BANK SIZE AND INTEREST-RATE SENSITIVITY OF BANK STOCK RETURNS

BANK SIZE AND INTEREST-RATE SENSITIVITY OF BANK STOCK RETURNS Jianzhou Zhu Univeriy of Wiconin Whiewaer Morheda Haan Wiley College Wanli Li Xian Jiaoong Univeriy ABSTRACT Thi udy re-examine he exra-marke inere rae eniiviy of bank ock reurn over he period from January 1976 hrough December 2005 uing he wo-index model developed by Sone (1974). The evidence how ha alhough he marke rik i he primary deerminan of bank ock reurn, movemen in inere rae poe ignifican explanaory power for he remaining variabiliy of bank ock reurn afer he marke rik i conrolled. Conien wih previou udie, he reul indicae ha he oberved exra-marke inere rae rik of bank ock reurn i eniive o he elecion of he ample period and inere rae index in empirical udie. More imporanly, he udy repor weak evidence ha he cro-ecional difference of inere rae eniiviy of bank ock reurn i negaively relaed o bank ize meaured by marke capializaion. INTRODUCTION The influence of inere rae change on bank ock reurn ha been a ubjec of ubanial empirical inveigaion. A common heoreical framework ued in mo of hee udie i he wo-index capial ae pricing model developed by Sone (1974). The wo-index model augmen he radiional ingle-index marke model by including an index of reurn on deb inrumen a an addiional explanaory variable. Sone believe ha he marke model leave a ignifican porion of covariance in ecuriy reurn unexplained and he index of reurn on deb inrumen, or imply he inere rae index a uually called, can capure he porion of covariance in ecuriy reurn mied by he equiy marke index. Paricularly, he explanaory power of he inere rae index in he wo-index model hould be more pronounced for ock iued by firm in gold, uiliie, banking and oher financial ecor. Reul from empirical eing of he wo-index model uing banking ock reurn have no been enirely conien. The balance of he evidence, however, i clearly weighed oward he model favor. Early udie by Chance and Lane (1980), Gulekin and Rogalki (1979) and Sweeney and Warga (1986) repreen a few excepion in which he inere rae variable conribue lile o he reurn generaing proce of bank ock beyond wha i explained by he ock marke index. Mo udie find ha bank ock reurn exhibi ignifican eniiviy o inere rae movemen over and above heir eniiviy o ock marke reurn. Moreover, hi eniiviy exceed ha hown by mo nonfinancial firm, confirming he noion ha he paricular naure of bank ae and liabiliie make bank ock reurn epecially eniive o Inernaional Journal of Buine, Accouning, and Finance Volume 1, Number 1, 2007 1

change in inere rae. Sudie fall ino he laer caegory include early udie by Marin and Keown (1977), Lynge and Zumwal (1980), Flannery and Jame (1984), Booh and Officer (1985), Scoo and Peeron (1986) and Bae (1990). Akella and Chen (1990) aribue he conflicing reul of hee udie o he choice of proxie for marke inere rae and find ha bank ock reurn are eniive o long-erm bu no o hor-erm inere rae. However, Manur and Elyaiani (1995) and Elyaiani and Manur (2004) ue hor-erm, medium-erm, a well a long-erm inere rae a alernaive proxie for he marke inere rae variable and find evidence of ignifican negaive inere rae eniiviy in all cae, alhough heir finding reveal ha he long-erm and medium-erm inere rae affec bank ock reurn by a greaer exen han do he hor-erm inere rae. More recenly, Elyaiani and Mauur (2005) found evidence ha inere rae i only occaionally ignifican while marke reurn and exchange rae ha more yemaic effec on bank ock reurn. Joeph and Veo (2006) documened imilar reul uing high-frequency daa. Difference in ample period eleced in empirical udie are alo cied a one poible reaon for he conflicing reul. Kane and Unal (1988) employ a wiching regreion echnique o eimae he parameer of a wo-index model over he period of 1975-1985. They repor ha he inere rae eniiviy of bank and aving and loan ock reurn varie ignificanly over differen regime (ubperiod). Flannery and Jame (1984) aribue he exra-marke inere-rae eniiviy of bank ock o he mauriy mimach of nominal ae and liabiliie on and off he balance hee of hee financial iniuion. Afer eimaing a wo-index model on a cro ecion of bank ock reurn, hey relaed he eimaed inere rae coefficien from hi regreion o heir mauriy mimach meaure and found ha he mauriy mimach i ignificanly relaed o he oberved inere rik of he bank and hrif ock. Uing random coefficien model, Kwan (1991) alo find ha he effec of inere rae change on commercial bank ock reurn i poiively relaed o he mauriy mimach beween he bank' ae and liabiliie. Boh Flannery and Jame' (1984) and Kwan' (1991) reul clearly indicae ha he inere rae eniiviy of bank ock reurn i ime-varying, given ha bank mauriy profile invariably change over ime. Thi i conien wih he conjecure ha he difference in ample period eleced by many udie may be a lea parially reponible for he conflicing empirical reul. The inabiliy of he coefficien for inere rae in he wo-index model i uggeive ha oher facor beide he equiy and deb marke reurn may alo have ignifican impac on he ock reurn of financial iniuion in pecific ime period. He, Myer, and Webb (1996) augmened he radiional wo-index model by adding real eae marke reurn proxied by REIT ock reurn. They find ha REIT ock reurn i a relevan facor in explaining boh bank ock reurn and rik. He and Reicher (2003) meaured real eae marke reurn wih change in median ale price of new houe old naionwide and found imilar reul. In hi udy we examine he cro-ecional difference of bank ock in erm of heir inere rae eniiviie and explore poible reaon reponible for he cro-ecional difference. Mo udie in exiing lieraure examine he exra-marke inere rae eniiviy of bank ock a a group. The difference of he inere rae eniiviy acro individual bank ha received much le aenion. Flannery and Jame' (1984) and Kwan' (1991) provide wo excepion in which hey focu on he cro-ecional difference of inere rae eniiviy among individual bank ock and aribue he difference of inere rae eniiviy o he difference in he degree of mauriy mimach on he balance hee of individual bank. In hi udy we empirically inveigae he relaionhip beween ize of commercial bank a meaured by 2 Inernaional Journal of Buine, Accouning, and Finance, Volume 1, Number 1, 2007

