Chapter 11
Optimal Portfolio Choice
11-1.
a.
Let n i be the number of share in stock I, then
nG 
200, 000  0.5
 4, 000
25
200, 000  0.25
nM 
 625
80
200, 000  0.25
nV 
 25, 000
2
The new value of the portfolio is
p  30n G  60n M  3nv
 $232, 500
b.
Return 
232, 500
 1  16.25%
200, 000
c.
The portfolio weight are the fraction of value invested in each stock
n  30
GoldFinger: G
 51.61%
232, 500
Moosehead:
n M  60
 16.13%
232, 500
Venture:
nV  3
 32.26%
232, 500
11-2.
Both calculations of expected return of a portfolio give the same answer.
11-3.
If the price of one stock goes up, the other stock price always goes up as well.
Berk/DeMarzo • Corporate Finance
100
11-4.
a.
RA 
10  20  5  5  2  9
 3.5%
6
RB 
21  30  7  3  8  25
6
 12%
 0.1  0.035 2 



2
2


0.2

0.08

0.05

0.035





1

Variance of A  
2
2
5  0.05  0.035    0.02  0.035  


   0.09  0.035 2

 0.01123
Volatility of A = SD(R A )  Variance of A  .01123  10.60%
 0.21  0.12 2   0.3  0.12 2 
1
2
2
Variance of B   0.07  0.12    0.03  0.12 
5
 0.08  0.12 2   0.25  0.12 2

 0.02448






Volatility of B = SD(R B )  Variance of B  .02448  15.65%
b.
 0.1  0.035  0.21  0.4   
 0.2  0.035  0.3  0.12  



1  0.05  0.035  0.07  0.12   
Covariance  

5  0.05  0.035  0.03  0.12   
 0.02  0.035  0.08  0.12   


 0.09  0.035  0.25  0.12 

 0.00794
c.
Correlation

Covariance
SD(R A )SD(R B )
 0.479
11-5.
a.
The mean for KO is 2.02%; the mean for XOM is 0.79%.
The standard deviation (i.e., volatility) for KO is 8.24%; the standard deviation for XOM is 4.25%.
b.
The covariance is 0.00213.
c.
The correlation is 60.83%.
Chapter 11
Optimal Portfolio Choice
11-6.
covariance  con  SD  R D   SD  R AA   0.69  0.5  0.72  0.2484
11-7.
Variance   0.7  0.01123   0.3  0.02448  2  0.7  0.3  0.00794  0.011
2
Standard Deviation 
11-8.
101
2
0.011  10.5%
All three methods result have the same result: The standard deviation (i.e., volatility) is 5.90%.
0.085
0.08
0.075
0.07
0.065
0.06
0.055
0.05
0.045
0.04
10
0%
90
%
80
%
70
%
60
%
50
%
40
%
30
%
20
%
10
%
0%
-1
0%
-2
0%
Volatility
11-9.
KO Portfolio Weight
(see spreadsheet application)
11-10.
First calculate the standard deviation of each stock and multiply this by the stock’s portfolio weight and by
the stock’s correlation with the whole portfolio. The sum of these products is the portfolio’s volatility.
11-11.
Volatility of a very large portfolio is
average covariance

