Every answer and its explanation appears here once you have finished the path. Each one then links to the matching glossary entry, where the concept is set out in full with its worked example.
1. After the July 2021 floods in the Ahr valley, the press spoke of a hundred year event, and many concluded it would not recur for a long time. For a book exposed over thirty years to a peril with a hundred year return period, what is the probability of suffering at least one such event?
About a quarter, the return period being no more than a 1% annual probability
A return period is the inverse of an annual exceedance probability, and nothing more. A hundred years means one percent every year, independently of what happened the year before. The calculation runs through the complement: the probability of crossing thirty years with no event is zero point nine nine to the thirtieth power, about seventy four percent, so the probability of suffering at least one approaches twenty six percent. A so called hundred year event is therefore something a quarter of exposed books will meet within a working lifetime. The misreading comes from the vocabulary itself, which suggests a periodicity while describing a frequency, and it spreads all the more easily because it is reassuring. Two points complete it. Two hundred year events can occur two years apart without any statistical law being broken. And above all, the definition rests on an assumption of stationarity, the idea that the distribution of events does not move over time: where climate shifts that distribution, a return period computed on history describes a world that is no longer the one the current contract covers.
Glossary entry · periode-de-retour2. A property insurer goes through a year marked by numerous hailstorms, each of moderate size, whose annual total reaches an unusual level without any single event being remarkable. Which reading of its exceedance curve accounts for that year?
The aggregate curve, the year weighing through the sum of its losses rather than the largest of them
An exceedance probability is the probability that a loss crosses a given level within the year, and the curve representing it carries loss on the horizontal axis and probability on the vertical. It comes in two versions, easy to confuse and expensive to confuse. The occurrence curve answers the question of the year's largest loss: what is the probability that the worst event exceeds a given amount. The aggregate curve answers the question of the sum: what is the probability that the annual total exceeds a given amount. The two coincide as long as a year holds only one significant event, and they part company as soon as events multiply. That is precisely the situation described. A hail season made of twenty moderate episodes can load the aggregate curve heavily while never approaching, on the occurrence curve, the level a single major windstorm would reach. The consequence is structural and goes beyond reading a chart: protection triggered event by event does not see such a year, whereas a reading in annual aggregate sees all of it. Knowing which of the two curves is being quoted is therefore the first question to ask of any exceedance figure.
Glossary entry · probabilite-de-depassement3. In 2023, United States hailstorms alone caused more than fifty billion dollars of economic damage, more than that year's hurricanes. What makes this family of perils hard to handle with tools calibrated on cyclones and earthquakes?
Numerous events, moderate in unit size and geographically scattered, which accumulate without ever weighing on their own
Perils called secondary bear that name only by contrast with tropical cyclones and earthquakes, on which catastrophe modelling concentrated its calibration effort for decades. They gather severe convective storms with hail, lightning and tornadoes, local flooding, wildfire, snow and ice. Their statistical signature is the reverse of the primary perils: less violent taken one at a time, but far more frequent, and scattered across wide territories. That shape makes them awkward for instruments designed around a single massive event. An episode of one to five billion clears no high threshold, but twenty episodes of that size in one year load an annual account as heavily as a hurricane. The word secondary has become misleading, since these perils now make up the majority of loss generating events recorded in a year, one hundred and forty two in 2023 on the count published by Swiss Re. Two effects compound to push them up: intensifying convective storms and lengthening fire seasons on one side, and on the other the growing concentration of insured value in exposed areas. Recalibrating correlation matrices across those footprints is the work this shift imposes.
Glossary entry · perils-secondaires4. An analyst dismisses a loss scenario sitting five standard deviations from the mean, on the grounds that a normal law makes it practically impossible. On what kind of risk is that reasoning most dangerous?
Those with a fat tail, where the extreme is far less improbable than the bell curve suggests
A fat tailed distribution is one in which events far from the mean carry a probability markedly higher than a normal law would predict. The difference is not a calibration nuance, it shifts the probability assigned to extremes by orders of magnitude, and reasoning in standard deviations loses all its reassuring power. Many real risks sit here: natural catastrophes, market crashes, systemic cyberattacks. The most useful consequence concerns the descriptive tools themselves. On a fat tail, mean and standard deviation stop being reliable summaries, because they are dominated by the handful of extreme values in the sample and swing violently according to whether the series happens to contain one very large loss. A book can show ten quiet years and a flattering average while none of that informs anyone about what is coming. This is why measures built on the tail itself are preferred, and why Nassim Taleb popularised the image of two distinct statistical worlds, one where a single observation changes nothing in the average and one where a single event dominates the entire series. Cyber risk and algorithmic risk are precisely suspected of belonging to the second, which bears directly on whether they can be insured.
Glossary entry · fat-tail5. To model claim severity on a book, the lognormal law is often preferred to the normal law. What property justifies that choice?
