Why Adding Assets Cannot Worsen the Minimum-Variance Optimum

Summary

The note asks how shrinking an investment universe affects a mean-variance portfolio frontier and whether a lower global minimum variance proves that the entire larger-universe frontier dominates. The answer frames portfolio construction as a constrained minimization problem and observes that removing an asset is equivalent to retaining the original universe while fixing that asset’s weight at zero.

Because the larger optimization permits every portfolio available in the smaller universe, as well as additional choices, its minimum objective value cannot be worse under the same constraints. Equality occurs when the larger-universe optimum assigns zero weight to the removed asset; otherwise the expanded choice set can improve the optimum. This reasoning applies to each fixed optimization problem, including a fixed target return when comparing frontier points. It does not by itself establish strict dominance at every point, nor does a lower global minimum variance alone establish the ordering of the entire frontier. The result assumes matching constraints and objective definitions across universes.

Key ideas

• Removing an asset can be represented by constraining its portfolio weight to zero.
• The larger asset universe includes all portfolios feasible in the smaller universe, given matching constraints.
• The minimum objective value therefore cannot worsen when an asset is added.
• Equality is possible when an optimal larger-universe portfolio gives the added asset zero weight.
• A lower global minimum variance alone does not prove strict dominance across the full frontier.

Tags

# N asset covariance matrix vs N-1 asset covariance matrix


# N asset covariance matrix vs N-1 asset covariance matrix












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.

## Answer by Attack68 (score 3, accepted)

https://quant.stackexchange.com/a/70370

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.