Even more subtly, if there is one insurance company that makes a lot of extra return on their "float" (the money that has been paid in premiums but not yet paid out in claims), they may actually sell insurance that is a "bad bet", i.e. is expected to have underwriting losses, but that the time value of money makes profitable for them.
In that context other insurance companies can't compete unless they also are making huge returns on their float. It drives risky behavior by insurance companies in exactly the opposite way you'd prefer.
In some sense, the two sides of the insurance business are decoupled.
When insurance companies agree on a 'bad bet', that just means that they are essentially agreeing to pay a high interest rate on their float.
A profit driven insurance company should only agree to such an expensive float if other funding sources are also expensive. And that's independent of the risk appetite of the investing managers of the insurance company.
To be explicit: a profit driven insurance company should issue more equity or a bond to finance their investment, unless the float is cheaper for them.
(Of course, this all goes out of the window, the moment you have legal rules that restrict insurance companies from issuing more bonds or equity. Then they might rationally agree to an expensive float.)
Warren Buffet talked about this extensively during the run-up to the 2008 Global Financial Crisis. A large part of his portfolio is insurance companies. Insurance premiums had fallen so much in 2007 (or became so competitive/stupid), that it was seriously affecting the revenues of his insurance companies.
