Single Index and Portfolio Models for Forecasting Value-at- Risk Thresholds *

Transcription

1 Single Index and Porfolio Models for Forecasing Value-a- Risk Thresholds * Bernardo da Veiga and Michael McAleer School of Economics and Commerce Universiy of Wesern Ausralia January 2005 Absrac: The variance of a porfolio can be forecased using a single index model or he covariance marix of he porfolio. Using univariae and mulivariae condiional volailiy models, his paper evaluaes he performance of he single index and porfolio models in forecasing Value-a-Risk (VaR) hresholds of a porfolio. The LR ess of uncondiional coverage, independence and condiional coverage of he VaR forecass sugges ha he single index model leads o excessive and ofen serially dependen violaions, while he porfolio model leads o oo few violaions. The single index model also leads o lower daily Basel Accord capial charges. The univariae models which display correc condiional coverage lead o higher capial charges han models which lead o oo many violaions. Overall, he Basel Accord penalies may be oo lenien and favour models wih oo many violaions. Keywords and phrases: Single index, porfolio spillover, Value-a-Risk hresholds, Basel Accord penalies, mulivariae condiional volailiy, condiional correlaions. JEL classificaions: F37, C51, C53, C32 * The auhors wish o hank Dave Allen, Felix Chan, Alvaro Veiga, Marcelo Medeiros, and seminar paricipans a he Insiue of Economics, Academia Sinica, Taiwan, Griffih Universiy, Ling Tung Universiy, Taiwan, Macquarie Universiy, Queensland Universiy of Technology, Universiy of Queensland and Ponifical Caholic Universiy, Brazil, for helpful commens and suggesions. The firs auhor acknowledges a Universiy Posgraduae Award and an Inernaional Posgraduae Research Scholarship a he Universiy of Wesern Ausralia, and he second auhor is graeful for he financial suppor of he Ausralian Research Council. 1

2 1. Inroducion The need o model he variance of a financial porfolio accuraely has become especially imporan following he 1995 amendmen o he Basel Accord, whereby banks were permied o use inernal models o calculae heir Value-a-Risk (VaR) hresholds (see Jorion (2000) for a deailed discussion of VaR). This amendmen was in response o widespread criicism ha he Sandardized approach, which banks used o calculae 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. Alhough he amendmen o he Basel Accord 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 asses he qualiy of he inernal models. In cases where he inernal models lead o a greaer number of violaions han could reasonably be expeced, given he confidence level, he bank is required o hold a higher level of capial (see Table 7 for he penalies imposed under he Basel Accord). If a bank s VaR forecass are violaed more han 9 imes in any financial year, he bank may be required o adop he Sandardized approach. The imposiion of such a penaly is severe as i affecs he profiabiliy of he bank direcly hrough higher capial charges, has a damaging effec on he bank s repuaion, and may lead o he imposiion of a more sringen exernal model o forecas he bank s VaR hresholds. One of he main ingrediens required o produce he VaR hreshold of a porfolio is he condiional variance of he porfolio reurns. Condiional volailiy models can be used o esimae he condiional variance of he porfolio reurns eiher by: (1) fiing a univariae volailiy model o he porfolio reurns (hereafer called he single index model); or (2) using a mulivariae volailiy model o forecas he condiional variance of each asse in he porfolio, as well as he condiional correlaions beween all asse pairs, in order o calculae he forecased porfolio variance (hereafer called he porfolio model). The porfolio model has boh inuiive and empirical appeal as i enables he modelling of he 2

3 relaionship beween subses of he porfolio, and also allows for scenario and sensiiviy analysis. Bollerslev (1990) proposed a Consan Condiional Correlaion (CCC) mulivariae GARCH model which models he condiional variances and correlaions using a simple 2-sep procedure. In his approach, a univariae GARCH model is fied o each reurns series in he firs sep, and he condiional correlaion marix is calculaed using he sandardized residuals in he second sep. The CCC approach can be exended by using more general univariae condiional volailiy models in he firs sep. In his paper we compare he performance of he single index and porfolio models in forecasing VaR hresholds for a porfolio conaining he S&P500 (USA), FTSE100 (UK), CAC40 (France) and SMI (Swizerland) indexes. Six differen crieria are used o compare he forecasing performance of he various condiional volailiy models and mehods considered, namely: (1) he linear regression approach of Pagan and Schwer (1990); (2) he uncondiional coverage es; (3) he serial independence of violaions es; (4) he condiional coverage es; (5) he size of he average capial charge: and (6) he magniude of he average violaions which would arise from using each model o forecas he VaR hreshold. The ess given in (2)-(4) above are likelihood raio ess, as derived in Chrisoffersen (1998). The plan of he paper is as follows. Secion 2 discusses various calibraed and esimaed univariae and mulivariae condiional volailiy models ha will be used o forecas he volailiy and VaR hresholds of a porfolio (see McAleer (2005) for a comparison of alernaive univariae and mulivariae, condiional and sochasic, financial volailiy models ha are available in he lieraure. The daa are discussed in Secion 3, he forecasing performance of he alernaive models is analysed and compared in Secion 4, and some concluding remarks are given in Secion 5. 3

