Modelling and Forecasting the Volatility of the Daily Returns of Nigerian Insurance Stocks

Transcription

1 Inernaional Business Research Modelling and Forecasing he Volailiy of he Daily Reurns of Nigerian Insurance Socks Dallah Hamadu (Corresponding Auhor) Deparmen of Acuarial Science and Insurance Universiy of Lagos, Akoka-Yaba, Lagos, Nigeria Tel: Ade Ibiwoye Deparmen of Acuarial Science and Insurance Universiy of Lagos, Okoka-Yaba, Lagos, Nigeria Absrac This paper examines he volailiy of he daily reurns of Nigerian insurance socks. Using empirical analysis, he sudy shows ha he Exponenial Generalized Auoregressive Condiional Heeroskedasic (EGARCH) model is more suiable in modelling sock price reurns as i ouperforms he oher models in model-esimaion evaluaion and ou-of-sample volailiy forecasing. Given he cardinal role of insurance in Nigeria s risk managemen sysem he presen findings can be useful in undersanding insurance indusry s sock risk. The policy implicaions are also considered. Keywords: Insurance Socks Reurns, Volailiy modelling, GARCH, TARCH, EGARCH, Ou-of Sample Forecass. Inroducion Volailiy modelling and forecasing have araced much aenion in recen years in emerging sock markes. For insance, many asse-pricing models used volailiy esimaes as a simple acuarial risk measure. In Nigeria volailiy modelling and forecasing has no araced he deserved aenion possibly because he sock marke is largely under-developed. This phenomenon is more pronounced in he insurance secor where many of he players appear o deliberaely avoid lising on he sock exchange because no informaion would hen need o be disclosed o heir shareholders. However, changes are being observed as he las wo decades have seen acceleraed growh of insurance markes. Arena (006) repors ha emerging markes have recenly experienced significanly faser real growh of heir insurance secors han indusrialized counries reflecing liberalizaion and financial inegraion, usually following he implemenaion of srucural reforms. Recapializaion of he insurance indusry in Nigeria has no doub recorded a huge volume of business, he secor was able o pull an aggregae gross premium income of N90 billion in 006, over 8% more han 005. Moreover, Nigerian invesors aiude and percepion of insurance socks are dramaically changing posiively. In fac, discerning invesors have since idenified insurance socks as a very imporan invesmen line since mos of he insurance socks are having impressive reurns (Ibiwoye and Adeleke, 008). Hence, here is currenly a high level of invesors ineres for insurance socks in he marke and subsequenly a high level of volailiy. Therefore, hedging agains risk and for porfolio managemen, reliable risk volailiy esimaes and forecass of hese socks are quie useful and need o be invesigaed. In Nigeria, volailiy modelling and forecasing have no araced much aenion for he simple reason ha he sock marke is largely undeveloped. The few excepions have been he sudy by Ologunde e al (006) which fied a regression model o he relaionship beween marke capializaion and ineres rae, Ibiwoye & Adeleke (008) who analysed price movemens in insurance socks pre-and pos- 005 consolidaion and ha of Olowe (009) on he impac of he 005 re-capializaion of he insurance indusry on he sock marke. This paper fills he gap in he emerging economy lieraure by invesigaing he volailiy of Nigerian insurance socks reurns using heeroskedasic condiional volailiy models.. Lieraure Review The pervasive daily reurn volailiy in equiy sock markes has araced considerable aenion in he lieraure in recen imes (Galeoi and Schianarelli, 994; Mankiw e al 99; Kumar and Makhija, 986, Schwer, 989; Eraker, 004). Mahemaical models are usually employed o predic he fuure behavior of sock prices because mos ransacions in socks, wheher o buy or sell, are aciviies ha ake place in he fuure (Chauvin, 006). In he 06

