Wavelet-Based Beta Estimation: Applications to Indian Stock Market

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1 Wavee-Baed Bea Eimaion: Appicaion o Indian Soc Mare Maabia Deo** Aaif Shah* **Profeor and Head Deparmen of Commerce Schoo of Managemen Pondicherry univeriy India *Reearch Schoar Deparmen of Commerce Schoo of Managemen Pondicherry univeriy India Thi paper appie he mui-cae bea eimaion approach baed on wavee anayi o a oc compriing BSE-Senex. Bea are cacuaed baed on he wavee decompoiion from he Maxima overap dicree wavee ranform (DWT). I i hown ha he mui-cae bea eimaion approach i uefu in cerain cae. JEL caificaion: G12, C49 Key Word: Bea, Wavee 1. Inroducion The aemen of mare ri ha poed a grea chaenge o financia pundi and academic reearcher (ee, for inance, Granger, 2002 for an overview). Mare ri arie from he random unanicipaed change in he price of financia ae and meauring i i crucia for inveor. Ahough he meauremen of mare ri ha a ong radiion in finance, here i i no univeray agreed upon definiion of ri. The modern heory of porfoio anayi dae bac o he pioneering wor of Harry Marowiz in he The aring poin of porfoio heory re on he aumpion ha inveor chooe beween porfoio on he bai of heir expeced reurn on one hand, and he variance of heir reurn, on he oher. Laer on, he Capia Ae Pricing Mode (CAPM) emerged hrough he conribuion of Sharpe (1964) and Linner (1965a, 1965b). According o he CAPM, he reevan ri meaure in hoding a given ae i he yemaic ri, ince a oher ri can be diverified away hrough porfoio diverificaion. Severa udie raied he iue ha bea eimaion differ depending on he inerva ued in cacuaing reurn [Fama (1970), Pogue and Sonic (1974), Levhari and Levy(1977), Smih (1978), and Hawawini (1980)]. In a emina wor Franfurer, Leung, and Brocman (1994) ue US daa o how ha he mean and variance of bea increae a reurn inerva increae Correponding Auhor Emai: [email protected] Auhor acnowedge Oer Dere Aociae profeor Texa Tech Univeriy USA, for hi hepfu commen. 96

2 from daiy o yeary ime horizon. Bornon, Kim, and Lee (1999) ugge ha bea refec macroeconomic ri ha may have differen frequencie (high or ow) and hu he eniiviy of bea o differen ime inerva may refec he impac of hoe ri. In brief, empirica obervaion revea ha bea change, a invemen horizon are changed. Tradiiona mehodoogy of eimaing bea herefore wi be inappropriae aiicay, ince o of informaion wi be o abou he bea dynamic acro differen inerva. Baicay, a ecuriy mare coni of houand of rader and inveor wih differen ime horizon in heir mind regarding heir invemen deciion. Owing o he differen deciion-maing ime horizon among inveor, he rue dynamic of he reaionhip beween oc reurn and ri eimae are iey o vary depending on he ime horizon of he inveor. Therefore, heir percepion and meauremen of ri are no preumed o be he ame. Financia anay have ong recognized he need o incorporae differen ime cae in ine wih he inveor deciion maing in he financia mare. However, due o he ac of appropriae anayica oo o decompoe daa ino more han wo ime cae, he anayi wa rericed, uni receny, o wo ime cae i.e., hor-run and ong-run ony (In & Kim, 2006). A reaivey new approach nown a wavee anayi ha ae care of differen ime cae or horizon in deciion maing woud hopefuy addre ha gap. In hi paper, we herefore, re-examine ri meauremen hrough a nove approach, wavee anayi. Wavee anayi coniue a very promiing oo a i repreen a refinemen in erm of anayi in he ene ha boh ime and frequency domain are aen ino accoun. The paper organize a foow. Secion 2 brief abou he wavee echnique, decribe he mehodoogy and provide he daa and i decripion ued in hi udy. Secion 3 repor he reu wih inference and he fina ecion concude. 2. Daa and Mehodoogy The arice ue daiy daa of a he oc compriing BSE-30 for a period of 28 monh from 5 January 2010 o 31 march The baic moivaion o conider BSE-30 i due o he fac ha BSE-30 i a repreenaive index of Indian oc mare compriing arge hiry capiaied companie. For each ampe company, daiy reurn erie (562 obervaion) have been coeced. In addiion daiy mare indice for imiar period of ime ao have been coeced. Daa ha been coeced from officia webie of Bombay Soc Exchange (BSE). Afer cacuaing he reurn erie for every oc and he mare, wavee anayi i ued o eparae ou each reurn erie ino i coniuen mui reouion (mui-horizon) 97

