Question

Suppose that assets 1 and 2 are 24% correlated and have the following expected returns and standard deviations:

Asset	E(R)	σ
1	14%	9%
2	8%	4%

a) Calculate the expected return and standard deviation for a portfolio consisting of equal weights in assets 1 and 2.

b) What are the weights of a minimum variance portfolio consisting of assets 1 and 2? What is the expected return and standard deviation of this portfolio?

c) Has there been an improvement with respect to the risk-adjusted return as a result of allocating capital according to the minimum variance portfolio weights? You can assume a risk-free rate of 1.5% p.a. in answering this question.

Answer 1

1. Expected return = 0.5*0.14 + 0.5*0.08 = 0.07+0.04 = 0.11 =11%

Expected std.deviation = sqrt(0.5²*0.09²+0.5²*0.04²+2*0.5*0.5*0.09*0.04*0.24) = sqrt(0.002857) = 0.053451 = 5.345%

2. W1 = [(0.04²) - 0.09*0.04*0.24]/[0.09²+0.04²-2*0.04*0.09*0.24] = 0.000736/0.007972 = 0.0923 = 9.23%

W2=100-9.23 = 90.77%

expected return = 0.0923* 14+0.9077*8 = 8.554%

Expected std deviation = sqrt(0.001532) = 0.0391= 3.91%

3. Sharpe ratio is first case = (11-1.5)/5.345 = 1.777

Sharpe Ratio is 2nd case = (8.554-1.5)/3.91 = 1.804

2nd sharpe ratio is better than first case, thus there is an improvement in the risk adjusted returns