KIER DISCUSSION PAPER SERIES

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1 KIER DISCUSSION PAPER SERIES KYOTO INSTITUTE OF ECONOMIC RESEARCH Discussion Paper No.727 GFC-Robus Risk Managemen Sraegies under he Basel Accord Michael McAleer Ocober 2010 KYOTO UNIVERSITY KYOTO, JAPAN

2 GFC-Robus Risk Managemen Sraegies under he Basel Accord* Michael McAleer Economeric Insiue Erasmus School of Economics Erasmus Universiy Roerdam and Tinbergen Insiue The Neherlands and Insiue of Economic Research Kyoo Universiy Japan Juan-Ángel Jiménez-Marín Deparmen of Quaniaive Economics Compluense Universiy of Madrid Teodosio Pérez-Amaral Deparmen of Quaniaive Economics Compluense Universiy of Madrid Revised: Ocober 2010 * For financial suppor, he firs auhor wishes o hank he Ausralian Research Council, Naional Science Council, Taiwan, and he Japan Sociey for he Promoion of Science. The second and hird auhors acknowledge he financial suppor of he Miniserio de Ciencia y Tecnología and Comunidad de Madrid, Spain. 1

3 Absrac A risk managemen sraegy is proposed as being robus o he Global Financial Crisis (GFC) by selecing a Value-a-Risk (VaR) forecas ha combines he forecass of differen VaR models. The robus forecas is based on he median of he poin VaR forecass of a se of condiional volailiy models. This risk managemen sraegy is GFC-robus in he sense ha mainaining he same risk managemen sraegies before, during and afer a financial crisis would lead o comparaively low daily capial charges and violaion penalies. The new mehod is illusraed by using he S&P500 index before, during and afer he global financial crisis. We invesigae he performance of a variey of single and combined VaR forecass in erms of daily capial requiremens and violaion penalies under he Basel II Accord, as well as oher crieria. The median VaR risk managemen sraegy is GFC-robus as i provides sable resuls across differen periods relaive o oher VaR forecasing models. The new sraegy based on combined forecass of single models is sraighforward o incorporae ino exising compuer sofware packages ha are used by banks and oher financial insiuions. Key words and phrases: Value-a-Risk (VaR), daily capial charges, robus forecass, violaion penalies, opimizing sraegy, aggressive risk managemen sraegy, conservaive risk managemen sraegy, Basel II Accord, global financial crisis. JEL Classificaions: G32, G11, G17, C53, C22. 2

4 1. Inroducion The Global Financial Crisis (GFC) of has lef an indelible mark on economic and financial srucures worldwide, and caused a generaion of invesors o wonder how hings could have become so bad (see, for example, Borio (2008)). There have been many quesions asked abou wheher appropriae regulaions were in place, especially in he USA, o ensure he appropriae monioring and encouragemen of (possibly excessive) risk aking. The Basel II Accord 1 was designed o monior and encourage sensible risk aking, using appropriae models of risk o calculae Value-a-Risk (VaR) and subsequen daily capial charges. VaR is defined as an esimae of he probabiliy and size of he poenial loss o be expeced over a given period, and is now a sandard ool in risk managemen. I has become especially imporan following he 1995 amendmen o he Basel Accord, whereby banks and oher Auhorized Deposi-aking Insiuions (ADIs) were permied (and encouraged) o use inernal models o forecas daily VaR (see Jorion (2000) for a deailed discussion). The las decade has winessed a growing academic and professional lieraure comparing alernaive modelling approaches o deermine how o measure VaR, especially for large porfolios of financial asses. The amendmen o he iniial Basel Accord was designed o encourage and reward insiuions wih superior risk managemen sysems. A back-esing procedure, whereby acual reurns are compared wih he corresponding VaR forecass, was inroduced o assess he qualiy of he inernal models used by ADIs. In cases where inernal models led o a greaer number of violaions han could reasonably be expeced, given he confidence level, he ADI is required o hold a higher level of capia (see Table 1 for he 1 When he Basel I Accord was concluded in 1988, no capial requiremens were defined for marke risk. However, regulaors soon recognized he risks o a banking sysem if insufficien capial were held o absorb he large sudden losses from huge exposures in capial markes. During he mid-90 s, proposals were abled for an amendmen o he 1988 Accord, requiring addiional capial over and above he minimum required for credi risk. Finally, a marke risk capial adequacy framework was adoped in 1995 for implemenaion in The 1995 Basel I Accord amendmen provides a menu of approaches for deermining marke risk capial requiremens, ranging from a simple o inermediae and advanced approaches. Under he advanced approach (ha is, he inernal model approach), banks are allowed o calculae he capial requiremen for marke risk using heir inernal models. The use of inernal models was inroduced in 1998 in he European Union. The 26 June 2004 Basel II framework, implemened in many counries in 2008 (hough no ye in he USA), enhanced he requiremens for marke risk managemen by including, for example, oversigh rules, disclosure, managemen of counerpary risk in rading porfolios. 3

