Instructions Grading
Introduction to Portfolio Theory
Portfolio Theory
Lecture 1
Luděk Benada
Department of Finance - 402, benada.esf@gmail.com
February 23, 2016
Luděk Benada MPF_APOT
Q Instructions
Q Grading
Q Introduction to Portfolio Theory
Luděk Benada MPF_APOT
1.. Active work at seminar (max. 3 absence)
Luděk Benada MPF_APOT
Instructions Grading
Introduction to Portfolio Theory
Instructions
• 1.. Active work at seminar (max. 3 absence)
• 2.. Bloomberg 5 Stocks—)>Covar and Correl Matrix
)
/ V— r^i r\ I       -\ i— f-sL f~ r\    O /
/04/2016. 2nd test - lo/05/2016
Luděk Benada MPF_APOT
□ iS1
Instructions Grading
Introduction to Portfolio Theory
Instructions
• 1.. Active work at seminar (max. 3 absence)
• 2.. Bloomberg 5 Stocks—)>Covar and Correl Matrix
• 3.. Two tests (Z30 p., each 15 p., v0 min. 60 %)
ivi , ■    r |B_i_B "1    O '' P~ 7 |
1st test   4/04/2016, 2nd test 16/05/2016
Luděk Benada MPF_APOT
□ iS1
Instructions Grading
Introduction to Portfolio Theory
Instructions
• 1.. Active work at seminar (max. 3 absence)
• 2.. Bloomberg 5 Stocks—)>Covar and Correl Matrix
• 3.. Two tests (Z30 p., each 15 p., v0 min. 60 %)
• No satisfy condition 1-3—^"F"
1st test - 4/04/2016, 2nd test - 16/05/2016
Luděk Benada MPF_APOT
□ iS1
Instructions Grading
Introduction to Portfolio Theory
Instructions
• 1.. Active work at seminar (max. 3 absence)
• 2.. Bloomberg 5 Stocks—)>Covar and Correl Matrix
• 3.. Two tests (Z30 p., each 15 p., v0 min. 60 %)
• No satisfy condition 1-3—^"F"
• 1st test - 4/04/2016, 2nd test - 16/05/2016
Luděk Benada MPF_APOT
□ iS1
Instructions Grading
Introduction to Portfolio Theory
Instructions
• 1.. Active work at seminar (max. 3 absence)
• 2.. Bloomberg 5 Stocks—)>Covar and Correl Matrix
• 3.. Two tests (Z30 p., each 15 p., v0 min. 60 %)
• No satisfy condition 1-3—^"F"
• 1st test - 4/04/2016, 2nd test - 16/05/2016 o Correction test (30 points)
Luděk Benada MPF_APOT
□ iS1
• 1.. Active work at seminar (max. 3 absence)
• 2.. Bloomberg 5 Stocks—)>Covar and Correl Matrix
• 3.. Two tests (Z30 p., each 15 p., v0 min. 60 %)
• No satisfy condition 1-3—^"F"
• 1st test - 4/04/2016, 2nd test - 16/05/2016 o Correction test (30 points)
9 Literature:ELTON, E.; Modern portfolio theory and investment analysis
Luděk Benada MPF_APOT
Instructions Grading
Introduction to Portfolio Theory
Marks
Prerequisite 1 + 2y/
— 5, ^
)
Luděk Benada MPF_APOT
Instructions Grading Introduction to Portfolio Theory			
Marks			
• Prerequisite 1 + 2y/			
9 Score of both tests:			
• A: [27,30)			
• B: [25,27)			
• C: [23,25)			
• D: [21,23)			
• E: [18,21)			
• F: [0,18)			
	□ i5>		
Luděk Benada	MPF_APOT		
Instructions Grading
Introduction to Portfolio Theory
Introduction to Portfolio Theory
History:
• HICKS, J. application of Mathematical Methods of the Theory of Risk (1934)
iv a a r*» i x\ a li —i—7    i i    i~\     i r  i' /     ."       f -\ r\x~r\\
\A/     F " C
/ V
(1976)
'o
Luděk Benada MPF_APOT
Instructions Grading
Introduction to Portfolio Theory
Introduction to Portfolio Theory
History:
• HICKS, J. application of Mathematical Methods of the Theory of Risk (1934)
* MARKOWITZ, H.;Portfolio Selection (1952) -Founder of MPT(inovative rA<7^>Efficient frontier)
