MODEL SELECTION FOR VALUE-AT-RISK: UNIVARIATE AND MULTIVARIATE APPROACHES SANG JIN LEE

MODEL SELECTION FOR VALUE-AT-RISK: UNIVARIATE AND MULTIVARIATE APPROACHES By SANG JIN LEE Bachelor of Science in Mahemaics Yonsei Universiy Seoul, Republic of Korea 999 Maser of Business Adminisraion Yonsei Universiy Seoul, Republic of Korea 004 Submied o he Faculy of he Graduae College of he Oklahoma Sae Universiy in parial fulfillmen of he requiremens for he Degree of MASTER OF SCIENCE December, 005

COPYRIGHT By Sang Jin Lee December, 005 ii

MODEL SELECTION FOR VALUE-AT-RISK: UNIVARIATE AND MULTIVARIATE APPROACHES Thesis Approved: Timohy L. Krehbiel Thesis Adviser B. Wade Brorsen Zhigang Zhang A. Gordon Emslie Dean of he Graduae College ii

Acknowledgemen This hesis is meaningful o me in wo aspecs; firs, hrough his hesis, I have had a chance o summarize wha I learned in he Maser of Science in Quaniaive Financial Economics program a Oklahoma Sae Universiy. Second, my long-erm objecive has been achieved. I finished my undergraduae degree in 999 jus afer he Asian financial crisis. Afer ha crisis, everyhing in he Korean financial marke changed. Korea moved from a Japanese business model o an American model. The rules of he game changed. A ha ime I decided ha someday I would go o he US o learn he new rules. This hesis is a produc of beginning o learn hem. However, wihou people who suppored and helped me, I can no imagine ha I would have finished his hesis. I have had hree pillars o suppor his work: my hesis commiee, friends and co-workers, and my family. As an academic pillar, I m indebed so much o my commiee members; Dr. B. Wade Brorsen, Dr. Zhigang Zhang, and my advisor Dr. Timohy L. Krehbiel. Dr. Krehbiel iniiaed his sudy and helped me hroughou he enire process. Dr. Zhang gave me good knowledge and advice and Dr. Brorsen gave me a lo of producive criics in finalizing his projec. I am especially lucky o have had a chance o work wih Dr. Brorsen who is a well-respeced scholar. My friends and co-workers are anoher pillar. I hanks my friends, especially Yong Su Jang, Yoon Suk Sung, Dong Ho Choi, Sung Jun Kim, Woo Ju Jo, Jun Sung iii

Bae, who has been my friends since high school, Dr. Ki Bum Bhin, Su Jong Im, Seung Wan Kim, Jae Min Shin and his wife, who are undergraduae colleagues, Dr. Hong Jin Whang and his wife, Wei Sun and his wife, Arun Aravind, who are friends in Oklahoma Sae Universiy. I also hank o my co-workers in Korea who gave suppor and aenion; Myung Sook Jung, Chung Gi Wi, Hee Jun Yu, Won Sik Jung, Jung Hun Lee, Mi Hoe Kim, wih whom I enered he company in he same year, Dr. Jong Yong Kim, Mr. Hyung Bok Seo, Mr. Su Il Kim, Mr. Jae Ik Moon, Mr. Chang Rae Jo, who were my bosses in he company. I should express my special hanks o Mr. Kwang Sik Kim, who moivaed me and helped me hroughou my whole sudy. Besides hem, here are so many ha I feel sorry ha I can no lis all heir names individually because of he limiaion of paper. The final and mos imporan pillar is my family, my elder broher and his wife, grandmoher, moher, and faher. Wihou heir suppor, everyhing in my life including his work would be impossible. I would like o name Dr. Trish Macvaugh as my family. To me, she is moher in he US because during my sudy, she lisened o me, cheered me up, and encouraged me. As you can see, his is no a work compleed by myself; wihou any one of hem, i would have been impossible for me o finish his work. I also would like o express my graiude o my company, Financial Supervisory Service in Korea, for allowing me ime o sudy. Sang Jin Lee iv

TABLE OF CONTENTS Chaper Page. Inroducion.... Value-a-Risk...6. Single Asse...6. Porfolio...7 3. VaR Evaluaion Models...0 3. Univariae Model... 3.. Equally Weighed Moving Average Model... 3.. Exponenially Weighed Moving Average Model... 3..3 GARCH Model... 3..4 Hisorical Simulaion Model...3 3. Mulivariae Models...4 3.. Orhogonal GARCH Model...4 3.. Dynamic Condiional Correlaion Model...5 4. Backesing...7 4. Rolling Backesing Procedure...7 4. Proporion of Failures Tes...8 4.3 Runs Tes...9 5. Daa... v

6. Empirical Resul...5 6. Univariae Approach...5 6.. Dow Jones Indusrial Average...6 6.. Hypoheical Porfolio...3 6. Mulivariae Approach...36 6.. Correlaion Esimaion of Mulivariae Models...36 6.. Mulivariae Approach...40 6.3 Univariae Approach vs. Mulivariae Approach...4 7. Conclusion...50 REFERENCES...53 APPENDIX A. Univariae Approach vs. Mulivariae Approach Using he EWMA Model...56 B. Principal Componen Analysis (PCA)...58 C. GARCH Model Fiing Resuls Using Various Compuer Sofware Packages...60 D. The Lengh of Periods of Backesing Trials...64 vi

LIST OF TABLES Table Page 3. Number of Parameers Needed According o he Number of Asses... 5. Descripive Saisics of DJIA Daily Reurns... 3 5. Reurns Sored from he Bigges o he Smalles...4 6. Resul of he PF es and Runs es of Univariae Models for DJIA...30 6. Resul of he PF es and Runs es of Univariae Models for he Hypoheical Porfolio...36 6.3 Resul of he PF es and Runs es of Mulivariae Models for he Hypoheical Porfolio...4 6.4 Resul of he PF es and Runs es of Univariae and Mulivariae Models for he Hypoheical Porfolio...44 6.5 Mean and Sandard Deviaion of he VaR esimaes...46 C. Coefficiens Used o Simulae Daa Se...6 C. Descripive Saisics of Simulaed Daa Se...6 C.3 GARCH Model Fied Resuls...63 vii

LIST OF FIGURES Figure Page. VaR Esimae and Profi/Loss Disribuion...7 4. The Procedure of he Rolling Backesing...8 5. Hisograms and Plos of he DJIA and he Porfolio...3 5. QQ-Plo of he DJIA and he Hypoheical Porfolio...4 6. VaR Esimaes for DJIA when $00 was invesed (Lef Tail)...7 6. VaR Esimaes for DJIA when $00 was Invesed (Righ Tail)...7 6.3 VaR Esimaes for DJIA wih Realized Loss (Lef Tail)...8 6.4 VaR Esimaes for DJIA wih Realized Loss (Righ Tail)...9 6.5 VaR Esimaes for Univariae Hypoheical Porfolio when $00 was invesed (Lef Tail)...33 6.6 VaR Esimaes for Univariae Hypoheical Porfolio when $00 was invesed (Righ Tail)...33 6.7 VaR Esimaes for Univariae Hypoheical Porfolio wih Realized Loss (Lef Tail)...34 6.7 VaR Esimaes for Univariae Hypoheical Porfolio wih Realized Loss (Righ Tail)...35 6.9 Correlaions Using Various Models...39 6.0 Porfolio Sandard Deviaion and Scaled Average Correlaions and Volailiy using he DCC model...40 6. Porfolio Sandard Deviaion and Scaled Average Correlaions and Volailiy using he O-GARCH model...40 viii