It is right skewed and takes positive values only, which matches the real shape of claims
A variable follows a lognormal law when its logarithm follows a normal law, and that simple construction gives it exactly the shape real claims display. It takes positive values only, the least one expects of a cost distribution, whereas a normal law assigns non zero probability to negative amounts. It is right skewed, which captures the familiar reality of a book: many small claims, few large ones, but large ones possible and sometimes very large. Its tail is fatter than the normal's, hence more realistic on extremes, without going as far as the power type laws used for the heaviest tails. Two parameters describe it completely, a scale parameter and a shape parameter, and the usual risk measures compute there by formula rather than simulation, which explains much of its practical popularity. It turns up on those grounds in solvency work for non life underwriting risk and in modelling losses from computer incidents. Being convenient does not make it true: fitting a lognormal to data whose tail is heavier understates the extremes, and testing that fit is part of the job.
Glossary entry · loi-log-normale6. Two books show the same one year loss at 99.5%, one hundred million. Looking beyond that threshold, the conditional average loss comes out at one hundred and thirty million for one and two hundred and twenty million for the other. What does that gap reveal, and which measure made it visible?
That the threshold says nothing about what lies past it, and the average loss beyond the threshold shows it
Value at risk gives, for a confidence level and a horizon, the loss threshold that should be crossed only with small probability, and it is that figure, at ninety nine and a half percent over one year, which underpins European solvency capital. Its weakness is exactly the one the question stages: it describes a door and says nothing about what stands behind it. Two books can cross the same door and meet, beyond it, situations of wholly different scale. The average loss conditional on exceeding, also called expected shortfall, repairs that by answering the missing question: given that the threshold is crossed, how much is lost on average. It is therefore always at least equal to the threshold and completes it rather than replacing it. Two reasons explain why it is preferred on risks with high catastrophic potential. The first concerns fat tails, where almost all the useful information lies beyond the threshold. The second is mathematical and often decisive: this measure is sub additive, meaning the risk of a whole never exceeds the sum of the risks of its parts, so that it rewards diversification instead of penalising it, which value at risk does not guarantee.
Glossary entry · var-tvar7. On the same chemical plant, two technical notes put forward two very different maximum losses. One is drawn from a percentile of the loss distribution, the other from an engineering fire spread scenario. How should the gap be read?
The two conventions coexist and do not measure the same thing, the gap being an error on neither side
Probable maximum loss estimates the largest loss an insurer may suffer on a risk under realistic catastrophic conditions, setting aside absolutely extreme scenarios, which already distinguishes it from possible maximum loss, describing the absolute worst with no regard to probability. But in practice the term covers two methods that do not reduce to one another, and that is the ordinary source of misunderstanding. The probabilistic convention ties the loss to a percentile of the distribution, the two hundred and fifty year level for instance, and is statistical in nature: it answers a question about frequency. The engineering convention, often called estimated maximum loss, builds a deterministic scenario and prices it: realistic fire spread, compartmentation, structural resistance, effectiveness of protection systems. It answers a physical question. Nothing obliges the two to converge on a given site, and their divergences are often instructive. The practical lesson is to treat maximum loss as a common language between insurers and reinsurers, but a language with two dialects: quoting a figure without saying which convention produced it makes it unusable to whoever receives it.
Glossary entry · pml-eml8. An insurer pools a very large number of companies covered against business interruption. All of them host their systems with the same cloud provider. What becomes of the mathematical foundation of pooling?
It gives way, because the law assumes independent risks and that condition is no longer met
The law of large numbers states that the observed average of many realisations of the same random process converges to its expectation as the sample grows. Applied to insurance it explains the whole trade: the individual claim stays unpredictable, but the average cost per policy becomes steadily more stable, and that stability is what makes a premium computable and pooling possible. The essential condition is almost always left unsaid, and it is the one at stake here: the risks must be independent. Correlated risks can occur together, and the average then stops converging on anything useful. Numbers do not help, and that is the counterintuitive point: adding insureds to a book whose members all depend on the same point of failure brings no diversification, it enlarges a single exposure. A book of a thousand companies hosted with one provider is not a thousand risks, it is one risk written a thousand times. This is the deep reason a cloud outage or a systemic attack threatens insurability in a way no traditional peril does: they do not make claims more severe, they make them simultaneous.
Glossary entry · loi-grands-nombres9. In 1921 Frank Knight drew a distinction between two situations that everyday language conflates. Which of the two escapes classical actuarial calculation, and under what condition does a hazard tip from one into the other?
The one where the probability distribution is itself unknowable, for want of relevant history or because the phenomenon changes faster than it can be observed
The distinction Frank Knight drew in 1921 separates risk, where probabilities are known or estimable, from uncertainty, where the distribution of possible outcomes itself escapes us. That boundary is the boundary of the trade, since insurance lives by turning a hazard into a risk that can be given probabilities, by means of a history and the law of large numbers. What has no probabilistic base cannot be priced, not because the calculation would be hard but because it has no object. Three features tip a hazard from the first domain into the second, and they are worth naming, because none of them has to do with the size of the loss. Absence of relevant history is one. Non stationarity is another, and the most discreet: when a phenomenon transforms faster than data accumulates, the history describes an object that no longer exists. The third is endogeneity, where the environment reacts to the cover itself. The AlgoPolis dissertation puts this distinction to work on agentic artificial intelligence, whose failures depend on models updated continuously, on capabilities that appear without having been foreseen, and on threats that adapt. It is worth noting that the obstacle described is not financial: a very heavy but probabilisable risk transfers, while a modest but non probabilisable one resists.
Glossary entry · incertitude-knightienne