Risk measurement based on extreme value theory and copula theory : an empirical study of European stock markets

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Value Theory and Copula Theory-An

Empirical Study of European Stock

Markets

Bi Wu

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: Bi Wu

Student nr: 10828648

Email: wubiinholland@gmail.com Date: August 15, 2017

Supervisor: Z. (Merrick) Li Second reader: Roger Laeven

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Abstract

This study focuses on Value-at-Risk (VaR) estimation based on Ex-treme Value Theory (EVT) method and Copula approach. CAC 40 (French stock market index), DAX (German stock index), FTSE100 (Financial Times Stock Exchange 100), IBEX 35 (Spanish Exchange Index), Euro Stoxx 50 (Eurozone stock index) are investigated in the empirical study. The results are compared with historical simulation VaR and variance-covariance VaR

Keywords Risk, VaR, Extreme Value Theory, Copula, Generalized Pareto Distribution, His-torical simulation, Variance-Covariance method

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First of all, I would definitely like to thank for all the people who offer me the the support during this year. I would like to say thanks to my parents for supporting me to finish the study, say thanks to teachers in University of Amsterdam for all the knowledge you gave me, and say thanks to my friends who accompany me and help me during this year. It is an important experience in my life to take the master program of Actuarial Science and Mathematical Finance in University of Amsterdam. During the intensive study, I was largely influenced by many excellent teachers in our faculty. I met many nice people in our class and most importantly, I obtain deeper understanding of financial risk modeling, R programming and actuarial science theory. I strongly believe these knowledge and experience will help me a lot in my future career and life. I will definitely keep pursuing deeper understanding and thinking of risk management afterwards.

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Introduction

In financial market, Value-at-Risk (VaR) is the most popular method in risk measuring. There are many other alternative models such as historical simulation and variance-covariance method. However, the majority of the models are based on certain assump-tions when it was created. For example, historical simulation assumes the history will happen again in the future, variance-covariance method assumes the risk is normally distributed. Recent financial risk studies suggest that the financial risk deviates signif-icantly from normal distribution and most likely has a excess Skewness, which is also known as fat tail. Therefore, the normal distribution assumption would underestimate the extreme event and therefore underestimate the risk.

Two important theories are getting to be popular to recent researchers, one is Ex-treme Value Theory (EVT) and the other is Copula theory. EVT approach measures the excess risk above the threshold based on Generalized Pareto Distribution (GPD). And the copula approach provides a flexible method for multi-variate risk aggregation by modeling dependence structure with selected copula model. To investigate the char-acteristics of this two approaches in stock market returns risk measurement, this study includes five European market indices, which are CAC 40 (French stock market index), DAX (German stock index), FTSE100 (Financial Times Stock Exchange 100), IBEX 35 (Spanish Exchange Index), Euro Stoxx 50 (Eurozone stock index). The sample period ranges from January 2010 to April 2017.

The two approaches, historical simulation and variance-covariance method are used for comparison. The result shows the EVT and Copula approach can better describe the tail behavior of the risk compares to the traditional VaR approaches. The VaR measurement is based on 95%, 99% and 99.5%.

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Chapter 2

Definition and Concept

2.1 Value at Risk (VaR)

2.1.1 Definition

Value at Risk (VaR) is an important tool for financial risk management. J.P. Morgan published their RiskM etrics model at 1994, which contains VaR as the core risk man-agement tool, VaR gradually became a standard approach in financial risk manman-agement. VaR measures the risk by calculating the quantile of the given risk. Probability and time horizon are two most important elements in VaR measuring. VaR answers the question of what is the maximum loss would be under p% probability in N business days. The VaR can be calculated by:

V aRp(L) = F_L−1(p) = inf{x ∈ R|FL(x) ≤ p} (2.1) There are many advantages of using VaR. It captures the important aspect of risk by combining the probability and time horizon. Moreover since VaR is a quantile based approach, it is always easy to understand and to interpret and it always exist.

On the other hand, VaR approach has the following deficiencies:

• The first problem is that VaR fails to capture the tail behavior of the risk, which is an predominant problem in measuring financial market risk with excess Kurtosis. The company might suffer from those risk with very low probability but high impact.

• Sub-additive does not always hold for VaR method. When it comes to risk diver-sification in corporate environment, VaR is not always able to depict the diversi-fication benefit.

• Using VaR in constructing the optimal investment portfolio might encourage risk taking behavior. VaR gives higher risk exposure to more risky asset, which leads to higher potential loss.

There are three well-known classic VaR models: Historical Simulation, Variance-Covariance Method and Monte Carlo Simulation.

2.1.2 Non-Parametric Method

Historical simulation is an important non-parametric method of VaR modeling. The key assumption is that risk is independent and identically distributed and will repeat from history. To implement historical simulation, the historical data will be ranked by the

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worst to the best.

For example, 95% quantile will be taken from the ranked data if we want to investigate what loss amount will not go beyond, under 95% probability. The historical simulation is easy to implement as no specific distribution is assumed. However, this method is quite sensitive to time length, especially when there is extreme event happened in the past.

2.1.3 Parametric Method

Variance-covariance (Gaussian) method is a parametric approach, which assumes the risk distribution follows a normal distribution XT +1 ∼ N (µ, σ2). µ and σ2 are the mean and variance for the normal distribution. Based on the characteristic of normal distribution, Variance-covariance method allows calculation of aggregated multi-normal distribution.