cpiAppp \A/   p 'C^nif^l A<z<zpt Prirp<z' A Thpnrv nf hAzirkpf
/ V
(1976)
'o
Luděk Benada MPF_APOT
Instructions Grading
Introduction to Portfolio Theory
Introduction to Portfolio Theory
• History:
• HICKS, J. application of Mathematical Methods of the Theory of Risk (1934)
• MARKOWITZ, H.;Portfolio Selection (1952) -Founder of MPT(inovative rA<7^>Efficient frontier)
• SHARPE, W., F.; Capital Asset Prices: A Theory of Market Equilibrium under Condition of Risk (1964) - (Discoverer of CAPM)
9 ROSS S ■ The Arbitrage Theory of Caoital Asset Pricing (1976)
Luděk Benada MPF_APOT
Instructions Grading
Introduction to Portfolio Theory
Introduction to Portfolio Theory
• History:
• HICKS, J. application of Mathematical Methods of the Theory of Risk (1934)
• MARKOWITZ, H.;Portfolio Selection (1952) -Founder of MPT(inovative rA<7^>Efficient frontier)
• SHARPE, W., F.; Capital Asset Prices: A Theory of Market Equilibrium under Condition of Risk (1964) - (Discoverer of CAPM)
• ROSS, S.; 77?e Arbitrage Theory of Capital Asset Pricing (1976)
Luděk Benada MPF_APOT
Portfolio
• Portfolio:£"=1 i/V/A'/L/Li w;= 1; Xw;... weigh;A;... asset
,        oUdldUl I Ly y[J\ IL-cJ
f       >*"V I/-* I -I— I >""V I/-* >-v J- «—v x"* -I— ■ X"* 1/-*        I ^ I <—V ■ I ■ -
Assuming a investor. The aim of building; a portfolio is
Luděk Benada MPF_AP0T
Portfolio
• Portfolio:!^ w;A;L"=i wi= v- Xw> • ■ • weiSh<Ai ■ ■ •asset
9 Conditions of assets - identifiability, mesuarability (price)
a I n\/PQtmpnt ■ ffr CT 1)
o Assuming a investor. The aim of building a portto
Luděk Benada MPF_APOT
Portfolio
• Portfolio:£"=1 i/V/A'/L/Li wj= 1; Xw\... weigh] A i... asset
• Conditions of assets - identifiability, mesuarability (price)
• Investment:/7^ o,1)
Assuming a investor The aim of building a oortfo
to inq sucn a composition ot assets tnat corresponds nis
Luděk Benada MPF_APOT
Portfolio
• Portfolio:£"=1 i/V/A'/L/Li wj= 1; Xw\... weigh]Ai... asset
• Conditions of assets - identifiability, mesuarability (price)
• Investment:/^ o,1)
• Assuming arational investor. The aim of building a portfolio is to find such a composition of assets that corresponds his needs.
Luděk Benada MPF_APOT
• Return of an asset:
r ■ = n( Pj- i / i
■ / \    t-\-K)
\      * J
y a _ I
• r/c/ =
i
Luděk Benada MPF_AP0T
• Return of an asset:
• r.
ln(Pt+k)-ln(Pt)
y _ ___I
• r/c/ =
i
Luděk Benada MPF_APOT
• Return of an asset:
• r; =
• r; =
r/c/
ln(Pt+k)-ln(Pt)
Pt+k-Pt
Pt
_ r t+K
/
Luděk Benada MPF_APOT
• Return of an asset:
• r; =
• r; =
• r/c/
ln(Pt+k)-ln(Pt)
Pt+k-Pt
Pt - OL
_Pt+k-Pt + D;
Luděk Benada MPF_APOT
9 Return of an asset:
• n = ln{Pt+k)-ln{Pt) ' pt
9 x\Tota\ — ~pt
Luděk Benada MPF_APOT
Instructions Grading
Introduction to Portfolio Theory
Return as random variable
• Uncertainty in the future development=4>random variable X (discreet random variable)
^Characteristic of RV E(X),cr2(X)=4>Mean Variance Portfolio
/______.___'  . ;__„ r .____
Luděk Benada
MPF_APOT
Instructions Grading
Introduction to Portfolio Theory
Return as random variable
• Uncertainty in the future development=4>random variable X (discreet random variable)
^Characteristic of RV E(X),cr2(X)=4>Mean Variance Portfolio
Mean
a // — JLv^ y.