 0.4  0.5  0.5 
 31.62%
Berk/DeMarzo • Corporate Finance
102
11-12.
a.
If the two stocks are perfectly negatively correlated, they fluctuate due to the same risks, but in opposite
directions. Because Intel is twice as volatile as Coke, we will need to hold twice as much Coke stock as Intel
in order to offset Intel’s risk. That is, our portfolio should be 2/3 Coke and 1/3 Intel.
We can check this using Eq. 11.9:
Var(R P )  (2 / 3) 2 SD(R Coke ) 2  (1/ 3) 2 SD(R Intel ) 2  2(2 / 3)(1/ 3)Corr(R Coke , R Intel )SD(R Coke )SD(R Intel )
 (2 / 3) 2 (0.252 )  (1/ 3) 2 (0.50 2 )  2(2 / 3)(1/ 3)( 1)(.25)(.50)
0
b.
From Eq. 11.3, the expect return of the portfolio is
E[R P ]  (2 / 3)E[R Coke ]  (1/ 3)E[R Intel ]
 (2 / 3)6%  (1/ 3)26%
 12.67%
Because this portfolio has no risk, the risk-free interest rate must also be 12.67%.
11-13.
In this case, the portfolio weights are xj  xw  0.50. From Eq. 11.3,
E[R P ]  x j E[R j ]  x w E[R w ]
 0.50(7%)  0.50(10%)
 8.5%
We can use Eq. 11.9
SD(R P )  x j2SD(R j ) 2  x w 2SD(R w ) 2  2x j x w Corr(R j , R w )SD(R j )SD(R w )
 .502 (.162 )  .502 (.20) 2  2(.50)(.50)(.22)(.16)(.20)
 14.1%
11-14.
In this case, the total investment is $10,000 – 2,000 = $8,000, so the portfolio weights are xj  10,000 8,000
 1.25, xw  2,000 8,000  0.25. From Eq. 11.3,
E[R P ]  x j E[R j ]  x w E[R w ]
 1.25(7%)  0.25(10%)
 6.25%
We can use Eq. 11.9
SD(R P )  x j2SD(R j ) 2  x w 2SD(R w ) 2  2x j x w Corr(R j , R w )SD(R j )SD(R w )
 1.252 (.162 )  (0.25) 2 (.20) 2  2(1.25)(0.25)(.22)(.16)(.20)
 19.5%
Chapter 11
11-15.
x(J&J)
-50%
-40%
-30%
-20%
-10%
0%
10%
20%
30%
40%
50%
60%
65%
70%
80%
90%
100%
110%
120%
130%
140%
150%
x(Walgreen)
150%
140%
130%
120%
110%
100%
90%
80%
70%
60%
50%
40%
35%
30%
20%
10%
0%
-10%
-20%
-30%
-40%
-50%
x
SD
29.30%
27.32%
25.38%
23.50%
21.70%
20.00%
18.42%
16.99%
15.77%
14.79%
14.11%
13.78%
13.75%
13.82%
14.23%
14.97%
16.00%
17.27%
18.73%
20.34%
22.07%
23.88%
115, 000
 1.15
100, 000

ER
11.50%
11.20%
10.90%
10.60%
10.30%
10.00%
9.70%
9.40%
9.10%
8.80%
8.50%
8.20%
8.05%
7.90%
7.60%
7.30%
7.00%
6.70%
6.40%
6.10%
5.80%
5.50%

E  R   rf  x E  R j   r  4%  1.15 11%   16.65%
Volatility  x SD R j   1.15 25%  28.75%
b.
103
The set of efficient portfolio’s is approximately those portfolios with no more than 65% invested in J&J (this
is the portfolio with the lowest possible volatility, as shown in the chart).
11-16.
a.
Optimal Portfolio Choice
R 
115, 000(1.25)  15, 000(1.04)
100, 000
 1  28.15%
Berk/DeMarzo • Corporate Finance
104
c.
R 
115, 000(0.80)  15, 000(1.04)
 1  23.6%
100, 000
11-17.
Investors who want to maximize their expected return for a given level of volatility will pick portfolios which
maximize their Sharpe ratio. The set of portfolios that do this is a combination of a risk free asset a single
portfolio of risk assets --- the tangential portfolio.
11-18.
Required Return  4%  0.2 
80%
25%
 8%   9.12%
You should add some of the venture fund to your portfolio because it has an expected return that exceeds the
required return.
11-19.
11-20.
Your current portfolio is not efficient.
30%
sp
 K  0.15 
 0.3
15%
Required Return  4%  0.3  6%   5.8%
Chapter 11
Optimal Portfolio Choice
105