8. In a portfolio, let X -return per dollar spent on Stock 1 and Y return spent

Question

8. In a portfolio, let X -return per dollar spent on Stock 1 and Y return spent per dollar on stock 2 One definition of an optimal portfolio is to invest p (fraction of $1) in Stock 1 and 1-p in Stock 2 so as to minimize Var(Z) where Z- pX+(1-p)Y a. Assume that X and Y have the following joint probability distribution: 2 0 .20 0 x 1 .05 .50 .05 i) Find Cov(X,Y), Var(X), Var(Y) ii)Are X and Y independent? iii) If p=5, what are P(22 2)? E(Z)? V(Z)? NOTE: You can find M(Z) by calculating the value of Z for each pair (Xx, Y-y), for which we know the prob. iv) Ifxis 1, what are the mean and variance of Y? b. Suppose instead that our criterion was to maximize P(Z22) What would be the value of p that that maximizes P(Z 2 2)? Explain.

Explanation / Answer

i)

X = (0+1+2)/3 = 1 = E(X)

Y = (0+1+2)/3 = 1= E(Y)

Cov(x,y) = (0-1)*(0-1)*0.2 + (0-1)*(1-1)*0 + (0-1)*(2-1)*0 + (1-1)*(0-1)*0.05 + (1-1)*(1-1)*0.5 + (1-1)*(2-1)*0.05 + (2-1)*(0-1)*0 + (2-1)*(1-1)*0 + (2-1)*(2-1)*0.20 = 0.20+0.20 = 0.40

Var(x) = x2PX(x)(E(X))2 = 02*0.2 + 12 *0.6 + 22 * 0.2 - 12 = 0.4

Var(y) = y2PY(y)(E(Y))2 = 02*0.25 + 12 *0.5 + 22 * 0.25 - 12 = 0.5

ii)

X = Var(X) ^ (1/2) = 0.632

Y= Var(Y) ^ (1/2) = 0.707

Corr(X,Y) = Cov(X,Y)/(X*Y) = 0.4/((0.632)*(0.707)) = 0.4/0.3 = 0.895, So X & Y are not independent

iii)

y 0 1 2 0 0.2 0 0 x 1 0.05 0.50 0.05 2 0 0 0.20

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