Question

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% (σM = 16%). (All numbers are annual.) Assume the CAPM holds. *PLEASE HELP WITH E-H; INCLUDED ADDITIONAL QUESTIONS FOR REFERENCE*

1a. What are the expected returns on securities with the following betas: (i) β = 1.0, (ii) β = 1.5, (iii) β = 0.5, (iv) β = 0.0, (v) β = -0.5?

1b. What are the betas of securities with the following expect returns: (i) 12%, (ii) 20%, (iii) -4%?

1c. 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%.

1d. 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%.

1e. What are the correlations between the portfolios in (i) Q.1c(i) and Q.1c(iv), (ii) Q.1d(i) and Q.1d(ii)?

1f. 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%.

1g. A stock with a beta of 1 (β = 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 (β = 2.0), what are the expected prices under these 2 scenarios, i.e., no dividends or a dividend of $4

1h. For a moment (but just a moment) assume that the CAPM may not hold. In other words, alpha (α) is non-zero. If a non-dividend paying stock with a beta of 1 (β = 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 (α) of this stock? What if the beta is 2 (β = 2.0)? What if the beta is 3 (β = 3.0)?

Answer 1

a) As per CAPM,

Expected return on a security = Risk free rate+ Beta of security*(Expected return on market portfolio - Risk free rate)

So,

i) Expected return on a security with beta 1

= 4%+ 1*(12%-4%)=12%

ii) Expected return on a security with beta 1.5

= 4%+ 1.5*(12%-4%)=16%

iii) Expected return on a security with beta 0.5

= 4%+ 0.5*(12%-4%)=8%

iv) Expected return on a security with beta 0

= 4%+ 0*(12%-4%)=4%

i) Expected return on a security with beta -0.5

= 4%-0.5*(12%-4%)=0%

b) From CAPM,

i) for a security with expected return of 12%, beta(b) is given by

4%+b*8% =12%

b =1 , So the beta of the security is 1

ii) for a security with expected return of 20%, beta(b) is given by

4%+b*8% =20%

b =2, So the beta of the security is 2

i) for a security with expected return of -4%, beta(b) is given by

4%+b*8% =-4%

b =-1 , So the beta of the security is -1

c) As the expected return on a portfolio is the weighted average return on individual securities

Let w be the weight of risk free asset and (1-w) be the weight of Market portfolio

i) For efficient portfolios with expected returns of 8%

w*4%+(1-w)*12% = 8%

w = 0.5 and 1-w=0.5

So, portfolio weight in the Risk free Asset is 0.5 or 50%

and portfolio weight in the market portfolio is 0.5 or 50%

ii) For efficient portfolios with expected returns of 10%

w*4%+(1-w)*12% = 10%

w = 0.25 and 1-w=0.75

So, portfolio weight in the Risk free Asset is 0.25 or 25%

and portfolio weight in the market portfolio is 0.75 or 75%

iii) For efficient portfolios with expected returns of 4%

w*4%+(1-w)*12% = 4%

w = 1 and 1-w=0

So, portfolio weight in the Risk free Asset is 1 or 100%

and portfolio weight in the market portfolio is 0 or 0%

iv) For efficient portfolios with expected returns of 24%

w*4%+(1-w)*12% = 24%

w = -1.5 and 1-w=2.5

So, portfolio weight in the Risk free Asset is -1.5 or -150% (Amount corresponding to 150% must be borrowed at risk free rate)

and portfolio weight in the market portfolio is 2.5 or 250% (100% +150% borrowed amount to be invested in market portfolio)

d) Standard deviation of a portfolio comprising of a risky asset and risk free asset

= weight of risky asset * standard deviation of risky asset

i) For a portfolio with standard deviation of 4%

weight of market portfolio = 4%/16% =1/4 or 0.25

weight of Risk free asset = 1-0.25 = 0.75

So, portfolio weight in the Risk free Asset is 0.75 or 75%

and portfolio weight in the market portfolio is 0.25 or 25%

ii) For a portfolio with standard deviation of 20%

weight of market portfolio = 20%/16% =5/4 or 1.25

weight of Risk free asset = 1-1.25 = - 0.25

So, portfolio weight in the Risk free Asset is -0.25 or -25%(Amount corresponding to 25% must be borrowed at risk free rate)

and portfolio weight in the market portfolio is 1.25 or 125%(100% +25% borrowed amount to be invested in market portfolio)

iii) For a portfolio with standard deviation of 16%

weight of market portfolio = 16%/16% =1

weight of Risk free asset = 1-1 = 0

So, portfolio weight in the Risk free Asset is 0 or 0%

and portfolio weight in the market portfolio is 1 or 100%