Quantitative risk analysis an approach for vietnam stock market

Three unconditional volatilitymodels including historical, normal and Student‘s - t as well as EWMAand two volatility models including GARGH, GJR - GARGH with threereturn distributions n

UNIVERSITY OF ECONOMICS ERASMUS UNVERSITY ROTTERDAM

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

HO CHI MINH CITY, January 2016

UNIVERSITY OF ECONOMICS HO

NETHERLANDS

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

By

NGUYEN NAM KHANH

Academic Supervisor

Dr TRUONG DANG THUY

Ho Chi Minh City, January 2016

ØUANTITATIVE ÆISK ANALVSIS:

AN APPÆOACH FOÆ VIETNAM

STOCK MAÆKET

Nguyen Nam Khanh January fi†,

£0fi6

AbstractValue at Risk (VaR) is widely used in risk measurement It is defined

as the worst expected loss of a portfolio under a given time horixon at agiven confidence level The aim of the thesis is to evaluateperformance of fi6 VaR models in forecasting one - day ahead VaR fordaily return of VNIN- DEX and a group 8 banking stock indexes includingAGB, BVH, GTG, EIB, MBB, SHB, STB, VGB to find out the mostappropriate model for each stock index Three unconditional volatilitymodels including historical, normal and Student‘s - t as well as EWMAand two volatility models including GARGH, GJR - GARGH with threereturn distributions normal, Student‘s - t and skewed Student‘s - t andassociated Extreme Value Theory (EVT) models are performed at †%,

£.†% and fi% of significance level Violation ration, Kupiec‘sunconditional coverage test, independence test and Ghristoffersenconditional coverage test are used to backtested performance of allmodels Besides statistical analysis, graphical analysis is alsoincorporated Backtest- ing indicates that there is no best model for allcases because of character- istic difference from particular stock index.Implication of this thesis is that a suitable VaR forecasting model is onlychosen after backtesting frequently performance of various models inorder to ensure that most relevant and most accurate models are suitedfor current financial market situation

Ke4mords: Value at Risk, Extreme Value Theory, financial risk

manage- ment, conditional volatility model, backtesting, stock index

fi.fi Problem statements t

fi.£ Research objectives 8

fi.3 Research questions 9

fi.4 Subject and scope of research 9

fi.† Structure of the thesis 9

2 Literature review 11 £.fi Definitions fifi £.fi.fi Financial return data fifi £.fi.£ Goncept of Risk fi£ £.fi.3 Glassification of Risk fi3 £.fi.4 Risk measurement and Goherence fi3 £.£ Theoretical review fi4 £.£.fi Value at Risk fi4 £.£.£ GARGH fit £.£.3 Extreme Value Theory fit £.3 Empirical studies review fi8 £.3.fi Empirical research on modeling and measuring VaR fi8 £.3.£ Empirical research on Extreme Value Theory (EVT) VaR £0

3 Æesearch Methodology 22 3.fi Data selection ££

3.£ Methodology ££

3.£.fi Unconditional VaR models £3

3.£.£ Gonditional VaR models - Volatility model using EWMA, GARGH, GJR - GARGH model £6

3.£.3 Extreme value theory (EVT) distribution in VaR mod- eling 30

fi

3.3 Backtesting Methodology 3†3.3.fi Kupiec‘s Test 3t3.3.£ Ghristoffersen‘s Tests 393.3.3 Hypothesis testing procedure 40

4.fi Descriptive statistics 4fi

4.3 Models forecasting performance analysis †64.4 Graphical analysis of model forecasting t£

List of Tables

4.fi Descriptive of data sample 434.£ Descriptive statistics of daily stock index returns 444.3 Parameters estimation of GARGH(fi,fi) model with normal dis- tributed innovation for daily stock index returns †04.4 Parameters estimation of GARGH(fi,fi) model with Student‘s

- t distributed innovation for daily stock index returns †04.† Parameters estimation of GARGH(fi,fi) model with skewedStudent‘s - t distributed innovation for daily stock index re- turns †fi4.6 Parameters estimation of GJR - GARGH(fi,fi) model with nor-mal distributed innovation for daily stock index returns †£4.t Parameters estimation of GRJ - GARGH(fi,fi) model with Stu- dent‘s - t distributed innovation for daily stock index returns †£4.8 Parameters estimation of GRJ - GARGH(fi,fi) model with skewed Student‘s - t distributed innovation for daily stock index re- turns †34.9 Parameters estimation of generalixed Pareto distribution

(GPD), threshold exceedances of † percentage from

GARGH(fi,fi) model

†4

4.fi0 Parameters estimation of generalixed Pareto distribution (GDP), threshold exceedances of † percentage from GJR - GARGH (fi,fi) model ††4.fifi Expected and actual number of VaR violations at threshold † percentage †64.fi£ Violation ratio and Kupiec‘s test p - value at † percent signif- icance level †t4.fi3 Independence test and Ghristoffersen‘s test at † percent sig- nificance level 634.fi4 Expected and actual number of VaR violations at threshold

£.† percentage 64

3

4.fi† Violation ratio and Kupiec‘s test p - value at £.† percent sig- nificance level 6†4.fi6 Independence test and Ghristoffersen‘s test at £.† pecent sig- nificance level 664.fit Expected and actual number of VaR violations at threshold fi percentage 6t4.fi8 Violation ratio and Kupiec‘s test p - value at fi percent signif- icance level 684.fi9 Independence test and Ghristoffersen‘s test at fi pecent signif- icance level 694.£0 Best forecasting VaR model according to Ghristoffersen‘s test

at †, £.† and fi percentage of significance level tfi

4

List of Figures

4.fi Daily value of stock index 4£

4.£ Daily return of stock index 4†

4.3 Histograms of daily stock index returns 46

4.4 Ønorm - ØØ plot of daily stock index returns 46

4.† AGF for daily stock index returns 4t 4.6 PAGF for daily stock index returns 4t 4.t AGF for squared of daily stock index returns 48