•—\    J      ' * 1
a "=VAJ  -2^/=ix/ Plx/J
Characteristics of msan'
Luděk Benada MPF_APOT
Instructions Grading
Introduction to Portfolio Theory
Return as random variable
• Uncertainty in the future development=4>random variable X (discreet random variable)
^Characteristic of RV E(X),cr2(X)=4>Mean Variance Portfolio
Mean
9   t^AJ   ~Li=lXl P\Xl)
Characteristics of msan'
Luděk Benada MPF_APOT
Instructions Grading
Introduction to Portfolio Theory
Return as random variable
• Uncertainty in the future development=4>random variable X (discreet random variable)
^Characteristic of RV E(X),cr2(X)=4>Mean Variance Portfolio
Mean
• E(X) =ip=1X/l
* E(x) =L/Lix' p(
Characteristics of mean.
Luděk Benada MPF_APOT
Instructions Grading
Introduction to Portfolio Theory
Return as random variable
• Uncertainty in the future development=4>random variable X (discreet random variable)
^Characteristic of RV E(X),cr2(X)=4>Mean Variance Portfolio
Mean
• E(X) =iEiLiX/i
• E(X) =E?=iX/*p(x/) Characteristics of msan'
Luděk Benada MPF_APOT
Instructions Grading
Introduction to Portfolio Theory
Return as random variable
• Uncertainty in the future development=4>random variable X (discreet random variable)
^Characteristic of RV E(X),cr2(X)=4>Mean Variance Portfolio
Mean
• E(X) =iEf=iX/i
• E(X) =ir=1x/*p(x/)
• Characteristics of mean:
(c)
= c.b
(1
(X + Y)    E(X) + E(Y)
Luděk Benada MPF_APOT
Instructions Grading
Introduction to Portfolio Theory
Return as random variable
• Uncertainty in the future development=4>random variable X (discreet random variable)
^Characteristic of RV E(X),cr2(X)=4>Mean Variance Portfolio
Mean
• E(X) =iEf=iX/i
• E(X) =Ef=iX/*p(x/)
• Characteristics of mean:
• E(c) = c, where c is a constant
E(c*X) = c.E(X)
• E(X + Y) = E(X) + E(Y)
~ V       /       ~ V   / ■ ~ V /
□ iS> ►   4 S ►   4 >
Luděk Benada MPF_APOT
Instructions Grading
Introduction to Portfolio Theory
Return as random variable
• Uncertainty in the future development=4>random variable X (discreet random variable)
^Characteristic of RV E(X),cr2(X)=4>Mean Variance Portfolio
Mean
• E(X) =iEf=iX/i
• E(X) =Ef=iX/*p(x/)
• Characteristics of mean:
• E(c) = c, where c is a constant
• E(c*X) = c.E(X)
• E(X + Y) = E(X) + E(Y)
~ V       /       ~ V   / ■ ~ V /
□ [51 ►   4 S ►   4 >
Luděk Benada MPF_APOT
Instructions Grading
Introduction to Portfolio Theory
Return as random variable
• Uncertainty in the future development=4>random variable X (discreet random variable)
^Characteristic of RV E(X),cr2(X)=4>Mean Variance Portfolio
Mean
• E(X) =iEf=iX/i
• E(X) =ir=1x/*p(x/)
• Characteristics of mean:
• E(c) = c, where c is a constant
• E(c*X) = c.E(X)
