Quantitative risk analysis an approach for vietnam stock market

Three unconditional volatility models including historical, normal andStudent’s - t as well as EWMA and two volatility models including GARCH,GJR - GARCH with three return distributions

UNIVERSITY OF ECONOMICS ERASMUS UNVERSITY ROTTERDAM

HO CHI MINH CITY INSTITUTE OF SOCIAL STUDIES

VIETNAM THE NETHERLANDS

VIETNAM – THE NETHERLANDS PROGRAMME FOR M.A IN DEVELOPMENT ECONOMICS

QUANTITATIVE RISK ANALYSIS:

AN APPROACH FOR VIETNAM

STOCK MARKET

BY

NGUYEN NAM KHANH

MASTER OF ARTS IN DEVELOPMENT ECONOMICS

VIETNAM – NETHERLANDS PROGRAM FOR M.A IN DEVELOPMENT ECONOMICS

QUANTITATIVE RISK ANALYSIS:

AN APPROACH FOR VIETNAM STOCK MARKET

A thesis submitted in partial fulfillment of the requirements for the degree of

MASTER OF ARTS IN DEVELOPMENT ECONOMICS

QUANTITATIVE RISK ANALYSIS:

AN APPROACH FOR VIETNAM

STOCK MARKET

Nguyen Nam Khanh January 15, 2016

AbstractValue at Risk (VaR) is widely used in risk measurement It is de…ned asthe worst expected loss of a portfolio under a given time horizon at a givencon…dence level The aim of the thesis is to evaluate performance of 16VaR models in forecasting one - day ahead VaR for daily return of VNIN-DEX and a group 8 banking stock indexes including ACB, BVH, CTG, EIB,MBB, SHB, STB, VCB to …nd out the most appropriate model for each stockindex Three unconditional volatility models including historical, normal andStudent’s - t as well as EWMA and two volatility models including GARCH,GJR - GARCH with three return distributions normal, Student’s - t andskewed Student’s - t and associated Extreme Value Theory (EVT) modelsare performed at 5%, 2.5% and 1% of signi…cance level Violation ration,Kupiec’s unconditional coverage test, independence test and Christo¤ersenconditional coverage test are used to backtested performance of all models.Besides statistical analysis, graphical analysis is also incorporated Backtest-ing indicates that there is no best model for all cases because of character-istic di¤erence from particular stock index Implication of this thesis is that

a suitable VaR forecasting model is only chosen after backtesting frequentlyperformance of various models in order to ensure that most relevant and mostaccurate models are suited for current …nancial market situation

Keywords: Value at Risk, Extreme Value Theory, …nancial risk ment, conditional volatility model, backtesting, stock index

1.1 Problem statements 7

1.2 Research objectives 8

1.3 Research questions 9

1.4 Subject and scope of research 9

1.5 Structure of the thesis 9

2 Literature review 11 2.1 De…nitions 11

2.1.1 Financial return data 11

2.1.2 Concept of Risk 12

2.1.3 Classi…cation of Risk 13

2.1.4 Risk measurement and Coherence 13

2.2 Theoretical review 14

2.2.1 Value at Risk 14

2.2.2 GARCH 17

2.2.3 Extreme Value Theory 17

2.3 Empirical studies review 18

2.3.1 Empirical research on modeling and measuring VaR 18

2.3.2 Empirical research on Extreme Value Theory (EVT) VaR 20

3 Research Methodology 22 3.1 Data selection 22

3.2 Methodology 22

3.2.1 Unconditional VaR models 23

3.2.2 Conditional VaR models - Volatility model using EWMA, GARCH, GJR - GARCH model 26

3.2.3 Extreme value theory (EVT) distribution in VaR mod-eling 30

3.3 Backtesting Methodology 35

3.3.1 Kupiec’s Test 37

3.3.2 Christo¤ersen’s Tests 39

3.3.3 Hypothesis testing procedure 40

4 Empirical Results 41 4.1 Descriptive statistics 41

4.2 GARCH, GJR - GARCH and EVT model estimation 49

4.3 Models forecasting performance analysis 56

4.4 Graphical analysis of model forecasting 72

5 Conclusion 76 5.1 Main …ndings 76

5.2 Implications 78

5.3 Limitation and further studies 79

List of Tables

4.1 Descriptive of data sample 434.2 Descriptive statistics of daily stock index returns 444.3 Parameters estimation of GARCH(1,1) model with normal dis-tributed innovation for daily stock index returns 504.4 Parameters estimation of GARCH(1,1) model with Student’s

- t distributed innovation for daily stock index returns 504.5 Parameters estimation of GARCH(1,1) model with skewedStudent’s - t distributed innovation for daily stock index re-turns 514.6 Parameters estimation of GJR - GARCH(1,1) model with nor-mal distributed innovation for daily stock index returns 524.7 Parameters estimation of GRJ - GARCH(1,1) model with Stu-dent’s - t distributed innovation for daily stock index returns 524.8 Parameters estimation of GRJ - GARCH(1,1) model with skewedStudent’s - t distributed innovation for daily stock index re-turns 534.9 Parameters estimation of generalized Pareto distribution (GPD),threshold exceedances of 5 percentage from GARCH(1,1) model54

4.10 Parameters estimation of generalized Pareto distribution (GDP),threshold exceedances of 5 percentage from GJR - GARCH(1,1) model 554.11 Expected and actual number of VaR violations at threshold 5percentage 564.12 Violation ratio and Kupiec’s test p - value at 5 percent signif-icance level 574.13 Independence test and Christo¤ersen’s test at 5 percent sig-ni…cance level 634.14 Expected and actual number of VaR violations at threshold2.5 percentage 64

4.15 Violation ratio and Kupiec’s test p - value at 2.5 percent ni…cance level 654.16 Independence test and Christo¤ersen’s test at 2.5 pecent sig-ni…cance level 664.17 Expected and actual number of VaR violations at threshold 1percentage 674.18 Violation ratio and Kupiec’s test p - value at 1 percent signif-icance level 684.19 Independence test and Christo¤ersen’s test at 1 pecent signif-icance level 694.20 Best forecasting VaR model according to Christo¤ersen’s test

sig-at 5, 2.5 and 1 percentage of signi…cance level 71

List of Figures

4.1 Daily value of stock index 42

4.2 Daily return of stock index 45

4.3 Histograms of daily stock index returns 46

4.4 Qnorm - QQ plot of daily stock index returns 46

4.5 ACF for daily stock index returns 47

4.6 PACF for daily stock index returns 47

4.7 ACF for squared of daily stock index returns 48

4.8 PACF for squared of daily stock index returns 48

4.9 EWMA and unconditional VaR models forecasting performance for daily return of EIB at 5% signi…cance level 72

4.10 GARCH VaR model forecasting performance for daily return of ACB at 5% signi…cance level 73

4.11 GJR - GARCH VaR model forecasting performance for daily return of MBB at 5% signi…cance level 74

4.12 EVT GARCH VaR model forecasting performance for daily return of CTG at 5% signi…cance level 74

4.13 EVT GJR - GARCH VaR model forecasting performance for daily return of BVH at 5% signi…cance level 75

I would like to send special thanks to my academic supervisor Dr TruongDang Thuy, for his patience guidance, enthusiasm and support during mythesis writing process