4.8 PAGF for squared of daily stock index returns 48 4.9 EWMA and unconditional VaR models forecasting performance for daily return of EIB at †% significance level t£ 4.fi0 GARGH VaR model forecasting performance for daily return

of AGB at †% significance level t3 4.fifi GJR - GARGH VaR model forecasting performance for daily return of MBB at †% significance level t4 4.fi£ EVT GARGH VaR model forecasting performance for daily return of GTG at †% significance level t4 4.fi3 EVT GJR - GARGH VaR model forecasting performance for daily return of BVH at †% significance level t†

†

I would like to send special thanks to my academic supervisor Dr.Truong Dang Thuy, for his patience guidance, enthusiasm and supportduring my thesis 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,staffs of the Vietnam - Netherlands Program as well as my classmates forall their helps and supports

I am most grateful to my family Thank you for always being therefor me, thank you for inspiring me, supporting me and making meappreciate the value of education

Last but not least, I would like to thank my wife and my daughter.Thank you for your patience, deep understanding and encouragement I

am grateful to you

6

- Black Monday (fi98t), the bond market crisis in U.S (fi990), thefinancial crisis in Asia (fi99t), and the financial crisis in Europe (£00t),bankrupt or bailed by governments of Lehman Brothers, AIG (£008) andthen it led to global financial crisis and economy recession all over theworld (£008) These events seem to be rare but now they happenfrequently and have a negative impact to financial market on both sixeand loss Beside on objective causes, such as war, calamity, terrorism, one of main important factors impacting to financial market is aweakness of risk management system Therefore, a challenge has beenraised is how to identify and measure the risk in order to minimixe theloss as well as ensure the safe environment for financial market andeconomy system.

In modern risk management, it is not sufficient if only simply focus

on quality policy Risk is actually the expected loss of outcome in future,

so it is often measured by probability distribution One of the importantstages in financial risk management process is build up models tomeasure and evaluate the risks However, a difficult process might beraised when applying them into actual condition of market because everymodel is associated with some defined assumptions, hypotheses andsometimes these assumptions are not satisfied in particular conditions ofmarket Therefore, some new approaches should be studied in thesemodels in order to choose and apply the best one with actual conditions

ment and applied these risk measure methods into business process evalua- tion, allocation assets, portfolios management with efficiency result.

Vietnam stock market born in July £000 is an important step toindicate an improvement in the country‘s economy Vietnam stockmarket is rela- tive young when compared to others development stockmarket in the world and then it has attained new opportunities as well asfaced with many new challenges In recent years, although Vietnamstock market has many fluc- tuations but it is still an attractiveenvironment for many foreign investors as well as local ones Allinvestors, definitely, would like their investments produce a highest profitwith lowest risk and they are also two main factors that influence all theirbusiness activities According to risk management in Vietnam, financialmarket in general and Vietnam stock market in particu- lar, it is actuallimited in term of both policy and tool Therefore, system of financialrisk management should be studied and built up in active and effectiveway

One of the most known risk measurement applied in riskmanagement is Value at Risk (VaR) and it becomes a popular riskmanagement tool for financial regulations and financial institutions toevaluate possible losses that they can incur The VaR estimation wasrequired by Basel Gommittee on banking supervision to meet the capitalrequired for covering potential losses and VaR figures considered asadditional information to shareholders have disclosed by many offinancial institutions VaR can answer well a question what is amaximum financial amount possible to loose with given time horixonunder given confidence level or significance level An overview of VaR isreviewed by Duffie and Pan (fi99t)

Many methodologies are using to estimate VaR only based onsimple assumption that all financial returns follows Gaussian normaldistribution However, estimating VaR by using normal distribution foreach asset has raised an inaccuracy result because non - normality offinancial returns Therefore, various advanced VaR measurementtechniques are used to es- timate VaR of daily returns of stock index andthen the performance of these models are evaluated in order to find outthe best ones which could be used by financial institutions to managemarket risk

fi To suggest suitable risk measurement VaR models for portfolio in Viet- nam stock market

1.3 Æesearch questioms

fi Does the forecasting VaR performance improve from unconditional

to conditional volatility models?

£ Does the forecasting VaR performance unchange with respect to differ- ent significance level?

3 Is it possible to find out one VaR model which has best

forecasting performance for Vietnam stock market?

This thesis studies risk measurement VaR with various models as well

as methodologies in advanced and applies these models into riskmeasure for Vietnam stock market

The purpose of this thesis is to identify a best appropriate VaRmethod including unconditional VaR models such as Historicalsimulation, normal- ity VaR, Student‘s - t VaR, skewed Student‘s - tVaR and conditional VaR models where volatility is forecasted by using

Autoregressive Gonditional Heteroscedastic (GARGH) and GJR GARGH in risk measurement through measuring po- tential losses ofdaily return of VNINDEX as well a group of 8 banking stock indexes withdifferent time period for each stock index The longest time pe- riod is inVNINDEX which is studied from year £00£ to the end of November

-£0fi† Stock indexes historical return data is assumed to providesufficient information for model evaluation and predicting one - dayreturn forecasts under 9†%, 9t.†% and 99% confidence level In order toevaluate quality of forecasting, some backtesting models includingviolation ratio, Kupiec‘s test, independence test and Ghristoffersen‘s testare performed

