An identity of hitting times and its application to the valuation of guaranteed minimum withdrawal benefit

In this paper we explore an identity in distribution of hitting times of a finite variation process (Yor's process) and a diffusion process (geometric Brownian motion with affine drift), which arise from various applications in financial mathematics. As a result, we provide analytical solutions to the fair charge of variable annuity guaranteed minimum withdrawal benefit (GMWB) from a policyholder's point of view, which was only previously obtained in the literature by numerical methods. We also use complex inversion methods to derive analytical solutions to the fair charge of the GMWB from an insurer's point of view, which is used in the market practice, however, based on Monte Carlo simulations. Despite of their seemingly different formulations, we can prove under certain assumptions the two pricing approaches are equivalent.Comment: 25 pages, 2 figure

A Generalization of the Discounted Penalty Function in Ruin Theory

As ruin theory evolves in recent years, there has been a variety of quantities pertaining to an insurer's bankruptcy at the centre of focus in the literature. Despite the fact that these quantities are distinct from each other, it was brought to our attention that many solution methods apply to nearly all ruin-related quantities. Such a peculiar similarity among their solution methods inspired us to search for a general form that reconciles those seemingly different ruin-related quantities. The stochastic approach proposed in the thesis addresses such issues and contributes to the current literature in three major directions. (1) It provides a new function that unifies many existing ruin-related quantities and that produces more new quantities of potential use in both practice and academia. (2) It applies generally to a vast majority of risk processes and permits the consideration of combined effects of investment strategies, policy modifications, etc, which were either impossible or difficult tasks using traditional approaches. (3) It gives a shortcut to the derivation of intermediate solution equations. In addition to the efficiency, the new approach also leads to a standardized procedure to cope with various situations. The thesis covers a wide range of ruin-related and financial topics while developing the unifying stochastic approach. Not only does it attempt to provide insights into the unification of quantities in ruin theory, the thesis also seeks to extend its applications in other related areas

Actuarial Applications of Epidemiological Models

The risk of a global avian flu or influenza A (H1N1) pandemic, and the emergence of the worldwide SARS epidemic in 2002–03 have led to a revived interest in the study of infectious diseases. Mathematical models have become important tools in analyzing the transmission dynamics and in measuring the effectiveness of controlling strategies. Research on infectious diseases in the actuarial literature only goes so far as to set up epidemiological models which better reflect the transmission dynamics. This paper attempts to build a bridge between epidemiological and actuarial modeling and set up an actuarial model which provides financial arrangements to cover the expenses resulting from the medical treatments of infectious diseases. Based on classical epidemiological compartment models, the first part of this paper proposes insurance policies and models to quantify the risk of infection and formulates financial arrangements, between an insurer and insureds, using actuarial methodology. For practical purposes, the second part employs a variety of numerical methods to calculate premiums and reserves. The last part illustrates the methods by designing insurance products for two well known epidemics: the Great Plague in England and the SARS epidemic in Hong Kong

Cyber Risk Assessment for Capital Management

Cyber risk is an omnipresent risk in the increasingly digitized world that is known to be difficult to manage. This paper proposes a two-pillar cyber risk management framework to address such difficulty. The first pillar, cyber risk assessment, blends the frequency-severity model in insurance with the cascade model in cybersecurity, to capture the unique feature of cyber risk. The second pillar, cyber capital management, provides informative decision-making on a balanced cyber risk management strategy, which includes cybersecurity investments, insurance coverage, and reserves. This framework is demonstrated by a case study based on a historical cyber incident dataset, which shows that a comprehensive cost-benefit analysis is necessary for a budget-constrained company with competing objectives for cyber risk management. Sensitivity analysis also illustrates that the best strategy depends on various factors, such as the amount of cybersecurity investments and the effectiveness of cybersecurity controls.Comment: This paper was first presented on July 5, 2021, at the 24th International Congress on Insurance: Mathematics and Economic

Optimal Dividend Payments for the Piecewise-Deterministic Poisson Risk Model

This paper considers the optimal dividend payment problem in piecewise-deterministic compound Poisson risk models. The objective is to maximize the expected discounted dividend payout up to the time of ruin. We provide a comparative study in this general framework of both restricted and unrestricted payment schemes, which were only previously treated separately in certain special cases of risk models in the literature. In the case of restricted payment scheme, the value function is shown to be a classical solution of the corresponding HJB equation, which in turn leads to an optimal restricted payment policy known as the threshold strategy. In the case of unrestricted payment scheme, by solving the associated integro-differential quasi-variational inequality, we obtain the value function as well as an optimal unrestricted dividend payment scheme known as the barrier strategy. When claim sizes are exponentially distributed, we provide easily verifiable conditions under which the threshold and barrier strategies are optimal restricted and unrestricted dividend payment policies, respectively. The main results are illustrated with several examples, including a new example concerning regressive growth rates.Comment: Key Words: Piecewise-deterministic compound Poisson model, optimal stochastic control, HJB equation, quasi-variational inequality, threshold strategy, barrier strateg

Privacy-Enhancing Collaborative Information Sharing through Federated Learning -- A Case of the Insurance Industry

The report demonstrates the benefits (in terms of improved claims loss modeling) of harnessing the value of Federated Learning (FL) to learn a single model across multiple insurance industry datasets without requiring the datasets themselves to be shared from one company to another. The application of FL addresses two of the most pressing concerns: limited data volume and data variety, which are caused by privacy concerns, the rarity of claim events, the lack of informative rating factors, etc.. During each round of FL, collaborators compute improvements on the model using their local private data, and these insights are combined to update a global model. Such aggregation of insights allows for an increase to the effectiveness in forecasting claims losses compared to models individually trained at each collaborator. Critically, this approach enables machine learning collaboration without the need for raw data to leave the compute infrastructure of each respective data owner. Additionally, the open-source framework, OpenFL, that is used in our experiments is designed so that it can be run using confidential computing as well as with additional algorithmic protections against leakage of information via the shared model updates. In such a way, FL is implemented as a privacy-enhancing collaborative learning technique that addresses the challenges posed by the sensitivity and privacy of data in traditional machine learning solutions. This paper's application of FL can also be expanded to other areas including fraud detection, catastrophe modeling, etc., that have a similar need to incorporate data privacy into machine learning collaborations. Our framework and empirical results provide a foundation for future collaborations among insurers, regulators, academic researchers, and InsurTech experts