Garched investment decision making with real risk

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1 Inernaonal Journal of Busness and Publc Managemen (ISSN: -644) Vol. (): -7 Avalable onlne a: hp//: MKU Journals, Aprl 0 Full Lengh Research Paper Garched nvesmen decson makng wh real rsk Emma Anyka Shleche, Parck Oloo Weke, Thomas Ocheng Acha School of Mahemacs Unversy of Narob, Narob, Kenya School of MahemacsUnversy of Narob, Narob, Kenya School of MahemacsUnversy of Narob, Narob, Kenya Correspondng Auhor: Emma Anyka Shleche Receved: November 8, 00 Acceped: December 5, 00 Absrac Acual fuure marke rsks (sysemac or non-dversfable) of nvesmen porfolos are deermned n hs paper. Fuure reurns are frs forecased usng pas reurns and GARCH (General Auoregressve Condonal Heeroskedasc) models. A Real Rsk Weghed Prcng Model (RRWPM) s used o esmae fuure sysemac rsk among oher parameers and deermnes he fuure coss of he porfolos. Forecased random error s hen calculaed as a random varable and used o deermne probably densy esmaes of porfolos marke rsk. Ths enables fuure acual marke rsks of porfolo nvesmens o be derved hence faclang proper fuure nvesmen decson makng. Keywords: Marke rsk; GARCH; Probably Densy Esmaes; Random Error. JEL Classfcaon: G.0 Inroducon Recen years have seen a surge of neres n economerc models of changng condonal varance. Probably he mos wdely used bu by no means he only such models, are he famly of ARCH (Auoregressve Condonal Heeroskedasc) models nroduced by Engle (98). Researchers have frufully appled he new ARCH mehodology n asse prcng models. For example, Engle and Bollerslev (986) used GARCH (,) o model he rsk premum on he foregn exchange marke and Bollerslev e al (988) exended GARCH (,) o a mulvarae conex o es a condonal CAPM (Capal Asse Prcng Model) wh me varyng covarance. However her resuls show ha shocks may perss n one norm and de ou n anoher, so he condonal momens of GARCH (, ) may explode even when he process self s srcly saonary and ergodc Nelson (990). Acha e al (008) revealed ha he GARCH (, ) model provded a good explanaon of he dynamcs of he marke reurns bu faled o obey he effcen marke prncple ndcang ha here s marke rsk. Ths paper uses a RRWPM as deermned by Anyka e al (00a) ha avods he exploson of condonal momens of GARCH (, ). Wh hs model he relaonshp beween he acual and esmaed values wh GARCH forecased me seres daa s almos perfec. Wh he deermnaon of oal forecased rsk usng he RRWPM he assumpon of an effcen marke need no be upheld. Secon oulnes how reurns of a porfolo of socks are forecased usng he GARCH (, ) model, secon uses forecased reurns n secon and RRWPM o deermne forecased fuure cos and oal rsk. Secon 4 calculaes esmaes of whe nose usng an esmaor derved by Anyka e al (00b) and deermnes probably densy esmaes of he porfolo sysemac rsk usng he Gaussan kernel, secon 5 surveys he process of usng he forecased reurns wh he RRWPM o resul no fuure real cos of capal and oher parameers. Probably esmaes of fuure porfolo rsks are esmaed as well as acual marke rsk. Fnally secon 6 summarzes wha has been done and concluded based on he resuls..0 Forecasng Reurns usng Garch (, ) ARCH models make he condonal varance of he me predcon error a funcon of me sysem parameers, exogenous and lagged endogenous varables, and pas predcon errors. For each neger, le be a model s (scalar) predcon error, b a vecor of parameers, x a vecor of predeermned

