more knowledge about the European financial market and risk management. Based on my huge interest in risk, I extended to the European stock market this year and developed from GARCH models to copula-GARCH models.

Firstly and most importantly, my deepest gratitude would go to my supervisor, Dr. Lu Yang, for the encouragement and guidance she has offered during this project. I am not an expert in R programming or in English writing, and she has provided invaluable support in these areas, without which the quality of my thesis would have suffered. Secondly, I would like to owe my heartfelt gratitude to Dr. S. Umut Can, for his kind help and instruction with copula theory and VaR during the study.

I would also like to offer my sincere thanks to my beloved family for their supporting and being confident in me throughout my studies, and for their generosity in financing my study abroad. I am also grateful to my friends and peers on my course for their patience in listening to my problems and for the time and insight they offered in helping me solve them along the course of this thesis.

This is not the end, but a new start for the rest of my life. Stay hungry and stay foolish!

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Chapter 1

Introduction

1.1 Introduction and Literature Review

Since the financial crisis in 2008, and a progressively globalizing economy, the European financial market has gradually recovered in recent years, while also becoming more volatile and vulnerable. Because of this, the study of risk management has attracted attention, becoming one of the core factors in business and financial management. In the financial market, effective risk management can help to reduce volatility of financial returns, increase the speed of price adjustments, narrow the bid-ask spread of financial assets, accelerate market trading volume and so forth. Moreover, good command of risk management is beneficial as it can add value to shares, and increase shareholders con-fidence in investing. Competent risk management can improve operational efficiency of financial markets and enhance competitiveness of financial institutions, as well as im-proving their ability to resist risk. In fact, through effective risk-management practices, an optimal capital allocation can be achieved.

The basis of risk management is quantifying risk, namely risk estimation. Financial risk estimation methods have become increasingly complex in order to account for the scale and dynamics of financial transactions. However, with developments in financial theory and engineering, they are also more comprehensive. According to previous aca-demic studies, the main risk measurements methods include volatility method, sensitiv-ity analysis, stress test, Value-at-Risk (VaR) and Extreme Value Theory (EVT). Among them, the VaR method is an international standard and can effectively measure the risk of stock portfolios. Compared with the other more traditional methods, VaR can cover a variety of market factors affecting financial assets [28]. Meanwhile, it can also mea-sure non-linear risk with greater adaptation, and summarize in one monetary value the (market) risk associated with any portfolio, for example, loss associated with a given probability [31]. Although VaR is a simple measure, it is not easily estimated. The key challenge is constructing a model which can accurately estimate the VaR of financial assets. VaR estimation can be approached in from several different directions, includ-ing the variance-covariance method, Monte Carlo simulation and historical simulation. The variance-covariance approach usually assumes a specific distribution, for example a normal distribution, on returns and thus calculates the corresponding VaR. However, in practice, this is often an invalid assumption, with non-normality commonly found in returns of financial assets [7]. Hence, research into empirical finance has largely shown that VaR cannot be adequately estimated with the multivariate normal distribution, which often underestimates VaR of financial asset portfolios. Additionally, the calcula-tion of VaR can sometimes become complicated, especially when using the Monte Carlo method. For VaR to be calculated, the statistical distribution, or the function of prob-ability density, of the portfolio returns must be determined.

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In reality, time series of financial market data generally exhibits time-varying volatil-ity, leptokurtosis and clustering. The GARCH model family has a strong advantage in modeling and predicting the volatility of financial returns, and adequately describes the heteroscedasticity of the residual of the return series, so it is currently universally used in financial metrology. In 1982, Engle proposed the Autoregressive Conditional Heteroskedasticity(ARCH) model to deal with persistent variance and the heavy-tail problem [17]. Later, Bollerslev proposed the GARCH model, which is a generalized ARCH model. This can explore and study the dynamic correlation structures between financial time series by quickly capturing clustering and heteroscedasticity. In 1986, Taylor solely proposed GARCH(1,1) [41], which has become the most popular model for practical applications. Later, in 1993, Ding, Granger and Engle constructed a para-metric model for the power of the conditional variance, called the power-GARCH model [15]. Since the analysis of VaR is related to the shape of the tail of the probability distri-bution, risk will be easily overestimated in practice. The proposed GARCH model can not only solve this problem, but also describe the clustering of volatility of return series. Therefore, applying the GARCH model to predict how volatile a time series is before estimating VaR is a widely-used approach to predicting the risks existing in financial markets. Mabrouk et al. found that the accuracy of prediction of VaR was affected by volatility models through estimating the VaR of the seven stock markets [32]. Alouid also proved that use of the GARCH model with skewness t distribution can improve the stability of VaR and ES prediction [5].

As a guideline for predicting macroeconomic trends, the stock market can not only rationally optimize the allocation of social resources, but also improve the efficiency of financial capital. Recently, the harmonization of the financial markets of the European Union created new opportunities, which the European Union aimed to take advantage of by integrating their operations. In order to disperse and defuse financial risks, multiple assets are typically combined so that risks can be evaded. By studying the interde-pendence between stocks, investors can optimize investment decisions based on their relevance, and appropriate departments can improve the efficiency of the stock mar-ket to provide an important reference for supervision. Consequently, the dependence in different assets becomes a significant issue when quantifying portfolio risk, for financial economists and investment practitioners alike [8]. However, measurement of dependency is challenging; for example, Embrechts, Lindskog, and McNeil demonstrated that the conventional Pearson correlation can be too limited as a criterion [28]. To effectively avoid and hedge risk caused by price fluctuation and financial assets, establishing a reliable and precise mathematical model to measure the risk of financial portfolios is now extremely important.

In the field of finance, there is a lot of research into dependence in assets, covering, for example, asset pricing, market risk prediction, and portfolio hedging. In reviewing this literature, we found that most scholars have focused on methods such as linear correlation coefficient, Granger causality analysis, infection model, vector autoregres-sive model(VAR) and GARCH class model to measure the dependence. Li studied the Chinese and other worldwide stock markets using the the ergodic analysis method of Granger causality test, and found that the time-varying dependence between them grad-ually changed from strong to weak [44].

Although these methods have achieved some success, the problems of non-normality and leptokurtosis in financial data and the possibility of non-linear dependence remain. Therefore, a new statistical tool, the theory of copula, was introduced to comprehen-sively study the correlation between random variables. In this framework, the assump-tion of joint normality is not necessary and linear, non-linear and tail dependencies

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Copula-GARCH Model — Bangchi Wang 3

can be described, making this a robust tool for modeling joint distributions. Copula functions allow break down of n-dimensional joint distributions into their n marginal distributions and a copula function [38]. On the contrary, a copula function can also gen-erate a multivariate joint distribution incorporating inter-variable dependence as well as the marginal distributions [38]. Sklar first proposed the concept of copula in 1959, followed by a comprehensive and systematic overview of the copula theory by Nelsen [39]. Embrechts applied copula in the financial field for the first time four years later. Through discussing the shortcomings of the practical application of traditional linear correlation coefficients, he confirmed the validity of copula theory in dependence study. Subsequently, copula theory was broadly used in financial research and statistical liter-ature. For instance, the normal copula was used by Georges to model derivative pricing and options time of exercise [22]. Fortin and Kuzmics estimated the VaR associated with a portfolio comprising the FTSE and the DAX stock indices using convex linear combinations of copula [20], while Embrechts et al. modeled extreme value and risk limits by copula [16]. Breymann and others provided results supporting a t-copula as superior to a normal copula for characterizing the amount of dependence in the tail. There data was taken from the crisis period, and modeled using alternative GARCH-type specifications. They found that the main results were unchanged with different copulas, but the t-copula was also sensitive to the use of raw returns [43].

