MODELLING AND FORECASTING DAILY RETURNS VOLATILITY OF NIGERIAN BANKS STOCKS

Transcription

1 European Scienific Journal May 4 ediion vol., No.5 ISSN: (Prin) e ISSN MODELLING AND FORECASTING DAILY RETURNS VOLATILITY OF NIGERIAN BANKS STOCKS C. E. Onwukwe T.K. Samson Z. Lipcsey Deparmen of Mahemaics/Saisics and Compuer Science Universiy of Calabar, Cross River Sae, Nigeria Absrac This sudy models and forecas daily reurn volailiy of Nigerian bank socks. Daa on daily closing prices for fifeen Nigerian banks were colleced beween 4 h January, 5 and 3 s Augus,. Daily reurns series were hen compued for each bank from price, saionariy of he resuling series and normaliy were esed. Differen auoregressive models were fied for he mean equaion. From he mean equaion, ARCH effec was esed using Lagragian Muliplier es. To capure he volailiy paern, hree symmeric models which are ARCH(), ARCH() and GARCH(,) and wo asymmeric models EGARCH(,) and TARCH(,) were considered.. Pos esimaion and performance evaluaion meric was done using he RMSE, MAE and MAPE. The resuls showed ha he reurn series were saionary bu no normally disribued wih presence of ARCH effec. Furhermore, resuls of pos esimaion revealed ha hese models were compeiive. However, EGARCH (, ) prediced daily reurn volailiy of majoriy of Nigerian bank socks compare o oher volailiy models considered. This is an indicaion of he suiabiliy of asymmeric volailiy models compared o symmeric models. Keywords: Heeroscedasiciy, volailiy, Lagrange muliplier Inroducion Invesmen in sock is essenially a long erm invesmen and every invesmen carries is own risk. This exisenial realiy is more pronounced in he ques for wealh hrough invesmen in sock marke (Abdullahi and Lawal, ). The sock marke has given invesors opporuniy o inves in securiies of quoed companies and reward in form of moneary benefi has 449

2 European Scienific Journal May 4 ediion vol., No.5 ISSN: (Prin) e ISSN been he major objecive of any invesor. Reurns on hese invesmens are used as major indices o evaluae invesmen insead of prices. Despie his anicipaed reurn on invesmen by invesors, his reurn ofen exhibi volailiy ha is, i is someimes large or small depending on price variabiliy. The recapializaion of he banking indusry in Nigeria in July 4 boosed he number of securiies lised on he Nigerian Sock Marke hereby increasing public awareness and he confidence abou he Sock marke (Olowe, 9a). However, since April, 8, invesors have been worried abou he falling sock prices on he Nigerian sock marke alhough his problem has been aribued o he global economic meldown (Olowe, 9a). However, despie his problem, volailiy modelling and forecasing have no araced much aenion in Nigeria (Dallah and Ade, ). Alhough, several sudies in volailiy modelling have been carried ou, Ibiwoye and Adeleke (8) whose work cenred on he analysis of price movemens in insurance socks before and afer5 consolidaion. Olowe (9a) wroe on Sock Reurn Volailiy and he Global Financial Crisis in an Emerging Marke: The Nigerian Case. Onwukwe e al () modelled he volailiy of four Nigerian Firms lised on he Nigerian Sock Exchange. Olowe (9b) also conduced anoher sudy which focused on he impac of he 5 re capializaion of he banking and insurance indusry on he sock marke. Also, worhy of noe is anoher sudy conduced by Dallah and Ade (); heir sudy was on modellng and forecasing of daily reurns of he Nigerian insurance socks. Despie hese scholarly sudies on volailiy modelling, none of hese sudies model daily reurn volailiy of he each of he Nigerian banks socks. This serves as a moivaion for his sudy. Review of volailiy models Several volailiy models have been used o sudy sock reurn volailiy, one of hem was he radiionally measure of volailiy which was carried ou hrough sudies of variance of an asses. This measure of uncondiional volailiy does no accoun for imevarying and clusering properies of sock volailiy. The became a challenge o analysis of financial ime series unil he ground breaking work of Engle which brough abou revoluion o analysis of financial ime series wih he inroducion of an Auoregressive Condiional Heeroscedasiciy Model in 98 (Engle, 98). In he ligh of his, he generalized ARCH (GARCH) model as a naural soluion o he problem wih he high ARCH orders was proposed by Bollerslev (Bollerslev, 986). In Bollerslev s GARCH model (Generalized Auoregressive Condiional Heeroscedasiciy model), in GARCH model, he condiional variance is usually expressed in erms of linear funcion of pas squared innovaions and earlier calculaed condiional variance. Some oher volailiy models include he sandard deviaion GARCH model 45