• E(X + Y) = E(X) + E(Y)
V       /        V   / ■   V /
□ iS> ►   4 S ►   4 >
Luděk Benada MPF_APOT
Instructions Grading
Introduction to Portfolio Theory
Return as random variable
• Uncertainty in the future development=4>random variable X (discreet random variable)
^Characteristic of RV E(X),cr2(X)=4>Mean Variance Portfolio
Mean
• E(X) =iEf=iX/i
• E(X) =Ef=iX/*p(x/)
• Characteristics of mean:
• E(c) = c, where c is a constant
• E(c*X) = c.E(X)
• E(X + Y) = E(X) + E(Y)
• E(X*Y) = E(X).E(Y)
Luděk Benada MPF_APOT
Instructions Grading Introduction to Portfolio Theory	
Dispersion of RV	
Variace(discete case... (D, var,<72, s2) D(X) = E[X - E(X)]2= E(X)2- [E(X)]2 D(X) =LtiE[X/ - E(X)]2*p(X()
> n >
	( VM
[Xj E	
n < 30
Luděk Benada MPF_APOT
Instructions Grading Introduction to Portfolio Theory	
Dispersion of RV	
• Variace(discete case... (D, var,<72, s2) D(X) = E[X - E(X)]2= E(X)2- [E(X)]2 D(X) =E?=1E[X, - E(X)]2*p(X,)	
• Property of var:	
D(c+X) = D(X), D(c) = D(c /x) — c D(/x)	
. D(X+Y) = D(X) + D(Y). V                /                 V     /                 V /	• • i   0*rnx/fv vi      HpnpnHpnt RV =4> n >
30=>* er = - y"/_i E\X; E(	/   1 J
^^^^ i_i i n < 30=4>Sample variance	~ n—i^/=l L ' '-v'vJ
Luděk Benada	MPF_APOT
Instructions Grading Introduction to Portfolio Theory		
Dispersion of RV		
• Variace(discete case... (D, var,<72, s2) D(X) = E[X - E(X)]2= E(X)2- [E(X)]2 D(X) =E?=1E[X, - E(X)]2*p(X,) • Property of var: • D(c+X) = D(X), D(c) = 0 D(c*X) = c2*D(X) • D(X+Y) = D(X) + D(Y)... r\/y i \y\      DfX^ -1- DfY^ -1- 9*rn\/fX           HpnpnHpnt RV		
	=4> n >	
30=^> G = - Y!i=i E[X, — E{ n < 30=4>Sample variance	v.       J1 ' n2_    1 Yn   Ffy ■ FfX^l^ ~ n—1^/=1   i   i '-v'vJ	
	«□►•«[^►«^►•«^^ j	
Luděk Benada	MPF_APOT	
Instructions Grading
Introduction to Portfolio Theory
Dispersion of RV
• Variace(discete case... (D, var,<7 , s ) D(X) = E[X - E(X)]2= E(X)2- [E(X)]2 D(X) =LtiE[X/ - E(X)]2*p(X()
• Property of var:
• D(c+X) = D(X), D(c) = 0
• D(c*X) = c2*D(X)
• D(X+Y) = D(X) + D(Y)...
> n >
	' VM
[Xj E(	
Luděk Benada MPF_APOT
Instructions Grading Introduction to Portfolio Theory	
Dispersion of RV	
• Variace(discete case... (D, var,<7 , s ) D(X) = E[X - E(X)]2= E(X)2- [E(X)]2 D(X) =LtiE[X/ - E(X)]2*p(X()
• Property of var:
• D(c+X) = D(X), D(c) = 0
• D(c*X) = c2*D(X)
• D(X+Y) = D(X) + D(Y)... independet RV
> n >
	' VM
[Xj E(	
Luděk Benada MPF_AP0T
Instructions Grading
Introduction to Portfolio Theory
Dispersion of RV
Variace(discete case... (D, var,<72, s2) D(X) = E[X - E(X)]2= E(X)2- [E(X)]2 D(X) =LtiE[X/ - E(X)]2*p(X()
Property of var:
• D(c+X) = D(X), D(c) = 0
• D(c*X) = c2*D(X)
