Modeling Risk: VaR Mehods for Long and Shor Trading Posiions Savros Degiannakis Deparmen of Saisics, Ahens Universiy of Economics and Business, 76, Paision sree, Ahens GR-14 34, Greece Timoheos Angelidis Deparmen of Banking and Financial Managemen, Universiy of Piraeus, 8, Karaoli & Dimiriou sree, Piraeus GR- 185 34, Greece Ahens Laboraory of Business Adminisraion. Ahinas Ave. & a Areos sree, Vouliagmeni GR- 166 71, Greece Absrac The accuracy of parameric, non-parameric and semi-parameric mehods in predicing he one-day-ahead Value-a-Risk (VaR) of perfecly diversified porfolios in hree ypes of markes (sock exchanges, commodiies and exchange raes) is invesigaed, boh for long and shor rading posiions. The risk managemen echniques are designed o capure he main characerisics of asse reurns, such as lepokurosis and asymmeric disribuion, volailiy clusering, asymmeric relaionship beween sock reurns and condiional variance and power ransformaion of condiional variance. Based on backesing measures and a loss funcion evaluaion mehod, we find ou ha he modeling of he main characerisics of asse reurns produces accurae VaR forecass. Especially for he high confidence levels, a risk manager mus employ differen volailiy echniques in order o forecas he VaR for he wo rading posiions. Keywords: Asymmeric Power ARCH model, Skewed- Disribuion, Value-a-Risk, Volailiy Forecasing. JEL: C3, C5, C53, G15. Tel.: +3-1-83-1. E-mail address: sdegia@aueb.gr. The usual disclaimer applies Tel.: +3-1-8964-736. E-mail address: aggel@unipi.gr. 1 of 11
1. Inroducion Value-a-Risk (VaR) a a given probabiliy level a, is he prediced amoun of financial loss of a porfolio over a given ime horizon. Given he fac ha he asse reurns are no normally disribued, since hey exhibi skewness and excess kurosis, i is plausible o employ volailiy forecasing echniques ha accommodae hese characerisics. The one-sep-ahead volailiy of daily reurns is esimaed by a se of Auoregressive Condiional Heeroskedasiciy (ARCH) models (assuming four condiional variance specificaions and hree disribuion assumpions), hisorical and filered-hisorical simulaions and he commonly used variance-covariance mehod. Our sudy sheds a ligh on he volailiy forecasing mehods under a risk managemen framework, since i juxaposes he performance of he mos well known echniques for differen markes (sock exchanges, commodiies and exchange raes) and rading posiions. Alhough, he normal disribuion produces adequae one-day-ahead VaR forecas a he 95% confidence level, models ha parameerise he leverage effec for he condiional variance, he lepokurosis and he asymmery of he daa, forecas accurae he VaR a he 99% confidence level. Moreover, shorrading posiions should be modelled using volailiy specificaions differen o ha of porfolios wih long rading posiions. The volailiy forecasing models and he VaR evaluaion mehods are presened in he nd and he 3 rd secions, respecively. The fourh secion illusraes he resuls of he sudy and he fifh secion concludes.. The Volailiy Forecasing Models Le ln( P P ) y 1 1 denoes he daily reurn series, where a day. The ARCH models can be presened in he following general framework: P is he price of an asse σ g z i. i ~. d. + ε ε z σ (,1) ({ ε }{ ; σ } i 1, j 1), i y c f j (1) of 11
