Expert estimates Portfolio (return and risk)
Portfolio Theory
Lecture 1
Luděk Benada
Department of Finance - 402, benada.esf@gmail.com
February 29, 2016
□ S1
Luděk Benada MPF APOT
Q Expert estimates
Q Portfolio (return and risk)
Luděk Benada MPF APOT
Expert estimates Portfolio (return and risk)
Estimates of market prices of assets at the time of realization
Pnt^
\r\ s\        w i s*> s\   y-vf       f— f— /-\+- Í r* 1    ir    Ln/MAr        +■   4- \r% r\    tri /-\ i     +■    y-vf   ki min/r    I r^^xllir^^r
Luděk Benada MPF APOT
Expert estimates Portfolio (return and risk)
o Estimates of market prices of assets at the time of realization
o N experts will provide estimates for all actives (considered for investment)
I n ^ \^        ^ I   111 ^ ^ ^ i   n  i c 11 c^^^J ^        i*^     1*^ ^ 1*^ ill ^\/      ^ 11     11
The orice of assetfs^ is know at the ooint of buvin£ (selling^
Luděk Benada MPF APOT
Expert estimates Portfolio (return and risk)
o Estimates of market prices of assets at the time of realization
o N experts will provide estimates for all actives (considered for investment)
• In the calculation is used the probability structure
L _    nA!nl    ~.£   L...,!n/r    (t- \ n rr-
=     vT) c\ o
Luděk Benada
MPF APOT
o Estimates of market prices of assets at the time of realization
o N experts will provide estimates for all actives (considered for investment)
• In the calculation is used the probability structure a No dividend payment considered
The orice of assetfs^ is know at the ooint of buvin£ (selling
Luděk Benada MPF APOT
Expert estimates Portfolio (return and risk)
Estimates of market prices of assets at the time of realization
N experts will provide estimates for all actives (considered for investment)
In the calculation is used the probability structure No dividend payment considered
The price of asset(s) is know at the point of buying (selling)
Luděk Benada MPF APOT
If the probability of price development is
/ N
= y Li=i{n- r)2
Luděk Benada MPF APOT
Expert estimates Portfolio (return and risk)
• If the probability of price development is known
• The mean of the security could be determined
• n = E/Li n * pi
ano
—\9
= A
=     vT) c\ o
Luděk Benada
MPF APOT
Expert estimates Portfolio (return and risk)
• If the probability of price development is known
• The mean of the security could be determined
• n = L/=1 n * pi
ano
—\9
= A
Luděk Benada
MPF APOT
Expert estimates Portfolio (return and risk)
• If the probability of price development is known
• The mean of the security could be determined
• n = L/=1 n * pi
9 ... and thus the risk of security
/v^ N
• o# — \ 1^;—i I H ' - r) * Pi
V ^i- J
Luděk Benada MPF APOT
Expert estimates Portfolio (return and risk)
• If the probability of price development is known
• The mean of the security could be determined
• n = L/=1 n * pi
9 ... and thus the risk of security
• a; = y/L?=i(n-T)2*Pi
Luděk Benada MPF APOT
Expert estimates Portfolio (return and risk)
Pit.. .Market price of an asset in the point of portfolio formation
M,j . . .
í r ■ i
• • Tr tne retur
r ■ . i
LI ic uc
riod
I d
Pyfc = 1! Piik
Luděk Benada MPF APOT
Expert estimates Portfolio (return and risk)
Pit.. .Market price of an asset in the point of portfolio formation
Nij.. .The number of total number of estimates for the future price (of i-th assets, done by j-th expert)
A/      Th    r b bilit        rdin    f " th x   rtx     tim t f
the return durins the period
I d
condition: lLp//7c = l!
... men musí De appnea ptJ    N p„k
ijk =
Luděk Benada MPF APOT
Expert estimates Portfolio (return and risk)
Pit.. .Market price of an asset in the point of portfolio formation
N;j.. .The number of total number of estimates for the future price (of i-th assets, done by j-th expert)
N;j... The probability according of j-th expert's estimates of the return during the period
I d
.. tnen must oe appnea pu    ^ *
P//7c = l!
Piik
ijk =
p.., - p..
Luděk Benada MPF APOT
Expert estimates Portfolio (return and risk)
Pit.. .Market price of an asset in the point of portfolio formation
N;j.. .The number of total number of estimates for the future price (of i-th assets, done by j-th expert)
N;j... The probability according of j-th expert's estimates of the return during the period
!ln accordance with the condition: Y^rLiPijk — 1!
.. t
ŽŇŽě *
ijk =
p.., -
Luděk Benada
MPF APOT
Expert estimates Portfolio (return and risk)
o Pit.. .Market price of an asset in the point of portfolio formation
• N;j.. .The number of total number of estimates for the future price (of i-th assets, done by j-th expert)
• N;j... The probability according of j-th expert's estimates of the return during the period
o !ln accordance with the condition: Y^rLiPijk — 1!
