1-in-200 Year Loss: Extreme Value Theory

Return period	POT, exact no simulation	Lognormal simulated	POT simulated	+ uncertainty simulated	In the data 25 years
1 in 10 years	29.0	17.7	27.8	27.9	25.6
1 in 25 years	45.1	23.2	43.9	44.4	51.0
1 in 50 years	61.5	28.3	58.9	59.8	–
1 in 100 years	82.7	34.1	83.0	82.8	–
1 in 200 years	110.5	42.0	106.3	113.0	–
1 in 250 years	121.1	43.8	115.4	125.7	–

The business question

A property insurer has kept a record of every loss event above US$1 million for 25 years — storms, floods and large fires, 189 events in all, the largest US$51 million.

For its capital and its catastrophe reinsurance the board asks two questions: how large is the event to expect once in 100 or once in 200 years, and how large is the year's total at the same return periods? With 25 years of history, a 1-in-200 year loss lies far beyond the data, so the answer depends on the model of the tail.

The model compares a Lognormal fitted to every event with extreme value theory, which says that the events above a high enough threshold follow a Generalised Pareto distribution ('peaks over threshold'). It then shows how much the answer widens when the uncertainty in the fitted tail — from only about 40 large events — is included.

Who it is for: Catastrophe and capital modellers, reinsurance buyers, pricing actuaries, risk managers working with extreme losses.
Main outputs: The largest event of a year (occurrence, OEP) and the year's total (aggregate, AEP) at return periods from 10 to 250 years, for three tail models · how often an event above US$25m, US$50m or US$100m occurs · the threshold diagnostic and the fitted tail.

How the model works

Five sheets after this one, read from left to right. Each one feeds the next.

Inputs sheet
The threshold above which the tail is modelled, and the reporting threshold below which no event was recorded.
Data sheet
Every loss event above US$1m for 25 years, in today's money; the number of events, the largest and the total in each year.
Tail sheet
The mean excess over a range of thresholds shows where the tail starts. A Generalised Pareto distribution is fitted to the events above the threshold (its location fixed at the threshold). Below it the data are used as they are (an Ogive). A Lognormal fitted to every event is the comparison.
Exact return periods and the frequency of very large events follow from the fitted distributions.
Year sheet
Next year's events are drawn under each model: Poisson numbers of events, the largest by VoseLargest and the total by VoseAggregateMC. The third model re-fits the tail to the data with parameter uncertainty in every iteration.
Results sheet
The largest event and the year's total exceeded once in 10 to 250 years, by model, against the exact values and the 25 years of data. How often a very large event occurs.

ModelRisk techniques

Peaks over threshold VoseGPDFitObject
Data below the threshold VoseOgiveObject
Largest event VoseLargest
Annual total VoseAggregateMC
Tail parameter uncertainty
Exact return periods VosePercentile

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The results in detail

The year's total

All the year's event losses exceeded once in T years — the aggregate exceedance (AEP).

Year's total — return periods

Chart: Lognormal, Peaks over threshold, + parameter uncertainty.

Return period	POT, exact no simulation	Lognormal simulated	POT simulated	+ uncertainty simulated	In the data 25 years
Average year	35.1	31.1	34.6	35.3	34.6
1 in 10 years	–	51.7	62.1	63.4	65.6
1 in 25 years	–	63.0	82.5	83.3	98.7
1 in 50 years	–	71.7	101.7	104.2	–
1 in 100 years	–	78.4	121.3	125.1	–
1 in 200 years	–	85.6	149.3	161.1	–
1 in 250 years	–	88.2	162.6	177.9	–

POT: peaks over threshold. POT, exact: from the fitted distributions without simulation (the Tail sheet) — the figure to use. The simulated columns rest, at 1 in 200, on the worst 50 of 10,000 years. In the data: from the years of history, which cannot reach further than about 1 in that many years.

How often is a very large event?

Expected years between events larger than each size.

