☰First-party risk is your own: your property, your liability, your people, your revenue. Third-party risk belongs to others connected to your business, the affiliated parties around it, meaning customers, contractors, tenants, and employees, that your business can choose to insure. A captive insuring the parent's property is writing first-party risk; a captive backing warranties sold to customers or tenant legal liability in an apartment portfolio is writing third-party risk.
The distinction matters for two reasons. Regulators and tax authorities look at the mix, since third-party premium strengthens the case that the captive is a real insurance company distributing real risk. And strategically, third-party risk is where a captive can accelerate into a profit-generating business: insurance sold to affiliated parties, priced with the knowledge only you have about them, is a revenue line the commercial market can't easily take back.