where c is a consan parameer, ε is he innovaion process, (,1) f is a densiy funcion of zero mean and uni variance, and g (.;.) is a funcional form of he pas innovaions and heir condiional sandard deviaion. Surveys of Bollerslev e al. (1994), Li e al. (1), Poon and Granger (3), Degiannakis and Xekalaki (4) cover a wide range of ARCH presenaions. Bollerslev (1986) proposed a generalizaion of Engle s ARCH model and inroduced he GARCH(1,1) specificaion: where α >, a 1 and b 1 σ, () a + a1ε 1 + b1σ 1. RiskmericsTM suggesed he exponenially weighed moving average, or EWMA, which is a special case of he GARCH(1,1), since a, a. 1 6 and b.94 1 : σ. (3).6ε 1 +.94σ 1 Alhough he GARCH(1,1) model capures he volailiy clusering phenomenon, i could no explain he asymmeric relaionship beween reurns and condiional variance. Nelson (1991) proposed he exponenial GARCH, or EGARCH(1,1), model: ln ( σ ) a + a ( ε σ E ε σ ) + γ ( ε σ ) b ln( σ ) 1 1 1 1 1 1 1 + 1 1, (4) where he parameer γ 1 accommodaes he asymmeric effec. Glosen e al. (1993) presened he TARCH(1,1) specificaion, where good news ( ε > ) and bad news ( < ) effec on he condiional variance: i ( a + γd ) ε b σ a + 1 1 + 1 1 ε have differen σ, (5) i for d denoing an indicaor funcion (i.e. d 1 if ε and d oherwise). Ding e al. 1 < (1993) inroduced he asymmeric power ARCH, or APARCH(1,1), model: δ δ ( ε γε ) b σ δ σ + a1 1 1 + 1 1 for a >, a 1, b 1, δ and γ < 1. a, (6) In he influenial paper of Engle (198), he densiy funcion of z, f (.), was considered as he sandard normal disribuion: 3 of 11
f 1 z ( ) z exp π. (7) Bollerslev (1987) proposed he Suden- disribuion in order o produce an uncondiional disribuion wih hicker ails: f ( z ; v) Γ Γ( ( v + 1) ) ( v ) π ( v ) z 1+ v v+ 1, v >, (8) where v denoes he degrees of freedom of he disribuion. Lamber and Lauren () suggesed ha no only he condiional disribuion of innovaions may be lepokuric, bu also asymmeric and proposed he skewed Suden- densiy funcion: f ( z ; v, g) Γ Γ( ( v + 1) ) ( v ) π ( v ) s g + g where g is he asymmery parameer, (.) oherwise, Γ( ( v 1) ) ( v ) Γ( v ) 1 sz + m 1+ g v d v+ 1, v >, (9) Γ is he gamma funcion, d 1 if z m / s, d 1 1 1 ( ) ( g ) m π g and s g + g m 1 are he mean and he variance of he non-sandardized skewed Suden- disribuion, respecively. as: Under he framework of he parameric echniques, he one-day-ahead VaR is compued ( z; a) + 1 VaR + σ, (1) 1 F ˆ where ( z a) F ; is he corresponding quanile of z disribuion and ˆ +1 σ is he one-day-ahead condiional sandard deviaion forecas given he informaion ha is available a ime. Under he i. ~ i. d N assumpion ha ε (,1), he calculaion of he VaR can be simplified: ( a) 1 VaR F ˆ + 1 ; σ + ε. (11) However, he conjecure of normaliy is no saisfied in acual financial reurns and, hence, his mehod, which we will refer o as Variance-Covariance (VC), usually underesimaes he rue VaR. 4 of 11
The Hisorical Simulaion (HS) mehod is a simple and inuiive non-parameric procedure, which relies on acual hisorical reurns o calculae he VaR as he corresponding percenile of he pas m reurns 1 : m ( y } a) VaR F { 1-1;. (1) + 1 + τ τ In he case of he parameric mehods, he disribuion choice is crucial; while in he nonparameric case here is no consisen approach in forecasing he volailiy. The Filered Hisorical Simulaion (FHS) mehod, which was presened in Hull and Whie (1998) and Barone-Adesi e al. (1999), combines he wo approaches in order o make he mos of hem. Given an adequae volailiy model, such as a GARCH(1,1), he VaR + 1 is compued based on he quanile