Analyzing the Diversification in the Optimal Portfolio Choice: an empirical study of the Moroccan Stock Market

Transcription

1 Ieraoal Joural of Scefc Research ad Iovave echolog ISS: Vol. 2 o. 6; Jue 205 Aalzg he Dversfcao he Opmal Porfolo Choce: a emprcal sud of he Morocca Sock Marke ************ LAIFA AIOUOUHE (correspodg auhor) Abdelmalek Essaad Uvers, Facul of Ecoomcs, ager, Morocco. Posal address: Aveue Omar Ibo Abdel Azz, rue dale, 6 bs, eoua, Morocco FARIS HAMZA, Abdelmalek Essaad Uvers, Poldscplar facul, eoua, Morocco. Posal address: Poldscplar facul, Marl, eoua, Morocco. ************ Absrac he Facal markes he emergg coures Asa, La Amerca ad Easer Europe have sparked a mpora leraure amed a udersadg how he work, her orgazao ad fuure prospecs, ad he dffere echques ad mehods of allocao of her porfolos. However, few sudes have bee devoed o he Morocca facal marke; especall hose cocered wh he problem of dversfcao he opmal porfolo choce. I hs sud, we wll r o udersad he porfolos wde dversfcao sraeg ha s based o he correlao of facal asses he Morocca facal marke. Smlarl, hs paper seeks o prese a crcal aalss of hs sraeg of he opmal dversfcao ad mpleme he med lear program wh absolue devao model of Koo & Yamazak Smplfed b Hamza & Jasse. hs program wll reduce he sze of he porfolo o be opmzed order o avod he maageme ad rasaco epeses of he facal asses geeraed b hs sraeg. Ke Words: dversfcao, coss of rasacos, absolue mea-devao model smplfed, med lear program, Casablaca Sock marke. Classfcao (JEL): G, G7, C6. Iroduco he choce of a opmal porfolo of he facal asses has bee, for a log me, a opc of a prmar eres he facal feld. he basc dea s o mamze he reur of he porfolo whle mmzg s rsk. I hs coe, several heorecal models of he asses allocao have bee suded sce he earl 950s wh he am of resolvg hs problem of he porfolo choce. he arcle of he ecooms Harr Markowz o he Ued Saes publshed 'he Joural of Face' 952 lauched he earl developmes of he moder heor of porfolo maageme. Afer had bee lmed o he academc feld for a log me, hs heor was eveuall mposed o he professoals of he facal world. Markowz suggesed epressg a secur s eres b he epecao of s reurs ad he rsk b s varace. he vesor, for he cosruco of hs porfolo, wll seek o make a full reur wh a mmum rsk. hs approach, called he mea-varace approach, led hm o w he obel Prze 990. He has also roduced he cocep of he effce border deduced from he mmum varace porfolo for a gve epecao of reur, whch represes he opmal combao of rsk ad reur. he opmzao s doe b defg a fuco of ul ha represe he vesors' preferece akg o 82

