Abstract Assessing systemic risk presents a significant challenge in finance and insurance, where conditional risk measures are essential for capturing contagion effects. This paper introduces two novel systemic risk measures – conditional interval value-at-risk (CoIVaR) and conditional interval expected shortfall (CoIES) – which extend traditional metrics by incorporating interval-based uncertainty. A formal theoretical framework is developed for both measures, offering a detailed characterization of their key properties and risk contributions. We then propose a comprehensive comparison methodology for systemic risk assessment, leveraging stochastic orders, dependence structures, and marginal distributions to establish conditions for ranking risk vectors. Finally, through numerical experiments and real-world stock market applications, we demonstrate the practical utility of CoIVaR and CoIES in quantifying systemic risk under uncertainty. The findings provide valuable insights into systemic risk propagation and establish a robust foundation for risk management in interconnected financial systems.

Abstract We revisit the recently introduced concept of return risk measures (RRMs) and extend it by incorporating risk management via multiple so-called eligible assets. The resulting new class of risk measures, termed multi-asset return risk measures (MARRMs), introduces a novel economic model for multiplicative risk sharing. We point out the connection between MARRMs and the well-known concept of multi-asset risk measures (MARMs). Then, we conduct a case study, based on an insurance dataset, in which we use typical continuous-time financial markets and different notions of acceptability of losses to compare RRMs, MARMs, and MARRMs and draw conclusions about the cost of risk mitigation. Moreover, we analyze theoretical properties of MARRMs. In particular, we prove that a positively homogeneous MARRM is quasi-convex if and only if it is convex, and we provide conditions to avoid inconsistent risk evaluations. Finally, the representation of MARRMs via MARMs is used to obtain various dual representations.

Abstract In today’s insurance market, numerous cyber insurance products provide bundled coverage for losses resulting from different cyber events, including data breaches and ransomware attacks. Every category of incident has its own specific coverage limit and deductible. Although this gives prospective cyber insurance buyers more flexibility in customizing the coverage and better manages the risk exposures of sellers, it complicates the decision-making process in determining the optimal amount of risks to retain and transfer for both parties. This article aims to build an economic foundation for these incident-specific cyber insurance products with a focus on how incident-specific indemnities should be designed for achieving Pareto optimality for both the insurance seller and the buyer. Real data on cyber incidents are used to illustrate the feasibility of this approach. Several implementation improvement methods for practicality are also discussed.

Abstract The additive reserving model assumes the existence of volume measures such that the corresponding expected loss ratios are identical for all accident years. While classical literature assumes these volumes are known, in practice, accurate volume measures are often unavailable. The issue of uncertain volume measures in the additive model was addressed in a generalization of the loss ratio method published in 2018. The derivation is rather complex and the method is computationally intensive, especially for large loss development triangles. This paper introduces an alternative approach that leverages the well-established EM algorithm, significantly reducing computational requirements.

Abstract In this paper, we explore the optimal risk sharing problem in the context of peer-to-peer insurance. Using the criterion of minimizing total variance, we find that the optimal risk sharing strategy should take a linear form. Although linear risk sharing strategies have been examined in the literature, our study uncovers a significant finding: to minimize total variance, the linear strategy should be applied to the residual risks rather than the original risks, as commonly adopted in existing studies. By comparing with the existing models, we demonstrate the advantage of the linear residual risk sharing model in variance reduction and robustness. Furthermore, we develop and study a number of new models by incorporating some constraints, to reflect desirable properties required by the market. With those constraints, the optimal strategies turn out to favor market development, such as incentivize participation and guarantee fairness. A relevant model is considered at last, which establishes the connection among multiple optimization problems and provides insights on how to extend the models into a more general setup.

Abstract The distribution-free chain ladder of Mack justified the use of the chain ladder predictor and enabled Mack to derive an estimator of conditional mean squared error of prediction for the chain ladder predictor. Classical insurance loss models, that is of compound Poisson type, are not consistent with Mack’s distribution-free chain ladder. However, for a sequence of compound Poisson loss models indexed by exposure (e.g., number of contracts), we show that the chain ladder predictor and Mack’s estimator of conditional mean squared error of prediction can be derived by considering large exposure asymptotics. Hence, quantifying chain ladder prediction uncertainty can be done with Mack’s estimator without relying on the validity of the model assumptions of the distribution-free chain ladder.

Abstract With recent epidemics such as COVID-19, H1N1 and SARS causing devastating financial loss to the economy, it is important that insurance companies plan for financial costs of epidemics. This article proposes a new methodology for epidemic and insurance modelling by combining the existing deterministic compartmental models and the Markov multiple state models to facilitate actuarial computations to design new health insurance plans that cover epidemics. Our method is inspired by the seminal paper (Feng and Garrido (2011)North American Actuarial Journal,15, 112–136.) of Feng and Garrido and complements the work of Hillairet and Lopez et al. in Hillairet and Lopez ((2021)Scandinavian Actuarial Journal,2021(8), 671–694.) and Hillairet et al. ((2022)Insurance: Mathematics and Economics,107, 88–101.) In this work, we use the deterministic SIR model and the Eyam epidemic data set to provide numerical illustrations for our method.

Abstract We propose a flexible lattice model to evaluate the fair value of insurance contracts embedding both financial and actuarial risk factors. Flexibility relies on the ability of the model to manage different specifications of the correlated processes governing interest rate, mortality, and fund dynamics, thus allowing the insurer to make the most appropriate choices. The model is also able to handle additional guarantees like a surrender opportunity for which explicit formulae are not available being it similar to an American derivative. The model discretizes mortality and interest rate dynamics through two different binomial lattices and then combines them into a bivariate tree characterized by the presence of four branches for each node. The probability of each branch is defined to replicate the correlation affecting the two processes. The bivariate model is useful to compute the value of survival zero coupon bond. When adding another source of risk, such as the fund dynamics for evaluating fund-linked insurance products, we model it through a bivariate tree that captures the influence of the interest rate on its drift term. Then, the mortality risk is embedded by defining a trivariate tree presenting eight branches emanating from each node with probabilities defined in order to capture the correlations of the processes. Extensive numerical experiments assess the model accuracy by considering some stylized policies, but the model application is not limited to them being it able to manage different contract specifications.

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