Event larger than (US$m)	In the data events	Lognormal once in … yrs	POT once in … yrs
10	19	2	1
25	4	30	7
50	1	519	31
100	–	17,537	157
200	–	1,186,886	864

How to read these results

The 25 years of data reach to about the 1-in-25 year event. Beyond that the answer comes from the model of the tail, and the models part company fast: the 1-in-200 year event is about US$40m with the Lognormal and about US$110m with peaks over threshold (the exact values, from the Tail sheet).
The Lognormal is shaped by the 146 moderate events and has a thin tail: it says an event above US$50m happens once in about 500 years, yet the history already holds one in 25 years. The Generalised Pareto is fitted only to the events above the threshold, where the tail is decided.
Choose the threshold where the mean excess starts to rise along a straight line, with enough events above it — here US$5m, with 43 events. Try US$3m or US$7.5m on the Inputs sheet: if the tail model is sound, the return periods change little.
With 43 large events the shape of the tail is uncertain. Including that uncertainty raises the 1-in-200 year figures again — by how much is itself hard to pin down at 10,000 samples: two runs put the 1-in-200 year event at about US$113m and US$158m. For capital and reinsurance it is part of the risk.
The 1-in-200 year total is far more than the 1-in-200 year event: in a bad year several large events come together (2006 had two above US$40m). Reinsurance per event is priced on the largest event. Capital and aggregate covers on the total.
Simulation error: with 10,000 simulated years a 1-in-200 value rests on the 50 worst years, and for a tail this heavy that is a rough estimate — here the simulated 1-in-200 event is about 4% below the exact value, and another seed put it 17% above. Use the exact values where they exist, and run 100,000 samples or more before relying on a simulated 1-in-200.

See it move. Open the workbook, change an input on the Inputs sheet and run the simulation: every table and chart on this page updates in Excel.

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Inputs and assumptions

Every assumption in the model, with its uncertainty and where it comes from. The figures are illustrative.

Tail 2 inputs

Where the tail model starts.

Assumption	Unit	Value
Threshold of the tailChosen with the mean excess on the Tail sheet; about 40 events lie above it.	US$m	5.0
Reporting threshold of the dataEvents below this were not recorded.	US$m	1.0

Units and conventions

Money: US$ million (US$m), in today's money.
Event: All the losses from one occurrence: one storm, one flood, one fire.
Return period: 'Once in T years': exceeded with probability 1/T in any one year. The 1-in-200 year loss is the 99.5th percentile of a year.
OEP and AEP: Occurrence exceedance: the largest single event of a year. Aggregate exceedance: the year's total of all events.
Mean excess: The average amount by which the events above a threshold exceed it.

Key assumptions and limitations

Events: Event numbers are Poisson; event losses are independent and identically distributed.
Data: Losses are in today's money and today's portfolio; trends in exposure and inflation are removed.
Tail: Above the threshold the losses follow a Generalised Pareto distribution — the basis of extreme value theory, not a certainty.
History: The 189 events are invented for this example.

How to use it

Paste your loss events on the Data sheet (year and loss, one event per row), in today's money.
Look at the mean excess on the Tail sheet and set the threshold on the Inputs sheet.
Run a simulation from the ModelRisk ribbon (10,000 samples with a fixed seed are set).
Read the 1-in-100 and 1-in-200 year event and total — and how far apart the tail models are.

Simulate this model with ModelRisk’s classic engine: it does not run in Turbo mode yet.

ModelRisk functions used

VoseGPDFitObject, VoseGPDFitP: The Generalised Pareto tail fitted to the events above the threshold, and its parameters (location, scale, shape).
VoseOgiveObject: The data themselves as a distribution, between the reporting threshold and the threshold.
VoseLognormalFitObject: A Lognormal fitted to every event, for comparison.
VoseLargest: The largest of n events drawn from a distribution.
VoseAggregateMC: The year's total of a random number of events.
VosePoissonFit, VoseProb, VosePercentile: Event numbers with uncertainty; exceedance probabilities and exact return periods.
VoseSimPercentile, VoseSimMean: Statistics of the simulation on the Results sheet.

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An illustrative example: the figures are invented for teaching and do not describe a real company, project or asset. Last updated 9 October 2026.

Catastrophe and Large-Loss Tail

The largest event of a year