2 Ieraoal Joural of Scefc Research ad Iovave echolog ISS: Vol. 2 o. 6; Jue 205 accou her averso o he rsk ad mamzg he oe, whch s gve he cosra represeed b he effce border. he works of Harr Markowz helped, as well, esablshg a heor of he opmal dversfcao of he sock marke porfolo. I fac, Markowz has esablshed he sraeg of dversfg hs porfolo ad reducg s rsk level whle maag a sasfacor reur. Furhermore, amog hs proposals, he suggess he use of all pes of asses o acheve a good dversfcao. he dversfcao ca occur a several levels: o ol secures, secors, coures, regos ad vesme pes, bu also sles ad sraeges. I ca reduce he rsk of a porfolo ecep whe he vesme compoes are weakl correlaed. he sadard heor of he choce of he facal porfolo posulaes ha he vesor arbraes amog all he asses ha es he marke so as o mamze he relao reur/rsk. hus, deermes a welldversfed ad effce porfolo. hs cofgurao volves rasacos coss. everheless, hese rasaco coss are o ake o cosderao; whereas he cao be gored real. I hs coe, Pogue showed 970 ha he rasaco coss could somemes eceed, ad eve furher, he epeced profabl of he vesor, especall for he large porfolos. Smlarl, Lelad showed 985 ha some freque adusmes of he porfolo o keep he asses close o her arge proporos, led o ver hgh rasaco coss. Cosequel, he reduco porfolo sze before eve carrg ou s opmzao s ver useful o avod heav epeses due o he rasaco coss. o acheve our goal ad deal wh our problem, we mus gve aswers o he followg derved quesos: Wha s he ecessar hreshold of dversfcao o elmae he specfc rsk? How ma asses wll be suffce o dversf sgfcal a porfolo of facal asses? How should we selec hem so as o opmze he dversfcao o reach a fed hreshold of asses? I hs regard, hs work wll focus mal o he presece of a cosra ha makes he dversfcao of our porfolo more effce wh he Morocca facal marke. I he facal dscple, s called he cosra of he mmum hreshold of vesme. I ams a reducg he rae of asses, whch make he opmal porfolo. hs work s orgazed as follows. Afer a geeral roduco. he secod seco s devoed o he revew of leraure mal he prcple of dversfcao ad opmal porfolo choce. Ideed preses a crcal aalss of dversfcao ad a program of resoluo, amel he med lear program formulaed b Hamza & Jasse (smplfg he model of Koo & Yamazak), whch akes o cosderao he cosras of he purchase hreshold. I he hrd oe, we llusrae our approach hrough a dgal applcao usg he hsorc weekl reurs of 74 facal secures ha buld he Morocca sock marke from Jauar 2, Jue 08, 204. We wll rel also o he ecoomerc sofware E-vews 7 for he ess of he dffere sascal parameers ad Malab sofware for he smulao of our resoluo program. I he las seco, we coclude hs work wh a dscusso abou he resuls ad a cocluso. 2. Leraure Revew 2.. Dversfcao ad opmal choce of a facal asses porfolo. As llusraed b Clauss (20), he prcple of dversfcao s based o a old sag: "Do o pu all oe s eggs oe baske." o dversf meas o compose a porfolo of secures from dffere secors order o dsrbue he overall rsk bewee hese dffere secures. A vesor, who cao ake rsk, wll buld a well-dversfed porfolo b vesg dffere asses he Formulao of dversfcao ad he mea-varace approach. he poeer of he moder face, Markowz has demosraed mahemacall he real of dversfcao. He esablshed ha he oal rsk of a group of secures s less ha he sum of rsks of he heor of rasacos coss, fouded b Coase 937 ad parcularl elaboraed b Wllamso sce 975, resuled a ver mpora emprcal developme sce 985. I 99, he Roal Academ Swede gave he obel Prze of Ecoom o Roald H.Coase for he dscover ad clarfcao of he role of he coss of rasacos 83

3 Ieraoal Joural of Scefc Research ad Iovave echolog ISS: Vol. 2 o. 6; Jue 205 hese dvdual secures 2. I oher words, vesg a group of secures reduces he rae of rsk whou losg he reur of he porfolos. he dversfcao cosss combg several srumes of vesme wh he same porfolo. Accordg o Harr Markowz, "a vesor ca reduce he rsk of hs porfolo smpl b holdg asses ha are o or ca be posvel correlaed, hus, dversfg hs vesme." Dversfcao s effce whe he rsk s decreased o he mamum, eher absoluel or for a gve level of reur. hus, he qual of dversfcao deps o wo parameers: he umber of secures cluded he porfolo. he level of correlao bewee he reurs of secures. he seleco of a secur o clude a porfolo s o made accordg o s dvdual characerscs, bu accordg o s behavor wh he porfolo. σ 2 p σ σ hus, he overall rsk of he porfolo s he weghed sum of corbuos o he rsk of each asse. Cosderg ha a uform weghed porfolo cosss of secures. he proporo vesed a secur s. hus he oal rsk of a porfolo ca be calculaed b he varace or sadard devao of s reurs: σ 2 p σ σ 2 p 2 σ ha s σ 2 p σ σ og ha s he larges varace. hus, we shall have 2 V σ 2 V hs rao s o zero whe srves for he f ( s he effec of o-correlao). So, σ s he average covarace bewee he secures, hus, he weghed sum of covarace wll be: 2 ( ) σ ( 2 2 σ ) σ he weghed sum of he covarace s o he average covarace whe he umber of secures he porfolo creases 3. Whe creasg he umber of secures, he decrease of rsk s a frs rapd, ad he slows dow sharpl. Beod a cera umber, becomes useless o keep dversfg because he margal beefs of dversfcao decrease whe he porfolo s dversfed, whle he margal coss rema hgh due o rasaco coss. he followg fgure llusraes hs resul: 2 H. Markowz (959), Porfolo Seleco: effce dversfcao of vesme. Yale Uvers Press. 3 Jacqulla ad Solk Les marchés facers e la geso de porefeulle, p.66, Duod