I would also like to thank Dr Pham Khanh Nam who also gave mevaluable advices for my thesis A special thank goes out to all lecturers,sta¤s of the Vietnam - Netherlands Program as well as my classmates for alltheir helps and supports

I am most grateful to my family Thank you for always being there for

me, thank you for inspiring me, supporting me and making me appreciatethe value of education

Last but not least, I would like to thank my wife and my daughter Thankyou for your patience, deep understanding and encouragement I am grateful

to you

to …nancial market is a weakness of risk management system Therefore, achallenge has been raised is how to identify and measure the risk in order tominimize the loss as well as ensure the safe environment for …nancial marketand economy system.

In modern risk management, it is not su¢ cient if only simply focus onquality policy Risk is actually the expected loss of outcome in future, so it isoften measured by probability distribution One of the important stages in

…nancial risk management process is build up models to measure and evaluatethe risks However, a di¢ cult process might be raised when applying theminto actual condition of market because every model is associated with somede…ned assumptions, hypotheses and sometimes these assumptions are notsatis…ed in particular conditions of market Therefore, some new approachesshould be studied in these models in order to choose and apply the best onewith actual conditions in various markets

Financial risk management in the world has attained some improvementprocess when changing the mind set from passive to active in risk manage-

ment and applied these risk measure methods into business process tion, allocation assets, portfolios management with e¢ ciency result.

evalua-Vietnam stock market born in July 2000 is an important step to indicate

an improvement in the country’s economy Vietnam stock market is tive young when compared to others development stock market in the worldand then it has attained new opportunities as well as faced with many newchallenges In recent years, although Vietnam stock market has many ‡uc-tuations but it is still an attractive environment for many foreign investors

rela-as well rela-as local ones All investors, de…nitely, would like their investmentsproduce a highest pro…t with lowest risk and they are also two main factorsthat in‡uence all their business activities According to risk management inVietnam, …nancial market in general and Vietnam stock market in particu-lar, it is actual limited in term of both policy and tool Therefore, system

of …nancial risk management should be studied and built up in active ande¤ective way

One of the most known risk measurement applied in risk management

is Value at Risk (VaR) and it becomes a popular risk management toolfor …nancial regulations and …nancial institutions to evaluate possible lossesthat they can incur The VaR estimation was required by Basel Committee

on banking supervision to meet the capital required for covering potentiallosses and VaR …gures considered as additional information to shareholdershave disclosed by many of …nancial institutions VaR can answer well aquestion what is a maximum …nancial amount possible to loose with giventime horizon under given con…dence level or signi…cance level An overview

of VaR is reviewed by Du¢ e and Pan (1997)

Many methodologies are using to estimate VaR only based on simpleassumption that all …nancial returns follows Gaussian normal distribution.However, estimating VaR by using normal distribution for each asset hasraised an inaccuracy result because non - normality of …nancial returns.Therefore, various advanced VaR measurement techniques are used to es-timate VaR of daily returns of stock index and then the performance of thesemodels are evaluated in order to …nd out the best ones which could be used

by …nancial institutions to manage market risk

1.2 Research objectives

1 To suggest suitable risk measurement VaR models for portfolio in nam stock market

1.4 Subject and scope of research

This thesis studies risk measurement VaR with various models as well asmethodologies in advanced and applies these models into risk measure forVietnam stock market

The purpose of this thesis is to identify a best appropriate VaR methodincluding unconditional VaR models such as Historical simulation, normal-ity VaR, Student’s - t VaR, skewed Student’s - t VaR and conditional VaRmodels where volatility is forecasted by using Exponential Weighted MovingAverage (EWMA), Generalized Autoregressive Conditional Heteroscedastic(GARCH) and GJR - GARCH in risk measurement through measuring po-tential losses of daily return of VNINDEX as well a group of 8 banking stockindexes with di¤erent time period for each stock index The longest time pe-riod is in VNINDEX which is studied from year 2002 to the end of November

2015 Stock indexes historical return data is assumed to provide su¢ cientinformation for model evaluation and predicting one - day return forecastsunder 95%, 97.5% and 99% con…dence level In order to evaluate quality offorecasting, some backtesting models including violation ratio, Kupiec’s test,independence test and Christo¤ersen’s test are performed

Findings of this study might be useful for …nancial institutions or …nancialregulatory, particularly risk managers in Vietnam stock markets

All data used in analysis are calculated by programming and statisticalsoftware called R

1.5 Structure of the thesis

Thesis contains 6 chapters First chapter introduces concept of risk and riskmeasurement through some tools such as Value at Risk (VaR), Extreme ValueTheory (EVT) Second chapter summarizes literature review and empirical

researches on VaR, EVT modeling Third chapter presents data and ology of VaR estimation as well as performance comparison through severalbacktesting models Fourth chapter discusses empirical results, analysis andbacktesting Finally, conclusions and implications as well as further studiesare last part of this thesis.

method-Chapter 2

Literature review

This literature review includes 3 parts The …rst part describes some itions of concepts used in this thesis The second part is theoretical reviewcontaining some models used to study risk management And the last partmentions empirical studies

de…n-2.1 De…nitions

Because stock indexes are mostly not stationary and often integrated at order

1 so it is modeled to changes of prices or log - return series of prices Daily

…nancial return data have some characteristics which are known as stylizedfacts According to McNeil et al (2005), these properties can be extended

to scope of time interval including shorter such as intra - day and longer such

to a fat - tails or leptokurtic Compared to normal distribution,

leptokur-tic distribution has an excess kurtosis indicating that the tail is fatter thanpredicted by the normal distribution.

