Estimating Stock Returns Volatility of Khartoum Stock Exchange through GARCH Models

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1 Esmang Sock Reurns Volaly of Kharoum Sock Exchange hrough GARCH Models Sharaf Obad Al, Abdalla Sulman Mhmoud. College of Compuer Scence, Alzaem alazhar Unversy, Sudan Deparmen of Mahemacs, College of Scences, Shaqra Unversy, kngdom of Saud Araba. Deparmen of Sascs, College of Economcs and Polcal Scences,Omdurman Islamc Unversy, Sudan Deparmen of Mahemacs, College of Ars and Scences, Taf Unversy, kngdom of Saud Araba Absrac: Ths sudy modeled and esmaed sock reurns volaly of Kharoum Sock Exchange (KSE) Index usng symmerc and asymmerc GARCH famly models, namely: GARCH(,), GARCH-M(,), EGARCH(,) and GJR-GARCH(,) models. The sudy was carred ou based on daly closng prces over he perod from nd January 6 o 3 Augus.The emprcal resuls reveals ha a hgh volaly process s presen n KSE Index reurns seres. The resuls also provde evdence on he exsence of rsk premum and ndcaes he presence of he leverage effec n he KSE ndex reurns seres. Our fndngs ndcae ha Suden- s he mos favored dsrbuon for all models esmaed. [Sharaf Obad, Abdalla Sulman. Esmang Sock Reurns Volaly of Kharoum Sock Exchange hrough GARCH Models. J Am Sc 3;9():3-44]. (ISSN: 545-3).. 9 Key words: volaly, GARCH, cluserng volaly, leverage effec, rsk premum,. Inroducon Fnancal me seres analyss s dreced o he undersandng of he mechansm ha drves a gven me seres of daa, or, n oher words: fnancal me seres analyss focuses on he ruh behnd he daa so ha one can fnd physcal models ha explan he emprcally observed feaures of real lfe daa. Wh such models one can make dsrbuonal forecass for fuure values n me seres Karlsson (). Volaly permeaes fnance and s a key varable used n many fnancal applcaons such as nvesmen, porfolo consrucon, opon prcng and hedgng as well as marke rsk managemen. Good forecass of volaly, herefore, become exremely mporan n makng fnancal decsons. Frs, knowledge of volaly could gude raders on he rsk of holdng an asse or he value of an opon, and also provdes reasonable forecasng confdence nerval. Secondly, a relable volaly model sheds furher lgh on he daa -generang process of he reurns. Thrdly, esmaes of fnancal markes volaly may be he elescope of envsagng how robusness of he economy s and he drecon of moneary and fscal polces. These mgh be he reasons why volaly modelng has ganed consderable populary n leraures of fnancal engneerng.(bassey and Issac, ). I s well known ha fnancal me seres daa, ncludng sock marke reurns, ofen exhb he phenomenon of volaly cluserng, meanng ha a perod of hgh volaly ends o be followed by perods of hgh volaly, and perods of low volaly end o be followed by perods of low volaly. Sock reurns also exhb lepokuross, meanng ha he dsrbuon of he fnancal daa has heavy aled, nonnormal dsrbuons. In addon, daa on sock marke reurns s expeced o show a so called leverage effec or asymmerc volaly. Ths means ha he effec of bad news on sock marke volaly s greaer han he effec nduced by good news Onour (7). The man objecve of hs paper s o model and esmae sock reurns volaly of Kharoum Sock Exchange (KSE) ndex, symmerc GARCH models wll be employed o capure he naure of volaly and rsk premum, whle asymmerc GARCH models o capure leverage effecs. Ths paper s organzed as follows: follows secon revew he nroducon, secon presens a bref revew of Kharoum Sock Exchange models. Secon 3 presens a bref revew of relevan leraure. Secon 4 expounds daa and mehodology. Secon 5 provdes dscusson of resuls and secon 6 s a concluson. - Kharoum Sock Exchange: Locaed n Kharoum, he Kharoum Sock Exchange (usually abbrevaed o KSE) s he prmary sock exchange of Sudan. The Kharoum Index s he man sock of KSE. The orgns of KSE go back o 96 when he Mnsry of Fnance, Bank of Sudan and Inernaonal Fnancal Corporaon conceved he dea. Ths was followed by esablshmen whn The bank of Sudan n 964 of a deparmen for governmen bonds. In fac, he frs governmen bonds were ssued n 966 wh a bar value of 5 mllon Sudanese pounds and lfe cycle of years. However, a subsequen 98 ACT o esablsh KSE was a falure. More serous effor n 99 culmnaed n seng up KSE board. The KSE self was recognzed n 994 as 3

