Optimal Combination of Trading Rules Using Neural Networks

Transcription

1 Vol. 2, No. Inernaonal Busness Research Opmal Combnaon of Tradng Rules Usng Neural Neworks Subraa Kumar Mra Professor, Insue of Managemen Technology 35 Km Mlesone, Kaol Road Nagpur , Inda Tel: E-mal: Absrac A large number of radng rules based on echncal analyss of prces are beng used by nvesng communy for generang radng sgnals for shor erm nvesmens. As profably of hese radng rules vary, s no easy o judge whch parcular rule really works. Insead of a sngle radng rule, combnaon of rules are lkely o offer he porfolo benefs of beer rsk adjused reurn and hence, an expermen s carred ou o combne sgnals generaed from of movng averages of dfferen wndow sze usng an arfcal neural nework. I s observed ha he rsk adjused performance measure of he arfcal neural nework based radng model s beer han ha of smple Buy and Hold sraegy. Keywords: Tradng Rules, Techncal Analyss, Neural Neworks. Inroducon The nvesors no only look for long erm capal gans from he marke, bu also lke o maxmze reurns explong opporunes from shor erm prce movemens. A large number of radng rules based on echncal analyss of prces are used for generang buy and sell sgnal for shor erm nvesmens. From he numerous radng rules beng used by he radng communy, s no easy o judge whch parcular rule really works. Invesmen professonals wh long years of experence may ake very good radng decson form her experse. Bu here s no guaranee ha such decsons wll always work and herefore he use of a sysemac procedure o generae radng decsons becomes mporan. A sysemac decson makng approach can also help o overcome varous lmaons ha are nheren n human professonals. Furher, as dfferen praconers have dfferen vews on he same nformaon se, sysemac evaluaon mehods wll reduce personal bas. In he area of nvesmen managemen, where decsons nvolve very large amouns of money, somemes he lfelong savngs of he clens, reassurance of he soundness of he nvesmen decson-makng process s necessary. Techncal radng rules are ncreasngly beng used n fnancal markes for over a cenury ever snce was popularzed by Charles Dow n 900. Bu analyss of radng rules have sared drawng more aenon n 990s and several auhors have expressed ha fnancal prces and reurns are predcable o some exen, eher from her own pas or from some oher publcly avalable nformaon. For example, Bessembnder & Chan (995), Blume, Easley & O.hara (994), Brock, Lakonshok & Lebaron (992), Ramazan (998), Jegadeesh & Tman (993, 2000), Lo & MacKnlay (988), Nefc (99), Ready (997) es varous radng rules based on echncal analyss and repored ha echncal analyss provdes nformaon beyond ha already ncorporaed n he curren prce. However, here canno be a fxed radng rule as excessve usage of a parcular radng rule wll reduce effcacy of ha rule. If everybody sars usng a parcular rule, ha rule wll no work any more and hence, he radng rules need o be connuously upgraded based on changng marke dynamcs. Analyses made he leraure are based on performance of a specfc radng rule used n solaon. Insead of relyng on a sngle radng rule, raders ofen use a varey of radng rules and somemes a combnaon. Combnng of rules are lkely o offer he porfolo benefs of beer rsk adjused reurn. Markowz (952, 959) has shown ha n porfolo conex unsysemac rsk can be reduced by dversfcaon; possbly smlar benef arses when mulple radng rules are combned for akng a radng decson. 2. Usng Movng s for Generang Tradng Sgnal The movng average (MA) mehod s one of he mos wdely used mehods of generang radng rules. I ncludes numeral versons and dfferen levels of complexy. A movng average s an average of observaons from several 86

