L thuyet Harry Markowitz 1 M hnh HarryMarkowitz: xydngmtmhnhdanhmcut,ccnhutphixcnh lng ri ro c th c, v c lng ri ro c th cho tng khon mc. M hnh danh mc u t c bn c Harry Markowitz pht trun , Markowitz pht hin t sut sinh li mong i i vi danh mc ca cc ti sn v mt c lng ri ro mong i. Markowitz ch ra rng phng sai ca t sut sinh li l mt c lng c ngha ca ri ro danh mc vi mt tp hp nhng gi nh, v ng tm cng thc tnh ton phng sai ca danh mc. Cng thc tnh ton phng sai danh mc ny ch ra s quan trng ca vic a dng ho u t gim thiu ri ro ro tng th v cng ch ra cch a dng ho u t hiu qa. Cc gi nh nn tng: Nh u t lun mong mun ti a ho t sut sinh li t vic u t ca mnh vi mt mc ri ro cho php. Do cn c nhng gi nh nn tng Ccnhutxemmikhonutkhcnhaucidinchomts phn phi xc sut ca t sut sinh li mong i ln mt vi thi k nm gi. Cc nh u t lun ti a ha li ch mong i trong mt thi k nht nh. Cc nh u t nh gi ri ro ca danh mc da trn c s phng sai ca t sut sinh li mong i. Cc nh u t cn c trn nhng quyt nh c lp ca t sut sinh li v ri romong i, vvy ng cong hu dng ca h lmt phng trnh ca t sut sinh li mong i v phng sai (hoc lch chun) ca t sut sinh li. Vimtmc ri ro cho trc, cc nh u t a thch t sut sinh li cao hnlmttsutsinhlithp.Tngt,vimtmctsutsinhli mong i cho trc, cc nh u t li thch t ri ro hn l nhiu ri ro. Vi nhng gi nh ny, mt ti sn ring l hay danh mc ca nhng ti sn c xem nh c hiu qa: - Nu khng c nhng ti sn no khc. - Hoc nu danh mc ca nhng ti sn em li t sut sinh li mong i cao hn vicngmchoctriro,haytrirohnvimctsutsinhlimongi cng mc hoc cao hn. L thuyet Harry Markowitz 2 1. Mi quan h gia ri ro v t sut sinh li: Khi a ra mt s l chn gia hai ti sn c cng t sut sinh li, nh u t s chn ti sn c mc ri thp hn. Tuy nhin c nhiu nh u t li chp nhn ri ro c c mt t sut sinh li cao hn. Nh vy thi ri ro l s khc bit trong t sut sinh li mong i, nh u t lun yu cu mt t sut sinh li cao hn c th chp nhn mt mc ri ro cao hn. Trong khi c nhiu thi ng x khc nhau, th gi nh c bn ca chng ta cho rng hu ht cc nh u t u chuyn mt khon tin ln pht trin danh mc u tmvntchiriro,vvychngtamongimiquanhngthungiat sut sinh li mong i v ri ro mong i. c bit y cng l iu ma nhn chung chng ta lun tm kim trong tng lai. 1.1 Ri ro Rirolnhngtnhhungmtigyranhngsckhngttchodoanh nghip. N em n cc tn tht cc thit hi v mt ti chnh cho nh u t. mc cao n c th a nh u t vo con ng ph sn. Ri ro thhin tnhbt n, tnhkhng chcchn hay nhycm v nhng kt qu d tnh t c trong tng lai ca nh u t ( cc ch tiu: EBIT, doanh thu, EPS, CF,NPV, IRR, PI, ty sut sinh li) C 2 cch phn loi ri ro: Phn loi ri ro thnh ri ro kinh doanh v ri ro ti chnh Phn loi ri ro thnh ri ro h thng v ri ro khng h thng 1.2 T sut sinh li: Tsutsinhlicamtchngkhoncolngnhltngcckhonthu nhphaylcachshutrongmtthik.Nichung,nchnhlsthayi trong gi tr ca u t cng thm vi tt c thu nhp bng tin. T sut sinh li mong i: Tsutsinhlimongicamtchngkhoncolngnhltngcc khon thu nhphoc ; ca ch s hu trong mt thi k. ni chung, n chnh l s thay i trong gi tr ca