Modelling Stock Market Volatility Using Univariate GARCH Models: Evidence from Sudan and Egypt

Transcription

1 Inernaional Journal of Economics and Finance; Vol. 4, No. 8; 01 ISSN X E-ISSN Published by Canadian Cener of Science and Educaion Modelling Sock Marke Volailiy Using Univariae GARCH Models: Evidence from Sudan and Egyp Suliman Zakaria Suliman Abdalla 1 & Peer Winker 1 Deparmen of Quaniaive Analysis, College of Business Adminisraion, King Saud Universiy, Riyadh, Kingdom of Saudi Arabia Deparmen of Economics, Jusus-Liebig Universiy of Giessen, Germany Correspondence: Suliman Zakaria Suliman Abdalla, Deparmen of Quaniaive Analysis, College of Business Adminisraion, King Saud Universiy, Riyadh, Kingdom of Saudi Arabia. Tel: sulabdalla@ksu.edu.sa Received: May 5, 01 Acceped: June 14, 01 Online Published: July 11, 01 doi: /ijef.v4n8p161 URL: hp://dx.doi.org/ /ijef.v4n8p161 Absrac Sock marke volailiy in wo African exchanges, Kharoum Sock Exchange, KSE (from Sudan) and Cairo and Alexandria Sock Exchange, CASE (from Egyp) is modelled and esimaed. The analysis is based on using daily closing prices on he general indices in he wo markes over he period of nd January 006 o 30 h November 010. The paper employs differen univariae specificaions of he Generalized Auoregressive Condiional Heeroscedasic (GARCH) model, including boh symmeric and asymmeric models. The empirical resuls show ha he condiional variance (volailiy) is an explosive process for he KSE index reurns series, while i is quie persisen for he CASE index reurns series. The resuls also provide evidence on he exisence of a posiive risk premium in boh markes, which suppors he hypohesis of a posiive correlaion beween volailiy and he expeced sock reurns. Furhermore, he asymmeric GARCH models find a significan evidence for asymmery in sock reurns in he wo markes, confirming he presence of leverage effec in he reurns series. Keywords: volailiy, sock reurns, GARCH models, heeroscedasiciy, volailiy clusering, leverage effec 1. Inroducion Over he las few years, modelling and forecasing volailiy of financial ime series (i.e. asse reurns) has become a ferile area of research in finance, and has been receiving considerable aenion from academics and praciioners. This is because volailiy is an imporan concep for many economic and financial applicaions, like porfolio opimizaion, risk managemen and asse pricing. A special feaure of volailiy, which according o Tsay (010) is he condiional variance of he underlying asse reurns, is ha i is no direcly observable. Consequenly, financial analyss are especially keen o obain good esimaes of his condiional variance in order o improve porfolio allocaion, risk managemen or valuaion of financial derivaives. Since he 1980s a number of models has been developed ha are especially suied o esimae he condiional volailiy of financial asses. Well-known and frequenly applied models of his ype are he (generalized) condiional heeroscedasic models. Among hese models, he Auoregressive Condiional Heeroscedasic (Noe 1) (ARCH) model proposed by Engle (198) and is exension, he Generalized Auoregressive Condiional Heeroscedasic (GARCH) model developed independenly by Bollerslev (1986), and Taylor (1986) have been he firs models inroduced ino he lieraure and have become very popular (Enders, 004). Since hen, here have been a grea number of empirical applicaions of modelling he condiional variance (volailiy) of financial ime series by employing differen specificaions of hese models and heir many exensions (Noe ). For example, Chou (1988), Nelson (1991), Bollerslev e al. (199), Engle and Paon (001), and Brooks and Burke (003) provide an exended mehodological framework ha can be applied o various problems in finance. On he empirical level, several applicaions have been found on boh developed and developing sock markes, see for example De Sanis and Imrohoroglu (1997), Husain and Uppal (1999), Bekaer and Wu (000), McMillan e al. (000), Engle and Paon (001), Poshakwale and Murinde (001), Brooks and Burke (003), Balaban e al. (004), Ogum e al. (005), Shin (005), Chukwuogor (006), Bali (007), Edel and Brian (007), Floros (007), Leaon (007), Ocran and Biekes (007), Alberg e al. (008), Samouilhan and Shannon (008), Shamiri and Isa (009), Olowe (009), Kalu (010) and Mishra (010)). These models were designed o explicily model and forecas he ime-varying condiional second order momen 161

