Skip to content

Cramér–Lundberg ruin model

An insurance surplus model with premium income and compound Poisson claims. Separate positive average profit from ruin, and compare an exact probability with Lundberg’s bound.

The Cramér–Lundberg ruin model describes an insurer’s reserve as initial capital plus steady premium income minus random claims arriving through a compound Poisson process. It measures the chance that claims exhaust the reserve. Positive expected profit alone does not keep every possible claim sequence affordable.

Write

U(t)=u+ct−∑i=1N(t)Xi.U(t)=u+ct-\sum_{i=1}^{N(t)}X_i.
Here u≥0u\ge0 is initial surplus, c>0c>0 is premium income per unit of time, and N(t)N(t) is a homogeneous Poisson process with rate λ>0\lambda>0. Claim amounts XiX_i are positive, independent and identically distributed, independent of the entire arrival process, with finite mean μ\mu. The reserve rises linearly between claims and jumps downward when each claim is paid. Embrechts, Klüppelberg and Mikosch, §1.1 set out this classical insurance model.

Which event counts as ruin?

Use strictly negative reserve as the ruin convention:
τ=inf⁡{t≥0:U(t)<0},ψ(u)=P(τ<∞),\tau=\inf\{t\ge0:U(t)<0\},\qquad \psi(u)=\mathbb{P}(\tau<\infty),
with inf⁡∅=∞\inf\varnothing=\infty. The probability ψ(u)\psi(u) is ultimate ruin probability: the chance of crossing below zero at any future time. Over a fixed horizon TT, the corresponding probability is ψ(u,T)=P(τ≤T)\psi(u,T)=\mathbb{P}(\tau\le T), which cannot exceed ψ(u)\psi(u).
Neither is generally the probability of a negative reserve at the endpoint TT. The mathematical reserve can recover after an earlier crossing; the insurer has already experienced ruin under the chosen definition. This model treats ruin as a capital shortfall event, without prescribing a legal insolvency procedure.

The expected unstopped reserve is

E[U(t)]=u+(c−λμ)t.\mathbb{E}[U(t)]=u+(c-\lambda\mu)t.
Thus the net profit condition is c>λμc>\lambda\mu, often written c=(1+θ)λμc=(1+\theta)\lambda\mu with safety loading θ>0\theta>0. It gives positive average drift. It does not rule out a large early claim or an unusually dense cluster of claims. These definitions and the safety-loading interpretation are developed in §1.1 of Modelling Extremal Events.

A profitable model with a 24.53% chance of eventual ruin

Consider a fictional model in which time is measured in years and amounts in monetary units:
InputValue
Claim arrival rate λ\lambda1 per year
Mean claim μ\mu1 monetary unit
Premium rate cc1.5 units per year
Initial surplus uu3 units
Assume claim amounts have an exponential distribution with mean μ=1\mu=1. The safety loading is θ=0.5\theta=0.5, so expected reserve grows by 0.5 units per year. For exponential claims and c>λμc>\lambda\mu, the exact ultimate probability is
ψ(u)=λμcexp⁡[−(1μ−λc)u].\psi(u)=\frac{\lambda\mu}{c} \exp\left[-\left(\frac{1}{\mu}-\frac{\lambda}{c}\right)u\right].
This special-case formula appears in the Charles University risk-theory summary, page 1. Substituting the fictional inputs gives
ψ(3)=23e−1=0.2452529608≈24.53%.\psi(3)=\frac{2}{3}e^{-1}=0.2452529608\approx24.53\%.
That is a probability over an infinite horizon. It is not a one-year default forecast, an expected monetary loss, or an estimate for an actual insurer or trading strategy.
Fictional reserve path starts at 3 monetary units and first becomes negative at 2.2 years, reaching minus 0.4. For the same model, initial capital 3 gives exact ultimate ruin probability 24.53%, below the Lundberg bound 36.79%.
Fictional reserve path starts at 3 monetary units and first becomes negative at 2.2 years, reaching minus 0.4. For the same model, initial capital 3 gives exact ultimate ruin probability 24.53%, below the Lundberg bound 36.79%.
Original illustration and exact model curves. The supplied path stops at ruin; the dashed mean reserve describes the unstopped mathematical process. The probability panel covers infinitely many years and all possible paths, not just the illustrated path.

The illustrated claim sequence reaches reserve −0.4-0.4 at 2.2 years despite positive average drift. It is a supplied scenario, not a simulated estimate of the 24.53% probability. In the exact formula, adding another 3 units of initial capital multiplies ruin probability by e−1e^{-1}, holding every other input fixed.

At u=0u=0, this strict-negative convention gives ψ(0)=2/3\psi(0)=2/3: premium income starts building before the first claim. If ruin instead means U(t)≤0U(t)\le0 and includes time zero, starting at zero means immediate ruin. The boundary convention matters.

The adjustment coefficient gives a bound, not usually an exact answer

Let MX(r)=EerX]M_X(r)=\mathbb{E}e^{rX}] be the claim [moment-generating function. If a positive solution RR exists with a finite exponential moment, the adjustment coefficient solves
λ(MX(R)−1)=cR.\lambda\bigl(M_X(R)-1\bigr)=cR.
The zero solution is excluded; Hald and Schmidli, §2 give this defining equation and its connection to Lundberg’s inequality. Under the net profit condition, Lundberg’s inequality gives
ψ(u)≤e−Ru,u≥0.\psi(u)\le e^{-Ru},\qquad u\ge0.

For the exponential example, MX(r)=1/(1−μr)M_X(r)=1/(1-\mu r) for r<1/μr<1/\mu, so the positive solution is R=1/μ−λ/c=1/3R=1/\mu-\lambda/c=1/3 per monetary unit. The exponent RuRu is dimensionless. At initial surplus 3,

ψ(3)=0.2452529608<e−R⋅3=0.3678794412.\psi(3)=0.2452529608 \quad<\quad e^{-R\cdot3}=0.3678794412.
Calling 36.79% the exact ruin probability would discard the factor 2/32/3. More generally, the Cramér–Lundberg asymptotic estimate has the form ψ(u)∼Ce−Ru\psi(u)\sim C e^{-Ru} as capital grows, under additional integrability and regularity conditions. A bound, an asymptotic estimate and an exact formula serve different purposes; Theorem 1.2.2 distinguishes them.

The model itself can also use heavy-tailed claim amounts. However, distributions such as Pareto have no finite positive exponential moment, so this positive-RR calculation cannot be applied to them. Ruin asymptotics for subexponential distributions require different tail arguments; replacing a heavy tail with an exponential one merely to obtain the bound changes the risk model.

For the claim-aggregation machinery, the membership lesson Superposition, Thinning, and Compound Poisson Processes derives the mean and variance of random claim totals; a public preview is available. Turning that aggregate-loss model into a reserve model adds premium inflows and the requirement to track the first shortfall. Constant claim intensity, independent severities and deterministic premium income are substantive assumptions: catastrophe dependence, changing exposure, investment returns, expenses and reinsurance need explicit extensions.

Sources

Last updated
  • ruin theory
  • insurance risk
  • compound poisson
  • adjustment coefficient
Keep connecting the ideas
← Back to the glossaryBack to top ↑