2 Inernaional Business Research Vol. 3, No. ; April 00 pas, much modelling aenion had been focused on he predicable componen of he sock reurn series. Laer aenion shifed o he error erm whereby i is assumed ha he laer is normally disribued. Schwar (989) found ha he ampliude of he flucuaions in aggregae sock volailiy is difficul o explain using simple models of sock valuaion and ha here is a srong residual auocorrelaion using leas squares hence he applied ARMA (, 3) model for he errors. Eraker (004) developed an approach based on Markov Chain Mone Carlo (MCMC) simulaion, which allows he invesigaion o esimae he poserior disribuions of he parameers as well as he unobserved volailiy and jump processes. Rydberg (000) reviewed some models ha have been used o describe he mos imporan or sylized feaures of financial daa. These include fac ools, asymmery-symmery, volailiy clusering, aggregaion Gaussianiciy, quasi-long-range dependence and seasonaliy. Rydberg (000) classified he models ino wo broad caegories: mahemaical finance models and economeric models. Since he goal of he laer is usually forecasing i requires less rigorous probabiliy heory han he previous and ends o focus more on he correlaion srucure of he daa. Models ha assume normally disribued log reurns like he Black & Scholes model had been exensively used in he mahemaical finance lieraure bu his assumpion has been dispued (Rydberg, 000). More recenly, aenion has shifed owards modelling financial-marke asse reurns by processes oher han normal error disribuion. I has been esablished ha he variances of he error erms in ordinary leas square (OLS) esimaes are no equal, and are indeed larger for some poins or ranges of daa han for ohers (Engle, 00). This incidence of heeroskedasiciy in which he usual procedures for esimaing sandard errors and confidence inervals fall shor are bes addressed by ARCH/GARCH models (Engle, 00). The ground breaking work of Engle (98) inroduced a means of capuring he propery of ime-varying volailiy. Furher research, however, has shown ha in pracical applicaions of he ARCH (q) model, large q s are usually required hereby necessiaing he need for many parameers (Rydberg, 000). To overcome his difficuly, Bollerslev (986) and Taylor (986) modified he basic ARCH model as Generalized Auoregressive Condiional Heeroskedasiciy (GARCH) model. GARCH has since gained widespread accepance in he lieraure and is ofen used for modelling sochasic risk volailiy in financial ime series. Floros (007) used various GARCH models wih boosrapped ou-of-sample period daa o evaluae he performance of minimum capial risk requiremen (MCRR) esimaes. The models show ha higher capial requiremens are necessary for a shor posiion, since a loss is hen more likely. David (997) classified he models for describing he properies of sock marke reurns ino wo he fas learning model and he slow learning model. Exploring he properies of exponenial GARCH model for measuring he asymmery beween reurns and volailiy, David (997) found ha he fas learning model generaes a negaive relaionship while he slow model generae reurns ha exhibi greaer excess kurosis. Oher ARCH/GARCH based sudies include Amin and Ng (997); Baillie and DeGennaro (990); Chahal and Wang (998) and Chan e al (99). Amin and Ng (997) argue ha implied volailiy dominaes he GARCH erms and herefore include an enire lag srucure hrough GARCH persisence erms in heir sudy. However, as Rydberg (000) had observed, neiher he ARCH nor he GARCH models consider boh asymmery and leverage (he fac ha volailiy negaively correlaed wih changes in sock