Cramer-Lundberg model
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 flow of premium, minus claims. Claims arrive as a homogeneous Poisson process, with independent, identically distributed amounts. The central result is the Lundberg bound: ruin probability is capped by an exponential that decays with initial capital. The decay rate is called the adjustment coefficient, and it solves an equation involving the Laplace transform of the severity distribution. In this setting, doubling capital does not halve ruin probability; it cuts it by an exponential factor. That only works if the severity distribution has a finite moment generating function. Once severities are heavy-tailed, Pareto or lognormal, the adjustment coefficient no longer exists. Ruin probability then falls no faster than the tail itself, which completely changes the marginal return on capital.
Textbook case: a book averages 500 claims a year, with exponential severities averaging 20K EUR, and collects 10.5M EUR of premium a year, a 5% safety loading. The adjustment coefficient equals 0.05 / (1.05 x 20,000), roughly 2.38 per million. Ruin probability is therefore bounded by exp(minus 2.38 times ten to the minus six times u), about 9% with 1M EUR of capital and 0.9% with 2M EUR. With Pareto severities of index 1.5 and the same mean, the same reduction takes tens of millions of capital.
classical risk process, adjustment coefficient, Lundberg bound, ruin theory
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