2 Inernaional Journal of Economics and Finance Vol. 4, No. 8; 01 (variance) of a series by using pas unpredicable changes in he reurns of ha series, and have been applied successfully in economics and finance, bu more predominanly in financial marke research. For he case of Egyp here are some sudies concerned wih he issue of modelling sock marke volailiy (see for example Tooma (003), Mecagni and Sourial (1999), Ebeid and Bedeir (004), and Floros (008)), bu o he bes of our knowledge, here are no such empirical sudies for he Sudanese sock marke. Thus, one of he conribuions of his paper is o provide empirical evidence on he fi of condiional volailiy models for he Sudanese sock marke. The main objecive of his paper is o model sock reurns volailiy in wo African markes; he Sudanese sock marke (Kharoum Sock Exchange, KSE), and he Egypian sock marke (Cairo and Alexandria Sock Exchange, CASE) by employing differen univariae specificaions of GARCH ype models for daily observaions on he index reurns series of each marke over he period of nd January 006 o 30h November 010, as well as describing special feaures of he markes in erms of rading aciviy and index componens and calculaions. The volailiy models employed in his paper include boh symmeric and asymmeric GARCH models. While he CASE migh be considered as one of he mos advanced sock markes wihin he African coninen, he KSE has been esablished only recenly and exhibis a limied rack record. Therefore, i migh be of paricular ineres o compare he volailiy on boh markes. The remaining of his paper is organized as follows: Secion provides a general overview of he sock exchange in Sudan and Egyp. Secion 3 describes he daa and provides summary saisics. In he fourh secion he GARCH mehodology is presened, while he esimaions resuls are discussed in Secion 5. Finally, Secion 6 concludes he paper.. Lieraure Review There has been a large amoun of lieraure on modelling and forecasing sock marke volailiy in boh developed and developing counries around he world. Many economeric models have been used o invesigae volailiy characerisics. However, no single model is superior. Pindyck (1984) demonsraes ha he increases in variance of sock reurns can explain much of he decline in sock prices. Whielaw (1994) offers empirical evidence for a posiive relaion beween a lagged volailiy measure and fuure expeced reurns. For Asian sock markes, Koumos (1999) and Koumos and Saidi (1995) found ha he condiional variance is an asymmeric funcion of pas innovaions. Posiive pas reurns are on average 1.4 imes more persisen han negaive pas reurns of an equal magniude. Lee e al. (001) examined ime-series feaures of sock reurns and volailiy in four of China s sock exchanges. They provided srong evidence of ime-varying volailiy and indicaed volailiy is highly persisen and predicable. Moreover, evidence in suppor of a fa-ailed condiional disribuion of reurns was found. By employing eleven models and using symmeric and asymmeric loss funcions o evaluae he performance of hese models, Balaban, Bayar, and Faff (003) forecased sock marke volailiy of foureen sock markes. According o symmeric loss funcions he exponenial smoohing model provides he bes forecas. However, when asymmeric loss funcions are applied ARCH-ype models provide he bes forecas. Balaban and Bayar (005) used boh symmeric and asymmeric ARCH-ype models o derive volailiy expecaions. The oucome showed ha here has a posiive effec of expeced volailiy on weekly and monhly sock reurns of boh Philippines and Thailand markes according o ARCH model. The resul is no clear if using he oher models such as GARCH, GJR-GARCH and EGARCH. For emerging African markes, Ogum, Beer and Nouyriga (005) invesigae he marke volailiy using Nigeria and Kenya sock reurn series. Resuls of he exponenial GARCH model indicae ha asymmeric volailiy found in he U.S. and oher developed markes is also presen in Nigerian sock exchange (NSE), bu Kenya shows evidence of significan and posiive asymmeric volailiy. Also, hey show ha while he Nairobi Sock Exchange reurn series indicae negaive and insignifican risk-premium parameers, he NSE reurn series exhibi a significan and posiive ime-varying risk premium. By using asymmeric GARCH models, Alberg e al. (006) esimae sock marke volailiy of Tel Aviv Sock Exchange indices, for he period They repor ha he EGARCH model is he mos successful in forecasing he TASE indices. Various ime series mehods are employed by Tudor (008), including he simple GARCH model, he GARCH-in-Mean model and he exponenial GARCH o invesigae he Risk-Reurn Trade-off on he Romanian sock marke. Resuls of he sudy confirm ha E-GARCH is he bes fiing model for he Buchares Sock Exchange composie index volailiy in erms of sample-fi. 3. Overview of Sock Marke in Sudan and Egyp 3.1 Sudanese Sock Marke The Kharoum Sock Exchange (KSE) is he principal sock exchange of Sudan locaed in Kharoum. The KSE sared is aciviies officially in January 1995 wih he assisance of he Common Marke for Easern and Souhern Africa (CoMESA) (Noe 3), wih he objecive of regulaing and conrolling he issuance of securiies, and 16

3 Inernaional Journal of Economics and Finance Vol. 4, No. 8; 01 mobilizing privae savings for invesmen in securiies. Securiies raded in he KSE are ordinary shares and invesmen unis (Noe 4). Furhermore, a subsanial number of muual funds and Governmen Invesmen Cerificaes (GICs) (Noe 5) are also raded (KSE Annual repor, 010). Orders are handled hrough brokers during rading hours and share prices are quoed in Sudanese Pound (SDG). Trading is processed manually by coninuous aucion from Sunday o Thursday for one hour from am o am. Thereby, buy and sell orders are passed on o floor-based represenaives of regisered brokers for execuion. Trading in securiies is aking place in wo markes, he so called primary and secondary markes (Noe 6). As a par of he financial sysem of Sudan, KSE operaes on he basis of Islamic Shariaa and is supervised and regulaed by he Cenral Bank of Sudan (Noe 7). The key feaure of Islamic Shariaa pracices in Kharoum Sock Exchange is ha i is aimed o offer invesmen porfolios from common socks of lised companies which ideally saisfy hree basic crieria: (i) legiimae field of economic aciviy; (ii) ineres-free dealings in boh asses and liabiliies, and (iii) he dominance of real asses. Thus, e.g., a company mus no be engaged in he producion of illegiimae goods like alcoholic drinks; i mus no deal wih ineres rae financing as a means o leverage is capial srucure hrough fixed deb liabiliies, or generae ineres income from invesmen securiies; and since a company s shares represen equiy righs in is asses, he laer should be real asses, no liquid money or receivable deb as hey canno be sold freely a a profi like real goods, real esae and machinery (Hassan and Lewis, 007). As consequences of hese rules, he composiion of asses raded a he KSE differs subsanially from oher sock markes. In paricular, due o he regulaions imposed by Islamic Shariaa (Noe 8) pracices a separae class of invesmen vehicles on he KSE is provided by he so called Governmen Musharakah (Noe 9) Cerificaes (GMCs), which represen an Islamic equivalen o convenional bonds (also known as Shahama bonds). Shahama bonds offer a way for he governmen o borrow money in he domesic marke insead of prining more banknoes. Afer one year, holders of GMCs can eiher liquidae hem or exend heir duraion. These bonds are backed by he socks of various companies owned by he Minisry of Finance. Consequenly, hey migh be considered as asse-backed securiies. The profiabiliy of GMCs depends on he financial resuls of he companies in he underlying porfolio. I can reach up o 33 per cen per annum. Hence, he profi of GMCs is variable raher han fixed. The governmen issues hese bonds on a quarerly basis and heir placemen on he marke is done usually very fas- in jus six days. The overall performance of he Kharoum sock marke is measured by he KSE Index, which is a marke capializaion-weighed index. In Sepember 003, he KSE index was esablished and lised in he Arab Moneary Fund daabase. A he end of he firs monh he index closed a poins. In December 005, he index closed a he highes level of poins. In November 010, i was flucuaing around an average value of Egypian Sock Marke In conras o he KSE, he Egypian exchange is one of he oldes sock markes esablished in he Middle Eas and Africa. Egyp s sock exchange has wo locaions: he main locaion is in Cairo (esablished 1903) and he oher one is in Alexandria (esablished 1883).These wo exchanges were compeing wih each oher before hey merged in recen years. Today, boh exchanges are governed by he same chairman and board of direcors. They are commonly referred o as he Cairo and Alexandria Sock Exchange (CASE) and share he same rading, clearing and selemen sysems, so ha marke paricipans have access o socks lised on boh exchanges. The overall performance of he Egypian sock marke is measured by he Capial Marke Auhoriy (CMA) Index, which covers all lised socks weighed in relaion o heir marke capializaion. I can be viewed as an all share index ha covers he broades base of socks. I is calculaed and released daily by he CMA (Noe 10). 3.3 Key Numbers of he KSE and CASE Table 1 provides some key figures of boh exchanges. I is obvious ha considering any of he indicaors used like number of lised companies or marke capializaion, he Kharoum sock exchange represens a much smaller marke compared o he Cairo and Alexandria sock exchange. While he number of lised companies comes close o one quarer of he CASE in 01, he number of ransacions falls shor of 0.1 percen and marke capializaion is below 5 percen of he corresponding values of CASE. The relaively large number of lised companies a he KSE appears o be due raher o a massive decline of he number of lised companies on he CASE since 006 han o an increase of aciviies on he KSE. Considering he volume of rading relaive o marke capializaion, boh markes exhibi some similariies, i.e. he yearly rading volume reaches abou 30 percen of marke capializaion in recen years. 163