2 Inernaonal Busness Research January 2009 consecuve me perods. To compue a movng average sequence, we compue successve averages of a gven number of consecuve observaons. The objecve underlyng he MA mehod s o smooh ou seasonal varaon n he daa. The mos wdely used movng average (MA) s he n-day smple MA gven by: SMA = P n = n+, where SMA s he smple n-day movng average a perod and P s he closng prce for perod. In he smple MA procedure, a buy sgnal s generaed when he closng prce rses above he MA and a sell sgnal s generaed when he closng prce falls below he MA. If here s a clear rend, hs mehod wll work well. If, however, he marke move sdeways or f here s excessve volaly, here wll be a lo of false sgnals. A modfcaon of smple movng average s exponenal movng average (EMA) ha gves more wegh o he mos recen me perods. I s descrbed recursvely as: EMA = α. P + ( α). EMA, where α s a value beween 0 and. For example, f α = 0.5, he mos recen value P s gven 50% wegh and all oher pas values are gven remanng 50% wegh. When he compuaon begns, he curren prce s se o EMA and as more prces are avalable, he averagng process s connued. EMA = P EMA2 = α. P2 + ( α). EMA EMA3 = α. P3 + ( α). EMA2 The exponenal movng average performs well for many busness applcaons, usually producng resuls superor o he movng average. (Kranardzc, 200) A movng average summarzes he recen pas daa, furher; spong he change n he rend of daa may addonally mprove forecasng performances. Some of he measures ha compare curren prce wh he movng averages are P MA : he dfference beween he curren prce and s movng average; MA MA (-k) : he dfferences beween wo movng averages, of same wndow sze; MA (,n) MA (,m) : he dfferences beween wo movng averages, of dfferen wndow sze; and P : he rao beween he curren value and s movng average. MA In he presen sudy he curren prce wll be compared wh s movng average n rao forma. When curren prce P s be more han s Movng average MA, s consdered as an ndcaon of uprend and a buy sgnal s generaed and P vce versa a sell sgnal s generaed when curren prce s less han s movng average. Numercally, he rao > MA P wll generae a buy sgnal and he rao < wll generae a sell sgnal. MA The value of he rao P MA wll neverheless depend on he wndow sze of he movng average. A movng average P havng less wndow sze, say 3 days, wll follow curren prce closely and he rao wll change from more han MA o less han more frequenly generang a large number of buy and sell radng sgnals. Whereas a movng average havng a large wndow sze, say 200 days wll generae radng sgnal less frequenly. More radng arsng ou of less wndow sze may capure he mnor movemens of prces well, bu consequenly he ransacon coss wll also ncrease. The success of a movng average based radng mehod clearly depends on selecon of a proper wndow sze, bu here s no known mehod o deermne he wndow sze. Therefore, an expermen s carred ou o combne sgnals generaed from of movng averages of dfferen wndow sze usng an arfcal neural nework. 3. An Inroducon o Arfcal Neural Nework The arfcal neural nework based echnques are an nformaon processng model derved from funconng of human bran. Ths s a smple nformaon processng devce ha acceps many npus, combnes hem, and produces an oupu. The basc elemen of a neural nework s a neuron. The oupu of one neuron becomes npu o oher neurons. A neural nework s a srucure of many such neurons conneced n a sysemac way. In he sudy, he neural neworks used are feed-forward neural neworks, where nformaon processng moves only n forward drecon as shown n fgure -. 87

