Forecasting Value-At-Risk with a Parsimonious Portfolio Spillover GARCH (PS-GARCH) Model *

Transcription

1 Forecasing Value-A-Risk wih a Parsimonious Porfolio Spillover GARCH (PS-GARCH) Model * Michael McAleer and Bernardo da Veiga School of Economics and Commerce Universiy of Wesern Ausralia Updaed: March 2007 Absrac: Accurae modelling of volailiy (or risk) is imporan in finance, paricularly as i relaes o he modelling and forecasing of Value-a-Risk (VaR) hresholds. As financial applicaions ypically deal wih a porfolio of asses and risk, here are several mulivariae GARCH models which specify he risk of one asse as depending on is own pas as well as he pas behaviour of oher asses. Mulivariae effecs, whereby he risk of a given asse depends on he previous risk of any oher asse, are ermed spillover effecs. In his paper we analyse he imporance of considering spillover effecs when forecasing financial volailiy. The forecasing performance of he VARMA-GARCH model of Ling and McAleer (2003), which includes spillover effecs from all asses, he CCC model of Bollerslev (1990), which includes no spillovers, and a new Porfolio Spillover GARCH (PS-GARCH) model, which accommodaes aggregae spillovers parsimoniously and hence avoids he so-called curse of dimensionaliy, are compared using a VaR example for a porfolio conaining four inernaional sock marke indices. The empirical resuls sugges ha spillover effecs are saisically significan. However, he VaR hreshold forecass are generally found o be insensiive o he inclusion of spillover effecs in any of he mulivariae models considered. Keywords and phrases: Volailiy, Value-a-Risk (VaR) hresholds, mulivariae GARCH, condiional correlaions, parsimonious porfolio spillovers, forecasing VaR. JEL classificaions: C32, C51, C53, F37, G15, G21 * The auhors wish o hank wo referees, Dave Allen, Manabu Asai, Felix Chan, Clive Granger, Suhejla Hoi, Marcelo Medeiros, Timo Terasvira, Alvaro Veiga, seminar paricipans a he Insiue of Economics, Academia Sinica, Taiwan, Chiang Mai Universiy, Fondazione Eni Enrico Maei - Milan, Griffih Universiy, Hiroshima Universiy, Kyoo Universiy, La Trobe Universiy, Ling Tung Universiy, Macquarie Universiy, Ponifical Caholic Universiy - Rio de Janeiro, Brazil, Queensland Universiy of Technology, Reserve Bank of New Zealand, Universiy of Canerbury, Universiy of New Souh Wales, Universiy of Padua, Universiy of Queensland, Universiy of Venice Ca Foscari, and Yokohama Naional Universiy, and paricipans a he Inernaional Conference on Simulaion and Modeling, Bangkok, Thailand, January 2005, Economeric Sudy Group Meeing, Chrischurch, New Zealand, March 2005, Inaugural Meeing of he Economeric Sociey of Thailand, June 2005, and Financial Engineering and Risk Managemen Workshop, Shanghai, July 2005, for helpful commens and suggesions. The firs auhor is graeful for he financial suppor of he Ausralian Research Council. The second auhor acknowledges a Universiy Posgraduae Award and an Inernaional Posgraduae Research Scholarship a he Universiy of Wesern Ausralia. 1

2 1. Inroducion Accurae modelling of volailiy (or risk) is of paramoun imporance in finance. As risk is unobservable, several modelling procedures have been developed o measure and forecas risk. The Generalised Auoregressive Condiional Heeroskedasiciy (GARCH) model of Engle (1982) and Bollerslev (1986) have subsequenly led o a family of auoregressive condiional volailiy models. The success of GARCH models can be aribued largely o heir abiliy o capure several sylised facs of financial reurns, such as ime-varying volailiy, persisence and clusering of volailiy, and asymmeric reacions o posiive and negaive shocks of equal magniude. This has also conribued o he modelling and forecasing of Value-a-Risk (VaR) hresholds. As financial applicaions ypically deal wih a porfolio of asses and risks, here are several mulivariae GARCH models which specify he risk of one asse as depending dynamically on is own pas risk as well as on he pas risk of oher asses (see Li e al. (2002) for a survey of recen heoreical developmens for condiional volailiy models, and McAleer (2005) for a discussion of a variey of univariae and mulivariae, condiional and sochasic, financial volailiy models). A volailiy spillover is defined as he impac of any previous volailiy of asse i on he curren volailiy of asse j, for any i j. A similar definiion applies for reurns spillovers. da Veiga and McAleer (2005) showed ha he mulivariae VARMA-GARCH model of Ling and McAleer (2003) and VARMA-Asymmeric GARCH (or VARMA-AGARCH) model of Hoi e al. (2003) provided superior volailiy and VaR hreshold forecass han heir nesed univariae counerpars, namely he GARCH model of Bollerslev (1986) and he GJR model of Glosen, Jagannahan and Runkle (1992), respecively. Mulivariae exensions have grea inuiive and empirical appeal as hey enable modelling of he relaionship beween subses of he porfolio and allow for scenario and sensiiviy analyses. Moreover, heir srucural and asympoic properies have been well esablished, especially for mulivariae GARCH models (for furher deails, see Ling and McAleer (2003) and Hoi e al. (2003), which exend he resuls for a range of univariae 2