In addition, Aloui et al. used a copula-GARCH model to study the composition of conditional dependencies between U.S. dollar exchange rates and the cost of crude oil [6]. Polara and Hotta applied the copula theory with GARCH model to calculate an estimate of the VaR of a portfolio consisting of the Nasdaq stock index and S&P500 stock index, finding that the best results, with reliable VaR limits, came from SJC cop-ula which allowed varying dependence in the tails [35]. In the research launched by Lu et al., Student-t copula was shown to provide a good model fit of the dependence struc-ture between naturual gas and crude oil fustruc-tures [31]. It was found by Hsu et al. that, compared to other dynamic hedging models, the copula-GARCH models perform better in in-sample as well as out-of-sample tests, with fully flexible distribution specifications [24].

1.2 Structure of this paper

From the results summarized above, we can see that by using a GARCH model to de-scribe the marginal distribution in portfolios, and copula to obtain joint distribution, the Value at Risk is easily estimated [37]. The present paper will apply a copula-GARCH model in order to estimate the VaR of a European stock portfolio with non-normal returns. The paper will be structured as described below:

Chapter 1: Discuss the present status and significance of the research, research pur-poses and objects, as well as the structure and processes of this paper.

Chapter 2: Define methodology including copula theory and GARCH-type models with different marginal distributions, introduce the characteristics of dependence coefficient and selection criterion.

Chapter 3: Introduce Value-at-Risk and discusses the advantages and disadvantages of three estimation method.

Chapter 4: Apply the method provide an estimation of the Value-at-Risk of a port-folio made up of two equally weighted assets: Cotation Assiste en Continu(CAC) 40 index and Financial Times Stock Exchange (FTSE) 100 indices. We first use

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GARCH-type model with two marginal distributions to model the assets respectively. Secondly, we apply three copula models to compute the dependencies between random variables. Then select the optimal method and compute the portfolio VaR by Monte Carlo Sim-ulation. Finally by comparing the results obtained under copula-GARCH model and univariate GARCH model, we evaluate the effect of copula on VaR estimation.

Chapter 5: Conclude the results, point out the shortcomings of the research and the direction of future research.

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Chapter 2

Multivariate Modeling using

Copula

In recent years, frequent financial fluctuations have made risk management and multi-variate financial time series an important global focus. With the development of eco-nomic globalization, the relationship between financial markets has become increas-ingly complex, and the non-linear and non-stacking characteristics have become in-creasingly prominent. Therefore, the traditional linear correlation-based multivariate financial models can no longer fully meet the needs of financial development. Multivari-ate GARCH models offer a possible solution to this problem.

Multivariate GARCH models are often used to analyze multiple markets and to an-alyze and forecast volatility of time series data. Volatility is a crucial feature of financial markets, as financial time series are characterized by time-variation and clustering and so forth. Understanding these mechanisms is essential to risk analysis, to account for the evolution of financial markets over time. However, within the field of risk analy-sis, there are some challenges yet to be overcome, for example dependence parameter estimation and multivariate distribution assumptions, especially when the data series show non-linear correlation and conditional heteroscedasticity. To address these issues, a copula method based on the GARCH model was used, where copula offers a function for connecting joint distributions and marginal distributions. An introduction to some necessary models follows.

2.1 Copula Theory

In 1995, Sklar introduced copula into statistics. He decomposes the N-dimensional joint distribution function into N marginal distribution functions and copula functions. This function summarizes the inter-variable relationships [38]. Following this, Nelson pro-vided an introduction to copulas, with a comprehensive description of the definition, classification, construction methods, and dependencies of this fuction [33]: multi ran-dom variables X1, X2, . . . XN has the joint distribution function F (x1, x2, . . . xN) and

respective marginal distribution functions FX1(x1), FX2(x2), . . . FXN(xN), then copula

C(u1, u2, . . . uN) is given by:

F (x1, x2, . . . xN) = C[FX1(x1), FX2(x2), . . . FXN(xN)]

At the same time, in 1998, He and Joe [27] extended to the conditional copula function which can provide the theoretical basis for analyzing the correlation between actual economic and financial time series.

For simplicity, throughout the research we will solely concentrate on the bivariate cop-ula, but it relatively straightforward to extend this theory to higher dimensions (see, for

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instance, Nelsen [33]). According to Nelsen [33] and Joe[27], the definition of a bivariate unconditional copula is as written below:

Definition 2.1 [33] A bivariate copula function (henceforth referred to as a copula, for brevity) is a function C with these listed characteristics:

1. Domain I = [0,1] × [0,1];

2. For random u,v from domain I, we have: C(u, 1) = u, C(1, v) = v; 3. C is grounded and 2-increasing.

Unary distribution function F (x1) and F (x2), if F (x1) and F (x2) are continuous,

u = F (x1) and v = F (x2) are standard uniform distributions, then C(u, v) is binary

distribution function with range [0,1].

In other words: a 2-dimensional copula is a function C : [0, 1]2 → [0, 1] of random vector (U, V )0 such that:

C(u, v) = P (U ≤ u, V ≤ v), (u, v) ∈ I.

According to Definition 2.1, we can subsequently infer the properties below:

1. For u, v from [0,1], C(u, v) maintains non-decreasing regardless of the change in u,v. To be more specific, if v is unchanged, the value of C(u, v) depends on u, if u is increasing then C(u, v) will increase or stay the same, and vice versa.

2. C(u, 0) = C(0, v) = 0; C(u, 1) = u, C(1, v) = v for any (u, v) ∈ I; which means that if one of (u, v) become 0, C(u, v) will be 0; if one of (u, v) become 1, then C(u, v) will depend on the other variable which is not 1.

3. for all u1, v1, u2, v2 from [0,1], u1 ≤ v1 and u2 ≤ v2 , we have:

VC([u, v]) = C(u2, v2) − C(u1, v2) − C(u2, v1) + C(u1, v1) ≥ 0

4. For any variables (u, v) from I, we have:

max(u + v − 1, 0) ≤ C(u, v) ≤ min(u, v). 5. For any variables u1, v1, u2, v2 from [0,1], we have:

|C(u2, v2) − C(u1, v1)| ≤ |u2− u1||v2− v1|.

6. C(u, v) = uv when u, v are independent.

Theorem 2.1 (Sklar Theorem) [38] Let F be a bivariate joint distribution function and F1, F2 the marginal distributions. Then, there is a copula C makes:

F (x, y) = C(F1(x), F2(y)).

Copula C can be uniquely determined when F1, F2 are continuous. Inversely, if F1, F2

are continuous and C is a copula function, then F (x, y) which is determined by the equation above is a bivariate joint distribution, with margins F1 and F2.

According to Theorem 2.1, by using the inverse functions of two random variable distri-butions, and the joint distribution function of them, it is possible to estimate a copula function that can describe the correlation between them. Conversely, a copula func-tion can separate the marginal funcfunc-tions corresponding to each random variable and the related structures between them. This not only effectively increases the accuracy of modeling multiple random variables, but also facilitates the analysis of correlations between variables.

