A Dynamic Extreme Value Theory Approach

Estimating the Value at Risk: A Dynamic Extreme Value Theory Approach

Author: Supervisor: Lauren McGeever Dr. S. U. Can 10464832 Second Reader: [email protected] Prof. R. J. A. Laeven

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 December 17, 2015

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Abstract

Since first being introduced globally in 1996 by the Basel committee, the value at risk has become “the first line of defence against financial risk” (Jorion 1996). Although the Value-at-Risk is undeniably a useful tool for estimating risk, it still has several shortcomings. This paper will examine these criticisms and proposes extreme value theory as a more sophisticated approach to calculating the Value- at-Risk. We will also examine the role of stochastic volatility models in scaling estimates by the volatility, in order to improve the prediction of the models.

The data to be examined belongs to three of the most well-known stock indices in the world. We will be using: FTSE 100, S&P 500 and Nikkei 225. We will be using two periods, an in-sample period of September 4th, 1984 to September 3rd, 2010 in which to calculate the VaR estimates and an out-of-sample period from 4th September 2010 to the 3rd September 2015 which will be used to back test our estimates. From this investigation we can conclude that use of Extreme Value The- ory provides better estimates for the Value-at-Risk, but only for higher conﬁdence levels. We also conclude that models ﬁtted with a stochastic volatility model offer better predictions. iv

Contents

Abstract iii

1 Introduction1

2 The Value at Risk3 2.1 Introducing the Value at Risk...... 3 2.2 Traditional VaR Methods...... 3 2.2.1 The Normal based VaR approach...... 3 2.2.2 Historical simulation approach...... 4 2.3 A Coherent Risk Measure...... 4

3 An Introduction to Extreme Value Theory6 3.1 Introduction to Extreme Value Theory...... 6 3.1.1 Extremal Types Theorem...... 7 3.2 Classical Extreme Value theory (The block maxima approach)8 3.3 The Peak-over-the-Threshold approach...... 9 3.3.1 The Generalized Pareto distribution...... 9 3.3.2 Threshold selection and Parameter estimation for the GPD...... 9 3.3.3 The Return level...... 10 3.4 Selecting an EVT method...... 11

4 Data and Methodology 12 4.1 The data...... 12 4.1.1 Using R Studio for analysis...... 13 4.2 Data transformation...... 13 4.3 Fitting a stochastic volatility model...... 14 4.4 Fitting a generalized Patero model...... 15 4.4.1 Checking the goodness of ﬁt for parameter estimates 16

5 Results and Back-testing 19 5.1 Examining Return levels...... 19 5.1.1 Return level estimates for static and dynamic models 21 5.2 VaR estimates from the in sample period...... 21 5.3 Back-Testing...... 23 5.3.1 The Portion of Failures Unconditional Coverage test 23 5.3.2 The Duration Based Conditional Coverage test.... 24

6 Conclusion 26

7 References 28

A Tables of results 30

B R code 34 v

C Plots 37 vi

List of Figures

4.1 Plot of the NIKKEI 225 log returns against observation days 14 4.2 Mean residual life plot for the NIKKEI 225, and 95% conﬁ- dence interval...... 15 4.4 Diagnostic plots for static EVT model of the NIKKEI 225.. 17 4.5 Diagnostic plots for dynamic EVT model of the NIKKEI 225 18

5.1 Return level plot the FTSE 100 with static EVT...... 19 5.2 Return level plot the FTSE 100 with dynamic EVT...... 20

C.3 Fitting the GPD over a range of thresholds for the S&P 500. 37 C.4 Fitting the GPD over a range of thresholds for the FTSE 100. 38 C.5 Fitting the GPD over a range of thresholds for the NIKKEI 225 38 C.6 Diagnostic plots for S&P 500...... 39 C.7 Diagnostic plots for FTSE 100...... 39 C.8 Return levels plot for the S&P 500 and NIKKEI 225...... 40 vii

List of Tables

4.1 Threshold choices and number of exceedances for: FTSE 100, S& P 500, NIKKEI 225...... 16 4.2 Maximum likelihood estimates of the shape and scale parameters...... 16

5.1 Return level results for Static EVT models...... 21 5.2 Return level results for dynamic EVT models...... 21 5.3 S&P 500 VaR estimates...... 22 5.4 S& P 500 VaR Violations...... 23 5.5 S&P 500 Conditional Coverage Back-test...... 25

A.1 NIKKEI 225...... 30 A.2 FTSE 100...... 30 A.3 VaR Violations FTSE 100...... 31 A.4 VaR Violations for NIKKEI 225...... 31 A.5 FTSE 100...... 32 A.6 NIKKEI 225...... 33 viii

List of Abbreviations

ARCH Autoregressive Conditional Heteroskedasticity

CDF cumulative distribution function

CVaR Conditional Value at Risk

EVT Extreme Value Theory

iid independent and identically distributed

GARCH Generalized Autoregressive Conditional Heteroskedasticity

GEV Generalized Extreme Value distribution

GPD Generalized Pareto distribution

MRL Mean Residual Life plot

POT Peak-Over-the-Threshold

P-P plots Probability-Probability plots

Q-Q plots Quantile-Quantile plots

r.v. Random Variable

VaR Value-at-Risk 1

Chapter 1

Introduction

Over the last century financial crises have occurred in greater frequency and magnitude, highlighting the need for stricter risk management. Some of largest the and most devastating crashes include: the Wall Street crash of 1929; Black Monday of 1987; the Asian crisis of 1997; the credit crisis of 2007; and even more recently the Chinese stock market crash of 2015. Primarily due to advances in modern technology, financial institutions are more interconnected than ever. This often creates a domino effect in which other financial institutions are affected by the failure of another. Looking at the credit crisis of 2007 this is very apparent. The bursting of the Ameri- can housing bubble was chiefly facilitated by sub-prime mortgages and the relaxation of lending standards. After this bubble burst large financial institutions such as Merrill Lynch, Citigroup and UBS incurred huge losses because of their large positions in the securitization of these sub-prime mortgages. Governments were obliged to rescue many of these financial institutions, however Lehman Brothers was allowed to fail. This created a ripple effect, which was felt by other institutions world wide. The formation of the Basel committee on banking supervision was the first attempt in creating an international regulatory framework; their aim was to implement risk management policy to reduce financial crises. In 1987 they proposed the Value-at-risk as a model for market risk, this was finally implemented in the 1996 amendment of the Basel accord. It required banks to hold capital for market and credit risk. VaR models are essential to risk management. Jorion, 1996 describes them as “the first line of defense against financial risk” and sees them as "improving transparency and stability in financial markets". This is because they have the capability to summarize the total risk in a single number. A VaR model can be interpreted as fullfilling the requirement: At X% certainty there will be no loss of more than Y over N days. Other favorable points are that this number is easy to interpret and that standard methods of calculation are easy to use. Value-at-Risk may appear a very favorable risk measure for downside risk however it is widely accepted that VaR may regularly underestimate risk. This underestimation can have damaging effects on institutions, leav- ing them unprepared for adverse market movements. Traditional methods of calculating VaR often rely on the unrealistic assumption that the underlying distribtuion is normal. For average returns (center of the distribution) this is a reasonable assumption. Unfortunately it has been observed by many that market returns often have heavier tails and are more leptokurtic than the normal distribution. Research by Mandelbrot, 1963 and Fama, 1965 did much in the way of proving this. The underestimation of the tails is a major problem because “financial solubility of an investment is likely to 2 Chapter 1. Introduction be determined by extreme changes in market conditions rather than typical changes” (Coles, 2001). These returns found in the tails are often referred to as extreme values because they are very large losses and occur rarely; though not as rare as we would like. Understanding that the normal distribution is inadequate for modeling the tails of financial returns, we can ask the question what distribution would be a better fit? One of the more popular solutions leads us to extreme value theory, “Letting the tails speak for themselves” (Coles, 2001) . This allows us to create an asymptotic distribution for the tails. EVT has been used in numerous fields of science for many years. These include: Ther- modynamic of earthquakes; Ocean wave modeling and even food science. Since the 1980’s EVT has also grown in popularity in the field of financial risk management, especially after the discovery of EVT’s second theorem: peak over the threshold by Pickands–Balkema–de Haan, 1974. This is because “ the generalized Pareto threshold model provides a direct method for the estimation of value-at-risk.” (Coles, 2001). In this thesis we will compare the standard ways of modeling VaR with the newer peak over the threshold EVT approach with and without the help of a stochastic volatility model to deal with changes in the volatility. We are aiming to prove that EVT gives more accurate results for value at risk than the standard methods and even better estimates when combined with a GARCH(1,1) model. We will be able to judge this by applying these methods to three of the main stock indices from around the world: the FTSE 100, the S&P 500 and the NIKKEI 225. To analyze whether EVT was indeed successful we will examine diagnostic plots including: p-p plots, q-q plots and return level plots to check the fit of the model. Finally we will back-test the results to determine their success. The structure of this thesis will be as follows: chapter 2 will give a brief description of value-at-risk and its problems in financial practice; chapter 3 will present a detailed description on the underlying assumptions and definitions of EVT; chapter 4 will introduce the data and the methodology behind estimating the VaR; chapter 5 will involve back-testing and determine the results; finally chapter 6 will offer a conclusion to the results. 3