reurns). Alhough GARCH (p, q) models give adequae fis for mos equiy-reurn dynamics, hese models ofen fail o perform well in modelling he volailiy of sock reurns because GARCH models assume ha here is a symmeric response beween volailiy and reurns. GARCH models are hus unable o capure he "leverage effec" of sock reurns. For equiies, i is ofen observed ha downward movemens in he marke are followed by higher volailiies han upward movemens of he same magniude. To accoun for his, Zakoian (990) and Glosan, Jagannahan, and Runkle (993) inroduced he hreshold GARCH (TGARCH) o ake care of exising leverage effec. During he same period Nelson (99) proposed he Exponenial GARCH (EGARCH) models in order o model asymmeric variance effecs. 3. Maerial and Mehods 3. Daa for he Sudy The daa for his sudy are from daily closing prices of insurance companies socks raded on he floor of he Nigerian Sock Exchange (NSE). The ime series daa cover almos eigh years saring from 5 h of December 000 o 9 h of Sepember 008 and coincidenly he period corresponds o Nigeria s recen sable marke economy and civil democraic governance. Alhough abou weny-six insurance companies are lised on he floor of NSE, some of hem did no survive he consolidaion exercise. We are considering only he daa of nine major insurance companies which daily lised sock prices are available for he period considered in he sudy. We used he daily daa from 5 h of December 000 o June 9 h 008 as raining daa se, and he daa from 0 June 008 o 9 h Sepember 008 as evaluaion es se or ou-of-sample daases (parial daa ses excluding holidays). Deails on 07

3 Inernaional Business Research hese companies can be found in Nigerian Insurance Diges (007) and he daa are available on hp:// 3.. Mehods 3.. Models Specificaion P Having observed P which is denoing he sock price a ime, le r = ln be he coninuously compounded P reurn series of ineres. Usually, he reurn series is decomposed ino wo pars, he predicable and he unpredicable as: r E + ε () r where ( ) ( ) = r E is he condiional mean of reurn a ime depending upon he informaion available a ime - and ε is he predicion error erm. Unforunaely, he condiional mean does have he abiliy o give useful predicions, hence, he recourse o mehods (addressing he volailiy of he error ern) such as ARCH and sochasic volailiy models in modern applied saisics and mahemaical finance. Assuming he unpredicable componen in () is an ARCH process, i be wrien as ε = z σ () where ~ N(0, ) and σ is he condiional variance. ARCH (p) Since he seminal paper of Engle (98) a rich lieraure has emerged for he modelling of heeroskedasiciy in financial ime series. Engle (98) inroduced he ARCH (p) model in which he condiional varianceσ is a linear funcion of lagged squared residuals z iid ε ε + β ε β p p σ = α + β ε, (3) Where, α > 0 and β i 0 and ε φ ~ N(0, σ ) and φ is he informaion se of all informaion up o ime. I is imporan o noe ha for ARCH models he uncondiional disribuion of ε is always lepokuric. In applicaions of he ARCH (p) model, i ofen urned ou ha he required lag p was raher large. In order o achieve a more parsimonious parameerizaion, hen, Bollerslev (986) inroduced he generalized ARCH (p, q) model (GARCH (p,q)). GARCH (p, q) hus, he volailiy model is now wrien as σ = α 0 + αε α pε p + βσ β qσ (4) q where, α i > 0 and β j > 0 for all i and j. In general, he value of p in (4) will be much smaller han he value of p in equaion (3). Imporan limiaions of ARCH and GARCH models are he non-negaiviy consrains of he α i s and β j s which ensure posiive condiional variances. Moreover, GARCH models assume ha he impac of news on he condiional volailiy depends only on he magniude, bu no on he sign, of he innovaion. As menioned above, empirical