marke capializaion and he inere rae eniiviy of bank ock reurn. I i reaonable o expec ha ock reurn of big bank are le eniive o change in inere rae relaive o ock reurn of mall bank. Big bank are generally more diverified in heir operaion and hu heir earning are le dominaed by inere income relaive o mall bank whoe opporuniie for diverificaion may be more limied. Similarly, a large bank may be able o exploi poible economie of cale in hedging again inere rik ha a mall bank canno. Thee difference will how up in he ae pricing model in erm of differen value of inere rae bea. Alhough everal udie provide rough comparion of inere rae eniiviy among ize-ored bank porfolio, he conjecure of he negaive correlaion beween bank ize and inere rae eniiviy of bank ock reurn ha no been formally eed. Furhermore, ome udie ha compare he inere rae eniiviie among ize-ored bank porfolio repor finding ha are inconien wih he above conjecure. For example, Neuberger (1991) found ha large bank ock were more eniive o inere rae change in a lea 3 of he 4 ubample period han mall bank ock. I i urpriing ha he auhor did no elaborae on hee finding given he fac ha he argued forcefully for he conjecure ha large bank ock hould be le eniive o inere rae change. The re of he paper i rucured a follow: Secion II decribe he daa e and analyi procedure. Secion III repor and inerpre he finding. We provide a brief concluion in Secion IV. DATA AND ANALYTICAL PROCEDURE Thi udy cover he period from January 1976 o December 2005. To our knowledge he ample period in he curren udy i much longer han hoe examined in exiing lieraure on he inere rae eniiviy of bank ock reurn. The lengh of he ample period allow u o divide he whole ample ino a reaonable number of ub-ample period while mainaining a ufficien number of obervaion in each ub-ample period o obain aiical robune. Thi i paricularly imporan for obaining a reliable concluion on he ime-varying naure of he inere rae eniiviy of bank ock reurn. We divided he 30-year ample period ino ix 5- year ub-ample period. Each ub-ample period ha 60 monhly daa poin. We obained monhly daa on ock reurn and marke capializaion for individual bank from CRSP (Cener for Reearch in Securiie Price) daa file. 36 bank have no miing daa over he enire ample period and are eleced ino he ample. For each ample bank, we calculaed i average marke capializaion over he whole ample period a well a he ix 5- year ub-ample period. For he whole ample period and each of he ix ub-ample period, hree ize-ored porfolio are conruced. The 10 bank wih he large average capializaion are deignaed o he Big-ize porfolio. The 10 bank wih he malle average capializaion are deignaed o he Small-ize porfolio. The remaining 16 bank are claified ino he Medium-ize porfolio. Finally, equal-weighed porfolio reurn are calculaed for he hree ize-ored porfolio a well a he porfolio including all ample bank. Thee monhly porfolio reurn are ued a he proxy for bank ock reurn (BK). The ock marke reurn (SP) are meaured by reurn on Sandard & Poor Compoie Index which are alo obained from CRSP dae file. A menioned in he inroducion, previou udie found ha bank ock reurn exhibied differen degree of eniiviy o change in hor-erm and long-erm inere rae. A number of udie found ha bank ock reurn are eniive only o change in long-erm rae Inernaional Journal of Buine, Accouning, and Finance Volume 1, Number 1, 2007 3

while oher udie found ha bank ock reurn are eniive o boh long-erm and hor-erm inere rae. Several udie find ha bank ock reurn are more eniive o long-erm rae han hey are o hor-erm rae while oher udie repor finding uggeing ju he oppoie. The conflicing reul in exiing udie may be parly caued by he differen combinaion of ample period covered and he inere rae index employed in hee udie. In curren udy, we ue hree alernaive marke inere rae indexe: 1-year Treaury bond yield (TB1) a he proxy for hor-erm marke rae, 7-year Treaury bond yield (TB7) a he proxy for mediumerm marke rae, and 10-year Treaury bond yield (TB10) a he proxy for long-erm marke rae. By applying each alernaive inere rae index o a common e of ample and ub-ample period covering a long a hiry year ince 1976 (January 1976 December 2005, January 1976 December 1985, January 1986 December 1995, January 1996 December 2005), he procedure allow u o addre he poible difference in he relaive eniiviy of bank ock reurn o hor-erm, medium-erm, and long-erm inere rae a well a wheher hi relaive eniiviy change over ime. Following mo udie in exiing lieraure, he following baic wo-index model i ued o meaure he conemporaneou effec of inere rae change on bank ock reurn while he reurn of he ock marke i conrolled. BK i, = β 0 + β msp + β dtb + ε (1) where BK i, i he holding period reurn on equal-valued bank porfolio i (i = 1, 2, 3, and 4 for All-bank porfolio, Big-ize porfolio, Medium-ize porfolio and Small-ize porfolio, repecively) from monh -1 o monh, SP i he reurn of broad marke in monh a meaured by Sandard & Poor Compoie Index, TB j, i he inere rae index j (j = 1, 2, and 3 for yield on 1-year, 7-year and 10-year Treaury bond, repecively) in monh ; ε i normally diribued and erially uncorrelaed diurbance erm from he model for inere rae index j. β 0, β m, and β d are eimaed inercep, ock marke bea, and deb marke bea repecively when inere rae index j i ued a he deb marke reurn erie. One poenial problem i eimaing he above model i he poible mulicollineariy beween he wo reurn erie ued a he explanaory variable in he equaion. In developing hi wo-index model, Sone (1974) poined ou ha reurn on deb are probably influenced by he ame facor ha deermine he reurn on he marke porfolio of ock. He uggeed orhogonalizing one of he erie by regreing i on he oher. The reidual erie from hi orhogonalizing regreion, which by definiion i uncorrelaed wih he oher explanaory variable, hen can be ued a a regreor in he bank ock reurn equaion. Gilibero (1985) demonraed ha he eimaed andard error of he econd-age regreion coefficien are unbiaed only for he erie ha wa ued a he dependen variable in he fir-age regreion. Thi mean udie uing orhogonalized ock marke reurn erie may produce biaed eimae of β d while udie uing orhogonalized deb marke reurn erie may produce biaed eimae of β m. To check he empirical implicaion of hi problem, we alo applied he following wo varian of Equaion (1): BK i, = β 0 + β msp' + β dtb + ε (2) BK i, = β 0 + β msp + β dtb' + ε (3) 4 Inernaional Journal of Buine, Accouning, and Finance, Volume 1, Number 1, 2007

All variable in Equaion (2) are defined imilarly a hey are in Equaion (1) excep SP' in Equaion (2) i he orhogonalized ock marke reurn erie; i.e., he reidual obained from he regreion SP β 0 + β jtb + υ =. (4) Similarly, TB ' j, in Equaion (3) i he orhogonalized deb marke reurn erie meaured by inere rae index j; i.e., he reidual obained from he regreion TB = β 0 + β j SP + κ. (5) Thi procedure i performed for he whole ample period a well a he ix ub-ample period. The reul are repored in he nex ecion. EMPIRICAL RESULTS Alhough we eimae he wo-index model in hree varian: wih boh ock marke reurn erie and deb marke reurn erie unorhogonalized (original daa erie), wih he ock marke reurn erie orhogonalized, and wih he deb marke reurn erie orhogonalized, he empirical reul are no qualiaively differen. Therefore, only he regreion reul uing original daa erie are repored o ave pace. Table 1 preen finding on he porfolio of all bank in he ample. The fir poin we can make baed on Table 1 i ha reurn on he marke porfolio of ock are indipuably he mo imporan deerminan of bank ock reurn. Regardle of wheher hor-erm, mediumerm or long-erm inere rae i ued in he equaion, he eimaed ock marke bea coefficien are alway ignificanly poiive for he whole ample period a well a he ix 5- year ub-ample period. In all cae, he -aiic on he eimaed ock marke bea coefficien i larger han he deb marke bea coefficien and he minimum value i 5.68, indicaing a level of ignificance beer han 1%. Thi i conien wih he capial ae pricing model. Regardle of he proxy for marke inere rae ued, he eimaed coefficien for ock marke bea i maller han 1 in all ime period excep for he period of 1991 1995 during which i i very cloe o 1, uggeing ha commercial bank ock are uually le riky han he broad marke. Thi i alo largely conien wih previou udie (Neuberger, 1991). Anoher concluion we can draw from Table 1 i ha bank ock reurn do exhibi exramarke eniiviy o change in inere rae. The eimaed bea coefficien for deb marke reurn i negaive in all cae when i i ignifican. Bu he effec of inere rae on bank ock reurn i much maller compared o he effec of ock marke reurn in erm of he eimaed value for bea coefficien. More imporanly, he inere rae bea demonrae more variabiliy acro differen ample and ub-ample period. No maer wha inere index i ued, he eimaed bea coefficien for deb marke lo i aiical ignificance in he la wo ubperiod and omeime even wrongly igned. Thi confirm he finding documened in previou udie ha he inere rae eniiviy of bank ock i ime-varying. Poible explanaion for he deb marke bea o loe i aiical ignificance include beer diverificaion and hedging by bank a well a he reduced variabiliy of marke inere rae. Finally, Table 1 ugge ha bank ock reurn may be more eniive o change in long-erm Inernaional Journal of Buine, Accouning, and Finance Volume 1, Number 1, 2007 5