4 Inernaional Journal of Economics and Finance Vol. 4, No. 8; 01 Table 1. Summary of Trading Aciviy in KSE and CASE, Marke No. Of Lised No. Of raded shares (In Volume of rading No. of ransacions Capializaion companies* Million) ($ millions) ($ millions) KSE CASE KSE CASE KSE CASE KSE CASE KSE CASE , NA NA 5,84 NA 3,91.61 NA , , , ,195 8,161,607 4, , , , ,177 1,31,53 3, , , , ,069 13,300,653, , , , ,66 9,606,668 3, ,75.96 Source: Compiled by he auhors based on daa from he KSE websie and AMF annual repors. 4. Research Mehods Auoregressive condiional heeroscedasiciy (ARCH) (Noe 1) and is generalizaion (GARCH) models represen he main mehodologies ha have been applied in modelling and forecasing sock marke volailiy (Noe 13) in empirical finance. In his paper differen univariae GARCH specificaions are employed o model sock reurns volailiy in Kharoum sock exchange and Cairo and Alexandria sock exchange, hese models are GARCH (1,1), GARCH-M (1,1), which will be used for esing symmeric volailiy and EGARCH(1,1), TGARCH(1,1) and PGARCH (1,1) for modelling asymmeric volailiy (Noe 14). These models will be shorly discussed in he following subsecions. For all hese differen models, here are wo disinc equaions, he firs one for he condiional mean and he second one for he condiional variance. We are mainly ineresed in he second equaion as i provides esimaes and condiional forecas of volailiy. 4.1 Symmeric GARCH Models The Generalized Auoregressive Condiional Heeroscedasic (GARCH) Model In his model, he condiional variance is represened as a linear funcion of a long erm mean of he variance, is own lags and he previous realized variance. The simples model specificaion is he GARCH (1,1) model: where 0, 0 and 0 1 Mean equaion Variance equaion 1 r = reurn of he asse a ime, = average reurn, = residual reurns, defined as:, and: r, (1) () z (3) where z are sandardized residual reurns (i.e. realizaion of an iid random variable wih zero mean and variance 1), and sands for he condiional variance. For GARCH (1,1), he consrains 1 0 and 1 0 are needed o ensure ha is sricly posiive (Poon, 005). The condiional variance equaion models he ime varying naure of volailiy of he residuals generaed from he mean equaion. This specificaion is ofen inerpreed in a financial conex, where an agen or rader predics his period s variance by forming a weighed average of a long erm average (he consan), he forecas variance from las period (he GARCH erm), and informaion abou volailiy observed in he previous period (he ARCH erm). If he asse reurn was unexpecedly large in eiher he upward or he downward direcion, hen he rader will increase he esimae of he variance for he nex period, while he GARCH-erm generaes persisence of volailiy The Generalized Auoregressive Condiional Heeroscedasic-in-Mean (GARCH-M) Model In finance, he reurn of a securiy may depend on is volailiy. To model such a phenomenon one may consider he GARCH-M model developed by of Engle, Lilien, and Robins (1987), where "M" sands for GARCH in he mean (Tsay 010). This model is an exension of he basic GARCH framework which allows he condiional mean of a 164

