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The answer:
1-.
First, determine the beta of the portfolio.
Total Amount Invested = $5,000 + $10,000 + $8,000 + $7,000 =
= $30,000
Weight of Stock A = $5,000 / $30,000 = 1/6
Weight of Stock B = $10,000 / $30,000 = 1/3
Weight of Stock C = $8,000 / $30,000 = 4/15
Weight of Stock D = $7,000 / $30,000 = 7/30
The beta of a portfolio is the weighted average of the betas of its
individual securities.
Beta_ Port = (1/6)(0.75) + (1/3)(1.1) + (4/15)(1.36) + (7/30)(1.88)
= 1.293
According to the CAPM:
E = rf + Beta_Port * [E_m ? rf]
where
E = the expected return on the portfolio
rf = the risk-free rate
E_m = the expected return on the market portfolio
In this problem:
rf = 0.04
Beta_Port = 1.293
E_m = 0.15
E = 0.04 + 1.293*(0.15 ? 0.04) =
= 0.1822
The expected return on the portfolio is 18.22%.
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2-.
Factor Beta of Factor Expected Value Actual Value
GNP 0.0042 $4,416 $4,480
Inflation -1.40 3.1% 4.3%
Interest Rate -0.67 9.5% 11.8%
a.
Systematic risk is risk related to economy/market-wide events like
interest rates, recessions and wars. These types of events affect all
stocks and cannot be diversified away. Generally, systematic risk
factors are those factors that affect a large number of firms in the
market. Note that those factors do not affect equally all the firms.
The systematic factors in the list are GNP, inflation and interest
rate.
Syst. Risk = 0.042(4,480? 4,416) ? 1.4(4.3%? 3.1%) ? 0.67(11.8% ?9.5%) =
= ? 0.53%
b.
Unsystematic risk is related to events that don't affect all
companies, only your company is affected. Unsystematic risk is the
type of risk that can be diversified away through portfolio formation.
Unsystematic risk factors are specific to the firm or industry.
Surprises in these factors will affect the returns of the firm in
which you are interested, but they will have no effect on the returns
of firms in a different industry and perhaps little effect on other
firms in the same industry. Examples include your plant burns down,
your product flops, or your product is a huge hit.
The unexpected bad news about the firm that dampens the returns by
2.6% is an unsystematic risk, in this case the only one
Unsystematic Risk = ? 2.6%
c.
Total Return = expected return + Syst. Risk + Unsyst. Risk =
= 9.5% ? 0.53% ? 2.6% =
= 6.37%
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3-.
Year Small-Company Stocks U.S. Treasury Bills
% %
1986 6.85 6.16
1987 -9.30 5.47
1988 22.87 6.35
1989 10.18 8.37
1990 -21.56 7.81
1991 44.63 5.60
a.
You will must to divide the sum of the returns by six to calculate the
average return over the six-year period.
Average Return on Stocks = (S1 + S2 + S3 + S4 + S5 + S6)/6 =
= (0.0685 -0.0930 + 0.2287 + 0.1018 -0.2156 + 0.4463)/6 =
= 0.0895
The average return on small-company stocks is 8.95%.
Average Return on Bills = (B1 + B2 + B3 + B4 + B5 + B6) / 6 =
= (0.0616 + 0.0547 + 0.0635 + 0.0837 + 0.0781 + 0.056)/6 =
= 0.0663
The average return on U.S. Treasury bills is 6.63%.
b.
The variance of each security is equal to the sum of the squared
differences between each return and the mean return [(R - )2],
divided by five (because the data are historical, the appropriate
denominator in the calculation of the variance is five (=T ? 1). The
standard deviation is equal to the square root of the variance.
Small-Company Stocks:
S (S - Av.S) (S - Av.S)^2
0.0685 -0.020950 0.000439
-0.0930 -0.182450 0.033288
0.2287 0.139250 0.019391
0.1018 0.012350 0.000153
-0.2156 -0.305050 0.093056
0.4463 0.356850 0.127342
------------------------------------------------
Total 0.273667
Variancie_Stocks = SUM[(S - Av.S)^2] / (T-1) =
= (0.273667) / (6 ?1) =
= 0.054733
The variance of small-company stocks is 0.0547 .
