I am trying to teach myself about MPT and optimization.

I understand that MPT solutions can be found using three equivalent optimization problems:

  1. Minimizing variance for given return limit
  2. Maximizing return for given variance limit
  3. Unconstrained minimization of linear combination of variance and return with a risk parameter.

My question is, how do I show algebraically the equivalence of these optimization problems?

E.g. given:

$max\, \,x^T\mu \\ s.t.\,\,x^T\Sigma x\le\sigma_*^2$

How do I get to this algebraically:

$min\,\,1/2 x^T\Sigma x-\gamma x^T\mu$

And how do I show that:

$\gamma = \sigma_* / (\mu^T\Sigma\mu)^{(-1/2)} $


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