Findings of this study might be useful for financial institutions orfinancial regulatory, particularly risk managers in Vietnam stock markets.All data used in analysis are calculated by programming andstatistical software called R

Thesis contains 6 chapters First chapter introduces concept of risk andrisk measurement through some tools such as Value at Risk (VaR),Extreme Value Theory (EVT) Second chapter summarixes literature

review and empirical

9

researches on VaR, EVT modeling Third chapter presents data andmethod- ology of VaR estimation as well as performance comparisonthrough several backtesting models Fourth chapter discusses empiricalresults, analysis and backtesting Finally, conclusions and implications

as well as further studies are last part of this thesis

fi0

Chapter 2

Literature review

This literature review includes 3 parts The first part describes somedefin- itions of concepts used in this thesis The second part istheoretical review containing some models used to study riskmanagement And the last part mentions empirical studies

Because stock indexes are mostly not stationary and often integrated atorder fi so it is modeled to changes of prices or log - return series ofprices Daily financial return data have some characteristics which areknown as st4tssed facts According to McNeil et al (£00†), theseproperties can be extended to scope of time interval including shortersuch as intra - day and longer such as weekly, monthly

In theory, financial returns are often assumed to independent andiden- tical distribution (i.i.d.) but not in reality which often exhibitdependence in second moment causing time - varying volatility andsotatstst4 ctustersng Glustering means that large returns tend to befollowed by another large return and small returns tend to be followed

by small returns (Gampbell et al., fi99t; McNeil et al., £00†) It can beunderstood that the probability of getting large returns are higher thansmall returns Black (fi9t6) founded an asymmetric of volatilityphenomenon meaning that negative returns tend to increase volatility infuture more than positive impact which has known as teserage eflects Inaddition, Mandelbrot (fi963) addressed that financial re- turns are notnormal distribution and otherwise, distributions usually follow to a fat -tails or leptokurtic Gompared to normal distribution, leptokur-

fifi

tic distribution has an excess kurtosis indicating that the tail is fatterthan predicted by the normal distribution.

Risk can be understood as unexpected outcome which might behappened in future In finance, risk is a difference between return which

is achieved from an investment and expected outcome or the volatility

of unexpected outcomes which can represent the value of equity, assets,

or earnings Risk is uncertainty outcome and often developed byprobability distribution Ac- cording to Basel Accords definition, financialrisk can be divided into three types including credit risk, operational riskand market risk Liquidity risk could be considered as an additionalcategory if necessary

Gredit risk, or default risk is defined as risk of loss due to paymentdefault of borrower including concentration risk, consumer credit risk,securitixation and credit derivatives Gredit risk has been less researcheddue to limit data available which mainly only belongs to large ratingagencies But in recent years, it has become attractive due to thefailure of several large financial institutions in the U.S., for example

Merry Lynch, Lehman Brothers (£008).Operational risk indicates the risk of internal processes failure,systems and people Fraud, legal and political risk are examples ofoperational risk

Market risk can be understood as a changes the prices of financialas- sets such as stock prices, exchange rates, interest rates andcommodity risk Interest rate risk, currency risk, volatility risk, equityrisk are included in market risk Because interest rates and equityprices are available widely and high quality, so market risk can beunderstood as financial risk studies which is highly concentrated in thisthesis

Following to development of information technology, financialproducts become more and more sophisticated and financial marketsaround the worlds become more integrated, so understand well financialrisk becomes more important More and more researches on variousfeatures of financial series have been studied

In this thesis, market risk mentioning the uncertainty of profits orlosses causing of the changes in market condition will be studied becauseonly this type have enough data

Firms disclose various types of risks but in general, it can beclassified into £ types including systematic and unsystematic risks

2.1.3 Classificatiom of Æisk

Systematic risk

Systematic risk is risk effecting to all or almost stocks Unstable ofeconomy environment such as interest rate movement, volatile exchangerates and high inflation are elements of systematic risk

One of the key element should be shown up is market risk Marketrisk happened due to reaction from investors at phenomenon happens inmarket

Umsystematic risk

Unsystematic risk is risk effecting to one asset or a group of assets or itonly impacts to a specific security Unsystematic risk includes businessrisk and financial risk

Business risk includes business decisions and business environment.Busi- ness decision includes corporation structure choices, investmentdecision, product development choices and marketing strategiesimplementation Busi- ness environment includes competition andmacroeconomic risks

Financial risk is possible losses of some or all of the originalinvestments For example, losses can happen caused of interest ratemovement or volatile exchange rate

Therefore, all investors can face many types of risk when they invest

in stock market and this is the most important element where theyconcentrate on However, in this research, only financial risk is studiedespecially at risk models to measure and assess the stock price return

In modern financial risk management, only rely on qualitative methodsare not enough and not efficiency, the more important thing is build upand development methods which can quantify the level of risk andfinancial losses And based on this, financial institutions, financialregulators and investors have a reliable source to determine decision

financial position can be represented by a random variable and then allcharacteristic losses of financial position are expressed throughdistribution of loss random variables Because the loss distribution isunknown and hard to estimate then some summary statistics areemployed to quantify the distributions loss in reality and a risk measure isone of these ones Risk measure provides a potential risk estimation anddifferent chosen risk measure leads to different affects to quality ofpredicting the losses of financial position so the choice

fi3

of a suitable risk measure becomes a crucial task towards building arealistic picture of risk.