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2.2 Dependence Measures

When we analyze the correlation of random variables, the value of correlations is always estimated by the following four coefficients: Pearson, Spearman, Kendall, and Tail de-pendence. Pearson is the coefficient to analyze the linear correlation between variables, while the other three are measurements based on copula. Spearman and Kendall are mainly measures of the dependence relationship in consistency, and the tail coefficient measures the correlation via EVT to analyze the possibility that the two variables also have extreme actual values simultaneously. For simplicity, we use copula to characterize tail dependence.

2.2.1 Pearson Linear Correlation Coefficient

Definition Assume random variables X Y , E(X) and E(Y ) are their respective ex-pectations and V ar(X) and V ar(Y ) are their respective variance, satisfying V ar(X) > 0 and V ar(Y ) > 0, and COV (X, Y ) is the covariance, then the Pearson coefficient can be defined as:

ρ(X, Y ) = COV (X, Y ) pV ar(X)V ar(Y ) =

E(XY ) − E(X)E(Y ) V ar(X)V ar(Y )

ρ(X, Y ) is a linear dependence measure between variables X and Y . The closer |ρ(X, Y )| is to 1, the stronger the correlation is. When ρ(X, Y ) equals 0, there is no correlation between variables.

2.2.2 Spearman’s Rank Correlation

Definition If X and Y have marginal cumulative distribution function (cdf) F1 F2

respectively, then Spearman’s rank correlation between X and Y is written as: ρs= ρ(F1(X), F2(Y )) =

COV (F1(X), F2(Y ))

pV ar(F1(X))V ar(F2(Y ))

If there exists a copula function C(u,v), and it can represent the correlation between X and Y, then ρs can be computed as follow [33] :

ρs(X, Y ) = 12 Z 1 0 Z 1 0 [C(u, v) − uv]dudv.

where u = F1(X), v = F2(Y ), u,v ∈ [0,1], and ρs ∈ [0, 1]. The bigger the value, the

stronger the correlation between variables.

2.2.3 Kendall’s Rank Correlation

Definition Assume that (X1, Y1) and (X2, Y2) are random variables and they follow

instinct distributions, and are independent of each other, then Kendall’s rank correlation can be defined as:

τ (X, Y ) = P [(X1− X2)(Y1− Y2) > 0] − P [(X1− X2)(Y1− Y2) < 0]

= 2P [(X1− X2)(Y1− Y2) > 0] − 1.

Also, τ can be computed from copula function as follow [33]: τ (X, Y ) = 4 Z 1 0 Z 1 0 C(u, v)dC(u, v) − 1.

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2.2.4 Tail Dependence

Tail dependence refers to how the values of variables change in relation to the value of a certain other variable changing. Tail dependence serves as a very important factor when studying the dependence structure of random variables. In the financial market, financial data generally exhibit the characteristics of leptokurtosis, but the tail dependence is often different from gradual independence shown by the tail of normal distributions. However, this feature can be easily described by copula functions.

Definition Assume that F1 and F2 are marginal distributions of random variables X

and Y , F is the joint distribution function of X and Y and C is the copula function, then the upper and lower tail dependence are defined as follows [29]:

λL= lim α→0P (Y < F −1 2 (α)|X < F −1 1 (α)) λU = lim α→1P (Y > F −1 2 (α)|X > F −1 1 (α))

If there exits a copula C between variables X and Y , then tail dependence can also be written as [10]: λL= lim α→0 C(α, α) α λ U _{= lim} α→1 1 − 2α + C(α, α) α

Where α refers to the probability, F₁−1(α), F₂−1(α) are quantiles and λL, λU∈ [0,1]. Tail dependence from copula mainly describe the dependence structure of tails of financial time series. It can reflect whether other stocks will change in the event of the occurrence of change in one stock, which makes a difference in the study of fluctuation in financial markets.

2.3 The Copula Family

Our work covers a class of copulas made up of frequently-used copulas: the Student-t copula, Gaussian copula, and Clayton copula. Other copulas include the Archimedean family, which includes the Plackett copula, Gumbel copula, Frank copulathe and so forth. This family of copulas was identified by Schweizer and Sklar [38] in their research of t-norms and later was named by Ling. These copulas will be studied in particular because they can be used to model many different dependence structures, and they are not elliptical. The one-parameter Archimedean are of interest in particular. The present paper investigates how well these three different copulas fit the financial data.

2.3.1 Gaussian Copula

For the sake of convenience, we set ui= Fi(xi). The Gaussian (or normal) copula, which

is the copula of the multivariate normal distribution, can be defined as follows [40]: CGaussian(u, v, ρ) = Φρ(Φ−1(u), Φ−1(v)

where Φρ is a bivariate standard normal joint distribution, Φ−1(u) and Φ−1(v) are the

inverse functions of Φ(u) and Φ(v). Φ(u), Φ(v) ∼ N(0,1). ρ is the linear dependence parameter between Φ−1(u) and Φ−1(v) and ρ ∈ [−1, 1].

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2.3.2 Student-t Copula

The same as Gaussian copula being derived from the multivariate normal distribution [25], the t copula, is from the multivariate t distribution. It is defined here as:

Ct(u, v; ρ, d) = td,ρ(t−1_d (u), t−1_d (v))

Where d is the degrees of freedom parameter. The t copula and the Gaussian copula all have the symmetry structure in their density functions. Both of the copulas can achieve better results in characterizing the symmetry dependence between financial assets. But the t copula performances more effectively in describing the tail dependencies between assets. The use of the t copula is increasing popular, because, by changing d, the degree of tail dependence can be adjusted. When the value of d equals 1, we can simulate a Cauchy distribution.

2.3.3 Clayton Copula

In 1978, Clayton introduced the Clayton copula ][12], and it is expressed as: C(u, v) = (u−θ+ v−θ− 1)−1/θ, θ ∈ (0, ∞)

To study the return series of the exchange rates of Mark-US dollar and Yen-US Dollar, Patton utilized a modified Joe-Clayton copula. This copula is not restricted by symmet-rical structure, but symmetry is included as a special case. The definition of Joe-Clayton copula is: CJ C(u, v | τU, τL) = 1 − ((1 − (1 − u)κ)−γ + (1 − (1 − v)κ)−γ− 1) 1/κ where κ = 1/log2(2 − τU) γ = −1/log2(τL) τU ∈ (0, 1), τL∈ (0, 1)

A slight asymmetry is still present in the Joe-Clayton copula when τU = τL, which is an

inconvenience for symmetrical dependence modeling. This problem can be overcome by using a modified form of the copula, the symmetrized Joe-Clayton copula (SJC), given by:

CSJ C(u, v | τU, τL) = 0.5 ∗ CJ C(u, v|τU, τL) + 0.5 ∗ CJ C(1 − u, 1 − v|τL, τU) + u + v − 1,

which is symmetric when τU = τL.

2.4 Copula Estimation

The maximum likelihood (MLE) methodEand inference functions for margins (IFM) method are used as estimation methods in the present paper.

2.4.1 Maximum Likelihood Estimation (MLE) Method

First we assume F is the joint distribution of variable vector (X, Y ), and F1, F2 the

respective marginal distribution functions. Each marginal distribution function depends only on θi(i = 1, 2). Denote the unknownevectorfoftparameters by ν = (θ1, θ2, θ), where

θ is thejparameters of the bivariate copula {Cθ, θ ∈ Θ} and Cθ is completely known

except for the parameter θ. Suppose that {ut, vt}Tt=1 is a sample of size T. Hence we

have by Sklars theorem:

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Thus the jointjdistribution function F is fully specifiedpby the parameter vector ν = (θ1, θ2, θ).

