Ruin probability

Classical Cramér-Lundberg model with exponential or gamma claims, adjustment coefficient, Lundberg bound and the initial capital needed for a target ruin probability.

Inputs

Mean claim μGamma shape (1 = exponential)Premium loading θ%Initial capital uTarget ruin probability%

Results

Ruin probability ψ(u)

15.740%

Exact, exponential claims

Lundberg bound e^(−Ru)

18.888%

Adjustment coefficient R

0.00001667

Capital for 0.5% ruin

306,960

Exact

Ruin probability by initial capital

ψ(u)Lundberg bound

Methodology

Conventions

Classical Cramér-Lundberg model: claims arrive as a Poisson process and premiums come in continuously at (1 + θ) times the expected claims. Ruin is over an infinite horizon.
Claims are gamma distributed with the given mean and shape; shape 1 is exponential, where the ruin probability is exact. Otherwise the Lundberg bound e^(−Ru) is an upper limit.

References

Klugman, Panjer and Willmot, Loss Models: From Data to Decisions, 5th ed.
Dickson, Insurance Risk and Ruin, 2nd ed.

Formulas

Adjustment coefficient: 1 + (1+\theta)\mu R = M_X(R)
Lundberg bound: \psi(u) \le e^{-Ru}
Exponential claims: \psi(u) = \frac{1}{1+\theta}\exp\!\Big(-\frac{\theta u}{(1+\theta)\mu}\Big)