Cramer-Lundberg model
/ KRAH-mer LOOND-bairg /
If you want to study whether an insurer survives, you need a clean toy world where you can actually calculate things — simple enough to solve, rich enough to be interesting. The Cramer-Lundberg model is that toy world: the textbook starting point for ruin theory, named after the Swedish actuaries who built it a century ago. It pictures premiums flowing in at a steady rate while claims arrive at random moments, each of a random size, and asks what becomes of the surplus.
Precisely, the model makes three clean assumptions. Premium comes in at a constant rate c. Claims arrive according to a Poisson process — meaning claims happen at random times at a constant average rate (call it lambda per year), independently of one another. And each claim's size is an independent draw from a fixed claim-size distribution. Put together, the total claims S(t) follow a compound Poisson distribution, and the surplus is U(t) = u + c*t - S(t). The premium rate is set with a positive loading, c = (1 + theta) * lambda * (mean claim size), which keeps the surplus drifting upward on average. From this tidy setup flow the model's famous results: the adjustment coefficient and the Lundberg inequality.
The Cramer-Lundberg model matters because it is the conceptual backbone of the whole subject — almost every more realistic model (claims that cluster, surplus that earns interest, dividends paid out) is described as a modification of it. Its great virtue is tractability: you can derive closed-form bounds on the ruin probability. Its great limitation is realism: real claim arrivals cluster (think a hailstorm hitting thousands of cars at once), real surplus earns investment income, and real claim sizes can be wildly heavy-tailed. So treat the model as a clarifying lens, not a literal forecast — its lessons about loadings and capital are sound even where its exact numbers are not.
Claims arrive Poisson at lambda = 100 per year, each averaging 10,000, so expected claims are 1,000,000 per year. With loading theta = 0.2, premium is c = 1.2 * 1,000,000 = 1,200,000 per year, and the surplus follows U(t) = u + 1,200,000*t - S(t) where S(t) is the running compound-Poisson total of claims.
Steady premium c, Poisson claim arrivals, random claim sizes — the classical setup behind ruin theory.
The model assumes claims arrive independently at a constant rate with no investment return. Real catastrophes cluster and real surplus earns interest, so the model clarifies thinking rather than predicting exact ruin numbers.