Input: mu, Sigma
Minimum Variance Portfolios: We show that portfolios on the Minimum Variance Frontier solve
with g(.) and h(.) being deterministic functions.
Efficient Portfolios are on the minimum variance frontier and fulfill mu_mv> mu_GMV
Minimum Variance Portfolios solve
With lambda and v being real valued scalar Lagrange multipliers.
As we assume n assets, there will be n + 2 first-order conditions for n+2 unknown parameters
We now apply some manipulations on the last equations. First, assuming sigma is invertible, we re-write the first FOC as
Second, we pre-multiply the last equation with mu' to arrive at
Third, we pre-multiply the expression for w_mv by 1’ to arrive at