Question

Suppose that many stocks are traded in the market and that it is possible to borrow at the risk-free rate, r_ƒ. The characteristics of two of the stocks are as follows:

Stock	Expected Return			Standard Deviation
A		10	%		25	%
B		18	%		75	%
Correlation = –1

a. Calculate the expected rate of return on this risk-free portfolio? (Hint: Can a particular stock portfolio be substituted for the risk-free asset?) (Round your answer to 2 decimal places.)

b. Could the equilibrium r_ƒ be greater than 12.00%?

Answer 1

a)

• Both the stocks are perfectly negatively correlated.
• As we are creating a risk-free portfolio, the rate of return of the portfolio would be equal to rf.
• Also, the standard deviation of the portfolio will be equal to 0.
• Cov(A,B) = -1

Wb = 1 - Wa

Standard deviation of portfolio = 0 = [(Wa * std dev of A)^2 + ((1-Wa) * std dev of A)^2 + 2 * Wa * Wb * std dev of A * std dev of B * Cov(A,B) ] ^0.5

=[(Wa * std dev of A)^2 + ((1-Wa) * std dev of A)^2 + 2 * Wa * Wb * std dev of A * std dev of B * (-1)] ^0.5

= [[(Wa * std dev of A) - (Wb * std dev of B)]^2 ]^0.5 = (Wa * std dev of A) - (Wb * std dev of B)

(Wa * std dev of A) - (Wb * std dev of B) = 0

(Wa * std dev of A) = (1- Wa) * std dev of B)

Wa * 0.25 = (1 - Wa) * (0.75)

Wa = 0.75 , Wb = 0.25

Expected Return = Wa * Ra + Wb*Rb = 0.75* 10 + 0.25* 18 = 12%

b) No.

As the Expected return (for risk-free portfolio) is 12%, the risk-free rate must be 12%. Answer

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