2 Inernaonal Journal of Busness and Publc Managemen (ISSN: -644) Vol. (): -7 varables, and he varance of gven nformaon a me, A unvarae ARCH model based on Engel ses z () Where, z.. d, wh E ( z ) 0, var( z ), (,,...,, x, b) ( z, z,...,, x, b) () Equaon () - () can be gven a mulvarae nerpreaon as suggesed by B rooks e al (00), n whch case z s a n by one vecor and s an n by n marx. We refer o any model of he form of equaon () - () wheher unvarae or mulvarae, as an ARCH model. The mos wdely used specfcaon of equaon () are he lnear ARCH and GARCH models nroduced by Engle and Bollerslev respecvely, whch make lnear n lagged values of z, by defnng p z () q p z (4) respecvely, where,, and are non negave. Snce equaon s a specal case of equaon 4 we refer o boh equaons and 4 as GARCH models, o dsngush hem as specal cases of equaon. The GARCH M model of Engle and Bollerslev adds anoher equaon R a b (5) n whch, he condonal varance of R, eners he condonal mean of R as well. For example f R s he reurn on a porfolo a me, s requred rae of reurn may be lnear n s rsks as measured by. Therefore he GARCH M model s used for forecasng n hs research..0 Garched-Real Rsk Weghed Prce Model The RRWPM s used o deermne he forecased cos of equy and real rsk hus he model wll be called Garched Real Rsk Weghed Prce Model (G RRWPM). To deermne he G - RRWPM we le he forecased weghed expeced reurns be l l l m E R a b E R (6) Where, w g w g w g g a w a, b w b, w s he wegh of forecased secury, a s he consan reurn unque o secury, b s a measure of he sensvy of he reurn of secury o he reurn on he marke ndex, E R l w g s he weghed expeced reurn of forecased secury, E R m g s he weghed expeced reurn of forecased marke ndex. Then ake weghed forecased dversfable rsk o be / c d hw g and weghed forecased non dversfable rsk as Where, e lg G w g c e lg (7) (8) c w e, d w w,, s he varance of he forecased marke ndex, varance of secury, varance of random error of secury. To fnd he wegh of nvesmen ha wll maxmze expeced reurns and mnmze oal varance we apply he classcal opmzaon mehod wh no consrans as gven by Rao ( ). We hus dfferenae he expresson; E R c d a b E R c d m g l w g l w g Wh respec o w, and dfferenae c d e lg Wh respec o w, where E R c d e (9) (0) are maxmum reurns (derved by subracng dversfable porfolo varance from porfolo expeced reurns), and c d e s he oal varance (derved by addng lg porfolo varance o non-dversfable varance) Noe: The second dervave of he dfferenal n 0 s equal o mplyng ha w obaned wll always maxmze reurns and ha n s equal o 4 mplyng ha w obaned wll always mnmze rsk. Equae he dfferenals n 9 o 0 o ge he value of w, a E ( R ) b c d c d c () l m g l l l l l l w g w g w g w g w g w g 6c 4 d a E ( R ) b l l l m g l w g w g w g w s smlarly derved. Thus w x Replacng n expresson gves he value of

3 Inernaonal Journal of Busness and Publc Managemen (ISSN: -644) Vol. (): -7 w For w a E R b m g 8 8 m g a E R b 8 8 () () Once hese weghs are deermned, hey are subsued n equaon 6 o gve maxmum reurns and n boh equaons 7 and 8 o gve mnmum oal rsk. The coss of capal are also deermned whch enable accurae fuure predcons. 4.0 Deermnng Forecased Whe Nose The non-dversfable varance esmaor x n e (4) as derved by Anyka e al (005) ndcaes he presence of random error n he rsk esmaor. Ths error s aken o be whe nose (wn) hus can be sad o be a random varable V, V, V,..., V whch s muually ndependen and dencally dsrbued. Ths s esmaed from a sample of daa by frs varyng he varance of ndvdual reurn values of r such ha: Where w nˆ r n (5) z z, z s he oal number of reurns and (5) s he predced random error. From (5) he acual value of w nˆ s gven by w n x (6) Where and are values represenng he scale (mean) and locaon (varance) parameers. These parameers are deermned such ha he bas and varance of he acual and predced values of wn are mnmzed as follows; var( w nˆ, w n ) = ˆ w nˆ w n w n w n (7) z z z The values of and whch wll mnmze varance are gven by he paral dervaves of and, respecvely. Afer several eraons; f = w n w n ˆ z f and f f = w nˆ w n z Whe nose of he real rsk s gven by he equaon 4 w n = w nˆ z (8) As deermned by Anyka e al (00b) where r G w are forecased sock reurns and z z The whe nose s esmaed as a random error from s defnon of beng muually ndependen dencally dsrbued random varable wh consan mean and varance and cov w n, w n 0 l Where l,, I s generally known ha he value of he bandwdh s of crcal mporance whle he shape of he kernel funcon has lle praccal mpac. The value of he bandwdh ha mnmzes he AMISE s gven by / 5 / 5 h R ( k ) / k R f " n A M ISE The Gaussan kernel by Sheaher and Jones (99) s used o deermne he probably esmaes. y I s gven by K ( v ) exp 5.0 Resuls 5.. Prelmnary Daa Tweny sock porfolos were pcked randomly from he New York Sock Exchange. The New York Share Index (NYSE) s used as he marke share ndex and he long erm Treasury bond as he rsk free asse. The monhly reurns of he weny socks, he NYSE and Treasury bond snce July Sepember 009 were forecased for dfferen perods and her reurns calculaed. A sample of he forecased parameers and reurns of Toyoa usng Malab forecasng sofware are gven by able 5.. Surveys The forecased reurns are subsued no equaon 6 o gve he forecased real rsk weghed expeced reurns, cos of equy, and he oal real rsk as shown n he able. Non dversfable rsk esmaed usng equaon 9 s used o calculae whe nose as an ndependen random varable as gven by equaon 4 and presened n able 4. Gaussan kernel s used o deermne he probably esmaes of non- dversfable rsk usng whe nose as an ndependen random varable and hus calculae acual non- dversfable rsk as abulaed n Table 5.

