How Short-Selling Rules Change Portfolio Weight Constraints
Summary
The document explains two possible budget constraints in mean-variance portfolio optimization when short sales are permitted. Requiring signed portfolio weights to sum to one treats short positions as negative weights whose proceeds can fund long positions. Requiring the sum of absolute weights to equal one instead limits total long and short exposure together, so a portfolio with offsetting positions uses less than the full amount on either side.
The answer connects the choice to different assumptions about capital and short-sale proceeds, noting that the signed-weight convention became common in practice while the absolute-weight convention reflects an approach associated with Markowitz's original work. It also says the appropriate constraint depends on investor circumstances, regulations, and broker policies; forbidding shorts with nonnegative weights makes the distinction moot. The discussion offers conceptual guidance rather than a formal treatment of margin, collateral, leverage, or jurisdiction-specific rules.
Key ideas
- A signed-weight sum of one allows short-sale proceeds to fund long positions.
- An absolute-weight sum of one constrains combined long and short exposure relative to equity.
- The constraints encode different assumptions about available capital and how short positions are financed.
- Regulation, broker policy, and the investor's intended portfolio determine which model is appropriate.
- Requiring every weight to be nonnegative removes the distinction between the two constraints.
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# Portfolio Optimization sum of weights constraint with short selling
# Portfolio Optimization sum of weights constraint with short selling
For mean-variance portfolio optimization with short-selling allowed I have seen 2 ways to specify the portfolio constraint.
In most resources I've seen, such as https://www.coursera.org/learn/financial-engineering-2/lecture/qwIYs/overview-of-mean-variance (week 1 first video), it is stated as:
$$ \sum_{i=1}^N x_i = 1 $$
However, in Tucker Balch's course https://classroom.udacity.com/courses/ud501/lessons/4432279076/concepts/44338591400923 (Lesson 02-04, Lecture 2), it is stated as:
$$ \sum_{i=1}^N |x_i| = 1 $$
Which one is correct? What is the reasoning behind it?
## Answer by nbbo2 (score 7, accepted)
https://quant.stackexchange.com/a/57085
In the early days of Portfolio Theory there were different views about short positions. Some authors modeled short positions as negative and required all weights to add up to 1 (first equation), others (including Markowitz himself) thought this was not realistic (he thought if you have 1 dollar you cannot both buy 1 dollar worth of stock and also short 1 dollar worth of stock) and required the second condition (if you have equity of 1 dollar you can buy half a dollar of stock(s) and short half a dollar of other stock(s)).
In time I believe the first view came to dominate, not only is it mathematically simpler but it is fairly realistic of how hedge funds really operate (at least under modern U.S. regulations). R. C Merton for example argued that this was correct (he should know as he eventually started a hedge fund). Markowitz I believe was never convinced. The second view may be more representative of how retail investors think about shorting (if indeed they take short positions at all).
You are free to choose whichever assumption you think is more appropriate for your situation (depending on local regulations and your broker's policy), or even to prohibit shorting entirely by requiring all $x_i \ge 0$ (in which case taking absolute values or not does not matter anymore). If you have no opinion and just want my recommendation I would say: use the first method. FWIW (and I don't want to advertise a specific firm) my account at Inter$**$tive Brokers allows me to take short positions corresponding to the first equation, and I am certainly not a big institutional investor. The only reason I see for teaching the second approach is if you want to stay consistent with Markowitz's original paper.
(Note: the first assumption is sometimes called "short selling with full use of the proceeds to buy other stocks" or words to that effect.)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.