3 European Scienific Journal May 4 ediion vol., No.5 ISSN: (Prin) e ISSN inroduced of Taylor (Taylor, 986) and Schwer (Schwer, 989). The EGARCH or Exponenial GARCH proposed by Nelson in 99(Nelson, 99). Threshold ARCH or TARCH and Threshold GARCH inroduced by Zakoian (Zakoian, 994) among ohers volailiy models. Mehodology of he sudy Daa for he sudy: Daa for his sudy were from daily closing prices of fifeen Nigerian bank socks raded on he floor of he Nigerian Sock Exchange (NSE). This ime series daa cover almos eigh years saring from 4 h January 5 o 3s Augus,. These daa are available on Cash Craf websie ( These banks are Access, Diamond, Eco Inernaional Incorporaed (ETI), Firs Ciy Monumen, Fideliy, Firs Bank of Nigeria, Guarany Trus Bank, IBTC, Skye, Serling, Unied Bank for Africa, Uniy, Wema, Zenih and Union Bank of Nigeria(UBN). The Economeric View Sofware (E view Version 7.) was used o enhance daa analysis. Model specificaion Compuaion of reurn series: The daily reurns were compued as he naural logarihm of he simple gross reurn which is given as: P R = ln = n P () where, P and P are he presen and he previous closing prices and n is he number of observaion ARCH models (Auoregressive Condiional Heeroscedasic model) The ARCH (p) as proposed by Engle (98) given by = α α α... α p p () α,α i, for i=,, p are he parameers of he model. α, α i > For ARCH () p=, hence ARCH () model can be specified as follows: = α α (3) α,α > Bu if p= ha is for ARCH (), we have α α α (4) = 45

4 European Scienific Journal May 4 ediion vol., No.5 ISSN: (Prin) e ISSN α, α α,, > GARCH (,) (Generalized Auoregressive Condiional Heeroscedasic) GARCH (, ) has proposed by Bollerslev (986) is given by = α α β (5) Where α,α, β are he parameers of he model, α,α, β are all non negaive., are he condiional and earlier calculaed condiional variances respecively. EGARCH (, ) (Exponenial Generalized Auoregressive Condiion Heeroscedasic) Insead of direcly performing he condiional variance, he EGARCH model is formed in logarihm of he condiional variance. The EGARCH (, ) as proposed by Nelson (99) is defined by: ln( = ) α β ln α γ (6) π α, α, γ β are he parameers, TARCH (, ) (Threshold ARCH) TARCH (, ) is an asymmeric model which allows for good and bad news. The hresholdarch process proposed by Glosen e al (993) allows differen effecs of good and bad news (negaive and posiive reurn shocks) on he volailiy. The condiional variance equaion in TARCH (p, q) model is now specified as follows: = α α β φλ (7) λ =, <, > Tes for ARCH effec and Model diagnosic check To es for ARCH effec (Heeroscedasiciy) he Lagragian Muliplier Tes of Engle was used. The null hypohesis is H : α =... = α m = Versus H a : α i for some i {,.., m} ( SSR SSR ) / m The es saisic F = (8) SSR ( n m ) 45

5 European Scienific Journal May 4 ediion vol., No.5 ISSN: (Prin) e ISSN Where, T = ( a ϖ ) = m T e = m SSR =, where ê is he leas square residual of he linear regression. T SSR, where ϖ = a is he sample mean of a. n = The es saisic is asympoically disribued as chi squared disribuion wih m degrees of freedom under he null hypohesis. The decision is o rejec he null hypohesis if F > χ ( α m ), where χ m ( α) is he upper (α ) h of he χ m or he p value of F less han 5%. Saionary es (Dickey Fuller Tes): The Dickey Fuller es was used in esing for he saionariy of he series. The es saisic, ˆ φ raio = = Sd( φ) n = ˆ p n = e p (9) The null hypohesis is rejeced if he calculaed value of is greaer han criical value. Goodness of fis crieria: Aikaike Informaion Crieria (AIC), Log likelihood and Swarz Crieria (SIC) are he mos commonly used model selecion crieria. These crieria were used in his sudy. The AIC values can be compued by he following simple equaion, RSS AIC = K ln( L) = K ln n () RSS = ê is he residual sum of squares. where, L is he maximized value of he Log Likelihood for he esimaed model and K is he number of independenly esimaed parameers in he model. Forecas performance evaluaion: The Roo Mean Square Error (RMSE), Mean Absolue Error (MAE) and Mean Absolue Percenage Error (MAPE) were used as performance evaluaion merics. The RMSE, MAE and MAPE are defined by: 453

