Catastrophe risk · RiskAtoms Academy

By the end you will be able to

1Compute the average annual loss and the occurrence exceedance curve from an event set, and read return-period losses off it.
2Tell the occurrence and aggregate views apart, and explain where each is used.
3Measure accumulation by zone and take the 1-in-200 event through the catastrophe reinsurance.
4Explain why the far tail is uncertain and how sensitive it is to the model's assumptions.

Why it matters

RiskAtoms Insurance SE's catastrophe losses average €160.8m a year. That figure goes into the price and is almost useless for anything else. The 1-in-200 windstorm costs €2,153m, 13.4 times the average, most of it in one region and in one night. No history of ten or fifty years can show that event. A catastrophe model can, and the insurer's capital, reinsurance and appetite all rest on what it says.

Core explanation

From an event set to the 1-in-200

Illustrative dataRiskAtoms Insurance SE's insurance book

The same composite European insurer as Modules 1.8 and 2.2, in € millions at 31 December 2025: four non-life lines with ten years of claims triangles, a 1.2 million vehicle-year motor pricing data set, a 17,000-event catastrophe set, its reinsurance, its life books, its assets and an outline of its Solvency II standard formula. Every policy, claim, event and figure is made up for teaching and seeded so it never changes; none is any insurer's data. Every lesson in Module 3.8 uses this book, so the figures agree.

This lesson uses the book's seeded event set: 17,000 events (windstorm, flood, hail and earthquake) across five zones with €700bn of sums insured, each with an annual rate and a gross loss to the property book, plus 50,000 simulated years. Losses follow Pareto tails, capped. The simplifications: events are independent, there is no secondary uncertainty or demand surge, earthquake sits in Northern Italy only, and the surplus treaty takes no catastrophe losses. The book's cat capital is the 1-in-200 net event from this set, not the standard formula's catastrophe factors.

How a catastrophe model works

L2.2.06: Insurance risk showed why catastrophes break pooling: one event, many claims. Claims history is too short to measure that, so insurers use a catastrophe model, usually licensed from a specialist vendor. It has four parts.

ExploreInside a catastrophe model

Click a box to see what happens there.

Hazard

A stochastic catalogue of tens of thousands of physically plausible events, each with a rate: storm tracks and wind fields, river flood footprints, hail swaths, earthquake ground shaking. It is calibrated on centuries of weather and geological records, not on insurance claims.

The event set and the average annual loss

Multiply each event's rate by its loss and add up: that is the average annual loss (M-C02: Expectation). Together the events occur 2.87 times a year on average.

€m, gross. Return-period columns: the occurrence loss of each peril alone.
Peril	Events	Rate a year	Tail index α	AAL	Share of AAL	1-in-100	1-in-200
Windstorm	10,000	1.50	1.30	111.9	70%	940.0	1,609.3
Flood	4,000	0.35	1.15	24.4	15%	330.8	602.3
Hail	2,000	1.00	1.70	19.0	12%	118.3	179.9
Earthquake	1,000	0.02	0.90	5.5	3%	85.8	186.3
All perils	17,000	2.87		160.8	100%	1,246.7	2,153.2

Windstorm is 70% of the AAL and dominates the tail. Each peril's event losses follow a Pareto distribution, whose tail index α measures how fat the tail is (M-C08: Skewness and fat tails): the smaller α, the more often the very large events. A Pareto has a finite mean only if α > 1, which is why windstorm's mean event before the cap is €86.7m even though most events cost less than half of that, and why earthquake, with α = 0.9, would have no finite mean at all without the book's cap.

Exceedance curves and return periods

Sort the events by loss. The occurrence exceedance probability at a loss x is the chance that the year's largest event reaches x; the aggregate exceedance probability is the chance that the year's total does. One over either is a return period, and the loss at a return period is a quantile (M-C09: Quantiles).

