Heuristic algorithm for portfolio selection with minimum transaction lots

Transcription

1 Heurisic lgorihm for porfolio selecio wih miimum rscio los Afri, Herm Mwegkg echer riig d Educio Sciece Fcul, Islmic Uiversi of Norh Sumer, Med, 0, Idoesi. Deprme of Mhemics, Uiversi of Norh Sumer, Med, 0, Idoesi. Correspodig uhor: fri@hoo.com Absrc. Porfolio selecio problem ws firs formuled i pper wrie b Mrkowiz, where ivesme diversificio c be rsled io compuig. Me-vrice model he iroduced hs bee used d developed becuse of i s limiios i he lrger cosris foud i he rel world, s well s i s compuiol complei which foud whe i used i lrge-scle porfolio. Qudric progrmmig model complei give b Mrkowiz hs bee overcome wih he developme of he lgorihm reserch. he iroduce lier risk fucio which solve he porfolio selecio problem wih rel cosris, i.e. miimum rscio los. Wih he Mied Ieger Lier models, proposed ew heurisic lgorihm h srs from he soluio of he relio problems which llow fidig close-o-opiml soluios. his lgorihm is buil o Mied Ieger Lier Progrmmig (MILP which formuled usig eres ieger serch mehod. Ke words: MILP, heurisics, porfolio opimizio, miimum rscio los, eres ieger serch. Iroducio he mi obecive i ficil ivesmes is o combie ceri sses io he porfolio which gives he opiml profi. Opiml porfolio provides blce bewee reur d risk. he mhemicl model ws firs cosruced b Mrkowiz i his ricle bou 50 ers go, is ofe difficul o use, due o i s limiios i use. Mrkowiz Me-Vrice model for porfolio selecio is oe of he bes kow models i he ficil field d is he foudio for moder porfolio heor. Bu is simplici i some ssumpios, i prcice, i c o work well, becuse his model lso igores he perceived rscio coss, liquidi cosris (d rscio coss h resul from o-lier, he miimum rscio los, d crdili cosris, mel resricios o porfolio o be ble o be priculr sse. If ll of he bove ssumpios pplied i his model, he will produce mied ieger o-lier progrmmig problems which subsill difficul o solve. Whe i pplied o lrge-scle problems, i s difficul o obi he ec soluio, eve is pproimio is lso becomes o simple d urelisic. I he origil Mrkowiz s model ssumes h sse clss reurs is mulivriive ormll disribuio. hus he reur o sses of porfolio c be described compleel b he firs wo momes, he epecio or he me of reur d he reur vrice (risk mesure. Opimizio is emp o fid he se of porfolios h hs he lowes level of risk for priculr re of reur or, lerivel, he highes reur re for priculr risk. he se is clled he efficie froier porfolios d c be deermied b qudric progrmmig. Which is usull preseed s plo curve epeced porfolio reur gis he sdrd deviio for ech predicio from he reur. here re wo criicl ssessme o he ssumpio of me-vrice, boh s cosumpio prefereces of qudric problem or sse prices re ormll disribued. he dowside of his model is h i ssumes o be ormll mulivriive. heoreicll, his mes h he firs wo momes, he epeced reur d vrice, is o sufficie o describe he overll porfolio. his model lso sed h ivesor c deermie he vlue or uili of welh o ives i porfolio bsed o epecios of reur d risk of he porfolios. here is ssumpio h he firs wo momes, he epeced reur d risk, sufficie o deermie he ivesor uili fucio, which is usull depiced wih idifferece curve. If reurs re o ormll disribued he sse s clss, ivesor s uili c be preseed wih ver differe disribuio bu hs me d sdrd deviio, 57

