Investment & Portfolio Management
Spring, 2009
Chung-Ang Business School
Problem Set #2
Do Not Hand In
1. Which statement about portfolio diversification is correct?
A. Proper diversification can reduce or eliminate systematic risk.
B. The risk-reducing benefits of diversification do not occur meaningfully until at least
50-60 individual securities have been purchased.
C. Because diversification reduces a portfolio's total risk, it necessarily reduces the
portfolio's expected return.
D. Typically, as more securities are added to a portfolio, total risk would be
expected to decrease at a decreasing rate.
E. None of the above statements are correct.
2. For a two-stock portfolio, what would be the preferred correlation coefficient
between the two stocks?
A. +1.00.
B. +0.50.
C. 0.00.
D. -1.00.
E. none of the above.
3. Which of the following is not a source of systematic risk?
A. the business cycle.
B. interest rates.
C. personnel changes
D. the inflation rate.
E. exchange rates.
4. Security X has expected return of 12% and standard deviation of 20%. Security Y has
expected return of 15% and standard deviation of 27%. If the two securities have a
correlation coefficient of 0.7, what is their covariance?
A. 0.038
B. 0.070
C. 0.018
D. 0.013
E. 0.054
Investment & Portfolio Management
Spring, 2009
Chung-Ang Business School
5. When two risky securities that are positively correlated but not perfectly correlated
are held in a portfolio,
A. the portfolio standard deviation will be greater than the weighted average of the
individual security standard deviations.
B. the portfolio standard deviation will be less than the weighted average of the
individual security standard deviations.
C. the portfolio standard deviation will be equal to the weighted average of the
individual security standard deviations.
D. the portfolio standard deviation will always be equal to the securities' covariance.
E. none of the above are true.
6. Given an optimal risky portfolio with expected return of 14% and standard deviation
of 22% and a risk free rate of 6%, what is the slope of the best feasible CAL?
A. 0.64
B. 0.14
C. 0.08
D. 0.33
E. 0.36
7. In words, the covariance considers the probability of each scenario happening and the
interaction between
A. securities' returns relative to their variances.
B. securities' returns relative to their mean returns.
C. securities' returns relative to other securities' returns.
D. the level of return a security has in that scenario and the overall portfolio return.
E. the variance of the security's return in that scenario and the overall portfolio variance.
8. When borrowing and lending at a risk-free rate are allowed, which Capital Allocation
Line (CAL) should the investor choose to combine with the efficient frontier?
I) with the highest reward-to-variability ratio.
II) that will maximize his utility.
III) with the steepest slope.
IV) with the lowest slope.
A. I and III
B. I and IV
C. II and IV
Investment & Portfolio Management
Spring, 2009
Chung-Ang Business School
D. I only
E. I, II, and III
9. Security X has expected return of 14% and standard deviation of 22%. Security Y has
expected return of 16% and standard deviation of 28%. If the two securities have a
correlation coefficient of 0.8, what is their covariance?
A. 0.038
B. 0.049
C. 0.018
D. 0.013
E. 0.054
Consider two perfectly negatively correlated risky securities K and L. K has an expected
rate of return of 13% and a standard deviation of 19%. L has an expected rate of return
of 10% and a standard deviation of 16%.
10. The weights of K and L in the global minimum variance portfolio are _____ and
_____, respectively.
A. 0.24; 0.76
B. 0.50; 0.50
C. 0.54; 0.46
D. 0.45; 0.55
E. 0.76; 0.24
11. The risk-free portfolio that can be formed with the two securities will earn _____
rate of return.