Suppose there exist multiple market risk series X1, X2, . . . , Xn, which are independent and identically distributed. If we assume the distribution of each single risk follows nor-mal distribution Xi ∼ N (µ, σ2), i = 1, 2, ..., n, the equally weighted aggregate risk will follows the normal distribution with the mean µ and the variance σ2

Xaggregate= (X1+ X2+ · · · + Xn)/n ∼ N (µ, σ2/n). (2.2)

Since the variance-covariance approach is based on normal distribution assumption, the main disadvantage is that tail behavior cannot be captured when there is fat tail. Therefore variance-covarience approach tends to underestimate the risk.

2.1.4 Monte Carlo Simulation

Monte Carlo simulation method is similar to historical simulation method. The main difference is that instead of using historical data and assuming the history pattern will repeat itself at the subsequent time interval, Monte Carlo simulation generates random variables, which follow certain theoretical distribution in order to estimate the loss in the future time period.

The data generation process will repeat a large number of times (N times), as a re-sult, N final loss value can be obtained. Therefore, the VaR value can be estimated by percentile value from the Monte Carlo simulation output. Monte Carlo simulation is sensitive to the underlying distribution and the simulation process is computation consuming.

2.2 Extreme Value Theory (EVT)

Extreme Value Theory is an important approach which mostly used to deal with the events with low frequency but high impact in financial risk management. Traditional VaR approach assumes that the stock market return follows the normal distribution. However, many recent studies suggest that the distribution of stock market return tends to follow a normal distribution with excess Kurtosis, which indicates that the stock mar-ket return exhibits a pattern with fatter tail compares to normal distribution.

The VaR base on normal distribution will underestimate the impact of extreme events because it fails to capture the behavior of the tail. Modern financial risk management researches are mainly based on Extreme Value Theory (EVT) Value at Risk (V aREV T_).

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4 Bi Wu — Master Thesis

The basic idea of Extreme Value Theory VaR is assuming the tail distribution of stock market Logarithmic return follows Generalized Pareto Distribution (GPD).

2.2.1 Block Maxima Method (BMM)

Generalized Extreme Value (GEV) Distribution

The generalized extreme value distribution is defined by the following distribution func-tion: Hξ,µ,σ(x) = ( exp(− 1 + ξx−µ_σ −1/ξ), ξ 6= 0, exp(−ex−µσ ), ξ = 0 (2.3) where 1 + ξx−µ_σ > 0,

the shape parameter ξ > 0, the location parameter µ ∈ R, the scale parameter σ > 0.

The value of shape parameter ξ gives different distribution. When ξ = 0, the distribution type is known as Gumbell distribution. When ξ < 0, the distribution type is known as W eibull distribution and when ξ > 0, the distribution type is belongs to F r´echet distribution. In general, ξ reflects the converge speed of the tail goes to 0. The larger ξ is, the slower the tail converge to 0. Therefore, F r´echet distribution has fattest tail among three different types of distribution.

Model Estimation

There are two main approaches in Extreme Value Theory to estimate the threshold: Block Maxima Method (BMM) and Peaks over Threshold (POT), which is also known as Threshold Exceedances.

The basic idea of Block Maxima Method (BMM) is to divide the sample into a number of blocks with equal size n, and then apply the distribution of Generalized Extreme Value (GEV) to model the maximum loss for each block.

There is a trade off between block size and block number. Having a larger block size increases the block GEV estimation but decrease the number of observations, which leads to lower bias but larger variance. In contrary, increasing the block number would lower the variance but higher the bias for GEV estimation.

2.2.2 Peak Over Threshold (POT)

Excess Distribution Function

Suppose there exists n independent and identically distributed random variables X1, ...Xn which has a distribution function F (x) = P r{X < x}. The excess distribution of (Xt) exceeds the certain threshold (Xt> u), we define it as follows:

Fu(x) = P r{X − u ≤ x|X > u} =

F (x + u) − F (u)

1 − F (u) , 0 ≤ x ≤ xF − u (2.4) where xF is the right endpoint of distribution function F (x).

According to the studies of Balkema and De Haan (1974) and Pickands (1975), the distribution Fu(x) over the threshold u can be approximately expressed by Generalized Pareto Distribution (GPD).

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Generalized Pareto Distribution (GPD)

The Generalized Pareto Distribution (GPD) is defined as:

Gξ,β(x) =    1 − 1 + ξ_βx −1/ξ , ξ 6= 0, 1 − exp(−x_β), ξ = 0 (2.5)

for a continuous x where:

ξ: is the shape parameter, ξ ∈ R β: is the scale parameter, β > 0

By definition, when ξ ≥ 0, x ≥ 0, and when ξ ≤ 0, 0 ≤ x ≤ −β/ξ, which is similar to Generalized Extreme Value (GEV) Distribution, different ξ indicates different converge rate. When ξ = 0, Gξ,β is known as Exponential distribution, when ξ < 0 Gξ,β is known as P aretotypeII distribution and when ξ > 0 Gξ,βis belongs to OrdinaryP areto distribution with α = 1/ξ, κ = β/ξ.

