Suppose X and Y denote the rates of return (in percent) on two stocks. (a) You a

Question

Suppose X and Y denote the rates of return (in percent) on two stocks.

(a) You are told X ~ N(16,25) and Y ~ N(6,4), and X and Y are independent. Suppose you want a combination of these stocks, W in your portfolio, such that W = 3X + 4Y . What is the probability distribution of the return on your portfolio? Please show your work.

(b) Suppose you are now told that X and Y are correlated, with the correlation coefficient being 0.8. There exists a combination of these stocks, Z = X + Y . Solve for V ar(Z). Please show your work.

Explanation / Answer

X~N(16,52) and Y~N(6,22)

so E[X]=16 V[X]=25 so SD(X)=5 E[Y]=6 and V[Y]=4 so SD(Y)=2

a) X and Y are independent

W=3X+4Y

W is a linear combination of normal variables. so W also follows a normal distribution.

with E[W]=3*E[X]+4*E[Y]=3*16+4*6=72

and V[W]=V[3X+4Y]=9*V[X]+16*V[Y] [as X and Y are independent there is no covariance term]

                              =9*25+16*4=289=172

so the distribution of W is

W~N(72,172) [answer]

b) X and Y are correlated. correlation coefficient being 0.8

so r(X,Y)=0.8

or COV(X,Y)/[SD(X)*SD(Y)]=0.8

or, COV(X,Y)/[5*2]=0.8

or COV(X,Y)=0.8*5*2=8

Z=X+Y

so Var(Z)=Var(X+Y)=Var(X)+Var(Y)+2COV(X,Y)=25+4+2*8=45 [answer]

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