A. 9.5%
B. 10.4%
C. 10.9%
D. 9.9%
E. none of the above
12. As diversification increases, the total variance of a portfolio approaches
____________.
A. 0
B. 1
Investment & Portfolio Management
Spring, 2009
Chung-Ang Business School
C. the variance of the market portfolio
D. infinity
E. none of the above
13. A single-index model uses __________ as a proxy for the systematic risk factor.
A. a market index, such as the S&P 500
B. the current account deficit
C. the growth rate in GNP
D. the unemployment rate
E. none of the above
14. According to the index model, covariances among security pairs are
A. due to the influence of a single common factor represented by the market index
return
B. extremely difficult to calculate
C. related to industry-specific events
D. usually positive
E. A and D
15. Analysts may use regression analysis to estimate the index model for a stock. When
doing so, the slope of the regression line is an estimate of ______________.
A. the  of the asset
B. the  of the asset
C. the  of the asset
D. the  of the asset
E. none of the above
16. If the index model is valid, _________ would be helpful in determining the
covariance between assets GM and GE.
A. GM
B. GE
C. M
D. all of the above
E. none of the above
Investment & Portfolio Management
Spring, 2009
Chung-Ang Business School
17. Assume that stock market returns do not resemble a single-index structure. An
investment fund analyzes 150 stocks in order to construct a mean-variance efficient
portfolio constrained by 150 investments. They will need to calculate _____________
expected returns and ___________ variances of returns.
A. 150, 150
B. 150, 22500
C. 22500, 150
D. 22500, 22500
E. none of the above
18. Consider the single-index model. The alpha of a stock is 0%. The return on the
market index is 16%. The risk-free rate of return is 5%. The stock earns a return that
exceeds the risk-free rate by 11% and there are no firm-specific events affecting the
stock performance. The
A. 0.67
B. 0.75
C. 1.0
D. 1.33
E. 1.50
19. Suppose you held a well-diversified portfolio with a very large number of securities,
of the portfolio would be approximately ________.
A. 1.34
B. 1.16
C. 1.25
D. 1.56
E. none of the above
20. 43. The index model for stock A has been estimated with the following result:
RA = 0.01 + 0.9RM + eA
If σM = 0.25 and R2A = 0.25, the standard deviation of return of stock A is _________.
A. 0.2025
B. 0.2500
C. 0.4500
D. 0.8100
Investment & Portfolio Management
Spring, 2009
Chung-Ang Business School
E. none of the above
21. The index model has been estimated for stocks A and B with the following results:
RA = 0.01 + 0.5RM + eA
RB = 0.02 + 1.3RM + eB
M = 0.25 (eA) = 0.20 (eB) = 0.10
The covariance between the returns on stocks A and B is ___________.
A. 0.0384
B. 0.0406
C. 0.1920
D. 0.0050
E. 0.4000
22. The index model has been estimated for stocks A and B with the following results:
RA = 0.01 + 0.8RM + eA
RB = 0.02 + 1.2RM + eB
M = 0.20 (eA) = 0.20  (eB) = 0.10
The standard deviation for stock A is __________.
A. 0.0656
B. 0.0676
C. 0.2561
D. 0.2600
E. none of the above
23. Consider the following data for securities A, B, and C:
R A  20% ; R B  10% ; R C  8% ;  A  4% ;  B  2% ;  C  2% ;
 AB  0.4 ;  AC  0.2 ;  BC  1.0
a. What is the expected return and standard deviation of a portfolio
constructed by placing 60% of your money in A and 40% in B?
Investment & Portfolio Management
Spring, 2009
Chung-Ang Business School
R P  0.6  0.2  0.4  0.1  0.16  16%
 P  0.6 2  0.04 2  0.4 2  0.02 2  2  0.6  0.4  0.4  0.04  0.02  0.0282  2.82%
b. If an investor had to place 100% of his or her money in only one of
the above three securities,
1) Which security would a risk-neutral investor pick?
Security A: A risk neutral investor only care about expected return.
2) What can you say about the preference ordering of the three
securities for a risk-averse investor?
A risk averse investor prefers Security B to Security C since Security
B has higher expected return with the same risk.
One cannot say anything about investor’s preference between
Security A and Security B since Security A has higher return and
risk than Security B.