Parameter Estimation

To estimate the distribution function of excess loss, the parameter ξ and β need to be estimated in Generalized Pareto Distribution (GPD) given a high threshold u. We denote Nu the number of data exceed threshold u. Yi is the excess amount calculated by Yi = ˜Xi− u, where ˜Xi is the observation above threshold u.

Therefore we obtain the independent and identically distributed series Yi, i=1,2...Nu to estimate Gξ,β. The parameter ξ and β can be estimated by the following log-likelihood function using maximum likelihood estimation:

lnL(ξ, β, Yi) = Nu X i=1 lngξ,β(Yi) = − Nulnβ − 1 +1 ξ Nu X i=1 ln 1 + ξYj β (2.6) Threshold Selection

The Peaks over Threshold approach(POT) models the risk by using the observations higher than certain threshold. The observations that higher than threshold will be mod-elled base on Generalized Pareto Distribution (GPD). Recent risk management research mostly applies the POT method. In this paper, we are focusing on the POT method. Selecting an appropriate threshold is a fundamental step in POT approach to estimate the tail behavior of stock market return. Similar to the block size in Block Maxima Method (BMM), the threshold u also has a trade off between bias and variance as shown following:

• A larger threshold u tends to has a fewer observations in the tail, and thus leads to higher variance.

• A lower threshold u allows more data in the sample, which lower the variance of the result.

Therefore, the optimal threshold selection for upper tail should be the smallest value when the tail approximation is accurate.

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E(X − u|X > u). This function gives the conditional average of excess value over the threshold, given the observation is higher than the threshold. The function is defined as

e(u) = PNu

i=1Yi Nu

(2.7) Let us assume ˆu is the optimal value for POT threshold. For observations higher than the threshold u, the mean excess function can be expressed as:

e(x) = β + ξ(x − û) 1 − ξ = ξ 1 − ξx + β − ξ û 1 − ξ , x > û (2.8) The expression indicates the mean excess function has a constant slope _1−ξξ when the data exceed the optimal threshold û. Therefore, theoretically optimal threshold can be detected visually by taking the start point of linear trend from the mean excess plot. By adopting this method, one can easily select a good threshold using the mean ex-cess plot. However, the accuracy of threshold selection cannot be guaranteed since the subjectivity in the visual selection. Thus some other techniques such as observing the stability of data need to be applied to verify the availability of the chosen threshold.

2.3 Copula definition and property

In most cases, it is impossible that an investment portfolio only affected by one single risk. Due to the complexity of the financial market, the loss of most financial products are generally related to a various types of risk. How to accurately calculate the potential loss of the aggregated risk is an important topic for the study of financial risk manage-ment.

2.3.1 Joint Distribution of Aggregated Risk

To measure the VaR of aggregated risk, the joint distribution of each single risk has to be calculated. The joint distribution is composed of both the marginal distributions of individual risks and the dependent structure among the risks. Therefore, even we know the marginal distribution of every individual risks, we are not always able to estimate the joint distribution of aggregated risk.

One exception is the multi-normal distribution, if the marginal distributions of each individual loss is given, we can calculate the joint distribution of total loss. However, since multi-normal distribution is not the common case in financial market and the nor-mal distribution does not give enough weight to tail distribution, multi-nornor-mal modeling is not a good method in financial risk aggregation.

Therefore, Sklar(1959) created the concept of Copula, which allows people to measure the joint distribution with a high flexibility.

2.3.2 Copula

Copula is a link function of multiple variables, which describes the dependent structure of the joint distribution and the marginal distribution. Copula has great importance of financial risk management as it can measure the dependent structure and build up multivariate models.

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Bivariate Copula

Suppose there are two bivariate random vectors (X, Y ), with the continuous marginal distribution F (x) = P (X ≤ x) and G(y) = P (Y ≤ y). The joint distribution H(x, y) = P (X ≤ x, Y ≤ y). Copula C is the joint distribution of the vector (X,Y), we can express bivariate copula C as:

C(u, v) =P (F (x) ≤ u, G(y) ≤ v)

=P (F←(u), G←(v)) (u, v) ∈ [0, 1]2 (2.9) where F←(u) is the generalized inverse function F←(u) = inf {x : F (x) ≥ u}

This equation can be re-arranged as:

H(x, y) = C(F (x), G(y)) (2.10)

From the above equation, one can observe that the joint distribution of vectors (X,Y) is composed of the marginal distribution of X and Y, and dependent structure C, which is represented by copula.