Chapter 2

The Value at Risk

2.1 Introducing the Value at Risk

Since first being introduced globally in 1996 by the Basel committee, the value at risk has become “the first line of defense against financial risk”(Jorion 1996). Its attractiveness as a risk measure comes from being able to summarize the total risk as a single number. Prior to its introduction many financial institutions relied on a delta-gamma-vega approach, it required many different risk measures being analyzed and was very time consuming. Extreme risk is found in the tail of the distribution, these risks are rare but can lead to huge losses for a financial institution. As a quantile function, VaR is able to focus on the tails of the distribution, though does not give any information beyond its confidence interval. (Hull ,2012) describes the VaR as answering one simple question: “how bad can things get?”. This is because the VaR can state that for p% certainity there will not be a loss of more than X in the next N days. Generally the Basel committee specifies a 10-day period and a confidence level of 99%. In this thesis however we will only be looking at a 1-day time horizon but looking a number of confidence levels. Higher confidence levels represents higher risk aversion; it also represents higher costs. Although the Value-at-risk is undeniably a useful tool for estimating risk, it still has several shortcomings . In this chapter we will look at: the VaR as a risk measure; its standard ways of calculation; and its criticisms.

Definition 2.1 The Value at Risk For a risk X, the value at risk at a confidence level p is defined as:

−1 V aR[X; p] = FX (p) = inf{x : FX (x) ≥ p} (2.1)

In other words, the VaR is the inverse cdf of X computed at the conﬁdence level p.

2.2 Traditional VaR Methods

2.2.1 The Normal based VaR approach The Normal based approach, like its name suggests, is a method of calculating the value at risk based on the assumption that the underlying distribution is normal. It is a parametric approach that uses the mean and standard 4 Chapter 2. The Value at Risk deviation of the data sample to estimate a quantile from the normal distribution. In the case of ﬁnancial data however the underlying distribution is rarely ever normal. Research done by Mandelbrot, 1963 and Fama, 1965 have indicated that ﬁnancial distributions are often leptokurtic and possess heavier tails than the normal distribution. This is a problem for VaR calculations as “VaR attempts to describe tail behavior precisely” (Jorion, 1996). The underestimation of the tails by the normal distribution leads to an underestimation of VaR.

2.2.2 Historical simulation approach The historical simulation is a non-parametric approach to calculating the value at risk. We are using past returns to predict future losses. This method involves using a sample of past data for a financial variable, selecting a confidence level then finding the quantile estimate at that level. Unlike the parametric approach it makes no distributional assumption but allows the past data to shape the distribution of the tails, however the past is not always representative of the future. There is great estimation risk in assuming that the future returns will mimic that of the past. The advance- ment of financial institutions in the way of technology and more complex financial derivatives means returns of the past are not necessarily a good benchmark for the future. Also with new financial products there might not be past data to analyse. Not having sufficient data is another problem of this method, a small sample is deemed “inadequate because of the small number of effective observations” (Jorion, 1996).

In this thesis we will compute values for VaR using these traditional approaches,to provide benchmarks to compare our EVT based VaR estimates against. As we consider both methods insufﬁcient of accurately describing the tails of the distribution we hope to attain more precise VaR results from a method that can describe the tails. Due to the simplicity of these approaches all calculations will be computed in Excel.

2.3 A Coherent Risk Measure

In a paper by Artzner et al (1998) they introduced a set of axioms, related to the axioms on acceptance sets, the risk measures that satisfy these require- ments are called a: coherent risk measure.

Deﬁnition 2.3 A Coherent Risk Measure

"Coherence: a risk measure satisfying the four axioms of translation invariance, subadditivity, positive homogeneity, and monotonicity, is called coherent." (Artzner et al, 1998)

Subaddivity : ρ(X + Y ) ≤ ρ(X) + ρ(Y ) 2.3. A Coherent Risk Measure 5

Monotonicity : X ≤ Y then ρ(X) ≤ ρ(Y )

Translation Invariance : ρ(X + a) = ρ(X) + a

Positive Homogeneity : ρ(λX) = λρ(X) for all λ ≥ 0

In the case of VaR, though it satisﬁes the other axioms, it does not necessarily satisfy the axiom of subadditivity. An example of how it does not satisfy this is that it is possible to reduce risk by splitting the total portfolio in to sub-portfolios. In other words, it may occur with VaR that:

ρ(X + Y ) > ρ(X) + ρ(Y )

This problem with VaR has led to many suggestions of other risk measures related to VaR which are in fact coherent. One of these is the Conditional Value at Risk, also known to many as the Tail Value at Risk. This risk measure considers the total loss in tails.

Deﬁnition 2.2 The Conditional Value at Risk

For a risk X, the conditional value at risk at level p ∈ (0, 1) is deﬁned as:

1 Z 1 CV aR[X; p] = V aR[X; t]dt; (2.2) 1 − p p

The CVaR is the arithmetic average of the VaRs of X beyond p.

