An offshore insurance company is usually referred to as a 'captive' insurance company. This has its rotes from the early 1960s, when a number of major US companies began to form their own insurance subsidiaries (which today are known as "captives,") in offshore financial centres.
Captive: A captive Insurance company is an insurance company owned by a non-insurance parent that insures or reinsures the risks of its parent or affiliated companies
Reasons to set up your own offshore captive insurance company can be because of a general dissatisfaction with existing insurance coverage or costs. The advantage is to save external costs and generating profits in a low tax jurisdiction.