Risk can be understood as unexpected outcome which might be happened

in future In …nance, risk is a di¤erence between return which is achievedfrom an investment and expected outcome or the volatility of unexpectedoutcomes which can represent the value of equity, assets, or earnings Risk

is uncertainty outcome and often developed by probability distribution cording to Basel Accords de…nition, …nancial risk can be divided into threetypes including credit risk, operational risk and market risk Liquidity riskcould be considered as an additional category if necessary

Ac-Credit risk, or default risk is de…ned as risk of loss due to payment default

of borrower including concentration risk, consumer credit risk, securitizationand credit derivatives Credit risk has been less researched due to limit dataavailable which mainly only belongs to large rating agencies But in recentyears, it has become attractive due to the failure of several large …nancialinstitutions in the U.S., for example Merry Lynch, Lehman Brothers (2008).Operational risk indicates the risk of internal processes failure, systemsand people Fraud, legal and political risk are examples of operational risk.Market risk can be understood as a changes the prices of …nancial as-sets such as stock prices, exchange rates, interest rates and commodity risk.Interest rate risk, currency risk, volatility risk, equity risk are included inmarket risk Because interest rates and equity prices are available widelyand high quality, so market risk can be understood as …nancial risk studieswhich is highly concentrated in this thesis

Following to development of information technology, …nancial productsbecome more and more sophisticated and …nancial markets around the worldsbecome more integrated, so understand well …nancial risk becomes moreimportant More and more researches on various features of …nancial serieshave been studied

In this thesis, market risk mentioning the uncertainty of pro…ts or lossescausing of the changes in market condition will be studied because only thistype have enough data

Firms disclose various types of risks but in general, it can be classi…edinto 2 types including systematic and unsystematic risks

2.1.3 Classi…cation of Risk

Systematic risk

Systematic risk is risk e¤ecting to all or almost stocks Unstable of economyenvironment such as interest rate movement, volatile exchange rates and highin‡ation are elements of systematic risk

One of the key element should be shown up is market risk Market riskhappened due to reaction from investors at phenomenon happens in market.Unsystematic risk

Unsystematic risk is risk e¤ecting to one asset or a group of assets or it onlyimpacts to a speci…c security Unsystematic risk includes business risk and

…nancial risk

Business risk includes business decisions and business environment ness decision includes corporation structure choices, investment decision,product development choices and marketing strategies implementation Busi-ness environment includes competition and macroeconomic risks

Busi-Financial risk is possible losses of some or all of the original investments.For example, losses can happen caused of interest rate movement or volatileexchange rate

Therefore, all investors can face many types of risk when they invest instock market and this is the most important element where they concentrate

on However, in this research, only …nancial risk is studied especially at riskmodels to measure and assess the stock price return

In modern …nancial risk management, only rely on qualitative methods arenot enough and not e¢ ciency, the more important thing is build up anddevelopment methods which can quantify the level of risk and …nancial losses.And based on this, …nancial institutions, …nancial regulators and investorshave a reliable source to determine decision making Loss of a …nancialposition can be represented by a random variable and then all characteristiclosses of …nancial position are expressed through distribution of loss randomvariables Because the loss distribution is unknown and hard to estimatethen some summary statistics are employed to quantify the distributions loss

in reality and a risk measure is one of these ones Risk measure provides apotential risk estimation and di¤erent chosen risk measure leads to di¤erenta¤ects to quality of predicting the losses of …nancial position so the choice

of a suitable risk measure becomes a crucial task towards building a realisticpicture of risk.

Coherence

Risk measure is a tool to estimate the potential loss of …nancial position

so it should be consistent with the basis theory in …nance called coherence.Let be a risk measure and is a coherence if four following conditions aresatis…ed for any two loss random variables X and Y (Artzner et al., 1999):

1 Subadditive: (X + Y ) (X) + (Y )

2 Monotonicity: If X Y then (X) (Y )

3 Positive homogeneity: (cX) = c (X), for any positive constant c

4 Transition invariance: (X + c) = (X) + c, for any positive constantc:

The subadditive property indicates that a combine of two positions isless risky than separate them individually This property relating to diver-si…cation mentions that a diversi…ed portfolio should not be greater thanindividual components in level of risk The monotonicity mentions that alower loss asset will generate a lower risk measure The positive homogeneityexpresses that doubling an asset should lead to double its risk The tran-sition invariance property shows that if one additional risk is added, it willgenerate more risky and adding one more constant to a random variable leads

to unchanged in its variability which is one of statistics properties

2.2 Theoretical review

Financial institutions can be lost billions of dollars due to a poor …nancialrisk management which are experienced in …nance crisis period In order tohave a quantitative …gures of risk, Value at Risk (VaR) was developed andnow it becomes a popular tool in risk management because this approachsummaries overall market risk through a single quantity, easy to understandand does not depend on a speci…c kind of distribution VaR can be used by

…nancial institutions to measure their risks as well as by a regulatory to setuprequirement VaR summarizes the worst expected loss of assets or portfolioover a target holding period with a given level of con…dence in normal marketcondition (Jorion, 1997, 2007) It could be estimated through the predictivedistribution in econometric modeling of the loss random variable

Let Vt; Vt+l be the value of a …nancial position at time t and t + l,respectively Let Lt(l) is the loss random variable of …nancial position for

the next l periods from the time index t and the cumulative distributionfunction of Lt(l) is denoted by Fl(xt)or Fl(x) For short expression, the timeindex t will be drop but it is understood that Fl(x) is a function depending

on the time index t Then, Lt(l) is either a negative or positive function of

Vt+l Vt

Because big loss is less frequently happened so small probability denoted

by p is used, for example, 5% or 1% or 0.1% to assess the loss After that,with a given time horizon l under probability p, VaR of the …nancial position

is de…ned as

V aR1 p= inffxjFl(x) 1 pg (2.1)From the de…nition, Fl(V aR1 p) 1 p is absolutely satis…ed, whichsays

P r(Lt(l) V aR1 p) 1 por P r(Lt(l) > V aR1 p) p (2.2)indicating with the probability 1 p, the potential loss of …nancial positionover the time period from t to t + l is less than or equal to V aR1 p or theprobability that the potential loss of …nancial position greater than V aR1 p

over the same time period is at most p

If VaR is a continuous loss random variable then it can be shown as

Given a probability density function of standardize return, VaR can becalculated by combination of volatilities and residual of distribution functionas:

where ^t is the conditional standard deviation at time t 1( ) is thequantile of a standardized normal variable, such at normal distribution, Stu-dent’s - t distribution, skew Student’s - t distribution or any assumed dis-tributions VaR is a coherent risk measure if the loss random variables arenormal distribution

