Gnc conrol appld o ass managmns. Jams Cunha Wrnr and Trnc C. Fogary SCISM Souh Ban Unvrsy 03 Borough Road London SE 0AA, UK {rnrjc,fogarc}@sbu.ac.u Absrac. Ths papr addrsss h problm of nvsmn opmzaon usng gnc conrol. Tm srs for soc valus ar oband from daa avalabl on h and ass prcs ar prdcd usng adapv algorhms. A porfolo s opmzd h h gnc algorhm basd on a rcursv modl of porfolo composon oband on-h-fly usng gnc programmng. Ths o sps ar ngrad no an auomac sysm - h fnal rsul s a ral-m sysm for updang porfolo composon for ach ass. Inroducon. IBM and ohr compans ar undrang massv suds on h applcaon of advancd compung chnologs o soc brorag and obanng br rsuls han h Yor soc mar s sharps radrs []. Evry nvsor nos ha hr s a rad off bn rs and rard: o oban a grar han xpcd rurn on nvsmn on mus b llng o a on a grar rs []. Porfolo opmzaon hory assums ha for a gvn lvl of rs, nvsors prfr hghr rurns o lor rurns. Smlarly, for a gvn lvl of xpcd rurn, nvsors prfr lss rs o mor rs. I s sandard o masur rs n rms of h varanc, or sandard dvaon, of rurn. Th porfolo opmsaon problm consss of obanng h bggs rurn on nvsmn h h las rs xposur ncssary undr h prvalng mar dynamcs. Ths dynamcs ar unprdcabl du o boh xognous facors such as govrnmn acons, mar rumors, unxpcd vns, c. and ndognous facors such as company and soc fundamnals. Mahfoud and Man [7] dvlopd a rul-basd sysm for managng ach ndvdual ass hr h ruls ar of h form: IF prc < lm and EPS > valu THE buy, hr prc s h ass prc, lm s h buy hrshold, and EPS s h arnng pr shar.
Chang al [8] usd h gnc algorhm o fnd h porfolo of asss h dffrn rs xposurs calld h ffcn fronr. Usually, hs problm s solvd h quadrac programmng, bu, for praccal purposs s dsrabl o lm h numbr of asss n a porfolo, as ll as h proporon of h porfolo dvod o any parcular ass. Kvn and Warmouh [9] proposd ha h porfolo vcor slf ncapsula h ncssary nformaon from h prvous prc rlavs. Thus, a h sar of day, h algorhm compus s n porfolo vcor + as a funcon of and h jus obsrvd prc rlavs x, usng a lnar rgrsson. Hlmbold al [4] slc a mor complx funcon and Pars and Hubrman [6] gnralz h da for nvsmn group modl for h porfolo slcon problm, adjusng hr porfolo as hy obsrv movmns of h mar ovr m and communca o ach ohr hr currn porfolo and s rcn prformanc. Invsors can choos o sch o any porfolo prformng br han hr on. In hs or h goal s o oban a rcursv mahmacal la usng gnc programmng and h gnc algorhm ha sablshs a rlaon bn h avalabl nformaon and h prcnag of varous asss o b hld n h porfolo. Th gnral framor, gnc conrol Wrnr [4], s rprsnd n fg.. I uss daa from xprmnal sup h smulad mar n hs cas o fd gnc programmng for h purpos of buldng a modl of h mar. Lar, h gnc algorhm adaps ral valus o oban h opmal prcnags of h asss n h porfolo ha ll fd gnc programmng, closng h loop. Fg.. Gnc conrol: obanng h srucur of h soluon h gnc programmng and adapng s paramrs h gnc algorhm. Th gnc algorhm. Th gnc algorhm GA mmc h voluon and mprovmn of lf hrough rproducon, hr ach ndvdual conrbus s on gnc nformaon o h buldng of n ons adapd o h nvronmn h hghr chancs of survval. Ths s h bass of gnc algorhms and gnc programmng [5], [6], and [7].