6 European Scienific Journal May 4 ediion vol., No.5 ISSN: (Prin) e ISSN RMSE = MAE = n T MAPE = T = T = T ( ˆ ).. n = () ( ˆ ). () ( ˆ ) (3) Resuls Resuls of summary saisics for he reurn series as shown in Table showed ha he mean reurns for majoriy of Nigerian banks were negaive which revealed ha hese banks incurred losses during he period under sudy. Resuls also showed ha he reurn series were no normally disribued for mos of he banks (Table) bu were saionary (Table 3). Table : Descripive Saisics showing he Reurns of Nigerian Bank Socks Banks Saisic Mean Minimum Maximum Sandard Skewness Kurosis Jacque Probabiliy deviaion Bera Access Diamond ETI FCMB Fideliy Firs bank GTB IBTC Sky Serling UBA UBN UNITY WEMA Zenih

7 European Scienific Journal May 4 ediion vol., No.5 ISSN: (Prin) e ISSN Table : Parameer esimae of he mean equaion, saionariy es and ARCH effec. Parameers Esimaes Banks Model Type Access AR() ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) Diamond AR() ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) ETI AR() ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) FCMB AR() ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) Fideliy AR() ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) IBTC AR() ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) Serling AR() ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) φ x 8 4.x 9 3.x 7.53x 5.87x 9.46x 7.5x 7.59x 6.4.3x 8.66x x 9.97x 7 4.3x 8 9.5x 8 7.6x 8.8x 7.9x x 9.x 7 6.4x x 5.9x 5 7.x 8.54x x 8 3.3x x 8.63x x 7.4 φ φ φ ADF es ARCH LM Tes p<.5 significan a 5%, p<., significan a %,,p<., significan a.%, ADF = Augmened Dickey Fuller es

8 European Scienific Journal May 4 ediion vol., No.5 ISSN: (Prin) e ISSN Banks Model Type UBN AR() ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) UBA AR() ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) Uniy AR() ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) WEMA AR() ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) Zenih AR() ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) Firs Bank Guaran y AR() AR() ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) Sky AR() ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ).76x 8 8.8x x 7.69 Table coninuaion Parameers Esimaes φ φ φ φ x x 9.75x 5.7x x 87.3x x x 8.3x 7 6.x x x 8 8.5x x 5 7.7x x 8.4x 8.3x x x x 6 7.x 8.5x 7.9x 7 4.x ADF es ARCH LM Tes p<.5 significan a 5%, p<., significan a %,,p<., significan a.%, ADF = Augmened Dickey Fuller es. 456

9 European Scienific Journal May 4 ediion vol., No.5 ISSN: (Prin) e ISSN Furhermore, in order o fi suiable mean equaion o he reurns series and o deermine is order, he plo of auocorrelaion (ACF) and parial auocorrelaion (PACF) were obained and he figure obained revealed ha he spikes of he ACF plo decay exponenially owards zero and he spikes of he PACF cu off afer lag for mos of he banks. Therefore, AR() was fied for he reurn series of Access, Diamond, Fideliy, Serling, Union Bank, Unied Bank for Africa, Uniy Bank, Wema, Zenih and Skye Bank while AR() model was fied for Eco Inernaional Corporaion(ETI), Firs Ciy Monumen Bank, IBTC and Guarany Trus Bank and AR(3) was fied for firs bank respecively.. The parameers of each of he AR models were significan for mos of he banks (Table ). Before eneraining hese models fied o he reurn series model diagnosic checking using he plo of he ACF and PACF of he residual was also conduced and he resuls suggesed ha hese models were appropriae. Afer obaining he mean equaion for differen bank socks, he residuals obained from he mean equaion for each bank were used o es for heeroscedasiciy or ARCH effec using he Lagrange Muliplier es. The p values of F were less han 5 % (p<.5) suggesing he presence of heeroscedasiciy (Table ). Moreover, haven esablished ha here is a presence of heeroscedasiciy in he residual based on he mean equaion; he parameers of he five differen heeroscedasic models were esimaed. For Access Bank, all heeroscedasic models fied had all heir parameers significan (p<.5). Similar resuls were obained for Fideliy Bank, IBTC Bank, Zenih and Skye Bank (p<.5). In addiion, for Diamond Bank, FCMB, ETI Serling Bank, UBN, UBA, Uniy Bank, Firs Bank and GTB all parameers esimaed were significan excep he leverage effec of he TARCH (, ) model (p>.5). For Wema Bank, he ARCH() erm, he GARCH(,) for EGARCH(,) as well as he leverage erm of boh EGARCH(,) and TARCH(,) were all insignifican(p>.5). Also, for GTB, he GARCH erm for boh GARCH (, ) and EGARCH (, ) were also no saisically significan (Table 3) 457

