It Pays to Violate: Model Choice and Critical Value Assumption for Forecasting Value-at-Risk Thresholds

Transcription

1 I Pays o Violae: Model Choice and Criical Value Assumpion for Forecasing Value-a-Risk Thresholds Bernardo da Veiga, Felix Chan and Michael McAleer School of Economics and Commerce, Universiy of Wesern Ausralia Keywords: Value-a-Risk, GARCH, Risk Managemen, Forecasing, Basel Accord penalies. EXTENDED ABSTRACT The inernals models amendmen o he Basel Accord allows banks o use inernal models o forecas Value-a-Risk (VaR) hresholds which are used o calculae he required capial banks mus hold in reserves as a proecion agains negaive changes in he value of heir rading porfolios. As capial reserves lead o an opporuniy cos o banks i is likely ha banks could be emped o use models ha underpredic risk and hence lead o low capial charges. In order o avoid his problem he Basel Accord inroduced backesing procedure whereby banks using models ha led o excessive violaions would be penalised hrough higher capial chares. This paper invesigaes he performance of five popular volailiy models ha can be used o forecas VaR hresholds under a variey of disribuional assumpions. The resuls sugges ha wihin he curren consrains and penaly srucure se ou in he Basel Accord he lowes capial charges arise when using models ha lead o excessive violaions, suggesing he curren penaly srucure is no severe enough. 304

2 Inroducion On June 6, 974 Hersa, a German bank, had received large paymens of DEM in Frankfur in exchange for USD paymens ha were o be made in New York laer ha day due o ime zone differences. However, before he USD paymens were made, Hersa was forced ino liquidaion by German regulaors. The Hersa fiasco led he G-0 counries o form a commied called he Basel Commiee on Banking Supervision which was iniially inended o deal wih he role of regulaors in cross-jurisdicional siuaions and invesigae ways of harmonizing inernaional banking regulaions. In 988 he Basel Commiee issued he Basel Capial Accord, which prescribe minimum capial requiremens ha Auhorized Deposi Taking Insiuions (ADI s) mus mee as a proecion agains credi risk, his became law in all G-0 counries by 99 wih he excepion of Japan ha was graned an exended ransiion period. In 993 he Basel Accord was amended o require ADI s o also hold capial in reserve agains marke risk based on he Value-a-Risk (VaR) approach. Value-a-Risk is a procedure designed o forecas he maximum expeced loss over a arge horizon, given a saisical confidence limi (see Jorion (000) for a deailed discussion of VaR mehods). Iniially, he Basel Accord sipulaed a sandardized approach which all insiuions were required o adop in calculaing heir VaR hresholds. This approach suffered from several deficiencies, he mos noable of which were is assumpion of no diversificaion benefis which led o conservaism (or los opporuniies) and is failure o reward insiuions wih superior risk managemen experise. In view of hese drawbacks a furher amendmen, called he Marke Risk amendmen, was proposed in 995 and subsequenly adoped in 996. The Marke Risk amendmen o he Basel accord allows Auhorized Deposi Taking Insiuions (ADI s) o use inernal models o measure and forecas marke risk The forecased marke risk, or volailiy, forms a basis for he calculaion of he Value-a-Risk. However, in order o mainain discipline and ensure ha ADI s have in place adequae models of marke risk a backesing procedure is used o coun he number of imes he acual losses exceeded he forecased VaR over he previous 50 business days. As VaR models are designed o provide 99% coverage (or lead o violaions % of he ime) he Basel accord specifies penalies ha increase he required capial charge if oo many violaions are deeced. A hree-zone approach is used o measure he accuracy of he forecasing model as shown in able. ADI s ha fall in he Green zone are deemed o have models ha are adequaely accurae and do no incur penalies from regulaors. Once in he Yellow zone regulaors will impose a penaly which will increase he required capial charge and will be required o jusify he excessive number of violaions, he greaer he number of violaion he more likely i is ha ADI s will be penalized and required o revise heir model. Finally, once an ADI eners he Red zone he model used is deemed o be unaccepably inaccurae and he ADI will be required o adop a more sringen model ha will lead o fewer violaions and larger capial charges. Wihin he consrains of he Basel accord ADI s should choose he model ha leads o he lowes possible capial charge, condiional on he model no leading o he ADI falling in he Red zone (or upper Yellow zone). Such approach will ensure ha he opporuniy cos associaed wih capial charges are minimized while maximizing he benefis associaed wih minimal regulaory inervenion, furhermore ADI s ha have good risk managemen sysems in place will benefi from a superior repuaion, lowed cos of deb and perhaps sronger demand for is deposi faciliies. Table : Basel Accord Penaly Zones Violaions Increase in Zone k Green 0 o Yellow