Volatility, time horizon and con…dence level or signi…cance level are tors determining VaR for portfolio or a certain asset The volatility is esti-mated through statistical models Depend on speci…c …nancial activity type,

fac-such as measured in one - day ahead VaR, one - year ahead VaR, time horizon

is chosen and a¤ects to volatility measure and then also a¤ect to VaR, where

a longer time period leads to a higher volatility and …nally, a higher VaR.Con…dence level chosen represents how often a loss on portfolio or speci…casset greater than VaR It can be set lower at 95% for short data periodand in case of very conservative approach in risk management; con…dencelevel can be set high as 99.9% or even 99.99% 95% and 99% are con…denceintervals which are commonly used in empirical studies (Danielson and deVries, 1997)

The oldest de…nition of VaR might be seen from the portfolio optimizationtheory by Markowitz (1952) However, it had become unsuitable measure-ment after stock market crashed in 1987 called Black Monday due to simpleassumptions in this methodology which does not appropriate to actual situa-tions A new suitable methodology should be studied and then well - knownRiskMetrics were developed and published by J.P Morgan in 1994, VaR hasbecome a popular measurement in risk management Research on VaR wasstrongly supported when Basel II Accord (BIS, 2006) (Basel Committee onBanking Supervision) clearly expressed that VaR is a preferred measurementfor market risk Since then, more and more studies on VaR have been devel-oped to improve the quality of risk management through providing a betterpredicted measure in future loss

The method to calculation VaR could be split into two groups under metric and nonparametric approaches In this study, EVT (Extreme ValueTheory) is a parametric approach focusing on tail distribution where rareevent existed Parametric methodology includes GARCH, Equally weightedmoving average (EqWMA), Exponential Weighted Moving Average (EWMA)

para-In the other hand, the Historical simulation belongs to nonparametric ology

method-The choice of risk measure is crucial step in order to build realistic …gures

of risk However, there is another essential element comes from …nancial turns which impacts to accuracy of risk …gures Numerous empirical studiespointed out that asset return exhibit volatility clustering, fat - tails and skew-ness; therefore these phenomena should be accounted for probabilistic model.Before EVT, lot of methodology researches in whole distribution mentioningfor entire sample of return of assets (McNeil & Frey, 2000) In this study, 16di¤erent models will be compared together in order to …nd out which model

re-is the best They are the variance - covariance with normal dre-istributionand Student’s t distribution, historical simulation, EMWA, GARCH, GJR

- GARCH and EVT combining with three distributed innovations normal,Studen’s - t and skewed Student’s - t; and then residuals extracted fromvolatility models GARCH, GJR - GARCH are modeled by Peak - Over -

Threshold model from Extreme Value Theory which only concentrates onthe tail.

In reality, …nancial series always volatile and in order to predict the ity of these return time series, several forecasting volatility models wereintroduced such as ARCH (Engle, 1982), a well - known models GARCH(Bollerslev, 1986) which is now widely applied in forecasting the volatility

volatil-of …nancial return time series In order to capture volatility clustering ture in …nancial time series, several alternative modes have been developed.For example, GARCH(1,1) is considered as a successful approach to accountcertain features of …nancial returns such as volatility clustering and excesskurtosis (Hansen & Lunde, 2005) However, GARCH is a model which isoften used to predict in short term and fail to capture asymmetric behav-ior (Baillie, Bollerslev & Mikkelsen, 1996; Davidson, 2004; Ding & Granger,1996) Therefore, several advanced models applied to particular condition

fea-of market have been produced such as Asymmetric Power AutoregressiveConditional Heteroscedastic (APARCH) (Ding, Granger and Engle, 1993),EGARCH (Nelson, 1991), GJR - GARCH (Glosten et al, 1993)

VaR is a well - known parametric methodology in risk measurement butcombining with a simple normal distribution assumption will raise an inac-curacy result According to this approximation, all risk measurement results

of high quantiles are underestimated, especially in fat - tails …nancial serieshappened frequently in empirical studies mentioned by Mandelbrot (1963)and Fama (1965) In order to cover this limitation, some studies have usedmore appropriate fat - tails distributions such as Student - t distribution ornormal distribution mixture but VaR actually only concentrate on central ofobservation which is also mean that they are studied under normal marketconditions In another side, nonparametric methods based on no assumption

of speci…c empirical distribution have been also given to pass this problem,they, however, are still faced with some limitation For example, non - para-metric method produces a problem on assuming all observations having thesame weight as well as it cannot be applied in out - of - sample quantiles

In contrast of forcing the entire return series in VaR, Extreme ValueTheory (EVT) only focus on tail areas of distributions where extreme or rareevents are taken into account Extreme value theory (EVT) is a useful tool

to support risk measurement because it could take over a better approach

to …t extreme rare events which are limited by mentioned simple assumptionbefore In a di¤erent way compared to VaR, EVT has no assumptions aboutthe original distribution of all the empirical observations It is a powerfuland robust framework in study of the tail behavior of distribution, especially

in fat - tails and it can be used to handle for a very high quantiles in dicting an extreme loss or crashes situations Although EVT is a popularmethodology which has been applied in climatology and hydrology long timeago, it has been only introduced comprehensive in …nance and insurance inrecent year by Embrechts et al (1997) Since its introduction to …nance,there are a signi…cant number of …nancial studies relating to extreme valueshave discovered in recent years and a comparison between their results withother VaR models are reviewed De Haan, Jansen, Koedijk, and de Vries(1994) gave the quantile estimation using Extreme Value Theory McNeil(1997, 1998) used the estimation of quantile risk measures and the tail ofloss by applying Extreme Value Theory in …nancial time series studies Em-brecht et al (1998) discussed a risk management tool through the ExtremeValue Theory McNeil (1999) provided an overview of Extreme Value Theoryfor risk manager McNeil and Frey (2000) estimated a risk measurement forheteroscedasticity at tail of …nancial time series

pre-EVT approach is de…nitely suitable for extreme quantiles than tional approach in heavy tail data In principle, there are two most well -known of extreme value models: block maxima model (BMM) and Peaks -Over - Threshold (POT) model The …rst one, BMM, requires a large ofobservations in sample with identically and independently distributed (i.i.d.)losses assumptions The second one, POT, is a more recent and modernmodel focusing on all return losses which exceeds some de…ned high thresh-old In practical applications, POT model is the most useful and often useddue to its simple assumption and e¢ cient use of return data on extreme val-ues which are very often limited (McNeil, Frey & Embrechts, 2005) In thisstudy, only POT model is used to support EVT in risk measurement

