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CHAPTER 7
Optimal Risky Portfolios
Covariance and Correlation
• Portfolio risk depends on the correlation between the returns of the assets in the portfolio
• Covariance and the correlation coefficient provide a measure of the way returns two assets vary
Two-Security Portfolio: Return
Portfolio Return
Bond Weight
Bond Return
Equity Weight
Equity Return
p D ED E
P
D
D
E
E
r
r
w
r
w
r
w wr r
( ) ( ) ( )p D D E EE r w E r w E r
= Variance of Security D
= Variance of Security E
= Covariance of returns for Security D and Security E
Two-Security Portfolio: Risk
2 2 2 2 2 2 ( , )P D D E E D E D Ew w w Cov r r
2D
2E
( , )D ECov r r
Two-Security Portfolio: Risk Continued
• Another way to express variance of the portfolio:
2 ( , ) ( , ) 2 ( , )P D D D D E E E E D E D Ew w Cov r r w w Cov r r w w Cov r r
D,E = Correlation coefficient of returns
Cov(rD,rE) = DEDE
D = Standard deviation of returns for Security DE = Standard deviation of returns for Security E
Covariance
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
Correlation Coefficients: Possible Values
Table 7.1 Descriptive Statistics for Two Mutual Funds
2p = w1
212 + w2
212
+ 2w1w2 Cov(r1,r2)
+ w323
2
Cov(r1,r3)+ 2w1w3
Cov(r2,r3)+ 2w2w3
Three-Security Portfolio
1 1 2 2 3 3( ) ( ) ( ) ( )pE r w E r w E r w E r
Table 7.2 Computation of Portfolio Variance From the Covariance Matrix
Table 7.3 Expected Return and Standard Deviation with Various Correlation Coefficients
Minimum Variance Portfolio as Depicted in Figure 7.4
• Standard deviation is smaller than that of either of the individual component assets
• Figure 7.3 and 7.4 combined demonstrate the relationship between portfolio risk
Figure 7.5 Portfolio Expected Return as a Function of Standard Deviation
• The relationship depends on the correlation coefficient
• -1.0 < < +1.0• The smaller the correlation, the greater the risk
reduction potential• If = +1.0, no risk reduction is possible
Correlation Effects
Figure 7.6 The Opportunity Set of the Debt and Equity Funds and Two Feasible CALs
The Sharpe Ratio
• Maximize the slope of the CAL for any possible portfolio, p
• The objective function is the slope:
( )P fP
P
E r rS
Figure 7.7 The Opportunity Set of the Debt and Equity Funds with the Optimal CAL and the
Optimal Risky Portfolio
Figure 7.8 Determination of the Optimal Overall Portfolio
Figure 7.9 The Proportions of the Optimal Overall Portfolio
Markowitz Portfolio Selection Model
• Security Selection– First step is to determine the risk-return
opportunities available– All portfolios that lie on the minimum-variance
frontier from the global minimum-variance portfolio and upward provide the best risk-return combinations
Figure 7.10 The Minimum-Variance Frontier of Risky Assets
Markowitz Portfolio Selection Model Continued
• We now search for the CAL with the highest reward-to-variability ratio
Figure 7.11 The Efficient Frontier of Risky Assets with the Optimal CAL
Markowitz Portfolio Selection Model Continued
• Now the individual chooses the appropriate mix between the optimal risky portfolio P and T-bills as in Figure 7.8
2
1 1
( , )n n
P i j i ji j
ww Cov r r
Figure 7.12 The Efficient Portfolio Set
Capital Allocation and the Separation Property
• The separation property tells us that the portfolio choice problem may be separated into two independent tasks– Determination of the optimal risky portfolio is
purely technical– Allocation of the complete portfolio to T-bills
versus the risky portfolio depends on personal preference
Figure 7.13 Capital Allocation Lines with Various Portfolios from the Efficient Set
Diversification and Portfolio Risk
• Market risk– Systematic or nondiversifiable
• Firm-specific risk– Diversifiable or nonsystematic
The Power of Diversification
• Remember:
• If we define the average variance and average covariance of the securities as:
• We can then express portfolio variance as:
2
1 1
( , )n n
P i j i ji j
ww Cov r r
2 21 1P
nCov
n n
2 2
1
1 1
1
1( , )
( 1)
n
ii
n n
i jj ij i
n
Cov Cov r rn n
Table 7.4 Risk Reduction of Equally Weighted Portfolios in Correlated and Uncorrelated
Universes
Figure 7.1 Portfolio Risk as a Function of the Number of Stocks in the Portfolio