6. Excepion Occurrence of DCC and GARCH (Normal)...48 6.3 Excepion Occurrence of O-GARCH and GARCH (Normal)...49 C. Plos of Simulaed Daa Se...6 D. The resul of he PF Tes (Lef Tail, Univariae)...64 D. The resul of Backesing Trials among GARCH Models (Lef Tail)...66 D.3 The resul of Backesing Trials among GARCH Models (Righ Tail)...66 ix

CHAPTER INTRODUCTION Value-a-Risk (VaR) is he mos popular ool in risk managemen because i is easy o communicae and easy o comprehend. The imporance of VaR is rapidly increasing because he inernaional agreemen in banking indusry he, so-called, Basel Accord heavily uses VaR mehodology. To manage marke risk, he Basel Accord requires a financial insiuion o have capial in proporion o he oal value of is riskadjused asse which is basically measured by VaR in he inernal models approach. This rule is acceped by he Group of Ten (G-0) counries and many oher counries. So, banks in hose counries evaluae heir risk exposure using his VaR mehodology. Jorion (000) inuiively defined VaR as he summary of he wors loss over a arge horizon wih a given level of confidence. For example, he chief financial officer of a financial company migh say ha he VaR of he bank is $0 million a a 95 percen confidence level over one day horizon, which means ha here is a 5% probabiliy for a loss greaer han $0 million o happen under normal marke condiions. So, since VaR is a single number summarizing he amoun of risk he company is exposed o, i is very easy o undersand and communicae. G-0 counries are Belgium, Canada, France, Germany, Ialy, Japan, he Neherlands, Sweden, Unied Kingdom, and he Unied Saes, plus Luxembourg and Swizerland.

However, he VaR mehodology requires disribuional assumpions for he relevan risk facors. Moreover, he VaR esimae depends on no only he asses class consiuing porfolio, bu also he model used o esimae he volailiy of hose asses. In his regard, i is valuable o invesigae which volailiy model produce superior risk measuremen for a given porfolio. In he pas sudy, Sarma e al (003) sudied he model selecion for VaR esimaion in he S&P 500 and India s NSE-50. Using a wo-sage model selecion procedure, hey compared he performance of candidae volailiy models such as equally weighed moving average(eqma) model, exponenially weighed moving average(ewma) model, he GARCH model, and he hisorical simulaion(hs) model. They found ha he EWMA model worked bes among hose models. Angelidis and Alexandros (004) analyzed he applicaion of several volailiy models o forecas daily VaR boh for single asses and porfolios. They considered models such as he GARCH model, he EWMA model, he exponenial GARCH model, he hreshold ARCH model, he exreme value heory (EVT), and he HS model. They found ha he bes model depended on porfolio. Some researchers warned of limiaions of he VaR approach in risk managemen. Bedder (995) examined eigh VaR esimaes for hree hypoheical porfolios. He found ha VaR is very effecive measure in risk managemen, bu since i depends on parameers, daa, assumpions, and mehodology, i migh be dangerous in is applicaion. Hendricks (996) compared differen VaR evaluaion approaches using simulaed daa from eigh foreign exchange markes. He concluded ha in almos all cases hese approaches covered he risk ha hey were inended o cover. However, he also observed

ha VaR esimaes from differen approaches were quie differen, which also implies ha VaR approach is overall good ool for risk managemen, bu needs some cauions in applicaion. The comparison of he univariae approach wih he mulivariae approach of VaR evaluaion migh also be a very ineresing quesion. In evaluaing he porfolio VaR, he mulivariae model can have some advanages over he univariae model. According o Bauwens e al. (004), one advanage of a mulivariae model is ha once we ge he covariance marix by he mulivariae approach, we do no need o calculae again he covariance marix even if he weighs of each asse are changed; under he univariae model, we should evaluae he variance of porfolio again whenever he weighs of each asse are changed. Anoher advanage is ha a mulivariae model may improve he evaluaion performance in updaing he variances and correlaions by considering he individual characerisics of he porfolio s componens and esimaing heir linear comovemen. According o he Longin and Solink (995), he markes become more closely relaed during periods of high volailiy. In his period, considering he individual correlaion among socks migh increase he model accuracy. So, i is a good research quesion wheher mulivariae models perform beer han univariae models. In he lieraure of he mulivariae VaR approach, Manfredo and Leuhod (999) invesigaed various VaR esimaion echniques for he agriculural enerprise porfolio; he EQMA model, he EWMA model, he GARCH model, he implied volailiy model, and he HS model as univariae model, and consan condiional correlaion model as mulivariae models. They found ha he EWMA model and he HS model provided reasonably good esimaes. Brooks and Persand (000) found ha he mulivariae 3

GARCH(,) model worked bes o ge a VaR esimae relaive o oher models like he HS model, he RiskMerics approach, and he modified RiskMerics approach using daily closing sock prices of five Souheas Asian counries. Engle (00) compared VaR esimaes from various mehods such as he BEKK model, he Dynamic Condiional Correlaion (DCC) model, and he orhogonal GARCH (O-GARCH) model, he mulivariae EQMA model, and he mulivariae EWMA model. He observed ha he DCC model overall performed bes o evaluae VaR under he various siuaions. Rombous and Verbeek (004) examined he usefulness of he mulivariae semiparameric GARCH models for porfolio selecion under a Value-a-Risk (VaR) consrain. They also examined several alernaive mulivariae GARCH models for daily reurns on he S&P 500 and NASDAQ indexes. To ell a good model from a bad model, we need some crieria. One obvious propery ha a good model should have is predicabiliy. Tha is, a good model should do a good job in predicing fuure risk exposure. Anoher crierion is wheher he model uses all informaion available. If a predicion model does no use all informaion available, is predicion abiliy will be lowered, which means ha he model is inferior. There are various saisical mehods based on hese ideas. We will review hem in he laer secion. So, his hesis will address wo quesions:. Which univariae models are appropriae o evaluae VaR of he Dow Jones Indusrial Average (DJIA).. Considering mulivariae volailiy models such as he DCC model and he O- GARCH model, which incorporae condiional correlaions among asses, as BEKK came from Baba, Engle, Kraf and Kroner who were conribuors o he model. 4

well as univariae models, which models are appropriae o evaluae VaR for a hypoheical porfolio. For he firs quesion, we will focus on he univariae model. Afer ha, we will urn our aenion o he mulivariae model for he second quesion. For hese quesions, we need some judging crieria, which will be inroduced laer. In he following secions, we will review he VaR conceps, which will be followed by a review of he various mehods o evaluae VaR. Afer ha, we will move o he model discerning crieria. The empirical resul of he univariae models will be presened and discussed firs. Then, he resul of he mulivariae models will follow, and he comparison of boh models will be discussed. Then, we will draw some conclusions and implicaions. This hesis is differen from he exising sudies from wo poins: daa and comparison. This hesis uses wo ses of daa: he DJIA and a hypoheical porfolio. Mos pas research used a porfolio consising of wo, hree or, a mos, five socks. Bu, in his hesis we use a hypoheical porfolio consising of 30 socks o es he performances of he mulivariae models. One oher poin is ha i seems ha lile sudy has been done abou he comparison of he mulivariae VaR esimae mehods wih he univariae VaR esimae mehods using he same porfolio in he VaR lieraure. So, he mos disincive poin of his hesis is ha comparison. This will allow us o deermine he value of condiional correlaion esimaion in his VaR applicaion. 5