3 GARCH models in Ling and McAleer (2002a, b)). However, he pracical usefulness of his resul can be affeced by he compuaional difficulies in esimaing he VARMA- GARCH and VARMA-AGARCH models for a large number of asses, as he number of parameers o be esimaed can increase dramaically wih he number of asses, and hence spillover effecs. Several parsimonious mulivariae models have been proposed o deal wih he overparameerizaion problem. The CCC model of Bollerslev (1990), he Dynamic Condiional Correlaion (DCC) model of Engle (2002), and he Varying Condiional Correlaion (VCC) model of Tse and Tsui (2002) use a wo-sep esimaion procedure o faciliae esimaion. McAleer e al. (2005) exended hese condiional correlaion models by specifying he shocks o reurns as being ime dependen, and esablished he srucural and asympoic properies of he more general model. The Orhogonal GARCH (O- GARCH) model of Alexander (2001) uses principal componen analysis o reduce he number of parameers o be esimaed. The need o develop volailiy models o esimae accuraely large covariance marices has become especially relevan following he 1995 amendmen o he Basel Accord, whereby banks were permied o use inernal models o calculae heir VaR hresholds. This amendmen was a reacion o widespread criicism ha he Sandardized approach, which banks were originally required o use in calculaing heir VaR hresholds, led o excessively conservaive forecass. Excessive conservaism has a negaive impac on he profiabiliy of banks as higher capial charges are subsequenly required. While he amendmen was designed o reward insiuions wih superior risk managemen sysems, a backesing procedure, whereby he realized reurns are compared wih he VaR forecass, was inroduced o assess he qualiy of he inernal models. Banks using models ha lead o a greaer number of violaions han can reasonably be expeced, given he confidence level, are required o hold higher levels of capial (see he discussion in Secion 5 and Table 4 for he penalies imposed under he Basel Accord). If a bank s VaR forecass are violaed more han 9 imes in a financial year, he bank may be required o adop he Sandardized approach. The imposiion of such a penaly is severe as i has an 3

4 impac on he profiabiliy of he bank direcly hrough higher capial charges, may damage he bank s repuaion, and may also lead o he imposiion of a more sringen exernal model o forecas he VaR hresholds. In his paper we invesigae he imporance of including spillover effecs when modelling and forecasing financial volailiy. We compare he forecased condiional variances produced by he VARMA-GARCH model of Ling and McAleer (2003), in which he condiional variance of asse i is specified o depend dynamically on pas squared uncondiional shocks and pas condiional variances of each asse in he porfolio, wih he forecased condiional variances produced by he CCC model of Bollerslev (1990), where he condiional variance of asse i is specified o depend only on he squared uncondiional shocks and pas condiional variances of asse i. We also develop a new Porfolio Spillover GARCH (PS-GARCH) model, which allows spillover effecs o be included in a more parsimonious manner. The parsimonious naure of he PS-GARCH model is of criical imporance o praciioners as he model can be esimaed for any number of asses, while several oher mulivariae models can be esimaed only for a reasonably small number of asses. This parsimonious naure avoids he so-called curse of dimensionaliy ha can render many mulivariae models impracical in empirical applicaions. This parsimonious model is found o yield volailiy and VaR hreshold forecass ha are very similar o hose of he VARMA-GARCH model. Using he axonomy proposed in Bauwens e al. (2006), boh he PS-GARCH and VARMA- GARCH models are nonlinear mulivariae exensions of he sandard univariae GARCH model. The plan of he paper is as follows. Secion 2 presens he new PS-GARCH model, discusses alernaive mulivariae GARCH models wih and wihou spillover effecs, and presens a simple wo-sep esimaion mehod for PS-GARCH. The daa for four inernaional sock marke indices are discussed in Secion 3, he volailiy and condiional correlaion forecass produced by alernaive mulivariae GARCH models are examined in Secion 4, he economic significance of he VaR hreshold forecass 4

