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. The average claim cost of a line moves from 2,800 euros in 2022 to 3,100 euros in 2023, a rise of 10.7 percent, while claim frequency is flat. What do these two figures together signal?
Inflation in unit costs, materials, labor or social inflation
Average claim cost is total claims incurred divided by the matching number of claims. It measures severity, and reading it alone teaches nothing: it speaks when paired with frequency. Severity up together with frequency up says the book has deteriorated. Severity up on flat frequency says something else: each claim costs more, which points to materials prices, labor costs, or social inflation in awards. Two reading cautions: the figure is sensitive to large losses and to shifts in portfolio mix, and paid average cost, computed on closed files, differs from estimated average cost, which folds in the reserves on files still open and inherits their uncertainty.
Glossary entry →2. The pure premium for a risk is 360 euros. The insurer targets a 30 percent expense ratio and a 5 percent profit margin. What is the gross premium?
554 euros, the pure premium divided by 0.65
The gross premium is the final price the policyholder pays. It starts from the pure premium, the expected cost of claims, and loads it for acquisition and administration expenses, the target profit margin, the cost of the regulatory capital tied up, and a risk loading for actuarial uncertainty. The formula is gross premium equals pure premium divided by one minus the target expense ratio minus the target margin, here 360 divided by 0.65, about 554 euros. The classic error, and it is worth making once so as never to make it again, is to load by 35 percent instead of dividing by 0.65: that gives 486 euros, and the 35 percent of loadings is then taken on the pure premium instead of on the final premium, which is what expenses and margin are actually drawn from. The shortfall is 68 euros per policy, invisible line by line and large across a book.
Glossary entry →3. A company has had a single cyber claim in five years. What does the actuary do with that individual experience when pricing?
Blends it with the sector's average experience, which is far more reliable
Credibility theory answers a question every pricer faces: how much weight should a risk's own experience carry against a broader benchmark? Plenty of data makes that experience statistically reliable and largely usable; little data makes it too unstable to take at face value. Credibility formalizes the trade-off by giving individual experience a weight between zero and one, rising with the volume and stability of the data, the remainder going to the collective benchmark. The final rate is thus a weighted average, and that weighting is what avoids the two symmetric mistakes: over-reacting to a handful of isolated claims, or ignoring a genuine specificity. The framework, formalized notably by Bühlmann, is everywhere in pricing. In cyber, credibility on own data is generally low for want of a long stable history, which shifts weight onto market benchmarks and expert judgment, at the price of coarser segmentation.
Glossary entry →4. A book of a thousand cyber SMEs looks well diversified, different sectors and countries, low linear correlation in normal conditions. Eight hundred of them run the same endpoint security tool. What does capital computed on a moderate correlation assumption miss?
Tail correlation, close to one for those eight hundred entities in a common-failure scenario
Tail correlation is the property whereby variables can show weak dependence in normal conditions and strong dependence in the tail. Formally, it is the probability that one risk exceeds its extreme quantile given that another exceeds its own. The decisive technical point is that Gaussian copulas have zero tail dependence: extreme events become asymptotically independent under them, and the Solvency II standard formula's aggregation through linear correlations implicitly makes the same assumption. Student copulas and Archimedean copulas, Gumbel or Clayton, do capture positive tail dependence. The 2008 crisis showed what the assumption costs: structured credit was priced with Gaussian copulas that gave defaults no tail dependence, while tail dependence in the underlying housing turned out to be very strong. Systemic cyber has the same shape: a common software failure hits every insured sharing that technology at once and in the same way, however far apart their sectors look. That is what cancels the diversification benefit normally granted, and leaves capital badly short.
Glossary entry →5. Two bonds correlated at 95 percent each carry an individual VaR of 100 million euros, and an aggregate VaR of 210 million. Which axiom does this violate?
Subadditivity
A risk measure is called coherent when it satisfies the four axioms set out by Artzner in 1999: monotonicity, translation invariance, subadditivity and positive homogeneity. Subadditivity requires that the risk of a sum never exceed the sum of the risks, that is, that diversification can only reduce the requirement. Here 210 million exceeds 200 million: aggregating two risks pushed required capital up. VaR is not subadditive in general and is therefore not coherent; TVaR, or expected shortfall, is. The consequence lands on capital allocation: an incoherent measure suggests a portfolio is worth more split than whole, which pushes firms to carve up entities to show less capital, and rewards an accounting reorganization rather than a real reduction in risk.
