Risk and Return Relationship in Stock Market and Commodity Prices: A Comprehensive Study of Pakistani Markets

Transcription

1 Iqra Universiy, Pakisan From he SelecedWorks of Ahmed Imran Hunjra Summer June 3, 2011 Risk and Reurn Relaionship in Sock Marke and Commodiy Prices: A Comprehensive Sudy of Pakisani Markes Ahmed Imran Hunjra Muhammad Azam Ghulam Shabbir Khan Niazi Babar Zaheer Bu Kashif Ur Rehman, e al. Available a: hps://works.bepress.com/ahmed_hunjra/12/

2 World Applied Sciences Journal 13 (3): , 2011 ISSN IDOSI Publicaions, 2011 Risk and Reurn Relaionship in Sock Marke and Commodiy Prices: A Comprehensive Sudy of Pakisani Markes Ahmed Imran Hunjra, Muhammad Azam, Ghulam Shabbir Khan Niazi, Babar Zaheer Bu, Kashif-Ur-Rehman and Rauf I Azam 1 Iqra Universiy Islamabad, Pakisan 2 Federal Urdu Universiy Ars, Science and Technology Islamabad, Pakisan 3 Quaid-I-Azam School of Managemen Sciences, Quaid-I-Azam Universiy Islamabad, Pakisan 4 Universiy Insiue of Managemen Sciences, Universiy of Arid Agriculure Rawalpindi, Pakisan Absrac: The objecive of his sudy is o deermine he risk and reurn relaionship on he basis of univariae modeling approach. This sudy is helpful o analyze he asymmeric naure of daa including he seasonal affec and non linear properies in risk and reurn relaionship scenario. In his sudy, monhly daa was used regarding gold price, coon prices and sugar price along wih KSE 100 index. The daa span of all variables cover he ime period from July 1998 o July The overall resuls indicae ha asymmeric and seasonal effec is presen in commodiies marke and sock marke. Bu he asymmeric properies and seasonal effec is mos dominan in sock marke prices comparaive o oher commodiies. Key words: Risk and reurn relaionship % Commodiies % Sock marke prices % Univariae modeling approach INTRODUCTION correlaion among condiional volailiy and reurns [1]. Commodiy marke reurns and sock marke reurns show Volailiy is he ambiguiy or risk abou he value of ime varying volailiy wih clusering and asymmeric securiy due o various marke forces. Volailiy can be effec. Empirical subsaniaion of effeciveness of risk higher or lower in erms of volume. Higher volailiy means managemen and price behavior of he markes is higher sandard deviaion of value of a paricular securiy depending upon he assumpion of volailiy of he over a shor period of ime and lower volailiy means reurns [2]. lower sandard deviaion of values of a paricular securiy The asymmeric volailiy has been elaboraed by he over he ime period. Volailiy is believed o play a vial leverage effec, i.e. an increases financial leverage cause role in accelerae pricing and hedging, opimal porfolio decline in he value of he sock, which resuls in making selecion and risk managemen predicing volailiy is he sock riskier and increases is volailiy [3, 4]. There supposed o be esing and challenging area of research has been a phenomenon of high level volailiy in he pas in he field of finance. This area of research highlighs few years in esablished and rising financial markes various dimensions of series of reurns of invesmen across he world. Financial expers and invesors consider securiies on various poins of ime like long he uncerainy, due o he volailiy in marke prices and deerminaion, volailiy clusering ec [1]. I has been he unseadiness of business performance, of he reurns noiced ha reurns of sock marke invesmen indicae on heir invesmen asses, recen developmens in various flucuaions along wih clusering effec. The financial economerics urge he use of quaniaive models auocorrelaion of volailiy ill long period of ime is of ha are capable o describe he inclinaion of invesors grea imporance. Volailiy clusering in reurn resuls owards risks, volailiy and expeced reurns. This calls shows ha small or large price changes follow small or for models ha are sufficien for dealing wih he volailiy large price changes of eiher signs. There is negaive of he marke (series) [5]. Corresponding Auhor: Ahmad Imran Hunjra, PhD: Scholar Iqra Universiy, Plo#5, H-9 Secor, Islamabad (44000), Pakisan.. Cell: , ahmedhunjra@gmail.com. 470

