Vehicle telematics provides granular data for dynamic driving risk assessment, but current methods often rely on aggregated metrics (e.g., harsh braking counts) and do not fully exploit the rich time-series structure of telematics data. In this paper, we introduce a flexible framework using continuous-time hidden Markov model (CTHMM) to model and analyse trip-level telematics data. Unlike existing methods, the CTHMM models raw time-series data without predefined thresholds on harsh driving events or assumptions about accident probabilities. Moreover, our analysis is based solely on telematics data, requiring no traditional covariates such as driver or vehicle characteristics. Through unsupervised anomaly detection based on pseudo-residuals, we identify deviations from normal driving patterns—defined as the prevalent behaviour observed in a driver’s history or across the population—which are linked to accident risk. Validated on both controlled and real-world datasets, the CTHMM effectively detects abnormal driving behaviour and trips with increased accident likelihood. In real data analysis, higher anomaly levels in longitudinal and lateral accelerations consistently correlate with greater accident risk, with classification models using this information achieving ROC-AUC values as high as 0.86 for trip-level analysis and 0.78 for distinguishing drivers with claims. Furthermore, the methodology reveals significant behavioural differences between drivers with and without claims, offering valuable insights for insurance applications, accident analysis, and prevention.

The special issue on risk sharing contains 12 papers. This editorial introduces these papers by briefly discussing their contents and contributions.

We present a dynamic pay-how-you-drive pricing scheme for motor insurance using telematics signals. More specifically, our approach allows the insurer to apply penalties to a baseline premium on the occurrence of events such as hard acceleration or braking. In addition, we incorporate a bonus-malus system (BMS) adapted for telematics data, providing a credibility component based on past telematics signals to the claim frequency predictions. We purposefully consider a weekly setting for our ratemaking approach to benefit from the signal’s high-frequency rate and to encourage safe driving via dynamic premium corrections. Moreover, we provide a detailed structure that allows our model to benefit from historical records and detailed telematics data collected weekly through an onboard device. We showcase our results numerically in a case study using data from an insurance company.

We propose an individual claims reserving model based on the conditional Aalen–Johansen estimator, as developed in Bladt and Furrer ((2023a) arXiv:2303.02119.). In our approach, we formulate a multi-state problem, where the underlying variable is the individual claim size, rather than time. The states in this model represent development periods, and we estimate the cumulative density function of individual claim sizes using the conditional Aalen–Johansen method as transition probabilities to an absorbing state. Our methodology reinterprets the concept of multi-state models and offers a strategy for modeling the complete curve of individual claim sizes. To illustrate our approach, we apply our model to both simulated and real datasets. Having access to the entire dataset enables us to support the use of our approach by comparing the predicted total final cost with the actual amount, as well as evaluating it in terms of the continuously ranked probability score.

The localized nature of severe weather events leads to a concentration of correlated risks that can substantially amplify aggregate event-level losses. We propose a copula-based regression model for replicated spatial data to characterize the dependence between property damage claims arising from a common storm when analyzing its financial impact. The factor copula captures the location-based spatial dependence between properties, as well as the aspatial dependence induced by the common shock of experiencing the same storm. The framework allows insurers to flexibly incorporate the observed heterogeneity in marginal models of skewed, heavy-tailed, and zero-inflated insurance losses, while retaining the model interpretation in decomposing latent sources of dependence. We present a likelihood-based estimation to address the computational challenges from the discreteness and high dimensionality in the outcome of interest. Using hail damage insurance claims data from a US insurer, we demonstrate the effect of dependence on claims management decisions.

In this paper, we explore a non-cooperative optimal reinsurance problem incorporating likelihood ratio uncertainty, aiming to minimize the worst-case risk of the total retained loss for the insurer. We establish a general relation between the optimal reinsurance strategy under the reference probability measure and the strategy in the worst-case scenario. This relation can further be generalized to insurance design problems quantified by tail risk measures. We also characterize distortion risk measures for which the insurer’s optimal strategy remains the same in the worst-case scenario. As an application, we determine the optimal policies for the worst-case scenario using an expectile risk measure. Additionally, we propose and explore a cooperative problem, which can be viewed as a general risk sharing problem between two agents in a comonotonic market. We determine the risk measure value and the optimal reinsurance strategy in the worst-case scenario for the insurer and compare the results from the non-cooperative and cooperative models.

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.

This paper introduces a method for pricing insurance policies using market data. The approach is designed for scenarios in which the insurance company seeks to enter a new market, in our case: pet insurance, lacking historical data. The methodology involves an iterative two-step process. First, a suitable parameter is proposed to characterize the underlying risk. Second, the resulting pure premium is linked to the observed commercial premium using an isotonic regression model. To validate the method, comprehensive testing is conducted on synthetic data, followed by its application to a dataset of actual pet insurance rates. To facilitate practical implementation, we have developed an R package called IsoPriceR. By addressing the challenge of pricing insurance policies in the absence of historical data, this method helps enhance pricing strategies in emerging markets.