24. You are in a world where there are only two assets: gold and stocks. You are
interested in investing your money in one or both of the assets. Consequently,
you collect the following data on the assets' returns over the past six years:
Average return
Standard deviation
Gold
8%
25%
Stock Market
20%
22%
Your estimate of the assets' correlation is 0.4.
a. If you were constrained to pick only one of the two assets, which
one would you choose?
Stock : Since it has higher expected return with lower standard
deviation.
Investment & Portfolio Management
Spring, 2009
Chung-Ang Business School
b. What is the average return and standard deviation of a portfolio
composed of equal proportions of gold and stocks?
RP  0.5  0.08  0.5  0.2  0.14  14%
 P  0.5 2  0.25 2  0.5 2  0.22 2  2  0.5  0.5  (0.4)  0.25  0.22  0.1293  12.93%
c. What is the average return and standard deviation of the portfolio
composed of gold and stocks that has the lowest risk?
 P2  X 2 G2  1  X 2  S2  2 X 1  X  GS
 0.25 2 X 2  0.22 2 1  X 2  2 X 1  X    0.4   0.25  0.22
 0.1549X 2  0.1408X  0.0484  0.1549X  0.454487 2  0.016404
The minimum variance portfolio can be constructed by assigning
45.4487% to gold and 54.5513% to stock. The expected rate of return
for the minimum variance portfolio is
17.2737%(=0.454487*0.08+0.545513*0.25).
d. You now learn that GPEC (a cartel of gold-producing countries) is
going to vary the amount of gold produced depending on stock
prices in the U.S. by producing less gold when the stock market is
up and more gold when the stock market is down. What effect will
this have on portfolios composed of gold and stock? Explain.
When stock prices go up (down), GPEC will be producing less
(more) gold. That will bring gold price up (down). Therefore, one
can infer that there exists a positive correlation between expected
returns of stock and gold. Consequently, the risk of portfolio
consisting of gold and stock will increase.
25. You are evaluating two risky investments, A and B, which have the
following distributions:
Probability
0.6
Return on A
20%
Return on B
30%
Investment & Portfolio Management
Spring, 2009
Chung-Ang Business School
0.4
10%
10%
a. What are the expected returns and standard deviations for A and B?
R A  0.6  0.2  0.4  0.1  0.16  16%
RB  0.6  0.3  0.4  0.1  0.22  22%
 A  0.6  0.2  0.16 2  0.4  0.1  0.16 2  0.0490  4.90%
 B  0.6  0.3  0.22 2  0.4  0.1  0.22 2  0.0980  9.80%
b. Suppose that an investor must pick either A or B to hold in some
combination with the riskless asset (RF = 8%). Which risky asset
should the investor choose?
Portfolio C: Asset A and riskless asset RC  XRA  1  X RF
RC  XR A  1  X RF  0.16X  1  X   0.08  0.08  0.08X
 C2  X 2 A2  0.0024X 2
Portfolio D: Asset B and riskless asset RD  XRB  1  X RF
RD  XRB  1  X RF  0.22X  1  X   0.08  0.08  0.14X
 D2  X 2 B2  0.0096X 2
Portfolio C has bother lower expected return and smaller variance
than portfolio D. therefore, it is not possible to determine which
portfolio to choose in terms of mea-variance comparison. However, if
we look into the probability distributions for the tow assets very
carefully, we can notice that asset A has lower return than asset B in all
contingencies. In other words, it is always better to hold asset B than
asset A. Therefore, one should not include the dominated asset such as
asset A into the portfolio.
Investment & Portfolio Management
Spring, 2009
Chung-Ang Business School
26. The stock returns for firm A and firm B have the following characteristics:
Firm
A
B
Expected return
10%
12%
Standard deviation
8%
20%
The correlation between the two stocks is 1.0.
a. If there are no restrictions on short sales or borrowing, what are the
portfolio weights, expected return and standard deviation on the
portfolio of these two assets with the minimum variance?
