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PORTFOLIO THEORY
Presented by
Prof. Eduardus Tandelilin, CWM
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ObjectivesTo understand the benefit of Diversification
To learn about constructing an efficient portfolio
To compare and contrast Diversifiable and
Non Diversifiable risk
To learn how to calculate Stocks eta and draw
!apital "arket #ine
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Put all your eggs in the onebasketand
WATCH THAT BASKET! -- Mark Twain
As we shall see in this chapter, this is probably not the best advice!
Intuitively, we all know that diversification is important when we aremanaging investments.
In fact, diversification has a profound effect on portfolio return andportfolio risk.
But, how does diversification work, exactly?
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PORTFOLIO RISK
Rate of Return Distribution
Two (W&M) stocks with perfect negative correlation (r = -1.0)
and for portfolio WM
a. Rates of Return
Stock W Stock M Portfolio WM
-10 -10
kw(%) kM(%) kP(%)
25 25 25
15 15 15
0 0 0
-10
2003 2003 2003
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b. Probability Distributions of Return
Probability
Density
Probability
Density
Probability
Density
kw kM kP
Stock WStock M Portfolio WM
0 0 015 15 15Percent Percent
Percent
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Year W kW(%) M - kM(%)Portfolio WM kP
(%)1999 40.0 -10.0 15.0
2000 -10.0 40.0 15.0
2001 35.0 -5.0 15.0
2002 -5.0 35.0 15.0
2003 15.0 15.0 15.0
Average return 15.0 15.0 15.0
Standard Deviation 22.6 22.6 0.0
Standard Deviations Portfolio ww2ww 2()2)2)22
22
2)
2) rp ++==
w1 * weighted fund invested in asset )
w2 * ) + w1* weighted fund invested in asset 2
r)(2 * correlation coefficient between asset ) and asset 2
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PORTFOLIO RISKRate of Return Distribution
Two (M&M) stocks with perfect positive correlation (r = +1.0)
and for portfolio MMc. Rates of Return
2003
Stock M Stock M Portfolio MM
kM(%) kM(%) kP(%)
25
15
0
-10
2003 2003
-10 -10
0 0
15 15
25 25
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d. Probability Distributions of Return
Probability Density
Percent
Stock M
Percent0 15
kM
Stock M Portfolio MM
Probability Density Probability Density
0 015 15Percent
kMkP
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Year M - kM(%) M kM (%)Portfolio MM - kP
(%)1999 -10.0 -10.0 -10.0
2000 40.0 40.0 40.0
2001 -5.0 -5.0 -5.0
2002 35.0 35.0 35.0
2003 15.0 15.0 15.0
Average Return 15.0 15.0 15.0
Standard Deviation 22.6 22.6 22.6
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PORTFOLIO RISK
Rate of Return Distribution
Two (W&Y) stocks with partial correlation (r = +0.65)
and for portfolio WY
e. Return
25
15
-10
Stock WkW(%)
0
Stock Stock Wk(%) kW(%)
25 25
15 15
0 02003 2003 2003
-10 -10
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f. Probability Distributions of Return
Stock W and Y
Probability Density
Portfolio WY
0 15 Percent
kP
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Year W - kW(%) Y kY(%)Portfolio WY - kP
(%)
1999 40.0 28.0 34.0
2000 -10.0 20.0 5.0
2001 35.0 41.0 38.0
2002 -5.0 -17.0 -11.0
2003 15.0 3.0 9.0
Average Return 15.0 15.0 15.0
Standard Deviation 22.6 22.6 20.6
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Correlation and Diversification
Suppose that as avery conservative, risk-averse
investor, you decide to invest all of your money in a bondmutual fund. Very conservative, indeed?
Uh,is this decision a wise one?
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The feasible set of portfoliosrepresents all portfolios
that can beconstructedfrom a given set of assets.
An efficient portfoliois one that offers the most returnfor a given amount of risk, or the least risk for a givenamount of return.