Occurrence (OEP)Aggregate (AEP)

€m, gross. OEP from the event set's rates; AEP from 50,000 simulated years.
Return period	Probability a year	Occurrence (OEP)	Aggregate (AEP)	OEP ÷ AAL
1 in 10	10.00%	194.2	304.6	1.2×
1 in 25	4.00%	409.7	532.2	2.5×
1 in 50	2.00%	720.4	853.5	4.5×
1 in 100	1.00%	1,246.7	1,384.0	7.8×
1 in 200	0.50%	2,153.2	2,507.2	13.4×
1 in 250	0.40%	2,660.2	2,891.5	16.5×
1 in 500	0.20%	3,607.0	4,062.5	22.4×
1 in 1,000	0.10%	5,683.8	6,000.0	35.3×

The two views answer different questions. Per-event reinsurance such as the cat XL responds to each occurrence, so it is designed on the OEP. The year's result, and capital, depend on all events together: the AEP. The curve is steep. Going from 1 in 100 to 1 in 200 adds €906m; the next 800 years of return period add €3,531m more. In this set, the €2,500m storm of L2.2.06: Insurance risk is a 1-in-232-year event and the €3,200m storm of its Lab Test 1 in 430.

Accumulation

Accumulation is how much one event can hit at once. Measure each zone alone, then the book as a whole:

€m. The whole book's 1-in-200 is €2,153.2m.
Zone	Sums insured (€bn)	AAL	1-in-200 (zone alone)
Northern Germany	180	48.8	761.8
Southern Germany	160	18.6	330.4
Benelux	140	45.8	696.1
Northern France	120	36.5	524.0
Northern Italy	100	11.1	209.1
Sum of the zones	700	160.8	2,521.4

The five zones' 1-in-200 losses add up to €2,521m; the book's is €2,153m. Spreading the book across five zones saves only 15%, because a European windstorm does not respect borders: one storm crosses Northern Germany, Benelux and Northern France in a night. Underwriters manage accumulation with zone limits on sums insured, by steering new business away from crowded zones, and with reinsurance.

Sort itDiversifies or accumulates?

RiskAtoms Insurance SE considers five ways to grow its property book by €20bn of sums insured. Which spread the catastrophe risk, and which pile it up?

More homes in Northern Germany's coastal towns.
More commercial property in Northern Italy.
More homes in Benelux.
A book of homes in a region the model shows exposed only to inland river flood.
Motor own damage policies for cars parked in Southern Germany.

Model uncertainty

Every figure above comes from a model whose tail is fitted to a few decades of data. Change windstorm's tail index, keeping the same random draws:

€m, gross, all perils.
Windstorm α	AAL	1-in-200 event	Change in 1-in-200
1.1	209.3	3,787.2	+76%
1.2	180.9	2,935.1	+36%
1.3 (book)	160.8	2,153.2	+0%
1.4	146.7	1,748.3	−19%
1.5	136.7	1,474.1	−32%

The AAL moves far less than the 1-in-200: from €136.7m to €209.3m as α falls from 1.5 to 1.1, while the 1-in-200 goes from €1,474.1m to €3,787.2m. Real models add more sources of doubt: secondary uncertainty in each event's loss, demand surge after big events, clustering of storms in the same winter, and climate change shifting the hazard itself. Firms compare vendor models, adjust them to their own claims experience, and treat the 1-in-200 as a range rather than a number.

The formal version

The pieces. For events $e$ with rates $\lambda_e$ and losses $L_e$ :

\begin{aligned} \text{AAL} &= \sum_e \lambda_e L_e = \text{€}160.8\text{m} \\ \text{OEP}(x) &= 1 - \exp\Big(-\sum_{e:\,L_e \ge x}\lambda_e\Big), \qquad x_T = \min\{x : \text{OEP}(x) \ge 1/T\} \\ \text{AEP}(x) &= P\Big(\sum_{k=1}^{N} L_{e_k} \ge x\Big), \quad N \sim \text{Poisson}\Big(\textstyle\sum_e \lambda_e\Big) \text{ (by simulation)} \end{aligned}

A check in closed form. For one peril with rate $\lambda$ and Pareto losses, $P(L \ge x) = (x_{\min}/x)^{\alpha}$ , so the rate of events above x is $\lambda (x_{\min}/x)^{\alpha}$ . Setting it equal to $r_T = -\ln(1 - 1/T)$ :

\begin{aligned} x_T &= x_{\min}\Big(\frac{\lambda}{r_T}\Big)^{1/\alpha} \\ x_{200}^{\text{wind}} &= 20 \times \Big(\frac{1.5}{0.0050125}\Big)^{1/1.3} = \text{€}1{,}605.7\text{m} \end{aligned}

The event set gives €1,609.3m for windstorm alone: the 10,000 stratified draws reproduce the distribution. The formula also shows why α matters so much: the 1-in-200 is the 1-in-1 loss scaled by $(\lambda/r_T)^{1/\alpha}$ , and a smaller α raises a large number to a larger power.

