Input:
# of stocks 76
Answer:
a. Estimates of expected returns 76
Problem 8-5
A portfolio management organization analyzes 60 stocks and constructs a mean-variance efficient portfolio using only these 60 securities.
a. How many estimates of expected returns, variances, and covariances are needed to optimize this portfolio?
Estimates of expected returns
b. If one could safely assume that stock market returns closely resemble a single-index structure, how many estimates would be needed?
Estimates
Stock ER Beta Firm-Specific SD
A 0.1 0.7 0.28
B 0.18 1.25 0.42
Risk-free Rate SD 0
Market Index SD 0.22
Risk-free Rate 0.07
b. WA0.35
WB0.35
WRF 0.3
Answer:
a. SD of Stk A 0.3196
Problem 8-6
The following are estimates for two stocks.
Stock Expected Return Beta Firm-Specific Standard Deviation
A13% 0.8 30% B
18 1.2 40
The market index has a standard deviation of 22% and the risk-free rate is 8%.
a. What are the standard deviations of stocks Aand B? (Do not round intermediate calculations.Round your answers to 2 decimal
places. Omit the “%” sign in your response.)
Stock A%
Stock B%
Variance =
SD = sqrt of variance
Nonsystematic Variance = 0.031213
0.0091
rf0.066
β0.65
Answers:
a. Stock A
Problem 8-8
Consider the two (excess return) index model regression results for Aand B:
a. Which stock has more firm-specific risk?
Stock A
Stock B
b. Which stock has greater market risk?
Stock A
Stock B
Stock A Stock B
Variance 0.338 0.853333
Inputs
Stock A Stock B
β1.3 1.6
σM0.2 0.2
R-square 0.2 0.12
Problem 8-9
Suppose that the index model for stocks Aand Bis estimated from excess returns with the following results:
R-squareB= 0.12
What is the standard deviation of each stock? (Do not round intermediate calculations. Calculate using numbers in decimal form,
not percentages. Round your answers to 2 decimal places. Omit the “%” sign in your response.)
Standard Deviation
Stock A%
Stock B%
Stock A Stock B
Variance 0.147456 0.384
β1.2 1.5
σM0.16 0.16
R-square 0.25 0.15
Answers: Risk for A Risk for B
Problem 8-10
Suppose that the index model for stocks Aand Bis estimated from excess returns with the following results:
RA= 3% + 0.7
Break down the variance of each stock to the systematic and firm-specific components. (Do not round intermediate calculations.
Calculate using numbers in decimal form, not percentages. Round your answers to 4 decimal places.)
Risk for A Risk for B
Stock A Stock B
Variance 0.11858 0.26136
SD 0.344354 0.5112
Answers:
Problem 8-11
Suppose that the index model for stocks Aand Bis estimated from excess returns with the following results:
RA= 3% + 0.7
Stock A Stock B
Variance 0.027788 0.078672
SD 0.166699 0.2805
β0.5 0.7
σM0.17 0.17
R-square 0.26 0.18
Problem 8-12
Suppose that the index model for stocks Aand Bis estimated from excess returns with the following results:
RA= 3% + 0.7
What is the covariance between each stock and the market index?(Calculate using numbers in decimal form, not percentages.
Do not round your intermediate calculations. Round your answers to 3 decimal places.)
Covariance
Stock A
Stock B
Stock A Stock B
Variance 0.084692 0.2016
SD 0.291019 0.4490
σM0.21 0.21
R-square 0.22 0.14
Covariance between returns of Stock A & Stock B
Answers: Portfolio
Problem 8-13
Suppose that the index model for stocks A and B is estimated from excess returns with the following results:
RA= 3% + 0.7
1. What is the standard deviation of the portfolio? (Calculate using numbers in decimal form, not percentages. Do not round
your intermediate calculations. Round your answer to 2 decimal places. Omit the “%” sign in your response.)
2. What is the beta of your portfolio? (Calculate using numbers in decimal form, not percentages. Do not round your
intermediate calculations. Round your answer to 2 decimal places.)
3. What is the firm-specific variance of your portfolio?(Calculate using numbers in decimal form, not percentages. Do not
round your intermediate calculations. Round your answer to 4 decimal places.)
4. What is the covariance between the portfolio and the market index? (Calculate using numbers in decimal form, not
percentages. Do not round your intermediate calculations. Round your answer to 3 decimal places.)
Stock A Stock B
Variance 0.039204 0.0729
Inputs
Stock A Stock B
β0.55 0.6
σM0.18 0.18
R-square 0.25 0.16
Cov(rA,rB ) = 0.0107
Cov(rP,rM ) = 0.010425
P0.5 Stock A 0.6
σP = 0.1758
Cov(rA ,rM ) = 0.01782
Cov(rB, rM) = 0.01944
Cov(rP,rM ) 0.010425
βp 0.57
βm 1
Answers: Tolerance βt-bill 0
1. SD of Portfolio Q 11.73% 0.1273
2. Portfolio Q Beta 0.59
3. Firm-specific Cov Port Q 0.0027 0.0037 0.0051
4. Covariance btwn Portfolio & Mkt Indx 0.01 0.11
Problem 8-14
Suppose that the index model for stocks A and B is estimated from excess returns with the following results:
Assume you create for portfolio Qwith investment proportions of 0.50 in P, 0.30 in the market index, and 0.20 in T-bills, portfolio Pis
composed of 60% Stock Aand 40% Stock B.