3 The Pakisan s economy is facing many challenges Financial ime series such as socks reurn or hese days like coninuously varying economic policies, exchange raes exhibi so called volailiy clusering. I deerioraing law and order siuaion, prevailing poliical means ha large flucuaions in hese series end o be uncerainy and insabiliy, food and energy crisis causing followed by large flucuaions and small flucuaions by increased producion and operaing cos of producions. small ones. Under his siuaion, he use of variance o A high rae of inflaion and he errorism aciviies in capure flucuaions in sock reurns is provided only differen par of counry, hese problems adversely affec gross volailiy. However, invesors and policy makers he commodiy markes and sock prices as well. Keeping may be ineresed o see he value of heir porfolio in in his view, i becomes an essenial o sudy he dynamic some fuure poin wih respec o risk if such rend naure of commodiy prices and sock marke prices. persisen in socks prices. In modeling his marke In he presen work, an effor has been made o phenomenon, Auoregressive Condiional undersand he advancemens of reurn and is volailiy Heeroscedasiciy (ARCH) approach is used. In for sock prices along wih he commodiy marke in anicipaion ha a high frequency daa migh resul in Pakisan. The objecive of his is o deermine he risk and improved accuracy of he volailiy and use pas daily reurn relaionship on he bases of univariae modeling reurns o predic monhly reurn volailiies. Firs mehod approach and o analyze he asymmeric naure including propose a simpler rolling-window monhly esimaor wih he seasonal affec and non linear properies of risk and equal weighs on pas squared reurns on he daily basis, reurn relaionship. hus can be called 'inuiive' in naure so i finds a The res of his paper is organized as follows: Secion reasonably insignifican risk and reurn relaion [7, 8]. II discusses review of lieraure, hypoheses saemens According o he porfolio heory, invesors demand a and concepual model and III secion is abou he higher reurn from he marke porfolio han expeced from mehodology. Empirical resuls and discussions of he reurn on risk free invesmens. This marke porfolio sudy are explained in secion IV. Finally, conclusions, reurn is dependen on risk, hus esablishing a posiive limiaions and fuure research are drawn in par V. relaionship [9]. The ARCH model of [10] is he commonly used Lieraure Review: The connecion beween risk and model in financial daa o capure he ime-varying reurn in he financial markes is widely sudied in volailiy along is various exensions like he GARCH [2] financial economics. The relaionship beween risk and and he EGARCH. The basic approach o all hese models reurns of porfolio invesmen has been of grea was o calculae condiional volailiy wihou no imporance, bu exising lieraure has no been able o sochasic funcion and no inclusion of excess lags in he bring an agreemen on he exisence of such relaionship equaion. Volailiy clusering approach normally capures in sock marke. Alhough various researchers have he effecs of sudden shocks or evens in financial daa widely examined he relaionship beween reurn and risk [11]. The GARCH-M model is mos useable model in heir research work. Human decision making process is proposed o capure he effec of volailiy in financial also composed of relaionship beween risk and reurn. daa [7, 12]. This has been known o every financial analys ha The relaionship of risk and reurn for Pakisani higher he risk, higher he reurn and lower he risk lower marke is he resul of GARCH-M model showing he he reurn. I is generally believed ha risk and reurn presence of srong volailiy clusers esablishing ha a relaionship is an imporan elemen for sock marke cyclical rend is followed by he ime pah of sock predicabiliy and volailiy. There are also some sae reurns. [13] sudied asymmeric asse pricing behavior variables ha predic boh risk and reurn. A rader ha and show ha he posiive shocks have a greaer impac normalizes he porfolio according o he sae variables, on he expeced volailiy han he negaive shocks in o maximize he condiional Sharpe raio, will generae a Pakisani marke [14]. The asymmeric behavior of sock porfolio wih ime varying risk. I is indicaed ha he marke can also be elaboraed hrough leverage effec spread, ineres raes and he defaul spread have and asse pricing model of porfolio heory [3, 4]. A predicing power for boh momens of reurns of bonds negaive relaionship in sock marke risk and reurn is and socks and he opimal asse allocaion for a mean also observed. Similarly risk and reurn relaionship has variance invesor ha realizes he forecasing power of widely been sudied in commodiy markes also [15]. hese variables show considerable ime variaion in The Generalized Auoregressive Condiional porfolio weighs and condiional momens [6]. Heeroscedasiciy (GARCH) model is mos preferred 471

