Unquantified dependence structure of the coupled earthquake-flood loss model underlying a public-private insurance scheme for Italian residential buildings

Perazzini, Gnecco and Pammolli proposed a public-private insurance scheme for seismic and flood risk on the Italian residential building stock in which multi-hazard policies are priced under three prudential but unverified dependence assumptions: earthquake and flood occurrences are treated as independent; the combined vulnerability of a building stricken by both perils is unknown, so multi-hazard expected loss is set equal to the sum of single-peril losses; and spatial dependence among municipal losses is bounded conservatively by grouping municipalities at least 50 km apart and applying Hoeffding's inequality for (r-1)-dependent variables. No work has tested seismic-flood statistical dependence in Italy, estimated a combined vulnerability for Italian residential typologies, or quantified the spatial dependence range of this portfolio; methodological templates exist only for other regions or other hazard pairs. The open problem is to quantify the inter-peril, vulnerability-interaction, and spatial components of the dependence structure of the coupled Italian earthquake-flood loss model and to determine the consequences for premiums, required public capital, and insolvency probabilities in the insurance scheme.

Background

Italy accumulates large losses from two natural perils with very different geographic footprints: earthquakes, whose hazard is described by peak ground acceleration (PGA, the maximum ground shaking amplitude recorded at a site during a quake) over most of the national territory, and river floods, which concern about two thirds of the national territory (versus the near-nationwide seismic exposure). Because only a small fraction of the Italian housing stock is insured against these perils, the financial burden of disasters falls largely on the government and on households. Perazzini, Gnecco and Pammolli (2020, published in journal form in 2022) studied a public-private insurance scheme in which property owners buy policies from an insurer that maximizes social welfare rather than profit, and the government backs the scheme by committing public capital $W_d$ to a fund that refills the insurer's reserves. Expected monetary losses per square metre, $l_j^h$, are estimated separately for each hazard $h$ and municipality $j$ by catastrophe risk modeling, i.e. by combining hazard (probability of an event of intensity $\zeta^h$), exposure (building stock and reconstruction cost), and vulnerability (a function $v^h(\zeta^h)$ giving the expected damage fraction at intensity $\zeta^h$; for floods the intensity is water depth, fitted in the source paper by a Gamma distribution chosen over statistically better-fitting generalized Gamma and generalized Beta alternatives for computational convenience). Homeowners are modeled as risk-averse expected-utility maximizers with utility $u(x)=\log(x+1)$, which pins down the maximum premium they will pay; the insurer is subject to a solvency constraint on the probability of default and of needing a further capital injection.

Two dependence problems make the insurer's aggregate loss distribution nontrivial. First, losses across municipalities are spatially correlated because a single event damages many nearby buildings. The source paper assumes there exists a distance $r$ beyond which municipal losses are independent, sets $r=50$ km as a precaution (in the 2020 technical report for all perils; the 2022 journal version retains 50 km for earthquakes but raises the flood and multi-hazard grouping to about $150$ km, the 98th percentile of inter-municipality flood distances), and partitions the covered municipalities (peril-specific portfolios of 6404 seismic, 7772 flood and 6217 multi-hazard municipalities) into groups whose members are at least $r$ km apart, and bounds the probability that the national claim amount exceeds a threshold using the Hoeffding (1963) exponential concentration bound for $(r-1)$-dependent sequences (sequences in which variables more than $r-1$ apart in the ordering are independent). This bound is distribution-free and therefore conservative, and the true spatial dependence range of Italian losses was left unquantified. Second, when the two perils are managed jointly, the losses interact. Earthquakes and floods are plausibly uncorrelated, but the source paper could verify this only by reference to pre-2020 studies (a European hazard-interaction mapping and an Italian insurance-coverage report), not from its own data. Following an actuarial multi-hazard model for Slovakian forests (Brunette et al. 2015), the paper distinguishes a mutual-exclusion scenario, in which the two perils never strike simultaneously and multi-hazard expected loss per square metre is exactly additive, $l^{MH}=l^s+l^f$, from a mutual-independence scenario, in which simultaneous occurrence is possible and the damage to a building hit by both an earthquake of intensity $\zeta^s$ and a flood of depth $\zeta^f$ is governed by an unknown combined vulnerability function $v^{s+f}(\zeta^s,\zeta^f)$. Writing the multi-hazard loss in the independence scenario gives $l^{MH}_{ind}=l^s+l^f+F^s(\zeta^s)F^f(\zeta^f)\,[v^{s+f}(\zeta^s,\zeta^f)-v^s(\zeta^s)-v^f(\zeta^f)]$, where $F^h$ is the event probability, so the two scenarios agree only if the interaction term vanishes. The authors state they are unable to define $v^{s+f}$ or even to determine its sign relative to $v^s+v^f$, and therefore adopt $l^{MH}=l^s+l^f$ as a prudential simplification.

