Asia Pacific Managemen Review 17(2) (2012) 113-126 Boosrapping Mulilayer Neural Neworks for Porfolio Consrucion Chin-Sheng Huang a*, Zheng-Wei Lin b, Cheng-Wei Chen c www.apmr.managemen.ncku.edu.w a Deparmen of Finance, Naional Yunlin Universiy of Science and echnology, aiwan b Graduae School of Managemen, Naional Yunlin Universiy of Science and echnology, aiwan c Deparmen of Agriculural Economics, Naional aiwan Universiy, aiwan Received 21 Augus 2010; Received in revised form 14 Sepember 2010; Acceped 17 February 2011 Absrac Despie having become firmly esablished as one of he major cornersone principles of modern finance, radiional Markowiz mean-variance analysis has, neverheless, failed o gain widespread accepance as a pracical ool for equiy managemen. he Markowiz opimizaion enigma essenially ceners on he severe esimaion risk associaed wih he inpu parameers, as well as he resulan financially irrelevan or even false opimal porfolios and asse allocaion proposals. We herefore propose a porfolio consrucion mehod in he presen sudy which incorporaes he adopion of boosrapping neural nework archiecure. In specific erms, a residual boosrapping sample, which is derived from mulilayer feedforward neural neworks, is incorporaed ino he esimaion of he expeced reurns and he covariance marix, which are hen, in urn, inegraed ino he radiional Markowiz opimizaion procedure. he efficacy of our proposed approach is illusraed by comparing i wih radiional Markowiz mean-variance analysis, as well as he James-Sein and minimum-variance esimaors, wih he empirical resuls indicaing ha his novel approach significanly ouperforms he benchmark models, in erms of various risk-adjused performance measures. he evidence provided in his sudy suggess ha his new approach has significan promise wih regard o he enhancemen of he invesmen value of Markowiz mean-variance analysis. Keywords: Boosrap, mulilayer feedforward, neural nework, porfolio consrucion 1. Inroducion * Given ha he opimal managemen of porfolio risk is an essenial componen of modern asse managemen, he benefis derived from efficien porfolio diversificaion are generally acknowledged by mos financial economiss and invesmen professionals. Markowiz-Sharpe porfolio heory provides he sandard normaive crieria for forming securiies porfolios, and alhough mean-variance efficiency already represens well received wisdom wihin he field of finance, i is difficul o undersand why, for almos half a cenury, mean-variance opimizaion mehodology has no me wih widespread accepance wihin he invesmen communiy. Michaud (1989) clearly demonsraes he Markowiz opimizaion enigma under many raionalized seings, sysemaically poining ou ha he major problem wih meanvariance opimizaion is is endency o maximize he effecs of errors wihin he inpu assumpions. In paricular, mean-variance opimizers are essenially esimaion-error maximizers, in he sense ha mean-variance opimizaion resuls in significan * Corresponding auhor. E-mail: huangcs@yunech.edu.w 113
overweighing of hose securiies wih large esimaed reurns, negaive correlaions and small variance. he end resul is he greaer likelihood of hese securiies being subjec o subsanial esimaion errors. he more severe esimaion risks involved in Markowiz mean-variance opimaliy were iniially documened by Jobson and Korkie (1981), who underook he analysis of 20 socks wih 60 rue monhly expeced reurns and covariance. heir resuls reveal a Sharpe raio of 0.34 for he rue opimal porfolio, 0.08 for he Mone Carlo simulaed opimal porfolio, and 0.27 for he naïve equally-weighed porfolio. In paricular, hey poin ou he inadmissibiliy of he hisorical sample of he mean esimaor; and hus, hey caegorically conclude ha he radiional implemenaion of a Markowiz diversificaion sraegy is an inferior sraegy. In an aemp o enhance he pracical implemenaion value of heir Markowiz porfolio consrucion, Jobson and Korke (1981) propose he alernaive James-Sein approach, under which i is assumed ha he grand mean of reurns for all securiies represens he bes esimaor for he respecive