Tap chf KHOA HOC DHSP TPHCM Nguyin Chi Long
MO HINH DINH GIA TAI SAN TU' BAN
NGUYEN CHI LONG"
TOM T A T
Moi nhd ddu tu trong thi trudng tdi chinh, khi phdi chgn lua cdc phuomg dn ddu tu khdc nhau, nhung chung cd ciing trung binh lgi tire, thi tiiy theo mice dg e nggi rui ro (thi hien qua hdm lgi ich) md lua chgn phucmg dn it riii ro nhdt, nghia Id phuang dn cd phuang sai be nhdt. Ndi dung ndy, dugc giai thiiu qua mo hinh dinh gid tdi sdn tu bdn.
Ddy la mgt trong nhimg kit qud nin tdng cua Todn tdi chinh.
Tit khda: ham lgi ich, ly thuyet diu tu hifn dai, md hlnh dinh gia tai san tu ban.
ABSTRACT
The capital asset pricing model
In financial markets, when the investor has a choice of different portfolios that have the same average return, depending on the level of risk aversion (presented by the utility function); he chooses the least risky portfolio; le. the portfolio that has the smallest
variance. This is presented through the capital asset pricing model This is one of the fundamental results of financial mathematics.
Keywords: utility function, modem portfolio theory, capital asset pricing model.
1. Lgi nhuan va ham lgi ich /. /. Mot sd khdi ni$m, dinh nghia
Xet md hinh tai chir^ mdt chu ky vdi thdi gian giao dich T = {0,1}. Thdi dilm / = 0 la thdi dilm hifn tai, bit diu giao dich va thdi dilm t= \\i thdi dilm dao han, kit thiic giao dich. Thi trudng tai chinh gdm A^ + / tai san nin tang dk diu tu, dd la mdt tai khoan tin dung trong ngan hang (hay trai phieu khdng nii ro) B^ t = 0,1; vdi lai suit cd dinh trong mdt chu ky la r va A'' chiing khoan
{s;},i = I,2,...,N;t = 0,I.
Ddi vdi tai khoan tin dung B„ gia thilt BQ = 1 don vj tiln tf giii vao ngan hang tai thdi diem / = 0 va se cd dugc Bj = l +r dan vi tiln tf khi / = 1.
Gia ciia N chiing khoan tai thdi dilm t = 0, Sl, S^,..., S^ thi dugc xac dinh, nhung gia chiing khoan tai thdi dilm r = 1 lai phu thudc vao mdt trong k kich ban tai chinh o)i,i=l,...,K thudc
n: = {(Dl, 0)2, ..., 0)^}
Gia Sli sir xuit hifn ciia mdi kich ban c^j e Dcd xac suit P((Di) > 0, / = 1, ...,K.
Ggi F = P(Q) la tap hgp tit ca cac tap con ciia Q thi F la trudng thdng tin ldn nhit ciia
TS, Trudng Dgi hoc Su- phgm TPHCM
thj trudng tai chinh dang x^t. Liic dd S[, i= 1, ..., N la cac biln nglu nhien xac dinh tren (Q, F, P) vi S\ (co) \i gia chiing khoan thii / tai thdi dilm / = 1 khi kich bin coeQ.
xuit hifn.
* Mgt phuang an diu tu (vilt tit PA) la mgt cap (x, tf) trong dd x \i tdng so tiln diu tiJ ban dau va ^ la danh myc chiing khodn diu tu, nd la v^cta gdm iV thanh phin ^:
= i(f>', ..., t/f) vdi ^ la sd dan vf c6 philu ciia chiing khoan thii / dugc mua tai thdi dilm r = 0. So tiln cdn lai sau khi mua N chiing khoan
se dugc giii vao tai khoan tin dyng (hay mua trai phieu khdng nii ro).
* Qua trinh gia cua PA (x, (ft) \i cap (VQ (X, (j)); V, (x, (j>)) Trong dd VQ (X, <l>)=x\i V, (x, (f>) \i biln nglu nhien
v,(x,(i>) = (fPBi + Z^'s;
Ggi R" la lgi tire ciia chiing khoan tiiii / (i = 1,...,N):
S'o
va i?° la lgi tiic ciia tai khoan tin dyng, day la hing sd xac dinh duang r:
Bo
* Qua trinh ldi G (x, (f>) ciia PA (x, 4>) \i biln nglu nhien G(x, (j>) = tfr + f^^'AS', vdi AS\- = 5,' - S',
1=1
vi khi bilu diln qua trinh ldi qua lgi tiic thi G(x,t/>)= ^'B'R' + f^^'S'R',
* Trong trudng hgp mgi hang hda ti-ong thj ti-udng phai chilt khiu tiii qua ti-inh gia chiing khodn da chilt khiu la
SQ = S, va -^i' = — sl; llic dd qua ti-inh gia da chilt khiu ciia PA (x, (j)) V, (x, ^)=xviV,ix,</>) = (fP+'^(/,' Si, va qua ti-inh ldi da chilt khiu la:
G (x, (|)) = J ] ^ ' A^', vdi AS'= Si - S-
Tap chi KHOA HQC DHSP TPHCM Nguyin Chi Long
* Tu cac khai nifm tren ta cd :
Vl (x, (|)) = Vo (x, (j)) + G (X, (})) (1) V, = 4 " - V t ; ( t = 0;l)va V, (x,(|))= To (x,(l))+ G (x, ())) (2)
* Thi trudng tai chinh la Idnh mgnh, nlu trong thi trudng khdng t^n tai PA (x, ^) nao thda man ca 3 dilu kifn sau:
i\)x = Vo(x,(l>)=0,
(2) Vi(x, (^)>0 (ho$c G ix, ^) >0),
i3)3(D€Q: Vi(x, (t>)(G)) > 0 (hoac G (x, (f>)((D) > 0).