 P2  X 2  0.08 2  1  X 2  0.2 2  2 X 1  X  AB A B
 X 2  0.08 2  1  X   0.2 2  2 X 1  X   1 0.08  0.2
2
 0.0144 X 2  0.048 X  0.04  0.0144 X  1.6667 
2
The minimum variance portfolio weights are given
by X A  1.6667, X B  0.6667 .: short sale of Firm B
Moreover, RP  1.6667  0.1   0.6667 0.12  0.0867 → 8.67%
And  P2  0 .
b. Susan is an officer of firm A. Under a company stock purchase
plan, she currently holds $200,000 worth of A's stock, and this
represents her total assets. This stock cannot be sold. Susan can
purchase additional amounts of stock A or stock B, and she can sell
stock B short. It is illegal for her to sell stock A short. How can Susan
eliminate the risk in her holding? Be specific (give numbers).
Suppose that Susan short sell firm B’s stock by $W and buy Stock A
with the proceeds. Then, Susan’s portfolio weight are given by
 20,000  W  W 
,

 . According to the result in part (a), Susan
 20,000 20,000 
should adjust the portfolio weights to 1.6667 and -0.6667,
respectively in order to fully eliminate the risk (variance=0). That is,
W=$13333.3333.
Investment & Portfolio Management
Spring, 2009
Chung-Ang Business School
27. Consider the following data for assets A and B:
R A  10% ; R B  19% ;  A  3% ;  B  5% ;  A  0.6 ;  B  1.4 ;  AB  0.4
Calculate the expected return, variance, and beta of a portfolio constructed by
investing 1/3 of your funds in asset A and 2/3 in asset B.
Note that R  (1/ 3) RA  (2 / 3) RB .
Hence, R  (1/ 3) * RA  (2 / 3) * RB  (1/ 3) * 0.1  (2 / 3) * 0.19  0.13  13%
 R2  (1 / 3) 2 *  A2  (2 / 3) 2 *  B2  2 * (1 / 3) * (2 / 3) *  AB
 (1 / 3) 2 * 0.03 2  (2 / 3) 2 * 0.05 2  2 * (1 / 3) * (2 / 3) * 0.4 * 0.03 * 0.05  0.001478
 R  (1/ 3) *  A  (2 / 3) *  B  (1/ 3) * 0.6  (2 / 3) *1.4  1.1333
28. Consider the following historical data for the returns on assets A and B and
the market portfolio:
Period
1
2
3
4
5
Asset A
10%
-3%
5%
2%
1%
Asset B
6%
Market Portfolio
4%
6%
2%
4%
2%
1%
5%
2%
1%
a. What is the covariance between asset A and asset B?
RA  (1 / 5) * (0.1  0.03  0.05  0.02  0.01)  0.03
RB  (1 / 5) * (0.06  0.06  0.02  0.04  0.02)  0.04
 AB 

1 T
R At  R A RBt  RB 
T  1 
t 1
1 0.1  0.030.06  0.04   0.03  0.030.06  0.04  0.05  0.030.02  0.04 

4  0.02  0.030.04  0.04   0.01  0.030.02  0.04

 0.5
Investment & Portfolio Management
Spring, 2009
Chung-Ang Business School
b. If the beta of asset B is 0.5, what is the systematic return and nonsystematic return for asset B in each period?
Note that RB   B   B RM  eB . Then, in period 1, the rate of return for
portfolio B is 0.06(=6%).
Systematic return in period 1 is given by  B RM  0.5 * 0.04  0.02 →
2%. The difference between realized return and systematic return is
unsystematic return, 4%(=6%-2%). One can do similar exercise for
other periods.