1. What is the standard deviation of portfolio Q? (Calculate using numbers in decimal form, not percentages.Do not round
intermediate calculations. Round your answer to 2 decimal places. Omit the “%” sign in your response.)
2. What is the beta of portfolio Q? (Do not round intermediate calculations. Round your answer to 2 decimal places.)
3. What is the “firm-specific” covariance of portfolio Q?(Calculate using numbers in decimal form, not percentages. Do not round
intermediate calculations. Round your answer to 4 decimal places.)
4. What is the covariance between the portfolio and the market index? (Calculate using numbers in decimal form, not percentages.
Do not round intermediate calculations. Round your answer to 2 decimal places.)
Inputs:
Estimated Beta 1.46
Problem 8-15
A stock recently has been estimated to have a beta of 1.24:
a. What will a beta book compute as the “adjusted beta” of this stock? (Do not round intermediate calculations. Round your
answer to 2 decimal places.)
b. Suppose that you estimate the following regression describing the evolution of beta over time:
βt= 0.3 + 0.7βt–1
Inputs:
Stock ER % Beta Residual SD %
A0.25 1.2 0.56
B0.19 1.6 0.70
C0.16 0.5 0.61
D0.13 1 0.53
σ
ER SD
αB-0.008 0.12 B -0.008 4900.00 -0.00016 -0.0519
αC0.05 0.09 C 0.05 3721.00 0.001344 0.4270
αD-0.02 0.06 D -0.02 2809.00 -0.00071 -0.2262
Total 0.003147 1.0000
Residual Variances
Stock
Answers:
Stk A Stk B Stk C Stk D
a. Excess Returns 18.00% 12.00% 9.00% 6.00%
0.3613
Alpha Values 8.40% -0.80% 5.00% –2.00%
Residual variances 3136.00 4900.00 3721.00 2809.00
b. Proportion 0.8046
Active portfolio 0.0308
Problem 8-17
A portfolio manager summarizes the input from the macro and micro forecasters in the following table:
Micro Forecasts
Asset Expected Return (%) Beta Residual Standard Deviation (%)
Stock A20 1.3 58
Stock B18 1.8 71
Stock C17 0.7 60
Stock D12 1.0 55
Residual variances
b. Compute the proportion in the optimal risky portfolio. (Do not round intermediate calculations. Round your answer to 4 decimal places.)
Proportion
c. What is Sharpe’s measure for the optimal portfolio and how much of it is contributed by the active portfolio? (Do not round intermediate calculations. Round
your answers to 4 decimal places.)
Sharpe’s measure
Passive EP 0.15 0.21
Worksheet
αA0.084 0.18 A 0.084 3136.0000 0.002679 0.8511
Inputs:
Stock ER % Beta Residual SD %
A 0.27 0.8 0.59
B 0.12 1.2 0.69
C 0.11 0.5 0.62
D 0.09 0.6 0.54
σ
ER SD
Asset
T-bills 0.06 0
A 3481.00 Forecast of Active Portfolio Weight of active portfolio
B 4761.00 Alpha (α) Expected Excess Return
Answers: C 3844.00 α = 0.1630
W0 = 0.31442
Tolerance D 2916.00 β = 0.75 W* = 0.2919
αA
a. Sharpe’s measure
0.3046 1%
σ2(e) = 3455.31 Information ratio of active portfolio αB
Sharpe’s measure =
σ(e) = 58.78 αC
b. A = 0.2772
αD
Unconstrained 8.78 0.00%
A2 =0.0769
Constrained 8.77 0.00% Optimal Risky Portfolio characteristics: Square of sharpe’s measure = 0.0928
Passive 7.50 0.00%
βP = 0.92 Market’s sharpe measure = 0.304572892
E(RP) = 0.1029 0.0842
B
D
Only positive alphas
A 0.162 3481.0000 0.004654 0.8994
σ2
C 0.02 3844.00 0.000520 0.1006
Total 0 0.00 0.005174 1.0000
Forecast for active portfolio
α = 0.1477
βP = 0.93
β = 0.77
E(RP) = 0.1028
σ2(e) = 2854.99 σ2
P634.886403
σ(e) = 53.43 σP25.19695226
A =
W0 = 0.3449 y = 0.5397
W* = 0.3196
Information ratio Final Positions
A = 0.002764647 M 0.3672
Unconstrained 0.0546 178.77
Unconstrained 8.78
Constrained 8.77
Problem 8-18
A portfolio manager summarizes the input from the macro and micro forecasters in the following table:
Micro Forecasts
Asset Expected Return (%) Beta Residual Standard Deviation (%)
Stock A20 1.3 58
Stock B18 1.8 71
Stock C17 0.7 60
Stock D12 1.0 55
Macro Forecasts
Expected Return (%) Standard Deviation (%) -T-
bills 8 0 Passive Passive
Passive EP 0.12 0.2
Worksheet