Master thesis in Finance (4210) Volatility and Value at Risk modelling using univariate GARCH models

Transcription

1 Sockholm School of Economics Maser hesis in Finance (410) Volailiy and Value a Risk modelling using univariae GARCH models Rishi Thapar (801) January 006 Tuor: Joel Reneby

2 Rishi Thapar (801) Maser Thesis-Volailiy and Value a Risk Modelling using univariae GARCH models Acknowledgemens I wish o hank Mr. Joel Reneby for his valuable guidance during he hesis. I am graeful o Mr. Måren Liljefors of Riksbank for his guidance and suppor during he hesis work. I wish o hank Mr. Johan Grönquis of Riksbank for inviing me o discuss an opporuniy o do hesis wih Riksbank. I also wish o hank Mr. David Brano of Riksbank for his suppor.

3 Rishi Thapar (801) Maser Thesis-Volailiy and Value a Risk Modelling using univariae GARCH models Table of Conens Acknowledgemens... Table of Conens Inroducion... 5 Theoreical background Sylized facs of marke volailiy Lepokurosis (heavy ails and sharp peaks) Volailiy Clusering Leverage effecs Modelling ime varying volailiy MA/EWMA ARCH GARCH Exensions o he basic GARCH model Value a Risk Relaed Research Overview Relaed Papers Wong Sham CM e al (003) Berkowiz and O Brien (00) Polasek and Pojarliev (003) Sarma e al (003) Daa Descripive Saisics Mehodology Evaluaion of predicive accuracy of Volailiy models Proxy for Acual Volailiy Error Saisics Non-Parameric ess on error saisics Mincer- Zarnowiz regression VaR evaluaion Basel Back-Tesing Kupiec Tes Chrisoffersen s Likelihood-Raio ess Loss Funcion ess Empirical Resuls Volailiy forecasing Models Volailiy Forecass Evaluaion Value-a- Risk Evaluaion... 7 Conclusions & Furher Suggesions References Appendix Appendix A Error Saisics for Volailiy Forecass Evaluaion

4 Rishi Thapar (801) Maser Thesis-Volailiy and Value a Risk Modelling using univariae GARCH models 9.1. Chrisoffersen s Likelihood-Raio ess Hypohesis ess Jarque-Bera es of normaliy Ljung-Box-Pierce Q-es Appendix B-Tables and Figures Lis of Tables Table 1: Descripive saisics (in-sample period) Table : Parameers for he esimaed GARCH_ (1, 1) Models... 0 Table 3: Volailiy Evaluaion... 1 Table 4: VaR Evaluaion... Table 5: Ljung-Box-Pierce Q-es Table 6: Log-Likelihood Values for Differen GARCH models Lis of Figures Figure 1: Log-reurns series Figure : Squared log-reurns Figure 3: Densiy and QQ plos... 3 Figure 4: Porfolio Losses and VaR

5 Rishi Thapar (801) Maser Thesis-Volailiy and Value a Risk Modelling using univariae GARCH models 1 Inroducion Value-a-Risk (VaR) has become he mos widely used marke risk measuremen mehodology in banks and financial insiuions. VaR for a porfolio is a funcion of volailiy of reurns of he porfolio. Therefore, he ask of VaR esimaion can be reduced o forecasing volailiy. Forecasing volailiy of financial ime series has been one of he mos acive areas of research in finance. In order o capure he empirically observed sylized facs of he financial ime series, mos imporan of hem being volailiy clusering and lepokurosis, various models have been proposed in lieraure. Generalised Auoregressive condiional heeroscedasiciy (GARCH) models have been very popular o model he ime varying variance of he reurns as a funcion of he lagged variance and lagged square reurns. Considering he prominence of VaR in risk managemen and presence of wide variey of alernaive VaR mehodologies, evaluaion of he predicive accuracy of VaR models is an imporan issue in risk managemen. The main challenge in VaR evaluaion is ha, like in he case of volailiy, VaR is a laen (unobserved) variable, herefore, i is no possible o calculae is acual realized value. This hesis seeks o answer he following wo quesions of Value-a-Risk Modelling -Do volailiy forecasing models based on GARCH lead o beer performance han a Naïve model ha measures volailiy by sandard deviaion of he pas reurns? -Wha evaluaion framework should be used o es he accuracy of he VaR esimaes? There has been large volume of lieraure on VaR modelling issues and approaches. There are many papers on evaluaing differen GARCH models for esimaion and forecasing of volailiy of financial asses reurns series. However, regarding evaluaion of VaR models, only a few papers have looked a VaR performance in pracice, mos imporan of hese are Berkowiz and O'Brien (00) and Jaschke, Sahl, and Sehle (003), using daily revenues and VaRs for U.S. and German banks respecively. Mos of oher papers have used simulaions or illusraive porfolios o evaluae differen VaR models. The reurns/p&l series daa and inernal VaR models of banks and financial insiuions are no publicly available. Therefore, here hasn been subsanial empirical work on VaR modelling of he acual porfolio reurns/profi & Loss disribuions and on evaluaion of VaR models, which are acually in use in he banks. In his hesis, volailiy and VaR modelling using he acual porfolio reurns of invesmen porfolios of he Cenral Bank of Sweden (Riksbank) is performed using univariae GARCH models. The hesis akes he reduced form approach of Berkowiz and O Brien (00) paper. In he reduced form model, volailiy or VaR of he porfolio is modelled direcly by fiing GARCH on daily reurns series raher han he common pracice of using GARCH o model risk facors of a porfolio in he conex of he popular variance-covariance approach of VaR modelling. The mehodology used in he hesis is also in line wih he Porfolio aggregaion mehod proposed in Riskmerics echnical paper (Zangari, P., 1997).The Porfolio Aggregaion approach esimaes VaR using he volailiy of he porfolio reurns raher han variance-covariance marix of he risk facors. Reduced form GARCH models, due o heir parsimony and flexibiliy, offer a simple alernaive o he srucural models. However, here is a need o empirically evaluae he performance of hese models. This hesis conribues in his direcion by empirically sudying he performance of reduced form GARCH modelling of he acual invesmen porfolios of a bank. A comprehensive evaluaion framework is employed o es he predicive accuracy of volailiy forecass and VaR esimaes. To es he accuracy of volailiy forecass, various error saisics and hypohesis esing are employed and o evaluae VaR esimaes, back esing mehodology recommended by Basel guidelines as well as advanced ess are performed. 5

