11-14. Using the data from Table 11.3, what is volatility of an equally weighted portfolio of
Microsoft, Alaska Air, and Ford Motor stock?
27.1% var-cov
MSFT
AA
Ford
0.1369
0.03515
0.03515 0.040404
0.1444 0.025536
0.040404 0.025536
0.1764
ave var ave cov
0.152567
0.033697
volatility 0.270777
0.270777
11-15. Suppose that the average stock has a volatility of 50%, and that the correlation between pairs of stocks is 20%. Estimate the volatility of an equally weighted portfolio with (a) 1 stock, (b) 30 stocks, (c) 1000 stocks.
Vol
Var
Corr
Covar
N
1
30
1000
Vol
50%
0.25
20%
0.05
50.0%
23.8%
22.4%
11-17. Consider an equally weighted portfolio of stocks in which each stock has a volatility of 40%, and the correlation between each pair of stocks 20%. a. What is the volatility of the portfolio as the number of stocks becomes arbitrarily large?

b. What is the average correlation of each stock with this large portfolio? a. Ave Covar = 40%

40%

20%=0.032
Limit Vol = (.032) 0.5
= 17.89% b. From Eq. 11.13
Corr = SD(Rp)/SD(Ri) = 17.89%/40% = 44.72%
11-20. Suppose Ford Motor stock has an expected return of 20% and a volatility of 40%, and
Molson Coors Brewing has an expected return of 10% and a volatility of 30%. If the two stocks are uncorrelated, a. What is the expected return and volatility of an equally weighted portfolio of the two stocks? b. Given your answer to (a), is investing all of your money in Molson Coors stock an efficient portfolio of these two stocks? c. Is investing all of your money in Ford Motor an efficient portfolio of these two stocks? a.
ER
Vol
XA
50%
A
XB
20%
40%
50%
B
Vol
10%
30%
25.0% b. No, dominated by 50-50 portfolio.
Corr
ER
0%
15.0% c. Yes, not dominated.
11-21. Suppose Intel’s stock has an expected return of 26% and a volatility of 50%, while Coca-
Cola’s has an expected return of 6% and volatility of 25%. If these two stocks were perfectly negatively correlated (i.e., their correlation coefficient is −1), a. Calculate the portfolio weights that remove all risk. b. If there are no arbitrage opportunities, what is the risk-free rate of interest in this economy? a.
If the two stocks are perfectly correlated negatively, 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
) (
Coke
) (


(2 / 3) (0.25 )
0

(1/ 3) (0.50 )
 
Intel
) b. From Eq. 11.3, the expect return of the portfolio is
E R
P
]

E R
Coke

E R
Intel
]


12.67%.

Because this portfolio has no risk, the risk-free interest rate must also be 12.67%.
For Problems 22–25, suppose Johnson & Johnson and the Walgreen Company have expected returns and volatilities shown below, with a correlation of 22%.
11-25. Using the same data as for Problem 22, calculate the expected return and the volatility
(standard deviation) of a portfolio consisting of Johnson & Johnson’s and Walgreen’s stocks using a wide range of portfolio weights. Plot the expected return as a function of the portfolio volatility. Using your graph, identify the range of Johnson & Johnson’s portfolio weights that yield efficient combinations of the two stocks, rounded to the nearest percentage point.
The set of efficient portfolios is approximately those portfolios with no more than 65% invested in
J&J (this is the portfolio with the lowest possible volatility).
20%
30%
40%
50%
60%
65%
70%
80% x(J&J) x(Walgreen)
-50% 150%
-40%
-30%
140%
130%
-20%
-10%
0%
10%
120%
110%
100%
90%
90%
100%
110%
120%
130%
140%
150%
80%
70%
60%
50%
40%
35%
30%
20%
10%
0%
-10%
-20%
-30%
-40%
-50%
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%
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%
11-26. A hedge fund has created a portfolio using just two stocks. It has shorted $35,000,000 worth of Oracle stock and has purchased $85,000,000 of Intel stock. The correlation between
Oracle’s and Intel’s returns is 0.65. The expected returns and standard deviations of the two stocks are given in the table below:

a. What is the expected return of the hedge fund’s portfolio? b. What is the standard deviation of the hedge fund’s portfolio? a. The total value of the portfolio is $50m (=-$35+$85). This means that the weight on Oracle is
–70% and the weight on Intel is 170%. The expected return is
Expected return
   

16.25%.
b.
Variance

0.7
 
0.45
   
0.40

2
2

0.7
    

0.283165

Std dev = (.283165)^.5 = 53.2%
11-33. You have $100,000 to invest. You choose to put $150,000 into the market by borrowing
$50,000. a. If the risk-free interest rate is 5% and the market expected return is 10%, what is the expected return of your investment? b. If the market volatility is 15%, what is the volatility of your investment? a. Er = 5% + 1.5