individual expeced reurns. heir resuls show ha he James-Sein esimaor can obain a Sharpe raio of up o 0.32 for he opimal porfolio. Fors and Savarino (1986) furher find ha he esimaion risk of he radiional Markowiz mean-variance efficien porfolio increases wih he size of he porfolio, and ha i can be driven down o a single share in hose cases where he risk olerance level is sufficienly high. Jorion (1986) presened a Bayes-Sein approach o he esimaion of porfolio analysis, noing ha a simulaion sudy demonsraed ha a simple James-Sein esimaor and a minimum-variance esimaor were boh capable of significanly ouperforming radiional Markowiz mean-variance opimizaion. Michaud (1998) mainains ha in pracice, he limiaions of mean-variance efficiency are generally derived from a lack of saisical undersanding of mean-variance opimizaion, wih a Bayesian ype saisical view of mean-variance opimizaion being capable of grealy enhancing he invesmen value of Markowiz mean-variance analysis, whils also reducing boh insabiliy and ambiguiy. Neural neworks are essenially saisical devices for performing wide areas of ineres such as inducive inference, classificaion, clusering, dimensional reducion, process conrol, machine vision, and speech recogniion. Whie (1989) indicaes ha from he poin of view of saisicians, hese neworks are analogous o non-parameric, non-linear regression models. Refenes e al. (1995) noed ha he novely surrounding neural neworks lies mainly in heir abiliy o model non-linear processes wih very few prior assumpions wih regard o he naure of he generaion process. his is paricularly useful in invesmen managemen wihin which lile is known, alhough a grea deal is assumed, abou he naure of he processes deermining asse prices. Franke and Klein (1999) employ feed-forward neural neworks in order o dynamically allocae capial o various componens of a currency porfolio, wih heir resuls showing ha he resulan porfolios economically ouperform he benchmark porfolios. Using recurren neural neworks in dynamic porfolio selecion, Lin e al. (2006) demonsrae ha his mehod provides greaer accuracy in dynamic porfolio selecion, whils also ouperforming he VAR model. Finally, Fernandez and Gomez (2007) uilize he Hopfield nework o ackle he issue of porfolio opimizaion in he conex of a mixed quadraic and ineger programming problem, whils Yu e al. (2008) use a radial basic funcion neural nework o solve heir mean-variance-skewness porfolio selecion model. he above prior lieraure deals mainly wih opimizaion algorihms or poin esimaion in porfolio consrucion; however, in he spiri of Michaud (1998), knowledge wih regard o he disribuion of expeced reurns is of significan imporance and direc 114
relevance o he Markowiz mean-variance saisical procedure. We herefore se ou in his sudy o design a noble mean-variance efficiency procedure uilizing empirical disribuion informaion drawn from a variaion of he boosrapping neural neworks iniiaed by Franke and Neumann (2000). We employ mulilayer feedforward neural neworks as he means of forecasing he consiuen sock reurns of a arge porfolio, wih he resampling of he prediced share reurns in our Markowiz mean-variance analysis forming he required inpu parameers for he risk and expeced reurns. he muli-layer perceprons are essenially he primary forecasing sysem in he proposed boosrapping neural neworks scheme. However, he informaion provided from muli-layer perceprons alone is no direcly applicable o he Markowiz porfolio consrucion procedure. Muli-layer perceprons mainly deliver he knowledge abou he fuure expeced reurns of individual socks in he porfolio bu lack of he expeced porfolio risk. he inpus of mean vecor and covariance marix of expeced reurns necessary for Markowiz porfolio consrucion can only be bes obained by adoping he proposed parameric boosrap approach on he expeced reurns forecased from muli-layer perceprons. he hrus of he proposed boosrapping neural neworks herefore ceners on inroducing he saisical boosrap approach o make muli-layer percepron forecasing applicable in he