3 Vol. 2, No. Inernaonal Busness Research The neurons n he nework are arranged n layers. Typcally, here s one layer for npu neurons (he npu layer), one or more layers of nernal processng uns (he hdden layers), and one layer for oupu neurons (he oupu layer). Each layer s eher parly or fully nerconneced o he precedng layer and he followng layer. The connecons beween neurons have weghs assocaed wh hem, whch deermne he srengh of nfluence of a neuron o oher neurons. Informaon flows from he npu layer hrough he hdden processng layer(s) o he fnal oupu layer o generae predcons. The connecon weghs are deermned by a ranng process, wheren known npu and known oupu daa s fed o he nework. The nework adjuss conneced weghs so ha a relaonshp beween npus and oupus can be esablshed wh ceran degree of accuracy. 3. Desgnng of Nework Srucure There are many parameers o desgn a feed forward neural nework. Decsons regardng number of npus n he npu layer, number of hdden layers and number of neurons n he hdden layers, nerconnecon of neurons among layers ec. are o be aken. Though some echnques are menoned n leraure for deermnng hese parameers, here s no unformy. Srucure of he nework largely remans a desgn ssue and leaves ample scope of nnovaon o he analys. 3.. Inpu Layer The npu layer o he neural nework s he medum hrough whch he npus are presened o he neural nework. When a se of npu s presened o he npu laer of he neural nework, he npus are processed and resulan nformaon s passed o he subsequen layer(s). Every npu neuron should represen some known varable ha has an nfluence over he oupu of he neural nework. As fnal oupu wll depend on npus nroduced o he nework, he qualy and relevance npus are very mporan Hdden Layers There are really wo decsons o be made wh regards o he hdden layers. The frs s how many hdden layers o have n he neural nework and hen how many neurons wll be n each of hese layers. Neural neworks wh wo or more hdden layers can represen funcons wh any knd and hence here s no heorecal reason o use neural neworks wh any more han wo hdden layers. Decdng he number of hdden neurons n layers s an mporan par of decdng he overall neural nework srucure. Hdden layers do no drecly nerac wh he exernal envronmen bu nfluences he fnal oupu. Hence, boh he number of hdden layers and number of neurons n each of hese hdden layers mus be carefully desgned. Usng lesser number of neurons n he hdden layers wll resul n under-fng. Under-fng occurs when few neurons n he hdden layers are unable deec relaonshps s complex scenaro. On he conrary, usng oo many neurons n he hdden layers may resul n over-fng. Over-fng occurs when he neural nework has so much nformaon processng capacy ha he lmed amoun of nformaon conaned n he ranng se s no enough o ran all of he neurons n he hdden layers. Anoher problem can occur even when ranng daase s very large. A large number of neurons n he hdden layers can ncrease he ranng me of he nework. Over-fng wh large ranng daa may f pas daa very well. The objecve of neural nework model s o exrac general relaonshp n he daa whch can be used n new envronmen. Thus generalzaon of nework relaons s more mporan han over-fng. There are few rule-of-humb mehods for decdng srucure of hdden layer. These rules-of-humb are only sarng pons o consder he nal srucure. Ulmaely he selecon of he archecure of he neural nework has o be fnalzed by expermenaon. The number of neurons should be n he range beween he sze of he npu layer and he sze of he oupu layer. The number of neurons may be 2/3 of he npu layer sze, plus he sze of he oupu layer. The number of neurons should be less han wce he npu layer sze. In he presen sudy, a hree layered nework wh: one npu layer havng hree npu nodes, one hdden layer havng hree processng nodes and fnally one oupu layer producng a sngle oupu s used, as shown n fgure. The srucure of he nework can ndeed be vared as per requremen of he analyss Oupu Layer The oupu layer of he neural nework presens oupu o he exernal envronmen. The oupu s derved from npus va complex relaonshps nbul n he neural nework srucure. 3.2 Inpu-Oupu Relaonshps Daa ranges of real-world npu parameers vary wdely. For example, one varable may have daa ha ranges beween 0 and, whle anoher varable can be a fve dg value. If boh of he varables are used n her naural scale, he second varable s lkely o be gven much more wegh n he model han he frs varable, smply because of s orgnal values (and herefore he dfferences beween records). To compensae for hs effec of scale, range felds are usually 88

