Optimal Risky Portfolios

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Chapter 8

• Optimal Risky Portfolios

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Risk Reduction with Diversification
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Two-Security Portfolio Return
rp W1r1 W2r2 W1 Proportion of funds in
Security 1 W2 Proportion of funds in Security
2 r1 Expected return on Security 1 r2
Expected return on Security 2
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Two-Security Portfolio Risk
?p2 w12?12 w22?22 2W1W2 Cov(r1r2)
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Covariance
Cov(r1r2) ?????1?2
?1,2 Correlation coefficient of
returns
?1 Standard deviation of returns for
Security 1 ?2 Standard deviation of
returns for Security 2
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Correlation Coefficients Possible Values
Range of values for ?1,2
1.0 gt ????gt ?-1.0
If ?? 1.0, the securities would be perfectly
positively correlated If ?? - 1.0, the
securities would be perfectly negatively
correlated
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Three-Security Portfolio
rp W1r1 W2r2 W3r3
?2p W12?12
W22???
W32?32
2W1W2
Cov(r1r2)
Cov(r1r3)
2W1W3
Cov(r2r3)
2W2W3
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In General, For an n-Security Portfolio
rp Weighted average of the n securities
?p2 (Consider all pairwise
covariance measures)
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Two-Security Portfolio
E(rp) W1r1 W2r2
?p2 w12?12 w22?22 2W1W2 Cov(r1r2)
?p w12?12 w22?22 2W1W2 Cov(r1r2)1/2
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Two-Security Portfolios withDifferent
Correlations
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Portfolio Risk/Return Two Securities Correlation
Effects

• Relationship depends on correlation coefficient
• -1.0 lt ? lt 1.0
• The smaller the correlation, the greater the risk
  reduction potential
• If??? 1.0, no risk reduction is possible

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Minimum-Variance Combination
1
??2
- Cov(r1r2)
2

W1
??2
??2
- 2Cov(r1r2)

1
2
W2
(1 - W1)
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Minimum-Variance Combination ? .2
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Minimum -Variance Return and Risk with ? .2
rp .6733(.10) .3267(.14) .1131
?
(.6733)2(.15)2 (.3267)2(.2)2
p
1/2
2(.6733)(.3267)(.2)(.15)(.2)
1/2
?
.0171
.1308
p
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Minimum -Variance Combination ? -.3
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Minimum -Variance Return and Risk with ? -.3
rp .6087(.10) .3913(.14) .1157
?
(.6087)2(.15)2 (.3913)2(.2)2
p
1/2
2(.6087)(.3913)(.2)(.15)(-.3)
1/2
?
.0102
.1009
p
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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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The Minimum-Variance Frontierof Risky Assets
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Extending to Include Riskless Asset

• The optimal combination becomes linear
• A single combination of risky and riskless assets
  will dominate

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Alternative CALs
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Portfolio Selection Risk Aversion
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Efficient Frontier with Lending Borrowing