6 Rishi Thapar (801) Maser Thesis-Volailiy and Value a Risk Modelling using univariae GARCH models The ouline of he hesis is as follows. In secion, heoreical background abou volailiy forecasing and VaR is presened. Deails abou some relaed previous papers are presened in secion 3. Daa used in he hesis is presened in secion 4. Deailed mehodology for model esimaion and volailiy forecasing, evaluaion of volailiy and VaR esimaes is presened in Secion 5. The empirical resuls and analysis of volailiy and VaR esimaes from he GARCH models are described in Secion 6 and secion 7 gives conclusions and also presens some suggesions for furher research. Theoreical background This secion inroduces various volailiy-forecasing models and discusses, in deails, GARCH based models. This secion also gives a brief overview of Value a Risk concep. The secion sars wih salien sylized facs abou he volailiy of financial asses reurns, which needs o be considered while specifying volailiy forecasing models..1 Sylized facs of marke volailiy A ypical rading porfolio of a bank or invesmen house is characerised by variey of differen ypes of financial asses and derivaives and hus has linear as well as non-linear exposure o a se of marke facors. Therefore, he porfolio reurn series usually exhibis non-normal and fa ail characerisics. I is imporan o consider empirically observed sylized facs of marke volailiy of a financial asse/porfolio before specifying a volailiy esimaion model. A model s abiliy o capure imporan empirical sylized facs is a desirable feaure. The imporan sylized facs abou of financial asses series, which have been documened in numerous sudies, are described below..1.1 Lepokurosis (heavy ails and sharp peaks) The disribuion of he financial asses reurns is lepokurosis, i.e., exhibi excess kurosis (heavyails) and sharp peaked. Typical kurosis esimaes for he financial reurn series are found o be in he range of 4 o 50. A normal disribuion has kurosis value equals 3. Therefore, kurosis value exceeding 3 indicaes heavy-ails. In a heavy-ailed disribuion, exreme oucomes are more frequen han wha he use of a normal disribuion would predic. Even afer correcing reurns for volailiy clusering (e.g. via GARCH-ype models and/or fiing fa-ailed disribuions), he residual ime series sill exhibi heavy ails. Therefore, his non-gaussian and heavy-ailed characerisic of financial ime series makes i necessary o use oher measures of dispersion han he sandard deviaion in order o capure he variance of he reurns..1. Volailiy Clusering Exreme reurns show high variabiliy, as eviden from he heavy ails and non-negligible probabiliy of occurrence of exreme values. Also exreme values appear in clusers, exreme reurns o be followed by oher exreme reurns, alhough no necessarily wih he same sign. The implicaion of volailiy clusering is ha he volailiy shocks oday influences he expecaion of volailiies of many fuure periods ahead. Reurn series are no sricly whie noise alhough hey show lile auocorrelaion especially in liquid markes. The absence of auocorrelaions in reurns series gives some empirical suppor for random walk heory in which he reurns are considered o be independen random variables. However, i has been shown empirically ha his absence of serial correlaion does no imply any nonlinear funcion of reurns will also have no auocorrelaion. Absolue or squared reurns exhibi significan posiive auocorrelaion or persisence (slow decay in auocorrelaions). Therefore, due o his nonlinear dependence, financial ime series have auo correlaion in volailiy of reurns bu no in he reurns hemselves. Also, if we increase he ime scale i.e. weekly and monhly reurn series end o exhibi serial correlaion. 6

7 Rishi Thapar (801) Maser Thesis-Volailiy and Value a Risk Modelling using univariae GARCH models.1.3 Leverage effecs I is found in many sudies ha here are leverage effecs (i.e. volailiy of reurns are negaively correlaed wih he reurns of he asses.) in financial ime series. A negaive shock leads o a higher condiional variance in he subsequen period han a posiive shock would do.. Modelling ime varying volailiy In forecasing volailiy, especially in modelling variance of shor-horizon asse reurns, usually mean reurn is assumed equal o zero. This is jusified by he argumen ha mean reurn of an asse is ypically several orders of magniude lower han is sandard deviaion. Therefore, he firs momen for he reurn series is usually defined as below. r The mos popular class of volailiy forecasing models, described below, are discree-ime parameric volailiy models, which explicily model he expeced volailiy, (h-sep ahead variance) as a non-rivial funcion of he hisorical ime informaion se, F. Therefore, hese models parameerize he firs wo condiional momens (mean and variance) of he reurns ime series. These models can be broadly classified ino hree caegories viz. MA/EMWA models, ARCH and Sochasic Volailiy (SV) models. MA/EMWA and ARCH models are described below...1 MA/EWMA Moving Average (MA) models are one of he simples models, where forecased volailiy is calculaed as moving average of he hisorical variance. In he case of exponenial weighed moving average (EWMA) models volailiy of he nex period is forecased as a MA process of weighed square deviaions from he mean and he weighs decay exponenially wih a decay facor. EWMA models are more responsive han he simple moving average o sudden changes in volailiy. Risk Merics, he mos commonly used model in pracice employs EWMA model o model he variance wih = 0.94 and can be represened by he following equaion. 1 1 a0 (1 )( r ) In fac, he Risk Merics model is a non-saionary version of GARCH (1, 1), where he persisence parameers sum o 1... ARCH In 198, R F Engle inroduced ARCH class of models in which ime-varying condiional variance is modelled wih he AuoRegressive Condiional Heeroscedasiciy (ARCH) processes ha use pas disurbances o model variance of he series. In oher words, oday s condiional variance is a weighed average of he pas squared disurbances. An ARCH (q) model is specified by r where ( r 1 F ) 1 and ( F and The variance q 1 0 j j1 j1 normal bu heeroscedasic...3 GARCH 1 ) h 1. Therefore, condiional on he pas, he ARCH model is Generalized Auoregressive Condiional Heeroscedasiciy (GARCH) model, firs suggesed by Tim Bollerslev in 1986 is obained by adding p auoregressive erms for o he ARCH (q) model. Therefore GARCH (p, q) has he following specificaion. 7

