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luckyluch
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Joined: January 16th, 2012, 3:53 pm

Portfolio Optimization for Short ETFs

November 17th, 2019, 10:52 am

I'm trying to use Lagrange's method to calculate the minimum variance solution for a portfolio consisting of assets which can be shorted, but do not provide leverage. An example of this would be a short ETF, which allows you to take a negative view, but the position has to be fully funded.

For example, if you have a non-margined account with 100$, and you want to spread two indices using ETFs, you would have to in practice pay 50$ for to buy one and $50 to short the other (vs a margined account, where you could offset the positions against each other and only pay your initial margin, leaving most of your 100$ intact - somewhat simplified but you get the idea ...)

I'm struggling however to select a budget equation to optimize. Typically you would restrict the sum of the weights to unity. However, in this case, it needs to be the absolute value of the sum of the weights. As the function needs to be smooth for the optimization to work, I've instead tried setting the squares of the weights to one i.e. $w^T w = 1$, but that leads to a messy equation, which I'm not sure I can solve.

So far I've settled for a hack: simply calculate the leveraged portfolio weights, and afterwards scale by the absolute value of the weights. This seems to do the trick, but my intuition tells me this is not equivalent to the solution I would get by solving for $w^T w = 1$.

My maths is pretty rusty and I would welcome any hints.