• D(X+Y) = D(X) + D(Y)... independet RV
» D(X+Y) = D(X) + D(Y) + 2*cov(X,Y)... dependent RV
> n >
	' VM
[X; E(	
n < 30
1 rn
Luděk Benada MPF_APOT
Instructions Grading
Introduction to Portfolio Theory
Dispersion of RV
• Variace(discete case... (D, var,<7 , s ) D(X) = E[X - E(X)]2= E(X)2- [E(X)]2 D(X) =LtiE[X/ - E(X)]2*p(X()
• Property of var:
• D(c+X) = D(X), D(c) = 0
• D(c*X) = c2*D(X)
• D(X+Y) = D(X) + D(Y)... independet RV
» D(X+Y) = D(X) + D(Y) + 2*cov(X,Y)... dependent RV
• ! Statistical population !=4> n > 30=^a2 = iEi'=1£[X/-E(X)]2,
n < 30^Sample variancecT2=^Ii:^1E[X/ - E(X)]2
Luděk Benada MPF_APOT
Instructions Grading
Introduction to Portfolio Theory
Risk
... the change in expected return - standard deviation^/D(X) ((7,s)
r, - - r)2... n > 30,
°i=\jlh*l<U{n-ř)2... n < 30,
V L*i=\Vi - -r) * Hi
• • •   L i i v.*      i \j u Ct u 111 u y   i <j   r\ i i \j v v
Luděk Benada MPF_APOT
Instructions Grading
Introduction to Portfolio Theory
Risk
... the change in expected return - standard deviation^/D(X) (<7,s)
• Gi=\lkn*Vj=An-ř)2... n > 30, ai=J7^I*Z"=i{n-ř)2... n < 30,
V
a it--    /Vn   (r- * n-
• • •   L i i v.*      i \j u Ct u 111 u y   i <j   r\ i i \j v v
Luděk Benada MPF_APOT
Instructions Grading
Introduction to Portfolio Theory
Risk
... the change in expected return - standard deviation^/D(X) (<7,s)
• Gi=\lkn*Vj=An-ř)2... n > 30,
• <yi=y/^*V=i(n-r)2... n < 30,
. . .   LI IC  L/l \JUd Ul 11 LV   1-3   r\l IL/VV
Luděk Benada MPF_APOT
Instructions Grading
Introduction to Portfolio Theory
Risk
• Oi=y/j;*IX=i{n-ř)2... n > 30,
• <yi=y/^*V=i(n-r)2... n < 30,
o a/ = y/^"=1(r/ — ř)2 * p\... the probability is known
Luděk Benada MPF_APOT
Instructions Grading Introduction to Portfolio Theory		
The relation between RVs		
a Covariance.. .(cov(X,Y),öx y) cov (X,Y) = E{[X-E(X)][Y-E(Y)]};		
		
w   KJjj- - n_-y L^\—\ ['/     '/J   ]} j     'j\' ' Prooertv of covsrisnce'		
		
	4  □  ►   < fip t   4 Š ► <	
Luděk Benada	MPF_APOT	
Instructions Grading Introduction to Portfolio Theory	
The relation between RVs	
Covariance.. .(cov(X,Y),öx y)
cov (X,Y) = E{[X-E(X)][Y-E(Y)]};
• °x,y =^=i[X,-E(X)]*[Y,-E(Y)]
^■^tttELi k,—ř^Wj-řj]...n < 30,
Luděk Benada MPF_APOT
Instructions Grading
Introduction to Portfolio Theory
The relation between RVs
Covariance.. .(cov(X,Y),öx y)
cov (X,Y) = E{[X-E(X)][Y-E(Y)]};
• ox,y =£E?=i[X/-E(X)]*[Yl-E(Y)] ^■=Jl?=1[r/-ri-]*[r/-ö-]...n >30,
0// = -^rLLirri—/Vl*fr,—ňl...n < 30,
Luděk Benada MPF_APOT
Instructions Grading
Introduction to Portfolio Theory
The relation between RVs
• Covariance.. .(cov(X,Y),öx y)
cov (X,Y) = E{[X-E(X)][Y-E(Y)]};
• °x,y =^=i[X,-E(X)]*[Y,-E(Y)]
• <yij=lnZUlri-ň]*[rj-řj]..M > 30,