Try it

Move along the curve from 1 in 10 to 1 in 1,000 years and watch where the losses leave the cat XL behind. Then take one peril alone and change its tail index: the AAL moves far less than the far tail does.

Try itExceedance curves and model uncertainty

Perils

Return period1 in 200 years

Windstorm tail index α (book 1.30)1.30

Occurrence (OEP): largest event in a yearAggregate (AEP): all events in a year

1-in-200 event (OEP)€2,153.2m

1-in-200 year (AEP)€2,507.2m

Average annual loss€160.8m

Event ÷ AAL13.4×

That event net of the cat XL€653.2m

Plus reinstatement premium€50.0m

The tail index redraws the same seeded events with x′ = x_min (x ÷ x_min)^(α ÷ α′), capped as in the book. The cat XL (1,500 xs 500) protects the whole property book, so its figures are shown for a single event of the chosen size. The occurrence curve is exact; the aggregate comes from 50,000 simulated years, and where sampling noise for a single peril would put it below the occurrence loss, which no year can be, it is shown at the occurrence loss. Gross losses, € millions, illustrative data.

Worked example

The 1-in-200 event, gross and net

The book's event set and cat XL, € millions.

Finding and pricing the 1-in-200 windstorm

Step 1 of 6

1
The rate to reach
$\text{OEP} = \tfrac{1}{200} \iff \sum_{L_e \ge x}\lambda_e = -\ln(1 - 0.005) = 0.0050125$

The retention is the tail

Above €2,000m the programme is exhausted and RiskAtoms Insurance SE keeps everything. The 1-in-200 event exceeds the top of the programme by €153.2m, which is why the net loss is more than the €500m retention. Buying another layer above 2,000 is one of the options in L3.8.10: Reinsurance in depth.

Regulatory lens

Solvency II charges catastrophe risk in its own sub-module, combined with premium and reserve risk at a correlation of 0.25 (Article 114 of Delegated Regulation (EU) 2015/35). The standard formula does not use the insurer's event set: it applies prescribed scenarios, with factors on sums insured by region for natural perils. Insurers may instead use an approved internal model, often built on vendor catastrophe models. Delegated Regulation (EU) 2026/269, which applies from 30 January 2027, amends the natural catastrophe risk factors and definitions in light of updated data and models. The scenarios and factors are in M4.6: Solvency II and insurance regulation.

Spark Atoms

Spark AtomHurricane Andrew and the models

Hurricane Andrew hit south of Miami on 24 August 1992 and cost insurers about $15.5bn at the time. Soon after landfall, Karen Clark's firm, one of the first catastrophe modellers, estimated $13bn from its model, at a time when few insurers used catastrophe models. At least 16 insurers failed in 1992–1993 according to the Insurance Information Institute, a new class of Bermuda reinsurers formed in 1993, and catastrophe models became standard equipment.

Spark AtomBernd, July 2021

The flash floods of July 2021 in the Ahr valley and neighbouring regions made 2021 the costliest year for natural hazards that German insurers had recorded, €12.5bn in all. The German insurance association GDV put the floods at €8.2bn in December 2021, later at about €8.4bn of insured property losses, and in July 2024 at €8.75bn in all. The book's own July 2021 river flood of €680m is a 1-in-229-year flood in its event set, but only 1 in 46 for a loss that size from any peril.

Key takeaways

A catastrophe model turns hazard, exposure, vulnerability and policy terms into an event loss table: one rate and one loss per event.
AAL = Σ rate × loss: €160.8m on the book, 70% of it windstorm. It prices the average year and says little about the tail.
OEP(x) = 1 − exp(−Σ rates of events ≥ x); the 1-in-200 occurrence loss is €2,153m (13.4 × AAL) and the 1-in-200 aggregate €2,507m.
Accumulation decides the tail: the five zones' 1-in-200s add up to €2,521m against €2,153m for the book, only 15% diversification.
Net of the cat XL the 1-in-200 event costs €703.2m, including €50m of reinstatement premium; the far tail is uncertain, with a 0.1 change in windstorm's α moving it by tens of percent.