conven-2.3 Empirical studies review

VaR

Since Basel II Accord (BIS, 2006) mentioned that VaR is preferred sure for market risk, VaR model have been exploded in research and thenmany methodologies were built with advantages and disadvantages as well

mea-as many advanced approaches were studied in order to improve quality of

predicting of di¤erent VaR models There are some advanced approaches,for example, Extreme Value Theory (EVT) or family of ARCH models such

as the autoregressive conditional heteroscedastic (ARCH) model of Engle(1982), the generalized autoregressive conditional heteroscedastic (GARCH)model of Bollerslev (1986), the exponential generalized autoregressive condi-tional heteroscedastic (EGARCH) model of Nelson (1991), the asymmetricpower autoregressive conditional heteroscedastic (APARCH) model of Ding,Granger and Engle (1993), the GJR - GARCH model of Glosten et al (1993)were used to study on various assets However, these studies provided dif-ferent results indicating that no existing best model for predicting, so it isvery useful contribution to test various method in particular asset using mostrecent data in market including recent …nancial crisis

Beder (1995) is considered as one of the …rst author studying on suring VaR accuracy of di¤erent assets including bonds, stock indexes andoptions as well as the mix of these ones by using two classical VaR modelswhich are Historical simulation (HS) and Monte Carlo simulation Authormentioned that VaR measures are very impacted by data, di¤erent parame-ters, assumptions and methodologies as well as provided 14 di¤erent VaRestimations in this study even though this paper only used two models Sothere are several shortcomings from these approaches The main advantage

mea-of Historical simulation methods is no parameters requirement However,one of the weakness of Historical simulation method is equally weighted as-sumption applied to all returns showing that very old returns have as sameimpact as recent ones to VaR estimation in case of long time period of datataken This weakness is mentioned in study of Boudoukh, Richardson andWhitelaw (1998) and they also provided suggestion about weight return fol-lowing to time from their sample to VaR estimation Hull & White (1998)also addressed the weight returns on volatility in order to account the recentvolatility changes

Distribution of …nancial returns is one of an important element impactingstrongly to VaR performance Using normal distribution in VaR estimationmight lead to an inaccuracy result because the …nancial returns are often not

…t with the normal distribution Mandelbrot (1963) addressed that …nancialreturns are not normal distribution and otherwise, distributions usually fol-low to a fat - tails or leptokurtic Bormetti et al (2007) presented a non

- Gaussian approach to measure market risk as well as discussed both itsadvantages and limitations In this study, author compared new approach toother well - known approaches in literature including normal VaR, Historicalsimulation and Monte Carlo simulation that are often used in …nancial analy-sis In order to capture the excess kurtosis, authors used …tted Student’s - tdistribution known as a better modeling for fat - tails characteristic of …nan-

cial returns Research investigated Italian stock market by using 1000 dailyreturn of two stocks and two indexes as well as con…rmed leptokurtic behav-ior of distribution returns because tail parameters fell in the range (2.9, 3.5).This study mentioned that at 95% con…dence level, the performance result ofthe Student’- t distribution and normal distribution are almost equivalent,however at a higher level such as 99%, the Student’s - t distribution outper-form the normal one and based on this results, they suggested this approachmight be a useful for practical applications in …nancial risk management.

However, both normal and Student’- t VaR measures are still have tion at assuming that volatility of …nancial returns is unchanged or constantinstead of clustering mentioned by Brooks (2008) Clustering means thatlarge returns tend to be followed by another large return and small returnstend to be followed by small returns It can be understood that the proba-bility of getting large returns are higher than small returns Based on ideasfrom ARCH models of Engle (1982), Bollerslev (1986) developed generalizedautoregressive conditional hetoroskedastic (GARCH) model which could cap-ture fat - tails and volatility clustering in …nancial returns Financial returnsdistribution are often change over time known as conditional volatility whichcould be captured by GARCH models and then VaR estimations can be de-pended on time or conditional One of model might be represented successfor this approach is exponential weighted moving average (EWMA) is em-ployed in RiskMetrics which is developed by corporation J.P Morgan (1994).However, this approach assumes that distribution of …nancial returns is nor-mal and this is not often happened in reality mentioned before as a weaknessassumption Furthermore, according to Black (1976), …nancial returns hasleverage e¤ects phenomenon or asymmetric e¤ect of volatility meaning thatnegative returns tend to increase volatility in future more than positive re-turns impact In order to capture this shortcoming, GJR - GARCH modelwas developed by Glosten et al (1993)

EVT has been applied in …nancial sector in recent years after occurring ofextreme event called Black Monday although it was applied widely in otherphysical sciences such as engineering, hydrology And up to now, EVT hasbecome popular in …nance because it can perform a good result in predictingworst situations McNeil (1997, 1998) used extreme value theory to esti-mate the tail of loss severity distributions and quantile risk measurementfor …nancial time series Embrechts et al (1999) demonstrated an overviewpicture about extreme value theory which is considered as risk managementtool Moreover, following to Embrechts et al (1999), McNeil (1999) gave

an extensive overview of this approach to risk managers McNeil and Frey(2000) estimated tail - related risk measurement for …nancial time series withheteroscedastic characteristic

Following is several studies using EVT method in VaR evaluation andcompare to other VaR models which will be discussed

Kuester, Mittik & Paolella (2006) compared out - of - sample performancebetween EVT and parametric VaR models using GARCH volatility modelingthrough investigating on 30 years of daily returns from time period February

8, 1971 to June 21, 2001 of NASDAQ Composite index This study used

1000 returns data moving window (in - of - sample) and model parameters ineach window were updated for every day forecast then it produced 6681 oneday VaR estimation which supported for comparing predictive performance

of the models Through descriptive statistics, it addressed asymmetric andleptokurtosis phenomena in distribution of returns and in order to accountfor this behavior, authors used skewed Student’s - t instead of normal distri-bution Authors pointed that Historical simulation fails the independent testand predictive performance of skew Student’s - t distribution is much betterthan normal return distribution They also found that volatility model haspositive e¤ect to predicting performance and once again, skew Student’s - tdistribution provides superior performance compared to normal and symmet-ric Student’s - t distribution assumption They concluded that a combinationbetween EVT approach and fat - tails GARCH volatility modeling providesbest results at various con…dence levels