Spcalzd Marov Chans undrln h horcal bass of hs algorhms chang of sas and sarchng procdurs. Each ndvdual of a gnraon rprsns a fasbl soluon o h problm, codng dsnc algorhms/paramrs o b valuad by a fnss funcon. GA opraors ar muaon h chang of a randomly chosn b of h chromosom and crossovr h xchang of randomly chosn slcs of h chromosom. Th bs ndvduals ar connuously bng slcd, and crossovr and muaon a plac. Follong a numbr of gnraons Fg., h populaon convrgs o h soluon ha prforms br. Fg.. Gnc algorhm: h squnc of opraors and valuaon of ach ndvdual. A gnralzaon of h Gnc Algorhm s Gnc Programmng GP Holland [5] and Goldbrg [6] hr ach ndvdual n a gnraon rprsns, h s chromosom, a fasbl soluon o h problm; n our cas, a mahmacal funcon o b valuad by a fnss funcon. Thr ar o nds of nformaon dfnd for h GP algorhm: rmnals varabl valus and random numbrs and funcons mahmacal funcons usd n h gnrad modl. Th vrual mar. Th frs problm hn sudyng mar nvsmn s n ha nvronmn o s h concps and ho o oban m srs of asss, socs and currncy, and mar ndx for a prod of yars a las. To solv hs problm xracd m srs valus from h hsory graphcs avalabl n h Yahoo Fnanc s [3]. W bul a daabas conans h follong nformaon [9] bn July 999 and July 00: FTSE00 soc quos, rad volum and quos by scor; Fx nrs; Europan and Amrcan ndcs; Soc xchangs ndcs around h orld: Argnna, Brazl, Canada, Chl, Pru, Vnzula, Ausrala, Chna, Hong Kong, Inda, Indonsa, Malaysa, Zaland, Pasan, Phlppns, Sngapura, Souh Kora, Sr Lana, Thaland, Taan, Ausra, Tchc Rpublc, Fnland, Grc, drland, Porugal, Russa, Slovaa, Span, Sss, Tury, Egyp, Isral; Commods: Gold, slvr, palado; Currncy convrgnc o pound: USD, Ausrala, Canada, Argnna, Brazl, Euro, Franca, Grmany, Hong Kong, Japan, Mxco, Russa and Sss.
Th bnchmar for porfolo rurn. Th rfrnc for rurn valuaon s h soc xchang ndx. In h cas of London hs s h FTSE 00 [0]. Th nvsmn opraon ould buld a porfolo h rflcs h sam consuon as h FTSE 00 ndx, and s prformanc s h sam as h mar. Ass forcas. To dvlop a forcas of asss prc or any ohr m srs valu hr ar o ncssars dfnons: h mahmacal funcon o b adjusd rmd flr and h adapaon algorhm rsponsbl for calcula h paramr valus of h flr follong mporal changs of h srs. Th FIR fn mpuls rspons flr of dmnson s a flr h rval pols z=0 n s ransfrnc funcon: Wz = 0 +.z - +.z - +... + n-.z -+ L W h flr coffcn vcor, and X las npus o h flr n nsan: W = [ 0 3... - ] T X = [x x- x- x-... x-+] T Flr oupu s dfnd as: y 0 x X T W Any soc/ass conans n s prc o componns: on dpndng of s fundamnals and ohr complly random, modld by h Bronan modl. Th nrnsc componn ould b adapd by Las Mans Squar LMS s [], hch cancls h random componn. L us dfn h prformanc funcon = W 4 a quadrac funcon h on mnmum pon. For any nal condon W, valua n valus of W no h conrary drcon of h prformanc hypr surfac gradn h ndca h maxmum drcon. Follong h conrary drcon, cranly ll h h mnmum. To valua h gradn of s ncssary do som approxmaons. L = E[ ] 5 Whr E s h avrag of sochasc varabl, h rror bn h ral valu d and h adapd by fng y. Thn: 3
0 0.. 6 and bcaus x W X d T 7 Thn X.. 8 LMS algorhms consss n ravly calculang h vcor flr coffcn W, h a ll vcor n h gradn conrary drcon:. W W 9 hr s an arbrary consan o conrol h raon sp. Rplacng no W rav quaon: X... W W 0 Ths s h LMS algorhm quaon, a vry smpl mahmacal modl asy o b mplmnd. LMS sably dpnds of h valu. If s [] 0 3 x hr x avrag por of x and s h flr ordr, h algorhm ll b convrgn and sabl. If h valu s oo bg ll b unsabl and f s oo ll convrgnc ll a oo much m. Asss daa pr procssng. L S b h quo of a soc. Th mar ors h h funcon S S Ln
