Has the Basel Accord Improved Risk Management During the Global Financial Crisis?*

Has he Basel Accord Improved Risk Managemen During he Global Financial Crisis?* º Michael McAleer Economeric Insiue Erasmus School of Economics Erasmus Universiy Roerdam and Tinbergen Insiue The Neherlands and Insiue of Economic Research Kyoo Universiy and Deparmen of Quaniaive Economics Compluense Universiy of Madrid 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 2012 * 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

Absrac The Basel II Accord requires ha banks and oher Auhorized Deposi-aking Insiuions (ADIs) communicae heir daily risk forecass o he appropriae moneary auhoriies a he beginning of each rading day, using one or more risk models o measure Value-a-Risk (VaR). The risk esimaes of hese models are used o deermine capial requiremens and associaed capial coss of ADIs, depending in par on he number of previous violaions, whereby realised losses exceed he esimaed VaR. In his paper we define risk managemen in erms of choosing from a variey of risk models, and discuss he selecion of opimal risk models. A new approach o model selecion for predicing VaR is proposed, consising of combining alernaive risk models, and we compare conservaive and aggressive sraegies for choosing beween VaR models. We hen examine how differen risk managemen sraegies performed during he 2008-09 global financial crisis. These issues are illusraed using Sandard and Poor s 500 Composie Index. Key words and phrases: Value-a-Risk (VaR), daily capial charges, violaion penalies, opimizing sraegy, risk forecass, aggressive or conservaive risk managemen sraegies, Basel Accord, global financial crisis. JEL Classificaions: G32, G11, G17, C53, C22. 2

1. Inroducion The global financial crisis of 2008-09 has lef an indelible mark on economic and financial srucures worldwide, and lef an enire generaion of invesors wondering how hings could have become so severe (see, for example, Borio (2008)). There have been many quesions asked abou wheher appropriae regulaions were in place, especially in he US, o permi he appropriae monioring and encouragemen of (possibly excessive) risk aking. The Basel II Accord was designed o monior and encourage sensible risk aking using appropriae models of risk o calculae Value-a-Risk (VaR) and forecas 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. 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 is 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 1998. 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 (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 only inroduced in 1998 in he European Union. The 26 June 2004 Basel II framework, implemened in many counries in 2008 (hough no ye formally in he USA) enhanced he requiremens for marke risk managemen by including, for example, oversigh rules, disclosure, managemen of counerpary risk in rading porfolios. 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

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 lead 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 capial (see Table 1 for he 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 1. This is one reason why financial managers may prefer risk managemen sraegies ha are passive and conservaive raher han acive and aggressive. 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 many imes ADIs should violae in any financial year (for furher deails, see McAleer and da Veiga (2008a, 2008b), McAleer (2008), Caporin and McAleer (2010) and McAleer e al. (2010)). However, in his paper we adop a differen approach based on an alernaive design of opimal sraegies. Since ADIs ypically wan o maximize heir profi wihin he rules of Basel II, choosing he forecasing model of VaR ha minimizes daily capial charges while keeping he number of violaions wihin he limis of Table 1, is required. We observe ha he risk model ha minimizes daily capial charges has changed before, during and afer he global financial crisis. Since no single model is opimal over ime, we have devised as an alernaive o single models, using combinaions of hem. Combining forecasing models is common in he ime series lieraure bu i has been rarely used for forecasing VaR for risk managemen purposes (Chiriac and Pohlmeier, 2010, which was released afer he firs version of his paper, propose o combine models bu heir benchmark crieria for model comparison is differen). 1 In he 1995 amendmen (page 16), a similar capial requiremen sysem was recommended, bu specific penalies were lef o each naional supervisor. We inerpre he penaly srucure conained in Table 1 of his paper as belonging only o Basel II, and no as par of Basel I or is 1995 amendmen. 4

