Weekly Premium Insurance
  
See: Insurance.
Premium insurance sounds very fancy. Like premium ice cream, or premium service. Who wouldn't want to get premium insurance every week?
That's not how the term works. It's not premium insurance delivered on a weekly basis. It's insurance where you pay the premium weekly.
Basically, this insurance coverage is targeted at low-income customers, or people with rough credit histories. The company isn't sure they can count on the client paying. So, rather than risk a month-long wait for their next check, they ask for payments on a weekly basis.