The optimal portfoliofor an investors is defined by the
tangency point between the efficient set of portfoliosand the investors highest indifference curve.
Efcient and OptimalPortolio
)%
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!ovariance and !orrelation !oefficient
=
==n
1i iBBiAAi
P)k-(k)k-(k(AB)CovCovarians
BA
AB
Cov(AB)r(AB)tCoefciennCorrelatio
==
)&
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Probability Distribution of Stocks E, F, G, Dan H
Probabilityof occurrence
!ate of !et"rn Distrib"tion
E F G H
0.1 10% 6. % 14% 2%
0.2 10 8 12 6
0.4 10 10 10 9
0.2 10 12 8 15
0.1 10 14 6 20
k = 10% 10% 10% 10%
= 0.00% 2.2% 2.2% 5%
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MEASURING THE PORTFOLIO RISK : EXAMPLE
COV (FG)=
= (6-10) (14-10) (0.1) + (8-10) (12-10) (0.2)
+ (10-10) (10-10) (0.4) + (12-10) (8-10) (0.2)
+ (14-10) (6-10) (0.1 = - 4.8
The negative sign indicates that the rates of returns on stocks F and G tendsto move in the opposite directions.
Correlation Coefficient F&G is:
=
5
1iiGGii
P)k-(k)k-(k
G
G
Cov(G)r
= /0)12022021
%0-+r34 =
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SCATTER DIAGRAM
a. Returns on E & F (r = 0) b. Returns on F & G (r = -1.0)
5 16
)/
)&
/
&
& )/ )& 2/
416
3 16
)/
)&
/
&
& )/ )& 2/ 316
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SCATTER DIAGRAM
a. Return F and H (r0.9) b. Return G and H (r-0.9)
Notes :a. Thered linesin each graph are called regression lines.
b. These graph are drawn as if each point had an equal probability of occurrence.
)/
&
)&
316
& )/ )& 2/ & )/ )& 2/
416
)&
)/
&
716 716
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THE TWO-ASSET CASE
A complicated looking but operationally simple equation can beused to determine the riskiness of a two-asset portfolio:
)1(!)1("#PortoolioBA
!!!!
p
ABrXXXX
BA ++==
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Average Return and Volatility For Portfolios
Portfolio Stan#ar# De$iation
Portfolio%&'e
cte#!et"rn
0000 0050 0100 0150 0200 0250 0300 0350
0025
0020
0015
0010
0005
0000
How Do Portfolios of These Stocks Perform?
100% !*S+!,*
.!/+P / ,M!,
100% MM+ /!P
100% W!SS
/M .!/+P
50% W!SS 50%
!*S+!,*
50% W!SS 50%MM+
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50% ireless ! 50% I""ucell
Ris# Increases it$ &'ected Return
50% ireless ! 50% Reinsurance
Ris# Decreases at First( T$en Increases as
&'ected Return Rises
Why Do Portfolios of Dierent Stocks Behave Dierently?