Differentiatingpwithkrespect to all variables, we obtain the density function f: f (u, v) = c(F1(u), F2(v))f1(u)f2(v),

where Fi is the marginal distribution, fi is the density function of Fi and c is the copula

density, given by

c(u, v) = ∂

2_{C(u, v)}

∂u∂v Then the log-likelihood function is derived:

L(ν) =

T

X

t=1

(logc(F1(ut; θ1), F2(vt, θ2); θ) + log(f1(ut; θ1)) + log(f2(vt; θ2)))

The maximum likelihood estimate νM LE is given by:

νM LE = arg max ν l(ν)

It is assumed throughout this section that, for the multivariate model and all its cop-ula and the univariate distributions, the generally regcop-ular qualifications for asymptotic maximum likelihood theory hold. Under these conditions, the existence of the maximum likelihood estimator is verified and the estimator is consistent and asymptotically valid. 2.4.2 Inference Functions for Margins (IFM) Method

The second approach for estimation is the IFM method. Different from MLE method, the parameters θ1 θ2 of marginal distributions and the unknown parameter θ of copula

are estimated separately in IFM, which makes the process computationally simpler. Firstly, the marginal parameters θ1 θ2 is estimated from the marginal distributions:

ˆ θ1 = arg max θ1 T X t=1 logf1(ut; θ1) ˆ θ2= arg max θ2 T X t=1 logf2(vt; θ1)

Secondly, given θ1 and θ2, we obtain the estimation of θ :

ˆ θ = arg max θ T X t=1 logc(F1(ut), F2(vt); θ, ˆθ1, ˆθ2)

Then we can define the IFM estimator as:

νIF M = ( ˆθ1, ˆθ2, ˆθ)0.

2.5 Copula Testing and Selection

Copula testing can be seen in two parts: first, the test of the margins and, second, the copula goodness of fit. Here we introduce several testing methods.

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2.5.1 K-S Test and Q-Q Plot

In the test of marginal distribution, the proposed evaluation method of the density dis-tribution model based on sequence probability integral transformation is applicable to the marginal distribution test. To be more specific, first probability integral transforma-tion is performed on the original sequence, and then we check whether the transformed sequence follows i.i.d (0,1) uniform distribution. If the transformed sequence obeys the independent and identical distribution, this indicates that the model is correct; if the transformed sequence obeys a uniform distribution, then the nulluhypothesis of the marginal distribution is correct. In general, the autocorrelation test is performed on the transformed sequence, and if the sequence does not have an autocorrelation, the sequence is considered to be independent, and a Kolmogorov-Smirnov test (K-S test) can determine if it follows uniform distribution.

Q-Q plots, or Quantile-Quantile plots, can also visually represent whether the fitted distribution fit well to actual data. In this paper,the marginal distributions goodness of fit is performed by K-S test.

Also, in oder to select the most acceptable model, the test of copula models goodness of fit ought to be evaluated.

2.5.2 Copula Selection

As found by Durrleman et al. [11], if the selected copula is unsuitable, obtained results might be significantly altered. Therefore, choosing correct copula is essential. The “em-pirical copula” proposed by Deheuvals, is the initial approach to selecting copula. The definition follows as:

ˆ C(t1 T t2 T) = 1 T T X t=1 1ut≤u_t1,vt≤v_t1

Where 1 is the indicator function. uti vti i = 1, 2 are the ti-th order statistics of the

variables and ti∈ [1, . . . T ].

In this framework, the best copula is defined as the copula which can minimize the distance from the hypothesizedicopula to the empirical copula. This distance is mea-sured with the discrete L2 norm. The quadratic distance between two copulas C1 and

C2 in a (finite) set of bivariate points A = a1, a2, ..., an is defined as[13]:

¯ d(C1, C2) = " _n X i=1 (C1(ai) − C2(ai))2 #1/2

Let {Cn}1≤n≤N be the set of copula under consideration. The criterion of selection is the

copula Cn which minimizes the quadratic distance between Cn, the estimated copula,

and the empirical copula ˆC in the region of interest.

Another option is to use the Akaikes information criterion (AIC) [4], and it is defined as:

AIC = −2L(ˆν) + 2k

where k is the number of estimated parameters and ˆν are the log-likelihood estimation. If the AIC value is quite small, then the copula performs a good fit to the sample data. We utilize AIC criterion to choose the best fitted copula in the empirical section.

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2.6 Marginal Distribution Modeling

Correctly fitting the marginal distributions of the log return series of two assets is the key to constructing the copula model. Therefore, it is imperative to choose the appropri-ate model of marginal distribution to ensure the correctness of the copula. In this paper, our marginal model is based on various autoregressive moving average (ARMA) mod-els with generalized autoregressive conditional heteroskedasticity (GARCH) processes, namely ARMA-GARCH models. In these models, the standard innovation is to obey the normal distribution and Student-t distribution respectively. ARMA-GARCH fam-ily models have been widely applied to fit univariate variables and to model the mean and volatility of time series data. This is because the main observed characteristics of financial markets can be captured by these models effectively.

2.6.1 ARMA Process

The ARMA model is a popular and effective approach to the study of time series. The AR, MA, and ARMA models are all designed to explain the intrinsic autocorrelation of time series, accordingly to predict the future values. Based on the ARMA model, there are extended ARIMA and SARIMA models.

The ARMA model is composed of two parts: an autoregressive (AR) and a moving average (MA). It is very easy to grasp the principle of the AR model. In the AR model, the current value of time series can be predicted by the linear combination of past value and white noise, which can be considered as a simple extension of random walk model. However, in the MA model, the current value is modeled from a linear combination of white noise occurring synchronously and at various times in the past. The biggest difference of the two models is that the historical white noise in the AR model indirectly affect the current predicted value (by affecting historical values of time series) [1]. The ARMA model is usually referred to as the ARMA(p,q) model. In this model, p refers to the order of the autoregressive part while q refers to the order of the moving average part. The BoxJenkins method can be used to estimate ARMA models.

Definition The process Xt is a zero-mean ARMA(p,q) process if it is stationary and

it satisfies for all t ∈ Z [30]:

Xt− φ1Xt−1− · · · − φpXt−p= t+ θ1t−1+ · · · + θqt−q

Where t∼ W N (0, σ2). Then ARMA(1,1) can be derived as:

Xt− µ = φ1(Xt−1− µ) + t+ θ1t−1

Due to the volatility clustering of financial time series, the volatility (second-order mo-ment) is not a constant anymore. The AR, MA and ARMA models cannot characterize the conditional heteroscedasticity of time series, on top of that, ARCH and GARCH model are needed in solving this problem.

2.6.2 ARCH Model

The ARCH model characterizes and models observable time series. Unlike the tradi-tional econometric models of time series that assume the variance to be stable, the ARCH model, regards the conditional variance or the current error term as a func-tion of the error term in the previous time period, which implies that the condifunc-tional variance varies with time. In 1982, Engle proposed using the autoregressive conditional heteroskedasticity model (ARCH model) to study the volatility [36]. In simple terms, the conditional variance of t is dependent on the magnitude of its previous value t−1

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Definition ARCH(q):

yt= σtzt

σt2 = α0+ α1t−12+ · · · + αq2t−q

where zt is an i.i.d random variable whose mean equals zero and variance equals one,

σt is the conditional variance. tis the residual of the white noise time series. α0 is the

mean, αi is the conditional variance and α0> 0 αi≥ 0, Σi=1qαi = 1.