For the purpose of studying different risk measures and their different methods of computation, we will also compute the values of CVaR on the data. Unlike the VaR which asks the question " what is the worst loss at conﬁ- dence level p?", CVaR asks " what is the total worst loss at conﬁdence level p ?". We intend to analyze whether or not CVaR is a more practical risk measure against our proposed EVT-VaR value. 6

Chapter 3

An Introduction to Extreme Value Theory

“Extreme value theory is unique as a statistical discipline in that it develops techniques and models for describing the unusual rather than the usual.” (Coles, 2001)

In simple words, EVT is the study of extreme events. By definition extreme events are low in frequency but large in magnitude. What is more interesting though is in general statistical practices extreme events are often ignored and disregarded as outliers. However in areas like finance they can be the most influential. Coles justifies this by saying “financial solubility of an investment is likely to be determined by extreme changes in market conditions rather than typical changes” (Coles, 2001). There are two main approaches of estimating EVT: the block maxima method and the peak over the threshold method. In this chapter we will be de- tailing: a brief history of EVT; classical block maxima EVT; Peak over the threshold EVT and why we choose to use this method rather than the other. General formulas and definitions are taken from Coles, 2001 and McNeil, 1999.

3.1 Introduction to Extreme Value Theory

First developed in 1928 by Fisher and Tippett and later extended by Gne- denko in 1943, “it is the study of the asymptotic distribution of extreme events” (Rocco, 2012). This is to say we disregard the center of the distribution and only focus on the tails, the maxima of the distribution.

Mn = max{X1 ...Xn }

Where X1 ...Xn is a sequence of iid rv.Mn represents the maximum of the process over n time units.

Theoretically we can derive a distribution function F for all values of n:

P r{MN ≤ z} = P r{X1 ≤ z , ...., Xn ≤ z }

= P r{X1 ≤ z }x....xP r{Xn ≤ z } = {F (z)}n 3.1. Introduction to Extreme Value Theory 7

It is not possible to use standard statistical techniques to estimate F. We can not estimate F from observed data; “very small discrepancies in the estimate of F can lead to substantial discrepancies for F n.” (Coles, 2001). This brings us to extreme value theory which allows us to choose an approximate family of models for F n.

Deﬁnition 3.1 Liner renormailization of the sample maxima

Mn − bn M*n = (3.1) an

We allow a linear re-normalization of Mn because for any z < Z+, where Z+ is the upper end-point of F, F n(z) → 0 as n → ∞ , so that the distribution of Mn degenerates to a point mass on Z+.

When Mn is stabilized by suitable sequences of an and bn , regardless of distribution F for the sample population, M*n has only 3 possible limiting distributions. They are the: Gumbel, Fréchet and the Weibull.

3.1.1 Extremal Types Theorem

If there exist sequences of constants {an > 0} and bn such that

P r{ Mn−bn ≤ z} → G(z) as n → ∞ an where G is a non-degenerate distribution function, then G belongs to one of the following families: z − b 1.G(z) = exp − exp[−( )] , −∞ < z < ∞; (3.2) a

( 0 z ≤ b, 2.G(z) = z−b −α (3.3) exp{−( a ) }, z > b;

( exp{−[−( z−b )α]}, z < b, 3.G(z) = a (3.4) 1, z ≥ b; for parameters a > 0, b and in the case of the 2 and 3, α > 0. Cases 1, 2 and 3 correspond to the: Gumbel, Fréchet and Weibull families of distributions.

Deﬁnition 3.3 The Generalized Extreme value distribution 8 Chapter 3. An Introduction to Extreme Value Theory

If there exist sequences of constants {an > 0} and bn such that

P r{ Mn−bn ≤ z} → G(z) as n → ∞ an where G is a non-degenerate distribution function, then G is member of the GEV family: z − µ G(z) = exp{−[1 + ξ( )]} (3.5) σ deﬁned on {z : 1 + ξ(z − µ)/σ > 0}, where −∞ < µ < ∞, σ > 0 and −∞ < ξ < ∞.

The model consists of three parameters: µ the location parameter; σ is a scale parameter and ξ is the shape parameter of the distribution. It is inference on ξ which determines the tail behavior:

• When ξ > 0 this corresponds to the Fréchet case, • When ξ < 0 this corresponds to the Weibull case, • When ξ = 0 this corresponds to the Gumbel case.

3.2 Classical Extreme Value theory (The block maxima approach)

This method is based on grouping data into n blocks with size m and ﬁtting the generalized extreme value distribution to the set of block maxima. For pragmatism, the block size m is usually set at a length of one year, however it is possible to use smaller block sizes. There is a trade off here between bias and variance. If block size is too small, leading to more blocks, data that may not necessarily be extreme will be included leading to a biased estimate. In the converse scenario, if block size is too big there will be fewer block maxima leading to estimation variance.

In regards to modeling ﬁnancial data by the block maxima approach this is not necessarily the best approach; due to volatility, extreme events often cluster together. This can cause one block to contain several extreme data points but only one will be included. This leads to a waste in data. Many papers including: McNeil(1997) and Coles(2001) believe the peak-over-the- threshold to be a far more valuable approach. “ Modeling only block maxima is a wasteful approach to extreme value analysis if other data on extremes are available.” (Coles, 2001). This problem leads us to disregard this approach. 3.3. The Peak-over-the-Threshold approach 9

3.3 The Peak-over-the-Threshold approach

In 1975 classical EVT was extended by Pickands–Balkema–de Haan, this approach is known as the Peak-over-the-Threshold method and is considered the second theory in EVT. Considering a iid sequence: X1 , X2 .... using the POT method we regard an event Xi as extreme if it exceeds a high threshold u. From this we can derive the following conditional probability:

Deﬁnition 3.4 The stochastic behavior of extreme events 1 − F (u + y) P r{X > u + y|X > u} = , y > 0. (3.6) 1 − F (u)

3.3.1 The Generalized Pareto distribution Pickands–Balkema–de Haan, 1975 found that, like the GEV distribution with block maxima, as the parent distribution of F is generally unknown we can use an approximate distribution for modeling the excesses over a threshold. They found that threshold excesses have approximate distribution within the generalized Pareto family.

Deﬁnition 3.5 The Generalized Pareto distribution

The generalized Pareto distribution has the distribution function:

ξy −1 H(y) = 1 − (1 + ) ξ (3.7) σe deﬁned on {y : y > 0 and (1 + ξy/σe) > 0}, where σe = σ + ξ(u − µ) .

3.3.2 Threshold selection and Parameter estimation for the GPD In order to fit a GPD we must first select a threshold. This can be done in many ways such as examining mean residual life plots and plotting the GPD over a range of thresholds then examining the fit; the method for this will be discussed in more detail in the next chapter. It is a simple procedure however, analogous to the block maxima approach, there is a trade off between bias and variance in selecting a threshold. Too low a threshold leads to non-extreme data being included in our model leading to a biased result. Too high a threshold, leads to fewer data points causing a large variance in our analysis.

Once a threshold is selected the parameters for the GPD can be estimated by maximum likelihood estimation:

Suppose that the values y1 ..., yk are the k excesses above a threshold u. 10 Chapter 3. An Introduction to Extreme Value Theory

For ξ 6= 0 the log-likelihood is derived from the generalized Pareto as:

k X `(σ, ξ) = −klogσ − (1 + 1/ξ) log(1 + ξyiσ), (3.8) i=1 −1 provided (1 + σ ξyi ) > 0 for i= 1,...,k; otherwise `(σ, ξ) = −∞ . Maximization of the log-likelihood with respect to ξ and σ give us our parameters for the GPD.

3.3.3 The Return level In order to retrieve estimates of value at risk from our GPD models we need to interpret the model in terms of quantiles or return levels. McNeil, 1999 advises this can be done by inverting the tail estimation formula.

