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

a)

Expected return on the portfolio = w(x)*E(x) + w(y)*E(y)

Standard deviation of portfolio =

where x and y are the securities

Expected return = 0.5*0.14+0.5*0.08 = 0.11 = 11%

Standard deviation = ((0.5*0.09)^2 + (0.5*0.04)^2 + 2*0.5*0.5*0.09*0.04*0.24)^0.5 = 0.0534

Standard deviation = 5.34%

b)

Minimum variance weights for 2 asset portfolio are given by

w1 = (0.04*0.04 - 0.24*0.09*0.04)/(0.09*0.09+0.04*0.04-2*0.24*0.04*0.09) = 0.0923

w2 = 1-w1 = 0.9076

Expected return = 0.0923*0.14+0.9076*0.08 = 0.08553= 8.553%

Standard deviation = ((0.0923*0.09)^2 + (0.9076*0.04)^2 + 2*0.9076*0.0923*0.09*0.04*0.24)^0.5 = 0.0391

Standard deviation = 3.91%

c)

Comparing part a) and part b), we observe that by varying the weights of the 2 portfolios, we get a standard deviation that is lesser than the standard deviation when the weights were equally distributed.

To check for risk-adjusted return, we use the Sharpe's ratio

Sharpe's ratio = (Portfolio return-Risk-free rate)/Standard deviation of the portfolio

Sharpe's ratio (part a) = (0.11 -0.015)/0.0534 = 1.77

Sharpe's ratio (part b) = (0.08553-0.015)/0.0391 = 1.80

Hence, there has been an improvement in the risk-adjusted return as a result of allocating capital according to the minimum variance portfolio weights