conen of Markowiz porfolio consrucion. he invesmen performance of he proposed boosrapping neural nework porfolio is sysemaically examined wihin his sudy, based upon various performance measures, and hen compared wih radiional Markowiz mean-variance analysis and wo ypes of Bayesian esimaors, he James-Sein and minimum-variance esimaors. he remainder of his paper is organized as follows. An explanaion of he boosrapping neural nework porfolio consrucion approach proposed in his sudy is provided in Secion 2. his is followed in Secion 3 by our presenaion and subsequen analysis of he empirical resuls. Finally, he conclusions drawn from his sudy are provided in Secion 4, along wih some suggesions for furher developmen of our approach. 2. Mehodology and daa 2.1 Mulilayer feedforward neural neworks Mulilayer feedforward neural neworks are generically nonlinear, nonparameric regression models. Wih sufficien hidden layer complexiy, a mulilayer feedforward nework can learn any unknown mapping from inpu space o oupu space. he archiecure of he mulilayer feedforward nework model employed in his sudy is consruced as shown in Figure 1. Figure 1. he mulilayer feedforward nework 115
where, 1 1 1 1 a f ( W p b ) (1) 2 2 2 1 2 a f ( W a b ) (2) he mulilayer feedforward nework is comprised of an inpu layer, a hidden layer, and an oupu layer. he connecion of he hidden layer beween he inpu layer and he oupu layer is expressed as weigh values W 1, W 2. he nework shown above has R inpu elemens, S 1 hidden layer neurons, S 2 oupu layer elemens, and bias vecors b 1, b 2. he f 1 adoped here is log-sigmoid ransfer funcion, and f 2 as linear ransfer funcion. According o Hornik e al. (1989), he above simple feedforward neural nework can have he universal approximaion propery. he Levenberg-Marquard algorihm (LM), implemened in Malab sofware, is used for nework raining o updae he weighs and bias values. he LM is a variaion of Newon s mehod ha was designed for minimizing funcions ha are sums of squares of oher nonlinear funcions, and very well suied o neural nework raining where he performance index is he mean square error. 2.2 Mean-variance opimizaion he Markowiz (1959) porfolio consrucion approach provides a mean-variance efficien porfolio, under shor-selling consrains and he risk olerance of individual invesors, as follows: Minimize: x Vx λx m (3) Subjec o: x e = 1, x 0 where x is an N vecor of porfolio weighs in he risky asses; V is he N x N covariance marix of he N risky asses; m is an N vecor of he expeced reurns of he N risky asses; and e is he N-column vecor of 1s, λ is he coefficien of risk olerance. Following Board and Sucliffe (1994), he values of λ are assumed o be 0.05 for low risk olerance, 0.50 for moderae risk olerance, and 2.00 for high risk olerance. 2.3 James-sein and minimum-variance esimaions his sudy follows he mehod adoped in Jorion (1986) o carry ou James-Sein esimaion on he Markowiz mean-variance porfolio, beginning wih he uilizaion of he radiional mean-variance efficien porfolio o compue he curren monhly expeced reurn, Y 0. he James-Sein esimaors are as follows: s 0 m (1 wˆ ) Y wy ˆ e (4) s 1 ee' V (1 ) (5) ˆ 1 ( 1 ) e' e where m s and V s are he mean and covariance marices for he densiy of he informaive prior of he vecor of he fuure reurns rae, where is number of observaions, 1 respecively; Y E[ r] r, where Y is sample mean reurn and r is he fuure rae of 1 1 reurn; ŵ is a shrinkage coefficien; S, where S is he sample covariance N 2 ( N 2) marix; and is a gamma disribuion wih mean,. 0 ˆ 1 0 ( Y Y e)' ( Y Y e) 116