5 arising from he alernaive mulivariae GARCH models is analysed in Secion 5, and some concluding remarks are given in Secion 6. Equaion Secion 1 2. Models and Esimaion This secion proposes a parsimonious and compuaionally convenien PS-GARCH model which capures aggregae porfolio spillover effecs, and discusses he srucural and saisical properies of he model. The new model is compared wih wo consan condiional correlaion models, one of which models spillover effecs from each of he oher asses in he porfolio and anoher which has no spillover effecs. 2.1 PS-GARCH Le he vecor of reurns on m ( 2) financial asses be given by Y = E( Y F ) + ε (1) 1 where he condiional mean of he reurns follows a VARMA process: Φ( L)( Y µ ) =Ψ ( L) ε (2) The reurn on he porfolio consising of he m asses is denoed as: m Y = E( x y F ) + ε (3) p, i, i, 1 p, i= 1 where y i, denoes he reurn on asse i a ime and x i denoes he porfolio weigh of asse i a ime, such ha: m xi, = 1. (4) i= 1 5

6 The porfolio spillover GARCH (PS-GARCH) model assumes ha he reurns on he porfolio follow an ARMA process, and ha he condiional volailiy of he porfolio can be approximaed by a GARCH process, as follows: Φ( L)( Y µ ) =Ψ ( L) ε (5) p, p p, ε = Dη (6) ε = h η (7) 1/ 2 p, p, p, h r s 2 p, = ω p+ α p, kε p, k + β p, lhp, l k= 1 l= 1 (8) r r H A C I B H G K h r r s r s 2 ˆ ˆ = ω+ kε k + k ( η k ) ε k + l l + kε p, k + l p, l k= 1 k= 1 l= 1 k= 1 l= 1 (9) where H = ( h 1,..., h m )', ω= ( ω1,..., ω m ) ', ( 1/ 2 ' r 2 2 ' D = diag ), η = ( η,..., η ), ε = ( ε,..., ε ), A k, = (,..., ' ), h i 1 m C k are diagonal, wih ypical elemens α, β ii ii and ' K = ( k,..., k ), I( ) = diag( I( )) is an m 1 m 2 and ˆp ε, k and hˆp, l are he fied values from and (5) and (8), respecively. The m m marices B l and γ ii, respecively, Gk g1 gm l 1 m η η m diagonal marix, i p q Φ ( L) = Im Φ1L... Φ pl and Ψ ( L) = Im Ψ1L... Ψ ql are polynomials in L, he lag operaor, F is he pas informaion available o ime, I m is he m m ideniy marix, and I ( η i ) is an indicaor funcion, given as: 1, εi 0 I( ηi ) = 0, εi > 0. (10) The indicaor funcion disinguishes beween he effecs of posiive and negaive shocks of equal magniude on condiional volailiy. Porfolio spillovers arise when are no null marices. G k and K l 6

7 Using (6), he condiional covariance marix for he PS-GARCH model is given by Q = D Γ D, for which he marix of condiional correlaions is given by E η η ) = Γ. ( The marix Γ is he consan condiional correlaion marix of he uncondiional shocks which is, by definiion, equivalen o he consan condiional correlaion marix of he condiional shocks. 2.2 VARMA-GARCH The VARMA-GARCH model of Ling and McAleer (2003), which assumes symmery in he effecs of posiive and negaive shocks on condiional volailiy, is given by: Y = E( Y F ) + ε (11) 1 Φ( L)( Y µ ) =Ψ ( L) ε (12) ε = Dη (13) r s r H = ω+ A ε + B H k k l l k= 1 l= 1 (14) 2 where H = ( hi,..., hm )', ω= ( ω1,..., ω m )', D ( 1/ = diag h i, ), η = ( ηi,,..., ηm )', r 2 2 ' ε = ( εi,..., ε m ), A k and B l are m m marices wih ypical elemens α ij and β ij, respecively, for i, j= 1,..., m, I η ) = diag( I( η )) is an m m marix, ( i p Φ ( L) = Im Φ1L... Φ pl and m 1 lag operaor, and q Ψ ( L) = I Ψ L... Ψ L are polynomials in L, he F is he pas informaion available o ime. Spillover effecs are given in he condiional volailiy for each asse in he porfolio, specifically where q A k and are no diagonal marices. Based on equaion (13), he VARMA-GARCH model also assumes ha he marix of condiional correlaions is given by Eη η ) = Γ. ( B l 7

8 An exension of he VARMA-GARCH model is he VARMA-AGARCH model of Hoi e al. (2002), which capures he asymmeric spillover effecs from each of he oher asses in he porfolio. The VARMA-AGARCH model is also a mulivariae exension of he univariae GJR model. 2.3 CCC The VARMA-GARCH, VARMA-AGARCH and PS-GARCH models have several popular consan condiional correlaion univariae and mulivariae models as special cases. If he model given by equaion (14) is resriced so ha marices, he VARMA-GARCH model reduces o: A k and B l are diagonal h r = ω + αε + β h i i i i, k i i, l k= 1 l= 1 s (15) which is he consan condiional correlaion (CCC) model of Bollerslev (1990). The CCC model also assumes ha he marix of condiional correlaions is given by Eη η ) =Γ. As given in equaion (15), he CCC model does no have volailiy spillover ( effecs across differen financial asses, and hence is inrinsically univariae in naure. Moreover, CCC also does no capure he asymmeric effecs of posiive and negaive shocks on condiional volailiy. 2.4 Esimaion The parameers in models (11), (14), (15) can be obained by maximum likelihood esimaion (MLE) using a join normal densiy, namely: n ˆ 1 ' 1 θ = arg min ( log Q + εq ε) (16) θ 2 = 1 8

