Lundberg inequality

Computing the exact probability that an insurer ever goes broke is usually hard — the surplus path can wiggle and jump in countless ways. The Lundberg inequality is a beautiful shortcut: it does not give the exact answer, but it gives a guaranteed ceiling, a simple upper limit you can compute in one line that the true ruin probability can never exceed. A safety ceiling you can always count on.

Precisely, in the classical Cramer-Lundberg model the inequality states that the infinite-time ruin probability satisfies psi(u) is at most e^(-R*u), where u is the initial surplus and R is the adjustment coefficient. Read it slowly: as you pile on more starting capital u, the right-hand side shrinks exponentially, so the ceiling on ruin drops fast. Double the capital and you square the bound (a bound of 1 percent becomes at most 0.01 percent). The inequality also makes the role of R vivid — a larger adjustment coefficient (fatter loading, tamer claims) steepens the decay and lowers the ceiling for any given u.

The inequality matters because it turns an intractable question into a usable rule of thumb and a capital target. If you want ruin probability below some level p, it suffices to choose u large enough that e^(-R*u) is at most p, namely u at least equal to (negative log p) divided by R — a clean back-of-envelope capital requirement. Two honest caveats. First, it is only an upper bound; the true ruin probability is usually smaller, so sizing capital to the bound is conservative. Second, and more importantly, the bound requires R to exist, which requires light-tailed claims; for heavy-tailed losses the exponential ceiling is simply false and ruin decays much more slowly than e^(-R*u).