Expected annual loss for a property, with a worked example

Definition: the four-part model

Expected annual loss (EAL), which catastrophe modelers call average annual loss (AAL), is the amount a property is expected to lose to a hazard in an average year, taking every possible event and its probability into account. It is the number an insurer prices on, because a premium is a fee for carrying an expected loss plus its uncertainty; a 0-to-100 hazard score is a ranking, not money. EAL is usually expressed twice: as a rate per mille of the insured value, which does not depend on the sum insured, and in euros per year, which does.

Every catastrophe model reaches it through the same four parts: hazard (how intense is the peril here: water depth, ground acceleration, gust speed), exposure (what stands here and what it costs to rebuild), vulnerability (what fraction of that value breaks at a given intensity, the damage function), and the financial step (integrate over probability, multiply by value, apply deductibles, limits and loadings). Two other quantities come out of the same integration. The probable maximum loss (PML) is the loss at a chosen rarity, typically the 1-in-100, 1-in-200 or 1-in-500 event, and is what a reinsurer prices. The technical premium is the EAL grossed up for expenses, capital and model uncertainty; one engine uses an indicative factor of 1.6.

Exceedance probability and return periods

A return period T is not a schedule; it is a probability. An event with a 100-year return period has an annual exceedance probability of 1/T = 1 percent, and over a 30-year mortgage the chance of seeing at least one such event is 1 − (1 − 0.01)^30, about 26 percent. Official flood maps publish extents and depths for a grid of return periods; the table shows the probabilities each implies and a representative floodplain depth per return period used where no address-specific depth exists.

Probabilities are arithmetic; representative depths are the Bytero design-depth table, interpolated in log(T) between anchors.

Depth-damage functions

A depth-damage function maps water depth above the finished floor to the fraction of a building's reconstruction value that is destroyed. The most cited pan-European set is the JRC 2017 database of global depth-damage functions (Huizinga, De Moel and Szewczyk), which gives a curve per continent and building use. Claims-calibrated curves for masonry and panel stock sit far lower at shallow depths: the German FLEMO family, derived from paid claims after the 2002 Elbe flood, reads roughly 2 to 6 percent below 21 cm, 6 to 9 percent at 21 to 60 cm, 9 to 12 percent at 61 to 100 cm, 12 to 17 percent at 101 to 150 cm and 17 to 25 percent above that, building only. The September 2024 flood in Czechia cost insurers about 19.7 billion CZK over roughly 100,000 property claims, around 5 to 8 percent of a typical sum insured per claim, which is the same order.

Curve anchors from the Bytero loss model (residential, JRC European, commercial and industrial curves); the JRC values follow Huizinga et al. 2017. Values between anchors are interpolated linearly.

Building, contents and business interruption

The curves above are building curves: the fraction of the structure's reconstruction value lost. Contents are insured on a separate sum with a separate curve, and they are more vulnerable at shallow depth, because floor-level furniture, appliances and stock are destroyed long before walls are. The JRC commercial and industrial curves already fold inventory into the building value, which is one reason they start higher than the residential curve; a residential model should keep the two sums apart, so a policy with building cover only is not overcharged for contents it does not insure.

Business interruption (BI) is the loss with no residential equivalent: gross profit lost while the premises cannot operate. Residential policies cover it as a small alternative-accommodation sublimit (one household policy carries 10,000 EUR). For commercial risks, downtime rises with severity. A HAZUS-based restoration model used by one engine assumes 17, 22 and 26 months for a commercial premises flooded to 1.2 m, 2.4 m and deeper, 10, 14 and 18 months for an industrial one, 24 and 20 months on a total loss, and caps the indemnity at the policy's maximum period, 12 months by default. The insured basis deliberately ignores the economic recapture that macro models apply, because the individual insured's shop is still shut whatever the wider economy does. The same model prices earthquake downtime from damage state: 0.3, 3, 9 and 12 months for slight, moderate, extensive and complete damage to a retail premises.

Integrating over return periods

The computation itself is short. A loss-exceedance curve is built from the return periods the address floods at, the damage ratio at each, and a dry anchor; the area under that curve on the probability axis is the expected annual damage ratio.

1. 1

   List the return periods that flood the address

   From the official maps or a modeled depth raster: for example flooded at Q10 and above, dry at Q5. The largest dry return period anchors the curve at zero damage: here p = 1/5 = 0.2, damage 0.