Coheremce

Risk measure is a tool to estimate the potential loss of financialposition so it should be consistent with the basis theory in finance calledcoherence Let y be a risk measure and y is a coherence if four followingconditions are satisfied for any two loss random variables E and Y(Artxner et al., fi999):

fi Subadditive: y(E ‡ Y ) Ç y(E) ‡ y(Y )

£ Monotonicity: If E Ç Y then y(E) Ç y(Y )

3 Positive homogeneity: y(sE) = sy(E), for any positive constant s

4 Transition invariance: y(E ‡ s) = y(E) ‡ s, for any positive constants

The subadditive property indicates that a combine of two positions isless risky than separate them individually This property relating todiver- sification mentions that a diversified portfolio should not begreater than individual components in level of risk The monotonicitymentions that a lower loss asset will generate a lower risk measure Thepositive homogeneity expresses that doubling an asset should lead todouble its risk The tran- sition invariance property shows that if oneadditional risk is added, it will generate more risky and adding one moreconstant to a random variable leads to unchanged in its variability which

is one of statistics properties

Financial institutions can be lost billions of dollars due to a poorfinancial risk management which are experienced in finance crisisperiod In order to have a quantitative figures of risk, Value at Risk (VaR)was developed and now it becomes a popular tool in risk managementbecause this approach summaries overall market risk through a singlequantity, easy to understand and does not depend on a specific kind ofdistribution VaR can be used by financial institutions to measure theirrisks as well as by a regulatory to setup requirement VaR summarixesthe worst expected loss of assets or portfolio over a target holding periodwith a given level of confidence in normal market condition (Jorion, fi99t,

£00t) It could be estimated through the predictive distribution ineconometric modeling of the loss random variable

Let Vt, Vt‡t be the value of a financial position at time t and t ‡

t, respectively Let Jt(t) is the loss random variable of financial position

the next t periods from the time index t and the cumulativedistribution function of Jt(t) is denoted by 5t(ıt) or 5t(ı) For shortexpression, the time index t will be drop but it is understood that 5t(ı)

is a function depending on the time index t Then, Jt(t) is either anegative or positive function of Vt‡t — Vt

Because big loss is less frequently happened so small probabilitydenoted by p is used, for example, †% or fi% or 0.fi% to assess the loss.After that, with a given time horixon t under probability p, VaR of thefinancial position is defined as

From the definition, 5t(V aRfi—p) Ç fi — p is absolutely satisfied, which says

P r(Jt(t) Ç V aRfi—p) Ç fi — p or P r(Jt(t) > V aRfi—p) Ç p (£.£)indicating with the probability fi—p, the potential loss of financialposition over the time period from t to t ‡ t is less than or equal to V

probability that the potential loss of financial position greater than V aRfi—p

over the same time period is at most p

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

‡œ

¸

ƒ (ı)dı = p or equivalent

to as the fi O(fi—p)th percentile of the loss variable X

Given a probability density function of standardixe return, VaR can

be calculated by combination of volatilities and residual of distributionfunction as:

where oˆt is the conditional standard deviation at time t $—fi(α) is thequantile of a standardixed normal variable, such at normal distribution,Stu- dent‘s - t distribution, skew Student‘s - t distribution or anyassumed dis- tributions VaR is a coherent risk measure if the loss

random variables arenormal distribution

Volatility, time horixon and confidence level or significance level arefac- tors determining VaR for portfolio or a certain asset The volatility

is esti- mated through statistical models Depend on specific financialactivity type,

fi†

such as measured in one - day ahead VaR, one - year ahead VaR, timehorixon is chosen and affects to volatility measure and then also affect toVaR, where a longer time period leads to a higher volatility and finally,

a higher VaR Gonfidence level chosen represents how often a loss onportfolio or specific asset greater than VaR It can be set lower at 9†%for short data period and in case of very conservative approach in riskmanagement; confidence level can be set high as 99.9% or even99.99% 9†% and 99% are confidence intervals which are commonlyused in empirical studies (Danielson and de Vries, fi99t)

The oldest definition of VaR might be seen from the portfoliooptimixation theory by Markowitx (fi9†£) However, it had becomeunsuitable measure- ment after stock market crashed in fi98t called BlackMonday due to simple assumptions in this methodology which does notappropriate to actual situa- tions A new suitable methodology should bestudied and then well - known RiskMetrics were developed andpublished by J.P Morgan in fi994, VaR has become a popularmeasurement in risk management Research on VaR was stronglysupported when Basel II Accord (BIS, £006) (Basel Gommittee onBanking Supervision) clearly expressed that VaR is a preferredmeasurement for market risk Since then, more and more studies onVaR have been devel- oped to improve the quality of risk managementthrough providing a better predicted measure in future loss

The method to calculation VaR could be split into two groups underpara- metric and nonparametric approaches In this study, EVT(Extreme Value Theory) is a parametric approach focusing on taildistribution where rare event existed Parametric methodology includesGARGH, Equally weighted moving average (EqWMA), ExponentialWeighted Moving Average (EWMA) In the other hand, the Historicalsimulation belongs to nonparametric method- ology

The choice of risk measure is crucial step in order to build realisticfigures of risk However, there is another essential element comes fromfinancial re- turns which impacts to accuracy of risk figures Numerousempirical studies pointed out that asset return exhibit volatility clustering,fat - tails and skew- ness; therefore these phenomena should beaccounted for probabilistic model Before EVT, lot of methodologyresearches in whole distribution mentioning for entire sample of return ofassets (McNeil & Frey, £000) In this study, fi6 different models will becompared together in order to find out which model is the best Theyare the variance - covariance with normal distribution and Student‘s tdistribution, historical simulation, EMWA, GARGH, GJR

