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1 Open Access Auhor Manuscrip SCI-PUBLICATIONS Auhor Manuscrip American Journal of Applied Sciences 5 (): 6-5, 8 ISSN Science Publicaions The Gaussianiy Evaluaions of Malaysian Sock Reurn Volailiy Chin Wen Cheong Faculy of Informaion Technology Mulimedia Universiy, 6 Cyberjaya, Selangor, Malaysia Absrac: We sudy he disribuion of sandardized reurns by using various frequencies daa. The empirical sandardized reurns are obained by using he unobserved and observable daily volailiy. Our empirical resuls evidence he realized-sandardized reurns follow neares o a Gaussian disribuion. On he oher hand, he sandardized reurns using daily closing and range-based daa are able o reduce bu no fully eliminae he excess kurosis condiion compare o he realized sandardized reurns. Keywords: financial ime series, realized volailiy, discree ime-domain modelling INTRODUCTION The financial ime varying volailiy is closely relaed o he risk managemen applicaions such as risk diversificaion, porfolio analysis and derivaive pricing predicion. One of he famous risk analysis applicaions is he immediae deerminaion of value-arisk (VaR) from he esimaed volailiy. The commercial applicaion of RiskMerics TM[] is successfully applied VaR in porfolio invesmens. Since he esimaed volailiy hinges crucially on is associaed condiional reurn disribuion, hus, an insigh undersanding of he condiional reurn disribuion is imporan in order o provide useful informaion in risk managemen analysis. In addiion, he idenified underlying disribuions of condiional reurns are imporan in he parameric discree-ime ARCH, sochasic volailiy and coninuous-ime diffusion processes modelling under a paricular underlying disribuion assumpion. In his paper, we have seleced he Malaysian sock exchange and as an emerging marke, he naure of he sock marke is characerized by low liquidiy, infrequen rading, low qualiy of informaion and rapid changes in regulaory framework. The daily closing reurns are adjused for infrequen rading behaviour which migh cause spurious correlaion. We laer focus on he disincion of sandardized reurn from fracional inegraed auoregressive moving average (ARFIMA) model, long-range dependence auoregressive condiional heoroscedasiciy (ARCH) models, rangebased volailiy and realized volailiy in erm of heir Gaussianiy. In his sudy, he sandardized reurn is defined as -sandardized reurn where is eiher he esimaed or realized volailiy sandard deviaion. In realized sandardized reurns, he - and -minue inerval shows he neares Gaussianiy compare o ohers. The Gaussianiy of hese reurn series are analysed by heir momens, empirical cumulaive disribuion funcion (CDF) plo, quanile-quanile (Q- Q) plos, Jacque-Bera es and a series of empirical disribuion ess. The overall resuls indicae he realized sandardized reurns are nearly Gaussian disribued compare o oher approaches modelling where improvemen of reducing bu no eliminae he lepokuric. DATA AND METHODOLOGY Daa Source: The Kuala Lumpur Composie Index (KLCI) ransacion prices cover he recovery period s o 6. This price index is weighed by marke capializaion wih he base year 977 of lised companies. Afer he bad experience of Asian Financial crisis, Malaysia implemens he selecive capial conrol in s Sepember 998 where RM is pegs a.8 o he USD. This acion sabilizes he RM where non-residens from Malaysia and abroad are resric o rade he RM. In addiion, Securiies Commission and he KLSE implemens recovery sraegic such as srenghening marke inermediaries, improving marke ransparency and improving liquidiy in corporae secors. In his recovery period, he Malaysia sock marke is speculaed by he RM-USD Corresponding Auhor: Chin Wen Cheong, Faculy of Informaion Technology, Mulimedia Universiy, 6 Cyberjaya, Selangor, Malaysia. Tel: , Fax:

