Question

Consider the monthly returns of two risky assets. The return of the first asset has a mean of 2% and standard deviation of 3%. The return of the second asset has a mean of 1.5% and standard deviation of 2%. The correlation coefficient of the two returns is 0.3. How can the minimum variance portfolio (MVP) be constructed? What are the mean and standard deviation of the return of the MVP? Consider a portfolio with 50% invested in asset 1 and 50% invested in asset 2. Is such a portfolio efficient?

Answer 1

Return on stock 1 = 2%

Return on stock 2 = 1.5%

Standard deviation of stock 1 = 3%

Standard deviation of stock 2 = 2%

Correlation co-efficient = 0.3

Let w1 and w2 be the weights of the stocks 1 and 2 respectively, in the Minimum variance portfolio, then

Substituting the values,

w1 = ((0.02^2)- 0.3*0.03*0.02) / (0.03^2 + 0.02^2 - 2*0.3*0.03*0.02)

w1 = 0.23404

w2 = 0.76595

Mean of the minimum variance portfolio =

Standard deviation of the minimum variance portfolio =

Mean return of the minimum variance portfolio = 0.23404*0.02 + 0.76595*0.015 = 0.01617

Mean return of the minimum variance portfolio = 1.617%

Standard deviation of the minimum variance portfolio = [(0.23404*0.03)^2 + (0.76595*0.02)^2 + 2* 0.23404*0.76595*0.3*0.03*0.02]^(0.5) = 0.01866

Standard deviation of the minimum variance portfolio = 1.866%

w1	w2	Expected return	Standard deviation
0.1	0.9	0.0155	0.019115439
0.2	0.8	0.016	0.018697593
0.3	0.7	0.0165	0.018777646
0.4	0.6	0.017	0.019349419
0.5	0.5	0.0175	0.020371549
0.6	0.4	0.018	0.021780725
0.7	0.3	0.0185	0.023507446
0.8	0.2	0.019	0.025487252
0.9	0.1	0.0195	0.027665863
1	0	0.02	0.03

Portfolio with 50% invested in asset 1 and 50% invested in asset 2

Using the above formulas and w1=w2=0.5

Mean return of the 50-50 portfolio = 0.5*0.02 + 0.5*0.015 = 1.75%

Standard deviation of the 50-50 portfolio = [(0.5*0.03)^2 + (0.5*0.02)^2 + 2* 0.5*0.5*0.3*0.03*0.02]^(0.5) = 2.037%An efficient portfolio is the one which provides highest return for a given level of risk. Since this portfolio lies on the efficient frontier, it is efficient.