There are some properties of bivariate copula:

C(u, 0) = C(0, v) = 0 ∀u, v ∈ [0, 1], (2.11)

C(u, 1) = u and C(1, v) = v ∀u, v ∈ [0, 1], (2.12)

C(u, v) + C(u1, v1) − C(u1, v) − C(u, v1) ≤ 0 ∀u, v, u1, v1 ∈ [0, 1] and u1 ≤ u, v1 ≤ v (2.13) N-dimensional Copula

The d-dimensional copula C(u) = C(u1, u2, ..., ud) with (u1, u2, ..., ud) ∈ [0, 1]dis defined as the dependent structure of joint distribution function:

F (u1, u2, ..., ud) =P (F1(x1) ≤ u1, F2(x2) ≤ u2, ..., Fd(xd) ≤ ud) =C(F1(u1), F2(u2), ..., Fd(ud))

∀(u₁, u2, ..., ud) ∈ [0, 1]d

(2.14)

where Fi(Xi < x), i ∈ (1, d) is the marginal distribution and C copula. The d-dimensional copula satisfies the following properties:

• C(1, ..., 1, u_i, 1, ...1) = ui ∀i ∈ [1, d]

• C(F1(u1), F2(u2), ..., Fd(ud)) is an increasing function of ui, ∀i ∈ [1, d]

• For the random vector (ai, bi) ∈ [0, 1]d, i ∈ [1, d] with ai ≤ bi∀ai, bi, the copula satisfies the following rectangle inequality:

2 X i1=1 ... d2 X id=1 (−1)i1+i2+...+id_C(u

1,i1, u2,i2..., ud,id) ≥ 0 (2.15)

where uj,1= aj, uj,2= bj ∀j ∈ {1, ..., d}

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Sklar’s Theorem

Consider the continuous bivariate joint distribution function H(x, y) = P (X ≤ x, Y ≤ y) and marginal distribution F (x) = P (X ≤ x) and G(y) = P (Y ≤ y), there is an unique copula which satisfy:

H(x, y) = C(F (x), G(y)), (x, y) ∈ R (2.16) On the other hand, if we have defined the marginal distribution F (x) = P (X ≤ x) and G(y) = P (Y ≤ y), with a given copula we can compute the joint distribution function H(x, y) = P (X ≤ x, Y ≤ y). In the other words, we can establish multiple joint distri-butions based on same marginal distridistri-butions but different dependent structure. Sklar’s theorem also applies to d-dimensional copula in multivarite joint distribution.

Special Case of Copula

The following are some important special bivariate copula cases:

• Comonotonic copula: The random vector (X,Y) has a comonotonnic copula if there is an increasing function between X and Y: X=f(Y). Comonotonic copula describes the perfect positive dependence, which can be expressed by:

C1(u, v) = min{u, v}, (u, v) ∈ [0, 1]2 (2.17) • Contercomonotonic copula: Conversely, contercomonotonic copula describes the perfect negative dependence between the random vector (X, Y ). If there exist an decreasing function X = f (Y ), the dependent structure can be express as:

C2(u, v) = max{0, u + v − 1}, (u, v) ∈ [0, 1]2 (2.18) • Independent copula: If random variable X and Y are independent, the vector (X,Y)

has a independent copula:

C3(u, v) = uv, (u, v) ∈ [0, 1]2 (2.19)

According to the F r´echet bounds theorem, comonotonic copula C1(u, v) is the upper bound of any bivariate copula function and countercomonotonic copula is the lower bound of any bivariate copula.

C3(u, v) ≤ C(u, v) ≤ C1(u, v), (u, v) ∈ [0, 1]2 (2.20)

F r´echet bounds theorem simply reflects the truth that the correlation of random vector (X,Y) is between -1 and +1. We can construct the any copula based on these special copulas above by:

C(u, v) =w1C1(u, v) + w2C2(u, v) + w3C3

=w1min{u, v} + w2max{0, u + v − 1} + w3uv ∀(u, v) ∈ [0, 1]2

(2.21)

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2.3.3 Copula Families

Normal (Gaussian) copula, student’s t copula, Gumbel copula and Clayton copula are very common copula in financial risk management studies. Gaussian copula and t copula are used for normal distribution data and student’s t distribution separately. Gumbel copula describes the right hand side fat tail while Clayton copula describes the left hand side fat tail.

Suppose we have the distribution function U = F (X) and V = G(Y ). Normal Copula (Gaussian)

Bivariate Normal copula describes the dependent structure of bivariate normal random vector (X,Y) with correlation coefficient r = cor(X, Y ) ∈ [−1, 1] :

CN(u, v; r) =P (F←(u), G←(v)) = Z Φ−1(u) −∞ Z Φ−1(v) −∞ 1 2π√1 − r2exp{− s2− 2rst + t2 2(1 − r2₎ }dsdt ∀ u, v ∈ [0, 1] (2.22)

When the correlation coefficient r equals to 1, the copula is comonotonic copula. When r equals to -1, the copula is countercomonotonic and when r equals to 0, the copula is independent copula. Normal copula can be used to construct the joint distribution without marginal normal distribution in order to obtain the characteristic of normal dependent structure.

Student’s t Copula

Bivariate Student’s t coupla is used to capture the dependent structure of bivarite student’s t random vector (X,Y) with correlation coefficient r = cor(X, Y ) and degree of freedom v: Ct(u, v; r, ν) =P (F←(u), G←(v)) = Z t−1ν (u) −∞ Z t−1ν (v) −∞ 1 2π√1 − r2{1 + s2_{− 2rst + t}2 ν(1 − r2₎ } −ν+2 2 dsdt ∀ u, v ∈ [0, 1] (2.23)

According to the study of Demarta, S. and McNeil,A. J. (2004), student’s t copula can better describe the extreme event than normal copula. Therefore, student’s t copula is more popular and widely used in financial market risk management research.

