light- vs heavy-tailed claims and ruin
Not all risks are alike in their extremes. Some lines of business have claims that stay close to typical — a dented car costs roughly what a dented car costs, never a thousand times more. Others can produce a single claim so enormous it dwarfs all the rest combined — one mega-lawsuit, one hurricane. The first kind is called light-tailed, the second heavy-tailed, and the difference utterly changes how an insurer can go broke and how much capital it needs.
Precisely, a claim-size distribution is light-tailed when its probability of very large values falls off fast (faster than any exponential), so its moment generating function exists; examples are the exponential, gamma, and normal. It is heavy-tailed when extreme values, though rare, are not vanishingly so — the tail falls off slowly, like a power law; examples are the Pareto and lognormal. For light tails, the adjustment coefficient R exists and the Lundberg bound gives ruin probability decaying exponentially, psi(u) about C*e^(-R*u). For heavy tails there is NO adjustment coefficient, and ruin decays only as slowly as the claim tail itself — roughly psi(u) is proportional to the probability that a single claim exceeds u. In plain terms: with heavy tails, ruin is overwhelmingly caused by one catastrophic claim, not by an unlucky accumulation of ordinary ones.
This distinction is one of the most practically important ideas in the whole subject. It tells you that the elegant exponential machinery (R, the Lundberg bound) is safe for many life and motor portfolios but dangerously optimistic for catastrophe, liability, and credit lines, where it can drastically understate required capital. It reshapes risk management: for heavy-tailed lines you do not chase a higher loading so much as cap the single worst loss through per-risk excess-of-loss reinsurance. The honest warning: assuming light tails because the math is nicer is a classic and costly modeling error — always check the data for fat tails before trusting an exponential ruin bound.
A motor book with exponential claim sizes is light-tailed: R exists and ruin probability falls like e^(-R*u), halving capital roughly squares the ruin chance. A catastrophe book with Pareto claim sizes is heavy-tailed: R does not exist, and doubling capital barely lowers ruin, because the threat is one giant claim that can exceed almost any buffer.
Light tails: ruin decays exponentially via R. Heavy tails: no R, ruin driven by a single huge claim.
Assuming light tails just because the exponential math is tidy is a classic, costly error. Heavy-tailed lines need a different toolkit — single-event caps and reinsurance — not a bigger loading.