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1 econstor Mke Your Publcton Vsble A Servce of Wrtschft Centre zbwlebnz-infortonszentru Econocs Etukudo I. A. Artcle Optl desgns pproch to portfolo selecton CB Journl of Appled Sttstcs Provded n Cooperton wth: The Centrl Bnk of ger Abu Suggested Ctton: Etukudo I. A. (00 : Optl desgns pproch to portfolo selecton CB Journl of Appled Sttstcs ISS The Centrl Bnk of ger Abu Vol. Iss. pp Ths Verson s vlble t: Stndrd-utzungsbedngungen: De Dokuente uf EconStor dürfen zu egenen wssenschftlchen Zwecken und zu Prvtgebruch gespechert und kopert werden. Se dürfen de Dokuente ncht für öffentlche oder koerzelle Zwecke vervelfältgen öffentlch usstellen öffentlch zugänglch chen vertreben oder nderwetg nutzen. Sofern de Verfsser de Dokuente unter Open-Content-Lzenzen (nsbesondere CC-Lzenzen zur Verfügung gestellt hben sollten gelten bwechend von desen utzungsbedngungen de n der dort gennnten Lzenz gewährten utzungsrechte. Ters of use: Docuents n EconStor y be sved nd coped for your personl nd scholrly purposes. You re not to copy docuents for publc or coercl purposes to ehbt the docuents publcly to ke the publcly vlble on the nternet or to dstrbute or otherwse use the docuents n publc. If the docuents hve been de vlble under n Open Content Lcence (especlly Cretve Coons Lcences you y eercse further usge rghts s specfed n the ndcted lcence.

2 Journl of Appled Sttstcs Vol. o. 53 Optl Desgns Approch to Portfolo Selecton I.A. Etukudo In order to obtn the best trdeoff between rsk nd return optzton lgorths re prtculrly useful n sset llocton n portfolo. Such lgorths nd proper soluton technques re very essentl to nvestors n order to crcuvent dstress n busness outfts. In ths pper we show tht by nzng the totl vrnce of the portfolo nvolvng stocks n two gern bnks whch s esure of rsk optl llocton of nvestble funds to the portfolo s obtned. A copletely new soluton technque odfed super convergent lne seres lgorth whch kes use of the prncples of optl desgns of eperent s used to obtn the desred optzer. Keywords: Portfolo selecton nu vrnce optl desgns optl llocton.. Introducton In every nvestent there s trdeoff between rsk nd returns on such nvestent. An nvestor therefore ust be wllng to tke on etr rsk f he ntends to obtn ddtonl epected returns. However there ust be blnce between rsk nd returns tht suts ndvdul nvestors eveu (985. Gret cre ust be tken by ny nvestor n the llocton of hs nvestble funds to lst of nvestents open to h n order to nze the totl rsk nvolved. A thetcl odel to sut proble of ths nture nd n prtculr qudrtc progrng odel for portfolo selecton ws developed by Mrkowtz ( A portfolo s set of nvestents tht n nvestor cn nvest n whle portfolo rsk refers to the rsk coon to ll securtes n the portfolo nd ths s equted wth the stndrd devton of returns Ebrh (008. The purpose of the nvestent of csh n portfolos of securtes s to provde better return thn would be erned f the oney were retned s csh or s bnk depost. The return y coe n the for of regulr ncoe by wy of dvdends or nterest or by wy of growth n cptl vlue or by cobnton of both regulr ncoe nd growth n cptl vlue Cohen nd Znbrg (967. Thus the rel obectve of portfolo constructon becoes tht of chevng the u return wth nu rsk Wever (983. Grubel (968 showed tht hgher returns nd lower rsks thn the usul re obtned fro nterntonl dversfcton. Arnott nd Copelnd (985 hve lso shown tht the busness cycle hs sgnfcnt effect on securty returns. On ther prt Chen Roll nd Ross (986 deterned tht certn croeconoc vrbles re sgnfcnt ndctors of chnges n stock returns. Contrbutng further Bun nd Mller (995 showed tht the evluton of portfolo perfornce should tke plce through coplete stock rket cycle becuse of dfferences n perfornce durng the rket cycle. Mcedo (995 deonstrtes tht swtchng between reltve strength nd reltve vlue strteges cn ncrese returns n n nterntonl portfolo. Snce portfolo selecton proble s qudrtc progrng proble whch nvolves nzton of rsk ssocted wth such nvestent by nzng the totl vrnce whch s esure of the rsk nvolved Frncs (980 sutble soluton technque should be dopted to obtn optl soluton. Etukudo nd Uoren (009 hve shown tht t s eser nd n fct better to use odfed super convergent lne seres lgorth (MSCLS Q whch uses the prncples of optl desgns of eperent n solvng qudrtc progrng probles rther thn usng the trdtonl soluton technque of odfed sple ethod. Ths pper therefore focuses on optl desgns pproch to optl llocton of nvestble funds n portfolo. Deprtent of Mthetcs/Sttstcs & Coputer Scence Unversty of Clbr Clbr ger nsedorenyn@gl.co

