Modeling Risk for Long and Short Trading Positions

Transcription

1 MPRA Munich Personal RePEc Archive Modeling Risk for Long and Shor Trading Posiions Timoheos Angelidis and Savros Degiannakis Deparmen of Banking and Financial Managemen, Universiy of Piraeus, Deparmen of Saisics, Ahens Universiy of Economics and Business 5 Online a hps://mpra.ub.uni-muenchen.de/8467/ MPRA Paper No. 8467, posed 3 July 7 :46 UTC

2 M o d e l i n g R i s k f o r L o n g a n d S h o r T r a d i n g P o s i i o n s Timoheos Angelidis, Deparmen of Banking and Financial Managemen, Universiy of Piraeus Savros Degiannakis, Deparmen of Saisics, Ahens Universiy of Economics and Business Absrac The accuracy of parameric, non-parameric and semi-parameric mehods in predicing he one-day-ahead Value-a-Risk (VaR) measure in hree ypes of markes (sock exchanges, commodiies and exchange raes) is invesigaed, boh for long and shor rading posiions. The risk managemen echniques are designed o capure he main characerisics of asse reurns, such as lepokurosis and asymmeric disribuion, volailiy clusering, asymmeric relaionship beween sock reurns and condiional variance and power ransformaion of condiional variance. Based on backesing measures and a loss funcion evaluaion mehod, we find ou ha he modeling of he main characerisics of asse reurns produces he mos accurae VaR forecass. Especially for he high confidence levels, a risk manager mus employ differen volailiy echniques in order o forecas accuraely he VaR for he wo rading posiions. Differen models achieve accurae VaR forecass for long and shor rading posiions, indicaing o porfolio managers he significance of modeling separaely he lef and he righ side of he disribuion of reurns. The behavior of he risk managemen echniques is examined boh for long and shor VaR rading posiions, while o bes of our knowledge, his is he firs sudy ha invesigaes he risk characerisics of hree differen financial markes simulaneously. Moreover, we implemen a wo-sage model selecion in conras of he mos commonly used backesing procedures in he aemp o idenify a unique model. Finaly, we employ parameric, non-parameric and semi-parameric echniques in order o invesigae heir performance in a unify environmen. Keywords: Asymmeric Power ARCH model, Evaluae Forecasing Abiliy, Skewed- Disribuion, Value-a-Risk, Volailiy Forecasing. JEL: C3, C5, C53, G5.

3 I n r o d u c i o n Value-a-Risk (VaR) a a given probabiliy level a, is he prediced amoun of financial loss of a porfolio over a given ime horizon. Given he fac ha asse reurns are no normally disribued, since hey exhibi skewness and excess kurosis, i is plausible o employ volailiy forecasing echniques ha accommodae hese characerisics in order o accurae esimae he rue bu unobservable VaR. A researcher can eiher implemen parameric, semi-parameric or non-parameric mehods in order o calculae he VaR number. In he case of he non-parameric echniques, he hisorical simulaion is he mos well known and simplifies he compuaion of he VaR as i does no make any disribuional assumpion abou porfolio reurns. Even if his mehod has been horoughly examined by several auhors, heir conclusions are conroversial. For example, Hendricks (996) and Daníelsson () argued ha he sample size affecs he precision of he VaR esimaes, wih he longer one producing he mos accurae esimaions. On he conrary, Hoppe (998) proposed he use of a smaller one, since i can accommodae he srucural changes of he rading behavior more efficienly. On he oher hand, many researchers prefer o parameerize he properies of he underlying disribuion. Venkaaraman (996) and Zangari (996) suggesed o he marke praciioners a mixure of normal disribuions, while Billio and Pelizzon () esimaed a mulivariae swiching regime model in order o calculae he VaR for Ialian socks. Their procedure is differen from ha of Zangari (996) as he VaR forecass were based on a wo sae Markov process insead of a Bernoulli. Alexander and Leigh (997) esimaed he exponenially weighed moving average (EWMA) and he auoregressive condiional heeroskedasiciy (ARCH) models and found ou ha he ARCH is preferable o EWMA. Guerma and Harris () exended he EWMA model allowing for ime-variaion in he higher momens of he reurn disribuion and inroduced he exponenially weighed maximum likelihood (EWML) model. In he case of US, UK and Japan equiy porfolios, he EWML model, compared o he GARCH(,) specificaion under boh he normal and he Suden s- disribuion, improved he esimaed daily VaR number a he higher confidence level. Minik and Paoella () sudied he exchange raes and inroduced he Asymmeric Power ARCH (APARCH) model wih an asymmeric generalized Suden s- disribuion o allow for ime varying skewness. Gio and Lauren (3a, 3b) considered a skewed Suden s- disribuion, in order o accommodae he lepokurosis and he observed skewness of he financial ime series. They focused on he join behavior of