Table 1 Regreion reul on he porfolio of all bank in he ample Panel A: TB10 a marke inere rae proxy Sample period Conan SP500 TB10 R 2 DW 1976-2005 0.011 (6.31)*** 0.789 (18.95)*** -0.170 (3.95)*** 0.53 1.98 1976-1980 0.015 (5.24)*** 0.627 (9.10)*** -0.414 (6.04)*** 0.73 1.82 1981-1985 0.020 (6.10)*** 0.695 (7.43)*** -0.377 (4.08)*** 0.70 1.82 1986-1990 0.002 (0.54) 0.821 (11.52)*** -0.034 (2.26)** 0.73 1.82 1991-1995 0.013 (3.24)*** 0.990 (6.95)*** 0.004 (0.04) 0.49 1.36 1996-2000 0.010 (1.28) 0.865 (5.677)*** -0.146 (0.82) 0.38 2.28 2001-2005 0.010 (2.79)*** 0.546 (6.36)*** 0.084 (1.23) 0.48 2.06 Panel B: TB7 a marke inere rae proxy Sample period Conan SP500 TB7 R 2 DW 1976-2005 0.011 (6.30)*** 0.791 (18.96)*** -0.144 (3.69)*** 0.53 1.96 1976-1980 0.014 (4.79)*** 0.621 (8.46)*** -0.326 (5.21)*** 0.70 1.78 1981-1985 0.020 (6.03)*** 0.705 (7.53)*** -0.349 (3.94)*** 0.70 1.83 1986-1990 0.002 (0.54) 0.826 (11.51)*** -0.201 (2.00)** 0.73 1.80 1991-1995 0.013 (3.25)*** 1.003 (7.00)*** 0.031 (0.29) 0.49 1.38 1996-2000 0.010 (1.29) 0.866 (5.74)*** -0.179 (1.07) 0.39 2.29 2001-2005 0.010 (2.78)*** 0.551 (6.47)*** 0.066 (1.14) 0.48 2.07 Panel C: TB1 a marke inere rae proxy Sample period Conan SP500 TB1 R 2 DW 1976-2005 0.012 (6.37)*** 0.794 (18.97)*** -0.095 (3.31)*** 0.52 1.93 1976-1980 0.015 (4.88)*** 0.615 (8.25)*** -0.198 (5.09)*** 0.70 1.62 1981-1985 0.019 (5.54)*** 0.789 (8.27)*** -0.156 (2.52)*** 0.66 1.86 1986-1990 0.002 (0.52) 0.850 (11.74)*** -0.100 (1.00) 0.71 1.76 1991-1995 0.013 (3.27)*** 0.970 (6.77)*** -0.026 (0.35) 0.49 1.35 1996-2000 0.010 (1.37) 0.864 (5.78)*** -0.266 (1.46) 0.40 2.29 2001-2005 0.010 (2.76)*** 0.566 (6.83)*** 0.038 (0.95) 0.48 2.10 ***Significan a abou 1%; **Significan a abou 5%; *Significan a abou 10%. inere rae. When he yield on 10-year Treaury bond i ued a he inere rae index, we obain he large -aiic on he eimaed deb marke bea. The -aiic are 6 Inernaional Journal of Buine, Accouning, and Finance, Volume 1, Number 1, 2007

monoonically reduced when he yield on Treaury bond wih horer mauriie are ued a he inere rae index. The regreion reul for ize-ored porfolio are repored in Table 2 hrough Table 4. Table 2 preen he reul uing yield on 10-year Treaury bond a he proxy for marke inere rae. Table 3 and Table 4 preen reul uing yield on 7-year Treaury bond and 1-year Treaury bond a he proxy for marke inere rae, repecively. Finding from he ize-ored regreion fir reinforce he concluion obained from regreion for he all-bank porfolio a repored in Table 1. Regardle of bank ize, inere rae proxy or ample period examined, he eimaed coefficien for ock marke reurn and heir aociaed -aiic are much larger in magniude a compared o hoe for he marke inere rae. For he all-bank porfolio a well a he hree ize-ored porfolio, he relaive change of he eimaed marke bea over ime are alo maller compared o hoe of he eimaed inere rae bea, uggeing ha he reurn on he ock marke porfolio have a ronger and more able role in he reurn-generaing proce of bank ock. Table 2 hough 4 alo how ha marke bea for big bank are uually larger and cloer o one compared o hoe for medium bank, which in urn are uually larger and cloer o one relaive o hoe for mall bank. Thi make ene becaue big bank generally allocae heir ae o more diverified economic ecor and heir rik hould herefore bear a greaer reemblance o he broad marke. Regreion for he hree ize-ored porfolio are alo conien wih reul repored in Table 1 regarding he ime-varying naure of he exra-marke inere rae eniiviy of he bank ock reurn. The eimaed coefficien for he marke inere rae index are found o be ignifican for ome period while no ignifican and wih even he wrong ign for oher period. Thi hold wihou excepion for all bank ize and all marke inere rae proxie. Finally regreion for he ize-ored porfolio uppor he concluion obained from Table 1 ha he inere rae eniiviy of bank ock reurn varie wih he elecion of marke inere rae proxy. Generally peaking, a we change he inere rae proxy from long-erm index o hor-erm index (or move from Table 2 o Table 3 and hen o Table 4), he eimaed coefficien for marke inere rae end o be maller in magniude and aociaed wih maller -aiic. For example, for big bank during he period from January 1986 o December 1990, he eimaed coefficien i -0.295 wih a -aiic of 1.99 for TB10 in Table 2. The equivalen coefficien are -0.258 wih a -aiic of 1.80 and -0.096 wih a -aiic of 0.68 for TB7 in Table 3 and TB1 in Table 4, repecively. A imilar paern i oberved for medium and mall ize bank. More imporanly, he regreion reul on ize-ored porfolio a repored in Table 2 hrough Table 4 ugge ha he inere rae eniiviy of bank ock reurn are negaively relaed o he ize of bank firm meaured by marke capializaion. Regardle of he yield on 10-year Treaury bond (Table 2), 7-year Treaury bond (Table 3) or 1-year Treaury bond (Table 4) ued, for hoe period during which he eimaed coefficien for marke inere rae are ignifican for big and medium bank, hey mu alo be ignifican for mall bank. Bu for ome period during which he eimaed coefficien for marke inere rae are ignifican for mall bank, hey are no ignifican for big and mall ize bank. For example, in Table 2 where he yield on 10-year Treaury bond i ued a marke inere rae index, he eimaed coefficien for marke inere rae during he la wo ub-ample period are no aiically ignifican a any convenional level for big and medium ize bank bu are ignifican a 10% level for mall bank. Similar reul are repored in Table 3 and Table 4. Inernaional Journal of Buine, Accouning, and Finance Volume 1, Number 1, 2007 7