* Mdt do do xac suit Q tren O dugc ggi la d^ do xdc suat trung hoa rdi ro nlu (1) Q^((D) > 0, Vco € n (Mdi kich ban xay ra vdi xac suit duang) va
(2) EQ [AS' J = 0 (Ky vgng ciia sd gia chiing khoan da chilt khiu liy theo do do Q thi bang 0).
* Mdt quyin tdi chinh (hay phdi sinh) la mdt biln nglu nhien X xac dinh tren khdng gian xac dinh (Q F,P) bilu diln mpt thu hogch tai thdi dilm dao ban t=\.
* Cho JTia mdt quyen tdi chinh. Mdt phuang an diu tu (x, tf)) dugc ggi \iphuong dn ddp umg (a replicating strategy) hay mgt bdo hp (hedge) cho X nlu Vj (x, (f>) = Xtai thdi diem/= 1.
* Mdt quyin tai chfnh X dugc goi la dgt duoc (attainable) hay mua bdn die^c (marketable) nlu cd mdt phuang an diu tu (x, cf)) bao hd cho X.
* Thi trudng tai chinh la ddy dU nlu mgi quyin tai chinh X dhi cd thi tim dugc mdt phuang an (x, (j)) bao hd cho X. Md hinh tai chinh khdng cd tinh chit nay ggi la md hinh tai chinh khdng day du.
2.2. Nguyen ly mpt gid trong thi trucmg tdi chinh dhy dd
Trong [4] va [2], chiing tdi da gidi thifu va chiing minh nguyen ly: Thi trudng tdi chinh Id lanh mgnh khi vd chi khi ton tgi mpt dp do xdc sudt trung hda rui ro.
v a nguyen ly: Trong thi tru&ng tdi chinh lanh mgnh thi thi trudng tdi chinh Id ddy dd khi vd chi khi ton tgi duy nhat mpt dp do xdc sudt trung hoa rui ro.
^ Ddi vdi nha diu Ui (vilt tit: NDT) tai chinh,vin dl quan tam chinh la: Cdch ndo Id tdi uu de ddu tu vdo thi trudng tdi chinh?
Ldi dap ciia cau hdi nay phy thugc vao md hinh tai chinh nao dang xet va chgn lua phuang an diu tu nao? Tfnh tdi uu dugc hiiu chinh xac nhu thi nao? Hay cy thi han la, xac dinh gia tri doi vdi moi each bilu diln phuang an diu tu nhu thi nao? Gia tri nay trong thi trudng tai chinh thudng bi chi phdi bdi ba dac trung sau:
1. NDT thich thu hogch cao hon la thu hogch thdp hon ddi vdi mpt phuong dn dau tu.
Dac trung nay la hiln nhi6n. Tuy nhien, trong thyc tl d thj trudng tai chinh, lgi ich thu dugc tu mgt phuang an diu tu cd tinh nglu nhien; ching han, phuang an dau tu 1 cd thi dat dugc thu ho^ch cao khi trang thai tai chfnh nay xay ra, nhung phuang an diu tu 2 lai dat dugc thu hoach cao khi trang thai tai chinh khac xay ra. Do dd, sg khdng cd y nghia khi so sdnh hai phuang an tren cung mgt tr^ng thai, ma phai x^t chung tren toan bg cac tr^g thai cd thi xay ra d thj trudng tai chinh, nghia la xet k^' vgng ciia nd, do dd dac trung thii hai la:
2. NDT xit gid tri trung binh hay ky vgng cda timg phuong dn ddu tu.
3. NDT thudng cd tdm ly e nggi rui ro.
D I lam rd dac trung nay ta x^t Vidgl:
Gia Sli NDT dugc mdi chgn mgt trong hai phuang an 1 vd 2, tuomg iing vdi thu hoach X^ \i X-^. Neu NDT chgn phuang an 1 sg thu dugc 100 trif u dong; cdn nlu chgn phuang an 2, thi phai tuan theo quy tic may nii sau: Nlu tung ddng xu (gdm 2 mat, mgt mat cd hinh Quoc huy ma ta ghi la H va mgt mat chi gia Trj dhn% xu ma ta ghi la T) ma mat H xuit hifn thi NDT thu dugc 200 trifu dong, cdn nlu m^t T xuat hifn thi NDT se khdng thu dugc ddng nao. Thdng thudng, nlu NDT khdng phai la ty phii, thi cd tam ly chgn phuang an 1 dl thu hoach chic chSn 100 ti-ifu ddng hem la chgn phuang an 2 cd thi xay ra tinh trang tring tay, nghia la NDT cd tam ly e ngai nii ro, mac du thu hoach trung binh ciia hai phuang dn Id nhu nhau:
Vi X, =100 la tit dinh va ky vgng ciia nd, £[A',] = 100. cdn ddi vdi X^ phu thudc nglu nhien vao T hoac H; X^il) = 0; ^'^(H) = 200; do dd nlu danh gia tiiu hoach theo ky vgng thi E[X.^ ] = i .0 + i .200 = 100 = E[X^ ].
Khai nifm e nggi rui ro thudng dugc sii dyng trong md hinh thdng qua cdc hdm Igri ich (utility functions).
Hdm lgi ich cho ta each do ludng sy chgn lya ciia NDT phy tiiudc vdo tdng von hifn cd va miic dg e ngai nii ro, ma NDT mong muon la d^t dugc tdng tai san vl sau ldn han. Do dd, ham lgi ich la hat nhan ciia ly thuylt diu tu tdi uu hifn dai.