4 2. Model Specificaions This secion describes alernaive models ha can be used o esimae he condiional variance of a porfolio direcly by modelling he hisorical porfolio reurns (namely, he single index model), or indirecly by modelling he condiional variance of each asse and he condiional correlaion of each pair of asses (namely, he porfolio model). Financial reurns are ypically modelled as a saionary AR(1) process, alhough his can easily be relaxed. In wha follows, he final wo models o be discussed, namely PS-GARCH and VARMA- GARCH, are valid only for he mulivariae approach. The models are presened in increasing order of complexiy. 2.1 Sandardized Normal (SN) The Sandardized Normal (SN) approach forecass he condiional variance a ime as he hisorical variance over a specified ime inerval. This approach is exremely simple and easy o implemen compuaionally. In his paper, he hisorical variance is calculaed using a rolling window for he previous 250 business days. 2.1 EWMA Riskmerics TM (1996) developed a model which esimaes he condiional variances and covariances based on he exponenially weighed moving average (EWMA) mehod, which is, in effec, a resriced version of he ARCH( ) model of Engle (1982). This approach forecass he condiional variance a ime as a linear combinaion of he lagged condiional variance and he squared uncondiional shock a ime 1. The EWMA model calibraes he condiional variance as: h = λh + (1 λε ) (1)

5 where λ is a decay parameer. Riskmerics TM (1996) suggess ha λ should be se a 0.94 for purposes of analysing daily daa. 2.3 ARCH Engle (1982) proposed he Auoregressive Condiional Heeroskedasiciy of order p, or ARCH( p ), model as follows: p 2 j j j= 1 h = ω + αε. (2) For he case p = 1, ω > 0, α1 > 0 are sufficien condiions o ensure a sricly posiive condiional variance, h > 0. The ARCH (or α 1 ) effec capures he shor run persisence of shocks. 2.4 GARCH Bollerslev (1986) generalized ARCH( p ) o he GARCH( p, q ) model, which is given by: h p q 2 = ω + αε j j + βh i j= 1 i= 1. (3) For he case p = 1, ω > 0, α1 > 0, β1 0 are sufficien condiions o ensure a sricly posiive condiional variance, h > 0. The ARCH (or α 1 ) effec capures he shor run persisence of shocks, and he GARCH (or β 1) effec indicaes he conribuion of shocks o long run persisence ( α1 + β1). In ARCH and GARCH models, he parameers are ypically esimaed using he maximum likelihood esimaion (MLE) mehod. In he absence of normaliy of he 5

6 sandardized residuals, η, he parameers are esimaed by he Quasi-Maximum Likelihood Esimaion (QMLE) mehod (see, for example, Li, Ling and McAleer (2002)). 2.5 GJR Glosen, Jagannahan and Runkle (1992) exended he GARCH model o capure possible asymmeries beween he effecs of posiive and negaive shocks of he same magniude on he condiional variance hrough changes in he deb-equiy raio. The GJR( p, q ) model is given by: p q 2 2 = ω+ α jε j + γ ( η 1) ε 1+ β i i j= 1 i= 1 (4) h I h where he indicaor variable, I( η ), is defined as: { 1, ε 0 0, ε > 0 I( η ) =. (5) For he case p = 1, ω > 0, α1 > 0, α1+ γ1 > 0, β1 0 are sufficien condiions o ensure a sricly posiive condiional variance, h > 0. The indicaor variable disinguishes beween posiive and negaive shocks, where he asymmeric effec ( γ 1 > 0 ) measures he conribuion of shocks o boh shor run persisence ( α /2 1+ γ 1 ) and long run persisence ( α + β + γ /2) Several imporan heoreical resuls are relevan for he GARCH model. Ling and McAleer (2002a) esablished he necessary and sufficien condiions for sric saionariy and ergodiciy, as well as for he exisence of all momens, for he univariae GARCH( p, q ) model, and Ling and McAleer (2003) demonsraed ha he QMLE for GARCH( p, q ) is consisen if he second momen is finie, E( ε 2 ) <, and 6