5 Inernaional Journal of Economics and Finance Vol. 4, No. 8; 01 sequence o depends on is condiional variance or sandard deviaion. A simple GARCH -M(1,1) model can be wrien as : Mean equaion r Variance equaion (5) The parameer in he mean equaion is called he risk premium parameer. A posiive indicaes ha he reurn is posiively relaed o is volailiy. In oher words, a rise in mean reurn is caused by an increase in condiional variance as a proxy of increased risk. Engle, Lilien, and Robins assume ha he risk premium is an increasing funcion of he condiional variance of ; in oher words, he greaer he condiional variance of reurns, he greaer he compensaion necessary o induce he agen o hold he asse (Enders 004). 4. Asymmeric GARCH Models An ineresing feaure of asse prices is ha bad news seems o have a more pronounced effec on volailiy han do good news. For many socks, here is a srong negaive correlaion beween he curren reurn and he fuure volailiy. The endency for volailiy o decline when reurns rise and o rise when reurns fall is ofen called he leverage effec (Enders, 004). The main drawback of symmeric GARCH models is ha he condiional variance is unable o respond asymmerically o rises and falls in, and such effecs are believed o be imporan in he behaviour of sock reurns. In he linear GARCH (p,q) model he condiional variance is a funcion of pas condiional variances and squared innovaions; herefore, he sign of reurns canno affec he volailiies (Knigh and Sachell, 00). Consequenly, he symmeric GARCH models described above canno accoun for he leverage effec observed in sock reurns, consequenly, a number of models have been inroduced o deal wih his phenomenon. These models are called asymmeric models. This paper uses EGARCH, TGARCH and PGARCH for capuring he asymmeric phenomena The Exponenial Generalized Auoregressive Condiional Heeroscedasic (EGARCH) Model This model capures asymmeric responses of he ime-varying variance o shocks and, a he same ime, ensures ha he variance is always posiive. I was developed by Nelson (1991) wih he following simple specificaion: 1 1 Ln( ) 1Ln( 1 ) 1, (6) 1 1 where is he asymmeric response parameer or leverage parameer. The sign of is expeced o be posiive in mos empirical cases so ha a negaive shock increases fuure volailiy or uncerainy while a posiive shock eases he effec on fuure uncerainy (Noe 15). 4.. The Threshold Generalized Auoregressive Condiional Heeroscedasic (TGARCH) Model Anoher volailiy model commonly used o handle leverage effecs is he hreshold GARCH (or TGARCH) developed by Zakoian (1994). In he TGARCH (1,1) version of he model, he specificaion of he condiional variance (Noe 16) is: 1 1 d , (7) where d 1 is a dummy variable, ha is: 1 if 1 0, bad news d 1 (8) 0 if 1 0, good news Again, he coefficien is known as he asymmery or leverage parameer. When 0, he model collapses o he sandard GARCH forms. Oherwise, when he shock is posiive (i.e., good news) he effec on volailiy is 1, bu when he news is negaive (i.e., bad news) he effec on volailiy is 1. Hence, if is significan and posiive, negaive shocks have a larger effec on han posiive shocks (Carer, 007) The Power Generalized Auoregressive Condiional Heeroscedasic (PGARCH) Model Ding, Granger and Engle (1993) also inroduced he Power GARCH (PGARCH) specificaion o deal wih asymmery. Unlike oher GARCH models, in his model, he sandard deviaion is modelled raher han he variance as in mos of he GARCH-family. In Power GARCH an opional parameer can be added o accoun 1 1 (4) 165

6 Inernaional Journal of Economics and Finance Vol. 4, No. 8; 01 for asymmery (Floros, 008). The model also offers one he opporuniy o esimae he power parameer insead of imposing i on he model (Ocran and Biekes, 007). The general asymmeric Power GARCH model specifies as of he following form: 1 1 1( ) (9) where 1 and 1 are he sandard ARCH and GARCH parameers, 1 is he leverage parameer and is he parameer for he power erm. When, equaion (9) becomes a classic GARCH model ha allows for leverage effecs, and when 1, he condiional sandard deviaion will be esimaed. I is possible o increase he flexibiliy of he PGARCH model by considering as anoher coefficien ha also has o be esimaed (see Zivo 008). 5. Daa and Empirical Resuls 5.1 The Daa and Basic Saisics The Daa Used for he Analysis The ime series daa used for modelling volailiy in his paper are he daily closing prices of he Kharoum Sock Exchange (KSE) index and he Capial Marke Auhoriy (CMA) index over he period from nd January 006 o 30h November 010, resuling in a oal of 136 observaions for he KSE index and 187 for he CMA index excluding public holidays. These closing prices have been aken from he KSE websie (hp:// and he CASE websie (hp:// Daily reurns r were calculaed as he coninuously compounded reurns corresponding o he firs difference in logarihms of closing prices of successive days: P r log (10) P 1 where P and P 1 denoe he closing marke index of KSE and CASE a he curren () and previous day (-1), respecively. I is very imporan o noe ha since Ocober 18, 009, he index on he Kharoum Sock Marke has been declining. In only 16 rading days, he sock marke index fell from Ocober 18, 009 o on November 10, 009. Since ha ime, he KSE index was reporing o flucuae around an average value of In order o see he impac of his sharp fall on he volailiy modeling, he full daa se is divided ino wo s: he firs covers Jan., 006 o Oc. 18, 009 wih 104 oal observaions, while he second ranges from Nov. 10, 009 o Nov resuling in 69 observaions. So, he resuls will be presened separaely for hree periods; for he period before he sharp fall, he period afer ha fall and for he whole daa se Descripive Saisics of KSE and CASE Reurns Series To specify he disribuional properies of he daily reurns series in KSE and CASE markes during he period of his sudy, some descripive saisics are repored in Table. Table. Descripive saisics of he KSE and CMAI reurn series Saisics KSE reurn series CMAI reurn series Firs Second Full sample period Mean 0.01% 0.00% -0.0% 0.01% Median 0% 0% 0.00% 0.01% Maximum 1% 1% 1.1% 6.55% Minimum -11% -1% % 17.43% Sandard Deviaion 1.47% 0.71% 1.37% 1.83% Skewness Kurosis Jarque-Bera Prob. of Jarque-Bera No. of observaions Sample period: January, 006 November 30,