29. You are the pension fund manager for a major university with $100 million
in an index fund that invests in the S&P 500 stocks. (The fund holds all the
stocks in the index in proportion to their market values.) Due to recent pressure
from student groups, the regents have decided to divest themselves of the
stocks of firms that invest in South Africa. You estimate that this will eliminate
100 of the 500 stocks in your portfolio. You have been asked to evaluate the
effect of the divestiture decision. You estimate that the correlation between
acceptable and eliminated stocks is 0.6. You also have the following data:
Number of Firms
Total Market Value
Average Beta
Standard Deviation
Acceptable Stocks
400
$3 billion
1.0
25%
eliminated stocks
100
$2 billion
1.25
30%
a. What will the effect of the divestment be on the beta of your
portfolio? (Report the beta before and after the divestment.)
Note that the beta for a portfolio is simply the weighted average if
beats of individual assets in the portfolio.
Before divestment:  Pbefore  (3 / 5) 1.0  (2 / 5) 1.25  1.1
After divestment:  Pafter  1.0
b. How will divestment affect the standard deviation of your portfolio?
(Report the standard deviation before and after the divestment.)
Investment & Portfolio Management
Spring, 2009
Chung-Ang Business School
Recall that for acceptable stocks: RA   A   A RM  eA and for
eliminated stocks RE   E   E RM  eE . We assume that the usual
conditions for single index model hold. Note that
R before  0.6RA  0.4RE and R after  R A .
Hence,
 Before  0.6 2   A2  0.4 2   E2  2  0.6  0.4   AE
 0.36  0.25 2  0.160  0.32  0.48  0.6  0.25  0.3  0.0585  0.2419
 after   A  0.25
Elimination of unacceptable stocks from the portfolio has lowered
expected return and increased risk of the portfolio.
c. Assume that the standard deviation of the overall market is 20%.
What is the effect of divestment on the proportion of your
portfolio's risk that is unsystematic? (Report the proportion before
and after the divestment.)
R before  0.6RA  0.4RE  0.6 A  0.4 E   0.6 A  0.4 E RM  0.6e A  0.4eE 


Systematic risk before divestment:  Pbefore  M2  1.12  0.2 2  0.0484
2
Unsystematic risk is the difference between total risk and systematic
risk.
Unsystematic risk before divestment: 0.2419-0.0484=0.1935
Then, 79.99%(=0.1935/0.2419) of total risk is attributable to unsystematic
risk.
R after  R A and RA   A   A RM  eA .


Systematic risk after divestment:  Pafter  M2  1.0 2  0.2 2  0.04 .
2
Unsystematic risk after divestment: 0.25-0.04=0.21.
84% (=0.21/0.25)of total risk is attributable to unsystematic risk.
Investment & Portfolio Management
Spring, 2009
Chung-Ang Business School
30. Suppose that an investor initially pays $6,000 toward the purchase of
$10,000 worth of stock (100 shares at $100 per stock), borrowing the
remaining $4,000 from the broker.
a. Calculate the initial margin.
Margin = ( equity in account / value of the stock ) = (6,000/10,000)=0.6: 60%
b. If the stock price declines to $70 per share, what is the percentage margin
now?
Margin = (7,000-4,000)/7,000=0.43: 43%
c. Suppose that the maintenance margin is 30%. What is the lowest price for
which the investor does not hear margin call form the broker?
Maintenance margin requires that
100  P  4,000  0.3  P  57.14 .
100  P
d. Let’s go back to the original situation where the stock price is $100 per share.
Suppose that the broker charges 6% interest on the money borrowed to finance
the margin purchase. If the stock price rises (declines) to $110 ($90) per share,
what is the rate of return of the investment?
$100 → $110 (10% increase in price)
Capital gain = (110-100)*100=1000
Interest payment = 4,000*0.06 = 240
Net gain = 1000-240 = 760
Rate of return = 760/6000 = 0.1267 → 12.67%
$100 → $90 (10% decrease in price)
Capital gain = (90-100)*100= -1000
Interest payment = 4,000*0.06 = 240
Net gain = -1000-240 = -1240
Rate of return = -1240/6000 = -0.2067 → -20.67%