Why is it the case that the weight vector of the global minimum variance portfolio the eigenvector of the covariance matrix with the smallest eigenvalue?

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  1. I know the relationship between PCA and eigenvector. If we want to compress data in 1 dimension, we should use Lagrangian with covariance matrix of the sampled data.
  2. By doing so, we can find the a vector that can produce maximum amount of varinace of data points when they are projected onto the vector. This vector is called the 1st principal component with the greatest eigenvalue.
  3. You can read more about the meaning of eigenvectors in a covarinace matrix in this blog post
  4. However, in global minimum variance portfolio, what we want to do is more of reverse to PCA. We want to find the vector that produces the least amount of variance when data points are projected onto the line. As such, I don't think we can use Lagrangian.
  5. However this blog post still argues that the weight vector in global minimum variance is the eigenvector of the covariance matrix with the smallest eigenvalue. Can anyone tell me why, please?

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