Convex Constraints for Limiting Portfolio Short Exposure

Summary

The document asks how to constrain the total short weight in a mean–variance portfolio when short selling is allowed. It defines short exposure as the sum of the negative parts of the asset weights and asks whether this constraint can be linearized with additional decision variables. The concern is that the maximum function is nonsmooth and may not fit a standard quadratic programming formulation.

The response points out that the resulting constraint is still convex, though nonsmooth. It also describes an alternative formulation involving a return objective, a fixed risk level, and a lower bound on the sum of weights; another approach penalizes risk in the objective and iterates a risk-aversion parameter to target gross market value. These alternatives relate short exposure to total and net portfolio weights under particular assumptions. The exchange does not provide the requested auxiliary-variable linearization or a complete derivation, so it is not a full implementation recipe.

Key ideas

• Total short exposure can be expressed as the sum of the negative parts of portfolio weights.
• The maximum-based short exposure constraint remains convex, though it is nonsmooth.
• The response presents a fixed-risk formulation with a bound on the sum of weights as an alternative approach.
• A risk-penalized objective can be iterated to target gross market value relative to a short exposure limit.
• The answer does not show the requested auxiliary-variable linearization in detail.

Tags

# Portfolio Optimization constrained to maximum N% of short selling portfolio weights


# Portfolio Optimization constrained to maximum N% of short selling portfolio weights












For mean-variance portfolio optimization with short-selling allowed, but restricted to a certain percentage of the portfolio weights (lets assume N), we can constrain it in the follwoing way:

(from j=1 to n) sum[max(-wj,0)] <= N

the problem is that it is not linear and so if we add it to our mean-variance problem formulation we will no longer have a convex quadratic program.

How would you linearize it with n new decision variables?

## Answer by Michael Isichenko (score 2)

https://quant.stackexchange.com/a/67874

First of all, your nonlinear problem involving $max(-w_i,0)$ is still convex albeit inconvenient for standard solvers due lack of smoothness. In addition to the slack variables mentioned in the comments, the problem of constrained short exposure can be cast in a convex form as follows: $$ w=argmax(\mu^Tw)\quad s.t.\quad w^T\Sigma w=R, $$ under the additional inequality constraint $\sum_i w_i>{\tt your\_threshold}$. Alternatively, the risk penalty can be subtracted from the PNL utility using a risk aversion coefficient. The latter can be iterated to meet the required ${\tt GMV}=\sum_i|w_i|$ expressed in the same units as your short exposure threshold.

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