The analysis of insurance and annuity products issued on multiple lives requires the use of statistical models which account for lifetime dependence. This paper presents a Dirichlet process mixture-based approach that allows to model dependent lifetimes within a group, such as married couples, accounting for individual as well as group-specific covariates. The model is analyzed in a fully Bayesian setting and illustrated to jointly model the lifetime of male–female couples in a portfolio of joint and last survivor annuities of a Canadian life insurer. The inferential approach allows to account for right censoring and left truncation, which are common features of data in survival analysis. The model shows improved in-sample and out-of-sample performance compared to traditional approaches assuming independent lifetimes and offers additional insights into the determinants of the dependence between lifetimes and their impact on joint and last survivor annuity prices.

We study Pareto optimality in a decentralized peer-to-peer risk-sharing market where agents’ preferences are represented by robust distortion risk measures that are not necessarily convex. We obtain a characterization of Pareto-optimal allocations of the aggregate risk in the market, and we show that the shape of the allocations depends primarily on each agent’s assessment of the tail of the aggregate risk. We quantify the latter via an index of probabilistic risk aversion, and we illustrate our results using concrete examples of popular families of distortion functions. As an application of our results, we revisit the market for flood risk insurance in the United States. We present the decentralized risk sharing arrangement as an alternative to the current centralized market structure, and we characterize the optimal allocations in a numerical study with historical flood data. We conclude with an in-depth discussion of the advantages and disadvantages of a decentralized insurance scheme in this setting.

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.

Yield curve extrapolation to unobservable tenors is a key technique for the market-consistent valuation of actuarial liabilities required by Solvency II and forthcoming similar regulations. Since the regulatory method, the Smith–Wilson method, is inconsistent with observable yield curve dynamics, parsimonious parametric models, the Nelson–Siegel model and its extensions, are often used for yield curve extrapolation in risk management. However, it is difficult for the parsimonious parametric models to extrapolate yield curves without excessive volatility because of their limited ability to represent observed yield curves with a limited number of parameters. To extend the representational capabilities, we propose a novel yield curve extrapolation method using machine learning. Using the long short-term memory architecture, we achieve purely data-driven yield curve extrapolation with better generalization performance, stability, and consistency with observed yield curve dynamics than the previous parsimonious parametric models on US and Japanese yield curve data. In addition, our method has model interpretability using the backpropagation algorithm. The findings of this study prove that neural networks, which have recently received considerable attention in mortality forecasting, are useful for yield curve extrapolation, where they have not been used before.

This paper develops a theoretical framework to examine the technology adoption decisions of insurers and their impact on market share, considering heterogeneous customers and two representative insurers. Intuitively, when technology accessibility is observable, an insurer’s access to a new technology increases its market share, no matter whether it adopts the technology or not. However, when technology accessibility is unobservable, the insurer’s access to the new technology has additional side effects on its market share. First, the insurer may apply the available technology even if it increases costs and premiums, thereby decreasing market share. Second, the unobservable technology accessibility leads customers to expect that all insurers might have access to the new technology and underestimate the premium of those without access. This also decreases the market share of an insurer with access to the new technology. Our findings help explain the unclear relationship between technology adoption and the market share of insurance companies in practice.

In recent decades, analysing the progression of mortality rates has become very important for both public and private pension schemes, as well as for the life insurance branch of insurance companies. Traditionally, the tools used in this field were based on stochastic and deterministic approaches that allow extrapolating mortality rates beyond the last year of observation. More recently, new techniques based on machine learning have been introduced as alternatives to traditional models, giving practitioners new opportunities. Among these, neural networks (NNs) play an important role due to their computation power and flexibility to treat the data without any probabilistic assumption. In this paper, we apply multi-task NNs, whose approach is based on leveraging useful information contained in multiple related tasks to help improve the generalized performance of all the tasks, to forecast mortality rates. Finally, we compare the performance of multi-task NNs to that of existing single-task NNs and traditional stochastic models on mortality data from 17 different countries.

This study introduces an innovative methodology for mortality forecasting, which integrates signature-based methods within the functional data framework of the Hyndman–Ullah (HU) model. This new approach, termed the Hyndman–Ullah with truncated signatures (HUts) model, aims to enhance the accuracy and robustness of mortality predictions. By utilizing signature regression, the HUts model is able to capture complex, nonlinear dependencies in mortality data which enhances forecasting accuracy across various demographic conditions. The model is applied to mortality data from 12 countries, comparing its forecasting performance against variants of the HU models across multiple forecast horizons. Our findings indicate that overall the HUts model not only provides more precise point forecasts but also shows robustness against data irregularities, such as those observed in countries with historical outliers. The integration of signature-based methods enables the HUts model to capture complex patterns in mortality data, making it a powerful tool for actuaries and demographers. Prediction intervals are also constructed with bootstrapping methods.

