finite- vs infinite-time ruin

There are two honest ways to ask 'will this insurer go broke?' One is bounded: 'will it go broke within the next year (or next ten years)?' The other is open-ended: 'will it ever go broke, even if we wait forever?' These are the finite-time and infinite-time views of ruin, and they can give very different-looking answers about the same company.

Precisely, the infinite-time (or ultimate) ruin probability psi(u) is the chance the surplus dips below zero at any time over an unbounded future. The finite-time ruin probability psi(u, T) is the chance it dips below zero at some point within a fixed horizon T — say one year. Because a longer window gives ruin more chances to occur, psi(u, T) increases toward psi(u) as T grows: a one-year ruin probability of 0.3 percent might rise to a ten-year figure of 1 percent and an infinite-horizon figure of 1.5 percent. The infinite-time version is mathematically cleaner (it has the famous Lundberg bound), while the finite-time version is harder to compute but closer to what a manager actually faces.

Which one you use depends on the question. Regulators and capital frameworks almost always think in finite terms — a one-year horizon is the industry default — because a firm reprices, raises capital, and resets its plan each year, so 'ruin sometime in the next thousand years' is not the relevant risk. Risk theorists love the infinite-time probability because it admits elegant closed-form bounds. The key caveat: never quote an infinite-horizon ruin probability as if it answered a one-year solvency question; it is systematically larger and answers a different, more pessimistic question.