Risk theory and ruin theory

Risk theory studies the distribution of the aggregate cost of an insurance portfolio; ruin theory studies what happens to the insurer's capital when that cost accumulates over time against a stream of premiums. Filip Lundberg set out the model in his 1903 thesis, and Harald Cramér gave it a rigorous form from 1930 onward. More than a century later, their results are still the best available intuition about the relationship between price, volatility and capital.

The model is deliberately simple: premiums arrive continuously at a constant rate, claims arrive according to a Poisson process with independent amounts, and there are no investments or expenses. That simplicity yields closed-form formulas in which one can read which variables matter. The first lesson is that without a safety loading ruin is certain; the second, that with a loading the probability of ruin decays exponentially with initial capital, at a speed that depends on the tail of the claims.

This guide expands on the summary in Actuarial mathematics and the Formula sheet, works through a complete numerical example with reinsurance, and compares the probability of ruin with the one-year value at risk approach used by the solvency capital requirement in Mexico.

The insurer's surplus at time t is the initial capital u plus premiums collected ct minus accumulated claims S(t). Claims arrive according to a Poisson process with intensity λ, and their amounts X₁, X₂, ... are independent and identically distributed with mean μ, and independent of the number of claims. The premium is set with a safety loading θ on the expected cost: c = (1 + θ)λμ.

Ruin occurs at the first instant the surplus becomes negative. The infinite-horizon probability of ruin, ψ(u), is the probability that this ever happens; the finite-horizon one, ψ(u, T), considers only the interval [0, T]. If θ is zero or negative, the law of large numbers guarantees that the surplus drifts to minus infinity or oscillates, and ψ(u) = 1 for all u. The condition θ > 0 is called the net profit condition.

A remarkable result, valid for any claim distribution, is that with no initial capital ψ(0) = 1/(1 + θ). With a 25% loading, an insurer that starts with no capital goes broke with probability 80%: the loading only protects if there is capital to absorb the first fluctuations.

The adjustment coefficient R is the unique positive root of Lundberg's equation, which equates the expected growth of the surplus with the moment generating function of the claims. Intuitively, R measures how fast premiums dominate claims: it increases with the loading θ and decreases with the variability and tail of X. For small loadings the approximation R = 2θμ/E[X²] holds, which shows that what matters is not only the mean of the claims but their second moment.

Lundberg's inequality bounds the probability of ruin for any initial capital by a decreasing exponential. It is an upper bound and therefore conservative; moreover, Cramér's theorem establishes that ψ(u) behaves asymptotically like C·e^-Ru for a constant C < 1. In practice, doubling capital squares the ruin factor: if e^-Ru = 1% with u, with 2u it is 0.01%.

The adjustment coefficient exists only if the moment generating function of X is finite in a neighborhood of zero, that is, for light-tailed claims (exponential, gamma, bounded distributions). With heavy tails such as Pareto or lognormal, R does not exist and ruin decays much more slowly, roughly in proportion to the integrated tail of the claim distribution. That is the technical reason catastrophe and liability risks require excess of loss reinsurance.

When claims are exponential with mean μ = 1/β, Lundberg's equation can be solved by hand and the probability of ruin has a closed form. It is the reference case for understanding the model and for validating simulation programs. The formula shows that the Lundberg bound overshoots by exactly the factor 1/(1 + θ).

For other distributions the general tool is the Pollaczek-Khinchine formula: the survival probability, 1 - ψ(u), is the distribution of a compound geometric sum of the surplus's record lows (ladder heights), whose amounts follow the equilibrium distribution of X. It is evaluated numerically with Panjer or simulation. De Vylder's approximation, which replaces the process with one having exponential claims by matching three moments, gives very accurate results for light tails.

The input to everything above is the distribution of the aggregate cost S over one period. In the individual model, losses are added up policy by policy; in the collective model, the portfolio is treated as a random sum of N claims, with N and the amounts X_i independent. The collective model is more tractable, accommodates Poisson, binomial or negative binomial frequency, and for large portfolios approximates the individual model very well.

