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— Comparison between Extreme Value

Theory and Copula Theory

Maikel Alles

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: Maikel Alles Student nr: 10791787

Email: Maikel.alles@upcmail.nl Date: July 12, 2018

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

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This document is written by Student Maikel Alles who declares to take full responsi-bility for the contents of this document. I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

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Abstract

This thesis estimated the Value-at-Risk for the SP500,DAX and a combination of these two market indices. The data was based on a sample period of may 2002 up till may 2012, with the financial crisis in the sample period. The Value-at-Risk is estimated with two dif-ferent approaches. The first approach that was used is the Extreme value approach. The second approach was the copula approach in which the parameters of the copula were able to vary overtime. The Value-at-Risk based on the Extreme Value approach was com-pared with the results based on the copula approach. The Extreme value theory made a good estimation of the Value-at-Risk based on the Christoffersen test. The Copula approach seems to overestimate the risk in the market and returns a higher level Value-at-Risk in comparison to the Extreme Value approach.

Keywords Copula, Value-at-Risk, Time series analysis, Extreme value theory, Gener-alized Pareto Distribution, Risk measures.

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Preface vi

1 Introduction 1

2 Theoretical background 3

2.1 Introduction to financial time series . . . 3

2.2 Risk measures . . . 4

2.3 Extreme Value Theory . . . 5

2.3.1 Generalized extreme value distribution . . . 5

2.3.2 Generalized Pareto distribution . . . 6

2.4 Estimation procedure for the Value-at-Risk . . . 7

2.5 Copula theory . . . 8

2.5.1 Introduction to copulas . . . 9

2.5.2 Time varying copulas . . . 10

2.5.3 The Symmetrized Joe Clayton copula . . . 10

2.5.4 Student’s t copula. . . 11

2.6 Estimating the Value-at-Risk with a copula . . . 12

3 Data statistics and summary 13 4 Results 16 4.1 Parameter estimation marginal distributions . . . 16

4.2 Extreme Value Theory . . . 18

4.2.1 Value-at-Risk SP500 . . . 18

4.2.2 Value at Risk DAX . . . 20

4.2.3 Value at Risk Portfolio . . . 21

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4.3 VaR estimation through a copula . . . 22

4.3.1 Symmetrized Joe Clayton Copula . . . 22

4.3.2 Student’s t copula. . . 25

4.3.3 Sensitivity Marginal distributions . . . 28

5 Conclusion 30

Appendix A: Stuff 32

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The last months I have worked on my master thesis with a lot of pleasure and curiosity. I especially want to thank my supervisor Dr. S.U.Can for his feedback and his guidance during the writing process of this thesis.

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Introduction

Risk managers are mainly concerned about the possible loss on a certain portfolio over a fixed time horizon. In order to determine the capital they need to set aside to cover possible losses, they use risk measures as a critical aid. By the introduction of Basel II, banks are required to hold sufficient capital to cover losses on their portfolio. One of the widely used risk measures, that is in line with Basel II, is the Value-at-Risk. Modelling the tail-distribution of losses for the estimation of the Value-at-Risk of the next day gives risk managers insight in the capital they need to put aside.

There are several ways to estimate the Value-at-Risk. One of the methods was proposed by McNeil & Frey in 2000. Their proposed method described the tail of the innovation distribution of a GARCH model using Extreme Value Theory. They used an AR-GARCH model to describe the behaviour of financial losses over time. They found that their model gives a better estimation for the Value-at-Risk, than methods that ignore heavy tails of the innovations distribution.

This thesis will follow the estimation procedure of McNeil & Frey (2000) for estimating the Risk through the Extreme Value Theory. The Value-at-Risk will be estimated for the stock market indices SP500, DAX and a combination of these two. We will compare the results of the Extreme Value Theory with the results of the copula theory. The Symmetrized Joe Clayton copula and the Student’s t copula will be used to derive the time dependent joint distribution for the portfolio. This derivation will allow the Value-at-Risk to be derived for the portfolio. The paper of Patton (2006) will be used as a guideline for the estimation of the copula

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parameters. Patton in his paper describes the evolution of the copula parameters over time.

The remainder of this thesis is structured as follows. Chapter 2 describes the theoretical background of financial time series and Extreme Value Theory, which includes the generalized extreme value distribution and the generalized Pareto dis-tribution. Furthermore in chapter 2 the copula theory is discussed with two different time varying copulas. Chapter 3 summarises the data and statistics of the SP500, the DAX and a combination of those two. Chapter 4 discusses the results of both the Extreme Value Theory and the copula theory. The final chapter discusses and concludes this thesis.

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Theoretical background

In this chapter the theoretical background of the Extreme Value Theory and time series analysis will be discussed. Section 2.1 gives an introduction into financial time series. Section 2.2 discusses the risk measure ”Value-at-Risk” and its properties. Section 2.3 and 2.4 discuss the generalized extreme value distribution and generalized Pareto distribution respectively. Section 2.5 gives a general introduction into copulas and discusses the copulas used in this thesis. The last section (2.6) explores the estimation procedure for the Value-at-Risk using copulas.