• <yij=^UVi-ř^j-řj\...^ < 30,
Luděk Benada MPF_APOT
Instructions Grading
Introduction to Portfolio Theory
The relation between RVs
• Covariance.. .(cov(X,Y),öx y)
cov (X,Y) = E{[X-E(X)][Y-E(Y)]};
• °x,y =^=i[X,-E(X)]*[Y,-E(Y)]
• <yij=ln^i=i^i-ň]*[rj-řj]..M > 30,
• <yij=^UVi-ř^j-řj\...^ < 30,
• Property of covariance:
cov(X,Y) = 0; E (X+Y) = OAE(X) = 0 = E(Y) = 0
/XX   \/\ /XX \/\
• cov(X,Yj cov(Y,XJ
• cov(X+a,Y+b) = cov(X,Y)
• cov(X*a,Y*b) = a*b*cov(X,Y)
range of covar (-oo;oo)
9 >S
Luděk Benada MPF_APOT
Instructions Grading
Introduction to Portfolio Theory
The relation between RVs
• Covariance.. .(cov(X,Y),öx y)
cov (X,Y) = E{[X-E(X)][Y-E(Y)]};
• °x,y =^=i[X-E(X)]*[Y,-E(Y)]
• <yij=lnZUlri-ň]*[rj-řj]..M > 30,
• criy=^LiLi[r/-ö]*[0-ö]...n < 30,
• Property of covariance:
• cov(X,Y) = 0; E (X+Y) = OAE(X) = 0 = E(Y) = 0
cov(X,Y) = cov(Y,X)
• cov(X+a,Y+b) = cov(X,Y)
• cov(X*a,Y*b) = a*b*cov(X,Y)
range of covar (-oo;oo)
Luděk Benada MPF_APOT
Instructions Grading
Introduction to Portfolio Theory
The relation between RVs
• Covariance.. .(cov(X,Y),öx y)
cov (X,Y) = E{[X-E(X)][Y-E(Y)]};
• °x,y =^=i[X-E(X)]*[Y,-E(Y)]
• <yij=ln^i=i^i-ň]*[rj-řj]..M > 30,
• <yij=^UVi-ř^j-řj\...^ < 30,
• Property of covariance:
• cov(X,Y) = 0; E (X+Y) = OAE(X) = 0 = E(Y) = 0
• cov(X,Y) = cov(Y,X)
• cov(X+a,Y+b) = cov(X,Y)
• cov(X*a,Y*b) = a*b*cov(X,Y)
range of covar (-oo;oo)
Luděk Benada MPF_APOT
Instructions Grading
Introduction to Portfolio Theory
The relation between RVs
• Covariance.. .(cov(X,Y),öx y)
cov (X,Y) = E{[X-E(X)][Y-E(Y)]};
• °x,y =^=i[X-E(X)]*[Y,-E(Y)]
• <yij=ln^i=i^i-ň]*[rj-řj]..M > 30,
• <yij=^UVi-ř^j-řj\...^ < 30,
• Property of covariance:
• cov(X,Y) = 0; E (X+Y) = OAE(X) = 0 = E(Y) = 0
• cov(X.Y) = cov(Y,X)
• cov(X+a,Y+b) = cov(X.Y)
• cov(X*a,Y*b) = a*b*cov(X,Y)
range of covar (-oo;<x>)
Luděk Benada MPF_APOT
Instructions Grading
Introduction to Portfolio Theory
The relation between RVs
• Covariance.. .(cov(X,Y),öx y)
cov (X,Y) = E{[X-E(X)][Y-E(Y)]};
• °x,y =^=i[X;-E(X)]*[Y-E(Y)]
• <yij=ln^i=i^i-ň]*[rj-řj]..M > 30,
• <yij=^UVi-ř^j-řj\...^ < 30,
• Property of covariance:
• cov(X,Y) = 0; E (X+Y) = OAE(X) = 0 = E(Y) = 0
• cov(X.Y) = cov(Y,X)
• cov(X+a,Y+b) = cov(X.Y)
a cov(X*a,Y*b) = a*b*cov(X,Y)
range of covar (-oo;oo)
Luděk Benada MPF_APOT
Instructions Grading
Introduction to Portfolio Theory
The relation between RVs
• Covariance.. .(cov(X,Y),öx y)
cov (X,Y) = E{[X-E(X)][Y-E(Y)]};
• °x,y =^=i[X-E(X)]*[Y,-E(Y)]
• <yij=lnZUlri-ň]*[rj-řj]..M > 30,
• <yij=^UVi-ř^j-řj\...^ < 30,
• Property of covariance:
• cov(X,Y) = 0; E (X+Y) = OAE(X) = 0 = E(Y) = 0
• cov(X.Y) = cov(Y,X)
• cov(X+a,Y+b) = cov(X.Y)