CHAPTER II Value-a-Risk Jorion (000) formally defines VaR as he descripion of he quanile of he projeced disribuion of gains and losses over he arge horizon. If c is he seleced confidence level, VaR corresponds o he c lower-ail levels. Mahemaically, i can be formulaed like his: VaR c = f ( x) dx () where x is a random variable of he profi/loss of porfolio, f(x) is he disribuion of x, and c is he seleced confidence level. If he profi/loss disribuion of porfolio is assumed o follow a normal disribuion wih zero mean, hen we can ge a VaR esimae in a very easy way as follows.. Single Asse If he profi/loss disribuion of an asse is assumed o follow a normal disribuion wih zero mean, hen VaR = -z c V 0 σ () where z c is he criical value a confidence level c, V 0 is he iniial value of he porfolio, and σ is he esimaed sandard deviaion of he porfolio s reurn. For insance, assume 6

ha he iniial value of an asse is $, he reurn of he asse follows a normal disribuion wih zero mean and sandard deviaion of σ. The omorrow s VaR esimae a 95% confidence level over one day horizon is.645σ in figure. Figure.: VaR Esimae and Profi/Loss Disribuion. Porfolio In he porfolio heory, he reurn ( r p ) of a porfolio is defined as follows: r p = n i= w r i i (3) where n is he number of asses in he porfolio, r i is he reurn of ih asse for i =..n, and w is he weigh of ih asse in he porfolio for i =..n. The variance ( σ ) of he i porfolio can be calculaed as follows: p 7

n i= n n σ = ( w σ ) + w w σ (4) p i i i= j= i j where σ i is he variance of ih asse, σ i, j is he covariance beween ih asse and jh asse. In equaion (4), he firs erm is called diversifiable risk or non-sysemaic risk which can be eliminaed hrough diversificaion and he second erm is called undiversifiable risk or sysemaic risk which can no be eliminaed hrough diversificaion. To invesigae he power of diversificaion 3, we consider a sraegy where weighs are equal o /n. Then n n n p = σ i + n i= n i= j= n i j i j σ σ (5) We define he average variance ( σ ) and average covariance ( C ) of he asses as i, j i, j σ = n n i= σ i C = n( n ) n n i= j= i j σ i, j (6), hen porfolio variance can be expressed as n σ p = σ + C. (7) n n Here, when he number of asses in he porfolio increases, he firs erm, non-sysemaic risk, will disappear, bu he second erm, sysemaic risk, will converge o C. In general, we can say ha he risk of a well-diversified porfolio comes from only sysemaic risk or covariance par in equaion (7). The variance of he porfolio can be expressed in he marix form as 3 You can see more deailed discussion on his in Chaper 8 of Bodie e al (00). 8

σ...... σ, n w [ ] w............ σ = p w w... wn (8).............. σ n,...... σ n wn or σ = w Σ w p where w is he vecor of he weighs of he porfolio, and Σ is he covariance marix. If risk facors follow a normal disribuion wih zero mean, he VaR of he porfolio can be calculaed as VaR p = -z V 0 σ p (9) We noe ha he relaion beween he mean and he sandard deviaion depends on he lengh of he ime horizon. Since he volailiy grows wih he square roo of ime and he mean wih ime for independen idenically disribued processes, he mean will dominae he volailiy over long horizons. Over shor horizons, such as a day, volailiy dominaes. This provides a raionale for focusing on volailiy ignoring expeced reurns or assuming ha hose are zero when we evaluae VaR measures using daily daa. In his hesis, since we will use he daily reurns of 30 socks, we also assume ha he expeced reurn of he daily reurn of each sock is zero. The expeced reurn of he porfolio can be assumed zero because he expeced reurn of he porfolio is he weighed average of he expeced reurns of he socks. This leads us o jus focus on he sandard deviaions of he socks and porfolio o evaluae VaR. 9

CHAPTER 3 VaR Evaluaion Models There are wo approaches o evaluae he VaR in a porfolio sense. The firs approach is o creae a univariae reurn series for he porfolio using he weigh of each asse, and hen we can use univariae models which will be reviewed. The oher approach is o esimae a mulivariae variance-covariance marix, and hen we can evaluae VaR by using equaion (7). We will use five univariae models and wo mulivariae models o evaluae VaR. Five univariae models are he EQMA model, he EWMA model, he GARCH model wih a normal disribuion, he GARCH model wih a -disribuion, and he HS model 4. As mulivariae models, we will use he O-GARCH model and he DCC model because hese can be easily applied o a porfolio consising of many asses. In fac, mos frequenly used and cied mulivarie volailiy models are he Vech 5 model, he BEKK model, he DCC model, and he O-GARCH model. However, as indicaed in Table 3., he Vech model and he BEKK model are pracically no available for a porfolio consising of many asses, which leads us o use jus he DCC model and he O-GARCH model. 4 Here, we do no consider he Mone Carlo simulaion mehod because we will use he linear porfolio, in which case he resul from he Mone Carlo simulaion should be he same as he resul from variancecovariance approach. 5 Vech is he name of a mahemaical operaor. 0

Table 3.: Number of Parameer Needed According o he Number of Asses #. Asses Vech Model BEKK Model DCC Model n n( n + ) n( n ) n( n + ) + + + n O-GARCH Model 3 n + 3 3 n 30 43,95 65 93 90 Noe ha in his able we assume ha he lags of all ARCH and GARCH parameers are 3. Univariae Models We will review five models: he EQMA model, he EQMA model, he GARCH model wih a normal disribuion, he GARCH model wih a -disribuion, and he HS model. 3.. Equally Weighed Moving Average Model The simples one is he equally weighed moving average model, where oday s volailiy is calculaed by he average of he volailiy over he given ime window. The mahemaical formula is 6 m ( ) σ = (0) r i m i= where σ is he sandard deviaion, n is he number of daily rae changes used o calculae sandard deviaion, r is daily reurn. In fac, his model gives equal weigh /m o each volailiy of he pas. So, ha is why his model is called equally weighed moving average model. 6 By using m insead of m- in he denominaor, we assume ha he volailiy esimae in equaion (5) is he maximum likelihood esimaor, no he unbiased esimaor. See Hull (003) chaper 7.

3.. Exponenially Weighed Moving Average Model I seems more reasonable o assume ha oday s volailiy is more affeced by he more recen evens. To incorporae his ino he model, we should give more weigh o he more recen evens and less weigh o he laer evens. One of hese weigh schemes is an exponenial scheme. The model using his exponenial weigh scheme is = ( ) + λ r λσ σ () where σ is he sandard deviaion for day n, r is he daily shock for day n-, and λ is he decay facor 7. In he ieraive way, we can easily show ha 8 m σ = ( λ) λ ( r i ) () i=, which shows ha he weighs for he r s decline a rae λ as we move back hrough ime. This model is called exponenially weighed moving average model. According o he echnical documen of he RiskMerics (996), i uses he EWMA model wih λ = 0.94 for updaing daily volailiy esimaes. In his hesis, we used λ = 0.94 because we used daily daa. 3..3 GARCH Model There is anoher weigh scheme called generalized auoregressive condiional heeroskedasiciy (GARCH) model, which is proposed by Bollerslev (986). The GARCH (p,q) process is hen given by 7 We noe ha since in his hesis he expeced mean of he price is assumed o be zero, he daily shock is equal o he daily reurn, ha is, ε n = r n for day n. So, hereafer he daily shock means he daily reurn, and vice versa. 8 Here, we assume ha he volailiies before ime (-m) are so small ha hey can be ignored.