2.1

Introduction to financial time series

Financial time series are well known for their non-stationary property. Stationarity is an important assumption in financial time series analysis, as non-stationarity will lead to inaccurate modelling of the data, which will then result in unpredictable future processes. To ensure that the data is stationary, McNeil & Frey (2000) trans-formed their data into log-return data, also known as log differences. As this thesis focuses on modelling loss distribution instead of profit distribution, the data will be transformed into negative log-returns. Let Ptbe the price of an asset at time t , then

the negative log return xt is defined as:

xt= − log  Pt Pt−1  = log(Pt−1) − log(Pt). (2.1)

Transforming the data can stabilize the mean and variance and as a result the 3

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process will become stationary. The stationary property can be verified using the Augmented Dicky-Fuller test (ADFT). The ADFT tests the null-hypothesis that the time series has a unit root against the alternative hypothesis that there is no unit root present and that the process is stationary.

To model the dynamics of xt, McNeil & Frey are using an AR(1)-GARCH(1,1)

model. These dynamic processes can be modelled as follows: xt = µt+ σtZt. The

AR(1) part of this model is used to describe the dynamics of the mean (µt) and the

GARCH(1,1) part of the model is used to describe the variance (σt). The variable

Zt can be considered as a white noise (0,1)-process. The AR(1)-GARCH(1,1) model can be expressed by the following three formulas:

xt= µt+ σtZt, (2.2)

µt = φxt−1, (2.3)

σ2t = α0+ α12t−1+ β1σ2t−1, (2.4)

where t = xt− µt.

To ensure the process is stationary, the model requires the restrictions α0 > 0,

α1 ≥ 0, β1 ≥ 0 and |φ| < 1. This model can be used to estimate the one step ahead

forecast with predicted future parameters σt+1 and µt+1 given the data up to and

including time t.

2.2

Risk measures

One of the most well known and widely used risk measures is the Value-at-Risk (VaR). The VaR was introduced in 1996 by J.P. Morgan and captures the possible portfolio losses in just in a single number. From a mathematical point of view, the VaR is just a quantile of the loss distribution FX at level p. The one-day VaR at

level p represents the expectation that, on average, out of a 100 days the VaR will be exceeded (1 − p) ∗ 100 times. The VaR-formula is defined as follows:

V aR[X; p] = FX−1(p) = inf{x : FX(x) ≥ p}. (2.5)

Artzner et al.(1999) states that a good risk measure P (X), where X is a stochas-tic loss, should satisfy the following axioms to be called a coherent risk measure:

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Definition 1. A risk measure P (X) is called a coherent risk measure if the following axioms are satisfied:

• Monotonicity : P (X1) ≤ P (X2) if X1 ≤ X2

• Translation invariance : P (X − I) = P (X) − I, ∀I ∈ R • Subadditivity : P (X1+ X2) ≤ P (X1) + P (X2)

• Positive homogeneity : P (λX) = λP (X) ∀λ ∈ R

The VaR is used worldwide but has a few shortcomings. One of the shortcomings is that the VaR does not satisfy all of the coherent axioms, in particular the subad-ditivity axiom which reflects the benefits of diversification. Another shortcoming is that the VaR says nothing about the size of the loss if that loss exceeds the VaR.

2.3

Extreme Value Theory

This section outlines the theoretical background of the Extreme Value Theory. The first subsection introduces the generalized extreme value distribution and the second subsection outlines the generalized Pareto distribution.

2.3.1

Generalized extreme value distribution

One of the approaches to model extreme outcomes is the block maxima approach. The block maxima approach is based on dividing the time interval into n-blocks of equal size and models the maximum outcome for each block. The maximum value of each block is denoted by Mn = max{X1, . . . , Xn}. The Extreme Value Theory

(EVT) can be used to describe the behaviour of mn. An important theorem in EVT

is the Fisher and Tippet theorem:

Theorem 1. Let {X1, ...Xn} be a sequence of random variables which are

indepen-dent and iindepen-dentically distributed. Let Mn = max{X1, ...Xn}. Suppose there exist a

sequence dn∈ R+ and a cn ∈ R such that:

lim n→∞Pr[ Mn− dn cn ≤ x] = lim n→∞F n(c ny + dn) → H(x). (2.6)

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If this holds true then F(x) lies in the maximum domain of attraction of H, F ∈ M DA(H). Fisher-Tippett states that for some value of ξ,Hξ(x) is equal to H(x).

The Generalized Extreme Value (GEV) distribution has the following structure:

Hξ,β,µ(x) =    exp(−(1 + ξx−µβ )−1ξ) if ξ 6= 0 exp(−e−x) if ξ = 0 (2.7)

The GEV has three different parameters: a location parameter µ ∈ R, a scale parameter σ > 0 and the parameter ξ also known as the shape parameter. In this model 1ξ is known as the tail index. The shape parameter distinguishes three different types of distributions:

• if ξ > 0, then Hξ,β,µ(x) belongs to Fr´echet distribution,

• if ξ = 0, then Hξ,β,µ(x) belongs to the Gumbel distribution,

• if ξ < 0, then Hξ,β,µ(x) belongs to the Weibull distribution.

The first case, ξ > 0, can be seen as a heavy-tailed distribution with infinite higher moments. This is common for financial data. The second case, ξ = 0, corresponds to flat-tailed distributions. The last case, ξ < 0, corresponds to short-tailed distri-butions.