Gumbel Copula

Bivariate Gumbel copula is defined as:

CGumbel(u, v) = exp{−[(−lnu)θ+ (−lnv)θ]1/θ}, θ ∈ [1, ∞) (2.24) When θ = 1, Gumbel copula is identical to independent copula, when θ → ∞, Gumbel copula approaches to comonotonic copula. The study of Nelsen (2007) indicates Gumbel copula can better describe heavy tail on right hand side.

Clayton Copula

Bivariate Clayton copula is defined as:

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When θ → 0, Clayton copula approaches to independent copula, when θ → ∞, Clayton copula approach to comonotonic copula. Opposite to Gumbel copula, Clayton copula can better describe the heavy tail on left hand side.

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Data and Statistic Summary

This study focuses on the daily data of stock market indices for five European countries. The daily indices data are obtained from Yahoo Finance. Sample period ranges from the first stock trading day of 2010 to the last stock trading of April 2017. Five European indices including CAC 40 (French stock market index), DAX (German stock index), FTSE100 (Financial Times Stock Exchange 100), IBEX 35 (Spanish Exchange Index), Euro Stoxx 50 (Eurozone stock index) are investigated in the empirical study. Figure 3.1 gives the plots of daily closing prices of these indices

Figure 3.1: Plot of Daily Prices

The figure plots the daily closed price of five European market indices. The data range from January 2010 to April 2017. The closed price of 1782 days are included in the sample. The indices are marked in different colors.

Due to the different stock trading day among these countries1, we merge the indices data with the same date to validate the comparison in the empirical study. There are 1747 observations for each index remained after deleting different date. Then we generate the

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CAC 40 had 1871 observations, DAX had 1860 observations, FTSE100 had 1849 observations, IBEX 35 had 1873 observations and Euro Stoxx 50 had 1782 observations

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log-return for each index by the equation(3.1): ut−1= log(St) − log(St−1) = log(

St St−1

), t = 2, 3, ..., 1747 (3.1) Figure 3.2 shows the plots for log-return of each index. We can observe volatility clus-tering in these daily stock return distribution. The large values concentrate on certain periods of time rather than distributed equally across the time, which indicates the dependency of consecutive time.

These five market indices daily returns have some similarities in the moving pattern. Among all indices, high volatility is observed in a rather similar period. This can be explained by the system risks in European market, i.e. European sovereign debt crisis and Brexis.

Figure 3.2: Plot of Stock Indices Log return

The figure plots the daily log-return of five indices. The sample period from January 2010 to April 2017. The volatility clustering effect is observed in each figure.

Statistical summary of log-return data can be found in table 3.1. According to the re-sult, we find the majority of the average daily returns are slightly larger than 0. Only the Spanish index has a negative return over the period. The Spanish index IBEX35 has worst performance since it has a negative average return and highest standard deviation. DAX index has most attracting average return and second lowest standard deviation. The FTSE100 index has lowest volatility and a positive average return.

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Table 3.1: Statistical Summary

The table reports the statistic summary of the daily returns for each market index. The mean, standard deviation, maximum value, minimum value, Skewness, Kurtosis and Jarque-Bera test statistics are given in the table. ∗ ∗ ∗ measures significance at 1%.

Stock Index Mean Std.Dev. Max Min Skewness Kurtosis Jarque-Bera CAC40 0.0156 1.3782 9.2208 -8.3843 -0.1520 3.5350 919.7∗∗∗ DAX 0.0413 1.3219 5.2104 -7.0673 -0.3416 2.5367 504.51∗∗∗ FTSE100 0.0398 1.0307 8.3680 -6.1994 -0.0131 4.8492 1717.1∗∗∗ IBEX35 -0.0072 1.6009 13.4836 -13.1852 -0.1719 6.6944 3279.8∗∗∗ E-Stoxx50 0.0094 1.4086 9.8465 -9.0110 -0.1408 3.7817 1050.5∗∗∗ The normality test can be performed by applying Jarque-Bera test, the test statistics is shown as follows: J B = 1 6[b + 1 4(k − 3) 2_] _(3.2) where, b = [(1/n) Pn i=1(Xi− ¯X)3]2 [(1/n)Pn i=1(Xi− ¯X)2]3 , k = (1/n) Pn i=1(Xi− ¯X)4 [(1/n)Pn i=1(Xi− ¯X)2]2 , (3.3)

and ¯X is the sample mean of Xi.

If the observations are normally distributed, the Jarque-Bera statistic will follow the χ2 distribution. The Jarque-Bera tests for five stock indices indicate none of the stock index follows normal distribution. Meanwhile, the negative Skewness of all stock index returns demonstrate a long tail in the lower tail. This result again verify the non-normality be-havior of financial market return.

Figure 3.3: Quantile-Quantile Plot

The figure provides the Quantile-Quantile plots for each market returns. The normal distribution is given by the red line while the empirical return data are shown in blue.

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Figure 3.3 gives the quantile-quantile plot (Q-Q plot) of each index, which provides an alternative way of normality test and a more direct reflection of tail behavior. The linear red lines from origin represent the plot of normal distribution and the blue lines give the empirical return data.

The Q-Q plots show that there are significant deviation from the normal distribution in the tail for all market indices. The S-shape Q-Q plots indicate that all indices have higher quantile than normal distribution in upper tail and lower quantile in lower tail, which demonstrate the heavy tail in market return distribution. Therefore, none of the index distribution follows normal distribution.