5 penalies imposed under he Basel II Accord). Penalies imposed on ADIs affec profiabiliy direcly hrough higher capial charges, and indirecly hrough he imposiion of a more sringen exernal model o forecas VaR. 2 This is one reason why financial managers may prefer risk managemen sraegies ha are passive and conservaive raher han acive and aggressive (for more on his, see below). Excessive conservaism can have a negaive impac on he profiabiliy of ADIs as higher capial charges are subsequenly required. Therefore, ADIs should perhaps consider a sraegy ha allows an endogenous decision as o how ofen ADIs should violae, and hence incur violaion penalies, in any financial year (for furher deails, see McAleer and da Veiga (2008a, 2008b), McAleer (2009), Caporin and McAleer (2010a), and McAleer e al. (2009)). Addiionally, ADIs need no resric hemselves o using only a single risk model. McAleer e al. (2009) propose a risk managemen sraegy ha consiss in choosing from among differen combinaions of alernaive risk models o forecas VaR. They discuss a combinaion of forecass ha was characerized as an aggressive sraegy, and anoher ha was regarded as a conservaive sraegy. 3 Following such an approach, his paper suggess using a combinaion of VaR forecass o obain a crisis robus risk managemen sraegy. The paper defines a crisis robus sraegy as an opimal risk managemen sraegy ha remains unchanged regardless of wheher i is used before, during or afer a significan financial crisis. Parameric mehods for forecasing VaR are ypically fied o hisorical reurns assuming specific condiional disribuions of reurns, such as normaliy, Suden-, or generalized normal disribuion. The VaR forecas depends on he parameric model, he condiional disribuion and can be heavily affeced by a few large observaions. Some models provide many violaions, bu low daily capial charges. Addiionally, hese resuls can change drasically from ranquil o urbulen periods. In shor, regardless of economic urbulence, is here a model o forecas VaR ha provides a reasonable number of violaions and daily capial charges? 2 In he 1995 amendmen (p. 16), a similar capial requiremen sysem was recommended, bu he specific penalies were lef o each naional supervisor. We consider ha he penaly srucure conained in Table 1 of his paper belongs only o Basel II, and was no par of Basel I or is 1995 amendmen. 3 This is a novel possibiliy. Technically, a combinaion of forecas models is also a forecas model. In principle, he adopion of a combinaion of forecas models by a bank is no forbidden by he Basel Accords, alhough i is subjec o regulaory approval. 4

6 We esimae several univariae condiional volailiy models o forecas VaR, assuming differen reurns disribuions (specifically, Gaussian, Suden- and Generalized Normal). Addiionally, we presen 12 new sraegies based on combinaions of sardard model VaR forecass, namely: lowerbound, upperbound (as defined in McAleer e al. (2009)), he average, and nine addiional sraegies based on he 10 h, 50 h, 90 h perceniles. Models are compared over hree differen ime periods o invesigae wheher we can find a risk managemen sraegy ha is robus over ime (ha is, crisisrobus). We provide evidence ha using he median of he poin VaR forecass of a se of univariae condiional volailiy models is a robus risk measure. A risk managemen sraegy based on he median forecas is found o be superior o alernaive single and composie model alernaives. The remainder of he paper is organized as follows. In Secion 2 we presen he main ideas of he Basel II Accord Amendmen as i relaes o forecasing VaR and daily capial charges. Secion 3 reviews some of he mos well-known models of condiional volailiy used o forecas VaR. In Secion 4 he daa used for esimaion and forecasing are presened. Secion 5 analyses he robus VaR forecass before, during and afer he GFC. Secion 6 presens some conclusions. 2. Forecasing Value-a-Risk and Daily Capial Charges The Basel II Accord sipulaes ha daily capial charges (DCC) 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 (3+k) for a violaion penaly, wherein a violaion involves he acual negaive reurns exceeding he VaR forecas negaive reurns for a given day: 4 { ( ) 60 } DCC = sup k VaR, - VaR -1 (1) where 4 Our aim is o invesigae he likely performance of he Basel II regulaions. In his secion we carry ou our analysis applying he Basel II formulae o a period ha includes he GFC, during which he Basel II Accord regulaions were no fully implemened. 5