4 Inernaonal Busness Research January 2009 ransformed so ha hey all have he same scale. In he sudy, range felds are made unform o have values beween and +0.5 by usng a rescaled sgmod funcon. The acvaon of each neuron from npu laer o hdden layer and agan from hdden layer o oupu layer s calculaed as a j = f ( w ) j x, where a j s he acvaon of neuron j, s he se of neurons n he precedng layer, w j s he wegh of he connecon beween neuron and neuron j, x s he oupu of neuron, f ( x) s a ransfer funcon used o scale he summaon values from -0.5 o In he sudy, we used sgmod or logsc ransfer funcon o scale he numercal values. The orgnal formula for sgmod converson s f SIGMOID ( x) = whch convers any x x + e value whn a range of 0 o. The sgmod values are furher rescaled by a deducng 0.5 so ha values reman whn -0.5 o Thus he converson funcon s f ( x) = x + e In he nework used (fgure -), x, x 2 and x 3 are he npu nodes, y, y 2 and y 3 are nodes n he hdden layer and z s he fnal oupu. The acvaon of each node n hdden layer s calculaed by usng followng rescaled sgmod funcon of weghed npus. y = ( ) 0.5, where w j s he connecon weghs beween y and x j nodes. Smlarly, he fnal oupu z. + w j x j e can be esmaed usng he ransformaon: z = ( ) w j y j e The fnal value (z) s used for generang radng sgnal. Values of hese rescaled sgmod funcons range beween -0.5 and +0.5 wh mean value of 0. If he value of z s found posve, s consdered as a sgnal of uprend and conversely, when he value of z s negave s aken as a sgnal of down rend. If buy, hold and sell decsons are represened by +, 0 and - respecvely, hen fnal buy and sell decsons are deermned usng he value of Sgn(z). Sgn(.) deermnes he sgn of a number: reurns f he number s posve, zero (0) f he number s 0, and - f he number s negave. 3.3 Tranng The oupu value z can be calculaed from he npus (x ) and connecon weghs w j usng relaonshps menoned n prevous secon. The values of x (npus) are known o us bu he values of connecon weghs (w j ) are no known. The ranng he nework s carred ou o fnd ou values of w j, so ha hese values can be used for generang fuure sgnals. The objecve s o forecas oupu z (+), whch wll mach wh fuure acual reurn r (+). However he fuure reurns can never be accuraely predced and any predcon wll always have some error. The purpose s o mnmze hese errors as much as possble so ha he forecas s of some praccal use. The oal error can be measured by addng absolue errors of each observaon ABS(r (+) z (+) ). A more accepable form s based on mnmzaon of Toal Squared Error (TSE): Mnmze: Σ (r (+) z (+) ) 2 The mnmzaon can be done usng any commercally avalable sofware. In he sudy, he opmzaon was carred ou usng Solver add-n avalable n Mcrosof Excel. Solver uses he Generalzed Reduced Graden (GRG2) nonlnear opmzaon code developed by Leon Lasdon, Unversy of Texas a Ausn, and Allan Waren, Cleveland Sae Unversy. When Solver reaches an accepable soluon, has mnmzed he oal squared error erm TSE by changng value of specfed cells (hese cells are weghs w j of he nework). The values of changed cells are he opmzed weghs and can be used n predcng radng sgnals n fuure. 4. Emprcal Tesng 4. Daa The sudy examnes he profably of echncal radng rules appled o hree Indan Sock Indces for he perod s Aprl 998 o 3 s December 2007, coverng a perod of 0 years. The daly closng values of followng ndces are analyzed n he sudy (deals on hese ndces can be obaned from S&P CNX Nfy CNX Nfy Junor CNX Defy 4.2 Converng Indces daa o Nework Inpus Inpus o he nework mus conan nformaon peranng o oupu (o predc prce movemens). A large number of academc sudes suppor usefulness of movng averages for deermnng rends n sock prce seres. The followng npus seleced n he sudy compares curren prce wh pas movng averages. The frs npu compares closng prce of he secury wh s 3 day movng average : 89

