The role of the SGT Density with Conditional Volatility, Skewness and Kurtosis in the Estimation of VaR: A Case of the Stock Exchange of Thailand

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1 Available online a Procedia - Social and Behavioral Sciences 4 ( ) The Inernaional (Spring) Conference on Asia Pacific Business Innovaion and Technology Managemen, Paaya, Thailand The role of he SGT Densiy wih Condiional Volailiy, Skewness and Kurosis in he Esimaion of VaR: A Case of he Sock Exchange of Thailand Golf Aaboonwongse a, * a Faculy of Economics, Thammasa Universiy, Prachan Road, Bangkok, Thailand Absrac One of primary ools used o assess he financial risk is Value-a-Risk (VaR). I urns o be a sandard measure of downward risk among financial inermediaries and regulaors recenly as i summarized he risk ino jus a single and easy-o-undersand number. Despie he simpliciy of VaR s concep, an accurae calculaion of VaR is sill challenging. This paper aims o propose an alernaive approach which is believed o provide more accurae VaR raher han he radiional ones. Insead of he convenional Gaussian disribuion, he more flexible skewed generalized (SGT) densiy funcion is assumed for reurn series. Is volailiy is characerized by eigh ypes of GARCH process. Meanwhile, condiional skewness and kurosis is modeled o exhibi ime-varying feaure by heir pas informaion se and auoregressive erm. Daily reurns on he SET index will be used o explore he performance of esimaed VaR. The finding shows ha his new approach can provide more accurae and robus esimaes of he acual VaR hreshold, especially wih TS-GARCH model, han any oher approaches ha have been applied earlier. Published by Elsevier Ld. Selecion and/or peer-review under responsibiliy of of he he Asia Pacific Business Pacific Business Innovaion Innovaion and Technology and Technology Managemen Managemen Sociey Sociey Open access (APBITM). under CC BY-NC-ND license. Keywords: Condiional value a risk; GARCH; Skewed generalized disribuion; Condiional skewness and kurosis. Inroducion One of primary ools used o assess financial risk is Value-a-Risk (VaR). I is defined as he wors loss over a arge horizon such ha here is a low, pre-specified probabiliy he acual loss will be larger. Is greaes advanage can be easily seen ha i summarizes he risk in consideraion ino a single and easy-o-undersand number. Despie he simpliciy of VaR s concep, an accurae calculaion of condiional VaR is sill saisically challenging. Many earlier applicaions of VaR assume ha he asse reurns are normally disribued. Hence, he reurns sandardized by he condiional mean and condiional sandard deviaion are sandard normal. This assumpion simplifies he compuaion of VaR quie considerably. However, here are many empirical sudies on reurn disribuions since 96s suggesing ha hey are no characerized by normaliy bu by he sylized facs of fa ails, high peakedness and skewness (Kon, 984; Badrinah and Chaerjee, 988; and Minik and Rachev, 993). * Corresponding auhor. Tel.: address: golf.aa@homail.com Published by Elsevier Ld. Selecion and/or peer-review under responsibiliy of he Asia Pacific Business Innovaion and Technology Managemen Sociey Open access under CC BY-NC-ND license. doi:.6/j.sbspro..3.58

2 Golf Aaboonwongse / Procedia - Social and Behavioral Sciences 4 ( ) This implies ha exreme evens are much more likely o occur in pracice han ha of he predicion from he symmeric hinner-ailed normal disribuion. Alhough subsequen research proposes many alernaive disribuions ha are more flexible han he normal for he sandardized reurns, hey are all assumed o be iid, implying ha he only feaures of he condiional reurn disribuion, which depend upon he condiioning informaion, are he mean and variance (Kadir e al., ). In fac, i seems more sensible ha oher feaures of disribuion such as skewness and kurosis will depend on he condiioning informaion as well. In his paper, he condiional densiy approach iniiaed by Hansen, 994; and exended by Jondeau and Rockinger, 3; and Bali e al., 8 will be adoped o esimae he VaR hreshold. In order o provide an accurae characerizaion of he shape and ails of he sandardized