7 Inernaional Journal of Economics and Finance Vol. 4, No. 8; 01 As we can see from Table, he average reurns for he CMA index is higher han he averagee of he KSE index. The disribuion of reurns differs markedly from normaliy given he observed skewness and excess kurosis. Consequenly, based on he Jarque-Bera (J-B) saisic, he null hypohesis of normaliy for he daily KSEE and CASE reurns has o be rejeced a he 1% significance level. Moreover, Figure 1 presens he paern of he price index series and is reurns for he CASE and he KSE during he sudy period. Figure 1. Daily prices and reurns for he CMA and KSE indices (Jan. 006 Nov. 010) While he ime series for CMA (upper par) resemblee qualiaively hose found for oher sock exchanges, he ime series for KSE exhibi differen paerns. In paricular, he dependence in he volailiy of reurns as shown in Figure 1 (lower par) appears much more pronounced, a leas up o he end of 009. Aferwards, he price series s becomes almos fla wih very small reurns. We will consider his obvious srucural break in our empirical analysis alhough i urned ou o be difficul o idenify a specific reason for his change Quanile-Quanile (Q-Q) Plos As a furher insrumen for analyzing he disribuional properies, we apply he Q-Q graphical examinaion o check wheher he KSE index and CMA index reurns series are normally disribued. The Q-Q plo is a scaerr plo of he empirical quaniles (verical axis) agains he heoreical quaniles (horizonal axis) of a given disribuion (Alexander, 001). If he sample observaions follow approximaely a normal disribuion wih mean equal o he empirical mean ( ) and sandard deviaion equal o he empirical sandard deviaion, hen he resulingg plo should be roughly scaered around he 45-degree line wih a posiive slope. The greaer he deparure from his line, he greaer he evidence agains he null hypohesis of a normal disribuion. The resulss of his graphical examinaion are provided in Figure. 167

8 Inernaional Journal of Economics and Finance Vol. 4, No. 8; Quaniles of Normal Quaniles of KSE Reurns Quaniles of Normal Quaniles of CMA Reurns Figure. Normal Quanile-Quanile Plos for he Daily Sock Reurns The QQ-plo in Figure confirms he findings from Table ha he KSE and CASE reurns daa do no follow a disribuion similar o a normal disribuion Tesing for Saionariy To invesigae wheher he daily price index and is reurns are saionary series, he Augmened Dickey Fuller (ADF) es (Dickey and Fuller, 1981) has been applied. Thereby, he lag lengh has been seleced auomaically based on he Schwarz informaion crierion wih a prese maximum lag lengh of. The resuls are repored in Table

9 Inernaional Journal of Economics and Finance Vol. 4, No. 8; 01 Table 3. ADF uni roo es oupu for he price index and reurns series in KSE and CASE ADF uni roo es for he price index series Index ADF saisic Criical values 1% 5% 10% KSE index Firs -.671(5)* Second (0)** Full period -.390(6) CMAI (1) ADF uni roo es for he reurn series Reurn ADF saisic Criical values 1% 5% 10% KSE index reurn Firs (1)** Second -0.35()** Full period (5)* CMA index (0)* Noes: 1- Figures in parenheses denoe he opimal lag lenghs, which were auomaically seleced based on he Schwarz Informaion Crierion (SIC). - Criical values for uni roo ess are aken from MacKinnon (1996). 3- * and ** Indicae ha he resuls are saisically significan a he 1% and 5% levels respecively. 4- ADF es includes a consan erm wihou rend. Table 3 repors he resuls of he ADF es for a lag lengh of 6 and 1 for he wo indices in levels and a lag lengh of 5 and 0 for he wo reurns series. The ADF ess for he level daa indicae ha hey have o be considered as non-saionary series for boh markes (Noe 11). When applying he same es for he reurns series, he resuls allow rejecing he null hypohesis of a uni roo a all convenional levels of significance for boh series. Therefore, we conclude ha he reurns series migh be considered as saionary over he specified period Tesing for Heeroscedasiciy Given ha we are ineresed in analyzing volailiy on boh markes, a firs sep consiss in esing for (condiional) heeroscedasiciy. To his end we apply he Lagrange Muliplier (LM) es proposed by Engle (198) o he residuals of simple ime series models of he reurns. In summary, he es procedure is performed by firs obaining he residuals e from he ordinary leas squares regression of he condiional mean equaion which migh be an auoregressive (AR) process, moving average (MA) process or a combinaion of AR and MA processes, i.e. an ARMA process. For example, in he ARMA (1,1) process he condiional mean equaion will be: r. (11) 1r Afer obaining he residuals e, he nex sep consiss in regressing he squared residuals on a consan and q lags as in he following equaion: e 1 1 e 0 e qe q... (1) The null hypohesis ha here is no auoregressive condiional heeroscedasiciy (ARCH) up o order q can be formulaed as: H :.. 0 agains he alernaive: 0 1 q H1 : i 0 for a leas one i = 1,,, q. The es saisic for he join significance of he q-lagged squared residuals is given by he number of observaions imes he R-squared ( TR ) of he regression (1). TR is evaluaed agains he ( q ) disribuion. This represens an asympoically locally mos powerful es (Rachev e al., 007, 94). 169