Glossary entry →6. Applied to cyber accumulation risk, one analytical framework surfaces three simultaneous failures: independence of losses compromised, frequency hard to estimate, maximum loss poorly bounded. Which framework is it?
The Berliner criteria on the limits of insurability
Baruch Berliner set out these criteria in Limits of Insurability of Risks, published by Prentice Hall in 1982. He identifies nine necessary conditions grouped in three dimensions. The actuarial dimension asks that losses occur randomly and independently, which is what makes pooling work, that average frequency and severity be estimable from sufficient history, that maximum loss stay bounded and financeable, and that the required premium remain economically viable. The behavioral dimension asks that moral hazard be controllable and adverse selection manageable. The social and regulatory dimension asks that no rule of public policy bar the cover. The most useful point is that the criteria are continuous, not binary: when a condition degrades, insurability does not vanish, it gets paid for in higher premiums, larger deductibles, sublimits and exclusions. What sets systemic cyber and physical climate risk apart is that several conditions degrade at once, and it is that convergence, not any single one, that explains the market withdrawals observed.
Glossary entry →7. A tsunami with a 500 year return period carries an annual probability of 0.2 percent. What is roughly the probability of at least one event over a hundred years of operating a coastal industrial site?
About 18 percent
The risk horizon is the period over which a probability of occurrence or of breaching a threshold is computed, and changing it transforms the answer without a single physical datum moving. Here, the probability of at least one event over a hundred years is one minus the probability of none, that is one minus 0.998 to the hundredth power, about 18 percent. The 20 percent answer is the linear approximation by multiplication: close enough here, but it drifts as the horizon lengthens and returns values above one hundred percent, which is enough to disqualify it as a method. The same effect appears elsewhere: a hundred year event has a one percent chance in any year, but about twenty-six percent over thirty years. In solvency the standard horizon is one year, the Solvency II SCR. For a property asset it may be the term of a loan. For climate or nuclear work it runs past fifty years. The choice of horizon therefore decides whether the risk analysis and the decision it informs are talking about the same thing.
Glossary entry →8. A board wants to detect reserve drift. Each annual reserve, taken alone, is defensible. Which question brings it into view?
How have prior years' estimates moved since they were first set?
Under-reserving starts as a valuation error and ends as the mechanism by which an insurer blinds itself to its own position. An insufficient reserve overstates the result, so dividends are paid out of profits that do not exist; it overstates equity, so apparent underwriting capacity is inflated; and above all it distorts the loss experience on which next year's rates are built. Under-pricing and under-reserving then validate each other in a closed loop where each error confirms the last. The drift is gradual and every single estimate stays within the defensible range, which is why a year-by-year review never catches it. It shows up only in the movement: by looking at how prior years' estimates have shifted since they were set, you stop judging a number and start judging a trajectory. Compounded over a decade it produces a large gap, usually recognized all at once, on a change of management or an external review.
Glossary entry →9. In Earth orbit, the owner of the destroyed satellite is indemnified, the party that caused the collision owes nothing, and the debris cloud that hampers every present and future user binds no one. Which instrument, a century old, internalizes exactly this kind of externality?
Compulsory liability insurance
The tragedy of the commons describes three things together: a shared and degradable space, costs borne collectively, and no mechanism returning to each actor the cost it imposes on the others. Earth orbit is a clean contemporary case, since each satellite earns a private return while the debris a collision creates harms everyone and belongs to no one. The instrument that internalizes this kind of externality is a century old and not exotic at all, it is liability insurance made compulsory: it turns harm done to others into a cost carried by whoever causes it, and the price of the cover then becomes a signal about behavior. The notable fact is therefore that nobody imposes it in orbit, though the remedy is known. Climate supplies the inverted variant, where insurers withdrawing concentrates the worst risk on a public pool funded afterwards by those who stayed.
Glossary entry →