2 a legal eny n he wake of endorsemen of he KSE ACT. The prmary marke acves were naed n 994 and were shorly followed by he secondary marke early n 995.he classfcaon of lsed companes was naed n 994, and n he Shahama cerfcaes were ssued. The Kharoum Index offcally come no beng n 3. A furher sep was jonng he Afrcan Marke Unon n 7. As a maer of fac he KSE has been growng seadly over he pas few years and now has 53 lsed companes worh 5 bllon dollars. However, he sock exchange s open for only one hour per day from Sunday hrough Thursday. Despe s rapd growh n erms of marke capalzaon KSE s characerzed as hghly concenraed marke as only op hree companes consue around 9% of he oal marke capalzaon. I s also, consdered an llqud marke as he shares of only hree companes are radable. Lke all fnancal nsuons n Sudan, KSE s regulaed by laws nspred from Islamc Sharaa. One of he mos popular fnancal nsrumens nroduced by Islamc Sharaa pracces n he KSE acves s he exsence of Governmen Musharakah Cerfcaes (GMCs), whch represens an Islamc equvalen of he convenonal bonds (also known as Shahama bonds). Through Shahama bonds he sae borrows money n he domesc marke nsead of prnng more banknoes. Afer one year, holders of GMCs can eher cash or exend hem. These bonds are backed by he socks and shares porfolo of varous companes owned by he Mnsry of Fnance and herefore are asse-backed. The profably of GMCs can reach 33 per cen per annum and depends on he fnancal resuls of he companes nvolved. Hence, he prof of a GMC s varable raher han fxed. The governmen ssues hese bonds on a quarerly bass and her placemen s done very quckly- n jus sx days. KSE s relavely small marke as compared o he sock markes of he developed counres or even o some counres n he Arab regon; he number of lsed companes s few and mos socks are nfrequenly raded, marke capalzaon and raded value are very low(elshehk,). 3.Leraure Revew The auoregressve condonal heeroscedascy (ARCH) model was frs elaboraed n a semnal paper by Engle(98). Snce hen, he opc of modelng volaly n fnancal me seres has been he focus of numerous researchers. Therefore, n hs secon we overvew a number of papers ha have nvesgaed he performance of GARCH models wh regard o non-normal error dsrbuon n maure sock markes. For nsance, a paper by Arab () has esmaed he volaly of exchange rae ha was caused by nconssen economc polces adoped by successve governmens whch faled o realze realsc exchange rae of he Sudanese pound. The consequences of hgh nflaon rae, deeroraon of he producve secors, connuous nernal and exernal defcs and deprecaon of he exchange rae. To esmae he volaly of he exchange rae, EGARCH (,) was used. The resuls show ha leverage effec erm s negave and sascally dfferen from zero, ndcang he exsence of he leverage effec (negave correlaon beween pas reurns and fuure volaly. Xn Zheng () esed wheher sock reurn dsrbuon s assumpons nfluence he performance of volaly forecasng. The mehodologes nclude emprcal analyss usng GARCH-, GARCH-Suden- and GARCH- Skewed Generalzed Error Dsrbuons. No only daly reurns, realzed weekly and monhly volales of S&P/ASX Index and ASX All Ordnares Index are calculaed over years, bu also he ou-of sample-volales are compared. Ther oupu ndcaes ha GARCH-Suden- s superor o ohers over shor-run forecas horzon whle GARCH-S performs beer han ohers over long-run forecas