This paper considers marke 2 risk managemen in erms of choosing sensibly and opimally from a variey of risk models. The main conribuion of his paper is o propose combining alernaive models of marke risk for purposes of risk managemen in he conex of Basel II. We also propose some examples of combinaions of sraegies, such as conservaive and aggressive sraegies, and discuss how o choose beween hem. From a pracical perspecive, he paper also examines how he new marke risk managemen sraegies performed during he 2008-09 global financial crisis and beyond. The paper hen forecass VaR and daily capial charges for he differen marke risk managemen sraegies considered. These issues are illusraed using Sandard and Poor s 500 Composie Index. The Basel II accord has been in operaion in Europe only from 2008. The effecs of he global financial crisis should probably no be aribued o any failings of Basel II as i was no implemened in he USA, which was he epicenre of he crisis (see, for example, Cannaa and Quagliariello (2009)). 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 volailiy ha are used o forecas VaR and calculae daily capial charges, and presens aggressive and conservaive bounds on risk managemen sraegies. In Secion 4 he daa used for esimaion and forecasing are presened. Secion 5 analyses he forecas values of VaR and daily capial charges before, during and afer he 2008-09 global financial crisis, and Secion 6 summarizes he main 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 2 Marke risk is defined as he risk of losses in he on and off-balance shee posiions arising from movemens in marke prices. The risks subjec o his requiremen perain o ineres rae relaed insrumens and equiies in he rading book, and foreign exchange risk and commodiies risk hroughou he bank (Basel Commiee on Banking Supervision (2006)). 5

(3+k) for a violaion penaly, wherein a violaion involves he acual negaive reurns exceeding he VaR forecas negaive reurns for a given day: 3 DCC = sup - 3+ k VaR, - VaR (1) 60-1 where DCC = daily capial charges, which is he higher of 60 VaR = Value-a-Risk for day, - 3+ k VaR and - VaR, -1 VaR Yˆ z ˆ, VaR 60 = mean VaR over he previous 60 working days, Yˆ = esimaed reurn a ime, z = 1% criical value of he disribuion of reurns a ime, ˆ = esimaed risk (or square roo of volailiy) a ime, 0 k 1 is he Basel II violaion penaly (see Table 1). [Table 1 goes here] The muliplicaion facor 4 (or penaly), k, depends on he cenral auhoriy s assessmen of he ADI s risk managemen pracices and he resuls of a simple back es. 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 he regulaors (for a deailed 3 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 2008-09 global financial crisis, during which he Basel II Accord regulaions were no fully implemened. 4 Formula (1) is conained in he 1995 amendmen o Basel I, while Table 1 appears for he firs ime in he Basel II Accord in 2004. 6

discussion of VaR, as well as exogenous and endogenous violaions, see McAleer (2009), Jiménez-Marin e al. (2009), and McAleer e al. (2010)). 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 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 proposed 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, a violaion will also end o be associaed wih lower daily capial charges. 5 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, Y, i could be decomposed as follows: Y E( Y F ). (2) 1 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 Y, and hence is disribuion, is deermined by he variabiliy of. If i is assumed ha follows a condiional disribuion such ha: 2 ~ D(, ) where and are he condiional mean and sandard deviaion of, 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 ), (3) 1 5 The number of violaions in a given period is an imporan guidance (bu no he only one) for he regulaors o approve a given VaR model. 7

where is he criical value from he disribuion of o obain he appropriae confidence level. I is possible for o be replaced by alernaive esimaes of he 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 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 (2009) and McAleer e al. (2010). 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 As discussed previously, 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 Gaussian, Suden, and Generalized Gaussian disribuions errors, where he degrees of freedom are esimaed. These models were chosen because hey are well known and are 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 (2010). As an alernaive o esimaing he parameers, we also consider he exponenial weighed moving average (EWMA) mehod by Riskmerics TM (1996) and Zumbach, (2007) ha calibraes he unknown parameers. 8

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 +φ2 y -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, 2-1 -1 (5) and 0, 0, 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 non-saionary 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 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: 9

2 h=ω+(α+γi(η -1 ))ε -1 + βh -1, (6) where 0, 0, 0, 0 are sufficien condiions for h 0, and I ) is an indicaor variable defined by: ( I 1, 0 0, 0 (7) as has he same sign as. The indicaor variable differeniaes beween posiive and negaive shocks, so ha asymmeric effecs in he daa are capured by he coefficien. For financial daa, i is expeced ha 0 because negaive shocks have a greaer impac on risk han do posiive shocks of similar magniude. The asymmeric effec,, measures he conribuion of shocks o boh shor run persisence, 2, and o long run persisence, 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, a leas in pracice. 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 logh = ω+α +γ + βlogh, β <1-1 -1-1 h-1 h-1 (8) where he parameers, and have differen inerpreaions from hose in he GARCH(1,1) and GJR(1,1) models. EGARCH capures asymmeries differenly from GJR. The parameers and in EGARCH(1,1) represen he magniude (or size) and sign effecs of he sandardized residuals, respecively, on he condiional variance, whereas and 10 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 wo ses of 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 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), 1 would seem o be a sufficien condiion for he exisence of momens; and (v) in addiion o being a sufficien condiion for consisency, 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) Suden, 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) As an alernaive o esimaing he parameers of he appropriae condiional volailiy models, 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. 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: 2 h=λh -1 +(1- λ)ε -1 (9) 11