Dinh nghia 1.
Mgt hdm (/:/?* x Q -> /? du(;rc ggi Id hdm lai ich niu nd thda man hai diiu kien sau:
/. Cd dinh <y e Q, thi hdm U(x, o)) Id tdng nggt theo biin x, nghia Id dgo hdm theo biin x ciia Uld U'(x, o)) > 0, vdi mgix > 0, vd
2. Cd dinh a)eQ, thi hdm U(x, co) Id ldm nggt theo biin x, nghia Id U(Ax + il-A,)y;co)> ?iV(x;(o) + i\-A)U(y;ct))
Hay tuang duang vdi U' '(x, co) < 0. vdi mgi x> 0.
Tap chf KHOA HOC DHSP TPHCM Nguyen Chi Long
D I dan gian each bilu diln, ta thudng viet ham lgi ich dang hifn theo bien tdng tai san X, U(x, (o) = U(X(Q})) = U(x), vi ngam hieu nd cdn phy thugc vao trang thdi co.
Bay gid ta xet mdt biln nglu nhien A'bieu diln thu hoach ciia NDT. Cd dinh hdm lgi ich U. Ta se do ludng thu hoach ciia NDT qua ky vgng
E[UiX)] = J^Pi(D,)UiXio),))
Su bilu diln thu hoach nay bao hdm ba ddc trung viia neu tren: Dac tnmg thii nhit phdn dnh qua hdm lgi ich thi tdng ng^t, ddc trung thii hai phan anh qua gid tri trung binh, cdn ddc trung thii ba, tinh e ngai nii ro, phan anh qua tinh ldm ngat ciia ham lgi ich.
Vidu 2:
•
Gid Sli NDT dang cd tdng tdi sdn la 5 trif u ddng vd thi trudng chi cd mdt cdch diu tu la mua mdt loai co philu: SQ=5 (trieu). Ciing gid sii, tai thdi dilm ddo han / = I, mdt trong hai kich ban trong thi trudng cd thi xdy ra gidng nhu vifc tung ddng xu hai mat HviT: Q := {H, T] vdi xdc suit ?(//) = PiT) = 0,5. Nlu kich ban H xdy ra (tinh hinh kinh tl phdt triln tdt) thi gid chiing khodn tang: S^iH) = 9 (trifu), nghia Id tang them 4 trifu; cdn nlu kich bdn T xdy ra (tinh hinh kinh tl khd khan) thi gid chiing khodn gidm: S^iT) = 1 (trifu), nghia la giam 4 trifu. (Trudng hgp nay cdn dugc ggi la trd chai cdng bang vi ky vgng lgi nhuan la
E[G] = 0,5. 4 + 0,5.(- 4)) = 0).
Xet ham lgi ich: U(x) = V^. Ta thii tim hiiu, tren quan dilm dap img nguyen ly cyc dai ky vgng hdm lgi ich, NDT se chgn phuang an diu tu hay khdng chgn?
NIU NDT tu chdi phuang an tren, giu nguyen 5 trieu ddng liic diu, thi din thdi dilm ddo han t = I,s6 tiln vin cdn nguyen 5 trifu; ddi vdi ham lgi ich thi tren: U(x) = U(5) = V5 (hing so) nen ky vgng cua nd EIV(5)J = U(5) = S= 2,24.
NIU NDT chgn phuang dn diu tu thi ky vgng ciia ham lgi ich E[U(x)] = P(H). U(x(H)) + P(T). U(x(T)) = 0,5. S + 0,5. VT = 2.
Vi ky vgng hdm lgi ich khi tir chdi phuang dn diu tu thi ldn hem ky vgng ham lgi fch khi chgn phuang dn (2,24 > 2), nen NDT se tu chdi phuang an.
Mdt cdch tdng c^udt, NDT e ngai nii ro thudng tii chdi trd chai cdng bdng vi ky vgng lgi tiic la 0%. Neu ky vgng lgi tiic ldn hom 0%, NDT cd thi chgn hay khdng chgn phuang dn ddu tu phy thudc vdo hdm lgi ich vd tdng vdn ban dau. Chang ban, neu xac suit xdy ra ciia kich bdn H, P(H) = 75% thay vi P(H) = 50%, thi ky vgng lgi ich la
EfU(x)J = 0,75. V9 + 0,25. VT = 2,5 > 2,24 nen NDT se chgn phuang dn diu tu.
Su dyng kit qua tren, tii vifc tim phuang an diu tu tdi uu trong thi trudng tdi
t r
chinh, chuyen sang tim phuang dn (x,^) sao cho E[UiViix,^))] dat gid tri tdi uu. Bdi
todn ndy dugc ggi Id bdi todn ddu tu tSi uu. Gid trj t6i uu dl nhign phy tiiuOc vdo tong vdn diu tu ban diu x. Khi v6n diu tu ban diu x cdng^ ldn, tiii ky vgng tiiu ho^h cdng cao, do dd ta xem von diu tu ban dau nhu m^t tiiam s6 ciia bdi todn,
Dinh nghia 2.
Mgt phuang dn ddu tu (x,^^') du(?c ggi Id mgt nghiem cua bdi todn ddu tu tdi uu, vdi vdn ddu tu ban dau IdV^^x vd hdm lgi ich Uniu
E[UiV, ix,tl>'))] = Max^,^,,E[UiV, ix, m Mfnh de 1.