2.3.2

Generalized Pareto distribution

A second approach to model the extreme outcomes is the Peaks-over-Threshold (POT) method. This approach is based on modelling extreme outcomes above a high level threshold u. Pickands-Balkema-de Haan presented the following definition in 1975:

Let x0 be the right endpoint of the distribution Fx, x0 = sup{x ∈ R : F (x) ≤ 1},

let Xi be the observation that exceed the threshold u and let Yi = Xi− u The excess

distribution over a high threshold u can be defined as follows: Fu(y) = Pr[Y ≤ y|X > u] = Pr[X − u ≤ y|X > u] =

F (x) − F (u)

1 − F (u) , (2.8) for 0 ≤ x ≤ x0− u.

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A result of the EVT is the Generalized Pareto Distribution (GPD) which can be used for the approximation of the excess distribution Fu.

Gξ,β(x) =    1 − (1 + ξxβ)−1ξ if ξ 6= 0 1 − exp(−xβ) if ξ = 0 (2.9)

where β > 0 and x ≥ 0 if ξ > 0. If ξ < 0 this leads to 0 ≤ x ≤ −βξ.

• if ξ > 0, then Gξ,β,µ(x) belongs to the the ordinary Pareto distribution,

• if ξ = 0, then Gξ,β,µ(x) belongs to the the exponential distribution,

• if ξ < 0, then Gξ,β,µ(x) belongs to a type II Pareto distribution.

The theorem of Balkema & de Haan (1974) and Pickands (1975) shows that typically the excess distribution will converge to a Generalized Pareto Distribution (GPD). The following theorem holds:

Theorem 2. If and only if F (x) ∈ M DA(H), then there exist a function β(u) such that: lim u→x0 sup 0≤x<x0−u |Fu(x) − Gξ,β(u)(x)| = 0. (2.10)

This theorem states that a function of the threshold β(u) exists, such that F u(x) can be approximated well by Gξ,β(u)(x) for some value of ξ.

2.4

Estimation procedure for the Value-at-Risk

This thesis follows the paper of McNeil & Frey (2000) for obtaining the VaR via a two stage approach. The first step is obtaining estimates for ˆµt+1 and ˆσt+1based on

the AR(1)-GARCH(1,1) model for negative log return data. Recall formula (2.3) and (2.4), the parameters ˆθ = { ˆφ, ˆα0, ˆα1, ˆβ1} can be estimated by Maximum Likelihood

(ML). The mean and variance at time t + 1 is predicted by the one step ahead forecast:

ˆ

µt+1= ˆφxt,

ˆ

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where ˆt = xt− ˆµt. This ends the first stage in the estimation procedure for the VaR.

The second stage is to model the tail of the residuals (Zt) using EVT to estimate

the quantile at level p, zp. Under formula (2.8) the cdf of the residuals, F (x) can be

written as follows:

F (x) = (1 − F (u))Fu(y) + F (u).

Let Nu be the number of residuals that exceed the threshold u and n be the

number of observations. F (u) can be approximated by the empirical estimate n−Nu

n .

Gξ,β(u)(x) can be used for the approximation of Fu(x), as discussed in theorem 3.

We are now able to derive the formula for the VaR.

F (x) = n Nu Gξ,β(u)(x) + n − Nu n = n Nu (1 − (1 + ξ β) −1 ξ) + n − Nu n = 1 − Nu n (1 − (1 + ξ β) −1 ξ).

The VaR can now be found by taking the inverse of F (Z), V aR[Z; p] = F−1(p). This will lead to the final result:

V aR[Z; p] = u + β ξ(

n Nu

(1 − p)−ξ − 1). (2.11) Substituting the estimated parameters in (2.11) gives an estimate for the VaR:

V aR[Z; p] = u + ˆ β ˆ ξ( n Nu (1 − p)− ˆξ− 1). (2.12) This can be used for estimating the VaR for the residuals, zp. The VaR for the

log return at time t + 1 is given by: xpt+1 = ˆµt+1+ ˆσt+1zp.

2.5

Copula theory

This section outlines the theoretical background of copulas. The first subsection introduces the general idea of copulas. The second subsection briefly summarises the copulas used in this thesis. The last section discusses estimating the VaR with the use of copulas.

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2.5.1

Introduction to copulas

Nowadays copulas are widely used in risk management. Copula theory is well known for its great flexibility in modelling the joint distribution. A copula is a function that binds univariate distributions into a single joint distribution and determines the dependence structures between random variables. Copulas are flexible in separating the univariate marginals and the dependence structure for the joint distribution. First let’s give a mathematical definition of a copula:

Definition 2. An n-dimensional distribution function C : [0, 1]n → [0, 1] is a copula

if the following properties hold true:

• C(u) = C(u1, ..., un) increases for each component ui,

• • ∀(a1, ..., an), (b1, ..., bn) ∈ [0, 1] with ai ≤ bi, 2 X i1=1 ... 2 X in=1 (−1)i1+...+inC(u 1i1, ..., unin) ≥ 0, where uj1 = aj and uj2 = bj∀j ∈ {1, ..., n}.

The first bullet point states that if the marginal distribution stays constant and the other marginally increases, then the joint probability will also increase. While the second bullet point states that the marginal distributions are standard uniform. The last and third bullet point ensures that the copula is a legitimate distribution function.

The well know Sklar’s theorem (1959) states that we can decompose any contin-uous multivariate distribution into a unique n-dimensional copula and n-marginal distribution. Conversely, any n-dimensional copula and n marginal distributions uniquely describe a joint distribution.