reurn disribuion, he more flexible skewed generalized (SGT) disribuion, which consiss of mean, variance, skewness, ail-hickness and peakedness parameers, is used in place of he normaliy assumpion. The esimaion of he condiional mean and volailiy of reurns is based on AR() and eigh variaions of he GARCH(,) process, respecively. Besides he firs wo momens (condiional mean and variance), he higher-order momens of he SGT disribuion are allowed o depend on he pas informaion se by defining he skewness, ail-hickness, and peakedness parameers of he densiy as an auoregressive process similar o he ARCH model of Engle, 98. The empirical analyses are based on he daily reurns on he Sock Exchange of Thailand (SET) value-weighed index during January 976 o December (8,65 observaions). The performance of he SGT-GARCH models wih ime-varying parameers in he esimaion of VaR will be assessed hrough he uncondiional coverage es of Kupiec, 995; and he condiional coverage es of Chrisoffersen, 998. The finding shows ha he condiional SGT-VaR approach inroduced in his paper provides quie accurae and robus esimaes of he acual VaR hreshold. This paper is organized as follows. Secion presens he condiional SGT-VaR models. Secion 3 describes assessmen of performance of he condiional SGT-VaR models. Secion 4 discusses he insample and ou-of-sample performance of he condiional SGT -VaR models. Secion 5 concludes he paper.. Condiional SGT -VaR models To compue he precise condiional VaR, his lieraure builds on Bali e al., 8. The aforemenioned condiional SGT-VaR models are defined as follows: r r u z () g( ), or ln( ) () g( ) h(, z ) g( ) (3) where r is reurns a ime ; and are, respecively he condiional mean and condiional sandard deviaion r based on pas informaion se up o ime -; z is he reurns innovaion a ime ; z ( r )/ is sandardized reurns, which is densiy funcion is given as: z fz z C (4) (( )/ ) sign z B C.5 (5) (6) B B (7) 3 ( 3 ) B B, (8) (9) The condiional sandard deviaion is assumed o follow various GARCH(,)-ype models hrough he funcional form g( ) as in expression () and (3). The condiional volailiy equaions g( ) for eigh variaions of GARCH(,) models are as follows: GARCH Model: z () IGARCH Model: ( ) z ()

3 738 Golf Aaboonwongse / Procedia - Social and Behavioral Sciences 4 ( ) EGARCH Model: ln( ) [ z E z ] z ln( ) () GJR-GARCH Model: z S z (3) QGARCH Model: z z (4) TGARCH Model: z S z (5) TS-GARCH Model: z (6) APGARCH Model: [ sign( z ) ] z (7) where S for z and S oherwise. The condiional high-order momen parameers of he SGT densiy are modeled as follows: / ( exp( )) z (8) exp( ) z (9) exp( ) z () Noe ha is resriced skewness parameer; and are resriced kurosis parameers according o he SGT definiion ha, and., and are unresriced ones which has ime-varying form. The condiional SGT-GARCH parameers are obained from he T maximizaion of he sample log-likelihood funcion L [ln( fz( z,, )) ln( )] wih respec o, and/or depending on each GARCH(,) specificaion and subjec o posiiviy and saionary consrains associaing wih each GARCH(,) * specificaion. Afer all condiional parameers of he reurn disribuion are esimaed, he r, which is he corresponding condiional hreshold for he reurn r a a given coverage probabiliy, can be a obained firsly from solving a from he equaion fz( z) dz and hen subsiue back ino equaion * r a. 3. Assessmen of he performance of condiional VaR 3.. Uncondiional coverage es Given independence, Kupiec, 995 consruced he uncondiional coverage es (LR UC ) under he null hypohesis ha he acual and expeced numbers of observaions falling below VaR hreshold UC N (called exceedence) are saisically he same as LR [ ln ( N )ln ], where N is he N NN number of sample observaions is he coverage probabiliy N and are he expeced and acual number of observaions falling below he VaR hreshold a. The LR UC is disribued by (). The accepance of null hypohesis refers ha he compued condiional VaR hreshold provides a good assessmen of risk exposure. 