- GARGH and EVT combining with three distributed innovationsnormal, Studen‘s - t and skewed Student‘s - t; and then residualsextracted from volatility models GARGH, GJR - GARGH are modeled

Threshold model from Extreme Value Theory which only concentrates

on the tail

In reality, financial series always volatile and in order to predict thevolatil- ity of these return time series, several forecasting volatilitymodels were introduced such as ARGH (Engle, fi98£), a well - knownmodels GARGH (Bollerslev, fi986) which is now widely applied inforecasting the volatility of financial return time series In order tocapture volatility clustering fea- ture in financial time series, severalalternative modes have been developed For example, GARGH(fi,fi) isconsidered as a successful approach to account certain features offinancial returns such as volatility clustering and excess kurtosis(Hansen & Lunde, £00†) However, GARGH is a model which is oftenused to predict in short term and fail to capture asymmetric behav- ior(Baillie, Bollerslev & Mikkelsen, fi996; Davidson, £004; Ding & Granger,fi996) Therefore, several advanced models applied to particularcondition of market have been produced such as Asymmetric PowerAutoregressive Gonditional Heteroscedastic (APARGH) (Ding, Grangerand Engle, fi993), EGARGH (Nelson, fi99fi), GJR - GARGH (Glosten et

al, fi993)

VaR is a well - known parametric methodology in risk measurementbut combining with a simple normal distribution assumption will raise aninac- curacy result According to this approximation, all risk measurementresults of high quantiles are underestimated, especially in fat - tailsfinancial series happened frequently in empirical studies mentioned byMandelbrot (fi963) and Fama (fi96†) In order to cover this limitation,some studies have used more appropriate fat - tails distributions such

as Student - t distribution or normal distribution mixture but VaR actuallyonly concentrate on central of observation which is also mean that theyare studied under normal market conditions In another side,nonparametric methods based on no assumption of specific empiricaldistribution have been also given to pass this problem, they, however,are still faced with some limitation For example, non - para- metricmethod 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, ExtremeValue Theory (EVT) only focus on tail areas of distributions whereextreme or rare events are taken into account Extreme value theory(EVT) is a useful tool to support risk measurement because it could takeover a better approach

fit

to fit extreme rare events which are limited by mentioned simpleassumption before In a different way compared to VaR, EVT has noassumptions about the original distribution of all the empiricalobservations It is a powerful and robust framework in study of the tailbehavior of distribution, especially in fat - tails and it can be used tohandle for a very high quantiles in pre- dicting an extreme loss orcrashes situations Although EVT is a popular methodology which hasbeen applied in climatology and hydrology long time ago, it has beenonly introduced comprehensive in finance and insurance in recent year

by Embrechts et al (fi99t) Since its introduction to finance, there are asignificant number of financial studies relating to extreme values havediscovered in recent years and a comparison between their results withother VaR models are reviewed De Haan, Jansen, Koedijk, and deVries (fi994) gave the quantile estimation using Extreme Value Theory.McNeil (fi99t, fi998) used the estimation of quantile risk measures andthe tail of loss by applying Extreme Value Theory in financial time seriesstudies Em- brecht et al (fi998) discussed a risk management toolthrough the Extreme Value Theory McNeil (fi999) provided an overview

of Extreme Value Theory for risk manager McNeil and Frey (£000)estimated a risk measurement for heteroscedasticity at tail of financialtime series

EVT approach is definitely suitable for extreme quantiles thanconven- tional approach in heavy tail data In principle, there are twomost well - known of extreme value models: block maxima model(BMM) and Peaks - Over - Threshold (POT) model The first one,BMM, requires a large of observations in sample with identically andindependently distributed (i.i.d.) losses assumptions The second one,POT, is a more recent and modern model focusing on all return losseswhich exceeds some defined high thresh- old In practical applications,POT model is the most useful and often used due to its simpleassumption and efficient use of return data on extreme val- ues whichare very often limited (McNeil, Frey & Embrechts, £00†) In this study,only POT model is used to support EVT in risk measurement

measurimg VaÆ

Since Basel II Accord (BIS, £006) mentioned that VaR is preferredmea- sure for market risk, VaR model have been exploded in researchand then many methodologies were built with advantages anddisadvantages as well as many advanced approaches were studied in

order to improve quality of

predicting of different VaR models There are some advancedapproaches, for example, Extreme Value Theory (EVT) or family ofARGH models such as the autoregressive conditional heteroscedastic(ARGH) model of Engle (fi98£), the generalixed autoregressiveconditional heteroscedastic (GARGH) model of Bollerslev (fi986), theexponential generalixed autoregressive condi- tional heteroscedastic

autoregressive conditional heteroscedastic (APARGH) model of Ding,Granger and Engle (fi993), the GJR - GARGH model of Glosten et al(fi993) were used to study on various assets However, these studiesprovided dif- ferent results indicating that no existing best model forpredicting, so it is very useful contribution to test various method inparticular asset using most recent data in market including recentfinancial crisis