• ...then must be applied PiJ = ^ * LJ^ piJk
Luděk Benada MPF APOT
Expert estimates Portfolio (return and risk)
o Pit.. .Market price of an asset in the point of portfolio formation
• N;j.. .The number of total number of estimates for the future price (of i-th assets, done by j-th expert)
• N;j... The probability according of j-th expert's estimates of the return during the period
o !ln accordance with the condition: Y^rLiPijk — 1! . ...then must be applied pff = £ *E& Pff*
• r,jk =
Luděk Benada MPF APOT
Expert estimates Portfolio (return and risk)
Return
Luděk Benada MPF APOT
Expert estimates Portfolio (return and risk)
Return
Normalization of the probability
Luděk Benada MPF APOT
Return
Normalization of the probability The return of portfolio
The risk of portfolio
Luděk Benada MPF APOT
Expert estimates Portfolio (return and risk)
• Return
• Normalization of the probability
• The return of portfolio
• The risk of portfolio
Luděk Benada MPF APOT
□
Expert estimates Portfolio (return and risk)
A portfolio based of n assets
LI I
L  i la J vv
í a n o r\
LI IU J   LI I       I ^lUI II   W I    LIMO   U Wl L I V7MU VV
be: rn =
W; * r,
LI I ^
tLUI II   \J\    I LI
L   VV III    VJ \^ I
• • »LI
U   I ^LUI I I   \J\    LI I       \J \J I LI
VV I DC,
rP : : L/=:
= 1
Luděk Benada MPF APOT
Expert estimates Portfolio (return and risk)
A portfolio based of n assets i-th asset has w/and r;
LI I
be: rn =
W; * r,
LI I ^
LU I I I   \J I    I    LI I   CJ JJL L   Will   VJ \^ I
• • »LI
u r c i. u r    u i l 11 c uuiliumu w i uc,
rP  : Li :
= 1
Luděk Benada MPF APOT
o A portfolio based of n assets
• i-th asset has w/and r;
• .. .thus the return of this portfolio will be: rp = L/^Li wi * H
trie cxucc^Lcu return cjt i-t.ii dbbci wi   uc t\
9 .. .tnus tne expecceu return ot tne portToiio wi ue.
rp = L/=i w; * ra
under condition: EJ?=1 = 1
Luděk Benada MPF APOT
o A portfolio based of n assets
• i-th asset has w/and r;
• .. .thus the return of this portfolio will be: rp = L/^Li wi * H 9 if the expected return of i-th asset will be T\
9 .. .tnus tne expecceo return ot tne portToiio wi ue.
rp = L/=i wi * Hi
under condition: EJ?=1 = 1
Luděk Benada MPF APOT
o A portfolio based of n assets
• i-th asset has w/and r;
• .. .thus the return of this portfolio will be: rp = L/^Li wi * H 9 if the expected return of i-th asset will be T\
• ...thus the expected return of the portfolio will be:
h = L/=i    * rounder condition: I?=1 = l
Luděk Benada MPF APOT
o A portfolio based of n assets
• i-th asset has w/and r;
• .. .thus the return of this portfolio will be: rp = L/^Li wi * H 9 if the expected return of i-th asset will be T\
• ...thus the expected return of the portfolio will be:
h = L/=i   * m
• under condition: I?=i = 1
Luděk Benada MPF APOT
o A portfolio based of n assets
• i-th asset has w/and r;
• .. .thus the return of this portfolio will be: rp = L/^Li wi * H 9 if the expected return of i-th asset will be T\
• ...thus the expected return of the portfolio will be:
h = L/=i   * m
• under condition: I?=i = 1
• Note: It is posible to have 1/1/; < 0, then sale short is executed
Luděk Benada MPF APOT
Expert estimates Portfolio (return and risk)
rn
op = y/D(X) = y/LiLiLiLiWi*Wj*aij = Vw'Vw
j    -1 ■ i j
-V   / J
Luděk Benada MPF APOT
Expert estimates Portfolio (return and risk)
rn
op = y/D(X) = y/LiLiLiLiWi*Wj*aij = Vw'Vw
AI     I A /
/V    ^i — .
jj * oř +
Luděk Benada MPF APOT
Expert estimates Portfolio (return and risk)		
■		
	rn	
• A special case (equal weights)
2_Iy/V    erf      N-lyN   yN °ij      ^ (j2 -
\_    =2 i  A/-1 _
Iv /V
Luděk Benada MPF APOT
Expert estimates Portfolio (return and risk)		
■		
	rn	
• ap = y/D(X) = yjL"=iLiLiWi*Wj*Gij = Vw'Vw
• ffp = I/li wf * of+ZiLxHjLl/i^j Wi * WJ * Of/
• A special case (equal weights)
Ärr2_lV/V i   N  lyN   VN °ij      ^    2 _
• °P — N^i=l N   *     N   Li=lLj=\/i^j A/*(A/-1)^°P ~
N      1   +   A/   * Q(/
Luděk Benada MPF APOT
The contribution of a partial risk to the total risk of the portfolio is decreasing to zero with growing number of securities
c   i a / i 4- h   4- h ä   rx~ vf\\ a/i n rx~   nil rv^ \r\ ,
of a
LUV
Luděk Benada MPF APOT
Expert estimates Portfolio (return and risk)
The contribution of a partial risk to the total risk of the portfolio is decreasing to zero with growing number of securities
The contribution to the portfolio risk flowing from covariance is with the growing number of assets approaching an average covariance
lUIVIUUd     IbK. Ul bcLU I I llco L.UUIU LK lUVcU L-UI 1L)I -Lcly.
Luděk Benada MPF APOT
Expert estimates Portfolio (return and risk)
• The contribution of a partial risk to the total risk of the portfolio is decreasing to zero with growing number of securities
• The contribution to the portfolio risk flowing from covariance is with the growing number of assets approaching an average covariance
• The individual risk of securities could be removed completely...
Luděk Benada MPF APOT