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Solutions for Assignment for Module 3

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					Solutions for the assignment for risk and return

1.     The following historical returns data for the last 10 years are gathered for stock
       XYTEL.

Year                                               Annual return(%)
1995                                               42.1
1994                                               -10.9
1993                                               20.4
1992                                               12.5
1991                                               10.3
1990                                               45.8
1989                                               -30.5
1988                                               11.4
1987                                               10.2
1986                                               -2.2

The stock price was $25.6 at the end of 1985, what would be the stock price at the end of
1989?
A.     $22.50
B.     $21.36
C.     $23.55
D.     $24.15
E.     $26.78

Answer: B
Solution
Closing price of 1989=(Closing price of 1985)(1+ return in 1986)(1+ return in 1987) (1+
return in 1988) )(1+ return in 1989)
= 25.6(1-0.022)(1+0.102)(1+0.114)(1-0.305)=$21.36


2.      Rank the historical volatility of the following portfolios in descending order:
small stocks, Treasury bills, long-term government bonds, and common stocks.
A.      Common stocks, long-term government bonds, small stocks, and Treasury bills
B.      Treasury bills, small stocks, long-term government bonds, and common stocks
C.      Small stocks, common stocks, long-term government bonds, and Treasury bills
D.      Long-term government bonds, small stocks, common stocks, and Treasury bills
E.      Small stocks, long-term government bonds, common stocks, and Treasury bills

Answer: C

3.      The standard deviation of the security A and B is 10% and 30%, respectively. The
correlation coefficient of A and B is precisely –1. What should be the portfolio weight of
the security A so that the portfolio of A and B has a zero variance?
A. 25%
B.   50%
C.   75%
D.   10%
E.   90%

Answer: C

Solution
Since the correlation coefficient between A and B is -1,
 P  wA A  wB B  wA (10)  wB (30)  0%
wA (10)  (1  wA )(30)  0%
Solving the above equation, the percentage weight of A is 75%.


4.      Answer the following two questions concerning portfolio risk/return measures
versus the risk/return measures of the individual securities that make up the portfolio.
Assume all weights are positive. 1) Can the return on the portfolio ever be less than the
lowest return on an individual security in the portfolio? 2) Can the variance of the
portfolio ever be less than the lowest variance of an individual security in the portfolio?
A.      1) yes, 2) yes
B.      1) no , 2) yes
C.      1) yes, 2) no
D.      1) no , 2) no

Answer: B

5.     You have a portfolio consisting of equal amounts of GM stock and Treasury bills.
If you replace one-third of the GM stock with more Treasury bills, the variance of the
expected portfolio returns will
A.     increase
B.     decrease
C.     remain unchanged

Answer: B


6.      Which of the following statements is/are true about the variance of the possible
future returns on a financial asset?

I. The variance is a weighted average of the squared deviations of the actual returns
from the expected returns.
II. The greater the dispersion in the possible returns on the firm's stock, the greater the
variance of the possible returns, all else equal.
III. The variance of the possible returns on a risk-free asset is zero.
A.      I only
B.      II only
C.      III only
D.      II and III only
E.      I, II, and III

Answer: E

7.      The following table shows the forecasted distribution of returns on the market
portfolio. What is the market risk premium if the risk-free rate is 8%?

State                             Prob. Of State                   Return for State
Boom                              0.35                             0.4
Good                              0.40                             0.2
Recession                         0.10                             0.1
Depression                        0.15                             -0.3

A.      .070
B.      .105
C.      .165
D.      .235
E.      .305

Answer: B

The expected return on the market portfolio is 18.5%.
The market risk premium =
The expected return on the market portfolio – the risk-free rate
= 18.5% - 8% = 10.5%


8.     What is the expected return of the asset B given that the portfolio return is 21%
and given the following information?

Asset                             Port. Weight                     Expected Return
A                                 0.3                              15%
B                                 0.3
C                                 0.3                              25%
D                                 0.1                              30%

A.      13%
B.      14%
C.      14.5%
D.      20%
E.      17%
Answer: D

wP  wA k A  wB k B  wC kC  wD k D
21%  (0.30)(15%)  (0.30)(k B )  (0.30)(25%)  (0.10)(30%)
k B  20%

9.      Given the following information on the portfolio and that the beta of the portfolio
is 0.881, what is the beta of E?

Stock                                   Investment ($)                       Beta
A                                       15,000                               0.65
B                                       10,000                               0.70
C                                       5,000                                1.10
D                                       12,500                               0.89
E                                       7,500

A.         1.22
B.         1.32
C.         1.42
D.         1.52
E.         1.62

Answer C

Solution

Question 9      portfolio beta problem
portfolio beta=                   0.881

security            amount invested($) security beta        portfolio weight weighted beta
A                               15000           0.65                     0.3         0.195
B                               10000            0.7                     0.2           0.14
C                                5000            1.1                     0.1           0.11
D                               12500           0.89                   0.25         0.2225
E                                7500                                  0.15
                                50000                                      1
                                                       weighted beta of E=          0.2135
                                                       beta of E=                      1.42




10.     Under the CAPM, the ratio of the risk premiums of two assets is the same as the
ratio of their betas.
A.      True
B.      False
Answer
A

Since
kA = kf + (A)(RPM)
kA - kf = (A)(RPM)

kB = kf + (B)(RPM)
kB - kf = (B)(RPM)

kA  k f        RP 
              A
                   
                   M    A

kB  k f        RP 
               B   M    B




11.     You hold three stocks in your portfolio - stock A, stock B, and stock C. The
portfolio beta is 1.50. Stock A constitutes 20 percent of the dollar value of your holdings
and has a beta of 1.00. If you sell all of your holdings in stock A, and replace them with
an equal investment in stock D (which has a beta of 1.25), your new portfolio beta will be
________ .
A.      .850
B.      1.250
C.      1.450
D.      1.550
E.      1.625

Answer: D

Solution
Let R stand for the remainder of the portfolio.

