บทที่ 5 : การจัดการพอร์ตการลงทุน
(Portfolio Management)
———Financial Mathematics 1/2019
N. Ekkarntrong, Ph.D.
\
5.1 ความเ&ยง (RISK)
➤ ลง#นใน bond ไ'ผลตอบแทน 8% ณ /น0นอา2ของ bond
➤ 4จารณา ➤ ผลตอบแทนการลง#นแบบ8 [1] 9อ 11% ห;อ 13% <น=บตลาด ➤ ผลตอบแทนการลง#นแบบ8 [2] 9อ 2% ห;อ 22% <น=บตลาด
➤ Risk[1]< Risk [2] ➤ ?าทราบแ@ ผลตอบแทน แAไBทราบความFาจะเIน8จะเJดผลตอบ
แทนKนๆ เราจะไBสามารถหา ความเOยง (risk) ไ'
risk so y .- -_
➤ เQน ➤ return rate 22% with prob. 0.99 for [1] and 0.5 for [2] ➤ return rate 2% with prob. 0.01 for [1] and 0.5 for [2]
➤ Risk[1]<risk[2] ➤ ผลตอบแทน บนการลง#นในRนทSพU8VความเOยง เIน)วแปร-ม
➤ เIน@ากลาง และ มองเIน ผลตอบแทน (return) ➤ เIนWวX/ด ความเ&ยง (risk) ➤ ห;อใY standard deviation เIนWวX/ด risk
K
E(K)
Var(K)
!k = Var(K)
➤ WวอZาง 5.1 ➤ ใ[ ผลตอบแทนบนการลง#นห\ง ห;อ 'วยความ
FาจะเIน 0.5 ➤ ➤
➤ ใ[ ผลตอบแทนบนการลง#น]น ห;อ 'วยความFาจะเIน 0.5 ➤ ➤
K1 = 3 % !1 %
E(K1) =Var(K1) =
K2 = 2K1 = 6 % !2 %
E(K1) =Var(K1) =
4--0.02=27.
10.5139. -1 Car) f - it.) = it .£
(3T - 17140.5) t f - It . - I F)40.57=0.0004
2%
0.0016 f dy = 0.04=47.
➤ ➤
➤ ➤ where is the log return in time step
Var(aK) = a2Var(K)!aK = !a !!K
Var (k(0,n)) = Var (k(1)) + " + Var (k(n))k(i) i = 1,…, n
5.2 สองห7กท9พ; (TWO SECURITIES)
➤ Risk && Expected return ➤ ตย. 5.2
➤ สมม^ใ[
➤ การลง#นในหลายห_กทSพUเIนการกระจายความเOยง
Scenario Prob. Return K_1 Return K_2
0.5 10% -5%0.5 -5% 10%
"1
"2
➤ ตย.5.3 ➤ สมม^ ➤ โดย
➤
S1(0) = 30, S2(0) = 40, V(0) = 1000
x1 = 20, x2 = 10
b b
w,= 30×202 =
607'
look£Wz=4qo = 40T '
weights,
= 35 , Sdn= 39
VG) = (207/35) t Clo) ( 397 = 1090
W,= (20/135)←
=64.22T '
, wz= 35.78J .
<ด>วนการลง?น (WEIGHTS)
➤ aดbวนการลง#น (weight) cยามโดย
➤ ,
➤
➤ หากVการ short sale จะไ'dาaดbวนบางWว^ดลบ และeกWวจะV@าเJน 100%
w1 = x1S1(0)V(0) w2 = x2S2(0)
V(0)w1 + w2 =Minutia .- fig
= I = loot.