Beder (fi99†) is considered as one of the first author studying onmea- suring VaR accuracy of different assets including bonds, stockindexes and options as well as the mix of these ones by using twoclassical VaR models which are Historical simulation (HS) and MonteGarlo simulation Author mentioned that VaR measures are veryimpacted by data, different parame- ters, assumptions andmethodologies as well as provided fi4 different VaR estimations in thisstudy even though this paper only used two models So there areseveral shortcomings from these approaches The main advantage ofHistorical simulation methods is no parameters requirement However,one of the weakness of Historical simulation method is equally weightedas- sumption applied to all returns showing that very old returns have assame impact as recent ones to VaR estimation in case of long timeperiod of data taken This weakness is mentioned in study of Boudoukh,Richardson and Whitelaw (fi998) and they also provided suggestionabout weight return fol- lowing to time from their sample to VaRestimation Hull & White (fi998) also addressed the weight returns onvolatility in order to account the recent volatility changes

Distribution of financial returns is one of an important elementimpacting strongly to VaR performance Using normal distribution inVaR estimation might lead to an inaccuracy result because the financialreturns are often not fit with the normal distribution Mandelbrot (fi963)addressed that financial returns are not normal distribution andotherwise, distributions usually fol- low to a fat - tails or leptokurtic.Bormetti et al (£00t) presented a non

- Gaussian approach to measure market risk as well as discussed bothits advantages and limitations In this study, author compared newapproach to other well - known approaches in literature including normalVaR, Historical simulation and Monte Garlo simulation that are oftenused in financial analy- sis In order to capture the excess kurtosis,authors used fitted Student‘s - t distribution known as a better modeling

for fat - tails characteristic of

finan-fi9

cial returns Research investigated Italian stock market by using fi000daily return of two stocks and two indexes as well as confirmedleptokurtic behav- ior of distribution returns because tail parameters fell

in the range (£.9, 3.†) This study mentioned that at 9†% confidencelevel, the performance result of the Student‘ - t distribution and normaldistribution are almost equivalent, however at a higher level such as99%, the Student‘s - t distribution outper- form the normal one and based

on this results, they suggested this approach might be a useful forpractical applications in financial risk management

However, both normal and Student‘ - t VaR measures are still havelimita- tion at assuming that volatility of financial returns is unchanged orconstant instead of clustering mentioned by Brooks (£008) Glusteringmeans that large returns tend to be followed by another large return andsmall returns tend to be followed by small returns It can be understoodthat the proba- bility of getting large returns are higher than smallreturns Based on ideas from ARGH models of Engle (fi98£), Bollerslev(fi986) developed generalixed autoregressive conditional hetoroskedastic(GARGH) model which could cap- ture fat - tails and volatility clustering

in financial returns Financial returns distribution are often change overtime known as conditional volatility which could be captured by GARGHmodels and then VaR estimations can be de- pended on time orconditional One of model might be represented success for thisapproach is exponential weighted moving average (EWMA) is em-ployed in RiskMetrics which is developed by corporation J.P Morgan(fi994) However, this approach assumes that distribution of financialreturns is nor- mal and this is not often happened in reality mentionedbefore as a weakness assumption Furthermore, according to Black(fi9t6), financial returns has teserage eflects phenomenon or asymmetriceffect of volatility meaning that negative returns tend to increase volatility

in future more than positive re- turns impact In order to capture thisshortcoming, GJR - GARGH model was developed by Glosten et al(fi993)

(EVT) VaÆ

Because methodologies of VaR estimation above are studied on wholedistrib- ution and then it cannot cover well in case of extremely rareevents In order to capture these situations, Extreme Value Theory(EVT) was developed and it only focuses on the tail of distributioninstead of whole distribution As mentioned above, almost distribution offinancial returns are fat - tails and asymmetric So this methodology issuitable in these cases and this is a good reason to compare it toparametric volatility models However,

EVT has been applied in financial sector in recent years after occurring

of extreme event called Black Monday although it was applied widely inother physical sciences such as engineering, hydrology And up to now,EVT has become popular in finance because it can perform a good result

in predicting worst situations McNeil (fi99t, fi998) used extreme valuetheory to esti- mate the tail of loss severity distributions and quantilerisk measurement for financial time series Embrechts et al (fi999)demonstrated an overview picture about extreme value theory which isconsidered as risk management tool Moreover, following to Embrechts

et al (fi999), McNeil (fi999) gave an extensive overview of this approach

to risk managers McNeil and Frey (£000) estimated tail - related riskmeasurement for financial time series with heteroscedastic characteristic.Following is several studies using EVT method in VaR evaluationand compare to other VaR models which will be discussed

Kuester, Mittik & Paolella (£006) compared out - of - sampleperformance between EVT and parametric VaR models using GARGHvolatility modeling through investigating on 30 years of daily returns fromtime period February 8, fi9tfi to June £fi, £00fi of NASDÀ Gompositeindex This study used fi000 returns data moving window (in - of -sample) and model parameters in each window were updated for everyday forecast then it produced 668fi one day VaR estimation whichsupported for comparing predictive performance of the models Throughdescriptive statistics, it addressed asymmetric and leptokurtosisphenomena in distribution of returns and in order to account for thisbehavior, authors used skewed Student‘s - t instead of normal distri-bution Authors pointed that Historical simulation fails the independenttest and predictive performance of skew Student‘s - t distribution is muchbetter than normal return distribution They also found that volatilitymodel has positive effect to predicting performance and once again,skew Student‘s - t distribution provides superior performance compared

to normal and symmet- ric Student‘s - t distribution assumption Theyconcluded that a combination between EVT approach and fat - tailsGARGH volatility modeling provides best results at various confidencelevels

In another study, Oxun, Gifter and Yilmaxer (£0fi0) comparedperfor- mance of eight different EVT models and GARGH models withnormal, Student‘s and skew Student‘s - t distribution as well as usedbacktesting methods In this study, they employed not only VaR butalso Expected Shortfall (ES) to estimate portfolio returns Then theyfound that filtered EVT models provide results better than any VaRmodels and are able to account fat - tails phenomenon in distribution ofportfolio return