4 Table 1: Augmened Dickey-Fuller Uni Roo Tess Commodiies Level Tau Pr<Tau Gold a level Coon a level Sugar a level Sock Index a level model because is auo regressive srucure of condiional variance may beer explain he serial correlaion properies of financial daa. [16] sudied firs he price volume in sock marke reurns hrough GARCH model approach and he incorporaed raded volume in variance par of equaion. O3n oher hand, [17] proposed he GARCH, EGARCH and sochasic volailiy models ha based upon he parameric volailiy model approaches. There is an exensive kind of lieraure on sock marke volailiy models in Pakisan [18-20]. Their empirical analysis was limied o sock marke only and sudies relaed o commodiy markes are scared in Pakisan. Daa Descripion and Mehodlogy Daa Descripion: In his sudy monhly daa was used regarding gold price, coon price, sugar price along wih KSE 100 index. The daa regarding monhly closing sock prices was colleced from websies of Karachi Sock Exchange, Gold from Forex.com, Coon from All Pakisan Texile Mills Associaion (APTMA) and Sugar was colleced from Minisry of Producion. The daa span of all variables cover he ime period from July 1998 o July We have used near monh fuures price of all commodiies as proxy for monhly daa. To capure he risk and reurn relaionship, we have esimaed percenage reurn of all commodiies and sock price. There was wide variaion of reurn in monhly daa of all commodiies and sock prices reurn. The price of all commodiies shows an increasing rend while he reurn graph was highly volaile 1 in naure. The serial and auo correlaion of differen series have been checked o idenify he saionary of series and randomness of series. The ADF es is applied o es he saionary of he reurn series of daa. The ADF es confirms ha all variables are saionary a level on he basis of criical and calculaed value crieria. The resuls are given below in Table 1. The effec of volailiy in reurn series can be confirmed hrough Pormaneau Q-Tes saisics. The effec of volailiy can be capured hrough ARCH ype s models. The Q-es is preformed up o 12 lags on all reurn series. The significance of Q-Tes confirmed ha all commodiies and sock prices reurn series confirmed he presence of volailiy clusering. The mean equaion is based on inercep and residual erm. Similarly he presence of ARCH effec can be esed on ime series daa by following Box Jenkins Mehodlogy. 2 The resuls are given in able 2 (Inser in appendix). Table 2: Pormaneau Q-Tes on commodiies marke reurn and sock Marke reurns Gold Coon Sugar Sock Index Order Q-Sa Pro Q-Sa Pro Q-Sa Pro Q-Sa Pro Figures of commodiies prices and sock reurn can be seen in appendix. 2 In firs sep, he ARMA models are designed afer his residual squared reurn series are regressed on heir corresponding lags. The significance of F-Tes as well as LM-Tes confirmed he ARCH effec and Precondiion owards GARCH family models. 472

5 Appendix Fig.1: Monhly Gold Price (Rs per 10 g) Fig. 2: Monhly Gold Price Reurn (Rs per 10 g) Fig. 3: Monhly Coon Price (Rs per 40 kg) Fig. 4: Monhly Coon Price Reurn (Rs per 40 kg) 473

6 Fig. 5: Monhly Sugar Price (Rs per kg) Fig. 6: Monhly Sugar Price Reurn (Rs per kg) Fig. 7: Monhly KSE 100 Index Price Fig. 8: Monhly KSE 100 Index price 474

7 Fig. 9: Volailiy of Gold price Squared Reurn GPR Fig. 10: Volailiy of Coon Price Squared Reurn CPR SPR2 Fig. 11: Volailiy of Sugar Price Squared Reurn STPR2 Fig. 12: Volailiy of Sock Price Squared Reurn 475