In another study, Ozun, Cifter and Yilmazer (2010) compared mance of eight di¤erent EVT models and GARCH models with normal,Student’s and skew Student’s - t distribution as well as used backtestingmethods In this study, they employed not only VaR but also ExpectedShortfall (ES) to estimate portfolio returns Then they found that …lteredEVT models provide results better than any VaR models and are able toaccount fat - tails phenomenon in distribution of portfolio return

perfor-Chapter 3

Research Methodology

The methodology of this empirical study is separated into 3 sections First,

in section 3.1, …nancial data i.e stock prices used in this study is brie‡ydiscussed Second, in section 3.2, risk measurement is employed and com-pared together in di¤erent methods Finally, backtesting methodology isimplemented and evaluated the performance of risk measurement methods

3.1 Data selection

The …nancial data used in this study is Vietnam stock prices including DEX, a group banking of 8 stock indexes are extracted from cophieu68.comwith the longest time period from January 02, 2002 to November 30, 2015

VNIN-In general, while stock prices are non – stationary or usually integrated oforder one, log –returns have expected properties such as stationary (Camp-bell et al., 1997) Hence following to empirical studies in …nance and riskmanagement, transferred logarithmic return of each …nancial position is used

3.2 Methodology

Forecasting performance of VaR measuring is impacted by stylized facts of

…nancial positions, such as volatility clustering, fat - tails, leverage e¤ects.Di¤erent chosen models give di¤erent VaR estimation result Therefore, theimportant task is to …nd out the most suitable model which can capture wellparticular characteristic of …nancial position and then provides accuracy VaRresults In this thesis, two models including unconditional and conditionalvolatility modes with various methodologies are studied EVT focusing onthe tail of distribution to measure extreme returns occur is also studied

3.2.1 Unconditional VaR models

This section demonstrates unconditional VaR models including one non parametric Historical simulation and two parametric models with well -known distribution returns such as normal, Student’s - t

-Historical simulation

Historical simulation (HS) is one of the most well - know non - parametricapproaches and attracting method because of simply concept and easy toimplement in VaR estimations This method uses empirical data to estimateVaR and assumes that all behaviors of historical returns will occur in thefuture returns It means that experiences in the past are a good proxyfor future forecasting Moreover, it does not require any statistical modelsmeaning that no required parameter estimates because it can use its data todetermine the distribution itself Assume we have t historical observations inour data sample, VaR forecast at time t + 1 is equivalent to quantile ofdistribution returns which is formulated as

V aRt+1 = Q (rt; rt 1; :::; r1) (3.1)where is signi…cance level and Q is a quantile of distributionreturns taken from sorted empirical sample

In this method, to ensure the quality of estimation, a large observableshould be available However, in practice, such large appropriate data sample

is not always feasible and even in case of enough observable, the historicaldata might not contain su¢ cient extreme value for VaR estimation

Normal VaR

In parametric approach, returns are assumed to follow some probability tribution functions, such as normal, Student’s - t, skewed Student’s - t dis-tribution Based on which probability distribution function is assumed, wehave an appropriate return de…ned as

dis-rt= t+ t= t+ tzt (3.2)where rt; t; t are return, mean and standard deviation, respectively zt

is independent and identical distributed (i.i.d) and it can be transformed tostandardize probability density function fz:

Because unconditional VaR models is assumed in this section, hence wehave t ; t :

In normal VaR models, …nancial return is assumed to follow normalitydistribution The density function of normal distribution can be expressed

Student’s t VaR

As we mentioned before, …nancial returns often have fat - tailed distributionand according to Bollerslev (1987), this characteristics can be modeled withStudent’s - t distribution denoted t( ) where is degree of freedom andprobability density function are shown as below

f (x; ) =

+1 2

p

x22

+1 2

; > 2 (3.14)where is Gamma function denoted as

Skewed Student’s t VaR

Skewed Student’s - t is used to account the leptokurtosis and asymmetric ofdistribution of innovations, the probability density function can be denotedas

f (zt; ; g) =

+1 2

; > 2(3.17)where is gamma function, g is asymmetric parameter, m is mean and

s is standard deviation

m =

1 2

p( 2)p

V aRt= ( + Qskewed student0 s t( ))Vt 1 (3.21)

3.2.2 Conditional VaR models - Volatility model using

EWMA, GARCH, GJR - GARCH model

In the previous section, volatility is assumed constant meaning that volatility

is no change overtime However, volatility of …nancial returns in reality isalways change and produces some characteristics called stylized fact, such asclustering, leverage e¤ect Unconditional VaR model described in previoussection cannot cover these characteristics because volatility in this assump-tion does not re‡ect the change in time In order to cover this shortcoming,volatility should be modeled as a time - varying variable Here are somemodels used to account these e¤ects are presented in following section.EWMA model

Moving average (MA) is one of the simplest linear model to calculate ditional volatility t In this method, average is calculated by taking somenumber of historical variances and in each calculation, the last variance will

con-be dropped out and replaced by the next one This is an arithmetic age then all observations have same weight which is also a limitation of thismodel Recent observations should have more e¤ect than older ones becausethey can re‡ect the current situation of volatility This model could be im-proved by setting di¤erent weight of each observation through exponentialcurve showing the most recent ones have higher e¤ect on the current volatilethan the older or earlier ones By using this curve, the power e¤ect is de-creasing exponentially when observations are far away from present Thismethod is called Exponential Weighted Moving Average (EWMA) In thismodel, variance 2t at time t is formulated by previous historical variances

aver-2

t = 2t 1+ (1 ) 2t 1 (3.22)where 0 < < 1 is decay parameter By applying recursive calculationfor previous variances, we have

is a disadvantage of this proposal because old observations and recent onesshould not have a same e¤ect to forecasted …gures Decay factor lambdacorresponding to each asset should be recalculated in order to re‡ect thereality behavior as much as possible.