Then,
Before rebalancing the portfolio, we have

wA  A  wR  R   P
(0.2)(1.00)  (0.8)  R  1.50
0.8  R  1.50  0.20  1.30
      1.30
 
  R
            1.625
       0.8

After rebalancing the portfolio, we have
wD  D  wR  R   P
(0.2)(1.25)  (0.8)(1.625)   p
  1.55
  p




12.     You hold five stocks in your portfolio - stock A, stock B, stock C, stock D, and
stock E. The portfolio beta is 1.00. Stock E constitutes 20 percent of the dollar value of
your holdings and has a beta of 1.50. If you sell all of your holdings in stock E, and use
half the funds to invest in stock F (which has a beta of .60), and the remainder to invest in
the risk-free asset, your new portfolio beta will be ________.
A.      .76
B.      .89
C.      .97
D.      1.18
E.      1.24

Answer: A

Solution
Let R stand for the remainder of the portfolio.

Then,
Before rebalancing the portfolio, we have

wE  E  wR  R   P
(0.2)(1.5)  (0.8)  R  1
0.8 R  1  0.3  0.7
      0.7
 
  R
           0.875
      0.8

After rebalancing the portfolio, we have

wF  F  w f  f  wR  R   P
(0.1)( 0.6)  (0.1)( 0.0)  (0.8)( 0.875 )   p
  0.06  0.7  0.76
  p




13.     The standard deviation of returns on Japanese stocks is 16.81 percent. The average
return is 12.73 percent. Assume that the frequency distribution of returns is essentially
normal. Then the lower limit of 95% confidence interval of returns on Japanese stocks is
A.      0%
B.      –4.08%
C.      -20.89%
D.      –37.70%
E.      -6.05%
Answer C
Solution
The lower limit of the 95 percent confidence interval
= average return – (2)(standard deviation)
= 12.73 – (2)(16.81)
= -20.89%

Use the following information to answer questions 14-16.
Returns in the next period for two stocks, A and B, and the market, M, are given by the
following probability distribution:

                                     Associated Rate of Return
State of the     Probability      of A            B                 M
Economy          the State
Boom             0.25                   40%             50%          40%
Normal           0.5                    0               5            15
Recession        0.25                  -10             -5           -15

14.    What is the expected rate of return for the portfolio AB, which consists of 50%
invested in A and 50% invested in B?

A.      6.625%
B.      7.625%
C.      8.625%
D.      9.625%
E.      10.625%

Answer: E
E[kAB]= wA E[kA]+ wB E[kB]
=(0.5)(7.5%) + (0.5)(13.75)
=10.625%

15.    What is the standard deviation of the portfolio AB, which consists of 50% invested
in A and 50% invested in B? The correlation coefficient between A and B is 1.
A. 17.26
B. 18.26
C. 19.26
D. 20.26
E. 21.26

Answer: D

Solution
Since the correlation coefficient between A and B is 1,
 P  wA A  wB B  (0.5)(19 .20 )  (0.5)( 21 .32 )  20 .26 %
16.     What is the beta of A?
A.      0.88
B.      0.98
C.      0.58
D.      0.68
E.      0.78

Answer: A

Solution
          ~ ~
      cov(k A , k M )       334
                              0.88
           
  P            2
                   M        380


17.    To estimate the beta of Ford’s common stock, you regressed the monthly returns
of Ford on the monthly returns of the S&P 500. The intercept of the regression line was
3% and the estimated slope was 0.6. If the S&P 500 is expected to earn 15% return this
year and the yield on the T bill is 5%, what is the expected annual return on the Ford
common stock this year according to the CAPM?
A.     11%
B.     7%
C.     8%
D.     9%
E.     10%


Answer: A

Solution
The estimated beta of the Ford stock is 0.6
Using the equation of the SML,

k  k RF   (k M  k RF )

k  5  0.6(15  5)  11%

18.     Investing only in a S&P 500 Index mutual fund that should have a beta of 1.0 and
the riskless security such as T-bills is probably the simplest asset allocation strategy.
Assume that the expected return of the S&P 500 Index is 12% and the expected return on
the treasury bills is 5%. If you want to get an expected return of 10% per year, what
should be the weight on the S&P 500 Index mutual fund?
A.      67.52%
B.      70.23%
C.      73.56%
D.     72.56%
E.     71.43%

Answer: E


E[kM,f]= wM E[kM]+ wf kf
=wM (12%)+ wf (5%)
=wM (12%)+ (1-wM)(5%)
7wM = 5
wM = 71.43%

				
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