However, while applying the ARCH model we found that when the lag order is large, ”αi

is non-negative” becomes difficult to satisfy, and the ARCH model cannot solve problems concerning volatility clustering. To overcome these problems, Bollerslev [36] modified the ARCH model to derive a generalized autoregressive conditional heteroskedasticity. (GARCH) model. This model extends the ARCH model to have a more flexible lag structure and longer memory. Under the conditions of the GARCH model, the condi-tional variance can depend on its own previous value, effectively avoiding overfitting and reflecting volatility clustering in the meantime.

2.6.3 GARCH model

Since the ARCH model limits the length of the lag, we can refer to a GARCH model to show more lasting fluctuations, which also allows the conditional variance αi to depend

on its previous lag value. In general, The GARCH(p, q) model is presented as follows. Definition GARCH(p,q):

yt= σtzt

σt2 = α0+ Σqj=1αjt−j2+ Σpi=1βiσ2t−i

Where zt is an i.i.d random variablewhose mean equals zero and variance equals one,

σt denotes the conditional variance, t−j is residue and satisfies α0 > 0, αj ≥ 0 (j=0,1,

. . . ,q), βi≥ 0 (i=1,2,. . . ,p), and Σp_i=1βi+ Σq_j=1αj < 1.

After comprising these two models we can see that, GARCH model has the same mean equation as the ARCH model, while GARCH has one more Σp_i=1βiσt−i2 than ARCH for

conditional variance equation. As a result, we define αj and 2t−j as ARCH term and βi

σ_t−i2 GARCH term.

Given a time series yt, the GARCH(1,1) model can be written as:

yt= c + σtzt

σ2_t = α0+ α2t−1+ βσt−12

Where zt is an i.i.d random variable whose mean equals zero and variance equals one,

σt denotes the conditional variance, t−j is residue and satisfies α0 > 0, α ≥ 0, β ≥ 0,

and β + α < 1.

Among them, α is known as the return coefficient and β is named the hysteresis coef-ficient. The GARCH(1,1) model is equivalent to an infinite ARCH model. β + α < 1 implies that the GARCH process is second-order weak stationary, only because the mean variance and covariance are limited and constant over time. GARCH model can make model estimation and recognition easier in contrast to the ARCH model, and the simpler low-order GARCH model can represent the high-order ARCH model. In real financial market, β is normally bigger than 0.7 while α is always smaller than 0.25. The

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value of α and β determine the shape of the wave sequence. A large hysteresis coeffi-cient β means that the impact on the conditional variance will disappear only after a considerable period of time, so the volatility is long-lasting; the large return coefficient α means that the volatility reacts quickly and sharply to market movements.

However, it is widely recognized that many financial time series are non-normal and tend to exhibit fat tail behavior [18]. Bollerslev proposes that the conditional Student-t distribution can better capture the fat-tailed characteristic. Alternatives to this include the skewed-t distribution [9], the generalized error distribution [34] and the skewed gen-eralized error distribution [42]. In the present study, two distributions for the error term twill be considered, namely the normal and Student-t distributions.

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Chapter 3

Value-at-Risk

Riskmetrics, a risk control method, was published by JP Morgan in 1994, and was based predominantly on a parameter named Value at Risk.

3.1 Introduction

The Value-at-Risk (VaR) is referred to the amount of loss of a financial asset or the portfolio in the future, over the specific time period δtand under a given percentile α.

Definition Given time t, and confidence level 1 − α, where α ∈ (0, 1), the VaR of a portfolio is defined as [35]:

V aRt(α) = −inf {x|Fp,t(x) ≥ α}.

Where inf {x|A} is the lower bound of all the constituent sets that make A. Fp,t is the

cdf of the portfolio returns Xp,t at time t (return from t − δt to t).

VaR is a left-continuous and non-decreasing function of α. From the definition of VaR, at time t we have an equivalent equation: V aRt(α) = Fp,t−1(α), where F

−1

p,t(x) is the inverse

of function of Fp,t(x) at a chosen value α. Then we derive that P (Xp,t≤ V aRt(α)) = α,

this equation suggests that we are 100 ∗ α% confident that in the period δt the loss will

be no larger than the VaR. Traditionally, a normal distribution is assumed. However, in this case the probability in the tail and, moreover, the VaR, is underestimated. Con-sequently, fat-tailed distributions, like Student t, is used. This captures the fat-tailness and skewness, and, therefore, better models the distribution of the portfolio.

In recent financial studies, VaR is widely used: it is common in risk reporting,internal asset allocation and performance measurement. Especially in investment banks, VaR modeling is commonly applied to firm-wide risk, as there is potential for independent trading desks to unintentionally expose the institution to highly correlated assets. VaR modelling can provide data which helps banks confirm that they have enough reserve capital available to cover potential losses [2]. In this case, adequate VaR estimation is prominent in risk forecasting and controlling. There are several methods to estimate the VaR, which will be illustrated in next section.

3.2 Estimation of VaR

Three main methods exist for VaR estimation: the historical simulation method for analyzing historical data, Monte Carlo modeling using statistical principles to obtain the distribution of assets, and the analysis method based on the variance and covari-ance. Historical simulation and the Monte Carlo method belongs to full-value estimation

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methods. They do not need to make assumptions about the statistical analysis of mar-ket factors, so they can to a certain extent deal with the leptokurtosis phenomenon of financial data.[21].

3.2.1 Historical Simulation

The historical simulation method belongs to the category of non-parametric estimation. By processing the frequency distribution of financial assets, it calculates the average value of returns, and obtains the VaR at the lowest confidence level.

The point of historical simulation is that this method is easily calculated and that no assumptions are needed throughout the computation, thus avoiding the fluctuations in the market. However, this method requires a significant quantity of data, which may lead to a poor effect when data is so volatile.

3.2.2 Monte Carlo Simulation

The Monte Carlo simulation was proposed in 1942. It infers the overall characteristics by generating a series of random numbers to predict the distribution of actual data. As with historical simulations, there is no need to presuppose throughout the calculation. The advantage of the Monte Carlo simulation method can be listed as follows: (1) It can generate a large amount of scenarios, and performs more reliably and produces more accurate data than a historical simulation, because there is less dependence on historical data. (2) It is a full value estimation method, able to deal with non linear, extreme and heavy tail problems. (3) It can model different behavior and different dis-tributions such as white noise, autocorrelation and bi-linearity of return series.

The difficulty in the Monte Carlo algorithm is (1) revaluing the portfolio in each sce-nario. The limiting element in determining that how many scenarios it is possible to generate is the time required to revalue a portfolio.[23] (2) The simulation outcome depends on how the simulation is built. Unlike in other methods of calculating VaR, in the Monte Carlo simulation, the user is required to construct a formula, typically built on observed data. This process is crucial to the utility of the measurement itself. The usefulness of the final obtained measurement is determined by the accuracy of the model assumptions, which is determined by the the skill and judgement of the model-builders. An important consideration when using this method is that the outcome is based on this process, which is complicated and error-prone. For example, sometimes the pseudo-random numbers generated may cause cluster effect. However, because the Monte Carlo method is flexible in application and is good for handling nonlinear and non-normal problems, Monte Carlo simulation is widely used in recent years.

3.2.3 Variance-Covariance Method

The Variance-Covariance method is mainly based on the assumption of the return series that it obeys a normal distribution. The method estimates the distribution parameters of risk factors, such as the correlation coefficient, mean and variance, from historical financial data, so as to calculate the VaR.