10 European Scienific Journal May 4 ediion vol., No.5 ISSN: (Prin) e ISSN Table 3: Parameer Esimaes of he heeroscedasic models, model selecion and model diagnosic checking Parameers Esimaes Model selecion Diagnos ic check Banks Models ω Access Diamo nd ETI FCM B Fideli y IBTC Serli ng ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) 6.45x x x x 8.x x 4.78x x 5 5.6x 9 7.3x 4.66x x 4.4x x 4 4.6x x 6 7.5x x 4.6x x x 4.6x 4.9x x x 4.5x x 6 α α β γ AIC RMSE P value p<.5 significan a 5%, p<., significan a %,,p<., significan a.%, AIC= Akaike Informaion Crieria, RMSE = Roo Mean Square Error. Bolded values are he leas AIC and RMSE respecively. for ARCH LM es

11 European Scienific Journal May 4 ediion vol., No.5 ISSN: (Prin) e ISSN Table 3 coninuaion Bank s UBN Model ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) ω 9.x 4 7.4x x 4.7.5x 5 Parameers Esimaes Model selecion Diagnosic Check α α β γ AIC RMSE ARCH Tes UBA Uniy WEM A ZENI TH Firs Bank ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) x 4.6x x x x x x 4 9.3x x x 4 9.x x x 4 3.8x x Sky Bank ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ).89x 4.5x x x GTB ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) p<.5 significan a 5%, p<., significan a %,,p<., significan a.%, AIC= Akaike Informaion Crieria, RMSE = Roo Mean Square Error. Bolded values are he leas AIC and RMSE respecively

12 European Scienific Journal May 4 ediion vol., No.5 ISSN: (Prin) e ISSN Also, he goodness of fi of hese heeroscedasic models was examined using Akaike Info Crieria (AIC). Model wih he leas AIC was considered o be mos suiable. Therefore, ARCH () proved o be he bes in erms of finess. (Table 3). Model diagnosic check was also performed o examine wheher he ARCH effec are sill presen. The resuls obained revealed ha he ARCH effec iniially presen has been successfully removed by all he fied heeroscedasic models.. Table 3: Parameer Esimaes of he heeroscedasic models, model selecion and model diagnosic checking Parameers Esimaes Model selecion Diagnosic check Banks Models ω Access Diamon d ETI FCMB ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) 6.45x x x x 8.x x 4.78x x 5 5.6x 9 7.3x 4.66x x 4.4x x 4 4.6x x 6 α α β γ AIC RMSE P value for ARCH LM es

13 European Scienific Journal May 4 ediion vol., No.5 ISSN: (Prin) e ISSN Fideliy IBTC Serling ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) 7.5x x 4.6x x x 4.6x 4.9x x x 4.5x x p<.5 significan a 5%, p<., significan a %,,p<., significan a.%, AIC= Akaike Informaion Crieria, RMSE = Roo Mean Square Error. Bolded values are he leas AIC and RMSE respecively. 46