9 where θ denoes he vecor of parameers o be esimaed in he condiional loglikelihood funcion, and Q denoes he deerminan of Q, he condiional covariance marix. When η does no follow a join mulivariae normal disribuion, equaion (16) is defined as he Quasi-MLE (QMLE). The models described above can also be esimaed using he following simple wo-sep esimaion procedure: (1) For each financial index reurn series, he univariae GARCH (1,1) model wih an AR(1) condiional mean specificaion is esimaed, and he uncondiional shocks and sandardized residuals of all m reurns are saved. (2) For he porfolio reurns, as defined by equaion (3), he univariae GARCH (1,1) model wih VARMA(1,1) condiional mean specificaion is esimaed, and he uncondiional shocks and sandardized residuals are saved. (3) For each financial reurns series, he univariae VARMA(1,1)-GARCH(1,1) model is esimaed, including he lagged squared uncondiional shocks and he lagged condiional variances of he remaining m-1 asses. The sandardized residuals of he m-1 financial reurns are saved. (4) For each financial reurns series, he VARMA(1,1)-PS-GARCH(1,1) model is esimaed, including he lagged squared uncondiional shocks and he lagged condiional variances from sep (2). The sandardized residuals of all m financial reurns are saved. (5) For each reurns series, he consan condiional correlaion marices of he VARMA(1,1)-GARCH(1,1) model are esimaed by direc compuaion using he sandardized residuals from sep (3). Bollerslev's (1990) CCC marix is esimaed direcly using he sandardized residuals from sep (1). Finally, he consan condiional correlaion marix of he PS-GARCH model is esimaed using he sandardized residuals from sep (4). The ess of spillover and asymmeric effecs are valid under he null hypohesis of independen (ha is, no spillovers) and symmeric effecs, so ha seps (3) and (4) are 9

10 valid under he join null hypohesis. The primary purpose of he srucural and asympoic heory derived in Ling and McAleer (2003) is o demonsrae ha such esing is saisically valid. Using exensions of he srucural ands asympoic properies derived in Ling and McAleer (2003), Hoi e al. (2002) and McAleer e al. (2005), i can be shown ha he QMLE of he parameers in he PS-GARCH model are consisen and asympoically normal in he absence of normaliy in he sandardized shocks η p, in (7) (he proof is available on reques). The VARMA-GARCH and VARMA-AGARCH models are available as preprogrammed opions in, for example, he RATS 6 economeric sofware package. In his paper, esimaion was underaken using he EViews 5.1 economeric sofware package, alhough he resuls were very similar using RATS Daa The daa used in he empirical applicaion are daily prices measured a 16:00 Greenwich Mean Time (GMT) for four inernaional sock marke indices (henceforh referred o as synchronous daa), namely S&P500 (USA), FTSE100 (UK), CAC40 (France), and SMI (Swizerland). New York and London are widely regarded as he wo mos imporan global markes, while Paris and Zurich are seleced for purposes of examining spillovers using synchronous daa. All prices are expressed in US dollars. The daa were obained from DaaSream for he period 3 Augus 1990 o 5 November 2004, which yields 3720 observaions. A he ime he daa were colleced, his period was he longes for which daa on all four variables were available. The raionale for employing daily synchronous daa in modelling sock reurns and volailiy ransmission is four-fold. Firs, he Efficien Markes Hypohesis would sugges ha informaion is quickly and efficienly incorporaed ino sock prices. While informaion generaed yeserday may be 10

