The Lundberg Adjustment Coefficient

Classical ruin theory models an insurer's capital as starting at some reserve $u$, growing steadily from premium income, and taking sudden downward jumps whenever a claim arrives (a Poisson process of claim times, each claim a random size). The central question is: what is the probability this capital ever goes negative — the probability of ruin? Exact ruin probabilities are hard to compute for most claim-size distributions, but there is a remarkably clean approximation built on a single number, the Lundberg adjustment coefficient $R$.

$R$ is defined as the positive root of an equation balancing the claim arrival rate, the premium income rate, and the moment generating function of claim sizes — intuitively, it measures how much "safety margin" the premium rate has over the actual claims risk, compressed into one number. Once $R$ is known, the Lundberg inequality gives a clean exponential bound on ruin probability:

$$\psi(u) \le e^{-Ru}$$

where $\psi(u)$ is the probability of eventual ruin starting from reserve $u$. This says ruin probability decays exponentially in the starting reserve, at a rate set entirely by $R$ — a bigger $R$ (meaning premiums are priced with a bigger safety margin over expected claims) makes that decay faster, so a modest increase in reserves buys a much bigger cut in ruin probability.

If $R = 0.02$ per dollar of reserve, doubling the reserve from $50 to $100 shrinks the ruin-probability bound from $e^{-1} \approx 0.37$ to $e^{-2} \approx 0.14$ — over twice the capital cushion, but a far larger than double reduction in ruin risk, because the relationship is exponential, not linear.