3 Red Noe: The number of violaions is given for 50 business days. 0 5 Figure : S&P500 Reurns This paper evaluaes he VaR forecasing performance of five popular condiional volailiy models namely he ARCH model of Engle (98), GARCH model of Bollerslev (986), EGARCH model of Nelson (99), he GJR model of Glosen Jagannahan and Runkle (993) and he resriced EWMA model proposed by Riskmerics TM (996) which is he indusry sandard (see McAleer (005) for a deailed discussion of condiional volailiy models). For he purpose of forecasing VaR hresholds i is necessary o assume a disribuion for he reurns so ha he appropriae criical values can be chosen. In his paper hree disribuional assumpions are made namely ha reurns are normally disribued, ha he reurns follow a - disribuion where he appropriae degrees of freedom are esimaed and ha he reurns follow a generalised error disribuion (GED) where he appropriae GED parameer is also esimaed. Finally, as an alernaive o assuming a paricular disribuion he criical values are also obained hrough boosrapping. Daa The daa used in his paper is a long series of he S&P500 index daily reurns ranging from 4 January 986 o 8 March 005. The S&P500 index was chosen as i is commonly regarded as he indusry proxy for US sock marke performance. Figure plos he S&P500 index reurns for he period. As can be seen he series displays considerable clusering ha needs o be capured by an appropriae model. Figure plos he hisogram of reurns and gives he descripive saisics. The S&P500 has a mean reurn of 0.035%, maximum of 8.709% and minimum of -.833% which occurred during he 987 sock marke crash. Furhermore, he series is negaively skewed, has exremely high excess kurosis and he Jarque-Bera saisic srongly rejecs he null hypohesis of normaliy Figure : S&P500 Reurns Hisogam and Descripive Saisics Models. EWMA Series: S&P500 Sample 4/0/986 8/03 /005 Observaions 500 Mean Median Maximum Minimum Sd. Dev Skewness Kurosis Jarque-Bera Probabiliy Riskmerics TM (996) 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 (98). This approach forecass he condiional variance a ime as a linear combinaion of he lagged condiional variance and he squared uncondiional shock a ime. The EWMA model calibraes he condiional variance as: h = λh + ( λ) ε () where λ is a decay parameer. Riskmerics TM (996) suggess ha λ should be se a 0.94 for purposes of analysing daily daa. 306

4 .3 ARCH Engle (98) proposed he Auoregressive Condiional Heeroskedasiciy of order p, or ARCH( p ), model as follows: p j j j = h = ω+ α ε. () p q = ω+ α jε j + γ ( η ) ε + β i i j= i= (4) h I h where he indicaor variable, I( η ), is defined as: {, ε 0 0, ε > 0 I( η ) =. (5) For he case p =, ω > 0, α > 0 are sufficien condiions o ensure a sricly posiive condiional variance, h > 0. The ARCH (or α ) effec capures he shor run persisence of shocks..4 GARCH Bollerslev (986) generalized ARCH( p ) o he GARCH(, pq) model, which is given by: p q = ω+ α jε j + βh i j= i= h. (3) For he case p =, ω > 0, α > 0, β 0 are sufficien condiions o ensure a sricly posiive condiional variance, h > 0. The ARCH (or α ) effec capures he shor run persisence of shocks, and he GARCH (or β ) effec indicaes he conribuion of shocks o long run persisence α + β ). ( In ARCH and GARCH models, he parameers are ypically esimaed using he maximum likelihood esimaion (MLE) mehod. In he absence of normaliy of he sandardized residuals, η, he parameers are esimaed by he Quasi-Maximum Likelihood Esimaion (QMLE) mehod (see, for example, Li, Ling and McAleer (00))..5 GJR Glosen, Jagannahan and Runkle (99) 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 debequiy raio. The GJR( p, q ) model is given by: For he case p =, ω > 0, α > 0, α+ γ > 0, β 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 > 0 ) measures he conribuion of shocks o boh shor run persisence ( α / + γ ) and long run persisence ( α + β + γ /). Several imporan heoreical resuls are relevan for he GARCH model. Ling and McAleer (00a) esablished he necessary and sufficien condiions for sric saionariy and ergodiciy, as well as for he exisence of all momens, for he univariae GARCH( pq), model, and Ling and McAleer (003) demonsraed ha he QMLE for GARCH( pq), is consisen if he second momen is finie, E( ε ) <, and asympoically normal if he fourh momen is 4 finie, E( ε ) <. The necessary and sufficien condiion for he exisence of he second momen of ε for he GARCH(,) model is α+ β <. Anoher imporan resul is ha he log-momen condiion for he QMLE of GARCH(,), 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 (995) and Jeanheau (998) for consisency, and in Boussama (000) for asympoic normaliy. In pracice, i is more sraighforward o verify he second momen condiion han he weaker logmomen 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 307