4 Inernaonal Journal of Busness and Publc Managemen (ISSN: -644) Vol. (): Concluson The r value for RRWPM averages for he weny forecased socks ndcang ha s almos a perfec esmaor of cos of equy. Ths s n comparson o he CAPM model whch averages 0.5. Ths shows ha a RRWPM avods he exploson of condonal momens of GARCH (, ) snce hs has no deerred he RRWAM o be a perfec esmaor of cos of equy. The acual non-dversfable rsk deermned usng derved whe nose enables one make fuure predcons on he varous porfolos. If we compare he marke rsk monhs afer he cred crunch n he Uned Saes of Amerca (US) economy as shown n able 7 and ha a he hegh of he crunch as shown n able 6 we see ha monhs laer he rsks are much lower as s rue wh he US economy rgh now. In parcular he companes whch needed fnancal asssance o say afloa monhs ago AIG, TM and FORD, had marke rsks of, , 7.57 and respecvely and monhs laer hey have marke rsks of , and respecvely. Thus hs research s a rue reflecon of he acual marke rsks. Also he leas rsky socks currenly (welve monhs laer) nclude BPH, TIF, AMC and VICL. Ths s a good predcon n relaon o oher mehods lke Value a rsk, Capal Asse Prcng Model snce s n comparson wh oher Porfolos. References Acha, T., Wangombe, A., and Anyka, E., 008. Tme-Seres Modelng of Reurns From he NSE 0 -Share Index: An Emprcal Sudy of he Impac of Polcal Clmae on Marke Volaly. Mbugua,W. M. (ed). 008, Arcle Eas Afrcan socey of Operaon Research Proceedngs, NAIROBI, Kenya., Ocober 4-6. Appendces Anyka, E., Waweru, R., Odhambo, R., 005. Nondversfable rsk n Invesmen Porfolos an Ad o Invesmen Decson Makng. Journal of Scence and Technology, 59, pp-7. Anyka, E., Weke, P., Acha, T., 00a. Real Rsk Weghed Prcng Model, n Muganda, N. (ed). 00, AIBUMA proceedngs, NAIROBI, Kenya., Augus 5-6, pp Anyka, E., Weke, P., Acha, T., 00b. Kernel Whe Nose Invesmen Decson Makng Under Fne Condons, n Muganda, N. (ed). 00, AIBUMA proceedngs, NAIROBI, Kenya., Augus 5-6, pp Bollersler,T., Engle, R. F., and Wooldrdge, J.M., 988 A Capal Asse Prcng Model wh Tme Varyng Covarances, Journal of Polcal Economy, 96, pp6- Brooks, C., Burke, S., and Ga Persand, G., 00. Mulvarae Garch Models, Sofware Choce and Esmaon Issues. Journal of appled Economerc,8, pp Engle, R. F., 98, Auoregressve Condonal Heeroskedascy wh Esmaes of he Varance of Uned Kngdom Inflaon, Journal of Economerca, 50, pp Engle, R. F., and Bollersler,T., 986, Modellng he Perssence of Condonal Varances, Journal of Economercs, 5, pp-50. Nelson, D. B., 990. Saonary and Persence n he GARCH (,), Journal of Model, Economerc Theory, 6, pp 8-4. Sheaher, S. J., and Jones M.C.. 999, A Relable Daa-based Bandwh Selecon Mehod for Kernel Densy Esmaon, J. Royal Journal of sascs, 5, pp Table : Condonal Probably Dsrbuon: Gaussan Parameer for Toyoa Forecass. Parameer Sandard Value T Error Sasc C MA() K GARCH() ARCH() Leverage() Where: Parameer refers o he Sandard value, he T error and he Sasc value. Sandard value = he deermned values of he unknowns. T Error = he error values n deermnng he sandard values. Sasc = he rao of Sandard value o T Error, C and K are he consan values used n esmang he MA (), GARCH () and ARCH (), Leverage () = he value ha compares he acual value and esmaed value. Table : 8 Monh Forecass of Toyoa Reurns These values are deermned usng he sandard values plus he prevous error erm. Table : A Table of Values for he Survey of RRWPM wh Monh Forecased Reurns 4