of he sandardized innovaions: m ( ˆ ε ˆ 1 σ 1 } 1; a) 1 VaR F ˆ + 1 { + τ + -τ τ σ + -. (13) 3. Evaluae he Forecasing Abiliy of Value a Risk Measures We employ a wo-sage procedure o evaluae he various risk managemen echniques. Two backesing crieria (uncondiional and condiional coverage) are implemened for examining he saisical accuracy of he models while, in a second sage, we employ a forecas evaluaion mehod o invesigae wheher he differences beween he VaR models, ha exhibied sufficien uncondiional and condiional coverage, are saisically significan. The simples mehod in deermining he adequacy of a VaR measure is o es he hypohesis ha he proporion of violaions is equal o he expeced one. Kupiec (1995) developed a likelihood raio saisic: LR uc N T -N N T-N N ln[1 - ) ( ) N] - ln[(1- p) p ] ~ X 1, (14) T T under he null hypohesis ha he observed excepion frequency, N / T, equals o he expeced one, p, where N is he number of days over a period T ha a violaion has occurred. Alhough his es can rejec a model ha eiher overesimaes or underesimaes he rue bu unobservable VaR, i canno examine wheher he violaions are randomly disribued hrough ime. 1 For more informaion abou HS mehod see Hendricks (1996), Van den Goorbergh and Vlaar (1999) and Daníelsson () among ohers. A violaion occurs if he prediced VaR is no able o cover he realized loss. 5 of 11
Chrisoffersen (1998) developed a condiional coverage es, which joinly invesigaes wheher i) he oal number of failures is equal o he expeced one and ii) he VaR violaions are independenly disribued hrough ime. Under he null hypohesis ha he failure process is independen and he expeced proporion of violaions is equal o p, he appropriae likelihood raio is: LR cc n π 1 1 11 T-N N n 1 n1 11 ln[(1 - p) p ] + ln[(1 - ) π (1 -π ) π X, (15) n 11 ] ~ where n is he number of observaions wih value i followed by j, for i, j, 1 and ij nij π ij are he corresponding probabiliies. i, j 1 denoes ha a violaion has been made, n j ij while i, j indicaes he opposie. Conrary o Kupiec's (1995) es, Chrisoffersen s procedure can rejec a VaR model ha generaes oo many or oo few clusered violaions. Lopez (1999) proposed o marke praciioners and regulaory auhoriies a procedure of evaluaing VaR models based on a loss funcion approach. According o he Basle Commiee on Banking Supervision (1996) proposal, he incorporaed boh he oal number of violaion and heir magniude erm. More formally, Lopez s loss funcion can be described as: Ψ + 1 1+ ( VaR + 1 - y ) + 1 if violaion else. occurs (16) Based on Diebold and Mariano (1995) and Sarma e. al. (3), we examine wheher he forecas accuracy of wo VaR models is saisically significan. Specifically, we es he null hypohesis of equivalen predicive abiliy of models A and B, agains he alernaive hypohesis ha model A is superior o model B. The Diebold-Mariano saisic is he "-saisic" for a regression of z on a consan wih heeroskedasic and auocorrelaed consisen sandard errors (HAC), where z A B Ψ - Ψ, and A Ψ and B Ψ are he loss funcions of models A and B respecively. A negaive value of z indicaes ha model A is superior o model B. 4. Empirical Resuls The framework in (1) is esimaed for (), (4), (5) and (6) condiional variance specificaions and (7) o (9) densiy funcions by maximum likelihood mehod. The EWMA model, he variance- 6 of 11