5 he exisence of he second momen of GJR(,) is α + β γ / + < (see Ling and McAleer (00b)). Moreover, he weak log-momen condiion for GJR(,), E(log[( α + γi( η)) η + β]) < 0, is sufficien for he consisency and asympoic normaliy of he QMLE (see McAleer, Chan and Marinova (00))..6 EGARCH Nelson (99) proposed he Exponenial GARCH (EGARCH) model, which is given as: ε log( h ) = +. p i ω αi i= h i r q ε k γk β j h j k= h k j= + + log( ) (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, 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 (005)). 4 Forecass A rolling window approach is used o forecas he -day ahead % VaR hresholds using he five condiional volailiy models described in Table : Forecas Resuls GARCH Disribuional Assumpion Normal -dis GED Boosrap Violaion 47 3 Capial Charge Proporion of Time Spen ou of he 4% 0% 9% 6% Risk Merics Disribuional Assumpion Normal -dis GED Boosrap Violaion Capial Charge Proporion of Time Spen ou of he 57% 0% 9% % EGARCH Disribuional Assumpion Normal -dis GED Boosrap Violaion Capial Charge Proporion of Time Spen ou of he 35% 0% 5% 8% ARCH Disribuional Assumpion Normal -dis GED Boosrap Violaion Capial Charge Proporion of Time Spen ou of he 67% 3% 4% 4% GJR Disribuional Assumpion Normal -dis GED Boosrap Violaion Capial Charge Proporion of Time Spen ou of he 30% 0% % 5% Noes: ) The daily capial charge is given as he negaive of he higher of he previous day s VaR or he average VaR over he las 60 business days imes (3+k), where k is he penaly. ) The expeced number of violaions is 30 a he % level 308

6 secion. A rolling window approach is one where he firs n observaions are used o esimae he model and forecas he n h + observaion. he sample is hen rolled forward by observaion so ha i ranges from he nd o he nh+ observaion and he n h + observaion s forecased. This process is repeaed unil he end of he sample. In order o srike a balance beween efficiency in esimaion and a viable number of forecass a rolling window size of 000 observaions is chosen, which leaves 300 observaions o be forecased. Table. gives he resuls of he forecasing exercise.as can be seen assuming a - disribuion always leads o he lowes number of violaions and he highes average capial charge, while assuming a normal disribuion always leads o he highes number of violaions and he lowes average capial charge. I is ineresing o noe ha he EGARCH model leads o he lowes average capial charge across all disribuional assumpions, while he ARCH model leads o he highes capial charge in all cases excep where he criical values are obained hrough boosrapping. I is ineresing o noe ha difference in VaR forecasing performance is much greaer across he various disribuional assumpions for a given model, han across he various models for a given disribuional assumpion. This resul suggess ha he disribuional assumpion is more imporan han he choice of condiional volailiy model. Finally, as can be seen assuming a -disribuion always leads o he lowes number of violaions and he highes average capial charge, while assuming a normal disribuion always leads o he highes number of violaions and he lowes average capial charge. As he VaR hresholds are esimaed assuming a 99% confidence level and here are 300 forecass he expeced number of violaions is approximaely 30. The resuls hen sugges ha he assumpion of normaliy is inadequae, as i leads o more violaions han could be reasonably expeced; while he assumpion ha he reurns follow a - disribuion is also inadequae as i leads o far fewer violaions han could be reasonably expeced. The resuls sugges ha assuming he reurns follow a GED disribuion or boosrapping he criical values lead o VaR hreshold forecass ha yield he correc number of violaions in mos cases. Wihin he framework se ou in he Basel Accord banks should choose he VaR model ha leads o he Lowes capial charge, while no yielding backesing resuls ha fall in he red zone. Ou of all he model/criical value combinaions considered in his paper only he ARCH-normal, ARCH-GED, ARCH-boosrap and GJR-normal lead o backesing resuls ha fall in he red zone over he enire forecasing period. Hence all oher models saisfy he Basel Accord consrains and hence are eligible o be used for he purpose of calculaing he banks capial charges. Two imporan observaions arise from his sudy. Firs, across all model/criical value combinaions ha saisfy he Basel Accord he EARCH-normal model gives he lowes daily capial charge a 7.536%, while he ARCH- gives he highes a.54%, suggesing ha he capial charges can be significanly reduced by choosing he appropriae model/criical value combinaion. Second, for each model considered he assuming a -disribuion always leads o he highes capial charges, while assuming ha he reurns follow a normal disribuion always leads o he lowes capial charges. This resuls has serious implicaions for regulaors as i suggess ha given he curren penaly srucure proposed in he Basel Accord, banks have an incenive o choose models ha lead o excessive violaions. 5 Conclusion This paper analysed he performance of five popular condiional volailiy models in forecasing VaR hresholds. The Basel Accord sipulaes ha banks mus hold capial in reserves o cover heir exposure o marke risk, or he risk ha a banks porfolio will experience a severe negaive reurn. The need for banks o have in place adequae risk managemen sysems sems from he inrinsically sysemic naure of he banking indusry, where bank failures can quickly spread and harm he enire financial sysem. Originally he marke risk amendmen se ou a sandardised model ha all banks were required 309