7 ( ) 60 DCC = daily capial charges, which is he higher of k VaR and - VaR -1, VaR = Value-a-Risk for day, VaR = Yˆ z σˆ, VaR 60 = mean VaR over he previous 60 working days, Ŷ = esimaed reurn a ime, z = 1% criical value of he disribuion of reurns a ime, σˆ = esimaed risk (or square roo of volailiy) a ime, 0 k 1 is he Basel II violaion penaly (see Table 1). [Inser Table 1 here] The muliplicaion facor 5 (or penaly), k, depends on he cenral auhoriy s assessmen of he ADI s risk managemen pracices and he resuls of a simple backes. I is deermined by he number of imes acual losses exceed a paricular day s VaR forecas (Basel Commiee on Banking Supervision (1996, 2006)). The minimum muliplicaion facor of 3 is inended o compensae for various errors ha can arise in model implemenaion, such as simplifying assumpions, analyical approximaions, small sample biases and numerical errors ha end o reduce he rue risk coverage of he model (see Sahl (1997)). Increases in he muliplicaion facor are designed o increase he confidence level ha is implied by he observed number of violaions o he 99 per cen confidence level, as required by regulaors (for a deailed discussion of VaR, as well as exogenous and endogenous violaions, see McAleer (2009), Jiménez-Marin e al. (2009), and McAleer e al. (2009)). In calculaing he number of violaions, ADIs are required o compare he forecass of VaR wih realised profi and loss figures for he previous 250 rading days. In 1995, he 5 The formula in equaion (1) is conained in he 1995 amendmen o Basel I, while Table 1 appears for he firs ime in he Basel II Accord in

8 1988 Basel Accord (Basel Commiee on Banking Supervision (1988)) was amended o allow ADIs o use inernal models o deermine heir VaR hresholds (Basel Commiee on Banking Supervision (1995)). However, ADIs ha propose using inernal models are required o demonsrae ha heir models are sound. Movemen from he green zone o he red zone arises hrough an excessive number of violaions. Alhough his will lead o a higher value of k, and hence a higher penaly, violaions will also end o be associaed wih lower daily capial charges. 6 Value-a-Risk refers o he lower bound of a confidence inerval for a (condiional) mean, ha is, a wors case scenario on a ypical day. If ineres lies in modelling he random variable,, i could be decomposed as follows: Y Y E( Y F 1) = + ε. (2) This decomposiion saes ha Y comprises a predicable componen, E(Y F 1 ), which is he condiional mean, and a random componen,. The variabiliy of, and hence is disribuion, is deermined by he variabiliy of. If i is assumed ha follows a condiional disribuion, such ha: ε ε Y ε 2 ε ~ D( μ, σ ) where μ and σ are he condiional mean and sandard deviaion of ε, respecively, hese can be esimaed using a variey of parameric, semi-parameric or non-parameric mehods. The VaR hreshold for Y can be calculaed as: VaR E( Y F 1) ασ =, (3) where α is he criical value from he disribuion of ε o obain he appropriae confidence level. I is possible for σ o be replaced by alernaive esimaes of he 6 The number of violaions in a given period is an imporan (hough no he only) guide for regulaors o approve a given VaR model. 7

9 condiional sandard deviaion in order o obain an appropriae VaR (for useful reviews of heoreical resuls for condiional volailiy models, see Li e al. (2002) and McAleer (2005), who discusses a variey of univariae and mulivariae, condiional, sochasic and realized volailiy models). Some recen empirical sudies (see, for example, Berkowiz and O'Brien (2001), Gizycki and Hereford (1998), and Pérignon e al. (2008)) have indicaed ha some financial insiuions overesimae heir marke risks in disclosures o he appropriae regulaory auhoriies, which can imply a cosly resricion o he banks rading aciviy. ADIs may prefer o repor high VaR numbers o avoid he possibiliy of regulaory inrusion. This conservaive risk reporing suggess ha efficiency gains may be feasible. In paricular, as ADIs have effecive ools for he measuremen of marke risk, while saisfying he qualiaive requiremens, ADIs could conceivably reduce daily capial charges by implemening a conex-dependen marke risk disclosure policy. For a discussion of alernaive approaches o opimize VaR and daily capial charges, see McAleer (2008) and McAleer e al. (2009). The nex secion describes several volailiy models ha are widely used o forecas he 1-day ahead condiional variances and VaR hresholds. 3. Models for Forecasing VaR ADIs can use inernal models o deermine heir VaR hresholds. There are alernaive ime series models for esimaing condiional volailiy. In wha follows, we presen several condiional volailiy models o evaluae sraegic marke risk disclosure, namely GARCH, GJR and EGARCH, wih normal, Suden- and Generalized normal disribuion errors, where he parameers are esimaed. These models are chosen as hey are well known and widely used in he lieraure. For an exensive discussion of he heoreical properies of several of hese models, see Ling and McAleer (2002a, 2002b, 2003a) and Caporin and McAleer (2010a). As an alernaive o esimaing he parameers, we also consider he exponenial weighed 8