covariance procedure and he echniques of hisorical and filered hisorical simulaion are applied, giving a oal of 16 volailiy-forecasing models. We generae ou-of-sample VaR forecass for wo equiy indices (S&P5, FTSE1), wo commodiies (Gold Bullion $/Troy Ounce, London Bren Crude Oil Index U$/BBL) and wo exchange raes (US $ o Japanese, US $ o UK ), obained from Daasream for he period of January 3 rd 1989 o June 3 h 3. For all models, we use a rolling sample of observaions in order o calculae he 95% and he 99% VaR + 1 for long and shor rading posiions. Under he framework of he loss funcion approach, we evaluae all he models wih p-value greaer han 1% for boh uncondiional and condiional coverage ess. A high cu-off poin is preferred in order o ensure ha he successful risk managemen echniques will no a) over or under esimae saisically he rue VaR and b) generae clusered violaions. In he case of a smaller cu-off poin, an incorrec model could no be easily rejeced, which migh urn o be cosly for a risk a manager. Table 1 summarizes he wo-sage model selecion procedure 3. In he firs sage (columns and 3) he models ha have no been rejeced by he saisical backesing procedures are presened, while in he second sage (column 4), he volailiy mehods ha are preferred over he ohers, based on he loss funcion approach, are exhibied. For example, in panel A, for he S&P5 index, he GARCH(1,1)-normal model achieves he smalles value of he loss funcion, while is forecasing accuracy is no saisically differen o ha of he EWMA, EGARCH(1,1) and APARCH(1,1) models wih normally disribued innovaions. The resuls can be summarized in he followings: a) The VC mehod underesimaes he "rue" VaR, since porfolio reurns exhibi skewness and excess kurosis. Thus, as i was expeced, i is no an appropriae echnique for risk managemen. b) More sophisicaed echniques ha accommodae he feaures of he financial ime series are needed, in order o calculae he one-day-ahead VaR. Brooks and Persand (3) poined ou ha models, which do no allow for asymmeries eiher in he uncondiional reurn disribuion or in he volailiy specificaion, underesimae he rue VaR. Gio and Lauren (3) proposed he 3 Tables wih deailed resuls are available upon reques. 7 of 11
skewed Suden- disribuion and poined ou ha i performed beer han he pure symmeric one, as i reproduced he characerisics of he empirical disribuion more accurae. These views are confirmed for boh confidence levels and rading posiions, as mos of he seleced models parameerise hese feaures. c) Specifically, a he 95% confidence level, specificaions wih normally disribued errors achieve he lowes loss funcion values. In mos of he cases, he echniques ha produce he mos accurae VaR predicions are he same for boh long and shor rading posiions. d) On he oher hand, he volailiy specificaions, ha parameerise he leverage effec for he condiional variance and he asymmery of he innovaions disribuion, forecas he VaR a he 99% confidence level more adequaely. However, he models ha mus be employed for modeling he shor and he long rading posiions are no he same. This finding is in conras wih ha of Gio and Lauren (3) who argued ha he APARCH model based on he skewed Suden- disribuion forecass he VaR adequae for boh rading posiions. e) For long posiion on OIL index (95% VaR), and shor posiion on GOLD index (99% VaR), here are no models ha produce adequae VaR forecass. Given he fac ha for hese cases hey have been rejeced by he condiional coverage es, here is evidence ha clusered violaions were generaed. 