horzon. Freed eal() n her sudy examned several sylzed facs (heavy-aleness, leverage effec and perssence) n volaly of sock prce reurns explong symmerc and asymmerc GARCH famly models for Saud Araba. Ther sudy was carred ou usng closng sock marke prces over 5 years coverng he perod January 994 o 3 March 9. The sample perod was dvded no hree sub-perods accordng o he local crss n 6. Ther fndngs revealed ha asymmerc models wh heavy aled denses mprove overall esmaon of he condonal varance equaon. Moreover, hey found ha AR ()- GJR GARCH model wh Suden- ouperform he oher models durng and before he local crss n 6, whle AR ()-GARCH model wh exhbs a beer performance afer he crss. Furhermore, her resuls revealed ha he exsence of leverage effec a percen sgnfcance level. They conclude ha volaly perssence n he samples durng and afer crses decreases n all models under varous dsrbuon assumpons. A paper by Vee and Gonpo () amed a evaluang volaly forecass for he US Dollar/Mauran Rupee exchange rae obaned va a GARCH (,) model under wo dsrbuonal assumpons: he generalzed Error Dsrbuon () and he Suden s- dsrbuon. They make use of daly daa o evaluae he parameers of each model and produce volaly esmaes. The forecasng ably was subsequenly assessed usng he symmerc 33

3 loss funcons whch are he Mean Absolue Error(MAE) and Roo Mean Square Error (RMSE). The laer show ha boh dsrbuons may forecas que well wh a slgh advanage o he GARCH(,)- for ou-of-sample forecass. Prashan() has saed ha volaly n he Indan and Chnese sock markes exhbs he perssence of volaly, mean reverng behavor and volaly cluserng. Hs sudy based on more han hree years of recen daly daa on Nfy and SSE o llusrae hese sylzed facs, and he ably of GARCH(,) o capure hese characerscs. Daly reurns n he sock markes exhb nonlneary and volaly cluserng whch are sasfacorly capured by he GARCH models. In boh markes, volaly ends o de ou slowly. In her fndngs revealed ha he volaly s more perssen n he Chnese sock marke han he Indan sock marke. A projec by Ladokhn (9) focused on he problem of volaly modelng n fnancal markes. I began wh a general descrpon of volaly and s properes, and dscussed s usage n fnancal rsk managemen. The research was dvded no wo pars: esmaon of condonal volaly and modelng of volaly skews. The frs one was focused on comparng dfferen models for condonal volaly esmaon. They examned he accuracy of several of he mos popular mehods: hsorcal volaly models (e.g., Exponenal Weghed Movng Average), he mpled volaly, and auoregressve condonal heeroskedascy models (e.g., he GARCH famly of models). The second par of he projec was dedcaed o modelng he mpled volaly skews and surfaces. Shamr and Isa (9) nvesgaed he relave effcency of several dfferen ypes of GARCH models n erms of her volaly forecasng performance. They compared he performance of symmerc GARCH, asymmerc EGARCH and nonlnear asymmerc NAGARCH models wh sx error dsrbuons (normal, skew normal, suden-, skew suden-, generalzed error dsrbuon and normal nverse Gaussan. In an nvesgaon by Kosapaarapm e al (8), employed sx smulaed sudes n GARCH (p,q) wh sx dfferen error dsrbuons are carred ou. In each case, hey deermne he bes fng GARCH model based on he creron and hen evaluae s ou of-sample volaly forecasng performance agans ha of oher models. The analyss was hen carred ou usng he daly closng prce daa from Thaland (SET), Malaysa (KLCI) and Sngapore (STI) sock exchanges. Ther Resuls show ha alhough he bes fng model does no always provde he bes fuure volaly esmaes he dfferences are so nsgnfcan ha he esmaes of he bes fng model can be used wh confdence. The emprcal applcaon o sock markes also ndcaed ha a non normal error dsrbuon ends o mprove he volaly forecas of reurns. They conclude ha volaly forecas esmaes of he bes fed model can