11 significan in explaining sock price changes oday, i is less likely ha news generaed las monh would have any explanaory power oday. Second, i has been argued by Engle e al. (1990) ha volailiy is caused by he arrival of unexpeced news and ha volailiy clusering is he resul of invesors reacing differenly o news. The use of daily daa may help in modelling he ineracion beween he heerogeneiy of invesor responses in differen markes. Third, sudies ha use close-o-close non-synchronous reurns suffer from he nonsynchroniciy problem, as highlighed in Scholes and Williams (1977). In paricular, hese sudies canno disinguish a spillover from a conemporaneous correlaion when markes wih common rading hours are analysed. Kahya (1997) and Burns e al. (1998) also observe ha, if cross marke correlaions are posiive, he use of close-o-close reurns for non-synchronous markes will underesimae he rue correlaions, and hence underesimae he rue risk associaed wih a porfolio of such asses. Finally, he use of synchronous daa allows he sysem o be wrien in a simulaneous equaions form, which can be esimaed joinly. Such join esimaion of he parameers eliminaes poenial economeric problems associaed wih generaed regressors, in which unobserved variables are obained (or generaed) hrough esimaion of auxiliary regression models (see, for example, Pagan (1984) and Oxley and McAleer (1993, 1994)), improves efficiency in esimaion, increases he power of he es for cross-marke spillovers, and analyses marke ineracions simulaneously. This allows all he relaionships o be esed joinly. Join esimaion is also consisen wih he noion ha spillovers are he impac of global news on each marke. The synchronous reurns for each marke i a ime R ) are defined as: ( i, R = log( P / P ) (17) i i, i, 1 11

12 where P i, is he price in marke i a ime, as recorded a 16:00 GMT. The descripive saisics for he synchronous reurns of he four indexes are given in Table 1. All series have similar means and medians a close o zero, minima which vary beween and , and maxima ha range beween and Alhough he four sandard deviaions vary slighly, he coefficiens of variaion (CoV) are quie differen, ranging from for S&P500 o for CAC40. The skewness differs among all four series, bu he kurosis is reasonably similar for all series. The Jarque-Bera es srongly rejecs he null hypohesis of normally disribued reurns, which may be due o he presence of exreme observaions. As each of he series displays a high degree of kurosis, his would seem o indicae he exisence of exreme observaions. Each of he reurns series exhibis clusering, which needs o be capured by an appropriae ime series model. [Inser Table 1 here] Several definiions of volailiy are available in he lieraure. This paper adops he measure of volailiy proposed in Franses and van Dijk (2000), where he rue volailiy of reurns is defined as: V R E R F 2 i, = ( i, ( i, 1)) (18) where F 1 is he informaion se a ime -1. The plos of he volailiies of he synchronous reurns are given in Figures 1a-d. Each of he series exhibis clusering, which needs o be capured by an appropriae ime series model. The volailiy of all series appears o be high during he early 1990 s, followed by a quie period from he end of 1992 o he beginning of Finally, he volailiy of all series appears o increase dramaically around 1997, due in large par o he Asian economic and financial crises. This increase in volailiy persiss unil he end of he 12

13 period, and is likely o have been affeced by he Sepember 11, 2001 erroris aacks and he conflics in Afghanisan and Iraq. [Inser Figures 1a-d here] The descripive saisics for he volailiy of he synchronous reurns of he four indexes, alhough no repored here, indicae ha CAC40 displays he highes mean (median) volailiy a (0.665), while FTSE100 has he lowes mean (median) volailiy a (0.425). The maxima of he four volailiy series differ subsanially, wih SMI displaying he highes maxima and S&P500 displaying he lowes. Alhough he four sandard deviaions vary, he coefficiens of variaion (CoV) are similar. All series are highly skewed. As each of he series displays a high degree of kurosis, his would seem o indicae he exisence of exreme observaions. 4. Value-a-Risk Formally, a VaR hreshold is he lower bound of a confidence inerval for he mean. Suppose ha ineres lies in modelling he random variable Y, which can be decomposed as follows: Y = E(Y F 1 )+ε (19) This decomposiion suggess ha Y is comprised of a predicable componen, E(Y F 1 ), which is he condiional mean, and a random componen, ε. The variabiliy of Y, and hence is disribuion, is deermined enirely by he variabiliy of ε. If i is assumed ha ε follows a disribuion such ha: ε D( µ, σ ) (20) 13

14 where µ and σ are he uncondiional mean and sandard deviaion of ε, respecively, hese can be esimaed using a variey of parameric and/or non-parameric mehods. The procedure used in his paper is discussed in Secion 3. The VaR hreshold for Y can be calculaed as: VaR = E( Y F ) ασ (21) 1 where α is he criical value from he disribuion of ε o obain he appropriae confidence level. Alernaively, σ can be replaced by alernaive esimaes of he condiional variance o obain an appropriae VaR (see Secion 2 above). 5. Forecass The purpose of his secion is o compare he volailiy and condiional correlaion forecass produced by he CCC model of Bollerslev (1990), he VARMA-GARCH model of Ling and McAleer (2003), and he new PS-GARCH model proposed in his paper. A rolling window approach is used o forecas he 1-day ahead condiional correlaions and condiional variances. The sample ranges from 3 Augus 1990 o 5 November In order o srike a balance beween efficiency in esimaion and a viable number of rolling regressions, he rolling window size is se a 2000 for all four daa ses, which leads o a forecasing period from 6 April 1998 o 5 November [Inser Figure 2 here] Figures 1a-d plo he forecased volailiies using he hree models for an equally weighed porfolio conaining S&P500, FTSE100, CAC40 and SMI. Table 2 shows he correlaions beween he hree ses of forecass. The volailiy forecass produced by all models are remarkably similar, wih correlaion coefficiens of he volailiy forecass ranging from o