where is a decay parameer. Riskmerics (1996) suggess ha should be se a 0.94 for purposes of analysing daily daa. As no parameers are esimaed, here is no need o esablish any momen or log-momen condiions for purposes of demonsraing he saisical properies of he esimaors. 4. Daa The daa used for esimaion and forecasing are he closing daily prices for Sandard and Poor s Composie 500 Composie Index (S&P500), which were obained from he Ecowin Financial Daabase for he period 3 January 2000 o 8 Augus 2012. Alhough i is unlikely ha an ADI s ypical marke risk porfolio racks only he S&P500 Composie Index, which is no a raded index (unlike is opions or fuures counerpars), he S&P Composie Index 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 ( R ) are defined as: R log P / P 1. (10) [Inser Figure 1 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 2008 onward, and have coninued well ino 2009 and during he European sovereign-deb crisis, which sared in May 2010. The mean is close o zero, and he range is beween +11% and -9.5%. 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 financial reurns daa. [Inser Table 2 here] 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) 12

where F 1 is he informaion se a ime -1. [Inser Figure 2 here] Figure 2 shows he S&P500 volailiy, as he square roo of V in equaion (11). The series exhibis 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 2007. 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 2008. In less han 4 weeks in Ocober 2008, he S&P500 index plummeed by 27.1%. In less han 3 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 an almos unbelievable 42.2% o reach a low on 20 November 2008. Since he end of 2008, here have been furher significan shocks, especially hose iniiaed by he European sovereign deb crisis, which sared in May 2010. However, hese shocks have no been as grea in magniude, alhough hey may have similar long lasing effecs, as he 2008-09 global financial crisis. An examinaion of daily movemens in he S&P500 index back o 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 18 more such days. On he downside, before 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 25 such days. As menioned previously, he changes since he end of 2008 have been significan hough less severe in magniude. This comparison is beween more han 99 monhs and a shorer period of 36 monhs. During his ime span of he global financial crisis, he 4% or more gain days chances increased five imes while he chances of 4% or more loss days muliplied by 18 imes. Such movemens in he S&P500 index are unprecedened. 13

Alernaive models of volailiy can be compared on he basis of saisical significance, goodness of fi, forecasing VaR, calculaion of daily capial charges, and opimaliy on a daily or emporally aggregaed basis. As he focus of forecasing VaR is o calculae daily capial charges, subjec o appropriae penalies, he mos severe of which is emporary or permanen suspension from invesmen aciviies, he goodness of fi crierion used is he calculaion of daily and mean capial charges, boh before and afer he 2008-09 global financial crisis. 5. Forecasing VaR and Calculaing Daily Capial Charges In his secion we conduc a hypoheical exercise o analyze he performance of exising saeof-he-ar and he proposed risk managemen sraegies, as permied under he Basel II framework, when applied o he S&P500 Composie Index. Before doing so, we will discuss briefly he performance of he hree major esimaed models, namely GARCH(1,1), GJR(1,1) and EGARCH(1,1), for he full sample period. The GARCH(1,1) esimaes under he hree densiies, namely Normal, Suden and Generalized Normal, in Table 4, are similar, wih he ARCH (or alpha) effec being around 0.8, and he GARCH (or bea) effec being around 0.91, such ha he sum exceeds 0.99. Similar resuls are obained for he asymmeric GJR(1,1) model in Table 5, wih he asymmery coefficien, gamma, being significan for all hree densiies, and wih a similar order of magniude. The esimaes for EGARCH(1,1) are also similar across he hree densiies for he full sample period. [Inser Tables 4-5 here] The forecas values of VaR and daily capial charges are analysed before, during and afer he 2008-09 global financial crisis considering alernaive risk managemen sraegies. In Figure 3, VaR forecass are compared wih S&P500 reurns, where he verical axis represens reurns, and he horizonal axis represens he days from 2 January 2008 o 3 Augus 2012. The S&P500 Composie Index reurns are given as he upper blue line ha flucuaes around zero. ADIs need no resric hemselves o using only one of he available risk models. In his paper we propose a risk managemen sraegy ha consiss in choosing from among differen combinaions of alernaive risk models o forecas VaR. We firs discuss a combinaion of 14

