Question

Suppose that there are many stocks in the security market and that the characteristics of stocks A and B are given as follows:

Stock		Expected Return				Standard Deviation
A			14	%			6	%
B			16				9
	Correlation = –1

Suppose that it is possible to borrow at the risk-free rate, r_f. What must be the value of the risk-free rate? (Hint: Think about constructing a risk-free portfolio from stocks A and B.) (Do not round intermediate calculations. Round your answer to 3 decimal places.)

Answer 1

Since Stock A and Stock B are perfectly negatively correlated, a risk-free portfolio can be created and the rate of return for this portfolio, in equilibrium, will be the risk-free rate. To find the proportions of this portfolio [with the proportion wA invested in Stock A and wB = (1 - wA) invested in Stock B], set the standard deviation equal to zero. With perfect negative correlation, the portfolio standard deviation is:

σP = Absolute value [wAσA- wBσB]

0 = 6 × wA − [9 × (1 - wA)]

wA = 0.6

The expected rate of return for this risk-free portfolio is:

E(r) = (0.6 × 14) + (0.4 × 16) = 14.800%

Therefore, the risk-free rate is 14.800%.