4 Ieraoal Joural of Scefc Research ad Iovave echolog ISS: Vol. 2 o. 6; Jue 205 Fgure : Rsk ad dversfcao hus, he oal rsk, whch affecs he epeced reur of a value, cosss of he ssemac ad specfc rsks. Ssemac or o-dversfable rsk: Ssemac rsk s arbuable o he geeral movemes of he marke ad ecoom 4. hs rsk cao be elmaed b dversfcao. Specfc or dversfable rsk 5 : he pecular of he specfc rsk s ha ca be dversfed wh oe porfolo. I srves for zero whe he umber of secures a porfolo s mpora eough (hs s he effec of dversfcao). ha s wh s o remueraed. I addo, all he emprcal sudes ha have bee coduced some facal markes showed ha he reur of dversfcao deps o he srucure of covarace 6 ad correlao 7 bewee he secures. Ideed, he specfc rsk of he porfolo decreases whe he umber of secures cosug he porfolo creases; b decreasg well as he oal rsk of he porfolo, ul o loger suppors he marke rsk. he umber of secures o hold s porfolo o acheve a oal dversfcao deps heavl o he correlaos bewee he reurs of secures he mea absolue devao approach ad dversfcao We wll epla he followg pages he effec of dversfcao usg he rsk fuco defed hrough he fuco Pece Wse Lear (PWL), sead of he Quadrac fuco. hs s he mea absolue dfferece of he porfolo reur compared wh s average. hus, we cosder a uform weghg porfolo cossg of secures. he proporo vesed a secur (,.) s /. We have: K( ) E [ R( ) E( R( )) ] E R E( R E R E( Koo & Yamazak showed ha her rsk measureme ad he sadard devao of porfolo reurs (he fuco of rsk based Markowz) are equvale o a cosa f asse reurs follow a mulvarae ormal dsrbuo. R ) 4 hs pe of rsk s geeraed from some uepeced macro-ecoomc eves lke flao, shock over he rae of eres, hgher rae of uemplome, recessos, he chage of govermes,, affecg he secures. 5 he specfc rsk or mcro-rsk s lked o he facors whch s fluece receves o frm or he group of frms, especall he srkes, chages he ase of he cosumers, errors of maageme ad udcal proceedgs. 6 σ cov( R, R ) E[( R E( R ))( R E( R )] 7 σ cov( R, R ) E[( R E( R ))( R E( R )] 85

5 Ieraoal Joural of Scefc Research ad Iovave echolog ISS: Vol. 2 o. 6; Jue 205 If he vecor of he secures reurs R... R ) s dsrbued accordg o a mulvarae ormal law, ( µ ) σ ( µ,..., µ ), Σ ( ). ( he R( ) ( µ ( ) µ, σ ( ) σ ) K( ) σ ( ), hus lm K( ) lm σ ( ) σ π π π herefore, he dversfcao has a lm ad he rsk, whch s measured b he absolue devao of he porfolo average reur o full dversfable Crcal aalss of dversfcao he dversfcao of a porfolo, f properl coduced, s of real eres. Bu also has s ow lmaos. Dversfcao helps o reduce he specfc rsk of he asses, bu b o meas ssemac rsk ha comes from he marke ad o whch dversfcao wll have o fluece. Smlarl, he oal dversfcao s o ver realsc for a porfolo maager, especall whe he resuls clude mmal amous of vesme. herefore, hs preves he maager from vesg he ver low opmal sum provded b he opmzao model. o remed he presece of small amous he opmal dversfed porfolo, vesors ca elmae secures wh a lower proporo of fed capal a a mmum level, ad be coe wh a porfolo heorecall resulg subopmal. hs s uforuael oe of he maor dffcules ecouered pracce b maagers' porfolos. o accou for hese praccal aspecs, s eresg o sud wheher rsk dversfcao ca be acheved b addg a purchase hreshold cosra o he model of Koo & Yamazak Smplfed b Hamza & Jasse (2000). hs amous o solvg a lear med program he case of usg he measure of meaabsolue rsk he Dsucve Programmg ad he Problem Modelg We propose hus a lear dsucve program as a ew modelg of hs problem. Usg he lear programmg, several crera ca be used for he opmum choce of a porfolo. Youg (998) proposed he orm L o measure he porfolo rsk. I s he "mma" crero for he porfolo choce. Aoher crero usg he orm L ca be used. I s he model of Koo ad Yamazak (99), whch measures he rsk hrough he absolue devao of porfolo reurs compared o s average:, where µ represes he mahemacal epecao of he radom varable. Usg he hsorc esmaor of 8 : ˆ ( ) ( r rˆ ) r r - K ˆ,...,, Ad (r r ˆ ) 8 Koo H & Yamazak H. (99), A mea Absolue Devao vesme Porfolo opmzao Model ad s applcaos o oko sock Marke, Maageme Scece. 86