I hould be emphaized ha he evidence for he negaive linkage beween bank ize and inere rae eniiviy of bank ock reurn hould be regarded a enaive. Due o he mall number of ample bank in hi udy we conruc only hree ize-ored porfolio o mainain a reaonable number of bank in each porfolio. Alhough hi i neceary for each porfolio o be repreenaive for he ock behavior of he paricular ize of bank, he porfolio number a mall a hree doe no allow u o dicover rong evidence for a reliable and yemaic relaion beween he ize of a bank firm and he inere rae eniiviy of i ock reurn. Alo, baed on he evidence repored in Table 2 hrough Table 4, he negaive aiical linkage beween bank ize and inere rae eniiviy of bank ock reurn can only be concluded wih he low confidence level (abou 90%) radiionally acceped. Neverhele, here are reaon o believe ha he rue relaion beween bank ize and he inere rae eniiviy of bank ock reurn i ronger han wha we uncovered in curren udy. Furher reearch i neceary o acerain hi empirical linkage. We will have more o ay on hi poin in he ecion of concluion and fuure reearch direcion. There are a lea wo poenial reaon why he evidence on he relaion beween bank ize and he level of inere rae eniiviy i weak. Alhough large bank end o be beer hedged again inere rae movemen, ome big bank may ake advanage of heir reource o peculae in inere rae derivaive and hereby increae heir expoure o marke inere rae movemen. Thi can reduce he difference beween he level of inere rae eniiviy for big and mall bank. Anoher reaon i ha large bank uually pay high dividend relaive o mall bank. Relaive o low dividend ock, he cah flow of high dividend ock bear a greaer reemblance o hoe of bond. The high dividend ock and hereby big bank ock are herefore ubjec o more influence from bond marke condiion. The poiive relaionhip beween dividend and inere rae eniiviy end o offe he negaive relaionhip beween ize and inere rae eniiviy for big ock bank. Therefore, he difference in he level of inere rae eniiviy beween big and mall bank i likely o be undereimaed wihou conrolling he influence of dividend yield. Exploring hi poibiliy hould be an inereing area for fuure udie. Table 2 Regreion reul on he ize-ored porfolio uing TB10 a inere rae proxy Panel A: Large-ize bank porfolio Sample period Conan SP500 TB10 R 2 DW 1976-2005 0.009 (4.29)*** 0.973 (19.37)*** -0.150 (2.90)*** 0.53 2.12 1976-1980 0.013 (4.12)*** 0.323 (7.82)*** -0.485 (6.14)*** 0.70 1.96 1981-1985 0.018 (3.95)*** 0.927 (7.05)*** -0.307 (2.36)*** 0.63 1.86 1986-1990 -0.002 (0.40) 1.006 (9.88)*** -0.295 (1.99)** 0.67 2.14 1991-1995 0.013 (2.65)*** 1.351 (7.67)*** 0.158 (1.08) 0.52 1.61 1996-2000 0.008 (1.05) 1.083 (6.48)*** 0.005 (0.03) 0.43 2.36 2001-2005 0.006 (1.52) 0.762 (8.16)*** 0.053 (0.72) 0.58 2.38 8 Inernaional Journal of Buine, Accouning, and Finance, Volume 1, Number 1, 2007

Panel B: Medium-ize bank porfolio Sample period Conan SP500 TB10 R 2 DW 1976-2005 0.012 (6.13)*** 0.720 (16.36)*** -0.181 (4.00)*** 0.46 2.09 1976-1980 0.013 (3.75)*** 0.617 (7.32)*** -0.379 (4.53)*** 0.63 1.72 1981-1985 0.200 (5.47)*** 0.597 (5.68)*** -0.449 (4.31)*** 0.64 2.14 1986-1990 0.003 (0.83) 0.742 (10.14)*** -0.201 (1.88)* 0.68 1.85 1991-1995 0.015 (3.73)*** 0.808 (5.77)*** -0.062 (0.53) 0.42 1.43 1996-2000 0.008 (1.12) 0.865 (5.45)*** -0.129 (0.70) 0.36 2.38 2001-2005 0.011 (2.66)*** 0.431 (4.44)*** 0.072 (0.35) 0.31 2.16 Panel C: Small-ize bank porfolio Sample period Conan SP500 TB10 R 2 DW 1976-2005 0.013 (6.69)*** 0.673 (15.04)*** -0.177 (3.84)*** 0.42 1.91 1976-1980 0.018 (4.71)*** 0.642 (6.84)*** -0.378 (4.06)*** 0.59 1.96 1981-1985 0.021 (5.67)*** 0.562 (5.13)*** -0.376 (3.48)*** 0.57 1.79 1986-1990 0.005 (1.28) 0.717 (10.00)*** -0.206 (1.97)** 0.67 1.88 1991-1995 0.011 (2.36)*** 0.807 (4.89)*** -0.083 (0.61) 0.34 1.54 1996-2000 0.011 (1.42) 0.646 (4.18)*** -0.310 (1.72)* 0.29 2.03 2001-2005 0.013 (3.37)*** 0.444 (4.73)*** 0.126 (1.69)* 0.37 1.89 ***Significan a abou 1%; **Significan a abou 5%. *Significan a abou 10%. Table 3 Regreion reul on he ize-ored porfolio uing TB7 a inere rae proxy Panel A: Large-ize bank porfolio Sample period Conan SP500 TB7 R 2 DW 1976-2005 0.009 (4.29)*** 0.975 (19.40)*** -0.130 (2.75)*** 0.53 2.11 1976-1980 0.013 (3.81)*** 0.613 (7.32)*** -0.384 (5.54)*** 0.67 1.93 1981-1985 0.018 (3.93)*** 0.932 (7.13)*** -0.289 (2.34)*** 0.63 1.86 1986-1990 -0.002 (0.38) 1.012 (9.90)*** -0.258 (1.80)* 0.66 2.13 1991-1995 0.013 (2.63)*** 1.361 (7.69)*** 0.156 (1.18) 0.52 1.61 1996-2000 0.008 (1.06) 1.078 (6.49)*** -0.038 (0.20) 0.43 2.39 2001-2005 0.006 (1.52) 0.766 (8.26)*** 0.043 (0.67) 0.58 2.38 Inernaional Journal of Buine, Accouning, and Finance Volume 1, Number 1, 2007 9

