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 μ₁, μ₂, σ₁ and σ₂, 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 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 μ₁, m2 for μ₂, s1 for σ₁, s2 for σ₂, and r for ρ. Use * for multiplication, / for division and ^ for power. For example, c*(2*m1 + w*m2)/(5*s1^2 + r*s2^2) means c(2 μ₁ + w μ₂)/(5 σ₁² + ρ σ₂²).

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).

Answer 1

ans::

as for given data

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]

c)

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)

thank you...and i dnt know d sorry...