Why Factor-Neutral Variance Maximization Is Unbounded

Summary

The document examines a portfolio problem that seeks to maximize variance while requiring zero exposure to each listed risk factor. It explains why those constraints alone do not produce a finite maximum: if a feasible portfolio exists, scaling all its weights by a factor greater than one preserves zero factor exposure while increasing variance quadratically.

The argument shows that the optimization problem is unbounded, so reframing the variance objective as a lower-bound constraint does not supply the missing limit on portfolio size. A practical formulation would need additional constraints, such as a normalization or exposure limit, but the document does not develop a revised model or discuss which constraints are suitable. Its conclusion relies on the stated homogeneous factor constraints and the usual nonnegative covariance-based variance; it does not address other portfolio restrictions or optimization methods.

Key ideas

• Zero factor-exposure constraints remain satisfied when portfolio weights are scaled.
• Scaling feasible weights upward increases portfolio variance quadratically.
• The stated maximization problem has no finite optimum without constraints on portfolio scale.
• Turning the objective into a variance threshold does not by itself bound the portfolio weights.

Tags

# Portfolio optimization - maximize variance with exposure to risk factors equal to zero


# Portfolio optimization - maximize variance with exposure to risk factors equal to zero












Optimize a portfolio such that the exposure to risk factors is zero and the variance is maximized (instead of traditional minimization problem).

so the optimization problem look like:

$$maximize\;w^T\,\Sigma\,w$$

With following constraints:

$$\beta_0\,w=0$$ $$\beta_1\,w=0$$ $$\beta_2\,w=0$$ $$\beta_3\,w=0$$

Where

$$\Sigma - covariance\,matrix$$ $$\beta_0..._3 - factor\,exposure$$

I am told that this is a non-convex problem. Can I convert this into an SDP with some relaxations? I could convert the objective function into a quadratic constraint such as below:

$$w^T\,\Sigma\,w > Min$$ Please advise.

## Answer by Chris Taylor (score 1, accepted)

https://quant.stackexchange.com/a/25416

There is no solution. If $w$ is a solution to the original problem, then consider $aw$ with $a>1$

$$\beta_i(aw) = a(\beta_i w) = 0$$

and

$$(aw)^T\Sigma(aw) = a^2 (w^T\Sigma w) > w^T\Sigma w$$

so the original solution $w$ was not a maximum.

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