Various measures have been introduced in the existing literature to evaluate extreme risk exposure under the effect of an observable factor. Due to the nice properties of the higher-moment (HM) coherent risk measure, we propose a conditional version of the HM (CoHM) risk measure by incorporating the information of an observable factor. We conduct an asymptotic analysis of this measure in the presence of extreme risks under the weak contagion at a high confidence level, which is further applied to the special case of the conditional Haezendonck–Goovaerts risk measure (CoHG). Numerical illustrations are also provided to examine the accuracy of the asymptotic formulas and to analyze the sensitivity of the risk contribution of the CoHG. Based on the asymptotic result in the Fréchet case, we propose an estimator for the CoHM via an extrapolation, supported by a simulation study.

We use recent advances in polynomial diffusion processes to develop a continuous-time joint mortality model for the actuarial valuation and risk analysis of life insurance liabilities. The model considers the stochastic nature of future mortality improvements and introduces a common subordinator for the marginal survival processes, resulting in a nontrivial dependence structure between the survival of pairs of individuals. Polynomial diffusion processes can be used to derive closed-form formulae for standard actuarial quantities. The model fits well with a classic dataset provided by a Canadian insurer and can be used to evaluate products issued to multiple lives, as shown through numerical applications.

Risk-sharing rules have been applied to mortality pooling products to ensure these products are actuarially fair and self-sustaining. However, most of the existing studies on the risk-sharing rules of mortality pooling products assume deterministic mortality rates, whereas the literature on mortality models provides empirical evidence suggesting that mortality rates are stochastic and correlated between cohorts. In this paper, we extend existing risk-sharing rules and introduce a new risk-sharing rule, named the joint expectation (JE) rule, to ensure the actuarial fairness of mortality pooling products while accounting for stochastic and correlated mortality rates. Moreover, we perform a systematic study of how the choice of risk-sharing rule, the volatility and correlation of mortality rates, pool size, account balance, and age affect the distribution of mortality credits. Then, we explore a dynamic pool that accommodates heterogeneous members and allows new entrants, and we track the income payments for different members over time. Furthermore, we compare different risk-sharing rules under the scenario of a systematic shock in mortality rates. We find that the account balance affects the distribution of mortality credits for the regression rule, while it has no effect under the proportional, JE, and alive-only rules. We also find that a larger pool size increases the sensitivity to the deviation in total mortality credits for cohorts with mortality rates that are volatile and highly correlated with those of other cohorts, under the stochastic regression rule. Finally, we find that risk-sharing rules significantly influence the effect of longevity shocks on fund balances since, under different risk-sharing rules, fund balances have different sensitivities to deviations in mortality credits.

This paper introduces a novel theoretical framework that offers a closed-form expression for the tail variance (TV) for the novel family of generalised hyper-elliptical (GHE) distributions. The GHE family combines an elliptical distribution with the generalised inverse Gaussian (GIG) distribution, resulting in a highly adaptable and powerful model. Expanding upon the findings of Ignatieva and Landsman ((2021) Insurance: Mathematics and Economics, 101, 437–465.) regarding the tail conditional expectation (TCE), this study demonstrates the significance of the TV as an additional risk measure that provides valuable insights into the tail risk and effectively captures the variability within the loss distribution’s tail. To validate the theoretical results, we perform an empirical analysis on two specific cases: the Laplace – GIG and the Student-t – GIG mixtures. By incorporating the TV derived for the GHE family, we are able to quantify correlated risks in a multivariate portfolio more efficiently. This contribution is particularly relevant to the insurance and financial industries, as it offers a reliable method for accurately assessing the risks associated with extreme losses. Overall, this paper presents an innovative and rigorous approach that enhances our understanding of risk assessment within the financial and insurance sectors. The derived expressions for the TV in addition to TCE within the GHE family of distributions provide valuable insights and practical tools for effectively managing risk.

This paper studies a long-standing problem of risk exchange and optimal resource allocation among multiple entities in a continuous-time pure risk-exchange economy. We establish a novel risk exchange mechanism that allows entities to share and transfer risks dynamically over time. To achieve Pareto optimality, we formulate the problem as a stochastic control problem and derive explicit solutions for the optimal investment, consumption, and risk exchange strategies using a martingale method. To highlight practical applications of the solution to the proposed problem, we apply our results to a target benefit pension plan, featuring the potential benefits of risk sharing within this pension system. Numerical examples show the sensitivity of investment portfolios, the adjustment item, and allocation ratios to specific parameters. It is observed that an increase in the aggregate endowment process results in a rise in the adjustment item. Furthermore, the allocation ratios exhibit a positive correlation with the weights of the agents.