5. Conclusion Assuming normaliy for he uncondiional reurn disribuion, we forecas accurae one-dayahead VaR a he 95% confidence level. However, gains in forecasing he 99% VaR wih models ha allow for asymmeries eiher in he uncondiional reurn disribuion or in he volailiy specificaion are subsanial. Differen models achieve accurae VaR forecass for long and shor rading posiions, indicaing o porfolio managers he difference of modeling he lef or he righ side of he disribuion of reurns. Using daa from hree ypes of financial markes (sock exchanges, commodiies, and exchange raes) here is evidence ha our resuls hold for differen ypes of markes. References Barone-Adesi, G., Giannopoulos, K., Vosper, L., 1999. VaR Wihou Correlaions for Nonlinear Porfolios. Journal of Fuures Markes, 19, 583-6. 8 of 11
Basle Commiee on Banking Supervision. 1996. Supervisory Framework for he Use of Backesing in Conjuncion wih he Inernal Models Approach o Marke Risk Capial Requiremens. Manuscrip, Bank for Inernaional Selemens. Bollerslev, T., 1986. Generalized Auoregressive Condiional Heeroskedasiciy. Journal of Economerics, 31, 37 37. Bollerslev, T., 1987. A Condiional Heeroskedasic Time Series Model for Speculaive Prices and Raes of Reurn. Review of Economics and Saisics, 69, 54-547. Bollerslev, T., Engle, R.F., Nelson, D., 1994. ARCH Models, In: Engle, R.F., McFadden, D. (Eds.), Handbook of Economerics, Volume 4, Elsevier Science, Amserdam, pp. 959-338. Brooks, C., Persand, G., 3. The effec of asymmeries on sock index reurn Value-a-Risk esimaes. The Journal of Risk Finance, Winer, 9-4. Chrisoffersen, P., 1998. Evaluaing inerval forecass. Inernaional Economic Review, 39, 841-86. Daníelsson, J.,. The emperor has no clohes: Limis o risk modeling. Journal of Banking and Finance, 6, 173-196. Degiannakis, S., Xekalaki, E., 4. Auoregressive Condiional Heeroscedasiciy Models: A Review. Qualiy Technology and Quaniaive Managemen, 1,, forhcoming. Diebold, F.X., Mariano, R., 1995. Comparing predicive accuracy. Journal of Business and Economic Saisics, 13, 3, 53-63. Ding, Z., Granger, C.W.J., Engle, R.F., 1993. A Long Memory Propery of Sock Marke Reurns and a New Model. Journal of Empirical Finance, 1, 83-16. Engle, R.F., 198. Auoregressive Condiional Heeroskedasiciy wih Esimaes of he Variance of U.K. Inflaion. Economerica, 5, 987 18. Gio, P., Lauren, S., 3. Value-a-Risk for Long and Shor Trading Posiions. Journal of Applied Economerics, 18, 641-664. Glosen, L., Jagannahan, R., Runkle, D., 1993. On he Relaion Beween he Expeced Value and he Volailiy of he Nominal Excess Reurn on Socks. Journal of Finance, 48, 1779 181. Hendricks, D., 1996. Evaluaion of Value-a-Risk Models Using Hisorical Daa. Economic Policy Review,, 39-7. 9 of 11
Hull, J., Whie, A., 1998. Incorporaing Volailiy Updaing Ino he Hisorical Simulaion Mehod for VaR. Journal of Risk, 1, 5-19. Kupiec, P.H., 1995. Techniques for Verifying he Accuracy of Risk Measuremen Models. Journal of Derivaives, 3, 73-84. Lamber, P., Lauren, S.,. Modelling Skewness Dynamics in Series of Financial Daa. Discussion Paper, Insiu de Saisique, Louvain-la-Neuve, Belgium. Li, W.K., Ling, S., McAleer, M., 1. A Survey of Recen Theoreical Resuls for Time Series Models Wih GARCH Errors. Discussion Paper 545, Insiue of Social and Economic Research, Osaka Universiy, Japan. Lopez, J.A., 1999. Mehods for Evaluaing Value-a-Risk