3.. Condiional coverage es Chrisoffersen, 998 argued ha he uncondiional coverage es is insufficien o assess he VaR hreshold when he assumpion of serial independence is violaed. The auhor developed he condiional coverage es o examine he serial independence of VaR esimaes by defining he indicaor I as I if exceedence occurs and I oherwise. The condiional coverage es saisic is consruced under null hypohesis of serial independence agains he alernaive of explici firs-order Markov dependence IND as LR [ n ln n ln n ln n ln ], where n ij is he number of observaions of indicaor variable I wih value i followed by j n /( n n) n /( n n), ( n n)/ N, and N n n n n. The LR IND is disribued by (). The accepance of null hypohesis indicaes ha he serial independence assumpion is held and i suffices o use he uncondiional coverage es o assess he VaR hreshold. 4. Risk measuremen of he condiional SGT-VaR models 4.. Assessmen of in-sample VaR performance Table presens saisics on he VaR hreshold of all models for he coverage probabiliies of %,.5%, %,.5%, and 5% using he sample beween January 976 and December for boh

4 Golf Aaboonwongse / Procedia - Social and Behavioral Sciences 4 ( ) esimaion and predicion (in-sample analysis). The firs row for each coverage probabiliy presens he average esimaed VaR hresholds of eigh GARCH(,) ypes. The second presens he acual and expeced (Acl/Exp) number of reurns ha fall below each hreshold. The hird row presens he uncondiional coverage es saisics (LR UC ) and he condiional coverage es saisics (LR IND ). The LR IND in all models and coverage probabiliies canno rejec he null hypohesis of he serial independen assumpion of he uncondiional coverage es, indicaing ha he assessmen of VaR hreshold can rely on he LR UC. The LR UC shows ha he APGARCH model is he mos inaccurae for predicing he VaR hreshold since hey rejecs he null hypohesis a all coverage probabiliy levels. The GARCH, IGARCH, and QGARCH models are all accurae only a high coverage probabiliy bu no he low one (excep he IGARCH model a % level). In conras, he EGARCH, GJR-GARCH, and TGARCH models do poorly for he high coverage probabiliies bu become beer when i goes furher o he ails of he reurn disribuion (low coverage probabiliies). The TS-GARCH model provides he bes assessmen of he risk exposure of a porfolio mimicking he SET index reurns since he null hypohesis canno be rejeced a all coverage probabiliy levels. I implies ha he VaR hreshold obained from he TS-GARCH model based on he SGT disribuion wih ime-varying volailiy, skewness and kurosis is accurae and robus regardless of coverage probabiliy chosen. Table : In-sample VaR performance of he condiional SGT-GARCH models GARCH IGARCH EGARCH GJRGRH QGARCH TGARCH TSGRCH APGRCH.% Acl/Exp 5/86 6/86 74/86 75/86 /86 78/86 9/86 33/86 LR UC /LR IND 5.64/. ** 9.5/.4 **.78 ** /.8 **.49 ** /.3 **.75/.35 **.78 ** /. **.4 ** /. ** 46.4/.88 **.5% Acl/Exp 75/9 69/9 8/9 6/9 7/9 7/9 7/9 359/9 LR UC /LR IND 4.96/.5 **.45/.4 **. ** /.53 **.7 ** /. ** 3.5/.9 ** 4.6 * /.9 **.3 ** /. ** 8./.9 **.% Acl/Exp 9/7 6/7 /7 9/7 8/7 34/7 6/7 38/7 LR UC /LR IND.4/.6 ** 6.43 * /. ** 4.4 * /.3 **.85 ** /. **.55/.5 ** 9.9/. **.6 ** /.43 ** 93.9/.57 **.5% Acl/Exp 73/5 56/5 8/5 69/5 7/5 77/5 4/5 4/5 LR UC /LR IND 4.76/.6 ** 7.54/.5 ** 8.38/. **.86/.4 ** 4.8/.3 ** 7.35/.3 **.59 ** /.7 ** 33./.98 ** 5.% Acl/Exp 475/43 45/43 67/43 6/43 474/43 385/43 39/43 53/43 LR UC /LR IND 4.77 * /. **.95 ** /.9 ** 75.75/.4 ** 78.7/. ** 4.56 * /. ** 5.6 * /.85 ** 3.86 * /.56 ** 9.79/.9 ** Noe: *, ** denoe ha he null hypohesis canno be rejeced a 5% and %, respecively. 4.. Assessmen of ou-of-sample VaR performance Table presens saisics on he VaR hreshold of all models for he coverage probabiliies of %,.5%, %,.5%, and 5% using he sample beween January and December 9 for esimaion, and he las quarer of December for predicion (ou-of-sample analysis). The resuls from he LR IND show ha all uncondiional coverage saisics are reliable and suffice o assess he performance of VaR hreshold (excep he TGARCH model a % level ha he LR IND canno be calculaed). The LR UC in all models canno be rejeced he null hypohesis a all coverage probabiliy levels. I srongly indicaes ha all models provide accurae and robus VaR hreshold in case of ou-of-sample analysis. 