2. 2

   Read the depth at each return period

   An address-specific depth from a raster where one exists; otherwise a representative floodplain depth for the return period, flagged as such in the output, so the reader knows which basis was used.

3. 3

   Subtract the floor elevation

   Depth rasters measure water above the terrain; damage curves are referenced to the finished floor. A residential ground floor sits about 0.30 to 0.45 m above grade on a plinth and threshold, so one engine subtracts a conservative 0.25 m (0.10 m for commercial and industrial floors near grade). A flat on the second floor adds two storeys of about 3 m each and drops out of the flood integral entirely.

4. 4

   Convert depth to a damage ratio

   Interpolate on the occupancy's depth-damage curve. Occupancy selects the curve: residential masonry, commercial or industrial.

5. 5

   Integrate by the trapezoid rule on the probability axis

   Sort the points by falling exceedance probability p and sum 0.5 × (damage_i + damage_i+1) × (p_i − p_i+1). Add a final point at p = 0 carrying the rarest damage ratio, so events rarer than the rarest mapped return period are not silently treated as harmless.

6. 6

   Multiply by exposure, then calibrate and load

   The result is a rate per unit of building value. Times the reconstruction value it becomes euros per year; a per-peril experience factor pins the model to observed claims; a loading turns it into a technical premium. The PML at 1-in-100 is simply the damage ratio at Q100 times the value.

Worked example: a floodplain house from Q10 to Q500

A masonry house with a reconstruction value of 200,000 EUR stands on a floodplain that is dry in a 1-in-5 event and wet from 1-in-10 upward. No address raster exists, so the representative depths apply, less the 0.25 m floor threshold. The table walks the six return periods and the tail.

Contributions are rounded to two decimals; the unrounded sum is 6.22 ‰. Curve and thresholds from the Bytero loss model.

Reading the example, and what changes the number

Three things stand out. The frequent, shallow events (Q10 to Q50) carry 3.59 ‰ of the 6.22 ‰, or 58 percent of the expected loss, even though each does little damage; expected loss is dominated by what happens often, not by the worst case. The tail beyond Q500 adds only 0.44 ‰. And the PML is a different animal: at Q100 the single-event loss is 0.112 × 200,000 = 22,400 EUR and at Q500 it is 44,000 EUR, more than seven times the annual expectation. The table shows the levers that move the result, computed on the same curve.

Recomputed from the worked-example curve; the flat share, calibration factor and pluvial cut-off are constants of the Bytero loss model.

From expected loss to technical premium

In euros, the base case is 6.22 ‰ × 200,000 EUR = 1,244 EUR per year before calibration, 510 EUR after the 0.41 river-flood factor, and 816 EUR per year as a technical premium at a 1.6 loading. That loading covers acquisition and administration expenses, the cost of capital for the tail, and the model's own uncertainty; a filed rate then adds the insurer's margin and any cross-subsidy between perils. The technical premium for a portfolio is the sum of its EALs times the loading, which is why an address-level EAL is the building block of a book-level view and why address-level hazard matters so much: the same house 40 m further from the river can sit outside the Q100 extent and carry a tenth of the expected loss.

Calibration is what separates a hazard-times-vulnerability arithmetic from a model an underwriter can use. One 2026 recalibration validated its physics against 993 properties with real premiums reconstructed from published tariffs and found that its earlier national target of 0.60 ‰ per year was four to six times the level implied by three independent sources, and that 45 percent of properties carried a natural-catastrophe EAL above their entire all-perils premium, which is structurally impossible. The fix was to correct the physics first (the masonry curve, the floor threshold, the ground-floor share) and then apply one residual factor per peril, derived from national damage series and neighboring-market claims rather than from the model itself. After recalibration the national mean sat at 0.185 ‰ per year within an independently derived band of 0.10 to 0.19 ‰, the average EAL was 32 percent of the average real premium, and the median EAL was 17 EUR per year on a 193,000 EUR home. Commercial catastrophe models differ by a factor of two to three between vendors on the same book, so a physics result within about 2.5 times of independent experience is the normal starting point, not a failure.

Bytero computes the per-peril hazard curve, the depth-damage integration, the experience calibration and the technical premium for a single address in the way described here, with every constant a named literal that an underwriter can read; the product is Bytero Shield and availability by market is on the coverage page.

Questions