Niu bdi todn dau tu tdi uu trong thf trudmg tdi chinh dang xet cd mgt nghiem, thi md hinh tdi chinh ndy Id lanh mgnh.
Chimg minh:
Ta cin chiing minh ring, nlu thj tixrdng tdi chinh khdng lanh m?nh tiii bdi todn dau tu toi uu vd nghif m.
Gia Sli till tiardng tai chinh la khdng lanh mgnh, nghia Id t^n t^ mgt phuong dn cd dg chenh If ch tiii gid (0, y/). D6i vdi m6i phuong dn diu tu (x,^) chiing ta phdi cd
F, ix, if, + y/)ico) = V, ix, ^)iQ)) + V, (0, i^)ico)
ti-ong dd: (x,^ +1^) Id phuang dn diu tu tong cua hai phuong dn (x,^) vd (0,y/), nghia Id phuong dn diu tu mua ^' + y/' dem vj cd phieu chiing khoan 5' Theo dinh nghia ciia dg chenh If ch thi gid, phuong dn ndy chi cin dau tu v6n ban dau Id x va bat ding thiic ti-en sg thda ngdt vdi it nhit mgt kjch ban <y E Q, do dd vdi mdi hdm lgi ich C/ta cd:
E[UiV,ix,^ + y/))]>E[UiV,ix,m
Dilu ndy chi ra ring, khi thj trudng tdi chinh khdng lanh m?mh, thi doi vdi moi phuang dn dau tu (x, ^), deu cd mgt phuang dn diu tu khdc, cd ciing s6 von dau tu ban diu vdi phuong dn (x,^) nhung thu hogch trung binh lgi cao hon. V|y bdi todn dau tu tdi uu khdng cd nghif m. Do dd, bo dl dd dugc chiing minh.D
Theo nguyen ly cdn bdn dinh gid phdi sinh, thi tinh chit Idnh mgnh cua th) tnrdng tdi chinh tuong duong vdi sy ton tgi mgt dg do xdc suit trung hda nii ro. M$t dg do xdc suit trung hda nii ro nhu vgy dugc tinh qua nghif m cua bdi todn dau tu toi uu qua mf nh de sau:
M^nhdi2.
Ggi ix,f) Id mgt nghiem cua bdi todn ddu tu tdi uu v&i tdng vdn ddu tu ban ddu la X vd hdm Igri ich U, thi dg do Q xdc dinh ben
^.y_P{(o)U\V,ix,tlt')ia>)) E[U\V,ix,f))\
Id mgt dg do xdc sudt trung hda rui ro. Trong dd U'ix) Id dgo hdm cda Utheo x.
Tap chi KHOA HQC DHSP TPHCM Nguyin Chi Long
Chimg minh:
Vi Qico) > 0 vdi mgi coeCl vi
tf M E[U\V^ix,<^ ))]
1
r--j;^Pi(D,)u\v,[x,ti>'\(o,))
E[U\V,ix,<l>'))]t
= 5 E[U\VAx,(l>'))]
E[U\V,ix,f))] ^ ^ '^ '^ ^^-i
= 1
Nen Q Id mdt dg do xdc suit xdc djnh trgn Q. Ta cin chiing minh Q thda them dieu kifn sau: EQ[AS^] = 0.
Do tinh chit ciia ky vgng, hdm hgp
^H^E[UiV,ix,m
vdi ^eR^ Id hdm khd vi vd dgt cyc trj tgi f Do dd, dao ham rigng ciia hdm ndy trift tieu tgi f
^E[UiV,ix,mU = 0 (3) Mat khac, tii (1) va (2), ta cd
V,ix,(l>) = B,V,ix,^) = B,ix + (^'AS'+... + ^''AS''), dodd
E[UiV,ix,^))] = E[UiB,ix + t^'AS' +... + (^"AS"))]
va tii (3), suy ra hf phuang trinh sau: Vdi J = 1,2,..., A^
B,Y,Pi.o)i'P\B,ix+ (/>''AS' +... + (zJ'^M'')(ty,.))AS^(ey,) = 0
M
Dodd
Y^Pi(D^'P\B,ix + (l>*'AS'+...-\-(l>*^AS'')ia),))AS\a),) = Q
i=\
Hay j;^Pico,JJ\V,ix,f)i(D,))AS\(D,) = Q (4) (vi 5, =l + r > 0 ) .
Suyra
E^[/^^]^Y.Q^(D,)AS\o),)
1=1
^ ^Pico,)U\V,ix,<i>')ico,))^^
tr E[U\V,ix,<f>'))]
^ ^7-t/'(^,)t/'(F,(x,/)(^,))Ai^(^,)
E[U\V,ix,tt> ))]t= 0 (do (4)). D Dg do xdc suit trung hda nii ro dugc xdc djnh trong mf nh dk tren cd the dung dk
tinh gid quyen tdi chinh. Do dd hai van dl cot ldi Id tim phucmg dn dau tu tdi uu vd djnh gid quyin tdi chinh lien hf chdc chg vdi nhau.
Trong thyc tl, vifc gidi hf phuang trinh trong (4) dk tim phuang dn diu tu thdng qua ^',i = l,...,N khdng hi dom gidn. Mgt ky thuat dl gidi bdi todn Id dya vdo dg do xdc suat trung hda nii ro vd phuang phdp nhan tii Lagrange; y tudng ciia phuang phap nay la phan tich bai toan dang xet thanh hai bai toan con theo hai budc sau:
Budc 1. Xdc dinh cue dgi F, ciia hdm V h-> E[UiV)] tren tgp hgp chdp nhdn duac cdc biin ngau nhiin V
Budc 2: Tim mdt phuang dn ddu tu md nd cd gid tri tgi thdi dii t = I, bdng gid tri cue dgi F, duac xdc dinh d budc I.
