Twenty years ago, the French created a so far unique insurance scheme to cover damages due to natural catastrophes. This so-called ''Cat-Nat system'' combines private insurance industry, a state-guaranteed public reinsurance and the Treasury. We provide a simple game-theoretic model which seems to capture the situation of the concrete scheme. We find that, depending on its expectations, the government modulates its policy to induce private insurers adopting one of those two opposite strategies: (i) transfer all the covered risks to the public reinsurer and therefore simply behave as indemnification intermediaries; (ii) conserve the largest share of high risks.