3 54 Optl Desgns Approch to Portfolo Selecton I.A. Etukudo. A qudrtc progrng odel for portfolo selecton For qudrtc progrng odel for portfolo selecton let n nuber of stocks to be ncluded n the portfolo nuber of shres to be purchsed n stocks n Y returns per unt of oney nvested n stocks t turty Assung the vlues of Y re rndo vrbles then E(Y Y ;... n ( [( Y Y ( Y Y ] V σ E ( where E(Y s the thetcl epectton of Y nd V s the vrnce covrnce tr of the returns. See Gruyter (987 Prsons (977 nd Etukudo et l (009. Hence the vrnce of the totl returns or the portfolo vrnce s gven by n n f ( XVX σ (3 whch esures the rsk of the portfolo selected. The non-negtvty constrnts re 0 n (4 Assung the nu epected returns per unt of oney nvested n the portfolo s B then n Y B (5. Mnzton of the totl rsk nvolved n the portfolo By nzng the totl vrnce f( of the portfolo the totl rsk nvolved n the portfolo s nzed. In order to obtn nu pont of equton 3 f( ust be conve functon Hller nd Lebern (006. Tht s f( f( f( f( f( (6 where f( 0 (7 M f( 0 where n. Strct nequltes of 6 nd 7 ply tht f( s strctly conve nd hence hs globl nu t. Fro equton 3 nd nequltes 4 nd 5 the portfolo selecton odel s gven by; Rerk Mn f ( subect to: The epected vlues n n σ n Y B ; 0 n Y nd the vrnce covrnce tr σ re bsed on dt fro hstorcl records.

4 Journl of Appled Sttstcs Vol. o Modfed super convergent lne seres lgorth (MSCLS Q Uoren nd Etukudo (009 The sequentl steps nvolved n MSCLS Q re gven s follows: Step : Let the response surfce be y c 0 + c + c + q + q + q 3 G k Select support ponts such tht 3k 4k where k 3 s the nuber of prttoned groups desred. By rbtrrly choosng the support ponts s long s they do not volte ny of the constrnts ke up the ntl desgn tr X M M M Step : Prtton X nto k groups wth equl nuber of support ponts nd obtn the desgn tr X k for ech group. Obtn the nforton trces M X X k nd ther nverses - M k such tht v v v - M v v v 3 v v v Step 3: Copute the trces of the ntercton effect of the vrbles for the groups. These re X I M M M where k nd the vector of the ntercton preters obtned fro f( s gven by q g q q 3 The ntercton vectors for the groups re gven by I M X X groups re v - M M + I I v v3 v v v 3 - I g v3 v 3 v33 nd the trces of en squre error for the Step 4: Copute the optl strtng pont fro u ; u > 0; u u - -. Step 5: The trces of coeffcent of conve cobntons of the trces of en squre error re

5 56 Optl Desgns Approch to Portfolo Selecton I.A. Etukudo H dg v v v33 dg{h h h 3 } k v v v33 By norlzng H such tht H I we hve H H dg h h h h h 3 h 3 The verge nforton tr s gven by M(ξ k H M H Step 6: Fro f( obtn the response vector z 0 z z z where z 0 f ( 3 ; z f ( 3 ; z f ( 3 33 Hence we defne the drecton vector d d0 d d - M (ξ z nd by norlzng d such tht d d we hve d d d d d + d d d + d Step 7: Obtn the step length ρ fro the qudrtc progrng proble. c - b ρ n c d where c b s the th constrnt of Step 8: Mke ove to the pont ρ d Step 9: Copute f( nd f(. Is f( - f( ε where ε then stop for the current soluton s optl otherwse replce by nd return to step 7. If the new step length ρ s neglgbly sll then n optzer hd been locted t the frst ove.