Phuang an diu tu tim dugc d budc 2, chinh la phuang dn tdi uu. Bdi todn con d budc 2 chinh Id bai todn tim phuang dn bdo hd, md nd tuomg duang vdi vifc gidi hf phuang trinh tuyln tinh.
Trudc tien ta xet md hinh tdi chfnh diy dii, nghia la ti-ong md hinh chi tdn tai mgt do do xdc suit gdc P vd mdt dg do xdc suit trung hda nii ro Q.
Dinh nghia 3.
Tdng tdi sdn dgt dugrc tir vdn ban ddu x> 0 dugrc dinh nghia Id tap W^:=\w^R':E,[Uv] = x\
KYiiW eiv^ thi cd mgt phuang an diu tu (x,^) sao cho F,(x,^) = W Bdi todn con d budc 1 chinh Id bdi todn tdi uu
Tim circ dgi E[UiW)]
V&i rdng bupc W eW^
Dung phucmg phdp nhan tii Lagrange, vdi hdm Lagrange
LiW, X) := E[UiW)] - XiE^l^W] - x) (5)
^ 1
T^p chi KHOA HOC DHSP TPHCM Nguyen Chi Long
MOt nghif m ciia bdi todn tdi uu cd rdng buOc tren Id nghif m ciia hf thiic cd dugc tir dgo hdm rieng ciia hdm Lagrange theo cdc biln W, s Wico^) vd X bing 0. Trong bilu thiic djnh nghia cua ham Lagrange (5), ta phdi tinh k;^ vgng theo hai do do khac nhau la P vd Q\ de tifn vifc tinh todn, ta djnh nghia mgt biln nglu nhien mdi
i W : = ^ (6) Pio)
va ggi la mgt dg gid trgng thdi.
Llic ndy hdm Lagrange cd thi vilt
LiW, A) = j ; Pico, )[UiWi0,)) - A(I(fi),) ^ Wioj,) - X)]
Dgo ham rieng cua hdm Lagrange theo cdc biln W, bing 0, cho ta
U'iWio>,)) = A ^ il)
B\
Ket hgp vdi do do xdc suit Q xdc djnh trong mf nh dk 2 thi
A = E[B^U'iW)] (8) Vi dgo ham U'(x) ciia hdm lgi ich Id tdng ng^t, do dd tdn tai hdm ngugc I(x) ciia
U'(x) sao cho U'(I(x)) =x^ I(U'(x)). do do tir (7) suy ra
Wi0) = lixi^) (9) Phucmg trinh tren cho ta nghif m ciia bdi todn tdi uu cd rdng budc khi ta bilt chinh
xac gid trj cua X,
Cdng thiic (9) khdng ^iiip ta tinh dugc X vi nd bilu diln thdng qua biln chua biet W, tiiy nhien ta Igi bilt rdng JTphdi thda man dilu kifn
EQ[^W] = X (10)
Thay PT trong (9) vdo (10), ta dugc
U^I^^h'^-^ (11) Gidi phucmg trinh (11) ta tim dugc X, rdi thay vdo (9) ta tim dugc nghif m ciia
bdi todn tdi uu cd rdng bugc. Trong trudng hgp hdm lgi ich U(x) trong djnh nghia 1. cd them tinh chat
(3) lim.^o ^' (^) = +«> vd lim,^^ [/• (x) = 0
thi nghifm X ciia phuomg trinh (11) ludn tdn tgi vd duy nhit, vi liic dd hdm hiX):=E[--IiX—)] Id hdm gidm ngdt, lien tyc vd thda mdn dilu kifn lim^^o hiX) = +00; lim^_,,„ hiX) = 0.
Trong tiudng hgp md hinh tdi chinh khdng diy dii, cd qud ldm Id huu han do do xdc suit tiimg hda nii ro Q„ i = 1,2,...,/ vd mOt quyin tdi chinhXld dgt dugc nlu vd chi neu ky vgng EQ[—X] cd cung mOt gid trj d6i vdi mgi dO do xdc suit trung hda nii ro
B\
Q = Q,,i = 1,2,...,/; do do ta cd thi t6ng qudt hda djnh nghia 3.
Dinh nghia 4.
Tgp hgp tdng thu hogch dgt dugrc tir vdn ddu tu ban ddu x > 0 trong thi tru&ng tdi chinh, cd thi khdng ddy dii, dugtc dinh nghia Id tgp:
W.:= WeR':EQ[jW] = x, VQ = Qr,i = \,...,l\
Bdi todn tdi uu ti-gn cd thi vilt lai nhu Id bdi todn toi uu vdi hOru hgn rdng budc:
nm cue dgi E[UiW)]
V&i rdng bupc EQ^ [^W] = x, v&i i =1, 2,..., I.
' Bi
vd hdm Lagrange tucmg iing
LiW, X) := E[UiW)] - 5] ^, iE[^ W]-x)
;=1 By
trongddZ,:=-^ vd A :=(;!„...,/l,) Id nghif m ciia bdi todn diu tu tdi uud budc 1
Wio) = Ii±X,^)
Dk xdc djnh nhan tii Lagrange X ta giai hf gom / phuang ti-mh E[LJi^^'^-'^^'^')] = x
BJ
2. Phan tfch trung binh phirotig sal ciia phvcmg dn dau tu*
Khi NDT phai chgn mgt trong hai phuang an diu tu ma chiing cd ciing trung bmh lgi tiic, thi NDT se chgn phuang dn ndo cd phuang sai be horn, nghia Id it nii ro han.