o cov(X*a,Y*b) = a*b*cov(X,Y) cov(X.X) =(72(X)
range of covar (-oo;<x>)
Luděk Benada MPF_APOT
Instructions Grading
Introduction to Portfolio Theory
The relation between RVs
• Covariance.. .(cov(X,Y),öx y)
cov (X,Y) = E{[X-E(X)][Y-E(Y)]};
• °x,y =^=i[X-E(X)]*[Y,-E(Y)]
• <yij=lnZUlri-ň]*[rj-řj]..M > 30,
• <yij=^UVi-ř^j-řj\...^ < 30,
• Property of covariance:
• cov(X,Y) = 0; E (X+Y) = OAE(X) = 0 = E(Y) = 0
• cov(X.Y) = cov(Y,X)
• cov(X+a,Y+b) = cov(X.Y)
o cov(X*a,Y*b) = a*b*cov(X,Y) cov(X.X) =(72(X)
range of covar (-oo;<x>)
• ^^standardization
Luděk Benada MPF_APOT
Instructions Grading Introduction to Portfolio Theory		
Pearosn 's correlation coefficient		
• The absolut dimension of covar is relativized «_____c°v(X,Y)		
PA Y	^\	
1 n   ^   1 ^"^^ ^    ^ ^ 1 ^™ J         ■     ^ ^^21 Iii n   ^ i ^51 n £C ^		
		
determination from OLS)	^                £ C i i	
Luděk Benada	MPF_APOT	
Instructions Grading Introduction to Portfolio Theory	
Pearosn 's correlation coefficient	
• The absolut dimension of covar is relativized	
Ä ^ cov(X,Y) • Oyy —--—-—-ry\r o~x*0~y	^\ f\
1 n   ^   1 ^"^^ ^    ^ ^ 12| i \^      " 1    (f"t^i Iii n   ^ i ^51 n £C ^	
	
determination from OLS)	^                    f i i
Luděk Benada	MPF_APOT
Instructions Grading
Introduction to Portfolio Theory
Pearosn's correlation coefficient
• The absolut dimension of covar is relativized
_ ^ cov(X,Y)
• Oyy —--—-—-
• Reflect the degree oflinear dependence
,-1-
Luděk Benada
MPF_APOT
Instructions Grading
Introduction to Portfolio Theory
Pearosn's correlation coefficient
• The absolut dimension of covar is relativized
_ ^ cov(X,Y)
• Oyy —--—-—-
• Reflect the degree oflinear dependence
o Interval for correlation <-l;l> (falling/rising)
*j I I ^\ if f*\   f\      f^ f\ if if f*\ I ^\    i f\       f^ f\ f*\   P i f^ i f*\ ( f | |
Luděk Benada MPF_APOT
Instructions Grading
Introduction to Portfolio Theory
Pearosn's correlation coefficient
• The absolut dimension of covar is relativized
_ ^ cov(X,y)
• Oyy —--—-—-
• Reflect the degree oflinear dependence
o Interval for correlation <-l;l> (falling/rising)
9 Pxy—       points lie on a straight line
determination from OLS)
Luděk Benada MPF_APOT
Instructions Grading
Introduction to Portfolio Theory
Pearosn's correlation coefficient
• The absolut dimension of covar is relativized
_ ^ cov(x,y)
• Oyy —--—-—-
ry\r o~x*0~y
• Reflect the degree oflinear dependence
o Interval for correlation <-l;l> (falling/rising) 9 Pxy—       points lie on a straight line
• Square of correlation coefficient.. .r2(coefficient of determination from OLS)
Luděk Benada MPF_APOT
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