8 Rishi Thapar (801) Maser Thesis-Volailiy and Value a Risk Modelling using univariae GARCH models q p 1 0 j j1 j j1 j1 j1 Wih 0 0, 0 j for all j = 1 o p, j 0 all j = 1 o q and j j < 1 Jus as an ARMA model ofen leads o a more parsimonious represenaion of he dependencies in he condiional mean han an AR model, he GARCH (p, q) model provides a similar added flexibiliy over he linear ARCH (q) model when paramerizing he condiional variance. The ARCH (q) model corresponds o a GARCH (0,q) model. GARCH (p,q) models are used in pracice because GARCH (p,q) model allows for parsimonious parameerizaion of an ARCH ( ) model. In pracice, low order GARCH models are widely used. GARCH (1,1) model is given by he following equaion. 1 0 q j1 p j1 The variance 1 is a weighed average of long erm variance 0 /(1- - ), prior variance wih weigh and squared disurbance erm wih weigh The resricions on he parameers and are 0 <= 0 <= 0, 0 <= <= 1, 0 <= <= 1 and + < 1. These resricions ensure ha he weighs are posiive and sum o 1. The magniude of and deermines he shor-erm dynamics of he forecased volailiy series. A large value of indicaes persisence, i.e. Shocks (exreme values) of he condiional variance will ake long ime o die ou. Large value of indicaes ha he volailiy reacs quie fas o he marke movemens. The above formula for GARCH (1,1) nicely demonsraes he essence of he volailiy clusering feaure in he GARCH model. If he marke has been volaile in he curren period, nex period's variance will be high, which is inensified or offse in accordance wih he magniude of he reurn deviaion of he curren period. If, on he oher hand, oday's volailiy has been relaively low, omorrow's volailiy will be low as well, unless oday's porfolio reurn deviaes from is mean considerably. The impac of hese effecs depends on he parameer values. For + < 1, he condiional variance exhibis mean reversion, i.e., afer a shock i will evenually reurn o is uncondiional mean. The condiion + < 1 also ensure ha model is covariance-saionary. Due o is abiliy o capure salien feaures of he reurn dynamics in very parsimonious and easily esimaed specificaions, GARCH has become he popular model in financial risk managemen...4 Exensions o he basic GARCH model The basic GARCH model is usually a good saring poin while modelling volailiy bu various exensions and varians o he basic GARCH(p,q) model have been proposed and used in finance. The developmen of hese exensions and varians aim o capure he sylized facs of he financial asses disribuion in a beer manner. One of he major resricions of he basic GARCH model is ha fails o capure he asymmeric or leverage effecs i.e. asymmerical response of volailiy o he marke moves. I is I is ofen noiced in he financial markes ha a negaive shock leads o a higher condiional variance in he subsequen period han a posiive shock would do. Exponenial GARCH or EGARCH, inroduced by Nelson in 1991, capures his asymmeric response by specifying he condiional variance as a funcion of no only of he magniudes of he lagged residuals and bu also heir signs. Many empirical sudies have also shown ha condiional disribuion for he error erm in he condiional mean equaion ofen has heavier ails han he Gaussian disribuion as assumed in he basic GARCH model. Alhough in he basic GARCH model, 8

9 Rishi Thapar (801) Maser Thesis-Volailiy and Value a Risk Modelling using univariae GARCH models condiionally normal disribuions produce heavy-ailed uncondiional disribuions, ofen his is no enough o capure he excess kurosis in he daa. Therefore, Symmeric bu fa-ailed disribuions like Suden- or generalized error disribuion (GED) has been used insead. In he hesis following varians of GARCH models are used. GARCH wih normal disribued errors (GARCH) z, z F ~ (0,1) N GARCH wih suden disribued errors (GARCH_) z, z F ~ ( ) v d Exponenial GARCH wih normal errors (EGARCH) As explained above, exponenial GARCH model can capure he asymmeric response of volailiy o he marke moves. Taking logarihms of condiional variances allows asymmery in response of volailiy o marke moves. Wih appropriae condiioning of he parameers, he EGARCH specificaion below capures he sylized fac of leverage effecs. z, z F ~ (0,1) N log E 1 log GARCH in mean (GARCHM) The GARCH-M model is defined simply by aking he condiional variance as a regressor in he mean equaion. r 1 1 z, z F ~ (0,1) N Asymmeric Threshold GARCH (ATGARCH) Anoher asymmeric specificaion for a GARCH model is ATGARCH model. The idea behind his model is ha asymmeric behaviour of he negaive shocks are sources for addiional risk. z, z F ~ (0,1) Where 1 N 1 0 1( 1) D ( 1) 1 D = 1 if 1 oherwise i is zero. 1 parameer. is asymmery parameer and is hreshold Asymmeric GARCH (AGARCH) Asymmeric model capures asymmerical response of volailiy o he marke moves. In his specificaion,, he hreshold parameer is se o zero. z, z F ~ (0,1) N 9

10 Rishi Thapar (801) Maser Thesis-Volailiy and Value a Risk Modelling using univariae GARCH models 1 0 1( 1) D ( 1) 1 Wih = 0 Threshold GARCH (TGARCH) Anoher asymmeric specificaion is he hreshold GARCH (TGARCH) model, which adds a dummy variable o he GARCH process. In his specificaion, 1, he asymmeric parameer is se o zero. z, z F ~ (0,1) N 1 0 1( 1) D ( 1) 1.3 Value a Risk Value a risk (VaR) approach has emerged as indusry sandard o measure marke risk, boh for capial requiremens and for inernal risk conrol, during he las few years. In he 1996 amendmen of Basel Accord, which oulined marke risk capial requiremen for he banks, advised he banks o use VaR approach for assessmen of heir marke risks and for calculaing regulaory capial requiremen. VaR of a porfolio is defined as he maximum loss on he porfolio ha can be expeced wih a cerain level of confidence over a cerain holding period. To inroduce some noaion, consider a porfolio of risky asses and assume V as he value of porfolio a ime. Assume ha we wan o calculae risk for he ime period [, +1]. We denoe he loss disribuion of he porfolio by L 1 ( V 1 V ) and he disribuion funcion for he loss series is F L such ha P( L x) FL ( x). Then VaR a ( (0,1) ) can be defined as he -quanile of F L. In oher words, VaR is produc of sandard deviaion of disribuion of L 1 series and -quanile of he sandardized disribuion wih uni variance ( ) and zero mean. VaR ( ) 1 q ( F L) q ( ) 1 Nominal value of VaR can is hen a produc of Value of he porfolio a, -quanile of he sandardized disribuion and volailiy of reurn series r V V ) / V VaR ( ) 1 V q ( ) ' 1 1 ( 1 Therefore, as eviden from he above expressions, VaR is a funcion of he volailiy forecas and is dependan on he assumpions of disribuion of loss/reurn series. Therefore, accuracy of VaR relies on accuracy of volailiy forecass. A common assumpion while calculaing VaR is ha he reurn series are normally disribued. However, as we have discussed in he earlier secions, he reurns show fa-ailed. Therefore, his assumpion of normaliy under-esimaes VaR and inroduce subsanial model risk. In conclusion, his secion begins wih an overview of salien sylized facs of financial ime series. The accuracy of a paricular volailiy model is dependan on he degree o which i is able o capures hese characerisics of financial ime series. Hence, i is imporan o consider he above-menioned sylized facs while modelling volailiy. In his secion, differen discree-ime parameric volailiy models were explained. The concep behind GARCH models was presened and various exensions of GARCH models were inroduced. GARCH models have become popular in financial risk managemen due o heir abiliy o capure salien feaures of he reurn dynamics in very parsimonious and easily esimaed specificaions. 10