Using a GPD to model the exceedances of a threshold u by a variable X, we have for x > u,

x − u −1 P r{X > x|X > u} = [1 + ξ( )] ξ σ the Nu /n = P r{X > u} , where n is the number of observations and Nu is the number of observations that exceeds the threshold u. From this it follows that:

Nu x − u −1 P r{X > x} ≈ [1 + ξ( )] ξ , n σ

To solve for the value at risk, we regard it as a solution of:

Nu xp − u −1 [1 + ξ( )] ξ = 1 − p, n σ

At a selected conﬁdence level 1-p, x1−p is exceeded on average once every 1/1-p observations. Rearranging this we have a direct calculation to obtain the VaR from our GPD model.

σ Nu VaR = x = u + [( (1 − p))−ξ − 1]. (3.9) p 1−p ξ n

Substituting parameters found from the GPD models allows us to directly calculate a VaR estimate. 3.4. Selecting an EVT method 11

3.4 Selecting an EVT method

For the purpose of this thesis, all extreme value models will be derived using the POT approach. From research we have determined that it is a more practical approach for ﬁnancial data. Often in ﬁnancial time series some years may contain more extreme data than others, therefore the block maxima method appears to be a wasteful approach. It is also very convenient that by calculating the return level from a GPD model gives a direct method into calculating VaR estimates. 12

Chapter 4

Data and Methodology

Aim of this thesis: We are aiming to calculate the value at risk using 4 different approaches: 1. A normal-based approach 2. Historical simulation 3. Static EVT 4. Dynamic EVT

These estimates will be calculated over a range of conﬁdence intervals to evaluate which model provides the approximation for the VaR.

We will also look at the Conditional Value-at-Risk to see if a coherent risk measure produces a better risk estimate: 1. Static CVaR 2. Dynamic CVaR

4.1 The data

The data that will be examined belongs to three of the most well-known stock indices in the world. We will be using: FTSE 100, S&P 500 and Nikkei 225. These are known as national indices as they represent weight-average of the largest stocks trading on the national exchange. National indices will give a good indication of market performance, as they comprise of many different industries from: fashion, media, property to commodities such as oil and gas.

FTSE 100: It consists of the 100 largest companies trading on the London stock exchange. It is weighted by market capitalization. This means that the highest capitalizing companies carry more weight in the index than smaller ones. Some of the most inﬂuential companies are: HSBC, Barclay, BP (gas and oil) and AstraZeneca (pharmaceuticals). The currency of the closing prices is Pounds Sterling.

S&P 500: This is the most diverse stock index we are using, it consists of 500 companies coming from the New York and the American/Canada (NAS- DAQ) stock exchange. These are the two largest stock exchanges in the world. Like the FTSE 100 they solely rely on market capitalization for how the companies within the index are weighted. It’s most weighted stocks are: Apple Inc., Google Inc. and Walmart stores Inc. The currency of the closing prices is US dollars. 4.2. Data transformation 13

Nikkei 225: The Nikkei is a price-weighted index. This means that out of 225 companies traded on the Tokyo stock exchange, the index allocates higher weights to stocks with higher prices. Some of the most well-known stocks incorporated in the index are: Honda Motors, Mitsubishi heavy in- dustry and Toshiba corp. The currency of the closing prices is in Japanese Yen.

We will be using two periods, an in-sample period of September 4th, 1984 to September 3rd, 2010 in which to calculate the VaR estimates. This is a wide time horizon but as extreme data is rare it will give enough data points for analysis. In this period there have been many ﬁnancial crises including: black Monday of 1987and the Asian crisis of 1997 which will provide extreme data within the sample. The second period is an out of sample period 4th September 2010 to the 3rd September 2015 which will be used to back test our estimates.

4.1.1 Using R Studio for analysis R studio is a free software program for statistical computing available around the world. We have chosen this program because of its easy availability and that it has a range of libraries especially for analyzing extreme data. In this thesis we will be using a range of R libraries including:

• extRemes • ismev • fExtremes • rugarch

All codes for calculations used in this thesis can be found in Appendix b.

4.2 Data transformation

Financial series are known for being non-stationary, this is because the mean and variance of the series changes over time. Stationary data is crucial as non-stationary data is unpredictable and can lead to inaccuracies in modeling. To ensure our data is stationary we calculate the daily log returns from the closing prices of each index. This method is known as differencing. “Transformations such as logarithms can help to stabilize the variance of a time series. Differencing can help stabilize the mean of a time series by removing changes in the level of a time series, and so eliminating trend and seasonality.” (Hyndman et al, 2013)

Xˆ = logxi+1 − xi Furthermore for presentational purposes we scale the data:

Xˆ → 100Xˆ 14 Chapter 4. Data and Methodology

FIGURE 4.1: Plot of the NIKKEI 225 log returns against observation days

From this we can see that the data is now reasonably close to stationarity, as the mean and variance appear almost constant. Another aspect we can determine from these graphs is the presence of volatility clusters; large losses are grouped together.

4.3 Fitting a stochastic volatility model

From our literature review we have chosen to ﬁt a GARCH (1,1) model to our data, this is so we can “scale our VaR estimates by the current volatility” (McNeil, 1999). This was developed by (Hull & White, 1998) and risk measures that achieve this are termed dynamic risk measurements. We term the other model static as it assumes volatility to be constant. There are a number of reasons why taking the volatility into account is important. Firstly,"An extreme value in a period of high volatility appears less extreme than the same value in a period of low volatility." (McNeil, 1999). Another is volatility clusters which are caused by serial correlation in the squared returns; (Dufﬁe & Pan, 1997) attributed to be "second major source of fat tails".

The Generalized Auto-Regressive Conditional Heteroskedastic model was ﬁrst extended from the Auto-Regressive Conditional Heteroskedastic model by (Bollerslev, 1986). An auto-regressive process with GARCH( 1,1) errors: 4.4. Fitting a generalized Patero model 15

Xt = µt + σtZt,

µt = λXt−1 (4.1) 2 2 2 σt = α0 + α1(Xt−1 − µt−1) + βσt−1, with α0, α1, β > 0 , β + α1 < 1 and |λ| < 1 . Where µt is the expected return, σt is the volatility of the return on day t and Zt is the noise variables which provides randomness to the process.

This process will be done in R studio using the rugarch package. Mc- Neil, 1999 advises to do this in two steps: ﬁrst ﬁt the data, in this case daily log returns of an index, with a GARCH (1,1) model; then extract the residuals from the model and apply EVT.

4.4 Fitting a generalized Patero model

FIGURE 4.2: Mean residual life plot for the NIKKEI 225, and 95% conﬁdence interval

From the literature review we have decided to fit our GPD model by maximum likelihood estimation. By selecting a threshold we are the able to solve for scale parameter σ and shape parameter ξ. To select a suitable threshold for our models we will use two methods to determine the optimum threshold. Firstly, using a mean residual life plot which plots thresholds against the mean exceedances, we look for where the graph is approximately linear to select our threshold. This first technique is often difficult to interpret but will give an approximation of a suitable threshold. The second technique involves fitting the GPD at a range of candidate thresholds; here we can check the stability of parameter estimates. 16 Chapter 4. Data and Methodology

From Table 4.1 we can see the plot is approximately linear around 3.5, verifying this by looking at the parameter estimates at that threshold leads us to accept this as our threshold for the NIKKEI 225. We investigate each threshold in same way for both static and dynamic EVT models with each index.

