1. 2. Building efficient portfolios
Chapter 7-8

2. Allocating Capital: Risky & Risk Free Assets
It’s possible to split investment funds between safe and risky assets.
Risk free asset: proxy; T-bills
Risky asset: stock (or a portfolio)

3. Allocating Capital: Risky & Risk Free Assets
Issues
Examine risk/return tradeoff.
Demonstrate how different degrees of risk aversion will affect allocations between risky and risk free assets.

4. Example Using Chapter 7.3 Numbers
rf = 7%
rf = 0%
E(rp) = 15%
p = 22%
y = % in p
(1-y) = % in rf

5. Expected Returns for Combinations
E(rc) = yE(rp) + (1 - y)rf
rc = complete or combined portfolio
For example, y = .75
E(rc) = .75(.15) + .25(.07)
= .13 or 13%

6. Possible Combinations
E(r)
E(rp) = 15% P
E(rc) = 13% C
rf = 7% F  c
22%

7. Variance For Possible Combined Portfolios=
Since rf y
= 0, then  *  
* Rule 4 in Chapter 6

8. Combinations Without Leverage
= .75(.22) = .165 or 16.5%
If y = .75, then
= 1(.22) = .22 or 22%
If y = 1
= (.22) = .00 or 0%
If y = 0   

9. Capital Allocation Line with Leverage
Borrow at the Risk-Free Rate and invest in stock.
Using 50% Leverage,
rc = (-.5) (.07) + (1.5) (.15) = .19
c = (1.5) (.22) = .33

10. CAL (Capital Allocation Line)
E(r) P
E(rp) = 15%
E(rp) -
rf = 8%
) S = 8/22
rf = 7% F 
p = 22%

11. CAL with Higher Borrowing RateP
) S = .27 9% 7%
) S = .36 
p = 22%

12. Risk Aversion and Allocation
Greater levels of risk aversion lead to larger proportions of the risk free rate.
Lower levels of risk aversion lead to larger proportions of the portfolio of risky assets.
Willingness to accept high levels of risk for high levels of returns would result in leveraged combinations.

13. Utility Function U = E ( r ) - .005 A s2 Where U = utility
E ( r ) = expected return on the asset or portfolio
A = coefficient of risk aversion
s2 = variance of returns

14. CAL with Risk Preferences
The lender has a larger A when
compared to the borrower P
Borrower 7%
Lender 
p = 22%

15. Optimal Risky Portfolios
Chapter 8

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

17. 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

18. 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

19. 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

20. 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

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

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

23. 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

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

25. 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.

26. 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)

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

28. Risk and Return: Minimum Variance
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

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

30. Risk and Return: Minimum Variance
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

31. 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.

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

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

34. 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

35. 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

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