Risk Theory and Ruin Probability
Risk theory tests collective risk models and the probability that an insurer's surplus ever falls below zero: the compound Poisson surplus process, the adjustment coefficient, Lundberg bounds, and modern risk measures.
Core rules
- Surplus U(t) = u + ct − S(t), with S(t) compound Poisson (rate λ, claim distribution X). The net profit condition c > λE[X] is required for ruin probability below one.
- The adjustment coefficient R > 0 solves cR = λ(M_X(R) − 1), equivalently λ + cR = λM_X(R); R is the smallest positive root.
- Lundberg inequality: ψ(u) ≤ e^{−Ru}. For exponential claims, ψ(u) = (λμ/c)e^{−Ru} exactly.
- The diffusion approximation gives ψ(u) ≈ exp(−2(c−λμ)u/(λE[X²])).
- VaR is not subadditive; TVaR/ES is coherent and captures tail severity.
Worked example
Exponential claims with mean 1,000, λ = 1, and premium c = 1,200 per unit time. Then 1200R = 1000R/(1 − 1000R), so 1 − 1000R = 0.8333 and R = 1.6667×10⁻⁴. Hence ψ(10,000) ≤ exp(−1.6667) = 0.189, and the exact exponential-claim value is (1000/1200)(0.189) = 0.157.
Traps
- Solving the Lundberg equation without checking the net profit condition.
- Taking the trivial root R = 0 or a negative root.
- Applying the exponential Lundberg bound to heavy-tailed claims, where it may fail.
- Confusing ruin probability (ever) with the probability of insolvency at a fixed horizon.
- Treating VaR as coherent or additive across lines.