TABLE 4.1: Threshold choices and number of exceedances for: FTSE 100, S& P 500, NIKKEI 225

Static EVT Dynamic EVT Threshold Number of Threshold Number of Choice Exceedances Choice Exceedances FTSE 100 2.5 132 2.5 140 NIKKEI 225 3.5 98 3.5 104 S&P 500 2.5 135 2.5 145

We can see from the table, that each threshold provides enough exceedances for ﬁtting the GPD model and therefore trade-off between bias and variance is sufﬁcient. If we had set the threshold any higher there would not be enough data points for inference. However if we had set the threshold any lower we run the risk of including data that is not necessarily extreme.

After determining a threshold for each index, we able to use extRemes package in R studio to ﬁt a generalized Patero model to the data. Using the command: fevd(....) it will use maximum likelihood estimation to ﬁnd estimates of the scale and shape parameter.

TABLE 4.2: Maximum likelihood estimates of the shape and scale parameters

Static EVT Dynamic EVT Scale, σˆ Shape, ξˆ Scale, σˆ Shape, ξˆ FTSE 100 0.8620123 0.2837375 0.8445027 0.2862725 NIKKEI 225 1.1538578 0.2165901 1.1119088 0.2276867 S&P 500 0.8832965 0.3721952 0.8504503 0.3827838

From the table we notice that all estiamtes of ξˆ > 0, this corresponds to the fact the distribution is heavy-tailed and that the end-point is inﬁnity.

4.4.1 Checking the goodness of fit for parameter estimates Verifying the goodness of fit of each model is a crucial step. This can be assessed graphically using many diagnostic plots such as: probability plots, quantile plots, density plots, return level plots and the profile likelihood for ξ. 4.4. Fitting a generalized Patero model 17

(A) Proﬁle likelihood of the (B) Proﬁle likelihood of the shape parameter for the NIKKEI shape parameter for the NIKKEI 225 using static EVT 225 using dynamic EVT

Comparing the profile likelihoods of ξ for static EVT model against the dynamic EVT model we can see that the dynamic model has a much smaller confidence interval. A smaller confidence interval indicates more accuracy in the dynamic model than the static. However both surfaces of the profile likelihood are slightly asymmetric, Coles states this reflects “ greater uncertainty about large values of the process”, Coles, 2001.

FIGURE 4.4: Diagnostic plots for static EVT model of the NIKKEI 225

Probability Plot Quantile Plot 1.0 16 14 0.8 12 0.6 10 Model Empirical 0.4 8 6 0.2 4 0.0 0.0 0.2 0.4 0.6 0.8 1.0 4 6 8 10 12 Empirical Model

Return Level Plot Density Plot 100 0.8 80 0.6 60 f(x) 0.4 40 Return level 0.2 20 0.0 0 1e−01 1e+00 1e+01 1e+02 1e+03 5 10 15 Return period (years) x 18 Chapter 4. Data and Methodology

FIGURE 4.5: Diagnostic plots for dynamic EVT model of the NIKKEI 225

Probability Plot Quantile Plot 1.0 16 14 0.8 12 0.6 10 Model Empirical 0.4 8 6 0.2 4 0.0 0.0 0.2 0.4 0.6 0.8 1.0 4 6 8 10 12 Empirical Model

Return Level Plot Density Plot 100 0.8 80 0.6 60 f(x) 0.4 40 Return level 0.2 20 0.0 0 1e−01 1e+00 1e+01 1e+02 1e+03 5 10 15 Return period (years) x

There is little visible difference between either set of diagnostic plots for the static or the dynamic model, however in the probability plots we can see that the plot for the dynamic model is closer to linearity than the static. A probability plot is a comparison between the fitted and the empirical distribution functions, the closer to linear the plot is the better the fit of the model. Overall, each probability plot is close to linear lending support for the fit of both.

The quantile plots are both linear to begin with but deviate at higher levels, this is not a convincing sign for the fit of the model, however the return level plot offers a more promising sign. For lower return periods the confidence intervals are close together and all data points are contained within. On the other hand, it can be seen that the confidence intervals widen considerably at higher return periods. From this we can interpret that our model is good fit for extrapolating to lower return periods but there is great uncertainty at higher levels, this however is to be expected.

From the diagnostic plots we can reaffirm that, though there is no substantial difference to be seen, the dynamic version seems to be a superior fit for the data. The same evidence can be found in the other diagnostic plots for the other indices in the appendix.Even with there being deviations from linearity to be seen in the quantile plots we can conclude overall each model is a relatively good fit for each data set. 19

Chapter 5

Results and Back-testing

5.1 Examining Return levels

From chapter 3 we found that the generalized Pareto threshold model can be used to directly estimate the value-at-risk. By rearranging the tail behavior equation, we extract a return levels; which is merely a quantile estimate. The return level plot can be interpreted as a graph of the VaR against risk. Extrapolating the GPD models we can use the return level plot to make predictions about future losses.

FIGURE 5.1: Return level plot the FTSE 100 with static EVT 20 Chapter 5. Results and Back-testing

FIGURE 5.2: Return level plot the FTSE 100 with dynamic EVT

Having a found a positive shape parameter, we expected the return level plots to be upward sloping and have an infinite endpoint. This shows there is a positive relationship between the VaR and risk; the larger the return level the higher the VaR. Both graphs are fitted with L-moments to define the 100th year return level, from the graphs we can visually determine the VaR estimate for this period. L- moments can be set at any return level and are a useful tool in graphically determining a quantile visually. Between the two return level plots for the FTSE 100 with static EVT and Dy- namic EVT little difference can be seen. This suggests the dynamic model does not offer a great improvement. The graphs are fitted with 95% confidence intervals and all data points are contained within, this shows the models are a good fit. The confidence intervals start to widen after the 10th year return level; this shows greater uncertainty about the accuracy of predictions for future periods.

In appendix C it is possible to see the return level plots for the NIKKEI 225 and the S&P 500. Both return level plots are similar to that of the FTSE 100 however S&P 500 has a much wider conﬁdence interval at later return periods than the other models; showing greater uncertainty in the accuracy. To conclude, as conﬁdence intervals widen so greatly after the 10th year, the models offer a good prediction for the VaR estimate but should not be consider reliable after the 10 year VaR. 5.2. VaR estimates from the in sample period 21

5.1.1 Return level estimates for static and dynamic models

TABLE 5.1: Return level results for Static EVT models

Return level FTSE 100 NIKKEI 225 S&P 500 2-year 5.913746 7.163566 6.634479 5-year 7.829365 9.136956 9.278186 10-year 9.647980 10.912571 11.970411 20-year 11.861860 12.975615 15.454509 50-year 15.543547 16.224008 21.680375 100- year 19.038796 19.146843 28.020499

TABLE 5.2: Return level results for dynamic EVT models

Return level FTSE 100 NIKKEI 225 S& P 500 2-year 5.964225 7.19733 6.722017 5-year 7.888029 9.18796 9.428754 10-year 9.718104 10.99521 12.208740 20-year 11.949854 13.11143 15.833280 50-year 15.668915 16.47404 22.367083 100-year 19.206782 19.52690 29.077703

Visually little improvement could be seen in the return level plots, looking at the VaR estimates for each return period we can observe that there are improvements in the dynamic estimates. They are slightly larger suggest- ing that the static estimates maybe underestimating the risk. This can be consistently seen at each conﬁdence level in every index. From our literature review we expected that implementing a stochastic volatility model before EVT analysis would improve estimates as the dynamic model can react to changes in the volatility.