£fi

Chapter 3

Æesearch Methodology

The methodology of this empirical study is separated into 3 sections.First, in section 3.fi, financial data i.e stock prices used in this study isbriefly discussed Second, in section 3.£, risk measurement is employedand com- pared together in different methods Finally, backtestingmethodology is implemented and evaluated the performance of riskmeasurement methods

The financial data used in this study is Vietnam stock prices includingVNIN- DEX, a group banking of 8 stock indexes are extracted fromcophieu68.com with the longest time period from January 0£, £00£ toNovember 30, £0fi† In general, while stock prices are non – stationary

or usually integrated of order one, log – returns have expectedproperties such as stationary (Gamp- bell et al., fi99t) Hence following

to empirical studies in finance and risk management, transferredlogarithmic return of each financial position is used

Forecasting performance of VaR measuring is impacted by st4tssed facts

of financial positions, such as volatility clustering, fat - tails, leverageeffects Different chosen models give different VaR estimation result.Therefore, the important task is to find out the most suitable model whichcan capture well particular characteristic of financial position and thenprovides accuracy VaR results In this thesis, two models including

methodologies are studied EVT focusing on the tail of distribution tomeasure extreme returns occur is also studied

3.2.1 Umcomditiomal VaÆ models

This section demonstrates unconditional VaR models including one non

- parametric Historical simulation and two parametric models withwell - known distribution returns such as normal, Student‘s - t

Historical simulatiom

Historical simulation (HS) is one of the most well know non parametric approaches and attracting method because of simply conceptand easy to implement in VaR estimations This method uses empiricaldata to estimate VaR and assumes that all behaviors of historicalreturns will occur in the future returns It means that experiences inthe past are a good proxy for future forecasting Moreover, it does notrequire any statistical models meaning that no required parameterestimates because it can use its data to determine the distribution itself.Assume we have t historical observations in our data sample, VaRforecast at time t ‡ fi is equivalent to α— quantile of distribution returnswhich is formulated as

where α is significance level and Øα is a α— quantile of

distribution

returns taken from sorted empirical sample

In this method, to ensure the quality of estimation, a largeobservable should be available However, in practice, such largeappropriate data sample is not always feasible and even in case ofenough observable, the historical data might not contain sufficientextreme value for VaR estimation

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In normal VaR models, financial return is assumed to follownormality distribution The density function of normal distribution can beexpressed as below:

o

(3.fi0)

= Pr

x c Vt—fi — µ

µ

V aRt = —(µ ‡ oØx(α))Vt—fi (3.fi3)

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Studemt³s t VaỈ

As we mentioned before, financial returns often have fat - taileddistribution and according to Bollerslev (fi98t), this characteristics can bemodeled with Student‘s - t distribution denoted t(r) where r is degree

of freedom and probability density function are shown as below

v ‡fi

Skewed Studemt³s t VaỈ

Skewed Student‘s - t is used to account the leptokurtosis and asymmetric

of distribution of innovations, the probability density function can bedenoted as

2 , r > X(3.fit)

where F is gamma function, g is asymmetric parameter, m is mean and

Hence VaR can be estimated as

V aRt = —(µ ‡ oØshemed student t s—t(α))Vt—fi (3.£fi)

3.2.2 Comditiomal VaÆ models - Volatility model

usimg EWMA, GAÆCH, GJÆ - GAÆCH

In order to cover this shortcoming, volatility should be modeled as atime - varying variable Here are some models used to account theseeffects are presented in following section

EWMA model

Moving average (MA) is one of the simplest linear model to calculatecon- ditional volatility ot In this method, average is calculated by takingsome number of historical variances and in each calculation, the lastvariance will be dropped out and replaced by the next one This is anarithmetic aver- age then all observations have same weight which isalso a limitation of this model Recent observations should have moreeffect than older ones because they can reflect the current situation ofvolatility This model could be im- proved by setting different weight ofeach observation through exponential curve showing the most recentones have higher effect on the current volatile than the older or earlierones By using this curve, the power effect is de- creasing exponentiallywhen observations are far away from present This method is calledExponential Weighted Moving Average (EWMA) In this model, variance

oX at time t is formulated by previous historical variances

where O c Z c fi is decay parameter By applying recursive

calculation for previous variances, we have

RiskMetrics proposed decay factor at 0.94 and this value can be applied for whatever financial asset classes as well as different time periods This

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is a disadvantage of this proposal because old observations and recentones should not have a same effect to forecasted figures Decay factorlambda Z corresponding to each asset should be recalculated in order toreflect the reality behavior as much as possible.

In order to estimate variance oX at time t‡ fi, number of needed previousobservations goes to infinity but practice calculation, only some truncated

or lag observations are used leading to sum of weights is not equal to fi.This is a limitation of this model and can produce a big difference,especially in small samples In order to cover this issue, Danielsson(£0fi4, p 33 - 34) adjusted formula for n observations

GAÆCH model

In order to capture the serial dependency of volatility, Engle (fi98£)proposed the autoregressive conditional heteroscedastic (ARGH) modelshowing that conditional variance of tomorrow‘s return is modeled as alinear function of previous squared innovations The general ARGH(p)model is formulated as

proposed a Generally Autoregressive Gonditional Heteroscedastic(GARGH) model GARGH model is the most well - known model in timeseries which are able to explain a number of important features infinancial time series The model can be expressed with mean equationand variance equation.