5 Vol. 2, No. Inernaonal Busness Research x = (p / Movng average of pas 3 days) The second npu compares closng prce of he secury wh s 7 day movng average: x 2 = (p / Movng average of pas 7 days) The hrd npu compares closng prce of he secury wh s 30 day movng average: x 3 = (p / Movng average of pas 30 days) 4.3 Fndng Value of Nework Weghs Tranng of nework refers o a mehod of deermnng he value of connecng weghs of he nework based on a ceran performance measure, such as cumulave prof. The performance of he radng sysems s usually deermned by opmzng over pas known daa, bu here s no consensus on how much pas daa o be used. A common procedure o assess he profably of echncal radng s o choose he opmal parameer usng he frs par of he avalable daa and hen es he parameer upon he remanng daa for ou-of-sample verfcaon. Ou-of-sample verfcaon s an mporan facor n esng he performance of echncal radng sraeges due o he danger of daa snoopng bases. For each fnancal seres, he ranng procedure s carred ou usng he pas one year s daa. The nework weghs (w j ), ha have shown he bes performance over a year, are used for he ou-of-sample radng n he nex year. A he end of he nex year, new opmal weghs for he year are agan calculaed, and hs procedure s repeaed durng he res of he sample perod. For example, he connecon weghs used for he year are raned weghs ha generaed he hghes cumulave reurn n he year The new connecon weghs for are seleced usng he daa for he year , and so forh. Ths procedure ensures ha he enre neural nework model s adapve and all he radng resuls are ou-of-sample. 4.4 Esmaon of (Loss) ably of a radng poson depends on he change of marke prce of he raded secury and he poson of he rader (eher long or shor). If rader has aken a long poson, he wll be benefed by a prce rse of he secury bu wll ncur loss by a prce decrease. Smlarly he rader can make prof n a declnng marke by akng a shor poson. The rader s presumed o ake a long poson whenever he nework oupu z (+) gves a posve value. Lkewse, a shor poson s aken whenever he value of z (+) s negave. The radng decson can be represened by he followng dummy varable., f z (+) > 0 d (+) = d (), f z (+) = 0 -, f z (+) < 0 The dummy varable d (+) s equal o one (negave one) when he rader goes long (shor) n raded asse. The reurn of he rader on a parcular day can be esmaed as follows: r (+) = d (+) ( P (+) P () ), where, r (+) denoes he reurn of he rader resulng from he decson aken (d (+) ) a he close of perod, whch depends on value of d (+) and change n he asse value whn perod and (+). When d (+) = d (), he exsng poson (long/shor poson n he asse) s mananed and no new ransacon need o be carred ou. Hence ransacon cos s no applcable. If d (+) d (), hen he poson held s reversed a he close of perod (+), necessang wo sngle ransacon (closng exsng poson and openng a new poson n oppose drecon). Takng ransacon cos no accoun, daly gross prof becomes: r (+) = d (+) ( P (+) P () ) c d (+) d () P (), where c s ransacon cos (n fracon of asse value) of a sngle ransacon and d (+) d () denoes absolue value of he dfference d (+) d (). Toal cumulave prof afer ransacon coss can be obaned addng daly profs. Abou 5 years ago, ransacon coss n Indan