(10% – 5%) = 12.5% b. Vol = 1.5

15% = 22.5%
11-34. You currently have $100,000 invested in a portfolio that has an expected return of 12% and a volatility of 8%. Suppose the risk-free rate is 5%, and there is another portfolio that has an expected return of 20% and a volatility of 12%. a. What portfolio has a higher expected return than your portfolio but with the same volatility? b. What portfolio has a lower volatility than your portfolio but with the same expected return?
Invest an amount x in the other portfolio and the expected return and volatility are
E [R ] x
  x(E[R ]
O
 r ) f

5%
 x(20%

5%)
S D(R )
 x SD(R )
 x(12%).
a. So to maintain the volatility at 8%, x

8% /12%

2 / 3, you should invest $66,667 in the other portfolio and the remaining $33,333 in the risk-free investment. Your expected return will then be 15%. b. Alternatively, to keep the expected return equal to the current value of 12%, x must satisfy 5%
+ x (15%) = 12%, so x = 46.667%. Now you should invest $46,667 in the other portfolio and
$53,333 in the risk-free investment, lowering your volatility to 5.6%
11-35. Assume the risk-free rate is 4%. You are a financial advisor, and must choose one of the funds below to recommend to each of your clients. Whichever fund you recommend, your clients will then combine it with risk-free borrowing and lending depending on their desired level of risk.

Which fund would you recommend without knowing your client’s risk preference?
Sharpe ratios of A,B and C are .6,.5 and 1, so you would choose C; it is the best choice no matter what your clients’ risk preferences.
11-37. In addition to risk-free securities, you are currently invested in the Tanglewood Fund, a broadbased fund of stocks and other securities with an expected return of 12% and a volatility of 25%. Currently, the risk-free rate of interest is 4%. Your broker suggests that you add a venture capital fund to your current portfolio. The venture capital fund has an expected return of 20%, a volatility of 80%, and a correlation of 0.2 with the Tanglewood
Fund. Calculate the required return and use it to decide whether you should add the venture capital fund to your portfolio.
Required Return
   
20%


10.4%
You should add some of the venture fund to your portfolio because it has an expected return that exceeds the required return.
11-44. Your investment portfolio consists of $15,000 invested in only one stock—Microsoft. Suppose the risk-free rate is 5%, Microsoft stock has an expected return of 12% and a volatility of
40%, and the market portfolio has an expected return of 10% and a volatility of 18%. Under the CAPM assumptions, a. What alternative investment has the lowest possible volatility while having the same expected return as Microsoft? What is the volatility of this investment? b. What investment has the highest possible expected return while having the same volatility as Microsoft? What is the expected return of this investment? a. Under the CAPM assumptions, the market is efficient; that is, a leveraged position in the market has the highest expected return of any portfolio for a given volatility and the lowest volatility for a given expected return. By holding a leveraged position in the market portfolio, you can achieve an expected return of
E R p
  r f
   m
  r f
 
5% x 5% .
Setting this equal to 12% gives 12 5 5 x x 1.4.
So the portfolio with the lowest volatility and that has the same return as Microsoft has
$15, 000 1.4

$21, 000 in the market portfolio and borrows $21,000 $15,000

$6, 000 ; that is, –$6,000 in the force asset.
 
   m

1.4 18

25.2%
Note that this is considerably lower than Microsoft’s volatility. b. A leveraged portion in the market has volatility

 
  
18%.
Setting this equal to the volatility of Microsoft gives
40% x 18%

x

40

18
2.222.
So the portfolio with the highest expected return that has the same volatility as Microsoft has
$15,000 2.2

$33,000 in the market portfolio and borrows 33,000 15,000

$18,333.33
, that is –$18,333.33 in the in force asset.
 p
 r
    r f


5% 2.222 5% 16.11%
Note that this is considerably higher than Microsoft’s expected return.
11-46. Suppose the risk-free return is 4% and the market portfolio has an expected return of 10% and a volatility of 16%. Johnson and Johnson Corporation (Ticker: JNJ) stock has a 20% volatility and a correlation with the market of 0.06. a. What is Johnson and Johnson’s beta with respect to the market? b. Under the CAPM assumptions, what is its expected return? a. b.

JJ

0.06

0.2
0.16

0.075
    
4.45%
11-47. Consider a portfolio consisting of the following three stocks:
The volatility of the market portfolio is 10% and it has an expected return of 8%. The riskfree rate is 3%. a. Compute the beta and expected return of each stock. b. Using your answer from part a, calculate the expected return of the portfolio. c. What is the beta of the portfolio? d. Using your answer from part c, calculate the expected return of the portfolio and verify that it matches your answer to part b.