5.2 VaR estimates from the in sample period

From the in sample period September 4th, 1984 to the September 3rd, 2010 we have used our different models to calculate the VaR estimate for each stock index. The table below displays VaR results for the S&P 500, tables of results for the other indices can be found in the appendix A: 22 Chapter 5. Results and Back-testing

TABLE 5.3: S&P 500 VaR estimates

Conﬁdence VaR VaR VaR VaR level Method Estimate Method Estimate Dynamic Normal-based EVT 95% 1.917249 1.90397361 99% 2.723617 3.28849077 99.50% 3.018813 4.20316878 99.90% 3.627473 7.54575133 Historical CVaR 95% 1.743926 2.619764 99% 3.152683 4.415435 99.50% 4.260993 5.315873 99.90% 8.374196 8.105525 Static EVT GARCH-CVaR 95% 1.8324382 2.550339 99% 3.23165322 4.330895 99.50% 4.14547826 5.229933 99.90% 7.44218 7.972556

Unsurprisingly,the lowest VaR estimates come from the parametric model based on the normal distribution. We have already established that ﬁnan- cial distributions have fatter tails than the normal distribution; therefore we expected that this model would severely underestimate risk further in to the tails.

The historical approach has generated what appear to be reasonably good VaR estimates, they are in line with the EVT estimates which we considered to be the more sophisticated method. The large in-sample period has obvi- ously provided enough effective observations to generate a prudent result, however it is not always possible to obtain so much data for an accurate result. It is also possible that the in sample period is more volatile than future periods which means the historical VaR will be an overestimation of risk. In the estimates for the S&P 500 however, the historical VaR gives a much larger value than the EVT estimates above the 99% level. This suggests the historical VaR is over estimating the risk. The historical data used for the S&P 500 contained very large losses, the largest being -22.8997The˙ historical simulation model expects the past data to be same as future and therefore puts more emphasis on large past losses which may not be representative of the present. Historical simulation only focuses on the data past the con- ﬁdence interval putting more weight on extreme values.

Similar to what seen in the results for the return periods, dynamic EVT offers consistently larger results than the static model. In the case of the FTSE 100 and the NIKKEI 225, found in appendix A, they offer more conservative results than the historical or parametric approach.

The largest estimates come from CVaR, by deﬁnition the CVaR is the arithmetic mean of the VaRs of the whole tail, therefore these estimates seem reasonable. It is also interesting to see that while implemented with a SV 5.3. Back-Testing 23 model, the CVaR estimates are scaled lower than without. As discussed before, large estimates do not necessarily mean a better risk measure. In order to judge these estimates we must back-test to review their practicality.

5.3 Back-Testing

In order to interpret whether the VaR estimates are indeed credible we will back-test against an out of sample period. This period will run from the period September 4th, 2010 to the September 3rd, 2015. We will use two back tests in order to judge the accuracy and effectiveness: the ﬁrst will be a standard back-test based on proportion of failures test by Kupiec (1995); the second we have chosen to use is the duration based approach by Christof- fersen and Pelletier (2004). The ﬁrst is often referred to as the unconditional coverage test and the second as the conditional coverage test.

5.3.1 The Portion of Failures Unconditional Coverage test One of the simplest back tests available is comparing the VaR estimates against the losses in an out of sample period. If there is a loss in the sample period greater than the value at risk, we count this as a VaR violation. Too many VaR violations indicate an underestimation of risk and therefore a poor model. We do expect however a certain number of violations this is generally taken as N(1-p), for a sample size of alpha and a conﬁdence level of p. An example of this is: if we are back testing a 1-day 95% VaR estimated over a period of 100 days we would expect an accurate model to have around 5 VaR violations. Too few violations indicate an over estimation of risk.

TABLE 5.4: S& P 500 VaR Violations

VaR Violations: Conﬁdence Static Dynamic GARCH- Expected Normal based Historical CVaR Level EVT EVT CVaR 95% 71.35 48 57 54 50 20 22 99% 14.27 19 12 11 9 4 4 99.50% 7.135 13 4 4 4 1 1 99.90% 1.427 7 0 0 0 0 0

Both dynamic and static EVT prove rather reasonable models over all stock indices in the back test. Though they seem to overestimate the risk at the 95%, this can be seen by having far fewer VaR violations than the expected. They perform well above the 99% confidence level having actual violations close to the number expected. Historical simulation preformed the best at the 95% level out of all the models and preforms consistently well, like the EVT estimates, above the 99% level. However, using a smaller 24 Chapter 5. Results and Back-testing in sample period or less extreme period for modeling would greatly dam- age the historical simulation’s predictive power. From the traditional models, the normal parametric VaR model preforms the worst in the back-test in confidence levels above 99%. Again, this proves that the assumption of normality in financial returns is completely incorrect for the tails of the distribution. It is worth noting that the out of sample period used for back testing happened to contain less extreme losses compared to the in sample period used to generate the VaR results. The largest loss in the daily log returns for the S&P 500 was 22.8997, compared to the largest loss in the out of sample period which was only 6.8958. This less extreme period is not the most ideal for back-testing however it also reminds us that overestimation of the risk is also an important factor to consider. Financial institutions are known for being conservative in measuring risk however it is not practical to hold excess capital in case of extreme market conditions, predicted by ineffective models. This leads us to look at our final risk measure, both the dynamic and the static CVaR models have the least violations at each confidence level. The violations at the: 95%, 99% and 99.5% are much lower than the expected; only at the 99.9% confidence level does it seem to be accurate. From the back test we can determine that CVaR, in this case, is a better risk measure than VaR.

5.3.2 The Duration Based Conditional Coverage test This test investigates the time between VaR violations to check whether they are independent. This test does not focus on the amount of VaR violations but the frequency in which they occur to test whether or not they are clustered. This is important because: “Large losses that occur in rapid succession are more likely to lead to disastrous events such as bankruptcy” (Christoffersen & Pelletier, 2004). To apply this test, we use the rugarch package in R which can implement the VaR duration test of Christoffersen and Pelletier, codes for this can be found in appendix B. 5.3. Back-Testing 25

TABLE 5.5: S&P 500 Conditional Coverage Back-test

Conﬁdence VaR Reject/ VaR Reject/ level Method Fail to reject Method Fail to reject Dynamic Normal-based EVT 95% Reject Reject 99% Reject Reject Fail to 99.50% Reject reject Fail to Fail to 99.90% reject reject Historical CVaR 95% Reject Reject 99% Reject Reject Fail to Fail to 99.50% reject reject Fail to Fail to 99.90% reject reject GARCH Static EVT CVaR 95% Reject Reject 99% Reject Reject Fail to Fail to 99.50% reject reject Fail to Fail to 99.90% reject reject

We can see with the duration based test that almost all the models are rejected at lower confidence levels; only the CVaR models for the NIKKEI are not rejected at levels lower than 95%. The test fails to reject the EVT models for higher confidence levels, as the EVT models also performed better at higher levels in the unconditional coverage test; this is an indication that EVT models are better for calculating the Value at Risk but only at higher confidence levels. In the case of CVaR it appears that it performed very well in the conditional coverage back test. This risk measure by definition accounts the whole tail. From this test it shows that considering a coherent risk measure instead of VaR maybe a better solution. 26

Chapter 6

Conclusion

In this thesis, the aim was to first look at whether EVT offered better results for calculating the value at risk when compared to other methods. There have been many criticisms over the suitability of traditional models for calculating VaR: the assumption that data follows a normal distribution is not practical for financial data; the other main approach, historical simulation relies on the assumption that future returns will follow the same pattern as the past, which is not always the case. These approaches seem to be hin- dered by the inability to correctly determine the underlying distribution, therefore by using a method that could correctly determine the distribution of extremes seemed a promising start. The second aim was to identify whether implementing a SV model, in this case a GARCH model, aided VaR calculations in respect to volatility. From research we identified the need for models to be able to react to volatility. Extreme values in periods of low volatility are considered far more ‘extreme’ than extreme values in periods of high volatility. The final aim was to compare our VaR risk estimates with the estimates of a coherent risk measure CVaR.

