You are managing a portfolio of n assets. Let Z be the n x n covariance matrix (positive definite, given) of asset returns, u be the n x 1 vector of expected asset returns (given), and w represent the vector of portfolio weights.
You wish to find the weights that minimize the variance for a target return. That is,
min w e R^n w^T Z w
subject to w^T 1 = 1 and w^T u = p,
where p e R is the target return (given), and 1 e R^n is a vector with 1 in each element.
Show that the vector of optimal weights w* is equal to
w* = (Z^(-1) (1 + 2u^T Z^(-1) 1))^(-1) Z^(-1) u,
where 1 = Z^(-1) 1, 2 = Z^(-1) u, and A = 1^T 2 - 1^T Z^(-1) u, B = 1^T 2, and C = n^(-1) 1^T Z^(-1) 1.
