Investments
CHAPTER 7
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Optimal Risky
Portfolios
Slides by
Richard D. Johnson
McGraw-Hill/Irwin
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
Figure 7.1 Portfolio Risk as a Function of the
Number of Stocks in the Portfolio
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Figure 7.2 Portfolio 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 = Expectedn return on Security 2
w 1
i 1
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Two-Security Portfolio: Risk
p2 = w1212 + w2222 + 2W1W2 Cov(r1r2)
12 = Variance of Security 1
22 = Variance of Security 2
Cov(r1r2) = Covariance of returns for
Security 1 and Security 2
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Covariance
Cov(r1r2) = 1,2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 >  > -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12 + W3232
+ 2W1W2 Cov(r1r2)
+ 2W1W3 Cov(r1r3)
+ 2W2W3 Cov(r2r3)
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Table 7.1 Descriptive Statistics for Two Mutual
Funds
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Table 7.2 Computation of Portfolio Variance
from the Covariance Matrix
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Table 7.3 Expected Return and Standard
Deviation with Various Correlation Coefficients
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Figure 7.3 Portfolio Expected Return as a
Function of Investment Proportions
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Figure 7.4 Portfolio Standard Deviation as a
Function of Investment Proportions
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Figure 7.5 Portfolio Expected Return as a
function of Standard Deviation
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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.
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Figure 7.6 The Opportunity Set of the Debt and
Equity Funds and Two Feasible CALs
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Figure 7.7 The Opportunity Set of the Debt and Equity Funds with
the Optimal CAL and the Optimal Risky Portfolio
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Figure 7.8 Determination of the Optimal Overall
Portfolio
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Figure 7.9 The Proportions of the Optimal
Overall Portfolio
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Figure 7.10 The Minimum-Variance Frontier of
Risky Assets
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Figure 7.11 The Efficient Frontier of Risky
Assets with the Optimal CAL
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Figure 7.12 The Efficient Portfolio Set
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Figure 7.13 Capital Allocation Lines with
Various Portfolios from the Efficient Set
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Table 7.4 Risk Reduction of Equally Weighted Portfolios
in Correlated and Uncorrelated Universes
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