he minimum-variance esimaor, as oulined in Jorion (1986), is a special case of he James-Sein esimaor wihin which we le ŵ = 1 and in order o obain m = Y 0 s 1 ee' andv ˆ 1 e' e. 2.4 Porfolio consrucion using boosrapping neural neworks A boosrapping neural nework is inherenly he saisical scheme by boosrapping he predicors of sock reurns drawn from muli-layer perceprons. Specifically, mulilayer perceprons are firsly implemened as he forecasing model of individual sock reurns in he porfolio. A parameric boosrap is hen employed o simulae he probabilisic srucure of he daa generaing process of he individual sock expeced reurns on he basis of he informaion provided by he muli-layer perceprons. he resuling simulaed disribuions of individual sock reurns are he primary source of he mean vecor and he covariance marix of expeced reurns which are necessary for he Markowiz porfolio consrucion. Proceeding along he lines of Franke and Neumann (2000), in his sudy we employ boosrapping mehodology as he means of rerieving useful informaion on equiy reurns based upon neural nework forecasing. he procedures involved in porfolio consrucion using a boosrapping neural nework are described as follows. Sep 1: he following auoregression, of order 4, is fed ino he mulilayer feedforward nework o esimae he condiional expecaions: m(r) = E(Y r r = r) (6) where, (r, Y ) is a raining se of independen and idenically-disribued random row vecors, as follows: = 1,,, r = {r 4, r 3, r 2, r 1 }, Y = r. (7) he noise variable, ε, can hen be approximaed by: ˆ Y mˆ ( r ), 1,,, (8) where, m ˆ ( r ) is he feedforward neural nework approximaor. Sep 2: Since we have he assumpion of E (ε ) = 0, in his sudy we follow Freedman (1982), cenering he esimaed residual as follows: ~ 1 ˆ ˆ k, 1,,, (9) k 1 Le F ~ be he sample disribuion given by ~ ~ ~ 1,...,. he resampling can hen be obained from F as: ~ * 1 k wih densiy, k 1,...,. (10) Sep 3: * he boosrapped ε and he esimaed mulilayer feedforward nework approximaor, m ˆ ( r ), can be used o generae he r *+ 1 as follows: * * * 1 r ˆ ( ) ˆ 1 m r k, 1,,, (11) k 1 117
Sep 4: Seps 2 and 3 are repeaed 500 imes o generae he resampling of he boosrapped expeced reurns, r *+ 1, for each of he componen socks. hus, as opposed o heir hisorical means, boosrapped expeced reurns are obained for he arge porfolio, along wih he covariance marix. In he final sage, hese boosrapping inpu parameers are hen fed ino he sandard Markowiz mean-variance opimizaion scheme, as illusraed in sub-secion 2.2 above, resuling in boosrapped neural nework porfolio opimizaion. 2.5 Empirical daa his sudy uses a sample of he consiuen socks of he aiwan Sock Exchange Corporaion (SEC) aiwan 50 index covering he period from January 1999 o December 2008. Since he SEC aiwan 50 index comprises of he 50 mos highly capialized blue chip socks, i represens almos 70 per cen of he aiwan sock exchange (SE). In forming he efficien porfolios in our sudy, we adop a five-year horizon of hisorical daa, as suggesed by Jobson and Kokie (1981), in order o srike an appropriae balance beween rapidly-changing marke condiions and saisical confidence. he enire sample is uilized, saring wih he iniial hisorical esimaion period which is used o forecas he immediae monhly expeced reurns; his begins wih he firs five years, wih a one-monh moving-window, hereby providing a oal of 60 monhly ou-ofsample ess. he performance of he experimenal boosrapped neural nework porfolio proposed in his sudy is compared wih our seleced daa using radiional Markowiz mean-variance analysis, as well as furher analysis by means of he James-Sein and minimum-variance esimaors. A schemaic illusraion of he porfolio consrucion and performance esing carried ou in his sudy is provided in Figure 2. Forecas Monh 1 (Jan 2004) Forecas Monh 2 (Feb 2004) Forecas Monh 60 (Dec 2008) Esimaion Period 1 (Jan 1999-Dec 2003) Esimaion Period 2 (Feb 1999-Jan 2004) 3. Empirical resul and analysis Figure 2. Esimaion and forecasing period imeframes he analysis of he overall performance of he proposed boosrapping neural nework porfolio in his sudy is underaken using an ou-of-sample daa period which runs from January 2004 o December 2008, using wo benchmarks indices, he aiwan Sock Exchange Capializaion Weighed Sock Index (AIEX) and he SEC aiwan 50 Index (50). Using he various performance measures, he relaive performance of our proposed mehod is subsequenly compared wih he performance of he radiional forms of analysis 118
