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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?

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    $\begingroup$ Could you provide a bit more context on the second equation? The link you provided needs an account to view. $\endgroup$
    – Bob Jansen
    Aug 2, 2020 at 9:42

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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.)

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    $\begingroup$ noob2 is absolutely right. The first one is what everyone uses in the industry (even Investopedia; the second one is worth mentioning only in its historical context, explaining that this is not how people write it now. $\endgroup$ Aug 2, 2020 at 14:46
  • $\begingroup$ There are managers who use the second form -- or the equivalent $\sum_i^N x_i = 1$ with $x_i\geq0~\forall i=\{1,\ldots,N\}$. Mutual funds, for example, are often prohibited from any short selling. The growth in "130-30" funds a decade ago was a reaction to concerns that no short selling unduly hurt performance. $\endgroup$
    – kurtosis
    Aug 3, 2020 at 3:50

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