policy limit

An insurer is happy to cover your house, but it does not want to write a blank cheque. So the contract says: we will pay your losses, but only up to a maximum — say 300,000. If your loss is smaller, you are fully covered; if it is larger, you receive the cap and bear the rest yourself. That ceiling is the policy limit, and it is how insurers keep a single claim from being unboundedly large.

With a policy limit u (and ignoring a deductible for a moment), the insurer pays min(X, u): the loss if it is below the limit, otherwise the limit. A 200,000 loss under a 300,000 limit pays 200,000; a 450,000 loss pays only 300,000. The quantity min(X, u) is called the limited loss, and its expected value, E[min(X, u)], the limited expected value, is a building block of severity calculations. When both a deductible d and a limit are present, the maximum the insurer can pay on one loss equals the limit minus the deductible region — the so-called maximum covered loss caps how far up the loss the policy responds.

Policy limits chop off the expensive right tail of the severity distribution, so they are the main tool for controlling exposure to giant losses and they make a heavy-tailed risk insurable at a sensible price. They also interact with reinsurance: an insurer that caps a policy may still buy excess-of-loss cover for the layer it does retain. The crucial honest point: a limit protects the insurer, but it transfers the tail risk back to the policyholder, who is unprotected above the cap. Buyers sometimes do not realise that a '300,000 limit' means a 450,000 loss leaves them 150,000 short — the limit is exactly where the protection stops.