Consider a one period economy consisting of only three assets: a risk free asset

Question

Consider a one period economy consisting of only three assets: a risk free asset and stocks of companies A and B. At date 0 (today), stocks A and B have expected returns of E[RA]=12% and E[RB]=18%, respectively, and their betas are A=0.5 and B =1.5, respectively. At date 1 (tomorrow), both companies will be liquidated and both companies will pay out a liquidating dividend. The liquidating dividends are expected to be $100 per share for both companies. Assume that the CAPM holds.

a) Calculate the prices of stocks A and B at date 0

b) What is the return of the risk free asset?

c) Suppose that an investor allocates 20% of his wealth in the risk free asset and the rest in the market portfolio of risky assets. What fraction of his wealth will be allocated to stock A?

d) Suppose returns on stock A and B are uncorrelated (i.e., AB = 0) and the standard deviation of the return on stock A is 0.25. What is the Sharpe ratio of the market porfolio?

Explanation / Answer

a. Price = D1/ Ke

Stock A = 100 / 0.12 = $833.33

Stock B = 100 / 0.18 = $555.56

b. Expected return = Rf+ BetaxRP

Stock A: 12 = Rf + 0.5RP [Eq.1]

Stock B: 18 = Rf + 1.5RP [Eq.2]

Eq.1:

Rf = 12 - 0.50RP

Putting this value in Eq. 2, we get

18 = 12 - 0.5RP + 1.5RP

RP = 6%

Rf = 12 - 0.50x6

Rf = 9%

Hence, Risk free return is 9%

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