Panel B: Medium-ize bank porfolio Sample period Conan SP500 TB7 R 2 DW 1976-2005 0.012 (6.13)*** 0.723 (16.39)*** -0.155 (3.77)*** 0.46 2.07 1976-1980 0.012 (3.47)*** 0.613 (6.96)*** -0.295 (3.94)*** 0.60 1.71 1981-1985 0.200 (5.38)*** 0.611 (5.78)*** -0.409 (4.08)*** 0.63 2.16 1986-1990 0.003 (0.83) 0.750 (10.15)*** -0.169 (1.65)* 0.67 1.83 1991-1995 0.015 (3.73)*** 0.814 (5.77)*** -0.0405 (0.38) 0.41 1.45 1996-2000 0.009 (1.13) 0.865 (5.50)*** -0.165 (0.94) 0.36 2.40 2001-2005 0.0105 (2.65)*** 0.437 (4.53)*** 0.054 (0.83) 0.31 2.16 Panel C: Small-ize bank porfolio Sample period Conan SP500 TB7 R 2 DW 1976-2005 0.013 (6.68)*** 0.675 (15.07)*** -0.146 (3.50)*** 0.42 1.90 1976-1980 0.017 (4.14)*** 0.640 (6.53)*** -0.290 (3.48)*** 0.56 1.91 1981-1985 0.021 (5.62)*** 0.571 (5.22)*** -0.349 (3.38)*** 0.56 1.79 1986-1990 0.005 (1.28) 0.722 (10.01)*** -0.175 (1.74)* 0.67 1.87 1991-1995 0.011 (2.36)*** 0.830 (4.98)*** -0.023 (0.19) 0.34 1.58 1996-2000 0.011 (1.44).0653 (4.27)*** -0.331 (1.95)** 0.30 2.02 2001-2005 0.013 (3.36)*** 0.451 (4.83)*** 0.103 (1.62)* 0.37 1.89 ***Significan a abou 1%; **Significan a abou 5%; *Significan a abou 10%. Table 4 Regreion reul on he ize-ored porfolio uing TB1 a inere rae proxy Panel A: Large-ize bank porfolio Sample period Conan SP500 TB1 R 2 DW 1976-2005 0.010 (4.36)*** 0.977 (19.41)*** -0.090 (2.58)*** 0.53 2.09 1976-1980 0.013 (3.78)*** 0.610 (7.02)*** -0.228 (5.05)*** 0.65 1.87 1981-1985 0.018 (3.77)*** 1.000 (7.84)*** -0.132 (1.59) 0.61 1.88 1986-1990 -0.002 (0.37) 1.044 (10.14)*** -0.096 (0.68) 0.65 2.07 1991-1995 0.013 (2.56)*** 1.313 (7.34)*** 0.038 (0.42) 0.51 1.56 1996-2000 0.009 (1.10) 1.071 (6.49)*** -0.148 (0.73) 0.44 2.44 2001-2005 0.006 (1.51) 0.778 (8.64)*** 0.016 (0.37) 0.58 2.40 10 Inernaional Journal of Buine, Accouning, and Finance, Volume 1, Number 1, 2007

Panel B: Medium-ize bank porfolio Sample period Conan SP500 TB1 R 2 DW 1976-2005 0.012 (6.25)*** 0.723 (16.42)*** -0.117 (3.88)*** 0.46 2.05 1976-1980 0.013 (3.81)*** 0.595 (6.92)*** -0.200 (4.44)*** 0.62 1.60 1981-1985 0.019 (4.93)*** 0.701 (6.53)*** -0.197 (2.84)*** 0.58 2.17 1986-1990 0.003 (0.83) 0.766 (10.45)*** -0.108 (1.07) 0.66 1.80 1991-1995 0.015 (3.92)*** 0.757 (5.47)*** -0.109 (1.56) 0.44 1.42 1996-2000 0.009 (1.21) 0.862 (5.54)*** -0.262 (1.38) 0.38 2.43 2001-2005 0.011 (2.64)*** 0.450 (4.81)*** 0.029 (0.63 0.31 2.19 Panel C: Small-ize bank porfolio Sample period Conan SP500 TB1 R 2 DW 1976-2005 0.013 (6.69)*** 0.681 (15.08)*** -0.078 (2.53)*** 0.41 1.88 1976-1980 0.017 (4.37)*** 0.639 (6.39)*** -0.166 (3.18)*** 0.55 1.78 1981-1985 0.021 (5.23)*** 0.666 (6.02)*** -0.138 (1.93)* 0.51 1.77 1986-1990 0.005 (1.26) 0.743 (10.29)*** -0.096 (0.96) 0.65 1.84 1991-1995 0.011 (2.36)*** 0.837 (5.02)*** -0.006 (0.07) 0.34 1.59 1996-2000 0.111 (1.53) 0.657 (4.33)*** -0.381 (2.05)** 0.30 1.97 2001-2005 0.013 (3.34)*** 0.470 (5.20)*** 0.071 (1.61) 0.37 1.90 ***Significan a abou 1%; **Significan a abou 5%; *Significan a abou 10%. CONCLUSIONS AND FUTURE RESEARCH DIRECTION In hi udy, we empirically re-examined he wo-index model which ae ha individual ock reurn are deermined by heir expoure o boh ock marke and deb marke variaion. Our reul are generally conien wih he model. The ock marke rik i found o be he mo imporan and conien yemaic rik priced in bank ock reurn. The eimaed coefficien for ock marke are aiically ignifican for all porfolio of bank ock and over all ample and ub-ample period. The magniude of ock marke coefficien are almo uniformly larger for big bank ock han mall bank ock regardle of he inere rae index and he ample period ued, uggeing ock marke rik ha relaively rong influence on big bank ock han i ha on mall bank. Thi i conien wih he fac ha big bank inve in a wider range of economic ecor and heir performance i herefore more rongly correlaed wih he broad marke. The more imporan finding of hi udy are relaed o he exra-marke inere rae eniiviy of bank ock. Fir, we find bank ock reurn are indeed eniive o change in marke inere rae meaured by he yield on hor-erm, medium-erm a well a long-erm U. S. governmen bond even afer he reurn on he marke porfolio of ock are conrolled. Bu he exra-marke inere rae eniiviy of bank ock i ime-varying and generally decline Inernaional Journal of Buine, Accouning, and Finance Volume 1, Number 1, 2007 11

