IAI Actuarial Core Principles · Economic Modelling

Ruin Theory: Surplus, Adjustment Coefficient and Lundberg's Inequality

Ruin theory asks whether an insurer's surplus, starting at u, ever falls below zero. Surplus is U(t) = u + ct − S(t). You find the ruin probability ψ(u) exactly in special cases, bound it with the adjustment coefficient R using ψ(u) ≤ e^(−Ru), or estimate it by simulation.

What this chapter covers

Ruin theory models an insurer's surplus over time. Money comes in as premium at a steady rate. Money goes out as random claims. Ruin happens if surplus ever drops below zero. The chapter gives you tools to measure that risk: the ruin probability ψ(u), the adjustment coefficient R, and bounds and exact formulas built on them.

The standard model is the compound Poisson (Cramér-Lundberg) model. Claims arrive as a Poisson process with rate λ. Claim sizes X are independent and identically distributed. Premium income is c per unit time, with c = (1 + θ)λE[X], where θ is the relative security loading. You need θ > 0 for ruin not to be certain. You then move to discrete-time versions and to simulation, where formulas stop working. The last step is reinsurance, which changes both the claim and premium streams and so changes R and ψ(u).

This chapter draws on the aggregate claims and claim size distributions you learn in risk modelling, and on moment generating functions. Ruin theory is not among the 2026 CM2 (Economic Modelling) topic weightings, so do not treat it as a CM2 chapter. Check the current IAI syllabus to confirm which core subject examines it before you study it in depth. The skills here (setting up a model, stating assumptions, solving an equation numerically) are useful in any modelling paper.

Ruin theory turns an insurer's solvency question into a calculation you can defend: how likely is it that claims outrun premium and capital?

The chapter follows a fixed method: set up the surplus process, solve for R, apply Lundberg's inequality, and then comment on the effect of reinsurance or a change in loading. Each part is mechanical once you know it. The chapter also makes you practise stating assumptions and interpreting results, which are useful skills in any modelling subject.

Ruin theory: topics in the order to study them

1Surplus Process and Ruin BasicsYou need the model, the notation and the premium loading rule before any result makes sense.
2Adjustment Coefficient and Lundberg's InequalityThis is the core result of the chapter, and you use R in everything that follows.
3Exact Ruin Probabilities and Special CasesOnce you know the bound, you can compare it with exact answers such as the exponential claims case.
4Discrete-Time Ruin and SimulationYou apply the same ideas when the model is in discrete steps or no formula exists, so it builds on the continuous-time results.
5Reinsurance and Ruin ProbabilityIt combines everything: you rebuild the surplus process after reinsurance, re-solve for R and judge the effect.

How to prepare Ruin theory

Treat this chapter as one method repeated with different claim distributions. Practise the method on paper until it is automatic, then add the variations.

Write the model from memory: U(t) = u + ct − S(t), c = (1 + θ)λE[X], and the definitions of ψ(u) and the time of ruin. Do this until you can state assumptions without thinking.
Learn the equation for R: λ + cR = λM_X(R), where M_X is the moment generating function of the claim size. Solve it by hand for exponential claims, and numerically for most other distributions.
Apply Lundberg's inequality ψ(u) ≤ e^(−Ru) to several values of u. Say what a larger R means: a smaller bound and lower risk.
Memorise the exact results you can state and use: ψ(0) = 1/(1 + θ) and, for exponential claims with mean μ, ψ(u) = (1/(1 + θ)) exp(−θu / ((1 + θ)μ)). Check that R = θ / ((1 + θ)μ) matches the exponent.
For discrete time and simulation, write down the recursion U_n = U_(n−1) + premium − claims, state how many paths you run and how you estimate ψ(u) as the proportion of paths that hit ruin. Note that finite-horizon estimates tend to understate ultimate ruin, subject to sampling error.
For reinsurance, redo the model with retained claims and net premium: find the new premium income after paying the reinsurer, re-solve for R, and compare. Finish each answer with a comment on cost against safety.
Finish with timed past-paper questions. Practise both the multiple-choice style, where you must pick the right formula fast, and the full written style with every step shown.

Common mistakes in Ruin theory

Using the wrong premium rate, such as c = λE[X] or forgetting the loading.
Fix: Always write c = (1 + θ)λE[X] first. With θ = 0 the net premium gives no margin, and over an infinite time horizon ruin is certain.
Solving the adjustment coefficient equation for R = 0.
Fix: State that you want the strictly positive root, and check that it lies within the range where M_X(R) exists.
Treating Lundberg's inequality as the exact ruin probability.
Fix: Say it is an upper bound. Use the exact formula only where it is available, such as exponential claims.
Forgetting that reinsurance reduces premium income as well as claims.
Fix: Subtract the reinsurance premium from income and recompute the net loading before re-solving for R.
Reading a simulated ruin proportion as the ultimate ruin probability.
Fix: State the time horizon. The finite-horizon ruin probability is a lower bound on ultimate ruin, so the estimate tends to understate it. Remember that sampling error means a single estimate can still land above the true ultimate ruin probability, and comment on this.
Giving a number without interpretation.
Fix: Add one sentence on what the result means for the insurer, such as how the capital, loading or reinsurance choice changes the risk.

Last-day revision: Ruin theory

Surplus: U(t) = u + ct − S(t), where S(t) is aggregate claims up to time t.
Premium rate: c = (1 + θ)λE[X], with θ > 0 the relative security loading.
Ruin probability ψ(u) is the probability that surplus ever falls below zero, starting from surplus u.
Adjustment coefficient R is the positive solution of λ + cR = λM_X(R).
Lundberg's inequality: ψ(u) ≤ e^(−Ru) for u ≥ 0.
Larger R means a smaller bound on ruin, so lower risk.
In the compound Poisson model, ψ(0) = 1/(1 + θ), whatever the claim size distribution.
Exponential claims with mean μ: R = θ / ((1 + θ)μ) and ψ(u) = (1/(1 + θ)) e^(−Ru).
Increasing θ or starting surplus u lowers ψ(u).
Simulation estimates ψ(u) as the proportion of simulated paths that reach ruin; more paths give a smaller standard error.
After reinsurance, use retained claims and net premium (premium less reinsurance cost), then re-solve for R.
State your assumptions: independent claims, Poisson arrivals, constant premium rate, and the time horizon used.

Ruin theory practice questions

Ruin theory: frequently asked questions

What is the adjustment coefficient in ruin theory?

It is the positive number R that solves λ + cR = λM_X(R) in the compound Poisson model. It measures how quickly the ruin probability falls as initial surplus rises. A larger R means less risk.

Is ruin theory part of the Economic Modelling paper?

The 2026 CM2 topic weightings list rational economic theory, measures of investment risk, asset valuations, liability valuations and option theory. Ruin theory is not among them. Check the current IAI syllabus to confirm which subject examines it before you study it in depth.

Do I need to memorise the exact ruin formula for exponential claims?

Yes, it is worth learning. For exponential claims with mean μ, ψ(u) = (1/(1 + θ)) exp(−θu / ((1 + θ)μ)). It also gives you a quick check on your value of R.

How do I answer a reinsurance question on ruin?

Find the claims the insurer keeps and the premium it keeps after paying the reinsurer. Then solve for the new R and compare it with the old one. Finish by commenting on whether the lower risk justifies the cost.