7 asympoically normal if he fourh momen is finie, E( ε 4 ) <. The necessary and sufficien condiion for he exisence of he second momen of ε for he GARCH(1,1) model is α1+ β1 < 1. Anoher imporan resul is ha he log-momen condiion for he QMLE of GARCH(1,1), which is a weak sufficien condiion for he QMLE o be consisen and asympoically normal, is given by E(log( αη + β )) < 0. The log-momen condiion was derived in Elie and Jeanheau (1995) and Jeanheau (1998) for consisency, and in Boussama (2000) for asympoic normaliy. In pracice, i is more sraighforward o verify he second momen condiion han he weaker log-momen condiion, as he laer is a funcion of unknown parameers and he mean of he logarihmic ransformaion of a random variable. The GJR model has also had some imporan heoreical developmens. In he case of symmery of η, he regulariy condiion for he exisence of he second momen of GJR(1,1) is α 1+ β γ / < (see Ling and McAleer (2002b)). Moreover, he weak logmomen condiion for GJR(1,1), E(log[( α + γ I( η )) η + β ]) < 0, is sufficien for he consisency and asympoic normaliy of he QMLE (see McAleer, Chan and Marinova (2002)). 2.6 EGARCH Nelson (1991) proposed he Exponenial GARCH (EGARCH) model, which is given as: ε ε log( h) log( ) p r q i k = ω+ αi + γk + β j h j i= 1 h i k= 1 h k j= 1. (6) As he range of log( h ) is he real number line, he EGARCH model does no require any parameric resricions o ensure ha he condiional variances are posiive. Furhermore, 7

8 he EGARCH specificaion is able o capure several sylised facs, such as small posiive shocks having a greaer impac on condiional volailiy han small negaive shocks, and large negaive shocks having a greaer impac on condiional volailiy han large posiive shocks. Such feaures in financial reurns and risk are ofen cied in he lieraure o suppor he use of EGARCH o model he condiional variances. Unlike he EWMA, ARCH, GARCH and GJR models, EGARCH uses he sandardized raher han he uncondiional shocks. Moreover, as he sandardized shocks have finie momens, he momen condiions of EGARCH are sraighforward and may be used as diagnosic checks of he underlying models. However, he saisical properies of EGARCH have no ye been developed formally. If he sandardized shocks are independenly and idenically disribued, he saisical properies of EGARCH are likely o be naural exensions of (possibly vecor) ARMA ime series processes (for furher deails, see McAleer (2005)). 2.6 PGARCH Ding, Granger and Engle (1993) generalized he sandard deviaion GARCH models of Taylor (1986) and Schwer (1989), and proposed an asymmeric Power GARCH (PGARCH) model. The PGARCH model is given by: p q δ δ δ = + i( i i i) + j j i= 1 j= 1, (7) σ ω α ε γ ε β σ in which he power parameer, δ, can be esimaed raher han imposed, and γ is included o capure he effecs of asymmeric shocks. Ling and McAleer (2002a) derived he necessary and sufficien condiions for he momens o exis for PGARCH. However, unlike he case of he EGARCH model, which also uses he absolue value funcion bu sandardized raher han uncondiional shocks, he PGARCH model uses uncondiional raher han sandardized shocks. As he disribuion of he absolue value of he 8

9 uncondiional shocks is presenly unknown, he saisical properies of he PGARCH model have ye o be developed. 2.7 PS-GARCH This secion describes he parsimonious porfolio spillover GARCH (PS-GARCH) model of McAleer and da Veiga (2005), which is inended o capure porfolio spillover effecs. Le Y = E( Y F ) + ε (8) 1 be a vecor of reurns on m financial asses, where F is he pas informaion available o 1 ime -1, and he condiional mean of he reurns follows a VARMA process: Φ( L)( Y µ ) =Ψ ( L) ε. (9) The reurn on he porfolio consising of he m asses is denoed as: m Y = E( x y F ) + ε (10) 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. (11) i= 1 The PS-GARCH model assumes ha he reurns on he porfolio follow a VARMA process, as follows: 9