sudies have shown ha changes in sock prices are negaively correlaed wih changes in volailiy. To overcome his TARCH (p, q) Model The hreshold GARCH, or TARCH (p, q), (Glosen e al. 993,) is σ = α 0 + ( α iε ) + γε d + ( β jσ j i= j= (5) where d = if ε < 0, and d = 0 oherwise. In his, good news ( ε < 0 ), and bad news ( ε > 0 ), have differenial effecs on he condiional variance. In his work we consider popular TARCH (, ). In his case, good news has an impac of α and bad news has an impac ofα + γ. EGARCH (p, q) Model Similarly, o o overcome he drawbacks, Nelson (99) inroduced he exponenial GARCH. As: q p 08

4 Inernaional Business Research Vol. 3, No. ; April 00 q p ε i ε i ln( σ = + + ) α 0 α i γ i + ( β j ln( σ j ), (6) i= σ i σ i j= The EGARCH (, ) used in he presen sudy is he EViews specificaion given by: ε ε ln( σ ) = α 0 + β ln( σ ) + α + γ σ σ The fac ha he EGARCH process is specified in erms of log of he condiional variance implies haσ is always posiive and, consequenly, here are no resricions on he sign of he model parameers. In fac, he leverage effec is exponenial, raher han quadraic, and ha he forecass can be esed by he hypohesisγ < Model Selecion Crieria In a holisic, model comparison approach he underlying goal is o selec he bes approximaing model from among compeing models under consideraion. Several model selecion crieria have been proposed based on differen consideraions. The mos prominenly used mehod is he Akaike Informaion Crierion (AIC) (Akaike, 978). The procedure selecs he bes model wih he lowes AIC. Fundamenally, AIC involves he noion of cross-validaion, bu only on heoreical sense. Given AIC values of wo or more models, he model saisfying minimum AIC is mos represenaives of he rue model and, on his accoun, may be inerpreed as he bes approximaing model among hose being considered (Dayon, 003). Le y, k, n and LL be response variable, he number parameers, he number of observaions and he maximised likelihood funcion respecively. The Bayesian Informaion Crierion is RSS AIC = K ln( LL) = K + ln (7) n where, RSS = n i eˆ is he residual sum of squares. The main reason for preferring he use of a model selecion procedure such as AIC in comparison o radiional significance ess is he fac ha, a single holisic decision can be made concerning he model ha is bes suppored by he daa in conras o wha is usually a series of possibly conflicing significance es. Moreover, models can be ranked from bes o wors suppored by he daa a hand, hus, enlarging he possibiliies of inerpreaion (for more insighs see Dayon, 003). Since, AIC serves only he purpose of model comparison; we consider hree diagnosic check mehods based on Ljung-Box Q saisics for pos-esimaion evaluaion analysis of he fied models. There are he sandardised residuals and squared residuals of he Auocorrelaion (AC) and parial auocorrelaion (PAC) funcions, and he auoregressive condiional heeroscedasic ARCH-LM es In addiion, we employ wo popular ou-of sample model selecion crieria o evaluae he predicive performance of he five compeing models considered in he invesigaion. The crieria are namely he Roo Mean Square Error (RMSE) and he Mean Absolue Error (MAE). Thus, we have where, =,..., m wih m, y and respecively. RMSE = m m ( yˆ y ) = MAE = m m = y yˆ ŷ denoing he number of forecass, he acual and he forecas The RMSE and he MAE can be joinly considered o diagnose he errors variaion in a se of forecass. The RMSE will always be larger or equal o he MAE; he greaer difference beween hem, he greaer he variance in he individual errors in he sample. 4. Analysis of Resuls 4. Preliminary Resuls Table 4. shows he preliminary analysis saisics of various insurance socks reurns. The mean reurn and sandard deviaion are repored, as well as he highes and lowes reurns observe for each sock. The sandard deviaion of sock reurns is he measure of dispersion of reurns around he average reurn over he period of sudy. (8) (9) 09