10 moving average (EWMA) mehod by Riskmerics (1996) and Zumbauch, (2007) ha calibraes he unknown parameers. We include a secion on hese models o presen hem in a unified framework and noaion, and o make explici he specific versions we are using. Apar from EWMA, he models are presened in increasing order of complexiy. 3.1 GARCH For a wide range of financial daa series, ime-varying condiional variances can be explained empirically hrough he auoregressive condiional heeroskedasiciy (ARCH) model, which was proposed by Engle (1982). When he ime-varying condiional variance has boh auoregressive and moving average componens, his leads o he generalized ARCH(p,q), or GARCH(p,q), model of Bollerslev (1986). I is very common o impose he widely esimaed GARCH(1,1) specificaion in advance. Consider he saionary AR(1)-GARCH(1,1) model for daily reurns, y : y=φ 1+φ2y -1+ε, φ 2 <1 (4) for = 1,..., n, where he shocks o reurns are given by: ε = η h, η ~iid(0,1) h=ω+αε + βh, (5) and ω > 0, α 0, β 0 are sufficien condiions o ensure ha he condiional variance h > 0. The saionary AR(1)-GARCH(1,1) model can be modified o incorporae a nonsaionary ARMA(p,q) condiional mean and a saionary GARCH(r,s) condiional variance, as in Ling and McAleer (2003b). 3.2 GJR In he symmeric GARCH model, he effecs of posiive shocks (or upward movemens in daily reurns) on he condiional variance, h, are assumed o be he same as he 9

11 negaive shocks (or downward movemens in daily reurns). In order o accommodae asymmeric behaviour, Glosen, Jagannahan and Runkle (1992) proposed a model (hereafer GJR), for which GJR(1,1) is defined as follows: 2 h=ω+(α + γi(η -1 ))ε -1 + βh -1, (6) where ω > 0, α 0, α + γ 0, β 0 are sufficien condiions for h > 0, and I η ) is an indicaor variable defined by: ( I ( η ) 1, ε < 0 = 0, ε 0 (7) as η has he same sign as ε. The indicaor variable differeniaes beween posiive and negaive shocks, so ha asymmeric effecs in he daa are capured by he coefficien γ. For financial daa, i is expeced ha γ 0 because negaive shocks have a greaer impac on risk han do posiive shocks of similar magniude. The asymmeric effec, γ, measures he conribuion of shocks o boh shor run persisence, α + γ 2, and o long run persisence, α + β + γ 2. Alhough GJR permis asymmeric effecs of posiive and negaive shocks of equal magniude on condiional volailiy, he special case of leverage, whereby negaive shocks increase volailiy while posiive shocks decrease volailiy (see Black (1976) for an argumen using he deb/equiy raio), canno be accommodaed (for furher deails on asymmery versus leverage in he GJR model, see Caporin and McAleer (2010b)). 3.3 EGARCH An alernaive model o capure asymmeric behaviour in he condiional variance is he Exponenial GARCH, or EGARCH(1,1), model of Nelson (1991), namely: ε ε -1-1 logh = ω+α +γ + βlogh, β <1 h-1 h-1-1 (8) 10

12 where he parameers α, β and γ have differen inerpreaions from hose in he GARCH(1,1) and GJR(1,1) models. EGARCH capures asymmeries differenly from GJR. The parameers α and γ in EGARCH(1,1) represen he magniude (or size) and sign effecs of he sandardized residuals, respecively, on he condiional variance, whereas α and α + γ represen he effecs of posiive and negaive shocks, respecively, on he condiional variance in GJR(1,1). Unlike GJR, EGARCH can accommodae leverage, depending on he resricions imposed on he size and sign parameers. As noed in McAleer e al. (2007), here are some imporan differences beween EGARCH and he previous wo models, as follows: (i) EGARCH is a model of he logarihm of he condiional variance, which implies ha no resricions on he parameers are required o ensure h > 0 ; (ii) momen condiions are required for he GARCH and GJR models as hey are dependen on lagged uncondiional shocks, whereas EGARCH does no require momen condiions o be esablished as i depends on lagged condiional shocks (or sandardized residuals); (iii) Shephard (1996) observed ha β < 1 is likely o be a sufficien condiion for consisency of QMLE for EGARCH(1,1); (iv) as he sandardized residuals appear in equaion (7), β < 1 would seem o be a sufficien condiion for he exisence of momens; and (v) in addiion o being a sufficien condiion for consisency, β < 1 is also likely o be sufficien for asympoic normaliy of he QMLE of EGARCH(1,1). The hree condiional volailiy models given above are esimaed under he following disribuional assumpions on he condiional shocks: (1) normal, and (2), wih esimaed degrees of freedom. As he models ha incorporae he disribued errors are esimaed by QMLE, he resuling esimaors are consisen and asympoically normal, so hey can be used for esimaion, inference and forecasing. 3.4 Exponenially Weighed Moving Average (EWMA) 11