Theorem 3. Sklar0s Theorem : Let F be a joint distribution with marginals F1, ..., Fn,

then there exists a copula function C : [0, 1]n→ [0, 1] such that ∀x

1, ..., xn ∈ R,

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If the marginals are continuous, the copula will be uniquely defined. If C is a copula and F1, . . . , Fn are univariate distributions, F will be a joint distribution with

margins F1, ..., Fn. The joint density can be found by taking the derivative of (2.13)

in respect to Xi ∀i. f (x1, ..., xn) = ∂nC(F 1(x1), ..., Fn(xn)) ∂x1...∂xn = c(F1(x1), ..., Fn(xn)) n Y i=1 fi(xi), (2.14)

where c is the copula density.

2.5.2

Time varying copulas

In order to model the dependence structure, many copulas have been introduced in statistical literature. This thesis will estimate two of these copulas. In this first stage the copula parameters will not change over time and in the second stage of the estimation the parameters will be allowed to change over time.

2.5.3

The Symmetrized Joe Clayton copula

The first copula that will be used is the Symmetrized Joe-Clayton (SJC) copula. The SJC copula is a modification of the Joe-Clayton copula to correct the tail dependence measures. The Joe-Clayton copula does not allow for symmetry if the tail dependence parameters are equal. If the upper and lower tail dependence measures are equal, then there is still some asymmetry in the copula. The Joe-Clayton copula is of the following form: CJ C(u, v|τU, τL) = 1 − (1 − {[1 − (1 − u)κ]−γ + [1 − (1 − v)κ]−γ − 1} −1 γ)1κ, (2.15) where κ = 1 log2(2−τU), γ = − 1 log2(τL), τ U ∈ (0, 1) and τL ∈ (0, 1).

The parameters τU and τL are known as the upper and lower tail dependence parameters. These parameters capture the behaviour of the tail in case of an extreme outcome. For example, given that one of the random variables is an extreme outcome, the tail dependence parameter gives the probability that the other outcome will also be a extreme value. A formal definition of tail dependence is given by:

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Definition 3. The lower tail dependence is given by: lim →0P [U ≤ |V ≤ ] = P [V ≤ |U ≤ ] = lim→0 C(, )  = τ L . Definition 4. The upper tail dependence is given by:

lim

→1P [U > |V > ] = P [V > |U > ] = lim→1

1 − 2 + C(, ) 1 −  = τ

U.

Patton proposed in 2006 the Symmetrized Joe-Clayton copula, that by construc-tion allows for symmetry if τU = τL.

CSJ C(u, v|τU, τL) = 0.5(CJ C(u, v|τU, τL+CJ C(1−u, 1−v|τU, τL)+u+v−1). (2.16)

The copula becomes more flexible by introducing time varying parameters. There are several ways to specify the evolution of these parameters over time. This thesis fol-lows the paper of Patton (2006) regarding the evolution of time varying parameters. The evolution of the time varying parameters are given by the following formulas:

τtU = Λ(ωU + βUτt−1U + αU 1 10 10 X j=1 |ut−j− vt−j|), (2.17) τtL= Λ(ωL+ βLτt−1L + αL 1 10 10 X j=1 |ut−j− vt−j|), (2.18)

where Λ(x) = (1 + e−x)−1 is the logistic transformation used to keep τU

t and τtL in

their domain, τU, τL∈ [0, 1].

2.5.4

Student’s t copula

Another frequently used copula for financial data is the Student’s t copula. The Student’s t copula preforms well in modelling extreme outcomes. The Student’s t copula consists of two different parameters ρ and ν and is given by the formula:

C(u, v|ρ, ν) = 1 2πp1 − ρ2 Z t−1ν (u) −∞ Z t−1ν (v) −∞  1 + x 2+ y2− 2ρxy (1 − ρ2  dxdy. (2.19)

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Patton (2004) proposed the following evolution of parameters ρ and ν: ρt= Λ1(ωρ+ βρρt−1+ αρ 1 10 10 X j=1 t−1ν (ut−j)t−1ν (vt−j)), (2.20) νt= Λ2(ων + βννt − 1 + αν 1 10 10 X j=1 t−1ν (ut−j)t−1ν (vt−j)), (2.21)

where Λ1(x) = 1−exp(−x)1+exp(−x) is used to keep ρt in (-1,1) and Λ2(x) = 2 + exp(−x) used

to keep νt in [2,∞).

In comparison with the SJC, the Student’s t copula is known for its symmetric structure in the tails. During extreme outcomes the upper and lower tail are being treated equal. The dependence is given by:

τtU = τtL = 2 ∗ T (−√ν + 1 √

1 − ρ √

1 + ρ; ν + 1), (2.22) where T is defined as the cumulative Student’s t distribution.

2.6

Estimating the Value-at-Risk with a copula

This thesis considers a portfolio St= Xt+ Ytwhich consists of Xtwith density gt(x),

cumulative distribution Gt(x), Yt with density ht(y) and cumulative distribution

Ht(y). Based on Sklar’s Theorem and the copulas defined in the previous section,

the time dependent bivariate cumulative distribution Ft(x, y) can be derived:

Ft(x, y) = Ct(Gt(x), Ht(y))

By using (2.14), we are able to derive the time dependent bivariate density: ft(x, y) = ct(Gt(x), Ht(y))gt(x)ht(y)

This now lets us derive the univariate cumulative distribution function of the portfolio S. FSt(s) = P [Xt+ Yt≤ s] = Z Z x+y≤s ft(x, y)dxdy = Z s −∞ Z s−y −∞ ft(x, y)dxdy

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Data statistics and summary

This thesis focuses on two different stock market indices: the German stock index (DAX) and the American stock index (SP500). Furthermore we look at a combi-nation of those two stock market indices, we will refer to this as the ”portfolio”. The daily stock indices are used from the sample period of May 1, 2002 until May 1, 2012, with the financial crisis within the sample period. Due to different dates of holidays between both countries and some unknown reasons the data sets are unequal of length. The data is merged in such a way that the non-matched data are dropped out. This leaves a sample that consists of 2485 observations per index. Figure 3.1 provides plots of the DAX and SP500 daily stock prices, where one can conclude that both stocks run parallel to each other.