be relably used for volaly forecasng. Moreover, her emprcal sudes demonsraed ha a skewed error dsrbuon ouperforms oher error dsrbuons n erms of ouof-sample volaly forecasng. Accordng o Engle eal(7) Volaly s a key parameer used n many fnancal applcaons, from dervaves valuaon o asse managemen and rsk managemen. Volaly measures he sze of he errors made n modelng reurns and oher fnancal varables. I was dscovered ha, for vas classes of models, he average sze of volaly was no consan bu changes wh me and s predcable. Auoregressve condonal Heeroscedascy (ARCH)/generalzed auoregressve condonal Heeroscedascy (GARCH) models and sochasc volaly models are he man ools used o model and forecas volaly. Movng from sngle asses o porfolos made of mulple asses, hey found ha no only dosyncrac volales bu also correlaons and covarance's beween asses are me varyng and predcable. Mulvarae ARCH/GARCH models and dynamc facor models, evenually n a Bayesan framework, were he basc ools used o forecas. Accordng o Karmakar(5) one of he objecves of he varous GARCH models s o provde good forecass of volaly whch can hen be used for a varey of purposes ncludng porfolo allocaon, performance measuremen, opon valuaon, ec. Invesors seekng o avod rsk, for example, may choose o adjus her porfolos by reducng her commmens o asses whose volales are predced o ncrease or by usng more sophscaed dynamc dversfcaon approaches o hedge predced volaly ncrease A sudy by Alshogea(988) was devoed o examnng wheher he volaly of he macroeconomc varables have any nfluence on Saud sock marke volaly. They presen descrpve sascs for Saud sock marke reurns. Then, hey esmaes Bollerslev s GARCH(p,q)-model wh no exogenous varables and checks weaher provdes an adequae model for he volaly of Saud sock reurns. Fnally, hey explore he mpac of macroeconomc varables on he volaly of he Saud sock marke reurn by examnng hree dfferen ses of he GARCH models namely AR()-GARCH-X(,), he AR()-GARCH-S(,, and he AR()-GARCH- G(,) model. 34

4 4.Daa and Mehodology 4. Daa The daa used n hs sudy s he daly closng prce ndex of he Kharoum Sock Exchange (KSE) over he perod from January 6 o 3 he Augus conssng of 7 observaons. The daly closng prces were convered o reurns seres as follows: p ln p r () - Where r s he connuously compounded daly reurns of KSE ndex a me, p and p - are he closng prce ndex of KSE a me and - respecvely. I s very mporan o noe ha snce Ocober 8, 9, he ndex on he Kharoum sock marke has been on declne. In a mere 6 days of radng (Ocober 8,9 o November,9),he sock marke ndex fell from 377. o Snce ha me,he KSE ndex was reporng o flucuae around an average value f (Zakra and Wnker, ). In he lgh of hs knowledge, we dvded he daly closng prces ndex no wo sub perods, he frs sub- perod coverng he perod from jan.,6 o Ocober.8,9 wh 4 observaons;he second sub perod whch s from Nov..9 o Aug.3, wh 9 observaons. 4.. Descrpve Sascs Descrpve sascs of daly reurns seres are presened n able o reach an undersandng of he naure and dsrbuon characerscs of he KSE ndex reurns seres. From able, we observe ha he average daly reurns s posve and close o zero for all perods, wh he values of sandard devaon reflecng a hgh level of dsperson from he average reurns n he marke. The hgh kuross values reveal ha he daly reurns seres of KSE ndex are clearly lepokurc. The skewness values are posve skewed for all perods mplyng ha he dsrbuon has a long rgh al and deparure from normaly. The deparure from normal dsrbuon n he KSE daly reurns seres was confrmed wh Jarque-Bera es as s assocaed sgnfcan level less han % confdence level. Engle (98) ARCH LM es sass ndcae he presence of ARCH process n he condonal varance for all he perods. Ths provde an evdence of he valdy of usng GARCH famly models. The Ljung Box Q sascs order6 n reurns