Optimal Risky
Portfolios
CHAPTER 7
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
rp

rP
 Portfolio Return
wr
D
D
 wE r E
wD  Bond Weight
rD
 Bond Return
wE  Equity Weight
rE
 Equity Return
E (rp )  wD E (rD )  wE E (rE )
Two-Security Portfolio: Risk
  w   w   2wD E Cov(rD , rE )
2
P
2
D
2
D
2
E
2
E

2
D
= Variance of Security D

2
E
= Variance of Security E
Cov(rD , rE )= Covariance of returns for
Security D and Security E
Two-Security Portfolio: Risk Continued
• Another way to express variance of the portfolio:
 P2  wD wDCov(rD , rD )  wE wE Cov(rE , rE )  2wD wE Cov(rD , rE )
Covariance
Cov(rD,rE) = DEDE
D,E = Correlation coefficient of
returns
D = Standard deviation of
returns for Security D
E = Standard deviation of
returns for Security E
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
Table 7.1 Descriptive Statistics for Two Mutual
Funds
Three-Security Portfolio
E (rp )  w1E (r1 )  w2 E (r2 )  w3 E (r3 )
2p = w1212 + w2212 + w3232
+ 2w1w2
Cov(r1,r2)
+ 2w1w3 Cov(r1,r3)
+ 2w2w3 Cov(r2,r3)
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
Correlation Effects
• 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
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:
SP 
E (rP )  rf
P
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 riskreturn 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
n
 
2
P
i 1
n
 w w Cov(r , r )
j 1
i
j
i
j
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:
n
 
2
P
i 1
n
 w w Cov(r , r )
i
j 1
j
i
j
• If we define the average variance and average
covariance of the securities as:
1 n 2
   i
n i 1
2
n
1
Cov 

n(n  1) j 1
j i
n
 Cov(r , r )
i 1
i
j
• We can then express portfolio variance as:
1 2
n 1
2
P   
Cov
n
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