Question: Denote by X and Y the (random) returns from projects 1 and 2, respectively, and let X and Y be jointly normally distributed: X is normally distributed with mean ?₁ and standard deviation ?₁, Y is normally distributed with mean ?₂ and standard deviation ? ₂, and Cor(X,Y)=?. Consider the return from a joint project a₁X+a₂Y, when the amount a₁ is invested in project 1 and the amount a₂ is invested in project 2.

a) What is the distribution of a₁X+a₂Y?

b) Suppose that an investor wants to meet a “target” return ?. Therefore, she needs to choose a₁ and a₂ such that E[a₁X+a₂Y]=?. Suppose that the investor is risk averse. Then, among all possible investments that yield return ?, the investor prefers an investment that has the minimal variance. In other words, the investor’s problem is to find a₁ and a₂ such that

i) E[a₁X+a₂Y]=?,

ii) For all b₁ and b₂, such that E[b₁X+b₂Y]=?, Var(a₁X+a₂Y)<Var(b₁X+b₂Y).

Find these optimal a₁ and a₂.