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.

a. Find an expression for the expected value of your investment after one month. Enter a formula below. For simplicity, use m1 for 1, m2 for 2, s1 for 1, s2 for 2, and r for . Use * for multiplication, / for division and ^ for power. For example, c*(2*m1 + w*m2)/(5*s1^2 + r*s2^2) means c(2 1 + w 2)/(5 12 + 22).

b. Find an expression for the variance of your investment after one month.

Explanation / Answer

a) X1 and X2 are random variable with expected values m1 and m2 respectively.

There to find an expression for the expected value,

P = c(wX1 + (1-w)X2) becomes

P = c(w*m1 + (1-w)*m2)

b) Portfolio variance is calculated by multiplying the squared weight of each security by its corresponding variance and adding two times the weighted average weight multiplied by the covariance of all individual security pairs.

(weight(1)^2*variance(1) + weight(2)^2*variance(2) + 2*weight(1)*weight(2)*covariance(1,2)

covariance = correlation*sd(1)*sd(2)

Variance with initial investment is given by,

Var = c^2[w^2*s1^2 + (1 - w)^2 * s2^2 + 2*w*(1-w)*r*s1*s2]

= c^2[w^2*s1^2 + (1+w^2 - 2*w) * s2^2 + 2*w*r*s1*s2 - 2*w^2*r*s1*s2]

= c^2[w^2(s1^2 + s2^2 - 2*r*s1*s2) + 2*w(r*s1*s2 - s2^2) + s2^2]

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