u t cng thm vi tt c thu nhp bng tin. r= 00t tp p cp rt :t sut sinh li trong sut k t. pt: gi ca chng khon thi im t. L thuyet Harry Markowitz 3 Ct :lu lng tin mt nhn c ca chng khon t t0 n t1 P0 :gi ca chng khon thi im 0 Khi hai hoc nhiu chng khon kt hp vi nhau trong mt danh mc,t sut sinh licadanhmcbngvibnhqungiaquyntsutsinhlicatngchng khon ring l, 1np j jjr xr== pr : t sut sinh li ca danh mc u t jr : t sut sinh li ca cc chng khon thnh phn jx: t trng u t vo cc chng khon thnh phn V d: Tnh ton t sut sinh limong i ivi danhmc ca nhng ti sn ri ro Ti sn Ttrng (wi) Tsutsinhli mong i (%) Tsutsinhli mong i (%) A B C D 0,20 0,30 0,30 0,20 10,0 11,0 12,0 13,0 2,00 3,30 3,60 2,60 E(Rp) = 11,50 Do t sut sinh li danh mc l 11,5% Bng cch gi nh t sut sinh li c phn phi chun Markowitz o lng ri ro thng qua phng sai hay lnh chun. 1.3 Phng php c lng ri ro: lnhchunlphngphpolngrngcasphntnsovigitr trung bnh. lch chun o lng s khng chc chn ca t sut sinh li Phng sai hay lch chun ca t sut sinh li ca mt khong u t c th. ( )221.ni iir r p o== L thuyet Harry Markowitz 4 ( )21.ni iir r p o== ( )211Niir rNo== 2o : phng sai ri: gi tr t sut sinh li thc t r: gi tr t sut sinh li k vng ca nh u t pi : kh nng xy ra t sut sinh li. lch chun Tuynhinvictnhtonlchchuncaccgitrtsutsinhlithc nghim th chng ta c th ly tng bnh phng cc khong chnh lch v chia cho N. Vi N l s mu thc nghim T bng trn ta tnh: riir r (ir r )2 pi (ir r )2 pi 8,0 10,0 12,0 14,0 3,20 1,20 0,80 2,80 10,24 1,44 0,64 7,84 0,20 0,30 0,20 0,30 2,048 0,432 0,128 2,352 4,960 lch chun = 2,227% Hipphngsaivhstngquanlhaikhinimcnphichiutrckhi ni v cong thc phng sai ca t sut sinh li i vi danh mc u t 1.4 Hip phng sai: i vi hai ti sn i v j, hip phng sai ca t sut sinh li c nh ngha l: CovAB = 1( )( )ni iA A iB Bip r r r r= L thuyet Harry Markowitz 5 ngha: Hip phng sai l mt c lng hai mc khc nhau tin li gn nhau nhm to thnh mt gi tr c ngha. Mt gi tr hip phng sai dng c ngha l t sut sinh li i vi hai khon utckhuynhhngdchchuynvcngmthngsovimctrung bnh ca chng trong sut mt khong thi gian.Mt gi tr hip phng sai m ch ra t sut sinh li i vi hai khon u t c khuynh hng dch chuyn v hai hng khc nhau. lncahipphngsaiphthucvophngsaivmiquanhgia nhng chui t sut sinh li. V du: Tnh hiep phng sai cua t suat sinh li: Ngy Coca-ColaHome Depot P ng caC tc T sut sinh li P ng caC tc T sut sinh li 12-04 01-05 02-05 03-05 04-05 05-05 06-05 07-05 08-05 09-05 10-05 11-05 12-05 60,938 58,000 53,030 45,160 46,190 47,400 45,000 44,600 48,670 46,850 47,880 46,960 47,150 0,18 0,18 0,18 0,18 -4,82 -8,57 -14,50 2,28 2,62 -4,68 -0,89 9,13 -3,37 2,20 -1,55 0,40 45,688 48,200 42,500 43,100 47,100 49,290 47,240 50,370 45,950 38,370 38,230 46,650 51,010 0,04 0,04 0,04 0,04 5,50 -11,83 1,51 9,28 4,65 -4,08 6,63 -8,70 -16,50 -0,36 22,16 9,35 = -1,81= 1,47 i vi v d trn: NgyT sut sinh li [ ( )iAr -(Ar )] 1211.12AB iA A iB BiCov r r r r=((= L thuyet Harry Markowitz 6 ( )iAr ( )iBrCoca-Cola ( )iAr -(Ar ) HomeDepot ( )iBr -(Br ) [ ( )iBr -(Br )] 01-05-4,825,50-3,014,03-12,13 02-05-8,57-11,83-6,76-13,3089,86 03-05-14,501,51-12,690,04-0,54 04-052,289,284,097,8131,97 05-052,624,654,433,1814,11 06-05-4,68-4,08-2,87-5,5515,91 07-05-0,896,630,925,164,76 08-059,13-8,7010,94-10,17-111,26 09-05-3,37-16,50-1,56-17,9727,98 10-052,20-0,364,01-1,83-7,33 11-05-1,5522,160,2620,695,43 12-050,409,352,217,8817,44 =-1,81= 1,4776,21 CovAB = 76,21/12 = 6,35 Hipphngsaibnhhngbitnhbinthincahaichuitsutsinhli ring l. Bnmunchunhaclnghipphngsainyavoxemxttnh bin thin ca hai chui t sut sinh li ring l. Hstngquangiatsutsinhlica2khonutthhinmongmun chun ha ny. 1.5 H s tng quan: Hstngquanlmtkhinimnilnmiquanhcnghnghayngc hng ca t sut sinh li hai chng kon theo thi gianH s tng quan ca nhng t sut sinh li

ngha ca h s tng quan H s tng quan ny ch c th thay i trong khong t -1 n +1.j iijijCovo o =L thuyet Harry Markowitz 7 Gi tr +1 ngha l t sut sinh li i vi hai c phiu cng thay i trong mt kiu tuyn tnh xc nh hon ton.Gi tr -1 khi t sut sinh li ca mt c phiu cao hn mc trung bnh, t sut sinhlicanhngcphiukhcsthphnmctrungbnhbngmts lng ln. Tnh ton lch chun cat sutsinhli i vi Coca-Cola v Home Depot: 2005 Coca-ColaHome Depot Ngy ( )iAr -(Ar )[ ( )iAr -(Ar )]2 ( )iBr -(Br )[ ( )iBr -(Br )]2 01-05-3,019,054,0316,26 02-05-6,7645,66-13,30176,82 03-05-12,69160,970,040,00 04-054,0916,757,8161,04 05-054,4319,653,1810,13 06-05-2,878,22-5,5530,77 07-050,920,855,1626,65 08-0510,94119,74-10,17103,38 09-05-1,562,43-17,97322,83 10-054,0116,10-1,833,34 11-050,260,0720,69428,18 12-052,214,907,8862,13 Tng cng:404,38Tng cng: 1241,54 Phng sai = 404,34/12 = 33,70 lch chun = 5,81 Phng sai = 1241,54/12 = 103,46 lch chun = 10,17 Phng sai - lch chun ca DMT gm hai chng khon: u tin c l bn s c suy ngh l lch chun t sut sinh li danh mc l bnh qungiaquynlchchuncaccchngkhon.thcrakhngphinhth,cc ktqangvisuynghcabnnukhigiccahailoichngkhonnyc ( )( )108 , 099 , 5835 , 617 , 10 81 , 535 , 6= = = =j iijijCovo oL thuyet Harry Markowitz 8 tuong7 quan xc nh hon ton vi nhau. Trong bt k trng hp no khc, a dng ha ri ro s lm gim ri ro dui con s ny. tnh ton ri ro danh mc gm hai chng khon, bn cn in vo 4 hp 2 2A Ax ocov( , )A B A B AB A Bxx A B xx o o =cov( , )A B A B AB A Bxx A B xx o o =2 2B Bx o Vi:xA = t trng u t vo chng khon A xB = t trng u t vo chng khon B =2Aophng sai ty suat sinh li cua chng khoan A =2Bo phng sai ty suat sinh li cua chng khoan B cov(A,B)= hiep phng sai cua ty suat sinh li cua co phan A va B (B A ABo o ) =AB he so tng quan gia t suat sinh li A va B Nu cc chng khon di chuyn cng chiu vi nhau th h s tng quan ABl dng v do hip phng sai cng dng. Nu cc chng khon khng c lin h vi nhau th h s tng quan v hip phng sai s bng 0. nu cc chng khon di chuyn ngc hng vi nhau th h s tng quan v hip phng sai nh hn 0. Sau khi hon tt cc hp trn, n gin bn cng li cc nhp lng nhn c phng sai ca danh mc.