Vgy NDT se giai bai todn trung binh phuang sai sau:
Bdi todn 1.
Tim cue tiiu Var[R]
Tap chf KHOA HOC DHSP TPHCM Nguyin Chi Long
V&i rdng bupc E[R] = p
trong dd R Id lgi tiic ciia phuang an diu tu dugc xdc dinh bdi:
Jg^i?(x,^):=^(^'^>-^°(^'^>
V,ix,(f>)
Khdi m^m phi rui ro irisk premium), ky hifu R-r, Id khdi nifm quan trgng trong lanh vuc diu tu, dugc xac djnh qua bo dk sau:
Bddil.
Phi riii ro ciia phuang dn ddu tu cd lgi tiirc R, trong thi tru&ng tdi chinh md ldi sudt ciia tdi khodn tin dung cd dinh r, duac xdc dinh b&i
R-r = -CoviR,L) (12) Trong do L Id mat do gid trgng thdi vd Gov, viit tdc ciia Covarian, chi hiep
phuang sai.
Chiimg minh:
r r
Xet mdt do do xac suat trung hda nii ro Q cd djnh trong thj trudng tai chinh, tii khai nifm gia chiing khodn chilt khiu, ta cd:
S, - B,Sn
Sy -SQ - •'i"o
Bl
_il + R')S',-il + R')S- l + R'
Liy ky vgng hai vl theo do do xdc suit Q, thi vl trdi bdng 0, nen
= ^^;^E,[R'-R']
l + R'
Suyra EQ[R'] = R°=r
Do do, CoyaiianiR',L) = CoviR',L) = E[R'L]-E[R']E[L]
= EQ[R']-Em
= r-R'
trong dd R' := E[R'] vi chii y ring do djnh nghia ciia mat do gid ti-gng thai L, thi EfLJ = 1.
Mat khdc tii djnh nghia ciia V^ vd R, thi
V ^ V
Dodd, R-r = -CoviR,L)
trong dd, R := E[R]. Vgy ta cd dilu cin chiing minh.D BSdi2.
Cho a, b Id hai sd th{rc vd b^O. Gid siir quyen tdi chinh a + b L dugrc sinh bdi mdt phuomg dn ddu tu bdo hd ndo do md nd cd Igri tuc R, thi phi rui ro ciia phuang dn ddu tu bdt ky cd Ign tire R, ty le v&i phi rui ro ciia phuong dn dau tu cd Igri tire R, theo hdng sd ty le la beta, v&i beta := ^^^ L \ hay ndi cdch khdc, phi rui ro thay ddi ty
VariR)
le v&i beta cua nd qua phep biin dSi tuyin tinh theo mgt dg gid trgng thdi:
R-r CoviR,R) R-r VariR) Chimg minh:
Xet mgt quyin tdi chinh mua bdn dugc cd dgng d^t bift a + b L, trong dd a, b Id hing sd vd b khdc 0, khi dd cd m^t phuang dn diu tu (x,^) sao cho qud trinh gid ciia nd ^ = V,ix,<^) vdi t = 0, 1 thda man
Vy=a + bL
Ggi R Id lgi tiic tuomg iing vdi phuang an ndy thi V,il + R) = a + bL
Phuang trinh nay cd nghif m L la
^_V,i\ + R)-a b
Ddi vdi mgt phuang dn dau tu bat k^' cd lgi tiic R thi CoviR, L) = Cov(/?,M±^lzf[)
b
= CoviR,^-^) b b ^
= CoviR,!^) V R
= ^ CoviR, R) b
Dodd (12) trd thdnh
Tap chi KHOA HOC DHSP TPHCM Nguyin Chi Long
R-r = -^-^CoviR,R) (14) b
Trudng hgp rieng khi R = R thi (14) trd thdnh R-r = - ^ CoviR, R) = - ^ VariR)
b b V R-r
Hay — - = ?;^ thay vdo (14), suy ra b VariR)
R-r CoviR,R) ..; x . , . . „
^=— = ~ , dieu can chung minh LJ R-r VariR)
De cd dugc kit qud cd diln md ngudi ta thudng ggi Id md hinh dinh gid tdi sdn tu bdn, ta chuyin bdi todn tim cyc tieu phuang sai lgi tiic sang bdi todn toi uu sau:
Bdi todn 2.
Tim cue tieu Variy^)
V&i rdng bupc E\y\ ] = x.(l + /?) va Fg = x
Rdng budc diu cua bdi todn 2 Id ddng thiic liy gid trj ciia qud trinh gid tai tiidi dilm ddo ban t = 1 ciia phuang dn diu tu dugc bd sung tdng von ban dau x, ma ky vgng ciia nd bing x.(l + p). Dilu kifn rdng budc Fg = x thi tuomg ducmg vdi
EQ[—F,] = x. Bdi todn 1 thi tuomg ducmg vdi bdi todn 2; that vay, neu F, la mdt 1
Bl
V — y
nghif m ciia bai todn 2, thi R = -^ thda mdn rang budc ciia bai todn 1, hem nua vdi
X
mgi phuang dn md lgi tiic R thda mdn rang budc ciia bdi todn 1 thi F, = x.(l + R) thda man E\yy ] = x.(l + p), nghia Id thda mdn rdng bugc ciia bdi todn 2,
Do dd VariR) = \variV,)<\variVy) = VariR), ^i^u ndy chiing td Id nghifm
X X
ciia bai todn 1, ngugc lai nlu R Id nghif m toi uu ciia bdi todn 1, thi dl thay Vy=x.i\ + R) Id nghif m tdi uu ciia bdi todn 2. Vdy bdi todn 1 vd 2 Id tucmg ducmg nhau.