for rurn and h h rcursv varaon: 3 j, j, r, r j, hr j, s h condonal varaon of h prod, = 0.94 s h dcay paramr of xponnal smoohng, r x, s h rurn of x n prod. Th nal condon s: r j,0 x T T T r j, r r r r, j, j 4 Th varaon marx of rs facors and corrlaon marx ar: 5 C j j j 6 Th dagonal rms of marx ar = Porfolo opmzaon. Th mhodology for sarchng for h consuon of a porfolo as sablshd by Maroz [3] 50 yars ago and has bn cnral o rsarch acvs n xndng, mprovng and rvsng hs approach n unforsabl and dynamc mars. Th am s o oban a mx of asss o maxmz h rlaon bn man rurn and rs, a modl opmzaon problm:
max.0 *,,, j j, j 7 hr ach composon s lmd bn h lms and. Th cas =0 rprsns maxmz xpcd rurn rrspcv of h rs nvolvd and h opmal soluon ll nvolv jus h sngl ass h h hghs rurn. Th cas = rprsns mnmz rs rrspcv of h rurn nvolvd and h opmal soluon ll ypcally nvolv a numbr of asss. Valus 0<< rprsns xplc rad off bn rs and rurn, gnrang soluons bn h xrms. W assum =0.3 n hs papr. A rcursv mahmacal modl for porfolo slcon. Th fulcrum pon of h problm of porfolo opmsaon consss n obanng a modl: F, rurn, rs, pas prc, prdcon prc, mar ndx of h fuur prcnag of ach ass avalabl, h a funcon dpndn only of h forcas and pas condon. A gnral hory for hs approach s avalabl n [4], [5] and [6]. 8 Sofar mplmnaon and rsuls h vrual mar. Th sofar consss n hr procdurs, runnng n squnc bfor rad or:. Gvn m srs valus up o las prod, oban h paramrs of an adapv flr for ach ass, hch modl s fundamnal bhavour, h a 0 h ordr approxmaon, manng ha all ffcs h a prod lss han 0 days ar modlld. Wh hs paramrs forcas h rurn for nx prod usng a FIR modl adapd by h LMS algorhm.. Gnc algorhms opmz h prcnags for ach ass, h h avalabl nformaon of h las prod, gvng, as a rsul, h opmum porfolo. For ach ass suppos ha s composon n h porfolo s lmd bn 0% and 30%, lmnang h ffcs of small flucuaons n h slcon. 3. Wh h nformaon of h voluon of, o approachs r appld: Us GP o oban quaon 8 ha prdcs h funcon h and hou consrans.
Us GP o oban quaon 8 ha gvs h bs rurn for h prod, on-h-fly. To possbls ar xplord: h and hou opmal composon of las prod oband by Gnc algorhm. Barng n mnd h assumpons dscrbd n h nroducon abou mar bhavor and forcas, h frs rsul consss n valuang h rror dsrbuon of forcas asss usng h adapv mhod, h an accuracy br han 5% for FTSE00 asss forcas ovr yars. Th nx sp consss n obanng h opmal porfolo h all asss avalabl usng pas nformaon. Th problm consss n solvng quaon 7 usng gnc algorhms, h a chromosom h bnary codd floa varabls for ach asss rprsnng h amoun no h porfolo. Ths valuaon uss rurn, sandard dvaon and varanc marx of h las 0 m srs valus avalabl. Th rurn sum for all prod nrvals gvs h ffcncy of h sragy. Fgur 3 and 4 sho h rsuls for fr and h consrans 0.<<0.3, hch fxs h prcnag of any ass bn 0% and 30%. Fg. 3. Daly rurn for unconsrand porfolo agans FTSE00 rurn -0.089. rurn=5.33 Fg. 4. Daly rurn for consrand porfolo agans FTSE00 rurn -0.089. rurn=9.49 Th consrand porfolo s ncssary du opraonal problms, ohrs h porfolo s formd of oo many asss. Th nx sp consss n volvng h soluon o quaon 8 usng gnc programmng o forcas h prcnag of ach ass n h porfolo. A r bul h h funcons: mulply *, sum+, subracon - and dvson Dv, and rmnals: ERC0.., prc forcas usng adapv algorhm, rurn n [-] and [-], man rurn for 0 las days prod, sandard dvaon of las 0 days, rad volum s valuad for h prod of yars, and oponally h las opmal prcnag of h asss oband by gnc algorhm.