Figure 3.1: Daily stock prices SP500 & DAX

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Following McNeil & Frey (2000), the data is transformed into negative log re-turns, seen in formula (2.1). Figure 3.2 shows the plots of the negative log return data of the SP500 and the DAX. From this figure it is visible that there is some volatility clustering over time, suggesting serial dependence

Figure 3.2: Daily negative log returns SP500 & DAX

Table 3.1 shows a statistical summary of the log returns for the SP500, DAX and the portfolio. The table shows that both SP500 and DAX have a mean that is slightly larger than zero, which is common for financial return data. Due to the negative log transformation, this results in a negative return on these indices. In addition, it seems that the DAX is more volatile than the SP500.

Table 3.1: Statistical summary

Stock index Mean Std.Dev Min Max 95%-Quantile 99%-Quantile SP500 0.0001 0.0139 -0.0947 0.l096 0.0195 0.0393

DAX 0.0001 0.0168 -0.0883 0.1080 0.0243 0.0490

Portfolio -0.0001 0.0154 -0.01082 0.07908 0.0248 0.0474

The data is split in two different groups. The first group consists of the first 1242 observations that are used as the ”observed” data-set. The second part consists of observations 1243 to 2485 and are used for backtesting the VaR. In order to backtest the VaR, a moving window technique with length 1242 is used. The first

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moving window consists of observation N=1 up to time M=1242, will be used to backtest observation M+1. The second moving window consists of observations 2 up to time M+1, which will be used to backtest observation M+2, and so on.

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Results

This chapter outlines the results found by the research done in this thesis. In section 4.1 the results from the time series are given. In Section 4.2 the results of the VaR with the extreme value will be discussed. Section 4.3 will discuss the VaR results of the copula Theory.

4.1

Parameter estimation marginal distributions

The maximum likelihood technique is used to calculate the parameters of the AR(1)-GARCH(1,1)-model, recall formula (2.2) up to (2.4), for the DAX,SP500 and the portfolio indices. The results of the parameter estimations of the first 1248 observa-tions are given in the table 4.1:

Table 4.1: Parameter estimation AR(1)-GARCH(1,1) model

SP500 DAX Portfolio φ -5.12×10−2(−2.76 × 10−2) -3.82×10−2(2.85 × 10−2) -8.19×10−3(2.76 × 10−2) α0 8.53×10−7(4.17 × 10−7) 1.43×10−6(5.73 × 10−7) 1.17×10−6(4.17 × 10−7) α1 4.78×10−2(1.07 × 10−2) 7.23×10−2(1.34 × 10−2) 6.67×10−2(1.07 × 10−2) β1 9.39×10−2(1.21 × 10−2) 9.18×10−1(1.33 × 10−2) 9.23×10−1(1.21 × 10−2) P-value 0.01 0.01 0.01

The table reports the estimated parameters of the AR(1)-GARCH(1,1) model for the first 1242 observations, between parentheses the standard errors are given.

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The Augmented-Dickey-Fuller test returns a small p-value for all three different AR(1)-GARCH(1,1) models. This implies that the null hypothesis of a unit root is rejected and that the three different AR(1)-GARCH(1,1) models are stationary.

We are now able to estimate the one step ahead prediction µt+1 and σt+1 used

to determine the VaR. The results of these prediction are given in the following two sets of figures:

Figure 4.1: One step ahead forecast for µ.

Figure 4.2: One step ahead forecast for σ.

Around the 400th observation there is a huge spike in the predicted sigma. This is the point where the financial crisis of 2007-2008 started.

For the distribution of gt(x) and ht(y), defined in chapter 2.6, there are a set of

assumptions for estimating the time dependent joint distribution of the portfolio. To derive the time dependent joint distribution, there needs to be set an assumption for the marginal distributions. A set of Q-Q plots will determine which distribution must

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be used. Figure 4.3 displays the Q-Q plot of the residuals of the AR(1)-GARCH(1,1) model for the first 1242 observation. From this figure it can be seen that the central bulk of the data is close to a straight line, but the tails show some deviation. It is visible that the normal distribution overestimates the left tail and underestimates the right tail of the loss distribution. The main focus is on the behaviour of the right tail. Based on these Q-Q plots, the assumption is made that the residuals, Zt

follows a standardized Student’s t distribution with mean 0 and standard deviation 1 for both indices.

(a) Q-Q-Plot Residuals DAX (b) Q-Q-Plot Residuals SP500

Figure 4.3: Q-Q plot marginal residual distributions

4.2

Extreme Value Theory

This section will look at the results found by the extreme value theory for the VaR for the SP500, DAX and the portfolio. This thesis will look at VaR at level 0.95 and 0.99 for the analysis. To use the EVT, the residuals are ordered from low to high and the threshold is set so that 85% of the observations within the moving window is lower than this threshold.