2 he sme. he cpil sse pricig model (CAPM d he rbirge pricig heor (AP hs show h he risk of porfolio is ssemic, h is some risk depeds ol o he mrke, wih he upper limi of he verge vrice of he porfolio of sses divided b he umber of sses i he porfolio, if he umber of sock icreses, he risk is growig drmicll. I his pper, will be showed h whe muliples of miimum rscios lo ke io ccou, he problem o deermie fesible soluio c be epressed s lier fucio of risk. he ew lgorihm proposed s soluio wih muliple models of he miimum rscio los. he proposed heurisic mehod bsed o he ide of buildig d problem solvig mied ieger b cosiderig subses of he vrious ivesme opios which re vilble. Subses re geered b eploiig iformio obied from relio heurisics problem. he reserch ws doe b buildig porfolio selecio model developed from he Mrkowiz model b ddig cosris miimum rscio los. he chge he qudric progrmmig problem (Qudric Progrmmig obied from he Mrkowiz model o be Mied Ieger Lier Progrmmig (MILP problem. Furhermore his model ws developed io heurisic lgorihm. Lierure Review he porfolio selecio process cosiss of wo sges. he firs sge srs wih observio d eperiece h ed up wih beliefs bou he performce of securiies h vilble i he fuure. he secod sge srs wih he relev beliefs bou fuure performce d eds wih selecig he porfolio. o mimize he reur epecios will be esier o use sic models of he use of ime series. As i he dmic cse if ivesors w o mimize he reur o he porfolio, he will pu ll his fuds io he securiies h hve mimum reur. Ivesors should diversif d he lso hd o mimize epeced reurs b disribuig fuds o ll securiies h provide mimum reur epecios. Porfolio wih mimum epeced reur does o eed o be porfolio wih mimum vrice. here is level where ivesors c obi he epeced reur b usig vrice or reduce vrice b reducig he epeced reur (Mrkowiz, 95. rdiiol porfolio opimizio problem is o fid pl o ives i he securiies echge h is ccepble bewee he level of reur d risk. he me-vrice model of Mrkowiz (95 is sic model of sigle period o ge porfolio h c be obied from he verge re of reur give he miimum risk. Mrkowiz e pper sugges umber of lerive models for he sme problem. Is mi obecive is o overcome he compuiol complei of he origil qudric progrmmig problems. For emple, he lier pproimio i pr, Me Absolue Deviio (MAD, Weighed Gol Progrmmig (WGP d miim models (MM (Kim, e l, 003. MAD models proposed b Koo d Ymzki (99 is model of lier progrmmig (LP where risk is mesured b sdrd deviio ised of he vrice. he showed h i is equivle o he Mrkowiz model if i s reur is mulivriive ormll disribuio. he Zeios d Kg (993 lze model for oher smmeric disribuios d foud h he model of MAD does o require specific pe of reur disribuio. Sperz gives he geerl form of MAD models usig weighed risk fucio. He showed h he model c be buil s compc equivle coefficie i he lier combiio which is chose ccordigl. hree subseque ppers iroduce more fleible model (Sperz, 996; Msii, 997; Msii d Sperz, 997. Furhermore Msii d Sperz (999 developed heir reserch beforehd o develop hree heurisics. Pper d Wike Koo (00 d Kellerer e l. (00 clcule his problem b creig relisic clculio feures such s fied rscio coss, d miimum rscio los. Msii d Ogrczk (003 iroduced ssemic view of he LP model h c be solved wih furher discussio of heir heor. 58

3 he lrges secio i he porfolio selecio models which widel recommeded i he lierure re bsed o he ssumpio of perfec divisio of he ivesme porfolio so h he disribuio of securiies o be preseed is rel vrible. I he rel world, egoible securiies rscios s muliple of he miimum lo (hereifer clled roud. B usig he roud, he porfolio selecio problem resoluio requires he deermiio of mied ieger progrmmig model soluio. Whe i pplied o he rel problems, i hs he rcbili h he ieger cosris model will be he vilble for lgorihm h is ble o fid good ieger soluio evehough i s o opiml i ccepble ime. A simple heurisic hs bee proposed d esed for he cse where here is miimum rscio los (Msii d Sperz, 997. Porfolio Opimizio s Model d Heurisic Algorihm Mrkowiz Model Suppose R s rdom vrible re of reur (per period from sses S, =,...,. s he mou of moe ivesed i fud S of ol C. Epecios reur (per period of hese ivesmes is deermied s: r(,..., = E R = E[ R ] ( = = A ivesor ws r(,..., s big s possible. A he sme ime he ws o mke he risk s smll s possible. he sdrd deviio of reur (per period: (,..., = E R E R ( = = sig he size d risk of he porfolio opimizio problem, i is formuled s qudric progrmmig problem prmeers: Subec o = mi = = C i= = r Where, r = E[R ] d = E R r ( R r ] i i ρ C 0,..., u = (3 i [( i d ρ is he miimum re of reur required b ivesors. u he mimum mou of moe h c be ivesed i S. his model is vlid if:. R mulivrie orml disribuio. b. Ivesors re risk verse i his cse i requires smller sdrd deviio. Usig he Mrkowiz model i lrge-scle porfolio opimizio (full covrice mri is cosidered imprcicl o ol becuse of he difficul i is clculios, bu lso becuse of he complei ssocied wih he implemeio of he obied soluio. Me Absolue Deviio Model Koo d Ymzki, (99 sugges lier progrmmig model of he clssic qudric models. heir pproch is bsed o he observio h he size of differe risks, volili 59