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Extreme Value Theory (EVT)

VaR Caculation

4.1 Threshold Selection

The Peak Over Threshold (POT) method will be used in this section to calculate the VaR under extreme value theory. Extreme value theory mainly focuses on the tail be-havior beyond the certain threshold. The threshold has to be selected in order to define the tail. Therefore how to properly select the threshold is an important step for the tail risk estimation.

Mean excess function is defined as E(X − u|X > u), which describes the average difference between the observed values and threshold, given the observations surpass the threshold. According to the theorem we introduced in the Chapter 2, the threshold should be chosen as the start point of linear trend in mean excess plot.

Figure 4.1: Mean Excess Plot

The figures provide the mean excess plot for market returns. The dots are calcu-lated as the average value above certain threshold in x-axis. The red lines give the reference for the chosen threshold.

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By using the M Eplot in R, we can obtain the mean excess plot shown in Figure 4.1. As we focus on the loss part of market return, the mean excess plot only use the negative value from the return data.

By visual observation, the following thresholds in Table 4.1 are selected for each in-dex. These thresholds are exhibited as red vertical lines as reference in the mean excess plot. Among all indices, IBEX has longest tail with higher number of observations and lower absolute value of threshold. FTSE100 has shortest tail with only 49 observations and higher absolute value of threshold.

Since there might not be an obvious and accurate start point of the linear trend and the threshold selection is rather subjective, the sensitivity test will be performed in next section.

Table 4.1: POT Thresholds

The table contains the selected threshold values, number of observations and proportion of data over the thresholds for each market index.

Stock Index CAC40 DAX FTSE100 IBEX35 EuroStoxx50

Threshold -2 -2.2 -2.2 -1.2 -2.5

Number of observaions 111 86 49 318 72

Proportion of data 0.064 0.493 0.0280 0.1821 0.0412

4.2 Goodness of Fit

Figure 4.2: GPD fitting over threshold range(CAC40)

The figures provide the sensitivity test for the selected threshold for GPD estima-tion. The threshold is examined using the maximum likelihood estimation in GPD model over a range of threshold. The upper part provide the test for scale parame-ter and the lower part provide the test for shape parameparame-ter. This figure only shows the results for CAC40.

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To test the goodness of fit for each threshold values, we fit GPD model over a certain range of thresholds using maximum-likelihood function. This method provides the sen-sitivity test over a certain threshold range for the scale and shape parameters in GPD modeling. Figure 4.2 shows the output of maximum likelihood estimates and confidence intervals of shape and modified scale for the market index CAC40.

There are 100 threshold values equally distributed between 0 and 5 are selected and tested in Figure 4.2. The upper figure provides the sensitivity test for scale parameter and the lower one gives the test for shape parameter. We can find in both estimated modified scale and shape, the likelihood estimation are stable around threshold 2. There-fore, we can conclude that by using 2 as threshold is appropriate for our GPD analysis. We perform the same sensitivity test for all other market returns, which shows sim-ilar patterns around the thresholds. But due to the length of the paper, the results of other market indices will be shown in Appendix. Given the results of the tests, we can conclude that the thresholds in Table 4.1 can be used for GPD estimation.

4.3 Generalized Pareto Distribution Estimation

Before the GPD estimation, we further inspect the GPD fit to the empirical distribution above the given thresholds. we plot the GPD fit of excesses over the thresholds as shown in Figure 4.3. The blue dots are the empirical observations while the curves represent GPD. All market indices show a good GPD fit. For DAX index data, some observations tend to deviate from GPD but then gradually turn back curve when it reach the end of tail. In general fit quite well for the empirical distribution for the observed excesses.

Figure 4.3: Fitting GPD to Excess Value

The grafts plot the GPD fit of excesses over threshold and the corresponding em-pirical distribution function for observed excesses

The observations over the given thresholds are modelled by Generalized Pareto Dis-tribution using maximum likelihood estimation. See equation (2.6) for the maximum likelihood estimation. The results of estimated parameters are given in the Table 4.2.

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Table 4.2: GPD Parameters

The table contains the description of GPD tail for each market index. The scale parameter β and shape parameter ξ are shown in last two rows. The scale and shape parameter are estimated based on the given threshold.

Parameters CAC40 DAX FTSE100 IBEX35 EuroStoxx50

Threshold -2 -2.2 -2.2 -1.2 -2.5 Number of data 111 86 49 318 72 Proportion of data 0.064 0.493 0.0280 0.1821 0.0412 β 1.0464 0.9060 0.5572 0.9764 1.0319 ξ -0.0036 0.7045 0.1686 0.1005 0.0185

4.4 VaR Calculation

The VaR function can be expressed as the equation (4.1) based on the GPD parameters: ˆ V aRp= ˆ β ˆ ξ (1 − p)n Nu −ξ − 1 ! + u (4.1)

where ˆβ is the sample shape parameter, ˆξ is the sample scale parameter, n is the number of sample and Nu is the number of observations exceed threshold. The VaR calculated are based on one trading day 95%, 99%, 99.5% VaR level.

Table 4.3: VaR Estimation

The table contains the VaR estimation results for historical simulation, variance-covariance method and extreme value theory approach.