➤ ตย.5.4 ➤ ใ[ ,
➤ portfolio จะประกอบ'วย
➤ ➤
➤ ไ'
V(0) = 1000, w1 = 120 % , w2 = ! 20 %S1(1) = 35, S2(1) = 39
x1 =x2 =V(1) =
5,101 = 30 , 52107=40
w,VIO ) = 40
towz =
-5
suo)x, 5,117 t xaszci
)
= 1205
➤ จาก ตย. 5.4 จะไ'
➤
➤
➤
V(1)V(0) = w1
S1(1)S1(0) + w2
S2(1)S2(0)
= w1 (w1(1 + K1) + w2(1 + K2))
KV =
ru, = x, sin + XssID
to Tio to
%,arms 3
PROPOSITION 5.1
➤ ➤ โดย 9อ aดbวนการลง#น ➤ 9อ ผลตอบแทนของห_กทSพUประเภท8 1 และ 2 ตาม
hiบ ➤ 4jจk
➤
KV = w1K1 + w2K2
w1, w2
K1, K2
(ahern (weights)
-
VIO) = X,5,101 t Xss, CO)
un -- x. sin + x. six- 9410' + Is"'
= x÷:S , 107 (t t Ky)
t 1125210) ( it kz)
= w,V 107 ( I t k , ) + wz Vio) ( ltkz)
= W,Clt Ky) t Wzlltkz) = It w,
K,+ Wzkz
➤ หมายเหl : Logarithmic returns
➤ ➤
ekV = w1ek1 + wek2
Ku -- V"fY = - I = w,K, thzkz #
2
5.2.1 ความเ&ยงและผลตอบแทนของพอBตการลง?น➤ Expected return :
➤ ตย. 5.5 ใ[พอmตการลง#น Vการลง#นใน 2 ห_กทSพU
➤ หา สมม^ใ[
E(KV) = w1E(K1) + w2E(K2)
E(KV) w1 = 60 % , w2 = 40 %
scenario Prob. Return K_1 Return K_20.3 10% 50%0.5 0% 20%0.2 -10% -30%
"1"2"3
Kv = W,K,t wzkz
E (Kv ) = (6041110.37 Got' ) -110.5710%7+10-2) f-"t)) =p,
+(40T) (lo.3) Hot ) + Co:5)(20T) + Co-2) C-307))
ทฤษEบท 5.2
➤ ความแปรปรวนของผลตอบแทนบนพอmตการลง#น 9อ ➤ Var(KV) = w2
1Var(K1) + w22Var(K2) + 2w1w2Cov(K1, K2)
Mn Kv = w, Ky t Wzkz bro: E
(Kv ) - hi, Elk,) t wz Elka)2
Gorm Var ( Kv) = El KY ) - ⇐ ( Kp)
= E ( wikf-wzkztawiyk.kz/-wffEC9D&-wfCElkzDI-2ywzEKDE(Kz)= Wielki ) t w! ELKE ) -12W,wz Elk, K2)- w,2( E (Kpk - way Elkz))
'd- 2W,wz ELKIE (KD
= w! Var ( K,) x wzd Var (kg) - aw,wz Cor (Kiska) #
➤ nหนดใ[ ➤
➤ ➤
➤ และ correlation coefficient 9อ
➤
➤ ไ' ➤ และ
#V = E(KV), !V = Var(KV)
#1 = E(K1), !1 = Var(K1)
#2 = E(K2), !2 = Var(K2)
$12 = Cov(K1, K2)!1!2
#V = w1#1 + w2#2
!2V = w2
1!21 + w2
2!22 + 2w1w2$12!1!2
END
Varlkv)
➤ ตย. 5.6
➤ ใ[ และ
➤ หา
w1 = 40 % , w2 = 60 % , !21 # 0.0184,!2
2 # 0.0024$12 # ! 0.96309
!2V
scenario Prob. Return K_1 Return K_20.4 -10% 20%0.2 0% 20%0.4 20% 10%
"1"2"3
←Risk# u
org = widget wind! -12441,902= 0.000736 L did
< of
➤ ตย. 5.7 ➤ 4จารณาพอmตการลง#น8V และ
➤ หา
w1 = 80 % , w2 = 20 %
!2V
scenario Prob. Return K_1 Return K_20.4 -10% 20%0.2 0% 20%0.4 20% 10%
"1"2"3
6,2=0.0184 , 022=0.0024
I
or = 0.009824 Iv for
2L I
62
PROPOSITION 5.3
➤ กรo8ไBอpญาตใ[r short sale!2V $ max{!2
1 , !22}
① maxton,
'
, a.4- or: or or.