11 Rishi Thapar (801) Maser Thesis-Volailiy and Value a Risk Modelling using univariae GARCH models 3 Relaed Research This secion presens a brief overview of he various academic papers ha have sudies volailiy forecasing and VaR models. Afer ha, a summary of he mehodology and resuls of research papers, ha are closely relaed o he hesis are presened. 3.1 Overview Over he years, researchers have approached volailiy forecasing from differen angles viz. hisorical ime series based models like AR/EWMA, GARCH, sochasic volailiy, implied volailiy models, non parameric models, geneic and neural neworks based models. The research paper by Poon and Granger (00) does an exensive survey of 93 differen papers ha sudied forecasing performance of volailiy models. Ou of hese abou 17 papers sudied performance of GARCH models. According o he paper, alhough, he conclusions from he papers vary a lo, hey have significan common characerisics e.g. They es a large number of very similar models all designed o capure volailiy persisence They use a large number of error saisics each of which has a very differen loss funcion They forecas and calculae error saisics for variance and no sandard deviaion, which makes he difference beween forecass of differen models even smaller They use squared daily, weekly or monhly reurns o proxy daily, weekly or monhly acual volailiy, which resul in exremely noisy volailiy esimaes. The noise in he volailiy esimaes makes he small differences beween forecass of similar models indisinguishable. Value-a-Risk (VaR) has become a well-known ool for measuring marke risk since he implemenaion of he Basel accord on Capial Requiremens (1996). There has been large volume of lieraure on VaR modelling issues and approaches. The inernal VaR models and VaR figures of banks and financial insiuions are, however, no publicly available. Many papers have used simulaions or illusraive porfolios o evaluae differen VaR models. Therefore, here hasn been subsanial empirical work on evaluaion of VaR models, which are acually in use in he banks and financial insiuions. 3. Relaed Papers 3..1 Wong Sham CM e al (003) This paper evaluaes performance of VaR forecass of nine univariae ime series models (random walk wih consan volailiy, AR/ARMA models wih consan volailiy, and AR/ARMA reurns model wih GARCH (1,1) volailiy), using Basel back-esing crieria. The paper has seleced a sock porfolio, Ausralia All Ordinary Index (AOI), as a proxy for he porfolio of a bank. The normaliy of he disribuion of he porfolio reurns is assumed. Therefore, VaR is calculaed by muliplying square roo of he forecased variance wih he Marke Value of porfolio and he criical value of he required confidence level. Relaive VaRs (which refers o he percenage of he porfolio value which may be los afer h-day holding period wih a specified probabiliy), for long and shor posiions of porfolio on AOI are also calculaed. The VaR models are esimaed using 4000 observaions and one-sep ahead forecass are produced. For evaluaing reproducive accuracy of he volailiy forecass, Mean Square Error (MSE) and Mean Absolue Error (MAE) are calculaed. In order o es he accuracy of VaR esimaes, failure raes and size of he forecas errors are calculaed. Dividing he sample period ino four sub periods ess he robusness of he resuls and he performance of he models are evaluaed across hese sub periods also. The key conclusion of he paper is ha ARCH and GARCH models consisenly fail back-esing whereas models of consan volailiy pass back-esing for mos of he sub periods. 11

12 Rishi Thapar (801) Maser Thesis-Volailiy and Value a Risk Modelling using univariae GARCH models 3.. Berkowiz and O Brien (00) This paper uses he acual daily Profi/Loss (P/L) and VaR of he rading books of six US banks o evaluae he inernal srucural VaR models of he Banks agains an ARMA-GARCH (1,,1) model. I employs Likelihood- Raio ess for comparing uncondiional as well as condiional coverage of he models. The inernal VaR models under evaluaions are parameric VaR models, based on variance and co-variance beween he various risk facors affecing he rading porfolios of he bank. Thus, hese models ake ino consideraion he effec of he change in he porfolio posiions. Whereas, he ARMA-GARCH (1,1) doesn akes his ino consideraion and model he VaR based on he condiional volailiy of he hisorical P&L. Therefore, he null hypohesis in he paper is ha he inernal VaR models perform beer han he ime series based ARMA-GARCH (1) model. The null hypohesis is rejeced for almos all banks and he GARCH model based on daily rading P&L ouperforms inernal VaR model for all he banks. The resuls show ha GARCH models generally provide for lower VaRs and are beer a predicing changes in volailiy. However, he mean violaion rae for he GARCH VaRs also is lower han ha of he banks VaRs. The inernal VaR models pass es of uncondiional coverage (wih mean violaion rae of 0,5% for 99% VaR), however he magniude of he failures (exceedence of losses over VaR) were high (beween -4 sandard deviaions beyond he mean VaR) and he failures end o be clusered. The clusering of failures indicaes ha he srucural VaR models are no able o capure ime-varying volailiy adequaely. The average GARCH VaRs are also lower han ha of inernal VaR models bu he sriking resuls are ha he violaions in he GARCH VaRs are no larger han ha in banks inernal VaRs models. Thus, he paper concludes ha GARCH models are beer because hey imply low level of regulaory capial requiremen wihou producing larger violaions. Alhough he GARCH models canno accoun for posiions sensiiviies o curren risk facor shocks or changes in curren posiions, hey are more parsimonious and accurae o model he dynamics of porfolio P&L Polasek and Pojarliev (003) This paper sudies he comparaive performance of ime series models based on Risk Merics s EWMA and differen GARCH models viz. GARCH wih normal errors, GARCH wih -disribuion errors, asymmeric GARCH and exponenial GARCH and Power GARCH. A hypoheic porfolio of 1 Million USD invesed in QQQ (a share ha racks NASDAQ 100 index) is used. The normaliy of reurns of NASDAQ 100 index is assumed. Therefore, one day 95% VaR is given by muliplying he square roo of he forecased variance by Various p and q values for GARCH (p,q) model were run and GARCH (1,1) was chosen on he basis of lowes AIC and BIC. Regressing he squared reurns on a consan and on he forecased variance compared volailiy-forecasing performance of he models. The performance of differen VaR models is evaluaed using failure rae. The likelihood raio ess of uncondiional, independence and condiional coverage for 1%-10% VaR range were done. Furher, loss funcion ha incorporaes penalies (a funcion of failure rae) and VaR cos (opporuniy cos due o overesimaion of VaR) was also used. GARCH model wih normal errors performs bes in erms of lowes number of failures and loss funcion. I passes condiional coverage es for %-5% VaR range. GARCH models in general performed beer han Risk Merics and consan volailiy models Sarma e al (003) This paper uses a comprehensive VaR model selecion framework, wih failure rae, likelihood-raio and regression based ess for condiional coverage, loss funcions and one-sided non parameric sign ess. 16 models based on EWMA (for 50,15,50,500 and 150 days window), Hisorical Simulaion (for 50,15,50,500 and 150 days window) and AR (1)-GARCH (1,1) for VaR esimaion (95% and 99%) of S&P 500 and Nify (India s NSE sock index) indices. For boh 95% & 99% VaR of S&P 500, no model was able o pass regression based condiional coverage ess. For 95% VaR of Nify Risk Merics model performed bes and survived all ess. For 99% VaR, AR (1)-GARCH (1) performed bes. 1