VaR CAC40 DAX FTSE100 IBEX35 EuroStoxx50

95% Historical Simulation VaR -2.2748 -2.1279 -1.6583 -2.6105 -2.3234 95 % Variance-Covariance VaR -2.2508 -2.1324 -1.6793 -2.6397 -2.3068 95 % EVT-VaR -2.2321 -2.1698 -1.7097 -2.5654 -2.2983 99 % Historical Simulation VaR -3.8600 -3.4695 -2.8035 -4.4678 -3.8457 99 % Variance-Covariance VaR -3.1899 -3.0331 -2.3815 -3.7304 -3.2665 99 % EVT-VaR -3.9286 -3.7289 -2.8281 -4.4897 -3.9812 99.5 % Historical Simulation VaR -4.7673 -5.0987 -3.1662 -5.2437 -4.8181 99.5 % Variance-Covariance VaR -3.5336 -3.3628 -2.6386 -4.1297 -3.6178 99.5 % EVT-VaR -4.6437 -4.4575 -3.4366 -5.4282 -4.7135

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The results are shown in Table 4.3, with the VaR calculated by historical simulation method and variance-covariance method. From the result, we can see that the VaR es-timation generated by EVT method is higher in absolute value than the VaR produced by two other methods, which indicates EVT method can better capture the risk of extreme values and generate a higher loss estimation. Variance-covariance method as-sumes normal distribution and therefore cannot capture the tail behavior of the market return.

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Chapter 5

Copula VaR Calculation

5.1 Correlation Matrix

The linear dependence between two variables can be calculated by Spearman correla-tion:

ρ(X, Y ) = cov(X, Y )

pvar(X)var(Y ) (5.1)

Table 5.1 shows the linear dependence of all five market index returns. These Euro-pean market returns in general have highly positive correlation between each other. The CAC40 has very strong linear correlation with all other four market indices while IBEX35 has weak correlation with other markets.

Table 5.1: Correlation Matrix

The table contains the correlation coefficients calculated by Spearman correlation.

CAC40 DAX FTSE100 IBEX35 ESTOXX50

CAC40 1.0000 0.9331 0.8663 0.8726 0.9834 DAX 0.9331 1.0000 0.8370 0.7999 0.9501 FTSE100 0.8663 0.8370 1.0000 0.7405 0.8553 IBEX35 0.8726 0.7999 0.7405 1.0000 0.9136 ESTOXX50 0.9834 0.9501 0.8553 0.9136 1.0000

5.2 Copula Estimation

Next we fit the multivariate normal copula model and t copula model to the empir-ical data, which generates the correlation matrix under each model. Table 5.2 panel A provides the correlation matrix produced by normal Copula while Table 5.2 panel B provide the correlation matrix under t Copula. By comparing the results from nor-mal copula and t copula, we can find t copula always has lower value than nornor-mal copula.

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Table 5.2: Copula Estimation

Panel A: Normal Copula Correlation Matrix

The table contains the correlation coefficients calculated by Gaussian copula and t copula. Panel A provides the correlation matrix under normal copula. Panel B provides the correlation matrix under t copula.

CAC40 1.0000 0.9299 0.8689 0.8691 0.9813

DAX 0.9119 1.0000 0.8370 0.8028 0.9501

FTSE100100 0.8570 0.7454 1.0000 0.7454 0.8570

IBEX35 0.9501 0.8028 0.8370 1.0000 0.9119

ESTOXX50 0.9813 0.8691 0.8689 0.9299 1.0000

Panel B: t Copula Correlation Matrix

CAC40 1.0000 0.9291 0.8671 0.8718 0.9820

DAX 0.9119 1.0000 0.8334 0.8023 0.9495

FTSE100 0.8548 0.7442 1.0000 0.7442 0.8548

IBEX35 0.9495 0.8023 0.8334 1.0000 0.9119

ESTOXX50 0.9820 0.8718 0.8671 0.9291 1.0000

The goodness of fit between normal copula and t copula can be compared by using their AIC and BIC. According to the criteria of AIC and BIC, the lower value indicates a better model. As shown in table 5.3, the t copula has lower AIC and BIC value and higher log likelihood value. Therefore we choose t copula for the further estimation.

Table 5.3: Copula Model Selection Normal Copula T Copula

Log likelihood 8102 8293

AIC -16185 -16565

BIC -16130 -16505

5.3 Marginal Distribution

To describe the joint distribution of the market indices return, we have to model the marginal distribution for each indices. Normal marginal distribution and student’s t marginal distribution are compared to each other based on AIC and BIC. Table 5.5 pro-vide the AIC and BIC value under normal marginal distribution and Table 5.6 propro-vides AIC and BIC values for student’s t distribution. For all indices, student’s t marginal dis-tribution gives lower AIC and BIC, which suggests that student’s t marginal disdis-tribution can better capture the pattern of market return.