6 Ieraoal Joural of Scefc Research ad Iovave echolog ISS: Vol. 2 o. 6; Jue he opmzao program of he porfolo of Koo ad Yamazak, obaed b he smplfed formulao of Hamza ad Jasse (998) 9, s wre as follows: M subec o ( ).,..., 0 0 ˆ,..., 0 ˆ r r r + ρ I he followg, he crero of absolue devao s reaed. For each secur, we mpose he followg codo: < 0, where he cosa represes he mmum perceage o ves he asse,...,. hs s a dsucve cosra. So f we roduce he dsucve cosra whch correspods o a purchase lm he opmzao program, becomes as follows M Subec o Uforuael, he opmal soluo cao be calculaed wh coveoal varas of he Smple algorhm for bouded varables Dversfcao ad Med lear programmg he program of opmzao [ ] D P s equvale o he followg lear med program 0 : M [ ] P LM 9 Koo H & Yamazak H (99), A mea Absolue Devao Porofolo opmzao Model ad s applcaos o oko sock Marke, Maageme Scece. 9 Fars HAMZA & Jaqcues Jasse Cho Opmal des Acfs Facers e Geso de Porefeulle, Hermes-Lavoser, Fars HAMZA & Jaques Jasse Cho Opmal des Acfs Facers e Geso de Porefeulle, Hermes-Lavoser, 2009 ( ).,..., 0,..., 0,..., 0 ˆ,..., 0 ˆ r r r < + ρ [ ] P D

7 Ieraoal Joural of Scefc Research ad Iovave echolog ISS: Vol. 2 o. 6; Jue 205 ( r rˆ ) rˆ q q q 0 ou 0, u 0 + ρ + u + v 0 0 0, v 0 0,...,,...,. Cosequel, solvg he dsucve program [ PD ] amous o solvg he med lear program [ LM ] P. 3. Mehodolog 3.. Hpoheses From he above-meoed lerar revew, we adop he followg hpoheses: Hpohess : he rasaco coss somemes ca be ver hgh compared wh he epeced profabl of he vesor, especall for he large porfolos. Hpohess 2: he maager of a porfolo cao amore fulfll he full dversfcao, especall whe he opmal soluos clude ver small amous of vesme. Hpohess 3: he lear med program appled o he mea- absolue-devao model (Koo & Yamazak) smplfed b Hamza & Jasse esure he reduco of he sze of he porfolo o be opmzed ad order o avod he rasaco coss of he facal asses Sample Our sud sample cludes daa correspodg o he weekl hsorcal reurs of 74 facal secures comg from he Casablaca sock marke ( makes he oal of he marke). he sock marke daa cover he perod of Jauar 2, 203- Jue 4, 204. For he smulao of he program, a epeced rae of reur of 0.05% per u of me (oe week) ad a acceped mmum hreshold of vesme of 5% for all,..., 74 are plaed. Our work s mal acheved b usg MALAB sofware. he ormal ess are performed o Evews7. he dsrbuo of he weekl reurs of our porfolo moves far from he dsrbuo of he ormal law (see app ). 4. Dscusso he ables below provde us wh he opmal composos of he porfolos obaed b he compared models. he opmal values of he obecve fuco ad he me of calculao are referred o he boom of he ables. he frs able provdes us wh he opmal composo of he porfolo obaed b he mea lear model - absolue devao gve b he program (see he heorecal par). Our opmal porfolo cosss of 3 shares dsrbued accordg o he dffere secors. 88