and s correspondng squared reflecs a hgh seral correlaon. sascs Mnmum Maxmum Mean Medan Sandard devaon Skewness Kuross Jarque- bera Prob. of Jarque- bera Q 6 Prob.Q6 Q 6 Prob.of Q 6 ARCH() Prob.of ARCH() Table. Descrpve Sascs of he KES reurns seres Frs sub-perod Second sub-perod Whole sample Fgure shows he daly reurns seres of he KSE ndex for he frs and second sub perods and for he whole sample perod. We can see from hs fgure small reurns end o be followed by smaller reurns and large reurns end o be followed by larger reurns. Ths behavor of sock reurns seres ndcaes ha here s a clear evdence of volaly cluserng n KSE ndex reurns. 35

5 Q4 Q Q Q3 Fgure : Daly reurns of KSE for frs and second sub- perods, and for he whole sample (respecvely) 4-- Un Roo Tes for he KSE daly Index Table presens resuls of he un roo es for boh daly closng prces ndex and s reurns seres usng Augmened Dckey Fuller (ADF) Tes sasc. The (ADF) es for KSE prce ndex n level form reveal ha s of saonary ype for frs and second sub perod, bu s non- saonary seres for he whole sample. However when applyng he same es for he reurns seres, we can rejec he null hypohess of a un roo for all perods. 36

6 perod Frs sub-perod second sub-perod Whole sample Table : Saonary Tes for daly closng prces and reurns seres KSE closng prces seres ADF sasc Crcal values ADF sasc % 5% % KSE reurns seres Crcal values % 5% % Mehodology To capure naure of volaly, rsk premum, and leverage effecs on KSE reurns seres, dfferen symmerc and asymmerc GARCH models were used.in he volaly modelng process usng GARCH models, he mean and varance of he seres are esmaed smulaneously. 4-- Exchange Rae Volaly Exchange rae volaly s a measure of he flucuaons n an exchange rae. I s also known as a measure of rsk, wheher n asse prcng, porfolo opmzaon, opon prcng, or rsk managemen, and presens a careful example of rsk measuremen, whch could be he npu o a varey of economc decsons. I can be measured on an hourly, daly, weekly, monhly or annual bass. Based on he assumpon ha changes n an exchange rae follow a normal dsrbuon, volaly provdes an dea of how much he exchange rae can change whn a gven perod. Volaly of an exchange rae, jus lke ha of oher fnancal asses, s usually calculaed from he sandard devaon of movemens of exchange. 4.. Volaly models volaly model should be able o forecas volaly. Vrually all he fnancal uses of volaly models enal forecasng aspecs of fuure reurns. Typcally a volaly model s used o forecas he absolue magnude of reurns. volaly models can be dvded no symmerc and asymmerc models.in hs paper we used wo symmerc GARCH models whch are GARCH(,) and GARCH M(,), and wo asymmerc GARCH models, namely EGARCH(,) and GJR-GARCH(,) ARCH Model ARCH models based on he varance of he error erm a me depends on he realzed values of he squared error erms n prevous me perods. The model s specfed as: y u () u N, h (3 h ) ~ q j u ( 4 Ths model s referred o as ARCH(q), where q refers o he order of he lagged squared ) reurns ncluded n he model. If we use ARCH() model becomes h u Snce h s a condonal varance, s value mus always be srcly posve; a negave varance a any pon n me would be meanngless. To have posve condonal varance esmaes, all of he coeffcens n he condonal varance are usually requred o be non-negave. Thus coeffcens mus be sasfy and GARCH Model The model allows he condonal varance of varable o be dependen upon prevous lags; frs lag of he squared resdual from he mean equaon and presen news abou he volaly from he prevous perod whch s as follows: h q p u h (5) In he leraure mos used and smple model s he GARCH(,) process, for whch he condonal varance can be wren as follows: h u h (6) Under he hypohess of covarance saonary, he uncondonal varance h can be found by akng he uncondonal expecaon of equaon 5. We fnd ha h h h (7) Solvng he equaon 5 we have h (8) For hs uncondonal varance o exs, mus be he case ha and for o be posve, we requre ha. Advanages of GARCH models compared o ARCH models The man problem wh an ARCH model s ha requres a large number of lags o cach he naure of he volaly, hs can be problemac as s dffcul o decde how many lags o nclude and produces a non-parsmonous model where he non- 37