15 [Inser Table 2 here] The forecased condiional correlaions and he correlaion of he condiional correlaion forecass are given in Figures 3-8 and Table 3, respecively. The condiional correlaion forecass are virually idenical for all hree models, wih correlaion coefficiens ranging from o This resul suggess ha for applicaions where he required inpus are he forecass of he condiional variances and/or he condiional correlaion marix, all hree models considered above yield very similar resuls. [Inser Figures 3-8 here] [Inser Table 3 here] 6. Economic Significance The 1988 Basel Capial Accord, which was originally concluded beween he cenral banks from he Group of Ten (G10) counries, and has since been adoped by over 100 counries, ses minimum capial requiremens which mus be me by banks o guard agains credi and marke risks. The marke risk capial requiremens are a funcion of he forecased VaR hresholds (see Secion 4 above). The Basel Accord sipulaes ha he daily capial charge mus be se a he higher of he previous day s VaR or he average VaR over he las 60 business days muliplied by a facor k. The muliplicaive facor k is se by he local regulaors, bu mus no be lower han 3. In 1995, he 1988 Basel Accord was amended o allow banks o use inernal models o deermine heir VaR. However, banks wishing o use inernal models mus demonsrae ha he models are sound. Furhermore, he Basel Accord imposes penalies in he form of a higher muliplicaive facor k on banks which use models ha lead o a greaer number of violaions han would reasonably be expeced given he specified confidence level of 1%. Table 4 shows he penalies imposed for a given number of violaions for 250 business days. 15

16 In cerain cases, where he number of violaions is deemed o be excessively large, regulaors may penalize banks even furher by requiring ha heir inernal models be reviewed. In circumsances where he inernal models are found o be inadequae, banks can be required o adop he sandardized mehod originally proposed in 1993 by he Basel Accord. The sandardized mehod suffers from several drawbacks, he mos noiceable of which is is sysemaic overesimaion of risk, which sems from he assumpion of perfec correlaion across differen risk facors. Overesimaing risk leads o higher capial charges which negaively impac boh he profiabiliy and repuaion of he bank. [Inser Table 4 here] The economic significance of he various models proposed above is highlighed by forecasing VaR hresholds using he PS-GARCH, VARMA-GARCH and CCC models (see Jorion (2000) for a deailed discussion of VaR). In order o simplify he analysis, i is assumed ha he porfolio reurns are normally disribued, wih equal and consan weighs. We conrol for exchange rae risk by convering all prices o a common currency, namely he US Dollar. We use he forecased variances and correlaions from Secion 4 o produce VaR forecass for he period 6 May 1998 o 5 November The backesing procedure is used o es he soundness of he models by comparing he realised and forecased losses (see Basel Commiee (1988, 1995, 1996) for furher deails). Figures 9-11 show he VaR forecass and realized reurns for each empirical model considered. Boh he CCC and PS-GARCH VaR forecass violae he hresholds 7 imes from 1720 forecass, while he VARMA-GARCH model leads o 6 violaions from 1720 forecass. Table 5 shows ha he mean daily capial charge, which is a funcion of boh he penaly and he forecased VaR, implied by PS-GARCH is he larges a 9.180%, followed by VARMA-GARCH a 9.051% and CCC a 9.009%. A high capial charge is undesirable, 16

17 oher hings equal, as i reduces profiabiliy. Table 5 also shows ha CCC leads o violaions ha are approximaely 10% greaer in erms of mean absolue deviaions, a 0.498, han he VARMA-GARCH and PS-GARCH models, a and 0.442, respecively. This is paricularly imporan because large violaions, on average, may lead o bank failures, as he capial requiremens implied by he VaR hreshold forecass may be insufficien o cover he realized losses. Finally, CCC also leads o he larges maximum violaion. [Inser Figures 9-11 here] [Inser Table 5 here] 7. Conclusion Accurae modelling of volailiy (or risk) is imporan in finance, paricularly as i relaes o he modelling and forecasing of Value-a-Risk (VaR) hresholds. As financial applicaions ypically deal wih a porfolio of asses and risks, here are several mulivariae GARCH models which specify he risk of one asse as depending dynamically on is own pas, as well as he pas of oher asses. These models are ypically compuaionally demanding, due o he large number of parameers o be esimaed, and can be impossible o esimae for a large number of asses. The need o creae volailiy models ha can be used o esimae large covariance marices has become especially relevan following he 1995 amendmen o he Basel Accord, whereby banks are permied o use inernal models o calculae heir VaR hresholds. While he amendmen was designed o reward insiuions wih superior risk managemen sysems, a backesing procedure in which he realized reurns are compared wih he VaR forecass, was inroduced o assess he qualiy of he inernal models. Banks using models ha lead o a greaer number of violaions han can reasonably be expeced, given he confidence level, are penalized by having o hold higher levels of capial. The imposiion of penalies is severe as i has an impac on he profiabiliy of he bank 17