8 Ieraoal Joural of Scefc Research ad Iovave echolog ISS: Vol. 2 o. 6; Jue 205 Share 89 Proporo SOKVIS ORD AFRIQUE 0, CERALE LAIIERE 0, BMCE BAK 0, CIMES DU MAROC 0, REBAB COMPAY 0, ASLIF 0, DIAC SALAF 0, HPS 0, UIMER 0, OULMES 0, DISWAY 0, AGMA LAHLOU-AZI 0, S.M MOEIQUE 0, EAKL 0, ZELLIDJA S.A 0, EQDOM 0, DELARE LEVIVIER MAROC 0, EXAS MAROC 0, MAROC LEASIG 0,0809 LESIEUR CRISAL 0, AFRIC IDUSRIES SA 0, REALIS. MECAIQUES 0, MIIERE OUISSI 0, COLORADO 0, SROC IDUSRIE 0, AFRIQUIA GAZ 0, FERIMA 0, CDM 0, PROMOPHARM S.A. 0, IVOLYS 0, AUO EJMA 0, Obecve Fuco 0, me calculao (m),32 able - he porfolo composo ams a a weekl reur of 0.05% accordg o he ( P ) program. he secod able gves us he opmal composo of he porfolo obaed b he mea lear model - absolue devao elmag all he shares, whch have a fraco of he order less ha , from he al porfolo (ha are: AFRIQUIA GAS FERIMA, CDM, PROMOPHARM A.C., IVOLYS, AUO EJMA). he opmzao leads o a opmal porfolo cossg of 28 secures. I hs able relaed o he cosra ha cacels he secures whose raes are below , we oe ha he secures CM COLORADO, EQDOM ad LYDEC, whch dd o ake par he opmal porfolo resulg from he frs opmzao program, have come ou wh lower perceages. So he program s sll rug ad does o solve he problem.

9 Ieraoal Joural of Scefc Research ad Iovave echolog ISS: Vol. 2 o. 6; Jue 205 Share Proporo SOKVIS ORD AFRIQUE 0, CERALE LAIIERE 0, BMCE BAK 0, REBAB COMPAY 0, DIAC SALAF 0, LESIEUR CRISAL 0, CIMES DU MAROC 0, OULMES 0, HPS 0, S.M MOEIQUE 0, AFRIC IDUSRIES SA 0, ASLIF 0, AGMA LAHLOU-AZI 0, ZELLIDJA S.A 0, UIMER 0, DISWAY 0, DELARE LEVIVIER MAROC 0, EAKL 0, EXAS MAROC 0, MAROC LEASIG 0, MIIERE OUISSI 0, CM 0, CARIER SAADA 0, SROC IDUSRIE 0, MICRODAA 0, COLORADO 0, EQDOM 0, LYDEC 0, Obecve Fuco 0,00028 me calculao (m),4 able 2- he Composo of he porfolo ams a a weekl reur of 0.05% akg o accou he cosra accordg o he ( P) program he resuls obaed b he opmzao program wh ad whou addoal cosra are drecl compared wh hose obaed b he lear med program he able below: Share Proporo UIMER 0,49897 SOKVIS ORD AFRIQUE 0,09544 LESIEUR CRISAL 0, EQDOM 0,06367 CERALE LAIIERE 0, OULMES 0, AGMA LAHLOU-AZI 0,05698 EAKL 0,

10 Ieraoal Joural of Scefc Research ad Iovave echolog ISS: Vol. 2 o. 6; Jue 205 SALAFI 0,0544 BMCE BAK 0,054 IMAR 0,0502 ASLIF 0,05006 EXAS MAROC 0,05 RISMA 0,05 CIMES DU MAROC 0,05 MAROC LEASIG 0,05 Obecve Fuco 0,00305 me calculao (m) 5,24 able 3- he Porfolo Composo ams a a weekl reur of 0.05% based o he lear med program (!) B mposg a mmum hreshold of he secures purchase of 5%, he obaed opmal porfolo cosss of 6 shares wh a rae above or equal o 5%. o solve hs lear med problem, we used a algorhm of separao ad evaluao (SE) or Brach ad Boud (B & B). 5. Cocluso he problem of reducg he sze of he facal asses porfolo ha eed o be opmzed order o avod rasaco epeses requres more eres from he researchers. hus, has become a maor cocer for he coures ha are aware of he mpora role of he facal markes he growh of her ecoomes. Lke mos developg coures, Morocco has made radcal chages cocerg he orgazao ad fucog of he facal markes. he reforms adoped sce 993 have bee srumeal he developme of s facal marke, bu hs has o led o a sgfca mproveme he maageme of he porfolos of facal asses; parcularl he prcple of dversfcao. hus, a raoal vesor wh our facal marke hopes o ow he "rgh" porfolo; ha s o sa, a porfolo ha offers solel a o-dversfable rsk ha he wll be pad for. he vesor mus elmae he dversfable rsk b opmzg a possble wa o buld a well-dversfed porfolo. everheless, he wll be lef wh heav epeses due o hs sraeg of dversfcao. he advaage of he used mehod our sock marke s ha esures a opmal porfolo of a lower umber of shares b elmag all secures wh a rae of vesed capal less ha a mmum perceage of vesme. App App : Graphc : Dsrbuo of he weekl reurs of he suded sample og ha hs algorhm s a geerc mehod o solve he problem of opmzao, parcularl he combaoral or dscree o-cove opmzao. I he mehods of separao ad evaluao, he separao esures he obame of a geerc mehod o localze all he opmal soluos; whereas, he evaluao avods he ssemac eumerao of all he soluos. 9