3 componen. For hi purpoe Maxima overap dicree wavee ranform (DWT) ha been appied o obain a muiage decompoiion of he reurn erie a differen cae crya () a foow: D1 (2 4 day), D2 (4 8 day) day, D3 (8 16 day), D4 (16 32 day), D5 (32 64 day) and S5 ( day). The wavee mehodoogy i ued o decompoe he mare reurn and company reurn in o differen ime cae. Wavee are imiar o a ine and coine funcion becaue hey ociae around zero, bu differ becaue hey are we ocaized boh in he ime and frequency domain. In conra o Fourier anayi wavee are compacy uppored, a a proecion of a igna ono he wavee pace are eeniay oca, no goba, and hu need no be homogeneou over ime. Wavee are fexibe in handing variey of non-aionary igna. Wavee, in oppoiion o ime and frequency domain anaye, conider nonaionariy an inrinic propery of he daa raher han a probem o be oved by preproceing he daa. There are wo baic wavee funcion: he faher wavee and he moher wavee. Formay he faher wavee can be repreened a, 2 Defined a non-zero over a finie ime engh uppor ha correpond o moher wavee given by 2 2 / 2. 2 (2) Where J=1,..., J in a J-eve decompoiion. The faher wavee inegrae o one and reconruc he rend componen (onge ime cae componen) of he erie. The moher wavee inegrae o zero and decribe a deviaion from he rend. In order o compue he decompoiion, wavee coefficien a a cae repreening he proecion of he ime erie ono he bai generaed by he choen famiy of wavee need o be cacuaed fir, hey are d, f ( ), / S, f ( ), (1) 98

4 Coefficien d. and S, are wavee ranform coefficien repreening he proecion ono moher and faher wavee repecivey. The erie or funcion f () in L 2 ( R ) can be hown in wavee repreenaion a f (,,,,,, 1, 1, ) S ( ) d ( ) d d ( ) (3) Here J refer o he number of cae (muireouion componen) and K range from 1 o he number of coefficien in he pecified componen. The origina ime erie muireouion decompoiion framewor can be given a f in f ( ) S D D D D (4) 1 1 S, D repreen S,, ( ) and d ( ),, repecivey wih 1,, J.The equenia e of erm ( S, D,... D,..., D1 ) in equaion (4) how he componen of origina unfiered erie repreened a differen reouion. Foowing Yamanda (2006) we eimae he foowing wo equaion: R R i, Rm, i, Rm. Rm, (6) Where, in equaion (5) meaure he ri aociaed wih a company oc price wherea, and are coefficien aociaed wih a hor periodiciy erie and a ong-periodiciy erie of mare reurn. (5) 3. Reu and Dicuion: Uing equaion (5) we eimae conveniona Bea and Equaion (6) i ued o eimae he hor-periodiciy componen and he ong-periodiciy componen of mare reurn. Thi i done by decompoing he daa ino hor and ong-periodiciy erie uing he maxima overap dicree wavee ranform (MODWT) 1. We chooe he Maxima Overap Dicree Wavee Tranform (MODWT) over he more conveniona orhogona DWT becaue, by 1 Shor periodiciy and ong periodiciy of mare reurn erie are defined a (D1+D2+D3) and (D4+D5+S5).repecivey. Since orhogonaiy condiion are o uing MODWT we eed he hor periodiciy and ong periodiciy componen for muicoineariy which wa found oo ow (0.06) o affec he eimaion procedure. 99

5 giving up orhogonaiy, he MODWT gain aribue ha are far more deirabe in economic appicaion. For exampe, he MODWT can hande inpu daa of any engh, no u power of wo; i i ranaion invarian ha i, a hif in he ime erie reu in an equivaen hif in he ranform; i ao ha increaed reouion a ower cae ince i overampe daa (meaning ha more informaion i capured a each cae); he choice of a paricuar wavee fier i no o crucia if MODWT i ued and, finay, exceping he a few coefficien, he MODWT i no affeced by he arriva of new informaion. Tabe.1 repor he conveniona bea eimae and he wavee-baed bea eimae for he 29 oc porfoio. The reu for he conveniona equaion (5) are repored in he fir, econd and hird coumn. The reu for he wavee-baed equaion (6) are repored from fourh o evenh coumn. The a coumn abuae he F -vaue for he nu hypohei H0: = 2. Thi abe indicae ha he Figure 1 Po from op o boom repreen unfiered, hor periodiciy and ong periodiciy mare reurn erie /Jan/10 05/Feb/10 05/Mar/10 05/Apr/10 05/May/10 05/Jun/10 05/Ju/10 05/Aug/10 05/Sep/10 05/Oc/10 05/Nov/10 05/Dec/10 05/Jan/11 05/Feb/11 05/Mar/11 05/Apr/11 05/May/11 05/Jun/11 05/Ju/11 05/Aug/11 05/Sep/11 05/Oc/11 05/Nov/11 05/Dec/11 05/Jan/12 05/Feb/12 05/Mar/12 2 F vaue i baed on wad e aic 100