over ime. During all ub-ample period afer 1990 he eimaed bea coefficien for he porfolio of all bank are no aiically ignifican regardle of he proxy ued for marke inere rae. The decline of inere rae eniiviy of bank ock reurn may be a reflecion ha bank become more diverified and beer hedged again inere rae rik. The decreae in inere rae volailiy afer 1980 may alo conribue o he decline in he inere rae rik of commercial bank. Second, when he model i eimaed uing ize-ored porfolio of bank ock, we find weak evidence ha mall bank ock end o be more eniive o change in marke inere rae han big and medium bank ock. The eimaed bea coefficien are aiically inignifican for he la hree ub-ample period for he porfolio of all bank and big bank. Bu hee coefficien are marginally ignifican and during he la wo ub-ample period for he porfolio of mall bank ock and ome ime he porfolio of medium bank ock. REFERENCES Akella, S. R. & Chen, S. (1990). Inere rae eniiviy of bank ock reurn: Specificaion effec and rucural change. Journal of Financial Reearch, 13, 147-154. Bae, S. C. (1990). Inere rae change and common ock reurn of financial iniuion: reviied. Journal of Financial Reearch, 13, 71-79. Booh, J. & Officer, D. (1985). Expecaion, inere rae, and commercial bank ock. Journal of Financial Reearch, 8, 51-58. Chance, D. & Lane, W. (1980). A re-examinaion of inere rae eniiviy in he common ock of financial iniuion. Journal of Financial Reearch, 3. 49-55. Elyaiani, A. & Manur, I. (2004). Bank Sock Reurn Seniiviie o he Long-erm and Shorerm Inere Rae-A Mulivariae GARCH Approach. Managerial Finance, 30, 32-55. Elyaiani, A. & Manur, I. (2005). The Aociaion beween Marke and Exchange Rae Rik and Accouning Variable-A GARCH Model of he Japanee Banking Iniuion. Review of Quaniaive Finance and Accouning, 25, 183-206 Flannery, M. & Jame, C. (1984). The effec of inere rae change on he common ock reurn of financial iniuion. The Journal of Finance, 39, 1141-1153. Gilibero, M. (1985). Inere rae eniiviy in he common ock of financial iniuion: a mehodological noe, The Journal of Financial and Quaniaive Analyi, 20, 123-126. Gulekin, N. B. & Rogalki, R. J. (1979). Commen: a e of Sone wo-index model of reurn. The Journal of Financial and Quaniaive Analyi, 14 (3), 629-639. He, L, T. & Reicher, A. K. (2003). Time variaion pah of facor affecing financial iniuion and ock reurn. Alanic Economic Journal, 31(1), 71-86. 12 Inernaional Journal of Buine, Accouning, and Finance, Volume 1, Number 1, 2007

He, L. T., Myer, N., & Webb, J. (1996). The eniiviy of bank ock reurn o real eae. Journal of real eae finance and economic, 12, 203-220. Joeph, N. L. & Vezo, P. (2006). The eniiviy of US bank ock reurn o inere rae and exchange rae change. Managerial Finance, 32(2), 182-199. Kane, E. J. & Unal, H. (1988). Change in marke aemen of depoi iniuion rikin. Journal of Financial Service Reearch, 201-29. Kwan, S. H. (1991). Reexaminaion of inere rae eniiviy of commercial bank ock reurn uing a random coefficien model. Journal of Financial Service Reearch, 61-76. Lynge, M. & Zumwal, K. (1980). An empirical udy of he inere rae eniiviy of commercial bank reurn: A muli-index approach. Journal of Financial and Quaniaive Analyi, 15, 731-742. Manur, I. & Elyaiani, E. (1995). Seniiviy of bank equiy reurn o he level and volailiy of inere rae. Managerial finance, 21, 57-77. Marin, J. & Keown, A. (1977) Inere rae eniiviy and porfolio rik. Journal of Financial and Quaniaive Analyi, 12, 181-196. Neuberger, J. A. (1991). Rik and reurn in banking: evidence from bank ock reurn. Federal Reerve Bank of San Francico Economic Review, 4, 18-30. Scoo, W. L. & Peeron, R. L. (1986). Inere rae rik and equiy value of hedged and unhedged financial inermediarie. Journal of Financial Reearch, 9, 325-329. Sone, B. K. (1974). Syemaic inere-rae rik in a wo-index model of reurn. The Journal of Financial and Quaniaive Analyi, 9, 709-721. Sweeney, R. J. & Warga, A. D. (1986). The pricing of inere-rae rik: Evidence from he ock marke. The Journal of Finance, 41, 393-410. Abou he Auhor: Jianzhou Zhu i an aian profeor of finance a he Univeriy of Wiconin-Whiewaer. Dr. Zhu ha auhored and co-auhored everal refereed journal aricle. Hi primary reearch area i in ae pricing. Morheda Haan i an aociae profeor of Quaniaive Analyi a Wiley College. She ha preened many reearch paper a regional, naional, and inernaional conference. She ha publihed everal aricle in refereed journal. Wanli Li i a profeor of accouning in he Deparmen of Accouning and Finance, School of Managemen, Xian Jiaoong Univeriy, Xian, China Inernaional Journal of Buine, Accouning, and Finance Volume 1, Number 1, 2007 13

CAN THE CLASSICAL MEAN-VARIANCE PORTFOLIO MODEL BE SAVED? Xiaolou Yang Humbold Sae Univeriy ABSTRACT The claical mean-variance porfolio model aume inveor know he rue expeced reurn. However, inveor have o eimae he expeced reurn from an unknown probabiliy diribuion. Uing he pa mean reurn a he eimae of he expeced reurn will no capure fuure uncerainie and rik, herefore a large eimaion error can be induced which yield poor ou-of-ample performance. Thi udy exend he claical mean-variance model by incorporaing he Geneic Algorihm ino a ae dependen ochaic porfolio deciion-making proce. Uing he U.S. ock marke daa, hi udy how ha he generalized mean-variance model wih he Geneic Algorihm echnique can ignificanly improve he accuracy of he expeced reurn eimaion and reduce he eimaion rik. The ou-of-ample performance of he generalized mean-variance model i much beer han ha of he andard mean-variance model and he radiional Bayeian approach. Moreover, hi udy e he effec of rik averion on he generalized model and found precauionary effec when fuure uncerainie increae. INTRODUCTION The claical mean-variance porfolio model (CMVPM) aume ha inveor know he rue expeced reurn. However, in realiy, inveor have o eimae he expeced reurn from an unknown probabiliy diribuion, which i exremely difficul o eimae preciely. The mo commonly ued mehod i o ue a hiorical mean a he eimae of he fuure expeced urn. However, a hiorical mean ha no way o incorporae fuure uncerainie ino he reurn eimaion, herefore generae huge eimaion rik (Michaud, 1998). Uing a hiorical mean a an eimae of he fuure expeced reurn ha a ignifican negaive impac on he mean-variance porfolio. For inance, he CMVPM generally overweigh hoe ae ha have high pa reurn, low variance and covariance o oher ae. Wihou conidering fuure uncerainy, hee ae are mo likely o have large eimaion error, which how udden hif in allocaion along he efficien fronier and are very unable acro ime. The CMVPM i generally lack of diverificaion and how poor ouof-ample performance. Therefore, eimaion rik i one of he primary reaon o make he CMVPM unfeaible in pracice. Chopra and Ziemba (1993) find ha error in mean are abou en ime a imporan a error in variance and covariance. Be and Grauer (1991) how ha opimal porfolio are very eniive o he value of expeced reurn. They argued ha a urpriingly mall increae in he mean of ju one ae drive half he ecuriie from he porfolio. Therefore, how o improve he echnique of mean reurn eimaion by incorporaing he conideraion of fuure uncerainie become a key iue for he porfolio opimizaion problem. Several approache were ued o inroduce uncerainie ino reurn eimaion o reduce he eimaion rik are uggeed in he lieraure. The mo popular one i he Bayeian 14 Inernaional Journal of Buine, Accouning, and Finance, Volume 1, Number 1, 2007