10 Inernaional Journal of Economics and Finance Vol. 4, No. 8; 01 In our case, we firs employ an auoregressive moving average ARMA (1,1) model for he condiional mean in he reurns series as an iniial regression, hen, es he null hypohesis ha here are no ARCH effecs in he residual series up o lag 5 corresponding o one rading week. The resuls of his examinaion are summarized in Table 4. Table 4. ARCH-LM Tes for residuals of reurns on he KSE and CASE markes KSE index reurn CMAI reurn ARCH-LM es saisic Prob. Chi-square (5) Noe: H 0 : There are no ARCH effecs in he residual series. The ARCH-LM es resuls in Table 4 provide srong evidence for rejecing he null hypohesis. Rejecing H 0 is an indicaion of he exisence of ARCH effecs in he residuals series of he mean equaion and herefore he variance of he reurns series of KSE and CASE indices are non-consan. 5. Empirical Resuls As repored in he daa descripion par, when he residuals were examined for heeroscedasiciy, he ARCH-LM es provided srong evidence of ARCH effecs in he residual series of boh markes. To model his condiional heeroscedasiciy, we proceed by applying he GARCH models. The resuls of esimaing differen GARCH specificaions for he KSE index and he CMA index reurns are presened in his secion. The models are esimaed using he maximum likelihood mehod under he assumpion of Gaussian disribued error erms. The log likelihood funcion is maximized using Marquard s numerical ieraive algorihm o search for opimal parameers (Noe 17). To accoun for he sharp decline of he KSE index in he second half of Ocober 009, a dummy variable (DUM) will be inroduced ino he mean equaion, which is se equal o 0 for he period before ha sharp decline and 1 hereafer. Thus, for he KSE, he mean equaion is specified as: Mean equaion r DUM (13) Beside he esimaion oupu of differen GARCH models, diagnosics es resuls for hese models are also provided, in paricular for esing wheher here are sill ARCH effecs lef in he residuals of he esimaed models (Noe 18). Table 5 and Table 6 show he parameer esimaes of differen GARCH models for he reurns of he KSE (Full sample period) and CASE indices for he period under sudy. Esimaion resuls of subperiods for he KSE reurns are repored in Table 1 in he Appendix. Table 5. Esimaion resuls of differen GARCH models for Kharoum sock exchange Coefficiens GARCH (1,1) GARCH-M (1,1) EGARCH (1,1) TGARCH (1,1) PGARCH (1,1) Mean equaion ** DUM ** E Variance equaion.69e-05*.99e-05* * 3.0E-05* ** * * * * * * * * ** * * * * Log likelihood ARCH-LM es for heeroscedasiciy saisic Prob Noe: * Denoes significance a he 1% level, and ** a 5% level 170

11 Inernaional Journal of Economics and Finance Vol. 4, No. 8; 01 Table 6. Esimaion resuls of differen GARCH models for Egyp sock marke Coefficiens GARCH (1,1) GARCH-M (1,1) EGARCH (1,1) TGARCH (1,1) PGARCH (1,1) Mean equaion ** ** Variance equaion.38e-06*.49e-06* * 3.67E-06* 3.70E * * * * * * * * * * * * * * Log likelihood ARCH-LM es for heeroscedasiciy saisic Prob Noes: * Denoes significance a he 1% level, and ** a 5% level In he resuls for he variance equaion repored in Tables 5 and 6, he firs hree coefficiens (consan), ARCH erm ( ) and GARCH erm ( ) for he GARCH (1,1) model are saisically significan and exhibi he expeced sign for boh markes. The significance of and indicaes ha, lagged condiional variance and lagged squared disurbance have an impac on he condiional variance, in oher words his means ha news abou volailiy from he previous periods have an explanaory power on curren volailiy. Moreover, Table 5 shows ha he sum of he wo esimaed ARCH and GARCH coefficiens (persisence coefficiens) in he GARCH (1,1) model for he KSE is larger han one, suggesing ha he condiional variance process is explosive. However, for he CASE reurns he sum of he ARCH and GARCH coefficiens is very close o one which is required o have a mean revering variance process, indicaing ha volailiy shocks are quie persisen, bu no explosive. Thus, for CASE he findings correspond o hose for many developed sock exchanges, while we find a deparure for he KSE. The GARCH-M (1,1) model is esimaed by allowing he mean equaion of he reurn series o depend on a funcion of he condiional variance. From esimaion resuls in Table 5 and Table 6, he esimaed coefficien (risk in he mean equaion is posiive for he wo markes, which indicaes ha he mean of he reurn premium) of sequence depends on pas innovaions and he pas condiional variance. In oher words, condiional variance used as a proxy for risk of reurns is posiively relaed o he level of reurns. These resuls show ha as volailiy increases, he reurns correspondingly increase wih a facor of 0.08 and 0.06 for KSE and CASE, respecively. These resuls are consisen wih he heory of a posiive risk premium on sock indices which saes ha higher reurns are expeced for asses wih higher level of risk. The effec urns ou o be significan for CASE. Furhermore, he asymmeric models EGARCH (1,1) and TGARCH (1,1) are used o invesigae he exisence of leverage effecs in he reurns of he KSE and he CASE indices during he sudy period. The main difference beween hese wo models is ha in he EGARCH model, here is no need of nonnegaive resricion of he parameers while in he TGARCH model parameers mus saisfy he posiive condiion (Irfan 010). The asymmerical EGARCH (1,1) esimaed for he reurns of he KSE index in Table 5 and he CASE index in Table 6 indicaes ha all he esimaed coefficiens are saisically significan a he 1% confidence level. The asymmeric (leverage) effec capured by he parameer esimae is also saisically significan wih negaive sign, indicaing ha negaive shocks imply a higher nex period condiional variance han posiive shocks of he same sign, which indicaes he exisence of leverage effecs in he reurns of he Kharoum sock marke index and Cairo and Alexandria Sock Exchange during he sudy period. An alernaive model o es for asymmery in KSE and CASE reurns is he TGARCH (1,1) model. From esimaion resuls of his model repored in Table 5 and Table 6, he coefficien for he leverage effec is significan and posiive for boh markes. The significance of his coefficien indicaes ha negaive shocks (bad news) have a larger effec on he condiional variance (volailiy) han posiive shocks (good news) of he same magniude. The oher version of asymmeric GARCH model applied in his paper is he (PGARCH). From he resuls for he PGARCH (1,1) in Table 5 and Table 6, he esimaed coefficien is significan and posiive for he case of Egyp, indicaing ha posiive shocks are associaed wih higher volailiy han negaive shocks. In case of Sudan, he esimaed coefficien is negaive, bu insignifican. 171