13 Rishi Thapar (801) Maser Thesis-Volailiy and Value a Risk Modelling using univariae GARCH models In conclusion, volailiy and VaR forecasing has been approached from differen angles in financial research wih GARCH based modelling being one of he imporan approaches. The four relaed papers, discussed in deails in his secion, have sudied GARCH models among he oher models. These papers differ from each oher paricularly in erms of how he predicive accuracy of he volailiy or VaR forecass is evaluaed. As seen in mos of he oher relaed research papers, hese papers (excep Berkowiz and O Brien (00).) use sock marke index or hypoheical porfolios. Berkowiz and O Brien (00) has analyzed he disribuion of hisorical rading P&L and he daily performance of VaR esimaes of six large U.S. banks. 13

14 Rishi Thapar (801) Maser Thesis-Volailiy and Value a Risk Modelling using univariae GARCH models 4 Daa The daa consis of log reurns of he hree invesmen porfolios primarily consising of governmenguaraneed securiies denominaed in he foreign currencies named as Porfolio A, Porfolio B and Porfolio C in his hesis. The log reurns daa of he respecive benchmark porfolios (named Benchmark A, Benchmark B and Benchmark C) for each invesmen porfolio are also used. These benchmark porfolios incorporae he Bank s preferences for liquidiy, risk and reurn and performance of he invesmen porfolios are evaluaed agains he respecive benchmark porfolio. Therefore, he volailiy of a benchmark porfolio can be considered as a good proxy for he volailiy of he corresponding invesmen porfolio and hus can be used o esimae VaR of he invesmen porfolio. The daa used consis of daily reurns of he hree invesmen porfolios and corresponding Benchmark porfolios. An ou of sample daa consising of 50 daily reurns are used for evaluaing volailiy and VaR forecass. 4.1 Descripive Saisics From Figure 1 (in Appendix B), i is eviden ha all he porfolio reurns series exhibi volailiy clusering. The plos of squared reurns in Figure also corroborae volailiy clusering. Similarly, from he densiy plos and Quanile-Quanile (QQ) plos in Figure 3, i is clear ha all he series show non-normal and fa-ailed behaviour. The Auocorrelaion Funcion (ACF) and Parial Auocorrelaion Funcion (PACF) plos of he reurn series don show significan auo-correlaion bu he squared reurns do exhibi auo-correlaion, up o lag lenghs more han 10. I can be followed from he Table 1 below, ha all he series exhibis heavy ails (excess kurosis values differen from zero). This indicaes he necessiy of fa-ailed disribuions o describe he reurns series condiional disribuion. I can be seen from able ha we can rejec he null hypohesis of normaliy in all reurns because he p-values are lower han 5%. Table 5 in Appendix B, presens he Jarque-Bera es saisic and associaed p-value of his es for lags 5, 10, 0, 30 and 50 for boh reurns and squared reurns. For mos of he reurn series, we can rejec he null hypohesis of no auocorrelaion as he p values are greaer han 0.05.For all squared reurn series, excep in case of BENCHMARK A and BENCHMARK C, ha we can rejec he null hypohesis of no auocorrelaion. Table 1: Descripive saisics (in-sample period) PORTFOLIO A BENCHMARK A PORTFOLIO B BENCHMARK B PORTFOLIO C BENCHMARK C N Mean 0,0001 0,0003 0,0006 0,0007 0,0005 0,0007 Sd. Dev. 0,00 0,00 0,001 0,001 0,0031 0,0031 Skewness -0,4133-0,4166-0,5195-0,4965-0,5709-0,6096 Excess Kurosis 0,7496 0,6975 0,8969 0,8395 0,8361 0,614 Jarque-Bera 14,011 (0,0009) 13,630 (0,0011) 18,64 (0,0001) 17,549 (0,000) 1,796 (0,0000) 6,743 (0,0000) 14

