Common mistake 1 with Ruin probability

Applying the adjustment-coefficient bound to heavy-tailed claims (Pareto, lognormal) where R does not exist and psi(u) decays only polynomially.

Common mistake 2 with Ruin probability

Forgetting that psi(u) is the probability of ruin at ANY future time, not within one year - the finite-horizon probability is smaller.

Common mistake 3 with Ruin probability

Reading a comfortable psi(u) as safety when the model ignores interest, expenses, catastrophe accumulation and reserving risk.

Common mistake 4 with Ruin probability

Comparing the infinite-horizon ruin probability with a one-year regulatory capital requirement. They are different quantities on different horizons, and the infinite-horizon figure is always the larger.

EconStatsCraft › Ruin probability

Ruin probability

The classical compound-Poisson surplus process: the premium rate from the safety loading, the adjustment coefficient R that solves the Lundberg equation, the exact probability of ultimate ruin for exponential claims, and the Lundberg upper bound e^(-Ru) as a function of the initial surplus.

The intuition

An insurer starts with capital, collects premium steadily, and pays claims at random moments. Ruin is the surplus hitting zero. The remarkable result is that if premium exceeds expected claims by any positive margin, the probability of ever being ruined falls exponentially in the starting capital - so a modest buffer buys a lot of safety, and the rate at which it does is a single number, the adjustment coefficient.

When to use it

Teaching the link between capital, premium loading and insolvency risk; setting a target surplus for a target ruin probability; understanding why a bigger safety loading buys a disproportionate reduction in ruin risk.

Where the estimator comes from

The Cramer-Lundberg surplus is U(t)=u+ct-\sum_{i=1}^{N(t)}X_i with N(t) Poisson(\lambda t) and premium c=(1+\theta)\lambda E[X] for loading \theta>0. The ADJUSTMENT COEFFICIENT R is the positive root of \lambda\big(M_X(r)-1\big)=cr, where M_X is the severity moment generating function. Lundberg\'s inequality then bounds the ultimate ruin probability, \psi(u)\le e^{-Ru}, and for exponential severities it is exact: \psi(u)=\frac{1}{1+\theta}e^{-Ru} with R=\frac{\theta}{(1+\theta)E[X]}. Note \psi(0)=1/(1+\theta), which depends only on the loading and not on the claim size distribution at all.

The model

c = (1+\theta)\,\lambda\,E[X],\qquad \lambda + cR = \lambda\,M_X(R),\qquad \psi(u) \le e^{-Ru},\qquad \psi(u) = \frac{1}{1+\theta}e^{-Ru}\ \text{(exponential claims)}

What it assumes

Claims arrive as a homogeneous Poisson process and claim sizes are independent and identically distributed with a moment generating function that exists (light-tailed) - so no Pareto or lognormal in the exact theory.
Premium income is received continuously and deterministically at rate c, with a strictly positive loading theta.
No interest on the surplus, no dividends, no reinsurance - the bare model.
The premium loading is strictly positive. With no loading, ruin is certain eventually regardless of initial capital, and the whole apparatus of adjustment coefficients assumes the loading rules that out.

Statistical properties

Ruin probability decays exponentially in initial capital. Each additional 1/R of capital divides the probability by e, which makes the capital-safety trade-off remarkably cheap at the margin.
With zero loading the process is a driftless random walk and ruin is CERTAIN eventually, whatever the starting capital. Positive loading is not a refinement, it is the whole thing.
psi(0) = 1/(1+theta) is independent of the severity distribution. A 20% loading gives an 83.3% chance of ruin from zero capital no matter what claims look like.
The adjustment coefficient requires the severity moment generating function to exist, so it does not exist for heavy-tailed severities such as Pareto or lognormal. For those, ruin decays only polynomially and Lundberg's bound simply does not apply.
These are INFINITE-horizon probabilities. Finite-horizon ruin is lower and is what a regulator with a one-year view actually cares about.
The model ignores interest on the surplus, reinsurance, expenses and dividends. Each has been added in the literature and each changes the answer materially.

How to read the output

R rises with the safety loading theta and with a less variable claim-size distribution. A larger R means ruin probability falls faster as capital is added.
psi(0), the ruin probability with no starting capital, is 1/(1 + theta) for exponential claims - it depends only on the loading, not on the claim-size scale.
The Lundberg bound e^{-Ru} is a genuine upper bound for any light-tailed severity; for exponential claims the exact answer is just that bound multiplied by 1/(1 + theta).
If the equation has no positive root R, the light-tail theory does not apply - the severity is too heavy-tailed and ruin behaves quite differently.

A worked reading

Exponential claims with mean 1, Poisson arrivals at rate 1 and a 20% premium loading give an adjustment coefficient R = theta/((1+theta) x mean) = 0.2/1.2 = 0.16667. From zero capital the probability of ultimate ruin is 1/1.2 = 0.83333 - eighty-three percent, on a book expected to make money every year, which is the fact that surprises people. Capital collapses it geometrically: 0.54937 at u = 2.5, 0.36217 at u = 5, 0.23875 at u = 7.5, 0.15740 at u = 10. Each 2.5 units of capital multiplies the ruin probability by about 0.66, and that constant ratio IS the exponential decay e^(-Ru) that Lundberg's bound describes - the bound is printed beside the exact figure and sits above it everywhere, which is what a bound should do. Note it is ULTIMATE ruin, over infinite time: the model never says when, and a 16% chance spread over a century is a different management problem from the same number over five years.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Ruin probability of 1 regardless of capital	Loading is zero or negative	Premium must exceed expected claims
No positive root for R	Heavy-tailed severity with no MGF	Lundberg does not apply; use a subexponential approximation
psi(0) not equal to 1/(1+theta)	An arithmetic error, for exponential claims	This is exact and is a good check
Probability barely falling with capital	R is small: thin loading or fat claims	Correct, and the useful warning
Simulated ruin above the Lundberg bound	A simulation error: it is an upper bound	Check the premium rate in the simulation
Finite-horizon result above the infinite-horizon one	Impossible	The infinite-horizon probability dominates by definition

Common mistakes

Applying the adjustment-coefficient bound to heavy-tailed claims (Pareto, lognormal) where R does not exist and psi(u) decays only polynomially.
Forgetting that psi(u) is the probability of ruin at ANY future time, not within one year - the finite-horizon probability is smaller.
Reading a comfortable psi(u) as safety when the model ignores interest, expenses, catastrophe accumulation and reserving risk.
Comparing the infinite-horizon ruin probability with a one-year regulatory capital requirement. They are different quantities on different horizons, and the infinite-horizon figure is always the larger.

Where to read this properly

Lundberg (1903) and Cramer (1930), the originals; see Asmussen and Albrecher for the modern treatment.
Asmussen and Albrecher, Ruin Probabilities (2010), ch. 4 and 5.
Embrechts, Kluppelberg and Mikosch, Modelling Extremal Events (1997), ch. 1, for the heavy-tailed failure.

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Ruin probability

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

See also

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