The standard deviation is equal to the square root of the variance:
SD_Stocks = Variancie_Stocks^(1/2) =
= 0.054733^(1/2) =
= 0.2340
The standard deviation of small-company stocks is 0.2340 .
U.S. Treasury bills:
B (B - Av.B) (B - Av.B)^2
0.0616 -0.004667 0.000022
0.0547 -0.011567 0.000134
0.0635 -0.002767 0.000008
0.0837 0.017433 0.000304
0.0781 0.011833 0.000140
0.0560 -0.010267 0.000105
-----------------------------------------
Total 0.000713
Variancie_Bills = SUM[(B - Av.B)^2] / (T-1) =
= (0.000713) / (6 ?1) =
= 0.000143
The variance of U.S. Treasury bills is 0.000143 .
The standard deviation is equal to the square root of the variance:
SD_Bills = Variancie_Bills^(1/2) =
= 0.000143^(1/2) =
= 0.0119
The standard deviation of U.S. Treasury bills is 0.0119 .
c.
The average return on Treasury bills is lower than the average return
on small-company stocks. However, the standard deviation of the
returns on Treasury bills is also lower than the standard deviation of
the small-company stock returns. There is a positive relationship
between the risk of a security and the expected return on a security.
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4-.
a.
Rp = SUM(Xi*Ri) (i= 1 to 5)
Where:
Rp = return of the portfolio
Ri = actual return observed on asset i (i=1 to 5)
Xi = proportion in asset i (i=1 to 5)
Because this is a two factors model:
Ri = Ei + beta_1*F1 + beta_2*F2 + e_i
Where:
Ei = expected return on asset i (i=1 to 5)
F1 and F2 denote the factors in this model
e_i = non-systematic or residual risk for asset i (i=1 to 5)
For this problem for each stock we have that:
Ri = 0.11 + 0.84*F1 + 1.69*F2 + e_i
Xi = 1/5 for all i
Rp = SUM(Xi*Ri) = (i= 1 to 5)
= SUM(1/5*Ri) =
= 1/5*SUM(0.11 + 0.84*F1 + 1.69*F2 + e_i) =
= 1/5*5*(0.11 + 0.84*F1 + 1.69*F2) + 1/5*SUM(e_i) =
= 0.11 + 0.84*F1 + 1.69*F2 + SUM(e_i)/5
b.
If you place in your equally weighted portfolio a very large number N
of stocks that all have the same expected returns and the same betas
you will have that:
Xi = 1/N for all i
Ei = E for all i
Rp = SUM(Xi*Ri) = (i= 1 to N)
= 1/N * SUM(Ri) =
= 1/N * SUM(E + beta_1*F1 + beta_2*F2 + e_i) =
= E + beta_1*F1 + beta_2*F2 + 1/N * SUM(e_i) =
= E + beta_1*F1 + beta_2*F2
In the last step we depreciate the term 1/N * SUM(e_i) because N is too large.
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5-.
a.
According to the CAPM:
E = rf + Beta_Port * [E_m ? rf]
where
E = the expected return on the portfolio
rf = the risk-free rate
E_m = the expected return on the market portfolio
Then:
E = 0.064 + 1.2 * (0.138-0.064) =
= 0.064 + 1.2 * 0.074 =
= 0.064 + 0.0888 =
= 0.1528
The expected return on the Solomon stock is 15.28%
b.
If the risk free rate decreases to 3.5 we have that:
E = 0.035 + 1.2 * (0.138-0.035) =
= 0.035 + 1.2 * 0.103 =
= 0.035 + 0.1236 =
= 0.1586
The expected return on the Solomon stock will be 15.86%
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I hope that this helps you. Feel free to request for a clarification
if you need it.
Best regards.
livioflores-ga |
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