6 – M4 L4 07 Leverage Constraint V5

Another important constraint is the leverage constraints. With this constraint, we limit the leverage ratio which, as you recall, is the sum of the absolute dollar value of all positions, long and short, divided by whatever actual capital we devote to supporting those positions. In our portfolio optimization problem, we work with portfolio weights which are percentages of total invested capital, so the leverage ratio is just the sum of the absolute value of the weights, long and short. Again, we discussed this when we discussed alpha factors, but the basic idea here is that when you short stocks, you borrow the stocks and sell them on the open market. You can use the cash you get to take long positions on other stocks. In general, any investment position is said to be leveraged if it is financed by a debt position. Whether it’s financed by borrowed cash or borrowed stocks. However, it would be much more risky to take very large short positions in order to finance very large long positions. These are large bets and shorting can be complicated. If the leverage ratio is allowed to be greater than one, it means the total magnitude of all year long and short positions is greater than the amount of initial capital that you have to work with. This means you’re borrowing money either by shorting more stocks or by borrowing cash. You have to borrow from somebody and they’ll have a maximum there willing to lend to you. So, you place a constraint on your leverage ratio which is the sum of the absolute values of all your positions to control the scale of this borrowing. This will keep you from ending up in a very extreme leverage position. If you’re using an objective function like this and you are not using a risk constraint like this, you need a constraint like the leverage constraint to keep your weights from growing to infinity. You can try to control leverage by just controlling risk, but the leverage constraint is usually a hard constraint that will also be necessary to enforce.

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