Matthias Scherer

Payments in parametric insurance are linked to an index and thus decoupled from policyholders’ true losses. While this principle has appealing operational benefits compared to indemnity coverage, i.e. being efficient and cost effective, a downside is the discrepancy between payouts and actual damage, called basis risk. We show that in an asymmetrically weighted mean square error framework, the basis risk-minimizing payment schemes for pure parametric and parametric index insurance contracts can be expressed as conditional expectiles of policyholders’ true loss given a compensation-triggering incident. We provide connections to stochastic orderings and demonstrate that regression approaches allow easy implementation in practice. The results are visualized in parametric coverage for cyber risks and agricultural insurance.

We demonstrate how the optimal portfolio, with respect to expected power utility, can numerically be derived if the underlying price processes are driven by geometric Lévy processes. According to a result by Nutz [ 26 ], the optimal portfolio weights are constant in this setting, both in the unrestricted case and if budget constraints are included. We make use of the Stochastic Gradient Descent Algorithm to numerically arrive at the optimal solution, even for a high-dimensional asset universe. We then showcase the approach, exploring a multivariate Merton model and a multivariate version of the Kou model, in which we investigate how the jumps and the dependence between them influence the optimal portfolio. The solution is compared to the standard Merton portfolio that is optimal in a multivariate Black–Scholes market. Finally, the presented methodology gives immediate access to the indifference pricing measure for the specific utility of our investor, and we demonstrate how to numerically obtain the corresponding utility-indifference price of non-vanilla derivatives within our framework.

Many 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.

A 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.

As 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.

A vector of bankruptcy times with Marshall–Olkin multivariate exponential distribution implies a simple, yet reasonable, continuous-time model for dependent credit-risky assets with an appealing trade-off between tractability and realism. Within this framework, the maximization of expected power utility of terminal wealth requires the optimization of a concave function on a polygon, a numerical problem whose complexity grows exponentially in the number of considered assets. We demonstrate how this seemingly impractical numerical problem can be solved reliably and efficiently in order to prepare the model for practical use cases. To this end, we resort to a specifically designed factor construction for the Marshall–Olkin distribution that separates dependence parameters from idiosyncratic parameters, and we develop a tailor-made stochastic gradient descent algorithm with random constraint projections for the model's numerical implementation. Finally, we explain a new method to include transaction costs and apply the model in a real-world, high-dimensional example.

Article When copulas and smoothing met: An interview with Irène Gijbels was published on January 1, 2023 in the journal Dependence Modeling (volume 11, issue 1).

We establish a one-to-one correspondence between (i) exchangeable sequences of random variables whose finite-dimensional distributions are minimum (or maximum) infinitely divisible and (ii) non-negative, non-decreasing, infinitely divisible stochastic processes. The exponent measure of an exchangeable minimum infinitely divisible sequence is shown to be the sum of a very simple “drift measure” and a mixture of product probability measures, which uniquely corresponds to the Lévy measure of a non-negative and non-decreasing infinitely divisible process. The latter is shown to be supported on non-negative and non-decreasing functions. In probabilistic terms, the aforementioned infinitely divisible process is equal to the conditional cumulative hazard process associated with the exchangeable sequence of random variables with minimum (or maximum) infinitely divisible marginals. Our results provide an analytic umbrella which embeds the de Finetti subfamilies of many interesting classes of multivariate distributions, such as exogenous shock models, exponential and geometric laws with lack-of-memory property, min-stable multivariate exponential and extreme-value distributions, as well as reciprocal Archimedean copulas with completely monotone generator and Archimedean copulas with log-completely monotone generator.