4 VaR models for long and shor rading posiions and argued ha for boh equiy indexes and commodiies he APARCH model had he bes overall performance. Huang and Lin (4) reached o he same conclusion, as hey argued ha he normal APARCH model is preferred a lower confidence level, while he Suden s- APARCH model is more accurae han eiher he RiskMerics TM or he normal APARCH models a higher confidence level. Furhermore, Brooks and Persand (3) also concluded ha he asymmery is an imporan issue in he VaR framework and herefore i mus be modeled eiher in he uncondiional mean reurn disribuion or in he volailiy specificaion. The filered hisorical simulaion approach was inroduced by Hull and Whie (998) and Barone-Adesi e al. (999). This mehod is a mixure of parameric and nonparameric saisical procedures as i forecass he variance hrough a parameric volailiy model bu i does no make any assumpion abou he disribuion of sandardized reurns. According o Barone-Adesi and Giannopoulos (), who compared he filered hisorical simulaion wih he hisorical one, he mixure of parameric and non-parameric saisical procedures produces more accurae VaR forecass. Our sudy sheds a ligh on he volailiy forecasing mehods under a risk managemen framework, since i juxaposes he performance of he mos well known echniques for differen markes (sock exchanges, commodiies and exchange raes) and rading posiions. Specifically, he 95% and 99% one day VaR number is esimaed by a se of ARCH models (assuming four condiional variance specificaions and hree disribuional assumpions), hisorical and filered-hisorical simulaions and he commonly used variance-covariance mehod. Under he framework of he parameric echniques, he differen disribuions will allow he selecion of a model for he reurn ails, while we have invesigaed hree differen markes in order he resuls no o be dependen on a specific financial marke. Moreover, we employ a wo-sage procedure o invesigae he forecasing power of each volailiy forecasing echnique. In he firs sage, wo backesing crieria are implemened o es he saisical accuracy of he models. In he second sage, we employ sandard forecas evaluaion mehods o examine wheher he differences beween models (ha have exhibied sufficien uncondiional and condiional coverage) are saisically significan. Alhough our analysis is similar o he presened papers, here are significan differences. Firs, we examine he behavior of he risk managemen echniques boh for long and shor VaR rading posiions, while mos of he research has been applied only on long ones. Therefore, we will be able o examine wheher an asymmeric model is able o 3

5 capure boh he characerisics of he wo ails. Second o bes of our knowledge, his is he firs sudy ha invesigaes he risk characerisics of hree differen financial markes simulaneously. Hence, we are able o infer wheher he financial markes of sock exchanges, commodiies and exchange raes share common feaures in he field of VaR forecasing. Third, we implemen a wo-sage model selecion in conras of he mos commonly used backesing procedures in he aemp o idenify a unique model. Las, we employ parameric, non-parameric and semi-parameric echniques in order o invesigae heir performance in a unify environmen, on he conrary o he exisen lieraure which focus only on one echnique a ime. Our sudy shows ha alhough here is no a specific model ha accurae esimaes he VaR number for all financial markes and rading posiions, here are some characerisics ha should be aken ino accoun in order for a risk manager o calculae he VaR accuraely. For all he financial markes under invesigaion, we infer ha he normal disribuion produces adequae one-day-ahead VaR forecass a he 95% confidence level. On he oher hand, models ha parameerise he leverage effec for he condiional variance, he lepokurosis and he asymmery of he daa, forecas accurae he VaR a he 99% confidence level. Moreover, shor-rading posiions should be modeled using volailiy specificaions differen from ha of porfolios wih long rading posiions, which implies ha even asymmeric models are no sufficienly asymmeric. The volailiy forecasing models and he VaR evaluaion mehods are presened in he nd and 3 rd secions, respecively. The fourh secion illusraes he resuls of he sudy and he fifh secion concludes. V o l a i l i y F o r e c a s i n g M o d e l s Le lnp P y denoe he daily reurn series, where P is he price of an asse a day. The ARCH models can be presened in he following general framework: g z c i. i ~. d. z, ; i, j, i y f j () 4

