Actually, if your investments fall in value to the point where it looks like your account is in danger of becoming insolvent, they'll likely immediately sell as

Actually, if your investments fall in value to the point where it looks like your account is in danger of becoming insolvent, they'll likely immediately sell as many of your assets as it takes to make up the difference, locking in your leveraged losses. This is known as a margin call, and it's very unpleasant to be on the receiving end of one. The actual risk to the broker is minimal -- they just have to make sure to trigger the sale at a point where the value of all of your holdings (not just what you purchased on margin) is still greater than what you owe them.