The branch of actuarial science that computes the probability an insurer's surplus falls below zero over a given horizon, from an initial capital, a premium inflow, and a claims process.
Ruin theory asks the only question that matters to an insurer: how much capital is needed to keep the probability of failure below a threshold. The base model follows a surplus that grows linearly with premium and drops at every claim, and derives a ruin probability as a function of initial capital. Only three levers push it down: more capital, a higher safety loading on premium, or a thinner claims tail, which points straight back to reinsurance. Its lasting contribution is conceptual rather than numerical: it established that required capital depends on the shape of the distribution and not only on its mean, and that ruin over an infinite horizon behaves very differently from ruin over one year. Solvency II took the idea and froze it at one year and one percentile, 99.5%, but the underlying mechanics are these.
Filip Lundberg defended his thesis at Uppsala in 1903 and founded the collective risk model; Harald Cramer formalized it in Stockholm during the 1930s. On a toy book with 1,000 expected claims a year, a 10K EUR average cost and a 5% safety loading, the adjustment coefficient gives a ruin probability around 1% for an initial capital near 9M EUR under an exponential tail, and several times that capital as soon as the tail becomes Pareto.
probabilité de ruine, ruin probability, risque de ruine, modèle de risque collectif