ε ψ ~ Ν(0, σ ), σ = α + 0 q i= α ε i i + p i= β σ i i (3) where p 0, q > 0, α 0 > 0, α 0, β 0. When p=q=0, ε is simply whie noise. I is i i generally acceped ha in mos cases GARCH(,) model is enough o model he volailiy of financial marke. So, we will use GARCH(,) model in his hesis, and hereafer GARCH model means GARCH(,) model. The meaning of GARCH model is ha oday s volailiy(σ ) is updaed by yeserday s volailiy(σ - ) and yeserday s shock(ε - ). We noe ha in fac he EWMA model is a paricular case of he GARCH(,) model where α 0 =0, α = - λ, and α = λ. Anoher variaion of he sandard GARCH model is o use he suden disribuion insead of normal disribuion as he condiional disribuion which he daily reurn follows. In fac, he condiional disribuion as well as he uncondiional disribuion of he daily reurn is generally considered o have faer ails han a normal disribuion. So, if we use he suden disribuion insead of a normal disribuion as he condiional disribuion he daily reurn follow, we are supposed o ge more realisic resul. 3..4 Hisorical Simulaion model The hisorical simulaion mehod uses hisorical daa o build he disribuion of he risk facor, and hen evaluae VaR from ha disribuion. In he case of he single index, we ge hisorical movemens or series of reurns ( r, =,.., n ) of he index. Based on hose movemens, we can ge he simulaed omorrow s value of he index V n+ as: 3

n+ = ( r + ) Vn (4) V * where =,.., n and Vn is he oday s porfolio value. Afer ha, we can consruc he disribuion of he change in he porfolio ( = Vn + Vn = ( r + ) * Vn Vn = r,,.., n ). = And, hen sor r ( =,.., n) observaions from he bigges loss o he bigges gain. This arrangemen can be considered as he disribuion of he risk facor. If we wan o ge 95% VaR, he 5h quanile of ha disribuion is wha we wan o ge. 3. Mulivariae Model. In mulivariae models, we migh consider weigh schemes ha are similar o weigh schemes in he univariae models like a moving average and GARCH. However, we noe ha if we use he mulivariae variance-covariance marix proposed by RiskMerics (00), equally and exponenial weighed schemes produce he same resul in boh approaches 9, ha is, he volailiy from he univariae reurn series is always equal o he volailiy from he mulivariae variance-covariance marix because hey use he same mehods in updaing he volailiy and he covariance. So, we will review jus wo GARCH ype models: he O-GARH model and he DCC model. 3.. Orhogonal GARCH Model According o Alexander (00), he orhogonal mehod uses principal componen analysis (PCA) approach o consruc covariance marices 0. In he orhogonal GARCH model, he ime-varying covariance marix H of he original sysem is approximaed by H = AD A' (5) 9 Comparison beween univariae approach and mulivariae approach is in appendix A. 0 See he appendix B for deails of PCA approach. 4

where A is he marix of rescaled facor weighs and D is he ime-varying diagonal marix of variances of he principal componens of original mulivariae series. The diagonal marix D of variances of principal componens is esimaed using a GARCH model. We noe ha basically PCA echnique is a linear ransformaion from one space measured on real-world basis ino he oher space measured on so-called principal componen basis. In he laer world, we can analyze real-world daa in a differen poin based on principal componens, which are muually orhogonal; hence we don need o pay aenion o correlaion beween componens. So, he correlaions in he real world are ransformed ino he variances of he principal componens. Hence, analyzing dynamics of variances of principal componens implicily incorporae dynamics of correlaions of real-world daa. 3.. Dynamic Condiional Correlaion model Engle (00) proposed a new class of mulivariae GARCH models named dynamic condiional correlaion model. The DCC model evolved from he consan condiional correlaion (CCC) model by Bollerslev (990). The CCC model esimaes condiional covariance marix H as: H = D RD where D = diag } (6) { h i, where R is a consan correlaion marix, ime series of asse i in he porfolio a ime. h i, is he condiional covariance of univariae 5

The DCC model assumes he correlaion marix is ime-varying, ha is, R insead of consan R and uses he GARCH scheme o incorporae ha ime-varying propery of he correlaion ino he model as follows: H = D R D where D = diag } { h i, R = diag{ Q } Q diag{ Q } (7) Q M N M N = ( α ) ( m β n Q + α m ε mε m ) + βnr m= n= m= n= where H is a ime-varying covariance marix, α m is he ARCH coefficien, β n is he GARCH coefficien, M is he order of ARCH parameer, N is he order of GARCH parameer, Q is he uncondiional covariance of he sandardized residuals, and ε m is he sandardized residual from univariae ime series, and diag {X} mean a diagonal marix of marix X. According o Engle (00), ifα 0, β 0, and ( M m= posiive, hen N α m β n ) 0, hen R will be posiive semi-definie. If any one of hem is n= R will be posiive definie. Engle and Sheppard (00) saed ha he DCC model was designed o allow for wo sage esimaion. In he firs sage, univariae GARCH models are esimaed for each residual series. Then, in he second sage, residuals, ransformed by heir sandard deviaion esimaed during he firs sage, are used o esimae he parameers of he dynamic correlaion. m n 6

CHAPTER 4 Backesing We can ge he various VaR esimaes depending on he model we used o measure volailiy. So, we need some crieria o decide which is beer. According o Campbell (005), a good model should have wo properies based on he resul of rolling backesing; uncondiional coverage propery and independence propery. 4. Rolling Backesing Procedure Firs, we will describe rolling backesing. The following example will bes explain he procedure of rolling backesing. Suppose we have 500 observaions of pas reurns of a porfolio and we use 000 observaions o esimae omorrow s VaR esimae. Using observaions from he firs o he 000h, we go omorrow s VaR esimae of $,000 and he omorrow s realized observaion or he 00s observaion is $,00, hen we say ha an excepion is realized or here is an excepion; if he 00s observaion is less han he VaR esimae, we say ha here is no excepion. Nex, using observaions from he second o he 00s, we can do he same comparison wheher he VaR esimae is exceeded by he 00nd observaions or nex day s realized loss of he porfolio. Coninuing his comparison from he 500h o he 499h observaion wih 500h Some auhors use a erm, exceedance insead of excepions because he realized loss exceeds he expeced loss or he VaR esimae. 7

observaion, each comparison is regarded as one backesing. Then we have A oal of 500 backesings. The procedure is described in figure. Wih his binomial sequence (excepion or no-excepion), we can do he uncondiional coverage propery and independence propery. Figure 4.: The Procedure of he Rolling Backesing 4. Proporion of Failures Tes Uncondiional coverage propery means ha he number of realized VaR excepions of rolling backesing wih pas daa mus be equal o he expeced number of VaR excepions indicaed by he VaR model wihin saisical olerance. For example, - day 99% VaR wih 500 backesing rials expecs 5 excepions (= ( - 99%) 500). If he realized excepion is ou of he range in which boh are saisically equal, hen we can conclude ha he model is inappropriae. Kupiec (995) proposed he proporion of failures (PF) es as he uncondiional coverage, which we will use in his hesis. The LR saisic for esing he null hypohesis ha he realized raio (p) of he excepions of VaR over he pas daa is equal o he probabiliy p* of he excepions of VaR is he following: * nx * x x nx x x PF = Ln [( p ) ( p ) ] + Ln[( ) ( ) ] (8) n n The null hypohesis of he saisical es is ha he number of excepions is 0. 8