Average Return and Volatility For Portfolios
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Expected Return For Portfolio
( )( ) ( ) ( ) 6%$0)6-%0/&0/6/$02&0/
111 22))
=+=
+= REwREwRE p
50% ireless ! 50% I""ucell
50% ireless ! 50% Reinsurance
( )( ) ( )( ) 6$'0)6'.0/&0/6/$02&0/
111 22))
=+=
+= REwREwRE p
E$pected %et&rn o Portolio 's e Avera*eO E$pected %et&rns O e +o "tocks
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Two-Asset Portfolio StandardDeviation
2))22)
2
2
2
2
2
)
2
)
22 wwwwp ++=
2DeviationStandard
p
=
)orrelation *et+een Stoc#s In,uencesPortfolio -olatility
What is Correlation Between Wireless and mm!cell?"#$"
What is Correlation Between Wireless and %eins!rance&ro!'?("#))
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Correlation of Reinsurance Group,Immucell, and Wireless
Relative Performance of Three Stocks
0
0(5
1
1(5
2
2(5
January 000 ! December 00
Stock
Price
Re
lative
toP
ri
January
000
!eins"rance .ro"' )44"cell -or'( Wireless eleco4
Wireless and mm!cell *ove To+ether, Wireless and %eins!rance *ovein -''osite Directions
When Stocks *ove To+ether. Com/inin+ Them Doesn0t %ed!ce %isk*!ch
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Average Return and VolatilityFor Portfolios
Portfolio Stan#ar# De$iation
Portfolio%&
'ecte#!et"rn
0000 0050 0100 0150 0200 0250 0300 0350
0025
0020
0015
00100005
0000
50% W!SS 50%
MM+
!*S+!,* .!/+P
/ ,M!,MM+ /!P
W!SS
/M .!/+P
50% W!SS 50%
!*S+!,*
Wireless and mm!cell Correlation1 "#$"
Wireless and %eins!rance &ro!'1 ("#))
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Correlation Coefficients And RiskReduction For Two-Asset Portfolios
10%
15%
20%
25%
0% 5% 10% 15% 20% 25%
Standard Deviation of Portfolio Returns
"#$ect
edReturnonthePortfolio
is !.0
/.0 .0
is /.0
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Portfolios of MoreThanTwo Assets Five-Asset Portfolio
11
1111
&&%%
$$22))
REwREw
REwREwREwRE p
++
++=
&'ected Return of Portfolio Is Still T$e1vera2e Of &'ected Returns Of T$e T+o
Stoc#s
How s The 2ariance of Portfolio n3!enced By 4!m/er -fAssets in Portfolio?
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5
4
3
2
1
54321Asset
T$e )ovariance Ter"s Deter"ine To 1 Lar2e &tentT$e -ariance Of T$e Portfolio
Asset 1 2 3 4 51
2
3
4
55
4
3
2
154321Asset
-ariance of Individual 1ssets 1ccount Only for 345t$of t$e Portfolio -ariance
Variance Covariance Matrix
2
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What Is a Stocks Beta?
*eta Is a easure of Syste"atic Ris#
2m
imi
=
What fBeta 5 6or Beta
76?
, e "tock -oves -ore an 1. onAvera*e /en te -arket -oves 1.(Beta 0 1)
, e "tock -oves ess an 1. on
Avera*e /en te -arket -oves 1.(Beta 2 1)
What fBeta 8 6?
, e "tock -oves 1. on Avera*e /ente -arket -oves 1.
, An 3Avera*e4 evel o %isk
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Diversifiable And Non-Diversifiable
Risk
As Number of Assets Increases, Diversification
Reduces the Importance of a Stocks Own Variance Diversifiable risk, unsystematic risk
Only an Assets CovarianceWith All Other AssetsContributes Measurably to Overall Portfolio Return
Variance Non-diversifiable risk, systematic risk
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How Risky Is an Individual Asset?
First 1''roac$ 6 1sset7s -ariance or StandardDeviation
What Really Matters Is Systematic Risk.How an Asset Covaries WithEverything Else
se Asset6s Beta
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The Impact Of Additional AssetsOn The Risk Of A Portfolio
%umber of Securities &'ssets( in Portfolio%umber of Securities &'ssets( in Portfolio
PortfolioRi
sk)
kp
%ondiversifiable Risk%ondiversifiable Risk
Diversifiable RiskDiversifiable Risk
Total riskTotal risk
1 5 10 15 20 251 5 10 15 20 25
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7al&in* %isk8 Assets "o&ld ake 'nto Acco&ntE$pected %et&rn and %isk
-ost 'nvestors 9 %isk Averse 9 #emandCompensation or Bearin* %isk
%isk Can Be #e:ned 'n -an8 /a8s
-arket "o&ld %e+ard Onl8 "8stematic %isk
Risk and Return
$%
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THE EFFICIENT PORTFOLIOS Anefficient portfoliois one that offers the most return for a given amount ofrisk, or the least risk for a given amount of return.