6 Tabe. 1 Conveniona and wavee-baed bea eimae 2 AdR 2 AdR BSE Senex Co F Cipa BHEL HDFC SBI HDFC Ban *** Hero Moor * Infoy *** ONGC Reiance TATA Power Hindaco TATA See * L&T Mahindra & Mahindra TATA Moor Hinduan Uniever ITC Serie Wipro Sun Pharama GAIL ICICI Jinda See Air Te ** Marui Suzui TCS NTPC DLF ** Baa Auo Average ** ***, **,* indicae ignifican f Saiic a10%, 5% and 1% repecivey. conveniona bea eimae are beween he wavee-baed bea eimae and are approximaey he average of hem. Second, he nu hypohei, H0: = i no reeced for 23 of he 29 companie a he 10% eve of ignificance. Third, he nu hypohei i reeced for companie uch a HDFC Ban, Hero Moor, Infoy, Aire and combined average reurn. To be precie, for HDFC Ban, Hero Moor and average porfoio < 101

7 wherea for Infoy, Aire and DLF >. Thee reu impy ha reurn on ecuriie in HDFC Ban, Hero Moor, Aire and from combined 30 oc porfoio are e voaie, in he hor-run han conveniona conideraion ugge. Thee empirica reu indicae ha whie conveniona bea eimae are uefu in mo cae, in ome cae he wavee baed bea eimae are uefu in underanding he eniiviy of he reurn of paricuar oc o he reurn on he mare index. Tabe.2 Mui-Scae Wavee Bea Eimae on each of he Recompoed Soc BSE Senex Companie D1 D2 D3 D4 D5 S5 CIPLA (0.15) (0.14) (0.17) (0.08) (0.33) (0.13) BHEL (0.06) (0.25) (0.18) (0.12) (0.09) (0.05) HDFC (0.07) (0.09) (0.06) (0.12) (0.18) (0.00) STATE BANK OF INDIA (0.48) (0.45) (0.47) (0.58) (0.03) (0.62) HDFC Ban (0.07) (0.05) (0.13) (0.23) (0.03) (0.27) Honda Moor (0.10) (0.05) (0.16) (0.09) (0.34) (0.10) Infoy (0.45) (0.40) (0.36) (0.34) (0.40) (0.42) ONGC (0.06) (0.06) (0.06) (0.07) (-0.00) (0.13) Reiance (0.56) (0.58) (0.55) (0.47) (0.65) (0.58) TATA Power (0.08) (0.03) (0.03) (0.16) (0.30) (0.08) Hindaco (0.48) (0.54) (0.51) (0.54) (0.65) (0.53) TATA See (0.56) (0.50) (0.46) (0.54) (0.70) (0.62) LARSEN AND TURBO (0.48) (0.50) (0.46) (0.62) (0.46) (0.63) Mahindra & Mahindra (0.10) (0.09) (0.21) (0.20) (0.17) (0.03) TATA Moor (0.11) (0.01) (0.17) (0.15) (0.09) (0.34) Hinduan Uniever (0.06) (0.00) (0.02) (0.06) (0.27) (0.02) ITC

8 (0.09) (0.06) (0.05) (0.05) (0.07) (0.00) Serie (0.13) (0.16) (0.24) (0.21) (0.23) (0.11) Wipro (0.18) (0.18) (0.13) (0.16) (0.16) (0.08) Sun Pharama (0.01) (0.07) (0.05) (0.01) (0.12) (0.12) GAIL (0.20) (0.11) (0.23) (0.26) (0.39) (0.23) ICICI Jinda See (0.41) (0.43) (0.55) (0.49) (0.64) (0.71) Air Te (0.26) (0.15) (0.11) (0.08) (0.28) (0.29) Marui Suzui (0.18) (0.23) (0.22) (0.30) (0.21) (0.55) TCS (0.38) (0.34) (0.36) (0.37) (0.38) (0.36) NTPC (0.34) (0.28) (0.41) (0.39) (0.46) (0.55) DLF (0.46) (0.40) (0.54) (0.54) (0.54) (0.69) Baa Auo (0.08) (0.07) (0.20) (0.05) (0.02) (0.00) AVERAGE (0.87) (0.87) (0.90) (0.90) (0.87) (0.06) Vaue in parenhei repreen Adued R 2 & D1 o S6 repreen cae-wie Bea of ampe companie In order o how weaher he Bea eimaion in India i a mui cae phenomena we decompoe boh mare reurn and company reurn ino ix cae uing (MODWT). Tabe. 2 repor he Mui cae bea eimaion reu of equaion (5). The bea of individua oc i evauaed a a ix cae. Since we empoy daiy daa in our anayi, wavee cae are uch ha cae one correpond o he period of 2-4 day, cae 2-4 o 8 day, Scae 3, 8 o 16 and o on dynamic. Scae S6 i he highe one a which we can ae he bea of each oc and i correpond o day dynamic i.e., approximaey one year. Our reu demonrae he mui-cae endency of he average bea coefficien in Indian oc mare. Thee finding direcy conradic wih he reu of Norworhy e a. (2000), Fernandez (2006), Maih (2010) who oberved he behaviour of R 2 decreaed monoonicay when moved o higher cae (onger inerva). In oher word heir reu concude ha mare reurn are more abe o expain individua oc reurn a ower cae (horer inerva) han 103