eimaor, developed by Jorion (1985, 1986). The idea of a Bayeian inference i o combine exra-ample, or prior, informaion wih ample reurn and herefore reurn are hrunk oward he prior. I hrink he opimal porfolio oward he minimum-variance porfolio. Therefore reduce he eniiviy o he expeced reurn eimaion. However, becaue i i compuaional expenive o olve muliple-uncerainie problem, Bayeian approach uually aume ha he deciion-maker ha only a ingle prior (Knigh, 1921). Due o he compuaional burden, Bayeian approach fail, i.e., hard o converge or unable o find a oluion, in many cae when uncerainy pace increae. Thi udy uilize an alernaive echnique, Geneic Algorihm (GA), o eimae he expeced reurn in he mean-variance porfolio model. For he purpoe of hi udy, hi model i called he generalized mean-variance model (GMVM). The udy alo will how how he GA can deal wih variou ource of economic uncerainie o reduce he eimaion rik and improve he porfolio performance. The GA i a probabiliic earch approach. I i an evoluionary opimizaion algorihm, which mimic operaion in naural geneic o earch for he opimal oluion, herefore can be ued a a ochaic opimizaion olver. The GA ha been widely ued in he area of yem engineering and environmenal cience for he opimaldeign problem (Dragan e al., 1999; Zhao e al. 2004; Zhou e al. 2000). In hi paper, I will how, in he fir ime, how o apply Geneic Algorihm ino a dynamic porfolio opimizaion yem wih variey of fuure uncerainie. The advanage of GA i ha i olve he model by forward-looking and backward-inducion, which incorporae boh hiorical informaion and fuure uncerainy when eimaing expeced reurn (Bauer, 1994; Berger, 1994; and Yang, 2006). I ignificanly improve he accuracy of he fuure expeced reurn eimaion and herefore he model performance. In addiion, GA doe well in handling a large variey of fuure uncerainie, which overcome he compuaional difficulie in he radiional Bayeian approach (pu a reference here). Thi udy compare he GMVM wih he andard mean-variance model (SMVM) and he radiional Bayeian approach, analye he effec of rik averion on GMVM. Uing he U.S. ock marke daa, he reul of hi udy reveal ha he GMVM ouperform boh SMVM and he radiional Bayeian approach in erm of ou-of-ample mean, variance and Sha raio. In paricular, he problem of a fund manager allocaing wealh acro riky ae i he uncerainy abou he expeced reurn on hee equiie. Thi udy alo characerize he properie of he opimal porfolio obained from he GMVM, he SMVM, and he Bayeian porfolio ha allow for uncerainy rik bu ha a ingle prior. The empirical reul howed ha uing CMVPM echnique could ignificanly improve he accuracy of he expeced reurn eimaion, reduce he eimaion rik and herefore improve he model performance of he andard mean-variance mehod. The porfolio weigh uing GA are more balanced and vary much le over ime han ha of he SMVM and he Bayeian approach. The ou-of-ample reurn generaed from he GMVM have a ubanially higher mean, a higher Sharp raio and a lower variance compared o he SMVM and Bayeian approach. Moreover, by conidering fuure uncerainie, he opimal porfolio wih GA exhibi a precauion effec. BACKGROUND OF THIS STUDY The Claical Mean-Variance Porfolio Model (CMVPM) Inernaional Journal of Buine, Accouning, and Finance Volume 1, Number 1, 2007 15

The (CMVPM) pioneered by Markowiz (1952, 1987) and developed by Sharpe (1970) i he claic paradigm of modern finance for allocaing capial among riky ae. According o he mean-variance model, he opimal porfolio of N riky ae,ω, i given by he oluion of he following opimizaion problem, δ maxω μ ω ω, (1) ω 2 Where μ i he N-vecor of he rue expeced reurn. i he N N covariance marix, and he calarδ i he rik averion parameer. The oluion of hi problem i 1 ω = 1 μ, (2) δ A fundamenal aumpion of he claical mean-variance model aume ha inveor know he rue expeced reurn. However, in realiy, inveor have o eimae he expeced reurn from an unknown probabiliy diribuion. The obained opimal porfolio baed on he eimaed expeced reurn i 1 ω = 1 ˆ μ. (3) δ Where, μˆ i he eimae of he expeced reurn. Equaion (3) coincide wih (2) if and only if ˆ μ = μ, or he expeced reurn are eimaed wih infinie preciion. To apply meanvariance model, people ue he pa mean reurn a he eimae of he expeced reurn. Thi can caue huge eimaion error by ignoring uncerainy rik. A a reul, opimal porfolio obained from equaion (3) conain exreme poiion. I i very unable over ime and normally yield poor ou-of-ample performance. The Tradiional Bayeian Approach The foundaion for he Bayeian approach wa propoed by Savage (1954), and developed by Jorion (1986), Paor (2000) and Paor e al. (2000). According o he Bayeian approach, an inveor maximize he expeced uiliy funcion by chooe porfolio weigh ω. The condiional expeced uiliy of he inveor i given by E[ U ( R) θ ] = U ( R) p( Rθ ) dr, (4) Where U () denoe he uiliy funcion. R i a vecor of fuure ae reurn and θ i a κ 1 vecor of expeced ae reurn. For above condiional expeced uiliy funcion, θ i known. p ( R θ ) i he condiional probabiliy deniy funcion (likelihood funcion) of ae reurn given parameerθ. However, in realiy, he rue value of θ i unknown and ha o be eimaed, denoed aθˆ. In general, θˆ i conruced from ample obervaion. Bayeian approach aume θ a a random variable. All informaion ha i known relaed o θ i ummarized in he poerior pdf (prior), denoed a p ( θ Y ). Therefore, he poerior pdf i obained uing he hiorical informaion from he pa reurn, where Y = ( y, K 1, y T ) i a vecor of pa reurn. Then he expeced uiliy funcion become E [ U ( R) Y ] = E[ E[ U ( R) θ ] Y ] = U ( R) p( Rθ ) p( θ Y ) drdθ, (5) p Where ( ) ( ) ( ) ( θ ) p( Y θ ) p Rθ p θ Y dθ = p Rθ dθ, (6) p θ p Y θ dθ ( ) ( ) 16 Inernaional Journal of Buine, Accouning, and Finance, Volume 1, Number 1, 2007

