Question

Consider the following information:

Bear Market Normal Market Bull Market
Probability 0.3 0.5 0.2
Return on Stock A -10% 0% 40%
Return on Stock B -5% 5% 50%

a) Calculate and comment upon the expected return and standard deviation of A and B.

b) Assuming that you have £20,000 to invest. You have decided to invest £10,000 in stock A and the remainder in stock B. Calculate and comment upon the expected return and standard deviation of your portfolio if the correlation between A and B is 0.5.

c) Does a fully diversified portfolio include any risk? Use and explain appropriate diagrams in your answer.

Answer 1

a) The expected return is:

$ER(A) = -0.1\times0.3 + 0\times0.5 + 0.4\times0.2 = 0.05 or 5$%

$ER(B) = -0.05\times0.3 + 0.05\times0.5 + 0.5\times0.2 = 0.11 or 11$%

Means for A and B are:

x(A) = (-0.1 + 0 + 0.4)/3 = 0.1,

x(A) = (-0.05 + 0.05 + 0.5)/3 = 0.167,

Standard deviation of A and B is:

$s(A) = (0.3\times(-0.1 - 0.1)^{2} + 0.5\times(0 - 0.1)^{2} + 0.2\times(0.4 - 0.1)^{2})^{0.5} = 0.187$ or 18.7%.

$s(B) = (0.3\times(-0.05 - 0.167)^{2} + 0.5\times(0.05 - 0.167)^{2} + 0.2\times(0.5 - 0.167)^{2})^{0.5} = 0.208$ or 20.8%.

b) Assuming that you have £20,000 to invest. You have decided to invest £10,000 in stock A and the remainder in stock B. Calculate and comment upon the expected return and standard deviation of your portfolio if the correlation between A and B is 0.5.

$ER = 0.5\times0.05 + 0.5\times0.11 = 0.08$ or 8%,

$s = (0.5^{2}\times0.187^{2} + 0.5^{2}\times0.208^{2} + 2\times0.5\times0.5\times0.5\times0.187\times0.208)^{0.5} = 0.171$ or 17.1%.

c) A fully diversified portfolio includes any risk too.