9 higher cae. However, he preen piece of wor find inereing reu. Ahough he bea eimae urn ignifican a each eve, he higher R 2 aociaed wih cae-3, cae-4, and cae-5 repecivey impie ha maor par of mare porfoio infuence on individua oc i a medium o higher frequencie. The finding indicae ha inveor wih medium and onger invemen horizon ha o repond o every fucuaion in reaized reurn. 4. Concuion: Uing wavee we examined he dynamic of oc reurn of 29 firm from Indian Soc mare. We cacuaed wo bea baed on he hor periodiciy and ong periodiciy of mare reurn baed on (MODWT). We have hown ha he conveniona bea eimae i an average of he wavee-baed bea eimae for mo of he cae. Thi mean ha he conveniona bea eimae are uefu in mo cae, bu in ome cae, bea for hor periodiciy and ong periodiciy mare reurn were ignificany differen uggeing wavee-baed bea eimae are uefu for underanding he eniiviy of he reurn of paricuar ecuriie o he reurn on he mare index. Furher we eimaed mui-cae bea on each oc compriing BSE-enex and oberved ha bea of a oc i no abe over ime, due o mui-rading raegie of inveor. In paricuar we found mare ri were concenraed by and arge, a he medium and higher frequencie of daa. In oher word he finding impy ha predicion of he Mare mode woud be more reevan a medium- o ong-run horizon a compared o hor ime horizon. 104

10 Refrence: Bornon, B., Kim, H. S & Lee, K (1999), Low and high frequency macroeconomic force in Ae pricing, Quarery review of economic and finance, 39, Daubechie, I. (1992), Ten Lecure on Wavee, CBSM-NSF Regiona Conference Serie in Appied Mahemaic. SIAM, Phiadephia. Franfurer, G., Leung, W., & Brocman, W. (1994), Compounding period engh and he mare mode, Journa Economic Buine 46, Genc ay, R., Sec u, F., & Whicher, B. (2005). Mui cae yemaic ri Journa of Inernaiona Money and Finance, 24, Granger, C. (2002). Some commen on ri, Journa of Appied Economeric, 17(5), Hawawini, G. A. (1980) Inerempora cro dependence in ecuriie daiy reurn and he hor-run inervaing effec on yemaic ri, Journa of Financia and quaniaive anayi,15, Maa, S. (1989). A heory for muireouion igna decompoiion: he wavee repreenaion, IEEE Tranacion on Paern Anayi and Machine Ineigence Vo. 11, pp Hirohi Yamada, Wavee-Baed Bea Eimaion and Japanee Induria Soc Price, Appied Economic Leer, 2005, 12, Sharpe, W. F. (1964) Capia ae price: a heory of mare equiibrium under condiion of ri. Journa of Finance, 19(3), In, F., & Kim, S. (2006) The hedge raio and he empirica reaionhip beween he oc and fuure mare: a new approach uing wavee The Journa of Buine, 79, K. (1999) Low and high frequency macroeconomic force in ae pricing, Quarery Review of Economic and Finance, 39, Linner, J. (1965) The vauaion of ri ae and he eecion of riy invemen in oc porfoio and capia budge Review of Economic and Saiic, 47, Maih Manur e a., Syemaic ri and ime cae: New evidence from an appicaion of wavee approach o he emerging Guf oc Mare, Inernaiona Review of Financia Anayi 19 (2010) 10 18, pp Pogue, G., & Sonic, B. (1974) The mare mode appied o European common oc: ome empirica reu, Journa of Financia and Quaniaive Anayi, 9, Quaniaive Anayi, 15, Rhaiem Naa e a., Eimaion of Capia ae pricing Mode a differen ime cae: Appicaion o French Soc mare, The Inernaiona Journa of Appie Economic and finance (1)2: 79-87,2007, pp

11 Rua Anonio and Nune Lui C., wavee-baed aemen of mare ri: The emerging mare cae, The Quarery Review of Economic and Finance 52 (2012) 84 92, pp