In 1981 Harry Panjer published a recursion that computes the distribution of S exactly when the frequency belongs to the (a, b, 0) class, that is, when Pr(N = k)/Pr(N = k - 1) = a + b/k. For Poisson, a = 0 and b = λ; for negative binomial, a > 0; for binomial, a < 0. Severity is discretized on a grid of span h (using the rounding or moment matching method), and the recursion returns the aggregate density at each grid point, at a computational cost on the order of the square of the number of points. For very large portfolios the fast Fourier transform is used.

The distribution of S yields loaded premiums, percentiles for capital, aggregate excess of loss (stop loss) reinsurance premiums, and the input for evaluating ruin in discrete time. It is the bridge between ruin theory and pricing practice.

When a percentile is all you need, the normal approximation uses only the mean and variance of S. It works for large portfolios with homogeneous claims, but underestimates the tail because S is right-skewed. The normal power (NP) approximation corrects the normal quantile with the skewness coefficient γ through a first-order Cornish-Fisher expansion. Another three-moment alternative is the translated gamma, which fits a shifted gamma with the same mean, variance and skewness.

The approximations are reasonable as long as γ is moderate, say below one. With greater skewness, typical of small portfolios or those with catastrophe exposure, NP deteriorates and Panjer, FFT or simulation are preferable. In the example on this page, NP reproduces the 99.5% quantile obtained by simulation with 400,000 scenarios to within 0.1%, while the normal underestimates it by 1.7%.

The ruin model offers three levers to control ψ(u): more capital, a higher loading or less retained risk. Reinsurance works on the third, but it has a cost: the reinsurer charges its own loading, so the cedent gives up more premium than expected cost and its net θ falls. The net effect depends on whether the reduction in variability makes up for the lost margin. With a proportional contract (quota share), retained claims are a fraction α of the original ones, which scales the second moment by α² but the mean only by α; excess of loss cuts off the tail, which is where the largest gain in the adjustment coefficient lies.

The optimal retention in the Lundberg sense is the one that maximizes the adjustment coefficient net of reinsurance. Classic results show that, for the same reinsurance premium calculated under the expected value principle, per risk excess of loss maximizes R among all contracts. In practice the decision is made with economic capital models, but Lundberg's intuition still holds: reinsuring the tail is more efficient than reinsuring proportionally.

The infinite-horizon probability of ruin is not the regulatory metric. Modern frameworks use one-year value at risk: capital is the difference between a high percentile of one year's losses and the available premiums and reserves. The two views complement each other: the one-year percentile measures immediate resilience to a bad year, and ruin measures the sustainability of the business model at a given loading. More on this in Solvency and Reinsurance.

The LISF (Insurance and Surety Institutions Law) does not use the infinite-horizon probability of ruin, but a solvency capital requirement (RCS) which, under its article 235, covers unexpected losses at a 99.5% confidence level over a one-year horizon, except for risks that require other periods. The RCS is calculated monthly with the CNSF's (National Insurance and Surety Commission) standard formula or with an approved internal model; for social security pensions and catastrophe risks only the standard formula is allowed. Eligible own funds must cover the RCS at all times.

The dynamic dimension, closer to ruin theory, lies in the dynamic solvency test under article 245 of the LISF: at least once a year, the institution projects its financial position under adverse scenarios to verify that its eligible own funds will continue to cover the RCS. It must be signed by an actuary holding a cédula profesional and current certification or accreditation by the CNSF, and it is governed by the standard of practice in Annex 7.3.1 of the CUSF (Single Insurance and Surety Circular).

Reinsurance is the regulated lever: article 258 empowers the CNSF to set procedures for maximum retention limits, and article 260 requires institutions to set them annually considering their volume of business and their risks. The logic is Lundberg's: retention must be consistent with the institution's capital and margin.