It looks like these are the companies that are really insuring you: Lloyd’s of London, Berkshire Hathaway’s National Indemnity, XL Catlin etc. Basically they are buying a policy from one of those companies adding 20% and selling it to you. A insurance company works by spreading risk over a large area. By selling everything in NY they are increasing their correlation which raises risk. One of the reasons for the sub p…
Reinsurance is basically insurance for the insurance co, for instance if a hurricane hits NYC and wipes out all of their policyholders at once.
(It's a bit complicated because there actually are many insurance co's that only sell insurance underwritten by a third party (although they may be able to offer lower price than buying from the third party directly because of how they target their customer base or handle claims), or sell insurance strictly on commission and outsource servicing claims, and there are varying forms of reinsurance that cover everything from huge tail risks to flat percentages of claims, or exotic circumstances like your corporate HQ burning down or massive lawsuits.)