where x is he realized number of excepions in he sample, n is he oal number of backesing rials. Under he null hypohesis, p = p*, he PF es has a chi-square disribuion wih degree of freedom. If he null hypohesis is rejeced because he realized raio is greaer han he expeced raio, we can say ha he model underesimaes VaR. On he oher hand, he null hypohesis is rejeced because he realized raio is less han he expeced raio, we can say ha he model overesimaes VaR. One possible reason of underesimaion is ha he disribuion of he reurn series of financial asse usually has a faer ail han he normal disribuion and he model fails o incorporae ha fa fail fully. 4.3 Runs Tes Independence propery means ha he excepions of he backesing should occur in a random way. If he occurrence of he excepions is no disribued randomly across ime, we can find some paerns, which a good model should incorporae wih is predicion schemes. In his hesis we will use he runs es o es a randomness of he excepions 3. Runs can be defined as a sequence wihin a series in which one of he alernaives occurs on consecuive rials. Using he example of a coin oss, if a series look like his: H H T H H T T T H T, hen HH, T, HH, TTT, H, and T are runs. The null hypohesis of a runs es is wheher he disribuion of a series of binary evens in a populaion is random. In order o calculae he es saisics, one mus deermine he number (n,n ) of imes each of he wo alernaives appears in he series and he 3 You can find more deails abou he runs es in Sheskin (003). 9

number(r) of runs in he series. The basic idea of a runs es is ha he number of runs should be wihin he appropriae range for he series o be random. In he above example, he number (n ) of heads is 5 and he number (n ) of ails is 5, and he number(r) of runs is 6. If he number of runs is oo small, say,, hen i migh be difficul o say ha he series is random because he series should be H H H H H T T T T T or T T T T T H H H H H. The normal disribuion can be employed wih a large sample size o approximae he exac disribuion of he runs es as he following: nn r [ + ].5 n + n z = (9) n n (n n n n ) ( n + n ) ( n + n where r is he number of run, n is he number represening alernaive which can be defined as non-excepion occurrence of backesing rials in his hesis, and n is he number represening alernaive which can be defined as excepion occurrence of backesing rials in his hesis. In he example above (n =5, n =5, r=), z-score is - 3.087, which obviously resuls in rejecing he null hypohesis or non-randomness. Possible reasons of he rejecion of null hypohesis are clusering of excepions and increase of he number of excepions. As excepions are clusered, he number of runs will decrease, and as he number of excepions increases, n will decrease and n will increase in equaion (9), which resuls in higher chance of he rejecion of null hypohesis. For example, when n is 353, n is 5, and r is 99, z-saisics is -.835(pvalue=.0684). When n is 353, n is 5, and r is 00, z-saisics is -.348(pvalue=.09). However, when n is 353, n is 5, and r is 99, z-saisics is -.986(pvalue=.0034). ) 0

CHAPTER 5 Daa We will use he DJIA index o address which univariae VaR model performs bes. The DJIA daa came from he Yahoo finance websie. The ime horizon of daa is from 0/3/986 o /3/004 and he oal number of sample of he DJIA closing price is 4584. We used he daily logarihmic reurn such as R = 00 [log( P ) log( P )] (0) where P is he closing price on day. For he mulivariae analysis, a hypoheical porfolio is considered o compare he performance of a univariae model wih ha of a mulivariae model. We used 30 socks of he DJIA componens a /3/004 4 and gave he same weigh /30 o each sock o consruc he porfolio. We go daa from Cener for Research in Securiy Prices (CRSP). The ime horizon of daa is from 0/3/986 o /3004 and he number of observaions is 4583 5. We use he daily logarihmic reurn for each sock as follows: R i = 00 [log( Pi, ) log( Pi, )] (), 4 The roser of DJIA has changed over ime. So, we ook a snap-sho a /3/004. The same company is idenified by he same PERMNO which is given by CRSP daabase. See daa descripion guide for he CRSP US sock daabase and he CRSP US indices daabase. Here is he lis of he companies: Alcoa Inc, AIG, American Express Inc, Boeing Co, Ciigroup Inc, Caerpillar Inc, Du Pon E I De Nem, Wal Disney- Disney C, General Elecric Co, General Moors, Home Depo Inc, Honeywell Inl Inc, Hewle Packard Co, IBM, Inel Cp, Johnson And Johns Dc, JP Morgan Chase Co, Coca Cola, Mcdonalds Cp, 3M Company, Alria Group Inc, Merck Co Inc, Microsof Cp, Pfizer Inc, Procer Gamble Co, SBC Communicaions, Unied Tech, Verizon Commun, Wal Mar Sores, Exxon Mobil Cp. 5 The 5-4-994 daa for Alria Group was no available. So, all he daa for ha day were removed.

where P i, is he adjused price for facors like sock spli and spin-offs, and includes dividend because we wan o focus on he price movemens which are no caused by corporae evens such as sock spli and spin-offs. Table 5. shows he descripive saisics. The hisograms and plos are in figure 5.. Here, we wan o noe several facs. Firs, boh means of he DJIA daily reurns and he hypoheical porfolio are so small relaive o sandard deviaion ha our assumpion o ignore he mean of daily reurns seems reasonable. Secondly, Ljung-Box Q es saisics show ha here are auocorrelaions also in boh cases. ARCH LM es saisics show ha here are ARCH effecs in boh cases. Thirdly, Jarque-Bera es saisics show ha he uncondiional disribuion of daily reurns is far from normal in boh cases. We can also confirm ha by he hisograms in figure 5.. Finally, we would like o pay aenion o he ail propery of boh porfolios. The lef ail and he righ ail of boh porfolios are faer han hose of a normal disribuion as indicaed in he QQ-plo of figure 5.. Also, we can observe ha he lef ails of boh are more deviaed from he normal disribuion han he righ ail. In Table 5. and figure 5., hough he bigges and he smalles reurn of he DJIA are respecively greaer and less han hose of he hypoheical porfolio, overall he hypoheical porfolio has a faer ail han he DJIA.

Table 5.: Descripive Saisics of DJIA Daily Reurns Variables DJIA Hypoheical Porfolio Observaions 4583 458 Mean 0.038 0.0558 Maximum 9.666 8.38 Minimum -5.63-3.5 Sd. Dev..83.84 Skewness -.76534 -.953 Kurosis 66. 4.6577 766,697.4 a 88,.9 Jarque-Bera (<.000) b (0.0000) 33.585 378.9437 Q() (<.000) (<.000) 86.6556 76.9 ARCH-LM (<.000) (<.000) a Bold means ha he number is saisically significan. b The number in parenhesis is p-value. Figure 5.: Hisograms and Plos of DJIA and Porfolio 3

Table 5.: Reurns Sored from he Bigges o he Smalles Percenile DJIA Hypoheical Porfolio Bigges 9.666 8.38 99%.8854 3.0996 98%.5.4435 97%.9949.03 95%.6469.7536.... 5% -.675 -.705 3% -.037 -.0808 % -.306 -.354 % -.8905 -.970 Smalles -5.63-3.5 Figure 5.: QQ-Plo of he DJIA and he Hypoheical Porfolio 4

