3 Assume the riskfree rate is 4 rf 4 the expected return on

3. Assume the risk-free rate is 4% (rf = 4%), the expected return on the market portfolio is 12% (E[rM] = 12%) and the standard deviation of the return on the market portfolio is 16% (sigmaM = 16%). (All numbers are annual.) Assume the CAPM holds.

a. What are the expected returns on securities with the following betas: (i) beta = 1.0, (ii) beta = 1.5, (iii) beta = 0.5, (iv) beta = 0.0, (v) beta = -0.5?

b. What are the betas of securities with the following expect returns: (i) 12%, (ii) 20%, (iii) -4%?

c. What are the portfolio weights (in the risk-free asset and the market portfolio) for efficient portfolios (portfolios on the efficient frontier) with expected returns of (i) 8%, (ii) 10%, (iii) 4%, (iv) 24%.

d. What are the portfolio weights (in the risk-free asset and the market portfolio) for efficient portfolios (portfolios on the efficient frontier) with standard deviations of (i) 4%, (ii) 20%, (iii) 16%.

e. What are the correlations between the portfolios in (i) Q.3c(i) and Q.3c(iv), (ii) Q.3d(i) and Q.3d(ii)?

f. Can securities or portfolios with the following characteristics exist in equilibrium, assuming the CAPM holds (yes or no): (i) expected return 0%, standard deviation 40%, (ii) expected return 9%, standard deviation 9%, (iii) expected return 34%, standard deviation 70%.

g. A stock with a beta of 1 (beta = 1.0) has a current price of $40/share. Assuming it pays no dividends, what is the expected price in 1 year? If it is expected to pay a dividend of $4/share at the end of the year, what is the expected price in 1 year (after the payment of the dividend)? If the beta of the stock is 2 (beta = 2.0), what are the expected prices under these 2 scenarios, i.e., no dividends or a dividend of $4?

h. For a moment (but just a moment) assume that the CAPM may not hold. In other words, alpha (alpa) is non-zero. If a non-dividend paying stock with a beta of 1 (beta = 1.0) has a current price of $50/share and an expected price in 1 year of $60/share (based on your personal analysis of the companies prospects), what is the alpha (alpa) of this stock? What if the beta is 2 (beta = 2.0)? What if the beta is 3 (beta= 3.0)?

Solution

3. Risk Free Return = 4%

Expected Market return = 12%

Standard Deviation of Rteurn on market = 16%

CAPM Model-

Expected Return = Risk Free return + Beta (Market return - Risk free return)

(i) If Beta = 1.0

Then,

Expected Return = Risk Free return + Beta (Market return - Risk free return)

= 4% + 1.0 (12%-4%)

= 12%

(ii) If Beta = 1.5

Then,

Expected Return = Risk Free return + Beta (Market return - Risk free return)

= 4% + 1.5 (12%-4%)

= 16%

(iii) If Beta = 0.5

Then,

Expected Return = Risk Free return + Beta (Market return - Risk free return)

= 4% + 0.5 (12%-4%)

= 8%

(iv) If Beta = 0.0

Then,

Expected Return = Risk Free return + Beta (Market return - Risk free return)

= 4% + 0.0 (12%-4%)

= 4%

(v) If Beta = -0.5

Then,

Expected Return = Risk Free return + Beta (Market return - Risk free return)

= 4% + (-0.5) (12%-4%)

= 0%

B. Betas of security if Expected Return are given-

(i) Expected Return = 12%

Expected Return = Risk Free return + Beta (Market return - Risk free return)

12% = 4% + Beta ( 12%-4%)

12%-4% = Beta * 8%

8% = Beta * 8%

Beta = 1.0

(ii) Expected Return = 20%

Expected Return = Risk Free return + Beta (Market return - Risk free return)

20% = 4% + Beta ( 12%-4%)

20%-4% = Beta * 8%

16% = Beta * 8%

Beta = 2.0

(iii) Expected Return = -4%

Expected Return = Risk Free return + Beta (Market return - Risk free return)

-4% = 4% + Beta ( 12%-4%)

-4%-4% = Beta * 8%

-8% = Beta * 8%

Beta = -1.0

C. Portfolio Weights-

In the above question the portfolio weight is if Risk free = 1 then Market return = 3. by using this concept we can calculate the below questions as follows. Assume Beta = 1

(i) Expected Return = 8%

Let Risk Free Return = X

Then, Market return = 3X

8% = X + 1 (3X -X)

8% = 3X

Risk Free return = 2.67%

and Market return = 2.67*3 = 8%.

(iv) Expected Return = 24%

Let Risk Free Return = X

Then, Market return = 3X

24% = X + 1 (3X -X)

24% = 3X

Risk Free return = 8%

and Market return = 8*3 = 24%

(ii) Expected Return = 10%

Let Risk Free Return = X

Then, Market return = 3X

10% = X + 1 (3X -X)

10% = 3X

Risk Free return = 3.33%

and Market return = 3.33*3 = 10%

(iii) Expected Return = 4%

Let Risk Free Return = X

Then, Market return = 3X

4% = X + 1 (3X -X)

4% = 3X

Risk Free return = 1.33%

and Market return = 1.33*3 = 4%

(i) Expected Return = 8%

Let Risk Free Return = X

Then, Market return = 3X

8% = X + 1 (3X -X)

8% = 3X

Risk Free return = 2.67%

and Market return = 2.67*3 = 8%

(i) Expected Return = 8%

Let Risk Free Return = X

Then, Market return = 3X

8% = X + 1 (3X -X)

8% = 3X

Risk Free return = 2.67%

and Market return = 2.67*3 = 8%

(i) Expected Return = 8%

Let Risk Free Return = X

Then, Market return = 3X

8% = X + 1 (3X -X)

8% = 3X

Risk Free return = 2.67%

and Market return = 2.67*3 = 8%

(i) Expected Return = 8%

Let Risk Free Return = X

Then, Market return = 3X

8% = X + 1 (3X -X)

8% = 3X

Risk Free return = 2.67%

and Market return = 2.67*3 = 8%

(i) Expected Return = 8%

Let Risk Free Return = X

Then, Market return = 3X

8% = X + 1 (3X -X)

8% = 3X

Risk Free return = 2.67%

and Market return = 2.67*3 = 8%

D. Portfolio Weights When standard deviations are given-

(i) Standard Devation = 4%

In the above question the portfolio weight is if Risk free = 1 then Market return = 3. by using this concept we can calculate the below questions as follows.

Standard Deviation = Risk free Return + Market Return

4% = X + 3X

4% = 4X

X = 1

Then Portfolio Weight =

Risk Free = 1 and Market Return = 3

(ii) Standard Devation = 20%

Standard Deviation = Risk free Return + Market Return

20% = X + 3X

20% = 4X

X = 5

Then Portfolio Weight =

Risk Free = 5 and Market Return = 15

(iii) Standard Devation = 16%

Standard Deviation = Risk free Return + Market Return

16% = X + 3X

16% = 4X

X = 4

Then Portfolio Weight =

Risk Free = 4 and Market Return = 12.