Post on 09-Apr-2018
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Chapter 8
PORTFOLIO THEORY
The Benefits of Diversification
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PORTFOLIO EXPECTED RETURN
n
E(RP) = w
i E(Ri)i=1
whereE(RP) = expected portfolio return
wi= weight assigned to security i
E(Ri) = expected return on security in = number of securities in the portfolio
Example A portfolio consists of four securities with expected returns
of 12%, 15%, 18%, and 20% respectively. The proportions of
portfolio value invested in these securities are 0.2, 0.3, 0.3, and 0.20respectively.
The expected return on the portfolio is:
E(RP) = 0.2(12%) + 0.3(15%) + 0.3(18%) + 0.2(20%)
= 16.3%
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PORTFOLIO RISK
The risk of a portfolio is measured by the variance (orstandard deviation) of its return. Although the expected
return on a portfolio is the weighted average of the
expected returns on the individual securities in the
portfolio, portfolio risk is not the weighted average of the
risks of the individual securities in the portfolio (except
when the returns from the securities are uncorrelated).
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MEASUREMENT OF COMOVEMENTS
IN SECURITY RETURNS
To develop the equation for calculating portfolio risk we
need information on weighted individual security risks
and weighted comovements between the returns of
securities included in the portfolio.
Comovements between the returns of securities are
measured by covariance (an absolute measure) and
coefficient of correlation (a relative measure).
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COVARIANCE
COV (Ri,Rj) =p1 [Ri1E(Ri)] [Rj1E(Rj)]
+p2 [Ri2E(Rj)] [Rj2E(Rj)]
+
+pn [RinE(Ri)] [RjnE(Rj)]
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ILLUSTRATION
T
S R R
-
9
4
7
T
R - 7
TR 9 4
T
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State of
nature
(1)
Probability
(2)
Return on
asset 1
(3)
Deviation of
the return on
asset 1 from its
mean
(4)
Return on
asset 2
(5)
Deviation of
the return on
asset 2 from
its mean
(6)
Product of the
deviations
times
probability
(2) x (4) x (6)
1 0.10 -10% -26% 5% -9% 23.4
2 0.30 15% -1% 12% -2% 0.6
3 0.30 18% 2% 19% 5% 3.0
4 0.20 22% 6% 15% 1% 1.2
5 0.10 27% 11% 12% -2% -2.2
Sum = 26.0
Thus the covariance between the returns on the two assets is 26.0.
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CO EFFIENT OF CORRELATION
Cov (Ri , Rj)
Cor (Ri , Rj) orVij=W(Ri ,Rj)
Wij
WiW
j
Wij= Vij. Wi . Wj
where Vij = correlation coefficient between the returns on
securities iandj
Wij = covariance between the returns on securities
iand j
Wi , Wj = standard deviation of the returns on securities
iandj
=
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PORTFOLIO RISK : 2 SECURITY CASE
Wp = [w12 W12 + w22 W22 + 2w1w2V12 W1 W2]
Example : w1 = 0.6 , w2 = 0. ,
W1 = 10%, W2 = 16%
V12 = 0.5
Wp = [0.62 x 102 + 0. 2 x 162 +2 x 0.6 x 0. x 0.5 x 10 x 16]
= 10. %
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PORTFOLIO RISK : n SECURITY CASE
Wp = [ wiwjVijWiWj]
Example : w1 = 0.5 , w2 = 0.3, and w3 = 0.2
W1 = 10%, W2 = 15%, W3 = 20%
V12 = 0.3, V13 = 0.5, V23 = 0.6
Wp = [w12 W1
2 + w22 W2
2 + w32 W3
2 + 2 w1 w2 V12 W1 W2
+ 2w2 w3 V13 W1 W3 + 2w2 w3 V23W2 W3]
= [0.52 x 102 + 0.32 x 152 + 0.22 x 202
+ 2 x 0.5 x 0.3 x 0.3 x 10 x 15
+ 2 x 0.5 x 0.2 x 05 x 10 x 20
+ 2 x 0.3 x 0.2 x 0.6 x 15 x 20]
= 10. %
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DOMINANCE OF COVARIANCE
As the number of securities included in a portfolio
increases, the importance of the risk of each individual
security decreases whereas the significance of the
covariance relationship increases
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PORTFOLIO OPTIONS
AND THE EFFICIENT FRONTIER
12%
20%
20% 0%
Risk, Wp
Expected
return , E(Rp)
1 (A)
6 (B)
23
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EFFICIENT FRONTIER FOR THE
n SECURITY CASE
Standard deviation, Wp
Expected
return ,E(Rp)
A O
N
M
X
F
B
D
Z
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OPTIMAL PORTFOLIO
P*
Q*
A
O
N
M
X
Standard deviation, Wp
Expected
return ,E(Rp)
IQ3IQ2
IQ1
IP3 IP2 IP1
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A
O
N
M
X
Standard deviation, Wp
Expected
return ,E(Rp)
Rf
u
D
E
VG
B
CF Y
I
II
S
RISKLESS LENDING AND
BORROWING OPPORTUNITY
Thus, with the opportunity of lending and borrowing, the efficient frontier changes. It is no
longerAFX. Rather, it becomes RfSG as it domniates AFX.
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SEPARATION THEOREM
Since RfSG dominates AFX, every investor would do well to
choose some combination ofRf and S. A conservative investor
may choose a point like U, whereas an aggressive investor may
choose a point like V.
Thus, the task of portfolio selection can be separated into twosteps:
1. Identification of S, the optimal portfolio of risky securities.
2. Choice of a combination ofRf
and S, depending on ones
risk attitude.
This is the import of the celebrated separation theorem
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SINGLE INDEX MODEL
INFORMATION INTENSITY OF THE MARKOWITZ MODEL
n SECURITIES
n VARIANCE TERMS & n(n 1) /2
COVARIANCE TERMS
SHARPES MODEL
Rit = ai + biRMt + eitE(Ri) = ai + biE(RM)
VAR (Ri) = bi2 [VAR (RM)] + VAR (ei)
COV (Ri,Rj) = bibjVAR (RM)
MARKOWITZ MODEL SINGLE INDEX MODEL
n (n + 3)/2 3n + 2
E(Ri) & VAR (Ri) FOR EACH bi, bjVAR (ei) FOR
SECURITY n( n 1)/2 EACH SECURITY &
COVARIANCE TERMS E(RM) & VAR (RM)
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SUMMING UP
Portfolio theory, originally proposed by Markowitz, is the
first formal attempt to quantify the risk of a portfolio anddevelop a methodology for determining the optimal portfolio.
The expected return on a portfolio ofn securities is :
E(Rp) = wiE(Ri)
The variance and standard deviation of the return on an
n-security portfolio are:
Wp2 = wiwjVijWiWj
Wp = wiwjVijWiWj
A portfolio is efficient if (and only if) there is an alternative
with (i) the same E(Rp) and a lower Wp or (ii) the same Wp anda higher E(Rp), or (iii) a higher E(Rp) and a lower Wp
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