CHAPTER 6 Empirical Resul We examine he univariae reurn series of he DJIA index and he hypoheical porfolio of 30 socks using univariae models. Then, we examine he mulivariae reurn series of 30 socks using mulivariae models. For each examinaion, we invesigae he lef ail of he disribuion of a porfolio value which is relevan o he holder of a long posiion. We also invesigae he righ ail which is relevan o he holder of a shor posiion. 6. Univariae Approach We will consider wo univariae reurn series; one is from he DJIA for he firs research quesion and he oher is from he hypoheical porfolio consising of 30 socks for he second quesion. 6.. Dow Jones Indusrial Average We backesed he appropriaeness of 99% -day VaR calculaed from each univariae model in he secion 3.. We used 000 observaions o calculae he VaR esimae a each backesing rial, so he oal number of backesing rials is 3583 [= 5

4583(he oal number of observaions) 000(observaions o ge one VaR backesing rial)]. The VaR esimae for each model is in figure 6. and figure 6. for each ail when we invesed $00 for each rial. In figure 6., he resuls using hree models (he EWMA model, he GARCH model wih a normal disribuion, and he GARCH model wih a - disribuion) seem very similar o each oher. Overall, he VaR esimae using EQMA is he smalles. In figure 6.3 and 6.4, you can compare he VaR esimae wih he realized loss. Though realized losses are smaller han VaR esimaes in mos cases, here are some cases ha realized losses are greaer han VaR esimaes, which cases are hough of as excepions. The resul of he PF es and runs es are presened in Table 6.. In he lef ail, all models excep he EQMA model were no rejeced wih he PF es a a 99% confidence level 6. All models excep he EWMA model and he GARCH model wih a normal disribuion were no rejeced wih he runs es a a 99% confidence level. As a resul, wo models, he GARCH model wih a -disribuion and he HS model were no rejeced wih boh ess. Among hese, he HS model show he neares number of excepions o he expeced number of excepions 36 which is % of he number of backesing rials 358. In he righ ail, all models excep he EQMA model were no rejeced wih he PF es a a 99% confidence level. All models were no rejeced wih he runs es a a 99% confidence level. So, all models excep he EQMA model were no rejeced wih he PF es and he runs es. 6 Sricly speaking, we should say ha he null hypohesis of he PF es (or he runs es) relaed o a model was rejeced or no rejeced wih he PF es (or he runs es). Bu, in his hesis, if we have no problem in communicaion, for convenience, we would like o say ha a model was rejeced or no rejeced wih he PF es (or he runs es) o mean he same hing. 6

As a resul, in boh righ and lef ails he GARCH model wih a -disribuion and he HS model were no rejeced wih boh he PF es and he runs es. Oher hree models were rejeced or inappropriae o evaluae VaR of he DJIA index porfolio wih respec o eiher he PF es or he runs es. Figure 6.: VaR Esimaes for DJIA When $00 was Invesed (Lef Tail) Figure 6.: VaR Esimaes for DJIA When $00 was Invesed (Righ Tail) 7

8 Figure 6.3: VaR esimaes for DJIA wih Realized Loss (Lef Tail) 8

9 Figure 6.4: VaR esimaes for DJIA wih Realized Loss (Righ Tail) 9

Table 6.: Resul of he PF es and Runs es of Univariae Models for DJIA PF Tes Runs Tes Tail Model Name Excepions F-saisics p-value No Excepions Excepions Runs z- saisics p-value Equally weighed(eqma) 59.664 a 0.0004 354 59 5 -.343 0.854 Exponenially weighed(ewma) 5 6.4695 0.00 353 5 99 -.986 0.0034 Lef GARCH (Normal Dis.) 5 6.4695 0.00 353 5 99 -.986 0.0034 GARCH ( Dis.)* b 47 3.03 0.0735 3536 47 9 -.9 0.034 Hisorical Simulaion* 38 0.308 0.783 3545 38 75 -.3574 0.747 30 Equally weighed(eqma) 58.67 0.0006 355 58 3 -.383 0.67 Exponenially weighed(ewma)* 44.7544 0.853 3539 44 89 0.40 0.6876 Righ GARCH (Normal Dis.) * 40 0.474 0.499 3543 40 8 0.993 0.7647 GARCH ( Dis.) * 4 0.7003 0.396 354 4 83 0.356 0.7447 Hisorical Simulaion* 44.7544 0.8533 3539 44 85 -.368 0.079 a Bold means we rejec he null hypohesis in each es or model does no work well in each es a a 99% confidence level. b Aserisk means model work well in boh ess a he same ime a a 99% confidence level. 30

We noe ha as indicaed in he daa analysis of he DJIA, he disribuion of he DJIA reurns has a faer ail han a normal disribuion. And, he lef ail shows more deviaion from he normal disribuion han he righ ail (i.e. negaive skewness), which means ha he lef ail have more exreme evens or sronger volailiy clusering han he righ ail. In he lef ail, he EWMA model and he GARCH model wih a normal disribuion migh fail o incorporae volailiy clusering compleely and excepions were more likely o be clusered, so ha he null hypoheses of he runs ess of hose models were rejeced. However, in he lef ail he models such as he GARCH model wih a - disribuion and he HS model, which are more like o incorporae exreme evens or volailiy clusering, were no rejeced wih he runs es as well as he PF es. In he righ ail which has less exreme evens or weaker volailiy clusering han he lef ail, no only he GARCH model wih a -disribuion and he HS model, bu also he EWMA model and he GARCH model wih a normal disribuion were no rejeced wih he runs es, which means ha hese models can handle appropriaely exreme evens or volailiy clusering in he righ ail. 6.. Hypoheical Porfolio We will compare he univariae models wih he mulivariae models using he same porfolio because we wan o know wheher he mulivariae models are more appropriae o evaluae VaR han he univariae models or condiional covariance esimaion improves risk measuremen. For he univariae models, we creaed he single 3

porfolio reurns using 30 socks giving /30 weigh o each sock as menioned in he daa secion. So, here, we will analyze he univariae hypoheical porfolio firs. We backesed he appropriaeness of 99% -day VaR calculaed from each univariae model in he secion 3.. We used 000 observaions o evaluae VaR a each backesing rial, so he oal number of backesing rials is 358 [= 458(he oal number of observaions) 000(observaions o ge one VaR backesing rial)]. The VaR esimaes for each model are in figure 6.5 and 6.6 for each ail when we invesed $00 for each rial. In hose figures, he resul using hree models (he EWMA model, he GARCH model wih a normal disribuion, and he GARCH model wih a - disribuion) seem very similar o each oher. Overall, he VaR esimaes using he EQMA model are he smalles. In figure 6.7 and 6.8, you can compare VaR esimaes wih realized losses. Though realized losses are less han VaR esimaes in mos cases, here are some cases ha realized losses are greaer han VaR esimaes, which cases are hough of as excepions. The resul of he PF es and he runs es are in Table 6.. In he lef ail, all models excep he EQMA model were no rejeced wih he PF es a a 99% confidence level. However, only EWMA model was no rejeced wih he runs es a a 99% confidence level. As a resul, in lef ail he only EWMA model was no rejeced wih boh ess. In he righ ail, only HS model was no rejeced wih he PF es a a 99% confidence level. However, all models were no rejeced wih he runs es a a 99% confidence level. So, in he righ ail only HS model was no rejeced wih boh ess. 3

In sum, all models were rejeced wih he PF es or wih he runs es in he lef or righ ail, which means no model is appropriae o evaluae VaR of he univariae hypoheical porfolio of 30 socks for boh ails. Figure 6.5: VaR Esimaes for Univariae Hypoheical Porfolio when $00 was Invesed (Lef Tail) Figure 6.6: VaR Esimaes Univariae Hypoheical Porfolio when $00 was Invesed (Righ Tail) 33

34 Figure 6.7: VaR Esimaes for Univariae Hypoheical Porfolio wih Realized Loss (Lef Tail) 34

35 Figure 6.8: VaR Esimaes for Univariae Hypoheical Porfolio wih Realized Loss (Righ Tail) 35

Table 6.: Resul of he PF es and Runs es of Univariae Models for he Hypoheical Porfolio PF Tes Runs Tes Tail Model Name Excepions F-saisics p-value No Excepions Excepions Runs z- saisics p-value Equally weighed(eqma) 60 3.7060 a 0.000 35 60 3-3.3056 0.0009 Exponenially weighed(ewma)* b 50 5.0484 0.046 353 50 97 -.8935 0.0583 Lef GARCH (Normal Dis.) 49 4.3938 0.036 3533 49 9-4.4555 0.0000 GARCH ( Dis.) 45.978 0.38 3537 45 83-4.9904 0.0000 Hisorical Simulaion 39 0.77 0.5986 3543 39 73-4.47 0.0000 36 Equally weighed(eqma) 64 8.530 0.0000 358 64 3 -.035 0.044 Exponenially weighed(ewma) 60 3.7060 0.000 35 60 5 -.869 0.0 Righ GARCH (Normal Dis.) 56 9.807 0.007 356 56-0.4084 0.6830 GARCH ( Dis.) 58.6840 0.0006 354 58 5-0.375 0.7433 Hisorical Simulaion* 43.366 0.45 3539 43 85 -.0398 0.985 a Bold means we can rejec he null hypohesis in each es a a 99% confidence inerval. b Aserisk means model work well in boh ess a he same ime a a 99% confidence level. 36

We noe ha he resuls of he hypoheical porfolio are differen from he DJIA in wo aspecs; one is ha he EWMA model and wo GARCH models are rejeced wih he PF es in he righ ail, which may be caused by he fac ha in he righ ail he disribuion of he hypoheical porfolio reurn has a faer ails han ha of he DJIA reurn as indicaed a he daa analysis secion. The faer ail makes i difficul for models o adequaely esimae VaR wih respec o boh he PF es. The oher is ha in he lef ail he GARCH model wih a -disribuion and he HS model were rejeced wih he runs es hough hese models were no rejeced wih he PF es. We noe ha in he DJIA case he number of runs is 9(GARCH model wih - disribuion) and 75(HS model), bu in he hypoheical porfolio ha number dropped o 83(GARCH model wih -disribuion) and 73(HS model), which means ha excepions are more clusered in he hypoheical porfolio, which resuled in he rejecion of randomness null hypohesis. 6. Mulivariae Approach The correlaion beween reurns of asses in a porfolio is an essenial characerisic of mulivariae models. So, firs we will discuss he correlaion esimaion of mulivariae models. Then, we will discuss he empirical resul of mulivariae approach in calculaing he VaR esimae. 6.. Correlaion Esimaion of Mulivariae Models Figure 6.9 shows average correlaion using various models; he firs panel is calculaed by he DCC model, he second panel by O-GARCH model, and he ohers are 37

calculaed over sliding windows of 6(6 monh), 500( years), and 000(4 years). As he ime window increases in he panel 3 o 5 of figure 6.9, he graph of he esimaed correlaion is less erraic and correlaions are more cenered o he overall mean. The overall level of correlaion using he DCC model based on he pas 000 observaions is analogous o he correlaion using sliding windows of 000 observaions, bu he graph using he DCC model is more erraic han ha of he graph using sliding windows of 000 observaions. Overall, he correlaion esimaed by he DCC model seems reasonable. Figure 6.0 and 6. show porfolio sandard deviaion and he average volailiies of 30 socks and average correlaion among 30 socks which are calculaed using he DCC model and he O-GARCH model over he enire horizon of daa respecively, where for he comparison purpose we muliplied 6 o he average correlaion. Firs, we noe ha he Longin and Solnik s (995) observaion ha he correlaion rises in periods of high volailiy seems o hold over he enire period. Second, he average volailiy of socks represens a diversifiable risk or non-sysemaic risk of he porfolio and he average correlaion represens an undiversifiable or sysemaic risk of he porfolio 7. In his regard, in figure 6.0 and 6. we can also observe how he sandard deviaion of he porfolio esimaed by he DCC model and he O-GARCH model incorporae he non-sysemaic risk and sysemaic risk. In figure 6.0 and figure 6., we also noe ha sysemaic risk (=porfolio sandard deviaion average volailiy) is a more dominan componen in porfolio variance han variances of each sock, non- 7 Rigorously speaking, sysemaic risk of a porfolio is he average covariance of socks. However, since correlaion is he mos inegral componen of covariance, we can say ha correlaion represens sysemaic risk. 38

sysemaic risk because he porfolio consiss of 30 socks, so his is a relaively welldiversified porfolio. Figure 6.9: Correlaions using Various Models 39

Figure 6.0: Porfolio Sandard Deviaion and Scaled Average Correlaions and Volailiy Using he DCC Model Figure 6.: Porfolio Sandard Deviaion and Scaled Average Correlaions and Volailiy Using he O-GARCH Model 40

6.. Mulivariae Approach In his sub-secion, we will analyze he mulivariae approach of evaluaing he VaR esimae of he porfolio consising of 30 socks, he resul of which is in Table 6.3. The comparison of his resul wih he resul from he univariae hypoheical porfolio consising of he same 30 socks will le us address he second research quesion. We backesed he appropriaeness of 99% -day VaR calculaed from mulivariae model in he secion 3.. We used 000 observaions o evaluae VaR a each backesing rial, so he oal number of backesing rials is 358 [= 458(he oal number of observaions) 000(observaions for geing one VaR backesing)]. The resul of he PF es and he runs es are in Table 6.3. In he lef ail, boh he DCC model and he O-GARCH model were no rejeced wih he PF es a a 99% confidence level. Bu, boh were rejeced wih he runs es a a 99% confidence level. As a resul, boh mulivariae models were rejeced wih he PF es or wih he runs es. In he righ ail, boh he DCC model and he O-GARCH model were no rejeced wih he PF es a a 99% confidence level. Boh also were no rejeced wih he runs es a a 99% confidence level. So, boh models were no rejeced wih boh ess ime in he righ ail. In sum, wo mulivariae models were rejeced wih he PF es or wih he runs es in he lef or righ ail, which means ha boh models are inappropriae o evaluae VaR of he hypoheical porfolio for boh ails hough boh models are appropriae o evaluae VaR of he hypoheical porfolio jus for righ ail. One possible reason of his resul is ha hose models failed o incorporae volailiy clusering in he lef ail as in he univariae hypoheical porfolio case. 4

Table 6.3: Resul of he PF es and Runs es of Mulivariae Models for he Hypoheical Porfolio Tail Model Name Excepions PF Tes Runs Tes F-saisics p-value No Excepions Excepions Runs z- saisics p-value Lef DCC 49 4.3938 0.036 3533 49 93-3.08 0.003 a O-GARCH 48 3.7806 0.059 3534 48 89-4.589 0.0000 4 Righ DCC* b 49 4.3938 0.036 3533 49 95 -.966 0.0493 O-GARCH* 47 3.096 0.073 3535 47 93-0.85 0.44 a Bold means we can rejec he null hypohesis in each es a a 99% confidence inerval. b Aserisk means model work well in boh ess a he same ime a a 99% confidence level. 4

6.3 Univariae vs. Mulivariae Table 6.4 represens he resuls of he PF es and he runs es of univariae and mulivariae models for he hypoheical porfolio, which are reproduced from Table 6. and Table 6.3. In he lef ail, he EQMA model was rejeced wih boh he PF es and he runs es a a 99% confidence level. The oher models excep he EWMA model and he EQMA model were no rejeced wih he PF es, bu were rejeced wih he runs es a a 99% confidence level. Only EWMA model was no rejeced wih boh he PF es and he runs es in he lef ail. In he righ ail, he EQMA model, he EWMA model, and he GARCH models were rejeced wih he PF es a a 99% confidence level hough hese models were no rejeced wih he runs es a a 99% confidence level. The HS model and wo mulivariae models were no rejeced wih boh ess a a 99% confidence level. Overall, in he righ ail he HS model and wo mulivariae models were no rejeced wih boh ess a a 99% confidence level. If we consider boh lef and righ ail a he same ime, all models could be rejeced wih he PF es or wih he runs es. The EWMA model could be rejeced wih he PF es in he righ ail hough ha model was no rejeced wih boh ess in he lef ail. On he oher hand, he HS model and wo mulivariae models were no rejeced wih boh ess in he righ ail, bu could be rejeced wih he runs es in he lef ail. So, we could rejec all models considered in his hesis wih he PF es or wih he runs es in he lef or righ ail, which means ha no univariae and mulivariae models are appropriae o evaluae VaR of he hypoheical porfolio for boh ails. 43

Table 6.4: Resul of he PF es and Runs es of Univariae and Mulivariae Models for he Hypoheical Porfolio * Resuls are reproduced from Table 6. and Table 6.3 Tail Model Name Excepions PF Tes Runs Tes F-saisics p-value z- saisics p-value Equally weighed(eqma) 60 3.7060 a 0.000-3.3056 0.0009 Exponenially weighed(ewma)* b 50 5.0484 0.046 -.8935 0.0583 Univariae model GARCH (Normal Dis.) 49 4.3938 0.036-4.4555 0.0000 Lef GARCH ( Dis.) 45.978 0.38-4.9904 0.0000 44 Mulivariae model Univariae model Hisorical Simulaion 39 0.77 0.5986-4.47 0.0000 DCC 49 4.3938 0.036-3.08 0.003 O-GARCH 48 3.7806 0.059-4.589 0.0000 Equally weighed(eqma) 64 8.530 0.0000 -.035 0.044 Exponenially weighed(ewma) 60 3.7060 0.000 -.869 0.0 GARCH (Normal Dis.) 56 9.807 0.007-0.4084 0.6830 Righ GARCH ( Dis.) 58.6840 0.0006-0.375 0.7433 Mulivariae model Hisorical Simulaion* 43.366 0.45 -.0398 0.985 DCC* 49 4.3938 0.036 -.966 0.0493 O-GARCH* 47 3.096 0.073-0.85 0.44 a Bold means we can rejec he null hypohesis in each es a a 99% confidence inerval. b Aserisk means model was no rejeced wih boh ess a he same ime a a 99% confidence level. 44

We noe ha wo mulivariae models were no rejeced in boh ails if we consider only PF es, which means ha he way he DCC model and he O-GARCH model incorporae he condiional correlaion movemens of he individual socks as well as he condiional variance can improve a leas he uncondiional propery of models compared wih he way of parameric univariae models such as he EWMA model and he univariae GARCH models. Since he inernaional sandard, Basel Accord, in he banking indusry considers only uncondiional coverage propery in is risk managemen regulaory mandaes, wo mulivariae models could be useful in ha applicaion. In addiion, we would like o discuss he paern of excepions of he PF es resul using he DCC model, he O-GARCH model, and he GARCH model wih a normal disribuion. Figure 6. compares binomial sequences of excepions occurring in rolling backesing of he VaR esimaes using he GARCH model wih a normal disribuion and he DCC model. Figure 6.3 shows he same hing in he case of he O-GARCH model insead of he DCC model. In he sem diagrams of figure 6. and 6.3, means ha excepion occurs a ha poin. In boh figures, he average correlaion is calculaed using he DCC model. The circle means ha excepion occurs a ha poin in ha ail and ha model, bu excepion does no occur in he same ail in he oher model. In figure 6., he overall paerns of excepions resuling from he DCC model look ineresing; more excepions occur when correlaion increases. However, in he case of GARCH model wih a normal disribuion, more excepions occur when correlaion decreases. The VaR esimae using he DCC model is above he VaR esimae using he GARCH model when correlaion decreases. On he oher hand, when correlaion increases, he siuaion is reversed; he VaR esimae using he GARCH model is above 45

he VaR esimae using he DCC model, which means ha he change of he VaR esimaion using he DCC model is smaller han he change of he VaR esimaion using he GARCH model when correlaion changes. In Table 6.5, he VaR esimae using he DCC model has he leas sandard deviaion among hree models. The reason is ha as you can see in he specificaion of he DCC model in equaion (7), he DCC model uses he GARCH specificaion in modeling he condiional correlaion, which resuls in he long memory of correlaion or slow response o he change of correlaion. Table 6.5: Mean and Sandard Deviaion of he VaR esimaes Variables GARCH(Normal) DCC model O-GARCH model Mean.408.436.4403 Sandard Deviaion 0.8890 0.657 0.873 As a resul, we noe ha he performance improvemen of DCC models relaive o univariae models occurs in he period when correlaion decreases, which is he opposie observaion we expec; in fac, we expeced ha he performance improvemen would occur when correlaion increased. Comparing he O-GARCH model wih he univariae GARCH model, our calculaion resul indicaes ha he VaR esimaes using he O-GARCH model seem more sensiive o marke caasrophic even or marke risk because only he seveneenh (backesing rial 303 in figure 6.3) ou of he bigges hiry VaR esimaes using he 46

univariae GARCH model is greaer han he VaR esimaes using he O-GARCH model on he same day. However, in figure 6.3 he occurrence paern of excepion resuling from he O-GARCH model is no much differen from he occurrence paern of excepion resuling from he GARCH model in caasrophic evens (around backesing rials 000 hrough 3000) Compared wih he DCC model wih respec o correlaion, he O-GARCH model seems o have shorer memory han he DCC model. The early backesing horizon of he VaR esimae using O-GARCH model in figure 6.3 seems o remember he pas big correlaion caused by he Black Monday sock crash, which you can check in figure 6.0. So, in ha horizon he VaR esimae using he O-GARCH model is greaer han he VaR esimae using he univariae GARCH model. However, afer ha period, he O-GARCH model looks more ap o respond o he change of correlaion, even more han he univariae GARCH model. As a resul, all univariae models and mulivariae models were rejeced wih he PF es or wih he runs es in he lef or righ ail. However, if we consider only he PF es, which is more imporan han he runs es wih respec o real applicaion, he mulivariae models, he DCC models and he O-GARCH model, were no rejeced in boh ails. We noe ha he performance improvemen of DCC models relaive o univariae models wih respec o he PF es occurred when correlaion decreased. The performance improvemen of O-GARCH model also occurred when correlaion decreased, bu overall VaR esimaes using O-GARCH model were greaer han he VaR esimae using he univariae GARCH model when correlaion increased. 47

48 Figure 6.: Excepion Occurrence of DCC and GARCH (Normal) 48

49 Figure 6.3: Excepion Occurrence of O-GARCH and GARCH (Normal) 49