Suppose that you wish to invest in two stocks which both have a current price of

Question

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, X1 and X2. Suppose that the expected values and standard deviation of X1 and X2 are 1, 2, 1and 2, respectively. We also assume that the correlation between the stocks is given by .

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 X1 + (1 – w) X2), where 0 w 1.

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.)

Explanation / Answer

The variance of the investment after one month is given by
c^2*(w^2*(s1^2 + s2^2 - 2*r*s1*s2) + 2*w*(r*s1*s2 - s2^2) + s2^2)

To minimize the risk, the above equation is differentiated with respect to w and equated to 0.

d/dw = 2*w*c^2(s1^2 + s2^2 - 2*r*s1*s2) + 2*c^2(r*s1*s2 - s2^2) = 0

take all 2*c^2 to the other side and we get,

w(s1^2 + s2^2 - 2*r*s1*s2) + (r*s1*s2 - s2^2) = 0

w = (s2^2 - r*s1*s2)/(s1^2 + s2^2 - 2*r*s1*s2)

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