In regards to methods of calculating VaR, EVT is a practical method. A convenient aspect of this model is it provides a direct method to estimate the VAR. By looking at the return level graph we can estimate the VaR for future periods; though we found great uncertainty in the estimates for periods above ten years. In order to determine whether these results were reasonable two back-tests were implemented. The portion of failure test was used to compare the amount of VaR violations against the expected number. In this test the EVT models narrowly out preformed the historical simulation approach, though we attribute this to the large amount of in-sample data used and that the out of sample period seemed to be less volatile than the in-sample period. For conﬁdence levels above 99% EVT had a similar number of VaR violations as the expected. Therefore, from this test we can conclude that EVT seems to outperform the other methods. The results for CVaR had far less violations compared to the expected number of violations leading to the conclusion that CVaR is a good alternative risk measure.

The second back-test used was the duration based test by Christoffersen and Pelletier, it examined the frequency of violations occurring. In this test EVT performed well but only at high confidence levels, similar to the other test. From this we can make the conclusion that EVT is a better approach for estimating VaR than traditional methods but only at higher confidence levels. The results of this test indicated that CVaR was a better risk measure as Chapter 6. Conclusion 27 it had the least rejections; therefore considering an alternative risk measure to VaR may be an appropriate solution for risk management. Again, we could contribute this to the fact this method examines the whole tail above the confidence level.For future analysis it would be worth examining CVaR estimated by an EVT approach.

Looking at the differences between the static and dynamic EVT estimates we can see that the dynamic version deﬁnitely offers a small improvement. In regards to CVaR, the SV model scaled down the estimates which we also consider an improvement. Therefore we can determine that dynamic risk measures offer a more accurate estimate than static risk measures. As the improvement was only small with a GARCH(1,1) model it would be worth investigating whether a more sophisticated SV model, such as an exponen- tial GARCH model, would give a larger improvement. 28

Chapter 7

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Appendix A

Tables of results

TABLE A.1: NIKKEI 225

Conﬁdence VaR Method VaR Estimate VaR Method VaR Estimate level Parametric Dynamic EVT 95% 2.43967006 2.398731087 99% 3.44948656 4.072751211 99.50% 3.81916021 5.005529959 99.90% 4.58138622 7.833321393 Historical CVaR 95% 2.37256956 3.315078 99% 4.03153497 5.350217 99.50% 5.06810609 6.473645 99.90% 7.19766682 10.02342 Static EVT GARCH-CVaR 95% 2.296917003 3.248083 99% 4.017019844 5.2855 99.50% 4.963718645 6.408653 99.90% 7.796052954 9.941546

TABLE A.2: FTSE 100

Conﬁdence VaR Method VaR Estimate VaR Method VaR Estimate level Parametric Dynamic EVT 95% 1.81240005 1.83990801 99% 2.57315367 3.18005862 99.50% 2.85165037 3.97680264 99.90% 3.42587965 6.56755624 Historical CVaR 95% 1.64680245 2.464263 99% 3.11488169 4.117955 99.50% 4.00433723 5.042463 99.90% 5.99451132 7.630797 Static EVT GARCH-CVaR 95% 1.78636494 2.410589 99% 3.1317173 4.064757 99.50% 3.92932379 4.987402 99.90% 6.51499645 7.562086 Appendix A. Tables of results 31

TABLE A.3: VaR Violations FTSE 100

VaR Violations: Conﬁdence Static Dynamic GARCH- Expected Parametric Historical CVaR Interval EVT EVT CVaR 95% 64.5 47 56 48 46 18 18 99% 12.9 15 6 6 6 3 3 99.50% 6.45 11 3 3 3 0 0 99.90% 1.29 6 0 0 0 0 0

TABLE A.4: VaR Violations for NIKKEI 225

VaR Violations: Conﬁdence Static Dynamic GARCH- Expected Parametric Historical CVaR Interval EVT EVT CVaR 95% 62.05 39 43 45 41 15 16 99% 12.41 13 9 9 8 4 4 99.50% 6.205 11 5 5 5 3 3 99.90% 1.241 6 2 1 1 1 1

. 32 Appendix A. Tables of results

TABLE A.5: FTSE 100

Conﬁdence VaR Reject/ VaR Reject/ level Method Fail to reject Method Fail to reject Dynamic Parametric EVT 95% Reject Reject Fail to 99% Reject reject Fail to Fail to 99.50% reject reject Fail to Fail to 99.90% reject reject Historical CVaR 95% Reject Reject Fail to 99% Reject reject Fail to Fail to 99.50% reject reject Fail to Fail to 99.90% reject reject GARCH Static EVT CVaR 95% Reject Reject Fail to 99% Reject reject Fail to Fail to 99.50% reject reject Fail to Fail to 99.90% reject reject Appendix A. Tables of results 33

TABLE A.6: NIKKEI 225

Conﬁdence VaR Reject/ VaR Reject/ level Method Fail to reject Method Fail to reject Dynamic Parametric EVT 95% Reject Reject Fail to Fail to 99% reject reject Fail to Fail to 99.50% reject reject Fail to 99.90% Reject reject Historical CVaR Fail to 95% Reject reject Fail to Fail to 99% reject reject Fail to Fail to 99.50% reject reject Fail to Fail to 99.90% reject reject GARCH Static EVT CVaR Fail to 95% Reject reject Fail to Fail to 99% reject reject Fail to Fail to 99.50% reject reject Fail to Fail to 99.90% reject reject 34

Appendix B

R code

Example of R code used for EVT calculations:

Data <- read.table("c:/Data.csv", header=TRUE, sep=",") attach(Data) length(Adj.Close)

#log returns F<- log(Adj.Close[2:6784])-log(Adj.Close[1:6783]) FTSE<-100*F

#Log return plot plot(FTSE, type=’l’, main=’Log returns of the FTSE 1000’, xlab=’Observations’, col="dark red")

#MRL mrl.plot(FTSE, conf=0.95 ) # Fititing threshold gpd.fitrange(FTSE,umin = 2,umax = 5,nint=100 ) Th=2.5 length(FTSE) length(FTSE[FTSE>Th])

# Fitting GDP TradingDays <- 252 FTSE.gpd<- gpd.fit(FTSE, Th, npy=TradingDays) gpd.diag(FTSE.gpd) # OR FTSE.mle <- fevd(FTSE, method = "MLE", type="GP", threshold=Th) FTSE.mle

#Profile log likelihood for shape parameter gpd.profxi(FTSE.gpd,0,0.8,nint=100)