8 Esimaion Mehodlogy: The condiional volailiy models as well as ARCH and GARCH models are described in his secion. ARCH ype family models consider he lags in condiional variance and he volailiy effec is capured hrough ime varying models. These models include ARCH, GARCH and for furher analysis EGARCH models have been used [1].Similarly he seasonal effec is also capured in Arch family models. ARCH Model: The basic specificaion of ARCH model is given as From above equaion; he sock marke monhly prices, condiional mean and error erm are represened. The final equaion for measuremen of condiional variance in reurn series can be wrien as: Where Y = ε + δ 1 2 δ N(0, σ ) q 2 2 = 0 + i 1 i= 1 σ β βδ In his condiional variance equaion, he parameers mus be β0 0andβi 0. In he above menioned ARCH model Y represen he monhly reurn of series and u is error erm ha is uncorrelaed and process zero mean value. In his condiional variance equaion, he parameers mus be β0 0andβi 0. he lags esimaion is no well defined in ARCH models variance equaion. y = ε + δ δ = w σ 1 * = wf GARCH Model: [2] proposed GARCH (p,q) model ha measure volailiy which is effeced by pas prices and pas lags denoed(q,p) respecively. GARCH (p,q) model is used o overcome he problem of pas lags esimaion as in ARCH model.the volailiy in GARCH model is also he funcion of pas reurn and pas lags a same ime (q,p) respecively. The GARCH model specificaion is given as: Where 2 δ N(0, σ ) The final equaion of condiional variance can be wrien as: The condiional variance is linear funcion of q and p lags of pas values or GARCH erms. EGARCH Model: Exponenial GARCH (EGARCH) model is preferred over GARCH model because i consider he asymmeric propery as well as size, leverage effec and lags effec. The EGARCH model is exension of GRACH model because i incorporaes posiive and negaive in model hrough logarihmic form. The specificaion of E GARCH models given as y = ε 1 + δ δ = wσ Where 2 δ N(0, σ ) δ δ 2 log σ β βδ ρ( ) ω α[ ( ) ] = σ 1 σ 1 π The resricion of non negaiviy is no required in E- GARCH model. The parameer D capures he asymmeric effec. Negaive value of rho capure he asymmeric effec ha is used for indicaing he higher volailiy while is for size effec of risk and reurn series. Empirical Resuls GARCH _ Mean Resuls: The volailiy of commodiies markes and sock marke reurn is analyzed by specificaion of ARCH models. Afer he specificaion of ARCH model for differen series, he GACH -mean family models are used.in GARCH -mean frame work he upper par of equaion specified he mean effec and lower par capure he variance effec of series. Differen GARCH- Mean(p,q) have been esimaed for differen series.the bes model is seleced on he basis of Akaike Informaion Crierion (AIC).For Gold price reurn,he GARCH-mean (1,2 )is bes fied model. The sandard coefficien of SQR (GARCH) is insignifican ha can be explained if here is effec of risk on he mean reurn i is beer capured by variance equaion in Gold price reurn. The Resuls of Gold price reurn are given below in Table 3. For coon price reurn, he GARCH-mean (2, 1) is bes fied model on he basis of AIC crieria. The sandard coefficien of SQR (GARCH) is also insignifican ha can be explained if here is effec of risk on he mean reurn i is also beer menioned by variance equaion in coon price reurn. The Resuls of coon price reurn are given below in Table 4. For sugar price reurn, he GARCH-mean (2, 1) is bes fied model on he basis of Akaike crieria. The sandard coefficien of SQR(GARCH) is significan ha can be explained as if here is effec of risk on he mean reurn i is may be beer menioned by boh he mean par of and variance par in case of sugar price reurn. The Resuls of sugar price reurn are given below in Table

9 Table 3: Resuls from GARCH (p=1, 2 and q=1, 2) model on Gold reurns Variable GARCH (1,1) GARCH (2,1) GARCH (1,2) GARCH (2,2) SQR(GARCH) *** 0.25 Inercep ARCH *** ARCH GARCH * GARCH AIC *, **, *** significan a 1%, 5% and 10% respecively Table 4: Resuls from GARCH (p=1, 2 and q=1, 2) model on Coon reurns Variable GARCH (1,1) GARCH (2,1) GARCH (1,2) GARCH (2,2) SQR(GARCH) *** Inercep ARCH1 0.60* 0.40* 0.54* 0.32** ARCH GARCH1 0.40* 0.89* * GARCH2-0.34** -0.34** AIC *, **, *** significan a 1%, 5% and 10% respecively Table 5: Resuls from GARCH (p=1, 2 and q=1, 2) model on Sugar reurns Variable GARCH (1,1) GARCH (2,1) GARCH (1,2) GARCH (2,2) SQR(GARCH) *** Inercep * * ARCH * ARCH ** GARCH * * GARCH2 0.41* -0.96* AIC *, **, *** significan a 1%, 5% and 10% respecively Table 6: Resuls from GARCH (p=1, 2 and q=1, 2) model on Sock Index reurns Variable GARCH (1,1) GARCH (2,1) GARCH (1,2) GARCH (2,2) SQR(GARCH) 1.007* * 1.04* Inercep * 96.78* ARCH ARCH2-0.12** -0.13* GARCH1-0.93* * 0.56 GARCH AIC *, **, *** significan a 1%, 5% and 10% respecively Table 7: Resuls from EGARCH (2, 2) model on Gold reurns Variable E-GARCH (2, 2) C 5.96* RES /SQR[GARCH](1) RES/SQR[GARCH](1) -0.10* EGARCH(1) 0.01 EGARCH(2) -0.98* AIC 5.95 *, **, *** significan a 1%, 5% and 10% respecively 477