15 Rishi Thapar (801) Maser Thesis-Volailiy and Value a Risk Modelling using univariae GARCH models 5 Mehodology This secion presens volailiy &VaR forecasing and evaluaion mehodology used in he presen hesis. VaR modelling of a porfolio based on ime-series models involves specifying a parameric disribuion for he porfolio reurns and esimaing he parameers of he disribuion using hisorical daa. The VaR of he porfolio can be hen calculaed by muliplying he square-roo of he condiional variance wih he appropriae criical value for he sandardized disribuion. In he presen hesis, GARCH models are used o forecas volailiy for he reurns of he invesmen porfolios of he Riksbank. Following ypes of GARCH (1,1) models are esimaed for all six series. GARCH wih normal disribued errors (GARCH), GARCH wih suden disribued errors (GARCH_), Asymmeric GARCH (AGARCH), Asymmeric hreshold GARCH (ATGARCH), Exponenial GARCH wih normal errors (EGARCH), GARCH in mean (GARCHM) and Threshold GARCH (TGARCH). The bes model is seleced on he basis of log-likelihood value and AIC. The seleced model is hen used o produce one-day ahead variance forecas. The sample is hen rolled one-day ahead and he model is re-esimaed and again a one-day ahead forecas is generaed and so on. In his way, 50 ouof-sample one-day ahead forecass are generaed. Ou of sample period wih 50 observaions is used o evaluae predicive accuracy of he variance forecass. The volailiy forecass from he GARCH models are compared wih a Naïve model, in which he volailiy calculaed using rolling sandard deviaion of he pas n observaions (n equals he number of observaions in he in-sample daa of he GARCH model). The volailiy evaluaion mehodology is explained in deail in secions below. Daily VaR forecass are esimaed for he ou-of-sample period. Daily VaR for a porfolio for ime can be calculaed on day -1 by using he volailiy forecas for day done on he day -1 Three VaR forecass for an invesmen porfolio are calculaed for each day, one using volailiy forecas of he reurns of he invesmen porfolio. Second, using he volailiy forecas of he corresponding benchmark porfolio. Third, using he volailiy forecas of he Naïve model. VaR forecass evaluaion is done using muliple ess, which are deailed, in secion below. 5.1 Evaluaion of predicive accuracy of Volailiy models In order o evaluae VaR models, i is imporan o compare he accuracy of he volailiy forecasing process underlying in he VaR model. Evaluaing volailiy forecass however, poses a challenge. Since, volailiy is a laen (unobserved) variable, herefore, i is no possible o calculae he acual rue volailiy. Therefore, he ex-pos evaluaion of volailiy forecas accuracy mus conen wih he fundamenal error-in-variable problem due o his issue. In mos of he empirical sudies of volailiy forecasing, daily squared reurns are used as a proxy for acual volailiy. This hesis also akes same approach, as explained below Proxy for Acual Volailiy Consider he following specificaions for he reurns r and z Where z is i.i.d. r F F because 1 0 Therefore, F 1 1 F z F z F

16 Rishi Thapar (801) Maser Thesis-Volailiy and Value a Risk Modelling using univariae GARCH models Because 1 z F 1 z ~ N(0,1) and Alhough is an unbiased esimaor for z ~ 1, i is a noisy esimaor due o is asymmeric disribuion. Therefore, some sudies have used inra-day high frequency reurns o consruc a beer proxy for he rue realized volailiy. True volailiy is esimaed by he sum of inraday squared reurns a shor inervals such as fifeen minues. Such a volailiy esimaor has been shown o provide an accurae esimae of he laen process ha defines volailiy. In his hesis, squared daily reurns are used as a proxy for he rue volailiy because he inraday reurns daa was no available Error Saisics Differen error saisics and hypohesis ess based on regression and quadraic loss funcions are used o assess he predicive accuracy of he models.. To evaluae performance of he differen models in forecasing condiional variance, models, error saisics used are as follows - Mean Square Error (MSE) - Median Square Error (MedSE) - Mean Absolue Error (MAE) - Adjused Mean Absolue Error (AMAPE) - Mean Mixed Error for under-predicions (MME(O)) - Mean Mixed Error for over-predicions (MME(U)) - Theil- Inequaliy Coefficien (TIC) These compuaions of AMAPE, MME and TIC are explained in secion A1 of Appendix A Non-Parameric ess on error saisics I is usually no sufficien o compare wo or more compeing models by aking inro considering he average error saisics like MSE, MAE ec. In order o es he superioriy of one model over oher, i is also imporan o see if he specified error loss funcions (e.g. MSE ec.) are saisically significanly beer in one model han in oher. One of he ways o do is o employ non- parameric sign and/or rank ess. In his hesis, one-sided non parameric sign es is used, as used in Sarma e al (003). The null hypohesis of his es is ha boh models under consideraion have same forecasing accuracy agains a one-sided alernaive hypohesis of superioriy of one model over he oher. H 0 : 0 H1 : 0 Where is defined as he median of he differenial loss funcion disribuion, dl dl, where li l j, wih l i and l j are loss funcion for model i and j respecively for day. Furher, define an process, s, where s = 1 if dl 0 0 oherwise 0 oherwise The sign saisic is hen given by S 50 ij s 1 Under null hypohesis, he sandardized S, as given below is asympoically sandard normal. a ij S ( S 0.5* 50) / 0.5* 50 ~ N (0,1) ij ij 16

17 Rishi Thapar (801) Maser Thesis-Volailiy and Value a Risk Modelling using univariae GARCH models a ij If S 1. 66, we can rejec he null hypohesis a 5% confidence level. Rejecion of null hypohesis means ha ha model i is significanly beer han model j in erms of he given loss funcion. The advanage of his sign es is ha he disribuion of he sign saisic is agnosic o he loss funcion disribuion under consideraion Mincer- Zarnowiz regression In addiion o he above error saisics a regression based performance measure,, known as Mincer- Zarnowiz regression, is used o evaluae condiional bias in he volailiy forecasing models. I has been largely used for evaluaing economics forecass. However, many sudies have also used i and is varians for he condiional variance evaluaion. The forecased condiional variance is regressed on a consan and on he ex-pos rue variances( proxied by squared reurns) for he ou-of-sample period and where r The necessary condiion for o be condiionally unbiased is 0 and 1. The forecasing performance of a model can be measured using 5. VaR evaluaion R of he regression. Considering he prominence of VaR in risk managemen and presence of wide variey of alernaive VaR models, assessmen of VaR esimae is an imporan issue in risk managemen. The risk arising from he fauly forecass i.e. model risk, is an imporan issue in risk managemen in financial insiuions. The financial regulaory organizaions need o make sure ha he VaR models used by he banks are no sysemaically biased. Alike evaluaion of volailiy models, evaluaion of VaR forecas is no sraighforward because acual VaR is unobservable. Various mehodologies for VaR evaluaion, which are used in he hesis, are discussed below Basel Back-Tesing Back-esing of a VaR model, as recommended by on Basel guidelines for marke risk capial requiremens, requires he model o be accurae( a model is accurae if he acual loss is smaller han he VaR forecas) a leas a leas on 99%( for VaR wih 1% significance level) and 95% (for VaR wih 5% significance level) of he ime. There should be a leas 50 days( around 1-year daa) for back esing he daily VaR. However, his simple approach of evaluaing a VaR model is neiher powerful nor accurae because o pass he back-es, a VaR model needs only o be correc on average and also his es doesn ake ino accoun he magniude of failures and independence of failures. 5.. Kupiec Tes More sophisicaed ess have been proposed in lieraure o es he saisical accuracy of he VaR forecass. The Kupiec es is based on a likelihood-raio es- saisic. To fix noaion, consider a series of one day ahead VaR forecass which are esimaed a confidence level 1 p ( e.g. for 95% VaR, p = 0,05). 1,T and We can define a failure process, f wih f = 1 if VaR Acual loss 0 oherwise 17