5 Inernaional Business Research This is no always he bes indicaor of risk variabiliy. Clearly, he reurn series displayed in Figure 4. has oo many exremes values o be generaed by a normal curve. Sample deparures from he normal disribuion are summarized by he coefficiens of skewness and Kurosis. The excess kurosis coefficiens are very large and saisically significan for all he socks. All he socks are negaively skewed. GARCH models allow he volailiy, or condiional heeroskedasciy, o vary over ime; herefore i can easily ake care of he fourh momen or kurosis o he daa. Nerveless, clusering condiional volailiy has a limied effec on he very high skewness. 4. Analysis of Main Resuls The main empirical resuls are summarized in he following Tables 4., 4.3, and 4.4 respecively. Table 4. gives he saisics of model variance parameers esimaes and Aikake performance evaluaion crierion under various fiing echniques. The resuls indicae wih a good level of confidence he suiabiliy of various fiing mehods. In fac, he ARCH coefficien(s) of he esimaed models are significan excep in he cases of he second order of ARCH () resuls of UNIC and WAPIC insurance socks price reurns. Similarly, he bea coefficiens of all he socks are also significan a 99% confidence level, which indeed shows he presence of ime-clusering volailiy of insurance socks in Nigerian marke. On he oher hand, he asymmery gamma coefficiens of he EGARCH are significan in for all he nine companies whereas he TGARCH are significan in six ou of he nine socks. Bu even in he case of he laer hree companies, he non linear asymmeric TGARCH model is very compeiive as expeced based on he kurosis and skewness coefficiens resuls given in Table 4.. Moreover, he EGARCH (, ) is in general he preferred model for hese socks using he AIC model selecion crierion resuls as shown in Table 4.. The pos-esimaion evaluaion using he Lung-Jung and Box-Pierce saisics are quie informaive in assessing he diagnosic checks of various simulaed models. Table 4.3 resuls show ha, he Auocorrelaion (AC) and Parial Auocorrelaion (PAC) Q(6) saisics are significan for all he fied models in case Crusader, Guinea and UNIC insurance socks whereas only ARCH() is significan for Cornersone and Presige Insurance companies socks. Similarly, in he case of Niger Insurance only ARCH () and EGARCH (, ) are no significan. On he oher hand, he Q(6) saisics for he sandardized squared residuals are significan for all he simulaed models only in he case of UNIC Insurance. In fac, he siuaion remains persisen even when we ried higher level Lags. This can be seen also easily from he ARCH-LM es Q saisics resuls. In fac, he ARCH es is only significan in he case of UNIC wih even higher order ARCH, GARCH, TGARCH and EGARCH. This corroboraes Chahal and Wang (998) findings ha ime-varying condiional volailiy has a limied effec on he hird momens or skewness. Table 4.4 resuls show he RMSE and MAE ou-of sample forecass comparison. The resuls sugges ha, he non-linear mehods perform beer han oher compeing mehods. In fac, he Exponenial GARCH (, ) model proves o be very compeiive as i performs beer han oher compeing mehods using he RMSE and MAE model forecas evaluaion crieria. This is closely follows by he TGARCH (, ) and as disan hird he GARCH (, ) model. However, i is imporan o noe he closeness amongs he magniudes of all he mehods in boh (RMSE and MAE) model performance saisics measures, which on a nushell confirmed he adequacy of hese condiional volailiy models in modelling Nigerian insurance sock prices reurns. 