1. Optimal Risky Portfolios
Chapter 8
Optimal Risky Portfolios

2. Risk Reduction with Diversification
St. Deviation
Unique Risk
Market Risk
Number of Securities

3. 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

4. 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

5. 1,2 = Correlation coefficient of returns
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

6. Correlation Coefficients: Possible Values
Range of values for 1,2
> r > -1.0
If r= 1.0, the securities would be perfectly positively correlated
If r= - 1.0, the securities would be perfectly negatively correlated

7. Three-Security Portfolio
rp = W1r1 + W2r2 + W3r3
2p = W1212
+ W2212
+ W3232
+ 2W1W2
Cov(r1r2)
+ 2W1W3
Cov(r1r3)
+ 2W2W3
Cov(r2r3)

8. In General, For an n-Security Portfolio:
rp = Weighted average of the
n securities
p2 = (Consider all pairwise
covariance measures)

9. 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

10. Two-Security Portfolios with Different Correlations
13%
 = -1
 = .3
 = -1
 = 1 %8
St. Dev
12%
20%

11. Portfolio Risk/Return Two Securities: Correlation Effects
The relationship depends on correlation coefficient.
-1.0 <  < +1.0
The smaller the correlation, the greater the risk reduction potential.
If r = +1.0, no risk reduction is possible.

12. Minimum-Variance Combination2
E(r2) = .14
= .20
Sec 2 12
= .2
E(r1) = .10
= .15
Sec 1   1
r22 - Cov(r1r2) = W1 s2
s2
- 2Cov(r1r2) + 1 2 W2
= (1 - W1)

13. Minimum-Variance Combination:  = .2W1 =
(.2)2 - (.2)(.15)(.2)
(.15)2 + (.2)2 - 2(.2)(.15)(.2)
= .6733 W2
= ( ) = .3267

14. Minimum -Variance: Return and Risk with  = .2
rp = .6733(.10) (.14) = .1131 
= [(.6733)2(.15)2 + (.3267)2(.2)2 + p
1/2
2(.6733)(.3267)(.2)(.15)(.2)] s
1/2
= [.0171]
= .1308 p

15. Minimum -Variance Combination:  = -.3W1 =
(.2)2 - (.2)(.15)(.2)
(.15)2 + (.2)2 - 2(.2)(.15)(-.3)
= .6087 W2
= ( ) = .3913

16. Minimum -Variance: Return and Risk with  = -.3
rp = .6087(.10) (.14) = .1157 s
= [(.6087)2(.15)2 + (.3913)2(.2)2 + p
1/2
2(.6087)(.3913)(.2)(.15)(-.3)] s
1/2
= [.0102]
= .1009 p

17. 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.

18. The Minimum-Variance Frontier of Risky Assets
Efficient
frontier
Individual
assets
Global
minimum
variance
portfolio
Minimum
variance
frontier
St. Dev.

19. Extending to Include Riskless Asset
The optimal combination becomes linear.
A single combination of risky and riskless assets will dominate.

20. Alternative CALs E(r) CAL (P) CAL (A) M M P P CAL (Global
minimum variance) A A G F  P
P&F M
A&F

21. Portfolio Selection & Risk Aversion
U’’ U’
U’’’
E(r)
Efficient
frontier of
risky assets S P Q
Less
risk-averse
investor
More
risk-averse
investor
St. Dev

22. Efficient Frontier with Lending & Borrowing
CAL
E(r) B Q P A rf F
St. Dev