A catastrophe model output almost always comes as a two-column table: return periods on one side, amounts on the other. That table exists in two versions, and many people who read it every week do not know which one they are looking at. The first answers the question of the year's largest event; the second, that of the year's total. On a portfolio exposed to a frequent peril, they diverge enough to have a program sized wrongly.
The occurrence exceedance curve, or OEP, reads as follows: at the two-hundred-year return period, the amount shown is the one the LARGEST event of the year exceeds with a probability of half a percent. It knows nothing of the number of events: a year with a single 80 million event and a year with four events whose largest is 80 million are identical to it. It is the right curve for anything that triggers event by event.
The aggregate exceedance curve, or AEP, answers the other question: what is the year's TOTAL, all events combined. It is always greater than or equal to the OEP at the same point, and the gap between the two measures exactly one thing, the peril's propensity to strike several times in one year. That gap is small for earthquake, considerable for hail or windstorm, and that is why the question of which one is being read does not arise the same way for every peril.
The classic error has two symmetric forms and they cost in both directions. Sizing a structure that responds per event by reading an aggregate curve leads to overstating what that structure will have to absorb at once, therefore to buying a height part of which will never serve on any one event. Sizing an annual structure by reading a per-event curve leads to the reverse, and that is the more dangerous of the two: the protection looks sufficient, it is sufficient on the isolated loss, and it turns out short in the year when three medium events follow one another.
One must add a subtlety that governs the reading and is forgotten every other time: the definition of an event is not given by nature, it is given by a text. An hours clause says that all damage occurring within a window of seventy-two or one hundred and sixty-eight hours counts as one. The model catalog has its own clustering convention. If the two do not coincide, the OEP curve being read does not describe the same thing as the structure it serves to size, and the gap appears on the day of a loss, not before.
These curves finally carry a limitation neither of them states: they describe the portfolio as it was submitted to the model, on a date, with that day's values and addresses. A portfolio growing by fifteen percent during the year, or whose concentration shifts because a large account has come in, is no longer the one the curve describes. Rereading a March curve in November to arbitrate a renewal amounts to measuring a building on a plan that is eight months old.
In practice, the discipline comes to four questions asked before quoting an amount. Is this an OEP or an AEP, which should appear on the document and does not always. On which event definition, and does it coincide with that of the contract concerned. On which portfolio perimeter and at which extraction date. And is the amount gross or net of what already exists. A figure quoted without those four attributes is not a measurement, it is a number, and two numbers of that kind do not compare.
A householders insurer holds a portfolio exposed to windstorm and hail. The modeling note returns, for the hundred-year return period, 140 million euros on the OEP and 205 million on the AEP. The annual retention of the reinsurance plan is set at 30 million. The team prepares a per-event structure 110 million high above 30, explaining that this covers the hundred-year event. A committee member observes that the 65 million gap between the two curves has been discussed nowhere. What does that observation reveal, and what must be established before the structure is settled?
The 65 million gap is not a measurement imprecision, it is the very content of the missing information: it says that on this portfolio, the share of the hundred-year burden coming from the REPETITION of events is of the same order as the share coming from the largest of them. On a hail and windstorm peril that is expected, and it is precisely what the proposed structure does not see. Sized on the OEP, it answers correctly on the isolated loss: a 140 million event is absorbed. It does not answer on the year with four events of 45 million each, where every event stays under the retention or barely crosses it, where the structure hardly triggers, and where the insurer carries one hundred and eighty million of net burden with its protection intact. That is the dangerous form of the error, because nothing in the figures presented flags it. Three things therefore have to be established before anything is settled, and none consists in picking an amount. The first is the event definition used by the model and the one the wordings will carry: if the catalog clusters on seventy-two hours and the treaty on one hundred and sixty-eight, the OEP curve being read does not describe the structure it serves to size. The second is the distribution of the number of events per year on this portfolio, which is the quantity the OEP-AEP gap summarizes, and which alone says how often the repetition year occurs. The third is to redo the reading on the perimeter and extraction date of the portfolio as it will be at inception, not as it was when the note was produced.
- 01The OEP curve answers the year's largest event, the AEP curve the year's total: they are not two refinements of the same measurement.
- 02The gap between them measures the peril's propensity to strike several times a year: small for earthquake, considerable for hail and windstorm.
- 03Reading a per-event curve to size an annual protection is the dangerous form of the error: nothing flags it before the repetition year.
- 04The event definition comes from a wording and not from nature: if the model's convention and the hours clause differ, the curve does not describe the structure.
- 05An amount quoted without its curve, its event definition, its perimeter and its extraction date is not a measurement and compares to nothing.