4.2.1

Value-at-Risk SP500

This thesis backtests the VaR of the log return xpt+1 with historical data xt+1. By

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1242 observations. The VaR is exceeded if xpt+1 < xt+1. Figure 4.4 gives the VaR for

the SP500 at all time points,as estimated by Extreme Value Theory

Figure 4.4: Value-at-Risk SP500

In Figure 4.4 the VaR is graphed at level 0.95 and 0.99. The Christoffersen coverage test can be used to test if the number of violations is in line with the expected number of exceedings at the different quantile levels. The Christoffersen test tests the hypothesis that the number of exceedances is correct in comparison to the expected level. A good model is supposed to pass this test.

Table 4.2: Backtesting the Value at Risk SP500 SP500 Times Exceeded Expected Christoffersen Test

95% 80 62 Reject H0 (0.025)

99% 22 12 Reject H0 (0.014)

The table reports the times the estimated VaR is exceeded at level 95% and 99% for the SP500. Between parentheses the p-value of the Christoffersen test is given.

Table 4.2 shows the times the VaR is exceeded for both quantiles and gives the times the VaR is expected to be exceeded. It can be seen that the Christoffersen test returns a p-value which is smaller than the 5% significance level for both quantiles.

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This rejects the null hypothesis of correct exceedances. This implies that the Extreme Value model seems to underestimate the risk for the SP500. One of the reasons that the EVT could underestimate the risk is the financial crisis of 2007-2008.

4.2.2

Value at Risk DAX

The following figure gives the VaR for the DAX index:

Figure 4.5: Value-at-Risk DAX

Table 4.3 shows the results for the DAX index. For the DAX the Christoffersen test returns a p-value larger than the significance level of 5%, indicating that for the EVT returns a correct number of exceedances. Furthermore, it is remarkable that at 99% quantile the times the VaR is exceeded is less then expected. Thus, we can conclude that EVT gives a good estimation for the VaR.

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Table 4.3: Backtesting the Value at Risk DAX DAX Times Exceeded Expected Christoffersen Test

95% 75 62 Fail to reject H0 (0.103)

99% 10 12 Fail to reject H0 (0.475)

The table reports the times the estimated VaR is exceeded at level 95% and 99% for the DAX. Between parentheses the p-value of the Christoffersen test is given.

4.2.3

Value at Risk Portfolio

The following figure (4.6) gives the VaR for the portfolio:

Figure 4.6: Value-at-Risk portfolio

One of the positive characteristics of the EVT is that there is no need to specify the dependence structure between the DAX and the SP500 to derive the VaR for the portfolio. The time values of the different indices are added up and the estimation procedure is done in the same way as for a single index. The Christoffersen test returns a p-value larger than 0.05 and thus the null hypothesis is not rejected, meaning that a combination of SP500 and DAX results in a good model. Remarkable is, even if the individual model of one of the two indices is rejected, the EVT could still work for a combination of those models. Furthermore, for high level quantiles

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the EVT seems a better estimation for the VaR.

Table 4.4: Backtesting the Value at Risk SP500 Portfolio Times Exceeded Expected Christoffersen Test

95% 76 62 Fail to reject H0 (0.080)

99% 14 12 Fail to reject H0 (0.659)

The table reports the times the estimated VaR is exceeded at level 95% and 99% for the portfolio. Between parentheses the p-value of the Christoffersen test is

given.

4.3

VaR estimation through a copula

As previously described in section 4.1, one of the assumptions is that the residuals follow a standardized Student’s t distribution. The VaR was estimated for the two different copulas, and the parameters of the copulas were constant and time varying. The one-step-ahead prediction from the time series analysis and two different copulas were used to determine the time dependent joint distribution of the portfolio.

4.3.1

Symmetrized Joe Clayton Copula

Figure 4.7 shows the parameter estimation of the constant SJC copula. Recall chap-ter 3 where the moving window technique was used where the constant SJC param-eters are assumed to stay constant within the moving window.

From this figure it is clear that the time dependent joint distribution is asym-metric, since these tail dependence parameters are mostly not equal. Furthermore, from this figure it appears that the lower tail dependence is slightly higher, and sometimes symmetric for the first 300 observations. This implies that joint negative outcomes happen more frequently during this period. It is remarkable is that there is a switch at the start of the financial crisis: the upper tail dependence is now higher than the lower tail dependence. The higher dependence implies that joint positive outcomes are to occur more often. Towards the end of the observations the upper and lower tail dependence seems to get closer to each other indicating symmetry

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Figure 4.7: Parameters constant Symmetrized Joe Clayton copula

in the long run. The average of the upper tail is equal to 0.429 and the average of the lower tail is equal to 0.396. The upper tail average indicates that if one of the variables has an extreme value in the upper tail, that the other variable has a 42.9% chance to be in the upper tail too.

Figure 4.8 shows the parameter estimations of the time varying SJC copula. The tail dependence parameters of this time varying SJC copula fluctuate more over time in comparison to the constant copula. This indicates that, by introducing the evolution of the time varying parameters by Patton, the copula becomes more flexible, especially during times of economic distress.

Furthermore, this figure shows that on average the difference between the upper and lower tail is positive, which is also found by the constant SJC copula, indicating that on average joint positive outcomes are more often to happen then joint negative outcomes. On average the upper tail is equal to 0.448 and the lower tail is equal to 0.427, which is a little higher than for the constant parameter estimation. In addition, towards the end of the observations the difference between the upper and lower dependence comes close to zero, which indicates symmetry.