14 European Scienific Journal May 4 ediion vol., No.5 ISSN: (Prin) e ISSN Table 3 coninuaion Parameers Esimaes Model selecion Diagno sic Banks Model ω Check α α β γ AIC RMSE ARCH Tes UBN UBA Uniy WEMA ZENIT H Firs Bank Sky Bank GTB ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) ARCH() ARCH() GARCH(, ) E GARCH(, )TARCH(, ) ARCH() ARCH() GARCH(, ) E GARCH(, ) TARCH(, ) ARCH() ARCH() GARCH(, ) E GARCH(, )TARCH(, ) ARCH() ARCH() GARCH(, ) E GARCH(, )TARCH(, ) ARCH() ARCH() GARCH(, ) E GARCH(, )TARCH(, ) ARCH() ARCH() GARCH(, ) E GARCH(, )TARCH(, ) ARCH() ARCH() GARCH(, ) E GARCH(, )TARCH(, ) 9.x 4 7.4x x 4.7.5x x 4.6x x x x x x 4 9.3x x x 4 9.x x x 4 3.8x x 5.89x 4.5x x x p<.5 significan a 5%, p<., significan a %,,p<., significan a.%, AIC= Akaike Informaion Crieria, RMSE = Roo Mean Square Error. Bolded values are he leas AIC and RMSE respecively. Forecasing performance of hese esimaed models were invesigaed using sample daa and saisics like Roo Mean Square Error was compued. Model wih he leas Roo Mean Square Error was considered o he mos suiable in erms of forecasing performance. This is because, he good performance in parameers esimaes models and goodness of fi saisics like Aikaike Informaion crieria (AIC), Swarz Crierion (SC) and oher

15 European Scienific Journal May 4 ediion vol., No.5 ISSN: (Prin) e ISSN crieria do no guaranee accurae of forecas of any volailiy model bu raher forecas evaluaion saisics like Roo Mean Square Error (RMSE), Mean Absolue Error should be used(lopez, ). Hence, EGARCH (, ) was recommended o be mos suiable for forecasing daily reurns volailiy of Nigerian bank socks. The EGARCH (, ) proved o mos suiable for all he Nigerian bank socks wih he excepion of socks like ETI, FCMB where he TARCH (, ) proved o be mos suiable and also ARCH () for Skye Bank and ARCH() for Wema Bank FCMB Access Bank Diamond Bank Firs Bank Fideliy Bank IBTC Skye Bank UBN Uniy Bank Wema Bank Serling Bank ETI Bank UBA.4.6 GTB Zenih Bank Graph of variance forecas for he fifeen Nigerian bank Socks (a) Access Bank

16 European Scienific Journal May 4 ediion vol., No.5 ISSN: (Prin) e ISSN = π ln ) ln( (b) Diamond Bank = λ > < =,, λ (c) ETI = λ > < =,, λ (d) FCMB = π ln ) ln( (e) Fideliy Bank = π ln ) ln( (f) IBTC Bank = π ln ) ln( (g) Serling Bank = π ln.8374 ) ln( (h) UBN = π ln ) ln( (i) UBA = π ln 6.77 ) ln( (j) Uniy Bank

17 European Scienific Journal May 4 ediion vol., No.5 ISSN: (Prin) e ISSN ln( ) = ln (k) Wema Bank (l) Zenih Bank =.679 ln( ) = ln (m) Firs Bank ln( ) = ln (n) GTB ln( ) = ln (o) Skye Bank = Discussion of findings The presence of leverage effec observed in he daily reurns paern of mos of he Nigerian bank socks is an indicaion ha he disribuion of he daily reurn paern of Nigerian bank socks is asymmeric. These resuls emphasized he impac of he good and bad news on he reurns of Nigerian bank socks (Table ). This finding is suppored by oher similar sudies in Nigeria (Dallah and Ade,, Olowe, 9a). This resul is also consisen wih sudies in oher emerging capial markes in oher counries of he world (Suliman,, Zako, 8, Mousafa, ). Furhermore, he resul of model forecasing abiliy which favoured EGARCH (,) for mos of he banks as he bes of he five compeing models(table 3) is in agreemen wih sudy by Dallah and Ade(8) whose sudy observed ha EGARCH(,) performed beer han ARCH(), ARCH(), GARCH(,) and TARCH(,) in modelling daily reurns volailiy of Nigerian Insurance socks. Similar resuls have been obained in oher counries of emerging capial marke like Egyp (Mousafa, ). The resul of his sudy was no in agreemen wih ha by Hien (8).Sudy by Hien (8) favoured GARCH (, ) as he bes models for modelling volailiy of Vienam socks. This variaion could have been as a resul of he ime her sudy was conduced because as a 8 π π π π 465

