I'd be surprised if an insurance company would determine fault without a police report.

Fair enough on minimal accidents. As a very young driver (< 6 months experience - i was reading a paper map propped up on the steering wheel as I was waiting for a light/coasting to a stop; bad idea) many years ago I did have a very minimal accident (low speed bumper to bumper), with no visible damage, but I insisted we not involve the police/insurance and agreed I'd pay for everything. That cost me $800 bucks (which was more than a paycheck for me at the time, had to post date a check to her), because I think the lady I bumped into got had by a mechanic.