2 SCI-PUBLICATIONS Auhor Manuscrip un-pegged regulaion(implemened a year middle of 5 where he RM was expeced undervalued by approximaely 6.5%), he merged of MESDAQ in KLSE besides he Main board and Second board previously sared in year, he flucuaing of perol prices, ec. We inend o sudy he reacion of marke paricipans and marke volailiy o he good and bad news. Adjused reurn: Emerging markes are ofen relaed o he infrequenly raded shares aciviies. This phenomena occurs when socks marke do no rade a every consecuive inerval. We adop he mehod proposed by Miller e al [] as follows: r = a + a r - + ε () ε r (adj) =. () ( a) The model assumes ha he non-rading adjusmen required o adjused reurns is consan hroughou he periods in mos of he high raded markes. Le assumes ha he reurn series afer adjusmen are decomposed as r ( adj) = ε where ε is independenly and idenically disribued wih mean zero and uni variance. Therefore, he -sandardized reurn can be rearranged become, r ε =. () Based on his equaion, if given he, he disribuion and srucure of ε can be deermined sraighforward. However, he condiional sandard deviaion is no direcly observable and has o be esimaed from ARCH-ype models, range-based volailiy or realized volailiy respecively. Unobservable esimaed ARCH-ypes volailiy: Componen GARCH: Ding and Granger [] and Engle and Lee [] decomposed volailiy ino wo componens wih one componen capures he shor-run innovaion impac and he oher capures he long-run impac of an innovaion as follows: = +, q, s = γ, q = ω + γ s q -, s, s -, q + γ ( a s + γ ( a q ) - - ) - Am. J. Applied Sci., 5 (): 6-5, 8 () Fracionally Inegraed GARCH: The condiional variance of FIGARCH(p,d,q) inroduced by Baillie e.al [5] can be expressed as: d α ϕ( ( = + a β ( β (, (5) 7 wih d. if d=, he model will become β = α( a, which is a GARCH model. ( If d=, he model β ( ( = α( a, will follow a IGARCH model. And when d is < d <.5, he erm (- d has an infinie binomial disribuion for nonineger powers. ARFIMA-FIGARCH: The ARFIMA(p,d,q )- FIGARCH(p,d,q ) model is able o capure he possibiliy of long-range dependence in boh he condiional reurn and is volailiy as follows: (- d (-β ()r = (-α ()a, a = ε, d ( ) ( ) α α B B = + a. (6) β( β( The shock erm, a, follows a condiional ime-varying variance and he ε ~iid, N(, ). Davidson [6] and Caporin [7] argued ha he d in ARFIMA is srucural differen from d where he persisence is increase when d approaches.5 compare o d approaches. The reverse behaviour may be due o he parameer acs direcly on he squared errors bu no on he condiional variance. Unobservable range-based volailiy: We have adoped he Parkinson [8] and Garman and Klass [9] approaches wih he assumpion of he expeced reurn is equal o zero. The mean reurn is no saisically differen from zero a 5% level under he -es(saisic.75). In addiion, he seleced range-based volailiy model is wihou he inclusion of bid-ask informaion. This is due o he limiaion of daa source. The Garman and Klass is he exension of Parkinson wih he inclusion of opening and closing price. Boh he volailiy esimaors can be expressed as: ( ), park = H L ; (7) ln, =. 5( H L ) GK ( C ( H L ) H L ).8C.9 (8) where he definiions are followed he Yang and Zhang [] wih H, L and C are he normalized high, low and closing prices respecively. Observable realized volailiy: In sock marke, he inraday reurns are obained by summing he rading hours wih he absence of overnigh rading. However, we are able o observe he close-o-open reurn for he overnigh period. The sock marke encouners a shor break in he afernoon and an overnigh non-rading

3 SCI-PUBLICATIONS Auhor Manuscrip period. Under hese condiions, we expec relaive larger changes in he sock index price during he closing period compared o he n-minue reurns observed during rading hours. Therefore he overnigh and afernoon break will provide a disoring effec on he volailiy esimaion. A beer alernaive by using only inraday reurns are proposed by Marens [] and Hansen and Lunde []. Similarly, hey suggesed o use a scaled sum of squared inraday reurns as follow: T M T A n n M A RV = ( + ) ( ) + ( ) c R, a R, b ; (9) a= b= oc + co where c =. The OC and CO represen closeo-open and