6 Journl of Appled Sttstcs Vol. o A uercl Eple An nvestor hs u of to nvest by purchsng shres n Ocenc Bnk nd Frst Cty Monuent Bnk. Below s the hstorcl dt of prces per shre n the bnks for 5 dys. We re requred to obtn optl llocton of the nvestble funds for purchse of shres n the portfolo n order to nze the totl rsk n the portfolo. Fro the dt on tble 3. the en prces per shre for Frst Cty Monuent Bnk nd Ocenc Bnk re 8.3 nd 8.9 respectvely. Tble 3.: Prce per shre Dy FCMB (Y Ocenc Bnk (Y Dy FCMB (Y Ocenc Bnk (Y Source: The gern Stock Echnge Y Y Tble 3.: Men devton Dy (Y Y (Y Y Dy (Y Y (Y Y The epected return per shre s the dfference between the en prce of tht shre nd ts prce on the 5 th dy. The nvestor ssues tht hs epected returns would be t lest Snce hs obectve s to nze hs totl rsk the proble nvolves obtnng optl portfolo where the nvestent s done t the 5 th dy prces. The shre prce devtons re obtned fro tble 3. s shown n tble 3. whle the vrnce covrnce tr tble for the shre prce re obtned fro tble 3. s shown n tble 3.3. Fro the tble 3.3 the vrnce- covrnce tr s gven by

7 58 Optl Desgns Approch to Portfolo Selecton I.A. Etukudo σ σ σ σ Hence the odel for nzng the totl rsk of the portfolo s Mn f( ( Subect to: where nd re respectvely the nuber of shres purchsed fro Frst Cty Monuent Bnk nd Ocenc Bnk n the Portfolo. Tble 3.3: Vrnce- covrnce tr vlue Dy (Y Y (Y Y(Y Y (Y Y Dy (Y Y (Y Y(Y Y (Y Y ( Y Y(Y Y σ Test for Convety Snce f( f( f( > 0 f( f( > 0 nd > 0 f( s strctly conve functon nd ts globl nu pont s obtned by solvng the bove portfolo selecton proble. 6. Soluton to the portfolo selecton proble by optl desgns pproch Mnze f(

8 Journl of Appled Sttstcs Vol. o. 59 Subect to: Let X ~ be the re defned by the constrnt. Hence X ~ { } Step : Select support ponts such tht 3 k 4 k where k 3 s the nuber of prttoned groups desred. By choosng k we hve 6 8 Hence by rbtrrly choosng 8 support ponts s long s they do not volte the constrnts (wthn the fesble regon the ntl desgn tr s Step : Prtton X nto groups such tht nd the desgn trces for the two groups re { ; } { ; } G G X X The respectve nforton trces re M XX nd M X X Step 3: The trces of the ntercton effect of the vrbles re X I nd X I nd the vector of the ntercton preters obtned fro f( s gven by

9 60 Optl Desgns Approch to Portfolo Selecton I.A. Etukudo g The ntercton vectors for the groups re I M XXIg nd I M X X Ig The trces of en squre error for the groups re respectvely M M + I I M M + I I Step 4: ow Obtn the optl strtng pont 0 u ; u > 0; u u [ 5 0] [ ] [ 00 0] [ ] [ 30 ] [ ] [ 5 ] [ ] Snce u...

10 Journl of Appled Sttstcs Vol. o u 0.55 u u u u u u 0.50 u Hence the optl strtng pont s 8 u Step 5: Obtn the trces of coeffcents of conve cobntons fro M nd M s follows: H dg dg{ } H I H dg { } nd by norlzng H nd H such tht H H + H H we hve H dg H dg dg { } dg { } The verge nforton tr s gven by

11 6 Optl Desgns Approch to Portfolo Selecton I.A. Etukudo M ( ξ H X X H + H X X H z 0 Step 6: Fro f( obtn the response vector z z z z f( ( (448(450 Therefore z z ( ( f( (50 46 f( ( z Here we defne the drecton vector d d d M (ξ z d nd by norlzng d such tht d d we hve (5 046( ( (50408( d d d Step 7: Obtn the step length ρ fro where c b For c nd b 8.75 c b n cd ρ s the th constrnt of the portfolo selecton proble we hve.4350 [ ] ρ [ ] 0.680