7 negavy consran could be faled. The GARCH model s usually much more parsmonous and ofen a GARCH(,) model s suffcen, hs s because he GARCH model ncorporaes much of he nformaon ha a much larger ARCH model wh large numbers of lags would conan GARCH-M Model The mporance of he relaonshps beween marke rsk and expeced reurns s crucal nfnance heory. The dea from Engle e al.(987) was consequenly used o esmae he condonal varances n GARCH and hen he esmaons wll be used n he condonal expecaons' esmaon. Ths s he so called GARCH n Mean (GARCH-M) modelas: r h u (9) h p q u j j h () j Where: s he volaly coeffcen (rsk premum) for he mean. P s he order of he ARCH componen model q s he order of he GARCH componen model A posve rsk-premum (.e. ) ndcaes ha daa seres s posvely relaed o s volaly. Furhermore, he GARCH-M model mples ha here are seral correlaons n he daa seres self whch were nroduced by hose n he volaly h process. The mere exsence of rsk-premum s, herefore, anoher reason ha some hsorcal socks reurns exhb seral correlaons GJR GARCH Model The GJR model s a smple exenson of GARCH wh an addonal erm added o accoun for possble asymmeres (Brooks, 8:45).Glosen, Jagananhan and Runkle (993) develop he GARCH model whch allows he condonal varance has a dfferen response o pas negave and posve nnovaons. h q p u u d where s a dummy varable ha s: d f u f u,, bad news good news h j j () In he model, effec of good news shows her mpac by, whle bad news shows her mpac by. In addon f news mpac s asymmerc and leverage effec exss. To sasfy non-negavy condon coeffcens would be,, and. Tha s he model s sll accepable, even f, provded ha (Brooks, 8:46) Exponenal GARCH Model Exponenal GARCH (EGARCH) proposed by Nelson (99) whch has form of leverage effecs n s equaon. In he EGARCH model he specfcaon for he condonal covarance s gven by he followng form: q p u logh j logh j j h r uk k () k h k Two advanages saed n Brooks (8) for he pure GARCH specfcaon; by usng log h even f he parameers are negave, wll be posve and asymmeres are allowed for under he EGARCH formulaon. In he equaon k represen leverage effecs whch accouns for he asymmery of he model. Whle he basc GARCH model requres he resrcons he EGARCH model allows unresrced esmaon of he varance (Thomas and Mchell5:6). If k ndcaes leverage effec exs and f k mpac s asymmerc. The meanng of leverage effec bad news ncrease volaly. 5- Emprcal Resuls In hs secon we esmae and dscuss dfferen GARCH models for he frs and second sub- perod of KSE reurns seres. The models are esmaed usng maxmum lkelhood mehod under hree errors dsrbuons namely normal, Suden- and generalzed error dsrbuon (). The lkelhood funcon s maxmzed usng Marquard erave algorhm o search for opmal parameers. Tables 3,5,7,and 9 presens parameers esmaes of GARCH(,), GARCH-M(,), EGARCH(,) and GJR- GARCH(,) models respecvely. In he varance equaon from able 3, all coeffcens (Consan), ARCH erm ( )and GARCH erm( ) are hghly sgnfcan a convenonal levels and wh expeced sgn for he wo perods. The sgnfcance of ndcaes he nformaon abou volaly from he prevous day and explanaory power on curren volaly. In he same way, sascal sgnfcance of he GARCH 38