13 As an alernaive o esimaing he parameers of he appropriae condiional volailiy models, Riskmerics (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. This approach forecass he condiional variance a ime as a linear combinaion of he lagged condiional variance and he squared uncondiional shock a ime EWMA model calibraes he condiional variance as: 1. The 2 h = -1-1 λh +(1-λ)ε (9) where λ is a decay parameer. Riskmerics (1996) suggess ha λ should be se a 0.94 for purposes of analysing daily daa. As no parameers are esimaed, here are no momen or log-momen condiions. 4. Daa The daa used for esimaion and forecasing are he closing daily prices for Sandard and Poor s Composie 500 Index (S&P500), which were obained from he Ecowin Financial Daabase for he period 3 January 2000 o 16 March Alhough i is unlikely ha an ADI s ypical marke risk porfolio only racks he S&P500 index, i is used as an illusraion of he broad movemens of profis and losses of he equiy porfolios of ADIs. If P denoes he marke price, he reurns a ime ( ) are defined as: ( ) R R = log P / P 1. (10) [Inser Figures 1-2 and Table 2 here] Figure 1 shows he S&P500 reurns, for which he descripive saisics are given in Table 2. The exremely high posiive and negaive reurns are eviden from Sepember 12

14 2008 onward, and have coninued well ino The mean is close o zero, and he range is beween -9.5% and +11%. The Jarque-Bera Lagrange muliplier es rejecs he null hypohesis of normally disribued reurns. As he series displays high kurosis, his would seem o indicae he exisence of exreme observaions, as can be seen in he hisogram, which is no surprising for daily financial reurns daa. Several measures of volailiy are available in he lieraure. In order o gain some inuiion, we adop he measure proposed in Franses and van Dijk (1999), where he rue volailiy of reurns is defined as: ( ( )) 2 1 V = R E R F, (11) where F 1 is he informaion se a ime -1. Figure 2 shows he S&P500 volailiy, as he square roo of V in equaion (11). The series exhibi clusering ha needs o be capured by an appropriae ime series model. The volailiy of he series appears o be high during he early 2000s, followed by a quie period from 2003 o he beginning of Volailiy increases dramaically afer Augus 2008, due in large par o he worsening global credi environmen. This increase in volailiy is even higher in Ocober In less han four weeks in Ocober 2008, he S&P500 index plummeed by 27.1%. In less han hree weeks in November 2008, saring he morning afer he US elecions, he S&P500 index plunged a furher 25.2%. Overall, from lae Augus 2008, US socks fell by a scarcely unbelievable 42.2% o reach a low on 20 November An examinaion of daily movemens in he S&P500 index from 2000 suggess ha large changes by hisorical sandards are 4% in eiher direcion. From January 2000 o Augus 2008, here was a 0.31% chance of observing an increase of 4% or more in one day, and a 0.18% chance of seeing a reducion of 4% or more in one day. Therefore, 99.5% of movemens in he S&P500 index during his period had daily swings of less han 4%. Prior o Sepember 2008, he S&P500 index had only 7 days wih massive 4% gains, bu since Sepember 2008, here have been a furher 12 such days. On he downside, before 13

15 he curren sock marke meldown, he S&P500 index had only 4 days wih huge 4% or more losses whereas, during he recen panic, here were a furher 17 such days. This comparison is beween more han 99 monhs and less han 6 monhs. During his shor ime span of financial panic, he 4% or more gain days chances increased 80 imes while he chances of 4% or more loss days increased 32 imes. Such movemens in he S&P500 index are unprecedened. 5. Robus Forecasing of VaR and Evaluaion Framework As seen in McAleer e al. (2010), he global financial crisis has affeced he bes risk managemen sraegies by changing he opimal model for minimizing daily capial charges. The objecive here is o provide a robus risk managemen sraegy, namely one ha does no change over ime, even in he conex of a GFC. This robus risk managemen sraegy also has o lead o daily capial charges ha are no excessive, and violaion frequencies ha are compaible wih he Basel II Accord. ADIs need no resric hemselves o using only a single risk model. We propose a risk managemen sraegy ha consiss in choosing a forecas from among differen combinaions of alernaive risk models o forecas VaR. McAleer e al. (2010) developed a risk managemen sraegy ha used combinaions of several models for forecasing VaR. I was found ha an aggressive risk managemen sraegy (namely, choosing he supremum of VaR forecass, or upperbound) yielded he lowes mean capial charges and larges number of violaions. On he oher hand, a conservaive risk managemen sraegy (namely, by choosing he infinum, or lowerbound) had far fewer violaions, and correspondingly higher mean daily capial charges. In his paper, we forecas VaR using combinaions of he forecass of individual VaR models, namely he rh percenile of he VaR forecass of a se of univariae condiional volailiy models. Alernaive single models wih differen error disribuions and several combinaions are compared over hree differen ime periods o invesigae which, if any, of he risk managemen sraegies may be robus. 14