-no
,
'
talons short sale lol w, ,wz 710
dorm wad, t Wak f ( W, twz) of = dz
no: of = wind, 't wish.-law,wzfz4k
← win,'+ wirftaw.wzd.dz f : -' EE ')
= (wi, + wields of e- max 44314
② maxki.my -
- of.
: #
➤ ตย. 5.8 ➤ 4จารณาพอmตการลง#น8V และ
➤ หา
w1 = ! 50 % , w2 = 150 %
!2V
scenario Prob. Return K_1 Return K_20.4 -10% 20%0.2 0% 20%0.4 20% 10%
"1"2"3
of = win,'t wiof -12 wire,Eik
PROPOSITION 5.4
➤ [[1]] ?า แsว เtอ
➤
➤ [[2]] ?า แsว เtอ
➤
➤
$12 = 1 !V = 0 !1 % !2
w1 = ! !2!1 ! !2
, w2 = !1!1 ! !2
$12 = ! 1 !V = 0 !1 % !2
w1 = !2!1 + !2
, w2 = !1!1 + !2
on! = win, 't win,'-12W
,w
,oik
= ( w,d,t wash )
&
or!-- O €7 Yd, twicea :÷÷÷.
⑦ on! -- O , wed, -1 wzoh -- O
W,-- - wing wz=
- wyd,- s -
or orI 2
Mn w,-1W,= 1
in -war! + - war,'
= or,q
win , aft - you? = go,-
Hw, d, for, - or, I - foraw,
= Nz =- Nz--
oh - r, of -Nz
270.2070in)
f n
,du -
- two, - will q¥ of -- lei -sir, -1541
Mr = Wfm+ Waltz Mv -
- G -SIM ,t SMz
f: s.. , lyse . ⇒ www.tanMy = ( I -S ) fly 1- Sitz
whimo en ⇒Tudor
short selling
⇒ an portfolio air ominous etowngn oratory, z ,- I I
✓
✓
¥ ①de:) = o fan of -- a - sik,
't s's.
'-1211-7%44 )
-
d s Sav s = wz
211 -Sl th did + 2544 (as tht Cn -s) 1211%42 = 0
- 24-
tze,'+Idea'- zsfze.dz -1%44-2%11=0
*spies'- shirt = af,
'-air)
i.
s = 6,7%451/6,'to! - ay,i, ra ) idognonsgn
② an 9%081=242+24'
-4%41=214442-2%6,4)721444' - 244 )= 214 - org )
' Zo
unwon s ndidsrpnongnIN d4d:( si ) > o
DT2
÷ .qadngnniihrorfddiofngnsrun.lv3 Arrl
Ii offs rid min oil opts.N1 solo nm optom so > 1 - off 1
56,
!. . I tf a tshirt selling
.
Li- ' TT '
-
-
PI ① 9W -if,z< I62
4244 so,d
an so = or,
'-f,
or,oh r.hr So 70
⇒ik"" ÷
: ::c .' "÷9,9.
② f.z -- II rid 144 -
- rid .
'
. so -- o
minor . :oqi So -- o sinus ohhh -sifts!'