18 European Scienific Journal May 4 ediion vol., No.5 ISSN: (Prin) e ISSN he effec of he global recession has no been fel. Also, Nigerian bank socks migh no exhibi he same volailiy as Vienam socks. Conclusion This sudy examined he volailiy behaviour of he Nigerian bank socks. Forecas performance of several varians of condiional heeroscedasic volailiy models were evaluaed using model evaluaion performance measures like he Roo Mean Square Error, Mean Absolue Error and Mean Absolue Percenage Error. The resuls of pos esimaion evaluaion carried ou revealed ha he asymmeric condiional heeroscedasic models are more suiable for modelling daily reurns volailiy of Nigerian bank socks as compared wih symmeric condiional heeroscedasic models. Recommendaion The resuls of his sudy had proven o be consisen wih oher similar sudies conduced in oher emerging capial markes, bu hese resuls should be reaed wih cauion as his sudy covers he mos widely used volailiy models. Therefore, furher sudy on oher volailiy models like a sochasic volailiy models and mulivariae volailiy models wih a more updaed daa is recommended. This will beer inform invesors and invesmen analys in Nigerian as volailiy is he major index used o evaluae asse performance and in sock pricing sraegy. References: Abdullahi and Lawal (). Analysis of he Risk Reurn Characerisics of he Quoed Firms in Nigeria Sock Marke. Inernaional Journal of Business and Social Science Vol. (7). Bollerslev,T.(986). Generalized Auoregressive Condiional Heroscedasiciy.. Journal of Economerics Dallah and Ade (). Modelling and Forecasing he Volailiy of he Daily Reurns of Nigerian Insurance Socks. Inernaional Business Reaserch, 3(), 64. Engle, R. F.(98). Auoregressive condiional heeroscedasiciy wih esimaes of he variance of Unied Kingdom inflaion, Economerica 5: Glosen, L.R. R. Jagannahan and D. Runkle. (993). On he Relaion beween he Expeced Value and he Volailiy of he Nominal Excess Reurn on Socks. Journal of Finance. 48, Hein (8). Modeling and Forecasing Volailiy by Garch Type Models. An MA Disseraion in Finance and Invesmen. 466

19 European Scienific Journal May 4 ediion vol., No.5 ISSN: (Prin) e ISSN Lopez J.A. (), Evaluaing he Predicive Accuracy of Volailiy Models. Journal of Forecasing,, 879 Mousafa, A. A(). Modelling and Forecasing Time Varying Sock Reurn Volailiy in he Egypian Sock Marke. Inernaional Research Journal of Finance and Economics Nelson, D.B. (99). Condiional Heeroskedasiciy in Asse Reurns. A New Approach. Economerica, 59, Ologunde, A. O.; Elumilade, D. O. & Asaolu, T. O. (6). Sock Marke Capializaion and Ineres Rae in Nigeria: A Time Series Analysis. Inernaional Research Journal of Finance and Economics, 4, Olowe, R.A (9a). Sock Reurn Volailiy And The Global Financial Crisis in An Emerging Marke: The Nigerian Case: Inernaional Research Review of Business Research Papers, 5(4), Olowe, R.A (9b). The Impac of he Announcemen of he 5 Capial Requiremen for Insurance Companies on he Nigerian Sock marke. The Nigerian Journal of Risk and Insurance, 6(), Onwukwe, C.E, Bassey, B.E. and Isaac, I.O (). The Volailiy of Nigerian Socks on Modeling he Volailiy of Nigerian Sock Reurns using Garch Models. Journal of Mahemaics Research,vol(), Schwer, G. W (989). Why does sock marke volailiy change over ime? Journal of Finance, 44, 553. Sulimon, Z.S(). Modelling Sock Reurn Volailiy: Empirical Evidence from Saudi Sock Exchange: Inernaional Journal of Finance and Economics. Euro Journal Publishing Inc., issue 85,6679. Taylor, S. J. (986). Modeling Financial Time Series, Chicheser: Wiley. TSay L.S (5). Analysis of Financial ime series, nd ed. Hoboken, N.J: Wiley. Zakoïan, J. M. (994). Threshold Heeroskedasic Models,.Journal of Economic Dynamics and Conrol, 8, Zlako, J.K (8). Forecasing Volailiy. Evidence from he Macedonian Sock Exchange: Inernaional Research Journal of Finance and Economics, Issue 8, 8:. 467