Reaction

Check what stuck

Question 1 · objective 1

An event loss table has three events: rates 0.01, 0.002 and 0.0005 a year, losses €300m, €1,000m and €2,500m. What is the average annual loss, in €m?

€m

Question 2 · objective 1

For the same three events, what is the return period of a loss of at least €1,000m from a single event, in years (to the nearest year)?

years

Question 3 · objective 2

On the book the 1-in-200 occurrence loss is €2,153m and the 1-in-200 aggregate loss €2,507m. Why is the aggregate larger?

AA bad year can hold several events; the aggregate adds them up, the occurrence takes only the largestBThe aggregate includes non-catastrophe claimsCThe aggregate is net of reinsuranceDThe aggregate uses a longer return period

Question 4 · objective 3

A windstorm costs €1,800m gross. The cat XL is 500 xs 500 and 1,000 xs 1,000. What does RiskAtoms Insurance SE keep, before reinstatement premium, in €m?

€m

Question 5 · objective 4

Lowering windstorm's Pareto tail index from 1.3 to 1.2, a modest change, moves the book's 1-in-200 event from €2,153m to €2,935m. What does that say?

AThe far tail rests on a parameter that a few decades of history cannot pin down: model uncertainty is as large as the risk being measuredBThe event set is wrong and should be discardedCThe AAL moves just as much, so pricing is equally uncertainDTail indices above 1 are always safe

Glossary

Catastrophe model: Software that simulates thousands of possible natural catastrophes and estimates what each would cost a particular portfolio.
Event set (event loss table): The list of simulated events, each with an annual rate and a loss to the portfolio. Sorted by loss, it is an event loss table.
Average annual loss (AAL): Average annual loss: the long-run average catastrophe loss a year, Σ rate × loss. The starting point for the cat load in the price.
Occurrence exceedance probability (OEP): Occurrence exceedance probability: the chance that the largest single event in a year costs at least a given amount.
Aggregate exceedance probability (AEP): Aggregate exceedance probability: the chance that all events in a year together cost at least a given amount.
Return period: One over an exceedance probability. A 1-in-200 loss has a 0.5% chance of being reached in any year.
Accumulation: The build-up of insured values in one area that a single event can hit at once.
Secondary uncertainty: Uncertainty in the size of the loss given that an event happens: the same storm can cost more or less.
Demand surge: A temporary rise in the cost of materials and labour after a large catastrophe, which makes claims cost more than usual.
Reinstatement: Restoring an excess-of-loss layer's cover after a claim, for a further premium, so it can pay again in the same year.

Deep Dives & Origins

Deep Dives

Society of Actuaries, An Overview of Catastrophe Modeling Output, study note, May 2025: event loss tables, AAL, OEP and AEP, with a worked example of the AAL and OEP. Intermediate.
C. Hemenway, “‘The phones started ringing right away’: cat modeling pioneer Karen Clark recalls Hurricane Andrew”, Insurance Journal, 1 June 2022. Introductory.

Origins

Society of Actuaries, An Overview of Catastrophe Modeling Output, GI 301 study note, 2025 (ELT, AAL, OEP, AEP, return period, secondary uncertainty, demand surge).
C. Hemenway, Insurance Journal, 1 June 2022 (Andrew's date, $15.5bn, the $13bn estimate, Bermuda reinsurers in 1993).
A. O'Connor, “25 years later: how Florida's insurance industry has changed since Hurricane Andrew”, Insurance Journal, 4 September 2017 (at least 16 insurer failures, per the Insurance Information Institute; cat models not yet widely used).
Artemis, “Natural disasters in Germany cost insurers €12.5bn in 2021”, 29 December 2021 (GDV: €8.2bn for the July flood).
autohaus.de, GDV long-term natural hazards review, 27 November 2023 (Bernd: about €8.4bn of insured property losses).
GDV, “Flutkatastrophe von 2021: 7,5 Milliarden Euro ausbezahlt”, gdv.de, updated 11 July 2024 (total loss €8.75bn).
Lloyd's Market Association, Catastrophe Modelling: Guidance for Non-Catastrophe Modellers, June 2013 (exposure, hazard, vulnerability and financial modules; exposure data quality).
Commission Delegated Regulation (EU) 2015/35, Article 114; Debevoise & Plimpton, “European Commission adopts amendments to the Solvency II Delegated Regulation”, 2 March 2026.

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