models ha can be characerized as an aggressive sraegy and anoher ha can be regarded as a conservaive sraegy, as given in Figure 3. 6 The upper red line represens he infinum of he VaR calculaed for he individual models of volailiy, which reflecs an aggressive risk managemen sraegy, whereas he lower green line represens he supremum of he VaR calculaed for he individual models of volailiy, which reflecs a conservaive risk managemen sraegy. These wo lines correspond o combinaions of alernaive risk models. [Inser Figure 3 here] As can be seen in Figure 3, VaR forecass obained from he differen models of volailiy have flucuaed, as expeced, during he firs few monhs of 2008. I has been relaively low, a below 5%, and relaively sable beween April and Augus 2008. Around Sepember 2008, VaR sared increasing unil i peaked in Ocober 2008, beween 10% and 15%, depending on he model of volailiy considered. This is essenially 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 global financial crisis, and has remained relaively high afer he crisis, especially during he European sovereign deb crisis from May 2010. Figure 4 includes daily capial charges based on VaR forecass and he mean VaR for he previous 60 days, which are he wo lower hick 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, and he green line corresponds o he conservaive risk managemen sraegy based on he infinum of he VaR forecass of he alernaive models of volailiy 7. Before he global financial crisis, here is a subsanial difference beween he wo lines corresponding o he aggressive and conservaive risk managemen sraegies. However, a he onse of he global financial crisis, he wo lines virually coincide, which suggess ha he averaged rolling window erm in he Basel II formula, which ypically dominaes he calculaion of daily capial charges, is excessive. 6 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 an ADI o produce a combined forecas, is no forbidden by he Basel Accords, alhough i is subjec o regulaory approval. 7 Noe ha VaR figures are negaive. 15

Afer he global financial crisis had begun, here is a subsanial difference beween he wo sraegies, arising from divergence across he alernaive models of volailiy, and hence beween he aggressive and conservaive risk managemen sraegies. [Inser Figure 4 here] I can be observed from Figure 4 ha daily capial charges always exceed VaR (in absolue erms). Moreover, immediaely afer he global financial crisis had sared, a significan amoun of capial was se aside o cover likely financial losses. This is a posiive feaure of he Basel II Accord, since i can have he effec of shielding ADIs from possible significan financial losses. The Basel II Accord would seem o have succeeded in covering he marke losses of ADIs before, during and afer he global financial crisis for a porfolio ha replicaes S&P500. Therefore, i is likely o be useful when exended o counries o which i does no currenly apply. [Inser Figure 5 here] Figure 5 shows he accumulaed number of violaions for hree sraegies over a period of 250 days. Table 3 gives he percenage of days for which daily capial charges are minimized, he mean daily capial charges, he oal number of violaions, he normalized number of violaion rae (ha is, he raio of NoV*250 o number of days) and he accumulaed losses 8 (AcLoss) for he alernaive models of volailiy. The upper red line in Figure 5 corresponds o he aggressive risk managemen sraegy, which yields a normalized number of violaions of 10.24, hereby exceeding he limi of 10 in 250 working days. The lower green line corresponds o he conservaive risk managemen sraegy, which gives only 2.09. This small number of violaions is well wihin he Basel II limis and will keep he ADIs in he green zone of Table 1, bu may lead o higher daily capial charges. This conservaive sraegy may be opimal in our case, if one decides o say in he green zone. I may be useful o consider oher sraegies ha lie somewha in he middle of he previous wo, such as he DYLES sraegy, developed in McAleer e al. (2010), which seems o work well in 8 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: 1 R 1 VaR 1 if R 1 0 and R 1 VaR 1 0 he oal loss value, T 1 1. oherwise 16, a preferred VaR model is he one ha minimizes