12 Journl of Appled Sttstcs Vol. o For c nd b Step 8: Mke ove to the pont 00 we hve.4350 [ ] ρ [ ] snce ρ d [ ] ρ s the nu step length. Step 9 f( f( ( ( ( ( (4.44( (.4350( snce f( f( Mke second ove by replcng.4350 by [ ] 00 The new step length s obtned s follows: ρ [ ] Snce the new step length s neglgble the optl soluton ws obtned t the frst ove nd hence 4.44 nd f( The portfolo selecton proble whch s nzton of portfolo vrnce ws solved usng odfed super convergent lne seres lgorth whch gve 4 0 s the nuber of shres to be purchsed fro Ocenc Bnk nd Frst Cty Monuent Bnk respectvely n order to obtn nu rsk or nu vrnce. 7. Sury nd Concluson In ths pper we ssued tht the portfolo hs lredy been selected by the nvestor fro lst of vlble nvestents. Usng hstorcl dt prces (5 dys of stocks fro Frst Cty Monuent Bnk nd Ocenc Bnk we showed how optl lloctons of nvestble funds could be de to ech Bnk s stocks by nzng the portfolo vrnce thereby nzng the totl rsk usng optl desgns pproch. The pproch dopted n obtnng optl soluton s recoended for use by potentl nvestors s wy out of busness collpse.

13 64 Optl Desgns Approch to Portfolo Selecton I.A. Etukudo References Arnott R. D. nd Copelnd W. A. (985: The Busness Cycle nd Securtes Selecton. Fnncl Anlysts Journl V4( 6 3. Bun W. S. nd Mller R. E. (995: Portfolo Perfornce Rnkngs n Stock Mrket Cycles. Fnncl Anlysts Journl V5( Chen Fu Roll R. nd Ross S. (986: Econoc Forces nd the Stock Mrket. Journl of Busness V59 ( Cohen J. B. nd Znbrg E. D. (967 Investent Anlyss nd Portfolo Mngeent Irwn.. Ebrh Zn (008: Modern Portfolo Theory 008 ( Etukudo I. A Effng E. O. Onwukwe C. E. nd Uoren M. U. (009: Applcton of Portfolo Selecton Model for Optl Allocton of Investble Funds n Portfolo M. Scent Afrcn Fculty of Scence Unversty of Port Hrcourt Vol. 8 o Etukudo I. A. nd Uoren M. U. (009: A Coprson of Modfed Super Convergent Lne Seres Algorth nd Modfed Sple Method for Solvng Qudrtc Progrng Probles ICASTOR Journl of Mthetcl Scences Kolkt Ind Vol. 3 o Frncs J. C. (980: Investent Anlyss nd Mngeent (Thrd Edton. McGrw Hll Inc. U.S.A. Grubel H. G. (968: Interntonlly Dversfed Portfolos Welfre Gns nd Cptl Flows. Aercn Econoc Revew V58 ( Gruyter W. D. (987: Opertons Reserch. Theory Technques nd Applctons. Berln ew York. Hller F. S. nd Lebern G. J. (006: Introducton to Opertons Reserch. Eghth Edton Tt McGrw-Hll ew Delh. Mcedo R. (995: Vlue Reltve Strength nd Voltlty n Globl Equty Country Selecton. Fnncl Anlysts Journl V5( Mrkowtz H. M. (95: Portfolo Selecton. Journl of Fnnce Mrkowtz H. M. (959: Portfolo Selecton. Effcent Dversfcton of Investents. Cowles Foundton Monogrphs. Yle Unversty Press ew Hven. eveu R. P. (985: Fundentls of Mngerl Fnnce. South Western Publshng Co. Cncntve Oho Prsons J. A. (977 The Use of Mthetcs n busness. Alender Hlton Insttute Inc. U. S. A. Uoren M. U. nd Etukudo I. A. (009: A Modfed Super Convergent Lne Seres Algorth for Solvng Qudrtc Progrng Probles Journl of Mthetcl Scences Kolkt Ind Vol. 0 o Wever D. (983: Investent Anlyss Longn In Assocton Wth The Socety of Investent Anlyss. Gret Brtn.