18 direcly hrough higher capial charges, may damage he banks repuaion, and may also lead o he imposiion of a more sringen exernal model o forecas he VaR hresholds. This paper examined various condiional volailiy models for purposes of forecasing financial volailiy and VaR hresholds. Two consan condiional correlaion models for esimaing he condiional variances and covariances are he CCC model of Bollerslev (1990) and he VARMA-GARCH model of Ling and McAleer (2003). Alhough he VARMA-GARCH model accommodaes spillover effecs from he reurns shocks of all asses in he porfolio, which are ypically esimaed o be significanly differen from zero, he forecass of he condiional volailiy and VaR hresholds produced by he VARMA-GARCH model are very similar o hose produced by he CCC model. Finally, he paper also developed a new parsimonious and compuaionally convenien Porfolio Spillover GARCH (PS-GARCH) model, which allowed spillover effecs o be included parsimoniously. The PS-GARCH model was found o yield volailiy and VaR hreshold forecass ha were very similar o hose of he CCC and VARMA-GARCH models. Therefore, alhough he empirical resuls sugges ha spillover effecs are saisically significan, he VaR hreshold forecass are generally found o be insensiive o he inclusion of spillover effecs in he mulivariae models considered. References Alexander, C.O. (2001), Orhogonal GARCH, in C.O. Alexander (ed.), Masering Risk, Volume II, FT-Prenice Hall, pp Basel Commiee on Banking Supervision, (1988), Inernaional Convergence of Capial Measuremen and Capial Sandards, BIS, Basel, Swizerland. Basel Commiee on Banking Supervision, (1995), An Inernal Model-Based Approach o Marke Risk Capial Requiremens, BIS, Basel, Swizerland. 18

19 Basel Commiee on Banking Supervision, (1996), Supervisory Framework for he Use of Backesing in Conjuncion wih he Inernal Model-Based Approach o Marke Risk Capial Requiremens, BIS, Basel, Swizerland. Bauwens, L., S. Lauren and V.K. Rombous. (2006), Mulivariae GARCH: A Survey, Journal of Applied Economerics, 21, Bollerslev, T. (1986), Generalised Auoregressive Condiional Heeroscedasiciy, Journal of Economerics, 31, Bollerslev, T. (1990), Modelling he Coherence in Shor-Run Nominal Exchange Rae: A Mulivariae Generalized ARCH Approach, Review of Economics and Saisic, 72, Burns, P, R. Engle and J. Mezrich (1998), Correlaions and Volailiies of Asynchronous Daa, Journal of Derivaives, 5, Engle, R.F. (1982), Auoregressive Condiional Heeroscedasiciy wih Esimaes of he Variance of Unied Kingdom Inflaion, Economerica, 50, Engle, R.F. (2002), Dynamic Condiional Correlaion: A Simple Class of Mulivariae Generalized Auoregressive Condiional Heeroskedasiciy Models, Journal of Business and Economic Saisics, 20, Engle, R.F., T. Io and W. Lin (1990), Meeor Showers or Hea Waves? Heeroskedasic Inra-Daily Volailiy in he Foreign Exchange Marke, Economerica, 58, Franses, P.H. and D. van Dijk (2000), Nonlinear Time Series Models in Empirical Finance, Cambridge, Cambridge Universiy Press. Glosen, L.R., R. Jagannahan, and D.E. Runkle (1992), On he Relaion beween he Expeced Value and Volailiy of he Nominal Excess Reurn on Socks, Journal of Finance, 46, Hoi, S., F. Chan and M. McAleer (2002), Srucure and Asympoic Theory for Mulivariae Asymmeric Volailiy: Empirical Evidence for Counry Risk Raings, invied paper presened o he Ausralasian Meeing of he Economeric Sociey, Brisbane, Ausralia, July Jorion, P. (2000), Value a Risk: The New Benchmark for Managing Financial Risk, McGraw-Hill, New York. Kahya, E. (1997), Correlaion of Reurns in Non-Conemporaneous Markes, Mulinaional Finance Journal, 1,