using he Markowiz mean-variance approach and boh he James-Sein and minimumvariance esimaors. he descripive saisics of he monhly reurns for he wo benchmark indices, and for he four opimal porfolios are repored in able 1, based upon low, moderae and high levels of risk olerance amongs invesors. able 1. Descripive saisics of monhly reurns Variables Mean S.D. Median Min. Max. Panel A: Benchmarks AIEX 0.3782 6.2955 0.6200 20.8600 11.2000 50 0.0390 5.7610 0.8900 18.4700 10.2000 Panel B: = 0.05 M-V 0.3280 5.2910 1.3018 17.7410 12.7554 J-S 0.3262 5.2901 1.3016 17.7366 12.7627 Min 0.3265 5.2902 1.3011 17.7377 12.7613 Nework 0.1293 6.5002 0.4277 118.7732 21.1896 Panel C: = 0.50 M-V 0.3465 5.3038 1.3240 17.7846 12.6747 J-S 0.3298 5.2925 1.3053 17.7411 12.7523 Min 0.3328 5.2939 1.3017 17.7527 12.7341 Nework 0.6044 7.3072 0.8176 25.1598 22.8121 Panel D: = 2.00 M-V 0.3853 5.3192 1.2990 17.6929 12.4041 J-S 0.3403 5.3014 1.3134 17.7561 12.7163 Min 0.3526 5.3073 1.3333 17.7919 12.6430 Nework 0.7464 7.4955 0.7574 26.0019 20.0993 Noes: a All daa are expressed as monhly figures in percenages, wih he ou-of-sample period running from January 2004 o December 2008. b M-V, J-S and Min indicae ha he porfolio weigh measures are esimaed from he respecive radiional Markowiz mean-variance analysis, James-Sein esimaor and minimum variance esimaor. As a resul of he global financial crisis which began in July 2007 of our ou-of-sample period, boh he AIEX and 50 experienced heir wors monhly negaive reurns of 20 per cen and 18 per cen, respecively; indeed, he AIEX even exhibis a negaive mean reurn of 0.38 per cen, alhough he median monhly reurn sill reaches a relaively posiive posiion of 0.62 per cen. As shown in Panels B, C and D of able 1, our proposed neural nework scheme generaes significanly higher mean reurns for he cases of λ = 0.5 and 2, however, resuls in moderaely lower mean reurn in he case of λ = 0.05. In specific erms, he reurns for our neural nework scheme enails he monhly reurns for ascending risk-olerance levels reaching 0.13 per cen, 0.60 per cen and 0.75 per cen. Meanwhile, almos he same mean and median invesmen reurns are exhibied by Markowiz mean-variance analysis and boh he James-Sein and minimum-variance esimaors. o summarize, he evidence presened in able 1 indicaes ha as compared o he more radiional porfolio 119
consrucion schemes, our proposed nework scheme exhibis valuable invesmen conen for he cases of moderae and high risk-olerance levels, wih all of he porfolio consrucion procedures of he presen sudy ouperforming he benchmark indices. 3.1 he sharpe raio he Sharpe raio uses he capial marke line as a benchmark o measure he deph and breadh of performance, wih a higher Sharpe raio being beer han a lower Sharpe raio. In paricular, a negaive Sharpe raio indicaes a siuaion of ani-skill, since he performance of he riskless asse is clearly superior. I is eviden from he empirical resuls presened in able 2 ha in erms of Sharpe raio, for he moderae and high levels of risk olerance, our proposed nework scheme ouperforms he oher hree porfolio consrucion mehods. able 2. Sharpe index Variables AIEX 50 M-V J-S Min Nework = 0.05 0.0639 0.0032 0.0586 0.0583 0.0583 0.0169 = 0.50 0.0639 0.0032 0.0620 0.0589 0.0595 0.0806 = 2.00 0.0639 0.0032 0.0692 0.0608 0.0631 0.0977 Noes: a All daa are expressed as monhly figures in percenages, wih he ou-of-sample period running from January 2004 o December 2008. b M-V, J-S and Min indicae ha he porfolio weigh measures are esimaed from he respecive radiional Markowiz mean-variance analysis, James-Sein esimaor and minimum variance esimaor. In specific erms, for ascending risk-olerance levels, he Sharpe raio in he new nework scheme reaches 0.017, 0.081, and 0.098. By conras, he Sharpe raios of he James-Sein esimaor, for example, have relaively smaller magniudes of jus 0.058, 0.059 and 0.061. Again, he four porfolio consrucion mehods are found o be superior o he benchmark indices in erms of he magniude of heir Sharpe raios. In paricular, he AIEX experiences a negaive Sharpe raio, which clearly indicaes a disinc lack of invesmen value during he curren global financial crisis period. Alhough originally referred o by reynor and Black (1973) as he appraisal raio, he informaion raio is he raio of relaive reurns o relaive risk, and whils he Sharpe raio examines he reurns relaive o a riskless asse, he informaion raio is based upon reurns relaive o a risky benchmark. he empirical resuls on he invesmen performance of he benchmark AIEX and 50 indices are repored in able 3, based upon heir informaion raio. 120