10 Φ( L)( Y µ ) =Ψ ( L) ε (12) p, p p, ε = Dη (13) ε = h η (14) 1/2 p p p h r s 2 p = ωp + αplε p, l + βplhp, l l= 1 l= 1 (16) r r s s s (16) H = W + A ε + C I( η ) ε + BH + G ˆ ε + K hˆ l l l l l l l l p, l l p, l l= 1 l= 1 l= 1 l= 1 l= 1 where H = ( h 1,..., h m )', W = ( ω1,..., ω m )', ( 1/ D = diag h i ), η = ( η1,..., ηm), ε = ( ε1,..., εm), and 2 ˆp ε, l and ˆp, l h are he fied values from and (12) and (15), respecively. The marices Al, Bl, C l, G l and K l are diagonal, wih ypical elemens α ii, β ii, γ ii, λ ii and δ, respecively, for i = 1,..., m, I η ) = diag( I ( η )) is an m m diagonal marix, ii Φ... ( i p q ( L) = I m Φ1 L Φ pl and m q Ψ ( L) = I Ψ1 L... Ψ L are polynomials in L, he lag operaor, F is he pas informaion available o ime 1, 1 I m is he marix, and I ( η i ) is an indicaor funcion, given as: m m ideniy I ( η ) i, { 1, ε 0 i, = 0, εi, > 0. (17) The indicaor funcion disinguishes beween he effecs of posiive and negaive shocks of equal magniude on condiional volailiy. Using (13), he condiional covariance marix for he PS-GARCH model is given by Q = D Γ D, in which i is assumed ha he condiional correlaions of he uncondiional 10

11 shocks are 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. 2.8 VARMA-GARCH The VARMA-GARCH model of Ling and McAleer (2003), which assumes symmery in he effecs of posiive and negaive shocks on mulivariae condiional volailiy, is given as follows: Y = E( Y F ) + ε (8) 1 Φ( L)( Y µ ) =Ψ ( L) ε (19) ε = Dη (20) r s = + l ε l + l l l= 1 l= 1 (21) H W A BH where H = ( h1,..., hm)', W = ( ω 1,..., ω m )', D 1/2 = diag( hi ), η = ( η1,..., ηm)', ε = ( ε,..., ε ), A l and m 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 ( L) = I m Φ1 L Φ pl and Φ... q ( L) = I m Ψ1 L Ψq L are polynomials in L, he Ψ... lag operaor, and F is he pas informaion available o ime. Spillover effecs are given in he condiional volailiy for each asse in he porfolio. Based on (20), he VARMA- GARCH model also assumes ha he condiional correlaion marix of he uncondiional shocks is given by E η η ) = Γ. ( 11

12 An exension of he VARMA-GARCH model is he VARMA-AGARCH model of Hoi, Chan and McAleer (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 GJR model. 3. Daa The daa used in he empirical applicaion are daily prices measured a 16:00 Greenwich Mean Time (GMT) for four inernaional sock marke indexes (henceforh referred o as synchronous daa), namely S&P500 (USA), FTSE100 (UK), CAC40 (France), and SMI (Swizerland). All prices are expressed in US dollars. The daa were obained from DaaSream for he period 3 Augus 1990 o 5 November 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 (see McAleer and da Veiga (2005)): (i) The Efficien Markes Hypohesis would sugges ha informaion is quickly and efficienly incorporaed ino sock prices. While informaion generaed yeserday may be significan in explaining sock price changes oday, i is less likely ha news generaed las monh would have any explanaory power oday. (ii) I has been argued by Engle, Io and Lin (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. (iii) 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, Engle and 12

13 Mezrich (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. (iv) The 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 (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, Ri = log( Pi, / Pi, 1), where P i, is he price in marke i a ime, as recorded a 16:00 GMT. The plos of he synchronous reurns are given in Figures 1a-d. Each of he reurns series exhibis clusering, which will be capured by an appropriae ime series model. The descripive saisics for he synchronous reurns of he four indexes are given in Table 1. All series have similar means and medians, which are 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. 13

