CHapter 8 Q & A - stuuf PDF

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4.

The following are estimates for two stocks. Sto ck

Expected Return 13%

A

Bet a

Firm-Specific Standard Deviation

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 A and B? b. Suppose that we were to construct a portfolio with proportions: Stock A:

. 3 0

Stock B:

. 4 5

T-bills:

. 2 5

Compute the expected return, standard deviation, beta, and nonsystematic standard deviation of the portfolio. 2

a. The standard deviation of each individual stock is given by:

b.

2

1/2

σ i =[ β i2 σ M +σ (e i )]

The expected rate of return on a portfolio is the weighted average of the expected returns of the individual securities: E(rP ) = wA × E(rA ) + wB × E(rB ) + wf × rf

The beta of a portfolio is similarly a weighted average of the betas of the individual securities: βP = wA × βA + wB × βB + wf × β f

 2 (eP )  wA2  2 (eA )  wB2  2 (eB )  w f2  2 (e f )

The variance of this portfolio is: 2 2 2 2 σ P =β Pσ M +σ ( e P )

5.

Consider the following two regression lines for stocks A and B in the following figure.

a. b. c. d. e.

Which stock has higher firm-specific risk? Which stock has greater systematic (market) risk? Which stock has higher R2? Which stock has higher alpha? Which stock has higher correlation with the market?

a. Deviation from the SCL b. Slope of the line 2

R = c.

β i2 σ 2M 2 β2i σ M + σ 2( ei )

= Variation of stock return/ Total Variance

d. .. e. Correlation Coefficient = Square root of R2

Micro Forecasts

Asset

Expected Return (%)

Beta

Residual Standard Deviation (%)

Stock A

20

1.3

58

Stock B

18

1.8

71

Stock C

17

0.7

60

Stock D

12

1.0

55

Macro Forecasts Expected Return (%)

Asset

T-bills

Passive equity portfolio

Standard Deviation (%)

8

0

16

23

Use the following data for Problems 9 through 14. Suppose that the index model for stocks A and B is estimated from excess returns with the following results:   RA=3%+.7RM+eA RB=−2%+1.2RM+eB σM=20%;  RsquareA=.20;  RsquareB=.12

9.

Que 9: What is the standard deviation of each stock?

The standard deviation of each stock can be derived from the following equation for R2: 2 β i2 σ M 2 Ri = 2 = σi

For stock A and B: Find Total Firm risk

10. Break down the variance of each stock into its systematic and firm-specific components. The systematic risk for A is:

 2A  M2  0.702  202  196

The firm-specific risk of A (the residual variance) is the difference between A’s total risk and its systematic risk:

The systematic risk for B is:

B’s firm-specific risk (residual variance) is

11.

What are the covariance and the correlation coefficient between the two stocks? 2

Cov(r A ,rB )=β A β B σ M =0.70 ×1. 20×400=336

ρ AB =

12.

Cov(r A ,r B ) 336 =0 .155 = 31. 30×69. 28 σ A σB

What is the covariance between each stock and the market index?

Note that the correlation is the square root of R2:

¿

Square root of R^2 is correlation coefficient

Cov( rA , rM )   A M  0.201/2 31.30 20  280 Cov( rB , rM )   B M  0.121/2 69.2820  480

13.

For portfolio P with investment proportions of .60 in A and .40 in B, rework Problems 9, 10, and 12.

For portfolio P we can compute: a. σP b. βP 2

c.

2 2

2

2 σ (e P )=σ P −β P σ M=1282 . 08−( 0 .90 ×400)=958 . 08

d. Cov(rP,rM ) = βP

σ 2M

This same result can also be attained using the covariances of the individual stocks with the market: e. Cov(rP,rM ) = Cov(0.6rA + 0.4rB, rM )

14.

Rework Problem 13 for portfolio Q with investment proportions of .50 in P, . 30 in the market index, and .20 in T-bills.

Note that the variance of T-bills is zero, and the covariance of T-bills with any asset is zero. Therefore, for portfolio Q:

σ Q = [ w 2P σ2P+w M2 σ 2M +2× w P ×w M × Cov(r P , r M )]

1/2 1/2

¿ [ (0 . 52 ×1 ,282 . 08)+( 0. 32 ×400 )+( 2×0 .5×0 .3×360 )] =21 . 55 %...