6 where c is a consan parameer, of zero mean and uni variance, and heir condiional sandard deviaion. is he innovaion process,, f is a densiy funcion g.;. is a funcional form of he pas innovaions and Surveys of Bollerslev e al. (994), Li e al. (), Poon and Granger (3), Degiannakis and Xekalaki (4) cover a wide range of ARCH presenaions. Bollerslev (986) proposed a generalizaion of Engle s (98) ARCH model and inroduced he GARCH(,) specificaion: a a b, () where, a and b. Riskmerics TM suggesed he exponenially weighed moving average, or EWMA, which is a special case of he GARCH(,), since a, a.6 and b. 94:. (3).6.94 Alhough he GARCH(,) model capures he volailiy clusering phenomenon, i could no explain he asymmeric relaionship beween reurns and condiional variance. Nelson (99) proposed he exponenial GARCH, or EGARCH(,), model: ln a a E b ln, (4) where he parameer accommodaes he asymmeric effec. Glosen e al. (993) presened he TARCH(,) specificaion, where good news and bad news have differen effec on he condiional variance: i a d b a, (5) i for d denoing an indicaor funcion (i.e. d if and d oherwise). Ding e al. (993) inroduced he asymmeric power ARCH, or APARCH(,), model: b a a, (6) for a, a, b, and. In he influenial paper of Engle (98), he densiy funcion of considered as he sandard normal disribuion: z, f., was f z z exp. (7) Bollerslev (987) proposed he Suden s- disribuion in order o produce an uncondiional disribuion wih hicker ails: 5

7 f z ; v v v v z v v, v, (8) where v denoes he degrees of freedom of he disribuion. Lamber and Lauren () suggesed ha no only he condiional disribuion of innovaions may be lepokuric, bu also asymmeric and proposed he skewed Suden s- densiy funcion: f z ; v, g v v v where g is he asymmery parameer, s g g. sz m g v d v, v, (9) is he gamma funcion, d if z m / s, d oherwise, m v v v g g and s g g m are he mean and he sandard deviaion of he non-sandardized skewed Suden s- disribuion, respecively. compued as: Under he framework of he parameric echniques, he one-day-ahead VaR is where z a z; a VaR F ˆ, () F ; is he corresponding quanile of z disribuion and is he one-dayahead condiional sandard deviaion forecas given he informaion ha is available a ime i. ~ i. d N. Under he assumpion ha, ˆ, he calculaion of he VaR can be simplified: ; a VaR F ˆ. () However, he conjecure of normaliy is no saisfied in financial reurns and, hence, his mehod, which we will refer o as Variance-Covariance (VC), usually underesimaes he rue VaR. The Hisorical Simulaion (HS) mehod is a simple and inuiive non-parameric procedure, which relies on hisorical reurns o calculae he VaR as he corresponding percenile of he pas m reurns [i] : m VaR F { y - } ; a. () In he case of he parameric mehods, he disribuion choice is crucial; while in he non-parameric case here is no consisen approach in forecasing he volailiy. The Filered Hisorical Simulaion (FHS) mehod, which was presened in Hull and Whie (998) and Barone-Adesi e al. (999), combines he wo approaches in order o make he 6

8 mos of hem. Given an adequae volailiy model, such as he GARCH(,), he VaR is compued based on he quanile of he sandardized innovaions: m { ˆ ˆ - } ; a VaR F ˆ -. (3) E v a l u a e h e F o r e c a s i n g A b i l i y o f V a l u e a R i s k M e a s u r e s Our objecive is o es hese differen volailiy forecasing echniques under a risk managemen environmen. Therefore, we employ a wo-sage procedure o evaluae he various risk managemen echniques. In he firs sage, wo backesing crieria (uncondiional and condiional coverage) are implemened o examine he saisical accuracy of he models while, in a second sage, we employ a forecas evaluaion mehod o invesigae wheher he differences beween he VaR models, ha exhibied sufficien uncondiional and condiional coverage, are saisically significan. The simples mehod in deermining he adequacy of a VaR measure is o es he hypohesis ha he proporion of violaions [ii] is equal o he expeced one. Kupiec (995) developed a likelihood raio saisic: LR uc N T-N N T-N N ln[ - ) ( ) N] - ln[(- p) p ] ~ X, (4) T T under he null hypohesis ha he observed excepion frequency, N / T, equals o he expeced one, p, where N is he number of days over a period T ha a violaion has occurred. Alhough he uncondiional coverage es can rejec a model ha eiher overesimaes or underesimaes he rue bu unobservable VaR, i canno examine wheher he violaions are randomly disribued. Chrisoffersen (998) developed a condiional coverage es, which joinly invesigaes wheher i) he oal number of failures is equal o he expeced one and ii) he VaR violaions are independenly disribued. Under he null hypohesis ha he failure process is independen and he expeced proporion of violaions equals o p, he appropriae likelihood raio is: LR cc T-N N n n n n ln[( - p) p ] ln[( - ) ( -) ] ~ X, (5) where n is he number of observaions wih value i followed by j, for i, j, and ij nij ij are he corresponding probabiliies. i, j denoes ha a violaion has been n j ij 7

9 made, while i, j indicaes he opposie. Conrary o Kupiec's (995) es, Chrisoffersen s procedure can rejec a VaR model ha generaes oo many or oo few clusered violaions. However, in mos of he cases, here are more han one risk models ha saisfy boh he backesing measures and herefore a risk manager can no selec a unique volailiy forecasing echnique. Hence, in order o selec one model among he various candidaes, we compare he bes performed models via a loss funcion. Lopez (999) proposed o marke praciioners a procedure of evaluaing VaR models based on a loss funcion approach. According o he Basle Commiee on Banking Supervision (996) proposal, he incorporaed boh he oal number of violaion and heir magniude erm. More formally, Lopez s loss funcion can be described as: ( VaR - y ) if violaion occurs (6) else. The magniude erm ( VaR - y ) ensures ha he larger he failure is he more he penaly is added o a model, while a score of one is added, similar o Kupiec's es, whenever a violaion occurs. According o Lopez's loss funcion, a model, which minimizes T he oal loss,, is preferred over he ohers. Based on Diebold and Mariano (995), Sarma e al. (3) and Angelidis e al. (4), we examine wheher he forecas accuracy of wo VaR models is saisically significan. Specifically, we es he null hypohesis of equivalen predicive abiliy of models A and B, agains he alernaive hypohesis ha model A is superior o model B. The Diebold-Mariano saisic is he "-saisic" for a regression of z on a consan wih heeroskedasic and auocorrelaed consisen sandard errors (HAC), where A B z, - and A and B are he loss funcions of models A and B, respecively. A negaive value of z indicaes ha model A is superior o model B. E m p i r i c a l R e s u l s Table I summarizes he basic descripive saisics of he 6 series, while he daily logreurns graphs are presened in Figure. Volailiy clusering is clearly visible in Figure, which suggess he presence of heeroskedasiciy. Moreover, based on Jarque-Bera 8

10 saisic, he null hypohesis of normaliy is rejeced a any level of significance, as here is evidence of excess kurosis relaive o ha of he normal disribuion and non-zero skewness. The preliminary descripive saisics indicae ha he characerisics of he wo ails are differen and herefore, i is ineresing o evaluae he risk models for differen rading posiions. <<Take in Table I>> We generae ou-of-sample VaR forecass for wo equiy indices (S&P5, FTSE), wo commodiies (Gold Bullion $/Troy Ounce, London Bren Crude Oil Index U$/BBL) and wo exchange raes (US $ o Japanese, US $ o UK ), obained from Daasream for he period of January 3rd 989 o June 3h 3. For all models, we use a rolling sample of observaions in order o generae, approximaely, 6 forecass and calculae he 95% and he 99% VaR + for long and shor rading posiions. <<Take in Figure >> The framework in () is esimaed for (), (4), (5) and (6) condiional variance specificaions and (7) o (9) densiy funcions by adoping he maximum likelihood mehod. The EWMA model, he variance-covariance procedure and he echniques of hisorical and filered hisorical simulaion are applied, giving a oal of 6 volailiy-forecasing models. Under he framework of he loss funcion approach, we evaluae all he models wih p-value greaer han % for boh uncondiional and condiional coverage ess. A high cuoff poin is preferred in order o ensure ha he successful risk managemen echniques will no a) over or under esimae saisically he rue VaR, as in he former case, he financial insiuion does no use is capial efficienly, while in he laer case i can no cover fuure losses and b) generae clusered violaions, since an adequae model mus wide he VaR forecass during volaile periods and narrow hem oherwise. In he case of a smaller cu-off poin, an incorrec model could no be easily rejeced, which migh urn o be cosly for a risk manager. Table II summarizes he wo-sage model selecion procedure [iii]. In he firs sage (columns and 3) he models ha have no been rejeced by he saisical backesing procedures are presened, while in he second sage (column 4), he volailiy mehods ha are preferred over he ohers, based on he loss funcion approach, are exhibied. For example, in panel A, for he S&P5 index, he GARCH(,)-normal model achieves he smalles value of he loss funcion, while is forecasing accuracy is no saisically 9

11 differen o ha of he EWMA, EGARCH(,) and APARCH(,) models wih normally disribued innovaions. <<Take in Table II>> The VC mehod underesimaes he "rue" VaR, since porfolio reurns exhibi excess kurosis relaive o ha of he normal disribuion. For example, he average excepion rae a he 99% confidence level for long (shor) rading posiions is.67% (.8%). Therefore, in mos of he cases, he p-values of he backesing measures are close o zero. Examining he 95% confidence level we reach o similar conclusion, hus his mehod is no an appropriae echnique for risk managemen. On he oher hand, he RiskMerics TM mehod is more appropriae echnique han he VC one, as for he 95% confidence level he excepion raes are saisically equal o he heoreical values. However, in some cases his mehod generaes clusered violaions indicaing ha he risk model is misspecified. A he higher confidence level i underesimaes he rue value of VaR, since he average excepion rae is 68% greaer han i is expeced. More sophisicaed echniques ha accommodae he feaures of he financial ime series are needed, in order o calculae he one-day-ahead VaR. ARCH models based on he normal disribuion (GARCH(,), EGARCH(,), TARCH(,) and APARCH(,)) perform beer han he VC and he RiskMerics mehods. Especially, for he 95% confidence level he failure raes are saisically equal o he heoreical values, irrespecively of he rading posiion. However, hey underesimae he VaR a he higher confidence level, even if his underesimaion is smaller han ha of he RiskMerics TM. Thus he degree of lepokurosis induced by he ARCH process does no capure all he lepokurosis presened in he daa. Hence, in order o model more adequaely he hickness of ails, we use wo differen disribuional assumpions for he sandardized residuals: Suden s- and skewed Suden s- disribuions. Brooks and Persand (3) poined ou ha models, which do no allow for asymmeries eiher in he uncondiional reurn disribuion or in he volailiy specificaion, underesimae he rue VaR. Gio and Lauren (3a) proposed he skewed Suden s- disribuion and argued ha i performed beer han he pure symmeric one, as i reproduced he characerisics of he empirical disribuion more accuraely. These views are confirmed for boh confidence levels and rading posiions, as mos of he seleced models parameerise hese feaures.

12 The volailiy specificaions, which parameerise he leverage effec for he condiional variance and he asymmery of he innovaions disribuion, forecas he VaR a he 99% confidence level more adequaely. However, he models ha mus be employed for he shor and he long rading posiions are no he same. This finding is in conras wih ha of Gio and Lauren (3a) who argued ha he APARCH model based on he skewed Suden s- disribuion forecass he VaR adequaely for boh rading posiions. Conrary o he findings for he 99% confidence level, he ARCH models under he Suden s- and he corresponding skewed disribuion overesimae he 95% VaR numbers for boh rading posiions, a resul ha is also documened by Guerma and Harris () and Billio and Pelizzon () among ohers. Therefore, even if he lepokuric disribuional assumpion seems o be a beer choice overall for he 99% confidence level, i should no be applied for he lower confidence inerval as i produces higher han exceped VaR forecass. Turning he discussion o he non-parameric mehods, he HS mehod underesimaes oal risk, as for mos of he cases he excepion raes are greaer han he expeced ones. The inadequae performance of he HS may is due o he fac ha he underlying disribuion does no remain consan. In erms of he coverage ess, he FHS procedure combined wih a GARCH(,) updaing volailiy echnique offers a major improvemen over boh he parameric and he non-parameric mehods, as he excepion raes are oo close o he heoreical ones for boh rading posiions. For example, a 95% confidence level, he average proporion of failures for he long (shor) rading posiion is 5.55% (5.78%). This is also he case for he 99% confidence level, as he corresponding percenages are.96% and.3%, respecively. However, he FHS mehod does no yield he bes VaR forecass, as, for example, i underesimaes he risk for he FTSE index a he 95% confidence level. Furhermore, for long posiion on OIL index (95% VaR) and shor posiion on GOLD index (99% VaR), here are no models ha produce adequae VaR forecass. Given he fac ha for hese cases he models have been rejeced by he condiional coverage es, here is evidence ha clusered violaions were generaed. So, all he models are very slow a updaing he VaR number when marke volailiy changes rapidly. Finally, we can no compare direcly he models based on he backesing measures, as a greaer p-value of a model does no indicae is superioriy among is compeiors. However, under he framework of he loss funcion, his is possible as we evaluae saisically he differences beween he various risk models. No model seems o

