Minimum Variance Portfolio
Suppose there are N stocks in the investmentable universe and we have a fully invested portfolio investing 100% of the capital. The Covariance matrix is denoted as ∑. We are interested in finding the portfolio with minimum variance. An investor choosing this portfolio is only concerned about the risk of the portfolio. Denoting a vector of ones by i=(1,….,1)^' , we have the following optimization problem:
$$Subject to : w’ .i = w_1 + w_2 +……+ w_N =1$$
I would like to solve above problem using Lagrangian multipliers.