Convexity of a Portfolio Weight Cap Based on Total Absolute Exposure

Summary

The document asks how to impose a diversification cap of the form that each portfolio position's absolute value cannot exceed a fixed fraction of the portfolio's total absolute exposure. It presents the constraint for a three-stock example and mentions a quadratic constrained quadratic programming approach as a possible formulation. The stated motivation is to prevent any single position from dominating the portfolio's overall exposure.

The responses do not provide a complete convex reformulation. One response sketches a QCQP construction, while another observes that, for a particular two-asset parameter choice, the feasible set is a union of unbounded sectors and is nonconvex. This illustrates that the absolute-value ratio constraint is not automatically convex simply because it is expressed as an inequality. The document leaves open how to handle general dimensions and parameters, and whether additional portfolio assumptions could yield a tractable convex model.

Key ideas

• The proposed cap limits each absolute position relative to total absolute portfolio exposure.
• The stated goal is to encourage diversification by limiting concentration in an individual stock.
• A response sketches a quadratic constrained formulation but does not complete it.
• A separate example identifies a nonconvex feasible set as a union of sectors.
• The document does not establish a general convex reformulation.

Tags

# How do I reformulate this max GMV ratio constraint in convex way?


# How do I reformulate this max GMV ratio constraint in convex way?












Let $n$ be the number of stocks in my portfolio. I would like to have the following inequality constraint in my optimization problem

$$ |x_i| \le \alpha \sum_{j=1}^n | x_j | $$

where $\alpha$ is known, say, $\alpha = 0.6$. The intuition here is the global minimum variance (GMV) of every optimized stock can't be more than $\alpha$ of the optimized overall GMV, to achieve a more diversified portfolio.

Assuming $n = 3$ and $\alpha = 0.6$, we have:

$$ \begin{aligned} | x_0 | &\le 0.6 \left( | x_0 | + | x_1 | + | x_2 | \right) \\ | x_1 | &\le 0.6 \left( | x_0 | + | x_1 | + | x_2 | \right) \\ | x_2 | &\le 0.6 \left( | x_0 | + | x_1 | + | x_2 | \right) \end{aligned} $$

How do I re-formulate this in a convex way? I am using the Mosek solver.

## Answer by Attack68 (score 0)

https://quant.stackexchange.com/a/79413

The closest I have come so far is to engineer a the constraint into a quadratically constrained quadratic program (QCQP):

where in your case, $q_i=0$ and $r_i=0$ and, $ P_i = \mathbf{I} - \alpha \mathbf{1} $ (where $\mathbf{1}$ is a matrix of ones)

I am assuming that your objective function is probably quaudratic as well according to most portfolio optimization problems.

## Answer by Rodrigo de Azevedo (score 0)

https://quant.stackexchange.com/a/83647

For $n = 2$ and $\alpha = \frac35$, we have the union of $2^2 = 4$ (unbounded) sectors, which is a non-convex set.

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