Suppose that you wish to invest in two stocks which both have a current price of $1. The values of these two stocks in one month are described by two random variables, say, X 1 and X 2 . Suppose that the expected values and variances of X 1 and X 2
are μ 1 , μ 1 , σ 12 , and σ 2 , respectively. We also assume that the correlation between the stocks is given by ρ.
Let c denote your initial investment, which is to be invested in the stocks, and assume that shares can be bought up to any percentages. Let w denote the percentage of your investment in stock 1. Finally, let P denote the value of your portfolio (investment) after a month. Then we have that P = c ( w X 1 + ( 1 - w ) X 2 ), where 0 ≤ w ≤ 1.
a) Find an expression for the expected value of your investment after one month.
b) Find an expression for the variance of your investment after one month.
c) Find the weights that minimize the risk of your investment.
( Hint: in the classical portfolio theory the risk is simply quantified by the variance. )
d) Find the correlation which minimizes the risk of the equally weighted portfolio (i.e., w = 0.5).