open-o-close oc respecively. RESULTS AND DISCUSSION Descripive saisics: Table, he unadjused reurn exhibis excess kurosis and he Box-Ljung Q() saisic indicaes he presence of serially correlaion. The highly significan value for he firs-order serial correlaion is caused by he infrequen rading of emerging marke. Afer he correcion of hin rading effec, he adjusmens appear o have eliminaed he apparen serial correlaion of he reurn series where he Q() shows insignifican serially correlaion a % significan level. Comparison of sandardized reurns: The disribuions of each of he sandardized-reurns series are examined relaive o a sandard Gaussian disribuion. We look ino heir momens, he empirical cumulaive disribuion funcion (CDF) plos, he Q-Q plos and empirical disribuion es respecively. Geomerically, he preliminary Gaussianiy analyses are illusraed by he CDF plos and Q-Q plos respecively in Figure and Figure. Mos of he ARCH-sandardized reurns move closely o a simulaed Gaussian disribuion. On he oher hand, especially he range-based sandardized reurns series show higher and lower cumulaive probabiliies in he early and end ails respecively relaive o he simulaed Gaussian CDF. In addiion, he range-based sandardized reurns series also indicae relaive wider range in he volailiy aribue axis. The Q-Q plo for adjused reurn exhibi s-shaped paern wih symmeric and heavy ailed a boh he ends. A few poins fall on he end of he line indicae ha he exreme values in he reurns series. For ARCH, Am. J. Applied Sci., 5 (): 6-5, 8 8 Table : Reurn and adjused reurn reurn adjused reurn Mean.9 -. Sd. Dev Skewness.6 c (.).55 c (.8) Kurosis.578 c (8.78). c (7.6) Jarque-Bera c (.) 6.5 c (.) Normaliy es D.99 c (.).7 c (.57) A.888 c (.).5 c (.) W.689 c (.).759 c (.) Auocorrelaion lag lag.56. lag..5 lag.6. lag Q() p-value. b.9 Noe: he indicaions are he same in noes Table normal ARFIMA cgarch figarch garch Fig. : CDF plos normal Fig. : CDF plos GK park RV RV

4 SCI-PUBLICATIONS Auhor Manuscrip Am. J. Applied Sci., 5 (): 6-5, 8 Table : -sandardized adjused reurn Comparison -sandardized adjused reurn (GARCH) (CGARCH) (FIGARCH) (ARFIMA) (park) (GK) (RV-) (RV-) Mean Sd. Dev Skewness.56 c.87 c.8 c.76 c.99 c.55 c.96 b. c Kurosis Jarque-Bera EDT D A W (.6).79 c (5.8) c (.).7 c (.57).5 c (.).759 c (.) (.57).98 c (5.57) 7.59 c (.) (.).977 c (5.5) 5.7 c (.). c.99 c (.99) c (.7). c.57 c (.7) (.).688 c.856 c (.) (.) (.5).758 (.977).7 c (.59). (>.).6 a (.96).798 (.8) (.).558 c (8.655) 8.7 c (.) (5.8) 5.5 c (8.7) 6.9 c (.) (.8).9 (.6) 7.6 b (.6).78 c (.) c.86 c (.) c.5 a (.95) c 6.55 c.865 (.) (.) (.7). c. c.555 (.) (.) (.5) (.77).6 (.5) 7.86 b (.7).5 (.5).6 (.79).87 (.5) ACF lag lag lag lag lag Q() p-value Noes: () -es for Gaussian skewness and kurosis. The sandard error for Gaussian skewness and kurosis are 6 T =.9 and T =.8. The parenheses indicae he -saisics. The null hypohesis indicaes S ~ = and K ~ = respecively. () Empirical disribuion es(edt): The es saisics are Kolmogorov-Smirnov(D), Anderson-Darling(A ) and Cramer-von Mises(W ) respecively. The parenheses indicae he p-values. H : The reurn series follows he Gaussian disribuion. H : The reurn series does no follow he Gaussian disribuion. a, b and c denoe %, 5% and % level of significance. range-based and realized -sandardized reurns series, a nearly linear line indicaing ha Gaussian disribuion provides a beer approximaion o he series. However, a series of saisical ess have o be implemened o examine he Gaussianiy of he sandardized reurns series. In Table, he sandardized reurns by long-range dependence ARCH-ype esimaions show a beer approximaion o Gaussian disribuion wih nearly uniy sandard deviaion and smaller skewness compare o he adjused and unadjused reurns series. The complee ARCH-ype esimaions are provided upon reques. However, he ARCH-sandardized reurns sill exhibi lepokuric even hough overall hey show smaller excess kurosis. On he oher hand, he rangebased-sandardized reurn show similar resuls compare o ARCH. However, he