13 sysemaically produce globally accepable VaR esimaes for all securiies, rading posiions and confidence levels. However, based on he proposed model selecion procedure, we manage o conclude o a smaller se of models and in some cases we idenify a unique risk managemen model. C o n c l u s i o n In his paper we examined he mos recenly developed VaR mehods for sock exchanges, commodiies, and exchange raes. In an ou-of-sample sudy we compared parameric, non-parameric and semi-parameric echniques boh for long and shor rading posiions. As he backesing ess do no idenify a unique model for each porfolio, we define a loss funcion o evaluae he models ha have me he prerequisie of he correc uncondiional and uncondiional coverage. Under he new framework, a model ha minimizes he oal loss is preferred over he remaining ones, while by implemening a es for he differences of he forecas error we provide saisical inference for he forecasing abiliy of he models. Assuming normaliy for he condiional reurn disribuion, we forecas accurae he one-day-ahead VaR a he 95% confidence level. However, gains in forecasing he 99% VaR wih models ha allow for asymmeries eiher in he condiional reurn disribuion or in he volailiy specificaion are subsanial. Differen models achieve accurae VaR forecass for long and shor rading posiions, indicaing o porfolio managers he significance of modeling eiher he lef or he righ side of he disribuion of reurns. Using daa from hree ypes of financial markes (sock exchanges, commodiies, and exchange raes) here is evidence ha our resuls hold for differen ypes of markes. An ineresing issue for furher research would be he implemenaion of he described wo-sage model selecion procedure for he expeced shorfall risk measure, which is he value of he loss condiioned ha a VaR violaion has occurred and has been considered as an alernaive downside risk measure. R e f e r e n c e s Angelidis, T., Benos, A. and Degiannakis, S. (4), "The Use of GARCH Models in VaR Esimaion," Saisical Mehodology, Vol., No., pp Alexander, C.O. and Leigh, C.T. (997), "On he Covariance Models Used in Value a Risk Models," Journal of Derivaives, Vol. 4, pp. 5-6.

14 Barone-Adesi, G. and Giannopoulos, K. (), "Non-parameric VaR Technics. Myhs and Realiies," Economic Noes by Banca Mone dei Paschi di Siena Spa, Vol. 3, pp Barone-Adesi, G., Giannopoulos, K. and Vosper, L. (999), "VaR Wihou Correlaions for Nonlinear Porfolios," Journal of Fuures Markes, Vol. 9, pp Basle Commiee on Banking Supervision (996), "Supervisory Framework for he Use of Backesing in Conjuncion wih he Inernal Models Approach o Marke Risk Capial Requiremens," Manuscrip, Bank for Inernaional Selemens. Billio, M. and Pelizzon, L. (), "Value-a-Risk: A Mulivariae Swiching Regime Approach," Journal of Empirical Finance, Vol. 7, pp Bollerslev, T. (986), "Generalized Auoregressive Condiional Heeroskedasiciy. Journal of Economerics," Vol. 3, pp Bollerslev, T. (987), "A Condiional Heeroskedasic Time Series Model for Speculaive Prices and Raes of Reurn," Review of Economics and Saisics, Vol. 69, pp Bollerslev, T., Engle, R.F. and Nelson, D. (994), "ARCH Models," in: Engle, R.F. and McFadden, D. (Ed.), Handbook of Economerics, Elsevier Science, Amserdam, Vol. 4, pp Brooks, C. and Persand, G. (3), "The Effec of Asymmeries on Sock Index Reurn Value-a-Risk Esimaes," The Journal of Risk Finance, Winer, pp Chrisoffersen, P. (998), "Evaluaing Inerval Forecass," Inernaional Economic Review, Vol. 39, pp Daníelsson, J. (), "The Emperor Has no Clohes: Limis o Risk Modeling," Journal of Banking and Finance, Vol. 6, pp Degiannakis, S. and Xekalaki, E. (4), "Auoregressive Condiional Heeroscedasiciy Models: A Review," Qualiy Technology and Quaniaive Managemen, Vol., No., pp Diebold, F.X. and Mariano, R. (995), "Comparing Predicive Accuracy," Journal of Business and Economic Saisics, Vol. 3, No. 3, pp Ding, Z., Granger, C.W.J. and Engle, R.F. (993), "A Long Memory Propery of Sock Marke Reurns and a New Model," Journal of Empirical Finance, Vol., pp Engle, R.F. (98), "Auoregressive Condiional Heeroskedasiciy wih Esimaes of he Variance of U.K. Inflaion," Economerica, Vol. 5, pp Gio, P. and Lauren, S. (3a), "Value-a-Risk for Long and Shor Trading Posiions," Journal of Applied Economerics, Vol. 8, pp

15 Gio, P. and Lauren, S. (3b), "Marke risk in Commodiy Markes: A VaR approach," Energy Economics, Vol. 5, pp Glosen, L., Jagannahan, R. and Runkle, D. (993), "On he Relaion Beween he Expeced Value and he Volailiy of he Nominal Excess Reurn on Socks," Journal of Finance, Vol. 48, pp Guerma, C. and Harris, R.D.F. (), "Forecasing Value-a-Risk Allowing for Time Variaion in he Variance and Kurosis of Porfolio Reurns," Inernaional Journal of Forecasing, Vol. 8, pp Hendricks, D. (996), "Evaluaion of Value-a-Risk Models Using Hisorical Daa," Economic Policy Review, Vol., pp Hoppe, R. (998), "VAR and he Unreal World," Risk, Vol., pp Huang, Y.C and Lin, B-J. (4), "Value-a-Risk Analysis for Taiwan Sock Index Fuures: Fa Tails and Condiional Asymmeries in Reurn Innovaions," Review of Quaniaive Finance and Accouning, Vol., pp Hull, J. and Whie, A. (998), "Incorporaing Volailiy Updaing Ino he Hisorical Simulaion Mehod for VaR," Journal of Risk, Vol., pp Kupiec, P.H. (995), "Techniques for Verifying he Accuracy of Risk Measuremen Models," Journal of Derivaives, Vol. 3, pp Lamber, P. and Lauren, S. (), "Modelling Skewness Dynamics in Series of Financial Daa," Discussion Paper, Insiu de Saisique, Louvain-la-Neuve, Belgium. Li, W.K., Ling, S. and McAleer, M. (), "A Survey of Recen Theoreical Resuls for Time Series Models Wih GARCH Errors," Discussion Paper 545, Insiue of Social and Economic Research, Osaka Universiy, Japan. Lopez, J.A. (999), "Mehods for Evaluaing Value-a-Risk Esimaes," Economic Review, Federal Reserve Bank of San Francesco, Vol., pp Minik, S. and Paoella, M. (), "Condiional Densiy and Value-a-Risk Predicion of Asian Currency Exchange Raes," Journal of Forecasing, Vol. 9, pp Nelson, D. (99), "Condiional Heeroskedasiciy in Asse Reurns: A New Approach," Economerica, Vol. 59, pp Poon, S.H. and Granger, C.W.J. (3), "Forecasing Volailiy in Financial Markes: A Review," Journal of Economic Lieraure, XLI, pp Sarma, M., Thomas, S. and Shah, A. (3), "Selecion of VaR Models," Journal of Forecasing, Vol., No. 4, pp

16 Van den Goorbergh, R.W.J. and Vlaar, P. (999), "Value-a-Risk Analysis of Sock Reurns. Hisorical Simulaion, Variance Techniques or Tail Index Esimaion?," DNB Saff Repors 4, Neherlands Cenral Bank. Venkaaraman, S. (996), "Value a Risk for a Mixure of Normal Disribuions: The Use of Quasi-Bayesian Esimaion Techniques," Economic Perspecives, Federal Reserve Bank of Chicago (March/April), pp. -3. Zangari, P. (996), "An Improved Mehodology for Measuring VAR," RiskMerics Monior, Reuers/JP Morgan. 5

17 Table I. Descripive Saisics of S&P5, FTSE, Gold Bullion $/Troy Ounce, London Bren Crude Oil Index U$/BBL, US $ o UK and US $ o Japanese, for he period of January 3rd 989 o June 3h 3. S&P 5 FTSE GOLD OIL US_UK US_YEN Mean.34%.% -.5%.6% -.%.% Median.39%.43% -.5%.8%.5% -.4% Maximum 5.573% 5.93% 7.38%.556% 3.58% 6.574% Minimum -7.3% % -7.8% -.5% -3.8% % Sd. Dev..48%.7%.786%.7%.586%.75% Skewness Kurosis Jarque-Bera

18 Figure. Daily log-reurns for he period of January 3rd 989 o June 3h 3. S&P5 Gold Bullion $/Troy Ounce US $ o UK FTSE London Oil Index U$/BBL US $ o Japanese