The advantage of the Variance-Covariance method is that the principle of it is simple and the calculation is time-saving. The shortcomings are manifested in three aspects. Firstly, the estimation is based on historical data, which can not predict the risk of emer-gencies; Secondly, the assumption on normal distribution is being questioned; Lastly, it only reflects the first-order linear impact of the risk factor on the portfolio. The risk of

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non-linear financial instruments (such as options) cannot be fully measured [3].

Our object is to work with one-day period VaR. Consider the a portfolio composed of two financial assets, with log returns at (t − δt) day, denoted as R1,t and R2,t

respec-tively. The portfolio log return, denoted as Rp,tis approximately equal to ω1R1,t+ω2R2,t,

where ω1 and ω2 are the portfolio weights of assets 1 and 2. For the VaR estimation, we

could study the distribution of the univariate portfolio return series Rp,t, or the bivariate

distribution of the vector (R1,t, R2,t). In this paper, we utilize the copula-GARCH model

to fit the equally weighted stock portfolio, consisted of the FTSE 100 index and CAC 40 index, to evaluate the use of the copula-GARCH model in estimating the portfolio risk. Due to the lack of analytic and easily-used formula, which can transform the conditional mean and volatility to Value-at-Risk in the portfolio, Monte Carlo simulation is selected to forecast the VaR estimated under copula models [19].

3.3 Chapter Summary

Value at Risk is expressed as the maximum expected loss on the portfolio over a certain period for the given level of confidence. It is important for the financial industry in gaging the amount of assets required to cover possible losses. Three main methods have been used to estimate the VaR in recent years: historical simulation, Monte Carlo simu-lation and variance-covariance method. In this paper, we select Monte Carlo method to forecast the VaR of log return series over the equally weighted European stock portfolio consisting of the CAC 40 stock index and FTSE 100 stock index between 2008 and 2018 with three different level of confidence, 95%, 99% and 99.9%, based on copula-GARCH model.

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Empirical Results

The data is obtained from Wind and all modeling is programmed in R.

4.1 Data Description

The theory presented is applied to a portfolio composed of CAC 40 index and FTSE 100 stock index. The data includes stock daily closing prices for 2610 days, from 26th May 2008 to 26th May 2018, shown in Figure 4.1. The sample size is large enough and it avoids anomalies in the financial market caused by financial crisis.

Figure 4.1: Daily closing price of CAC and FTSE stock indices

We take the log-returns of CAC and FTSE separately, where you can find their plots in Figure 4.2. Figure 4.2 also presents the plots of absolute log returns of both series, Figure 4.3puts the two assets together and Table 4.1contains descriptive statistics.

In Figure 4.2we can observe evidence of the fact referred to as volatility clustering, where large absolute log returns has the tendency to follow large absolute log returns and small absolute log returns tend to follow small absolute log returns. Daily returns appear to be fairly stable over the pre-financial crisis period (before 2008), causing several outliers. After the crisis, there seems to be more instability present in all return series. In Figure 4.3 we can see that the log returns of two assets (1) have the rough same trend, (2) both do not follow a normal distribution, (3) have leptokurtosis and (4) are obviously dependent.

Descriptive statistics and characteristics of log return series are shown in Table4.1. The annualized means of both series are positive, with FTSE presenting the larger mean. Skewness and excessive kurtosis are exhibited in the log returns. This finding indicates

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Figure 4.2: Daily log returns and absolute log returns of CAC and FTSE stock indices Table 4.1: Descriptive statistics of daily log-returns of CAC and FTSE stock indices

Statistic CAC FTSE Mean 0.00004428 0.00009158 Standard Deviation 0.01460776 0.01183289 Minimum -0.09472 -0.09266 Median 0.0001529 0.0001223 Maximum 0.1059 0.09384 Skewness 0.01639495 -0.121489 Kurtosis 6.576438 9.090875

that returns not follow normal distribution.

In the following chapters, the copula-GARCH modeling will be done in the sequence below:

(1) Apply ARMA-GARCH type model to the two log return series and select the best-fitted marginal distribution by AIC criterion.

(2) Compute the pseudo-observations for the standardized residuals from (1). Those pseudo-observations (ut, vt) meet the requirement of uniform distribution. Then apply

three different copulas: Normal, student-t and Clayton to them, select the best-fitted copula by AIC.

(3) Based on the fitted copula-GARCH model, obtain one simulated log returns( ˆr1,t, ˆr2,t):

a). Simulate random variables ( ˆut, ˆvt) from the copula[14].

b). Get the simulation of standardized residuals (η1,t, η2,t) from the inverse functions of

the estimated marginals F_1,t−1 , F_2,t−1:

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Figure 4.3: CAC returns(black) vs FTSE returns(red)

where ˆθ1,t and ˆθ2,t are marginal correlations.

c). Use standardized residuals in step b). and the estimated means (µ1,t, µ2,t) and

stan-dard deviation(σ1,t, σ2,t) from ARMA-GARCH model in step (2), we obtain the

simu-lated log-returns of two assets from the mean function of GARCH model:

( ˆr1,t, ˆr2,t) = (µ1,t+ η1,tσ1,t, µ2,t+ η2,tσ2,t).

(4) Take the mean of log returns of the two assets for each day since the portfolio is equally weighted:

ˆ

rp,t= 0.5 ˆr1,t+ 0.5 ˆr2,t

(5) Simulate N = 5000 Monte Carlo scenarios over the holding period [0,T] and calculate the VaR of log return with 95%, 99% and 99.9% for each day.

(6) Repeat the procedure to calculate the VaR based on simulations from univariate ARMA-GARCH model.

(7) Compare the portfolio log return to the real data with the VaR of each day, calculate the number of days that the real portfolio log return exceed their VaR and compute the percentage of out-of-sample performance.

4.2 Marginal Distribution Modeling

As mentioned in Section 2, in order to apply the copula models, the conditional marginal distributions must be specified. Therefore, it is necessary to pretest the data. By ex-amining the Augmented Dickey-Fuller Test shown in Table 4.2, we find that the log return series are both stationary. The LjungBox test in table4.2indicates the presence of autocorrelation in all log return series at order 11. From table4.3 and4.4, Lagrange Multiplier and Portmanteau-Q tests show the existence of ARCH effects in both series, which supports the use of a GARCH-based approach.

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Table 4.2: ADF test and LM test for two log-return series

TEST CAC FTSE

ADF Test

Statistic -15.322 -14.766 Probability 0.000 0.000

LB Test

df 11 11

p-value 2.868e-05 8.567e-07 Table 4.3: Arch test for CAC log-return series order Portmanteau-Q statistics p-value

4 534 0

8 962 0

12 1422 0

16 1698 0

20 2011 0

order Lagrange-Multiplier statistics p-value

4 2254 0

8 839 0

12 511 0

16 357 0

20 264 0

Table 4.4: Arch test for FTSE log-return series order Portmanteau-Q statistics p-value

4 902 0

8 1599 0

12 2269 0

16 2775 0

20 3255 0

order Lagrange-Multiplier statistics p-value

4 1843 0

8 671 0

12 400 0

16 284 0

20 213 0

This implies that it is appropriate to fit these series with ARMA-GARCH-type models. ARMA-GARCH models are used by many authors to model financial return series and in this paper we will use ARMA(1,1)-GARCH(1,1) as marginal modeling. We fit for the log return series rf and rcwith normal and student-t distributions.