4 d risk-reled L re close eough d h he lerive risk mesures re lso suible for porfolio opimizio. Volili of he porfolio reur is Where R ses rdom sse reurs, µ is he me. he risk-l of porfolio s reur is defied s: ( = E ( ( R µ (4 = w( = E ( R µ (5 = heorem (Koo d Ymzki If (R, R,, R re mulivrie ormll disribued rdom vribles, he w( = (. π Proof: Le ( µ, µ,..., µ re me of ( R, R,..., R. Le = ( i R s covris mri of ( R, R,..., R. he R i i ormll disribued where is me is µ i i d is sdrd deviio is = hus, w = E[ U ] i (. ( where U ~ N (0,. i i u ( w( = u e du π ( u ( = ue du π ( = ( π his heorem impl h miimizig ( equivle o miimizig w( whe R, R,..., R re mulivrie ormll disribued. B his ssumpio, Mrkowiz model c be formuled s: mi w ( = E R E R = = Subec o: = = E[ ] ρ C = C 0 u, =,..., 60 ( (6 Eiher R, R,..., R re mulivriive ormll disribued or o, me-bsolue deviio ( model bove will remi o form efficie porfolio o mesure risk-l. le r s he relizio of rdom vrible R durig period, for =,, which ssumed vilble

5 from he hisoricl dum or of some fuure proecios. I is lso ssumed h epecio vlue of rdom vribles c be pproimed b he verge obied from hese dum. Especill, le Ne, w( = E = r = E[ R ] = r (7 ( R µ = ( r µ = = (8 he bsolue vlue i his equio mkes i o-lier. Bu, i c be lierizes b usig ddiiol vribles. Le = ( r r = ( r r = = + ( r r = Furher, i c be wrie s: + ( r r = ; 0, he obied: 0 d ( r r he le = r r, =,,, =,, he (6 c be mde s he followig miimizio problems: subec o: mi = = r = = = C = ρ C 0 u, =,..., hus, from (9 d (0 obied he followig pproch: subec o: mi = 0 (9 (0 = 6

6 + = = 0, =,..., 0, =,..., = = r = C ρ C 0 u, =,..., ( his pproch is lier progrmmig. herefore i c be used o solve he lrge scle of porfolio opimizio problems. Me-semi-bsolue deviio model If he risk is mesure b usig me-semi-bsolue deviio ised of me-bsolue deviio, he he obecive fucio is: mi 0, ( r r ( Ad c be wrie s: mi = ( r r + 0, =,..., (3 i.e wih he smller cosris. So, i c be see from ( h (0 is equivle wih (. hus, vrice is uder ssumpio h is reur is mulivrie ormll disribued. Becuse he model is bsed o semi-bsolue deviio, he he risk fucio is lier, so h i c be iroduced spesificio h obied from he mrke srucure such pplied cosris. Porfolio Selecio Model wih Miimum rscio Lo Briefl, he followig is he oio for mied ieger model wih míimum rscio lo. he purchse Price for securiies wih míimum lo is deoed s c. hus, for ever securiies, míimum lo c be described s Moe d is equivle wih c = N p. Where p is he mrke Price for securiies eeded s míimum quiies. So, c = p whe he sse is rded wihou míimum lo. Ne, C 0 d C s he míimum mou of moe d he máimum h is vilble o be ivesed b ivesor. Ieger vribles, S showed he míimum lo quiies for ever securiies which bécme prs of he opimum porfolio. he quiies c showed prs of he ol moe o be ivesed i securiies. he coss d re vries ccordig o mrke codiios d pes of he greemes, showed he rscio cos proporio of purchsig, becuse of he rscio cos proporio c be direcl icluded io he price. Ne, ssume h he price c is icluded ll he possible rscio cos proporio. hus, he mied ieger lier progrmmig for porfolio selecio problems wih miimum rscio los is formuled s follows: mi = 6

7 c 0, =,..., Subec o + ( r r C = c c C r ρ C C 0 C 0 u, S 0, =,..., (4 Porfolio Selecio wih he Neres Ieger Serch Mehod Accordig o he soluio obied from he model bove, e heurisic is developed o selec he porfolios wih he bsic ides is follows: Assume mied ieger lier progrmmig (MILP:] Mi P = C Subec o ieger, for ll Noe h he fesible bsis vecor of MILP wich is solved s he coiuous problems, d i c be wrie s: ( B k = βk αki ( N i K αk* ( N * K αk, m( N m. Le ( B k be url vlued vribles, β k is priioed io ieger d frcio compoes β = [ β ] + f. If ( B k is icresed o he closes ieger, ([ β ] + c be k k k icresed o be o bsis vribles, le ( N * bove di upper limi s log s k*, i.e. oe of he eleme vecor α *, is egive. ke * oe of o-bsis vribles ( N * he umericl d sclr vlue of ( B k is ieger, he * c be sed s: * fk, he oher o-bsis vribles is remi α k* zero. So, fer i is subsied io * for ( N * obied ( = [ β ] +. Now ( B k α is ieger. I is ow cler h he o-bsis vribles is impor i roudig he bsis vribles vlues. his bsic ide is lso used o solve he mied ieger sochsic problems. Heurisic lgorihm of he fesible soluio serch mehod Afer solvig he relio problems wih he mehods before for he lier sochsics progrm, he serchig procedure for ieger regio soluios c be described s follows: Le = [ ] + f, 0 f <, he coiuous soluio of he relio problems re: δ = Sep choose bsis he smlles ifesible ieger, so h * mi{ f, f } Sep Do he pricig operios, i.e. compue v =l B l i Sep 3 Compue i = v wih obied from mi i I. for o-bsis lower limied B i i i k 63

