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 [FONT=MathJax_Math-italic-Web]k[/FONT][FONT=MathJax_Math-italic-Web]X[/FONT] ). 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
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 [FONT=MathJax_Math-italic-Web]P[/FONT][FONT=MathJax_Main-Web]([/FONT][FONT=MathJax_Math-italic-Web]Z[/FONT][FONT=MathJax_Main-Web]≤[/FONT][FONT=MathJax_Main-Web]0[/FONT][FONT=MathJax_Main-Web])[/FONT] .

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 [FONT=MathJax_Math-italic-Web]V[/FONT][FONT=MathJax_Math-italic-Web]A[/FONT][FONT=MathJax_Math-italic-Web]R[/FONT][FONT=MathJax_Main-Web]([/FONT][FONT=MathJax_Math-italic-Web]Z[/FONT][FONT=MathJax_Main-Web])[/FONT][FONT=MathJax_Main-Web]≤[/FONT][FONT=MathJax_Main-Web]4[/FONT]



My attempt for a. To find E(Z) do I just use the fact that [FONT=MathJax_Math-italic-Web]E[/FONT][FONT=MathJax_Main-Web]([/FONT][FONT=MathJax_Math-italic-Web]X[/FONT][FONT=MathJax_Main-Web])[/FONT][FONT=MathJax_Main-Web]=[/FONT][FONT=MathJax_Main-Web]1.05[/FONT] and [FONT=MathJax_Math-italic-Web]E[/FONT][FONT=MathJax_Main-Web]([/FONT][FONT=MathJax_Math-italic-Web]Y[/FONT][FONT=MathJax_Main-Web])[/FONT][FONT=MathJax_Main-Web]=[/FONT][FONT=MathJax_Main-Web]1.02[/FONT] thus [FONT=MathJax_Math-italic-Web]E[/FONT][FONT=MathJax_Main-Web]([/FONT][FONT=MathJax_Math-italic-Web]X[/FONT][FONT=MathJax_Main-Web])[/FONT][FONT=MathJax_Main-Web]+[/FONT][FONT=MathJax_Math-italic-Web]E[/FONT][FONT=MathJax_Main-Web]([/FONT][FONT=MathJax_Math-italic-Web]Y[/FONT][FONT=MathJax_Main-Web])[/FONT] [FONT=MathJax_Main-Web]2[/FONT] [FONT=MathJax_Main-Web]=[/FONT][FONT=MathJax_Math-italic-Web]E[/FONT][FONT=MathJax_Main-Web]([/FONT][FONT=MathJax_Math-italic-Web]Z[/FONT][FONT=MathJax_Main-Web])[/FONT] ? The same goes for [FONT=MathJax_Math-italic-Web]V[/FONT][FONT=MathJax_Math-italic-Web]A[/FONT][FONT=MathJax_Math-italic-Web]R[/FONT][FONT=MathJax_Main-Web]([/FONT][FONT=MathJax_Math-italic-Web]Z[/FONT][FONT=MathJax_Main-Web])[/FONT] ?
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.

Covariance Matrix Suggestion.

Hey mahjk17.

For Var[Z] one recommendation is that you create a covariance matrix for random variables X and Y where it will be in the form:

[ Var(X), Cov(X,Y)]
[Cov(X,Y), Var(Y)]

and then you multiply this by a vector which contains a linear combination of these random variables to get your final variance.

For example if you used 8X + 2Y then your vector would be [8.2]^T or the transpose of the row vector [8,2].