5 Inernaonal Journal of Busness and Publc Managemen (ISSN: -644) Vol. (): -7 COMPANY BETA ALPHA r s ey E ( r ) w n TM HMC PARD VICL DWCH BP STI PNC AIG F AMR BPH CTL PFE RTI GSK BCE SBGI YAH TIF Where: TM = Toyoa Moors, HMC = Honda moors, PARD = Ponard pharmaceucals, VICL = Vcal, DWCH = Daa wach, BP = Brsh power, STI = Sun Trus Bank, PNC = PNC Fnance servces, AIG = Amercan Inernaonal group, F = Ford, AMR = Amr company BHP = BHP Bllon, CTL = CENTURY TEL, PFE = Pfzer, RTI = RTI Inl Meals, GSK = GlaxoSmhKlne, BCE = BCE Company, SBGI = Snclar Broadcas Group, YH = Yahoo group, TF = Tffany. r = The coeffcen of deermnaon, s = The sandard error for he y esmae and ey E ( r ) = Cos of equy. Table 4: A Table of Esmaed Whe Nose of Forecased Reurns COMPANY TM HMC PARD VICL DWCH BP w n STI PNC AIG F AMR BPH CTL PFE RTI GSK BCE SBGI YAH TIF Whe nose n able 4 s deermned usng equaon8 on page 8 Table 5: A Summary of he Resuls of a Gaussan Kernel Densy Esmaon of Forecased Whe Nose X y Mn. :-.967 Mn. : s Qu.: s Qu.:0.044 Medan: Medan: Mean : Mean:0.668 rd Qu.: rd Qu.:0.99 Max. :.95 Max. :0.044 Ths able show daa dvded no four quarers 5

6 Inernaonal Journal of Busness and Publc Managemen (ISSN: -644) Vol. (): -7 Fgure I: A Plo of he Densy Esmaes of a Gaussan Kernel Densy Esmaon of Forecased Whe Nose Densy N = 0 Bandwdh =. N represens he oal number of companes beng nvesgaed Table 6: Fnal Resuls of a Gaussan Kernel Densy Esmaon of Forecased Whe Nose and Acual Non-Dversfable Rsk Company F Probably n Esmaes * TM HMC PARD VICL DWCH BP STI PNC AIG F AMR BPH CTL PFE RTI GSK BCE SBGI YAH TIF * Is calculaed by mulplyng he probably esmaes wh esmae real non-dversfable rsk. TABLE 7: Fnal Resuls of Whe Nose and Kernel Densy Esmaon of Porfolos of Socks w n F Probables Acual n YH TIF TM HM PONARD VIC DAWT BP SUNTB PNC AIG FORD

7 Inernaonal Journal of Busness and Publc Managemen (ISSN: -644) Vol. (): -7 AMR BPH CTL PFE RTI GSK BCE STGI The las wo values n Table 6 are he frs wo n Table 7. 7