Introduction to actuarial mathematics and quantitative risk management

• Stéphane Loisel (CNAM)

Room: Amphithéâtre Yvonne Choquet-Bruhat In this talk, we start by explaining the basics of actuarial mathematics and quantitative risk management. We then provide a few results concerning risk theory, quickest detection problems as well as impact of climate change on insurance risks.

Cyber risk frequency modeling using Hawkes process with external excitation

• Yousra Cherkaoui Tangi (CREST-ENSAE - Milliman France)

Room: Amphithéâtre Yvonne Choquet-Bruhat “Modeling cyber attack frequency and spread is crucial for the insurance industry to evaluate risk and adapt mitigation strategies. Hawkes processes have been widely adopted in the cyber literature for their ability to model self-exciting behavior of cyber attacks. However, such a model, in its standard linear form, may seem limited to account for external factors driving the dynamics of cyber attacks, such as vulnerabilities which, once discovered, can trigger cascading attacks. We therefore consider a larger class of Hawkes processes by adding external excitation to capture vulnerabilities discoveries and stochastic marks to account for the magnitude and variability of their impacts on the contagion dynamics. We develop and compare parametric estimation methods for the externally excited marked Hawkes process for cyber attack modeling without resorting to stationarity assumptions. We tackle the likelihood estimation and expectation calculation for both exponential and Erlang kernels, the latter being key to account for delays in the contagion mechanism. We test estimation methods, comparing deterministic and Monte Carlo likelihoods. Simulations show that the deterministic method performs well in low variance settings, while the Monte Carlo one excels in high variance, emphasizing the need to choose suitable calibration techniques based on the underlying contagion variability. We compare different methods to tackle the inference problem based on public datasets containing features of cyber attacks found in the Hackmageddon database and cyber vulnerabilities from the Known Exploited Vulnerability database and the National Vulnerability Dataset. By refining the external excitation database selection, the degree of endogeneity of the model is nearly halved.”

An optimal Index insurance framework for extreme losses

• Daniel Nkameni (CREST ENSAE (Polytechnic Institute of Paris), Insurance and Finance, France and Detralytics France)

Room: Amphithéâtre Yvonne Choquet-Bruhat The modern insurance landscape is increasingly shaped by the emergence of new risks with destructive potential, challenging the feasibility of developing sustainable financial protection mechanisms. Many administrations and regulators frequently emphasize the potential uninsurability of certain risks, such as those stemming from natural disasters linked to climate change or cyber-attacks associated with rapidly advancing artificial intelligence. To address these challenges, index-based insurance is often highlighted as a technical tool capable of providing coverage in situations that might otherwise seem intractable. This paper proposes the design of index insurance coverage for extreme losses, focusing specifically on cases where these losses exhibit heavy-tailed distributions. The index coverage is structured to activate above a predetermined threshold, while traditional indemnity-based insurance remains in effect below this threshold. This approach seeks to combine the advantages of both coverage types. The proposed methodology begins by adapting the standard utility of wealth framework to accommodate heavy-tailed losses. An approximation of this utility is then developed to account for the scarcity of economic loss data that is often encountered in practice. Finally, the study concludes with an empirical verification, demonstrating that the proposed insurance framework remains effective even in scenarios of limited loss data, provided sufficient data on the index is available.

A compound Hawkes process with dependencies for insurance applications

• Thomas Peyrat (Institut de Mathématique de Toulouse - CREST-ENSAE - Exiom Partners)

Room: Amphithéâtre Yvonne Choquet-Bruhat In insurance, one of the processes commonly used to model risk is the compound Poisson process. Assuming independence between the counting process (Poisson Process) and the claims (independent and identically distributed random variables), the calculation of the first two cumulants is immediate. However, these assumptions limit the scope of application of this process to certain risks. In order to remove some of these assumptions, we will present a similar model in which the counting process is a Hawkes process whose intensity is impacted by claims. We will also briefly present a methodology for calculating the first two cumulants of this process.