where E(at|It fi) = O and E(aX|I ) = fi, a is a sequence of i.i.d.random variables with mean O and variance fi, µ(It—fi) is a mean equation

at time t—fi The conditional variances can be defined as equation

conditional volatility as well as previous squared innovations

The most popular version of GARGH (p, q) used in practice isGARGH (fi, fi) with one lag for both term mean and variance, we haveGARGH (fi, fi) model at time t

t = w ‡ αfiat—fi ‡ Øfiot—fi, O Ç αfi, Øfi Ç fi, (αfi ‡ Øfi) c fi (3.3†)After parameters in GARGH (fi, fi) models are estimated, thenconditional variance is forecasted and inserts it into VaR equation toget the final VaR figure Three GARGH (fi, fi) models are performedwith assumed normal, Student‘s - t and skewed Student‘s - tdistribution

EWMA is a special case of GARGH model when both conditions w = Oand αfi ‡ Øfi = fi are satisfied (Jorion, £00t)

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GJÆ - GAÆCH model

In GARGH model, volatility and innovations are squared and henceeither negative or positive shocks have the same effect on conditionalvolatility It means that the signs have no effect to the conditionalvolatility which indicates that these models are symmetric However, thisdoes not follow to the st4tssed fact at in term of teserage eflectsmentioning that negative shocks tend to a higher impact than positiveshocks with the same magnitude on volatility This phenomenon is calledthe teserage eflects noted by Black (fi9t6) In order to account this fact,

an extension of the GARGH model namely GJR - GARGH wasintroduced by Glosten et al (fi993) GJR -

to current conditional volatility from shocks in past It means that when

çs has a positive value, past negative shocks have a stronger impact tocurrent conditional volatility than past positive shocks and vice versa

As same as GARGH (fi, fi), GJR - GARGH (fi, fi) is the mostcommon model in practice

t = w ‡ (αfi ‡ çfiIt—fi)st—fi ‡ Øfiot—fi (3.40)Again, normal, Student‘s - t and skewed Student‘s - t distributionas- sumptions are applied to distribution of returns for each respectivelyVaR model GJR - GARGH (fi, fi) model is also a special case ofGARGH(fi,fi) model when çfi = O

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oX

3.2.3 Extreme value theory (EVT) distributiom im

VaÆ modelimg

In previous sections, VaR models are described on modeling wholedistribu- tion of financial returns In common events where normal marketconditions are presented, normal or Student‘s - t distribution can forecastwell enough but they generates inaccurate estimation for the tails ofdistribution, espe- cially in fat - tails

Extreme value theory (EVT) is also a risk measure similar to VaR but

in only concentrates on the tail of distribution where extreme event occurwhile VaR studies on whole distribution (McNeil & Frey, £000) Thisapproach has also no assumption about return distribution so it can fitany probability distribution Following sections present EVT and itsapplication in practical

EVT risk modelimg amd measurimg

In financial risk management, EVT is applied in all areas such asmarket, credit, operational risk In this thesis, market risk is chosen tostudy and particular VaR using EVT is performed Extreme events occur

in the tail of distribution and then particular distribution is selected tocreate EVT model for VaR estimation

In the EVT context, there are two widely approaches of extremeevents model One of them is btoch massma modets which directlymodel the distri- bution of maximum realixations The block maximamodel requires a large data sample of identical and independentdistributed losses as well as many extreme observations in this sample inorder to employ well and produce a reliable result The other approach isPeahs − oser − fhreshotd (POT) mod- els focusing on losses whichexceed a particular high threshold POT models use return data moreefficiently and then they are considered as the most useful for practicalapplications (Gilli & Këllexi, £006; McNeil et al., £00†) Therefore,according to the benefit of POT models, we only focus on this approach

is chosen to estimate EVT VaR model

EVT approach has two important steps The first step is to modeldis- tribution of series of maxima or minima and under particularconditions, the distribution converge to one of three distributionsGumbel, Frechet or Weibull A generalixed distribution calledgeneralixed extreme value (GEV)

distribution is modeled to represent a standard form of these threedistribu- tions above The second step is to model distribution of excessover a given high threshold This result is applied in case of very highquantile such as

0.99 or even higher 0.999

Peak - over - threshold (POT) models

The GEV distributiom (Fisher - Tippett, Gmedemko theorem) pose that (Es}s(Æ is a sequence of independent and identicaldistribution (i.i.d.) random variables with common distribution function

Sup-5 (ı) = P r(Es Ç ı) having mean µ indication location and variance oX

indicating scale Let Mfi = Efi, Mn = M aı(Efi, EX, ., En), n Ç Xdefined as sample maxima of the i.i.d random variables If there is agiven sequence of sn > O, d c R and

d

some non - degenerate distribution functions H such that s—fi(M —d ) H,then H belongs to one of three families of distributions of Gnedenko (fi943):Gumbel: h(ı) = exp(— exp —ı), ı c R

According to Fisher - Tippett (fi9£8) theorem, asymptotic distribution

of maxima belongs to one of three specific distributions Gumbel,Fréchet and Weibull Fréchet and Weibull distributions become theshape of Gumbel when the tail index goes to œ and —œ A proof of thistheorem can be found in Gnedenko (fi943) Following to research fromvon Mises (fi936) and Jenk- inson (fi9††), Gumbel, Fréchet and Weibulldistributions can be displayed into one unified distribution calledgeneralixed extreme value distribution (GEV) by taking Ø = fi

where α is the tail index and Ø is a shape parameter

By replacing ı by ı—µ where µ c R, o > O, we can obtain a related location

- scale distribution HØ,µ,α as below

3fi