Assigned Risk

Has nothing to do with the requirement of playing the Parker Bros board game every Thanksgiving with relatives who hate you. Instead, Assigned Risk comes about when an insurance company is forced to cover someone they otherwise wouldn't.
See, insurance companies don't get to choose all of their clients. For the most part, the actuarial wizards that insurance companies keep locked up in their magical towers use complicated math to figure out which insurance policies represent profitable bets for the company. Sometimes, though, the government steps in and forces the companies outside their comfort zones.
The main example of this is car insurance. States often require drivers to carry some sort of vehicle insurance. For good drivers, this is no problem. They can go to the open market and shop for the best insurance they can find.
However, bad drivers are often left out. Because they pose a bigger risk, these ditsy lead-footed traffic-school recidivists have trouble finding anyone to cover them. So the government steps in.
In these cases, the government will facilitate pools of insurance companies, who will take on the more suspect drivers. The additional risk these represent is known in insurance circles as "assigned risk." As in, the companies have been assigned the risk, rather than choosing to take it on. It also comes up a lot in worker's compensation, another insurance field with a broad mandate of obligatory coverage.