8 parameer ( ) no only ndcaes explanaory power on curren volaly bu also suggess volaly cluserng n he daly reurns of KSE volaly. Furhermore, he sum of parameers and of second sub- perod s less han one and close o uny,ndcang ha volaly perssence s presen n KSE ndex reurns seres. In conras, he sum of hese parameers for he second sub -perod s larger han one,ndcang ha he condonal varance process s explosve. Perod Frs sub-perod Second sub-perod Table 3: parameer esmaon of GARCH(,) model Suden- Mean Equaon.89.3 (Consan) (.434) (.56) (.944) Varance Equaon (Consan) () () () ( ARCH effec) () () () ( GARCH effec) () () () Mean Equaon (Consan) (.8664) (.8356) (.998) Varance Equaon Consan)( () () () ( ARCH effec) () () () ( GARCH effec) () () () Perod Frs sub-perod Second sub-perod Table 4: Model dagnoscs of GARCH(,) mod Suden- Q () Q () LM () Q () 7.79 (.6).389 (.).64 (.9984) (.).387 (.).37 (.) (.846).374 (.) (.9999) (.456) 5.3 (.758) (.59) Q () (.99) (.) (.) LM () (.696) (.9999) (.989) Fgures n he parenheses are p-values. Q () and Q () are respecvely he Box-Perce sascs a lag of sandardzed and squares sandardzed resduals.,, are he Akake Informaon Creron, Schhwarz Creron and Log lkelhood value respecvely. 39

9 Perod Frs sub-perod Second subperod (Consan) (rsk (Consan) ( ARCH ( GARCH (Consan) (rsk Table 5: parameer esmaon of GARCH-M(,) model premum) effc) effec) premum) ( ) Consan) ( ARCH ( GARCH effc) effec) Suden- Mean Equaon () (.4) () (.55) Varance Equaon () () () () () () Mean Equaon -48 (.7945).4697 (.863) Varance Equaon 8 () ().638 () 56 (.5966).973 (.64) 5 () () () -.93 (.59).3979 (.5) 7 () () () -5 (.9463).763 (.9949) 73 ().3547 ().5985 () Perod Frs sub-perod Second sub-perod Table 6: Model dagnoscs of GARCH -M(,) model for KSE Suden- Q () Q () LM () Q () Q () LM () (.77).68 (.).986 (.9998) (.657).94 (.).946 (.9967) )(.794 (.).3938 (.) (.9).64 (.).69 (.) (.794).3945 (.).846 (.9999) (.863).3963 (.).4653 (.999)

10 From esmaon resuls n able 5,he esmaed coeffcen (rsk premum) of condonal varance n he mean equaon of he GARCH- M(,) model for he frs sub perod s posve and sgnfcan, ndcang ha KSE reurn s rsky "accep". In oher words, when rsk goes up volaly also goes up, so he more he volaly he more he rsky "accep" s. These resuls underscore ha hgh and low of KSE ndex are assocaed wh he rse and fall of he reurns volaly, ha s, an ncrease n he rsk leads o an ncrease n he amoun of he rsk premum demanded by nvesors o compensae for he addonal amoun of rsk o whch hey are exposed. For he second sub- perod, he esmaed of rsk premum s posve, bu nsgnfcan. The resuls of he varance equaon for he GARCH- M(,) model shows ha all coeffcens are hghly sgnfcan a radonal levels. The esmaed resuls of EGARCH(,) and GJR -GARCH(,) models n able 7 and 8 reveal ha all esmaed coeffcen n he varance equaons are Sascally sgnfcan a convenonal levels. Moreover, he esmaes of he leverage effec erm( ) n each of asymmerc models(egarch and GJR -GARCH) are sgnfcan accep bu s nsgnfcan n GJR -GARCH under normal dsrbuon for he second perod. The sgn of s negave n EGARCH model and posve n GJR- GARCH model, whch s conssen wh he normal condons. These resuls sgnfy ha bad news or shocks marke or dsurbng nformaon has more effec on condonal varance han good news, ndcang ha he exsence of leverage effec s observed n reurns of he KSE ndex. Tables 4,6,8 and presen he dagnoscs ess for dfferen GARCH models (GARCH, GARCH-M, EGARCH and GJR -GARCH). From he resuls of hese ables he Ljung Box Q sascs of order on boh sandardzed resduals and squared sandardzed resduals are all non sgnfcan a 5% levels, ndcang ha no seral correlaon exs n he sandardzed resduals of he models. The LM es for presence of ARCH effecs a lag for all GARCH models dd no exhb addonal ARCH effecs. The concluson drawn from hese resuls s ha he GARCH models consdered n smlar nvesgaons are all adequae for descrbng he volaly of Kharoum Sock Exchange(under smlar condons). Perod sub- Frs perod Second subperod (Consan) (Consan) ( ARCH ( GARCH (Leverage (Consan) (Consan) ( ARCH ( GARCH (Leverage Table 7: parameer esmaon of EGARCH(,) model effc) effec) effec) effc) effec) effec) Suden- Mean Equaon () (.776) Varance Equaon () () () () () () () () Mean Equaon -765 (.837) Varance Equaon ().943 () () (.8) - (.64) ().886 ().94 () (.4) 557 (.6676) ().73 ().6476 () () 44 (.93) () () )( -.73 (.3) 4