7 o use when calculaing heir VaR hresholds. This model was heavily criicised by indusry paricipans as being oo conservaive and hence leading high capial charges. Furhermore i was argued ha his approach did no reward insiuions wih superior risk managemen and did no promoe research ino more sophisicaed VaR models. The inernal modes amendmen o he Basel accord was inended o allow bans o use inernal models, provided a series of quaniaive and qualiaive crieria were me. An obvious concern of regulaors was ha his amendmen would encourage banks o pick models ha underprediced risk and hence led o lower capial charges han models ha correcly prediced risk. Hence a backesing procedure was developed o assess he performance of each model and o penalise models ha underprediced risk hrough higher capial charges. The resuls sugges ha he EGARCH model dominaes all oher models, as i gives he lowes capial charge while never enering he red zone. The ARCH model is always he wors performing model, giving he highes capial charge and almos always falling in he red zone. Of all he disribuional assumpions, he assumpion of normaliy always leads o he mos number of violaions, which are much higher han expeced given he confidence level chosen, and he lowes capial charges. While assuming a -disribuion always leads o he lowes number of violaions, which are much lower han expeced given he confidence level chosen, and he highes capial charges. When he criical values are obained hrough boosrapping or he assumpion ha he reurns follow a Generalised Error disribuion he resuls lead o he correc number of violaions and capial chares ha are higher han under he assumpion of normaliy bu lower han under he assumpion ha he reurns follow a - disribuion. These resuls sugges ha he penaly srucure proposed under he Basel Accord is no severe enough o discourage banks from choosing models and criical values ha clearly underpredic risk. 6 Acknowledgemens The firs auhor acknowledges a Universiy Posgraduae Award and an Inernaional Posgraduae Research Scholarship a he Universiy of Wesern Ausralia. The second and hird auhors are graeful for he financial suppor of he Ausralian Research Council. References Basel Commiee on Banking Supervision, (988), Inernaional Convergence of Capial Measuremen and Capial Sandards, BIS, Basel, Swizerland. Basel Commiee on Banking Supervision, (995), An Inernal Model-Based Approach o Marke Risk Capial Requiremens, BIS, Basel, Swizerland. Basel Commiee on Banking Supervision, (996), 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. (986), Generalised Auoregressive Condiional Heeroscedasiciy, Journal of Economerics, 3, Engle, R.F. (98), Auoregressive Condiional Heeroscedasiciy wih Esimaes of he Variance of Unied Kingdom Inflaion, Economerica, 50, Glosen, L.R., R. Jagannahan, and D.E. Runkle (99), On he Relaion beween he Expeced Value and Volailiy of he Nominal Excess Reurn on Socks, Journal of Finance, 46, Jorion, P. (000), Value a Risk: The New Benchmark for Managing Financial Risk, McGraw-Hill, New York. McAleer, M. (005), Auomaed Inference and Learning in Modeling Financial Volailiy, Economeric Theory,,

8 Nelson, D. (99), Condiional Heeroskedasiciy in Asse Reurns: a New Approach, Economerica, 59, Riskmerics TM (996), J.P. Morgan Technical Documen, 4 h ediion, New York. J.P. Morgan. 3