Esimaes., Economic Review, Federal Reserve Bank of San Francesco,, 3-17. Nelson, D., 1991. Condiional Heeroskedasiciy in Asse Reurns: A New Approach. Economerica, 59, 347-37. Poon, S.H., Granger, C.W.J., 3. Forecasing Volailiy in Financial Markes: A Review. Journal of Economic Lieraure, XLI, 478-539. Sarma, M., Thomas, S., Shah, A., 3. Selecion of VaR Models. Journal of Forecasing,, 4, 337-358. Van den Goorbergh, R.W.J., Vlaar, P., 1999. Value-a-Risk Analysis of Sock Reurns. Hisorical Simulaion, Variance Techniques or Tail Index Esimaion? DNB Saff Repors 4, Neherlands Cenral Bank. 1 of 11
Table 1. The wo-sage model selecion procedure. Column presens he models ha have no been rejeced by he uncondiional coverage backesing crierion (Kupiec 1995), Column 3 presens he models ha have no been rejeced by he condiional coverage backesing crierion (Chrisoffersen 1998), Column 4 presens he models ha are preferred over he ohers based on he loss funcion approach. In Column 4, he model wih he lower value of he loss funcion is bold faced. Series Uncondiional Coverage Condiional Coverage Loss Funcion 95% VaR Long Posiions S&P 5 EWMA, G-N, E-N, A-N, FHS EWMA, G-N, E-N, A-N, FHS EWMA, G-N, E-N, A-N FTSE1 G-T, T-T, G-ST, E-ST, T-ST G-T, T-T, G-ST, E-ST, T-ST G-ST, T-ST OIL VC, EWMA, G-N, E-N, T-N, A-N, FHS - - GOLD EWMA, G-N, E-N, T-N, A-N, FHS EWMA, G-N, E-N, T-N, A-N, FHS G-N, E-N, T-N US_UK EWMA, G-N, E-N, T-N, A-T, A-ST, FHS EWMA, G-N, E-N, T-N, A-T, A-ST, FHS EWMA, G-N, E-N, T-N, A-T, A- ST, FHS US_YEN EWMA, G-N, E-N, T-N, HS, FHS EWMA, G-N, E-N, T-N, HS, FHS G-N, E-N, T-N Shor Posiions S&P 5 EWMA, G-N, E-N, T-N, A-N G-N, A-N G-N, A-N FTSE1 EWMA, G-N, E-N, T-N, A-N, A-ST EWMA, G-N, E-N, T-N, A-N, A-ST EWMA, T-N, A-ST OIL VC, EWMA, G-N, E-N, T-N VC, EWMA, G-N, E-N, T-N G-N, E-N, T-N GOLD EWMA, G-N, E-N, T-N, A-N, FHS EWMA, G-N, E-N, T-N, FHS G-N, T-N US_UK G-N, E-N, T-N, A-ST, FHS G-N, E-N, T-N, A-ST, FHS E-N US_YEN VC, EWMA, G-N, E-N, T-N, A-N, HS, FHS VC, EWMA, G-N, E-N, T-N, A-N, HS, FHS 99% VaR Long Posiions VC, G-N, E-N, T-N, HS, FHS S&P 5 E-T, T-T, A-T, A-ST, FHS E-T, T-T, A-T, A-ST, FHS E-T, T-T, A-T, A-ST, FHS FTSE1 G-T, E-T, T-T, A-T, G-ST, E-ST, T-ST, A-ST G-T, E-T, T-T, A-T, G-ST, E-ST, T-ST, A-ST G-T, E-T, G-ST, E-ST, T-ST, A-ST OIL E-N, A-N, G-T, E-T, T-T, A-T, G- A-N, A-T, G-ST, FHS A-T, G-ST ST, E-ST, T-ST, HS, FHS GOLD G-N, FHS G-N, FHS FHS US_UK VC, EWMA, G-N, E-N, T-, A-T, A-ST VC, EWMA, G-N, E-N, T-, A-T, A-ST VC, G-N, E-N, A-T, A-ST US_YEN VC, G-N, T-N, HS, FHS VC, G-N, T-N, HS, FHS T-N, HS, FHS Shor Posiions S&P 5 G-N, T-N G-N, T-N G-N, T-N FTSE1 A-N, FHS A-N, FHS A-N OIL VC, EWMA, G-N, E-N, T-N, A-N, VC, EWMA, G-N, E-N, T-N, A-N, VC, E-N, A-N, HS, FHS HS, FHS HS, FHS GOLD A-T, A-ST, FHS - - US_UK VC, A-T, A-ST, FHS VC, A-T, A-ST, FHS VC, A-T, A-ST, FHS US_YEN G-T, E-T, T-T, A-T, G-ST, T-ST, A-T, HS, FHS G-T, E-T, T-T, A-T, G-ST, T-ST, A-T, FHS G-T, E-T, A-T, G-ST, T-ST, A-T Models: G-N (GARCH(1,1)-normal), G-T (GARCH(1,1)-suden-), G-ST (GARCH(1,1)-skewed-), E-N (EGARCH(1,1)-normal), E-T (EGARCH(1,1)-suden-), E-ST (EGARCH(1,1)-skewed-), T-N (TARCH(1,1)-normal), T-T (TARCH(1,1)-suden-), T-ST (TARCH(1,1)- skewed-), A-N (APARCH(1,1)-normal), A-T (APARCH(1,1)-suden-), A-ST (APARCH(1,1)-skewed-), EWMA (RiskMerics), VC (Variance Covariance Mehod), HS (Hisorical Simulaion Technique), FHS (Filered Hisorical Simulaion Technique). 11 of 11