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Figure 4.8: Parameters time varying Symmetrized Joe Clayton cop-ula

The VaR can be derived based on the estimation of the parameters, as defined in section 2.6. Figure 4.9 shows the graph of the VaR at level 95% and 99& for the time varying SJC copula. Based on figure 4.9 it is clear that the copula model overestimates the risk. As can be seen in table 4.5, at 95% quantile the VaR is 4 times exceeded, where 62 was expected. At 99% quantile the VaR is 1 time ex-ceeded, where 12 was expected. A clarification for overestimating this risk could be the assumption that the marginal distribution follows a standardized Student’s t distribution. Furthermore, the Christoffersen test returns a p-value equal to zero for both quantiles, indicating the model does not estimate the VaR correctly. The constant SJC copula returns similar results which can be found in appendix.

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Figure 4.9: Value-at-Risk time varying Symmetrized Joe Clayton copula with Student’s t marginals

Table 4.5: Backtesting the Value-at-Risk portfolio Time varying SJC copula SP500 Times Exceeded Expected Christoffersen Test

95% 4 62 Reject H0 (0.000)

99% 1 12 Reject H0 (0.000)

The table reports the times the VaR is exceeded at level 95% and 99% for the Portfolio. Between Parentheses the P-value of the Christoffersen test is given.

4.3.2

Student’s t copula

One of the big differences between the Student’s t copula and the SJC copula is that there is no difference between the upper and lower tail dependence for the Student’s t copula, implying that joint positive and joint negative outcomes have similar probabilities. The estimation of the constant Student’s t copula parameter ρ, the correlation parameter, is given in figure 4.10. The correlation of the SP500 and the DAX lies between 0.45 and 0.65. Figure 4.10 shows that the correlation between the SP500 and DAX decreases as the financial crisis of 2007-2008 comes

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closer. After this point, the correlation increases.

Figure 4.10: Parameter ρ constant Student’s t copula

The time varying Student’s t copula correlation parameter ρ is given in figure 4.11. This figure shows that the correlation during stressful periods quickly adapts to a lower level. The correlation is at its lowest around the 400th observation, which is during the financial crisis. In addition, the correlation seems to be around 0.6 during relatively quiet periods. On average the correlation for the time varying parameter is equal to 0.54 and the constant copula is equal to 0.56, which seems to be rather low. A reason for this low correlation between the SP500 and DAX could be the different geographical locations of the indices.

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The results of the evolution of the degree of freedom parameter, ν, for the con-stant and time varying Student’s t copula are given in the appendix.

The tail dependence parameters, as given in formula 2.22, for the constant Stu-dent’s t copula on average equals to 0.375. For the time varying StuStu-dent’s t copula the tail dependence parameter is on average equal to 0.367. When comparing the tail dependence of the Student’s t copula with the SJC copula, it seems that the values are close to each other, but slightly higher for the SJC copula. In addition, joint extreme outcomes happen more often with the SJC copula.

The VaR at level 95% and 99% for the time varying Student’s t copula is given in figure 4.12. The results of the time varying Student’s t copula can be found in table 4.6. For this model, the VaR at level 95% is 9 times exceeded and for the 99% quantile the VaR is 1 time exceeded. Also, the null hypothesis of the Christoffersen test is rejected for both quantiles. The conclusion is that this model overestimated the risk and appears not to be a good model to estimate the VaR. The results of the constant Student’s t copula can be found in the appendix.

Figure 4.12: Value at Risk time varying Student’s t copula with Student’s t marginals

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Table 4.6: Backtesting the Value at portfolio Time varying Student’s t copula SP500 Times Exceeded Expected Christoffersen Test

95% 9 62 Reject H0 (0.000)

99% 1 12 Reject H0 (0.000)

The table reports the times the VaR is exceeded at level 95% and 99% for the Portfolio. Between Parentheses the P-value of the Christoffersen test is given.

4.3.3

Sensitivity Marginal distributions

Based on the Q-Q plot in chapter 3, an assumption was made that the residuals follow a standardized Student’s t distribution. In this subsection we assume that the residuals follow a standard normal distribution, thus gt+1 ∼ N (µt+1(x), σt+1(x))

and ht+1 ∼ N (µt+1(y), σt+1(y)). In table 4.7 the results of the VaR are given for the

copula with normal distributed marginals.

Table 4.7: Table with the results for normal distributed marginals

Quantile Times exceeded Expected Christoffersen Test Constant SJC copula

95% 5 62 Reject H0 (0.000)

99% 1 12 Reject H0 (0.000)

Time varying SJC copula

95% 4 62 Reject H0 (0.000)

99% 1 12 Reject H0 (0.000)

Constant Student’s t copula

95% 5 62 Reject H0 (0.000)

99% 1 12 Reject H0 (0.000)

Time varying Student’s t copula

95% 7 62 Reject H0 (0.000)

99% 1 12 Reject H0 (0.000)

The table reports the times the VaR is exceeded at level 95% and 99% for the normally distributed marginals. Between Parentheses the p-value of the Christoffersen test is

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Table 4.7 shows that the copulas with normally distributed marginals do not give a good fit for the estimation of the VaR. The very small p-values of the Christoffersen test indicate that the model is not a good fit for the data. Based on these results, the model described in chapter 2.6 overestimates the risk for both different marginal distributions. In comparison to the EVT, the copula theory does not provide a good fit for the VaR of the stock market data.