5. Conclusion We have examined he volailiy behaviour of he Nigerian insurance socks price. Several varians of heeroskedasic condiional volailiy models were evaluaed using model evaluaion performance merics. The pos esimaion evaluaion revealed ha mos of he models sudied were compeiive. However, he resuls show ha he EGARCH is a more preferred modelling framework for evaluaing risk volailiy of Nigerian insurance socks. These findings are subsaniaed by using AIC, RMSE, and MAE evaluaion informaion measures. The presen findings are relevan o he invesing communiy as a whole who inves heir hard-earned money on corporae insurance business expecing reasonable reurns. Keeping in mind ha insurance sock reurns are exponenially volaile, and paricularly because he Nigerian financial sysem is currenly undergoing reforms, invesors are beer informed on insurance socks in heir porfolio profile. References Amin, K. I. & Vicor K. Ng. (997). Inferring fuure volailiy from he informaion in implied volailiy in Eurodollar opions: A new approach. The Review of Financial Sudies, Vol. 0, No., pp Arena, Marco. (006). Does Insurance Marke Aciviy Promoe Economic Growh? World Bank Policy Research Working Paper 4098, December 006 Available: hp:// Akaike, H. (978). A Bayesian analysis of he minimum AIC procedure. Annals of he Insiue of Saisical Mahemaics, 30,

6 Inernaional Business Research Vol. 3, No. ; April 00 Baillie, Richard T. and Ramon P. DeGennaro. (990). Sock reurns and volailiy The Journal of Financial and Quaniaive Analysis, Vol. 5, No., pp Bollerslev, T. (986). Generalised Auoregressive Condiional Heeroskedasiciy. Journal of Economerics, 5, Chahal, Mandeep S. & Jun Wang. (998). Jump diffusion processes and emerging bond and sock markes: An invesigaion using daily daa. Mulinaional Finance Journal, ( 3), Chan, Kalok, Chan, K. C. & Karolyi G. A. (99). Inraday volailiy in he sock index and sock index fuures markes. The Review of Financial Sudies, 4(4), David, Alexander. (997). Flucuaing confidence in sock markes: Implicaions for reurns and volailiy. The Journal of Financial and Quaniaive Analysis, 3(4), Dayon, C. Michell. (003). Model Comparisons Using Informaion Measures. Journal of Modern Applied Saisical Mehods, (), 8-9. Engle, R. F. (98). Auoregressive Condiional Heeroscedasiciy wih esimaes of he variance of he Unied Kingdom inflaion. Economerica, 50, Engle, R. (00). Garch 0: The use of Arch/Garch models in applied economerics. Journal of Economic Perspecives, 5( ), Eraker, Bjorn. (004). Do Sock Prices and Volailiy Jump? Reconciling Evidence from Spo and Opion prices. The Journal of Finance, 59(3), Floros, Chriso. (007). The use of GARCH models for he calculaion of minimum capial risk requiremens: Inernaional evidence. Inernaional Journal of Managerial Finance, 3(4), Galeoi, Marzio & Fabio Schianarelli. (994). Sock Marke Volailiy and Invesmen: Do Only fundamenals Maer? Economica, 6(4), Glosan, L., R. Jagannahan, & D. Runkle. (993). On he Relaionship beween he Expeced Value and he Volailiy of he Nominal Excess Reurn on Socks. Journal of Finance, 48, Ibiwoye, Ade and I. A. Adeleke. (008). Is Insurance Nigeria s Nex Capial Marke Honey Po? African Journal of Business Managemen, (9), Kumar Raman & Makhija Anil.K. (986).Volailiy Of Sock Prices and Marke Efficiency. Managerial and Decision Economics, 7( ), 9-. Mankiw, Gregory N.; David Romer & Maew D. Shapiro. (99). Sock Marke Forecasabiliy and Volailiy: A Saisical Appraisal. The Review of Economic Sudies, 58(3), Nelson, Daniel B. (99). Condiional Heeroskedasiciy in Asse Reurns: A New Approach. Economerica, 59(), Ologunde, A. O.; Elumilade, D. O. & Asaolu, T. O. (006). Sock Marke Capializaion and Ineres Rae in