I
64101=6, Foon minimum + ash-5%44
i. senior, A
⑦ IN Ecf,z
' t
Gz
44%4 E Isr s.co
od short selling or portal oils,e. his short selling airport n' Koh
6, coz " r " Azz = VOjo.LI = 0.5821 4,241
as>Tv case② Tv Cor . 5.6
our portfolio riannwiodvsofnfanhrrwahrrnissisd short selling
so = 4%2402 =-1.1663 L O
-
65+42-29245gm;n=o=w (
no short selling)2
So = - 1.1663 = Wz
sho,
4=2.1663 / -
i w,= y
#"
selling Id"
2
of s w for : t wa'
og 't 2 Ww, wah, 442
⇒ as Iwaki #
5. 3 tenurednaked
① Risk & Expected Return on a Portfolioweights"
wi = Xi Silo)
,
i = 1,2, . . . ,h
¥tinitial investment
Tai w = [ w, wz -- - Wn ]
wi = 1 Ufo n = UWT Ido us [ 11 . . - I ]
in
attainable set doinoutdoor portfolio aid weight ooaaaio,Av M
Hattainable portfolio
TW K, .kz , . . . , kn od . return our dunker n Junker
expected return Mi e E ( ki )
M = [ µ , Mz - - - Mn ]
Cij =Cov ( Ki , Kj )
covariance matrix C- fed:&:-
: :&? ) , ciisvarfki )= dic'"m dm -
'
i ' Cnn
Assume C ' exists .
✓✓
Mr - Elkv) = E ( ¥,
wiki ),
=
.
w ;Elk ;
) = MW
-
ni Hnk - "
MY!n
of = Var ( Kv ) = Var ( II,
wi ki )
= Cov ( ¥! wiki , ¥, Yi Kj )h
= I w i wj Ci j
s
' '
ci em win:: I
PI airmen minimum no , of =wCwTominous uwt - 1
I Lagrange multipliers
TW Ftw,X ) - w Cat - Xuwt
-
+Lagrange multiplier
'
www.wi-mp:÷÷÷÷÷÷H÷:Lt.E.widi.E.im.ci . - ' ' Eiwidinlf?= w
, Ii wi Ci , + wz .widizt - - - t Wn
;wi din
= W, ( W, d, , t Wzcz , t - - i t Wn dm )
+ wz (W, Caz + wz Czz t - - - + Wndnz)f- ✓ - '
1- Wh (W, dm t wzczn t - - i t Wy Cnn )
xuwt .- Ali i - .
.iq?/=HEiwilFCw,X7--wdwT-XuwT2F= 2W
,C, ,+ wzcz , t - - - t hunch ,
2W? t wz Gz t w , C , ,
t - -it wyd , n - X
= 2 w,
C, ,
+ 2 Wg Cz ,t 2 wzdz , t . -- + Iwm Cny
- Xn
= 2 2- wid ; ,- A = O
is ,
2F
Tq,
= 2.wi dig - X -
- o
= . wi:c.
. . .} ""
2Wh
saw . . . . " ,i ?÷÷ ?:/. " . . . . . .(ng dhz - i - Cnn
= [ o o . . . o ]
LI = 2 w C - Xu = O
2W
W = Xu C- I
-
2
misdate uwt - I = wut, ¥uT=i
" I
•: A =2-
ud- 'ut
Boi - i
w = u d#
(Mv -- Mut
- --
-
¥ Gcw ,Xp ) = Wcw"- Xuwtyumwt
21=02W;
2¥ = 2¥? wifi - AH -MM,= o
mmwt-ulm.ms - - -Mn] =M.¥Miwi
2£,
=2 II wi diz - 7117 -MM, =0
I
<
',
f÷n = 2.II. wi din - Xin -man = 0
fwd =IWC - Xu -
mm = o
- Hmm . - - wtf!! ?: ÷?:/dh, dnz - - - dm
- X Cr i - . . it ya fu , Mz - - thy ] - O
Toi w = 7uC"tmm2
an UW! I = WUT bro : Mr = m w
! W MT
hi wut - i = Azul-ii.
AgmC-
'ut - ④
bid !
wmT=µ✓ = Tzu C- '
m't Mz mC- '
MT - ②
on X, M
÷÷÷÷÷÷÷÷¥ii÷÷÷÷÷÷:i. w.li . no:/
'
nlni-
i: #
1
Outpatient97W
,of = ?