5. Conclusion Wih complexiy in he curren financial marke, Value-a-Risk (VaR) is one of primary ool used o assess he financial risk. Despie he simpliciy of is concep, an accurae calculaion of condiional VaR is sill saisically challenging. This paper proposes an alernaive o compue condiional VaR called condiional SGT-VaR approach. The radiional normaliy assumpion has been relaxed o he more flexible skewed generalized (SGT) disribuion. The condiional volailiy is assumed o follow 8 ypes of GARCH(,) process including symmeric and asymmeric ones. Furhermore, he convenional assumpion in condiional VaR calculaion ha disribuion of sandardized reurn is iid is also relaxed. We allow higher-order momens of he SGT densiy o rely on he pas informaion se by

5 74 Golf Aaboonwongse / Procedia - Social and Behavioral Sciences 4 ( ) defining he skewness, ail-hickness and peakedness parameers of he SGT densiy as an auoregressive form similar o he ARCH process. The role of condiional skewness and kurosis in he esimaion of he condiional VaR is invesigaed by using he uncondiional coverage es and condiional coverage es o evaluae he performance of he condiional SGT-VaR approach. The in-sample performance resuls indicae ha he condiional SGT-VaR approach wih ime-varying skewness and kurosis in case of he TS- GARCH provides very good predicion of marke risks regardless of coverage probabiliy chosen. However, he performance resuls for ou-of-sample analysis are sill unclear. The SGT-VaR approach wih condiional volailiy, skewness and kurosis in all GARCH-ype can provide accurae VaR hreshold. There is no superior GARCH specificaion among ohers. Fuure research should exend he predicion sample size for he ou-of-sample analysis. Table : Ou-of-sample VaR performance of he condiional SGT-GARCH models GARCH IGARCH EGARCH GJRGRH QGARCH TGARCH TSGRCH APGRCH.% Acl/Exp /.6 /.6 /.6 /.6 /.6 /.6 /.6 /.6 LR UC /LR IND.9 ** /.3 **.96 ** /.4 **.9 ** /.3 **.9 ** /.3 **.9 ** /.3 **.5 ** /NA.9 ** /.3 **.9 ** /.3 **.5% Acl/Exp /.93 /.93 /.93 /.93 /.93 /.93 /.93 /.93 LR UC /LR IND. ** /.3 **.94 ** /.4 **. ** /.3 **. ** /.3 **. ** /.3 **. ** /.3 **. ** /.3 **. ** /.3 **.% Acl/Exp /.4 /.4 /.4 /.4 /.4 /.4 /.4 /.4 LR UC /LR IND.5 ** /.3 **.4 ** /.4 **.5 ** /.3 **.5 ** /.3 **.5 ** /.3 **.5 ** /.3 **.5 ** /.3 **.5 ** /.3 **.5% Acl/Exp /.55 /.55 /.55 /.55 /.55 /.55 /.55 /.55 LR UC /LR IND. ** /.3 **. ** /.4 **. ** /.3 **. ** /.3 **. ** /.3 **. ** /.3 **. ** /.3 **. ** /.3 ** 5.% Acl/Exp /3. 4/3. /3. /3. /3. /3. /3. /3. LR UC /LR IND.47 ** /.4 **.5 ** /.56 **. ** /.3 **. ** /.3 **.47 ** /.4 **.47 ** /.4 **. ** /.3 **.47 ** /.4 ** Noe: *, ** denoe ha he null hypohesis canno be rejeced a 5% and %, respecively. Acknowledgemens The auhor hanks Prof. Siisak Leelahanon, Prof. Tare Janarakolica, and Prof. Thanomsak Suwannoi for heir exraordinary helpful commens and suggesions. References [] Kon, S. Models of sock reurns: A comparison. Journal of Finance 984, 39, [] Badrinah, S.G., and Chaerjee, S. On measuring skewness and elongaion in common sock disribuion: The case of he marke index. Journal of Business 988, 6, [3] Minik, S., and Rachev, S.T. Modeling asse reurns wih alernaive sable disribuions. Economeric Reviews 993,, [4] Hansen, B.E. Auoregressive condiional densiy esimaion. Inernaional Economic Review 994, 35, [5] Jondeau E., and Rockkinger, M. Condiional volailiy, skewness and kurosis: Exisence, persisence, persisence and comovemens. Journal of Economic Dynamics & Conrol 3, 7, [6] Bali, T.G., Mo, H., and Tang, Y. The role of auoregressive condiional skewness and kurosis in he esimaion of condiional VaR. Journal of Banking & Finance 8, 3, [7] Engle, R.F. Auoregressive condiional heeroscedasiciy wih esimaes of he variance of Unied Kingdom inflaion. Economerica 98, 5, [8] Kupiec, P.H. Techniques for verifying he accuracy of risk measuremen models. Journal of Derivaives 995, 3, [9] Chrisoffersen, P.F. Evaluaing inerval forecass. Inernaional Economic Review 998, 39, [] Hazlina Abdul Kadir, Reza Masinaei, Nasim Rahmani (), Long-Term Effecs of Bank Consolidaion Program in a Developing Economy, Journal of Asia Pacific Business Innovaion and Technology Managemen.Volume, No., P-3