GNGTS 2021 - Atti del 39° Convegno Nazionale

285 GNGTS 2021 S essione 2.2 or part of it, from insolvency risk. The definition of this region has a strong implication on the pricing process, since distribution parameters are calibrated from events occurred or simulated in this specific area. Secondly in this first stage, target losses covered with CAT bonds have to be defined. 2. CAT bond zonation : when the portfolio is significantly scattered over a wide region, different risk levels can be observed within the same region. For this reason, a common practice is to tailor CAT bonds associated to different risk levels, in order to meet the needs of different types of investors, via the subdivision of the region of interest in smaller zones. A region with high-impact and frequent events leads to calibrating high- risk CAT bonds with related high gains for risk-seeking investors; on the contrary, a zone with rare and lowly impacting losses leads to low-risk CAT bonds, more attractive for risk-averse investors. 3. Calibration of distribution parameters : the third step consists in the computation of the Poisson process and loss distribution parameters, which are at the base of the mathematical procedure for computing first the default probability and then the CAT bond price. Regarding the loss distribution, rarely enough historical data of extreme events are available, and thus computer simulations are needed to predict potential losses that can arise for the portfolio of interest. 4. CAT bond price computation : lastly, CAT bond price is computed. Among the most common techniques, stochastic processes are adopted for CAT bond pricing; in this case, one common method is to model the credit default probability which follows the way of pricing credit derivatives in finance, and to assume the time to be continuous. The catastrophe process is thus modelled as a compound doubly stochastic Poisson process, where the potentially catastrophic events follow a doubly stochastic Poisson process, and the associated losses are assumed independent and generated from a common probability distribution. The CAT bond’s default occurs when the accumulated losses L(t) exceed the money threshold level D before the expiration time T . Under these assumptions, the price for zero-coupon V ZC (i.e. debt security that does not pay interest but renders profit only at maturity) and coupon V C CAT bond (i.e. debt security that includes attached coupons and pays periodic interest payments during its lifetime and its nominal value at maturity), can be computed as discounted expected value of the future payoff. The general mathematical formulation is detailed in Hofer et al. 2019. Case study The exposed framework is applied to design a coverage scheme for the entire residential building asset of Italy considering seismic events as relevant natural hazard. In this application, the Italian Government is taken as the issuing entity, which adopts CAT bonds for a full risk-transfer, considering as lower bound seismic events with magnitude M ≥ 4.5. The region of interest is represented by the Italian peninsula, and the target losses are represented by the potential direct costs to be sustained for repairing seismic damage to the Italian residential building stock. First, Italy is divided in three zones based on the seismic risk maps developed by Zanini et al. 2019. This zonation (Fig. 1a), based on the seismic risk map and adopting administrative borders, assures an almost constant combination of events frequencies and amount of losses within each zone, and the exact attribution of each event to the corresponding zone. The calibration of the Poisson process and loss distribution parameters is based on the numerical simulation of 100’000 years of seismicity within the national territory, because of the limited number of real losses and claim data. For the generation of 100’000 years of seismic events, the seismogenic source zone model ZS9 of Meletti et al. 2008 is adopted, together with the seismogenic zone parameters of Barani et