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 ⇒ nawaek@kku.ac.th .

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