14 [Inser Figures 1a-d here] [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 (1999), where he rue volailiy of reurns is defined as: V = ( R E( R F )) i, i, i, 1 2 where F 1 is he informaion se a ime -1. The plos of he volailiies of he synchronous reurns are given in Figures 2a-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 period, and is likely o have been affeced by he Sepember 11, 2001 erroris aacks and he conflics in Afghanisan and Iraq. [Inser Figures 2a-d here] The descripive saisics for he volailiy of he synchronous reurns of he four indexes are given in Table 2. The 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. 14

15 [Inser Table 2 here] 4. Forecass In his secion he forecasing performance of he various models described in he previous secion is compared. For purposes of he empirical analysis, i is assumed ha he porfolio weighs are equal and consan over ime, bu hese assumpions can be relaxed. Exchange rae risk is conrolled by convering all prices o a common currency, namely he US Dollar. The models described in Secion 2 are used o esimae he variance of he porfolio direcly for he single index model, and o esimae he condiional variances and correlaions of all asses and asse pairs o calculae he variance of he porfolio for he porfolio model. Apar from he Sandardized Normal and Riskmerics TM models, all he condiional volailiy models are esimaed under he following disribuional assumpions of he uncondiional shocks: (1) normal; and (2), wih en degrees of freedom. A rolling window is used o forecas he 1-day ahead condiional correlaions, condiional variances and VaR hresholds. 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 Six differen approaches are used o evaluae he porfolio variance and forecass of he VaR hresholds, as described below. 4.1 Linear Regression Approach Pagan and Schwer (1990) proposed a procedure whereby he volailiy forecass are regressed on he realized volailiy. In his paper, he squared porfolio reurns are used as a proxy for he realized volailiy. The auxiliary regression equaion is given by: 15

16 RV = α + βfv + ε where RV is he realized volailiy and FV is he forecased volailiy. In his auxiliary equaion, he inercep, α, should be equal o zero and he slope, β, 2 should be equal o 1. The coefficien of deerminaion, R, is a measure of forecasing performance, and he -raio of he coefficiens is a measure of he bias. Tables 3-4 give he esimaes and es saisics for he single index and porfolio models. On he basis of 2 2 he R crierion, he porfolio PGARCH- model performs he bes, wih R = The wors performing models are he single index and porfolio Sandardized Normal 2 models, boh of which have R = I is ineresing o noe ha models esimaed using a -disribuion end o display marginally superior forecass, he only excepion being he porfolio CCC model. In all cases, he porfolio models ouperform he single 2 index models based on R, which suggess ha he porfolio model approach leads o superior forecass of he condiional variance of he porfolio compared wih heir single index counerpars. [Inser Tables 3-4 here] 4.2 Tess of Uncondiional Coverage, Serial Independence and Condiional Coverage Chrisoffersen (1998) derived likelihood raio (LR) ess of uncondiional coverage, serial independence and condiional coverage. Subsequenly, Lopez (1998) adaped hese ess o evaluae VaR hreshold forecass. An adequae VaR model should exhibi he propery ha he uncondiional coverage (which is calculaed as he number of observed violaions divided by T ) should equal α, where α is he level of significance chosen for he VaR, and T is he number of rading days in he evaluaion period. The probabiliy of observing x violaions in a sample of size T, under he null hypohesis, is given by: 16

17 250 x T-x Pr(x) = C x (0.01) (0.99) Therefore, he LR saisic for esing wheher he number of observed violaions, divided by T, is equal o α is: x N x x N x LR = 2[log( α (1 α) ) log((0.01 )(0.99 ))] UC where α = x / N, x is he number of violaions, and N is he number of forecass. The LR saisic is asympoically disribued as χ 2 (1) under he null hypohesis. However, a model ha leads o he correc uncondiional coverage may sill be subopimal if he violaions are serially dependan, as hey may lead o bank failures. The es of independence is he LR saisic for he null hypohesis of serial independence agains he alernaive of firs-order Markov dependence. Finally, Chrisoffersen (1998) proposed he condiional coverage es, which is a join es of uncondiional coverage and independence. The condiional coverage LR saisic is given as he sum of he uncondiional coverage LR saisic and he independence LR saisic, which is asympoically disribued as χ 2 (2) under he join null hypohesis. Tables 5-6 presen he resuls of he LR ess of uncondiional coverage, serial independence and condiional coverage. Wih he excepion of Riskmerics TM and EGARCH, all he single index models which assume normaliy fail he uncondiional coverage es, in ha hey lead o a significanly greaer number of violaions han expeced. The single index models esimaed under he assumpion ha he reurns follow a -disribuion perform far beer, wih he ARCH model being he only model o fail he uncondiional coverage es due o an excessive number of violaions. The porfolio models perform quie poorly, wih en of he sixeen models considered failing he uncondiional coverage es. I is worh noing ha he porfolio models fail he uncondiional coverage es because hey lead o an insufficien number of violaions. 17