8 Sep 4 Sep 5 Sep 6 II. < 0 d δ = f i he compue ( δ If i > 0 d δ = - f i he compue ( δ If i < 0 d δ = - f i he compue If i > 0 d δ = f i he compue If i for o-bsis upper limied i δ i δ If i < 0 d δ = - f i he compue If i > 0 d δ = f i he compue If i > 0 d δ = - f i he compue If i < 0 d δ = f i he compue i i ( δ ( δ i δ δ i Else go o he upper o-bsis of he e superbsis (if here is eis. So h colum * is icresed from is lower limi or decresed from is Upper limi. Else, go o he e. Compue α = B i.e. fid for * * O he fesibili es, here re 3 possible fied fesible vribles sice he relesig of he o-bsis from is limi. if * lower limied he ke B l i ' i ' A = Mi i ' α > 0 α i ' Bi ' B = Mi i ' α < 0 α C = Máimum moveme of * deped o θ * = mi( A, B, C if * upper limied, he ke B l i ' i ' A ' = M i i ' α > 0 α i * B ' i ' Bi ' = M i i ' α < 0 α i * C ' = Máimum moveme of * deped o θ * = mi( A', B ', C ' keep he bsis for hese 3 possibili. if A or A, he become o-bsis lower limied l i ' B i ' i i 64

9 * become bsis (replcig sill bsis (o ieger.. if B or B B i ' Bi ' become obsis upper limied i ' * 3. if C or C become bsis (replcig sill bsis (o ieger. B i ' Repe from sep * become bsis (replcig become ieger superbsis. Coclusios his pper cocludes h wih he developed model, he covris mri re o loger eeded o build porfolio selecio model s i i he clssicl model of Mrkowiz. I is esier o solve lier progrmmig h o-lier progrmmig, so h i would reduce he compuio ime o solve i. Chges i he ipu d would o mke sigific chge o he whole model. he vrible c be used s corol vrible o limi he mou of sses i he porfolio. his mehod c be pplied o porfolio coiig pes of sses, s log s he reur d he risk forecsig re vilble. Refereces Ceci M., Filippii F., (005, Porfolio Selecio wih Miimum rscio Los: A Approch wih Dul Epeced Uili, Jourl of Operiol Reserch, Vol. 0, 9-5 Corueols G., uucu R., (006, Opimizio Mehods i Fice, Preice Hll, Ic., New Jerse Elo E., Gruber M., (987, Moder Porfolio heor d Ivesme Alsis, Joh Wile & Sos, New York Frrel J. L., (983,Guide o Porfolio Mgeme, McGrw-Hill, New York Krlof J. K., (006, Ieger Progrmmig, heor d Prcice, CRC Press, Florid Kim J.S., Kim Y.C., Shi K.Y., (003, A Algorihm for Porfolio Opimizio Problem, Jourl of Iformic, 005, vol 6, No., Koo H., Wike A., (999, Me-Absolue Deviio Porfolio Opimizio Model Uder rscio Coss, Jourl of Operios Reserch Socie of Jp, Vol. 4, No.4, Koo H., Ymzki H., (99, Me-Absolue Deviio Porfolio Opimizio Model d Is Applicios o oko Sock Mrke, Mgeme Sciece, Vol. 37, No. 5, Msii R., Sperz MG. (997. Heurisic Algorihms for he Porfolio Problem Wih Miimum rscio Los, Jourl of Operiol Reserch, Vol.4, 9-33 Mrkowiz H. (95. Porfolio Selecio, Jourl of fice Vol.7, 77-9 Mrkowiz H., (959, Porfolio Selecio, Efficie Diversificio of Ivesmes, Joh Wile & Sos, New York. Nsh S.G. d Sofer A., (996, Lier d No Lier Progrmmig, McGrw Hill Compies, Ic., New York Pphrisodoulou C., (004, Opiml Porfolio Usig Lier Progrmmig Models, Jourl of Operiol Reserch Socie, Vol.55, Ro S.S., (978, Opimizio, heor d Applicios, d ed., Wile Eser Limied, Idi. Shrpe W.F., Aleder G.J., Bile J.V., (995, Ivesme, 5 h ed., Preice Hll, Ic., New Jerse 65