I ( minimum variance.
Toi m = [ 0.10 0.15 0.20 ],U = [ 7 2 I ]
" %=a÷÷is me . E:&:/↳ , Czz (33
am Cij -- Covlki, Kj ) =L; did; ←-40.240.28710.at)
o -" ''"
&
.co?yg20.0l75f=fo.zo) (0.241/0.25)
c.fi?.::i:::::::..:l....=fo.1o)(0.28/10.24) g. 0175 0.0120
"" I:÷÷÷÷:÷÷:÷:÷:L- 4
= 2.47×10 = 0.000247
c-'
=L adj CC )detlc)
0576 0120
g
00670.0176 )÷.fi::i÷÷÷÷÷÷÷÷:::i÷÷÷÷÷÷÷:÷÷::::i÷÷÷÷÷÷:p3h
" is:*: :: "
"
t::: "
"
l ::*: :::
i:÷÷÷÷i÷÷. :÷÷÷i
- l
W = ud-
Ud - 'utth
ur a C- '
= ?⇒ w
u C-'
ut = ?
•
.
.W = [0.3lb 0.439 0.29T)
µ✓ = M WT = 0.746
or -- wcwt.o.az ) prop .
".
inurn µ✓ prop . 5.10
W = [7.578 - 8.614µV 0.84T - 2.769µV
- 1.422 -111.384µV ]
of =wCw[ = . -
inuvohkrrnrw now ) !Y:!!.Mw prop .
5.9 Irn : 5. I
fi mail ⇒ [email protected] .
Wz =/
I
Coiymininunrarianaline .
f wz-- 7
190 4,07 Kowitz bullet
Iii""
nauniaain
W,- I - wz - W, @r.Mv ) V
t
1
' dl.
.
I
n2
-
.
f) minimum variance line .I
•
¥ . Tai w srnudadwmoiqusuwofnnv.o.gr (emfs minimum variance portfolio)
✓ ghlrionomw (offer efficient frontier ) covariance matrix nxn
" ÷÷÷÷÷' """"
bro :discount 80
u wt = IT[ I 7 - i . I ]
Few," = ;:÷ -
" 71. . . . .
ng:II = o shot
2¥ = Imi - Im wt.nl/ziEwlwc+cwY/-xu=o#
÷÷ - I :÷H÷*. -
an = o
m - f;Y÷) wC - an = o
m - Inge) wC -
a un-
- o
Wv -- Xu wt = Xli r -- - I ] (waging
= Aw,+ Tweet " '
+ Awn
2hL = X X2W, X
÷.
..
. ! )2W, =
X- ha2Wh
T
AU
Flw,7) = m WT -µ - Xu wtTT✓wcw
= CM, Mz - --Mnlf!!) - M
F--"""%
win'i÷÷÷÷÷= TIM ; Wi -M
n
F-' Ewi
-2W; E. wicijjet
W, (wyd , , -1 Wzcz , t - " +Whdh) wid ,, -12W,Wzdz ,+ W2¢W, Cyztwzdzz t . .
.
t Wndnz ) -124441:
1- . .. -12W ; Whdh ,
+ Wn Kw , cant wzdznt ' ' ' + whdnn)
¥,
Elm) - I . mini -all'x¥4in'DE.im?.iwic.;TIi#
- X =O
2¥.
-Elm) - t.E.niwi.nl/Ee?iwidizl-x# 11
II. wj.IT icij ?s
.