Suppose, we assume two securities A, kA= 5% with SD A,A= 4%, and, kB= 8%
andB= 10%.
If x equals 0.75, then kp= 5.75% kp= xAkA + xBkB
= 0.75 (5%) + 0.25 (8%) = 5.75%
Next we can determineP.Substitute the given values forA,B, and rAB, and then
solve forPat different values of x. For example, in the case where rAB= 0 and x
=0.75, thenP= 3.9%.
=(0.5625) (16) + 0.0625) (100) + 2 (0.75) (0.25) (0) (4) (10)
=9.00 + 6.25 =15.25 = 3.9%
)12)1 82222
p ABrXXXX BA ++=
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Proportion ofPortfolio inSecurity 8
19alue of :
Proportion ofPortfolio inSecurity
19alue of )+:
!ase ;1r8* ;=? 5=@5 5=5 5=@5 ?= 5=@5 ;=5
;=5 ;=5 =5 @ =5 5=< =5 ?
;=? ;=> @=!5 >=5 @=!5 @= @=!5 =5
; 1 > 1; > 1; > 1;
An Example
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Illustration of Portfolio Returns, Risk,
and the Attainable Set of Portfolios c* 'ttainable Set of Risk+Return ,ombinations
,ase -.
r'/ 1*0
,ase --.
r'/ 0
,ase ---.
r'/ !1*0
k'5
a* Returns b* Risk
1002' Portfolio
'llocation1002/
Percent
k:;9kP
k'5
k'5
kP
kP
k/3
k/3
1002'
1002'
Portfolio
'llocation
Portfolio
'llocation
1002/
1002/
'4
P
/10
/10
P
'4
1002'
1002'
Portfolio
'llocation
Portfolio
'llocation
1002/
1002/
1002/1002' Portfolio
'llocation
'4P
/10
"#$ected
Return)
&2(kP
rP&2(
rP&2(
3
5 '
:
'
:
3
5
4 10
Risk) P &2(4 10
Risk) P &2(
5
3
'
:
Risk) P &2(4 10
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)8OOSI9: T8 OPTI1L PORTFOLIO
T8 FFI)I9T ST OF I9-ST9T
"#$ected PortfolioReturn) kP
/
'
,
D "
6
78easible) or
'ttainable Set
"fficient Set &/,D"(
Risk) P &2(
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RISK3RT;R9 I9DIFFR9) );R-S
=s >isk Premium 1>P for>isk * P* $0$6? >P=* 20&6
/ ) 2 $ % & ' , - .
@s >isk Premium 1>P for
>isk * P
* $0$6? >P@
* )0/6
;=
;@
5Apected >ate of >eturn( rP
&
'
,
-
.
)/
%
$2
)
>isk( p16
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SL)TI9: OPTI1L PORTFOLIO OF RISKisk( p16
B
A
-
,02
'
%2 ' - )/%02 ,0)
'D? 'D! 'D1 '! '1
'?
,
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T8 )1PIT1L 1RKT LI9Investors Equilibrium: Combining the Risk Free Asset with the Market Portfolio
>isk( P
Increasing utility
"
N
8
7
4
5
@
k"
kP
k>3
P "
;$ ;2 ;)
5Apected >ate of
>eturn( kP
>
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CAPITAL MARKET LINE (CML)
P
"
>3">3p
B
2k+kC1kkC!"#
+==
"kC
>3kC
" >isk( P/
5Apected >ate of
>eturn( kP
"
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)1L);L1TI9: *T1 )OFFI)I9T
>ealied >eturn on Stock EkE16
>ealied >eturnon the market( k"
aE * ;ntercept * +-0.6
'0))/
)'F
F
==
==
m
j
j
k
k
run
riseb
kE * aE
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Thank You..Thank You..