Table4.5 displays parameter estimates for both marginal distributions. The results of AIC and BIC illustrate that the ARMA-GARCH model with student-t distribution is always effective than that with normal innovations in both stock indices. Thus, we select the ARMA(1,1)-GARCH(1,1) with t innovation to model the marginal distribution of the two stock index series in the following modelings. Table4.6represents the values of test statistics for ARMA-GARCH residuals.

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Figure 4.4: QQ-plot of CAC index against normal(left) and student-t(right) distributions

Figure 4.5: QQ-plot of FTSE index against normal(left) and student-t(right) distribu-tions

Figure 4.6 and 4.7 show the dependency among the residuals of the two assets with the ARMA-GARCH model. Dependency modeling will be explained in the following part.

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Table 4.5: Parameter estimates of ARMA-GARCH models and standard errors of FTSE and CAC stock indices.

parameter ARMA-GARCH-N ARMA-GARCH-t CAC

mu1 1.772e-05(1.624e-05) 2.660e-05(2.443e-05) ar1 9.666e-01(2.360e-02) 9.561e-01(3.384e-02) ma1 -9.841e-01(1.750e-02) -9.746e-01(2.571e-02) omega1 2.293e-06(5.970e-07) 1.509e-06(5.971e-07)

alpha1 1.057e-01(1.312e-02) 1.037e-01(1.596e-02) beta1 8.866e-01(1.320e-02) 8.961e-01(1.455e-02) shape 6.005e+00(7.318e-01) AIC1 -5.976897 -6.020122

BIC1 -5.963405 -6.004381

FTSE

mu2 1.753e-05(8.822e-06) 1.996e-05(9.642e-06) ar2 9.570e-01(1.741e-02) 9.551e-01(1.738e-02) ma2 -9.755e-01(1.299e-02) -9.732e-01(1.294e-02) omega2 1.935e-06(4.562e-07) 1.606e-06(5.124e-07)

alpha2 1.058e-01(1.388e-02) 1.141e-01(1.739e-02) beta2 8.785e-01(1.536e-02) 8.787e-01(1.723e-02) shape 6.218e+00(8.255e-01) AIC2 -6.486518 -6.518957

BIC2 -6.473026 -6.503216

Table 4.6: Values of test statistics for ARMA-GARCH residuals. Test Statistics ARMA-GARCH-N ARMA-GARCH-t

CAC LM Arch Test(T R22) 6.168085 6.534284 p-value 0.9073728 0.8867956 QW(10) for R 6.802552 6.805709 p-value 0.7439445 0.7436511 QW(10) for R2 _6.078729 _6.424053 p-value 0.808606 0.7784675 FTSE LM Arch Test(T R22) 5.078978 4.407514 p-value 0.9552888 0.974911 QW(10) for R 5.566463 5.526357 p-value 0.8502782 0.8533649 QW(10) for R2 4.667559 4.012707 p-value 0.9122513 0.9467719

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Figure 4.6: Dependency under ARMA-GARCH-N

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4.3 Copula Modelling

The common copula family used in modeling is: Frank, Clayton, Joe, Gumble, T-student and Normal. Copula selecting is partly based on previous analysis. In this section, three copula are modeled to fit all assets residuals with their marginal distributions according to the IFM method.

4.3.1 Estimation of Parameters

ut= F1(x1,t); vt= F2(x2,t),

where x1,t,x2,t are standardized residuals of two assets from GARCH modeling, and

F1, F2 are marginal distributions. If both ut vt series are standard uniform then the

models were correctly specified. We have two ways to select the marginal distributions: the parametric method and the non-parametric method. In order to save time and to provide a better fit, we chose the non-parametric method to transform residuals into uniform variables. Here, asset 1 is taken as an example:

F1(X1) = 1 n n X j=1 1x1,j≤x1,i

where 1x1,j≤x1,i is an indicator function:

1x1,j≤x1,i =

1, if x1,j ≤ x1,i

0, if x1,j > x1,i

By KS test we can see that for both utand vtseries the p-values are 1. Together with the

CDF plots below (Figure4.8), we can tell that the fit seems quite good. While Figure

4.9shows the correlation of utand vt.

Figure 4.8: Empirical distribution of transformed series ut and vt.

4.3.2 Copula Fitting

We apply Gaussian copula, tcopula and Clayton copula to utand vtseries respectively,

resulting in test statistic values shown in Table 4.5. The quality of the adjustment can be assessed by the minimal quadratic distance method or AIC criterion, as mentioned in Section 2. We chose the common way to select copula with the minimal AIC. The analysis of the results in Table 4.7 shows that t-copula shows lowest AIC and BIC. Table 4.8 shows the different dependence measure of three copulas: Spearman’s Rank

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Figure 4.9: Dependency of ut and vt

Correlation ρ, Kendall’s Rank Correlation τ and lower and upper tail dependence λs (Section 2.2). From the two tables we can tell t copula offers the best adjustment and also the symmetric structure of tail dependence, indicating that the two stock indices will synchronously increase and decrease.

Table 4.7: Summary of three copulas.

copula parameter AIC BIC tcopula ρ=0.8428(0.006) -3289.084 -3277.351 Gaussian copula ρ=0.8386(0.005) -3155.746 -3149.879 Clayton copula α=2.352(0.061) -2614.075 -2608.209

Table 4.8: Dependence measures of three copulas.

copula tau rho lower lambda upper lambda t copula 0.8307669 0.6381887 0.5425335 0.5425335 Gaussian copula 0.8263467 0.6332509 0.0000000 0.0000000 Clayton copula 0.7265617 0.5404412 0.7447509 0.0000000

4.4 Estimation of VaR

In this step we are going to compute the Value-at-Risk of the European stock portfolio using a copula-GARCH model. We arbitrarily consider a equally weighted stock portfo-lio, however, this is not a limit and the weights can be various. Hence, considering that the returns are small, we have approximately portfolio log return Rp,t≈ 1₂R1,t+ 1₂R2,t.

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99% and 99.9% confidence level.

Following this, we compute the number of days that the practical portfolio log return on that day exceeds the VaR estimated under copula-GARCH model. This is repeated under univariate GARCH model to evaluate the effect of copula-GARCH model on VaR. Results are displayed in Figure4.10 and Table 4.9.

From Table4.9, We can see that for a higher level of confidence, α = 0.01 or α = 0.001, the percentage of forecasting performance provided by the tcopula-GARCH model is almost close to α; for lower level of confidence such as α = 0.5, both models do not provide a good fit. This indicates that tcopula-GARCH model performs better in VaR estimation for higher levels of confidence(1 − α). Figure4.10 show that VaR estimated under tcopula-GARCH model are higher than that under univariate GARCH model, and with higher level of confidence the VaR estimated under univariate GARCH model are more volatile.

Table 4.9: Out-of-Sample Performance(percentage) 1 − α tcopula-GARCH model univariate GARCH model

0.05 0.064(168) 0.069(180)

0.01 0.009(24) 0.007(19)

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Figure 4.10: Value at Risk estimation under copula-GARCH model(up) and GARCH model without copula(down) with different level of confidence 95%(blue), 99%(red), 99.9%(green)

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Chapter 5

Conclusion and Further Work

This paper represented that conditional copula theory provides a robust approach for estimating the VaR of an equally weighted portfolio in the European stock market. Our model is particularly useful in enabling simultaneous consideration of the assymmetic characteristics of financial data which make it not conform to a normal distribution, for example skewness, leptokurtosis, and fat tails. This asymmetry and excess kurtosis can be captured by the marginals when using GARCH-type models, while copulas can fit the asymmetry and leptokurtosis in the dependencies. Additionally, copulas are flexi-ble, which enables estimation of the joint multivariate distribution, as the dependence structure and marginal distribution models are kept separate. This separation means that parameters in the marginals and copulas can escape the curse of dimensionality. The empirical study of the real European stock portfolio by copula-GARCH model results in the following findings:

(1) The log return series of both stocks have the characteristics of non-normality, asym-metry and leptokurtosis.