able 3. Informaion raio Variables M-V J-S Min Nework Panel A: AIEX = 0.05 0.1846 0.1841 0.1841 0.1289 = 0.50 0.1897 0.1851 0.1859 0.1968 = 2.00 0.2005 0.1880 0.1913 0.2171 Panel B: 50 = 0.05 0.0704 0.0700 0.0701 0.0213 = 0.50 0.0750 0.0709 0.0716 0.1135 = 2.00 0.0847 0.0735 0.0765 0.1360 Noes: a All daa are expressed as monhly figures in percenages, wih he ou-of-sample period running from January 2004 o December 2008. b M-V, J-S and Min indicae ha he porfolio weigh measures are esimaed from he respecive radiional Markowiz mean-variance analysis, James-Sein esimaor and minimum variance esimaor. Firsly, we find ha across moderae and high risk olerance levels, and for boh benchmark indices, our proposed nework scheme demonsraes, in paricular for 50, higher informaion raios as compared o hose of he oher porfolio mehods. In specific erms, based upon he ascending risk-olerance levels, our proposed nework scheme reveals informaion raios of 0.021,0.114,and 0.136 wih respec o he 50, whereas he comparaive informaion raios obained by a ypical porfolio consrucion mehod, for example, he James-Sein esimaor, are only 0.07, 0.071, and 0.074. Secondly, similar o he finding of Jorion (1986), a all levels of risk olerance, and for boh benchmarking indices, here are virually no differences whasoever beween he performance levels of he hree benchmark porfolio schemes, in erms of heir respecive informaion raios. racking errors are calculaed as he relaive sandard deviaion of reurns beween a porfolio and a benchmark. A racking error is a useful performance measure relaive o a benchmark since i is measured in unis of asse reurns. he comparaive empirical racking errors of he porfolio consrucion schemes wih respec o he benchmark indices are repored in able 4. I is of no real surprise o find ha for boh benchmark indices, our proposed nework scheme exhibis significanly larger racking errors han hose of he oher porfolio schemes. In specific erms, we find ha he larger he racking errors of he proposed scheme, he higher he invesmen value involved in acive equiy managemen. Furhermore, we also find ha for boh benchmark indices, he racking error measures of he radiional porfolio procedures are approximaely he same across all risk-olerance levels. 121
able 4. racking error Variables M-V J-S Min Nework Panel A: AIEX = 0.05 3.8264 3.8270 3.8269 3.9360 = 0.50 3.8207 3.8256 3.8245 4.9937 = 2.00 3.8081 3.8210 3.8196 5.1810 Panel B: 50 = 0.05 4.1028 4.1032 4.1031 4.2361 = 0.50 4.1007 4.1024 4.1018 4.9819 = 2.00 4.0891 4.1007 4.1011 5.2013 Noes: a All daa are expressed as monhly figures in percenages, wih he ou-of-sample period running from January 2004 o December 2008. b M-V, J-S and Min indicae ha he porfolio weigh measures are esimaed from he respecive radiional Markowiz mean-variance analysis, James-Sein esimaor and minimum variance esimaor. 3.2 he risk adjused reurns his sudy furher analyzes he invesmen performance of he porfolio consrucion schemes in he conen of risk-adjused expeced reurns. According o he capial asse pricing model (CAPM), he marke porfolio and he riskless asse are specific poins on a securiy marke line (SML). he SML provides he measure of risk-adjused reurns based upon he differences in reurns, given he same level of risk, beween he porfolio and he SML. he Jensen s Alpha provides quie a robus measure of he abnormal reurns ha are generaed by he porfolio as compared o a passive combinaion of he risk-free asse and a marke index wih exacly he same risk characerisics as he porfolio. Modigliani and Modigliani (1997) propose M 2 performance measure by using reurn per uni of oal risk as measured wih he sandard deviaion. he invesmen porfolio s sandard deviaion is adjused o reflec he sandard deviaion of he marke benchmark porfolio. he reurn premiums of he adjused invesmen porfolio and he marke index porfolio are hen compared. In addiion, Graham and Harvey (1997) design wo performance measures o compare newsleer performance wih a benchmark risk-adjused reurn. he firs (GH1) is based on a leveraged/unleveraged benchmark porfolio ha has he same volailiy as he managed porfolio over he evaluaion period. he second (GH2) indicaes he difference beween he reurn on he leveraged/unleveraged managed porfolio ha has he same volailiy as he benchmark and he reurn on he benchmark. As noed in Graham and Harvey (1997), M 2 is similar o GH2, bu no allowing for curvaure in he efficien fronier. he observaion above is laer confirmed in our empirical resuls shown in ables 5 and 6. Focusing on he SML risk-adjused excess reurns, as compared o he oher porfolios, our proposed nework scheme exhibis subsanially larger excess reurns, wih his resul remaining uniform across all risk-olerance levels for he wo benchmark indices. his phenomenon even holds firm in he unadjused excess reurns for he cases of λ = 0.5 and 2; moreover, he picure obained even more brillian wih regard o he Jensen s Alpha. 122
As repored in able 1, boh he AIEX and he 50 experienced heir wors respecive monhly negaive reurns of 20 per cen and 18 per cen as a resul of he global financial crisis which began in July 2007. Furhermore, during he 60-monh ou-of-sample period, he AIEX even exhibied a negaive mean reurn of 0.38 per cen. Given ha in he presen sudy, he risk-free asse is found o significanly ouperform boh marke indices and individual socks; our resuls show ha he radiionally consruced porfolios have negaive Jensen s Alphas. able 5. Excess reurns wih he AIEX benchmark Variables M-V J-S Min Nework Panel A: = 0.05 Excess Reurn (No risk adjusmen) 0.7062 0.7044 0.7047 0.5075 Risk-Adjused Excess Reurns (SML) 0.2889 0.2867 0.2871 0.5925 Jensen's Alpha 0.1060 0.1083 0.1079 0.1054 MM 0.3795 0.3770 0.3774 0.5679 GH1 0.3244 0.3222 0.3226 0.5852 GH2 0.3831 0.3806 0.3810 0.5671 Panel B: = 0.50 Excess Reurn (No risk adjusmen) 0.7246 0.7080 0.7109 0.9825 Risk-Adjused Excess Reurns (SML) 0.3127 0.2913 0.2948 1.4028 Jensen's Alpha 0.0820 0.1035 0.0999 0.6274 MM 0.4064 0.3823 0.3863 1.1825 GH1 0.3478 0.3268 0.3303 1.3667 GH2 0.4097 0.3858 0.3898 1.1816 Panel C: = 2.00 Excess Reurn (No risk adjusmen) 0.7635 0.7184 0.7308 1.1246 Risk-Adjused Excess Reurns (SML) 0.3579 0.3055 0.3202 1.6231 Jensen's Alpha 0.0352 0.0891 0.0745 0.7955 MM 0.4582 0.3981 0.4150 1.3331 GH1 0.3924 0.3406 0.3552 1.5803 GH2 0.4611 0.4015 0.4182 1.3324 Noes: a All daa are expressed as monhly figures in percenages, wih he ou-of-sample period running from January 2004 o December 2008. b M-V, J-S and Min indicae ha he porfolio weigh measures are esimaed from he respecive radiional Markowiz mean-variance analysis, James-Sein esimaor and minimum variance esimaor. Neverheless, as indicaed in able 5, our proposed nework scheme is he unique procedure produces posiive alphas uniform across all risk-olerance levels for he wo 123
benchmark indices. In specific erms, based upon he ascending risk-olerance levels, our proposed nework scheme reveals Jensen s Alphas of 0.1054, 0.6247, and 0.7955 wih respec o he AIEX, whereas he comparaive Jensen s Alphas obained by a ypical porfolio consrucion mehod, for example, he James-Sein esimaor, are -0.1079, - 0.0999, and -0.0745. he empirical resuls even become sronger in erms of MM, GH1 and GH2 ha our proposed scheme uniformly ouperforms oher schemes across all riskolerance levels. A similar conclusion is drawn from able 6. able 6. Excess reurns wih he 50 benchmark Variables M-V J-S Min Nework Panel A: = 0.05 Excess Reurn (No risk adjusmen) 0.2890 0.2872 0.2875 0.0903 Risk-Adjused Excess Reurns (SML) 0.1111 0.1089 0.1093 0.3703 Jensen's Alpha -0.3865-0.3887-0.3883-0.1877 MM 0.1362 0.1339 0.1343 0.3086 GH1 0.1274 0.1252 0.1256 0.3444 GH2 0.1378 0.1355 0.1359 0.3058 Panel B: = 0.50 Excess Reurn (No risk adjusmen) 0.3075 0.2908 0.2938 1.5654 Risk-Adjused Excess Reurns (SML) 0.1344 0.1135 0.1169 1.1509 Jensen's Alpha 0.3634 0.3841 0.3806 0.4130 MM 0.1608 0.1387 0.1424 0.8710 GH1 0.1503 0.1297 0.1331 1.0967 GH2 0.1623 0.1403 0.1440 0.8697 Panel C: = 2.00 Excess Reurn (No risk adjusmen) 0.3463 0.3013 0.3136 0.7074 Risk-Adjused Excess Reurns (SML) 0.1790 0.1273 0.1418 1.3643 Jensen's Alpha 0.3167 0.3704 0.3563 0.5707 MM 0.2082 0.1532 0.1686 1.0058 GH1 0.1944 0.1432 0.1576 1.3035 GH2 0.2095 0.1547 0.1701 1.0078 Noes: a All daa are expressed as monhly figures in percenages, wih he ou-of-sample period running from January 2004 o December 2008. b M-V, J-S and Min indicae ha he porfolio weigh measures are esimaed from he respecive radiional Markowiz mean-variance analysis, James-Sein esimaor and minimum variance esimaor. In summary, our proposed nework scheme coninually exhibis robusly superior invesmen performance over each of he oher mehods examined here in erms of he conen of risk-adjused excess reurns. 124
4. Conclusions As indicaed by Michaud (1989), as opposed o he numerical procedure involved in Markowiz mean-variance analysis, a more saisically-based approach can largely avoid error magnificaion and financially-irrelevan porfolios. In he presen sudy, we design a boosrapping neural nework porfolio consrucion scheme which proves capable of providing and uilizing useful knowledge on he disribuion of expeced reurns of he consiuen shares of a porfolio. he empirical resuls of our sudy are in general agreemen wih he viewpoin of Michaud. Firsly, we find ha our proposed boosrapping neural nework scheme generaes subsanially larger reurns and moderae risks, as compared o radiional Markowiz meanvariance analysis and boh he James-Sein and minimum-variance esimaors. Secondly, for all risk-olerance levels, as an acive managemen ool, our proposed nework scheme ouperforms all of he oher approaches in erms of various risk-adjused measures. Finally, our resuls sugges ha he performance of he radiional Markowiz meanvariance analysis and boh he James-Sein and minimum-variance esimaors is indisinguishable. Based upon our empirical resuls and all plausible raionales, we conclude ha as an acive approach o invesmen managemen, our proposed boosrapping neural nework mean-variance analysis mehod shows real promise. References Board, J., Sucliffe, C. (1994) Esimaion mehods in porfolio selecion and he effeciveness of shor sales resricions: UK evidence. Managemen Science, 40(4), 516-534. Fernandez, A., Gomez, S. (2007) Porfolio selecion using neural neworks. Compuers and Operaions Research, 34(4), 1177-1191. Franke, J., Mahias, K. (1999) Opimal porfolio managemen using neural neworks: A case sudy. Universiy of Kaiserslauern, Deparmen of Mahemaics echnical Repor, Germany. Franke, J., Neumann, M. (2000) Boosrapping neural neworks. Neural Compuaion, 12(8), 1929-1949. Freedman, D.A. (1982) Boosrapping regression models. Annual Saisics, 9(6), 1218-1228. Fros, P.A., Savarino, J.E. (1986) An empirical bayes approach o efficien porfolio selecion. Journal of Financial and Quaniaive Analysis, 21(3), 293-305. Graham, J.R., Harvey, C.R. (1997) Grading he performance of marke-iming newsleers. Financial Analyss Journal, 53(6), 54-66. Hornik, K., Sinchcombe, M., Whie, H. (1989) Mulilayer feedforward neworks are universal approximaors. Neural Neworks, 2, 359-366. Jobson, J.D., Korkie, B. (1981) Puing markowiz heory o work. Journal of Porfolio Managemen, 7(4), 70-74. Jorion, P. (1986) Bayes-sein esimaion for porfolio analysis. Journal of Financial and Quaniaive Analysis, 21(3), 279-292. Lin, C.M., Huang, J.J., Gen, M., zeng, G.H. (2006) Recurren neural neworks for dynamic porfolio selecion. Applied Mahemaics and Compuers, 175(2), 1139-1146. Markowiz, H.M. (1952) Porfolio selecion. Journal of Finance, 7(1), 77-91. 125
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