11 Ieraoal Joural of Scefc Research ad Iovave echolog ISS: Vol. 2 o. 6; Jue 205 App 2: he code uder Malab a fucoopmsao() [S,, ab] lsread('prce.ls'); [0,] sze(s); for :, for :0-, r(,)(s(+,)-s(,)) )/S(,);, ; dsp(r); [,] sze(r); for :, rm() mea(r(:,)); ; accordg o he [ ] P program D Y(:,)/(-); Y(+:+,)0; for : for :, (,) r(,)-rm(); ; for : for : f() A(,) ; else A(,) 0; 92

12 Ieraoal Joural of Scefc Research ad Iovave echolog ISS: Vol. 2 o. 6; Jue 205 for : A(,+)(,); ; A(+,:) 0 ; for : A(+,+) rm(); for :+ f(<) aeq(,)0; else aeq(,); beq; for : b(,)0; b(+,) ; Aa-A; bb-b; m ma(sze(y)); L zeros(m,); [,fval] LIPROG(Y,Aa,bb,aeq,beq,L); for : ()(+); dfd(<0^-4); d2fd(>0.0065); (d)0; lswre('res_sage.ls',[' um2cell(')]); dfd(<0.0065); d3fd((d)>0); d2fd(>0.0065); lswre('el_sage.ls',[(d(d3))' um2cell((d(d3))')]); lswre('d.ls',d(d3)+ ) lswre('value_obecve_c_y.ls',fval) App 3: he code uder Malab accordg o he program (P) wh a addoal cosra fuco opmsao2() [S,, ab] lsread('prce.ls'); dlsread('d.ls'); [0,] sze(s); for :, 93

13 Ieraoal Joural of Scefc Research ad Iovave echolog ISS: Vol. 2 o. 6; Jue 205 for :0-, r(,)(s(+,)-s(,))/s(,); %r()ds/s0, ; dsp(r); [,] sze(r); %rm s he average of reur for :, rm() mea(r(:,)); ; %dsp(rm); for : Y(,)/(-); for +:+ Y(,)0; for : for :, (,) r(,)-rm(); ; for : for : f() A(,) ; else A(,) 0; for : A(,+)(,); ; A(+,:) 0 ; for : A(+,+) rm(); for :+ f(<) aeq(,)0; else aeq(,); [k,h]sze(d); for :h aeq(+,:)0; 94

14 Ieraoal Joural of Scefc Research ad Iovave echolog ISS: Vol. 2 o. 6; Jue 205 aeq(+,d()); beq(); beq(2:h+)0; %I ee(); %for :, % A((+3+),:) I(,:); %; %dsp(3); %U oes(,); %dsp(u); for : b(,)0; b(+,) ; [k,h]sze(a); %for :, % b(+3+,) 0.0; %; %dsp(b); % ma(sze()); %L zeros(,); %U 0^0*oes(,); %dsp(sze()); %dsp(sze(a)); %dsp(sze(b)); %XLP(,A,b) Aa-A; bb-b; % Lmes f?reures e sup?reures de : m ma(sze(y)); %L zeros(,); L zeros(m,); %dsp(l); U oes(m,); % % Opmsao : [,fval] LIPROG(Y,Aa,bb,aeq,beq,L); for : ()(+); dfd(<0^-4); d2fd(>0.0065); (d)0; 95