(2) The results of the analysis reinforce how important it is to specify the right marginal distributions. ARMA(1,1)-GARCH(1,1) with student-t innovation provides a better fit of the log return series of both assets than that with normal innovation. The skewness parameter and the degree of freedom parameter of the student-t distribution indicate the significance of asymmetry and kurtosis in the univariate time series. This implies that assumption of normality is not fulfilled, and traditional models based on this as-sumption are, therefore, inappropriate for fitting the univariate series.

(3) Copula functions are very important for constructing joint bivariate distributions with marginal distributions. After applying three different copulas, we found that Stu-dent t copula serves as an effective method to fit the dependence structure between European stock futures according to AIC criterion.

(4) This paper introduced three main methods to estimate Value-at-risk. We used Monte Carlo simulation because it has powerful computational ability and it is useful in making no assumptions during calculation. The results showed that the tcopula-GARCH model performs better in VaR simulation under higher levels of confidence, thus indicating that dependence among different stocks influences the prediction of VaR.

However, due to limited time and knowledge, possible extensions for future research remain, described below:

(1) The conditional variance in our GARCH model is a symmetric function, which is in-consistent with reality. The actual yield movement of financial asset exhibits a leverage effect, to be more specific, the rise and fall of the stock price may asymmetrically affect the subsequent fluctuations of price. The decline of the stock price provides greater impact on subsequent fluctuations compared with the increase of price, even if they have the same magnitude. This means that a better model which reacts asymmetrically

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to both positive and negative residuals, such as the Exponential GARCH (EGARCH) model, could result in improvements in forecasting performance [19].

(2) SJC copula embodies symmetric and asymmetric dependencies. Throughout our study, asymmetry found in the dependence structure has little impact on performance in forecasting of portfolio VaR. In this case, symmetric copulas such as SJC can provide a closer fit of the structure of dependencies between stock return series, and, conse-quently, better performances in forecasting the portfolio VaR [31].

(3) Bayesian analysis can be considered as a method for estimation of the parameters of copula. This analysis is independent of parameter choices, and is easily numerically implemented. Even for small samples, reliable identification is possible. Additionally, Bayesian analysis is conceptually advantageous because it is genuine model selection method, independent of the choice of an optimal parameter. Furthermore, Bayesian analysis can be used with any copula, provided that it is possible to numerically com-pute the copula density and Kendalls tau [26].

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Appendix: R code

install.packages("MASS") install.packages("copula") install.packages("GAS") install.packages("TSA") install.packages("fGarch") library("TSA") library("copula") library("fGarch") library("tseries") require(rugarch) library("GAS") ##data description Data<- read.csv("C:/Users/Susan/Desktop/thesis/data.csv",header=TRUE) Date <- as.Date(Data$Date) FTSE <- Data$FTSE CAC <- Data$CAC Data <- cbind(FTSE,CAC) cor(FTSE,CAC) plot(Date,FTSE) plot(Date,CAC) #log return r_f <- diff(log(FTSE)) r_c <- diff(log(CAC)) cor(r_f,r_c) plot(Date[-1],r_f) plot(Date[-1],r_c) plot(Date[-1],abs(r_f)) plot(Date[-1],abs(r_c)) plot(Date[-1],r_f,ylab="return",type = "l") lines(Date[-1],r_c, type = "l" , col = "red" ) ##descriptive statistics return <- cbind(r_f,r_c) plot(density(r_f),ylim=c(0,60)) lines(density(r_c),col="red") summary(return) sd(r_f) sd(r_c) 31

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skewness(r_f) skewness(r_c) kurtosis(r_f) kurtosis(r_c) ##garch modelling #ADF adf.test(r_f) adf.test(r_c) ##archlm install.packages("aTSA") library("aTSA") mod <- arima(r_f,order = c(1,0,0)) arch.test(mod) LB.test(mod)

mod1 <- arima(r_c, order = c(1,0,0)) arch.test(mod1)

LB.test(mod1) #FTSE

#norm

r_f.vec <- as.vector(r_f)

fit_f.1 <- garchFit(formula = ~arma(1,1)+garch(1,1),data=r_f.vec,include.mean=TRUE,cond.dist="norm",trace=F) residuals_f <- residuals(fit_f.1, standardize= TRUE)

acf(residuals_f^2,na.action=na.omit) pacf(residuals_f^2,na.action=na.omit) qqnorm(residuals_f); qqline(residuals_f) summary(fit_f.1)

#std

fit_f.2 <- garchFit(formula = ~arma(1,1)+garch(1,1),data=r_f.vec,include.mean=TRUE,cond.dist="std",trace=F) residuals_f.1<- residuals(fit_f.2, standardize= TRUE)

acf(residuals_f.1^2,na.action=na.omit) pacf(residuals_f.1^2,na.action=na.omit) qqnorm(residuals_f.1);qqline(residuals_f.1) summary(fit_f.2) #cac #norm r_c.vec <- as.vector(r_c)

fit_c.1 <- garchFit(formula = ~arma(1,1)+garch(1,1),data=r_c.vec,include.mean=TRUE,cond.dist="norm",trace=F) residuals_c <- residuals(fit_c.1, standardize= TRUE)

acf(residuals_c^2,na.action=na.omit) pacf(residuals_c^2,na.action=na.omit) qqnorm(residuals_c); qqline(residuals_c) summary(fit_c.1) #std r_c.vec <- as.vector(r_c)

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residuals_c.1 <- residuals(fit_c.2, standardize= TRUE) acf(residuals_c.1^2,na.action=na.omit)

pacf(residuals_c.1^2,na.action=na.omit) qqnorm(residuals_c.1); qqline(residuals_c.1) summary(fit_c.2)

##norm

plot(residuals_f,residuals_c,type = "p", xlab = "FTSE", ylab = "CAC") ##std

plot(residuals_f.1,residuals_c.1,type = "p", xlab = "FTSE", ylab = "CAC")

### uniform ##using std db #non-parametric method u_f <- pobs(residuals_f.1) u_c <- pobs(residuals_c.1) hist(u_f) hist(u_c) plot.ecdf(u_f,xlab="u_f") plot.ecdf(u_c,xlab="u_c")

plot(u_f,u_c, xlab = expression(hat(u_f)), ylab = expression(hat(u_c))) ks.test(u_f,"punif") ks.test(u_c,"punif") #parametric method mean = mean(u_f) sd=sd(u_f) u_f.1 <- pt(residuals_f.1,df = 4) mean = mean(u_c) sd=sd(u_c) u_c.1 <- pt(residuals_c.1,df = 4) hist(u_f.1) hist(u_c.1)

plot(u_f.1,u_c.1, xlab = expression(hat(u_f)), ylab = expression(hat(u_c))) #

U <- cbind(u_c,u_f) ##copula

fitcop_f.1 <- fitCopula(ellipCopula("t", dim = 2), data = U, method = "mpl") summary(fitcop_f.1) AIC(fitcop_f.1) BIC(fitcop_f.1) tC <- tCopula(0.8428) rho(tC) tau(tC) lambda(tC)

Estimating the value-at-risk of European stock portfolio using Copula-GARCH model