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Table 5.4: Marginal Distribution Parameter

The table contains the results of marginal distribution estimation by normal distribution and t distri-bution. In the left side, the marginal distribution is estimated by normal distribution and in the right side it is estimated by student’s t distribution

normal distribution t distribution

Mean sd AIC BIC Mean sd df AIC BIC

CAC40 0.0155 1.2779 6078 6089 0.0409 0.9873 3.8338 5875 5892 (0.0329) (0.0233) (0.0280) (0.0313) (0.3927) DAX 0.0419 1.3215 5932 5943 0.0776 0.9375 3.6329 5749 5766 (0.0316) (0.0223) (0.0267) (0.0314) (0.3761) FTSE100 0.0154 1.0303 5063 5074 0.0386 0.7263 3.7217 4833 4849 (0.0246) (0.0174) (0.0207) (0.0231) (0.3712) IBEX35 -0.0071 1.6004 6601 6612 0.0221 1.1570 4.0841 6368 6384 (0.0382) (0.0271) (0.0326) (0.0359) (0.4331) ESTOXX50 0.00945 1.4082 6154 6165 0.0345 1.0151 3.8973 5956 5972 (0.0337) (0.2383) (0.0287) (0.0325) (0.4096)

5.4 Copula VaR

Base on the results from previous sections of this chapter, we decide to use student’s t copula as dependent structure and student’s t marginal distribution to describe the market returns.

Therefore, VaR can be estimated Monte Carlo simulation with student’s t copula and marginal distribution. The five market returns are aggregated by equally weighted. The result shows the one day 99% VaR equals to -3.5007 and one day 99.5% VaR equals to -4.5020. We can also compute the equal weighted market return using historical sim-ulation method and variance-covariance method. The one day 99% VaR by historical simulation is -3.0215 and -2.4707 by variance-covariance method.

This result shows that the VaR estimated by t copula method is higher in absolute value than the VaR produced by historical simulation method and variance-covariance method. The main reason is t-copula approach can better describe the dependent struc-ture of the stock returns as well as the extreme events. The variance-covariance method underestimate the risk since it assume the normal distribution.

Table 5.5: VaR Estimation

The table provides the VaR estimation base on historical simulation, variance-covariance method and copula method in 95%, 99% and 99.5% VaR level.

Method 95% VaR 99% VaR 99.5% VaR Historical Simulation -1.7323 -3.0215 -3.5984 Variance-Covariance -1.7433 -2.4707 -2.7370 t Copula-VaR -2.0023 -3.5007 -4.5020

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Conclusion

In this study, we investigate two theories in risk measurement: EVT and copula ap-proach. The EVT approach is investigated for each individual stock market index and copula approach is applied to risk aggregation. Both of the approaches are compared to historical simulation approach and variance-covariance approach. The VaR measure-ment is based on 95%, 99% and 99.5%.

The results show that under EVT approach, the estimated VaR value is closer to his-torical simulation but higher than variance-covariance method. The copula approach produces a significant higher risk measure compares to both historical simulation and variance-covariance method under all VaR level. Based on the characteristics of EVT approach and copula approach, we believe the risk measurement is more accurate since EVT reflects the tail behavior of the risk and copula can better capture the dependence structure of the risk.

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References

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Wipplinger E. Philippe Jorion: Value at Risk-The New Benchmark for Managing Fi-nancial Risk[J]. FiFi-nancial Markets and Portfolio Management, 2007, 21(3): 397.

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Figure 6.1: GPD fitting over threshold range(DAX)

The figures provide the sensitivity test for the selected threshold for GPD estima-tion. The threshold is examined using the maximum likelihood estimation in GPD model over a range of threshold. The upper part provide the test for scale parame-ter and the lower part provide the test for shape parameparame-ter. This figure only shows the results for DAX.

Figure 6.2: GPD fitting over threshold range(FTSE100)

The figures provide the sensitivity test for the selected threshold for GPD estima-tion. The threshold is examined using the maximum likelihood estimation in GPD model over a range of threshold. The upper part provide the test for scale parame-ter and the lower part provide the test for shape parameparame-ter. This figure only shows the results for FTSE100.

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Figure 6.3: GPD fitting over threshold range(IBEX35)

The figures provide the sensitivity test for the selected threshold for GPD estima-tion. The threshold is examined using the maximum likelihood estimation in GPD model over a range of threshold. The upper part provide the test for scale parame-ter and the lower part provide the test for shape parameparame-ter. This figure only shows the results for IBEX35.

Figure 6.4: GPD fitting over threshold range(EuroStoxx50)

The figures provide the sensitivity test for the selected threshold for GPD estimation. The threshold is examined using the maximum likelihood estimation in GPD model over a range of threshold. The upper part provide the test for scale parameter and the lower part provide the test for shape parameter. This figure only shows the results for Euro Stoxx 50.

Risk measurement based on extreme value theory and copula theory : an empirical study of European stock markets

Value Theory and Copula Theory-An

Empirical Study of European Stock

Markets

Bi Wu

Contents

Introduction

Chapter 2

Definition and Concept

2.1

Value at Risk (VaR)

2.2

Extreme Value Theory (EVT)

2.3

Copula definition and property

Data and Statistic Summary

Extreme Value Theory (EVT)

VaR Caculation

4.1

Threshold Selection

4.2

Goodness of Fit

4.3

Generalized Pareto Distribution Estimation

4.4

VaR Calculation

Chapter 5

Copula VaR Calculation

5.1

Correlation Matrix

5.2

Copula Estimation

5.3

Marginal Distribution

5.4

Copula VaR

Conclusion

References