In order to estimate variance 2

t+1at time t+ 1, number of needed previousobservations goes to in…nity but practice calculation, only some truncated orlag observations are used leading to sum of weights is not equal to 1 This

is a limitation of this model and can produce a big di¤erence, especially insmall samples In order to cover this issue, Danielsson (2014, p 33 - 34)adjusted formula for n observations

GARCH model

In order to capture the serial dependency of volatility, Engle (1982) proposedthe autoregressive conditional heteroscedastic (ARCH) model showing thatconditional variance of tomorrow’s return is modeled as a linear function ofprevious squared innovations The general ARCH(p) model is formulated as

2

However, ARCH(p) model requires p to large in order to capture tire volatility structure In order to cover this limitation, Bollerslev (1986)

en-proposed a Generally Autoregressive Conditional Heteroscedastic (GARCH)model GARCH model is the most well - known model in time series whichare able to explain a number of important features in …nancial time series.The model can be expressed with mean equation and variance equation.

rt = (It 1) + at (3.31)where E(atjIt 1) = 0and E(a2

tjIt 1) = 1, atis a sequence of i.i.d randomvariables with mean 0 and variance 1, (It 1)is a mean equation at time t 1.The conditional variances can be de…ned as equation below Let t =

rt t be the innovation at time t, GARCH (p, q) model for conditionalvolatility 2t at time t is formulated as

t is modeled as a linear function of previous squaredconditional volatility as well as previous squared innovations

The most popular version of GARCH (p, q) used in practice is GARCH(1, 1) with one lag for both term mean and variance, we have GARCH (1,1) model at time t

2

t = ! + 1a2t 1+ 1 2t 1; 0 1; 1 1; ( 1+ 1) < 1 (3.35)After parameters in GARCH (1, 1) models are estimated, then conditionalvariance is forecasted and inserts it into VaR equation to get the …nal VaR

…gure Three GARCH (1, 1) models are performed with assumed normal,Student’s - t and skewed Student’s - t distribution

EWMA is a special case of GARCH model when both conditions ! = 0and 1+ 1 = 1 are satis…ed (Jorion, 2007)

After that, conditional volatility is forecasted and input into VaR formula

to estimate VaR using GJR - GARCH model Pagan & Schewert (1990)mentioned that sign of i determines how much impact to current conditionalvolatility from shocks in past It means that when i has a positive value,past negative shocks have a stronger impact to current conditional volatilitythan past positive shocks and vice versa

As same as GARCH (1, 1), GJR - GARCH (1, 1) is the most commonmodel in practice

2

t = ! + ( 1+ 1It 1) 2t 1+ 1 2t 1 (3.40)Again, normal, Student’s - t and skewed Student’s - t distribution as-sumptions are applied to distribution of returns for each respectively VaRmodel GJR - GARCH (1, 1) model is also a special case of GARCH(1,1)model when 1 = 0:

3.2.3 Extreme value theory (EVT) distribution in VaR

modeling

In previous sections, VaR models are described on modeling whole tion of …nancial returns In common events where normal market conditionsare presented, normal or Student’s - t distribution can forecast well enoughbut they generates inaccurate estimation for the tails of distribution, espe-cially in fat - tails

distribu-Extreme value theory (EVT) is also a risk measure similar to VaR but inonly concentrates on the tail of distribution where extreme event occur whileVaR studies on whole distribution (McNeil & Frey, 2000) This approachhas also no assumption about return distribution so it can …t any probabilitydistribution Following sections present EVT and its application in practical.EVT risk modeling and measuring

In …nancial risk management, EVT is applied in all areas such as market,credit, operational risk In this thesis, market risk is chosen to study andparticular VaR using EVT is performed Extreme events occur in the tail ofdistribution and then particular distribution is selected to create EVT modelfor VaR estimation

In the EVT context, there are two widely approaches of extreme eventsmodel One of them is block maxima models which directly model the distri-bution of maximum realizations The block maxima model requires a largedata sample of identical and independent distributed losses as well as manyextreme observations in this sample in order to employ well and produce areliable result The other approach is Peaks - over - Threshold (POT) mod-els focusing on losses which exceed a particular high threshold POT modelsuse return data more e¢ ciently and then they are considered as the mostuseful for practical applications (Gilli & Këllezi, 2006; McNeil et al., 2005).Therefore, according to the bene…t of POT models, we only focus on thisapproach in this thesis

POT models can be classi…ed into two types which are Hill estimator(semi - parametric models) and generalized Pareto distribution or GPD (fullyparametric models) In order to measure extreme or rare events, simpleparametric methods is rarely used, so in this study, GDP is chosen to estimateEVT VaR model

EVT approach has two important steps The …rst step is to model tribution of series of maxima or minima and under particular conditions,the distribution converge to one of three distributions Gumbel, Frechet orWeibull A generalized distribution called generalized extreme value (GEV)

dis-distribution is modeled to represent a standard form of these three tions above The second step is to model distribution of excess over a givenhigh threshold This result is applied in case of very high quantile such as0.99 or even higher 0.999.

distribu-Peak - over - threshold (POT) models

The GEV distribution (Fisher - Tippett, Gnedenko theorem) pose that fXigi2N is a sequence of independent and identical distribution(i.i.d.) random variables with common distribution function F (x) = P r(Xi

Sup-x) having mean indication location and variance 2 indicating scale Let

M1 = X1; Mn = M ax(X1; X2; :::; Xn); n 2 de…ned as sample maxima ofthe i.i.d random variables If there is a given sequence of cn > 0; d2 R andsome non - degenerate distribution functions H such that c 1

(x) = exp[ ( x )]; x 0; < 0

According to Fisher - Tippett (1928) theorem, asymptotic distribution

of maxima belongs to one of three speci…c distributions Gumbel, Fréchetand Weibull Fréchet and Weibull distributions become the shape of Gumbelwhen the tail index goes to 1 and 1 A proof of this theorem can be found

in Gnedenko (1943) Following to research from von Mises (1936) and inson (1955), Gumbel, Fréchet and Weibull distributions can be displayedinto one uni…ed distribution called generalized extreme value distribution(GEV) by taking = 1

Jenk-H (x)=

(exp [1 + x] 1 ; 6= 0; 1 + x > 0

where is the tail index and is a shape parameter

By replacing x byx where 2 R; > 0, we can obtain a related location

- scale distribution H ; ; as below

H; ; (x) =

(exp [1 + x ] 1 ; 6= 0; 1 + x > 0

where is scalar, is tendency, 1 is tail index indicating the tail thickness

of distribution - a larger tail index, the thicker tail Depending on tail indexvalue, the distribution H becomes a speci…c type For example, if the tailindex value equal zero, the distribution H is Gumbel type; if the tail index

is negative, the distribution H is Weibull and if the tail index is positive,

H is Fréchet distribution The Frechet distribution is fat - tails distributionwhich is frequently discovered in empirical …nancial returns Unlike para-metric VaR approach requiring a distribution of returns assumption, the Hdistribution of maximum or minimum can be estimated without assuming ofthe original distribution of the observations In other words, H distributionalways belongs to one of these three distributions type although original dis-tribution is unknown i.e whatever the original distribution This result isvery important and signi…cant because we can model asymptotic distributionregardless of the original distribution of the observed data