Phng sai cua mot danh muc gom 2 chng khoan: B A AB A B B A A px x x x o o o o oB22 2 2 2 2+ + = Phng sai ca DMT gm nhiu chng khon: Phngphptnhtonrirodanhmccthddngcmrngchodanh mc 3 hoc nhiu chng khon hn na. chng ta ch phi in vo nhiu hp hn. Khi ch c hai chng khon, chng ta c s lng bng nhau ca cc hp phng sai v cc hp hip phng sai. Khi c nhiu chng khon th s lng hip phng sai nhiu hn s lng phng sai. Kt qu l mc a dng ha ca mt danh mc phn nh ch yu c hip phng sai. 2 2 21 1 1n n np i i i j iji i ii jx x x Cov o o= = === + L thuyet Harry Markowitz 9 2. Kt hp cc c phn v danh mc u t: Vi mot nhom cac chng khoan cosan, bang cach nao chung ta xac nh c cach ket hp tot nhat cac chng khoan o vao cac danh muc au t ? Chung ta se tm nhng danh muc au t ma c d kien se tao ra li nhuan toi a vi rui ro tng ng. Vung qua trng v la cac ket hp co the co cua ty suat sinh li ky vong va o lech chuan neu ban au t vao mot danh muc cac co phan Cac iem c anh dau tren hnh bieu th nhng v tr cua cacvon au t co phieu ca the.Trong thc te, ban co kha nang la chon vao mot danh muc nhieu hn hailoaicophan.Changhan,motiemnamtrongvungquatrngvcothetieu bieu chomotdanhmucgom 40 chng khoan, 80 chng khoan,hoac co the la 80 chng khoan khac iem khac nhng vi ty trong khac. Ro rang cac ket hp nay la bat tan va tao ra hnh giong nh qua trng vCac co phieu nay co the ket hp vi nhau trong cac danh muc au t. L thuyet Harry Markowitz 10 pproAATat ca cac v tr danh muc au t co the at c nay phan anh tap hp cac c hoi au t san co cho chung ta. D nhien, chung ta se a chuong mot so v tr trong so cac v tr nay hn so vi cac v tr khac. Vi mc rui ro, hoac o lech chuan cho san,chung ta se a chuong cac v tr co ty suat sinh li mong i cao hn, vi mc li nhuan c tnh cho san, chung ta se a chuong nhng v tr co rui ro thap hn Nhaautlucnaocungmongmuontangtysuatsinhlivagiamolech chuanchonenhochautvaonhngdanhmucnamtrongngam(ng cong AB hnh tren) o la cac danh muc hieu qua Vay ng bien hieu qua: ng bien hieu qua mieu ta tap hp nhng danh muc au t co ty suat sinh li ln nhat cho moi mc o rui ro, hoac rui ro tha p nhat cho moi mc ty suat sinh li Moiiemnamtrenngbienhieuquaphananhmotdanhmucautvi cacquyensodanhmuccphanbochotngloaicophieucuacaccophieu trong tong the o doc cua ng cong hieu qua giam dan khi ban di chuyen hng len. ieu nay co ngha la rui ro cang cao ty suat sinh li cang ln Cong thc tnh toan o doc: Trong khi, ng cong hu dung cua moi nha au t rieng le ch ro s anh oi gia ty suat sinh li mong i va rui ro ma anh ta at c. L thuyet Harry Markowitz 11 Do o danh muc au t hieu qua se la s ket hp cua ng bien hieu qua va ng hu dung cua nha au t. 3. Vay n phi ri ro v cho vay: Gisnhutcthchovayvivaytinvilisutphirirorf (tng ng vi li sut tri phiu kho bc). Nu nh u t u t mt s tin ca mnh vo tri phiu kho bc (cho vay) v t phn cn li vo danh mc c phn thng S, nh u t s nhn c tt c cc kt hp t sut sinh li v ri ro nm dc trn ng thng ni lin rf v S . Nu nh u t i vay n vi li sut rfv u t s tin ny vo danh mc c phn thng S, nh u t c th ko di chui cc kh nng v pha bn phi ca S. Vay v cho vay ko di chui cc kh nng u t. Nu bn u t vo danh mc S v vay hoc cho vay ti mc li sut phi ri ro rf bn cth chn bt cimno nm trnngthngtrf n S. Lc nybn s c cmt t sut sinh li d kin cao nht (so vi bt k ri ro no nu bn ch u t vo c phn thng.) Phng trnh ngCML: (%) (%) L thuyet Harry Markowitz 12 m pf fm mf pr r r r r ro o oo o o*) ( * * * ) 1 (=+ = + = Trong : rp(%): t sut sinh li d kinca danh mc u t rf (%)

: t sut li nhun khi u t vo chng khan khng ri ro rm (%)

