so I have been using a M-V framework to form M-V efficient portfolios. I have noticed that every time I make my investment universe smaller the minimum variance frontier moves to the right. This effect seems to come directly from the covariance matrix. My question is, how could you mathematically describe this effect if you wanted to prove that a minimum variance frontier composed of N assets dominates a frontier composed of N-1 assets. I consider that N contains all N-1 assets. My intuition tells me that if the global minimum variance portfolio gmv_N<gmv_(N-1) then it is sufficient for me to assume that the rest of the curve generated by N assets will dominate the N-1 curve. I might be understanding this in a wrong way.
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The M-V framework is a minimisation problem of the form:
$$\min_\mathbf{x_n} f(\mathbf{x_n}, \mathbf{u})$$
where $\mathbf{x_n^*}$ solves the $N$ assets weights subject to fixed parameters $\mathbf{u}$, such that this is a minimum.
By definition, if one asset is removed, say $x_n=0$ then the minimisation problem becomes,
$$\min_{\mathbf{x_{n-1}}} f(\mathbf{x_{n-1}}, \mathbf{u}, x_n), \quad x_n=0$$
By definition,
$$f(\mathbf{x_n^*}, \mathbf{u}) \leq f(\mathbf{x_{n-1}^*}, \mathbf{u}, x_n=0) $$
with equality only when $x_n=0$ in the solution to the original minimisation. This is true for all asset removals.