18 Rishi Thapar (801) Maser Thesis-Volailiy and Value a Risk Modelling using univariae GARCH models The process f is a binomial process wih independen draws of 1s and 0s. Under he null hypohesis, VaR esimaes are accurae and hus probabiliy of occurrence of failures ( when f 1) on each draw equals p. T n n Since he probabiliy of occurrence of n number of failures is ( 1 p) p, he LR saisic is given by T n n T n n LR ln (1 p) p ln (1 n / T ) ( n / T ) uc Under null hypohesis, he above es saisic has a disribuion wih one degree of freedom. This es has some limiaions. Firsly, since, he failures occur rarely (by design), Kupiec es has poor power characerisics, which become worse as he confidence inerval being esed increases. We need a large sample size for he es o have significan power. Secondly and more imporanly, his es assumes ha he occurrences of failures are uncondiional because his es provides average and uncondiional (i.e. wihou reference o he informaion available a each ime poin) coverage by simply couns he failures over he enire period and his es lacks power agains he dependence beween he failures i.e. he zeros and ones come clusered ogeher in a ime-dependen fashion Chrisoffersen s Likelihood-Raio ess Since, VaR are inerval forecass (i.e. one-sided inerval forecass of he porfolio reurns), here is more informaion available in he failure process raher han jus average coverage. Also, due o he presence of persisence and condiional heeroscedasic volailiy of porfolio reurns, condiional probabiliies of failures should also be esed. For example, in he periods of high volailiy of porfolio reurns, he VaR forecass should be larger han over-all average value and vice-verse. A VaR model ha ignores he ime-varying dynamics of he reurns, migh produce correc uncondiional coverage, bu i may fail o accoun for persisence and ime-varying aribues. Chrisoffersen developed Likelihood-Raio ess for evaluaing uncondiional coverage, independence and correc condiional coverage. These ess are described in deails in Appendix A Loss Funcion ess Apar from he hypohesis based ess of Kupiec and Chrisoffersen, VaR models can also be evaluaed using loss funcions, ha es economic significance raher han saisical significance and ake consideraion he specific ineress (in oher ohers uiliy funcion) of he risk managers. Lopez(1999) inroduced regulaory loss funcions ha assign a numerical score, which reflecs specific regulaory concerns, o VaR esimaes. A model ha has minimum value of he loss funcion is he beer one. One example of such a loss funcion, which is used in he hesis, is L = 1+(Acual Loss VaR ) if VaR Acual loss 0 oherwise The above loss funcion akes ino consideraion he magniude of he failure, i.e. by how much he acual loss exceeds VaR esimae and hus penalises he model ha produces higher magniude. The abiliy o inroduce exra informaion, i.e. abou he magniude of he failure and flexibiliy o define specificaion of he loss funcion are wo main advanages of his loss funcion. Two differen VaR models can be easily compared by designing a simple hypohesis es based on he above menion loss funcion and by performing a one-sided sign es, which is described in deail in he previous secion. In conclusion, in his hesis, seven candidae GARCH models are examined o forecas volailiy and a final model is chosen according o of highes log-likelihood values. The one-day ahead volailiy 18

19 Rishi Thapar (801) Maser Thesis-Volailiy and Value a Risk Modelling using univariae GARCH models forecass are hen obained from he final chosen model and accuracy of his forecas is deermined by uilizing comprehensive crieria ha include error saisics calculaion, Mincer- Zarnowiz regression and sign ess on he error saisics. Daily VaR forecass are using volailiy forecas of he porfolio, of he benchmark porfolio and by using naïve forecas.esimaed for he ou-of-sample period and hese forecass are evaluaed using a comprehensive VaR evaluaion framework. 19

20 Rishi Thapar (801) Maser Thesis-Volailiy and Value a Risk Modelling using univariae GARCH models 6 Empirical Resuls 6.1 Volailiy forecasing Models GARCH (1, 1) models were evaluaed on all of he six log-reurns series and log-likelihood values are abulaed in able 6 in Appendix B. In some cases, no convergence was reached while fiing GARCH models. In hose cases, no value of log-likelihood value was repored in his able. Table-6 in appendix B shows GARCH_ (1, 1) i.e. GARCH models wih suden disribuion errors performed bes in erm of highes log-likelihood values for all series. Also, i performed bes in erms of lowes AIC values for almos all series. Therefore, GARCH_ (1, 1) model is used o perform oneday ahead volailiy forecass. Table gives he parameers of he GARCH_ (1, 1) models fied o he differen series. Table : Parameers for he esimaed GARCH_ (1, 1) Models PORTFOLIO BENCHMARK PORTFOLIO BENCHMARK PORTFOLIO BENCHMARK A A B B C C µ 0,0005 0,0007 0,0003 0,0003 0,0004 0,00041 á0 3,E-07 4,9E-06,4E-07,4E-07 8E-07 6,9E-06 á1 0, ,015 0, ,0479 0,0788-5E-05 â1 0,8909 0,015 0,9014 0,8984 0, ,704 df 11,737 10,7789 9, ,76 8, ,33711 As inferred from he esimaed values of he coefficiens of he fied GARCH models, in all porfolio series, excep for BENCHMARK A and BENCHMARK C, he value of 1 1, is more han 0,9. This implies ha hese series exhibi high volailiy persisence and ha he response funcion of volailiy of shocks decays a a relaively slow rae. As he sum ends o 1 he higher is he insabiliy in he variance and shocks end o persis insead of dying ou. For example, in case of PORTFOLIO A series, 0, 93, meaning ha 93% of a variance shock remains he nex day. 1 1 /( 1 ) The long-erm seady sae variance in a GARCH (1,1) model is given by If we compare he value implied by he each of he models above, i comes ou o be approx. equal o he square of he sandard deviaion of he sample series. The degrees of freedom for he suden- error erm, df, can be used o infer abou he degree of heavy-ails in he series. The df value can be used o calculae he fourh momen of he implied by 4 he model ( 3( df ) /( df 4). The calculaed value is an inference abou he kurosis of he sample series. For example, in case of he PORTFOLIO A model wih esimaed df value of approx. 11, he value of implied kurosis is 3,85 and hus he implied excess kurosis equals 0,85, which is sufficienly close o he kurosis value of he sample, 0,75 as given in he able Volailiy Forecass Evaluaion Based on he GARCH_ (1,1) models, as specified in Table, one-day ahead forecass were calculaed using rolling sample. In oal 50 ou of sample forecass were obained. The volailiy forecass from he GARCH models are compared wih a Naïve model, where he volailiy calculaed using rolling sandard deviaion of he pas n observaions (n equals he number of observaions in he in-sample daa of he GARCH model). The square of he log reurns was used as a proxy for he rue volailiy. In order o compare he predicive accuracy of GARCH models vis-à-vis Naïve models, differen error saisics were calculaed. One-sided sign ess were also performed for he error saisics, in order o es he saisical significance of he difference in he performance of GARCH and Naïve models. The resuls are presened in Table 3. 0