In Elm)- I VIII. hidin) - at 11
JET wj.IE ,widij
°
IF ¥2 [ µ , Mz " ' Mn )
Tw=
- w ;-MEH.
wifi.
wifi - " widin )E
- [ x x - .. X )
-
- Cmm - -- nd - iEniI t.E.hn?iniciz--iEwidinl
- ( x x . . . 7) FE
= m - ftp.n-lw?-xudr=oI
m -Mu - jwc #
(Mjf )wd = m - torn
an min : " Mg = 870
*a.4W µ = Adv
add ywc =m -Mu ,
7- g.MERr >o
#
isfind efficient frontier .Ensign
① dno.gr Iroh N,y,µ inner
⑦ airmails efficient frontier tanto
⑦ 180 no >gunshots
I¢µ=mot C
Hmmm)"
re =return on risk - free asset.
of -- o
Doiron portfolio ( risky , , risk- free)p t
Mi , 6,70 Me -- re , of
-0
M = tf + µµ-rfm→rBkpremium
6M
Exampled GV prop . 5,12 phaser market portfolioidasownrwrnrriskbro : I risk -free
µ , e-0.10 ,
N,-- 0.28
, 9,2--92 ,= - O - 10
,
µz= 0.15 , 02=0.24 , fz, =L ,= 0.20
,
I tf =5%
µ,= 0.20 , dy = 0.25
, jpg , =L, ,= 0.2T
,
ICM, Maths ]
Tau w roomieryw
= (m -mu) C- I
Can prop .
TT[ 7 I 7 ]
¥
6=0, µ - re T
give[ 0.293 7.347 2.067 ] on UW =L
ur phd do y = 3.694
.
'
.We [ 0.079 0.363 0.558 ]
is^ Ki --Pikmin
g.Ku =p km 'd I
do :mN Kv
E = Kv- 43 Km -14 )
Tactual .
Dorm on line of best fit
E ( 5) = Effkv - 43km HIM ]
E ( E) = E ( K Z - 2 Kv ( 13km +d) + (13495+213%2+27)= Elk f ) - 2/3 Elkvkpg ) - 2L El Kit
+ PIE ( km2 ] + 2pxE[Kµ ] + L2
Farm
22,7¥ =- 2E Ckvkm ] -1213 El Kpi
] + 2dECKµ ] -0
Elkvkm) = PECK; ) + LEC km ]- ①
2E(E =- 2 EC Kv ) t 213 Elka ] t
2x = O
• & Elks = 13 Elka ] + x
L = E ( Kv ) - PEC km ] - ②
sonar ② Iv ① Tor
Elkukn) =p Elkpi ] + ELK DECK] - PFEIKMD'
=p ( Elkin ] - EKD) 1- ELK] Elkin ]
Elkvkmt = 13 off + Elk ) Elkin ]
an Covlx,Y ) = EIXY] - ELXIEIY )
( Cov IX. x ) = Varix) )Elkvkn) - Elkv ] Elkm
] = 134!
Covlknkm ) =p = pvg-
Mywar ph @ Td
L = El Kul- Covckv, km ) Elk ,y )a-of
= Mv - Br MM
Bmw- bosun
pv = Covckr , km )
q-
in beta factor
"'m portfolio
{Ja mN§WlNmwonoUrnvVv portfolio MoronaMv - pram tarNr pv ④ t.in MMM ,MutM PvE Ko : Mnt
,Mvt
or? = Var ( Kv )= Var ( Ev t (L +13km )) (one -- Ku-ktpkn)
= Var ( Ev ) + Var (dtpkm)
= Varied + p&VarlKm )= Var ( Ev ) + p&gyy2 systematic risk .
←- on
risk rsoniuuoon Tsystematic
risk. qzmodyn.noWurman
ison prop . 5,12
ywµC =m -
Mu , r >0 , µ
pv -
- Cov£km = Wu Chin = wmcw!- -
Wmc wht Wn Cwf
= :÷÷÷= i :÷an µ = ?
an risk - free 8 Mus tf , pv -Bf-
- O
O =rf .
'
. µ= re
Mn -M
13 =Mv :.Mr --p(Mm - re) -17
Mn - re
Mr = re 1- (Mn- tf )Pv
demand = supply .
① Mv ) re+ ( Mn - Hpv
,② Mv Cre t (Mn
- Hpv
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