adjustment (Lundberg) coefficient

Imagine you want one number that captures how safe an insurer's whole setup is — its premium, its claim frequency, the spread of its claim sizes — rolled into a single dial. The bigger the dial, the faster the ruin probability falls as you add capital. The adjustment coefficient, usually written R, is that dial. It is the key quantity from which the famous Lundberg bound on ruin is built.

Precisely, R is a positive number defined as the solution of an equation built from the claim-size distribution and the premium loading. The intuition is what matters: R is large when premiums carry a fat loading and when claim sizes are tightly bunched (light-tailed); R is small when the loading is thin or claim sizes are wildly variable. In the Lundberg inequality the ruin probability satisfies psi(u) is at most e^(-R*u) — so a bigger R makes that exponential decay steeper, meaning ruin probability collapses faster as initial surplus u grows. Roughly, with a small loading theta and claims that are not too wild, R is approximately 2*theta/(second moment ratio of claim sizes) — bigger loading, bigger R.

The adjustment coefficient matters because it converts the messy details of a risk into a single comparison number: between two books of business, the one with the larger R is the safer one for a given amount of capital. It also shows up directly in pricing and reinsurance decisions — choosing a reinsurance arrangement that raises R is choosing to make ruin less likely. The crucial caveat: R only exists when the claim-size distribution is light-tailed, meaning its moment generating function exists. For genuinely heavy-tailed claims (like some catastrophe or liability losses) there is no adjustment coefficient at all, the exponential Lundberg bound fails, and ruin probability decays far more slowly — a warning the model itself signals.