10 For sock price reurn, he GARCH-mean (1, 1) is bes fied model on he basis of AIC crieria. The sandard coefficien of SQR (GARCH) is significan ha can be explained as if here is effec of risk on he mean reurn i is may be beer menioned by boh he mean par of and variance par in case of sock price reurn. The GARCH value coefficien represen ha sock marke reurn series is long mammary daa.the Resuls of sock price reurn are given below in Table 6. E-GARCH Model Specificaion Resuls: The asymmeric effec volailiy of differen series is capured hrough he esimaion of E-GARCH model.the asymmeric and herocedasiciy effec of all series is esed. The mos appropriae E-GARCH model of each commodiy price reurn and sock price reurn are repored below. The significan erms indicae ha RES /SQR[GARCH] indicae ha bad news has large effec on he volailiy of he series han any good news. The resuls of he E-GARCH model indicae ha RES /SQR [GARCH] represen he asymmeric marke paern or variance. The negaive sign and significan sign in E-GARCH model represen ha any bad news has large effec on volailiy of reurn series comparaive o good marke news. The mos appropriae E-GARCH models confirm he asymmeric volailiy effec for differen series. Resuls of Seasonal Effec Models: The seasonaliy in risk and reurn series is capured by inroducing he ime dummies in men and variance par of univariae modeling series.the dummies represen he seasonal effec in mos appropriae GARCH-Mean framework in differen series. The value of mean coefficien is posiive for all series indicaing he posiive relaionship beween risk and reurn of each series. The sock models also represen a posiive relaionship beween marke volailiy and is reurn [15, 1]. The significance of differen seasons indicaes ha he seasonal effec is presen in series. The resuls of seasonal effec of differen series are given below The above menioned resuls of Gold price reurn indicaing ha seasonal dummies have asymmeric effec on Gold Marke. The seasonal dummies in risk and reurn par of gold price are combinaion of posiive and negaive sign.the reurn par of Gold indicaing no seasonal effec while he negaive seasonal effec is observed in volailiy par is observed in monh of June. Similarly, he naure of seasonal effec can be observed in coon price, gold price reurn. The asymmeric naure in sock price reurn is also observed.the seasonal effec in reurn par is posiive and significan for Seasonal dummies and negaive effec is also observed o heir corresponding monhs in volailiy par of sock price reurn series. Table 8: Resuls from EGARCH (3, 3) model on Coon reurns Variable E-GARCH (3, 3) C 1.29* RES /SQR[GARCH](1) 0.52** RES/SQR[GARCH](1) EGARCH(1) 0.60* EGARCH(2) 0.74** EGARCH(3) AIC 6.52 *, **, *** significan a 1%, 5% and 10% respecively Table 9: Resuls from EGARCH (2, 2) model on Sugar reurns Variable EGARCH (2, 2) C 1.41** RES /SQR[GARCH](1) 0.07 RES/SQR[GARCH](1) 0.47* RES /SQR[GARCH](2) 0.01 RES/SQR[GARCH](2) -0.69* EGARCH(1) EGARCH(2) 0.63* AIC 5.59 *, **, *** significan a 1%, 5% and 10% respecively 478

11 Table 10: Resuls from EGARCH (5, 5) model on Sock Price reurns Variable EGARCH (5, 5) C 5.01** RES /SQR[GARCH](1) 0.65* RES/SQR[GARCH](1) 0.07 RES /SQR[GARCH](2) RES/SQR[GARCH](2) 0.05 RES /SQR[GARCH](3) RES/SQR[GARCH](3) 0.07 RES /SQR[GARCH](4) 0.16 RES/SQR[GARCH](4) 0.16 RES /SQR[GARCH](5) 0.64** RES/SQR[GARCH](5) EGARCH(1) 0.24 EGARCH(2) EGARCH(3) 0.41** EGARCH(4) -0.80* EGARCH(5) AIC 7.18 *, **, *** significan a 1%, 5% and 10% respecively Table 11: Risk-reurn relaionship and seasonaliy in reurn and risk Gold Coon Sugar Sock Index Variable Esimae Pr>[] Esimae Pr>[] Esimae Pr>[] Esimae Pr>[] SQR (GARCH) C D D D D D D D D D D D C ARCH(1) ARCH(2) GARCH(1) GARCH(2) D D D D D D D D D D D