In Bayeian oluion, an opimal porfolio i defined in erm of he predicive pdf. The predicive pdf of fuure reurn i obained by aking he expecaion overθ wih repec o he poerior diribuion of θ. Subiue he above equaion ino he objecive funcion and eimae he N-dimenional vecor of he porfolio weigh o ge ω = λω + λ ω (7) BS MIN ( ), 1 MV Where ωmin are he minimum-variance porfolio weigh and ω MV i he mean-variance porfolio weigh. Geneic Evoluion Proce THE APPLICATION OF THE GENETIC ALGORITHM (GA) The concepion of he GA in i curren form i generally aribued o Holland (1975). GA ar wih a populaion of randomly generaed oluion called candidae o explore he oluion pace of a problem. Then GA earche for beer oluion hrough a number of ieraion, which i called generaion. The performance of each oluion i evaluaed by a fine crierion, ypically repreened by an objecive funcion. In each generaion, relaively good oluion, in erm of he fine crierion, have a higher chance o be eleced o reproduce offpring by geneic operaor croover and muaion. Croover convey informaion from paren o he offpring while muaion provide mall randomne o he original candidae o generae populaion wih beer fine. By doing hi, he algorihm idenifie candidae wih opimal fine value, and dicard hoe wih poor fine value. Thi procedure coninue unil he maximum number of ieraion i me or here i no furher fine improvemen occur. Geneic Algorihm coni of four main ep: evaluaion, elecion, croover and muaion. In hi ecion, he udy illurae how o apply Geneic Algorihm ino he CMVPM o improve he accuracy of reurn eimaion and hu porfolio model performance. 1. Evaluaion. GA ar wih a e of randomly generaed candidae in he iniial period. The evaluaion operaor meaure he fine of each candidae oluion in he populaion. A GA earche for beer oluion hrough a number of ieraion and aign each of hem a relaive value baed on he fine crieria, i.e., an objecive funcion. To apply GA, I ue he claical mean-variance model a he objecive crierion funcion. In paricular, an agen maximize he um of he expeced porfolio reurn minu he porfolio variance. 1 2 Max π EU[ W σ ] 2.. : W i i ϖ π = i, i, N i= 1 = 1, = 1, r ϖ, i, i, where, π i he probabiliy ha ae occur a ime period W i he wealh a ime under ae (8) Inernaional Journal of Buine, Accouning, and Finance Volume 1, Number 1, 2007 17

r, i reurn for ae i a ime period when ae occur i ϖ i, i he porfolio weigh for ae i a ime period when ae occur 2 σ i he variance of he porfolio U i he uiliy funcion Thi objecive funcion will be ued a he fine crierion o evaluae candidae in each ieraion. 2. Selecion. The elecion operaor i o elec candidae of he curren e of populaion for developing he nex generaion. Variou mehod have been propoed bu all follow he idea ha he candidae wih he be fine value have a greaer chance of urvival. The eleced candidae, called paren, will be ued o produce offpring for he nex ieraion. In hi udy, I ue a variaion of he claic Roulee Wheel Selecion Operaor (Michalewicz, 1996). By hi variaion, I pick up he op wo be candidae ouide he populaion o guaranee he be candidae pae o he nex generaion. 3. Croover. The croover operaor ake he eleced candidae and combine hem abou a croover poin hereby creae a new candidae. In hi udy, afer he op wo paren are eleced baed on he fine funcion, he offpring i generaed uing he weighed average of he wo paren for he uage in he nex ieraion. 4. Muaion. The muaion operaor modifie he gene of a candidae ubjec o a mall muaion facor and inroduce furher randomne ino he e of populaion in order o reul a ubequen e of populaion wih beer fine. I e he muaion rae a 0.05 randn(randn i a random number generaed by a compuer in he inerval of [0,1]). Thi ieraive proce coninue unil he erminaion crieria me. For inance, a number of generaion wihou fine improvemen occur, which implie ha convergence low o he opimal oluion. How o chooe he erminaion crieria (i.e., he number of generaion wihou furher fine improvemen occur) i very imporan, which ignificanly influence he convergence of he algorihm. Yang (2006) conduced a e of experimen o e he ufficien number of he parameer eing in he ue of GA including he number of ieraion and ize of candidae o guaranee he opimal oluion. Thi udy will apply he finding of Yang (2006) and e he opimal ieraion a 30 and he ize of candidae a 40 when apply GA approach. DYNAMIC GENETIC ALGORITHM DESIGN The aic ingle-ae GA i fir dicued here. The opimal porfolio problem for a ingle-ae GA i ha an inveor allocae hi wealh o a porfolio wih weny riky ae, N = { r1, K, r20}, where r i, i = 1, K20 i he riky ae. Inveor know he curren and previou ae reurn. The opimal porfolio i decided baed on he expecaion of he fuure reurn. In he iniial period, here are 30 randomly generaed candidae for he opimal porfolio, ω1, K, ω30. Each candidae repreen he percenage of he wealh allocaed o each of hee weny ae. The op wo candidae wih he be performance, evaluaed by he objecive funcion in (8), urvive a he paren o creae a new group of candidae by croover and muaion in he nex ieraion o explore he opimal oluion. Thi procedure coninue unil he maximum number of ieraion i reached. Single-ae model, however, ha drawback of rik inconiency over ime. Moreover, he ingle-ae GA i unable o capure he uncerainie in he economy, which i a key feaure 18 Inernaional Journal of Buine, Accouning, and Finance, Volume 1, Number 1, 2007

of he financial marke. Thu, i i vial o develop a ochaic GA proce for he porfolio opimizaion problem, which could capure he dynamic apec of he ae allocaion problem. The ochaic naure i deigned by incorporaing muliple ae ino he model. Each ae repreen a cenario indicaing he marke index, indicaion differen realizaion of he economy repone o he uncerainie. Becaue differen marke index i aociaed wih differen ae reurn realizaion, herefore i a good way o idenify uncerainie and rik. There are oal T=10 period in he ime horizon ={1, 2, 3,, T}. Each period ha differen ae/cenario. Invemen deciion are made a beginning of he each period. To apply he GA ino a muliple ae porfolio opimizaion yem, he udy dicree he choice pace ino muliple ubpace. Each ubpace repreen one ae/cenario wih pecific rik/uncerainy. Then he udy ued a random generaing proce o generae muliple poibiliie in each ime period, while guaraneeing he um of he poibiliie equal o 1 in each period. Inveor do no know which ae will occur. They only know he probabiliy of each ae and he ae reurn in each ae. For boh ingle-ae and muli-ae GA, he ae reurn are randomly generaed uing he pa mean reurn and variance. Therefore, he reuling fuure expeced reurn obained by GA baed on boh hiorical informaion and fuure uncerainie. For comparion, he variance-covariance marix of he ample daa wa ued and i decribed in he nex ecion, o calculae he opimal porfolio for boh mean-variance model including GMVM and SMVM, and he Bayeian approach. DATA COLLECTION The daa come from he Cener for Reearch in Securiy Price (CRSP) daabae. The porfolio i formed by weny ock, which are randomly eleced from he enire marke daabae. The daa range hrough January1990 o December 2004. Summary aiic for he indexe of he daa are provided in Table 1. Table 1 Mean and Variance of Good and Bad Sock Reurn Mean Variance Mean Variance 1 0.98 46.35 11 0.34 1.38 2 **0.56 **1.28 12 0.32 0.76 3 0.53 6.37 13 0.31 0.69 4 0.48 3.82 14 0.28 0.52 5 **0.47 **1.43 15 0.26 0.38 6 *0.42 *13.95 16 0.24 0.32 7 0.41 2.26 17 0.15 0.21 8 **0.39 **0.63 18 *0.13 *1.86 9 *0.37 *3.91 19 0.12 0.35 10 0.36 1.39 20 *0.05 *1.96 Noe: he ock wih ** ign denoe a good ock; he ock wih * ign denoe a bad ock. The ock were ored from he highe reurn o he lowe reurn. Table 1 how ha he highe ock reurn i 0.98%, wih he highe variance of 46.35. The lowe ock reurn i 0.05%, wih relaive low variance of 1.96. Noe ha he invere relaionhip beween mean and variance i no conien over all ock. There are ome oulier (abnormal) in he porfolio Inernaional Journal of Buine, Accouning, and Finance Volume 1, Number 1, 2007 19