15 Ieraoal Joural of Scefc Research ad Iovave echolog ISS: Vol. 2 o. 6; Jue 205 lswre('res_sage2.ls',[' um2cell(')]); dfd(<0.0065); d3fd((d)>0); d2fd(>0.0065); lswre('el_sage2.ls',[(d(d3))' um2cell((d(d3))')]); lswre('d2.ls',d(d3)+ ) lswre('value_obecve_c_y2.ls',fval ) App 4: he code uder Malab accordg o he lear med program or Broud & Bruch. Fuco BB() %dae [S,, ab] lsread('cours.ls'); [0,] sze(s); %calculus of reur of a share for :, for :0-, r(,)(s(+,)-s(,))/s(,); dsp(r); [,] sze(r); %rm s he of average of reur for :, rm() mea(r(:,)); ; Y(:)/(-); Y(+:+4*)0; for : for :, R(,) r(,)-rm(); A(:,:)ee(); A(:,+:+)-R; A(:,++:+4*)0; A(+,:)0; A(+,+:+)rm; A(+,++:+4*)0; b(:)0; b(+) ; 0.05; Aeq(:+2*,:+4*)0; Aeq(,+:+); Aeq(2:+,+:+)ee(); Aeq(2:+,++:+2*)-ee(); Aeq(2:+,+2*+:+3*)ee(); Aeq(2+:2*+,+:+)-ee(); 96

16 Ieraoal Joural of Scefc Research ad Iovave echolog ISS: Vol. 2 o. 6; Jue 205 Aeq(2+:2*+,++:+2*)*ee(); Aeq(2+:2*+,+3*+:+4*)ee(); beq(); beq(2:2*+)0; A-A; b-b; m ma(sze(y)); L zeros(m,); for : beqbeq; Aeq(+,++)0; Aeq(++,++)0; beq(+); beq(++)-; [,v,eflag] LIPROG(Y,A,b,Aeq,beq,L); [2,v2,eflag2] LIPROG(Y,A,b,Aeq,beq,L); f(v2<v) beqbeq; 2; fvalv2; else ; fvalv; for : ()(+); for : q()(++); for : ()(+); for : q()(++); dfd(>0^-0); [(d)' um2cell((d)')] sum((d)) lswre('bb.ls',[(d)' um2cell((d)')]); lswre('value_obecve_bb.ls',fval ) 97

17 Ieraoal Joural of Scefc Research ad Iovave echolog ISS: Vol. 2 o. 6; Jue 205 Refereces AMBLER Seve, (200), ECO 6080 : Ecoome facère Rsque, dversfcao e froère effcee, Uversé du Québec ` Moréal. BREALEY Rchard, MYERS Sewar, ALLE Frakl, Prcpes de geso facère, specalzed Edo, 8 h edo Pearso, Educao. BERARD Phlppe, (Aprl 2006) La héore du porefeulle : ue roduco, Pars-Dauphe Uvers. COBRAU Broque ad GILLE Va de BERG, (2007) Geso de porefeulle, 4 h edo of Boeck. D ARVISEE Phlppe, (2008) Face Ieraoale, Duod, 2 d edo. El BOUHADI Abdelhamd, (2008), Cosruco d u porefeulle effce : Applcao emprque à parr d u échallo de valeurs coées à la Bourse des Valeurs de Casablaca Uversé Cad Aad, Marrakech. FARBER Adré, Mare-Paule LAURE, OOSERLICK Km, PIROE Hugues, (2008), L esseel de la face de marché e de la face d ereprse, 2 d edo, She colleco. JACQUILLA Berrad ad SOLIK Bruo, (986) Les marchés facers e la geso de porefeulle, Duod. HAMZA Fars & Jacques JASSE, (2009) Cho Opmal des Acfs Facers e Geso de Porefeulle, Hermes-Lavoser. HULL Joh, GODLEWESKI Chrsophe, MERLI Mame, (200), Geso des rsques e suos facères, 2 d edo. Pearso. LEUEGGER Mare-AGES, (999) Geso de porefeulle e héore des marchés facers, 2 d edo. Ecoomca. MARKOWIZ.H. (959), Porfolo Seleco: effce dversfcao of Ivessemes. Yale Uvers Press. POGUE.G.A. (970), A eeso of he Markowz Porfolo Opmzao, Maageme scece, p VIVIAI Jea Laure, (200) Geso de porefeulle, Duod, 2 d edo