The reference model of ruin theory, where claims arrive as a Poisson process and ruin probability decays exponentially with capital, provided the tail is not too heavy.
The Cramer-Lundberg model describes an insurer's surplus as initial capital plus a constant premium inflow minus a sum of claims arriving as a homogeneous Poisson process, with independent and identically distributed amounts. Its central result is the Lundberg bound: ruin probability is bounded above by an exponential decay in initial capital, whose decay rate is the adjustment coefficient, the solution of an equation involving the Laplace transform of the severity distribution. The practical consequence is sharp: in this setting, doubling capital does not halve ruin probability, it cuts it by an exponential factor. But that exponential only exists if the severity distribution has a finite moment generating function. As soon as claims follow a heavy-tailed law, Pareto or lognormal, the adjustment coefficient no longer exists and ruin probability decays only as fast as the tail itself, which completely changes the marginal return on capital.
A book with an intensity of 500 claims a year, exponential severities averaging 20K EUR, premium income of 10.5M EUR a year, that is a 5% safety loading. The adjustment coefficient equals 0.05 / (1.05 x 20,000), roughly 2.38 per hundred thousand: ruin probability is bounded by exp(minus 2.38 times ten to the minus five times u), which gives about 9% for 1M EUR of capital and 0.9% for 2M EUR. Under a Pareto law with index 1.5 and the same mean, the same reduction requires tens of millions.
processus de risque classique, coefficient d'ajustement, borne de Lundberg, classical risk process