19 Table II. Exhibi 4. The wo-sage model selecion procedure. Column presens he models ha have no been rejeced by he uncondiional coverage backesing crierion (Kupiec 995), Column 3 presens he models ha have no been rejeced by he condiional coverage backesing crierion (Chrisoffersen 998), Column 4 presens he models ha are preferred over he ohers based on he loss funcion approach. In Column 4, he model wih he lower value of he loss funcion is bold faced. Series Uncondiional Coverage Condiional Coverage Loss Funcion 95% VaR Long Posiions S&P 5 EWMA, G-N, E-N, A-N, FHS EWMA, G-N, E-N, A-N, FHS EWMA, G-N, E-N, A-N FTSE G-T, T-T, G-ST, E-ST, T-ST G-T, T-T, G-ST, E-ST, T-ST G-ST, T-ST OIL VC, EWMA, G-N, E-N, T-N, A-N, FHS - - GOLD EWMA, G-N, E-N, T-N, A-N, FHS EWMA, G-N, E-N, T-N, A-N, FHS G-N, E-N, T-N US_UK EWMA, G-N, E-N, T-N, A-T, A-ST, EWMA, G-N, E-N, T-N, A-T, A-ST, EWMA, G-N, E-N, T-N, A-T, A- FHS FHS ST, FHS US_YEN EWMA, G-N, E-N, T-N, HS, FHS EWMA, G-N, E-N, T-N, HS, FHS G-N, E-N, T-N Shor Posiions S&P 5 EWMA, G-N, E-N, T-N, A-N G-N, A-N G-N, A-N FTSE EWMA, G-N, E-N, T-N, A-N, A-ST EWMA, G-N, E-N, T-N, A-N, A-ST EWMA, T-N, A-ST OIL VC, EWMA, G-N, E-N, T-N VC, EWMA, G-N, E-N, T-N G-N, E-N, T-N GOLD EWMA, G-N, E-N, T-N, A-N, FHS G-N, E-N, T-N, FHS G-N, T-N US_UK G-N, E-N, T-N, A-ST, FHS G-N, E-N, T-N, A-ST, FHS E-N US_YEN VC, EWMA, G-N, E-N, T-N, A-N, VC, EWMA, G-N, E-N, T-N, A-N, VC, G-N, E-N, T-N, HS, FHS HS, FHS HS, FHS 99% VaR Long Posiions S&P 5 E-T, T-T, A-T, A-ST, FHS E-T, T-T, A-T, A-ST, FHS E-T, T-T, A-T, A-ST, FHS FTSE G-T, E-T, T-T, A-T, G-ST, E-ST, T-ST, A-ST G-T, E-T, T-T, A-T, G-ST, E-ST, T-ST, A-ST G-T, E-T, G-ST, E-ST, T-ST, A-ST OIL E-N, A-N, G-T, E-T, T-T, A-T, G- A-N, A-T, G-ST, FHS A-T, G-ST ST, E-ST, T-ST, HS, FHS GOLD G-N, FHS G-N, FHS FHS US_UK VC, EWMA, G-N, E-N, T-, A-T, A-ST VC, EWMA, G-N, E-N, T-, A-T, A-ST VC, G-N, E-N, A-T, A-ST US_YEN VC, G-N, T-N, HS, FHS VC, G-N, T-N, HS, FHS T-N, HS, FHS Shor Posiions 8

20 S&P 5 G-N, T-N G-N, T-N G-N, T-N FTSE A-N, FHS A-N, FHS A-N OIL VC, EWMA, G-N, E-N, T-N, A-N, VC, EWMA, G-N, E-N, T-N, A-N, VC, E-N, A-N, HS, FHS HS, FHS HS, FHS GOLD A-T, A-ST, FHS - - US_UK VC, A-T, A-ST, FHS VC, A-T, A-ST, FHS VC, A-T, A-ST, FHS US_YEN G-T, E-T, T-T, A-T, G-ST, T-ST, A-T, HS, FHS G-T, E-T, T-T, A-T, G-ST, T-ST, A-T, FHS G-T, E-T, A-T, G-ST, T-ST, A-T Models: G-N (GARCH(,)-normal), G-T (GARCH(,)-Suden s-), G-ST (GARCH(,)-skewed-), E-N (EGARCH(,)-normal), E-T (EGARCH(,)-Suden s-), E-ST (EGARCH(,)-skewed-), T-N (TARCH(,)-normal), T-T (TARCH(,)-Suden s-), T-ST (TARCH(,)- skewed-), A-N (APARCH(,)-normal), A-T (APARCH(,)-Suden s-), A-ST (APARCH(,)-skewed-), EWMA (RiskMerics), VC (Variance Covariance Mehod), HS (Hisorical Simulaion Technique), FHS (Filered Hisorical Simulaion Technique). [i] For more informaion abou HS mehod see Hendricks (996), Van den Goorbergh and Vlaar (999) and Daníelsson () among ohers. [ii] A violaion occurs if he prediced VaR is no able o cover he realized loss. [iii] Exhibis wih deailed resuls are available upon reques. 9