20 Li, W.K., S. Ling and M. McAleer (2002), Recen Theoreical Resuls for Time Series Models wih GARCH Errors, Journal of Economic Surveys, 16, Reprined in M. McAleer and L. Oxley (eds.), Conribuions o Financial Economerics: Theoreical and Pracical Issues, Blackwell, Oxford, 2002, pp Ling, S. and M. McAleer (2002a), Saionariy and he Exisence of Momens of a Family of GARCH Processes, Journal of Economerics, 106, Ling, S. and M. McAleer (2002b), Necessary and Sufficien Momen Condiions for he GARCH(r,s) and Asymmeric Power GARCH(r,s) Models, Economeric Theory, 18, Ling, S. and M. McAleer (2003), Asympoic Theory for a Vecor ARMA-GARCH Model, Economeric Theory, 19, McAleer, M. (2005), Auomaed Inference and Learning in Modeling Financial Volailiy, Economeric Theory, 21, McAleer, M., F. Chan, S. Hoi and O. Lieberman (2005), Generalized Auoregressive Condiional Correlaion, unpublished paper, School of Economics and Commerce, Universiy of Wesern Ausralia. Oxley, L. and M. McAleer (1993), Economeric Issues in Macroeconomic Models wih Generaed Regressors, Journal of Economic Surveys, 7, Oxley, L. and M. McAleer (1994), Tesing he Raional Expecaions Hypohesis in Macroeconomeric Models wih Unobserved Variables, in L. Oxley e al. (eds.), Surveys in Economerics, Blackwell, Oxford, 1994, pp Pagan, A.R. (1984), Economeric Issues in he Analysis of Regressions wih Generaed Regressors, Inernaional Economic Review, 25, Scholes, M., and J. Williams (1977), Esimaing Beas from Nonsynchronous Daa, Journal of Financial Economics, 5, Tse, Y.K. and A.K.C. Tsui (2002), A Mulivariae Generalized Auoregressive Condiional Heeroscedasiciy Model wih Time-Varying Correlaions, Journal of Business and Economic Saisics, 20, da Veiga, B. and M. McAleer (2005), Single Index and Porfolio Mehods for Modelling and Forecasing Value-a-Risk, unpublished paper, School of Economics and Commerce, Universiy of Wesern Ausralia. 20

21 Table 1: Descripive Saisics for Reurns Saisics S&P500 FTSE100 CAC40 SMI Mean Median Maximum Minimum Sd. Dev Skewness Kurosis CoV Jarque-Bera Table 2: Correlaions Beween Condiional Volailiy Forecass for he Porfolio CCC VARMA-GARCH PS-GARCH

22 Table 3: Correlaions of Rolling Condiional Correlaion Forecass Beween Pairs of Indexes S&P500 and FTSE100 S&P500 and CAC40 CCC VARMA- VARMA- PS-GARCH CCC GARCH GARCH PS-GARCH S&P500 and SMI FTSE100 and CAC40 CCC VARMA- VARMA- PS-GARCH CCC GARCH GARCH PS-GARCH FTSE100 and SMI CAC40 and SMI CCC VARMA- VARMA- PS-GARCH CCC GARCH GARCH PS-GARCH

23 Table 4: Basel Accord Penaly Zones Zone Number of Violaions Increase in k Green 0 o Yellow Red Noe: The number of violaions is given for 250 business days. Model CCC VARMA-GARCH PS-GARCH Table 5: Mean Daily Capial Charge and AD of Violaions Number of Violaions Mean Daily Capial Charge Maximum AD of Violaions Mean Noes: (1) The daily capial charge is given as he negaive of he higher of he previous day s VaR or he average VaR over he las 60 business days imes (3+k), where k is he penaly. 23

24 35 Figure 1a: S&P500 Volailiy 70 Figure 1b: FTSE100 Volailiy Figure 1c: CAC40 Volailiy 90 Figure 1d: SMI Volailiy

25 16 Figure 2: Porfolio Volailiy Forecas CCC VARMA-GARCH PS-GARCH 25

26 Figure 3: Rolling Condiional Correlaion Forecass Beween S&P500 and FTSE CCC VARMA-GARCH PS-GARCH Figure 4: Rolling Condiional Correlaion Forecass Beween S&P500 and CAC CCC VARMA-GARCH PS-GARCH Figure 5: Rolling Condiional Correlaion Forecass Beween S&P500 and SMI.52 Figure 6: Rolling Condiional Correlaion Forecass Beween FTSE100 and CAC CCC VARMA-GARCH PS-GARCH CCC VARMA-GARCH PS-GARCH Figure 7: Rolling Condiional Correlaion Forecass Beween FTSE100 and SMI Figure 8: Rolling Condiional Correlaion CAC40 and SMI CCC VARMA-GARCH PS-GARCH CCC VARMA-GARCH PS-GARCH 26

27 6 Figure 9: Realized Reurns and CCC VaR Forecass Porfolio Reurns CCC 27

28 Figure 10: Realized Reurns and VARMA-GARCH VaR Forecass Porfolio Reurns VARMA-GARCH 28

29 6 Figure 11: Realized Reurns and PS-GARCH VaR Forecass Porfolio Reurns PS-GARCH 29