De gidi bdi todn 2 bdng phuang phdp nhan tii Lagrange, ta dua vao tham sd p vi tim cyc tilu ham myc tieu VariV^)- y6!£[F,] vdi rang budc Fg = x, nhung
VariVy) = E[Vy^]-iE[Vy]f, nen hdm myc tieu sg dugc xet dudi dang E[^V,'-PV,].
Vgy ta xet bdi todn tdi uu sau:
Bdi todn 3.
Tim gid tri cifc dgi ciia E[--Vy^ + /3V, ] V&i rdng bupc FQ = x.
Hdm Uix) := —x^ +/3x khdng phdi Id hdm lgi fch dom difu ngdt theo djnh nghia 1, tuy nhien nd Id hdm ldm vi dgo hdm b$c 2 dm. Vi U'ix) = -x + /? ngn hdm ngugc cd thi dong nhit /(x) = -x + /3. Gidi phuang trinh (11) ta tlm dugc nhdn tii Lagrange
^_ ix.B,-P)B,
EQ[L]
TIT cdng thiic (9), ta tim dugc nghifm t6i uu, k^ hifu Id V^ (thay cho Strong (9))
'^=:ife'^«ti]-i)...B,^ (.5)
Dod6,£[f>] = - ^ ( £ , M - , ) . . . B , _ L - (16) Bay gid ta mu6n V^ thda mdn dilu kifn rdng budc ciia bdi todn 2 nen
£[FJ = x.(l + p) (17) Y^iP^Q thi EQ[L] > 1 vd tii (16), (17) ta tim dugc nghifm p
x.iE,[L]i\^p)-B,)
^~ E,[L]-\ (^^>
Chii y ring p Id ham tdng ngdt theo p vd bdng x.(l + r) = x.5, khi p = r Hon nua khi p = r thi nghifm toi uu cua bdi todn 3 sg Id F, = x.(l + r), Id hing sd da bilt.
Vdi sy chgn lya gid trj ndy ciia p ti-ong (18) tiii nghifm F, ciia bdi todn 3 se thda man rdng bugc ciia bdi todn 2. Nlu V^ Id mgt biln nglu nhien ndo dd md nd thda man rdng huge ciia bdi todn 2, thi
E[V,] = x.i\ + p) = E[V{\
Vddodd, E[-]-V^ + pv,]>E[--V,^ + pv,]
« E[U^,']<E[U^^]
O Var[V,]<Var[V]
Tap chf KHOA HOC DHSP TPHCM Nguyin Chi Long
Mat khdc, bing ly lg ngugc lgi vdi y tren, ta thiy nghifm ciia bdi todn 3, ciing la nghifm ciia bdi todn 2; do dd hai bdi todn ndy la tucmg nhau vdi p vi p xdc djnh nhu tren.
Bddi3.
Ggi R Id lgi tuc cua mdt phuang dn ddu tu cd phuang sai eyre tiiu so v&i tdt cd cdc phuang dn ddu tu md lgi tuc cua nd cd ciing ky vgng p, thi R Id hdm tuyin tinh theo mat do gid trgng thdi L, eu thi han R xdc dinh b&i
^^pE^{L]-r _ip-r)L
EQ[L]-\ EQ[L]-\ ^ ^
Chimg minh:
Vdi p xdc djnh trong (18), thay vdo nghifm tdi uu F, trong (15), ta cd x[i\-^p)EQ[L]-B,] _
iEQ{L]-\)EQ[L] ^"^'"' " ' " ^ " ' £ ^ [ Z ]
V,-^Z\ZT'::\EJL]-L).xBy
x[(l + p)£e[Z]-(l + /-)]
(£g[I]-Z) + x(l + r)-
iEQ[L]-\)EQ[L] ' ^' ' 'EQ[L]
= (^(l±0_^i^(PzZi -jy^ )^EJL]-L)^xil^r)-J^
^EQ[L]-\ EQ[L]-\ iEQ[L]-\)EQ[Lr ^' ' ' ^ 'EQ[L]
^ (^a^o^.[^]-^(iH^^)^^<£ziix^,[z]-z)+x(i+.)-A-
iEQ[L]-l)EQ[L] EQIL]-^ eL J ^ V ^^^j^j
= (^^^4^)(E,[L]-L)^xil^r). ^
EQ[L] EQ[L]-r ^^ ^ ^ - 'EQ[L]. ., - x(l + r)Z xip-r).,„ rTT T^ /i X ^
Sy tucmg duang ciia bdi todn 2 vdi bdi todn 1 cho ta
^ r(£g[Z]-l) ^ pEQ[L]-rEQ[L] ip-r)L
EQ\L-\-\ EQ\L\-\ EQ\L-\-\
^ pEQ[L]-r ip-r)L
EQ[L]-1 EQ[L]-1
Day la dieu cin chiing minh.D 3. Mo hinh djnh gia tdi san tu ban
Dilm danp quan tam ciia bd dt 3 la nghiem R ciia bai todn trung binh phuang sai la mOt ham tuyen tinh theo L, mgt dp gia trang thai. Cdng thiic (13) cung cho ta hf thiic lien lac giOa ky vgng lgi tiic ciia mpt phuomg dn bit ky vdi ky vpng lgi tiic ciia mpt phuomg dn phy thupc tuyln tinh vao mgt dp gid trang thdi. Rd rdng nghifm R ciia bdi toan 1, vl trung binh phuang sai, la dgt dugc. Djnh ly cd ten Md hinh djnh gid tdi sdn tu ban sau la hf qua trirc tiep ciia cdc bd de tren.