Under these assumptions the model reaches policy-relevant conclusions: seismic expected losses at national level are about six times flood losses, yet the public capital needed to manage the two perils is almost the same; joint management of the two perils lowers the required public capital (a diversification benefit, since independent risks are being pooled) and makes premiums more geographically homogeneous; but neither single- nor multi-hazard policies are sustainable for a private insurer alone, because spatial correlation drives the capital requirement above what individuals are willing to pay. These conclusions inherit their quantitative content from the unverified dependence assumptions: if seismic and flood losses are positively dependent, the diversification benefit shrinks; if $v^{s+f}$ exceeds the sum of the single-peril vulnerabilities, the additive simplification understates expected multi-hazard loss and the computed insolvency probabilities are optimistic rather than prudential; and the conservative Hoeffding grouping may overstate the public capital requirement, so the case for government involvement rests partly on an uncalibrated bound.

The multi-hazard context is an established research field: Kappes et al. (2012) reviewed the general challenges of multi-hazard risk analysis, and Marzocchi et al. (2012) performed one of the first multi-risk assessments in Italy, though at the scale of a single municipality and for societal rather than financial risk. What remained open after the source paper is the dependence-aware calibration of the Italian earthquake-flood insurance model: an empirical test of inter-peril independence, an estimate or bound of the combined vulnerability for Italian residential typologies, and a quantification of the spatial dependence range, together with their consequences for the scheme's outputs.

Problem Statement

For the public-private insurance model of seismic and flood risk on the Italian residential building stock of Perazzini, Gnecco and Pammolli, quantify the dependence structure of the coupled multi-hazard loss model that the scheme's three prudential assumptions currently stand in for, and determine its consequences for the scheme's outputs. Concretely: (1) test and quantify, or bound, the statistical dependence between earthquake and flood occurrence and losses in Italy, replacing the unvalidated independence assumption (including the mutual-exclusion versus mutual-independence scenario distinction and the role of any physical coupling such as coseismic hydrological changes or earthquake-induced landslide damming); (2) estimate or rigorously bound the combined vulnerability function $v^{s+f}(\zeta^s,\zeta^f)$ for Italian residential structural typologies, replacing the additive-loss simplification $l^{MH}=l^s+l^f$ and establishing whether co-occurrence interaction amplifies or mitigates damage relative to additivity; (3) estimate the spatial dependence range of municipal losses and its effect on the tail of the insurer's aggregate claim distribution, replacing or tightening the conservative Hoeffding grouping at $r=50$ km (about $150$ km for floods and multi-hazard in the 2022 journal version). All three components are required, and they serve one overarching objective: a dependence-aware recalibration of the insurance scheme. The answer must therefore also re-evaluate, under the estimated dependence structure, the model's policy-relevant outputs, namely risk-based premiums, the required government capital $W_d$, and the insolvency and fund-refill probabilities for single- and multi-hazard policies, establishing whether the original conclusions (diversification benefit from joint management; unsustainability of purely private provision) survive.

References

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