: t sut li nhun d kin khi u t vo danh mc th trng o p (%) : lch chun ca t sut sinh li danh mc u t o m(%) : lch chun ca t sut sinh li o: t trng vn u t vo danh mc th trng 1-o: t trng vn u t vo chngkhon khng ri ro Ta xt v d: Gi s danh mc S crs = 15%vos = 16%. Tri phiu kho bc rf = 5%,of = 0 % Trng hp cho vay: nh u t mua tri phiu khng ri ro v u t vo danh mc th trng S vi t l l1 :1 (o = 0.5) th li nhun v ri to nhn c l: rp = 0,5 * 5% + 0,5 * 15% = 10% o p = 0,5 *16% = 8% Nh vy trong trng hp ny s gim i mt na ri ro th nh u t s phi chp nhn ng thi s st gim t sut sinh li l 5%. Trnghp i vay: nh u t quyt nh i vay li sut phi ri ro 5% mt s tin tng ng vi vn t c ban uca mnh v sau u t tt c vo danh mc S (lc ny ta c o = 2 ), th li nhun v ri ro nhn c l: rp = 2 *15% - 1*5% = 25% o p= 2*16% - 1*0 = 32% Nhvynhutgiatngctsutsinhlivngthiphignh chu mt ri ro ln hn. Do vy: Ta thy ng CML (ng th trng vn)l tip tuyn vi tp hp hiu qa ca cc chng khon c ri ro. Khng c l do g m nh u t chn dnah mc T bi v vi bt k im no m nh u t c th nhn c trn ng gi tr li sut ca tri phiu chnh ph th nh u t cng c th nhn c mt im vi lch chun tng t nhng t sut sinh li cao hn ng CML, ti y nhu t c c hi u t tt nht (ti im S). CML l tp hp hiu qa ca trt c ti sn ri ro v phi ri ro. L thuyet Harry Markowitz 13 Bt chp mc ri ro , bn c th tc t sut sinh li cao nht bng mt cu trc bao gm danh mc S v vay hoc cho vay. Kt qa ny hnh thnh nn nhng g m cc nh u t ti chnh gi l nguyn l phncch(seperationprinciples)nghalccnhutphithchinhaiquyt inh ring bit. Lachndanhmccccphnttnht,trongvdchngtalcphnS. im ny c xc nh hon ton t nh gi ca nh u t v t sut sinh li phngsaivhipphngsai.Khngcnhngtnhcmcnhnvthi khng thch ri ro xem xt trong quyt nh ny. Nh u t by gi phi xc nh kt hp im S l danh mc cc ti sn c ri rovitisnphirironhthnonhncnhycmiviriro tng xng vi khu v ca tng ngi. 4. Kt lun: M hnh qan tr danh mc u t c bn ca Markowitz xc nh t sut sinh li mong i i vi mt danh mc ca nhng ti sn v mt c lng ri ro mong i,llchchuncatsutsinhlimongi.Markowitzchothytsut sinh li camt dnahmc u t l gai1tr trung bnh theo t trng ca t sut sinh limong i cho nghng ti sn ring l trong danhmc u t. lch chun ca mt danh mc u t l mt cng thc khng ch lch chun vi nhng n mc u t c th m cn ca hip phng sai ca t sut sinh li i vi tt c cc cp ti sn trong danh mc u t. Trong mt danh mc u t vi s lng ln ti sn th nhng hip phng sai ny l mt nhn t quan trng. Nhng t trng khc nhau hay mt danh mc u t vi s lng ln ti snnm gi khc nhau th hin bng s kt hp ng cong hu dng. h s tng quan gia nhng ti sn l nhn t then cht cn phi xem xt khi la chn u t bi v c th duy tr t sut sinh li t c trong khi vn gim thiu c mc ri ro ca cc danh mc u t bng cch kt hp nhng ti sn hoc nhng danh mc u t c tng quan khng nh thp hay tng quan ph nh. Vic gi nh mt s lng ln nhng ti sn v v s ng cong kt hp, ng bin hiuqa- ngcong baogm tt c cc kt hp tt nht. N xc nh nhng danh mc u t c t sut sinh li cao nht i vi mc ri ro cho trc hoc ri ro thp nht i vi mc t sut sinh li cho trc. t vic nh ra danh mc u t ti u,talcchnmtdnahmcutnmtiimtiptuyngiangbinhiu qa v ng hu dng cao nht. Bi v li ch ri ro - t sut sinh li khc nhau gia nhng nh u t, nn vic la chn danh mc u t cng s khc nhau gia nhng nh u t. T vic nh ra danh mc u t ti u, bn s la chn mt danh mc u t nm ti im tip tuyn gia ng bin hiu qu v ng ng hu dng cao nht ca bn. Cc nh u t u thch t sut sinh li cao v lch chun thp. Khng phi c adnghautlcththucktqunhmongicanhut.Thnh L thuyet Harry Markowitz 14 phn ca danh mc u t hiu qu nht ph thuc vo nh gi ca nh u t v t sut sinh li, lch chun v h s tng quan. Bn cnh khu v ca nh u t cng nh hng khng nh n vic la chn mt danh mc u t nh mong mun cah;iunybaohmcsnhinunhutmongimttsutli nhun cao hnth phi bit chp nhn mt mc ri ro cao hn v ngc li.