markes used o be very hgh. Bu he scenaro s changed now. Brokerage raes on Indan bourses have crashed o hsorc lows due o compeon. Apar from compeon, susaned reforms n he fnancal markes have led o lower ransacon cos. Brokers can no longer jusfy hgher ransacon cos afer nroducon screen based radng, elecronc ransfer of shares hrough deposores, launch of nerne drven radng and subsanal ncrease of radng volume. The brokerage raes n he marke durng he perod have plummeed from around 2% for delvery bases ransacons o less han 0.05% of urnover for fuure rades. All profably calculaon n he sudy s carred ou a 0.05% ransacon cos analogous o cos applcable n fuures marke. 4.5 Tradng s The radng resuls of usng Neural Nework model s calculaed for hese hree fnancal seres and he same are compared wh he profably of Buy and Hold sraegy. In Buy and Hold Sraegy, he secury s bough a he sar of he sudy perod and sold a he end of he perod. No ransacon s carred ou durng he perod and no ransacon cos s ncurred. Whereas n radng model usng neural neworks, ransacons were many causng hgh ransacon cos. The 90

6 Inernaonal Busness Research January 2009 profably of Neural Nework model s compared wh Buy & Hold sraegy for he seleced hree seres. The resuls are gven n ables, 2 & 3. I may be seen from he ables ha oal ne prof usng Neural Nework Model s generally hgher han ha of Buy and Hold Sraegy even afer ransacon coss ndcang usefulness of echncal radng rules. 4.6 Sascal ess The mos wdely used rsk adjused nvesmen performance measure s developed by Prof. Wllam F. Sharpe; hs measure s no only wdely used n academa bu also by marke praconers. The Sharpe Rao (SR) can be calculaed as follows. µ ( r ) r f SR = σ Where r s he reurn over perod, s he mean and s he sample sandard devaon over he n perods observed and r f s he rsk-free rae of neres. Orgnally, he benchmark for he Sharpe Rao was aken o be a rsk-less secury, where he dfferenal reurn s equal o he excess reurn of he fund over a one-perod rsk-less rae of neres. The usefulness of he Sharpe Rao s based on he premse ha a dfferenal reurn represens he resul of a zero-nvesmen sraegy. Bu n case of radng n a forward or fuure conrac, one need no fnance he asse by makng full paymen, ofen such conracs can be purchased by provdng a small margn paymen or provdng some shor of guaranee. Therefore raded conracs of sock ndex fuures can be consdered as zero-nvesmen sraeges. Sharpe rao n zero-nvesmen sraeges can be calculaed omng rsk free rae as follows. µ ( r SR ) = σ The Sharpe rao of an nvesmen can be compared wh any benchmark by compung he Sharpe Rao for he benchmark and compare wh he nvesmen model. The Sharpe Rao of Neural Nework model and Buy and Hold Sraegy are compared for he hree fnancal seres n Tables 4, 5 and 6. In he ables, can be found ha Sharpe Rao of Neural Nework model s hgher han ha of Buy and Hold Sraegy n mos of he cases ndcang ha he model has gven beer rsk adjused reurn. The Sharpe Rao s also drecly relaed o he -sasc for measurng he sascal sgnfcance of he reurn. The Sharpe Rao, when mulpled by he square roo of n (he number of reurns used for he calculaon) s equvalen o -sasc. µ ( r ) = σ value. value = SR. The -sasc as defned above for he full fnancal seres s calculaed n able -7. Snce -sascs of all he hree fnancal seres are no only posve n bu also sascally sgnfcan a % level, use of he arfcal neural nework based radng model may be consdered as a beer alernave o Buy and Hold Sraegy. 