12 The basic purpose of his sudy is o observe risk and reurn relaionship in commodiy markes as well as sock marke on he basis of univariae modeling approach. The asymmeric and nonlinear relaionship beween risk and reurn is observed on he basis of GARCH-MEAN and E -GARCH modeling approach. The mos appropriae models for commodiies and sock markes are repored. The overall resuls indicae ha Asymmeric and seasonal effec is presen in commodiies marke and sock markes. Bu he asymmeric properies and seasonal effec is mos dominan in sock price risk and reurn relaionship. The GARCH-M model is seleced on he bases of Akakia crieria. The mos appropriae model for gold price reurn is GARCH (1,2); coon price reurn is GARCH (2,1), while sugar price reurn is GARCH (2,2) and sock marke price reurn is GARCH (1,1) which is consisen regarding sudies, [21, 1]. The mos appropriae E-GARCH specificaion for gold price reurn, coon price reurn, sugar price reurn and sock price reurn EGARCH (2, 2), EGARCH (3, 3), EGARCH (2, 2) and EGARCH (5, 5) are respecively. Seasonaliy affec is absorbed in risk and reurn relaionship of each commodiy and sock price reurns. The asymmeric properies of seasonal affec is mos dominan in sock price risk and reurn relaionship, ha implies, ha bad or good news highly affec sock marke reurn posiive or negaive. Fuure Research: This work has go norms aenion in financial economic lieraure. A dynamic way of analysis can be adoped for he in-deph exploraion he opic. The fuures sudies on his opic can be carried ou inform of bidirecional causaliy and mulivariae relaionship among risk and reurn model. The impac of differen shocks in economy on sock marke price can be analyzed. World Appl. Sci. J., 13 (3): , 2011 CONCLUSION 4. Chrisie, A.A., The Sochasic behavior of common sock variances-value, leverage and ineres rae effecs. J. Financial Economics, 10: Boomgaard, R. Griffin and Mark, Enerprise Risk and Reurn Managemen for Financial Insiuions. Norh American Acuarial Journal Sociey of Acuaries-Schaumburg, Illinois. 3(2): Brand, M., P.S. Clara and R. Valkanov, Parameric Porfolio Policies: Exploiing Characerisics in he Cross Secion of Equiy Reurns. The Review of Financial Sudies, 22(9): French, K., W. Schwer and R. Sambaugh, Expeced sock reurns and volailiy. J. Financial Economics, 19: Ghysels, E., P.S. Clara and R. Valkanov, There is a risk-reurn radeoff afer all, J. Financial Economics. 76: Markowiz, H., Porfolio Selecion. Journal of Finance, 7: Engle, R.F., Auoregressive condiional heeroscedasiciy wih esimaes of he variance of U.K. inflaion. Economerica, 50: Nelson, D.B., Condiional Heeroskedasiciy in Asse reurns: A new approach. Economerica, 59(2): Engle, R.F., D.M. Lilien and R.P. Robins, Esimaing Time-varying Risk Premia in he Term Srucure: The ARCH-M Model. Economerica, 55: Ahmad, E. and M.A. Qasim, Sock Marke Volailiy in Pakisan: An Empirical Analysis. The Middle Eas Business and Economic Rev., 16(2): Ahmad, E. and U.B. Zaman, Volailiy and Sock Reurn a Karachi Sock Exchange. Pakisan REFERENCES Economic and Social Rev., 37(1): Baillie, R.T. and R.P. DeGennarro, Sock reurns and volailiy. J. Financial and Quaniaive Analysis, 25: Lamoreaux C.G. and W.D. Lasrapes, Heeroskedasiciy in sock reurn daa: Volume versus GARCH effecs. The J. Finance, 45: Harvey, C.R., The specificaion of condiional expecaions. J. Empirical Finance, 8(5): Choudhry, T., Sock marke volailiy and he crash of 1987: Evidence from six emerging markes. J. Inernaional Money and Finance, 15(6): Kumar, B. and P. Singh, Volailiy Modeling, Seasonaliy and Risk-Reurn Relaionship in GARCHin-Mean Framework: The Case of Indian Sock and Commodiy Markes. Indian Insiue Of Managemen Ahmedabad W.P. No , April. 2. Bollerslev, T., Generalized auoregressive condiional heeroscedasiciy. J. Economerics. 31: Black, F., Sudies of sock price volailiy changes. Proceedings of he 1976 Meeing of Business and Economics Saisics Secion of he American Saisical Associaion, 27:

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