A power of 7.8 accounts for nearly all the reduction in this trigger's basis risk. Across the same 20 storms, weighting each zone's gust by its exposure share raises R-squared from 0.7293 to 0.7390. Add the power transform and multiply by absolute sums insured, and the figure reaches 0.9307. Those sums insured rise from EUR 4.494tn in 1999 (extrapolated) to EUR 16.918tn in 2020, carrying the loss trend alongside the convexity. The paper presents hazard, vulnerability and exposure as a combined construction. Its R-squareds leave the vulnerability exponent doing most of the work.
A cost field in place of a cat model
Ery and Koch propose a cat bond trigger that investors can calculate independently, using data owned by neither side. Event cost is approximated with a field covering 95 CRESTA-zone centroids in Germany. CRESTA zones are postcode-level units used by the insurance industry for exposure reporting. For each zone, the trigger applies a damage function to the gust and multiplies the result by sums insured.
Hazard comes from the PERILS/UK Met Office peak three-second gust at 10 metres within a 72-hour window. Each storm therefore contributes one figure per zone. The sample contains 100 events between February 1999 and February 2020. Exposure comes from PERILS industry sums insured by zone, spanning residential, commercial and industrial lines across building, contents and business interruption. The annual trigger adds the cost field across every zone and every event during the year.
The hazard field is modeled as a max-stable random field, the usual family for joint extremes, and fitted by pairwise composite likelihood. CLIC, the composite likelihood information criterion, determines the model choice. Extremal-t with a Cauchy correlation wins at 3,368,504. Brown-Resnick records 3,432,668, while Smith records 3,424,245.
Event counts follow a negative binomial distribution with mean 4.546 (standard error 0.666). The fit uses 22 annual observations, from zero in 2003 and 2016 to twelve in 2014. Two million simulated years generate 9,090,933 events. Pricing applies a Wang transform to the empirical annual loss distribution, loading its tail, together with a Vasicek short rate. The financial and catastrophe filtrations are assumed independent.
The paper says realizations require only publicly available information from reputed and unbiased third parties. The intended advantage is clear: no vendor model licence, with settlement based on gust data available to both sides. In practice, PERILS licenses the per-zone gust series, the loss records and the exposure database. The further claim is lower basis risk than the wind-speed triggers used by existing European windstorm bonds.
How much does exposure weighting add?
The paper compares its trigger with three alternatives over the same 20 loss events. PERILS reports these events because each produced more than a EUR 200mn multi-country market loss. A simple sum of gusts across zones reaches 0.7293. Weighting gusts by each zone's exposure share, normalized so the weights sum to one, lifts the result to 0.7390.
The third alternative multiplies total industry exposure by a power of Germany's single maximum gust. It performs worst at 0.6923. Parameters are fitted on the same sample by regressing log(L_i/E_i) on the logarithm of the maximum gust (log k1 = -35.416, standard error 3.885; k2 = 6.953, standard error 1.093). The first two alternatives resemble earlier European windstorm parametric deals, including Pylon and Green Valley.
The convexity carries the result.
An exponent of 7.8 is far above the cubic scaling suggested by kinetic energy dissipation. It falls just below the 8 to 12 range that the paper cites from Prahl et al. for personal lines. Ery and Koch search for the exponent on a grid using the same 20 losses behind the reported 0.9307, a choice they disclose.
They address part of that dependence with ten-fold cross-validation, leaving out two storms in each fold. The optimum returns to 7.8 in four folds. It is 7.6 in three, 7.9 in two and 8.1 in one. Slope coefficients run from 1.383e-16 to 1.507e-16, compared with 1.442e-16 for the full-sample fit. The exponent looks stable. Even so, the headline 0.9307 remains an in-sample result, while the max-stable family and its correlation function were selected from the same 100 events.
Xavier high, Friederike low
Two of the 20 loss events expose misses in opposite directions. Xavier, in October 2017, had a peak gust of 39.95 m/s and an adjusted industry loss of EUR 378.847mn. Friederike, in January 2018, peaked at 37.23 m/s yet produced an adjusted loss of EUR 1,264.908mn. Their exposure bases were similar, at 14.580tn versus 15.457tn.
The trigger overshoots Xavier and undershoots Friederike. Ery and Koch connect the first miss to the speed of Xavier's passage across the region, and the second to Friederike's track through larger parts of Germany. Wind speed is the hazard component's sole input. The paper says that capturing such events "would require adjusting the trigger to include more than just the wind speed as input variable in the physical hazard component". With an exponent of 7.8, a 7% difference in peak gust becomes a 73% difference in modeled damage.
The full-model check makes this weakness matter. The paper says both the trigger and the benchmark "underestimate losses in the right tail of the distribution," precisely where attachment and exhaustion are set. The comparison has 22 years of annual trigger realizations. According to the authors, the confidence bounds are "so large that the empirical quantiles still fall within the 95% confidence interval." They also write that "owing to the lack of data, we were not able to perform an out-of-sample validation." Only the damage exponent receives the ten-fold treatment.
Pricing a 7% spread at a 1% attachment
In the pricing example, tranche A has a modeled 100-year attachment of EUR 7.861bn and exhausts at the 200-year EUR 11.008bn. The 7% spread is supplied exogenously. Lambda, the Wang skewness parameter, is solved so tranche A prices at 100.00 per 100 of face. Tranche B attaches at 9.593bn and prices at 100.69, equivalent to 6.28% at par. Because Lambda is chosen to force tranche A to par, its proximity to the 0.7 estimated by Galeotti et al. from historical spreads is coincidence.
A modeled 100-year attachment and 200-year exhaustion imply annual exceedance probabilities of 1% and 0.5%. The layer's expected loss is therefore bounded between 0.5% and 1% of limit. Against that range, a 7% spread represents a multiple somewhere between 7 and 14. The paper says the selected spread "is in line with the spread for outstanding cat bonds covering European windstorm" without quantifying the comparison. Since the spread enters as an input, the exercise values a spread already given.
The appendix calibration changes both the hazard source and the fitting window. PERILS/UKMO per-event zone gusts are replaced with ERA5 hourly reanalysis gusts at 0.25 degree resolution from 1979 to 2020. Monthly maxima are then fitted over October to March and averaged within the 89 CRESTA zones that contain at least one grid point. This calibration selects extremal-t with a Whittle-Matérn correlation.
Under that version, tranche A prices at 102.84. The main fit gives 100.00, while Brown-Resnick gives 100.11. The two calibrations differ by 2.84 points per 100 of face. By comparison, tranche A at 100.00 and tranche B at 100.69 are separated by 0.69 points.
We could not reproduce any of this. PERILS licenses the per-zone gust series, industry loss records and exposure database. We also lack a cat bond price or contract terms for comparison. Listed equities, futures and options provide no substitute for a German windstorm footprint.
Attachment and exhaustion are based on German industry losses across all lines and coverages, an arrangement the authors describe as essentially an aggregate industry loss warranty. The 0.9307 result compares the trigger with an industry index. A sponsor must still translate that index to its own book through market-share factors by zone, peril and line. The resulting second-stage error is not measured here. A sponsor-level version of the regression that held near 0.9 would change my assessment of the construction's value.