12 Inernaional Journal of Economics and Finance Vol. 4, No. 8; 01 The resuls of diagnosic ess (es for ARCH effecs) are repored below he esimaion resuls in Table 5 and Table 6. The ARCH-LM es saisic for all GARCH models (where ARCH and GARCH erms are aken o be of order 1) did no exhibi any addiional ARCH effec remaining in he residuals of he models. This shows ha he variance equaions are well specified for he wo markes. (excep for he EGARCH (1,1) model for he KSE). 6. Summary and Concluding Remarks Modelling and forecasing volailiy of reurns in sock markes has become a ferile field of empirical research in financial markes, because volailiy is considered as an imporan concep in many economic and financial applicaions like asse pricing, risk managemen and porfolio allocaion. In his paper we have modeled and esimaed sock reurn volailiy in wo African markes; he Sudanese sock marke (Kharoum Sock Exchange, KSE), and he Egypian sock marke (Cairo and Alexandria Sock Exchange, CASE) by applying differen univariae specificaions of GARCH ype models for daily observaions on he index series of each marke over he period of nd January 006 o 30h November 010, as well as describing special feaures of he markes in erms of rading aciviy and index componens and calculaions. A oal of five differen models were considered in his paper. The volailiy of he KSE and CASE reurns have been modelled by using univariae Generalized Auoregressive Condiional Heeroscedasic (GARCH) models including boh symmeric and asymmeric models ha capure mos common sylized facs abou index reurns such as volailiy clusering and leverage effecs. These models are GARCH(1,1), GARCH-M(1,1), exponenial GARCH(1,1), hreshold GARCH(1,1) and power GARCH(1,1). The firs wo models imply a symmeric effec of pas shocks whereas he second group of models allows capuring asymmeric effecs. Based on he empirical resuls presened, he following can be concluded: Firs, he paper finds srong evidence ha daily reurns could be characerized by he above menioned models for he wo markes, KSE and CASE daa showed a significan deparure from normaliy and he exisence of heeroscedasiciy in he residuals series. Second, he parameer esimaes of he GARCH (1,1) models ( and ) indicae ha he condiional volailiy of sock reurns on he Kharoum Sock Exchange is an explosive process, while i is quie persisen for he CASE index reurns series. Third, he parameer describing he condiional variance in he mean equaion, measuring he risk premium effec for GARCH-M(1,1), is saisically significan in he wo markes, and he sign of he risk premium parameer is posiive. The implicaion is ha an increase in volailiy is linked o an increase of reurns, which is an expeced resul. Fourh, based on asymmerical EGARCH (1,1) and TGARCH(1,1) esimaion, he resuls show a significan evidence for he exisence of he leverage effecs in he wo markes, he same resul is confirmed only for he CASE by using he PGARCH(1,1) model. I is lef o fuure research o sudy in more deail he causes of he srucural break in he KSE ime series and how i can be aken ino accoun explicily in he volailiy equaions. Furhermore, i migh be sudied o wha exen volailiy forecass based on he presen models are useful in he conex of risk managemen for he sock markes considered. Acknowledgemen Financial suppor provided by he German Academic Exchange Service (DAAD) is graefully acknowledged. References Alberg, D., Shali, H., & Yosef, R. (008). Esimaing sock marke volailiy using asymmeric GARCH models. Applied Financial Economics, 18, hp://dx.doi.org/ / Alexander, C. (001). Marke Models: A Guide o Financial Daa Analysis. Chicheser. John Wiley and Sons Ld. Aly, H., Mehdian, S., & Perry, M. (004). An Analysis of he Day-of-he-Week Effecs in he Egypian Sock Marke. Inernaional Journal of Business, 9(3), Ayub, M. (007). Undersanding Islamic Finance. Chicheser, England, John Wiley & Sons Ld. Balaban E., & Bayar A. (005). Sock reurns and volailiy: empirical evidence from foureen counries. Applied Economics Leers, 1, hp://dx.doi.org/ / Balaban E., Bayar, A., & Faff, R. (003). Forecasing Sock Marke Volailiy: Evidence from 14 Counries. 10h Global Finance Conference 003, Frankfur/Main, June 15-17, 003. Balaban, E., Bayar, & A. Faff, R. (004). Forecasing Sock Marke Volailiy: Furher Inernaional Evidence. ASAC, Quebec, Canada. Bali, T. G. (007). Modeling he dynamics of ineres rae volailiy wih skew fa-ailed disribuions. Annals of Operaions Research, 1, hp://dx.doi.org/ /s