range-based approach exhibis sronger violaion from Gaussian disribuion which indicaed by he smaller p-value compare o ARCH approach. Finally, he realized-sandardized reurns show closes o a Gaussian disribuion wih decreased posiive skewness and kurosis approximae ha are.9 and.6 for and -minue inerval respecively. The -saisics for gaussianiy skewness and kurosis ess in Table confirm ha only he realized sandardized reurns wih -minue inerval are insignifican a 5% significan level. Finally, formal ess by Jacque-Bera and empirical disribuion ess are implemened o furher analyze he presence of Gaussianiy of he reurn series. From Table, only he realized sandardized reurns wih -minue and - minue inerval are normally disribued a 5% significan level. These findings are inline wih he work by Andersen e.al. [] and Andersen e.al. [] who repored ha he sock realized sock reurns are approximaely normally disribued. 9

5 SCI-PUBLICATIONS Auhor Manuscrip Normal Quanile Normal Quanile Normal Quanile adjused reurn sigma(garch) sigma(arfima) Am. J. Applied Sci., 5 (): 6-5, 8 Normal Quanile Normal Quanile Normal Quanile sigma(parkinson) sigma(cgarch) sigma(rv-) Noe: The symmeric red line indicaes a normal-normal QQ-plo Fig. : Quanile-quanile plos for he overall sandardized reurns series Normal Quanile Normal Quanile Normal Quanile sigma(gk) sigma(figarch) sigma(rv-) CONCLUSION In his paper, our empirical resuls evidence ha he realized-sandardized reurn series wih - and - minue inerval show remarkably near o a Gaussian disribuion compare o ARCH and range-based sandardized reurn series. The resul suggess ha for he underlying Gaussianiy disribuion assumpion in volailiy modelling, he realized volailiy approach provide a beer heoreical modelling framework. Our findings may offer some saisical implicaions in he disribuion of reurns series for any furher heoreical modelling and predicion of ohers marke financial ime series. ACKNOWLEDGEMENTS The auhor would like o graefully acknowledge he financial suppor from Mulimedia Universiy and 5 he Research Design and Developmen and Innovaion Gran in publishing his research. REFERENCES. Morgan, J.P., 996. RiskMericsTM Technical Documen, New York.. Miller M.H., Muhuswamy, J. and Whaley, R.E., 99. Mean reversion of Sandard and Poor 5 index basis changes: Arbirage-induced or saisical illusion? Journal of Finance, 9: Ding Z., and C.W.J. Granger, 996. Modelling volailiy persisence of speculaive reurns: a new approach. Journal of Economerics 7: Engle R.F., and G.G.J. Lee, 999. A long-run and sor-run componen model of sock reurn volailiy, in coinegraion, causaliy and forecasing, ed. By Engle RF, and Whie H, Oxford Universiy Press.

6 SCI-PUBLICATIONS Auhor Manuscrip 5. Baillie R. T., Bollerslev T., and Mikkelsen H.O., 996. Fracionally inegraed auoregressive condiional heeroscedasiciy, Journal of Economerics 7:. 6. Davidson, J.,. Momen and memory properies of linear condiional heeroscedasiciy models, Manuscrip, Cardiff Universiy. 7. Caporin,. Esimaion and idenificaion of FIGARCH, Universia ca Foscari di Venezia, Venezia, I. 8. Parkinson, M., 98. The exreme value mehod for esimaing he variance of he rae of reurn, Journal of Business, 5: Garman M., and Klass M., 98. On he esimaion of securiy price volailiies form hisorical daa. Journal of Business 5: Yang, D., & Zhang, Q.,. Drif independen volailiy esimaion based on high, low, open and close prices. Journal of Business, 7: 779 Am. J. Applied Sci., 5 (): 6-5, 8. Marens, M.,. Measuring and forecasing S&P 5 index-fuures volailiy using high-frequency daa. The Journal of Fuures markes, : Hansen, P.R., & Lunde, A.,. A forecas comparison of volailiy models: Does anyhing bea a GARCH(,)? Journal of Applied Economerics, : Andersen, T.G., & Bollerslev, T., 998. Answering he skepics: Yes sandard volailiy models do provide accurae forecass. Inernaional Economic Review, 9: Andersen, T.G., Bollerslev, T., Diebold, F.X., & Labys, P.,. Modeling and forecasing realized volailiy. Economerica, 7: