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8/6/2019 Optimal Risk Portfolio
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Optimal Risky Portfolios
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Risk Reduction with
Diversification
Number ofSecurities
St. Deviation
Market Risk
Unique Risk
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Risk Reduction with
Diversification
http://../Demo/Diversification%20effect.xlsx8/6/2019 Optimal Risk Portfolio
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rp = W1r1 + W2r2W
1= Proportion of funds in Security 1
W2 = Proportion of funds in Security 2
r1 = Expected return on Security 1
r2 = Expected return on Security 21
n
1i
iw
Two-Security Portfolio: Return
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p2 = w1212 + w2222 + 2W1W2 Cov(r1r2)12 = Variance of Security 122 = Variance of Security 2
Cov(r1r2) = Covariance of returns for
Security 1 and Security 2
Two-Security Portfolio: Risk
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1,2
= Correlation coefficient of
returns
Cov(r1r2) = 1,212
1 = Standard deviation ofreturns for Security 12 = Standard deviation ofreturns for Security 2
Covariance
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Range of values for 1,2+ 1.0 > > -1.0
If= 1.0, the securities would be perfectlypositively correlated
If= - 1.0, the securities would beperfectly negatively correlated
Correlation Coefficients: Possible
Values
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2p = W1212 + W2212+ 2W1W2
rp = W1r1 + W2r2 + W3r3
Cov(r1r2)
+ W3232
Cov(r1r3)+ 2W1W3
Cov(r2r3)+ 2W2W3
Three-Security Portfolio
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rp = Weighted average of the
n securitiesp2 = (Consider all pairwise
covariance measures)
In General, For An N-Security
Portfolio:
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E(rp) = W1r1 + W2r2
p2 = w1212 + w2222 + 2W1W2 Cov(r1r2)p = [w1212 + w2222 + 2W1W2 Cov(r1r2)]1/2
Two-Security Portfolio
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Portfolios with Different
Correlations
= 1
13%
%8
E(r)
St. Dev12% 20%
= .3
= -1
= -1
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Correlation Effects
The relationship depends on correlation
coefficient.
-1.0 < < +1.0 The smaller the correlation, the greater the
risk reduction potential.
If = +1.0, no risk reduction is possible.
http://../Demo/Portfolio%20Standard%20Deviation%20as%20Correlation%20Changes.xls8/6/2019 Optimal Risk Portfolio
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1
1 2
22 - Cov(r1r2)W
1
=
+ - 2Cov(r1r2)
W2 = (1 - W1)
2
2E(r2) = .14 = .20Sec 212 = .2
E(r1) = .10 = .15Sec 1
2
Minimum-Variance Combination
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W1 =(.2)2 - (.2)(.15)(.2)
(.15)2 + (.2)2 - 2(.2)(.15)(.2)
W1 = .6733
W2 = (1 - .6733) = .3267
Minimum-Variance Combination:
= .2
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rp = .6733(.10) + .3267(.14) = .1131
p = [(.6733)2(.15)2 + (.3267)2(.2)2 +
2(.6733)(.3267)(.2)(.15)(.2)]1/2
p = [.0171]1/2
= .1308
Risk and Return: Minimum
Variance
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W1 =(.2)2 - (.2)(.15)(.2)
(.15)2 + (.2)2 - 2(.2)(.15)(-.3)
W1 = .6087
W2 = (1 - .6087) = .3913
Minimum - Variance Combination:
= -.3
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rp = .6087(.10) + .3913(.14) = .1157
p = [(.6087)2(.15)2 + (.3913)2(.2)2 +
2(.6087)(.3913)(.2)(.15)(-.3)]1/2
p= [.0102]1/2
= .1009
Risk and Return: Minimum
Variance
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Extending Concepts to All
Securities
The optimal combinations result in lowest
level of risk for a given return.
The optimal trade-off is described as the
efficient frontier.
These portfolios are dominant.
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Minimum-Variance Frontier of Risky Assets
E(r) Efficient
frontier
Global
minimum
variance
portfolio Minimumvariance
frontier
Individual
assets
St. Dev.
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Alternative CALs
M
E(r)
CAL (Global
minimum variance)
CAL (A)CAL (P)
P
A
F
P P&F A&FM
A
G
P
M
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Efficient Frontier with Lending & Borrowing
E(r)
Frf
A
P
Q
B
CAL