Question

3. Assume two perfectly negatively correlated securities X & Y. X has an expected return = 12% and a risk = 20%. Y has an expected return = 9% and a risk = 18%. Based on this information, what is the risk free rate? Hint: you need to find the minimum variance portfolio of X & Y. Since correl = -1, you know that the minimum variance portfolio has risk = 0.

Answer 1

Security	Return	Std Deviation
X	12%	20%
Y	9%	18%

Since Stocks X & Y are perfectly negatively correlated, a risk-free portfolio can be constructed and its rate of return in equilibrium will be the risk-free rate. To find the proportions of this portfolio ( wX invested in Stock X and wY = 1 – wX in Stock Y), set the standard deviation to zero.

With perfect negative correlation, the portfolio standard deviation reduces to:

σp = | wXσX - wYσY| => 0 = | 20wX - 18(1-wX) | =>

20wX-18+18wX=0

32wX=18

wX =0.474

WY = 0.526

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

= 0.474*12%+0.526*9%

=5.688%+4.734%

= 10.422%

So the risk free rate with the risky portfolio X & Y is 10.42%.