pracice, or he median sraegy, which was found o be opimal, in a differen conex, in McAleer e al. (2011). I is also worh noing from Table 3 and Figure 6, which gives he duraion of he minimum daily capial charges for hree alernaive models of volailiy, ha wo models of risk, including he conservaive risk managemen sraegy, do no minimize daily capial charges for even one day. On he oher hand, he aggressive risk managemen sraegy minimizes he mean daily capial charge over he year relaive o is compeiors, and also has he second highes frequency of minimizing daily capial charges. The Riskmerics and EGARCH model wih disribuion errors also minimize daily capial charges frequenly. [Inser Figure 6 here] In erms of choosing he appropriae risk model for minimizing DCC, he simulaions resuls repored here would sugges he following: (1) Before he global financial crisis from 3 January 2008 o 6 June 2008, he bes model for minimizing daily capial charges is GARCH (coinciding wih he Upperbound). For he period 6 June 2008 o 16 July 2008, GJR was bes and, for only 5 days, EGARCH was he bes. This is a period wih relaively low volailiy and few exreme values. (2) Riskmerics is he bes model during he beginning of he crisis, from 16 July 2008 o 15 Sepember 2009. The S&P500 reached a peak on 12 Augus 2008, afer which i sared o decrease. In he second half of Sepember 2008, he volailiy on reurns began o increase considerably. (3) From 24 Sepember 2008 o he end of 2009, he bes model was EGARCH_T. This is a period wih considerably high volailiy and a large number of exreme values of reurns. EGARCH can capure asymmeric volailiy, hereby providing a more accurae measure of risk during large financial urbulence. (4) During he res of he sample, he Upperbound seems o be he sraegy ha minimizes he DCC mos of he days, bu no he only one. The global financial crisis has affeced he bes risk managemen sraegies by changing he opimal model for minimizing daily capial charges. Here we proposed combinaions of models o accommodae his siuaion. Our resuls sugges ha some of hese combinaions migh have 17

provided adequae coverage agains marke risk of a porfolio ha replicaes S&P500, during he period 2008-09, which includes he global financial crisis. 6. Conclusion Alernaive risk models were found o be opimal in erms of minimizing daily capial charges before and during he global financial crisis. Volailiy increased four-fold during he 2008-09 global financial crisis and he European sovereign deb crisis saring from May 2010, and remained relaively high afer he crisis, as illusraed using he S&P500 Composie Index. As he risk model ha opimizes daily capial charges has been changing during ha period, his suggess ha, as in ime series, he forecass of VaR could be improved using a combinaion of models raher han a single model. In his paper we proposed he idea of consrucing risk managemen sraegies ha used combinaions of several models for forecasing VaR. I was found ha, in our case, an aggressive risk managemen sraegy yielded he lowes mean capial charges, and had he highes frequency of minimizing daily capial charges hroughou he forecasing period, bu which also ended o violae oo ofen. Such excessive violaions can have he effec of leading o unwaned publiciy, and emporary or permanen suspension from rading as an ADI. On he oher hand, a conservaive risk managemen sraegy would have far fewer violaions, and a correspondingly higher mean daily capial charges. This sraegy will be he preferred one if he ADIs wan o say in he green zone of he Basel II Accord penalies. The area beween he bounds provided by he aggressive and conservaive risk managemen sraegies would seem o be a ferile area for fuure research. A risk managemen sraegy ha used combinaions of alernaive risk models for predicing VaR and minimizing daily capial charges, namely he median, was found o be opimal in McAleer e al. (2012a, 2012b). A risk model ha uses he DYLES sraegy esablished in McAleer e al. (2010) may also be a useful risk managemen sraegy. The recommended policy changes o pracice by ADIs are sraighforward as he mehods suggesed in his paper are pracical, are simple o undersand and implemen, are easy o monior and regulae, are leads o accurae forecass, in general. 18

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Table 1: Basel Accord Penaly Zones Zone Number of Violaions k Green 0 o 4 0.00 Yellow 5 0.40 6 0.50 7 0.65 8 0.75 9 0.85 Red 10+ 1.00 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. 22

Table 2. Descripive Saisics for S&P500 Reurns 3 January 2000 03 Augus 2012 1,400 1,200 1,000 800 600 400 200 0-10 -8-6 -4-2 0 2 4 6 8 10 Series: S&P500 Reurns Sample 3/01/2000 3/08/2012 Observaions 3283 Mean -0.000184 Median 0.017670 Maximum 10.95792 Minimum -9.469733 Sd. Dev. 1.340072 Skewness -0.154533 Kurosis 10.57462 Jarque-Bera 7861.464 Probabiliy 0.000000 23

Table 3. Percenage of Days Minimizing Daily Capial Charges, Mean Daily Capial Charges, Number of Violaions, Normalized Number of Violaions and Accumulaed lossses for for Alernaive Models of Volailiy MODEL % of days minimizing DCC Mean DCC NoV Norm. NoV AcLoss RSKM 18.4 % 12.10 31 6.48 21.30 GARCH 1.0% 12.02 35 7.32 21.84 GJR 3.0% 11.82 33 6.90 18.18 EGARCH 11.7% 11.35 41 8.57 28.11 GARCH_ 0.0% 12.97 10 2.09 7.60 GJR_ 12.1% 12.38 14 2.93 8.08 EGARCH_ 15.8% 11.76 17 3.55 11.23 GARCH_g 5.9% 12.38 21 4.39 13.58 GJR_g 9.3% 11.45 20 4.18 12.67 EGARCH_g 6.2% 11.75 27 5.64 19.09 Lowerbound 0.0% 13.43 10 2.09 5.88 Upperbound 16.7% 11.14 49 10.24 33.29 24