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Conclusion

This thesis backtests the VaR using two different approaches: the extreme value ap-proach and the copula apap-proach. These two different techniques are used to estimate the VaR of the SP500, DAX and the portfolio. The data used was from May 1, 2002 to May 1, 2012, with the financial crisis in the middle of the data set.

For the first approach, the extreme value approach, we followed the paper of McNeil Frey (2000), in which the two stage estimation procedure for the VaR was used. The first step was the assumption that the marginals follow an AR(1)-GARCH(1,1) distribution for both market indices. We used this to calculate the one step ahead prediction for the mean and variance. The second step of the estimation procedure was to model the tail of the innovation distribution using EVT methods. After this two-stage approach the VaR was derived.

The individual model for the SP500 index rejected the null hypothesis of the Christoffersen test at 5% significance, but not at 1% significance. This indicates that the EVT does not provide a very plausible VaR estimate. The individual model for the DAX provides a good fit based on the Christoffersen test. Furthermore, without modelling the dependence structure between the DAX and the SP500, the null hypothesis of the Christoffersen test of the portfolio model was not rejected, indicating that the EVT does give a solid estimate of the VaR for a combination of those indices.

The second approach was based on the copula theory. In this thesis two different copulas are used: the Symmetrized Joe Clayton copula and the Student’s t copula. Based on the well known Sklar’s theorem, the time dependent joint distribution of

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the portfolio was derived. After this derivation we were able to compute the VaR. We looked at two different approaches for the parameter estimation of the copula. At first, the parameters of these copulas were set to be constant over time. After that, the parameters were able to vary over time, based on the papers of Patton from 2004 and 2006. To use this technique to compute the VaR, we had to make some assumptions for the marginal distributions. The two marginal distribution that were considered were the normally distributed marginals and Student’s t distributed marginals.

The method used for the second approach overestimated the market risk. The Christoffersen test returned really small p-values for all constant and time varying copulas. This indicates that the copula based model for the VaR does not preform well. Based on the set of assumptions, this model preforms worse than the EVT model.

Lastly, this thesis looked at the SJC and Student’s t copulas. For possible further research, other copulas can be tested in different markets. In addition, since we assumed that the marginal distribution was normally or Student’s t distributed, other distributions could be considered for further research.

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Figure 5.1: Value at Risk Constant Symmetrized Joe Clayton Cop-ula Student’s t marginals

Table 5.1: Backtesting the Value-at-Risk portfolio constant SJC copula SP500 Times Exceeded Expected Christoffersen Test

95% 5 62 Reject H0 (0.000)

99% 1 12 Reject H0 (0.000)

The table reports the times the VaR is exceeded at level 95% and 99% for the Portfolio. Between Parentheses the P-value of the Christoffersen test is given.

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Figure 5.2: Value at Risk constant Student’s t Copula with Stu-dent’s t marginal distributions

Table 5.2: Backtesting the Value-at-Risk portfolio constant Student’s t copula SP500 Times Exceeded Expected Christoffersen Test

95% 6 62 Reject H0 (0.000)

99% 1 12 Reject H0 (0.000)

The table reports the times the VaR is exceeded at level 95% and 99% for the Portfolio. Between Parentheses the P-value of the Christoffersen test is given.

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Figure 5.3: Degree of freedom parameter constant Student’s t cop-ula

Figure 5.4: Degree of freedom parameter time varying Student’s t copula

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References

Artzner, P., Delbaen, F., Eber, J., Heath D. (1998), Coherent Measures of Risk, Mathematical Finance, 9(3), 203-228.

Balkema, A. and de Haan, L. (1974), Residual life time at great age, Annals of Probability, 2, 792-804.

Christoffersen, P.F. (1998), Evaluating interval forecasts, International Economic Review, 39, 841-862

Hafner, C.M. and Manner, H. (2008), Dynamic stochastic copula models: Estima-tion, inference and applications. Journal of Applied Econometrics, 27, 269-295 Kaas, R., Goovaerts, M., Dhaene, J. and Denuit, M. (2008), Modern Actuarial Risk

Theory: Using R, Springer, Berlin

McNeil, A.J. and Frey, R. (2000), Estimation of Tail-Related Risk measures for Hetroscedastic Financial Time Series: an Extreme Value Approach, Journal of Empirical Finance 7, 271-300.

McNeil, A. J., Frey R. and Embrechts, R.(2005), Quantitative Risk Management, Princeton University Press, Princeton.

Patton, A.J., (2004), On the Out-of-Sample Importance of Skewness and Asymmet-ric Dependence for Asset Allocation, Journal of Financial EconometAsymmet-rics, 2(1), 130-168.

Patton, A.J., (2006), Modelling asymmetric exchange rate dependence, International Economic Review, 47-2, 527-556

Patton, A.J., (2012), A review of copula models for economic time series, Journal of Multivariate Analysis, 110, 4-18.

Pickands, J. (1975), Statistical Inference Using Extreme Order Statistics, The Annals of Statistics, 3, 119-131

Sklar, A. (1959), Fonctions de repartition a n dimensions et leurs marges, Publica-tions de lInstitut Statistique de lUniversite de Paris 8 , 229-231.

Tsay, R.S (2010), Analysis of Financial Time Series, Third edition, New Jersey: Wiley.

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