#return level plot rl <- return.level(FTSE.mle, conf = 0.05, return.period= c(2,5,10,20,50,100)) Appendix B. R code 35

FTSE.lmom <- fevd(FTSE, method = "Lmoments", type="GP", threshold=Th) plot(FTSE.lmom) FTSE2.lmom <- return.level(FTSE.lmom, conf = 0.05, return.period= c(2,5,10,20,50,100))

################ # return level plot w/ MLE plot(FTSE.mle, type="rl", main="Return Level Plot for FTSE 100 with static EVT", ylim=c(0,200), pch=16) loc <- as.numeric(return.level(FTSE.lmom, conf = 0.05, return.period=100)) segments(100, 0, 100, loc, col= ’midnightblue’,lty=6) segments(0.01,loc,100, loc, col=’midnightblue’, lty=6)

# Assessing for staionarity time<-matrix(1:6783,ncol=1) FTSE.gpd2<-gpd.fit(FTSE, threshold = Th, npy = 252, ydat=time, sigl=1, siglink=exp) gpd.diag(FTSE.gpd2)

######################################### ##### DYNAMIC EVT##############

# Fitting GARCH(1,1) model spec <- ugarchspec(variance.model = list(model = "sGARCH", garchOrder = c(1, 1), submodel = NULL, external.regressors = NULL, variance.targeting = FALSE), mean.model = list(armaOrder = c(1, 1), external.regressors = NULL, distribution.model = "norm", start.pars = list(), fixed.pars = list())) garch <- ugarchfit(spec=spec,data=FTSE,solver.control=list(trace=0)) FTSE.dy<-garch@fit$residuals length(FTSE.dy)

# MRL plot for dynamic data mrl.plot(FTSE.dy, conf=0.95 )

# Fitting threshold gpd.fitrange(FTSE.dy, 2,5, nint=100)

Th <- 2.5 length(FTSE.dy[FTSE.dy>Th])

# Fit GPD FTSE.dy.gdp <- fevd(FTSE.dy, method = "MLE", type="GP", threshold=Th) FTSE.dy.gdp

## OR from ismev 36 Appendix B. R code

FTSEgdp<- gpd.fit(FTSE.dy, Th, npy=252) gpd.diag(FTSEgdp)

# Profile log liklihood of shape parameter gpd.profxi(FTSEgdp,-0.01,0.75,nint=100)

## return level plot FTSE.rl <- return.level(FTSE.dy.gdp, conf = 0.05, return.period= c(2,5,10,20,50,100)) FTSE.rl FTSE.lmom.dy <- fevd(FTSE.dy, method = "Lmoments", type="GP", threshold=Th) plot(NIKKI.lmom.dy) FTSE2.lmom.dy <- return.level(NIKKI.lmom.dy, conf = 0.05, return.period= c(2,5,10,20,50,100)) plot(FTSE.dy.gdp, type="rl", main="Return Level Plot for FTSE 100 with dynamic EVT", ylim=c(0,200), pch=16) loc <- as.numeric(return.level(FTSE.lmom.dy, conf = 0.05, return.period=100)) segments(100, 0, 100, loc, col= ’midnightblue’,lty=6) segments(0.01,loc,100, loc, col=’midnightblue’, lty=6)

#### Assessing stationarity time<-matrix(1:6783,ncol=1) FTSE.gpd2.dy<-gpd.fit(FTSE.dy, threshold = Th, npy = 252, ydat=time, sigl=1, siglink=exp) gpd.diag(FTSE.gpd2.dy)

### Tail VaR

CVaR( x = FTSE, alpha = 0.05) CVaR( x = FTSE, alpha = 0.01) CVaR( x = FTSE, alpha = 0.005) CVaR( x = FTSE, alpha = 0.001)

#### GARCH Tail VaR

CVaR( x = FTSE.dy, alpha = 0.05) CVaR( x = FTSE.dy, alpha = 0.01) CVaR( x = FTSE.dy, alpha = 0.005) CVaR( x = FTSE.dy, alpha = 0.001)

#### Conditional coverage backtest VaRDurTest(0.05,actual = data, VaR = ...) VaRDurTest(0.01,actual = data, VaR= ....) VaRDurTest(0.005,actual = data, VaR= ...) VaRDurTest(0.001,actual = data, VaR= ...) 37

Appendix C

Plots

(A) Mean residual life plot with (B) Mean residual life plot with static EVT for the S&P 500 dynamic EVT for the S&P 500

(A) Mean residual life plot with (B) Mean residual life plot with Static EVT for the FTSE 100 dynamic EVT for the FTSE 100

FIGURE C.3: Fitting the GPD over a range of thresholds for the S&P 500 3 2 1 −1 Modified Scale −3

2.0 2.5 3.0 3.5 4.0 4.5 5.0 Threshold 0.6 Shape 0.2 −0.2 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Threshold 38 Appendix C. Plots

FIGURE C.4: Fitting the GPD over a range of thresholds for the FTSE 100 4 2 0 −2 Modified Scale −4

2.0 2.5 3.0 3.5 4.0 4.5 5.0 Threshold 1.0 0.5 Shape 0.0

2.0 2.5 3.0 3.5 4.0 4.5 5.0 Threshold

FIGURE C.5: Fitting the GPD over a range of thresholds for the NIKKEI 225 3 2 1 0 −2 Modified Scale −4

2.0 2.5 3.0 3.5 4.0 4.5 5.0 Threshold 1.0 0.6 Shape 0.2

−0.2 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Threshold Appendix C. Plots 39

FIGURE C.6: Diagnostic plots for S&P 500

Probability Plot Quantile Plot 1.0 20 0.8 15 0.6 Model Empirical 0.4 10 0.2 5 0.0 0.0 0.2 0.4 0.6 0.8 1.0 2 4 6 8 10 12 14 Empirical Model

Return Level Plot Density Plot 1.2 300 1.0 0.8 200 0.6 f(x) Return level 0.4 100 50 0.2 0 0.0 1e−01 1e+00 1e+01 1e+02 1e+03 5 10 15 20 Return period (years) x

FIGURE C.7: Diagnostic plots for FTSE 100

Probability Plot Quantile Plot 1.0 12 0.8 10 0.6 8 Model Empirical 0.4 6 0.2 4 0.0 0.0 0.2 0.4 0.6 0.8 1.0 4 6 8 10 12 Empirical Model

Return Level Plot Density Plot 1.2 1.0 100 0.8 80 0.6 f(x) 60 Return level 0.4 40 0.2 20 0 0.0 1e−01 1e+00 1e+01 1e+02 1e+03 2 4 6 8 10 12 14 Return period (years) x 40 Appendix C. Plots

FIGURE C.8: Return levels plot for the S&P 500 and NIKKEI 225

Return Level Plot for S&P 500 with static EVT Return Level Plot for S&P 500 with dynamic EVT 200 200 150 150 100 100 Return Level Return Level 50 50 0 0

2 5 10 20 50 100 200 500 2 5 10 20 50 100 200 500 Return Period (years) Return Period (years)

Return Level Plot for NIKKEI 225 with static EVT Return Level Plot for NIKKEI 225 with dynamic EVT 200 200 150 150 100 100 Return Level Return Level 50 50 0 0

2 5 10 20 50 100 200 500 2 5 10 20 50 100 200 500 Return Period (years) Return Period (years)