# Hazard Model: Epidemic-Type Aftershock Sequence (ETAS) Model for Hungary
## 1 Introduction  
### Motivation  
The origin of the study presented in this article is an ongoing effort by the UNIQA Insurance Group (www.uniqagroup.com, accessed on 5 February 2023) to build a proprietary earthquake model for Hungary. Earthquake models in insurance are used to measure the risk of a set of objects, such as buildings, civil engineering structures, and means of transport, with a focus on assessing the potential losses caused by severe earthquakes.  
One key application of an earthquake model or, more generally, a natural catastrophe model in insurance is the determination of capital requirements. According to the Solvency II regulatory standards applicable in European Union member countries [1,2], the solvency capital requirement (SCR) of an insurance company shall correspond to the 200-year loss; that is, the insurer must hold their own funds to cover the risk of a loss whose probability of occurrence within a year is 0.5%. The SCR is calculated either by the Solvency II standard formula or by an internal model. For natural perils, the standard formula is a deterministic approximation of the risk based on fixed factors applied to insured amounts of the objects by geographical area. On the other hand, an internal model is based on the stochastic simulation of the peril, and its output is not just an SCR single value but a full probability distribution forecast of the loss. The use of an internal model by an insurance company for calculating the SCR is subject to case-by-case regulatory approval by the competent supervisory authorities, including a comprehensive and lengthy validation process. The development of an internal model requires a significant effort; however, the benefits includea more accurate and more detailed evaluation of the risk, and a range of model uses in risk management tasks beyond the calculation of the SCR. Another typical and important use of stochastic catastrophe models is the optimization and pricing of protection covers for the purpose of risk transfer. Reinsurance risk transfer is an instrument for risk mitigation, and it can reduce the SCR of the ceding insurer.  
The general architecture of a stochastic earthquake model for insurance is the following: (1) the first fundamental module is a statistical model of earthquake occurrences, which is used to generate a synthetic event catalogue via stochastic simulation. This is followed by two equally important ones: (2) the ground motion attenuation model, translating the catalogue into event footprints, and (3) the vulnerability model, translating the event footprints into the loss of the insured objects in financial terms. All three modules require a full study of their own. In this article, only the first module is discussed, that is, a model of earthquake occurrences.  
Earthquake models used in the field of insurance typically focus on the mainshocks only. This simplifies the modelling effort drastically since the mainshocks are assumed to follow a stationary process. This simplification is justified if aftershock losses are low compared to the mainshock loss. Furthermore, customer claims arising from aftershock losses are often not reported separately from the mainshock; therefore, it is a straightforward approach to model them only implicitly via conservative parameterization. Nevertheless, the recent nearby Petrinja 2020 event in Croatia and its aftershocks highlighted some properties of earthquake sequences [3], suggesting that clustering cannot necessarily be safely ignored even in a Central European model: aftershock activity after such a major event may last for months or even years, and some aftershocks (or foreshocks) are significant events on their own. Moreover, there are historical records of complex earthquake sequences (e.g., Romania 1991) lasting for several months and including multiple significant shocks [4]. Large events are still possible late in the sequence, and large aftershocks can trigger second-generation sequences, epidemic style.