Table 4 GARCH(1,1) Esimaes Densiy Parameer All Sd. Error Normal + 0.083** 0.0068 0.909** 0.0072 0.992 Densiy Parameer All Sd. Error Suden- (6.85) + 0.083** 0.0098 0.914** 0.0093 0.998 Densiy Parameer All Sd. Error Generalized Normal + 0.083** 0.0105 0.913** 0.0105 0.996 Noes: All denoes he full sample period. The enries in parenheses for he Suden- disribuion are he esimaed degrees of freedom. ** These esimaes are saisically significan a he 1% level. 25

Table 5 GJR(1,1) Esimaes Densiy Parameer All Sd-error Normal + + /2-0.024** 0.0054 0.152** 0.0102 0.9366** 0.0059 0.989 Densiy Parameer All Sd-error Suden- (8.37) + + /2-0.027** 0.0075 0.154** 0.0138 0.940** 0.0070 0.990 Densiy Parameer All Sd-error -0.026** 0.0081 Generalized Normal 0.153** 0.0147 0.938** 0.0080 + + /2 0.989 Noes: All denoes he full sample period. The enries in parenheses for he Suden- disribuion are he esimaed degrees of freedom. ** These esimaes are saisically significan a he 1% level. 26

Table 6 EGARCH(1,1) Esimaes Densiy Parameer All Sd-error Normal 0.101** 0.0107-0.123** 0.0078 0.982** 0.0017 Densiy Parameer All Sderror 0.096** 0.0139 Suden- (7.72) -0.129** 0.0108 0.987** 0.0021 Densiy Parameer All Before 0.101** 0.0156 Generalized Normal -0.128** 0.0114 0.986** 0.0024 Noes: All denoes he full sample period. The enries in parenheses for he Suden- disribuion are he esimaed degrees of freedom. ** These esimaes are saisically significan a he 1% level. 27

Figure 1. Daily Reurns on he S&P500 Index 3 January 2000 3 Augus 2012 12% 8% 4% 0% -4% -8% -12% 00 01 02 03 04 05 06 07 08 09 10 11 12 28

Figure 2. Daily Volailiy in S&P500 Reurns 3 January 2000 3 Augus 2012 12% 10% 8% 6% 4% 2% 0% 00 01 02 03 04 05 06 07 08 09 10 11 12 29

Figure 3. VaR for S&P500 Reurns 2 January 2008 3 Augus 2012 12% 8% 4% 0% -4% -8% -12% -16% 2008 2009 2010 2011 2012 S&P500 Reurns EGARCH_T GARCH_T GJR_T RSKM Lowerbound Upperbound Noe: The upper blue line represens daily reurns for he S&P500 index. The upper red line represens he infinum of he VaR forecass for he differen models described in Secion 3. The lower green line corresponds o he supremum of he forecass of he VaR for he same models. 30

Figure 4. VaR and Mean VaR for he Previous 60 Days o Calculae Daily Capial Charges for S&P500 Reurns, 3 January 2008-3 Augus 2012 20% 10% 0% -10% -20% -30% -40% -50% 2008 2009 2010 2011 2012 S&P500 Reurns VAR_EGARCH_T VAR_GARCH_T VAR_GJR_T VAR_RSKM VAR_Lowerbound VAR_Upperboudn DCC_EGARCH_T DCC_GARCH_T DCC_GJR_T DCC_RSKM DCC_Lowerbound DCC_Upperbound 31

Figure 5. Number of Violaions Accumulaed Over 250 Days, 3 January 2008-3 Augus 2012 16 14 12 10 8 6 4 2 0 2008 2009 2010 2011 2012 RSKM Lowerbound Upperbound 32

Figure 6. Duraion of Minimum Daily Capial Charges for Alernaive Models of Volailiy, 3 January 2008-3 Augus 2012 Lehman Brohers bankrupcy Sepember 15, 2008 European sovereign-deb crisis May 9, 2010 1 0 2008 2009 2010 2011 2012 RSKM (18%) EGARCH_T (16%) Upperbound (17%) Noe: One in he figura means ha he model minizmizes DCC in ha day. 33