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November 19th, 2012, 08:34 AM   #1
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Expectation, Variance, Covariance

a.) Let X denote the price of a stock in one year from now. (More precisely, X denotes the value in one year of the amount of stock we can buy today with 1 dollar. So if today I invest k dollars in that stock, then in one year from now the value will be kX ). Let Y denote the price of another stock in one year from now. (Again the value in a year from now of 1 dollar invested today). Assume that
E(X)=1.05
E(Y)=1.02
VAR(X)=0.16
VAR(Y)=0.04
COV(X,Y)=0.02
Assume that you invest 8 dollars into X and invest 2 into Y . So, the value of our investment in one year from now is Z. You keep the investment for the whole year without changing it. Calculate E(Z), VAR(Z) and assuming that Z is normal calculate P(Z?0) .

b.) Find the best investment strategy if you are given 3 dollars to invest into X and Y. You can distribute those 3 dollars however you want. The goal is to maximize the expectation of Z. You are given the constrain to keep the risk under a certain level. That level is given by VAR(Z)?4



My attempt for a. To find E(Z) do I just use the fact that E(X)=1.05 and E(Y)=1.02 thus [E(X)+E(Y)]/2 =E(Z) ? The same goes for VAR(Z) ?
For b. I know that the standard deviation for Z will be 2 since the variance is 4, but how will I be able to maximize the expectation.
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