11 Perod Frs subperod Second subperod Table 8: parameer esmaon of GJR -GARCH(,) model (Consan) (Consan) ( ARCH ( GARCH (Leverage effc) Mean Equaon (Consan) Consan)( ( ARCH ( GARCH (Leverage effec) effec) effc) effec) effec) 77 (.89) 355 () ().3999 ().3999 () -36 (.87) 64 ().9639 (.) ().3477 (.43) Suden- Mean Equaon 4 (.548)) Varance Equaon 4 ().8363 ().67 ().5463 () 85 (.9696) Varance Equaon 73 ().5759 (6).655 ().8 (.53) 564 (.9785) 57 () () ().4596 (.).4 - (.347) 53 ().9579 (.54).7868 )(.3744 (.3) Perod Frs sub-perod Second sub-perod Table 9: Model dagnoscs of EGARCH (,) model Suden- Q () Q () LM () Q () Q () LM () (.596).57 (.).636 (.9985) (.447) (.997) (.7679) )(.843 (.).4864 (.) (.889).797 (.).994 (.) (.857).79 (.).78 (.) (.54) 3.53 (.).794 (.9859) Tables 4,6,9 and presens he dagnoscs ess for dfferen GARCH models (GARCH, GARCH-M, EGARCH and GJR- GARCH). From he resuls of hese ables he Ljung Box Q sascs of order on boh sandardzed resduals and squared sandardzed resduals are all non 4

12 sgnfcan a 5% levels, ndcang ha here s no seral correlaon n he sandardzed resduals of he models. LM es for presence of ARCH effecs a lag for all GARCH models dd no exhb addonal ARCH effecs. The concluson drawn from hese resuls s ha he all GARCH models consdered n hs sudy are all adequae for descrbng he volaly of Kharoum Sock Exchange The resuls of hree selecon crera(,, ) presened n ables 4,6,9 and reveal ha suden- s he mos favored dsrbuon for all models esmaed n hs sudy. Perod Frs sub-perod Second sub-perod Table : Model dagnoscs of GJR -GARCH (,) model Suden- Q () Q () LM () Q () Q () LM () 6.85 (.577).8359 (.).84 (.9996) (.469) (.994) (.77) )(.68 (.).33 (.) (.87).595 (.) (.) (.836).769 (.).5934 (.) (.489) (.998).794 (.9859) Conclusons In hs paper we have modeled and esmaed sock reurns volaly of KSE usng dfferen GARCH models ncludng symmerc and asymmerc models, GARCH(,), GARCH M(,), EGARCH(,) and GJR GARCH(,). Daly reurns seres of KSE ndex reveal some paerns such as posve skewness, lepokuross, sgnfcan deparure from normaly, volaly cluserng and exsence of Heeroscedascy, all of whch are commonly experenced n oher sock markes. Emprcal resuls s also show ha explosve volaly process s presen n KSE ndex reurns over he sample perod. The rsk premum erm for GARCH M(,) s sascally sgnfcan wh posve sgn for he frs sub-perod,mplyng ha an ncrease n volaly s assocaed wh an ncrease n reurns. Moreover, he resuls reveal he exsence of leverage effec n EGARCH and TGARCH models. Our fndngs also show ha Suden- s found o be he mos favored dsrbuon for all models esmaed n our sudy. References: -Alshakh, A. M. E, Sulman, Z. S, (), "Modelng Sock Marke Volaly usng GARCH Models Evdence from Sudan", Inernaonal Journal of Busness and Socal Scence Vol. No. 3. -Alshogeahr,M.A M. A, (998)" Macroeconomc Deermnans of Sock Marke Movemen: Emprcal Evdence from Saud Sock Marke",M.S,Kng Saud Unversy. 3- Arab, K. A, (), "Esmaon of Exchange Rae Volaly va GARCH Model Case Sudy Sudan (978 9)", Inernaonal Journal of Economcs and Fnance; Vol. 4, No.. 4- Al Freed I. A, Shamr. A and Isa. Z, (), "A Sudy on he Behavor of Volaly n Saud Araba Sock Marke Usng Symmerc and Asymmerc GARCH Models", Journal of Mahemacs and Sascs 8 (): Engle,R.F., Llen,D.M., and Robns, R.P.(987) " Esmang Tme Varyng Rsk Prema n he Term Srucure The ARCH-M Model " Economerca, 55, pp

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