16 We conduc an exercise o analyze he performance of exising VaR forecasing models, as permied under he Basel II framework, when applied o he S&P500 index. Addiionally, we analyze welve new sraegies based on combinaions of he previous sandard single-model forecass of VaR, namely: lowerbound (0 h percenile), upperbound (100 h percenile), average, and nine addiional sraegies based on he 10 h hrough o he 90 h perceniles. I is inended o deermine wheher we can selec a robus VaR forecas irrespecive of he ime period, and o provide reasonable daily capial charges and number of violaion penalies. 5.1 Evaluaing Crisis-Robus Risk Managemen Sraegies In Table 3 he performance of he differen VaR forecasing models is evaluaed using several sandard crieria ha are relevance for he risk manager, namely: daily capial charges (DCC), number of violaions (NoV), he failure rae (ha is, he raio of NoV o number of days) (FailRa), accumulaed losses 7 (AcLoss), and he value of he asymmeric linear ick loss funcion 8 (AlTick) ha allows a comparison of model performance. As Giacomini and Komunjer (2005) argue, he ick loss funcion is he implici loss funcion whenever he arge is a forecas of a paricular α-quanile. This loss funcion akes ino accoun he magniude of he implici cos associaed wih VaR forecasing errors. As VaR esimaes are used frequenly o assis in sraegic financial decision-making and managing marke risk, VaR forecasing errors can imply financial disress, such as incorrecly esimaing daily capial charges subjec o regulaory conrol. The performance crieria are calculaed for each model and error disribuion, and for each of he hree sub-samples, before, during, and afer he GFC, where before is from 2 January 2008 o 11 Augus 2008, during is from 12 Augus 2008 o 9 March 2009, and afer is from 10 March 2009 o 16 March Table 3 shows he 7 López (1999) suggesed measuring he accuracy of he VaR forecas on he basis of he disance beween he observed reurns and he forecased VaR values if a violaion occurs: R < + < Ψ + = 1 VaR 1 if R 1 0 and R 1 VaR +1 1, a preferred VaR model is he one ha minimizes 0 oherwise he oal loss value, Ψ T + 1 = = 1Ψ. 8 The ick loss funcion of order α defined as ( ) ( α 1 ( 0 ) ) + + +, where e = R VaR. L α e 1 = e 1 < e

17 performance saisics for comparing he VaR forecass for each sub-period. We also include 12 new forecasing sraegies based on combinaions of individual model VaR forecass, namel : lowerbound, upperbound, average, and nine addiional sraegies based on he 10 h hrough o he 90 h perceniles, including he median. No risk model is found always o be superior o is compeiors as here is no sraegy ha opimizes every evaluaion saisic for he hree sub-periods. Noneheless, he 50 h percenile sraegy (namely, he median - shaded row of Table 3) is found o be robus, as i produces adequae VaR forecass ha exhibi sable resuls across differen periods relaive o he oher risk models. Furhermore, we compue for each sub-sample and each saisic he values of he 1 s, 2 nd and 3 rd quariles. In Table 4, he median sraegy saisics fall in almos all cases in he 2nd quarile. The EGARCH- model performance is close o he median sraegy, excep ha he former has average daily capial charges, (AvDCC) slighly higher for he hree periods. In general, he median sraegy provides a robus VaR forecas, regardless of wheher here is a GFC. [Inser Tables 3-4 here] As can be seen in Table 3, before he GFC, AvDCC based on he Riskmerics VaR forecas is lower (9.03%) han ha based on he median (9.77%). During he GFC, AvDCC based on he median sraegy increased drasically (20.57%), alhough no as much as when he Riskmerics model is used (22.51%). Finally, afer he GFC, AvDCC decreased, becoming smaller for he median sraegy (10.95%) han for he Riskmerics model (11.19%). Figure 3 shows he evoluion of he evaluaion saisics for he sraegies based on he 10 h hrough o he 90 h perceniles in he hree sub-samples. GFC-robusness, in our conex, implies a rade-off beween he number of violaions and daily cos of capial charges. In general, when he chosen percenile increases, i is found ha DCC decreases and NoV increases, all he more so when he seleced percenile is greaer han he median. Figure 4 shows VaR forecass using four models (Riskmerics, median, upperbound and lowerbound) and S&P500 reurns, where he verical axis represens reurns, and he 16

18 horizonal axis represens he period from 2 January 2008 o 16 March The S&P500 reurns are given as he upper blue line ha flucuaes around zero. The upper red line represens he supremum of he VaR calculaed for he individual volailiy models, which reflecs an aggressive risk managemen sraegy, whereas he lower green line represens he infimum of he VaR calculaed for he individual volailiy models, which reflecs a conservaive risk managemen sraegy. These wo lines correspond o a combinaion of alernaive risk models. The brown line is he median of he VaR, calculaed for he individual models of volailiy. As a benchmark, he black line denoes he Riskmerics model. [Inser Figures 3-4 here] As can be seen in Figure 4, VaR forecass obained from he differen models of volailiy have flucuaed, as expeced, during he firs few monhs of I has been relaively low, a below 5%, and relaively sable beween April and Augus Around Sepember 2008, VaR sared increasing unil i peaked in Ocober 2008, beween 10% and 15%, which is almos a four-fold increase in VaR in a maer of one and a half monhs. In he las wo monhs of 2008, VaR decreased o values beween 5% and 8%, which is sill wice as large as i had been jus a few monhs earlier. Therefore, volailiy has increased subsanially during he GFC, and has remained relaively high hereafer. Figure 5 includes daily capial charges based on he median VaR for he previous 60 days, which are he lower lines. The red line corresponds o he aggressive risk managemen sraegy based on he supremum of he daily VaR forecass of he alernaive models of volailiy, he green line corresponds o he conservaive risk managemen sraegy based on he infinum of he VaR forecass, and he brown line denoes he robus sraegy based on he median of he VaR forecass. As a benchmark, we include DCC based on he Riskmerics model. Table 3 shows ha he Riskmerics AvDCC and FailRa are higher han he median AvDCC, excep for before he GFC. This can be seen in Figure 5 when comparing he DCC Riskmerics and DCC median lines. The DCC median line is close o he upperbound DCC line virually hroughou he sample, bu FailRa for he median sraegy is always much lower and is falling wihin he Basel Accord limis han when he upperbound sraegy is considered. 17

19 [Inser Figure 5 here] 6. Conclusion In his paper we proposed robus risk forecass ha use combinaions of several condiional volailiy models for forecasing VaR. Differen sraegies for combining models were compared over hree differen ime periods, using S&P500 o invesigae wheher we can deermine a GFC-robus risk managemen sraegy. Backesing provided evidence ha a risk managemen sraegy based on VaR forecas corresponding o he 50 h percenile (median) of he VaR forecass of a se of univariae condiional volailiy models is robus, in ha i yields reasonable daily capial charges, number of violaions ha do no jeopardize insiuions ha migh use i, and more imporanly, is invarian before, during and afer he GFC. I is worh noing ha, as in McAleer e al. (2010), he VaR model minimizing DCC before, during and afer he GFC changed frequenly. Alhough he median model is no derived as he bes model for minimizing DCC and he number of violaion penalies, i is neverheless a model ha balances daily capial charges and violaion penalies in minimizing DCC. The idea of combining differen VaR forecasing models is enirely wihin he spiri of he Basel II Accord, alhough is use would require approval by he regulaory auhoriies, as for any forecasing model. This approach is no compuaionally demanding, even hough several models have o be specified and esimaed over ime. Furher research is needed o compue he sandard errors of he forecass of he combinaion models, including he median forecas. 18

20 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. Basel Commiee on Banking Supervision, (2006), Inernaional Convergence of Capial Measuremen and Capial Sandards, a Revised Framework Comprehensive Version, BIS, Basel, Swizerland. Berkowiz, J. and J. O'Brien (2001), How accurae are value-a-risk models a commercial banks?, Discussion Paper, Federal Reserve Board. Black, F. (1976), Sudies of sock marke volailiy changes, in 1976 Proceedings of he American Saisical Associaion, Business and Economic Saisics Secion, pp Bollerslev, T. (1986), Generalised auoregressive condiional heeroscedasiciy, Journal of Economerics, 31, Borio, C. (2008), The financial urmoil of 2007-?: A preliminary assessmen and some policy consideraions, BIS Working Papers No 251, Bank for Inernaional Selemens, Basel, Swizerland. Caporin, M. and M. McAleer (2010a), The Ten Commandmens for managing invesmens, Journal of Economic Surveys, 24, Caporin, M. and M. McAleer (2010b), Model selecion and esing of condiional and sochasic volailiy models, o appear in L. Bauwens, C. Hafner and S. Lauren (eds.), Handbook on Financial Engineering and Economerics: Volailiy Models and Their Applicaions, Wiley, New York (Available a SSRN: hp://ssrn.com/absrac= ). Engle, R.F. (1982), Auoregressive condiional heeroscedasiciy wih esimaes of he variance of Unied Kingdom inflaion, Economerica, 50, Franses, P.H. and D. van Dijk (1999), Nonlinear Time Series Models in Empirical Finance, Cambridge, Cambridge Universiy Press. 19