5. Concluson In he sudy, nvesmen decsons were aken usng echncal analyss based radng rules and esed on hree sock ndex seres n Indan sock marke. Insead of relyng on a sngle radng rule, he rules are combned usng an arfcal neural nework model. The heorecal profs from he model are esmaed and compared wh profs obanable from Buy and Hold sraegy. I s observed ha he rsk-adjused performance of he Neural Nework based radng model s generally beer han Buy and Hold sraegy. Lke many prevous sudes, he presen sudy also demonsraes ha s possble o earn posve reurn by usng echncal radng rules. However one of he major mpedmens of radng prof s ransacon cos. The sudy s carred ou akng a relavely low ransacon cos of 0.05%, usually applcable for fuures radng where rades are squared off whou delvery. Wherever he delvery s nvolved, he brokerage fees are sgnfcanly hgher. Thus nvesor has o pay more aenon n mnmzng ransacon cos for radng success. In he sudy, only movng averages are used as npus o he nework. Many oher ndcaors; boh echncal analyss ndcaors and fundamenal analyss raos can also be used as npus o he arfcal neural nework model o mprove he nvesmen performance. Confguraon of he neural nework model and node relaonshps can also be alered for furher developmen. References Bessembnder, H., Chan, K. (995). The ably of Techncal Tradng Rules n he Asan Sock Markes. Pacfc-Basn Fnance Journal, 3 (2/3), n n 9

7 Vol. 2, No. Inernaonal Busness Research Blume, Lawrence, Davd Easley & Maureen O.hara. (994). Marke Sascs and Techncal Analyss: The Role of Volume, Journal of Fnance, 49, Brock, W., Lakonshok, J. & Lebaron, B. (992). Smple Techncal radng rules and he sochasc properes of sock reurns. Journal of Fnance, 47, Gencay, Ramazan. (998). The Predcably of Secury Reurns wh Smple Techncal Tradng Rules. Journal of Emprcal Fnance, 5, Jegadeesh, Narasmhan & Sherdan Tman. (993). Reurns o Buyng Wnners and Sellng Losers: Implcaons for Sock Marke Effcency. Journal of Fnance, 48, Jegadeesh, Narasmhan. (2000). Foundaons of Techncal Analyss: Compuaonal Algorhms, Sascal Inference, and Emprcal Implemenaon: Dscusson. Journal of Fnance, 55, Kanardzc M. (200). Daa Mnng Conceps, Models, Mehods, and Algorhms. Wley-Inerscence. Lo, Andrew W. & A. Crag MacKnlay. (988). Sock Marke Prces Do No Follow Random Walks: Evdence from A Smple Specfcaon Tes. Revew of Fnancal Sudes,, Markowz, Harry M. (952). Porfolo selecon, Journal of Fnance, 7, Markowz, Harry M. (999). The early hsory of porfolo heory: , Fnancal Analyss Journal, 55, 5-6. Nefc, Salh N. (99). Nave Tradng Rules n Fnancal Markes and Wener-Kolmogorov Predcon Theory: A Sudy of.techncal Analyss. Journal of Busness, 64, Ready, M. (997). s from Techncal Tradng Rules, Workng paper. Unversy of Wsconsn-Madson, Madson, WI. Table. Comparng profs of Neural Nework model and Buy and hold sraegy (For Nfy) No of Tradng s Gross Transacon per Transacon Cos Ne from Buy-Hold Sraegy

8 Inernaonal Busness Research January 2009 Table 2. Comparng profs of Neural Nework model and Buy and hold sraegy (For Junor-Nfy) No of Tradng s Gross Transacon per Transacon Cos Ne from Buy-Hold Sraegy Table 3. Comparng profs of Neural Nework model and Buy and hold sraegy (For Defy) No of Tradng s Gross Transacon per Transacon Cos Ne from Buy-Hold Sraegy

9 Vol. 2, No. Inernaonal Busness Research Table 4. Comparson of Sharpe Rao (For Nfy) Neural Nework Model per Sandard Devaon of Daly Sharpe Rao Buy & Hold Sraegy per Sandard Devaon of Daly Sharpe Rao Table 5. Comparson of Sharpe Rao (For Junor-Nfy) Neural Nework Model Buy & Hold Sraegy Sandard Sandard Devaon Sharpe Devaon Sharpe per per of Daly Rao of Daly Rao

10 Inernaonal Busness Research January 2009 Table 6. Comparson of Sharpe Rao (For Defy) Neural Nework Model Buy & Hold Sraegy Sandard Sandard Devaon Sharpe Devaon Sharpe per per of Daly Rao of Daly Rao Table 7. Sharpe Rao and -sasc of Fnancal Seres Sl. No. Fnancal Seres per Sandard Devaon of Daly Sharpe Rao -sasc Nfy * 2 Junor-Nfy * 3 Defy * * Sgnfcan a %. Fgure. Schemac dagram of an Arfcal Neural Nework 95