Bounding Short Positions in Mean-Variance Portfolio Optimization
Summary
The document formulates the Markowitz minimum-variance portfolio for a target expected return, with weights summing to one. The question asks how to permit short selling while limiting the proceeds raised from short positions relative to initial wealth. The response points out that the stated unconstrained weight formulation already permits negative weights, then proposes lower bounds on asset weights as linear constraints.
This captures a standard way to impose per-asset short-sale limits, which a solver supporting box constraints can handle. It does not fully resolve the requested aggregate cash restriction: placing the same lower bound on every weight limits each short position individually, but may not enforce a cap on total short proceeds relative to wealth. Translating that financial cap requires defining how positions, prices, and initial wealth map to portfolio weights and adding a suitable aggregate constraint. The document gives no numerical example or solver results.
Key ideas
- Negative portfolio weights represent short positions in the stated Markowitz formulation.
- Linear lower bounds on weights can cap individual asset short positions.
- An aggregate limit on short-sale proceeds may require an additional constraint beyond per-asset bounds.
- A wealth-relative constraint depends on how portfolio weights map to position values.
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# Portfolio optimization
# Portfolio optimization
first I just hope that this question is in the right place.
I have started working on portfolio optimization and the formulation of the problem and their solution : For example in the Markowitz problem lowest variance of the portfolio for a certain level of return can be express as for n risky assets
min $\ \frac{1}{2} \theta^TQ\theta$
s.t. $\ \mu^{T}\theta = \mu^{*}$
and $\ \mathbb{1}^{T}\theta = 1$
where $\ \theta \in \mathbb{R}^{n} $ is the the weight of the risky asset in our portfolio $\ \mu \in \mathbb{R}^{n}$ is the expected return of the risky asset and $\ \mu^{*} \in \mathbb{R}^{+} $ is the objective function.
Now and it is my problem, I would like to formulate in terms of optimization problem the following problem/ let's say I have initial wealth X, I still want to determine the portfolio with the lowest variance for a certain expected return $\ \mu^{*} $, I allow short selling but I also want that the money raised by short selling stay under a certain level that depends on X for example that the level of money raised by short selling be inferior than X/5.
Thank you for any hints or suggestions, I hope I did not forget some hypothesis and I made myself clear.
## Answer by Richi Wa (score 3)
https://quant.stackexchange.com/a/23061
The problem as you formulate it above already allows for short-selling. You just have to add the constraint: $$ \theta_i \ge l $$ where $l$ is the lower bound. This is equivalent to $$ -\theta_i \le -l $$ which if often the way linear constraints are formulated. Any solver that is able to work with box-constaints can solve this.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.