suppose I have a portfolio, say the assets X_1, X_2 and Z where Z is risk free. Also these are all independent.
Then I want to maximize $aE(X_1) + bE(X_2) + cE(X_2)$ subject to: (1) $a + b + c = 1$, and (2) Std(portfolio) = M, so Var(portfolio) = M^2 meaning $a^2 Var(X_1) + b^2 Var(X_2) = M^2$.
How do I do this? Do I consider the Lagrangian $L(x,y,z) = a E(X_1) + b E(X_2) + c E(Z) - \lambda(a + b + c -1) - \mu(a^2 Var(X_1) + b^2 Var(X_2) - M^2)$?
Then I get $\lambda = E(Z)$ but then i don't know how to proceed. Is this correct?
Thanks in advance.