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AbstractMany insurers have started to underwrite cyber in recent years. In parallel, they developed their first actuarial models to cope with this new type of risk. On the portfolio level, two major challenges hereby are the adequate modelling of the dependence structure among cyber losses and the lack of suitable data based on which the model is calibrated. The purpose of this article is to highlight the importance of taking a holistic approach to cyber. In particular, we argue that actuarial modelling should not be viewed stand-alone, but rather as an integral part of an interconnected value chain with other processes such as cyber-risk assessment and cyber-claims settlement. We illustrate that otherwise, i.e. if these data-collection processes are not aligned with the actuarial (dependence) model, naïve data collection necessarily leads to a dangerous underestimation of accumulation risk. We illustrate the detrimental effects on the assessment of the dependence structure and portfolio risk by using a simple mathematical model for dependence through common vulnerabilities. The study concludes by highlighting the practical implications for insurers.

AbstractAs the cyber insurance market is expanding and cyber insurance policies continue to mature, the potential of including pre-incident and post-incident services into cyber policies is being recognised by insurers and insurance buyers. This work addresses the question of how such services should be priced from the insurer's viewpoint, i.e. under which conditions it is rational for a profit-maximising, risk-neutral or risk-averse insurer to share the costs of providing risk mitigation services. The interaction between insurance buyer and seller is modelled as a Stackelberg game, where both parties use distortion risk measures to model their individual risk aversion. After linking the notions of pre-incident and post-incident services to the concepts of self-protection and self-insurance, we show that when pricing a single contract, the insurer would always shift the full cost of self-protection services to the insured; however, this does not generally hold for the pricing of self-insurance services or when taking a portfolio viewpoint. We illustrate the latter statement using toy examples of risks with dependence mechanisms representative in the cyber context.

AbstractAfter scrutinizing technical, legal, financial, and actuarial aspects of cyber risk, a new approach for modelling cyber risk using marked point processes is proposed. Key covariates, required to model frequency and severity of cyber claims, are identified. The presented framework explicitly takes into account incidents from malicious untargeted and targeted attacks as well as accidents and failures. The resulting model is able to include the dynamic nature of cyber risk, while capturing accumulation risk in a realistic way. The model is studied with respect to its statistical properties and applied to the pricing of cyber insurance and risk measurement. The results are illustrated in a simulation study.

AbstractEstablishing a standard formula (SF) for the regulation of European insurance companies is a Herculean task. It has to acknowledge very different business models and national peculiarities. In addition, regulatory authorities—as a stakeholder on their own—have a number of supervisory objectives the SF should incentivize. With the intervention of the SF in economic activities, the principle of equal treatment must be maintained. The large circle of users makes its procedural simplicity indispensable to ensure that it is applied and implemented in a proportionate manner. Above all, the SF should be risk-sensitive. Compared to Solvency I, the SF of Solvency II is considered a significant improvement, as many of the aforementioned desiderata have been much better realized. The following analysis and survey of model-theoretical aspects of the SF shows that these improvements could be achieved above all with regard to epistemic uncertainties. The stochastic model underneath the SF is still subject to considerable uncertainties; so that the probability functional of the SF is exposed to significant model risk. As part of the Own Risk and Solvency Assessment (ORSA), insurance companies must prove the adequacy of the SF for their company. The vague prior knowledge represented by the stochastic component of the SF is not sufficient for an SF intrinsic validation of the aleatoric component.

This is a succinct guide to the application and modelling of dependence models or copulas in the financial markets. First applied to credit risk modelling, copulas are now widely used across a range of derivatives transactions, asset pricing techniques and risk models and are a core part of the financial engineer's toolkit.

AbstractA Neural Network (NN) approach for the modelling of mortality rates in a multi-population framework is compared to three classical mortality models. The NN setup contains two instances of Recurrent NNs, including Long Short-Term Memory (LSTM) and Gated Recurrent Units (GRU) networks. The stochastic approaches comprise the Li and Lee model, the Common Age Effect model of Kleinow, and the model of Plat. All models are applied and compared in a large case study on decades of data of the Italian population as divided in counties. In this case study, a new index of multiple deprivation is introduced and used to classify all Italian counties based on socio-economic indicators, sourced from the local office of national statistics (ISTAT). The aforementioned models are then used to model and predict mortality rates of groups of different socio-economic characteristics, sex, and age.