21 Rishi Thapar (801) Maser Thesis-Volailiy and Value a Risk Modelling using univariae GARCH models Table 3: Volailiy Evaluaion PORTFOLIO A BENCHMARK A GARCH NAIVE GARCH NAIVE MSE 1.80E E E-08.0E MedSE 3,67E-09 7,09E-09 3,018E-09 6,10E-09 MPE,97E-03 3,61E-03,99E-06 0, ,348-8,095 AMAPE 0,6139 0,649 0,6186 0,6508-8, -7,463 MME(U) 0, , , , ,167-1,376 MME(O) 0, , , , ,831-6,451 TIC 0, , , ,00158 R 0,7% 0,759% 0,110% 0,801% PORTFOLIO B BENCHMARK B GARCH NAIVE GARCH NAIVE MSE 1.54E E-08 1,57E E-08-9,360-9,866 MedSE.34E E-09,47E-1 4.7E-09 MPE.51E E-03.54E-03 3,4E-06-9,360-9,866 AMAPE 0,6007 0,6545 0,6060 0,6564-8,981-8,854 MME(U) 0, , , , ,914-14,546 MME(O) 0, , , , ,831-7,463 TIC 0, , , ,00163 R 0,081% 0,619% 0,036% 0,598% PORTFOLIO C BENCHMARK C GARCH NAIVE GARCH NAIVE MSE 1,03E E E E-07 -,783-1,518 MedSE.9E E E E-08 MPE 6.44E E E ,783-1,518 AMAPE 0,5805 0,5931 0,5841 0,595 -,783-1,01 MME(U) 0, , , , ,060-3,668 MME(O) 0, , , , ,403-1,65 TIC 0,0006 0, , ,00053 R 0,53% 0,000% 0,831% 0,000% 1

22 Rishi Thapar (801) Maser Thesis-Volailiy and Value a Risk Modelling using univariae GARCH models GARCH model for all series have lower MSE, MedSE, MAE, AMAPE, MME (O) and MME (U) values compared o Naïve model. All one-sided sign es values, excep in case of BENCHMARK C are less han Therefore, we can rejec he null hypohesis a 5% confidence level and may conclude ha GARCH models are beer han Naïve model on hese error loss funcions. We observe ha in all porfolios, Naïve models have lower TIC values, herefore, indicaing ha Naïve models are beer han GARCH on he basis of his error funcion. However, one-sided sign ess for his error funcion shows he opposie (excep in BENCHMARK C). We may conclude ha alhough he mean TIC values for Naïve models are lower compared o GARCH models, he GARCH models are beer han Naïve models saisically. I is difficul o make a consisen conclusion based on Mincer-Zarnowiz regression regarding he relaive performance of GARCH and Naïve models. Naïve models seem o perform beer for PORTFOLIO A, BENCHMARK A, PORTFOLIO B and BENCHMARK B. In conclusion, GARCH models for almos all porfolios seems o have beer predicive accuracy as compare o Naïve models on he basis of mos of error measures. 6.3 Value-a- Risk Evaluaion 5% Value a Risk esimaes using GARCH and GARCH_B models are calculaed as produc of he volailiy forecas, porfolio reurn and he criical value (which is obained from he disribuion able corresponding o he degrees of freedom of he errors of he fied GARCH model). To calculae 5% Value a Risk based on he naïve model, criical value is aken as Please refer o figure 4 (in Appendix B) for plos of acual losses and forecased VaR from he GARCH and Naïve models. Table-4 summarizes resuls from VaR accuracy ess for he differen porfolios. Table 4: VaR Evaluaion PORTFOLIO A PORTFOLIO B PORTFOLIO C GARCH GARCH-B NAIVE GARCH GARCH-B NAIVE GARCH GARCH-B NAIVE Nr of failures fail. Rae 4,00% 3,60% 1,60% 3,0% 3,0%,00% 1,0% 1,0% 1,0% kupiec es 0,563 1,138 8,185 1,944 1,944 6,071 10,81 10,81 10,81 Uncodiional Coverage(LRu) 0,563 1,138 8,185 1,944 1,944 6,071 10,81 10,81 10,81 Independence (LRind) 7,136 7,97 0,130 7,4 7,4 8,173 0,073 0,073 0,073 Uncodiional Coverage(LRcc) 7,699 8,435 8,315 9,366 9,366 14,44 10,885 10,885 10,885 Loss Fn 10,048 9,044 4,011 8,44 8,15 5,105 3,63 3,537 3,533 S garch,garch-b 15,558 15,811 15,811 S garch,naive 15,685 15,685 15,558 The criical value for LR uc (Likelihood Raio for uncondiional coverage) and LR ind (Likelihood Raio for independence) is 3, 8415 and for LR cc (Likelihood Raio for condiional coverage) is 5, We may conclude from he above resuls ha all he hree VaR models (GARCH, GARCH-B and Naïve) in he case of all hree porfolios pass Basel back es because he failure rae is less han 5%. In he case of PORTFOLIO A porfolio, boh GARCH models (GARCH and GARCH-B) pass Kupiec/ uncondiional coverage es whereas he Naïve model fails. However, opposie is rue for he es of independence. For es of correc condiional coverage, boh GARCH and Naïve models fail he es. In he case of PORTFOLIO B porfolio, boh GARCH models (GARCH and GARCH-B) pass Kupiec/ uncondiional coverage es whereas he Naïve model fails. Tes of independence and correc condiional coverage is failed on all hree models.