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15 Inernaional Journal of Economics and Finance Vol. 4, No. 8; 01 Taylor, S. J. (1986). Modelling financial ime series. John Wiley & Sons, Chicheser. Tooma, E. A. (003). Modeling and Forecasing Egypian sock marke volailiy before and afer price limis. Working Paper 0310, Economic Research Forum, Cairo, Egyp. Tsay, R. S. (010). Analysis of Financial Time Series. 3rd Ediion New York, Unied Saes of America, John Wiley & Sons, Inc. hp://dx.doi.org/10.100/ Tudor, C. (008). An Empirical Sudy on Risk-Reurn Tradeoff Using GARCH-Class Models: Evidence from Buchares Sock Exchange. Inernaional Conference on Business and Economy ICBE Consana - Romania, November 6-8. Usmani, T. (1998). An Inroducion o Islamic Finance. Idaraul Maarif, Karachi. Venardos, Angelo M. (010). Curren Issues in Islamic Banking and Finance: Resilience and Sabiliy in he Presen Sysem. Singapore: World Scienific Publishing. hp://dx.doi.org/10.114/ Whielaw, R. (1994). Time variaions and covariaions in he expecaion and volailiy of sock marke reurns. Journal of Finance, 49, Zakoian, J. M. (1994). Threshold Heeroscedasic Models. Journal of Economic Dynamics and Conrol, 18, hp://dx.doi.org/ / (94) Zivo. E. (008). Pracical Issues in he Analysis of Univariae GARCH Models. Handbook of Financial Time Series, Springer, New York. Noes Noe 1. A ime series is said o be heeroscedasic if is variance changes over ime, oherwise i is called homoscedasic. Noe. To menion only a few of he mos frequenly used: GARCH-M model by Engle, Lilien, and Robins (1987), IGARCH model by Engle and Bollerslev (1986), Exponenial GARCH model by Nelson (1991), Threshold GARCH model by Zakoian (1994) and Glosen e al. (1993) and Power ARCH model by Ding e al. (1993). Noe 3. Member saes are: Burundi, Comoros, Democraic Republic of Congo, Djiboui, Egyp, Erirea, Ehiopia, Kenya, Libya, Madagascar, Malawi, Mauriius, Rwanda, Seychelles, Sudan, Swaziland, Uganda, Zambia and Zimbabwe. Noe 4. An invesmen uni is a proporional accouning share in he oal ne asses of an open end invesmen fund (Invesmen funds are he insiuions of collecive invesmen which serve as framework for collecion of money funds. Colleced money funds are hen invesed in various asses). The invesmen uni value is an indicaor of how successful a fund is, and he changes of his value depend on he flucuaion of prices of securiies and oher propery ha he fund has invesed in. Noe 5. Governmen invesmen cerificaes (GICs) are medium-erm securiies, based on various conracs financed by he Minisry of Finance of Sudan via he isisna, murabaha and ijara ools. Issuance of hese sukuk is similar o he convenional securiizaion, where he Minisry of Finance acs as he originaor. GICs are based on a limied mudarabah, which means ha he raised money is invesed solely in he projecs sipulaed in he original conrac. Noe 6. The Primary Marke deals wih he rading of new securiies. When a company issues securiies for he firs ime (i.e. IPO), hey are raded in he Primary Marke hrough he help of issuing houses, dealing /brokerage firms, invesmen bankers and or underwriers. The acronym IPO sands for Iniial Public Offering, which means he firs ime a company is offering securiies o he general public for subscripion. Once he securiies (shares) of a company are in he hands of he general public, hey can be raded in he Secondary Marke o enhance liquidiy amongs holders of such financial securiies. Thus, he Secondary Marke faciliaes he buying and selling of securiies ha are already in he hands of he general public (invesors). Noe 7. For more explanaions abou he ideas of Islamic banking see for example, Venardos (010). Noe 8. For a deailed discussion of he Islamic Shariaa principles and is pracices on sock exchange see for example, El-Gamal (006) and Ayub (007). Noe 9. 'Musharakah' is a word of Arabic origin which lierally means sharing. In he conex of business and rade i means a join enerprise in which all he parners share he profi or loss of he join venure. I is an ideal alernaive o he ineres-based financing wih far reaching effecs on boh producion and disribuion (Usmani, 1998). 175

16 Inernaional Journal of Economics and Finance Vol. 4, No. 8; 01 Noe 10. For a deailed discussion of he Egypian Sock Marke see for example, Sourial and Mecagni (1999) and Aly, e al. (004). Noe 11. I is very imporan o poin ou ha, here migh be some bias owards acceping he null hypohesis of a uni roo for he index series in level form for he case of Sudan, his simply because of he clear exisence of he break poins in he series a he end of Ocober 009 (see Figure1). ADF es fails in case of srucural break and i has low power. As one way o accoun for hese srucural breaks, Perron (1989) inroduced a dummy variable o he ADF es. For a deailed discussion of he srucural breaks in uni roo es see for example Mills and Markellos (008). In order o check he robusness of our finding, we repea he es for KSE for he wo subperiods. Noe 1. The main feaure of ARCH model is o describe he condiional variance as an auoregression process. However, mos empirical ime series require using long-lag lengh ARCH models and a large number of parameers mus be esimaed. As a poenial soluion of he problem, GARCH models have been proposed (see Engle and Bollerslev 1986; Nelson 1991) which exhibi higher persisence. Noe 13. Volailiy can be defined as a saisical measure of he dispersion of reurns for a given securiy or marke index. Volailiy can eiher be measured by using he sandard deviaion or variance beween reurns from ha same securiy or marke index. Commonly, he higher he volailiy, he riskier he securiy. Noe 14. In he symmeric models, he condiional variance only depends on he magniude, and no he sign, of he underlying asse reurn, while in he asymmeric models shocks of he same magniude, posiive or negaive, migh have differen effecs on fuure volailiy. Noe 15. This is in conras o he sandard GARCH model where shocks of he same magniude, posiive or negaive, have he same effec on fuure volailiy. Noe 16. The model uses zero as is hreshold o separae he impacs of pas shocks. Oher hreshold can also be used; see (Tsay, 010) for he general conceps of hreshold models. Noe 17. For poenial issues regarding he numerical soluion of he maximum likelihood esimaors for GARCH models, he ineresed reader migh consul Maringer and Winker (009). Noe 18. If he variance equaion of GARCH model is correcly specified, here should be no ARCH effec lef in he residuals. Appendix. Esimaion resuls of differen GARCH models for Kharoum sock exchange (Sub-periods) Coefficiens GARCH (1,1) GARCH-M (1,1) EGARCH (1,1) TGARCH (1,1) PGARCH (1,1) Firs Second Firs Second Firs Second Firs Second Firs Second Mean equaion E * -7.09E-05* * -1.90E E E E * * Variance equaion 3.5E-05* 6.00E-08* 3.70E-05* 6.05E-08* * * 3.55E-05** 5.8E-08** 8.64E E * * * 0.188* * * ** ** 0.865** 0.1** * * * * * * ** ** 0.37** 0.640** * * 0.549** * 0.190** 0.058**.794**.057** Log likelihood ARCH-LM es for heeroscedasiciy saisic Prob * and ** indicae significan a 5% and 1% respecively. 176