Nigeria: A Time Series Analysis. Inernaional Research Journal of Finance and Economics, 4, Olowe, Rufus Ayo. (009). The Impac of he Announcemen of he 005 Capial Requiremen for Insurance Companies on he Nigerian Sock marke. The Nigerian Journal of Risk and Insurance, 6(), Rydberg, Tina Hviid (000). Realisic saisical modelling of financial daa. Inernaional Saisical Review, 68(3), Schwer, G. William. (989). Why does sock marke volailiy change overime? The Journal of Finance, 44(5), Taylor, S. J. (986). Modelling Financial ime Series, John Wiley & Sons, Chicheser. Zakoian, J. M. (994). Threshold Heeroscedasic Models. Journal of Economic Dynamic and Conrol. 8,

7 Inernaional Business Research Table 4.. Summary of Descripive Saisics of Socks Reurns N Minimum Maximum Mean Sd. Deviaion Higher Momens Saisic Saisic Saisic Saisic Saisic Skewness Kurosis AIICO Cornersone Crusader GUINEA Law Union NIGER Presige UNIC WAPIC

8 Inernaional Business Research Vol. 3, No. ; April 00 Figure 4.. Socks Prices and heir Raes of Reurn Plos 3

9 Inernaional Business Research Table 4.. Summary of Model Parameers Esimaion Saisics and Goodness of Fi Insurance Socks Reurn AIICO Cornersone Crusader Guinea LawUnion Niger Presige Unic Wapic Coefficiens Model α ARCH() * ARCH() 0.830*.449* GARCH (,) * * TARCH(,) * * -0.60* EGARCH(,) * 0.800* 0.54* ARCH() * ARCH() * GARCH (,) 0.856* 0.88* TARCH(,) 0.94* 0.84* * EGARCH(,) 0.456* * -0.3* ARCH().089* ARCH() ** * GARCH (,) * * TARCH(,) 0.457* * -0.73* EGARCH(,) 0.85* * 0.00* ARCH().08* ARCH() 0.635* 0.904* GARCH (,) 0.870* 0.664* TARCH(,) 0.774* * EGARCH(,) * * ** ARCH() 0.395* ARCH() 0.394* 0.04* GARCH (,) 0.6* 0.955* TARCH(,) 0.80** 0.968* EGARCH(,) * * ** ARCH() * ARCH() 0.475* 0.548* GARCH (,) 0.375* 0.85* TARCH(,) 0.30* 0.849* 0.06 EGARCH(,) 0.938* * ** ARCH() * ARCH() * * GARCH (,) * 0.60* TARCH(,) 0.4* 0.676* * EGARCH(,) 0.530* 0.55* * ARCH() 0.486* ARCH() * 4.49E-05 GARCH (,) 0.486* 0.79* TARCH(,) * 0.463* * EGARCH(,) 0.56* 0.877* * ARCH() * ARCH() * GARCH (,) * 0.80* TARCH(,) 0.455* * -0.40* EGARCH(,) * 0.639* 0.038* α β γ * Significan a %; ** is significance a 5%; AIC is he Aikake Informaion Crierion Model Fi AIC

10 Inernaional Business Research Vol. 3, No. ; April 00 Table 4.3. Pos Esimaion Model Evaluaion of Various Fiing Mehods Insurance Reurn AIICO Cornersone Crusader Guinea Law Union Niger Presige Unic Wapic Socks Lung-Jung-Box Saisics Sandardized Residuals Squared Residuals Model Q(6) Q(6) ARCH() ARCH() GARCH (,) TARCH(,) EGARCH(,) ARCH() 6.885** 9.95 ARCH() GARCH (,) TARCH(,) EGARCH(,) ARCH() 6.85*.33 ARCH() 4.49* 0.66 GARCH(, ) * 9.94 TARCH(,) 50.08* 3.7 EGARCH(,) 68.73* ARCH() 63.93* * ARCH() 9.69* 444.8* GARCH(,) 4.35* TARCH(,) *.746 EGARCH(,) 39.9*.98 ARCH() ARCH() GARCH (,) TARCH(,) EGARCH(,) ARCH() ARCH() 5.03**.703 GARCH (,) * TARCH(,) * EGARCH(,) ARCH() ** ARCH() GARCH (,) TARCH(,) EGARCH(,) ARCH() 49.43* * ARCH() 49.5* 34.96* GARCH (,) 4.3* * TARCH(,) 48.57* * EGARCH(,) 4.8* * ARCH() ARCH() GARCH (,) TARCH(,) EGARCH(,) * Significan a % ** significance a 5% ARCH-LM Tes L-Jung Q P-value

11 Inernaional Business Research Table 4.4. Ou-of Sample Forecas Performance of Fied Volailiy Models Insurance Socks Reurn Model RMSE MAE AIICO ARCH() Cornersone ARCH() Crusader GARCH (,) Guinea TARCH(,) Law-Union EGARCH(,) Niger ARCH() Presige ARCH() Unic GARCH (,) Wapic TARCH(,) EGARCH(,) ARCH() ARCH() GARCH (,) TARCH(,) EGARCH(,) ARCH() ARCH() GARCH (,) TARCH(,) EGARCH(,) ARCH() ARCH() GARCH (,) TARCH(,) EGARCH(,) ARCH() ARCH() GARCH (,) TARCH(,) EGARCH(,) ARCH() ARCH() GARCH (,) TARCH(,) EGARCH(,) ARCH() ARCH() GARCH (,) TARCH(,) EGARCH(,) ARCH() ARCH() GARCH (,) TARCH(,) EGARCH(,) RMSE = Roo Mean Square Error, MAE = Mean Absolue Error 6