Ch. 7

```CHAPTER 7
Optimal Risky
Portfolios
Investments, 8th edition
Bodie, Kane and Marcus
Slides by Susan Hine
McGraw-Hill/Irwin
Diversification and Portfolio Risk
• Market risk
– Systematic or nondiversifiable
• Firm-specific risk
– Diversifiable or nonsystematic
7-2
Figure 7.1 Portfolio Risk as a Function of the
Number of Stocks in the Portfolio
7-3
Figure 7.2 Portfolio Diversification
7-4
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
7-5
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 )
7-6
Two-Security Portfolio: Risk
  w   w   2wDw
EE 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
7-7
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 )
7-8
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
7-9
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
7-10
Table 7.1 Descriptive Statistics for Two
Mutual Funds
7-11
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)
7-12
Table 7.2 Computation of Portfolio
Variance From the Covariance Matrix
7-13
Table 7.3 Expected Return and Standard
Deviation with Various Correlation
Coefficients
7-14
Figure 7.3 Portfolio Expected Return as
a Function of Investment Proportions
7-15
Figure 7.4 Portfolio Standard Deviation
as a Function of Investment Proportions
7-16
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
7-17
Figure 7.5 Portfolio Expected Return as
a Function of Standard Deviation
7-18
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
7-19
Figure 7.6 The Opportunity Set of the
Debt and Equity Funds and Two
Feasible CALs
7-20
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
7-21
Figure 7.7 The Opportunity Set of the
Debt and Equity Funds with the Optimal
CAL and the Optimal Risky Portfolio
7-22
Figure 7.8 Determination of the Optimal
Overall Portfolio
7-23
Figure 7.9 The Proportions of the
Optimal Overall Portfolio
7-24
Markowitz Portfolio Selection Model
• Security Selection
– First step is to determine the risk-return
opportunities available
– All portfolios that lie on the minimumvariance frontier from the global minimumvariance portfolio and upward provide the
best risk-return combinations
7-25
Figure 7.10 The Minimum-Variance
Frontier of Risky Assets
7-26
Markowitz Portfolio Selection Model
Continued
• We now search for the CAL with the highest
reward-to-variability ratio
7-27
Figure 7.11 The Efficient Frontier of
Risky Assets with the Optimal CAL
7-28
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
7-29
Figure 7.12 The Efficient Portfolio Set
7-30
Capital Allocation and the Separation
Property
• The separation property tells us that the
portfolio choice problem may be separated
– Determination of the optimal risky portfolio
is purely technical
– Allocation of the complete portfolio to Tbills versus the risky portfolio depends on
personal preference
7-31
Figure 7.13 Capital Allocation Lines with
Various Portfolios from the Efficient Set
7-32
The Power of Diversification
n
• Remember:   
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
7-33
Table 7.4 Risk Reduction of Equally
Weighted Portfolios in Correlated and
Uncorrelated Universes
7-34
Risk Pooling, Risk Sharing and Risk in
the Long Run
• Consider the following:
p = .001
Loss: payout = \$100,000
No Loss: payout = 0
1 − p = .999
7-35
Risk Pooling and the Insurance Principle
• Consider the variance of the portfolio:
1 2
  
n
2
P
• It seems that selling more policies causes
risk to fall
• Flaw is similar to the idea that long-term
stock investment is less risky
7-36
Risk Pooling and the Insurance Principle
Continued
• When we combine n uncorrelated
insurance policies each with an expected
profit of \$ , both expected total profit and
SD grow in direct proportion to n:
E (n )  nE ( )
Var (n )  n Var ( )  n 
SD(n )  n
2
2
2
7-37
Risk Sharing
• What does explain the insurance business?
– Risk sharing or the distribution of a fixed
amount of risk among many investors
7-38
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