18 Five single index models fail he serial independence es, four of which assume ha he reurns follow a -disribuion. However, only wo of he porfolio models fail he serial independence es, boh of which assume normaliy. Finally, eigh of he welve single index models fail he condiional coverage es. Of he porfolio models considered, eigh of he sixeen fail he condiional coverage es. I is ineresing o noe ha rejecion of he null hypohesis in each case was more likely under a -disribuion han under normaliy. [Inser Tables 5-6 here] Overall, he empirical resuls presened in his secion offer some mixed evidence on he relaive performance of he VaR hreshold forecass produced by he single index and porfolio models. Based on he uncondiional coverage es, i seems ha he single index model leads o an excessive number of violaions, while he porfolio model has oo few violaions. Furhermore, he serial independence es favours he porfolio model. From a regulaory viewpoin, he porfolio model may be preferred as i is likely o lead o fewer bank failures, while banks are likely o favour he model which leads o he lowes coss. The nex secion aemps o quanify he coss o boh banks and regulaors ha are associaed wih he use of each ype of model. 4.3 Daily Capial Charge and Magniude of Violaions 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 hresholds. However, banks wishing o use inernal models mus demonsrae ha heir models are sound. The backesing procedure is used 18

19 o es he adequacy of he models by comparing he realised and forecased losses (for furher deails, see Basel Commiee (1988, 1995, 1996)). 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 7 shows he penalies imposed for a given number of violaions for 250 business days. [Inser Table 7 here] Tables 8-9 give he mean daily capial charges for each model. The wors performing models are he porfolio and single index Sandardized Normal models, which would lead o average daily capial charges of 12.92% and 12.33% respecively. The bes performing models are he single index EGARCH and PGARCH models, which would lead o average daily capial charges of 7.97% and 8.04%, respecively. Wih he excepion of he Riskmerics TM model, all single index models lead o lower daily capial charges han he corresponding porfolio models. Furhermore, all he single index models which are esimaed assuming a -disribuion lead o higher capial charges han he equivalen models esimaed under a normal disribuion These resuls sugges ha he penalies imposed under he Basel Accord may no be sufficienly severe, as virually all of he normally disribued single index models are found o lead o a greaer number of violaions han would be expeced on he basis of he uncondiional coverage es. Thus, i would seem ha he Basel Accord ends o favour models which lead o an excessive number of violaions. However, such an inference may be overly simplisic because, 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, he offending banks may be required o adop he sandardized mehod originally proposed in 1993 under he Basel Accord. The sandardized mehod suffers from several drawbacks, he mos noiceable of which is is 19

20 sysemaic overesimaion of risk, which sems from he assumpion of perfec correlaion across differen risk facors. This penaly would lead o higher capial charges, which would have a negaive impac on boh he profiabiliy and repuaion of he affeced banks. The cenral idea underlying capial reserve requiremens is o minimize he possibiliy of bank failures, so ha banks are required o hold sufficien capial o cover a leas hree imes he forecased wors poenial loss. The magniude of he average violaion is, herefore, an imporan consideraion, because models ha perform well according o all he relevan crieria may sill be inadequae if hey lead o an excessively large number of violaions. Tables 8-9 give he maximum and average absolue deviaions of violaions from he VaR forecass. The wors performing models are he single index and porfolio Sandardized Normal models, respecively, as hey lead o boh he larges maximum absolue deviaions a and 3.509, respecively, and he highes average absolue deviaions a and 0.617, respecively. The bes performing models on he basis of he maximum absolue deviaions are he porfolio PGARCH- and EGARCH- models, which lead o he lowes maximum absolue deviaions a and 0.727, respecively, while he bes performing models on he basis of he average absolue deviaions are he single index EGARCH and PGARCH models, which lead o he lowes average absolue deviaions a and 0.362, respecively. Alhough hese resuls migh seem o offer suppor for he use of hese wo single index models, i is imporan o noe ha he single index EGARCH and PGARCH models lead o 30 and 31 violaions, respecively, which is large even in comparison wih heir - disribuion counerpars. Overall, he numbers of violaions are considerably greaer for he single index models han for heir porfolio counerpars. [Inser Tables 8-9 here] 20

21 5. Conclusion The variance of a porfolio can be forecased using a single index model, or from he forecased variances and covariances of all he asses in he porfolio. Using alernaive univariae and mulivariae condiional volailiy models, including he simple Sandardized Normal (SN) model, he exponenially weighed moving average (EWMA) model proposed by Riskmerics TM (1996), and alernaive univariae and mulivariae, symmeric and asymmeric, condiional volailiy models, his paper evaluaed he performance of he single index and porfolio models in forecasing he Value-a-Risk (VaR) hresholds. In he empirical example, he porfolio comprised four inernaional sock marke indexes, namely he S&P500, FTSE100, CAC40 and SMI financial indexes, for he period 3 Augus 1990 o 5 November On he basis of he empirical resuls, he porfolio approach was found o yield superior porfolio volailiy forecass based on he linear regression approach of Pagan and Schwer (1990). The likelihood raio ess of uncondiional coverage, independence and condiional coverage of he VaR forecass, as proposed in Chrisoffersen (1998), offered mixed evidence, wih a large proporion of he single index and porfolio models failing he uncondiional coverage es, he former due o an excessive number of violaions and he laer due o an insufficien number of violaions. Furhermore, i was found ha he single index models led o lower daily capial charges based on he Basel Capial Accord penalies, as compared wih heir porfolio counerpars. Finally, wihin he class of single index models, he models which display correc condiional coverage led o higher capial charges han models which led o excessive violaions. These resuls seemed o sugges ha he penalies imposed under he Basel Capial Accord may be oo lenien, and ended o favour models ha led o an excessive number of violaions, as well as models ha were found o be sub-opimal based on various performance crieria. 21

22 References 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. 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. 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 Saisics, 72, Boussama, F. (2000), Asympoic Normaliy for he Quasi-Maximum Likelihood Esimaor of a GARCH Model, Compes Rendus de l Académie des Sciences, Série I, 331, (in French). Burns, P., R. Engle and J. Mezrich (1998), Correlaions and Volailiies of Asynchronous Daa, Journal of Derivaives, 5, Chrisoffersen (1998), Evaluaing Inerval Forecass, Inernaional Economic Review, 39, Ding, Z., C.W.J. Granger and R.F. Engle (1993), A Long Memory Propery of Sock Marke Reurns and a New Model, Journal of Empirical Finance, 1, Elie, L. and T. Jeanheau (1995), Consisency in Heeroskedasic Models, Compes Rendus de l Académie des Sciences, Série I, 320, (in French). Engle, R.F. (1982), Auoregressive Condiional Heeroscedasiciy wih Esimaes of he Variance of Unied Kingdom Inflaion, Economerica, 50, 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 (1999), Nonlinear Time Series Models in Empirical Finance, Cambridge, Cambridge Universiy Press. 22

23 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 Jeanheau, T. (1998), Srong Consisency of Esimaors for Mulivariae ARCH Models, Economeric Theory, 14, 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, 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), 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 (2002b), Saionariy and he Exisence of Momens of a Family of GARCH Processes, Journal of Economerics, 106, Ling, S. and M. McAleer (2003), Asympoic Theory for a Vecor ARMA-GARCH Model, Economeric Theory, 19, Lopez, J.A (1998), Mehods for Evaluaing Value-a-Risk Esimaes, Federal Reserve Bank of New York Economic Policy Review, McAleer, M. (2005), Auomaed Inference and Learning in Modeling Financial Volailiy, Economeric Theory, 21, McAleer, M., F. Chan and D. Marinova (2002), An Economeric Analysis of Asymmeric Volailiy: Theory and Applicaion o Paens, invied paper presened o he Ausralasian Meeing of he Economeric Sociey, Brisbane, July 2002, o appear in Journal of Economerics. 23

24 McAleer, M. and B. da Veiga (2005), Spillover Effecs in Forecasing Volailiies and VaR, unpublished paper, School of Economics and Commerce, Universiy of Wesern Ausralia. Nelson, D. (1991), Condiional Heeroskedasiciy in Asse Reurns: a New Approach, Economerica, 59, 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, D.A.R. George, C.J. Robers and S. Sayer (eds.), Surveys in Economerics, Basil Blackwell, Oxford, 1994, pp Pagan, A.R. (1984), Economeric Issues in he Analysis of Regressions wih Generaed Regressors, Inernaional Economic Review, 2, Pagan, A.R. and W. Schwer (1990), Alernaive Models for Condiional Sock Volailiy, Journal of Economerics, 50, Riskmerics TM (1996), J.P. Morgan Technical Documen, 4 h ediion, New York. J.P. Morgan. Scholes, M. and J. Williams (1977), Esimaing Beas from Nonsynchronous Daa, Journal of Financial Economics, 5, Schwer, W. (1989), Sock Volailiy and Crash of 87, Review of Financial Sudies, 3, Taylor, S. (1986), Modelling Financial Time Series, New York, Wiley. 24

25 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: Descripive Saisics for Volailiies Saisics S&P500 FTSE100 CAC40 SMI Mean Median Maximum Minimum Sd. Dev Skewness Kurosis CoV

26 Table 3: Linear Regression Tes for he Single Index Model Model α -raio β -raio 2 R Sandardized Normal Riskmerics TM ARCH ARCH GARCH GARCH GJR GJR PGARCH PGARCH EGARCH EGARCH Noes: (1) The null hypohesis α = 0 is esed agains he alernaive 0 (2) The null hypohesis β = 1 is esed agains he alernaive 1 α. β. 26

27 Table 4: Linear Regression Tes for he Porfolio Model Model α -raio β -raio 2 R Sandardized Normal Riskmerics TM ARCH ARCH CCC CCC GJR GJR EGARCH EGARCH PGARCH PGARCH VARMA-GARCH VARMA-GARCH PS-GARCH PS-GARCH Noes: (1) The null hypohesis α = 0 is esed agains he alernaive 0 (2) The null hypohesis β = 1 is esed agains he alernaive 1 α. β. 27

28 Table 5: Likelihood Raio Tes of Uncondiional Coverage, Serial Independence and Condiional Coverage for he Single Index Model Model UC SI CC Sandardized Normal * * Riskmerics TM ARCH * * ARCH * GARCH GARCH GJR GJR PGARCH PGARCH * * EGARCH EGARCH * * Noe: Enries in bold denoe significance a he 5% level; * denoes significance a 1%. 28

29 Table 6: Likelihood Raio Tes of Uncondiional Coverage, Serial Independence and Condiional Coverage for he Porfolio Model Model UC SI CC Sandardized Normal 6.857* * * Riskmerics TM ARCH ARCH CCC CCC * * GJR GJR * EGARCH EGARCH * * PGARCH * * PGARCH * * VARMA-GARCH VARMA-GARCH * * PS-GARCH PS-GARCH * * Noes: Enries in bold denoe significance a he 5% level; * denoes significance a 1%. 29

30 Table 7: 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. 30

31 Table 8: Mean Daily Capial Charge and AD of Violaions for he Single Index Model Model Number of Violaions Mean Daily AD of Violaions Capial Charge Maximum Mean Sandardized Normal Riskmerics TM ARCH ARCH GARCH GARCH GJR GJR PGARCH PGARCH EGARCH EGARCH Noes: (1) The daily capial charge is given as he negaive of (3+k) imes he greaer of he previous day s VaR or he average VaR over he las 60 business days, where k is he penaly. (2) AD is he absolue deviaion of he violaions from he VaR forecas. 31

32 Table 9: Mean Daily Capial Charge and AD of Violaions for he Porfolio Model Model Number of Violaions Mean Daily AD of Violaions Capial Charge Maximum Mean Sandardized Normal Riskmerics TM ARCH ARCH CCC CCC GJR GJR EGARCH EGARCH PGARCH PGARCH VARMA-GARCH VARMA-GARCH PS-GARCH PS-GARCH Noes: (1) The daily capial charge is given as he negaive of (3+k) imes he greaer of he previous day s VaR or he average VaR over he las 60 business days, where k is he penaly. (2) AD is he absolue deviaion of he violaions from he VaR forecas. 32

33 33

34 34