Dinh ly.
Niu R Id mgt nghiem ciia bdi todn I ve trung binh phuang sai v&i p>rvd niu R Id lgi tire ciia mdt phuang dn bdt ky, thi
E[R] - r = ^"""'^^'/^ iE[R] - r) (20) VariR)
Hf thiic tren quan trpng d chd, trong gidi cdc nhd diu tu dua vdo phdn tich trung binh phucmg sai, thi thudng tdn tai phuang an ma nd cd the dugc xem nhu la nghifm ciia bai todn 1 (ching han, nhu chi sd chiing khodn), vd tu dd cd thi udc lugng dugc ky vpng lgi tiic cua mot phuang an bit ky thdng qua hf thiic (20).
A
H^ qud. Gid suR Id mdt nghiim cua bdi todn trung binh phucmg sai I v&i tham sd p>r vd tdng vdn ddu tu ban ddu Id x. Ldy p>r vd p^ p Id mdt tham sd khdc, thi
R:=Xr + i\-X)Rv&i X:=^^
p-r
Ld mdt nghiim eiia bdi todn trung binh phuang sai I, v&i tham sd p vd tdng vdn ddu tu ban ddu x.
Hf qua dugc kilm chiing de dang vi E[R] = ;ir + (1 -X)E[k\ = Xr + i\-X)p = p vd vdi mdt phuang an diu tu bit ky R tiida man E[R] = p tiii VariR) ^ VariR) •
Hf qua quan trpng ciia djnh ly chi ra ring, ta chi cin tim nghifm ciia bai toan trung binh phuang sai ddi vdi bai toan cd tiiam sd p rdi suy ra cdc nghifm khdc ddi vdi bdi todn cd tham sd khac bing cdch diu tu theo td hgp ldi ciia trdi philu khdng nii ro vd sd lugng cd philu tuomg iing vdi nghifm cd djnh cua bdi todn tiling binh phuang sai ndy. Nguyen ly ndy dugc ggi Id nguyin ly quy hd tuong ddu tu (Mutual Fund Principle), dugc phdt bilu qua mf nh dl sau:
M^nh di.
Gid sir ta cd dinh mdt phuang dn ddu tu md Igri tiirc ciia nd Id nghiem cua bdi todn I vi trung binh phuang sai, tuang iing vdi mdt tham sd lgi tire trung binh p Thi nghiem ciia bdi todn trung binh phuang sai tuang ung v&i mdt tham sd lgi tire trung binh khde, ed the tim dugc bdng mdt phuang dn ddu tu bao gdm sir ddu tu vdo tdi
Tgp chf KHOA HOC DHSP TPHCM Nguyin Chi Long
khoan tiit kiem ngdn hdng (hay trdi phieu khdng rui ro) vd phicang dn ddu tu cd dinh ban ddu.
De ldm rd hom tu md hinh dinh gid tdi sdn tu bdn, ta xet mpt quyen tai chinh cd giaXtai thdi diem dao ban / = /, ma ta mudn dinh gid ciia quyen tdi chinh ndy tai thdi diem t = 0. Gid sii lgi tiic R la nghiem cua bai toan 1 ve tmng binh phuang sai. Theo
X - x
dinh nshia, lai tiic ciia quyen tai chinh A' Id R.= -—-, thay vdo (20), ta duac
X
X VariR)
Giai phuang trinh ndy ta dugc nghiem .r, chinh Id ciia quyen tdi chinh tai thdi diem t = 0:
E[X]
x = —
l + r + I iE[R]-r) VariR)
Vdy tu R ngudi ta cd the xdc djnh gia ciia mdt tai sdn tu ban X.
TAI LIEU THAM KHAO
1. Nguyen Van Hihi, Vuang Quan Hodng (2007), Cdc phuang phdp todn hoc trong tdi chinh, Nxb Dai hpc Qudc gia Ha Noi.
2. Nguyen Chi Long (2011), "Bd dl Fakas vd dp dung trong thi trudng tdi chinh", Tgp chi Khoa hgc DHSP TPHCM, 27(61), tr. 41-53.
3. Nguyen Chi Long (2011), "Dinh gid tai san trong md hinh nhj thiic", Sd chuyen de cua DH Sdi Gdn: Hdi thdo Khoa hgc Qudc ti Gidi tich vd Todn Ung dung, DHSG TPHCM, tr. 513-525.
4. Nguyen Chf Long (2010), "Nguyen ly can ban djnh gia tdi san trong thj trudng tai chinh", Tgp chi Khoa hgc DHSP TPHCM, 21 (55),"tr. 3 8 - 5 1 .
5. Nguyen Chi Long (2008), Xdc sudt thdng ke vd qud trinh ngdu nhien, Nxb Dai hpc Qudc gia TPHCM.
6. Tran Hung Thao (2004), Nhgp mon todn hgc tdi chinh, Nxb KHKT Hd Ndi.
7. Robert J. Elliott and P E. Kopp (2005), Mathematics of Financial Markets, Springe Finance, Second Edition.
8. Hans Foellmer and Alexander Schied (2002), An Introduction in Discrete time, Walter de Gruyter.
9. G. Pennacchi (2008), Theory of Asset Pricing, Pearson Education, Increase affect.
10. Pliska (1997), Introduction to Mathematical Finance, Blackwell Publishing.
(Ngdy Tda soati nhin 6wgc bii: 23-3-2011; ngiy chip nh$n ding: 16-8-2011)
