Usually the constraint is that all weights sum up to 1. But in a crisis when all assets are falling in prices, intuitively, all the weights should be negative in the optimization.

But it contradicts with the constraint. Should I adjust the constraint to something like the sum is greater than -x and smaller than 1?

Thanks in advance!

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    $\begingroup$ You could add cash as an asset and let that weight grow above 1. $\endgroup$ – Bob Jansen Jan 26 at 16:41

Mean-Variance (MV) optimal simply gives you MV optimal. This is bounded between negative infinity to positive infinity. Any constraints on this you night wish to apply to this are at your supplemental discretion.

Sum(W) = +1 is simply the traditional constraint to enforce a long-only portfolio; or applied to a long-short where shorts have to be financed by longs.

Imagine I was in positioning cash to different currencies: looking at EURUSD (risky), USDJPY (conservative), GBPUSD (risky) and AUDUSD (risky). I am long-only and unlevered, allocating my cash between these five currencies. I would then be negative across the board, summing to -1, across the board if I was risk-seeking!

First-time buyers are habitually 3-500% long of their home, their biggest asset on their household balance sheet. Before they even start looking at and adding in their investment portfolio, their weights do not equal 1. They massively exceed that (imaginary) constraint.

So why the constraint of sum(W) = 1? This Wizard of Oz applies if you have 1 capital to allocate between competing - and unlevered, no shorting - demands on your capital. Absent those conditions, sum(W) is bounded by plus or minus infinity ;-)

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