Excess over a threshold (Pickands, Balkema - in Haan theorem)

In general, not only the maxima of observations are interested but also thelarge observations exceeding a given high threshold behavior should be con-sidered Hence, the second step of EVT approach is estimate the conditionaldistribution of the excess over a given high threshold

Let X is a random variable with distribution function F and a thresholdgiven u An excess over u is de…ned by y = Xi u Then the excess distri-bution over the threshold u has distribution function expressing probabilitythat a loss exceeds the threshold u by at most an amount y (Mc Neil, Frey

& Embrechts, 2005) This conditional probability can be written as

for 0 y xF u where xF 1 is right end point of F:

Since x = y + u for X > u, then

F (x) = [1 F (u)]Fu(y) + F (u) (3.48)According to second theorem of Pickands (1975) and Balkema & de Haan(1974), the conditional distribution Fu of the excess may be approximated

by a generalized Pareto distribution (GPD) because when the threshold ubecomes large enough, the Fu(y) converges to the GPD G ; (x):

Fu(y) G ; (x), u ! 1 (3.49)where the GPD is de…ned generally as

of Pareto II type distribution corresponding to a bounded i.e short - tailedwhen < 0:When > 0;it takes a version of heavy - tailed namely ordinaryPareto distribution This version is most suitable in …nancial risk manage-ment since it has a heavy - tails or fat - tails In case > 0, kth momentE[Xk]is in…nite when k > 1= : For example, when = 0:5;it has an in…nitesecond moment i.e variance and an in…nite four moment for = 0:25:Again, we can extend the location distribution G ; ; (x) by adding alocation parameter 2 R

distri-to McNeil (1999) study, threshold u should be chosen high enough in order

to provide a result can be considered accuracy, in the other hand, it should

be chosen su¢ cient low in order to enough data sample for estimating rameters The choice of the threshold is a trade - o¤ in term of varianceand bias At a low threshold, the number of observations supported to esti-mating parameters increase, however, some observations from the centre ofdistribution are also accounted and then the index of tail is less variance i.e.more precise but biased In the other side, the bias is reduced by choosingthe high threshold but generates more volatility in estimator due to fewerobservations.

pa-Excess losses are approximated by …tting GPD and parameters are tained by using maximum log - likelihood method Number of excess losses

ob-nu are counted by number of losses over threshold u When having GPDmodel for excess losses, the tail of underlying distribution F can be esti-mated and then VaR can be obtained as EVT measure approach With asu¢ cient high enough u, Fu(y) converges to the GPD distribution and since

x = y + u for X > u, then we have

F (x) = [1 F (u)]G ; ;u(x u) + F (u) (3.53)The last term F (u) can be calculated by (n nu)=nwhere n is the samplesize and nu is the number of exceedances over threshold u Hence we havethe following estimator which is only valid when X > u

^

F (x) = 1 n nu

n nun

G ; ;u(x u) = 1 1 + x u

1

(3.57)where ^ and ^ are estimators obtained by maximum likelihood method

Exploratory data analysis Before using statistical modeling to evaluate,preliminary data analysis should be used in order to see the overview pictureabout data In this thesis, Q - Q plots are used to explore data analysis

Q - Q plots In all analyses, histogram of the data is usually studied

…rstly As discussed above, …nancial returns are often fat - tails instead of thenormality distribution which is assumed in VaR methodology Q - Q plotsprovide a graph of quantile in order to assess the goodness of …t of data to theparametric model This graph has a linear trend if the parametric model …tsthe data In EVT application, Q - Q plots is used to measure the fat - tails bycomparing normal distribution If points on the graph lie along the straightline, data is from normal distribution In case of convex departure, short -tail distribution is presented; wherever, fat - tail distribution is introduced

if there is a concave behavior The more linear in the graph, the moreappropriate the model in goodness of …t

EVT VaR model

After using maximum likelihood method to estimate the shape and scale, VaR estimation can be obtained through utilizing the EVT According

to Smith (1987), the GDP distribution with parameters and has densityfunction f

f (x) = 1 1 + x

1 1

(3.58)The the log - likelihood function is

3.3 Backtesting Methodology

The accuracy of future risk predictions represents the quality of VaR els In order to evaluate the quality of VaR estimation, the models should be

mod-backtested with suitable methods through statistical procedure when paring actual pro…ts and losses to corresponding VaR estimations Back-testing methodology is a procedure which is used to compare various VaRmodels in order to choose the best model in forecasting risk measurement.This procedure compares VaR forecast and realized historical return in term

com-of VaR violation happening when actual return exceeds predicted VaR

In order to evaluate the quality of model, the best approach is forecast

a future returns based on currently available data and wait for the comingactual result then a comparison result between predicted …gures and realityones provides the quality of VaR estimation However, this way has an issuerelating to how long it will take to generate su¢ cient future returns in ordertest the signi…cance of violations because violations happen rarely To coverthis limit, another better and simpler approach is use the historical data ofreturns to backtest the performance of VaR estimation The main idea of thisapproach is separate whole historical data sample into two parts - …rst partcalled in - sample or estimation window is used to forecast and second partcalled out - of - sample or testing window is used to compare the forecasted

…gures generated from data sample in part one with the actual ones In sample, out - of - sample and whole data sample size are denoted by WI, WO

-and N , respectively, such that

In this study, Vietnam stock index (VNINDEX) returns is taken around

13 years from 2002 to 2015 with 250 trading days in average of each year and

in total we have 3439 observations First 1000 observations of data sampleare put in in - sample and the rest is 2439 observations are put into out - of

- sample in order to forecast VaR and then these forecasted …gures will becompared to the actual ones Rolling window approach is used to forecastall data size of out - of - sample First forecast …gure, we use all 1000observations of in - sample data to estimate …rst future value for day 1001.Second forecast continue use 1000 observations but the …rst observation isdropped and the latest one i.e actual …gure at 1001 is included In order

to get total 2439 one - day predicted …gures supported for backtesting, thisrolling windows having 1000 observations is shifted to 2439 times A group

of 8 banking stock indexes with shorter time period are also studied withthe same approach but only …rst 500 observations are chosen in windowestimation

After generating whole one - day ahead forecasted sample, these valuesare used to compare with the actual …gures and see the di¤erent betweenthem VaR violations happen when actual …gures exceeds VaR estimation