21 Giacomini, R., and Komunjer, I. (2005), Evaluaion and combinaion of condiional quanile forecass. Journal of Business & Economic Saisics, 23, Gizycki, M. and N. Hereford (1998), Assessing he dispersion in banks esimaes of marke risk: he resuls of a value-a-risk survey, Discussion Paper 1, Ausralian Prudenial Regulaion Auhoriy. Glosen, L., R. Jagannahan and D. Runkle (1992), On he relaion beween he expeced value and volailiy of nominal excess reurn on socks, Journal of Finance, 46, Jimenez-Marin, J.-A., McAleer, M. and T. Pérez-Amaral (2009), The Ten Commandmens for managing value-a-risk under he Basel II Accord, Journal of Economic Surveys, 23, Jorion, P. (2000), Value a Risk: The New Benchmark for Managing Financial Risk, McGraw-Hill, New York. Li, W.K., S. Ling and M. McAleer (2002), Recen heoreical resuls for ime 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, (2003a), Asympoic heory for a vecor ARMA-GARCH model, Economeric Theory, 19, Ling, S. and M. McAleer (2003b), On adapive esimaion in nonsaionary ARMA models wih GARCH errors, Annals of Saisics, 31, Lopez, J. A., (1999), "Mehods for evaluaing value-a-risk esimaes," Economic Review, Federal Reserve Bank of San Francisco, pages McAleer, M. (2005), Auomaed inference and learning in modeling financial volailiy, Economeric Theory, 21, McAleer, M. (2009), The Ten Commandmens for opimizing value-a-risk and daily capial charges, Journal of Economic Surveys, 23,

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23 Figure 1. Daily Reurns on S&P500 Index 3 January March % 8% 4% 0% -4% -8% -12% Figure 2. Daily Volailiy in S&P500 Reurns 3 January March % 10% 8% 6% 4% 2% 0%

24 Figure 3. Crieria for Comparing Percenile Sraegies BEFORE CRISIS DURING CRISIS AFTER CRISIS 23

25 Figure 4. VaR for S&P500 Reurns 2 January March 2010 Peak_value Dae Trough_value dae S&P Reurns /8/ /3/ Before During Afer M M M M01 S&P Reurns VaR RSKM VaR Lowerbound VaR Upperbound VaR Median Noe: The upper blue line represens daily reurns for S&P500 index. The upper red line represens he supremum, he lower green line he infinum, he bold brown line he median of he VaR forecass of he alernaive models (see Secion 3), and he black line is he Riskmerics VaR forecas. 24

26 Figure 5. DCC and S&P500 Reurns Before During Afer M M M M01 S&P Reurns DCC Median DCC Upperbound DCC Lowerbound DCC RSKM 25

27 Table 1: Basel Accord Penaly Zones Zone Number of Violaions k Green 0 o Yellow Red Noe: The number of violaions is given for 250 business days. The penaly srucure under he Basel II Accord is specified for he number of violaions and no heir magniude, eiher individually or cumulaively. 26

28 Table 2. Descripive Saisics for S&P500 Reurns (%) 3 January March ,200 1, Observaions 2660 Mean Median Maximum Minimum Sd. Dev Skewness Kurosis Jarque-Bera Probabiliy

29 Table 3. Comparing Alernaive Models of Volailiy BEFORE CRISIS DURING CRISIS AFTER CRISIS Model AvDCC NoV FailRa AcLoss AlTick AvDCC NoV FailRa AcLoss AlTick AvDCC NoV FailRa AcLoss AlTick RSKM % % % GARCH % % % GJR % % % EGARCH % % % GARCH_ % % % GJR_ % % % EGARCH_ % % % GARCH_g % % % GJR_g % % % EGARCH_g % % % Inf % % % Sup % % % Mean % % % h Per % % % h Per % % % h Per % % % h Per % % % h Per. (Median) % % % h Per % % % h Per % % % h Per % % % h Per % % % Noe: Percenage of Days Minimizing Daily Capial Charges (%DmDCC), Average of Daily Capial Charges (AvDCC), Number of Violaions (NoV), Failure Rae (FailRa), Accumulaed Loses (AcLoss), and Asymmeric Linear Tick loss funcion value (AlTick) for alernaive models of volailiy. 28

30 Table 4. Quariles for Evaluaion Saisics Before During Afer Quariles AvDCC NoV FailRa AcLoss Alick DCC NoV FailRa AcLoss Alick. DCC NoV FailRa AcLoss Alick % % % % % % % % % Models RSKM EGARCH_ Median Noe: The op par of he able gives he value of he rh quarile for he evaluaion saisics in Table 3 when considering all he GARCH-family models and Riskmerics. The boom par of he able is he quarile where he evaluaion saisics fell for he RSKM, EGARCH_ and Median models. For example, for he RSKM model, an enry of one in he firs column means ha he corresponding AvDCC fell in he firs quarile. 29