Th bs modl gvs h rurn shon n fgs 5 and 6, usng, h or hou, h las asss opmal prcnag h consrans of prcnag bn 0% and 30% for any asss n h porfolo. Fg. 5. Porfolo hou las opmal prcnag h consran agans FTSE00 rurn -0.089. rurn=0.66 Fg. 6. Porfolo h las opmal prcnag h consran agans FTSE00 rurn - 0.089. rurn=7.53 Th rurns ar valuad agans h nx m prod s rurn, o sablsh h ffcacy of mahmacal modl. Th ass slcon modl usng quaon 8 oband by GP forcas as good rsuls as h opmzaon usng quaon 7, h =0.3. Th dffrnc bn fgurs 3 and 4, and 5 and 6 s ha n h frs cas, gnc algorhm uss only avalabl daa o opmz h problm, and h scond cas uss GP o forcas a fuur soluon basd n asss forcas usng adapv algorhm. Conclusons. Gnc programmng for producng a prdcv modl for porfolo asss prcnag assocad h gnc algorhm o oban porfolo opmal valus obans good rsuls hn comparng h FTSE00 ndx, and o h sam lvl as oband by gnc algorhms calculang h avalabl daa. Ths framor could b appld n asss managmn, ang car h xognous nflunc n h mar. Th nx sp of h projc consss n apply h algorhm o dffrn scnaros, o vrfy h adapably of h prdcv modl.
Th sofar concp s adqua o ral m applcaon n asss managmn, h adqua ss and adapaon of man machn nrfac and broadcas daa acquson. Rfrncs:. Mro s; Wall Sr s ban by robos Thursday, Augus 9,00 pag 7.. Argonn aonal Laboraory EOS; Th Porfolo Slcon Problm: An Inroducon hp://-fp.mcs.anl.gov/oc/gud/cassuds/por/nroducon.hml 3. hp://u.fnanc.yahoo.com/?u 4. Wrnr,J.C.; Acv nos conrol n ducs usng gnc algorhm PhD. Thss- São Paulo Unvrsy- São Paulo-Brazl-999. 5. HOLLAD,J.H. Adapaon n naural and arfcal sysms: na nroducory analyss h applcaons o bology, conrol and arfcal nllgnc. Cambrdg: Cambrdg prss 99 rdção 975. 6. GOLDBERG,D.E. Gnc Algorhms n Sarch, Opmsaon, and Machn Larnng. Radng,Mass.: Addson-Whsly, 989. 7. KOZA,J.R. Gnc programmng: On h programmng of compurs by mans of naural slcon. Cambrdg,Mass.: MIT Prss, 99. 8. hp://amy.h-ho.n.jp/jbaba/gf.hm 9. hp://u.bz.yahoo.com/quo/ovrv.hml 0.hp://.fs.com/.B.Wdro & S.Sarns, Adapv sgnal procssng, Prnc-Hall Inc; S.Kuo & D.Morgan, Acv nos conrol sysms. Algorhms and DSP mplmnaon, John Wly & sons..bellager, M. G. Adapv dgal flrs and sgnal analyss Yor, Marcl Dr,987 Chapr 4. 3.H. Maroz; Porfolo slcon ; Chang,T-J;Mad,.; Basly,J.E.; Sharaha,Y.M.; Hurscs for cardnaly consrand porfolo opmsaon Compurs & opraons rsarch 70007-30. 4.Hlmbold,D.P.; Schapr,R.E.; Sngr,Y.; Warmuh,M.K.; On-ln porfolo slcon usng mulplcav updas Machn Larnng: Proc. Of h 3 h Inrnaonal Confrnc 996 5.Covr,T.M.; Unvrsal Porfolos, Mahmacal fnanc 99-9. 6.Pars,D.C.; Hubrman,B.A.; Adapv porfolo slcon by nvsmn groups IFAC Symposum on Compuaon n Economcs, Fnanc and Engnrng CEFES 98, Cambrdg England, 998. 7.Mahfoud,S.; Man,G.; Fnancal forcasng usng gnc algorhms Appld Arfcal Inllgnc, 0 996 543-565. 8.Chang,T.J.; Mad,.;Basly,J.E.; Sharaha,Y.M.; Hurscs for cardnaly consrand porfolo opmzaon Compurs & Opraons Rsarch 7 0007-30. 9.Kvn,J.; Warmuh,M.K.; Exponnal gradn vrsus gradn dscn for lnar prdcons Tchncal rpor UCSC-CRL-94-6, Unvrsy of Calforna, Sana Cruz, Jun 994 fp.cs.ucsc.du/pub/ml/ucsc-crl-94-6.ps.z